Higgs vacuum (in)stability during inflation: the dangerous relevance of de Sitter departure and Planck-suppressed operators
HHiggs vacuum (in)stability during inflation
The dangerous relevance of de Sitter departure andPlanck-suppressed operators
Jacopo Fumagalli, S´ebastien Renaux-Petel and John W. Ronayne
Institut d’Astrophysique de Paris, GReCO, UMR 7095 du CNRS et de Sorbonne Universit´e, 98bisboulevard Arago, 75014 Paris, France
E-mail: [email protected] , [email protected] , [email protected] Abstract:
The measured Standard Model parameters lie in a range such that the Higgspotential, once extrapolated up to high scales, develops a minimum of negative energy den-sity. This has important cosmological implications. In particular, during inflation, quantumfluctuations could have pushed the Higgs field beyond its potential barrier, triggering the for-mation of anti-de Sitter regions, with fatal consequences for our universe. By requiring thatthis did not happen, one can in principle connect (and constrain) Standard Model parameterswith the energy scale of inflation. In this context, we highlight the sensitivity of the fate ofour vacuum to seemingly irrelevant physics. In particular, the departure of inflation froman exact de Sitter phase, as well as Planck-suppressed derivative operators, can, already andsurprisingly, play a decisive role in (de)stabilizing the Higgs during inflation. Furthermore, inthe stochastic dynamics, we quantify the impact of the amplitude of the noise differing fromthe one of a massless field, as well as of going beyond the slow-roll approximation by usinga phase-space approach. On a general ground, our analysis shows that relating the period ofinflation to precision particle physics requires a knowledge of these “irrelevant” effects. a r X i v : . [ h e p - ph ] M a y ontents H – 1 – Introduction
One of the main surprises after the discovery of the Higgs boson [1, 2] was the fact that themeasured values of the Standard Model (SM) parameters lie exactly within the boundaryregion that separates where the SM would develop, or not, a true vacuum of negative energydensity once extrapolated up to the Planck scale (see [3–8] and [9–11] for studies before andafter the Higgs discovery). The energy scale at which this instability takes place is extremelysensitive to the boundary conditions measured at the electroweak (EW) scale. In particular,the central values of the measured top and Higgs masses hint that our vacuum is metastable,i.e. it is not the true vacuum but its lifetime is larger than the age of our Universe. As it was phrased in a recent review [13]: a metastable vacuum, by definition, hasimplications that can only be studied in the context of the cosmological history. Even iftoday the lifetime of our vacuum is much larger than the age of our Universe, assuming aperiod of inflation implies that, in the very early Universe, the Higgs (when it is not theinflaton) behaved as a test scalar field in a (quasi) de Sitter background. There, stochastickicks could have pushed it beyond the potential barrier towards the true vacuum, until thepoint of forming anti de Sitter (AdS) regions fatal for our universe [14, 15]. For good reasons(e.g. you reading this sentence), we have to enforce that no such regions formed in our pastlight-cone. This request brings interesting implications. The shape of the Higgs potentialdepends on the measured SM parameters, while the size of the stochastic kicks is of order ofthe Hubble rate H , which is directed linked to the energy scale of inflation. Thus, the twocan be related: given the measured SM parameters one can constrain H [14–21]. Conversely,assuming a detection of primordial gravitational waves one can constraint the SM parameters[19, 20, 22]. Furthermore, after inflation, the oscillations of the inflaton induce tachyonicexcitations of the Higgs field that can as well trigger a vacuum instability [23–29].These various studies share two main simplifying assumptions: a constant Hubble scaleduring inflation, and no new physics between the Standard Model and very high scales. Thelatter assumption is motivated by minimality, while the former approximation is motivatedby the (typically) slight departure of inflation from a de Sitter phase, and it also has theadvantage of making the analysis model-independent. The purpose of this work is to showthat considering departures from a perfect de Sitter background, as well as including Planck-suppressed derivative operators in the analysis, can play a significant role in determining thefate of the Higgs vacuum during inflation.Consider, as an example, a dimension-six operator of the form O = C H † H M ( ∂φ ) , (1.1)where H is the Higgs doublet and φ a generic inflaton. This operator preserves the would-be(quasi)-shift symmetry of the inflationary sector. In particular, the request of preserving the The addition of Planck-suppressed operators can significantly influence the tunneling rate from the falseto the true vacuum [11]. However, it has been shown that a small value of the non-minimal coupling is enoughto wash out the effect of these higher-order operators [12]. – 2 –atness of the inflationary potential entails no constraints on C and M . However, higher-orderoperators like the one in Eq. (1.1), which are allowed by symmetry and hence compulsory froman effective field theory point of view, generate a non-trivial geometry in the Higgs-inflatonfield space manifold; the curvature of this manifold induces an effective mass for the Higgsthat can stabilize (or further destabilize) it during inflation, similarly to what occurs in thegeometrical destabilization of inflation [30]. As we will see, the effect is already significant forthe conservative choice M = M Pl , and it would be completely determinant for M even slightlysmaller than M Pl . From Eq. (1.1) it is easy to see that the absolute value of the inducedmass is proportional to the first slow-roll parameter, i.e. − ( ∂φ ) (cid:39) ˙ φ ∝ (cid:15) = − ˙ H/H . Hence,our analysis cannot avoid considering an evolving background in which the time dependenceof the Hubble parameter H is taken into account. More generally, irrespective of the impactof higher-order derivative operators, we will show that the inevitable time dependence ofthe inflationary background influences in a non-trivial manner both the classical and thestochastic dynamics of the Higgs, and hence its cosmological fate. Eventually, in the presenceof operators that induce an effective mass for the Higgs, be they derivative operators like inEq. (1.1) or non-minimal couplings like in Refs. [14, 34], taking into account the fact that thestochastic noise of a light field differs from the one of an exactly massless field, as it is usuallydone, has an important impact on the final results.Summarizing, we study the relevance of the following and previously neglected physicson the Higgs (in)stability during inflation: • H being not exactly constant. Inflation takes place in a quasi de Sitter background.Different models will determine different evolutions of H and in turn different stochasticdynamics of the Higgs field. • The variance of the noise deviating from the almost massless case. Parameterizing therandom kicks with H/ π is accurate only when there is a large hierarchy between themass of the field and H . • Planck-suppressed derivative operators.
Couplings like the one in Eq. (1.1), as well asmore general ones that respect the shift symmetry of the inflationary sector, modifythe effective mass of the Higgs, for instance by inducing a non-trivial geometry in theHiggs-inflaton target space. • Considering a stochastic approach in phase space.
Deviations of inflation from a strictslow-roll phase is communicated to the spectator Higgs, notably at the end of inflation.We therefore take into account stochastic effects beyond the slow-roll regime.While the improvement coming from considering the stochastic approach in phase space hasa minor impact on the final results, we show explicitly that the first three effects in generalplay a crucial role in determining the fate of the instability. As a result, for fixed boundary See [31–33] for studies of the fate of this instability. – 3 –onditions, i.e. measured Standard Model parameters and scale of inflation H , we obtain, fordifferent inflationary models and Planck-suppressed operators, outcomes for the fate of theinstability that are different, sometimes by orders of magnitude, from the benchmark analysisin which the aforementioned effects are neglected.Moreover, the time dependence of the background and of the various effects contributingto the Higgs dynamics prompts us to introduce important conceptual novelties in the analysis.A non-static effective potential combined with the stochastic diffusion of the Higgs leads toa new procedure (explained in Sec. 3.2) to compute the fraction of Hubble patches in AdSin our past light cone. In particular, when matching the stochastic and classical dynamics,we pay attention to the fact that patches already in AdS cannot be rescued, together withthe finite time it takes for them to form when the Higgs backreaction cannot be neglectedanymore.Our main numerical results are displayed in figures 5-6 and 7-8. In the first two, we showthe impact of our analysis in shaping the constraints on the Hubble scale, for two differentbackground evolutions, each with and without derivative Planck-suppressed operators. Inshort, the effects we studied, that might be considered negligible, are instead often crucial tocorrectly estimate the bounds on H . From a different perspective, in the latter two figures, wealso present our results by looking at how a given Hubble scale constrains the top and Higgsmasses within their experimental error bars. Remarkably, there exists a degeneracy betweenvalues of H separated by different orders of magnitude on one side, and effects coming fromthe time-dependent background or derivative Planck-suppressed operators on the other side.Therefore, even with the assumption of the SM being valid up to the Planck scale, it appearsunlikely that a future detection of primordial gravitational waves would, on its own, enableus to constrain SM parameters like the top mass.The paper is organized as follows. In Sec. 2.1 we describe the various effects thatcontribute to the classical dynamics of the Higgs during inflation. In particular, we highlightthe effects of derivative Planck-suppressed operators. We introduce stochastic effects and theprobability distribution function (PDF) of the Higgs in Sec. 2.2, before studying each of theeffects listed above in Sec. 2.3. In Sec. 3.1 we define the criteria required to avoid patches ofAdS in our past light cone, while in Sec. 3.2 we outline the procedure used to estimate theamount of these patches in a time-dependent setup, and give analytical estimates in Sec. 3.3.The last part 4 is devoted to our (numerical) results. There we show explicitly the sensitivityof the fate of the Higgs instability to the different effects studied in this work. Conclusionand outlooks are provided in Sec. 5. Falling in the AdS vacuum (or a brief story of an AdS patch)
The fate of a spacetime region in which a scalar field is falling towards a negative energyvacuum, as well as the fate of an AdS patch embedded in a de Sitter geometry, representnon-trivial general relativity problems. As a matter of fact, these issues were not well un-derstood until very recently [14, 15]. Thus, before starting, we briefly brush over the currentunderstanding of these phenomena. As mentioned, for central values of the SM parameters,– 4 – lassical regimeStochastic regime
AdS EW Figure 1 : Dynamics of the Higgs field during inflation. In blue: sketch of the SM potentialwhich, for the current best measurement of the SM parameters, develops an instability atlarge field values. We denote the position of the potential barrier by h max , and the Higgsvalue beyond which the classical dynamics dominates over the quantum jumps by h cl (Eq. (3.5) ). This “point of classicality” evolves in a time-dependent background. In green the PDF(at three different times) that gives the probability of finding a given Hubble patch with Higgsvalue h after N e -folds of inflation. We trust the PDF up to h cl , for larger values we evolvethe tail classically to compute the fraction of Hubble patches that are in AdS at the end ofinflation, see Sec. 3.2. the effective quartic coupling of the Higgs potential becomes negative, hence developing aninstability, at field values much less than the Planck scale. We call h max the value of theHiggs field at which the SM potential has its maximum. For large enough field values, theclassical motion, led by the SM potential, dominates over the stochastic quantum contribu-tions. From this point on, that we label as h cl (defined in Eq. (3.5)), the Higgs starts to rolldown classically towards the true vacuum. Initially, when the Higgs experiences the negativepart of its potential, the total energy density is still completely dominated by the positivecontribution coming from the inflationary background V ( φ ) (cid:39) M H . Once the energydensity of the Higgs sector becomes large enough, it strongly backreacts on spacetime, tothe point of eventually generating an anti-de Sitter (AdS) patch. In this respect, the resultsfirst obtained analytically in [14] were later substantially confirmed with full GR simulations– 5 –n [15]: in Hubble patches in which the Higgs field exceeds h cl , after a finite time inflationterminates locally, leading to a crunching region surrounded by a causally disconnected one ofnegative energy density, i.e. an AdS patch. The latter persists during inflation and, as a firstapproximation, expands comovingly with the ambient de Sitter spacetime. Conversely, afterinflation (in an approximately Minkowski background), the AdS patch expands at the speedof light, engulfing the surrounding space-time which therefore cannot be in the electroweakvacuum today. This is the reason to demand that there was not a single AdS Hubble patchin our past light cone, see Eqs. (3.1)-(3.4). Furthermore, since the generation and evolutionof an AdS bubble is not fatal to the ambient inflationary spacetime, it is possible to countthe fraction of volume transmuting into AdS by using a probability density function (PDF)following a Fokker-Planck equation, augmented with necessary precautions listed in Sec. 3. Here we describe the various effects that we take into account and that determine the clas-sical dynamics of the Higgs during inflation, postponing to the next section the inclusion ofstochastic effects. The radial SM Higgs h is taken as a spectator field with (initially) no rolein the inflationary dynamics, the latter being driven by an inflaton field φ endowed with apotential V ( φ ). For simplicity, the other operators that we consider are taken to respect theshift symmetry φ → φ +const . , hence we do not include Higgs-inflaton couplings in the poten-tial. However, we take into account the non-minimal coupling of the Higgs to the spacetimecurvature R , and two-derivative higher-order operators. The resulting total Lagrangian reads L = − G IJ ∂ϕ J ∂ϕ I − V ( h ) − V inf ( φ ) , ϕ I = { φ, h } , (2.2)where the higher-order operators are included in the inflaton-Higgs field space metric G IJ , towhich we come back below, and V ( h ) is defined as V ( h ) = V SM − ξh R, V SM = λ eff ( µ ( h ))4 h . (2.3)Here, V SM is the renormalization group (RG) improved Standard Model Higgs potential.In our analysis, it has been computed at NNLO [9], using the two-loop effective potential For instance, operators like O n +2 = C n +2 φ n h M n − (2.1)induce an effective mass for the Higgs (see for instance [35, 36]). However, contrary to shift-symmetric deriva-tive operators that we study, these operators are tightly constrained by the request of not spoiling at loop levelthe flatness of the inflationary potential. In fact, even if the operator (2.1) does not influence the dynamicsat tree level for h = 0 on the background, it generates a φ dependent one-loop contribution to the potential.Thus, all the Wilson coefficients in (2.1) have to be rather small, with bounds depending on the particularform of the inflationary potential. Reasoning as in [37], one can easily show that for V ( φ ) = 1 / m φ forinstance, one has C n +2 (cid:46) − (4+ n ) . – 6 –38], two-loop matching conditions at the EW scale [10] and three-loop beta-functions [39–41]. λ eff is the two-loop effective quartic coupling in the Landau gauge defined in Ref. [10],with the contribution from the anomalous dimension already absorbed in a field redefinitionof h , which facilitates the analysis and significantly reduces the gauge dependence of thepotential [14]. The optimal choice for the renormalization scale has been chosen to take intoaccount the (quasi) de Sitter background, i.e. to keep higher-order terms under control weuse µ (cid:39) h +12 H [42], where sub-leading slow-roll corrections are neglected, although it hasbeen shown that considering other linear combinations such as µ = αh + βH has negligibleimpact [34]. We consider generic values of the non-minimal coupling as ξ = 0 is not a fixedpoint of the renormalization group flow, i.e. if it is set to zero at one scale it will be differentfrom zero at any other scale. Note that in our convention, a negative ξ tends to stabilize theHiggs. Before discussing the kinetic terms, it is instructive to use a multifield point of view,in which the Higgs direction is identified with the entropic direction in a two-field model(albeit a special one in which the Higgs is not contributing to the background dynamics). Inthis context, its (super-Hubble) fluctuations are known to acquire the effective mass (see e.g.[43, 44]) V ; ss + 3 H η ⊥ + (cid:15)R fs H M , (2.4)where semicolons stay for covariant derivatives with respect to the field space metric G IJ , thesubscript s indicates a projection along the entropic direction, η ⊥ is a dimensionless parameterthat measures the deviation of the trajectory from a field space geodesic, and R fs denotesthe curvature of the field space. This formula makes it clear that the effective mass of theHiggs during inflation is not simply given by V (cid:48)(cid:48) ( h ) = V (cid:48)(cid:48) SM − ξR . In particular, non-standardkinetic terms contribute to it in general, for instance through the curvature of the field space,but also through non-trivial Christoffel symbols in the covariant derivative V ; ss (cid:39) V ; hh . Inwhat follows, we confirm these expectations in a simple EFT parameterization of the kineticterms.As mentioned above, we consider kinetic terms that respect the (approximate) shift-symmetry of the inflaton, i.e. with G IJ independent of the inflaton. Taking into account the SU (2) gauge symmetry of the electroweak sector further restricts the allowed kinetic terms.Momentarily using the Higgs doublet H , they are of the form L kin = − D µ H † D µ H − a ( H † H )( ∇ µ φ ) − b ( H † H ) (cid:16) H † D µ H∇ µ φ + h . c . (cid:17) (2.5) We work with signature ( − + ++), such that R = (12 H + 6 ˙ H ). We sometimes compare our results tothe ones of Refs. [14, 22], which use signature (+ − − − ) such that R = − (12 H + 6 ˙ H ). However negativevalues of ξ still correspond to a stabilizing effect since we use a sign flip in the definition of the non-minimalcoupling in (2.3). This multifield formalism is usually formulated in the Einstein frame, but relative corrections comparedto the Jordan frame used in this paper are in h /M , so they are completely negligible. This is the implicitpoint of view we also use when incorporating the non-minimal coupling in the Higgs potential (2.3). – 7 –here D µ denotes the gauge derivative, a and b are generic functions, and the kinetic terminvolving only the Higgs can always be put in a canonical form. As usual, we parameterizethe effect of high-scale physics by expanding a and b in powers of H † H /M , where M denotesthe cutoff of the theory. In terms of the radial Higgs field, we thus write L kin = −
12 ( ∂h ) − (cid:32) − C (cid:18) hM (cid:19) + . . . (cid:33) ( ∂φ ) − C hM ∂h∂φ (cid:32) O (cid:18) hM (cid:19) + . . . (cid:33) , (2.6)where all coefficients are thought to be of order one, and the terms in C and C correspondrespectively to dimension 5 and 6 operators. Other higher-order operators, correspondingto higher powers of h/M , can be kept, but have negligible impact for our purposes, i.e. weonly keep dangerous irrelevant operators. Eventually, note that the field space defined by thekinetic terms (2.6) is curved for generic values of the parameters C and C .The classical background equations of motion deduced from (2.2) read ¨ φ I + Γ IJK ˙ φ J ˙ φ K +3 H ˙ φ I + G IJ V ,J = 0, where Γ IJK denotes the Christoffel symbols of the metric G IJ , i.e.¨ φ − C C h M ˙ φ − C hM ˙ φ ˙ h + C M ˙ h + 3 H ˙ φ + V inf ,φ − C hM V ,h (cid:39) , (2.7)¨ h + 2 C hM ˙ φ + 4 C C h M ˙ φ ˙ h − C hM ˙ h + 3 H ˙ h + V ,h − C hM V inf ,φ (cid:39) , (2.8)where we kept for each term only its dominant part in h/M . Using these equations, one caneasily show the self-consistency of the regime where the Higgs is considered as a spectatorfield, with no backreaction on the inflaton, and where the dynamics obeys:3 H ˙ φ + V inf ,φ (cid:39) , (2.9)¨ h + 3 H ˙ h + V ,h + (cid:32) − C V inf ,φ M + 2 C ˙ φ M (cid:33) h (cid:39) , (2.10)3 H M (cid:39) V inf ( φ ) + 12 ˙ φ . (2.11)For this, note that the typical velocity of the Higgs is ˙ h ∼ H (as the analysis below willconfirm), so that the kinetic energy is completely dominated by the inflaton: ˙ h / ˙ φ ∼ H / ( (cid:15)M ) ∼ P ζ ∼ − , where P ζ denotes the amplitude of the primordial curvaturepower spectrum, and one consistently used the fact that (cid:15) is directly related to the velocityof the inflaton: (cid:15) ≡ − ˙ HH (cid:39) ˙ φ H M . (2.12)All terms neglected in (2.9)-(2.12) are hence suppressed compared to leading-order ones by(combination of) powers of h/M , h/M Pl , H/M and P ζ .Summarizing: derivative operators in (2.6) have a negligible impact on the inflaton, butthey modify the dynamics of the Higgs, whose evolution (2.10) can be intuitively understood– 8 –s the one of a canonical field subject to the time-dependent effective potential V eff = V SM ( h ) + H h (cid:18) − ξ (cid:16) − (cid:15) (cid:17) − C sign( V inf ,φ ) √ (cid:15) M Pl M + 4 C (cid:15) M M (cid:19) . (2.13)Here, we used Eqs. (2.9) and (2.12) to express the effective potential in terms of the firstslow-roll parameter (cid:15) . The three terms in parenthesis correspond to the effects of the non-minimal coupling, and of the dimension 5 and 6 derivative operators. Each generate quadraticcontributions to the potential, whose effective mass in Hubble units are set respectively by ξ , √ (cid:15)M Pl /M and (cid:15)M /M . Hence, although the effects of the derivative operators may seeminnocuous at first sight, as they are slow-roll suppressed, a second thought reveals that theycan play an important role in the dynamics of the Higgs field, in the same way as a smallvalue of ξ can modify the fate of the Higgs instability during inflation [14, 34]. Furthermore,it is important to stress that, motivated by minimality, we will focus on operators suppressedby the Planck scale, i.e. M = M Pl , but effects are obviously even more important if oneconsiders values of M even slightly smaller than M Pl . Eventually, let us note that the effectsof the derivative operators are tied to the non-zero value of (cid:15) , or equivalently to the slightbreaking of the inflationary shift symmetry by the potential. In general, one expects thisbreaking to be communicated through loops to the kinetic sector, i.e. one expects derivativecouplings that also slightly break the shift symmetry. We leave the study of such a generalsetup to future works, and content ourselves with assessing the impact of the operators in(2.6).Following the interpretation of Eq. (2.13) as the effective potential governing the dynamicsof the spectator Higgs field, it is natural to define its effective mass as M ≡ ∂ V eff ∂h = V (cid:48)(cid:48) SM + H (cid:18) − ξ (cid:16) − (cid:15) (cid:17) − C √ (cid:15) M Pl M + 4 C (cid:15) M M (cid:19) , (2.14)where we chose sign( V inf ,φ ) = 1, which one can always do for monotonous potentials byredefining φ → − φ . It is instructive to compare this to the effective mass (2.4). Un-der the same approximations as above, one can easily show that the two expressions co-incide at leading-order in h/M , with a negligible contribution from the bending (the secondterm in (2.4)), V ; ss (cid:39) V ,hh − C V inf ,φ /M reproducing the first three terms in (2.14), and (cid:15)R fs M (cid:39) C (cid:15)M /M giving rise to the last term. While the effect of the dimension-sixoperator can thus be explained by its contribution to the field space curvature, as mentionedin the introduction, the effect of the dimension-five operator comes from its contribution tothe covariant Hessian of the potential. As inflation proceeds, initially sub-Hubble fluctuations of the Higgs field exit the Hubbleradius and feed its infrared dynamics [45, 46]. This stochastic evolution is usually modeledby the simple Langevin equation dhdN + 13 H ∂V eff ∂h = η ( N ) , (2.15)– 9 –here here and in the remainder of this paper, h denotes the super-Hubble coarse-grainedpart of the Higgs field. N is the number of e -folds of inflation, and η is a Gaussian whitenoise with variance the power spectrum of the Higgs fluctuations when they join the IR sector(more about the factor f below in section 2.3.2): (cid:104) η ( N ) η ( N (cid:48) ) (cid:105) = (cid:18) Hf π (cid:19) δ ( N − N (cid:48) ) . (2.16)Stochastic effects have received a renewed attention in the past years (see e.g. [47–62]).However, despite substantial progress, a general theory quantifying the theoretical errors ofEqs. (2.15)-(2.16) is still lacking, concerning for instance the approximations of a Markoviandynamics or the Gaussianity of the noise. Given the scope of this paper, we will very conser-vatively use Eqs. (2.15)-(2.16) (and a phase-space generalization in section 2.3.5). Below wediscuss in detail their practical implementation and consequences, but for the moment, it isenough to mention their main characteristics.In particular, long-wavelength fluctuations are substantially generated, corresponding to f (cid:39) M in(2.14) verifying M (cid:28) H , whereas fluctuations are exponentially suppressed if M > / H .In the relevant range of values of h that we will be led to consider, the effect of the SM potentialis negligible, as we will discuss in more detail in the next section. Considering for the momentonly the non-minimal coupling, like in the current literature, M (cid:39) − ξH and several caseshave to be distinguished. For ξ < − / ≡ ξ the Higgs fluctuations are suppressed and thereare no stochastic kicks. Thus, if the Higgs starts below the instability scale, i.e. | h | (cid:46) h max ,a non-minimal coupling ξ < ξ is enough to ensure stability during inflation [34]. However, ξ additionally has to obey ξ (cid:38) −O (1) to ensure stability during (p)reheating [23, 25, 29]. In order to check if values of ξ > ξ are compatible with our universe, one has to take intoaccount stochastic effects. The mass terms generated by the higher-order operators that weconsider in this paper are proportional to √ (cid:15) or (cid:15) and hence are negligible at the beginningof inflation, at least for M (cid:39) M Pl . Thus, if the value of ξ is such that stochastic effects areinefficient, then our terms will not change this drastically. Their inclusion may modify theupper bound derived from studying the post-inflationary evolution but this is beyond thescope of this work. However, for a given value of ξ for which stochastic effects are important,the fate of the Higgs does depend on the higher-order operators, which become increasinglyimportant as inflation proceeds and (cid:15) grows.From the Langevin equation (2.15)-(2.16), one can write the Fokker-Planck (FP) equationfor the probability distribution function (PDF) P ( h, N ) that gives the probability (given some During preheating the Ricci scalar rapidly oscillates about zero. When the induced mass term is negative,the associated tachyonic instability can lead to efficient particle production triggering the vacuum instability.The higher-order operators considered in this work could potentially have a similar effect, and a careful studyof the preheating phase might constraint the size of the coefficients C , C . However, since O ∝ C ˙ φ and ˙ φ is always positive, we can already argue that, for C >
0, no constraint would arise from this effect. For 0 < ξ < O (1), the effective potential acquires another minimum, but the Higgs is still light, so thatthe stochastic approach is valid, see e.g. [22]. – 10 –nitial conditions) that in a particular Hubble volume the Higgs acquires the value h after Ne -folds of inflation : ∂P∂N = ∂∂h (cid:18) ∂V eff /∂h H P (cid:19) + ∂ ∂h (cid:18) H π f P (cid:19) . (2.17)We label any finite integral of the PDF with the notation P ( | h | < Λ , N ) ≡ (cid:90) Λ − Λ P ( h (cid:48) , N ) dh (cid:48) . (2.18)Given that our initial conditions at N = 0 will consist of the Hubble patch that is theprogenitor of our observable universe, P ( | h | < Λ , N ) can be equivalently interpreted as thefraction of corresponding volume at time N in which | h | < Λ. The next section is dedicatedto the study of the evolution of P ( h, N ), which constitutes the building block of our analysis,and can be used in other contexts. However, owing to the backreaction of the Higgs onspacetime when the former falls towards the true vacuum, it is worth stressing at this stagethat the study of the cosmological fate of the Higgs requires additional efforts beyond thecomputation of the PDF, which will be the focus of section 3. A fact that considerably simplifies the stochastic analysis is that the contribution from V SM to the effective potential (2.13) is negligible in the regime where stochastic effects play animportant role. This is obviously not true anymore for large values of h such that the potentialis steep and the Higgs classically fall towards the AdS vacuum, which is the object of Sec.3. Neglecting the running of λ for the sake of the argument, the ratio between the SMcontribution to the mass term and the ξ one is ( λ/ ξ )( h/H ) , and the ratio between theSM contribution to the drift and the amplitude of the noise is (2 πλ/ h/H ) . As stochasticeffects lead to values of h of order H , these ratios are of order λ (cid:46) − . We will thus be ableto neglect the first term in the mass (2.14), and most importantly, the SM contribution to thedrift term in the Langevin equation (2.15), which is thus linear. Hence, assuming Gaussianinitial conditions, the PDF remains Gaussian. This is indeed the case in what follows, aswe take as initial conditions for the Higgs values a Dirac delta centered in zero, so that thePDF is centered and only described by its variance. This choice is the one often made in theliterature and can be thought of as “the most optimistic approach”, with initial conditionstaken N (cid:63) e -folds before the end of inflation, when the largest scales observed today exitedthe Hubble radius (this number depends on the reheating history, but for definiteness, weconservatively use N (cid:63) = 60 in numerical applications). It is worth stressing that even if the instability scale h max or other intermediate quantities are gaugedependent quantities, the probabilities derived from the FP equation are not [14]. More quantitatively, we will see that, neglecting the SM contribution, typical values of h /H are of order3 H / (8 π M ) (see Eq. (2.22)), so that the above first ratio is of order 10 − λ/ξ , so indeed well negligible. – 11 –enoting the variance by σ ≡ (cid:104) h (cid:105) , one deduces from the FP equation (2.17) that itevolves as dσ ( N ) dN = − M H σ + H f π , (2.19)whose solution with initial condition σ (0) = 0 is given by σ ( N ) = 14 π (cid:90) N dN (cid:48) H ( N (cid:48) ) f ( N (cid:48) ) exp (cid:18) − (cid:90) NN (cid:48) dN (cid:48)(cid:48) M ( N (cid:48)(cid:48) ) H ( N (cid:48)(cid:48) ) (cid:19) , (2.20)where we remind that we label by N = 0 the time at which the cosmological pivot scale exitsthe Hubble radius, with H ( N = 0) ≡ H (cid:63) . To better appreciate the effects that we study inthis paper, let us first consider the benchmark solution of Eq. (2.20) under the simplifyingassumptions that H is constant and f = 1, i.e. a pure de Sitter phase and stochastic kicks of anexactly massless field. With (cid:15) = 0, the mass term (2.14) simplifies to M = − ξH = const,and the solution (2.20) becomes σ = 3 H π M (cid:20) − exp (cid:18) − M H N (cid:19)(cid:21) . (2.21)In particular, for the interesting situation of a positive mass term ξ <
0, the distributionrelaxes towards the steady “de Sitter equilibrium” in a typical time-scale given by N relax (cid:39) H / M = − / ξ . Thus, for N (cid:38) N relax , the variance reaches a constant value given by σ = 3 H π M . (2.22)Obviously, this can occur within the last N (cid:63) e -folds of inflation that we consider only if N relax (cid:46) N (cid:63) , i.e. if | ξ | (cid:38) − for N (cid:63) (cid:39)
60. For somewhat smaller values, one can formallyconsider the limit ξ → N relax → ∞ ), in which case the Higgs simply undergoes freediffusion, with a variance linearly growing with time: lim ξ → σ = ( H / π ) N . In the following subsections, we discuss one by one the different effects that make ourresults differ from the benchmark one (2.22).
When taking into account stochastic effects, a split should be performed between the infraredscales described by the theory, which are sufficient larger than the Hubble radius, and theultraviolet modes. Incorporating this splitting via a smooth window function is physicallymotivated but results in a colored noise, which render the analysis technically more involved,and with results hardly depending on details of the window function as it becomes sharp (seee.g. [64–66]). As a result, a hard cutoff is usually used, with the introduction of a smallparameter w such that only modes with k ≤ waH are described by the stochastic theory. Wefollow this procedure, and for the amplitude of the noise, we use the analytical approximation Ref. [63] takes into account the possibility of reaching a static distribution in this massless case due to theeffects of boundary conditions. – 12 – = - w = - w = - - - - M / H f Figure 2 : Dependence on M /H of f in Eq. (2.23) , governing the amplitude of the stochas-tic noise in (2.16) , for different values of w . The dashed line represents the zeroth-orderdescription that is often used, with f = 1 for M /H < / and f vanishing for largervalues. of the power spectrum of a test scalar field of mass parameter M in de Sitter space, givingrise to the noise power spectrum (2.16) with f = (cid:112) π w / (cid:12)(cid:12)(cid:12) H (1) ν ( w ) (cid:12)(cid:12)(cid:12) , ν = (cid:112) / − M /H for M /H ≤ / (cid:112) π w / e − µ π (cid:12)(cid:12)(cid:12) H (1) iµ ( w ) (cid:12)(cid:12)(cid:12) , µ = (cid:112) M /H − / M /H ≥ / , (2.23)where H (1) ν is the Hankel function of the first kind. The dependence of f on M /H isdisplayed in figure 2 for three different values of w . For light enough scalar fields with M /H (cid:46) − , one recovers the standard amplitude of the noise usually considered in thestochastic formalism, i.e. f (cid:39)
1, with only a percent level deviation for all values of w .Naturally, the almost independence on w comes from the fact that such almost massless fieldsacquire an almost constant amplitude on super-Hubble scales. For M /h > /
4, fluctuationsdecay rapidly on super-Hubble scales, hence the strong dependence on w , and the very smallvalue of f (cid:46) − , which is well consistent with the zeroth-order description in which suchmassive fields are considered not to give rise to stochastic fluctuations. The intermediateregime 0 . (cid:46) M /H (cid:46) f depends on the arbitrary choice of w . This limitation of the currentformulation of the stochastic formalism motivates further studies, which are however beyondthe scope of this work. In the rest, we simply use w = 10 − , noting that our procedure hasthe advantage of not overestimating the noise in these “quasi-massive” situations comparedto the zeroth-order description often used, represented in figure 2 by the dashed line, Forinstance, in the same de Sitter approximation as in the previous section 2.3.1, the equilibrium– 13 – igure 3 : Impact of the time dependence of the Hubble scale H on the evolution of thevariance of the Higgs field. The three curves correspond to the solution (2.20) with f = 1 , fordifferent background evolutions: constant H , Starobinsky inflation and quartic inflation, allhaving the same value of the Hubble scale H (cid:63) e -folds before the end of inflation, ξ = − . ,and C = C = 0 in (2.14) . While in the plateau case the system almost relaxes towards thecorresponding de Sitter equilibrium, the final variance differs by one order of magnitude inthe quartic model. result (2.22) for the variance is multiplied by f , which, for values of ξ as small as 0 . H In comparison to previous works, we distinguish ourselves by evaluating the variance of theHiggs field on a time-dependent background. To emphasize its impact, here we consider thetime dependence of H alone, without including the effects of higher-order operators. Themass term M /H = − ξ (1 − (cid:15)/
2) induced by the non-minimal coupling receives a small (cid:15) correction, but it becomes important only in the last e -folds of inflation. A much moreimportant effect comes from the explicit time dependence of the noise term in (2.19): as H decreases during inflation, the amplitude of the stochastic kicks also decreases, and the cumu-lative effects on the variance (2.20) may be important, depending on the inflationary model.Obviously, one expects little deviation compared to the idealized description of constant H in plateau models of inflation, in which (cid:15) = − ˙ H/H is very small during the bulk of theinflationary evolution, to substantially grow only in the last e -folds. On the contrary, effectsare more pronounced in models with a steady decrease of H , like in monomial inflation.Eventually, note that taking the de Sitter equilibrium result (2.22), with its parametersevaluated at time N , is not in general a good approximation to the full time-dependent result.As already noted before in a general context, when H is evolving, this adiabatic equilibrium– 14 –s a good approximation only if the relaxation time is smaller than the time scale over which H varies [53]. The latter is given in slow-roll inflation by N H = 1 /(cid:15) so that the condition forapproximate equilibrium becomes N rel (cid:39) H M < (cid:15) = N H , (2.24)i.e. (cid:15) (cid:46) M /H , where the right-hand side is (cid:46) O (1) in situations with non-negligiblestochastic effects. For single-field plateau models, this is not very constraining, given theirvery small values of (cid:15) in the bulk of the inflationary evolution. However, with (cid:15) = ( H end /H ) /p it is easy to show that the above condition is never satisfied in the relevant range M (cid:46) H and for monomial inflation with an exponent p > N ∗ , i.e.the one corresponding to the plateau reached in the approximation of constant H .These expectations are confirmed by explicit numerical results in figure 3, where we showthe exact solutions (2.20) for the variance, in two examples which are representative of theabove classes: Starobinsky and quartic inflation, normalized with the same initial value H (cid:63) for the Hubble scale. We used f = 1 to focus on the effects of the time-dependence, we chose ξ = − .
01, and for comparison we display the corresponding solution with constant H = H (cid:63) .The differences between Starobinsky inflation and H = const are minor as they accumulateonly in the last e -folds, whereas the time-dependence of the inflationary background has animportant impact for quartic inflation, in which the final variance is comparatively decreasedby one order of magnitude. We stress that such kind of effects is all the more important asthe fate of the Higgs is exponentially sensitive to its variance, as we will see in section 3.3.The reader may wonder why we consider the model of quartic inflation, which is ruledout, for instance because it generates primordial gravitational waves with amplitude exceedingby far the observational constraints r < .
07 [67]. The reason is that quartic inflation isruled out in the sense of a single scalar field both driving inflation and generating primordialfluctuations. Here, on the contrary, we are only interested in the background dynamics, whichgoverns the time-dependence of the Hubble rate, and hence the amplitude of stochastic effects.Curvaton-type or more general multifield models may well have the same time-dependence of H as quartic inflation, without being ruled out by constraints on n s and r , which depend onthe precise mechanism at the origin of primordial fluctuations. Let us now incorporate the effects of the derivative operators (2.6), which contribute to theeffective mass of the Higgs as M /H (cid:39) − ξ (cid:16) − (cid:15) (cid:17) − C √ (cid:15) M Pl M + 4 C (cid:15) M M . (2.25)As mentioned above, we concentrate on the minimal case of Planck-suppressed operators, i.e. M = M Pl . With C and C order one numbers, the dimension-five and 6 operators induce– 15 – igure 4 : Impact of the dimension 6 derivative operator in (1.1) on the evolution of thevariance of the Higgs (normalized to H (cid:63) ). The three curves are for Starobinsky inflation, ξ = − . and C = 1 , , − , corresponding respectively to positive, vanishing and negativecurvature of the field space manifold. For each case the full versus dashed lines representthe evolution determined by the conventional Fokker-Planck equation (2.17) versus the phase-space one (2.27) discussed in 2.3.5. All curves are almost coincident for N ≤ . contributions to M /H of order √ (cid:15) and (cid:15) respectively. These contributions are small N (cid:63) e -folds before the end of inflation, although their contributions can already be similar to the oneof the non-minimal coupling, depending on parameters and models. More importantly, theirimportance increases as inflation proceeds, with O (1) contributions by the end of inflation,at which (cid:15) = 1. Depending on the inflationary model and the signs of C and C , the inducedmass term can be positive or negative, and with a specific time-dependence, resulting in variedresults. For simplicity, we only show in figure 4 the evolution of the variance in a situationwhere effects are expected to be the least pronounced: with the dimension 6 operator only(with C = ± (cid:15) substantially grows only in the last e -folds of inflation (see figure 6a for the effect of the dimension-five operator). One can seethat even in this situation, the effects of derivative operators are important, resulting in anincrease (respectively decrease) of the final variance for a negatively (respectively positively)curved field space, with respect to the situation without these operators. We note also thatany contribution to the mass term, like the one of the derivative operators, has two combinedeffects, one deterministic and one stochastic, which go in the same direction: a positivecontribution to M induces a steeper effective potential in the Langevin equation (2.15), anda decrease of the amplitude of stochastic kicks, both further stabilizing the Higgs (a negativecontribution acting in the other direction). We have checked that both of these (related)effects contribute substantially to the evolution of the variance.– 16 – .3.5 Stochastic formalism in phase space Since the effects of derivative operators emphasized in the previous section become increas-ingly important in the last e -folds of inflation, the reader might wonder if the assumption ofa slow-roll evolution, or more precisely of an overdamped evolution, hidden in the Langevinequation (2.15), is consistent. As we are going to show, taking into account the stochasticevolution in phase space does not modify significantly previous results.In phase space, the evolution is described by two coupled Langevin equations, one for h and one for its momentum π (cid:39) ˙ h = Hdh/dN , which can be written in general as dX a dN = h a + g aα ξ α , X a = { h, π } , (2.26)where ξ α are independent normalized Gaussian white noises, verifying (cid:104) ξ α ( N ) ξ β ( N (cid:48) ) (cid:105) = δ αβ δ ( N − N (cid:48) ). In the situation of interest here, a test scalar field with quadratic poten-tial, the amplitudes of the noises g aα do not depend on the X a ’s, i.e. the noises are notmultiplicative. There is no Itˆo versus Stratonovich ambiguity then [60, 68], the h a describethe deterministic dynamics (2.10), i.e. h a = { π/H, − (3 π + ∂ h V eff /H ) } , and the generalisedFP equation for the probability distribution in phase space W ( h, π, N ) reads ∂W∂N = L ( X ) · W, L ( X ) ≡ − ∂∂X a h a + 12 ∂ ∂X a ∂X b D ab , (2.27)where the diffusion coefficients D ab = δ αβ g aα g bβ are nothing else than the correlation functionsof the UV modes of h and π when they reach the IR sector at k = waH . Similarly as above,we take as initial conditions a Dirac distribution in phase space W ( h, π,
0) = δ ( h ) δ ( π ). Asthe dynamics is still linear, W subsequently follows a centered Gaussian distribution in phasespace, whose evolution of the variances is simply obtained from Eq. (2.27) as ∂ (cid:104) h (cid:105) ∂N = 2 H (cid:104) hπ (cid:105) + D hh ,∂ (cid:104) hπ (cid:105) ∂N = − M H (cid:104) h (cid:105) − (cid:104) hπ (cid:105) + 1 H (cid:104) π (cid:105) + D hπ ∂ (cid:104) π (cid:105) ∂N = − M H (cid:104) hπ (cid:105) − (cid:104) π (cid:105) + D ππ . (2.28)Different prescriptions for the diffusion coefficients have been discussed in the literature [52,54, 56, 60, 69–72]. In figure 4, we used the simple one D hh = ( Hf / π ) , with the othercoefficients vanishing, corresponding to neglecting the stochastic kicks of the momentum π .It is apparent that the effects of derivative operators and of the time-dependence of H weare interested in are well described by the conventional stochastic approach (2.17) of previoussections. In other words, considering the difference between the conventional field-spaceapproach and the phase-space one as a measure of the theoretical uncertainty of phase-space– 17 –ffects, we can see that the latter is negligible for our purpose. Eventually, while figure4 considers the effect of the dimension-six operator, the same conclusion is reached for thedimension-five operator, see figure 6a. As a result, in the rest of the paper, we stick to theconservative stochastic approach described by the Fokker-Planck equation (2.17).
In this section we explain our procedure to extract, from the evolution of the PDF P ( h, N )studied in section 2, the fraction of AdS patches that can reside in our past light cone. A fewprecautionary words: the word fraction is used in a probabilistic sense here, and the approachused in this paper, as any study of the Higgs stability during inflation, relies on samplingsomething that is by definition unique, i.e. our observable universe. As a consequence, if notsatisfied, the inequality in Eq. (3.1) below would not necessarily imply that our universe cannot exist with these initial conditions and parameters, but would tell us instead that it isvery unlikely. We label with F AdS the fraction of patches in AdS at the end of inflation. Following Ref. [14],the existence of our universe as we know it, with the Higgs in the electroweak vacuum, requiresthe following bound to be satisfied: F AdS × N < , (3.1)where N represent the number of Hubble patches present at the end of inflation in the volumegiving rise to our observable universe today, i.e. N = H − / ( a H − /a end ) (cid:39) e N (cid:63) . We mightas well be interested in the fraction of patches that can potentially lead to AdS regions,despite being still safe at the end of inflation. The fate of Hubble patches with values ofthe Higgs greater than the location h max of the potential barrier depends on the details ofthe post-inflationary dynamics (see Ref. [14] for details and Ref. [21] for a scenario in whichreheating does not happen instantaneously).After inflation, the Higgs potential receives thermal corrections from the SM bath, con-tributing to the Higgs mass as [73]: M T (cid:39) T ( a ) e − h / (2 πT ) , (3.2)with T ( a ) = 1 . T m a − / (1 − a − / ) / and T m = 0 .
54 (0 . H end M Pl T RH ) / , where T RH is thereheating temperature, and we set a = 1 at the end of inflation. The thermal contribution While we do not fully understand the motivations from the authors for this prescription, we have alsochecked that we obtain very similar results when using D ππ = (3 H f/ π ) , with other coefficients put to zero,which is advocated in [52, 56, 70] (with f = 1). This can be seen as a further proof that the phase-spacetheoretical uncertainty is negligible for our purpose. – 18 –dds to the one coming from the mass M induced by the non-mimal coupling and thehigher-order operators. This term after inflation becomes M (cid:39) (cid:16) − ξ − √ C + 2 C (cid:17) H a , (3.3)where we have assumed that, while the inflaton is oscillating about the minimum of itspotential, the Universe experiences a matter dominated phase so that H = H end /a / and (cid:15) = 3 /
2. These equalities have to be thought as the result of averaging over many oscillations.Thus, the rescuing ability of these corrections depends on the reheating temperature T RH aswell as on C , C and ξ . The interplay between the decay of the mass contributions (3.2)and (3.3) and the dynamics of the Higgs during this stage determines if a patch with a givenvalue h is brought back to the safe region | h | < h max before M + M T becomes negligible. Inshort: any given set of parameter corresponds to a maximum value of h , usually labelled as h end , that can be rescued and brought back safely to the EW vacuum. As can be seen from(3.3), our operators during the post-inflationary phase are relevant, shifting ξ by an orderone number, i.e. ξ → ξ eff = ξ + √ C − C . One could thus follow the same procedure as in[14], but now with a new effective ξ , to determine h end .For sufficiently large reheating temperature the thermal corrections always dominate andfor approximately T RH (cid:38) Gev, patches (which are not yet in AdS) with arbitrary largevalues of the Higgs can be rescued. Conversely, for low reheating temperature T RH (cid:46) GeV,and no induced mass coming from the additional operators, i.e. ξ = C = C = 0, any patchin which | h | > h max will end up forming an AdS region. Thus, less conservative bounds canbe derived by asking that there was no patch of that type at the end of inflation: F | h | > h max × N < , (3.4)with F | h | > h max the fraction of Hubble regions where | h | > h max .It is worth mentioning that even for negligibly small thermal corrections, but ( ξ, C , C ) (cid:54) =0, the maximum value of the Higgs that can be rescued thanks to the post-inflationarydynamics ( h end ) is indeed different from h max . On top of that, the exact determination of h end also depends on how the reheating phase is modelled. Thus, given that the main interestof this work concerns the dynamics during inflation, we show results only for the two bounds(3.1) and (3.4), corresponding to cases where the impact of the post-inflationary dynamics ismaximal and minimal respectively. The picture of a spectator Higgs field undergoing a stochastic motion and subject to aquadratic potential is (obviously) not the right description in a patch that is falling towards In particular, in Ref. [14], it is shown, for the illustrative case ξ ∼ − . T m ∼ h max and H end /h max (cid:38) (see their Fig. 9). In our case, we expect the rescuing effect to be amplifiedand to become relevant even for higher reheating temperature, or analogously, to provide the same effect inabsence of thermal corrections as the one given by higher reheating temperature. This expectation is motivatedby the shift of order one that the higher-order operators induced on ξ in (3.3). – 19 –he true vacuum and forming an AdS region. At large enough field values, the effect of thequadratic SM potential is not negligible anymore, so that the PDF becomes non-Gaussian,and more importantly, the backreaction of the Higgs on spacetime becomes important. Giventhe complexity of the system, all approaches used to model it should rely on some approx-imation schemes. Thus, before illustrating our procedure, we find it convenient to discussdifferent ones present in the literature.First, let us define h cl as the Higgs value at which the dynamics becomes classicallydominated. It can be estimated by requiring that the deterministic part driven by the effectivepotential in the Langevin equation (2.15) overcomes the noise term: h c l : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ h V eff H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Hf π , (3.5)and h cl is such that if h (cid:38) h cl > h max the Higgs will classically roll towards the true vac-uum. As a first approximation, it was assumed in Ref. [14] that once the Higgs reaches h cl ,it instantaneously forms an AdS region. This way of proceeding brings some important sim-plifications. Since up to | h | (cid:46) h cl the contribution coming from the SM quartic potential canbe neglected to a good approximation, one can model the PDF with a Gaussian in the bulkregion [ − h cl , h cl ], and cut its tails at | h | ≥ h cl . Then, the fraction of patches in AdS can beestimated by computing F AdS = 1 − P ( | h | < h cl ) at the end of inflation. Later, in Ref. [15],the finite time to fall in AdS from h cl has been taken into account in the following way: theFP equation (2.17) was used beyond the value h cl , although the noise becomes negligible then,which enables one to capture the non-Gaussian tails induced by the classical effects of the SMpotential. This procedure was applied up to the value h ¯s¯r where the backreaction of the Higgswas estimated to generate very rapidly an AdS region. Similarly as above, the fraction ofAdS patches was then computed as F AdS = 1 − P ( | h | < h ¯s¯r ) at the end of inflation, returningresults similar to the Gaussian approximation of Ref. [14], which is expected in situationswhen a steady PDF is reached much before the end of inflation.We are now in the position to highlight one of the key differences compared to previousworks. In all previously studied setups (at least to the best of our knowledge), the effectivepotential is a static function during inflation. As a consequence, the fraction of patchestransmuting in AdS can only increase during inflation, i.e. the PDF initially peaked at zerocan only flatten while its tails become fatter. As we discussed in section 2.3, the effects we areinterested in, from higher-order derivative operators and de Sitter departure, are genuinelytime-dependent, and they can act as rescuing processes shrinking the PDF while inflationproceeds. Therefore, we have to carefully take into account that patches once in AdS cannotbe brought back to the safe region for the Higgs field. In other words, we have to model adistribution that is losing part of its tails, and which later cannot be re-introduced in the In Ref. [14], h cl is determined in an almost equivalent way by considering where the deterministic termovercomes the stochastic one in the FP equation. This gives a slightly different result which does not affectour conclusions. We prefer to use Eq. (3.5) to determine h cl as it is independent of the Gaussian ansatz forthe PDF, ansatz that precisely breaks down around h cl . – 20 –ulk when the distribution following the FP equation is shrinking. Furthermore, the timedependence introduced by our analysis modifies on a case by case basis the time to fall inthe AdS vacuum, as well as the value of h cl computed from (3.5). Thus, for any given setup(model for the evolution of H , value of the non-minimal coupling and values of the Wilsoncoefficients for the derivative operators) we proceed in the following way: • We model the evolution of the PDF with a Gaussian satisfying the FP equation (2.17)up to the value | h | = h cl . The point of classicality computed from Eq. (3.5) changes overtime in a different way for each framework. For instance, the decrease of H dynamicallyextends the range of the classical region. Implicitly we can write h c l ( N ) ≡ h c l ( H ( N ) , ξ, C , C , V SM ) . (3.6) • In the classical region, we numerically trace the evolution of the full two-field inflaton-Higgs system, including the backreaction of the Higgs on the background. We considerthe fall in AdS unavoidable when the Hubble parameter (in the Einstein frame) becomesnegative, i.e. H E < At each time N of the evolution we compute the number of e -folds N AdS necessary to fall in AdS with initial conditions given by h = h cl ( N ). Wesay that h cl ( N ) ∈ AdS if N AdS < N (cid:63) − N , (3.7)where N (cid:63) − N is the number of e -folds left before the end of inflation when the Higgsstarts its classical dynamics at the value h cl ( N ). • We estimate the fraction of patches in AdS at the end of inflation by computing themaximum area of the distribution under the tails, namely for | h | > h cl ( N ), amongst thetimes N such that h cl ( N ) ∈ AdS, i.e. that leave enough time before the end of inflationfor a patch of value h cl ( N ) to fall in AdS. In order to exclude the possibility of AdSpatches in our past light cone, we impose the bound from Eq. (3.1): F AdS = max { N : h cl ( N ) ∈ AdS } [ P ( | h | > h cl ( N ) , N )] = P ( | h | > h cl ( N m ) , N m ) < e − N (cid:63) , (3.8)where we call N m the time at which the maximum is evaluated. Because of the timedependence of h cl ( N ) and the finite time to fall in AdS from there, it is worth stressingthat the maximum is not necessarily reached at the time when the variance of the PDFhas grown to its largest value. In a static situation, an early work on the subject analytically solves the FP equation with boundaryconditions that act as sinks at some given Higgs values [16]. This is not possible in our case, because of thetime dependence of the background, together with the fact that the point where the Higgs is trapped in AdSchanges with time. Attacking this problem fully numerically is an interesting option, but goes beyond thescope of this work. In our simulations this happens less than one e -fold after the time where the full potential crosses zero,i.e. V inf ( φ ) + V SM (cid:39) – 21 – For the purpose of estimating the fraction of patches where | h | > h max at the end ofinflation, we take the PDF evolved until then, i.e. P ( h, N end ). This can be used on thecondition that P ( | h | < h max , N end ) < P ( | h | < h cl , N m ) ≡ − F AdS . (3.9)This is imposed to exclude (approximately) patches that are judged safe at the end ofinflation according to the FP evolution alone, but that have actually classically fallenin AdS before. As an approximate way of taking into account these AdS tails that cannot be rescued, and following Eq. (3.4), we therefore impose the bound F | h | >h max = max [ P ( | h | > h max , N end ) , F AdS ] < e − N (cid:63) (3.10)in order to exclude the possibility of patches with | h | > h max in our past light cone.Before moving on, it is worth mentioning another possibility one can in principle follow toperform the analysis. The reader may indeed wonder why one does not simply sample a largenumber of evolutions using the Langevin equation with initial conditions given by h = 0. Thenfit the distribution of the final Higgs values at the end of inflation with a PDF and with thatcompute the survival probability. Unfortunately, proceeding in this way would overestimatethe impact of the effects studied in this paper. Indeed, given the smallness of the probabilitieswe are considering (see Eq. (3.1)), any reasonable number of realizations would always probethe central part of the distribution. This can be correctly and safely extrapolated to computethe tails in a static case, like in Ref. [22]. However, in our situations, it would return the PDFevolved with the FP up to the end of inflation. As already mentioned, this distribution ignoresthe important fact that patches which are in AdS at a given time during inflation cannot lateron be brought back to the safe region for the Higgs, and therefore is not trustworthy. Before moving to the full numerical results it is useful to draw a few analytical considera-tions about the bounds (3.8)-(3.10). As discussed in the previous section, the PDF can beapproximated with a Gaussian distribution. Thus, from Eq. (3.8), we obtain F AdS ≡ P ( | h | > h cl ( N m ) , N m ) = 1 − erf( x ) (cid:39) e − x √ πx < e − N (cid:63) , x ≡ h cl ( N m ) √ σ ( N m ) . (3.11)Given the very small probability e − N (cid:63) we are considering, with N (cid:63) (cid:39)
60, the approximationof the error function is robust, and N m can be estimated by minimizing h cl ( N ) /σ ( N ) withinthe domain where the Higgs has enough time to fall in AdS before inflation ends, see Eq.(3.7). If the time dependence of H was ignored, h cl would be constant, and N m would occurat the maximum of σ . This is not our case though, and different possibilities can arise. Forinstance, in situations where h cl decreases and σ grows, N m is simply the latest time at which h cl ( N ) ∈ AdS. If both h cl and σ decrease, the competing effect between the evolution of the– 22 –lassicality point and the one of the variance determines whether a given setup alleviates orworsens the Higgs instability.To estimate h cl from Eq. (3.5) we first approximate the SM potential as [14] V SM (cid:39) − b ln (cid:18) h h √ e (cid:19) h , (3.12)with b (cid:39) . / (4 π ) for central values of the SM parameters. When only the SM potential ispresent, Eq. (3.5) can be solved exactly: h (4) ≡ αf H W (cid:104) αH h (cid:105) / , (3.13)with α (cid:39) · π/ .
16 and where W is the Lambert function (or product logarithm) functiondefined as the inverse function of f ( y ) = ye y , i.e. z = W ( z ) e W ( z ) . In the presence of aquadratic term in the effective potential, that we write schematically as V eff = V SM + V (2) ,Eq. (3.5) has no exact solution. However, h cl is well approximated by the value ˜ h at which | ∂ h V SM | = ∂ h V (2) = ⇒ ˜ h = M b W (cid:104) M bh (cid:105) / , (3.14)if ˜ h > h (2) , where h (2) is such that ∂ h V (2) = 3 H f / π , which means that the quadratic termdominates at small field values. Analogously if ˜ h < h (2) , which happens for very small massesso that V (2) is negligible, we can use (3.13) to approximate h cl . Considering for definitenessthe case where h cl (cid:39) ˜ h in (3.11), we arrive at the boundNo AdS patch : H ( N m ) h max < √ N (cid:63) ˆ σ e β/ ˆ σ , ˆ σ ≡ σ ( N m ) H ( N m ) , (3.15)with β = M / (12 H N (cid:63) b ) | N = N m . This generalizes the one used in Ref. [14], to which itreduces when assuming a constant Hubble scale and the variance given by the equilibriumsolution in Eq. (2.22). It is worth stressing the exponential factor in the right-hand side,which renders the bound above very sensitive to even small changes in ˆ σ and M /H . Thisis indeed very important as we have seen that the various effects studied in this paper havein general a substantial impact on these quantities, and notably on the (normalized) varianceof the Higgs.Finally we discuss the bound (3.10) coming from the request of avoiding patches with | h | > h max . With precautions spelled out in Sec. 3.2, one may use for this purpose thevariance evaluated at the end of inflation, which, analogously to Eq. (3.11), gives rise toNo patch over the barrier : H (cid:63) h max < (cid:113) N (cid:63) ˆ σ , ˆ σ ≡ σ ( N end ) H (cid:63) , (3.16)– 23 –here we normalized the variance with respect to H (cid:63) , the value of the Hubble scale when thepivot scale exits the Hubble radius. This bound is less sensitive to changes of the variableˆ σ . Nevertheless, we have seen that the latter may vary by orders of magnitude whenvarying the inflationary model (at fixed H (cid:63) ), or details of Planck-suppressed operators, whichrenders this bound also an interesting probe of these aspects. When approximating inflation with an exact de Sitter phase, and neglecting possible con-tributions from Planck-suppressed operators, the fate of the Higgs only depends on the SMparameters measured at the EW scale, the energy scale during inflation H (cid:63) , and the value ofthe non-minimal coupling ξ . Motivated by our study in Sec. 2, our aim in this section isto exemplify the additional sensitivity of the fate of the Higgs on the time dependence of theinflationary background, and on derivative Planck-suppressed operators.For the latter, we vary the two Wilson coefficients C and C in Eq. (2.6) in the range {− , , } . To model the time evolution, we use two different background dynamics. Thefirst is given by a plateau type inflationary potential `a la Starobinsky [74], which gives riseto an evolution for the Hubble rate as H ( N ) (cid:39) H ∗ exp[ O (1) / ( N − N end )]. In that case, (cid:15) and H are nearly constant when the inflaton evolves along the plateau of the potential, and H changes only by an order one factor in the last e -folds. In contrast, in monomial potentialsthese quantities have a non-negligible evolution throughout the inflationary phase (as wehave seen, with the consequence of the system never reaching the de Sitter equilibrium formonomial inflation with exponent greater than 2), e.g. the Hubble rate evolves as H ( N ) = H ∗ (1 + N end − N ) / (1 + N end ) for a dynamics `a la quartic. In our numerical results, we thusmake use of the inflationary potentials V ( φ ) = Λ (1 − e − √ / φ/M Pl ) and V ( φ ) = λφ . In allscenarios, we take into account the deviation of the amplitude of the stochastic noise fromthe massless limit, as set by the function f in Eq. (2.23). Different results obtained for thetwo background evolutions highlight one of our main points: the bounds are sensitive to thede Sitter departure and are therefore inherently model-dependent.Following the procedure explained in Sec. 3.2, we numerically computed the fractionof AdS patches at the end of inflation, and the fraction of patches in which the Higgs hasfluctuated above the potential barrier, for different scenarios. We present these results byshowing constraints on the energy scale of inflation in section 4.1, and on the SM parametersin section 4.2, both coming from the requirement of not having a single patch in AdS in ourpast light cone. As already discussed, we are not interested in the post-inflationary dynamics, which would introduce atleast one extra parameter dependence through the reheating temperature. In this respect, the two bounds(3.1) and (3.4) should be thought of as the two extreme limits of the effects of thermal corrections to the Higgspotential after inflation, one where there is no rescuing effect ( T RH (cid:46) GeV), and the other with maximumrescuing effects ( T RH (cid:38) Gev). – 24 – .1 Bounds on the energy scale of inflation
For each scenario that we consider, i.e. a given evolution of the scale factor during inflationand precise Planck-suppressed couplings, we compute F AdS and F | h | > h max for different valuesof the non-minimal coupling ξ and of H (cid:63) /h max . The results are displayed in figure 5, wherethe three coloured regions, following Ref. [14], are defined as follows: • The red region is the part of parameter space where there is (on average) at least oneHubble patch in AdS at the end of inflation, hence the corresponding model cannotdescribe our observable universe. • In the orange region, at least one patch has fluctuated above the potential barrier atthe end of inflation, but without falling into AdS. These patches may or may not berescued depending on the post-inflationary dynamics, and in particular the reheatingtemperature. Thus, we label this region as potentially unsafe . • In the green region, not a single patch has fluctuated above the barrier during inflation,i.e. both bounds (3.1) and (3.4) are satisfied, and the Higgs safely rolls towards ourelectroweak vacuum after the end of inflation.
In figure 5a, we first look at the time-dependent effects alone, so that we set to zero both C and C in the Higgs effective potential. In black dashed lines we highlight the boundaries ofthe green-orange-red regions under the assumptions of a de Sitter background ( H = constant)and noise amplitude given by the one of an exactly massless scalar field, i.e setting f = 1. Ourmotivation is to provide a direct comparison between our results and previous ones present inthe literature [14]. In this respect, one can notice the significant shift of the boundary at largenegative values of ξ . This comes from the increased mass of the Higgs fluctuations, whichresults in the suppression of the amplitude of the noise, i.e. f <
1. Hence, the transition tothe region in parameter space where the stochastic noise is irrelevant now appears smoothed.The dashed red line indicates the boundary of the would-be red region if we had ignoredthe finite time to fall into AdS, i.e. if we had simply considered the probability to fall to AdSat its maximum value during the evolution. The difference with the actual boundary of thered region highlights the importance, in our approach, to consider the classical evolution of thetwo-field system. Since the time to fall into AdS is a non-trivial quantity in a time-dependentbackground, and given that it is highly setup-dependent, we provide further details about itin appendix A.In order to better understand the differences between the two models in figure 5a, letus recall that the time dependence of H leads to two effects that play a role in determiningthe fate of the vacuum instability, and that may compete: the decrease of the variance (seeSec. 2.3.3), and the time dependence of h cl , the point after which the dynamics becomesclassically dominated (see Eq. (3.5)). When the variance decreases, the probability of being– 25 – a) Fate of the Higgs without derivative opera-tors, for Starobinsky-like and quartic-like inflation-ary evolutions. (b)
Impact of the dimension-six operator on the fateof the Higgs for quartic inflation.
Figure 5 : Cosmological fate of the Higgs for different evolutions of the scale factor andparameters. The green region represent scenarios where there is not a single Hubble patch inour past light cone in which the Higgs has fluctuated above the barrier. The orange regionrepresents the potentially unsafe scenarios, in which there is no patch in AdS at the end ofinflation, but there exist patches above the barrier that can potentially become AdS regionsdepending on the post-inflationary dynamics. In red: the region in which there exists atleast one Hubble patch in AdS at the end of inflation, and the corresponding model cannotdescribe our observable universe. The red dashed curved on the left plots highlights the would-be boundary of the red region if the finite time to fall into AdS was not taken into account.All plots are obtained considering the noise in the stochastic process given by the function f in Eq. (2.23) . The dashed black lines on the top left plot mark off the boundaries between thethree regions for the benchmark analysis assuming H constant and f = 1 . beyond an arbitrary fixed value of the Higgs diminishes. Yet, h cl is a dynamical quantity thatmay decrease at such a rate that compensates for this effect, resulting in a net increase of P ( | h | > h cl ).For Starobinsky-like evolution, as discussed in Sec. 2.3, for sizeable enough values of ξ , thevariance initially reaches the value of the de Sitter equilibrium associated to H (cid:63) . Afterwards,the variance and h cl decrease with the net effect that the probability of being beyond h cl increases. However, this occurs only in the last e -folds of inflation, so that patches withvalues around h cl do not have the time to fall in AdS. Hence, in this situation, our carefulway (3.8) of computing F AdS , which determines the position of the red-orange boundary, doesnot lead to an important difference compared to de Sitter zeroth-order result (dashed blackline). The only notable difference concerns the boundary between the allowed (green) and– 26 –otentially unsafe (orange) region: it is lifted due to the suppression of the noise coming fromthe function f , and the decrease, as H diminishes, of the probability of being beyond thefixed value h max .The larger rate of change of H in quartic-like evolution, despite reducing the varianceat a greater rate, results in an expansion of the disallowed (red) part of parameter spacecompared to plateau-like models. Conversely, as h max is a fixed point, the quartic evolutionacts as rescuing when we look at the boundaries between the green-orange regions. There, theshrinking of the variance leads to recovering patches that would have been otherwise beyondthe potential barrier at the end of inflation.More intuitively, one can also understand the above physical consequences of the time-dependence of H as follows. The decrease of H determines a smaller size of the randomkicks of the Higgs field. Thus, regions where the Higgs is just above the potential barriercan be more easily rescued under the same positive mass term (as the one induced by thenon-minimal coupling ξ ). This leads to a larger green region for quartic inflation in figure5a. At the same time, regions where the Higgs has fluctuated far away beyond the potentialbarrier could not be rescued anymore given the smallness of the quantum jumps. Theseregions become effectively classically dominated, with the only option to fall into AdS. Thisleads to a larger red region for quartic inflation in figure 5a. In figure 5b, we show the effects of the dimension-six Planck-suppressed operator for quarticinflation, contrasting the cases of a positive and negative curvature of the inflaton-Higgs field-space manifold, corresponding respectively to C = 1 (top) and C = − ∝ C (cid:15)h . At the same time, the corresponding increase (resp.decrease) of the mass of the Higgs’ fluctuations reduces (resp. enhances) the amplitude ofthe stochastic noise. Both effects act in the same direction in the two cases. They tend todecrease the variance of the PDF for positive curvature, and to increase it for negative one.As a consequence, the overall (de)stabilizing effect is visible in figure 5: for positive curvatureof the field-space manifold (top right plot), the red region shrinks and the red region expands,while a negative curvature has the opposite effect (bottom right plot). Quantitatively, thisimplies that, for given values of ξ and SM parameters, varying the details of Planck-suppressedcouplings between the Higgs and the inlaton modifies the constraint on the Hubble scale byorders of magnitude, which is rather remarkable.Let us also highlight the impact on our results caused by considering the finite time to fallinto AdS. To compute the fraction of patches in AdS, we take the maximum of P ( | h | > h cl )in the domain (3.7), which is cut at a given e -fold by requiring that h cl ( N ) has enough time– 27 – a) Evolution of the variance of the Higgs distri-bution (normalized to H (cid:63) ) for ξ = − . . Foreach case the full versus dashed lines represents theevolution determined by the conventional Fokker-Planck equation (2.15) versus the phase-space one (2.27) discussed in 2.3.5. (b) Like in figure 5, green, orange and red regionscorrespond respectively to safe, potentially unsafeand excluded scenarios.
Figure 6 : Impact of the dimension-five operator in (2.6) on the fate of the Higgs for Starobin-sky inflation. to fall into AdS before inflation ends. For positive curvature, this maximum always occursbefore the time when there would not be enough e -folds left to fall into AdS. Hence includingour cut does not affect the final results. Conversely, for negative curvature, P ( | h | > h cl ) keepsgrowing until the end of inflation, and it is therefore crucial to take into account this cut inorder not to significantly overestimate the effect of the derivative Planck-suppressed operator.Finally, it is worth pointing out how the impact of derivative higher-order operators onthe bounds is tight to the underlying background evolution. For instance, the dimension-six operator has tiny effects on Starobinsky-like evolutions (indeed fig. 5a would slightlychange only for the case of negative curvature), for analogous reasons to those implying thatplateau models lead to bounds close to the ones found in the de Sitter approximation, i.e.the change in P ( | h | > h cl ) does not happen early enough during inflation. On the contrary,the dimension-five operator, as it gives a contribution to the effective potential ∝ C √ (cid:15)h (instead of ∝ (cid:15) ), already influences significantly Starobinsky-like models, and has an even moredramatic impact on quartic inflation. For completeness, we illustrate explicitly the effect ofthe dimension-five operator on Starobinsky-like inflation in figure 6. In figure 6a we showthe evolution of the variance for a given value of ξ and Wilson coefficients C = { +1 , , − } .In comparison to figure 4 (where deviations from the situation with no Planck-suppressed– 28 – a) Effect of the dimension-5 operator onStarobinsky-like models. (b)
Effect of the dimension-6 operator on quartic-like models for H = 10 GeV . Figure 7 : Bounds on the top/Higgs masses from the energy scale of inflation for ξ = − . , α s = 0 . and T RH (cid:46) GeV (equivalent to demanding no Hubble patch with | h | > h max at the end of inflation), for different setups listed in the legend. Any given scenario marks aline separating the excluded region in parameter space (above) from the allowed one (below).In green, the stability region where the quartic Higgs coupling stays positive up to the Planckscale. The dashed black line on the right plot stays for the benchmark analysis done assuming H = 10 GeV = const and f = 1 . operator occur only in the last 10 e -folds for Starobinsky-like inflation), deviations occurearlier and also lead to a variance that is orders of magnitude different at the end of inflation.Indeed, the overall (de)stabilizing effects, represented in figure 6b for C = ±
1, are importantthroughout parameter space, with magnitudes similar to the effects that the dimension-6operator has on quartic inflation (which is expected, as (cid:15) quartic ∼ √ (cid:15)
Starobinsky ). The study of the Higgs instability during inflation can be equivalently applied to constrainthe SM parameters within their experimental error bars. In fact, as already mentioned, therunning of the Higgs quartic coupling is highly sensitive to the EW boundary conditions.Thus, it is instructive to look at the outcomes of our analysis from this different perspective.The SM parameters over which our results are sensitive to are the top and Higgs masses( M t , M h ) and the strong coupling constant α s . The biggest uncertainty (both experimentaland theoretical) comes from determining M t (see [75, 76] for recent discussions on the sub-ject). For illustrative purposes, in order to compare our results with experimental data, we fix α s to its central value α s = 0 . M t , M h ) within the five-sigma boundariesfrom their best current estimate: M h = 125 . ± .
14 GeV and M t = 172 . ± . a) Effect of the dimension-5 operator onStarobinsky-like models. (b)
Effect of dimension-6 operator on quartic-likemodels for H = 10 GeV . Figure 8 : Bounds on the top/Higgs masses from the energy scale of inflation for ξ = − . , α s = 0 . and T RH (cid:38) GeV (equivalent to demanding no Hubble patch in AdS at theend of inflation), for different setups listed in the legend. Any given scenario marks a lineseparating the excluded region in parameter space (above) from the allowed one (below). Ingreen, the stability region where the quartic Higgs coupling stays positive up to the Planckscale. The dashed black line on the right plot stays for the benchmark analysis done assuming H = 10 GeV = const and f = 1 . M t in particular we take the quoted direct measurements value from the PDG [77]. For eachvalue of ξ , a given H (cid:63) , hopefully given in the future by a detection of primordial B-modes,marks a line dividing the ( M t , M h ) plane in two regions: below, the allowed (safe) region inparameter space (no patches in which the Higgs has fluctuated above the potential barrier),above the (potentially) unsafe region in which dangerous patches have formed (meaning re-gions in which h > h max in figure 7, or AdS regions in figure 8). In the same manner as in theprevious section, for a few parameters of interest, we study how these bounds change oncethe various effects considered in this work are taken into account.In figure 7 we fix ξ = − .
05 and consider F | h | >h max , so that each setup provides a lineseparating the safe region below from the potentially unsafe one (above), equivalently to theboundary between the green and orange regions in figure 5. Two scales of inflation are usedin figure 7a, namely H (cid:63) = 10 GeV and H (cid:63) = 10 GeV, for Starobinsky-like inflationaryevolutions. For each of them, we considered the dimension-five operator (alone) by varying C = 0 , ±
1. In the right figure 7b, we consider the dimension-6 operator alone, C = 0 , ± H ∗ = 10 GeV. This energy scale correspondsto a tensor-to-scalar ratio of order ∼ − , the lowest one that can be observationally probedin the near future [78].Unsurprisingly, increasing H (cid:63) always shrinks the allowed region. This has been shownpreviously in [19, 20, 22] and is easy to understand; larger H (cid:63) imply larger stochastic kicks.Thus, under the same conditions, it is more likely that the Higgs ends up beyond the potentialbarrier. More interestingly, depending on the sign of the Wilson coefficient, for plateau-likemodels the dimension-5 operator has an important stabilizing or destabilizing effect, similarto the one the dimension-6 operator has on quartic-like models (even if the latter is larger)for the reason mentioned in the previous section. Indeed, one can see in figure 7a that achange of H (cid:63) by two orders of magnitude can be otherwise mimicked by simply consideringthe effects of a Planck-suppressed operator (see for example the solid and dashed blue linesversus the solid blue and solid brown). Eventually, in figure 7b, the dashed black line marksthe boundary for the benchmark study, in which the time dependence of the backgroundand the deviation from massless noise ( f (cid:54) = 1) are not taken into account. The appreciabledifference between the dashed black and the blue solid line thus highlights the importance toinclude these effects in the analysis.In figure 8 we consider F AdS , the fraction of patches already in AdS at the end of inflation,with the same setup as in figure 7 but here for ξ = − .
01. The various lines, equivalent to theboundary of the red regions in figure 5, split the parameter space between the allowed region(below) and the excluded one (above), which cannot be rescued by any post-inflationarydynamics. Given the exponential sensitivity of the bound (3.8) to models’ parameters, thatwe understood analytically in Sec. 3.3, for larger values of ξ , the shift of the various lines canbe as pronounced as to exit the five-sigma contours, meaning a complete rescuing effect. Inthe current example of ξ = − .
01 (chosen for illustrative purposes), one can already see howeasily the effects studied in this work can alleviate possible tensions between the central valuesof measured SM parameters and typical expected values for the energy scale of inflation.
We revisited the important question of the stability of the Higgs vacuum during inflation,by taking into account features of realistic models that have been hitherto overlooked: theunavoidable time-dependence of the Hubble scale during inflation, and the generic presenceof derivative operators coupling the Higgs and the inflaton. A motivation for looking atthe latter aspect is the well known fact that higher-order operators suppressed by a highenergy scale can have a critical impact on effective masses of scalar fields during inflation, asexemplified by the eta-problem and the geometrical destabilization of inflation.We studied these aspects in a simple but rather generic manner. We considered differentinflationary backgrounds and enlarged usual setups by considering two-derivative higher-orderoperators that are inflaton shift-symmetric, keeping track of the effects of dangerous irrele-vant operators. We focused for simplicity on operators suppressed by the Planck scale, as– 31 –e demonstrate that even these ones have significant consequences. We showed that one caninitially neglect the backreaction of the Higgs on the inflaton, and consider that the formerundergoes a stochastic motion subject to a time-dependent effective potential. This comprisesnot only to the SM potential and quadratic potential induced by the non-minimal coupling ofthe Higgs, like in previous studies. It also has two additional quadratic contributions gener-ated by specific dimension-5 and dimension-6 operators. The corresponding induced massessquared, in Hubble units, can assume any sign and are proportional respectively to √ (cid:15) and (cid:15) = − ˙ H/H , the usual slow-roll parameter, as a consequence of their kinetic origin. There-fore, their quantitative impact depends on the specific evolution of the Hubble scale duringinflation, and is inevitably tied to the other aspect that distinguishes our work from previousones, i.e. considering the time-dependence of the inflationary background. We stress thatdespite the apparent smallness of these mass terms, they can have a crucial impact on thecosmological fate of the Higgs vacuum.We considered the Fokker-Planck equation that governs the evolution of the distributionof Higgs’ values in Hubble-sized regions. We showed explicitly that the effects caused bythe time dependence of the background, Planck-suppressed derivative operators, and thestochastic noise of light fields differing from the one of exactly massless ones, have importantconsequences for the distribution of Higgs values, and hence for the fate of the Higgs.Previous works showed that not a single Hubble patch in our observable universe at thebeginning of the radiation era should be such that the Higgs reached sufficiently large valuesas to form an AdS patch, i.e. a crunching region surrounded by a causally disconnectedone of negative energy density. However, not all patches in which the Higgs has fluctuatedabove the potential barrier share this fate. Depending on the reheating temperature, thermalcorrections to the Higgs potential can go from rescuing regions (which are not yet in AdS)with arbitrarily large Higgs values, to rescuing none. Therefore, we used two different criteria,corresponding to these two extreme situations, to qualify each model either as excluded,allowed, or potentially unsafe. By doing so, and owing to the inherent time-dependence ofour effective potential, we had to pay attention to the fact that patches already in AdS cannotbe rescued, as well as to the finite time it takes for them to form when the Higgs backreactioncannot be neglected anymore, resulting in a new procedure explained in section 3.In our numerical analysis, we considered two different inflationary backgrounds, corre-sponding to Starobinsky and quartic inflation, meant as representative of models with re-spectively negligible and appreciable time dependence of the Hubble scale in the bulk of theinflationary phase. We also varied the Wilson coefficients of the dangerous dimension-5 and-6 operators, the value of the non-minimal coupling and the overall Hubble scale. As forthe purely Standard Model sector, we varied the Higgs and top masses measured at the elec-troweak scale within their experimental error bars. An obviously important parameter for thefate of the Higgs is the ratio H (cid:63) /h max between the Hubble scale, setting the overall amplitudeof stochastic kicks, and the location of the potential barrier. This shows that results can beseen from two complementary perspectives: as bounds on the energy scale of inflation, or as– 32 –ounds on SM parameters governing the location of the potential barrier. We adopted thetwo viewpoints and summarized our main numerical results in figures 5 and 6 (first perspec-tive) and 7 and 8 (second perspective), contrasting them with previous similar figures in theliterature that do not take into account aspects developed in this work.Besides the precise understanding that we gained of how the different effects we tookinto account affect the fate of the Higgs, we can draw two general lessons from these results.The first is that, for given SM parameters and scale of inflation, different time-dependence ofthe Hubble scale and Planck-suppressed couplings between the Higgs and the inflaton, whichmay appear as unimportant details, can lead to radically different outcomes for the fate of theinstability, turning an allowed model into an excluded one and vice-versa. The second relatedlesson is that, with the existence of a degeneracy between values of the Hubble scale separatedby several orders of magnitude on one side, and effects coming from Planck-suppressed cou-plings on the other side, it appears unlikely that a future detection of primordial gravitationalwaves would, on its own, enable one to constrain efficiently SM parameters.Our work offers natural avenues for future studies in different directions. For the first timein the study of the Higgs vacuum instability, we have taken into account the time-dependenceof the background and the amplitude of the noise differing from the one of exactly masslessfields. We did so in simple motivated manners, but given the important impact of theseaspects, it would be useful to develop a more thorough theoretical understanding of them,and more generally of the theoretical uncertainties of the stochastic formalism (see discussionsin section 2). It would also be worthwile to go beyond the Gaussian approximation for thePDF in our time-dependent background (see [15, 20, 21] for such studies in de Sitter). Forinstance, challenging as it is, one can envisage to solve numerically the Fokker-Planck equationwith the quartic potential taken into account, and with suitable time-dependent boundaryconditions that incorporate the transmutation of inflationary patches into AdS ones at largeHiggs values. Eventually, it would be interesting to revisit the generation of primordial blackholes in the Standard Model [79] by taking into account the aspects developed in this work.In particular, the various setups that we studied lead to different times to form an AdS region,which could alleviate the fine-tuning problem behind this proposal [80, 81].– 33 – a) Time needed to form an AdS region versus thenumber of e -folds. For each scenario, the dashedvertical line indicates the time relevant to computethe fraction of patches in AdS at the end of inflation(see Eq. (3.8) ). (b) Evolution of h cl normalized to h max , in time-dependent backgrounds. h cl , defined in Eq. (3.5) , isthe point beyond which the classical dynamics dom-inates over the stochastic one, and the Higgs startsto fall towards the true vacuum. Figure 9 : Time to form an AdS region starting from the “point of classicality” h cl . Acknowledgments
We are grateful to V. Branchina, P. Carrilho, J.R. Espinosa, G. Franciolini, T. Markkanen,D. Mulryne, L. Pinol, M. Postma, D. Racco, A. Riotto, A. Shkerin, T. Terada, V. Vennin, L.Witkowski for interesting and helpful discussions, as well as to the referee who helped improv-ing our paper. J.F, S.RP and J.W.R are supported by the European Research Council underthe European Union’s Horizon 2020 research and innovation programme (grant agreementNo 758792, project GEODESI).
A Time to fall into AdS
As explained in Sec. 3.2, it is particularly important in a time-dependent setup to take intoaccount, at any step of the evolution, the finite time to form an AdS region starting from h = h cl ( N ), the point after which stochastic kicks are not relevant anymore. Around h cl ,the energy is still dominated by the inflaton sector and hence, it is a good approximation toconsider the Higgs as a spectator field until that point. However, the Higgs backreaction onthe background cannot be neglected anymore when the Higgs falls towards the true vacuum.Thus, from h cl onwards we evolve the full two-field system classically. We consider the– 34 –ormation of an AdS patch unavoidable when the Hubble scale in the Einstein frame becomesnegative. This corresponds to the onset of the development of a shrinking region, whichhas been shown in [15] to lead to an AdS region (see the introduction). As an aside, thisprescription is even more optimistic than the “most optimistic possibility” in Ref. [80], i.e. V eff ( h ) + V ( φ ) = 0.In figure 9 we plot the time required to fall into AdS from h cl onwards (left), and the timeevolution of h cl itself (right), for a given ξ , two different background evolutions and differentchoices of Wilson coefficient for the six-dimensional operator. For simplicity and given theillustrative scope of this appendix, we restrict to the approximate form of the potential (3.12)for the central values of SM parameters. The dashed vertical lines correspond to the timeat which the maximum of P ( | h | > h cl ( N ) , N ) is taken (at N = N m , see Eq. (3.8)) in thedomain h cl ∈ AdS defined in (3.7), i.e. where there is still enough time left before the end ofinflation to form an AdS patch starting from h cl ( N ).As an example, consider Starobinsky inflation with C = 1 (the green line in the topleft panel). In that case, the variance grows and subsequently decreases, and the time N m in (3.8) coincides with the time of the maximum of P ( | h | > h cl ( N ) , N ) during the wholeinflationary evolution: 21.2 e -folds before the end of inflation, sufficiently long enough for therequired 14.7 e -folds to fall into AdS. In contrast, for C = − P ( | h | > h cl ( N ) , N ). In thiscase, N m does not coincide with the time of the maximum of P ( | h | > h cl ( N ) , N ) during thewhole inflationary evolution; it occurs 11 e -folds before the end of inflation, when there is stillenough time to fall into AdS. This underlines the importance of considering the finite timeto fall into AdS to properly estimate the fraction of AdS patches at the end of inflation.In figure 9b we plot the values of h cl as a function of time. This gives the reader anindication of the initial field value of the Higgs at the start of the classical evolution. As forthe initial velocity of the Higgs, we set ˙ h = 0 (even if it is almost instantaneously attractedtowards the slow-roll velocity). The initial conditions for the field φ at the start of the classicalevolution are the same as the ones it would have had at that time during inflation with theHiggs as a spectator field. Something important to note: one might naively expect that thesmaller h cl , the larger the time to fall into AdS. However, this deceptive intuition does nottake into account the full two-field evolution. In this framework, the drop in overall energydensity due to the inflationary field rolling towards the end of inflation dominates over thedecrease of h cl . This is rather remarkable, as it can change the time to fall into AdS by a We use as a criterion H E < The difference between the analytical approximation and the full NNLO numerical potential is that theapproximate one has a steeper drop-off after h cl . Classically evolving from h cl to the point where the energydensity becomes negative means the Higgs field gathers more kinetic energy and arrives at that point earlier(a shift of about 3 e -folds). – 35 –ubstantial amount, as can be seen by comparing the two models in figure 9a. References [1]
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