HiggsDilaton Cosmology: an effective field theory approach
Abstract
The HiggsDilaton cosmological model is able to describe simultaneously an inflationary expansion in the early Universe and a dark energy dominated stage responsible for the present day acceleration. It also leads to a nontrivial relation between the spectral tilt of scalar perturbations and the dark energy equation of state . We study the selfconsistency of this model from an effective field theory point of view. Taking into account the influence of the dynamical background fields, we determine the effective cutoff of the theory, which turns out to be parametrically larger than all the relevant energy scales from inflation to the present epoch. We finally formulate the set of assumptions needed to estimate the amplitude of the quantum corrections in a systematic way and show that the connection between and remains unaltered if these assumptions are satisfied.
I Introduction
The shortcomings of the hot big bang model can be solved in an elegant way if we assume that the Universe underwent an inflationary period in its early stages. The easiest way for this paradigm to be realized is by a scalar field slowly rolling down towards the minimum of its potential history ().
As discussed in Ref. Bezrukov:2007ep (), inflation does not necessarily require the existence of a new degree of freedom. The role of the inflaton can be played by the Standard Model (SM) Higgs field with its mass lying in the interval where the SM can be considered a consistent effective field theory up to the inflationary scale. More precisely, if the Higgs boson is nonminimally coupled to gravity and the value of the corresponding coupling constant is sufficiently large, the model is able to provide a successful inflationary period followed by a graceful exit to the standard hot Big Bang theory Bezrukov:2008ut (); GarciaBellido:2008ab (). The implications of this scenario have been extensively studied in the literature Bezrukov:2008ej (); DeSimone:2008ei (); Barvinsky:2009fy (); Barvinsky:2008ia (); Bezrukov:2009db (); Clark:2009dc (); Barvinsky:2009ii (); Barvinsky:2009jd (); Lerner:2010mq (); Lerner:2009na (); Giudice:2010ka (); Burgess:2009 (); Barbon:2009ya (); Burgess:2010zq (); Hertzberg:2010dc (); Buck:2010sv (); Lerner:2011it (); Greenwood:2012aj (). Earlier studies of nonminimally coupled scalar fields in the context of inflation can be also found in Refs. Spokoiny:1984bd (); Salopek:1988qh (); Fakir:1990eg ().
When the Higgs inflation model described above is rewritten in the socalled Einstein frame, where the gravity part takes the usual EinsteinHilbert form, it becomes essentially nonpolynomial and thus nonrenormalizable, even if the gravity part is dropped off. Therefore, it should be understood as an effective field theory valid only up to a certain “cutoff” scale. One should distinguish between two different definitions of the “cutoff”. Quite often the cutoff of the theory is understood as the energy at which the tree level unitarity in highenergy scattering processes is violated. A second definition of the cutoff is the energy associated to the onset of new physics. As it was recently stressed in Ref. Aydemir:2012nz (), the breaking of tree level unitarity does not imply the appearance of new physics or extra degrees of freedom right above the corresponding energy scale; it just signals that the perturbation theory in terms of lowenergy variables breaks down. For the case of Higgs inflation, the treelevel scattering amplitudes above the electroweak vacuum appear to hit the perturbative unitarity bound at energies Burgess:2009 (); Barbon:2009ya (); Burgess:2010zq (); Hertzberg:2010dc (). Whether the theory requires an ultraviolet completion at these energies or simply enters into the nonperturbative strongcoupling regime with onset of new physics at higher energies (which could be as large as the Planck scale) is still an open question. Nevertheless, the Higgs inflation scenario is selfconsistent. As shown in Ref. Bezrukov:2010jz () (see also Ferrara:2010in ()), the beginning of the strong coupling regime (i.e. the cutoff scale according to the first definition which will be used in this article) depends on the dynamical expectation value of the Higgs field, which makes the theory weakly coupled for all the relevant energy scales in the evolution of the Universe. In other words, the SM with a large nonminimal coupling of the Higgs field to gravity represents a viable effective theory for the description of inflation, reheating, and the hot Big Bang theory.
The Higgs inflation scenario can be easily incorporated into a larger framework, the HiggsDilaton model Shaposhnikov:2008xb (); GarciaBellido:2011de (). The key element of this extension is scaleinvariance (SI). No dimensional parameters such as masses are allowed to appear in the action. All the scales are instead induced by the spontaneous breaking of SI. This is achieved by the introduction of a new scalar degree of freedom, the dilaton, which becomes the Goldstone boson of the broken symmetry and remains exactly massless. The coupling of the dilaton field to matter is weak and takes place only through derivative couplings, not contradicting therefore any 5th force experimental bounds Kapner:2006si ().
Although the dilatation symmetry described above forbids the introduction of a cosmological constant term, the everpresent cosmological constant problem reappears associated to the finetuning of the dilaton selfinteraction Shaposhnikov:2008xb (). However, if the dilaton selfcoupling is chosen to be zero (or required to vanish due to some yet unknown reason), a slight modification of general relativity (GR), known as Unimodular Gravity (UG), provides a dynamical dark energy (DE) stage in good agreement with observations. The scaleinvariant UG gives rise to a “runaway” potential for the dilaton Shaposhnikov:2008xb (), which plays the role of a quintessence field. The strength of such a potential is determined by an integration constant that appears in the Einstein equations of motion due to the unimodular constraint on the metric determinant. The common origin of the inflationary and DE dominated stages in HiggsDilaton inflation allowed to derive extra bounds on the initial inflationary conditions^{1}^{1}1The finetuning needed to reproduce the present dark energy abundance is transferred into the initial inflationary conditions for the fields at the beginning of inflation., as well as potentially testable relations between the early and late Universe observables GarciaBellido:2011de ().
Some of the properties of the HiggsDilaton model described above were previously noted in the literature. The first attempt to formulate a viable SI theory nonminimally coupled to gravity was done by Fujii in Ref. Fujii:1982ms (), although without establishing any connection to the SM Higgs. The role of dilatation symmetry in cosmology was first considered by Wetterich in Refs. Wetterich:1987fk (); Wetterich:1987fm (). In these seminal papers, the dynamical dark energy, associated with the dilaton field, appears as a consequence of the dilatation anomaly and is related to the breaking of SI by quantum effects. The present paper has a number of formal analogies and similarities regarding the cosmological consequences for the late Universe with Refs. Wetterich:1987fk (); Wetterich:1987fm (). At the same time, our approach to the source of dark energy is different from the one adopted in Refs. Wetterich:1987fk (); Wetterich:1987fm (), as we assume that SI is an exact (but spontaneously broken) symmetry at the quantum level, leading therefore to a massless dilaton. In Ref. Wetterich:1987fk (); Wetterich:1987fm (), both the cases of exact and explicitly broken dilatation symmetry were considered. Our theory with exact dilatation symmetry is different from that of Wetterich:1987fk (); Wetterich:1987fm () in two essential aspects. First, in our work the Higgs field of the SM has nonminimal coupling to gravity (it is absent in Ref. Wetterich:1987fk (); Wetterich:1987fm ()), which is important for the early Universe and leads to Higgs inflation. Second, the unimodular character of gravity (as opposed to standard general relativity used in Wetterich:1987fk (); Wetterich:1987fm ()) leads to an automatic and very particular type of dilatation symmetry breaking, which results in dynamical dark energy due to the dilaton field (absent in Wetterich:1987fk (); Wetterich:1987fm () for the case of exact scale invariance).
Our purpose here is to study, following the approach of Ref. Bezrukov:2010jz (), the selfconsistency of the HiggsDilaton model by adopting an effective field theory point of view. We will estimate the fielddependent cutoffs associated to the different interactions among scalars fields, gravity, vector bosons and fermions. We will identify the lowest cutoff as a function of the background fields and show that its value is higher than the typical energy scales describing the Universe during its different epochs. The issue concerning quantum corrections generated by the loop expansion is also addressed. Since the model is nonrenormalizable, an infinite number of counterterms must be added in order to absorb the divergences. It is important to stress at this point that, in the lack of a quantum theory for gravity, the details of the regularization scheme to be used cannot be univocally fixed. This means that the predictions of the model will be sensitive to the assumptions about the UVcompletion of the theory (corresponding to different regularization prescriptions). We will adopt a “minimal setup” that keeps intact the exact and approximative symmetries of the classical action and does not introduce any extra degrees of freedom. Within this approach, the relations connecting the inflationary and the dark energy domination periods hold even in the presence of quantum corrections.
The structure of the paper is as follows. In Section II we briefly review the HiggsDilaton model. In Section III we calculate the cutoff of the theory in the Jordan frame and compare it with the other relevant energy scales in the evolution of the Universe. In Section IV we propose a “minimal setup” which removes all the divergences and discuss the sensitivity of the cosmological observables to radiative corrections. Section V contains the conclusions.
Ii HiggsDilaton cosmology
We start by reviewing the main results of Refs. Shaposhnikov:2008xb (); GarciaBellido:2011de (), where the HiggsDilaton model was proposed and studied in detail. The two main ingredients of the theory are outlined below. The first one is the invariance of the SM action under global scale transformations, which leads to the absence of any dimensional parameters or scales. Denoting by the field content of the theory in a metric , these trasformations can be written as^{2}^{2}2For a theory invariant under all diffeomorphisms, this is equivalent to
(1) 
with the socalled scaling dimension and an arbitrary constant. In order to achieve invariance under these transformations, we let the masses and dimensional couplings in the theory to be dynamically induced by a field. The simplest choice would be to use the SM Higgs, already present in the theory. Note however that this option is clearly incompatible with the experiment. As discussed in Refs. Salopek:1988qh (); CervantesCota:1995tz (), the excitations of the Higgs field in this case become massless and completely decoupled from the SM particles.
The next simplest possibility is to introduce a new scalar singlet under the SM gauge group. We will refer to it as the dilaton . The coupling between the new field and the SM particles, with the exception of the Higgs boson, is forbidden by quantum numbers. The corresponding Lagrangian is given by
(2) 
where is the SM Higgs field doublet and are respectively the nonminimal couplings of the Higgs and dilaton fields to gravity GarciaBellido:2011de (). The term is the SM Lagrangian without the Higgs potential, which in the present scaleinvariant theory becomes
(3) 
with the selfcoupling of the Higgs field.
In order for this theory to be phenomenologically viable, we demand the existence of a symmetrybreaking ground state with nonvanishing background expectation value for both^{3}^{3}3If the Higgs field is massless, and if there is no electroweak symmetry breaking. the dilaton () and the Higgs field in the unitary gauge (). This is given by
(4) 
All the physical scales are proportional to the nonzero background value of the dilaton field. For instance, the SM Higgs mass is given by
(5) 
with the effective Planck scale in the Jordan frame. The same happens with the effective cosmological constant
(6) 
which depending on the value of the dilaton selfcoupling , gives rise to a flat (), deSitter () or antideSitter () spacetime. It is important to notice however that physical observables, corresponding to dimensionless ratios between scales or masses, are independent of the particular value of the background field . In order to reproduce the ratio between the different energy scales, the parameters of the model must be properly finetuned. As shown in Eq. (5), the difference between the electroweak and the Planck scale is encoded in the parameter^{4}^{4}4Note that the alternative choice is not compatible with CMB observations, cf. Eq. (24) and Fig.5. . Similarly, the hierarchy between the cosmological constant and the electroweak scale, cf. Eq. (6), implies . The smallness of these parameters, together with the tiny value of the nonminimal coupling , gives rise to an approximate shift symmetry for the dilaton field at the classical level, . As we will show in Section IV, this fact will will have important consequences for the analysis of the quantum effects.
The second ingredient of the HiggsDilaton cosmological model is the replacement of GR by Unimodular Gravity, which is just a particular case of the set of theories invariant under transverse diffeomorphisms. These theories generically contain an extra scalar degree of freedom on top of the massless graviton (for a general discussion see for instance Ref. Blas:2011ac () and references therein). In UG the number of dynamical components of the metric is effectively reduced to the standard value by requiring the metric determinant to take some fixed constant value, conventionally . As shown in Ref. Shaposhnikov:2008xb (), the equations of motion of a theory subject to that constraint
(7) 
coincide with those obtained from a diffeomorphism invariant theory with modified action
(8) 
Note that, from the point of view of UG, the parameter is just a conserved quantity associated to the unimodular constraint and it should not be understood as a cosmological constant.
Since the two formulations are completely equivalent^{5}^{5}5As usual, there are some subtleties related to the quantum formulation of (unimodular) gravity. However, these will not play any role in the further developments. The interested reader is referred to the discussion in Ref. Blas:2011ac () and references therein., we will stick to the diffeomorphism invariant language. Expressing the theory resulting from the combination of the above ideas in the unitary gauge we get
(9) 
where the potential includes now the UG integration constant
(10) 
Notice that the Lagrangian given by Eqs. (9) and (10) bears a clear resemblance with the models studied in Ref. Wetterich:1987fk (); Wetterich:1987fm (). In particular, it coincides (up to the nonminimal coupling of the Higgs field to gravity) with the BransDicke theory with cosmological constant studied in Wetterich:1987fk (). However, the interpretation of the term is different. In our case this constant is not a fundamental parameter associated with the anomalous breaking of SI Wetterich:1987fm (), but an automatic result of UG.
The phenomenological consequences of Eq. (9) are more easily discussed in the Einstein frame. Let us then perform a conformal redefinition of the metric with conformal factor . Using the standard relations Fujii:2003pa ()
(11) 
we get
(12) 
where
(13) 
is the potential (10) in the new frame. The noncanonical kinetic term in Eq. (12) can be written as
(14) 
where the quantity
(15) 
can be interpreted as the metric in the twodimensional field space in the Einsteinframe. Note that, unlike the simplest Higgs inflationary scenario Bezrukov:2007ep (), Eq. (14) cannot be recast in canonical form by field redefinitions. In fact, the Gaussian curvature associated to (15) does not identically vanish unless , which, as shown in Ref. GarciaBellido:2011de (), is not consistent with observations. Nevertheless, it is possible to write the kinetic term in a quite simple diagonal form. As shown in Ref. GarciaBellido:2011de (), the whole inflationary period takes place inside a field space domain in which the contribution of the integration constant is completely negligible. We will refer to this domain as the “scale invariant region” and assume that it is maintained even when the radiative corrections are taken into account (cf. Section IV). In this case, the dilatational Noether’s current in the slowroll approximation, , is approximately conserved, which suggests the definition of the set of variables
(16) 
The physical interpretation of these variables is straightforward. They are simply adequately rescaled polar variables in the plane. Expressed in terms of and , the kinetic term (14) turns out to be
(17) 
with
(18) 
The potential (13) is naturally divided into a scaleinvariant part, depending only on the field, and a scalebreaking part, proportional to and depending on both and . These are respectively given by
(19) 
where we have safely neglected the contribution of and in Eq. (13). Note that the nonminimal couplings of the fields to gravity with naturally generate a “runaway” potential for the physical dilaton, similar to those considered in the pioneering works on quintessence Wetterich:1987fk (); Wetterich:1987fm (); Ratra:1987rm ().
The inflationary period of the expansion of the Universe takes place for field values . From the definition of the angular variable in Eq. (16), this corresponds to^{6}^{6}6Strictly speaking, the condition holds beyond the inflationary region and includes also the reheating stage. . In that limit, we can neglect the term in the kinetic term (17) and perform an extra field redefinition
(20) 
where
(21) 
The variable is periodic and defined in the compact interval , with the value of the field at the beginning of inflation. In terms of these variables the Lagrangian (12) takes a very simple form^{7}^{7}7Note that the definition of the angular variable used in this work is slightly different from that appearing in Ref. GarciaBellido:2011de (). The new parametrization makes explicit the symmetry of the potential and shifts its minimum to make it coincide with that in Higgsinflation.
(22) 
with .
The potential (19) becomes
(23) 
whose scaleinvariant part resembles the potential of the simplest Higgs inflationary scenario Bezrukov:2007ep (), cf. Fig. 1. The analytical expressions for the amplitude and the spectral tilt of scalar perturbations at order can be easily calculated to obtain GarciaBellido:2011de ()
(24) 
where denotes the number of efolds between the moment at which the pivot scale exited the horizon and the end of inflation. Note that for , the expression for the tilt simplifies and becomes linear in
(25) 
An interesting cosmological phenomenology arises with the peculiar choice^{8}^{8}8Some arguments in favour of the case can be found in Ref. Shaposhnikov:2008xi (); GarciaBellido:2011de (); Blas:2011ac (). . In this case, the DE dominated period in the late Universe depends only on the dilaton field , which give rise to an intriguing relation between the inflationary and DE domination periods. Let us start by noticing that around the minimum of the potential the value of is very close to zero. In that limit, , which prevents the use of the field redefinition (20). The appropriate redefinitions needed to diagonalize the kinetic term (17) in this case turn out to be
(26) 
Using Eqs. (17) and (19) it is straightforward to show that the part of the theory associated to the Higgs field simplifies to the SM one. The resulting scaleinvariance breaking potential for the dilaton is still of the “runaway” type
(27) 
making it suitable for playing the role of quintessence. Let us assume that is negligible during the radiation and matter dominated stages but responsible for the present accelerated expansion of the Universe. In that case, it is possible to write the following relation between the equation of state parameter of the field and its relative abundance Scherrer:2007pu ()
(28) 
For the present DE density , the above expression yields
(29) 
Comparing Eqs. (25) and (29), it follows that the deviation of the scalar tilt from the scaleinvariant one is proportional to the deviation of the DE equation of state from a cosmological constant^{9}^{9}9Outside this region of parameter space, the relation connecting is somehow more complicated GarciaBellido:2011de ()
(30) 
The above condition is a nontrivial prediction of HiggsDilaton cosmology, relating two a priori completely independent periods in the history of the Universe. This has interesting consequences from an observational point of view^{10}^{10}10Similar consistency relations relating the rate of change of the equation of state parameter with the logarithmic running of the scalar tilt can be also derived, cf. Ref. GarciaBellido:2011de (). The practical relevance of those consistence conditions is however much more limited than that of Eq. (30), given the small value of the running of the scalar tilt in Higgsdriven scenarios. and makes the HiggsDilaton scenario rather unique. We will be back to this point in Section IV, where we will show that the consistency relation (9) still holds even in the presence of quantum corrections computed within the “minimal setup”.
Iii The dynamical cutoff scale
Following Ref. Bezrukov:2010jz (), we now turn to the determination of the energy domain where the HiggsDilaton model can be considered as a predictive effective field theory. This domain is bounded from above by the fielddependent cutoff , i.e. the energy where perturbative treelevel unitarity is violated Cornwall:1974km (). At energies above that scale, the theory becomes stronglycoupled and the standard perturbative methods fail. In order to determine this (background dependent) energy scale, two related methods, listed below, can be used.

Expand the generic fields of the theory around their background values
(31) such that all kind of higherdimensional nonrenormalizable operators
(32) with appear in the resulting action. These operators are suppressed by appropriate powers of the fielddependent coefficient , which can be identified as the cutoff of the theory. This procedure gives us only a lower estimate of the cutoff, since it does not take into account the possible cancelations that might occur between the different scattering diagrams.

Calculate at which energy each of the Nparticle scattering amplitudes hit the unitarity bound. The cutoff will then be the lowest of these scales.
In what follows we will apply these two methods to determine the effective cutoff of the theory. We will start by applying the method to compute the cutoff associated with the gravitational and scalar interactions. The cutoff associated to the gauge and fermionic sectors will be obtained via the method .
iii.1 Cutoff in the scalargravity sector
We choose to work in the original Jordan frame where the Higgs and dilaton fields are nonminimally coupled to gravity^{11}^{11}11A similar study in the Einstein frame can be found in the Appendix A.. Expanding these fields around a static background^{12}^{12}12Note that, in comparison with the analysis performed in Ref. Lerner:2011it () for generalized Higgs inflationary models, both the dilaton and the Higgs field acquire a nonzero background expectation value, cf. Section II. As we will see below, this will give rise to a much richer cutoff structure.
(33) 
we obtain the following kinetic term for the quadratic Lagrangian of the gravity and scalar sectors
(34)  
The leading higherorder nonrenormalizable operators obtained in this way are given by
(35) 
Note that these operators are written in terms of quantum excitations with nondiagonal kinetic terms. In order to properly identify the cutoff of the theory, we should determine the normal modes that diagonalize the quadratic Lagrangian (34). After doing that, and using the equations of motion to eliminate artificial degrees of freedom, we find that the metric perturbations depend on the scalar fields perturbations, a fact that is implicit in the Lagrangian (34). The gravitational part of the above action can be recast into canonical form in terms of a new metric perturbation given by
(36) 
The cutoff scale associated to purely gravitational interactions becomes in this way the effective Planck scale in the Jordan frame
(37) 
The remaining nondiagonal kinetic term for the scalar perturbations is given in compact matrix notation by
(38) 
where is the Jordan frame analogue of Eq. (15) and depends only on the background values of the fields, i.e.
(39) 
In order to diagonalize the above expression we make use of the following set of variables
(40)  
Note here that this is precisely the change of variables (up to an appropriate rescaling with the conformal factor ) needed to diagonalize the kinetic terms for the scalar perturbations in the Einstein frame. To see this, it is enough to start from Eq. (14) and expand the fields around their background values . Keeping the terms with the lowest power in the excitations, , it is straightforward to show that the previous expression can be diagonalized in terms of
(41)  
Written in terms of the canonically normalized variables (36) and (40) these operators read
(42) 
where the different cutoff scales are given by
(43)  
(44)  
(45) 
The effective cutoff of the scalar theory at a given value of the background fields will be the lowest of the previous scales. We will be back to this point in Section III.3.
iii.2 Cutoff in the gauge and fermionic sectors
Let us now move to the cutoff associated with the gauge sector. Since we are working in the unitary gauge for the SM fields, it is sufficient to look at the treelevel scattering of nonabelian vector fields with longitudinal polarization. It is well known that in the SM the “good” high energy behaviour of these processes is the result of cancellations that occur when we take into account the interactions of the gauge bosons with the excitations of the Higgs field^{13}^{13}13In the absence of the Higgs field, the scattering amplitudes grow as the square of the centerofmass energy, due to the momenta dependence of the longitudinal vectors . Lee:1977yc (); Lee:1977eg ().
In our case, even though purely gauge interactions remain unchanged, the graphs involving the Higgs field excitations are modified due to the noncanonical kinetic term. This changes the pattern of the cancellations that occur in the standard Higgs mechanism, altering therefore the asymptotic behaviour of these processes. As a result, the energy scale where this part of the theory becomes strongly coupled becomes lower.
To illustrate how this happens, let us consider the scattering in the channel. The relevant part of the Lagrangian is
(46) 
where . After diagonalizing the kinetic term for the scalar fields with the change of variables (40), the above expression becomes
(47) 
where the effective coupling constants are given by
(48) 
From the requirement of tree unitarity of the matrix, it is straightforward to show that the scattering amplitude of this interaction hits the perturbative unitarity bound at energies
(49) 
It is interesting to compare the previous expression with the results for the gauge cutoff of the simplest Higgs inflationary model Bezrukov:2010jz (). In order to do that, let us consider two limiting cases: the inflationary/highenergy period corresponding to field values and the lowenergy regime at which . In these two cases, the above expression simplifies to
(50) 
in agreement with the Higgs inflation model.
To identify the cutoff of the fermionic part of the HiggsDilaton model, we consider the chirality nonconserving process . This interaction receives contributions from diagrams with and exchange (channel) and from a diagram with fermion exchange (channel). In the asymptotic highenergy limit, the total amplitude of these graphs grows linearly with the energy at the center of mass. Once again, the channel diagram including the Higgs excitations unitarizes the associated amplitude Chanowitz:1978uj (); Chanowitz:1978mv (); Appelquist:1987cf (). Following therefore the same steps as in the calculation of the gauge cutoff, we find that this part of the theory enters into the strongcoupling regime at energies
(51) 
where is the Yukawa coupling constant. The above cutoff is higher than that of the SM gauge interactions (49) during the whole evolution of the Universe.
iii.3 Comparison with the energy scales in the early and late Universe
In this section we compare the cutoffs found above with the characteristic energy scales in the different periods during the evolution of the Universe. If the typical momenta involved in the different processes are sufficiently small, the theory will remain in the weak coupling limit, making the HiggsDilaton scenario selfconsistent.
Let us start by considering the inflationary period, characterized by . As shown in Fig. 2, the lowest cutoff in this region is the one associated with the gauge interactions . The typical momenta of the scalar perturbations produced during inflation are of the order of the Hubble parameter at that time. This quantity can be easily estimated in the Einstein frame, where it is basically determined by the energy stored in the inflationary potential (23). We obtain . When transformed to the Jordan frame () this quantity becomes , which is significantly below the cutoff scale in that region. The same conclusion is obtained for the total energy density, which turns out to be much smaller than . Moreover, the cutoff exceeds the masses of all particles in the Higgs background, allowing a selfconsistent estimate of radiative corrections (cf. Section IV).
After the end of inflation, the field starts to oscillate around the minimum of the potential with a decreasing amplitude, due to the expansion of the Universe and particle production. This amplitude varies between and , where is the asymptotic Planck scale in the low energy regime. As shown in Fig. 1, the curvature of the HiggsDilaton potential around the minimum coincides (up to corrections) with that of the Higgsinflation scenario. All the relevant physical scales, including the effective gauge and fermion masses, agree, up to small corrections, with those in Higgsinflation GarciaBellido:2012zu () . This allows us to directly apply the results of Bezrukov:2008ut (); GarciaBellido:2008ab (); Bezrukov:2011sz () to the HiggsDilaton scenario. According to these works, the typical momenta of the gauge bosons produced at the minimum of the potential in the Einstein frame is of order , with the mass of the gauge bosons in the Einstein frame and the curvature of the potential around the minimum. After transforming to the Jordan frame we obtain , with the weak coupling constant. The typical momentum of the created gauge bosons is therefore parametrically below the gauge cutoff scale (78) in that region.
At the end of the reheating period, , the system settles down to the minimum of the potential , cf. Eq. (23). In that region the effective Planck mass coincides with the value . The cutoff scale becomes . This value is much higher than the electroweak scale (cf. Eq. (5)) where all the physical processes take place. We conclude therefore that perturbative unitarity is maintained for all the relevant processes during the whole evolution of the Universe.
Iv Quantum corrections
In this section we concentrate on the radiative corrections to the inflationary potential and on their influence on the predictions of the model.
Our strategy is as follows. We regularize the quantum theory in such a way that all multiloop diagrams are finite, whereas the exact symmetries of the chosen classical action (gauge, diffeomorphisms and scale invariance) remain intact. Moreover, we will require the regularization to respect the approximate shift symmetry of the dilaton field in the Jordan frame, cf. Section II. Then we add to the classical action an infinite number of counterterms (including the finite parts as well) which remove all the divergences from the theory and do not spoil the exact and approximate symmetries of the classical action. Since the theory is not renormalizable, these counterterms will have a different structure from that of the classical action. In particular, terms that are nonanalytic with respect to the Higgs and dilaton fields will appear Shaposhnikov:2009nk (). They can be considered as higherdimensional operators, suppressed by the fielddependent cutoffs. For consistency with the analysis made earlier in this work, we demand these cutoffs to exceed those found in Section III.
An example of the subtraction procedure which satisfies all the requirements formulated above has been constructed in Ref. Shaposhnikov:2008xi () (see also earlier discussion in Englert:1976ep ()). It is based on dimensional regularization in which the ’t HooftVeltman normalization point is replaced by some combination of the scalar fields with an appropriate dimension, (we underline that we use the Jordan frame here for all definitions). The infinite part of the counterterms is defined as in prescription, i.e. by subtracting the pole terms in , where the dimensionality of spacetime is . The finite part of the counterterms has the same operator structure as the infinite part, including the parametric dependence on the coupling constants.
Although the requirement of the structure of higherdimensional operators, formulated in the previous paragraphs puts important constraints on the function , its precise form is not completely determined Shaposhnikov:2008xi (); Shaposhnikov:2009nk (); Codello:2012sn (), and the physical results do depend on the choice of . This somewhat mysterious fact from the point of view of uniquely defined classical theory (2) becomes clear if we recall that we are dealing with a nonrenormalizable theory. The quantization of this kind of theories requires the choice of a particular classical action together with a set of subtraction rules. The ambiguity in the choice of the fielddependent normalization point simply reflects our ignorance about the proper set of rules. Different subtractions prescriptions applied to the same classical action do produce unequal results. Sometimes this ambiguity is formulated as a dependence of quantum theory on the choice of conformally related frames in scalartensor theories Flanagan:2004bz (). The use of the same quantization rules in different frames would lead to quantum theories with different choices of .
Among the many possibilities, the simplest and most natural choice is to identify the normalization point in the Jordan frame with the gravitational cutoff (37),
(52) 
which corresponds to the scaleinvariant prescription of Ref. Shaposhnikov:2008xi (). In the Einstein frame the previous choice becomes standard (fieldindependent)
(53) 
A second possibility is to choose the scaleinvariant direction along the dilaton field, i.e.
(54) 
When transformed to the Einstein frame it becomes
(55) 
and coincides with the prescription II of Ref. Bezrukov:2009db () at the end of inflation.
In what follows we will use this “minimal setup” for the analysis of the radiative corrections. It will be more convenient to work in the Einstein frame, where the coupling to gravity is minimal and all nonlinearities are moved to the matter sector. The total action in the Einstein frame naturally divides into an Einstein Hilbert (EH) part, a purely scalar piece involving only the Higgs and dilaton (HD) fields and a part corresponding to the chiral SM (CH) without the radial mode of the Higgs boson Bezrukov:2009db (); Dutta:2007st (); Feruglio:1992wf ()