Higgs-Dilaton Cosmology: an effective field theory approach
The Higgs-Dilaton 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 non-trivial relation between the spectral tilt of scalar perturbations and the dark energy equation of state . We study the self-consistency 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 cut-off 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.
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 non-minimally 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 non-minimally 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 so-called Einstein frame, where the gravity part takes the usual Einstein-Hilbert form, it becomes essentially non-polynomial and thus non-renormalizable, even if the gravity part is dropped off. Therefore, it should be understood as an effective field theory valid only up to a certain “cut-off” scale. One should distinguish between two different definitions of the “cut-off”. Quite often the cut-off of the theory is understood as the energy at which the tree level unitarity in high-energy scattering processes is violated. A second definition of the cut-off 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 low-energy variables breaks down. For the case of Higgs inflation, the tree-level 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 non-perturbative strong-coupling 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 self-consistent. As shown in Ref. Bezrukov:2010jz () (see also Ferrara:2010in ()), the beginning of the strong coupling regime (i.e. the cut-off 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 non-minimal 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 Higgs-Dilaton model Shaposhnikov:2008xb (); GarciaBellido:2011de (). The key element of this extension is scale-invariance (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 ever-present cosmological constant problem reappears associated to the fine-tuning of the dilaton self-interaction Shaposhnikov:2008xb (). However, if the dilaton self-coupling 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 scale-invariant UG gives rise to a “run-away” 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 Higgs-Dilaton inflation allowed to derive extra bounds on the initial inflationary conditions111The fine-tuning 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 Higgs-Dilaton model described above were previously noted in the literature. The first attempt to formulate a viable SI theory non-minimally 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 non-minimal 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 self-consistency of the Higgs-Dilaton model by adopting an effective field theory point of view. We will estimate the field-dependent cut-offs associated to the different interactions among scalars fields, gravity, vector bosons and fermions. We will identify the lowest cut-off 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 non-renormalizable, an infinite number of counter-terms 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 UV-completion 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 Higgs-Dilaton model. In Section III we calculate the cut-off 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 Higgs-Dilaton cosmology
We start by reviewing the main results of Refs. Shaposhnikov:2008xb (); GarciaBellido:2011de (), where the Higgs-Dilaton 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 as222For a theory invariant under all diffeomorphisms, this is equivalent to
with the so-called 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
where is the SM Higgs field doublet and are respectively the non-minimal 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 scale-invariant theory becomes
with the self-coupling of the Higgs field.
In order for this theory to be phenomenologically viable, we demand the existence of a symmetry-breaking ground state with non-vanishing background expectation value for both333If 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
All the physical scales are proportional to the non-zero background value of the dilaton field. For instance, the SM Higgs mass is given by
with the effective Planck scale in the Jordan frame. The same happens with the effective cosmological constant
which depending on the value of the dilaton self-coupling , gives rise to a flat (), deSitter () or anti-deSitter () 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 fine-tuned. As shown in Eq. (5), the difference between the electroweak and the Planck scale is encoded in the parameter444Note 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 non-minimal 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 Higgs-Dilaton 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
coincide with those obtained from a diffeomorphism invariant theory with modified action
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 equivalent555As 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
where the potential includes now the UG integration constant
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 non-minimal coupling of the Higgs field to gravity) with the Brans-Dicke 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 ()
where the quantity
can be interpreted as the metric in the two-dimensional field space in the Einstein-frame. 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 slow-roll approximation, , is approximately conserved, which suggests the definition of the set of variables
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
The potential (13) is naturally divided into a scale-invariant part, depending only on the field, and a scale-breaking part, proportional to and depending on both and . These are respectively given by
where we have safely neglected the contribution of and in Eq. (13). Note that the non-minimal couplings of the fields to gravity with naturally generate a “run-away” 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 to666Strictly 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
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 form777Note 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 Higgs-inflation.
The potential (19) becomes
whose scale-invariant 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 ()
where denotes the number of e-folds 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
An interesting cosmological phenomenology arises with the peculiar choice888Some 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
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 scale-invariance breaking potential for the dilaton is still of the “run-away” type
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 ()
For the present DE density , the above expression yields
Comparing Eqs. (25) and (29), it follows that the deviation of the scalar tilt from the scale-invariant one is proportional to the deviation of the DE equation of state from a cosmological constant999Outside this region of parameter space, the relation connecting is somehow more complicated GarciaBellido:2011de ()
The above condition is a non-trivial prediction of Higgs-Dilaton cosmology, relating two a priori completely independent periods in the history of the Universe. This has interesting consequences from an observational point of view101010Similar 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 Higgs-driven scenarios. and makes the Higgs-Dilaton 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 cut-off scale
Following Ref. Bezrukov:2010jz (), we now turn to the determination of the energy domain where the Higgs-Dilaton model can be considered as a predictive effective field theory. This domain is bounded from above by the field-dependent cut-off , i.e. the energy where perturbative tree-level unitarity is violated Cornwall:1974km (). At energies above that scale, the theory becomes strongly-coupled 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
such that all kind of higher-dimensional non-renormalizable operators
with appear in the resulting action. These operators are suppressed by appropriate powers of the field-dependent coefficient , which can be identified as the cut-off of the theory. This procedure gives us only a lower estimate of the cut-off, 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 N-particle scattering amplitudes hit the unitarity bound. The cut-off will then be the lowest of these scales.
In what follows we will apply these two methods to determine the effective cut-off of the theory. We will start by applying the method to compute the cut-off associated with the gravitational and scalar interactions. The cut-off associated to the gauge and fermionic sectors will be obtained via the method .
iii.1 Cut-off in the scalar-gravity sector
We choose to work in the original Jordan frame where the Higgs and dilaton fields are non-minimally coupled to gravity111111A similar study in the Einstein frame can be found in the Appendix A.. Expanding these fields around a static background121212Note 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 non-zero background expectation value, cf. Section II. As we will see below, this will give rise to a much richer cut-off structure.
we obtain the following kinetic term for the quadratic Lagrangian of the gravity and scalar sectors
The leading higher-order non-renormalizable operators obtained in this way are given by
Note that these operators are written in terms of quantum excitations with non-diagonal kinetic terms. In order to properly identify the cut-off 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
The cut-off scale associated to purely gravitational interactions becomes in this way the effective Planck scale in the Jordan frame
The remaining non-diagonal kinetic term for the scalar perturbations is given in compact matrix notation by
where is the Jordan frame analogue of Eq. (15) and depends only on the background values of the fields, i.e.
In order to diagonalize the above expression we make use of the following set of variables
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
where the different cut-off scales are given by
The effective cut-off 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 Cut-off in the gauge and fermionic sectors
Let us now move to the cut-off associated with the gauge sector. Since we are working in the unitary gauge for the SM fields, it is sufficient to look at the tree-level scattering of non-abelian 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 field131313In the absence of the Higgs field, the scattering amplitudes grow as the square of the center-of-mass 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 non-canonical 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
where . After diagonalizing the kinetic term for the scalar fields with the change of variables (40), the above expression becomes
where the effective coupling constants are given by
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
It is interesting to compare the previous expression with the results for the gauge cut-off of the simplest Higgs inflationary model Bezrukov:2010jz (). In order to do that, let us consider two limiting cases: the inflationary/high-energy period corresponding to field values and the low-energy regime at which . In these two cases, the above expression simplifies to
in agreement with the Higgs inflation model.
To identify the cut-off of the fermionic part of the Higgs-Dilaton model, we consider the chirality non-conserving process . This interaction receives contributions from diagrams with and exchange (channel) and from a diagram with fermion exchange (channel). In the asymptotic high-energy 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 cut-off, we find that this part of the theory enters into the strong-coupling regime at energies
where is the Yukawa coupling constant. The above cut-off 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 cut-offs 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 Higgs-Dilaton scenario self-consistent.
Let us start by considering the inflationary period, characterized by . As shown in Fig. 2, the lowest cut-off 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 cut-off scale in that region. The same conclusion is obtained for the total energy density, which turns out to be much smaller than . Moreover, the cut-off exceeds the masses of all particles in the Higgs background, allowing a self-consistent 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 Higgs-Dilaton potential around the minimum coincides (up to corrections) with that of the Higgs-inflation scenario. All the relevant physical scales, including the effective gauge and fermion masses, agree, up to small corrections, with those in Higgs-inflation GarciaBellido:2012zu () . This allows us to directly apply the results of Bezrukov:2008ut (); GarciaBellido:2008ab (); Bezrukov:2011sz () to the Higgs-Dilaton 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 cut-off 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 cut-off 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 multi-loop 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 counter-terms (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 counter-terms will have a different structure from that of the classical action. In particular, terms that are non-analytic with respect to the Higgs and dilaton fields will appear Shaposhnikov:2009nk (). They can be considered as higher-dimensional operators, suppressed by the field-dependent cut-offs. For consistency with the analysis made earlier in this work, we demand these cut-offs 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 Hooft-Veltman 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 counter-terms is defined as in prescription, i.e. by subtracting the pole terms in , where the dimensionality of space-time is . The finite part of the counter-terms has the same operator structure as the infinite part, including the parametric dependence on the coupling constants.
Although the requirement of the structure of higher-dimensional 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 non-renormalizable 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 field-dependent 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 scalar-tensor 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 cut-off (37),
which corresponds to the scale-invariant prescription of Ref. Shaposhnikov:2008xi (). In the Einstein frame the previous choice becomes standard (field-independent)
A second possibility is to choose the scale-invariant direction along the dilaton field, i.e.
When transformed to the Einstein frame it becomes
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 non-linearities 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 ()