Renormalization of Yukawa model with sterile scalar in curved space-time

We explore classical and quantum properties of sterile scalar field coupled to $N$ copies of Dirac fermions in an external gravitational field. The model can be useful for better understanding of the restrictions which quantum theory may impose on the frequently used in cosmology sterile scalars such as inflaton or quintessence. We find that the self interaction scalar potential of consistent model include odd (first and third) powers of scalar field, which may have phenomenologically relevant consequences. The expression for the effective potential restored using the renormalization group should be extended compared to the standard expression which was derived in 1980-ies. This conclusion is supported by direct calculation of effective potential using normal coordinates and covariant cut-off regularization.


Introduction
The unique scalar field of the Minimal Standard Model (MSM) of particle physics is the Higgs boson, which is complex and belongs to the fundamental representation of SU (2).
The extensions of MSM such as nonminimal, supersymmetric version, of Grand Unification Theories, typically have larger scalar sector, but the new scalars are always representations of the symmetry group of the corresponding particle physics model. The consistency of such models with respect to quantum field theory requirements are the main tool in restricting the extensions of MSM, and in particular the scalar sector.
At the same time there is another sort of scalar fields, which are intensively used in cosmology. Both inflaton and quintessence are real scalars which are not related to representations of the symmetry group of particle physics and which can be called sterile scalars. An interesting question concerns the restrictions which can be imposed in quantum theory on the self-interaction potential of such a scalar field. A practical realization of this program requires assuming the form of the interaction between the sterile scalar and elementary particles. In the present work we consider the simplest possible version of such interaction, that means we consider a sterile scalar coupled to the N copies of massive Dirac fermion through Yukawa interaction.
In what follows we start by using the standard heat kernel approach to derive the one-loop divergences in such a model in curved space-time. As a result of this calculation we arrive at the minimal form of consistent self-interacting potential which provides multiplicative renormalizability of the model. The main new feature of this potential in comparison with a usual scalar (e.g. Higgs) is the presence of third and first powers of scalar field in the classical potential. Let us note that in the previous existing calculations of similar divergences in Refs. [1,2,3,4] the presence of these terms was recognized, but the consequences of this aspect of the theory were never sufficiently well explored. We fill this gap in the present work, including the discussion of the role of odd terms for the induced action of gravity, when the scalar field is in the minimum of the potential of the sterile scalar.
Another relevant aspect of the odd terms concerns the form of effective potential. The standard expression for effective potential restored from the renormalization group equation for effective action in curved space [5] (see also further development of the renormalization group method, applied for other sectors of effective action in [6]) is valid only for the theory where divergences have only second and fourth powers of the scalar field. The effective potential in the model with Yukawa interaction was calculated recently in [3,4] for the special case of massless fermions, when the odd terms in the scalar potential are not necessary and the loop contributions to the potential can be derived on the basis of the standard general expressions in flat [7] and curved spacetime [5,2]. If fermions are massive and there are odd terms, these standard results are incomplete and the renormalization group derivation should be modified. As usual (see the discussion in [8]) the renormalization group derivation is based on the identification of the renormalization parameter µ, which enables one to easily go beyond the local potential approximation [6,2]. On the other hand, such identification represents an assumption, which is always good to check, at least in the simplest case of effective potential. Thus, in order to achieve an additional verification of the complete result, we perform the derivation of effective potential for a sterile scalar directly, using the method which was developed recently in [8] in the basis of Riemann normal coordinates and local momentum representation [9,10].
The paper is organized as follows. In the next Sec. 2 we describe the derivation of oneloop divergences in the model with a sterile scalar coupled to N-component fermion. The beta-and gamma-functions are calculated, and the one-loop effective potential restored from the renormalization group in Sec. 3 . In Sec. 4 we present the tree-level analysis of the curved-space analog of the spontaneous symmetry breaking, in the presence of odd scalar terms, using an approximation of small odd terms and weak gravitational field. The analysis of the gravitational induced action shows that in the presence of odd terms, for a very light scalar, there are unusual low-energy terms, which can be in principle phenomenologically interesting. Finally, in Sec. 5 we draw our conclusions and discuss the perspectives for further work.
2 Yukawa model with sterile scalar and one-loop divergences Consider Yukawa model with a single real sterile scalar coupled to the N copies of fermion field. It proves useful to choose the classical action of the form where m is a scalar fields mass, M is a spinor field mass, h is the Yukawa coupling constant, λ, g and τ are the coupling constants in the scalar sector which survive in the flat limit, while ξ and f are the non-minimal parameters of scalar field coupling to gravity. The terms with odd powers and correspondingly the parameters g, τ and f were not analysed in detail in the previous considerations of the model in [1,2,3,4], regardless the odd terms were identified in the one-loop divergences. The consistent theory should include these terms from the very beginning and this is the approach we are starting to pursue. Regardless of the calculation of one-loop divergences in this theory follows the standard procedure (see, e.g., [2] for a number of well-elaborated examples), we shall give some details below, in order to simplify possible verifications. Let us stress from the very beginning that gravity will not be quantized in this paper (indeed, the generalization to the case of quantum gravity can be found for similar models in the original papers [11] and book [2], where the one-loop calculations were done by means of the generalized Schwinger-deWitt technique, and the subsequent recent work [12], where similar calculations were performed by using Feynman diagrams (without much detail) and used for attempting to reconciliate the renormalizability of quantum gravity and the absence of higher derivative ghosts.
In the framework of the background fields method in the theory with classical gravitational field, we start by decomposing the fields into classical ϕ,Ψ, Ψ and quantum σ,η, η counterparts, The one-loop divergences are defined by the bilinear part of the expansion of the action, which involves the operatorĤ, or in the explicit form Here the quadratic in quantum fields action S (2) depends on the background gravitational field and the background fieldsΨ, Ψ, ϕ. After some algebra we get In order to reduce the problem of deriving ln DetĤ to the standard form, let us introduce the conjugated matrix operatorĤ * as follows, It is well-known that the one-loop effective action has the form ∼ Tr ln (Ĥ). To calculate the divergences of effective action we will write it as Tr ln (Ĥ) = Tr ln (ĤĤ * ) − Tr ln (Ĥ * ).
It is easy to see that Tr lnĤ * contributes only to the vacuum divergences, that are already known for an arbitrary model [13,2]. Therefore it is sufficient to calculate the divergences of the productĤĤ * , which has a standard form, According to (8), one can identifŷ The one-loop divergences can be derived by means of the Schwinger-De-Witt technique (see, e.g. [2], [14]) and are given by the general expression where ε = (4π) 2 (D − 4) and We give intermediate formulas in the Appendix, and here present only the final result, where the vacuum divergences were also included for completeness. A few general comments are in order at this point. First of all, the result (12) confirms our expectations. All odd terms which we included into the classical action (3) really emerge in the one-loop divergences. The reason is that in the theory with sterile scalar these terms are not protected by any kind of symmetry, and hence it was actually expected that they would show up. Second, as far as we have the odd-power divergences, one should expect the logarithmic contributions in the corresponding finite part of effective action, in particular in the effective potential of a sterile scalar. In the next sections we shall see that these expectation will be completely confirmed. Third, it is worth pointing out that the odd terms may affect on the form of the non-local form factors, similar to what we had in the Yukawa model for the even terms [15] and earlier for a self-interacting scalar [16]. The discussion of this issue goes beyond the framework of the present work and will be left for the future.
The renormalization relations between bare and renormalizable quantities have the form which directly follows from the divergences. For the fields we have The relations for masses have the form For the usual even couplings and nonminimal parameters we have And, finally, for the odd couplings and nonminimal parameters, Turn attention to non-trivial renormalization of the odd coupling parameters and in particular of the new non-minimal coupling parameter f .

Renormalization group and effective potential
In this section we consider the renormalization group equation for the effective potential and discuss its solution to derive the effective potential for the model under consideration up to first order in scalar curvature. The form of the equation is defined by the corresponding beta-and gamma-functions which are calculated on the basis of the renormalization relations for the parameters and fields (for the theories in curved space time see e.g. [17,18,2]). Let us begin with beta-functions . They are defined as follows where P are the renormalized parameters, P = {m, M, h, λ, ξ, g, τ, f }. The scheme of derivation is described in [2] and we will not repeat it here, but only present the results.
The calculation which is based on the relations (17), (20), (23) leads to The gamma-functions are defined as follows: where Φ are the renormalized fields, Φ = (ϕ, Ψ k ). The relations (15) lead to In the case of conformal invariant theory we should put all dimensional constants m 2 , M, g, τ and f vanish, and ξ = 1 6 . It is easy to see that in this situation the pole coefficient in the expression for the divergences (12) is also conformal invariant. Also, the beta functions have the corresponding conformal fixed point, as it has to be from the general perspective [17,2]. Now we briefly discuss how to find the one-loop effective potential from the MS renormalization group equation in curved spacetime. The starting point is the overall µindependence of effective action, which immediately leads to [17,2] µ ∂ ∂µ where we assume the sum over all parameters P and all fields Φ = (ϕ, Ψ k ). From now on we shall set D = 4. The effective potential is defined as zero-order approximation in the derivative expansion for the scalar sector of Γ, Since (37) is a linear homogeneous equation, we get Equation (39) means that the explicit functional dependence on µ in the effective potential is exactly compensated by the µ-dependence of the scalar field ϕ and parameters P . At the one-loop level the last dependence can be written in the simple form involving the first order logarithmic dependence, as one can figure out from the above expressions for beta-and gamma-functions.
We will search for the effective potential up to the terms linear in scalar curvature, where V 0 is the flat-space effective potential and RV 1 is the first curvaturedependent correction to V 0 . Both functions V 0 and V 1 satisfy equation (39). Before solving the equations for V 0 and V 1 , we will take into account that in the one-loop approximation each spin gives additive contribution to the effective action. Therefore we can write where the labels (0) and ( 1 2 ) mean contribution from quantum scalar and spinor fields, respectively. The quantities V The equation (40) is a complicated partial differential equation with non-constant coefficients. Before solving this equation, it proves helpful to bring qualitative considerations that simplify the solution. The parameter µ can enter to the solution for V (0) 0 only logarithmically. Since the argument of the logarithm must be dimensionless, the dependence on µ should be through the parameter t = 1 2 ln X µ 2 , where the quantity X has the mass dimension two. This quantity can be constructed only from the dimensional parameters of the classical action, i.e. from m, M, ϕ, g, τ with arbitrary dimensionless coefficients. In principle, these coefficients should be fixed with the help of appropriate renormalization conditions for the effective potential. However, it is natural to assume that the form of effective potential should be consistent with the form of the operatorĤ (9) in the scalar sector. It means that the most natural choice for X is X (0) = m 2 + gϕ + 1 2 λϕ 2 . Thus, we identify for the scalar field contribution to effective potential. The parameters M and h do not contribute in the scalar sector. Therefore V 0 (t, m 2 , g, τ, ϕ) and the equation (40) becomes Here the functions β where and Solution to the equation (43) is written as follows where is the classical potential and m 2 (t (0) ), λ(t (0) ), g(t (0) ), τ (t (0) ) and ϕ(t (0) ) are the running parameters P (t (0) ) and the scalar field, satisfying the equations with the initial conditions As before, here P = m 2 , λ, g, τ . Since we work in the one-loop approximation, all quantum corrections are linear in , hence we can set the quantity Q (45) equal to zero in the expressions for beta-and gamma-functions. Then the solutions of the equations (48) can be easily found where we took into account that γ (0) = 0. The relations (50) together with the explicit forms for the functions β can be done in a similar way, so we skip the details. The result has the following form f ). These relations together with (51) are final solutions for curvature dependent contribution to effective potential from quantum scalar field.
We now turn to finding the quantum contributionV separately taking into account that consistence with form of the operatorĤ in fermionic sector (5) motivates a natural choice for dimensionless parameter containing the logarithm of µ in the form All other considerations are analogous to one for V 1 . Thus, we present only the final results for quantum corrections, and where the running parameters and field have the form Here we have solved the equations for running parameters and field with zero initial conditions since the classical contribution to effective potential has been already found when we calculated V in the relations (55) are the β P and γ ϕ at non-zero M and h but with zero parameters m 2 , λ, g, τ, ξ. The relations (53) and (54) define the final contribution to effective potential from quantum spinor field.
Finally, let us note that the calculation of the one-loop effective potential using the renormalization group may be organized in a slightly different way. One can consider the effective potential as a sum of classical potential and quantum corrections. Obviously, the quantum corrections alone satisfy the equation (39). The solution of this equation is carried out in the same way as we have described above, but for all running couplings one has to use zero initial conditions. The final result will of course be the same.

Direct calculation of effective potential
The results of the previous sections have shown the importance of the terms which are odd in the scalar field. This is something we learned from the divergences derived in the framework of the Scwinger-DeWitt method. Due to the importance of this quantum calculation, it looks reasonable to control its output by qualitatively different method. This is done in the present section by deriving effective potential in the O(R)-approximation using the normal coordinates and local momentum representation, in a way similar to what was done recently in [8], where one can find many relevant technical details and further references.

Riemann normal coordinates and scalar contribution
These coordinates are related to the geodesic lines which link a fiducial point P ′ (x µ ′ ) with another point with the coordinates x µ = x µ + y µ . In order to use the local momentum representation we assume that g µν (P ′ ) = η µν . In the vicinity of this point, in the linear in curvature approximation, we have Then, the bilinear operator in the scalar sector can be written as where V ′′ is the second derivative of the classical potential Further calculation in this subsection will essentially repeat the one of [8], but with another potential (58). We include this short review part for making all the presentation more consistent.
We can expand (57) in the Riemann normal coordinates as where · · · denotes high order terms in curvature.
The main advantage of the local momentum representation is that the calculation can be performed in flat space-time and the result can be always presented in a covariant way. For instance, the equation for the propagator of a real scalar field has the form is a covariant Dirac delta function. Since we are going to make calculations around the flat metric, it is most useful to work with the modified propagatorḠ(x, x ′ ) wherê The explicit form ofḠ(x, x ′ ) is known [9,10] for the free case when V ′′ = m 2 and for the even potential at constant ϕ [8]. As far as it is sufficient to regard V ′′ = const for the derivation of effective potential, we can replace m 2 bym 2 = V ′′ and obtain, in first order of curvature expansion, the following expression: We can expandĤ andḠ(x, x ′ ) up to the first power of curvature scalar aŝ Starting from this point the O(R 2 ... ) terms will not be mentioned. As far as Tr lnĤ = − Tr ln G(x, x ′ ), we get Consider first the effective potential in flat space. The first term in the r.h.s of (62) includes the flat space contributionV 0 (ϕ), which can be defined as Tr ln S 2 (ϕ = 0).
Here S 2 is the bilinear form of the classical scalar field action in the background field formalism, As a result we haveV Unlike the previous sections, all the subsequent calculations will be performed in the cut-off regularization what helps to simplify the calculations and make them more explicit. The transition to dimensional regularization can be easily done, of course. By introducing the four-dimensional Euclidean momentum cut-off Ω and integrating over angular coordinates, we arrive at [8]V Taking the last integral, after some algebra we obtain In order to cancel divergences, we follow the minimal substraction scheme and introduce an appropriated counterterm in the form where µ is the dimensional renormalization parameter. In this way, the quadratic and logarithmic divergences are eliminated and the renormalized effective potential can be written as Let us consider the linear in curvature corrections. The first order contribution is due to the second term in the r.h.s. of Eq. (63). This term can be easily presented in the form and hence After taking the last integral, the final result reads whereV 0 is given by Eq. (69) andV 1 =V f in Similar to the flat space case, the potential must be modified by adding a counterterm, Thus, the renormalized expression is The full renormalized effective potential for the scalar sector of (1) is the sum of expressions (69) and (75), where we restored the first power of the loop parameter and added the cosmological constant term, ρ Λ .

Fermion contributions and overall expression
Let us consider now the fermion contribution to the effective potential of the sterile scalar.
In the case of potential the background field ϕ can be treated as a constant, hence we denoteM = M + hϕ. Taking the Grassmann parity of the quantum field into account, in the Euclidean notations we get whereĤ f = i(γ µ ∇ µ + iM )δ ij . As usual, we consider [21] Tr lnĤ f = 1 2 Tr ln (Ĥ fĤ * f ), After a small algebra this gives The fermion propagator is defined from the relation Following the same scheme which was used in the scalar case, one can define modified propagatorḠ which satisfies the equation From the paper by Bunch and Parker [9] we learn thatḠ(x, x ′ ) =Ḡ(y) is defined as in the first order in curvature. Thus, Using the same considerations as for the scalar field, we find that The first term in r.h.s. correspond to the flat space case and the second one is the first order in curvature contribution. We will first perform the calculation in the flat space, when In the momentum representation this gives which provides the flat space part of the one-loop effective potential, In order to renormalize this result, we introduce a counterterm of the form Thus, The contribution in first order of curvature is which can be written in momentum space as Therefore, while there is no remnant finite part in this case. The divergences can be eliminated by adding a counterterm Finally, the first order in curvature part of the renormalized effective potential has the form Summing up the scalar (76) and fermion (91) contributions, we arrive at the general expression for the effective potential of our model, which includes a single real sterile scalar and N copies of massive fermion fields, where the interacting and odd terms of the classical potential V (remember that we separated the term with scalar mass for the sake of convenience) are defined in Eq. (58). The effective potential (92) depends on an arbitrary parameter µ. To fix the value of this parameter, we should imposed the renormalization conditions in a usual way.

Vacuum state and induced action
In this section we discuss some interesting aspects of the model under consideration related with symmetry breaking.
In the scalar theory in flat space without odd terms in the scalar sector, the classical potential U = − 1 2 m 2 ϕ 2 + V with "untrue" sign at m 2 has a constant minimum position, which corresponds to the spontaneous symmetry breaking (SSB) of discrete symmetry. In curved space such a constant position of the minimum and corresponding vacuum state are impossible [19]. The most immediate reason is that the classical potential contains now the non-minimal ξR(x)ϕ 2 term and constant solution for is impossible in general case. Such a general analysis is beyond of this paper, see the discussion e.g. in Ref. [20].
Things get even more complicated in the case of the sterile scalar, since the odd powers of the scalar in the potential make the discrete symmetry impossible and hence there is no much sense to speak about its breaking. However, let us assume that odd terms and also the more traditional non-minimal term ξRϕ 2 are small and treat them as small perturbations. In this case we do have a SSB in the zero-order approximation, and this is the terminology which we shall adapt in what follows.
Let us consider the equation of motion for vacuum expectation value (VEV) of the scalar field.
where v is supposed to be constant and in many cases it is so. However, as we have just mentioned, the solution of this problem in curved space is quite nontrivial [19], because in general R is not constant and ξ = 0. As a result, the equation for the VEV can not be solved with constant v even neglecting the kinetic term. Following [19] we can expand the solution into power series in the curvature tensor or ξ, such that In the solution of the problem of VEV we shall treat both ξR and odd terms as small perturbations.
In the zeroth order we have Now, in order to solve Eq. (94) in the first order, consider the following approximations: After solving the perturbative problem independently for ξR and odd terms, and summing up the results we obtain where To obtain the induced low-energy action we substitute the solution (2a) into the classical action (1). The result has the form In the expression (100) As expected, these formulas show the small contributions of the odd parameters of the sterile scalar τ , f and g. Since these parameters are certainly small, the change in these induced quantities is irrelevant compared to the quantities induced, e.g., in the electroweak phase transition, where v 0 is the vacuum expectation value of the Higgs field and the mentioned induced quantities may be much larger that, for instance, for the quintessence field. At the same time the induced action (100) have two other details which may be in fact more significant. First of all, in the case of quintessence the mass scale should be very small, and hence the VeV value v 0 should be small too. Then the non-local terms in the second line of Eq. (100) may become phenomenologically relevant. It would be interesting to explore the phenomenological limits on the odd parameters, starting from v 0 from the experimental data on Newton law and observational data on the bending of light, and see whether these limits can produce some restrictions on the quintessence potential. This investigation is beyond the scope of the present work, since it is devoted to the quantum aspects of a sterile scalar, however even the possibility looks attractive.
The second aspect concerns the term (✷+µ 2 0 ) −1 R in the first line of Eq. (100). According to the recent discussion in Ref. [22] this term can be relevant in cosmology, in the periods when the inverse of the size of the horizon may be comparable with the cosmic scale. Up to some extent, the corresponding effect can lead to the change of observational predictions in both inflationary and late cosmology epochs.

Conclusions and Perspectives
As we have seen in the previous sections, in the model with sterile scalar interacting to fermions there is no symmetry protection from the terms which are odd in the field, as a result these odd terms are necessary for the renormalizability of the theory. Similar conclusion has been done recently in [3,4], but we were trying to make renormalization in a consistent way that requires including odd terms into the classical potential.
As far as we include these terms into the classical action, the induced action of gravity, including both cosmological and inverse Newton constants, starts to depend on the new terms. And the most dramatic effect is that the induced action (4) gains the non-local contributions with the very small mass (3) in the Green function. This illustrated the effect which the sterile scalar (such as, e.g., quintessence) can produce on the gravitational action. It would be interesting to explore the effect of these non-local terms at the phenomenological level.