Our electroweak vacuum may be metastable in light of the current experimental data of the Higgs/top quark mass. If this is really the case, high-scale inflation ...
arXiv:1602.00483v1 [hep-ph] 1 Feb 2016
UT 16-04 IPMU 16-0012
Fate of Electroweak Vacuum during Preheating Yohei Ema◊ , Kyohei Mukaida , Kazunori Nakayama◊, ◊
Department of Physics, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 133-0033, Japan
Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Abstract Our electroweak vacuum may be metastable in light of the current experimental data of the Higgs/top quark mass. If this is really the case, high-scale inflation models require a stabilization mechanism of our vacuum during inflation. A possible candidate is the Higgs-inflaton/-curvature coupling because it induces an additional mass term to the Higgs during the slow roll regime. However, after the inflation, the additional mass term oscillates, and it can potentially destabilize our electroweak vacuum via production of large Higgs fluctuations during the inflaton oscillation era. In this paper, we study whether or not the Higgs-inflaton/-curvature coupling can save our vacuum by properly taking account of Higgs production during the preheating stage. We put upper bounds on the Higgs-inflaton/-curvature coupling, and discuss possible dynamics that might relax them.
Contents 1 Introduction and Summary
1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Higgs-Inflaton Coupling
3
2.1 Preheating via Quartic stabilization: c 2 φ 2 h2 . . . . . . . . . . . . . . . . . . . . .
6
2.2 Preheating via Curvature stabilization: ξRh2 . . . . . . . . . . . . . . . . . . . .
11
3 Higgs-Radiation Coupling
15
3.1 Instant Preheating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2 Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4 After Preheating
19
4.1 Cosmic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.2 Turbulence and thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.3 Complete reheating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5 Conclusions and Discussion
24
A Thermalization after Inflation
26
1 1.1
Introduction and Summary Introduction
The current measurements of the Higgs and top quark masses suggest that the Higgs quartic coupling flips its sign well below the Planck scale if there are no new physics other than the Standard Model (SM) [1, 2, 3]. It indicates that our electroweak vacuum may be metastable, although the lifetime is much longer than the age of the universe for the best fit values of SM parameters. In the cosmological context, an interesting consequence of the Higgs metastability is that high-scale inflation has a tension with it [4, 5, 6]. This is because Higgs acquires fluctuations of the order of the Hubble parameter Hinf during inflation, and hence, as long as there is no new physics other than SM, the electroweak vacuum decays into the true one if the inflation scale is high enough. Therefore, some stabilization mechanism of Higgs is necessary for high-scale inflation to be consistent with the metastability. A leading candidate of such a stabilization mechanism may be the following Higgs-inflaton/-curvature interaction [6]: 1 2 2 2 − 2 c φ h , (1.1) Lint (φ, h) = 1 − ξRh2 , 2 where φ is the inflaton field, h is the Higgs field and R is the Ricci scalar. Indeed, Higgs acquires an effective mass term, and hence it can suppress the Higgs fluctuations during 2 2 inflation if the conditions c 2 φinf , ξRinf ¦ Hinf are satisfied. Thus, the interaction (1.1) can stabilize the electroweak vacuum during inflation. However in such a case, the interaction (1.1) itself makes the dynamics during the preheating stage♥1 highly nontrivial. The Higgs-inflaton quartic coupling causes the broad resonance [7, 8] due to the breakdown of the adiabaticity and the Bose enhancement, and hence the number density of Higgs grows exponentially. The Higgs-curvature coupling causes a tachyonic enhancement of Higgs [9, 10, 11] because the curvature-induced effective mass squared becomes negative for some period during one oscillation of the inflaton. In fact, the condition for the effective mass to work during inflation is almost equivalent to the condition for the broad/tachyonic resonance of Higgs to take place during the preheating regime. Thus, the exponential growth of Higgs fluctuations may force our electroweak vacuum to decay into the true one during the preheating stage, and hence the interaction (1.1) may eventually fail to save the electroweak vacuum. In this paper, we focus on the dynamics of Higgs during the preheating caused by the interaction (1.1). The main purpose of this paper is to investigate in what parameter space the interaction (1.1) does not trigger the electroweak vacuum decay during the preheating stage. We use both analytical and numerical methods to study the effects of the broad/tachyonic resonance on the vacuum stability. We also consider the interactions between Higgs and radiation composed of other SM particles, and clarify that they are less significant during the preheating, as long as the inflaton perturbatively produces other SM ♥1
A note on terminology. In this paper, the word “preheating" represents the epoch in which some resonant particle production processes occur due to the inflaton oscillation after inflation.
1
particles. In addition, we discuss possible dynamics of Higgs after the resonance shuts off, that is to say, after the preheating. In the next sub-section, we summarize main results of this paper for the convenience of the readers. We also give the organization of this paper there.
1.2
Summary
Here main results of this paper are summarized. For simplicity, it is assumed that inflaton oscillates with a quadratic potential. We study the preheating dynamics of Higgs caused by the interaction (1.1), and obtain the parameter region in which Higgs remains in the electroweak vacuum during the preheating. To be more specific, the electroweak vacuum survives the preheating stage if the couplings satisfy the following inequalities. Upper bounds on Higgs-inflaton coupling: −4
c ® 1.5 × 10
0.1
µqtc
mφ
1.5 × 1013 GeV
,
(1.2)
for the quartic coupling case, and
ξ®9×
2 2 p 2 Mpl , neff µcrv Φini 2
(1.3)
for the curvature coupling case.
Here mφ is the inflaton mass, Φini is the initial inflaton amplitude, Mpl is the reduced Planck mass, neff is an effective number of oscillation [See the text below Eq. (2.33)], µqtc ' 0.1 and µcrv ' 2. See Eqs. (2.24) and (2.35). It is assumed that the broad/tachyonic resonance is effective at the onset of the inflaton oscillation. The upper bounds are obtained from the requirement that the Hubble expansion should kill the broad/tachyonic resonance before Higgs rolls down to the true vacuum. See Sec. 2 for details. Classical lattice simulations are performed to confirm these upper bounds. We also discuss effects of radiation in Sec. 3 and also in the first part of Sec. 4.3, but they are less significant as long as the inflaton perturbatively decays into other SM particles. In order to suppress the Higgs fluctuations during inflation, the couplings c and ξ should satisfy c ¦ O (Hinf /Φini ) and ξ ¦ O (0.1) [4, 6]. Thus, our result indicates that the allowed values of the couplings lie in a rather narrow band, for the stability of Higgs during both in the inflation and the preheating stages. After the preheating, the dynamics of Higgs becomes quite complicated once we include the interactions among Higgs, top quark and SM gauge bosons. If we ignore this interaction, the bounds given in Eqs. (1.2) and (1.3) become severer, for the cosmic expansion reduces the Higgs effective mass induced from Eq. (1.1) much faster than the tachyonic mass induced from its self interaction [See Eqs. (4.4) and (4.5)]. Thus, it is essential to include this interaction so as to discuss the fate of electroweak vacuum after the preheating. A possible dynamics is that, after the electroweak vacuum survives the preheating stage, it 2
thermalizes with the other SM particles. Once it is thermalized, the life time of our vacuum can be estimated by means of the bounce method under a periodic Euclidean time [12, 13]. If this is really case, the electroweak vacuum does not decay after the preheating either for the central value of the top quark mass [14]. See the discussion in Sec. 4.2. Also, it is noticeable that other SM particles produced via the decay of inflaton, which is responsible for the complete reheating, would play essential roles both during and after the preheating. Typically, a larger reheating temperature tends to stabilize the electroweak vacuum, while a smaller one does not play the role. See the discussion in Sec. 4.3. Also, note that the resonant production of other SM particles, during the early stage of complete reheating, might affect the bounds given in Eqs. (1.2) and (1.3). A detailed study of the dynamics of Higgs, including Higgs-radiation and inflaton-radiation couplings, is left for a future work. The organization of this paper is as follows. In Sec. 2, we first shut off the Higgsradiation couplings and focus on the role of the Higgs-inflaton/-curvature coupling during the preheating regime. We study effects of the broad/tachyonic resonance on the electroweak vacuum stability both analytically and numerically there. Then, in Sec. 3, we turn on the Higgs-radiation coupling, but ignore the decay of inflaton into other SM particles, and investigate how it modifies results of the previous sections. In fact, we will see that it is less significant during the preheating. In Sec. 4, we discuss a possible fate of Higgs after the preheating. At the end of Sec. 4, we discuss how the decay of inflaton into other SM particles could change the results. The last section is devoted to the conclusion and the discussion.
2
Higgs-Inflaton Coupling
If the Higgs-inflaton/-curvature coupling is quite large, we can easily expect that Higgs soon rolls down to the true vacuum due to the resonance induced by the inflaton oscillation. Thus the resonance parameter or the coupling should be rather small to avoid the decay of the electroweak vacuum during the preheating stage. The main goal of our study here is to estimate the coupling value below which the electroweak vacuum survives the preheating stage. In particular, we will make it clear in what condition the electroweak vacuum does not decay albeit the resonance occurs at the beginning of the inflaton oscillation. In order to achieve this goal, we use both analytical and numerical methods for the study of the preheating stage. Here note that the coupling cannot be arbitrarily small in order to suppress the Higgs fluctuation during inflation. Now we start to explain our setup. First, let us shut off the Higgs-radiation couplings, i.e. gauge and top Yukawa couplings, so as to clarify the role of Higgs-inflaton coupling in the preheating stage. Hence, we study the following model throughout this section:♥2 L = Linf (φ) + LHiggs (h) + Lint (φ, h),
(2.1)
1 1 Linf (φ) = ∂µ φ∂ µ φ − m2φ φ 2 , 2 2
(2.2)
1 1 LHiggs (h) = ∂µ h∂ µ h − λ(µ)h4 . 2 4
(2.3)
where
♥2
For simplicity, we treat Higgs as a one component field here. However, our constraints derived from now are not sensitive to it because they depend only logarithmically on the number of components.
3
Here we have dropped the negative Higgs mass squared, which leads to the electroweak symmetry breaking, since it is irrelevant for our following discussion. Note that, although the simple chaotic inflation model with quadratic potential is excluded by observations [15], it is possible to modify the large filed behavior to make the inflationary prediction consistent with observations (see e.g., Refs. [16, 17, 18, 19, 20, 21]). We implicitly assume this in the following. As representative models for the stabilization of Higgs, we consider two different mechanisms: 1 2 2 2 − 2 c φ h · · · quartic, (2.4) Lint (φ, h) = 1 − ξRh2 · · · curvature. 2 For the running coupling constant of the Higgs potential λ(µ), we roughly approximate its form as ˜ ˜ ' 0.01. λ(µ) ' λsign hmax − µ ; with λ (2.5) Note that the scale, hmax , where the Higgs quartic coupling becomes negative, significantly depends on the current data of top quark mass. For its central value, the scale is approximately given by hmax ' 1010 GeV [22]. For a homogeneous Higgs field, the renormalization scale should be taken as its expectation value, µ ' h. In the following discussion, we have to deal with an inhomogeneous Higgs field owing to Higgs particle production events during the preheating stage. We take the renormalization scale to be a typical scale of preheating, p∗ , defined later v.s. a field value of Higgs at each space-time point, h(x): µ = Max[h, p∗ ]. The dynamics of preheating is well described by a classical equations of motion. Thus, all we have to do is to solve numerically the classical equation of motion, which is derived from the Lagrangian (2.1), with appropriate initial fluctuations. However, later, we would like to turn on the Higgs-radiation coupling and discuss how it affects the fate of electroweak vacuum. For that purpose, it is instructive to understand the dynamics qualitatively so that we can apply our understandings to more complicated systems. In the following, we study the stage of preheating in two representative cases separately. We first discuss qualitative behavior of the system, and in particular clarify the condition where the Higgs field escapes from our electroweak vacuum. Then, we show results of numerical simulations and confirm our qualitative understanding.
Preliminaries Before moving to each case, we summarize common features of this system. Right after the inflation, since there are no particles, we can safely neglect effects of Higgs fluctuations on the inflaton dynamics at first. Then, the inflaton obeys the following approximated solution: Φini , φ(t) ' Φ(t) cos mφ t ; Φ(t) = 3 a 2 (t)
(2.6)
where a(t) ∝ t 2/3 is the scale factor, and Φ(t) is an inflaton amplitude with Φini being its initial value. Correspondingly, the dispersion relation of Higgs oscillates with time, and the Higgs field acquires fluctuations. 4
To be concrete, let us start with the mode expansion of Higgs field: Z d3 k i k·x ˆ a h (t)e + H.c. , h(x) = k k [2πa(t)]3/2
(2.7)
where k represents the comoving momentum and a(t) is the scale factor. Here a rescaled wave function, hk , is introduced for later convenience. Obeying the usual procedure, we start from the quadratic action; take account of equation of motion of Higgs up to linear term in Higgs at first, and then include non-linear terms perturbatively. Up to quadratic action, the equation of motion for the wave function reads (2.8) 0 = ¨hk (t) + ω2k;h(t) + ∆(t) hk (t), ˙ where ∆2 ≡ −9H 2 /4 − 3H/2. Here the Hubble parameter is defined as H ≡ a˙/a, and ωk;h(t) represents the time dependent dispersion relation of Higgs. In this scheme, the equation leaves the Wronskian invariant since it is linear, hk˙h∗k − h∗k˙hk = i. Together with the canonical commutation relation, this normalization implies the following algebras for the creation/annihilation operator: [ˆ ak , aˆk† 0 ] = δ(k − k0 ) and [ˆ ak , aˆk0 ] = [ˆ ak† , aˆk† 0 ] = 0. Here note that there is redundancy of the expression in Eq (2.7). We can always rephrase ˆ ˜k = αk aˆk +β ∗ aˆ† and ˜hk = α∗ hk −βk h∗ un˜ k , ˜hk ) satisfying aˆ Eq. (2.7) by another set of (α k −k k −k ˆ ˜ k , ˜hk ), der |αk |2 −|βk |2 = 1. This is the well-known Bogolyubov transform, B : (ˆ ak , hk ) 7→ (α which leaves the Wronskian, commutators and the norm (hk , h∗−k ) · (ˆ ak , aˆ−† k ) t invariant. By using this redundancy, one can always take a basispwhich satisfies the initial condition:♥3 p hk (t → 0) → 1/ 2ωk;h(0) and ˙hk (t → 0) → −i ωk;h(0)/2. Then, the initial vacuum state is annihilated by the corresponding operator aˆk |0; in〉 = 0. In the following discussion, we take this basis. Here we have omitted contributions from the cosmic expansion ∼ O (H 2 /ω2k;h), that is, we keep the leading order WKB result with respect to the cosmic expansion. One can discuss particle production by means of Bogolyubov transformation as done in literature. Nevertheless, it is instructive to see the same physics in a different viewpoint, by only looking at correlators, since the relation with outcomes of classical lattice simulations can be seen clearly.♥4 It is convenient to define two correlators and their Fourier transforms [23, 24, 25, 26, 27]: Z ¬ e i k·(x−y) ¶ (2.9) h(x), h( y) + ≡ 3/2 G F ;h(x 0 , y0 ; k), k a(x 0 )a( y0 ) Z ¬ e i k·(x−y) ¶ h(x), h( y) − ≡ (2.10) 3/2 Gρ;h(x 0 , y0 ; k), a(x )a( y ) k 0 0 where [•, •]± stands for anti-commutator/commutator respectively, expectation values, 〈•〉, are taken by the initial state, and we have used a shorthanded notation for the R vacuum R 3 momentum integral, k ≡ d k/(2π)3 . These two propagators, G F /ρ , are referred to as the statistical/spectral functions respectively. The spectral function imprints the spectrum of theory and the statistical one encodes the occupation number. One can see that these two propagators can be expressed in terms of wave functions up to quadratic action: G F /ρ;h(x 0 , y0 ; k) = hk (x 0 )h∗k ( y0 ) ± h∗k (x 0 )hk ( y0 ) . (2.11) ♥3
Note here that the initial condition should be consistent with the normalization of Wronskian. Also, it may be conceptually clear especially when the effects of inflaton/Higgs particle production becomes relevant, though it is equivalent after an appropriate reinterpretation. ♥4
5
In classical lattice simulations, which we implement later, the left-hand-side of Eqs. (2.9) and (2.10) are computed, by regarding the field variables as classical ones. Thus, there is no counterpart of Eq. (2.10) in classical lattice simulations, and indeed the classical approximations are justified
for |G F | |Gρ | [28]. An important quantity is the expectation value of energy density, Tˆ 00 , and corresponding number density, which are defined via the statistical function, G F [29, 30]: 2 Z ¬ ¶ 1 H ∂ x 0 ∂ y0 + ω2k;h(x 0 ) G F ;h(x 0 , y0 ; k) y →x − (Vac.) + O Tˆh00 (x) = 0 0 4 ω2k;h k/a(x 0 ) (2.12) 2 Z i 2 2 H 1 h ˙h (x ) + ω2 (x ) h (x ) − ω (x ) + O = . (2.13) 0 k 0 k;h 0 k 0 k;h 2 ω2k;h k/a(x ) 0
Here, again, we only keep the leading order term in the WKB approximation with respect to the cosmic expansion. This motivates the following definition of the comoving number density in a momentum space: h i 1 1 ˙h (t) 2 + ω2 (t) h (t) 2 − , (2.14) nk;h(t) = k k k;h 2ωk;h(t) 2 at the leading order in H 2 /ω2k;h expansion. Importantly, it is an adiabatic invariant quanR R tity. The physical energy and number densities are given by k/a(t) ωk;h nk;h and k/a(t) nk;h respectively. The break-down of adiabaticity plays the key role in the following discussion. To see this, it is instructive to confirm that the initial condition for the wave function indeed corresponds to the initial vacuum. One can easilyp see that the number density coincides that p of vacuum for hk = 1/ 2ωk;h and ˙hk = −i ωk;h/2. Moreover, the WKB solution of Rt p Eq. (2.8), hk (t) = e−i dτωk;h (τ) / 2ωk;h(t), which is valid for the adiabatic change of ωk;h, ˙ k;h/ω2k;h| 1, also leaves the number density unchanged from that of vacuum up i.e. |ω ˙ k;h/ω2k;h|). Thus, the Higgs field acquires fluctuations for the nonto corrections of O (|ω adiabatic change of ωk;h(t) induced by the Higgs-inflaton coupling [Eq. (2.4)].♥5
2.1
Preheating via Quartic stabilization: c 2 φ 2 h2
Qualitative discussion In this case, the dispersion relation of Higgs is given by k2 ω2k;h(t) = c 2 Φ2 (t) cos2 mφ t + 2 + δm2self;h(t) a (t)
(2.15)
where δm2self;h represents a finite density correction to the Higgs mass term from the Higgs self interaction, which we discuss later. At first, since there are no Higgs particles right after the inflation, this term can be neglected. After several oscillations of inflaton, the cosmic expansion can be treated adiabatically, i.e. mφ a˙/a.♥6 Thus, we may focus on the effect of inflaton oscillation on the adiabaticity. ♥5
At the onset of inflaton oscillation, Higgs fluctuations are also produced because the Hubble parameter changes non-adiabatically [31] ♥6 Strictly speaking, a small oscillating term remains in the scale factor a [32].
6
The adiabaticity is broken near the origin of inflaton potential, φ ∼ 0, if the following inequality holds: p p∗ (t) > mφ ; p∗ (t) ≡ cmφ Φ(t), (2.16) where p∗ (= k∗ /a) is a characteristic physical momentum of Higgs particle production. We assume that this inequality is satisfied at the onset of the inflaton oscillation in the following discussion. Then, the quartic coupling should satisfy cΦini > mφ : p2 M m pl φ , (2.17) c > 4 × 10−6 13 1.5 × 10 GeV Φini where Φini is the initial inflaton amplitude. Note that this inequality implies cΦini > Hinf , with Hinf being the Hubble scale during (or at the end of) inflation, since mφ ¦ Hinf holds generically. Thus the stabilization of the Higgs field during inflation is ensured under the assumption (2.16).♥7 Hereafter we consider this situation. Below the momentum p∗ , the Higgs field acquires fluctuations for each passage of φ ∼ 0, Rt Rt p −i ωk;h i ωk;h and its wave function becomes hk (t) = [αk (t)e + βk (t)e ]/ 2ωk;h(t) for the outside of the non-adiabatic regime near φ ∼ 0. Since the Higgs is boson, its number density grows exponentially due to the Bose enhancement [7, 8]: Z Z Z 2 e2µk mφ t (2.18) nh(t) = nk;h(t) = βk (t) ' 2 k/a(t) k/a(t) k/a(t) r 1 π ∼ e2µqtc mφ t p∗3 (t), (2.19) 2 32π 2µqtc mφ t where µk is a momentum dependent function, and it has a maximum value, µqtc , at k ' k∗ /2. µqtc is a numerical factor of the order of O (0.1). Here we have used the steepest descent method to evaluate the integral, and estimated the second derivative of µk as µ00k∗ ∼ 2µqtc /δk2 with δk ∼ k∗ /2. While the Higgs fluctuation continuously grows, it induces effective mass corrections to the inflaton and the Higgs itself via the quartic interaction, c 2 φ 2 h2 , and the self interaction, λh4 , respectively. In our case, the self coupling λ is larger than c 2 , and hence we may just consider the latter effect. Note that since the typical scale, p∗ , should be larger than the inflaton mass so that the adiabaticity is broken, the running coupling constant is negative at least for mφ > hmax , which is fulfilled for the central value of top quark mass. Therefore, the self coupling induces the tachyonic mass term: Z Z 1 h (t) 2 ˜ ˜ G F ;h(t, t; k) = −3λ δm2self;h(t) = −3λ k 2 k/a(t) k/a(t) Z nk;h(t) ˜ ' −3λ ωk;h(t) k/a(t) ˜ ∼ −3λ
nh(t) ωk∗ ;h(t)
.
(2.20)
If there is a hierarchy mφ Hinf (or Φini Mpl ), it is possible to choose Hinf < cΦini < mφ . In such a case, the Higgs field is stabilized during inflation and also there is no violent Higgs production after inflation. ♥7
7
Once this term dominates over the other terms in the dispersion relation of Higgs, the Higgs field rapidly rolls down to its true minimum by the tachyonic instability. Let us quantify this condition. Given that the Higgs field is dominated by modes with momentum of p∗ , it is reasonable to expect the following inequality should be satisfied for the tachyonic instability not to occur:♥8 2 (t) δmself;h
φ∼0
®
p∗2 (t)
↔
˜ 3λ
π
r 2
16π
2µqtc mφ t
e2µqtc mφ t ® 1,
(2.21)
where the subscript φ ∼ 0 indicates that the effective mass is evaluated at the passage of φ ∼ 0. Eq. (2.21) implies that the electroweak vacuum decays at the time:♥9 ! 3 1 16π 2 t dec ∼ ln . (2.22) ˜ 2µqtc mφ 3λ However, we have to recall that the efficient Higgs production continues while Eq. (2.16) holds. One can easily see that the cosmic expansion kills the non-perturbative Higgs production at the time:♥10 p 2 6 cMpl . (2.23) t end ∼ 3mφ mφ Therefore, the electroweak vacuum survives the preheating stage for t end ® t dec , which yields the following upper bound for the Higgs-inflaton coupling:
c®
p 6 mφ 8µqtc Mpl
3
ln
16π 2 ˜ 3λ
!
' 1.5 × 10
−4
0.1 µqtc
mφ 1.5 × 1013 GeV
.
(2.24)
Numerical simulation Now, we show the results of three-dimensional classical lattice simulations for the quartic coupling case. The main purpose of this sub-section is to confirm the upper bound given in Eq. (2.24). We solve the classical equations of motion derived from the Lagrangian (2.1) in the configuration space. We take the grid number as N = 128 × 128 × 128 with a comoving edge size being L = 10/mφ . The time step is taken to be dt = 10−3 /mφ . The inflaton mass The time interval ∆t during which the inflaton-induced Higgs effective mass is negligible compared to δmself;h is given by m2φ q(mφ ∆t)2 ∼ |δm2self;h | where q ≡ c 2 Φ2 /(4m2φ ). We estimate the growth rate during this time interval as |δmself;h |∆t ∼ |δm2self;h |/p∗2 . Thus, Eq. (2.21) is needed for avoiding the enhancement of Higgs fluctuations. ♥9 ˜ 2 )1/4 mφ , t dec corresponds to the time at which the resonance If hmax is sufficiently large, say hmax (q/λ shuts off due to the positive Higgs quartic coupling. Here we assume hmax is sufficiently small: hmax ∼ 1010 GeV. ♥10 Strictly speaking, after the broad resonance, a narrow resonance takes place, but it is soon killed by the cosmic expansion at q2 mφ ' H. ♥8
8
Normalized by the initial inflaton amplitude Φini
Normalized by the initial inflaton amplitude Φini
a32 a ( - 2) a3
102
3
2
100 10-2 10-4 10-6 10-8 0
20
40
60
80
a32 a ( - 2) a3
102
3
100 10-2 10-4 10-6 10-8
100
0
10
20
Normalized by the initial inflaton amplitude Φini
Normalized by the initial inflaton amplitude Φini
mφt a32 a3( - 2) a3
102 100 10-2 10-4 10-6 10-8 0
20
40
60
80
100
2
30 mφt
40
50
60
50
60
a32 a3( - 2) a3
102 100 10-2 10-4 10-6 10-8 0
mφt
10
20
30 mφt
40
Figure 1: Numerical calculation of the time evolution of the inflaton expectation value (black), the inflaton dispersion (red) and the Higgs dispersion (blue). We take the parameters as N = 1283 , dt = 10−3 /mφ , L = 10/mφ and mφ = 1.5 × 1013 GeV. Upper left panel: c = 1 × 10−4 and p p Φini = 2 Mpl . Upper right panel: c = 2 × 10−4 and Φini = 2 Mpl . Lower left panel: c = 1 × 10−4 p p and Φini = 0.2 Mpl . Lower right panel: c = 2 × 10−4 and Φini = 0.2 Mpl . Higgs remains in the electroweak vacuum in the left panels, while it rolls down to the true vacuum in the right panels. is fixed as mφ = 1.5 × 1013 GeV. We show results for c = 1 × 10−4 and c = 2 × 10−4 . Some details of our numerical calculation are summarized below:♥11 • We start to solve equations of motion with the initial inflaton amplitude p the classical p being Φini = 2 Mpl or 0.2 Mpl . By changing the initial inflaton amplitudes, we test whether or not the inflation scale, Hinf , changes essential features of the preheating ˙ ini = 0, but we dynamics. We show the results with the initial velocity of inflaton Φ have checked that our results are not sensitive to the initial velocity. We also introduce gaussian initial fluctuations on the inflaton and Higgs field following Refs. [34, 35], which arise from the quantum vacuum fluctuations. • We take the metric as the Freedman-Lemaître-Robertson-Walker (FLRW) one in our numerical simulation. Fluctuations of the metric are neglected since their effects are suppressed by the Planck mass. Thus, there are two equations which describe the time ♥11
We have also confirmed our results by using LATTICEEASY [33].
9
103
102
Comoving number density nk;h + 1/2
Comoving number density nk;h + 1/2
mφt = 0 10 20 30 40 50 60 70 80
101
100
10-1
1
103
102
101
100
10-1
10 Comoving momentum k/mφ
mφt = 0 5 10 15 20 25 30
1
10 Comoving momentum k/mφ
Figure 2: Time evolution of the comoving number density of Higgs. We take p the parameters 3 −3 13
as N = 128 , dt = 10 /mφ , L = 10/mφ , mφ = 1.5 × 10 GeV and Φini = c = 1 × 10−4 . Right panel: c = 2 × 10−4 .
2 Mpl . Left panel:
evolution of the scale factor and the Hubble parameter in addition to the equations of motion for the inflaton and Higgs fields, and it is well-known that one of them is redundant. We have used this redundancy as a check of our numerical calculation. We have verified that our numerical calculation is consistent with the redundancy at least at O (10−3 ) precision. • In order to avoid numerical divergence, we add a hexic term to the Higgs potential to stabilize it in our calculation. We take the coefficient of the hexic term such that the ˜ 2 = 5 × 10−8 M 2 . Note that hmin hmax Higgs field value at the true minimum is λh min pl for our parameters. Moreover, at the time when Higgs rolls down to the true vacuum, ˜ 2 c 2 Φ2 is satisfied, and hence the hexic term is not expected to the inequality λh min affect the Higgs dynamics at least before Higgs rolls down to the true vacuum. We have verified that it is not affected whether the electroweak vacuum decays or not even if we change the coefficient of the hexic term. In Fig. 1, we show the time evolution of the inflaton vacuum expectation value squared 〈φ〉2 (black), the inflaton dispersion 〈φ 2 〉−〈φ〉2 (red) and the Higgs dispersion 〈h2 〉 (blue),
2 where 〈...〉 denotes the spatial average. Here note that 〈h 〉 corresponds to [h(x), h(x)]+ /2, and thus can be expressed by the statistical function, G F ;h. See Eq. (2.9). They are multiplied by the scale factor of the cube, where the initial scale factor is taken to be aini = 1. We take the Higgs-inflaton coupling as c = 1 × 10−4 for the left panels and c = 2 p × 10−4 for the right panels, respectively. The initial inflaton amplitudes are taken as Φini = 2 Mpl for the p upper panels and Φini = 0.2 Mpl for the lower panels. As it is clear from Fig. 1, Higgs stays at the electroweak vacuum for c = 1 × 10−4 , while it rolls down to the true vacuum for c = 2 × 10−4 , independent of the initial inflaton amplitude. It is consistent with our estimation (2.24). In fact, an interesting feature of Eq. (2.24) is that it does not depend on the initial inflaton amplitude or the inflation scale as long as the inequality (2.16) is good enough initially. This is mainly because the growth rate, µqtc , does not much depend on the inflaton amplitude for the broad resonance.♥12 Thus, the Higgs fluctuations are efficiently produced at the last epoch [Eq. (2.23)] independent of an ♥12
As we will see later, the situation is completely different for the Higgs-curvature coupling case.
10
initial inflaton amplitude, since the number of the inflaton oscillation is dominated by that epoch due to the decrease of the Hubble parameter. We alsopplot the time evolution of the comoving number density of Higgs [Eq. (2.14)] for Φini = 2 Mpl in Fig. 2. The left panel shows the case with c = 1 × 10−4 and the right panel does the case with c = 2 × 10−4 . In the left panel, we can see that Higgs is efficiently produced at the beginning of the oscillation, but the resonance ceases to be effective due to the Hubble expansion. After t ' 30/mφ , the comoving number density of Higgs almost remains constant. On the other hand, in the right panel, the comoving number density of Higgs continues to grow resonantly.♥13 As a result, Higgs rolls down to the true vacuum once p the condition (2.21) is satisfied. The time evolution of the number density for Φini = 0.2 Mpl are quite similar.
2.2
Preheating via Curvature stabilization: ξRh2
Qualitative discussion In this case, the dispersion relation of Higgs reads
k2 + δm2self;h(t) (2.25) a2 (t) with the Ricci curvature being R = 6 a¨/a + [˙ a/a]2 . In the inflaton-oscillation dominated era, the scale factor satisfies the following equalities: i a˙(t) 2 1 h 2 a¨(t) 1 1 2 2 2 2 2 φ˙ (t) + mφ φ (t) , = = m φ (t) − φ˙ (t) . (2.26) a(t) a(t) 3Mpl2 2 φ 6Mpl2 ω2k;h(t) = ξR +
Plugging Eqs. (2.6) and (2.26) into the dispersion relation, we get m2 2 2 ξm 1 3 k φ 2 φ 2 −ξ + Φ (t) cos 2mφ t + 2 + Φ (t) + δm2self;h(t). ω2k;h(t) + ∆(t) ' − 2 2 2 4 Mpl a (t) 2Mpl (2.27) In contrast to the case of the quartic interaction [Eq. (2.15)], the modes can be tachyonic in a one oscillation when ξ satisfies the following inequalities: ξ < 3/16 or 3/8 < ξ, otherwise they are stable. If the coupling ξ is not large, or ξΦ2 /Mpl2 < O (1), the production becomes indistinguishable from the narrow resonance.♥14 Thus, we concentrate on a slightly large coupling, or♥15 p 2 2Mpl , ξ> (2.28) Φini where an efficient particle production via the tachyonic preheating occurs. Note that if this inequality holds, the Higgs stability during inflation is ensured since Φini ® Mpl leads to ξ ¦ O (1).♥16 In this case, the growth rate of the number density X k (t) is estimated ♥13 We do not plot the comoving number density of Higgs for mφ t > 30 in the right panel because Higgs already rolls down to the true vacuum before that time. ♥14 In fact, it is almost stable for ξΦ2 /Mpl2 , Φ2 /Mpl2 < 1. See Ref. [10]. ♥15 Since the Higgs-curvature coupling is introduced to stabilize the Higgs field during inflation, we do not consider the case with −ξ ¦ O (1) here. ♥16 If Φini Mpl , it is possible to choose O (0.1) < ξ < (Mpl /Φini )2 . In such a case, Higgs is stable during inflation and also tachyonic resonance does not occur after inflation.
11
as [36]:♥17 x p X k (t) ' − p Ak + 2x q, q
(2.29)
1 Φ2 (t) ξΦ2 (t) 3 ξ− Ak (t) = 2 2 + , q(t) = , 4 4 a mφ 2Mpl2 Mpl2
(2.30)
where k2
and x ' 0.85 in our case. Recalling that the amplitude of inflaton is proportional to 1/t, one can see that the first oscillation takes the dominant role in tachyonic preheating contrary to the previous case. Note also that the typical physical momentum enhanced by the tachyonic resonance is given by 1 p∗(tac) (t) ≡ p mφ q1/4 (t), x
(2.31)
In terms of the typical momentum p∗(tac) , the condition for the efficient particle production via the tachyonic preheating has the same expression as Eq. (2.16): p∗(tac) (t) > mφ .
(2.32)
The number density of Higgs field produced after the j-th passage of φ ∼ 0 is estimated as Z Z Ç p Pj z π neff µcrv ξ ΦMini (tac)3 2 i=1 X k (t i ) pl p nh(t j ) = nk;h(t j ) ' e ∼ e (t j ), (2.33) ∗ 2 16π 2 k/a(t ) k/a(t ) j
j
where µcrv ' 2 and z ' 1 for ξ ¦ O (1), and Φini is the initial amplitude at the beginning of the inflaton oscillation. Here we include an effective number of oscillations neff in our formula. For Φini ¦ Mpl , the first one or two oscillations dominate the Higgs production since the amplitude drastically decreases within the first oscillation, and hence we estimate neff ' 1. On the other hand, for Φini ® Mpl , the later oscillations can also be important since p the decrease of the amplitude is rather slow. For Φini = 0.2 Mpl , for example, we roughly estimate neff ' 1.5-2. Now let us derive the condition where the electroweak vacuum is stable during the preheating. At least the tachyonic mass induced by the Higgs quartic coupling should be smaller than that induced by the curvature coupling: ξR ∼ qm2φ .♥18 Thus, we estimate the condition where Higgs is stable against the tachyonic mass as r p ˜ π neff µcrv ξ ΦMini 3λ 2 2 pl ® 1. e (2.34) ® qmφ ↔ δmself;h(t j ) ξR∼0 16π2 2q Hence, the electroweak vacuum survives the tachyonic resonance for♥19 This is understood as follows. The time interval during which the Higgs with momentum p = k/a becomes tachyonic in one inflaton oscillation is ∆t k ∼ m−1 (1 − p2 /ξR) for p2 ® ξR (here R should be φ p regarded as just a typical value). In one inflaton oscillation, these modes are enhanced as exp( ξR∆t k ) ∼ p p p exp( q − p2 /p∗(tac)2 ) ∼ exp( q − Ak / q), where p∗(tac) is defined in (2.31). ♥18 If hmax is sufficiently large, it corresponds to the condition that the positive Higgs quartic coupling shuts off the tachyonic resonance. ♥19 A similar study is performed for the Higgs-curvature coupling in Ref. [11]. However, the criteria for the stabilization of Higgs are different, and the bound we obtain here is a bit weaker than their result. ♥17
12
Normalized by the initial inflaton amplitude Φini
3
2
100 10-2 10-4 10-6 10-8 10-10
0
5
10
15 mφt
20
25
30 Normalized by the initial inflaton amplitude Φini
Normalized by the initial inflaton amplitude Φini Normalized by the initial inflaton amplitude Φini
a32 a ( - 2) a3
102
a32 a3( - 2) a3
102 100 10-2 10-4 10-6 10-8 10-10
0
5
10
15 mφt
20
25
30
a32 a ( - 2) a3
102
3
2
100 10-2 10-4 10-6 10-8 10-10
0
5
10
15 mφt
20
25
30
25
30
a32 a3( - 2) a3
102 100 10-2 10-4 10-6 10-8 10-10
0
5
10
15 mφt
20
Figure 3: Numerical calculation of the time evolution of the inflaton expectation value (black), the inflaton dispersion (red) and the Higgs dispersion (blue). We take the parameters as N =p1283 , dt = 10−3 /mφ and mφ = 1.5 × 1013 GeV. The comoving edge size is L = 20/mφ for Φini = 2 Mpl and p p L = 40/mφ for Φini = 0.2 Mpl . Upper left panel: ξ = 10 and Φini = 2 Mpl . Upper right panel: p p ξ = 20 and Φini = 2 Mpl . Lower left panel: ξ = 25 and Φini = 0.2 Mpl . Lower right panel: ξ = 30 p and Φini = 0.2 Mpl . Higgs remains in the electroweak vacuum in the left panels, while it rolls down to the true vacuum in the right panels.
ξ®
1 n2eff µ2crv
Mpl Φini
2 ln
16π2 ˜ 3λ
2 r !2 2 p 2 Mpl 2 2 '9× . π neff µcrv Φini
(2.35)
It is clear that q cannot be much larger than O (1) since otherwise the tachyonic growth is catastrophic. Thus we have taken q ' 1 in the logarithm in this estimation to derive the conservative bound.
13
103
102
101
Comoving number density nk;h + 1/2
Comoving number density nk;h + 1/2
mφt/π = 0 1 2 3 4 5 6 7 8 9
100
10-1
1 Comoving momentum k/mφ
10
103
mφt/π = 0 1 2
102
101
100
10-1
1 Comoving momentum k/mφ
10
Figure 4: Time evolution of the comoving number density of Higgs. p We take the parameters as 3 −3 13
N = 128 , dt = 10 /mφ , L = 20/mφ , mφ = 1.5 × 10 GeV and Φini = 2 Mpl . Left panel: ξ = 10. Right panel: ξ = 20. We evaluate the number density at the end points of the oscillations. We show the number density only before Higgs rolls down to the true vacuum for the right panel.
Numerical simulation Now, we show the results of classical lattice simulations for the curvature coupling case. The main purpose of this sub-section is to confirm the upper bound given in Eq. (2.35). We solve the classical equations of motion derived from the Lagrangian (2.1) in the configuration space. We take the grid number as N = 128 × 128 × p128 with a comoving edge size being L = 20/mφ for the initial inflaton amplitude Φini = 2 Mpl and L = 40/mφ p for Φini = 0.2 Mpl ♥20 and the time step as dt = 10−3 /mφ . We fix the inflaton mass as mφ = 1.5 × 1013 GeV. We summarize some details of our numerical calculation below (some are the same as those in the quartic coupling case): • We start to solve equations of motion with the initial inflaton amplitude p the classical p ˙ ini = 0. We introbeing Φini = 2 Mpl or 0.2 Mpl . We set the initial velocity as Φ duce gaussian initial fluctuations in the inflaton and Higgs fields which arise from the quantum fluctuations. • We have used the redundancy of the equations of motion for the metric as a check of our numerical calculation. We have verified that our numerical calculation is consistent with the redundancy at least at O (10−3 ) precision. • We add a hexic term to the Higgs potential to stabilize it in our calculation. We ˜ 2 = take the coefficient such that the Higgs field value at the true minimum is λh min 5 × 10−8 Mpl2 . • In the numerical calculation, we solve the equations of motion in the Einstein frame. We have taken only leading terms in ξh2 /Mpl2 and ξ2 h2 /Mpl2 . This treatment is justified since ξh2 /Mpl2 , ξ2 h2 /Mpl2 1 always holds in our numerical calculation. ♥20
The typical momentum we are interested in is smaller than that in the quartic coupling case, and hence we take the comoving edge size smaller than that in the quartic coupling case here.
14
In Fig. 3, we show the time evolution of the inflaton vacuum expectation value squared 〈φ〉2 (black), the inflaton dispersion 〈φ 2 〉−〈φ〉2 (red) and the Higgs dispersion 〈h2 〉 (blue), which are multiplied by the scale factor of the cube whose initial value is aini = 1. We take the as follows: ξ = 10 and p Higgs-curvature coupling and the initial inflaton amplitude p Φini = 2 Mpl for the upper left panel, ξ = 20 and Φini = 2 Mpl for the upper right panel, p p ξ = 25 and Φini = 0.2 Mpl for the lower left panel and ξ = 30 and Φini = 0.2 Mpl for the lower right panel, respectively. p For the Φini = 2 Mpl cases (the upper panels), Higgs remains in the electroweak vacuum for ξ = 10, while it rolls down to the true vacuum for ξ = 20. Thus, the condition (2.35) is consistent with our numerical calculation within a factor of two.♥21 On the contrary to the quartic coupling case, the critical value of the Higgs-curvature coupling dep pends on the initial inflaton amplitude. In our numerical calculation, for the Φini = 0.2 Mpl cases (the lower panels), Higgs remains in the electroweak vacuum for ξ = 25, while it rolls down to the true vacuum for ξ = 30. Again, it is consistent with our estimation (2.35) once we include the effect of neff ' 1.5-2. In Fig. 4, we also plot p the time evolution of the comoving number density of Higgs [Eq. (2.14)] for Φini = 2 Mpl . The curvature coupling is ξ = 10 for the left panel and ξ = 20 for the right panel, respectively. We have evaluated the number density at the end points of the oscillations since it is well-defined only at around these points for the tachyonic case. We can see that Higgs is created dominantly within the first few oscillations. It is a typical feature of the tachyonic resonance where the growth rate p depends on the inflaton amplitude. The time evolution of the number density for Φini = 0.2 Mpl are quite similar.
3
Higgs-Radiation Coupling
In the previous section, in order to illustrate the impacts of Higgs-inflaton coupling. we have neglected how interactions between Higgs and radiation via Yukawa and gauge couplings affect the dynamics of the Higgs field. However, this is obviously one-sidedness because zero-temperature quantum corrections from these interactions drive the Higgs four point coupling negative at a high scale. Therefore, we have to discuss finite density corrections from the Higgs-radiation coupling. In this section, we investigate whether or not these corrections could relax the bounds [Eqs. (2.24) and (2.35)] derived in the previous section. In contrast to the vacuum corrections, which can destabilize the Higgs potential via fermion loops (i.e. top quark), finite density corrections tend to stabilize the Higgs field to its enhanced symmetry point by making the effective potential deeper, once the top quarks and electroweak gauge bosons are produced. There are three ways to produce these particles in the course of reheating dynamics: Instant Preheating: For a relatively large Higgs-inflaton coupling, Higgs particles produced non-perturbatively at φ ∼ 0 may decay into top quarks within one oscillation at a large field value of inflaton [37]. If this decay is prompt, the efficiency of ♥21
The difference between Eq. (2.35) and our numerical calculation pmay be due to the fact that the inflaton amplitude decreases drastically within the first oscillation for Φini = 2 Mpl , while we assume that the amplitude is constant to derive Eq. (2.35). Our main purpose is, however, an order estimation of the critical value of the curvature coupling, and hence Eq. (2.35) is enough.
15
broad/tachyonic resonance is reduced and the decay products stabilize the Higgs at its enhanced symmetry point. Annihilation: If the number density of Higgs becomes large owing to the broad/tachyonic resonance, the annihilation of Higgs into top quarks and electroweak gauge bosons becomes significant [38, 39]. This reduces the efficiency of resonance, and also produced top quarks and gauge bosons may stabilize the Higgs at its enhanced symmetry point. Complete Reheating: Since the Higgs-inflaton/-Curvature coupling alone cannot lead to a complete decay of inflaton, an additional interaction which completes the reheating is required. Since the radiation is produced before the complete decay of inflaton via this interaction [40, 41, 42], the abundant top quarks and electroweak gauge bosons may stabilize the Higgs at its enhanced symmetry point. We discuss the impacts of first two effects in this section. To avoid complications, the effects of the complete reheating is explained in Sec. 4.3 together with the dynamics after the preheating, for the another unknown parameter, i.e. reheating temperature, has to be introduced to study its effect.
3.1
Instant Preheating
Since the Higgs boson couples with top quarks via the sizable Yukawa coupling, its decay might affect the early stage of the preheating dynamics in two ways: (i) reduce the efficiency of Higgs production, (ii) stabilize the Higgs potential by the screening mass term from decay products. As we will see soon, the second effect is irrelevant unless the Higgs-Inflaton coupling is quite large, like c ¦ O (10−1 ). Slow decay of Higgs As a first step, let us neglect the second effect (ii), and assume that the resonance is not terminated by the back-reaction of decay products. For clarity, we discuss the quartic stabilization as an illustration at first. Later, we show that a similar discussion leads to the same conclusion, in the case of the curvature stabilization. One may estimate a typical decay ¯ h→t ¯t = (3α t /2)m ¯ H;h ∼ rate of Higgs boson by taking an oscillation average, which yields Γ p (3α t /2 2)cΦ. Here the bar indicates the oscillation average, Γh→t ¯t is the decay rate of Higgs into top quarks, mH;h is the effective mass of Higgs, and α t ≡ y t2 /(4π) ∼ 0.1 is the top ¯ Yukawa coupling. The decay reduces the growth of Higgs fluctuations as nh ∝ e2µmφ t−Γh→t ¯t t . As a result, the decay time given in Eq. (2.22) becomes slightly longer: ! p 3 1 16π 2 3α t cMpl mφ t dec ∼ ln . (3.1) + ˜ 2µqtc 2µqtc mφ 3λ By comparing it with ˜t end given in Eq. (2.23), one may estimate the impacts of the Higgs decay on the upper bound [Eq. (2.24)] as follows: c ® 1.5 × 10−4
0.1 µqtc
mφ 13
1.5 × 10 GeV 16
1 − 0.5
α 0.1 −1 t
0.1
µqtc
.
(3.2)
Similarly, in the case of the curvature stabilization, one finds ξ®9×
2 2 p α 2 −2 2 Mpl t 1 − 0.2 . neff µcrv Φini 0.1 µcrv 2
(3.3)
Thus, the effect (i) does not change the upper bound by an large amount, rather within uncertainties of our estimation. Instant decay of Higgs The next step is to include the effect (ii). For the curvature stabilization, both the efficiency of the resonance and the decay is proportional to the coupling, ξ, and thus the above estimation does not depend on the value of ξ. However, for the quartic stabilization, this is not the case because the efficiency of the resonance becomes independent of a large coupling, c. Hence, the Higgs decay, which is proportional to the coupling c, dominates at the early epoch of preheating stage. In fact, we will see that, for a sizable coupling c, the resonance can be killed by the back-reaction of decay products of Higgs. If the quartic coupling, c, is large enough, the Higgs boson produced at φ ∼ 0, decays completely before the inflaton moves back to its potential origin φ ∼ 0. Its condition is ˜ h→t ¯t mφ /π ↔ πα t cΦ mφ . While this condition holds, the inflaton decays given by Γ with the following rate [43, 44]:♥22 Γinst ∼
c2 1
4π4 [3α t /2] 2
mφ for πα t cΦ(t) mφ .
(3.4)
Assuming that the conditions, πα t cΦ(t) mφ and p∗2 > δm2th;h, hold at Γinst ∼ H, one can estimate the “reheating temperature” as follows: Tinst ∼
90 π g∗ 2
1 4
p
Γinst Mpl ' 3 × 10 GeV 13
c 0.1 14 100 14 0.1
αt
g∗
mφ 13
1.5 × 10 GeV
1
2
,
(3.5)
where g∗ is the degree of freedom of plasma produced via this process. The amplitude of inflaton at that time is estimated as Φinst ∼ 4 × 1014 GeV [0.1/α t ]1/2 [c/0.1]2 . Both conditions, πα t cΦ mφ and p∗2 > δm2th;h, are satisfied for c ¦ 0.1 at that time. Moreover, for a large enough coupling c 0.1, the resonance may be terminated by the back-reaction, and the inflaton itself might participate in thermal plasma. If this is really the case, the coherence of inflaton is lost, and the resonance becomes inefficient. In this paper, we do not concretely estimate the coupling c above which the inflaton participates in the thermal plasma, for the following reasons. First of all, the quartic interaction yields the Coleman-Weinberg potential [45]: VCW =
c2φ2 4 φ ln . 64π2 m2φ c4
(3.6)
Here note that once the condition, πα t cΦ(t) mφ , is violated owing to the reduction of inflaton p amplitude, e.g. by the cosmic expansion, the effective decay rate of inflaton becomes ∼ Γinst × [( 3π/2)(α t cΦ/mφ )3/2 ], which decreases faster than the Hubble parameter. ♥22
17
Normalized by the initial inflaton amplitude Φini
Normalized by the initial inflaton amplitude Φini
104
a32 3 2 a ( - 2) a3 a3
102 100 10-2 10-4 10-6 10-8 10-10
0
10
20
30 mφt
40
50
60
104
a32 a ( - 2) a3 a3 3
102
2
100 10-2 10-4 10-6 10-8 10-10
0
5
10
15 mφt
20
25
30
Figure 5: Numerical calculation of the time evolution of the inflaton expectation value (black), the inflaton dispersion (red), the Higgs dispersion (blue) and the χ dispersion (orange). We plot the quartic coupling case for the left panel and the curvature coupling case for the right panel, respectively. We take the parameters as N = 1283 , dt = 10−3 /mφ , mφ = 1.5 × 1013 GeV, Φini = p 2 Mpl and ghχ = gχχ = 0.5. Left panel: c = 2 × 10−4 and L = 10/mφ . Right panel: ξ = 20 and L = 20/mφ . The annihilation processes cannot save the electroweak vacuum. If we stick to the potential of the quadratic form given in Eq. (2.2), the coupling c is bounded from above c ® 10−3 [46]. Thus a flattening mechanism of the inflaton potential may be required for such a large coupling with c ¦ 0.1; examples are the non-minimal coupling [16] or the modified kinetic terms [17, 20, 21]. In these cases, the inflaton potential is more complicated, and in particular, it becomes to be dominated by the φ 4 interaction below a threshold field value of φ that depends on model parameters. Since the energy density of inflaton behaves as radiation below this scale, the resonance does not take place once p∗2 ∼ δm2th;h is saturated. Eventually, the whole system, including inflaton, might be thermalized through thermal dissipations [47]. We postpone this issue to avoid model dependent discussions.
3.2
Annihilation
In the broad/tachyonic resonance, the number density of Higgs grows exponentially. If the number density is large enough, Higgs can annihilate into top quarks and electroweak gauge bosons. In particular, Higgs may rapidly excite the gauge bosons to exponentially large number densities [48]. If the number densities of these particles become comparable to that of Higgs before it rolls down to the true vacuum, they can stabilize Higgs since the gauge coupling is typically larger than the Higgs quartic coupling. In order to see whether they can save the electroweak vacuum, we consider the following simplified Lagrangian: L = Linf (φ) + LHiggs (h) + Lχ (χ) + Lint (φ, h) + Lann (h, χ),
(3.7)
where Linf (φ), LHiggs (h) and Lint (φ, h) are the same as those given in Eqs. (2.3) and (2.4),
18
and 1 1 2 4 χ , Lχ (χ) = ∂µ χ∂ µ χ − gχχ 2 4 Lann (h, χ) = −
1 2
2 2 2 ghχ hχ .
(3.8) (3.9)
Here the light field χ schematically represents the SM gauge bosons, and we model the gauge interactions as the quartic interactions. We take ghχ = gχχ = 0.5 in our calculation. We have solved the classical equations of motion derived from the Lagrangian (3.7) numerically. In Fig. 5, we show the time evolution of the inflaton vacuum expectation value squared 〈φ〉2 (black), the inflaton dispersion 〈φ 2 〉 − 〈φ〉2 (red), the Higgs dispersion 〈h2 〉 (blue) and the χ dispersion 〈χ 2 〉 (orange). They are multiplied by the scale factor of the cube, where the initial value is taken to be aini = 1. pWe take the parameters as N = 1283 , dt = 10−3 /mφ , mφ = 1.5 × 1013 GeV and Φini = 2 Mpl . In the left panel we show the result of the quartic coupling case with c = 2 × 10−4 and L = 10/mφ , while in the right panel we show that of the curvature coupling case with ξ = 20 and L = 20/mφ . As we can see from the figure, Higgs rolls down to the true vacuum well before χ is sufficiently produced for the present parameter set. The results do not change even if we take the couplings larger, say ghχ = gχχ = 1. Thus, we expect that the annihilation processes cannot save the electroweak vacuum during the preheating. There is one comment. It depends on the numerical value of the Higgs quartic coupling whether the annihilation processes can save the electroweak vacuum or not. In the present ˜ ' 0.01, and hence the annihilation case, the Higgs quartic coupling is rather large, λ processes cannot be effective. It is possible, however, that they save the electroweak vacuum if the Higgs quartic coupling is small enough in the high scale region. It may be worth studying further on this respect.
4
After Preheating
So far we have focused on the stability of Higgs during the preheating stage, i.e. the stage where the broad/tachyonic resonance is effective. And we have obtained the robust bound on the Higgs-inflaton coupling above which Higgs falls into its true vacuum. In this section, we consider the dynamics of Higgs after the preheating stage, in the case where the electroweak vacuum survives during the preheating stage, and study whether or not the dynamics after the preheating forces the electroweak vacuum to decay. For notational simplicity, we denote the effective mass of Higgs induced by the Higgs-inflaton/-curvature coupling as m2H;h. It is given by m2H;h = c 2 Φ2 (t)/2 for the Higgs-inflaton coupling and m2H;h = ξm2φ Φ2 (t)/2Mpl2 for the Higgs-curvature coupling. [See Eqs. (2.15) and (2.27).] Here we summarize basic properties of Higgs fluctuations just after the preheating stage. At t = t end , the inflaton induced mass mH;h is comparable to the inflaton mass, mH;h(t end ) ' mφ , and also the physical momentum scale of Higgs is estimated as p(t end ) ' mφ . Since we assume that Higgs does not go to the true vacuum during the preheating stage, the tachyonic effective mass term of Higgs induced by its self interaction is smaller than the typical momentum scale, which indicates the following condition, mH;h(t end ) ' p(t end ) > |δmself;h(t end )|, owing to mH;h(t end ) ' mφ ' p(t end ). Therefore, not only the short wave length mode of Higgs with p ∼ mφ but the long wave length mode is stable against the 19
tachyonic mass term just after the preheating.♥23 As one can guess from the numerical simulations shown in Figs. 4 and 5, the comoving number density in a momentum space at t = t end may be approximated with nk;h(t end ) =
f
for H k/aend ® mφ
k−n
for mφ ® k/aend
,
(4.1)
with n ¾ 4. The stability condition, m2φ > |δm2self;h(t end )|, puts a rough upper bound on a ˜ Since Higgs has interactions with the electroweak gauge constant, f ® 103 × (10−2 /λ). bosons and the top quark, a characteristic coupling of this system is roughly, α t , α ∼ 0.05– 0.1. Thus, the Higgs distribution mostly lies in the “over-occupied” regime, 0 < m < 1; or “extremely over-occupied” regime, 1 < m [49], at the boundary of the inequality given in Eqs. (2.24) and (2.35). Here we parametrize the typical value of Higgs distribution as f = α−m . A complete analysis of this system after the preheating stage is beyond the scope of paper, which is required to determine the fate of electroweak vacuum after the preheating quantitatively. Instead, we discuss its possible processes, and point out important ingredients which could change the dynamics qualitatively.
4.1
Cosmic expansion
First, let us start with the discussion on the effect of cosmic expansion. Since the inflaton harmonically oscillates with time, the mass term m2H;h decreases as m2H;h ∝ a−3 . On the other hand, the effective mass of Higgs induced by the its quartic coupling follows δm2self;h ∝ a−2 . This is because, as we saw earlier, the modes with the momentum p ∼ p∗ or p∗(tac) dominate the Higgs dispersion, and hence we can treat them as relativistic particles. As an illustration, let us neglect the effective mass of Higgs induced by other contributions (e.g. by Higgs-radiation interaction) except for the Higgs-inflaton coupling and the Higgs self interaction. Right after the end of preheating, the mass term m2H;h stabilizes the electroweak vacuum against the tachyonic mass term generated by Higgs self interaction. However, since the mass term m2H;h decreases faster than the tachyonic mass term δm2self;h, the long wave length mode of Higgs may be destabilized eventually. We estimate the typical time scale when the inflaton induced mass becomes comparable to the tachyonic mass by mH;h(t eq ) ' |δmself;h(t eq )|. For the quartic stabilization, we get 3
mφ t eq ∼ 6 × 10
c 7 4 10−4
exp −7.8 ×
c 10−4
−1 .
(4.2)
Here we focus on the coupling c dependence, and take µ = 0.1 and mφ = 1.5 × 1013 GeV. For the curvature stabilization, the time scale is estimated as ! r 1 2 ξ ξ Φ n µ ini eff crv mφ t eq ∼ 2 × 103 exp −6 × − 1 . (4.3) p 2 2 2Mpl 2 See the q < 1 and 0 < Ak < 1 region of the stability/instability chart of the Mathieu equation given in, e.g. Ref. [10]. ♥23
20
Here we take mφ = 1.5 × 1013 GeV. Interestingly, unless there exist other contributions to the Higgs effective mass term, we can show that almost all the parameters required to stabilize the electroweak vacuum during the slow roll regime results in the catastrophe. We derive the upper bounds by demanding 〈h2 〉 ® h2max at t = t eq as mφ hmax /1010 GeV 0.1 −5 1 + 0.4 ln , (4.4) c ® 3 × 10 µqtc 1.5 × 1013 GeV mφ /1.5 × 1013 GeV for the quartic stabilization, and ξ ® 0.5 ×
2 2 2 p 2 Mpl hmax /1010 GeV 1 + 0.3 ln , neff µcrv Φini mφ /1.5 × 1013 GeV 2
(4.5)
for the curvature stabilization.♥24 Thus, the inclusion of Higgs-radiation coupling is crucial.
4.2
Turbulence and thermalization
Next, let us discuss how the Higgs-radiation coupling could change these bounds qualitatively. In this section, we neglect radiation generated in the process of complete reheating, assuming that the reheating temperature is low enough. See also the discussion in the next subsection. The SM Higgs, which is responsible for the spontaneous electroweak symmetry breaking, inevitably couples with other particles; in particular, electroweak gauge bosons and top quark via sizable couplings, i.e. gauge coupling and top Yukawa. The initial Higgs distribution illustrated in Eq. (4.1) is eventually thermalized, producing SM particles, and its thermalization time scale may be characterized by these couplings. Once it is thermalized, the life time of our vacuum can be estimated by means of the thermal bounce as done in literature [50, 51]. For instance, it was shown in Ref. [52] that the life time of the electroweak vacuum is long enough if we adopt the central value of the top quark mass. Therefore, we expect that the electroweak vacuum may be stable if thermalization time scale is much shorter than t eq . As shown in Refs. [53, 54, 55], an initially over-occupied system, like Eq. (4.1), enters the turbulent regime at first, cascades self-similarly, and eventually attains thermal distribution. In addition, the IR cascade may develop the long wave length mode as pointed out in Ref. [55, 56], and might boost the vacuum decay. Here, however, as an illustration, we simply compare the typical time scale of elastic scatterings with the Hubble parameter. In fact, for a mildly over-occupied system, its thermalization may be dominated by the elastic scatterings [49, 54]. This is the case of c ® 10−4 for the quartic coupling and ξ ® 1 for the curvature coupling, though it strongly depends on which interaction dominates the thermalization. The thermalization time of elastic scatterings may be evaluated as α2 T (w.b.) (t th ) ∼ H(t th ) with T (w.b.) (t) being a would-be temperature when the system at that time t would be thermalized. For the relevant parameters of our interest, we find that the thermalization time scale, t th , is much longer than or at most comparable to t eq . Further studies on this case will be presented elsewhere. ♥24
It should be regarded as an illustration since the condition for the resonance to occur is only marginally satisfied at these values.
21
4.3
Complete reheating
Finally, we study the effective mass of Higgs from radiation generated during the process of the complete reheating. As already mentioned, since the Higgs-inflaton/-curvature coupling alone cannot lead to a complete decay of inflaton, an additional interaction which completes the reheating is required. We discuss how it affects the dynamics of Higgs both during and after the preheating in the following. To discuss the complete reheating process in detail [41, 42], we have to specify the interaction between inflaton and radiation. The way how inflaton reheats the Universe is one of the most unknown part of the thermal history after inflation, and thus the coupling between inflaton and radiation strongly depends on inflationary models. In the following discussion, we mainly focus on the case where inflaton reheats the Universe via Planck-suppressed operators. If inflaton is singlet, it can couple only with Higgs via the renormalizable operator among SM fields [See Eq. (2.4)]. It is natural to expect that inflaton couples with other SM fields via Planck-suppressed operators. At the end of the next sub-section, we briefly comment on the case with slightly larger couplings between inflaton and radiation. The “thermal mass” contribution to the Higgs effective potential may be parametrized as 1 2
δm2th;h h2 θ hth − h ,
(4.6)
where θ is the Heaviside step function, δmth;h is the “thermal mass” from radiation generated during the course of the complete reheating, and hth is a typical threshold field value above which the electroweak gauge bosons and top quark may not be produced efficiently owing to its large effective mass proportional to the field value of Higgs. Both δmth;h and hth depend on time. See Appendix A for details, and the concrete forms of δmth;h and hth for each regime, given in Eqs. (A.1) and (A.2). We expect p that the electroweak vacuum is 2 2 stabilized if both δmth;h(t eq ) > |δmself;h(t eq )| and hth (t eq ) > 〈h2 (t eq )〉 are fulfilled.♥25 During preheating Here we discuss whether or not the dynamics of the complete reheating would change the upper bound given in Eqs. (2.24) and (2.35). First of all, let us estimate a relevant time scale in the following discussion. For the quartic stabilization, the longest time scale is governed by Eq. (2.4), which characterizes the end of resonant amplification by the cosmic expansion: 1.5 × 1013 GeV c mφ t end ' 40 × . (4.7) 1.5 × 10−4 mφ For the curvature stabilization, the dynamics is almost determined by the first few oscillations. Hence, we have to deal with the time scale of O (100−2 ) times oscillations of inflaton. Since this time scale is rather short, we cannot rely on the instantaneous thermalization approximation, frequently assumed in literature [6, 40]. To see this, it is instructive to roughly estimate the two-to-two scattering rate because, at least, this interaction should This requirement is somewhat conservative, for the “thermal mass” from hard primaries [Eq. (A.1)] may reduce the Higgs dispersion at the stage of preheating stage in the case of |δmself;h | < p∗ . ♥25
22
be faster than the cosmic expansion to attain thermal equilibrium. A naive estimation may give the rate of (α2 /m2s ) × nrad ∼ αmφ , where α is the fine structure constant of SM gauge group and the screening mass is given by m2s ∼ αnrad /mφ . One can see that even the twoto-two scattering does not take place for mφ t ® α−1 ∼ O (10), which is comparable to the time scale of our interest. The above estimation is too naive, so it should be understood as an illustration. See Appendix A for details. Let us estimate the effects of radiation produced during the process of complete reheating. As can be seen from Eqs. (A.1) and (A.2), radiation induces the effective potential with the following approximate form, ∼ δm2th;hh2 θ (hth − h), with δm2th;h and hth depending on time t. One can show that this effective mass is always smaller than the inflaton mass, and thus the condition for efficient Higgs production given in Eq. (2.16) [Eq. (2.32)] holds: p∗ > mφ δmth;h. Also, the condition given in Eq. (2.21) [Eq. (2.34)] holds for p∗ > mφ ˜ 1/2 hth . Therefore, in the case of Planck-suppressed because of |δmself;h| > p∗ > mφ ¦ λ decay of inflaton, we expect that the process of complete reheating is not likely to change the upper bound given in Eqs. (2.24) and (2.35). However, note here that we have taken a ˜ φ so that the preheating before the complete reheating can be neglected. sufficiently small Γ ˜ φ might affect the obtained bound. See also discussion in Sec. 5. An enhanced Γ After preheating The dynamics of Higgs after the preheating strongly depends on the reheating dynamics. This is because radiation produced via the complete reheating generates additional Higgs effective mass term, which follows δmth;h ∝ a−3/8 before the complete decay of inflaton. Since it decreases much slower than the tachyonic mass term, this term eventually takes over the dominant contribution of the effective mass. First, let us discuss a case with TRH ' 1010 GeV, which is a typical example of dim. 5 Planck-suppressed decay. For the quartic stabilization with c ' 10−4 , a typical time scale given in Eq. (4.2) resides in mφ t eq ∼ 6×103 ⊂ [˜t max , ˜t RH ] for TRH ' 1010 GeV [See Eq. (A.2) and definitions below it]. The thermal mass of Higgs at that time is given by 1
δmth;h(t eq ) ∼ 1012 GeV
1 α 1 Γ ˜ φ 4 6 × 103 4 2 0.1
0.2
mφ t eq
,
(4.8)
with kmax ∼ 1012 GeV. The effective mass term of Higgs induced from its self interaction at that time is 10 δm self;h (t eq ) ∼ 7 × 10 GeV
10−4 c
3 4
exp 7.8 ×
c 10−4
−1 .
(4.9)
For a smaller coupling of c, the equality time, t eq , becomes longer, and it is more likely to be stabilized by the thermal contribution, for the thermal mass decreases slower than the effective mass of Higgs generated from its self interaction. Therefore, we expect that the reheating temperature of TRH ' 1010 GeV may save the electroweak vacuum in the case with a quartic coupling of c ® 10−4 , because the conditions, δmth;h > |δmself;h| and ˜ 1/2 , are satisfied. Also, for the curvature stabilization with ξ ' 2, hth = kmax /g > |δmself;h|/λ the typical time scale shown in Eq. (4.3) resides in mφ t eq ∼ 1.6 × 103 ⊂ [˜t el , ˜t soft ] for TRH ' 1010 GeV [See Eq. (A.2) and definitions below it]. The “thermal mass” of Higgs may 23
be given by δmth;h(t eq ) ∼ 1012 GeV
α
34
0.1
˜φ Γ
38
0.2
1.6 × 103 mφ t eq
14 ,
(4.10)
with kmax ∼ 6 × 1011 GeV. The effective mass term of Higgs induced from its self interaction is estimated as ! r ξ Φ n µ ini eff 10 δm − 1 . (4.11) p self;h (t eq ) ∼ 2 × 10 GeV exp 6 × 2 2Mpl 2 Hence, we expect that the reheating temperature of TRH ' 1010 GeV may save the electroweak vacuum in the case with a curvature coupling of ξ ® 2. Note, however, that we have assumed ξ 1 in our estimation, and hence the numerics, ξ ' 2, should be understood as an illustration. Next, let us estimate the lower bound of the reheating temperature below which the reheating dynamics cannot save the electroweak vacuum, utilizing the results given in [41, 42]. In order to estimate the lower bound conservatively, we demand that the “thermal mass” term, δm2th;h, is always much smaller than the dispersion of Higgs, 〈h2 〉, throughout the thermal history up to the equality time t eq . Moreover, strictly speaking, not only the thermal potential but the thermal dissipation might relax the Higgs to its enhanced symmetry point. A complete analysis is beyond the scope of this paper. Instead, we simply require that the thermal interactions are slow enough α2 T t eq 1, which is essentially the same as mφ t eq ˜t soft . Imposing these requirements, for both the quartic/curvature stabilization, we obtain a rather conservative bound on the reheating temperature below which the reheating dynamics cannot save the electroweak vacuum: TRH ® O (105 ) GeV. Anyway, we need further investigations to derive more precise lower bound on the reheating temperature.
5
Conclusions and Discussion
The current experimental data of the Higgs and top quark masses indicate that the electroweak vacuum is metastable in the context of no new physics other than SM. From the viewpoint of inflationary cosmology, its interesting consequence is that high-scale inflation requires some stabilization mechanism of Higgs during inflation. A possible candidate of such a mechanism is the Higgs-inflaton/-curvature coupling. After inflation, however, it causes an exponential enhancement of Higgs fluctuations due to the broad/tachyonic resonance, and hence the electroweak vacuum may eventually decay into the true one during the preheating stage. In this paper, we have focused on the preheating dynamics of Higgs induced by the Higgs-inflaton/-curvature coupling. We have clarified in what parameter space our electroweak vacuum decays into the true vacuum via the broad/tachyonic resonance. We have derived the criterion when Higgs rolls down to the true vacuum, and confirmed it by performing the three-dimensional classical lattice simulations. To be more specific, as long as the broad/tachyonic resonance is effective at the onset of the inflaton oscillation, the electroweak vacuum survives the preheating stage only if the couplings satisfy the following
24
inequalities: −4
c ® 1.5 × 10
0.1
µqtc
mφ 1.5 × 1013 GeV
,
for the quartic coupling case, and ξ®9×
2 2 p 2 Mpl , neff µcrv Φini 2
for the curvature coupling case. See Eqs. (2.24) and (2.35). These conditions claim that the Hubble expansion should kill the resonance before the Higgs quartic coupling becomes relevant. In order to suppress the fluctuation of Higgs during inflation, the couplings c and ξ should satisfy c ¦ O (Hinf /Φini ) and ξ ¦ O (0.1) [4, 6]. Thus, our results indicate that the Higgs-inflaton/-curvature coupling should be rather small to stabilize Higgs during both the inflation and the preheating stages. We have also seen that the Higgs-radiation coupling does not change the situation as long as the inflaton perturbatively reheats the universe. This is the main conclusion of this paper. Here we give some remarks. First of all, we comment on the dynamics of Higgs after the preheating, in the case where the electroweak vacuum survives during the preheating stage. As explained in Sec. 4.1, if one neglects the Higgs-radiation coupling, the cosmic expansion leads to the decay of our electroweak vacuum for almost all the parameters of our interest [See Eqs. (4.4) and (4.5)]. Thus, including the Higgs-radiation coupling is essential to discuss the fate of electroweak vacuum after the preheating. Such an over-occupied system, as illustrated in Eq. (4.1), may exhibit turbulence and cascade towards not only UV but IR, which might have implications on thermalization after the preheating. As a first step, we have simply compared the elastic scattering rate with the Hubble parameter and have seen that the cosmic expansion is faster. However, in order to estimate the conditions to avoid the catastrophe quantitatively, we might have to perform numerical simulations including Higgs-radiation coupling. Moreover, it was mentioned in Sec. 4.3, that the coupling between inflaton and radiation which leads to the complete reheating plays the crucial role. Importantly, the relevant time scale of Higgs dynamics is rather short, and the instantaneous thermalization assumption of radiation produced during the course of the complete reheating might be questionable, in particular for a low reheating temperature. We have roughly estimated its effect in two cases; typical reheating temperature of TRH ' 1010 GeV and low reheating temperature of TRH ' 105 GeV. It was shown that thermal effects may save the electroweak vacuum in the former case after the preheating, while in the latter case, the cosmic expansion kills almost all the parameters required for the stability of vacuum during inflation. Second, we have assumed that other SM particles are produced perturbatively via the decay of inflaton, and neglected their resonant production. For instance, in the case of a dimension 5 Planck-suppressed operator with an order one coupling, which yields TRH ¦ O (1010 ) GeV, and with Φini > Mpl ; the resonance takes place at the early stage of the complete reheating. Although its efficiency strongly depends on a coupling of inflaton with radiation and an amplitude of inflaton after the inflation, their resonant production might affect the stabilization of Higgs during the preheating stage. We leave thorough studies for a future work. Third, we discuss the hmax dependence of our result. We have used the value hmax = 10 GeV in this paper, but we expect that our result does not change much as long as hmax 10
25
˜ 2 pqm2 or qm2 for the quartic or curvature coupling case, respectively. satisfies λh φ max φ If the inequality is inverted, the resonance shuts off due to the positive Higgs quartic selfcoupling even for a quite large value of the resonance parameter. Thus, Higgs will be trivially stable against the preheating. Precise determination of the top mass in the future will make the situation more clear. Fourth, although we have treated only the Higgs-inflaton/-curvature coupling given in Eq. (1.1), we can generalize our study to the following Planck-suppressed interaction: h2 cK µ Lint = − ∂µ φ∂ φ + cV V φ , (5.1) 6Mpl2 2 where V φ is the potential of inflaton. If cK ' cV , the effective mass term of Higgs induced by this interaction does not oscillate much and hence the resonant Higgs production is expected to be suppressed, leading to weaker constraint.♥26 If cK 6= cV , the effective mass term oscillates with time during the inflaton oscillation regime, and we obtain similar constrains to the case studied in the main text. Note that the these couplings generically exist due to, e.g. radiative processes. For example, it is discussed in Ref. [57] how the Higgs-inflaton quartic coupling emerges from loop effects in various models. Finally, we comment on the shape of the inflaton potential. In this paper, we assumed that inflaton oscillates around the origin of the potential, which is typical in high-scale inflation models. However, it is possible that inflaton oscillates around some finite vacuum expectation value (VEV). The result does not change in the case of the Higgs-curvature coupling: we can just regard φ(t) as the displacement from the VEV. In the case of quartic coupling c 2 φ 2 h2 , the results depend on the value of inflaton VEV around which the inflaton oscillates. It is expected that the resonant Higgs production effect becomes weaker for larger VEV, although further detailed investigations will be necessary to derive precise constraint on the parameter.
Acknowledgments This work was supported by the Grant-in-Aid for Scientific Research on Scientific Research A (No.26247042 [KN]), Young Scientists B (No.26800121 [KN]), Innovative Areas (No.26104009 [KN], No.15H05888 [KN]). This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The work of Y.E. and K.M. was supported in part by JSPS Research Fellowships for Young Scientists. The work of Y.E. was also supported in part by the Program for Leading Graduate Schools, MEXT, Japan.
A
Thermalization after Inflation
In this appendix, we summarize basic properties of thermalization after inflation in the case of the reheating via a Planck-suppressed decay of inflaton. Here we assume that the inflaton decays perturbatively, and neglect the resonant production. The thermalization process in this case is investigated in Refs. [41, 42].♥27 We follow the discussion given there. Precisely speaking, even if cK = cV , the energy density of inflaton and the Hubble parameter have oscillating parts at the onset of the oscillation [32]. ♥27 See also Ref. [58]. ♥26
26
˜ φ m3 /M 2 Suppose that inflaton reheats the Universe via a Planck-suppressed decay, Γφ = Γ pl φ ˜ φ 1. In this case, the number density of radiation right after the decay of inflaton with Γ ˜ φ m3 (mφ t)−1 , which is always smaller than the thermay be given by nh ∼ Γφ nφ /H ∼ Γ φ mal one; so-called “under-occupied” primaries [49]. The bottleneck process is in-medium collinear splittings of hard primaries [49] with the momentum of p ∼ mφ . It is shown that, ˜ −3/5 , these hard primaries cannot participate in thermal plasma, for mφ t ® ˜t max ≡ α−16/5 Γ φ and remain intact. They may yield the following finite density corrections to the Higgs mass: −1 2 ˜ φ m2 mφ t δmth;h ∼ g 2Γ for geff |h| mφ . (A.1) φ hard
Here g ∼ y t denotes the electroweak gauge coupling and the top Yukawa collectively, to avoid unnecessary complications. Note that mφ t 1 is required since the inflaton should oscillate once at least so as to decay. Though the hard primaries dominate the energy and number densities, the soft population is produced via collinear splittings by them. A quantum destructive interference effect prevents emission faster than a time that it takes to resolve the overlaps between the parent and daughter, that is t ¦ k/k⊥2 ∼ 1/kθ 2 . In the medium, the daughter acquires transverse momentum by random collisions, k⊥2 ∼ qˆel t with qˆel ∼ α2 nh being the diffusion constant at that time. Thus, for a given time t, there is an upper bound on the momentum, k ® kform ≡ qˆel t 2 . While a medium induced cascade takes place below kform with a typical angle θ ® α1/2 , a vacuum cascade may become relevant above kform with a minimum angle θ ¦ 1/(kt)1/2 [59]. On the one hand, if the formation momentum is lower than the Hubble parameter, kform ® H, the finite density corrections to the Higgs mass may be dominated by the vacuum cascade spectrum. On the other hand, for kform ¦ H, the LPM-suppressed spec˜ 1/2 mφ provide dominant corrections trum and the thermal-like spectrum below kmax ∼ αΓ φ ˜ −1/2 , the soft sector to the Higgs mass. After a characteristic time scale, mφ t > ˜t soft ≡ α−3 Γ φ −16/5 ˜ −3/5 ˜ is thermalized among themselves. Eventually, for mφ t > t max ≡ α Γφ , the radiation, including hard primaries, is thermalized against the expansion of the Universe, and follows the standard evolution. Thus, the soft population may yield the following corrections to the Higgs mass term [41, 42]:♥28 ˜ φ m2 αΓ for g |h| kmax ∼ H; mφ t < ˜t el , φ 1 1 ˜t el 2 2 ˜ φ m2 ˜ α Γ for g |h| k ∼ α Γ m ; ˜t el < mφ t < ˜t soft , max φ φ φ mφ t 2 δm2th;h ∼ g 2 m t φ soft ˜ φ m2 ˜ φ mφ (mφ t); ˜t soft < mφ t < ˜t max , α2 Γ for g |h| kmax ∼ α4 Γ φ ˜ t soft 1 1 4 ˜t max 2 1 8 α 5 Γ ˜ 5 m2 ˜ 4 mφ (mφ t)− 4 ; ˜t max < mφ t < ˜t RH . for g |h| k ∼ Γ max φ φ φ mφ t (A.2) ˜ −1/2 ; Here the time after which medium effects dominate splittings is defined as ˜t el = α−1 Γ φ −1/2 −3 ˜ the time after which the soft sector is thermalized is ˜t soft = α Γφ ; the time after which ♥28
Owing to the Fermi-Dirac statistics, the soft sector is dominated by bosons, i.e. SM gauge bosons.
27
˜ −3/5 ; and the time when the reheating is the radiation is thermalized is ˜t max = α−16/5 Γ φ ˜ −1 M 2 /m2 . completed is ˜t RH = Γ φ pl φ
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