Higgs-inflaton coupling from reheating and the metastable Universe
HHIP-2015-39/TH
Higgs–inflaton coupling from reheating and the metastable Universe
Christian Gross, Oleg Lebedev, and Marco Zatta
Department of Physics and Helsinki Institute of Physics,Gustaf H¨allstr¨omin katu 2, FI-00014 Helsinki, Finland
Current Higgs boson and top quark data favor metastability of our vacuum which raises questionsas to why the Universe has chosen an energetically disfavored state and remained there duringinflation. In this Letter, we point out that these problems can be solved by a Higgs–inflaton couplingwhich appears in realistic models of inflation. Since an inflaton must couple to the Standard Modelparticles either directly or indirectly, such a coupling is generated radiatively, even if absent at treelevel. As a result, the dynamics of the Higgs field can change dramatically.
The current Higgs mass m h = 125 . ± .
24 GeV andthe top quark mass m t = 173 . ± . ± . H above the Higgs instability scale leads to desta-bilization of the EW vacuum, which poses a problem forthis class of inflationary models. Related issues have beenstudied in [7–9].The Higgs potential at large field values is approxi-mated by [10] V h (cid:39) λ h ( h )4 h , (1)where we have assumed the unitary gauge H T =(0 , h/ √
2) and λ h ( h ) is a logarithmic function of the Higgsfield. The current data indicate that λ h turns negative ataround 10 GeV [1], although the uncertainties are stillsignificant. In the early Universe, the Higgs potential ismodified by the Higgs–inflaton coupling V hφ with the fullscalar potential being V = V h + V hφ + V φ , (2) where V φ is the inflaton potential. Since the inflatonmust couple to the SM fields either directly or throughmediators as required by successful reheating, quantumcorrections induce a Higgs-inflaton interaction.In what follows, we consider a few representative ex-amples of reheating models. We focus on the Higgs cou-plings to the inflaton φ which are required by renormal-izability of the model. Such couplings are induced ra-diatively with divergent coefficients and necessitate thecorresponding counterterms. The dim-4 Higgs–inflatoninteraction takes the form V hφ = λ hφ h φ + σ hφ h φ , (3)where λ hφ and σ hφ are model–dependent couplings. Aswe show below, the range of λ hφ relevant to the Higgspotential stabilization is between 10 − and 10 − (seealso [2]). For definiteness, we choose a quadratic inflatonpotential [11] as a representative example of large fieldinflationary models, V φ = m φ + ∆ V − loop , (4)where m (cid:39) − M Pl and ∆ V − loop is the radiative cor-rection generated by various couplings of the model. Werequire this correction to be sufficiently small such thatthe predictions for cosmological observables of the φ –model are not affected, although some quantum effectscan be beneficial [12]. The divergent contributions to∆ V − loop are renormalized in the usual fashion and theresult is given by the Coleman–Weinberg potential [13].The leading term at large φ is the quartic coupling∆ V − loop (cid:39) λ φ ( φ )4 φ , (5)with λ φ being logarithmically dependent on φ .The energy transfer from the inflaton to the SM fieldsin general proceeds both through non–perturbative ef-fects and perturbative inflaton decay [14, 15]. In whatfollows, we make the simplifying assumption that thereheating is dominated by the perturbative inflaton de-cay such that the reheating temperature is given by T R (cid:39) . √ Γ M P l , where Γ is the inflaton decay rate. a r X i v : . [ h e p - ph ] D ec While this assumption is essential for establishing a cor-relation between λ hφ and T R , it does not affect the rangeof λ hφ consistent with the inflationary predictions. Weconsider three representative reheating scenarios whichassume no tree level interaction between the Higgs andthe inflaton, and compute the consequent loop–inducedcouplings.
1. Reheating via right–handed neutrinos.
Theinflaton energy is transferred to the SM sector via itsdecay into right–handed Majorana neutrinos ν R whichin turn produce SM matter. The added benefit of thismodel is that the heavy neutrinos may also be responsi-ble for the matter–antimatter asymmetry of the Universevia leptogenesis [16]. The relevant tree level Lagrangianreads − ∆ L = λ ν φν R ν R + y ν ¯ l L · H ∗ ν R + M ν R ν R + h . c . , (6)where l L is the lepton doublet, M is chosen to be realand we have assumed that a single ν R species dominates.These interactions generate a coupling between the Higgsand the inflation at 1 loop (Fig. 1). Since we are inter-ested in the size of the radiatively induced couplings, letus impose the renormalisation condition that they van-ish at a given high energy scale, say the Planck scale M Pl = 2 . × GeV. Then, a finite correction is in-duced at the scale relevant to the inflationary dynamics,which we take to be the Hubble rate H = mφ/ (cid:0) √ M Pl (cid:1) ,with other choices leading to similar results. We find inthe leading–log approximation, λ hφ (cid:39) | λ ν y ν | π ln M Pl H ,σ hφ (cid:39) − M | y ν | Re λ ν π ln M Pl H ,λ φ (cid:39) | λ ν | π ln M Pl H . (7)Here we have chosen the same renormalization conditionfor λ φ and λ hφ , σ hφ . Since the dependence on the renor-malization scale is only logarithmic, this assumption doesnot affect our results. The most important constraint onthe couplings is imposed by the inflationary predictions.Requiring λ φ φ / (cid:28) m φ / e -folds ofexpansion (see e.g. [17]), we find λ φ (cid:28) × − andtherefore λ ν < × − . The seesaw mechanism alsolimits the size of the Yukawa coupling y ν .The experimen-tal constraints on the mass of the active neutrinos requireapproximately ( y ν v ) /M < M < GeV,which in turn implies y ν < .
6. We therefore get anupper bound on the size of the Higgs–inflaton coupling, λ hφ < × − . (8)Note that λ hφ is positive and thus the inflaton creates apositive effective mass term for the Higgs. The trilinear FIG. 1. Leading radiatively induced scalar couplings via theright–handed neutrinos. (Diagrams with the same topologyare not shown). φh term is irrelevant as long as | λ ν | φ (cid:29) M , which is thecase for all interesting applications. (Similarly, the cubicterm φ is negligible.)During the inflaton oscillation stage, the magnitudeof φ decreases as 1 /t . When the effective masses of ν R and h turn sufficiently small, the decays φ → ν R ν R , φ → hh become allowed. The constraints above implyΓ( φ → ν R ν R ) (cid:29) Γ( φ → hh ) and therefore the total infla-ton decay width is Γ = | λ ν | π m, where we have neglectedthe ν R mass compared to that of the inflaton. Assum-ing that the right–handed neutrinos decay promptly andthe products thermalize (or ν R themselves thermalize) sothat T R (cid:39) . √ Γ M Pl , we find the following correlationbetween the Higgs–inflaton coupling and the reheatingtemperature T R , λ hφ (cid:39) × − | y ν | (cid:18) T R . × GeV (cid:19) , (9)where T R is bounded by 1 . × GeV. Note that thisrelation holds only under the assumption of perturbativereheating. Therefore, for the neutrino Yukawa couplingand the reheating temperature within one–two orders ofmagnitude from their upper bounds, the dynamics of theHiggs evolution change drastically. Similar conclusionsapply to models with multiple ν R species.
2. Reheating and non–renormalizable opera-tors.
A common approach to reheating is to assume thepresence of non–renormalizable operators that couple theinflaton to the SM fields. Let us consider a representativeexample of the following operators O = 1Λ φ ¯ q L · H ∗ t R , O = 1Λ φ G µν G µν , (10)where Λ , are some scales, G µν is the gluon field strengthand q L , t R are the third generation quarks. These cou-plings allow for a direct decay of the inflaton into theSM particles. It is again clear that a Higgs–inflaton in-teraction is induced radiatively. In order to calculatethe 1–loop couplings reliably, one needs to complete themodel in the ultraviolet (UV). The simplest possibilityto obtain an effective dim-5 operator is to integrate outa heavy fermion. Therefore, we introduce vector–like FIG. 2. Leading radiatively induced scalar couplings viathe vector–like quarks Q L,R and SM quarks q L,R . (Diagramswith the same topology are not shown). 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 . , (11)where the heavy quarks have the quantum numbers ofthe right–handed top t R , M is above the inflaton massand the couplings to the third generation are assumedto dominate. One then finds that O appears at treelevel with 1 / Λ = y Q λ Q / M , whereas O appears onlyat 2 loops with 1 / Λ ∼ y Q λ Q y t α s / (64 π M ) and can beneglected. Using the renormalization condition that therelevant couplings vanish at the Planck scale, we get inthe leading–log approximation (see Fig. 2) 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 ,λ φ (cid:39) | λ Q | π ln M Pl M , (12)where y t is the top Yukawa coupling and we have assumed M (cid:28) M Pl . Requiring smallness of the correction to theinflaton potential in the last 60 e –folds, we get | λ Q | < × − / (ln M Pl / M ) / and obtain the bound λ hφ < − (cid:18) ln M Pl M (cid:19) / , (13)where we have taken y t ( M ) (cid:39) .
5. For M in the allowedrange, this implies λ hφ < × − . We find again that λ hφ is positive and can be large enough to affect the Higgsevolution. Assuming no large hierarchy between λ Q and y Q , we have φ | λ Q | (cid:29) M| y Q | and the trilinear φh termis unimportant for the Higgs evolution.The trilinear interaction is however important for theinflaton decay. Taking for simplicity the couplings to bereal, we have Γ( φ → tth ) = λ Q y Q m / (512 π M ) andΓ( φ → hh ) = σ hφ / (32 πm ), which impliesΓ( φ → tth )Γ( φ → hh ) = π y t (ln M Pl / M ) m M (cid:28) M just above the inflaton mass. Thereforethe radiatively induced coupling dominates the inflaton decay. (This conclusion can be avoided by tuning thephases of λ Q and y Q such that Re( λ Q y Q ) (cid:39) T R < − M| y Q | (ln M Pl / M ) / forreal couplings. Taking | λ Q | M Pl as the upper bound on | y Q |M (see above) and allowing for the maximal valueof M to be 10 − M Pl , one finds T R < × GeV.An approximate correlation between λ hφ and T R can beexpressed as λ hφ (cid:39) − | λ Q || y Q | T R M . (15)
3. Reheating through dark matter production.
This somewhat more exotic scenario exhibits differentqualitative features. It assumes that the inflaton inter-acts mostly with dark matter or some other SM singlet,which then produces the SM fields through rescattering.The simplest renormalizable model of this type is basedon scalar DM s with the tree level interactions − ∆ L = λ φs φ s + σ φs φs + λ hs h s + λ s s + m s s . (16)In this case, DM is produced both through the non–perturbative effects and inflaton decay, while the SM par-ticles are generated via the Higgs field. Assuming thatDM is much lighter than the inflaton, the induced scalarcouplings in the leading–log approximation are λ hφ (cid:39) − λ φs λ hs π ln M Pl H ,σ hφ (cid:39) − λ hs σ φs π ln M Pl H ,λ φ (cid:39) − λ φs π ln M Pl H . (17)Unlike in the previous examples, we see that λ hφ canbe of either sign. It is positive for λ φs λ hs <
0, which is anadmissible possibility. The φ interaction gives a smallcontribution to the inflaton potential for | λ φs | < × − ,which implies | λ hφ | < × − | λ hs | . (18)Here λ hs is only restricted by perturbativity and can beas large as O (1) which results in even more significantinflaton–Higgs coupling than before. The trilinear termis unimportant for the Higgs field evolution for λ φs φ (cid:29) φφ hhss φ hhss φφ φφss FIG. 3. Leading radiatively induced scalar couplings viascalar dark matter. σ φs . Note that since the inflaton decay proceeds mostlythrough the σ φs coupling, at leading–log level there isno connection between the reheating temperature andthe size of the induced λ hφ . Finally, the model at handcan be viewed as a template for a class of models whichinvolve a scalar mediator between the inflaton and theSM or dark matter.The above examples show that a sizeable λ hφ can gen-erally be induced in realistic reheating models. It cantherefore make a crucial impact on the Higgs field evo-lution. Consider the typical situation that the trilinear φh term is small compared to the quartic φ h interac-tion. With positive λ hφ , the Higgs potential V h + V hφ ispositive for φ > (cid:115) | λ h | λ hφ h . (19)At larger inflaton values, the Higgs potential is convexand dominated by the Higgs–inflaton interaction termwhich creates an effective Higgs mass m h = φ (cid:112) λ hφ /
2. Ifsuch initial conditions are created and the effective massis sufficiently large, the Higgs field evolves to zero.In the reheating models above, we have obtained theupper bound λ hφ < − with some model–dependentvariations. Using | λ h | (cid:39) − at energies far above theinstability scale 10 GeV [1], we find that the initialvalue of the inflaton φ must exceed that of the Higgsfield h by at least two orders of magnitude. The use ofour renormalizable Higgs potential is meaningful as longas h (cid:28) M Pl so that in practice we take 0 . M Pl as theupper bound on h . In that case, the minimal value of φ is about 10 M Pl , which is typical for large–field inflationmodels.The evolution of the system at large field values is gov-erned by the equations¨ h + 3 H ˙ h + ∂V∂h = 0 , ¨ φ + 3 H ˙ φ + ∂V∂φ = 0 , (20)where 3 H M = ˙ h / φ / V and V (cid:39) m φ / λ hφ h φ /
4. Taking the initial values of ˙ h and ˙ φ to besmall, we find the following hierarchy m φ (cid:28) H (cid:28) m h , (21)where the effective inflaton mass is m φ = (cid:112) m + λ hφ h /
2. Therefore, the Higgs field evolvesquickly while the inflaton undergoes the usualslow roll. The magnitude of h decreases linearly, h ∼ (cos m h t ) /m h t , and within a few Hubble times H − the Higgs field value reduces by an order ofmagnitude [2]. After that the Hubble rate is dominatedby the inflaton mass term H (cid:39) mφ/ (cid:0) √ M Pl (cid:1) and theusual slow roll inflation begins. Since the effective massof the Higgs field is large and approximately constant, itevolves exponentially quickly to zero, | h ( t ) | ∼ e − Ht | h (0) | . (22) After 20 e –folds it becomes of electroweak size. Thismechanism is operative as long as m h > H/ λ hφ is10 − < λ hφ < − . (23)In this range, the quantum fluctuations of h during in-flation are also insignificant since (i) the Higgs field isheavy and (ii) the barrier separating the two vacua isat large field values h bar ∼ (cid:113) λ hφ | λ h | φ (cid:29) H . The lowerbound on λ hφ also guarantees that the classical evolu-tion of φ dominates, i.e. the initial inflaton value satisfies φ/M Pl < / (cid:112) m/M Pl [18]. The total number of e –foldsis about ( φ /M Pl ) /
4, with φ bounded by Eq. (19).The presence of a small trilinear term φh does not af-fect these considerations. As long as the effective Higgsmass term remains large and positive, the Higgs fieldevolves to zero. In that case, its effect is negligible. TheHiggs–inflaton interaction offers no solution to the cos-mological problems if the effective Higgs mass term istoo small or negative. In that case, h is overwhelminglylikely to end up in the catastrophic true vacuum.Since we introduce additional fields that couple to theHiggs, one may wonder how those affect the running ofthe Higgs quartic coupling. In the first two examples, thiseffect is small since the extra states are very heavy andthe (negative) leading contribution to the beta–functionis proportional to the fourth power of the Higgs–fermioncoupling. In the case of scalar mediators, the effect canbe significant depending on the scalar mass and its cou-pling to the Higgs. For m s ∼ TeV and λ hs ( H ) > ∼ . m s and/or smaller couplings the electroweak vacuum isstill metastable, while the stabilization mechanism de-scribed here is at work.In summary, reheating the Universe after inflation ne-cessitates (perhaps indirect) interaction between the in-flaton and the SM fields. As a result, a Higgs–inflatoncoupling is induced radiatively as required by renormal-izability of the model. Such a coupling can be sufficientlylarge to alter drastically the Higgs field dynamics in theearly Universe. In particular, it can hold the key to thequestion how the Universe has evolved to the energeti-cally disfavored state, given that the current data pointto metastability of the electroweak vacuum. Acknowledgments
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