SUT/Physics-nnn Electroweak Vacuum Stability and the Higgs Field Relaxation via Gravitational Effects Mahdi Torabian * Department of Physics, Sharif University of Technology, Azadi Ave, 1458889694, Tehran, Iran The measured values of the Standard Model (SM) parameters favors a shallow metastable elec- troweak (EW) vacuum surrounded by a deep global AdS or a runaway Minkowski minimum. Fur- thermore, fine-tuning is the only explanation for the Higgs relaxing in its present local minimum. In this paper, assuming no new physics beyond the SM, we study the universal effect of gravity on the Higgs dynamics in the early universe. A generic two-parameter model is considered in which the Higgs is non-minimally coupled to a higher-curvature theory of gravity. The coupling between the Higgs field and the Weyl field in the Einstein frame has genuine predictions. In a broad region in the parameter space, the effective Higgs mass is large and it initially takes over through fast oscillations. This epoch is followed by the Weyl field slowly rolling a plateau-like potential. This framework generically predicts that the Higgs self-coupling in the EW vacuum is enhanced, com- pared to the SM predictions, through couplings to the gravity sector. Moreover, when the Higgs is settled in the EW vacuum, all other scalar flat directions would be lifted via gravitational effects mediated by the Weyl field. I. Introduction The great achievement of the LHC has been the discov- ery of the Standard Model (SM) Higgs boson [1–3] with mass m h = 125.35±0.15 GeV [4]. The scalar sector of the SM is completed and all parameters are determined. In particular, the Higgs self-coupling parameter is deduced at the electroweak scale to be around λ(m EW ) ≈ 0.13. This is the only parameter of the SM which is not mul- tiplicatively renormalized. With the central value of top quark mass m t = 173.2 ± 0.9 GeV [5], the beta-function β λ (at low/intermediate scales) is dominated by the top Yukawa coupling and thus it is negative. The SM, as a renormalizable theory, can in principle be applied in an arbitrary high energy and make predictions. If fact, the LHC has found no trace of new physics and no significant deviation of the SM predictions are observed. Within the SM, the effective Higgs potential can be computed at desired loop orders. The self-coupling parameter is monotonically decreasing and it vanishes at an interme- diate energy around 10 11 GeV and subsequently turns negative (see Fig. 1) [6–8]. At higher scales the gauge interactions take over and make the beta function positive. Then, the quartic cou- pling is increasing as it develops a new minimum which will be the global one. The location of the global mini- mum is located at tens of m Pl and is sensitive to Planck suppressed operators. It is reasonable to abandon the naive extrapolation to an arbitrary high energy and limit the running up to the Planck scale. Thus, the potential would basically be seen ill as it is unbounded from be- low. The electroweak vacuum is a local minimum and the barrier separating it from the deep well is extremely small. If one computes the tunneling rate between the vacua and ignores Planck suppressed interactions, one finds that life-time of the present-day electroweak vac- uum is greater than the age of the Universe and thus the vacuum is metastable [9]. However, if one includes higher dimensional operators, the the lifetime would be much shorter [10, 11]. Consequently, due to a huge negative cosmological constant, it leads to a catastrophic gravita- tional collapse. Moreover, the Higgs potential raises issues in connec- tion to the early universe cosmology. In order to end up in the present-day electroweak vacuum and prevent the Higgs from rolling down to global AdS minimum, a fine- tuning at level of one part in a hundred million in the Higgs value is needed [12, 13]. Moreover, if that initial condition is prepared, the Higgs will not stick to that in the presence of Hubble-size quantum fluctuations during a high scale inflation. New physics beyond the SM, including new particles and/or new interactions, could possibly change this pic- ture and stabilize the Higgs potential. However, excellent agreement of the SM predictions with the experimental results puts tight constraints on new physics as it must have marginal effect on the electroweak fit. Moreover, 100 000 -50 000 0 50 000 100 00 246GeV mPl V(h) h -30 000 -20 000 -10 000 0 10 000 20 000 30 000 0 10 10 GeV FIG. 1. The shape of the Higgs potential in the SM up to the Planck scale. Inner panels show zoomed-in regions arXiv:2011.06537v1 [hep-ph] 12 Nov 2020
Transcript
SUT/Physics-nnn
Electroweak Vacuum Stability and the Higgs Field Relaxationvia Gravitational Effects
Mahdi Torabian∗
Department of Physics, Sharif University of Technology, Azadi Ave, 1458889694, Tehran, Iran
The measured values of the Standard Model (SM) parameters favors a shallow metastable elec-troweak (EW) vacuum surrounded by a deep global AdS or a runaway Minkowski minimum. Fur-thermore, fine-tuning is the only explanation for the Higgs relaxing in its present local minimum.In this paper, assuming no new physics beyond the SM, we study the universal effect of gravity onthe Higgs dynamics in the early universe. A generic two-parameter model is considered in whichthe Higgs is non-minimally coupled to a higher-curvature theory of gravity. The coupling betweenthe Higgs field and the Weyl field in the Einstein frame has genuine predictions. In a broad regionin the parameter space, the effective Higgs mass is large and it initially takes over through fastoscillations. This epoch is followed by the Weyl field slowly rolling a plateau-like potential. Thisframework generically predicts that the Higgs self-coupling in the EW vacuum is enhanced, com-pared to the SM predictions, through couplings to the gravity sector. Moreover, when the Higgs issettled in the EW vacuum, all other scalar flat directions would be lifted via gravitational effectsmediated by the Weyl field.
I. Introduction
The great achievement of the LHC has been the discov-ery of the Standard Model (SM) Higgs boson [1–3] withmass mh = 125.35±0.15 GeV [4]. The scalar sector of theSM is completed and all parameters are determined. Inparticular, the Higgs self-coupling parameter is deducedat the electroweak scale to be around λ(mEW) ≈ 0.13.This is the only parameter of the SM which is not mul-tiplicatively renormalized. With the central value of topquark mass mt = 173.2± 0.9 GeV [5], the beta-functionβλ (at low/intermediate scales) is dominated by the topYukawa coupling and thus it is negative. The SM, as arenormalizable theory, can in principle be applied in anarbitrary high energy and make predictions. If fact, theLHC has found no trace of new physics and no significantdeviation of the SM predictions are observed. Withinthe SM, the effective Higgs potential can be computedat desired loop orders. The self-coupling parameter ismonotonically decreasing and it vanishes at an interme-diate energy around 1011 GeV and subsequently turnsnegative (see Fig. 1) [6–8].
At higher scales the gauge interactions take over andmake the beta function positive. Then, the quartic cou-pling is increasing as it develops a new minimum whichwill be the global one. The location of the global mini-mum is located at tens of mPl and is sensitive to Plancksuppressed operators. It is reasonable to abandon thenaive extrapolation to an arbitrary high energy and limitthe running up to the Planck scale. Thus, the potentialwould basically be seen ill as it is unbounded from be-low. The electroweak vacuum is a local minimum andthe barrier separating it from the deep well is extremelysmall. If one computes the tunneling rate between thevacua and ignores Planck suppressed interactions, onefinds that life-time of the present-day electroweak vac-uum is greater than the age of the Universe and thus thevacuum is metastable [9]. However, if one includes higher
dimensional operators, the the lifetime would be muchshorter [10, 11]. Consequently, due to a huge negativecosmological constant, it leads to a catastrophic gravita-tional collapse.
Moreover, the Higgs potential raises issues in connec-tion to the early universe cosmology. In order to end upin the present-day electroweak vacuum and prevent theHiggs from rolling down to global AdS minimum, a fine-tuning at level of one part in a hundred million in theHiggs value is needed [12, 13]. Moreover, if that initialcondition is prepared, the Higgs will not stick to that inthe presence of Hubble-size quantum fluctuations duringa high scale inflation.
New physics beyond the SM, including new particlesand/or new interactions, could possibly change this pic-ture and stabilize the Higgs potential. However, excellentagreement of the SM predictions with the experimentalresults puts tight constraints on new physics as it musthave marginal effect on the electroweak fit. Moreover,
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FIG. 1. The shape of the Higgs potential in the SM up to thePlanck scale. Inner panels show zoomed-in regions
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generically new physics would inevitably introduce a nat-uralness problem to the scalar sector.
The scalar fields generically have unsuppressed cou-plings to other fields and provides sizable portals to thedifferent sectors (the very same feature that causes insta-bility and unnaturalness of the Higgs). In particular thescalar and tensor backgrounds have sizable interactions.In this paper, irrespective of presence or absence of newphysics beyond the SM, we study the ubiquitous effect ofgravity on the Higgs dynamics in the early universe. Weconsider a well-motivated framework in which the Higgsfield is non-minimally coupled to a higher-curvature the-ory of gravity (see [14–24] for recent studies of differentaspects of these couplings). There are two gravitationalfree parameters in this framework and different dynamicscan be found in different regions of the parameter space.Through a conformal transformation, we can move to theEinstein frame which has direct contact to observables.In these coordinates, there are genuine couplings betweenthe Higgs field and the emergent Weyl field. Both fieldshave a plateau-like potential. It has been know that theHiggs field itself can play the role of inflaton [25] . Inthis framework, it is definitely possible is a corner inthe parameter space. The Higgs inflation at tree-levelis in perfect agreement with the observation of the CMBspectrum as it accommodates the spectral index of scalarpower spectrum and the tensor-to-scalar ratio. However,there are debates that quantum loop effects might jeopar-dize predictions and make the scenario complicated [26–28]. These effects include the above mentioned instabilityof the potential and the violation of perturbative unitar-ity close to the inflationary scale [29–31] .
These complications can be avoided in other regionsof the parameter space which is the aim of this paper.The effective curvature of the Higgs potential for a broadrange of parameters and large field values is large. Ini-tially in the early universe, the Higgs field dominatesthe dynamics as it coherently oscillates about its min-imum. The universe is matter dominated and the en-ergy in the Higgs field is drifted away by cosmic ex-pansion. The Higgs-Weyl interactions alleviate the in-stability problem and eventually the Higgs field is set-tled close to its present-day electroweak values. Finally,the Weyl (inflaton) field takes over the dynamics and itsplateau-like potential derives cosmic inflation [32] whichis in great agreement with inflationary observables in re-cent Planck results [33]. Moreover in this framework, thestructure of the electroweak vacuum is modified gravi-tationally compared to the SM. In particular the Higgsself-coupling parameter receives contributions from thegravity sector. In general we observe that in this setup,through omnipresent Weyl-scalars interactions, all scalarfields develop non-flat potentials with masses and mayalso receive non-zero VEV’s.
The structure of the paper is as follows. In the nextsection we introduce the model via its classical action.Then we study the stability condition by analyzing thescalar potential in the Einstein frame. Next we numer-
ically solve the equations of motion. Then we studyphysics around the electroweak vacuum and computethe Higgs sector parameters. Next we show that in thisframework all moduli are lifted and there is no flat direc-tions. Finally, we conclude in the last section.
II. The Action
The dynamics of the Higgs field which is non-minimallycoupled to a higher-curvature theory of gravity is givenby the following action parametrized in the Jordan frame
S =
∫d4x(−gJ)1/2
[12
(m2 + ξφ2
)RJ + 1
4αR2J
− 12gµνJ ∂µφ∂νφ− VJ(φ)
], (1)
where φ2 = 2H†H. The action includes all the operatorsup to dimension four which respects gauge symmetries.Thus, they must be included in a consistent quantumtheory as they are needed based on perturbative renor-malization theory [32, 34]. In this framework, the param-eters ξ and α define a two-parameter family of models.Needless to say, physics is different in different regionsof the parameter space. The non-minimal scalar-gravitycoupling is studied in variety of models especially con-nected to cosmic inflation. Moreover, the term quadraticin the Ricci scalar is the simplest generalization to Gen-eral Relativity. Although it is a higher-derivative theoryof gravity, it is free from Ostrogradski classical instabilityor the presence of spin-2 ghost (and also spin-0 ghost forpositive α) in the spectrum [35].
To make direct contact with observables, we can moveto the Einstein frame through a conformal transforma-tion of the metric
gEµν = m−2Pl (m2 + ξφ2 + αR)gµν ≡ e
√23m
−1Pl χgµν . (2)
Then, the action is
SE =
∫d4x(−gE)1/2 1
2
[m2
PlRE − gµνE ∂µχ∂νχ
−e−√
23m
−1Pl χgµνE ∂µφ∂νφ− 2VE(φ, χ)
].(3)
The Ricci-squared term introduces a new propagatingscalar field, a.k.a. Weyl scalar. In fact, the higher deriva-tive term make a spin-0 degree of freedom propagatingwhich is not ghost-like. It is not seen in the Jordanframe and is transparent in the Einstein frame. Notethat the Weyl scalar has a canonical kinetic term whilethe Higgs field is non-canonical. In fact, the Weyl andthe Higgs fields interact via the derivative terms besidesthe scalar potential. The scalar potential in the Einsteinframe reads as
VE(φ, χ) = e−2χ[
14α−1(m2
Ple−χ −m2 − ξφ2)2 + VJ(φ)
]= e−2χ
[14α−1(m2
Pleχ −m2
)2+ 1
4 (λ+ ξ2α−1)φ4
2
+ 12
(µ2 −
(m2
Pleχ −m2
)ξα−1
)φ2], (4)
where χ =√
23m−1Pl χ. We also introduced the conven-
tional Higgs potential (with a mass parameter µ2 < 0)
VJ(φ) =1
2µ2φ2 +
1
4λφ4. (5)
As can be seen from the potential (4), the Higgs massparameter and the quartic coupling in the Einstein framereceived contributions from the gravitational sector.
There is an upper bound on the value of α aroundα . 1061 from gravitational experiments measuringYukawa correction to the Newtonian potential [37, 38].For greater values, the mass of the Weyl field is lessthan the present Hubble rate around 10−33 eV. Moreover,the parameter ξ basically normalizes the 4-dimensionalPlanck mass in the Einstein frame. An upper limit existsonly when it is positive ξ . (mPl/v)2 ∼ 1032. A muchtighter bound can be put via collider physics. As arguesabove, we need to rescale the Higgs field to make its ki-netic term canonical. Around the electroweak vacuumwe find that
ϕ ≡ e−χ0/2φ ≈ (1 + ξv2/m2Pl)φ. (6)
Therefore, the Higgs coupling to the SM particles is mod-ified. This modification has an observable effect at col-liders by suppressing or enhancing the decay modes ofthe Higgs particle. The combined analysis of the AT-LAS and CMS excludes |ξ| & 1015 at 95% C.L. [39].Thus, we find a large (gravitational) parameter spacefor 1 . α . 1061 and |ξ| . 1015 and different param-eters, different dynamics can be obtained. As argued inintroduction, we are interested in values through whichthe cosmic inflation is driven by the plateau-like poten-tial of the Weyl field. The Planck results on the CMBanisotropy log(1010As) = 3.044 ± 0.414 68% C.L. andthe primordial gravitational waves r < 0.11 95% C.L.[33] constraint the free parameter α as
α = (12π2rAs)−1 & 3.4× 107. (7)
The simplest manifestation of the Starobinsky inflationpredicts r ≈ 2.5 × 10−3 and therefor α ≈ 109. However,modifications to the model predict larger r and so smallerα works as well (see [36]). As we later see, the electroweakvacuum further constrains the parameter space.
Stability conditions
As can be seen from the scalar potential (4) stabilityat Planck field values can be obtained for
λ(mPl) + ξ(mPl)2α(mPl)
−1 ≥ 0. (8)
Assuming no new physics between the electroweak scaleand the Planck scale and applying the central values of
the measures SM parameters, the best-fit value for theHiggs self-coupling at the Planck scale is
λ(mPl) ≈ −0.0129. (9)
Then, the stability condition (8) implies that (α ≈ 109)
|ξ(mPl)| & few × 103. (10)
For negative ξ the Higgs potential is convex for any valueof the scalar fields. For positive ξ, further condition oninitial field values is imposed so that the quadratic Higgsterm does not take over the quartic term to destabilizethe potential
χini . 2 ln φini + ln(ξ/2). (11)
Similarly for negative ξ, if the initial Weyl field valuesatisfies
χini & 16.1 + 2 ln φ0 − ln(−ξ), (12)
then Higgs quadratic term takes over the quartic termand makes the Higgs potential stable in large field values.It helps to choose smaller value of |ξ|. In the rest of thepaper, for concreteness, we choose positive values of ξand study the evolution of the Higgs and the Weyl fields.
III. Dynamics in the Early Universe
Assuming spatial homogeneity, the dynamics of theHiggs field φ(t), the Weyl field χ(t) and the scale fac-tor a(t) in the Friedman metric
ds2 = −dt2 + a(t)d2x, (13)
is governed by the following equations of motion
χ+ 3Hχ+1√6e−χφ2 + V E,χ = 0, (14)
φ+ 3Hφ−√
23m−1Pl χφ+ V E,φ = 0, (15)
3H2m2Pl =
1
2χ2 +
1
2e−χφ2 + VE , (16)
−2Hm2Pl = χ2 + e−χφ2. (17)
In the above equations V E,φ and V E,χ are field derivative
of the scalar potential and H = a/a is the Hubble ex-pansion rate. These are coupled second-order differentialequations that can be solved by numerical methods. Thesolutions for scalar fields in Planck mass versus Plancktime are plotted in figure 2. The Higgs field initial valueis taken of order one in Planck mass. Its initial velocitycould also be chosen order one, however, it is found thatit has insignificant qualitative effect on the solutions. Onthe other hand, the Weyl field initial conditions are cho-sen such that the universe undergoes at least 60 e-foldsof exponential expansion.
The solutions are interpreted as follows. The Higgsand the Weyl fields are initially frozen for tens of Planck
ϕ0=1, χ0=5.5, dϕ0=0, dχ0=0FIG. 2. Time evolution of the Higgs (left panel) and the Weyl (right panel) fields. The initial conditions are given on the top.
time (mini-inflation) until they commence harmonic os-cillations about their local minima. The effective Higgsmass is large and the Higgs field oscillates with large am-plitudes. The energy density in its coherent oscillationstakes over the dynamics of the universe and it is red-shifted away by cosmic expansion. The Higgs field valuesis decreasing and relaxing toward its small values. It isimportant to emphasis that by this time the Higgs fieldamplitude is less than 10−8mPl so it later evolves to theelectroweak vacuum. When the Hubble rate is around10−4mPl, the Weyl field takes over the energy densityby its plateau-like potential and slowly rolls down. Theuniverse enters an epoch of inflation which lasts around60 e-folds. Then, the Weyl field oscillates about its min-ima and the universe is filled by the Bose condensates ofHiggs and Weyl particles. After many damped oscilla-tions fields settle down in their minima near the origin.Finally, they decay and reheat the universe. The fieldshave slightly different evolution, although qualitativelythe same, depending on the value of the non-minimalcoupling parameter as is plotted in figure 3.
IV. Predictions for the Electroweak Vacuum
At late times, the Higgs field is closed enough to theorigin of its potential and, through the symmetry break-ing mechanism, is settled to its local minimum. Via ap-propriately choosing the mass parameter, the Higgs fieldreceives a non-zero vacuum expectation value (VEV) andspontaneously breaks the electroweak symmetry. TheVEV and the curvature about the minimum are respec-tively computed as
v2 ≡ φ20 =
−µ2
λ+ ξ−µ2
m2
≈ −µ2
λ≈ (246 GeV)2, (18)
m2φ = −2µ2λ+ ξ2α−1
λ+ ξ−µ2
m2
≈ −2µ2(1 + ξ2λ−1α−1).(19)
The Higgs VEV in the first equation is fixed by the mea-sured masses of particles in the SM. As m2 � −µ2 thesign of the Higgs non-minimal coupling parameter is notrelevant in the symmetry breaking mechanism. Further-more, the Weyl field receives a non-zero VEV and massas follows
χ0 =√
23mPl ln
[ m2
m2Pl
+−µ2
m2Pl
ξ − α−µ2
m2
λ+ ξ−µ2
m2
]≈ ξ−µ
2
λm−1
Pl ≈ 10−5ξ eV, (20)
m2χ =
m2Pl
3α
[ m2
m2Pl
− µ2
m2Pl
ξ
λ− ξ(µ2/m2)
]− µ2 µ2
m2Pl
2− λλ−ξ(µ2/m2)
3(λ− ξ µ2
m2 )≈ m2
Pl
3α∼ 1013 GeV. (21)
The interactions between the Higgs and the Weyl fieldsbesides the Higgs receiving a VEV induces a non-zeroVEV for the Weyl field. It is sensitive to the sign of thenon-minimal coupling parameter. It further implies thatthe canonical Higgs field is obtained by a field redefinition
φc = e−χ0/2φ. (22)
As a result of Higgs-Weyl interactions, there is also anon-diagonal component in the scalar mass matrix
FIG. 3. solutions for different ξ parameter: ξ=1000(black), 5000(green), 10000(red), 15000(blue), 20000(yellow)
The former is the inflaton mass which is effectively decou-pled from low-scale physics. The lighter mass eigenvalueis the mass of the Higgs particle and is compared withits measured value. Then, the Higgs mass parameter andthe self- coupling are determined as follows
µ2(mEW) ≈ −(89.1 GeV)2, (27)
λ(mEW) ≈ 0.13. (28)
The mass-mixing angle is negligible and computed as
tan 2θ ≈ −√
6ξvm−1Pl ≈ −10−16ξ. (29)
The physical Higgs field is the mass eigenvector
h = cos θe−χ0/2φ+ sin θ χ. (30)
The couplings of the Higgs boson to other particles (com-pared to the SM) are suppressed due to gravitations ef-fetcs. Thus, the Higgs production and decay rates areaffected. It has an observable effect at colliders in termsof the global signal strength
µ ≡ σ·BrσSM·BrSM
= e−χ cos2 θ ≈ 1− 6m−2Pl ξ
2v2. (31)
The combined analysis of the ATLAS and CMS measure-ments implies µ = 1.07 ± 0.18 [1, 2, 39]. Therefore, thismeasurement excludes |ξ| & 1015 at 95% C.L. However, atighter bound is obtained from the strength of the Higgsself-coupling and is analyzed in the following.
Enhanced Higgs self-coupling
Interestingly, the scalar potential in the Einstein frame(4) indicates that the Higgs cubic and quartic self-coupling is modified by gravitation effects. Around theelectroweak vacuum and using (28) we find that
λeff(mEW) = 0.13 + ξ2(mEW)α(mEW)−1. (32)
It is a distinctive deviation from the SM prediction. Inparticular, the Higgs quartic self-interaction at the EWscale is controlled by the parameter
A sharp prediction of this mode is that the Higgs self-coupling is greater than the SM prediction. Presently,there is no direct measurement of the Higgs self-coupling.The next generation of particle colliders are commis-sioned to directly measure the Higgs self-coupling inmulti-Higgs processes. It is interesting to note that per-turbativity (i.e. λ4 . 1) imposes a strong upper boundon the value of the Higgs non-minimal coupling param-eter. For instance, if the Weyl field is responsible forcosmic inflation, then α ∼ 108 and we find an upperbound as |ξEW| . 104. This is the tightest bound onthe non-minimal coupling parameter in the literature. Ingeneral, as we could see in this framework with the SMinteracting with gravity, collider experiments constrainthe parameter space in the gravitational sector and favorregions where ξ2α−1 . O(1).
Gravitationally uplifted flat directions
Another distinctive prediction in this framework isthat, with the gravitational effects and a Higgs-like mech-anism, all scalar fields receive non-zero masses. More-over, depending on the sign of the non-minimal couplingparameters, they might also develop a non-zero VEV.Different scenarios are studied in this part.
We consider a generic scalar field ψ with (essential)non-minimal coupling parameter ξψ. It is interesting tonote that, assuming no other potential term, then therewill be a non-trivial potential for ψ in the Einstein frame
VE ⊃ e−2χ[
14ξ
2ψα−1ψ4− 1
2ξψα−1(m2
Pleχ−m2
)ψ2]. (34)
5
Thus, the gravitational effects induce an effective self-coupling and mass parameter.
We observed previously that the Wely field receives anon-zero VEV as the Higgs field develops a VEV (20).Then, the field ψ receives an effective mass parameter
µ2ψ = µ2
φ
ξφ − α−µ2
m2
λ+ ξφ−µ2
m2
ξψα−1 ≈ µ2
φ
ξφξψαλ
. (35)
We recall that µ2φ < 0 and α, λ > 0. Depending on the
sign of ξφξψ two scenarios can be imagined.
i. ξφξψ < 0 : The field ψ is settled at zero field value.However, its flat direction is lifted and it receives a mass
m2ψ = µ2
ψ ≈ − 12α−1ξψξφ(125GeV)2. (36)
ii. ξφξψ > 0 : The field ψ is tachyonic around theorigin and its receives a non-zero VEV
ψ20 = −µ2
ψξ−2ψ α ≈ ξ−1
ψ ξφ(246GeV)2. (37)
The mass around this minimum is
m2ψ = −2µ2
ψ ≈ ξψξφα−1(125GeV)2. (38)
The gravitations effects induce a slight non-zero VEVand consequently break any symmetry under with thescalar field is charges.
In either cases, we observed that gravitational cou-plings induce a non-zero mass for scalar fields and liftthe flat directions in the Einstein frame (if there was anyin the Jordan frame). Furthermore, depending on thesign of non-minimal parameters, they might also developa non-zero VEV through gravitational effects and spon-taneously break symmetries if they are charged. Essen-tially, the non-zero VEV of the Weyl field, which itselfis induced by the non-zero VEV of the Higgs field, in-duces a non-zero VEV to any scalar field (non-minimallycoupled to gravity). As can be explicitly observed, thedominant Higgs-like mechanism in a sector sets the rele-vant scales in the other sectors. The Weyl field plays therole of a portal among different sectors. The significant
prediction is that there is no scalar field with flat direc-tion possibly no associated symmetry is preserved in atheory of gravity. It is a genuine gravitational effect.
V. Conclusion
In this note we proposed a general gravitational frame-work with scalars non-minimally coupled to a Ricci-squared theory of gravity as it is implied by renormal-ization theory. We found that there is a two-parameterfamily of models with rich dynamics. We were inter-ested in region of the parameter space where the emer-gent Weyl field is the inflaton. We found that, at earlytimes, the Higgs field in the Einstein frame had a largeeffective mass. It quickly relaxed to its small field valuesthrough damped oscillations prior to inflation. It alle-viates the metastability problem of the Higgs potentialand explains why the Higgs field could be trapped in theshallow electroweak vacuum.
This framework has two predictions. Firstly, we foundthat the Higgs self-coupling receives contribution fromthe gravitational sector and thus is enhanced. Any devi-ation in future measurements can be naturally explainedthrough omnipresent gravitational couplings with noneed of new physics. On the other hand, perturbationtheory in low scale physics constrains the gravitationalparameter space.
Secondly, we observed that natural gravitational cou-plings lift flat directions along moduli fields. Moreover,in some part of the parameter space, scalar fields de-velop non-zero VEVs through pure gravitational effects.If they are charged, they gravitationally spontaneouslybreak symmetries. They Weyl field plays the role ofa portal that carries the effect of the dominant Higgs-like mechanism in one sector to the other sectors. If anon-minimally coupled scalar receives a non-zero VEV,all other scalars with non-minimal couplings receive non-zero masses and possibly non-zero VEV’s. All the aboveobservations are universal and generic as are induced byubiquitous and natural extensions in the gravity sector.
Acknowledgments This work is supported by the re-search deputy of SUT.
Note added: This article is an expanded and a pub-lished version of [40] initially presented in IPM 1st Topi-cal Workshop on Theoretical Physics.
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