Postinflationary vacuum instability and Higgs-inflaton couplings
Kari Enqvist, Mindaugas Karciauskas, Oleg Lebedev, Stanislav Rusak, Marco Zatta
PPrepared for submission to JCAP
Postinflationary vacuum instability andHiggs–inflaton couplings
Kari Enqvist, a Mindaugas Karˇciauskas, b Oleg Lebedev, a Stanislav Rusak, a and Marco Zatta a a University of Helsinki and Helsinki Institute of Physics, P.O. Box 64, FI-00014, Helsinki,Finland b Department of Physics, University of Jyvaskyla, P.O. Box 35 (YFL), FI-40014 Universityof Jyv¨askyl¨a, FinlandE-mail: kari.enqvist@helsinki.fi, mindaugas.m.karciauskas@jyu.fi,oleg.lebedev@helsinki.fi, stanislav.rusak@helsinki.fi, marco.zatta@helsinki.fi
Abstract.
The Higgs–inflaton coupling plays an important role in the Higgs field dynamicsin the early Universe. Even a tiny coupling generated at loop level can have a dramatic effecton the fate of the electroweak vacuum. Such Higgs–inflaton interaction is present both atthe trilinear and quartic levels in realistic reheating models. In this work, we examine theHiggs dynamics during the preheating epoch, focusing on the effects of the parametric andtachyonic resonances. We use lattice simulations and other numerical tools in our studies.We find that the resonances can induce large fluctuations of the Higgs field which destabilizethe electroweak vacuum. Our considerations thus provide an upper bound on quartic andtrilinear interactions between the Higgs and the inflaton. We conclude that there exists afavorable range of the couplings within which the Higgs field is stabilized during both inflationand preheating epochs. a r X i v : . [ h e p - ph ] A ug ontents σ hφ A.1 Computation of the Floquet Exponent 18A.2 Boundary Between Stability and Instability Regions 19
The discovery of the Higgs boson at the LHC [1, 2] in July 2012 furnished the final pieceof the Standard Model (SM) of particle physics; however, it has also raised important newquestions. One of these relates to the issue of the electroweak vacuum stability and the fateof the Higgs field in the early Universe, particularly during the inflationary and reheatingeras [3].For the currently preferred values of the top quark mass and the strong coupling, theself-coupling of the Higgs field turns negative at a high energy scale of order µ c ∼ GeV [4–6] (see [7] for its gauge (in)dependence). This would suggest that there exists another,deeper vacuum state than the one we currently occupy. One finds then that the electroweakvacuum is metastable with the lifetime longer than the age of the Universe. Although thisdoes not pose an immediate problem, the existence of the deeper vacuum raises cosmologicalquestions. In particular, one must explain how the Universe ended up in an energeticallydisfavored state and why it stayed there during inflation [8]. Even if one fine–tunes the Higgsfield initial conditions before inflation, light scalar fields experience large fluctuations of orderthe Hubble rate H during the exponential expansion epoch [9]. Unless H is sufficiently small,the Universe is overwhelmingly likely to end up in the catastrophic vacuum [10].These problems can be solved by coupling the Higgs field to the scalar curvature [3] or bytaking into account the Higgs–inflaton coupling [8]. We focus on the latter possibility in this– 1 –aper and neglect the effect of the non-minimal coupling to gravity. As shown in [11], Higgs–inflaton interaction is inevitable in realistic models of reheating. Indeed, the inflaton energymust be transferred to the Standard Model fields which leads to a (perhaps indirect) couplingbetween the inflaton and the SM particles. The latter induces Higgs–inflaton interaction atloop level, V Hφ = λ hφ H † Hφ + σ hφ H † Hφ , (1.1)where H is the Higgs doublet and φ is a (real) inflaton. Here λ hφ and σ hφ typically receivelog-divergent loop contributions and thus require renormalization. In other words, thesecouplings are generated by the renormalization group (RG) evolution [11]. Their magnitudecan be large enough to alter the Higgs evolution completely, in particular, by inducing a largeeffective Higgs mass which drives the Higgs field to zero. This mechanism is operative in therange 10 − < λ hφ < − , (1.2)with the upper bound coming from the requirement that the Higgs–inflaton interaction pre-serve flatness of the inflaton potential, and the lower limit dictated by the condition that theHiggs effective mass be greater than the Hubble rate during inflation. The trilinear interac-tion should be subdominant, λ hφ φ (cid:29) σ hφ φ so that the effective mass term does not dependon the sign of the inflaton field. This is usually the case in explicit reheating models [11].In this work, we study the effect of the above couplings after inflation. Although theHiggs–inflaton interaction can stabilize the Higgs potential during inflation, during preheatingits effect can instead be destabilizing (see also [12]). The parametric resonance [13, 14] dueto the quartic interaction h φ and the tachyonic resonance [15, 16] due to the h φ term canlead to very efficient Higgs production. This causes large fluctuations and the Higgs variance (cid:104) h (cid:105) that can exceed the critical value beyond which the system becomes unstable. We findthat these considerations place important upper bounds on both λ hφ and σ hφ such that therange of favored couplings (1.2) reduces.The field of Higgs dynamics in the early Universe has been very active in the recentyears. Higgs field fluctuations during inflation in the metastable Universe have been studiedin [17, 18] and [19–21]. The Higgs condensate dynamics assuming stability of the Higgsvacuum were analyzed in detail in [22, 23]. These considerations are affected by the presenceof further Higgs interactions which are usually not included in the Standard Model. Theeffect of the non-minimal coupling Higgs to gravity on the Higgs dynamics was recentlyrefined in [24]. In this framework, it was also noted that the resonances during preheatingcan destabilize the electroweak vacuum [12]. The effect of the quartic Higgs–inflaton couplingon the Higgs production during preheating was considered in detail in [25] (see also [26, 27]).Our present work goes beyond these previous studies in that we consider a more realisticcase of both quartic and trilinear interactions present, which brings in new and importantqualitative features. We also refine the earlier analysis of the pure quartic case. Finally, wediscuss implications of our findings for realistic reheating models.This paper is organized as follows. In the next section, we present our setup. In section 3,we consider the effect of the quartic Higgs–inflaton interaction on Higgs production duringpreheating. Section 4 is devoted to the more realistic case of both trilinear and quarticinteractions present. The effect of this term is small close to the conformal limit. – 2 –
Framework
In this section, we present our inflationary setup. For concreteness, we study the Higgsproduction within the simple m φ chaotic inflation model with m = 1 . × − M Pl , M Pl = 1 . × GeV , (2.1)while our results easily generalize to other large field models. In the unitary gauge H =(0 , h/ √ T , the relevant Lagrangian is given by L = 12 ∂ µ φ∂ µ φ − m φ + 12 ∂ µ h∂ µ h − λ h ( h )4 h − λ hφ φ h − σ hφ φh , (2.2)where the self-coupling λ h ( h ) is determined by the RG equations of the Standard Model.During inflation, φ undergoes a slow–roll evolution. On the other hand, for λ hφ > − and a sufficiently large initial inflaton value φ (cid:29) M pl , the Higgs mass is dominated by theinflaton interaction, m eff h (cid:39) (cid:112) λ hφ / | φ | [8]. Then Higgs field evolves exponentially quicklyto zero.Not long after the end of inflation, the inflaton field undergoes oscillations φ ( t ) = Φ( t ) cos mt . (2.3)As long as the energy density of the Universe is dominated by the inflaton oscillations, thescale factor behaves as a = ( t/t ) / and the amplitude of oscillations decays asΦ( a ) = Φ a − / . (2.4)For concreteness, we assume that φ ( t ) in eq. (2.3) becomes a good approximation to theevolution of the inflaton at Φ (cid:39) . M Pl . (2.5)Soon thereafter we can accurately approximate the time dependence of Φ( t ) asΦ( t ) (cid:39) (3 π ) − / M Pl mt . (2.6)The inflaton induced Higgs mass term also oscillates which can lead to efficient Higgsproduction. As one can see in eq. (1.1) the second term grows with respect to the first one asΦ( t ) decreases due to the expansion of the Universe. Therefore the effect of the trilinear termbecomes important at some stage even though it was negligible during inflation. Since theconsequent effective Higgs mass term can have either sign ∝ σ hφ φ , the tachyonic resonancebecomes effective. Both of the resonances play an important role and will be studied in thenext sections.Before we proceed, let us clarify our assumption about the running coupling λ h ( µ ),where µ is the renormalization scale. During the resonances, the Higgs quanta are producedcoherently with the corresponding occupation numbers being very large. Thus we may treat h semi–classically. In this regime, we may take λ h ( µ ) = λ h (cid:16)(cid:112) (cid:104) h (cid:105) (cid:17) , (2.7) Here the effect of the trilinear term is negligible since we assume λ hφ φ (cid:29) | σ hφ | φ during inflation. – 3 –here (cid:112) (cid:104) h (cid:105) plays the role of the relevant energy scale at which the coupling should beevaluated. Since we are only interested in the high energy regime, in our numerical analysiswe use the step–function approximation λ h ( µ ) = 0 . × sign (cid:16) h SM c − (cid:112) (cid:104) h (cid:105) (cid:17) , (2.8)where h SM c ∼ GeV is the critical scale of the Standard Model at which λ h flips sign. Let us first consider the case where the trilinear interaction is negligible, σ hφ ≈
0. TheHiggs–inflaton interaction is quartic so that we recover the well–known parametric resonancesetting [14].The equations of motion for the Higgs field are quadratic in h apart from the quarticself–interaction. During the parametric resonance regime, the effect of the latter can beapproximated as h → h (cid:104) h (cid:105) , which is known as the Hartree approximation. In that case,the equations of motion for different momentum modes decouple. In terms of the rescaledHiggs momentum modes X k ≡ a / h k , where a is the scale factor, one has [14]¨ X k + ω k X k = 0 with ω k = k a + λ hφ cos ( mt ) + 3 λ h a − (cid:104) X (cid:105) + 32 wH . (3.1)In the last term, w = p/ρ = − (cid:16)
23 ˙ HH (cid:17) is the equation of state parameter of the Universe,which vanishes in the matter-like background. We thus neglect this term.If the Higgs–inflaton coupling λ hφ is substantial, the Higgs modes experience amplifi-cation due to broad parametric resonance. The parameter characterizing the strength of theresonance is q ( t ) = λ h Φ ( t )2 m (3.2)such that q (cid:29) (cid:104) h (cid:105) . The fluctuations can be so signif-icant that they exceed the size of the barrier separating the electroweak vacuum from thecatastrophic one at large field values. In this case, vacuum destabilization occurs. In whatfollows, we will estimate the corresponding critical size of λ hφ .As was shown in [14], in the broad resonance regime the Higgs modes evolve adiabaticallyaway from the inflaton zero-crossings and can be described by the WKB approximation X k (cid:39) α k √ ω k e − i (cid:82) ω k d t + β k √ ω k e i (cid:82) ω k d t , (3.3)where α k , β k are some constants. Adiabaticity is broken for certain modes near the inflatonzero-crossing, where the frequency ω k evolves very quickly. There the system can be treatedin analogy to the Schr¨odinger equation as a scattering of plane wave solutions. The adiabaticconstants α k and β k can be thought of as Bogolyubov coefficients. We assume a vacuuminitial condition for the Higgs modes with α k = 1 and β k = 0. The occupation number ofHiggs quanta after j (cid:39) mt/π zero crossings is then n j +1 k = | β j +1 k | (3.4)– 4 –nd can be written in terms of the corresponding Floquet index µ jk as [14] n j +1 k (cid:39) e πµ jk n jk . (3.5) µ jk can be calculated via scattering of plane waves in a parabolic potential [14], µ jk = 12 π ln (cid:20) e − πκ j + 2 sin θ j tot (cid:113) e − πκ j (1 + e − πκ j ) (cid:21) with κ j ≡ k √ q j a j m . (3.6)Here a j is the scale factor after j zero crossings. The term sin θ tot is determined from the phaseaccrued by the modes and behaves in a stochastic manner for different momenta (see [14]).We take it to be zero for our estimates and use the consequent average value of the Floquetindex. The occupation numbers at late times ( a j (cid:29)
1) can then be approximated by n j +1 k (cid:39) j e − ¯ µ j k m where ¯ µ j = M Pl √ πq Φ a j . (3.7)Here the factor of 1 / (cid:104) h (cid:105) (cid:39) (cid:90) d k (2 πa ) n k ω k (cid:39) j m κ / eπ / a √ q , (3.8)where κ = ¯ µ − j is the momentum in units of m which contributes most significantly. Herewe have assumed that ω k is dominated by the inflaton–induced term. Already after the firstzero-crossing (cid:104) h (cid:105) exceeds the critical scale ∼ GeV of the Standard Model and thereforethe Higgs self-coupling λ h = λ h (cid:16)(cid:112) (cid:104) h (cid:105) (cid:17) can be taken to be negative from the beginning.For our analysis, we take λ h = − − at large field values. Note that the fact that (cid:104) h (cid:105) exceeds the SM critical scale does not necessarily lead to vacuum destabilization since thepresence of the Higgs–inflaton coupling pushes the barrier separating the two vacua to largervalues of order h c ∼ (cid:115) λ hφ | λ h | | φ | . (3.9)However, the position of the barrier is modulated by | cos mt | so it is not immediately clearwhat vacuum stability would require.To derive the stability condition, one can use the following reasoning. Around each infla-ton zero crossing, the effective Higgs mass squared is dominated by the Higgs self-interactionterm λ h (cid:104) h (cid:105) . Such a tachyonic term leads to exponential amplification of the Higgs field bya factor of order e m eff h ∆ t , where m eff h is the modulus of the effective Higgs mass term and ∆ t is the (short) period during which the Higgs self-interaction term dominates. ∆ t is givenexplicitly by | ∆ t | < (cid:115) | λ h |(cid:104) h (cid:105) λ hφ Φ m . (3.10) This assumption does not have a significant numerical impact on our main results. – 5 – h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) Λ h Φ (cid:61) (cid:180) (cid:45) (cid:45) (cid:60) h (cid:62) (cid:144) (cid:60) h c (cid:62) Figure 1 : Time evolution of the Higgs fluctuations scaled by the location of the potentialbarrier h c = (cid:113) λ hφ | λ h | Φ, for different λ hφ . The λ h h term is treated in the Hartree approximation .The tachyonic amplification is insignificant as long as m eff h ∆ t does not exceed unity, that is, (cid:112) | λ h |(cid:104) h (cid:105) | ∆ t | < . (3.11)Clearly, this condition eventually gets violated since ∆ t grows as the inflaton amplitude Φdecreases. However, if the resonance ends before this takes place, no destabilization occurs.Using λ hφ Φ (cid:39) m at the end of the resonance [14], one finds that the stability conditioncan be written as λ hφ < π (ln 3) m M (cid:34) ln (cid:32) eπ / | λ h | (cid:33) + 32 ln (cid:32)(cid:114) λ hφ π M Pl m (cid:33)(cid:35) (cid:39) × − . (3.12)Here we have neglected a smaller additive constant in the square brackets. If this condition isviolated, the Higgs field grows explosively since the amplification factor e m eff h ∆ t increases with (cid:104) h (cid:105) itself. This leads to fast vacuum destabilization. On the other hand, if this condition issatisfied, it implies that the Higgs potential is dominated by the inflaton coupling term onthe average and h does not fluctuate beyond the barrier (3.9). This result is consistent withthe bound obtained in [25].Figure 1 shows our numerical evolution of the Higgs fluctuations for different values of λ hφ . To produce this plot we have solved the mode equations in the Hartree approximationusing Mathematica software. We see that for λ hφ greater than a few times 10 − , the Higgsfield grows above the critical value and blows up at mt >
40. The destabilization timehowever should not be taken at face value since the Hartree approximation turns out to berather crude for this purpose.We have also performed a more sophisticated lattice simulation which takes into accountthe Higgs self–interaction without resorting to the Hartree approximation. We used the– 6 – igure 2 : Vacuum destabilization time versus λ hφ (green curve) with LATTICEEASY.Points below the red line correspond to the active parametric resonance. Our theoreticalupper bound on λ hφ is marked by the vertical dashed line.LATTICEEASY package [28] for this purpose. In our simulations, we choose the box size of10 /m (the L parameter of LATTICEEASY) with 64 grid points per edge (the N parameter).We have checked that a larger and finer grid does not change the results significantly. InFigure 2, we plot the destabilization time versus λ hφ . The green curve shows mt at which thesystem is destabilized, that is, the Higgs field variance blows up. The red line marks the endof the resonance such that the points below it correspond to vacuum destabilization duringthe resonance as studied in this section. Our theoretical bound on λ hφ is marked by thevertical dashed line. We see that the latter describes the general situation reasonably welland λ hφ above 3 × − typically leads to vacuum destabilization during the resonance. Onthe other hand, we also see the limitations of our approach. In particular, Figure 2 shows thatthe strength of the resonance does not behave monotonically with λ hφ . This is expected sincewe have taken the term sin θ tot to be zero, whereas in reality it either enhances or suppressesthe resonance such that there can be certain values of λ hφ satisfying our bound yet leadingto an unstable configuration. Formally, the area around λ hφ ∼ × − appears to be stableduring the resonance and the destabilization occurs shortly after the resonance. However,one can classify this region as unstable since in reality the end of the resonance is not sharplydefined due to various approximations we have made. Apart from these complications, wefind that our simple considerations give a fairly good description of the system behaviorduring the parametric resonance.Comparing Fig. 1 and Fig. 2, one finds that the commonly used Hartree approximationoverestimates the destabilization time. This is to be expected since the quantity h experi-ences greater fluctuations than h (cid:104) h (cid:105) does. Nevertheless certain questions such as the effectof perturbative Higgs are easier addressed using our Mathematica routine which employs theHartree approximation. Hence we use both numerical approaches. The EW vacuum can be destabilized at later times as seen in Figure 2. This is however a differentphenomenon which we consider in subsequent sections. – 7 – h Φ (cid:61) (cid:45) (cid:72) (cid:71)(cid:61) (cid:76) Λ h Φ (cid:61) (cid:45) (cid:72) (cid:71)(cid:61) (cid:76) Λ h Φ (cid:61) (cid:45) (cid:72) (cid:71)(cid:185) (cid:76) Λ h Φ (cid:61) (cid:45) (cid:72) (cid:71)(cid:185) (cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) mt (cid:60) h (cid:62) a (cid:70) (cid:45) (cid:45) k (cid:144) m n k Figure 3 : Left: effect of the perturbative Higgs decay h → t ¯ t . Right: example of theoccupation number evolution for different momenta with λ hφ = 3 × − . Blue (red) curvescorrespond to early (late) times. (The Hartree approximation is employed).Figure 2 also shows that the late time behaviour (beyond the resonance) of the Higgsfluctuations is important, which we discuss in section 5.So far our discussion has ignored perturbative decay of the Higgs quanta, which reducesthe efficiency of the resonance and can potentially invalidate our conclusions. The maindecay channel is provided by the top quarks which are effectively massless for our purposes.The corresponding decay width is Γ( h → t ¯ t ) = 3 y t m eff h π . (3.13)Taking y t ( m eff h ) ∼ / m eff h (cid:39) (cid:112) λ hφ / | φ | and averaging | cos mt | , we find that the pertur-bative decay reduces the number of the Higgs quanta by a factor 2 or so in the region ofinterest ( λ hφ ∼ − ), see the left panel of Figure 3. Therefore it does not significantly affectour bound on λ hφ . On the other hand, for larger λ hφ ∼ − , the Higgs decay can reduce (cid:104) h (cid:105) by an order of magnitude thus delaying (but not avoiding) vacuum destabilization.For completeness, in the right panel of Figure 3, we present a typical example of theoccupation number evolution for different momenta. We find that at late times the Higgsfield is typically dominated by the modes with momenta k ∼ m . The trilinear Higgs–inflaton interaction brings in an additional effective mass term whosesign oscillates in time. This results in the tachyonic resonance [15] which amplifies the Higgsfluctuations. We find that the effect is important and cannot be neglected.The parametric and tachyonic resonances have been studied separately in detail. Inrealistic models, both of them are present at the same time, yet their combined effect isnot well understood (see however [16],[29]) . In particular, the Higgs field goes through asequence of exponential amplification periods and plateaus. In what follows, we study someof the important aspects of the system and obtain the corresponding bound on σ hφ . We also note that the range of parameters considered in these papers is very different from that of interesthere. – 8 – .1 Equations of Motion
The trilinear interaction introduces an additional oscillating contribution to the effectiveHiggs mass. The frequency of this contribution is half the frequency of the quartic interaction.In particular, the Higgs dispersion relation in eq. (3.1) becomes ω k = (cid:18) ka (cid:19) + σ hφ Φ ( t ) cos mt + 12 λ hφ Φ ( t ) cos mt + 3 λ h a − (cid:10) X (cid:11) , (4.1)where, as in eq. (3.1), we have neglected terms proportional to ˙ H ∼ H . These terms becomesmall, as compared to m , soon after the end of inflation. Let us introduce p ( t ) ≡ σ hφ Φ ( t ) m , (4.2) δm ( t ) ≡ λ h a − (cid:10) X (cid:11) . (4.3)and the q ( t ) parameter, which is defined in eq. (3.2). Then the equation of motion for the(rescaled) Higgs field can be written asd X k d z + (cid:20) A ( k, z ) + 2 p ( z ) cos 2 z + 2 q ( z ) cos 4 z + δm ( z ) m (cid:21) X k = 0 , (4.4)where z ≡ mt , (4.5) A ( k, z ) ≡ (cid:18) kam (cid:19) + 2 q ( z ) . (4.6)This differential equation reduces to the Whittaker–Hill equation if the Universe expansionand the Higgs self–interaction are neglected. Its solutions exhibit the resonant behaviorsimilar to those of the Mathieu equation, although the situation is more complicated dueto the presence of two parameters p and q . According to the Floquet theorem, a generalsolution of the Whittaker–Hill equation can be written as X ( z ) = ρ e µz y ( z ) + ρ e − µz y ( − z ) , (4.7)where ρ and ρ are integration constants, y ( z ) are periodic functions of period π and µ is acharacteristic exponent, or Floquet exponent, which in general is a complex number. When µ attains a real part, the solution grows exponentially. We discuss the most important proper-ties of these solutions in Appendix A. In particular, the stability chart of the Whittaker–Hillequation is quite different from that of the Mathieu equation in the parameter range ofinterest (see fig. 9).In reality, the Universe expansion cannot be neglected and leads to the end of theresonance. Hence the Whittaker–Hill equation only provides a simple approximation to theequations of motion. The duration of the resonance is essential for our considerations sinceit determines the size of (cid:104) h (cid:105) . Let us consider it in detail.– 9 – .2 Duration of the Resonance The essential difference between the solutions of eq. (4.4) in an expanding and static Universesis that in the former case the boundaries between the stability and instability regions areno longer clearly defined: they are smeared [14]. Despite this fact, we will use figure 9 asa helpful illustration. For that matter, the time dependence of A ( k, z ), p ( z ) and q ( z ) canbe introduced adiabatically: as they evolve, one can think of them as tracing a trajectoryin the three dimensional space, crossing through stable and unstable regions. Once theseparameters decrease substantially and the trajectory converges to the lowest stable region,the resonance ceases.The definition of the parameter A in eq. (4.6) contains two terms, which are timedependent. To compute the duration of the resonance, we first estimate the relative size ofthese two contributions at the end of the resonance. Using eq. (2.4) and the definition of q in eq. (3.2), we can write (2 k/a f m ) q f ∼ ( m/ Φ ) / ( k/m ) λ / hφ q / f , (4.8)where the subscript f refers to values at the end of the resonance. Typically, the excitedmodes towards the end of the resonance are k/m ∼
1. Since m/ Φ ∼ × − , we have(2 k/a f m ) q f (cid:39) − λ / hφ q / f . (4.9)In this work we are interested in the range of values of λ hφ given in equations (1.2) and (3.12).Taking also q / f ∼
1, the above ratio lies in the range 10 − ...
1. That is, at the end of theresonance, the q –term dominates and it suffices for our purposes to consider the evolutionof the k = 0 mode only. This restricts our parameter space to the plane A = 2 q . Thestability and instability regions for constant p , q can be obtained by the methods discussed inAppendix A. The result is shown in figure 4, where the labeled curves display the trajectories p ( t ) , q ( t ) for different λ hφ and σ hφ . The vertical line p = 0 corresponds to the parametricresonance and one recovers the standard results of ref. [14].The resonance stops when q ( z ) and p ( z ) reach the last stable region around p = 0 and q = 0 in figure 4. To estimate the time when this happens, we approximate the boundaryof the lowest stable region by a linear relation q = 0 . − . | p | . This approximation isshown by bold red lines in figure 4. One has to keep in mind however, that in the expandinguniverse the boundaries between stable and unstable regions are smeared. Hence, even whena trajectory in ( q, p ) parameter space reaches the last stable region, the resonance continuesfor some time, depending on the phase. Thus, the end of the resonance corresponds to q f = 0 .
48 (1 − δ ) − . | p f | , (4.10)where δ is a “fudge” factor to be determined from simulations. Our results show that δ variesfrom 0 to about 1 /
4. An analogous result for the parametric resonance was obtained in [14],in which case the resonance stops somewhere in the range 1 ≤ q f ≤ / In our parameterrange, we find that δ (cid:39) . Note that the definition of q in ref. [14] differs from ours by a factor of 1 / – 10 – igure 4 : Stability (shaded) and instability (white) regions of the Whittaker-Hill equationfor A = 2 q . The labeled curves describe evolution of p ( t ) , q ( t ) for different values of σ hφ and λ hφ : (1) σ hφ = 8 × − M Pl , λ hφ = 1 . × − ; (2) σ hφ = − × − M Pl , λ hφ = 1 . × − ;(3) σ hφ = 7 × − M Pl , λ hφ = 2 . × − ; (4) σ hφ = 1 × − M Pl , λ hφ = 3 × − . Theboundary of the last stability region around p = 0 is marked in red. Figure 5 : Evolution of the occupation numbers for the mode k/m = 0 .
63 with LAT-TICEEASY. Different color lines correspond to models with different values of σ and λ hφ (same as in fig. 4). The vertical dashed lines show the end of the resonance according toeq. (4.11), apart from model 4 for which the standard result q f = 1 holds [14]. The Higgsself–interaction is set to zero, λ h = 0.Using eqs. (4.2), (3.2) and eq. (2.6), we find mt f (cid:39) (cid:34) . | σ hφ | /M Pl λ hφ (cid:32)(cid:115) . λ hφ m σ hφ − (cid:33)(cid:35) − . (4.11)– 11 – igure 6 : Evolution of (cid:104) h (cid:105) for the models of figure 5 and λ h = 0. Due to the Universeexpansion, (cid:104) h (cid:105) decreases when the resonance is not active.In figures 5 and 6, we plot numerical LATTICEEASY computations of occupation numbers n k with k ≈ . m and (cid:10) h (cid:11) for several models. The values of t f from eq. (4.11) are shownby dashed vertical lines. We conclude that the agreement is quite good.Eq. (4.11) does not apply for very small values of σ hφ such that the tachyonic resonanceis inefficient. In particular, the amount of time the system spends in the last instabilityregion (just above the red line in fig. 4) is so small that no substantial amplification occurs.For such models, the dynamics of the resonance are close to those of the pure parametriccase [14]. It is also worth recalling that λ hφ in eq. (4.11) is not allowed to be too small sothat the ratio in eq. (4.9) is below unity. As in the parametric resonance case, the Higgs field fluctuations can grow large enough sothat the system moves over to the catastrophic vacuum. This transition is facilitated by thepresence of the trilinear term which results in very large Higgs occupation numbers. In whatfollows, we study the destabilization effect due to σ hφ . That is, we choose λ hφ for which thesystem is stable and analyze how large a σ hφ one can add without destabilizing the vacuum.As before, we focus on the destabilization during the resonance, i.e. before t f in eq. (4.11). σ hφ The analysis of the mixed trilinear–quartic case is substantially more complicated than thepure quartic case. As seen from the stability chart, the system goes through a series of stableand unstable regions with a varying exponent µ ( t ). We will thus content ourselves with onlyan order of magnitude estimate of the critical σ hφ .Towards the end of the resonance, the Higgs–dependent potential is dominated by thetrilinear term σ hφ φh since the quartic interaction decreases faster with time. The desta-bilization occurs when this term becomes overtaken by the Higgs self–interaction λ h h . Note that, as expected, n k starts growing when p ( t ) , q ( t ) reach the relevant instability region. In particular,for curve 2 the growth begins at mt ∼ – 12 –herefore, one can estimate the critical variance by (cid:104) h (cid:105) cr ∼ | σ hφ | Φ | λ h | , (4.12)where the Hartree approximation has been used and the oscillatory behavior of φ has beenignored.On the other hand, the Higgs variance as a function of time can be calculated via theoccupation numbers as in (3.8). The dominant contribution is given by modes around thecomoving momentum k ∗ which maximizes n k . For the parameter range of interest, we findthat k ∗ ∼ m towards the end of the resonance and the width of the k -distribution is of order k ∗ /
2. The corresponding n k ∗ is a rather complicated function of time containing sectionswhere it undergoes an exponential increase. For our purposes, we simply interpolate it by e µ ∗ mt with some effective exponent µ ∗ . We then obtain (cid:104) h (cid:105) (cid:39) ∆ k ∗ k ∗ a n k ∗ ω k ∗ ∼ m a e µ ∗ mt (cid:112) | σ hφ | Φ . (4.13)The destabilization occurs if (cid:104) h (cid:105) reaches the critical value during the resonance. Thelatter stops around 2 | σ hφ | Φ end (cid:39) m . Taking this into account and dropping order oneconstants, one finds | σ hφ | < m M pl × µ ∗ ln a | λ h | ∼ GeV , (4.14)with a end = (Φ / Φ end ) / being the scale factor at the end of the resonance. Here we take atypical value µ ∗ ∼ O (10 − ) (cf. fig. 9). Note that the main σ hφ –dependence of the resultcomes from the duration of the resonance, mt end ∼ σ hφ × M pl /m , while that of µ ∗ andln a end is milder.Although this estimate is very crude, we find that the bound is within a factor of afew from our numerical results. Here we have neglected both the λ hφ –dependence and thedependence on the sign of σ hφ .Note that both the λ hφ and σ hφ bounds do not appear to depend explicitly on thecritical scale of the Standard Model. This dependence is hidden in our assumption about λ h at the energy scales of interest. As long as λ h ∼ − − in that range, our bounds apply.Finally, we have considered a chaotic φ inflation model which fixes a large H infl ∼ GeV. For models with small H infl < GeV, the Higgs fluctuations during inflation arenot dangerous and the Higgs–inflaton coupling can be set negligibly small. Our LATTICEEASY simulations show that the bound on σ hφ depends both on λ hφ andthe sign of σ hφ . The latter is due to the fact that even though the Whittaker-Hill equationenjoys the symmetry z → z + π/ , p → − p , the time translation invariance is broken by theUniverse expansion. Figs. 7 and 8 display the bounds on σ hφ as a function of λ hφ . We seethat the upper bound varies between 10 GeV and 6 × GeV in the region of interest.We should note that these plots are somewhat simplified in that it is tacitly impliedthat | σ hφ | below the critical value leads to a stable system. In practice, this is not always thecase and the destabilization time can be a non-monotonic function of σ hφ . However, theseeffects do not change our results drastically. This is supported by our numerical analysis. Such models however do not solve the problem of the Higgs initial conditions at the beginning of inflation. – 13 – igure 7 : Upper bound on σ hφ > Figure 8 : Upper bound on | σ hφ | for negative σ hφ from LATTICEEASY simulations. In theshaded region, the Higgs vacuum is destabilized during the resonance. So far we have discussed vacuum destabilization during the resonance. The initial stage ofpreheating is dominated by a single process, that is, resonant Higgs production. At laterstages, other processes such as rescattering, thermalization, etc. become important.As seen in fig. 2, the Higgs vacuum can be destabilized much after the end of theresonance. The simple reason for it is that (cid:112) (cid:104) h (cid:105) and the position of the barrier h c ∝ Φ– 14 –cale differently in time. If only the quartic coupling is present, h c ∝ a − / , (cid:112) (cid:104) h (cid:105) ∝ a − α , (5.1)where α is between 1 and 3 /
4, depending on which k –modes dominate (cid:104) h (cid:105) . This can beseen from the first equality in (3.8) and the fact that the comoving occupation numbers areconstant after the end of the resonance, while the ω k scaling depends on the balance between k /a and the inflaton–induced mass term. In any case, (cid:112) (cid:104) h (cid:105) decreases slower in time than h c does so that after a sufficiently long period the Higgs fluctuations go over the barrier.Fig. 2 shows that the relevant time scale is of order 100 mt . Analogous considerations alsoapply to the mixed trilinear–quartic case.However, the true dynamics of the system on a larger time scale are complicated. TheHiggs interacts with other fields of the Standard Model which becomes important after theresonance. As noted in [25], thermalization effects can generate a thermal mass term forthe Higgs thereby stabilizing the vacuum. Also, non–perturbative production of particlesvia the Higgs couplings can reduce (cid:104) h (cid:105) [23]. These effects are subtle and require a carefulinvestigation which is beyond the scope of our present work. On the other hand, the resonanceregime is quite well understood and thus we believe our bounds on λ hφ and σ hφ are solid. In this section, we consider implications of our bounds for model parameters of representativereheating scenarios. We choose two examples considered in [11]: reheating via right-handedneutrinos and reheating via non–renormalizable operators.In general one expects the Higgs–inflaton couplings to be present already at the treelevel. However, if they are for some reason suppressed, λ hφ and σ hφ are generated by loopcorrections. Therefore, the loop–induced couplings can be regarded as the correspondinglower bound. In what follows, we consider two conservative scenarios in which λ hφ and σ hφ are entirely due to loop effects. In this model, the inflaton decays into heavy right–handed neutrinos which subsequentlydecay into SM particles. This option is attractive since the inflaton–neutrino coupling isallowed already at the renormalizable level. The relevant interaction terms are − ∆ L = λ ν φν R ν R + y ν ¯ l L · H ∗ ν R + M ν R ν R + h . c . , (6.1)where l L is the lepton doublet, the Majorana mass M is chosen to be real and we have assumedthat a single ν R species dominates. The quartic and trilinear Higgs-inflaton couplings aregenerated at 1 loop and the result is divergent. In other words, such couplings are requiredby renormalizability of the model. As the renormalization condition, we take λ hφ ( M Pl ) = 0, σ hφ ( M Pl ) = 0 such that at the inflationary scale the couplings are generated by loop effects.In the leading–log approximation, we find λ hφ (cid:39) | λ ν y ν | π ln M Pl µ ,σ hφ (cid:39) − M | y ν | Re λ ν π ln M Pl µ , (6.2)– 15 –here µ is the relevant energy scale. In what follows, we assume real couplings and take µ ∼ m since this is the typical momentum of the Higgs quanta towards the end of theresonance. In any case, the dependence on µ is only logarithmic.The value of λ ν is constrained by inflationary dynamics. In order not to spoil flatness ofthe inflaton potential, the coupling must satisfy λ ν < − [11]. Taking λ hφ < × − and | σ hφ | < GeV (see fig. 7), we find the following bounds on the neutrino Yukawa couplingand the Majorana mass, y ν < . ,M < × GeV . (6.3)Although these constraints are not particularly strong, they are non–trivial. In particular,they imply that the neutrino Yukawa coupling cannot be order one. A common approach to reheating is to assume the presence of non–renormalizable operatorsthat couple the inflaton to the SM fields. Let us consider a representative example of thefollowing operators, O = 1Λ φ ¯ q L · H ∗ t R , O = 1Λ φ G µν G µν , (6.4)where Λ , are some scales, G µν is the gluon field strength and q L , t R are the third generationquarks. These couplings allow for a direct decay of the inflaton into the SM particles. Itis again clear that a Higgs–inflaton interaction is induced radiatively. In order to calculatethe 1–loop couplings reliably, one needs to complete the model in the ultraviolet (UV). Thesimplest possibility to obtain an effective dim-5 operator is to integrate out a heavy fermion.Therefore, we introduce vector–like quarks Q L , Q R with the tree level interactions − ∆ L = y Q ¯ q L · H ∗ Q R + λ Q φ ¯ Q L t R + M ¯ Q L Q R + h . c . , (6.5)where the heavy quarks have the quantum numbers of the right–handed top t R , their mass M is taken to be above the inflaton mass scale and the couplings to the third generation areassumed to dominate. One then finds that O appears at tree level with 1 / Λ = y Q λ Q / M ,whereas O appears only at 2 loops with 1 / Λ ∼ y Q λ Q y t α s / (64 π M ) and can be neglected.Using the renormalization condition that the relevant couplings vanish at the Planck scaleand the fact that the heavy quarks contribute only at scales above M , we get in the leading–log approximation λ hφ (cid:39) | λ Q y t | π ln M Pl M ,σ hφ (cid:39) − M Re( λ Q y Q y t )2 π ln M Pl M , (6.6)where y t is the top Yukawa coupling and we assume M (cid:28) M Pl . As in the previousexample, one of the couplings is constrained by the inflationary dynamics, | λ Q | < × − / (ln M Pl / M ) / [11], since it generates a correction to the inflaton potential. The heavyquark mass must be well below the Planck scale, M (cid:28) M Pl , and the bound on λ Q depends Choosing a higher µ would result in slightly looser bounds. – 16 –ery weakly on M in the allowed range. Therefore, in practice one may take | λ Q | < − .Our results λ hφ < × − , | σ hφ | < GeV lead to a stronger bound. For real couplings,we get | λ Q | < × − , | y Q | < . , (6.7)where in the second inequality we took M ∼ m to obtain the most conservative bound and y t ( M ) ∼ /
2. This implies, in particular, that the minimal value of the suppression scaleΛ = M / | λ Q y Q | is around the Planck scale and the maximal reheating temperature is oforder 10 GeV (see [11] for details).
This work is devoted to an in–depth analysis of the Higgs–inflaton coupling effects in thereheating epoch. We have focussed in particular on the preheating stage when the para-metric and tachyonic resonances are active. Our framework includes both the quartic andtrilinear couplings since these are present simultaneously in realistic models. The resultingmixed parametric–tachyonic resonance is described by the Whittaker–Hill equation. Whileinheriting certain features of the two resonances, it brings in new effects which require athorough investigation.Within this framework, we have analyzed the issue of electroweak vacuum stabilityduring the preheating epoch assuming that the Higgs self–coupling turns negative at highenergies. Even though the Higgs–inflaton couplings can stabilize the system during inflation,resonant Higgs production thereafter can lead to vacuum destabilization. The relevant quarticand trilinear Higgs–inflaton couplings are generated by the renormalization group equationsin realistic models, and even their tiny values make a difference. Using both analyticalmethods and lattice simulations in a representative large field ( φ ) inflation model, we obtainupper bounds on the couplings from vacuum stability during preheating. These allow for arange of couplings, roughly 10 − < λ hφ < − and | σ hφ | < GeV, which ensure stabilityboth during inflation and preheating.Our analysis is limited to the timescale of the mixed resonance. This leaves out theissues of the late–time behavior of the Higgs fluctuations which can further limit the allowedrange for the couplings. The required analysis is highly involved and we leave it for futurework.
Acknowledgments
MK is supported by the Academy of Finland project 278722 and during the initial stages ofthis work was supported by JSPS as an International Research Fellow of the Japan Societyfor the Promotion of Science. O.L. and M.Z. acknowledge support from the Academy ofFinland project “The Higgs and the Universe”.– 17 –
The Whittaker-Hill Equation
A.1 Computation of the Floquet Exponent
The Whittaker-Hill equation is given by (cid:20) d d z + 2 p cos (2 z ) + 2 q cos (4 z ) (cid:21) X = − AX, (A.1)where a constant A can be thought of as an eigenvalue of the differential operator on theLHS. The analysis of this equation can be found in [29–32].According to the Floquet theorem, the solution can be written as a series of the form X ( z ) = e µz ∞ (cid:88) n = −∞ c n e inz . (A.2)Plugging this Ansatz into the above equation, we obtain a recursive relation γ n (cid:0) c n − + c n +1) (cid:1) + c n + ξ n (cid:0) c n − + c n +2) (cid:1) = 0 , (A.3)where γ n ≡ pA − ( iµ − n ) and ξ n ≡ qA − ( iµ − n ) . (A.4)For given values of A , q and p we can find the Floquet characteristic exponent µ by solvingfor the roots of the determinant∆ ( iµ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . ξ − γ − γ − ξ − ξ γ γ ξ
00 0 ξ γ γ ξ . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (A.5)It is possible to prove (see, e.g., refs [29, 30]) that this determinant can be written in acompact form as sin (cid:16) iµ π (cid:17) = ∆ (0) sin (cid:16) √ A π (cid:17) . (A.6)From this equation we can easily find µ = − iπ arccos (cid:104) (cid:16) cos (cid:16) √ Aπ (cid:17) − (cid:17)(cid:105) . (A.7)The advantage of this representation of solutions is that it can be evaluated numericallyvery efficiently. Indeed, as one can see from the definitions of γ n and ξ n in eqs. (A.4),the off-diagonal elements of ∆ ( iµ ) decrease as ∝ n − as they depart from the center of thematrix. – 18 – .2 Boundary Between Stability and Instability Regions The stability of the solution in eq. (4.7) is determined by the characteristic exponent µ . Ingeneral, µ is a complex number µ = α + iβ . If the real part of µ is non-zero, that is α (cid:54) = 0,the given solution is unstable. For stable solutions α = 0 and their periodicity is determinedby the value of the imaginary part β . If β is a rational fraction, the solution is periodic, whilefor irrational β the solution is non-periodic. Particularly interesting are the cases where β isan integer. If β = 2 l , where l ∈ Z , then solutions are either even or odd periodic functionswith a period π . For β = 2 l + 1 those solutions are even or odd periodic functions witha period 2 π . These solutions of period π and 2 π lie on the boundary between the regionswhere the families of stable and unstable solutions reside, that is, the so called stability andinstability regions in ( A, q, p ) space. To find the equations for these boundary surfaces, wecan use the following Ans¨atze y ( z ) = ∞ (cid:88) n =0 C n cos (2 nz ) , (A.8) y ( z ) = ∞ (cid:88) n =0 S n +1 sin ((2 n + 1) z ) , (A.9) y ( z ) = ∞ (cid:88) n =0 C n +1 cos ((2 n + 1) z ) , (A.10) y ( z ) = ∞ (cid:88) n =0 S n +2 sin ((2 n + 2) z ) . (A.11)The function y ( z ) describes even solutions of period π ; y ( z ) is an odd function of period2 π ; y ( z ) is an even function of period 2 π and y ( z ) is an odd function of period π . To findthe coefficients, one plugs these functions back into the Whittaker-Hill equation. This givesus four recursive relations which can be written in a matrix form, M J C J = A J C J , (A.12)where no summation over J is implied; C J stands for C = ( C , C , C , . . . ) T , C = ( S , S , . . . ) T ,etc. and A J represent eigenvalues A IJ ( p, q ) of an infinite square matrix M J , where I =0 , , . . . , ∞ . These eigenvalues define the boundaries between the stability and instabilityregions.To find A IJ ( p, q ), let us compute the four matrices M J explicitly. Plugging eq. (A.8)into eq. (A.1), we find M = p q p q − p q q p − p q p − n . . . , (A.13)where the first row corresponds to n = 0. Similarly, one obtains the other three matrices, M = − p − p − q q p − q − p qq p − p q p − (2 n + 1) . . . , (A.14)– 19 – igure 9 : Stability (shaded) and instability (white) regions of the Whittaker-Hill equationfor several values of q . The solid curves are contours of constant | Re µ | . The leftmost panel( q = 0) is the stability chart of the Mathieu equation. M = p − p + q q p + q − p qq p − p q p − (2 n + 1) . . . , (A.15) M = − q − p q p − p qq p − n + 1) p . . . . (A.16)We compute the eigenvalues of these matrices numerically by truncating them at some highvalue of n . Some solutions of eq. (A.7) and eigenvalues of these matrices are shown in figure 9.The leftmost panel ( q = 0) can be recognized as the familiar stability chart of the Mathieuequation. References [1]
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