Covariant quantum corrections to a scalar field model inspired by nonminimal natural inflation
aa r X i v : . [ g r- q c ] J un Covariant quantum corrections to a scalar field model inspired bynonminimal natural inflation
Sandeep Aashish ∗ and Sukanta Panda † Department of Physics, Indian Institute of ScienceEducation and Research, Bhopal 462066, India (Dated: June 2, 2020)
Abstract
We calculate the covariant one-loop quantum gravitational effective action for a scalar fieldmodel inspired by the recently proposed nonminimal natural inflation model. Our calculation isperturbative, in the sense that the effective action is evaluated in orders of background field, arounda Minkowski background. The effective potential has been evaluated taking into account the finitecorrections. An order-of-magnitude estimate of the one-loop corrections reveals that gravitationaland non-gravitational corrections have same or comparable magnitudes. ∗ [email protected] † [email protected] . INTRODUCTION A fully consistent quantum theory of gravity has remained elusive despite longstandingefforts to construct a gravity theory that is valid at the Planck scale (see Ref. [1] for acomprehensive review). A well known problem with quantizing gravity perturbatively isthe inability to consistently absorb the divergences, giving rise to non-renormalizability.However, in the past two decades or so, a more modern view has developed where generalrelativity is studied as a quantum effective field theory at low energies [2]. This treatmentallows separation of quantum effects from known low energy physics from those that dependon the ultimate high energy completion of the theory of gravity (see Ref. [3] for a review byDonoghue and Holstein).More recently, with the availability of high precision data from experiments probing theearly universe, especially inflation era, it has become important to consider quantum gravi-tational corrections in early universe cosmology [4–6]. This has motivated several studies ofaspects of quantum gravitational corrections in inflationary universe, see for example Refs.[7–15].One of the well-known methods employed in such studies is to compute the effectiveaction, which is known to be the generator of 1PI diagrams [16, 17]. An advantage of thistechnique is that one can directly obtain divergence structure at a given loop order, withoutgoing through the hassle of summing over individual Feynman diagram contributions. Otherapplications include the calculation of effective potential [18]. The computation of effectiveaction is most commonly carried out using the background field method, where small fluc-tuations about a classical background field are quantized, not the total field. In general, itturns out that the results consequently depend on the choice of background field [19–21]. Incase of gravity, which is treated as a gauge theory, it is therefore important to ensure thatthere are no fictitious dependence of conclusions on the choice of gauge and background.Hence, in this work, we use DeWitt-Vilkovisky’s covariant effective action approach thatsystematically yields gauge and background independent effective action [22, 23].We consider a recently proposed modification of the natural inflation (NI) model [24],wherein a periodic nonminimal coupling term similar to NI potential is added along witha new parameter, that eventually leads to a better fit with Planck results [25]. Thesephenomenological implications are in no way the only motivation for considering this model2n the present work. Rather, it serves as a toy model to achieve our mainly three objectives,which are as follows. First, to set up the computation using symbolic manipulation packagesto evaluate one-loop covariant effective action up to quartic order terms in the backgroundfield. As a starting point, we work in the Minkowski background. Second, we aim to recoverand establish past results. And third, we wish to estimate the magnitudes of quantumgravitational corrections from the finite contributions at least for the effective potential,since there are typically several thousands of terms one has to deal with.The organization of this paper is as follows. In Sec. II, we introduce and briefly reviewthe nonminimal natural inflation model. Sec. III covers a review of covariant effective actionformalism, notations, and the methodology of our calculations. Sec. IV constitutes a majorpart of this paper, detailing the calculations of each contributing term mentioned in Sec. III,along with the divergent part, loop integrals, and renormalization. Some past results andtheir extensions have also been presented. Finally, in Sec. V, we derive the effective potentialincluding the finite corrections from the loop integrals, and perform an order-of-magnitudeestimation of quantum corrections.
II. PERIODIC NONMINIMAL NATURAL INFLATION MODEL
Natural inflation was first introduced by Freese et al .[24] as an approach where inflationarises dynamically (or naturally ) from particle physics models. In natural inflation models,a flat potential is effected using pseudo Nambu-Goldstone bosons arising from breaking thecontinuous shift symmetry of Nambu-Goldstone modes into a discrete shift symmetry. As aresult, the inflation potential in a Natural inflation model takes the form, V ( φ ) = Λ (1 + cos( φ/f )) ; (1)where the magnitude of parameter Λ and periodicity scale f are model dependent. However,majority of natural inflation models are in tension with recent Planck 2018 results [26].However, it was shown in Ref. [27] that once neutrino properties are more consistentlytaken into account when analyzing the data, natural inflation does marginally agree withdata.This work concerns a recently proposed extension of the original natural inflation modelintroducing a new periodic non-minimal coupling to gravity [25]. The authors in [25] showed3hat the new model leads to a better fit with observation data thanks to the introduction of anew parameter in the nonminimal coupling term, with n s and r values well within 95% C.L.region from combined Planck 2018+BAO+BK14 data. An important feature of this modelis that f becomes sub-Planckian, contrary to a super-Planckian f in the original naturalinflation model [24], and thus addresses issues related to gravitational instanton corrections[10, 28–31].Our objective here is to study one-loop quantum gravitational corrections to the naturalinflation model with non-minimal coupling, using Vilkovisky-DeWitt’s covariant effectiveaction approach [23]. One of the first works considering one-loop gravitational correctionswere pioneered by Elizalde and Odintsov [32–35]. Vilkovisky-DeWitt method was used tostudy effective actions in Refs. [36–40]. Unfortunately, calculating the covariant effectiveaction exactly is highly nontrivial, though non-covariant effective actions can in principlebe evaluated using proper time methods. Hence, we take a different route by employing aperturbative calculation of one-loop effective action, in orders of the background scalar field.This requires us to apply a couple of approximations. First, we work in the regime wherepotential is flat, i.e. φ ≪ f , which is generally true during slow-rolling inflation. Second,the background metric is set to be Minkowski. This choice is debatable, since it does notaccurately represent an inflationary scenario, but has been used before [12, 41] as a first steptowards studying quantum corrections.The action for the nonminimal natural inflation in the Einstein frame is given by, S = Z d x √− g (cid:18) − Rκ + 12 K ( φ ) φ ; a φ ; a + V ( φ )( γ ( φ )) (cid:19) (2)where, γ ( φ ) = 1 + α (cid:18) (cid:18) φf (cid:19)(cid:19) , (3)and, K ( φ ) = 1 + 24 γ ′ ( φ ) /κ γ ( φ ) . (4) V ( φ ) is as in Eq. (1). Here, φ ; a ≡ ∇ a φ denotes the covariant derivative. In the region wherepotential is flat, φ/f ≪
1, and we expand all periodic functions in Eq. (2) up to quarticorder in φ followed by rescaling √ k φ → φ : S ≈ Z d x √− g (cid:18) − Rκ + m k φ + λk φ + (1 + k k φ ) φ ; a φ ; a (cid:19) + O ( φ ) (5)4here parameters m, λ, k and k have been defined out of α, f and Λ in from Eq. (2): m = Λ (2 α − α ) f ; λ = Λ (8 α − α + 1)(1 + 2 α ) f ; k = 11 + 2 α ; k = α ( κ f + 96 α + 48 α )2 κ f (1 + 2 α ) . (6)We have also omitted a constant term appearing in (5) because such terms are negligiblysmall in early universe. The action (5) is in effect a φ scalar theory with derivative coupling. III. EFFECTIVE ACTION FORMALISM
A standard procedure while calculating loop corrections in quantum field theory, is touse the well known background field method, according to which a field is split into aclassical background and a quantum part that is much smaller in magnitude (and hencetreated perturbatively) [19–21]. A by-product of this procedure is the background and gaugedependence of quantum corrections. We briefly review here the covariant effective actionformalism, that yields gauge-invariant and background field independent results, employedin this work. Interested reader is advised to see Ref. [23] (chap. 7) for a detailed review.Quantization of a theory S [ ϕ ] with fields ϕ i is performed about a classical background¯ ϕ i : ϕ i = ¯ ϕ i + ζ i , where ζ i is the quantum part. Here, ϕ i is the local coordinate of a pointin the ‘field space’ and represents any scalar or vector or tensor field(s) in the coordinatespace. The index i in field space corresponds to all gauge indices and coordinate dependenceof fields. This way of writing the field-space equivalent of a coordinate space quantity(such as a vector field) is called condensed notation [42]. In our case, ϕ i = { g µν ( x ) , φ ( x ) } ;¯ ϕ i = { η µν , ¯ φ ( x ) } where η µν is the Minkowski metric; and, ζ i = { κh µν ( x ) , δφ ( x ) } . Thefluctuations ζ i are assumed to be small enough for a perturbative treatment to be valid, viz. | κh µν | ≪ | δφ | ≪ | φ | . In this limit, the infinitesimal general coordinate transformationscan be treated as gauge transformations associated with h µν [2, 43]. In fact, for any metric g µν ( x ), this infinitesimal transformation takes the form, δg µν = − δǫ λ g µν,λ − δǫ λ ,µ g λν − δǫ λ ,ν g λµ . (7)5n the condensed notation, an infinitesimal gauge transformation of any field ϕ i is given by, δϕ i = K iα [ ϕ ] δǫ α , (8)where K iα is identified as the generator of gauge transformations, while δǫ α are the gaugeparameters. As with Einstein notations, repeated (or contracted) indices in the condensednotation represent a sum over all the associated gauge or tensor indices and integral over allcoordinate indices. The gauge fixing condition is given by fixing a functional χ α [ ¯ ϕ ] so that itintersects each gauge orbit in field space only once. Including the gauge-fixing condition(s)and corresponding ghost determinant(s), the covariant one-loop effective action is given by[44, 45]Γ = − ln Z [ dζ ] exp (cid:20) (cid:18) − ζ i ζ j (cid:16) S ,ij [ ¯ ϕ ] − Γ kij S ,k [ ¯ ϕ ] (cid:17) − α f αβ χ α χ β (cid:19)(cid:21) − ln det Q αβ [ ¯ ϕ ] , (9)as α −→ dζ ] ≡ Q i dζ . A few comments on Eq. (9) are in order.The first term inside the exponential is the covariant derivative of the action functional withrespect to ζ i in field space. Γ kij are the field-space connections defined with respect to thefield-space metric G ij , and are responsible for general covariance of Eq. (9). In general,the field-space connections have complicated, non-local structure especially in presence ofa gauge symmetry. However, they reduce to the standard Christoffel connections, in termsof G ij , when χ α is chosen to be the Landau-DeWitt gauge i.e. χ α = K αi [ ¯ ϕ ] ζ i = 0, alongwith α → f αβ is any symmetric, positive definite operator and makes no non-trivial contribution to effective action [23]. Note also that the contributions from connectionterms, and hence the question of covariance, is relevant for off-shell analyses, since S ,i = 0on-shell. det Q αβ is the ghost determinant term that appears during quantization. Thisterm is absorbed into the exponential by introducing Faddeev-Popov ghosts, c α and ¯ c α , sothat [23], ln det Q αβ = ln Z [ d ¯ c α ][ dc β ] exp (cid:0) − ¯ c α Q αβ c β (cid:1) . (10)As a result,Γ[ ¯ ϕ ] = − ln Z [ dζ ][ d ¯ c α ][ dc β ] exp (cid:20) − ζ i ζ j (cid:16) S ,ij [ ¯ ϕ ] − Γ kij S ,k [ ¯ ϕ ] (cid:17) − α f αβ χ α χ β − ¯ c α Q αβ c β (cid:21) . (11)The computation of Eq. (11) traditionally has involved the use of proper time method,such as employing the heat kernel technique. For Laplace type operators (coefficients of6 i ζ j in the exponential), of the form g µν ∇ µ ∇ ν + Q (where Q does not contain derivatives),the heat kernel coefficients are known and are quite useful because they are independentof dimensionality [42]. However, these operators in general are not Laplace type, as in thepresent case. A class of nonminimal operators such as the one in Eq. (11) can be transformedto minimal (Laplace) form using the generalised Schwinger-DeWitt technique [47], but inpractice the implementation is quite complicated and specific to a given Lagrangian. Exam-ples of such an implementation can be found in Refs. [48, 49]. We take a different approachhere, calculating the one-loop effective action perturbatively in orders of the backgroundfield. While one does not obtain exact results in a perturbative approach, unlike the heatkernel approach, it is possible to obtain accurate results up to a certain order in backgroundfields which is of relevance for a theory in, say, the early universe. Some past examples areRefs. [12, 41]. Moreover, our implementation of this method using xAct packages for Math-ematica [50, 51] is fairly general in terms of its applicability to not only scalars coupled withgravity, but also vector and tensor fields (see, for instance, Ref. [52]). A caveat at this time,is that the perturbative expansions are performed about the Minkowski background and nota general metric background. However, a generalization to include FRW background is partour future plans.For convenience, we write the exponential in the first term of Γ as,exp[ · · · ] = exp n − (cid:16) ˜ S [ ¯ ϕ ] + ˜ S [ ¯ ϕ ] + ˜ S [ ¯ ϕ ] + ˜ S [ ¯ ϕ ] + ˜ S [ ¯ ϕ ] (cid:17)o ≡ exp n − ( ˜ S + ˜ S + ˜ S + ˜ S + ˜ S ) o (12)˜ S yields the propagator for each of the fields ζ i . The rest of the terms are contributionsfrom interaction terms, which we assume to be small. Treating ˜ S , ..., ˜ S as perturbative,and expanding Eq. (12), Γ[ ¯ ϕ ] can be written as,Γ[ ¯ ϕ ] = − ln Z [ dζ ][ d ¯ c α ][ dc β ] e − S (1 − δS + δS · · · );= − ln(1 − h δS i + 12 h δS i + · · · ); (13)where δS = P i =1 ˜ S i and h·i represents the expectation value in the path integral formulation.Finally, we use ln(1 + x ) ≈ x to find the contributions to Γ at each order of backgroundfield. We only use the leading term in the logarithmic expansion, since all higher orderterms will yield contributions from disconnected diagrams viz-a-viz h δS i , etc which weignore throughout our calculation. Moreover, since we are interested in terms up to quartic7rder in background field, we truncate the Taylor series in Eq. (13) up to δS . With theseconsiderations, the final contributions to Γ at each order of ¯ ϕ is: O ( ¯ ϕ ) : h ˜ S i ; O ( ¯ ϕ ) : h ˜ S i − h ˜ S i ; O ( ¯ ϕ ) : h ˜ S i − h ˜ S ˜ S i + 16 h ˜ S i ; O ( ¯ ϕ ) : h ˜ S i − h ˜ S ˜ S i + 12 h ˜ S ˜ S i − h ˜ S i − h ˜ S i . (14)Also, we recall that the metric fluctuations have a factor of κ . Accordingly, the terms in Eq.(14) will also contain powers of κ . It turns out, as will be shown below, that all contributionsare at most of the order κ . Expecting O ( κ ) terms to be significantly suppressed, we onlytake into account the corrections up to O ( κ ). In what follows, we will detail the evaluationof terms in Eq. (14). IV. COVARIANT ONE-LOOP CORRECTIONSA. Setup
The first step towards writing Γ[ ¯ ϕ ] in Eq. (11) is to identify the field space metric, givenin terms of the field-space line element, ds = G ij dϕ i dϕ j (15)= Z d xd x ′ (cid:0) G g µν ( x ) g ρσ ( x ′ ) dg µν ( x ) dg ρσ ( x ′ ) + G φ ( x ) φ ( x ′ ) dφ ( x ) dφ ( x ′ ) (cid:1) . (16)A prescription for identifying field space metric is to read off the components of G ij fromthe coefficients of highest derivative terms in classical action functional [53]. For the scalarfield φ ( x ), the field-space metric is chosen to be, G φ ( x ) φ ( x ′ ) = p g ( x ) δ ( x, x ′ ); (17)For the metric g µν ( x ), a standard choice for field-space metric is [23, 46] G g µν ( x ) g ρσ ( x ′ ) = p g ( x ) κ (cid:26) g µ ( ρ ( x ) g σ ) ν ( x ) − g µν ( x ) g ρσ ( x ) (cid:27) δ ( x, x ′ ) , (18)where the brackets around tensor indices in the first term indicate symmetrization. As aconvention, we choose to include κ factor in Eq. (18) to account for dimensionality of8he length element in Eq. (16), although choosing otherwise is also equally valid as longas dimensionality is taken care of. The inverse metric can be derived from the identity G ij G jk = δ ki : G g µν ( x ) g ρσ ( x ′ ) = κ (cid:26) g µ ( ρ ( x ) g σ ) ν ( x ) − g µν ( x ) g ρσ ( x ) (cid:27) δ ( x, x ′ ); (19) G φ ( x ) φ ( x ′ ) = δ ( x, x ′ ) . (20)Next, using Eqs. (17)-(20), one can find the Vilkovisky-DeWitt connections Γ kij which hasan identical definition to the Christoffel connections thanks to the Landau-DeWitt gaugechoice. Out of a total of six possibilities there are three non-zero connections obtained asfollows:Γ g λτ ( x ) g µν ( x ′ ) g ρσ ( x ′′ ) = δ ( x ′′ , x ′ ) δ ( x ′′ , x ) (cid:20) − δ ( µ ( λ g ν )( ρ ( x ) δ σ ) τ ) + 14 g µν ( x ) δ ρ ( λ δ στ ) + 14 g ρσ ( x ) δ µ ( λ δ ντ ) + 14 g λτ ( x ) g µ ( ρ ( x ) g σ ) ν ( x ) − g λτ ( x ) g µν ( x ) g ρσ ( x ) (cid:21) (21)Γ g λτ ( x ) φ ( x ′ ) φ ( x ′′ ) = κ δ ( x ′′ , x ′ ) δ ( x ′′ , x ) g λτ ( x ) (22)Γ φ ( x ) φ ( x ′ ) g λτ ( x ′′ ) = 14 δ ( x ′′ , x ′ ) δ ( x ′′ , x ) g λτ ( x ) = Γ φ ( x ) g λτ ( x ′ ) φ ( x ′′ ) . (23)Note that upon substituting into Eq. (11), all calculations here are evaluated at the back-ground field(s) which in our case is the Minkowski metric and a scalar field ¯ φ ( x ). We alsorecall that this rather unrestricted choice of background is allowed because of the DeWittconnections that ensure gauge and background independence. As alluded to earlier, theLandau-DeWitt gauge condition, K αi [ ¯ ϕ ] ζ i = 0, is given in terms of the gauge generators K αi . Since there is only one set of transformations vi-a-viz general coordinate transforma-tion, there exists one gauge parameter which we call ξ λ ( x ). In the condensed notation, thiscorresponds to δǫ α where α → ( λ, x ). Gauge generator on the gravity side K g µν λ ( x, x ′ ) isread off from Eq. (7), while K φλ ( x, x ′ ) is read off from the transformation of φ : δ ξ φ = − ∂ µ φξ λ . (24)Substituting in the definition of χ α [ ¯ ϕ ] in coordinate space, we obtain χ λ [ ¯ φ ] = Z d x ′ (cid:0) K g µν λ ( x, x ′ ) κh µν ( x ′ ) + K φλ ( x, x ′ ) δφ ( x ′ ) (cid:1) = 2 κ (cid:18) ∂ µ h µλ − ∂ λ h (cid:19) − ω∂ λ ¯ φδφ. (25)9here ω is a bookkeeping parameter, which we adopt from Ref. [46]; a second such parameter ν (not to be confused with the tensor index) appears with all Vilkovisky-DeWitt connectionterms to keep track of gauge (non-)invariance. That is, we write S ; ij = S ,ij − ν Γ kij S ,k .As shown later, playing with these parameters reproduces past non-gauge-invariant results.Here and throughout, the indices of field-space quantities like the gauge generator are raisedand lowered using field-space metric defined in Eqs. (17) - (20). Lastly, we choose f αβ = κ η λλ ′ δ ( x, x ′ ) in Eq. (11) to determine the gauge fixing term. One last piece needed beforebackground-field-order expansions, the ghost term Q αβ . We use the definition [23], Q αβ ≡ χ α,i K iβ , to obtain Q µν = (cid:18) − κ η µν ∂ α ∂ α + ω∂ µ ¯ φ∂ ν ¯ φ (cid:19) δ ( x, x ′ ) . (26) B. Loop integrals and divergent parts
Substituting the connections, the gauge fixing term and the ghost term along with action(5) in Eq. (11), and employing the notations in Eq. (12), we obtain:˜ S = Z d x h m ( δφ ) k + δφ ,a δφ ,a − h ab h cc,a,b − c a c a,b,b κ + h ab h ac,b,c − h ab h ac,b,c α + h aa h bc,b,c + h aa h bc,b,c α − h ab h ab,c,c + h aa h bb,c,c − h aa h bb,c,c α i (27)˜ S = Z d x h m κδφh aa ¯ φ k − m κνδφh aa ¯ φ k − κδφh bb ¯ φ ,a,a + κνδφh bb ¯ φ ,a,a − κδφh bb,a ¯ φ ,a + κωδφh bb,a ¯ φ ,a α + κδφ ¯ φ ,a h ab,b − κωδφ ¯ φ ,a h ab,b α + κδφh ab ¯ φ ,a,b i (28)˜ S = Z d x h − m κ h ab h ab ¯ φ k + m κ h aa h bb ¯ φ k + λ ¯ φ ( δφ ) k − m κ ν ¯ φ ( δφ ) k + k ¯ φ δφ ,a δφ ,a k + 2 k δφ ¯ φ ¯ φ ,a δφ ,a k − κ h bc h bc ¯ φ ,a ¯ φ ,a + κ νh bc h bc ¯ φ ,a ¯ φ ,a + κ h bb h cc ¯ φ ,a ¯ φ ,a − κ νh bb h cc ¯ φ ,a ¯ φ ,a + k ( δφ ) ¯ φ ,a ¯ φ ,a k − κ ν ( δφ ) ¯ φ ,a ¯ φ ,a + κ ω ( δφ ) ¯ φ ,a ¯ φ ,a α + ωc a ¯ c b ¯ φ ,a ¯ φ ,b + κ h ac h bc ¯ φ ,a ¯ φ ,b − κ νh ac h bc ¯ φ ,a ¯ φ ,b − κ h ab h cc ¯ φ ,a ¯ φ ,b + κ νh ab h cc ¯ φ ,a ¯ φ ,b i (29)˜ S = Z d x h κλδφh aa ¯ φ k − κλνδφh aa ¯ φ k + k κνδφh bb ¯ φ ¯ φ ,a,a k + k κh bb ¯ φ ¯ φ ,a δφ ,a k + k κδφh bb ¯ φ ¯ φ ,a ¯ φ ,a k + k κνδφh bb ¯ φ ¯ φ ,a ¯ φ ,a k − k κh ab ¯ φ δφ ,a ¯ φ ,b k − k κδφh ab ¯ φ ¯ φ ,a ¯ φ ,b k i (30)10 S = Z d x " − κ λh ab h ab ¯ φ k + κ λh aa h bb ¯ φ k − κ λν ¯ φ ( δφ ) k − k κ h bc h bc ¯ φ ¯ φ ,a ¯ φ ,a k + k κ νh bc h bc ¯ φ ¯ φ ,a ¯ φ ,a k + k κ h bb h cc ¯ φ ¯ φ ,a ¯ φ ,a k − k κ νh bb h cc ¯ φ ¯ φ ,a ¯ φ ,a k − k κ ν ¯ φ ( δφ ) ¯ φ ,a ¯ φ ,a k + k κ h ac h bc ¯ φ ¯ φ ,a ¯ φ ,b k − k κ νh ac h bc ¯ φ ¯ φ ,a ¯ φ ,b k − k κ h ab h cc ¯ φ ¯ φ ,a ¯ φ ,b k + k κ νh ab h cc ¯ φ ¯ φ ,a ¯ φ ,b k (31)Here, the indices ( a, b, c, . . . ) and ( µ, ν, ρ, . . . ) are used interchangeably to denote the gaugeindices. ˜ S leads to the well known free theory propagators for gravity and massive scalarfield and the ghost field respectively, D ( x, x ′ ) = Z d k (2 π ) e ik · ( x − x ′ ) D ( k ) = h δφ ( x ) δφ ( x ′ ) i ; D αβµν ( x, x ′ ) = Z d k (2 π ) e ik · ( x − x ′ ) D αβµν ( k ) = h h αβ ( x ) h µν ( x ′ ) i ; (32) D Gµν ( x, x ′ ) = Z d k (2 π ) e ik · ( x − x ′ ) D Gµν ( k ) = h ¯ c µ ( x ) c ν ( x ′ ) i ;where, D ( k ) = 1 k + m k ; (33) D αβµν ( k ) = δ αµ δ βν + δ αν δ βµ − δ αβ δ µν k + ( α − δ αµ k β k ν + δ αν k β k µ + δ βµ k α k ν + δ βν k α k µ k ;(34) D Gµν ( k ) = 1 k δ µν . (35)Looking at the structure of rest of the terms ˜ S i , it is straightforward to conclude that allterms with odd combinations of h µν ( x ) and δφ ( x ) appearing in Eqs. (14) will not contributeto the effective action, since h h αβ ( x ) δφ ( x ′ ) i = 0. Therefore, h ˜ S i = 0 and there is nocontribution at O ( ¯ φ ) to the one-loop effective action. Similarly, h ˜ S i = h ˜ S ˜ S i = h ˜ S i = 0,and hence at O ( ¯ φ ) too, there is no contribution to the effective action. Hence, the onlynon-zero contributions in Eq. (14) come at O ( ¯ φ ) and O ( ¯ φ ). In the latter, we ignore h S i terms since they are relevant at O ( κ ) and above while we are interested in terms up to κ order. Expectation value of ˜ S i consists of local terms, and thus describes contributions fromtadpole diagrams. 11he ghost term appears only in ˜ S . However, it can be shown that at O ( ¯ φ ) it yields nonontrivial contributions, and as a result, has usually been ignored in past literature whereonly quadratic order corrections were considered [12, 41, 46]. Consider the ghost propagator(35). Because there is no physical scale involved, the term containing ghost in (29) yields, (cid:28)Z d xωc a ¯ c b ¯ φ ,a ¯ φ ,b (cid:29) = Z d xω ¯ φ ,a ¯ φ ,b h c a ¯ c b i = Z d xω ¯ φ ,a ¯ φ ,b Z d k (2 π ) δ ab k , (36)which in four dimensions gives no physical result. The only nontrivial ghost contributioncomes at quartic order in background field.Eventually, finding the one-loop corrections then boils down to evaluating up to κ order,the quadratic and quartic order corrections from the following:Γ = h ˜ S i − h ˜ S i + h ˜ S i − h ˜ S ˜ S i + 12 h ˜ S ˜ S i − h ˜ S i . (37)In principle, solving Eq. (37) broadly consists of two steps: (i) writing each term in termsof the Fourier space integral(s) of Green’s functions found in Eqs. (33) - (35); and (ii)solving the resulting loop integrals. In this section, we restrict ourselves to writing just thedivergent part of effective action, since there are already several thousand terms to deal withand writing their finite parts would introduce unnecessary complexity. We do consider finitepart in the subsequent section, where we evaluate the effective potential after assuming allderivatives of background fields to be zero.
1. Calculating h ˜ S i i We first deal with h ˜ S i and h ˜ S i . For convenience, we do not explicitly write the tensorindices of correlators, fields and their coefficients. First, the derivatives of field fluctuationsare transformed to momentum space: Z d xA ( x ) h ∂ m δ ( x ) ∂ n δ ( x ′ ) i −→ Z d x d p (2 π ) A ( x )( − ip ) m ( ip ) n h δ p ( x ) δ p ( x ′ ) i , (38)where, δ ( x ) and A ( x ) represent the field fluctuations and coefficients respectively. δ ( x ) hererepresents any of the fields ( δφ ( x ) , h µν ( x ) , c µ ( x ) , ¯ c µ ( x )), and is not to be confused with theDirac delta function δ ( x, x ′ ). h δ p δ p i represents the propagator(s) in momentum space. Then, h δ p δ p i is replaced with values of Green’s function to obtain the loop integrals. For solving12ntegrals here, we primarily use the results in Ref. [54] to evaluate the divergent termsin dimensional regularization, except for some higher rank two-point integrals that appearbelow, which we solve by hand using well known prescriptions [55, 56]. There are threetypes of loop integrals coming from Eqs. (29) and (31): Z d x d p (2 π ) A ( x ) 1 p ; Z d x d p (2 π ) A ( x ) p µ p ν p ; Z d x d p (2 π ) A ( x ) 1 p + m k . (39)The first two integrals are poleless, and vanish due to the lack of a physical scale [54]. Thethird integral is straightforward and contributes to the divergent part. See appendix A forvalues of all integrals appearing here, including finite parts for some integrals used in thenext section.
2. Calculating h ˜ S i ˜ S j i h ˜ S ˜ S i and h ˜ S ˜ S i contain terms of the form, R d x R d x ′ A ( x ) B ( x ′ ) h δφ ( x ) δφ ( x ′ ) ih h ( x ) h ( x ′ ) i (40)= R d x R d k (2 π ) d k ′ π d k ′′ π A ( x ) ˜ B ( k ′′ ) D φφ ( k ) D hh ( k ′ ) e − i ( k + k ′ ) · x δ (4) ( k + k ′ − k ′′ )= R d xA ( x ) R d k (2 π ) e − ik · x ˜ B ( k ) R d k ′ (2 π ) D φφ ( k − k ′ ) D hh ( k ′ ) (41)where A ( x ) , B ( x ′ ) are classical coefficients, and D φφ , D hh are scalar and gravity propagatorsrespectively; ˜ B ( k ) is the Fourier transform of B ( x ′ ). There are also the derivatives of Eq.(40) present, and are dealt with in a way similar to Eq. (38), leading to factors of k ′ µ in theloop integrals. Consequently, we encounter three types of loop integrals: Z d k ′ (2 π ) k ′ a . . . k ′ b ( k ′ − k ) + m k ; Z d k ′ (2 π ) k ′ a . . . k ′ b k ′ (( k ′ − k ) + m k ) ; Z d k ′ (2 π ) k ′ a . . . k ′ b k ′ (( k ′ − k ) + m k ) ;(42)which constitute standard one-, two- and three-point n -rank integrals ( n = 0 , , h ˜ S ˜ S i yields 4 − point correlators given by, h δφ ( x ) δφ ( x ) δφ ( x ′ ) δφ ( x ′ ) i ; h δφ ( x ) δφ ( x ) h ( x ′ ) h ( x ′ ) i ; h δφ ( x ) δφ ( x )¯ c ( x ′ ) c ( x ′ ) i ; h h ( x ) h ( x )¯ c ( x ′ ) c ( x ′ ) i ; h ¯ c ( x ) c ( x )¯ c ( x ′ ) c ( x ′ ) i ; h h ( x ) h ( x ) h ( x ′ ) h ( x ′ ) i (43)The second, third and fourth terms in (43) are of the form h δ ( x ) δ ( x ) δ ′ ( x ′ ) δ ′ ( x ′ ) i (again, δ ( x ) , δ ( x ′ ) denote the fields), thereby corresponding to disconnected tadpoles and hence donot give any meaningful contribution. The rest of 4 − point correlators in Eq. (43) are13esolved into 2 − point functions using Wick theorem [57, 58]. Fortunately, the last terminvolving only graviton propagators can be ignored since it only contains O ( κ ) terms.Moreover, h ¯ c ( x )¯ c ( x ′ ) i = h c ( x ) c ( x ′ ) i = 0. Therefore, after applying Wick theorem, the finalcontribution in Eq. (43) comes from, h δφ ( x ) δφ ( x ) δφ ( x ′ ) δφ ( x ′ ) i = h δφ ( x ) δφ ( x ′ ) ih δφ ( x ) δφ ( x ′ ) i + h δφ ( x ) δφ ( x ′ ) ih δφ ( x ) δφ ( x ′ ) i ;(44) h ¯ c ( x ) c ( x )¯ c ( x ′ ) c ( x ′ ) i = h ¯ c ( x ) c ( x ′ ) ih ¯ c ( x ) c ( x ′ ) i . (45)Using Eq. (44) and (45) in h ˜ S ˜ S i along with Eqs. (33)-(35), and Fourier transformingaccording to Eq. (38) gives rise to up to rank-4 two-point integrals: Z d k ′ (2 π ) k ′ a . . . k ′ b ( k ′ + m k )(( k ′ − k ) + m k ) ; (46)
3. Calculating h ˜ S i ˜ S j ˜ S k i The last term to be evaluated is h ˜ S ˜ S ˜ S i . It consists of six-point correlators given by, h h ( x ) h ( x ′′ ) δφ ( x ) δφ ( x ′′ )¯ c ( x ′ ) c ( x ′ ) i ; h h ( x ) h ( x ′′ ) δφ ( x ) δφ ( x ′′ ) δφ ( x ′ ) δφ ( x ′ ) i ; h h ( x ) h ( x ′′ ) δφ ( x ) δφ ( x ′′ ) h ( x ′ ) h ( x ′ ) i (47)Again, the last term can be ignored since it has no terms up to O ( κ ). And the first termcan be written as h h ( x ) h ( x ′′ ) δφ ( x ) δφ ( x ′′ ) ih ¯ c ( x ′ ) c ( x ′ ) i , which implies disconnected diagramsand thus can also be ignored. So, ghost terms only end up in h ˜ S ˜ S i . Hence, only the secondterm needs to be evaluated, which after applying Wick theorem similar to Eq. (44) turnsout to be, h h ( x ) h ( x ′′ ) δφ ( x ) δφ ( x ′′ ) δφ ( x ′ ) δφ ( x ′ ) i = h h ( x ) h ( x ′′ ) ih δφ ( x ) δφ ( x ′ ) ih δφ ( x ′ ) δφ ( x ′′ ) i + h h ( x ) h ( x ′′ ) ih δφ ( x ) δφ ( x ′ ) ih δφ ( x ′ ) δφ ( x ′′ ) i , (48)A typical scalar integral in h ˜ S ˜ S ˜ S i takes the form, Z d x Z d x ′ Z d x ′′ Z d k (2 π ) Z d k ′ (2 π ) Z d k ′′ (2 π ) A ( x ) B ( x ′ ) C ( x ′′ ) × e − ik · ( x ′ − x ) e − ik ′′ · ( x − x ′′ ) e − ik ′ · ( x ′′ − x ′ ) D φφ ( k ) D φφ ( k ′ ) D hh ( k ′′ )= Z d x Z d p (2 π ) Z d k (2 π ) A ( x ) ˜ B ( p ) e − ip · x ˜ C ( k ) e − ik · x Z d k ′ (2 π ) × D φφ ( k ′ − p − k ) D φφ ( k ′ − k ) D hh ( k ′ ) , (49)14esulting in scalar and tensor two-, three- and four-point integrals: Z d k ′ (2 π ) k ′ a . . . k ′ b d d d d ; Z d k ′ (2 π ) k ′ a . . . k ′ b d d d ; Z d k ′ (2 π ) k ′ a . . . k ′ b d d . (50)where, d = ( k ′ − k ) + m k ; d = ( k ′ − k − p ) + m k ; d = d = k ′ . (51)There are up to rank-3 four-point integrals in h ˜ S ˜ S ˜ S i , and hence have no divergent part[54].
4. Divergent part
In total, there are several thousand terms that eventually add up to give the divergentpart of Eq. (37). After solving all the above integrals and extracting their divergent partsusing dimensional regularization, we end up with Fourier transforms ˜ B ( k ) (and ˜ C ( p ) incase of six-point functions) with or without factors of k a and/or p a . These expressions aretransformed back to coordinate space as follows: Z d x Z d p (2 π ) Z d k (2 π ) A ( x ) ˜ B ( p ) ˜ C ( k ) e − ip · x e − ik · x k a . . . k b p µ . . . p ν → Z d x ( i∂ µ ) . . . ( i∂ ν ) B ( x )( i∂ a ) . . . ( i∂ b ) C ( x ) (52)15nd likewise for other cases including h ˜ S i ˜ S j i and h ˜ S i i . Substituting these results for thedivergent part in Eq. (37), we get, divp (Γ) = Z d xL " k m ¯ φ k + 3 m κ ¯ φ k − m λ ¯ φ k − m κ ν ¯ φ k + 3 m κ ν ¯ φ k + k m ¯ φ ¯ φ ,a,a k − m κ ¯ φ ¯ φ ,a,a k + 17 m κ ν ¯ φ ¯ φ ,a,a k − m κ ν ¯ φ ¯ φ ,a,a k + m κ ω ¯ φ ¯ φ ,a,a k + m κ νω ¯ φ ¯ φ ,a,a k − κ ν ¯ φ ¯ φ ,a,a,b,b + κ ν ¯ φ ¯ φ ,a,a,b,b − κ ω ¯ φ ¯ φ ,a,a,b,b − κ νω ¯ φ ¯ φ ,a,a,b,b − k m ¯ φ k π − k m κ ¯ φ k π + k m λ ¯ φ k π + m κ λ ¯ φ k π − λ ¯ φ k π + k m κ ν ¯ φ k π − m κ λν ¯ φ k π − k m κ ν ¯ φ k π + m κ λν ¯ φ k π − k m ¯ φ ¯ φ ,a,a k π + k λ ¯ φ ¯ φ ,a,a k π − κ λ ¯ φ ¯ φ ,a,a k π − k m κ ν ¯ φ ¯ φ ,a,a k π + 3 κ λν ¯ φ ¯ φ ,a,a k π − κ λν ¯ φ ¯ φ ,a,a k π + k m κ ω ¯ φ ¯ φ ,a,a k π − κ λω ¯ φ ¯ φ ,a,a k π − k m κ νω ¯ φ ¯ φ ,a,a k π + κ λνω ¯ φ ¯ φ ,a,a k π − k m ¯ φ ¯ φ ,a ¯ φ ,a k π − k m κ ¯ φ ¯ φ ,a ¯ φ ,a k π − k λ ¯ φ ¯ φ ,a ¯ φ ,a k π + 5 k m κ ν ¯ φ ¯ φ ,a ¯ φ ,a k π + κ λν ¯ φ ¯ φ ,a ¯ φ ,a k π − k m κ ν ¯ φ ¯ φ ,a ¯ φ ,a k π + 3 k m κ ω ¯ φ ¯ φ ,a ¯ φ ,a k π − κ λω ¯ φ ¯ φ ,a ¯ φ ,a k π + k m κ νω ¯ φ ¯ φ ,a ¯ φ ,a k π + 7 k ¯ φ ¯ φ ,b,b,a ¯ φ ,a k π + 5 k κ ω ¯ φ ¯ φ ,b,b,a ¯ φ ,a k π − k κ νω ¯ φ ¯ φ ,b,b,a ¯ φ ,a k π − k ¯ φ ¯ φ ,a ¯ φ ,b,a,b k π − k κ ω ¯ φ ¯ φ ,a ¯ φ ,b,a,b k π − k ¯ φ ¯ φ ,a,a ¯ φ ,b,b k π + k κ ¯ φ ¯ φ ,a,a ¯ φ ,b,b k π − k κ ν ¯ φ ¯ φ ,a,a ¯ φ ,b,b k π + 3 k κ ν ¯ φ ¯ φ ,a,a ¯ φ ,b,b k π + k κ ω ¯ φ ¯ φ ,a,a ¯ φ ,b,b k π − k κ νω ¯ φ ¯ φ ,a,a ¯ φ ,b,b k π + k ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k π + k κ ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k π − k κ ν ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k π + 3 k κ ν ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k π + 3 k κ ω ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k π − k κ νω ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k π + k ¯ φ ¯ φ ,a ¯ φ ,a,b,b k π − k κ ω ¯ φ ¯ φ ,a ¯ φ ,a,b,b k π − k κ νω ¯ φ ¯ φ ,a ¯ φ ,a,b,b k π − k ¯ φ ¯ φ ,a,a,b,b k π + k κ ν ¯ φ ,a ¯ φ ,a ¯ φ ,b ¯ φ ,b k π − k κ ¯ φ ¯ φ ,a ¯ φ ,a,b ¯ φ ,b k π + k κ ν ¯ φ ¯ φ ,a ¯ φ ,a,b ¯ φ ,b k π − k κ ω ¯ φ ¯ φ ,a ¯ φ ,a,b ¯ φ ,b k π − k ¯ φ ¯ φ ,a,b ¯ φ ,a,b k π − k κ ¯ φ ¯ φ ,a,b ¯ φ ,a,b k π − k κ ω ¯ φ ¯ φ ,a,b ¯ φ ,a,b k π (53)16here, L = − / π ǫ ( ǫ = n −
4) as the dimensionality n →
4. As expected, there are no α dependent terms. Although not explicitly shown here, factors of 1 /α appear in individualpieces in Eq. (37). However, when all contributions are added to evaluate Γ, these termscancel so that the final result is gauge-invariant. Final result for divergent part of Γ afterremoving bookkeeping parameters ( ω → ν →
1) in the Landau gauge ( α →
0) leads tothe covariant corrections: divp (Γ) = Z d x h k m ¯ φ k + 5 m κ ¯ φ k − m λ ¯ φ k + k m ¯ φ ¯ φ ,a,a k + 5 m κ ¯ φ ¯ φ ,a,a k − κ φφ ,a,a,b,b − k m ¯ φ k − k m κ ¯ φ k + k m λ ¯ φ k + 13 m κ λ ¯ φ k − λ ¯ φ k − k m ¯ φ ¯ φ ,a,a k + k λ ¯ φ ¯ φ ,a,a k − κ λ ¯ φ ¯ φ ,a,a k − k m ¯ φ ¯ φ ,a ¯ φ ,a k + 9 k m κ ¯ φ ¯ φ ,a ¯ φ ,a k − k λ ¯ φ ¯ φ ,a ¯ φ ,a k − κ λ ¯ φ ¯ φ ,a ¯ φ ,a k + 7 k ¯ φ ¯ φ ,b,b,a ¯ φ ,a k + 7 k κ ¯ φ ¯ φ ,b,b,a ¯ φ ,a k − k ¯ φ ¯ φ ,a ¯ φ ,b,a,b k − k κ ¯ φ ¯ φ ,a ¯ φ ,b,a,b k − k ¯ φ ¯ φ ,a,a ¯ φ ,b,b k + k κ ¯ φ ¯ φ ,a,a ¯ φ ,b,b k + k ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k + 23 k κ ¯ φ ¯ φ ,a ¯ φ ,a ¯ φ ,b,b k + k ¯ φ ¯ φ ,a ¯ φ ,a,b,b k − k κ ¯ φ ¯ φ ,a ¯ φ ,a,b,b k − k ¯ φ ¯ φ ,a,a,b,b k + k κ ¯ φ ,a ¯ φ ,a ¯ φ ,b ¯ φ ,b k − k κ ¯ φ ¯ φ ,a ¯ φ ,a,b ¯ φ ,b k − k ¯ φ ¯ φ ,a,b ¯ φ ,a,b k − k κ ¯ φ ¯ φ ,a,b ¯ φ ,a,b k i (54)If instead we turn off the DV connections by setting ν = 0 and choose α = 1 , ω = 1,we recover gauge-dependent results obtained in the past by Steinwachs and Kamenshchik[49], where they calculated the one-loop divergences for a general scalar-tensor theory thatin the single field limit (with the identifications U = 1 , G = K , and V = V /γ in theirnotations) encompasses the model (5). Similarly, in the case k = 0 , k = 1 we recover thegauge-invariant calculations of Mackay and Toms [46] (excluding cosmological constant andnonminimal coupling to gravity). C. Renormalization and Comparisons
Not all the divergences in Eq. (54) can be absorbed by renormalizing the parameters inthe classical action (5), particularly the quartic derivatives of ¯ φ ( x ), which are absent in theclassical action. However, we need not worry about these UV divergences since the current17ramework is an effective theory approach, and we assume that such divergences are resolvedby some high energy theory. For now, we consider only the counterterms for terms presentin the classical action functional so as to absorb corresponding divergent parts, which willin turn induce 1-loop corrections to the parameters m k , k k , λk of the theory (5).We start by re-writing Eq. (54) in the form, divp (Γ) = L Z d x ( A ¯ φ (cid:3) ¯ φ + B ¯ φ + C ¯ φ + D ¯ φ ∂ µ ¯ φ∂ µ ¯ φ ) (55)where we have ignored the terms not present in the classical background action. We notethat the terms of the form ¯ φ (cid:3) ¯ φ in Eq. (54) are transformed to − φ ∂ µ ¯ φ∂ µ ¯ φ after by-partsintegration. The coefficients A, B, C, D are read off from Eq. (54): A = 5 m κ k + k m k ; B = k m k + 5 m κ k − m λ k ; C = − k m k − k m κ k + k m λ k + 13 m κ λ k − λ k ; D = 23 k m k + 9 k m κ k − k λ k − κ λ k . (56)Taking into account the field Renormalization ¯ φ → Z / ¯ φ , the classical background La-grangian reads, L Z = − Z ¯ φ (cid:3) ¯ φ + 12 m k Z ¯ φ + λ k Z ¯ φ + 12 k k Z ¯ φ ∂ µ ¯ φ∂ µ ¯ φ (57)Suppose, the renormalized Lagrangian is given in terms of renormalized parameters as fol-lows, L r = −
12 ¯ φ (cid:3) ¯ φ + 12 (cid:18) m k (cid:19) r ¯ φ + 124 (cid:18) λk (cid:19) r ¯ φ + 12 (cid:18) k k (cid:19) r ¯ φ ∂ µ ¯ φ∂ µ ¯ φ (58)where ( · ) r represents the renormalized parameter. The counterterm Lagrangian is thendefined as δ L = L r − L Z . Accordingly, the counterterms for field and other parameters areas follows: δ Z = Z − δ (cid:18) m k (cid:19) = m k Z − (cid:18) m k (cid:19) r ; δ (cid:18) λk (cid:19) = λk Z − (cid:18) λk (cid:19) r ; δ (cid:18) k k (cid:19) = k k Z − (cid:18) k k (cid:19) r . (59)18hese counterterms are fixed by demanding that divp (Γ) = − R d xδ L . With some algebraicmanipulations, the counterterms read, δ Z = − A π ǫ ; δ (cid:18) m k (cid:19) = B π ǫ ; δ (cid:18) λk (cid:19) = 3 Cπ ǫ ; δ (cid:18) k k (cid:19) = D π ǫ . (60)Using Eq. (60) in (59), we find the one-loop corrections to coupling parameters in terms ofthe coefficients A, B, C, D , ∆ (cid:18) m k (cid:19) = m A π k ǫ + B π ǫ ;∆ (cid:18) λk (cid:19) = 3 Cπ ǫ + λA π k ǫ ; (61)∆ (cid:18) k k (cid:19) = k A π k ǫ + D π ǫ . For the sake of comparisons, and also as a crosscheck, we point out that upon choosing ν = 0 , α = 1 , ω = 0 in the case k = 0 , k = 1, the gauge-dependent one-loop quantumgravitational correction to φ theory first calculated by Rodigast and Schuster [59] is recov-ered: ∆ λ = κ π ǫ ( m λ − λ / κ ). Note that, all gravitational corrections in Eq. (61) appearwith a factor of κ , while the ones without it are nongravitational corrections that could inprinciple be obtained from flat space quantum field theory. Also, in the gauge covariantversion of the same case (viz. ν = 1 , α = 0 , ω = 1 with k = 0 , k = 1), our results matchthat of Pietrykowski [60].In a similar spirit, we would like to shed some light on the extensions of the work of Ref.[46]. There, a self-interacting scalar field with nonminimal coupling to gravity (of the form ξRφ /
2) was considered and the corresponding field and mass renormalizations were studied.The action in Ref. [46] matches ours if we put k = 0 , k = 1 and add ξRφ /
2. However,corrections to quartic coupling including contributions from the nonminimal coupling havenot been calculated so far. Without going into the details, partly because the process is moreor less unchanged, we present here the covariant one-loop corrections to quartic coupling λ so as to complete the analysis of Ref. [46], δλ = 3 λ π ǫ + κ π ǫ (cid:18) m λ + 218 m λξ − m λξ (cid:19) . (62)19 . EFFECTIVE POTENTIAL It is evident from the analysis so far that extracting any more information, in the formof finite corrections for example, is a cumbersome task. A resolution to this problem lies inmaking a reasonable compromise, wherein the derivatives of background fields are ignoredbasis the assumption that either the background field is constant due to a symmetry or itis slowly varying. The resulting effective action is known as effective potential. One of thefirst instances of this workaround is the well known Coleman Weinberg potential [18, 61].This approximation holds up especially during inflation, where the slow-rolling conditionrequires fields to be slowly varying. In this section, we evaluate the effective potential of thetheory (5) including finite terms and infer cosmological implications.We begin by substituting ∂ µ ¯ φ = 0 in Eqs. (28)-(31), resulting in,˜ S = Z d x h m κδφh aa ¯ φ k − m κνδφh aa ¯ φ k i ; (63)˜ S = Z d x h − m κ h ab h ab ¯ φ k + m κ h aa h bb ¯ φ k + λ ¯ φ ( δφ ) k − m κ ν ¯ φ ( δφ ) k + k ¯ φ δφ ,a δφ ,a k i ; (64)˜ S = Z d x h κλδφh aa ¯ φ k − κλνδφh aa ¯ φ k i ; (65)˜ S = Z d x h − κ λh ab h ab ¯ φ k + κ λh aa h bb ¯ φ k − κ λν ¯ φ ( δφ ) k i . (66)Using the above expressions in Eq. (37) and following the steps outlined in the Sec. IV B,we obtain the covariant effective potential,Γ eff [ ¯ φ ] = 18 π Z d x [ A ǫ ¯ φ + A ¯ φ + B ǫ ¯ φ + B ¯ φ ] (67)20here, A and B are the same as B and C from Eq. (56) respectively, and, A = ( γ + log( π ))( − k m k − m κ k + m λ k ) + 3 k m k + m κ k − m λ k +( − k m k − m κ k + m λ k ) log( m k µ ) − φ Z d k (2 π ) e − ik · x ˜¯ φ m κ log (cid:0) k k m (cid:1) k − m κ log (cid:0) k k m (cid:1) k k ! ; (68) B = − k m k − k m κ k + k m λ k + 25 m κ λ k − λ k +( γ + log( π ))( 3 k m k + 7 k m κ k − k m λ k − m κ λ k + λ k )+( 3 k m k + 5 k m κ k − k m λ k − m κ λ k + λ k ) log( m k µ ) − φ Z d k (2 π ) e − ik · x ˜¯ φ m κ λ log (cid:0) k k m (cid:1) k ! + 1¯ φ Z d k (2 π ) e − ik · x ˜¯ φ − m κ λ log (cid:16) (cid:0) m k k (cid:1) / − (cid:0) m k k (cid:1) / (cid:17)(cid:0) m k k (cid:1) / k + λ log (cid:16) (cid:0) m k k (cid:1) / − (cid:0) m k k (cid:1) / (cid:17)(cid:0) m k k (cid:1) / k − m κ λ log (cid:0) k k m (cid:1) k k + k arctan (cid:16) k / k (cid:0) m − k k (cid:1) / (cid:17) k (cid:0) m − k k (cid:1) / k / ! + 1¯ φ Z d p (2 π ) e − ip · x ˜¯ φ k m κ log (cid:16) (cid:0) m k p (cid:1) / − (cid:0) m k p (cid:1) / (cid:17)(cid:0) m k p (cid:1) / k . (69)The logarithmic terms appearing in expressions above are dealt with as follows. In thecontext of the present problem and the effective theory treatment, we restrict ourselvesto the condition k ≪ − M p so that k k m ≪ q m k k ≈ q m k k .For the arctan( · · · ) term, we use arctan( x ) ≈ x for small x . After these expansions, allterms with factors of k will vanish since we assume the derivatives of ¯ φ to be zero. Hence,all the integrands of momenta integrals in Eqs. (68,69) reduce to c-numbers times Fourier21ransforms of ¯ φ n . Using these simplifications, the coefficients A and B are obtained as, A = ( γ + log( π ))( − k m k − m κ k + m λ k ) + 3 k m k + m κ k − m λ k + 3 m κ k + ( − k m k − m κ k + m λ k ) log( m k µ ); B = − k m k − k m κ k + k m λ k + 25 m κ λ k − λ k − m κ λ k + λ k − m κ λ k + 3 k m κ k +( γ + log( π ))( 3 k m k + 7 k m κ k − k m λ k − m κ λ k + λ k )+( 3 k m k + 5 k m κ k − k m λ k − m κ λ k + λ k ) log( m k µ ) (70)The counterterms for quadratic and quartic terms have a similar form to Eq. (60), so thatthe effective potential can be written in terms of renormalized parameters which can becalculated from Eq. (61) with A = 0. The effective action takes the form, V eff = 12 m k ¯ φ + 14! λk ¯ φ + A ¯ φ + B ¯ φ . (71) A. Estimating the magnitude of corrections
Making a definitive statement about cosmological implications of quantum corrected po-tential requires an analysis in the FRW background, which unfortunately is out of scopeof the present work. However, we can get an order-of-magnitude estimate of the quantumcorrections to the effective potential using the values of parameters k , k , m , λ from theresults of Ref. [25].From the action (5), the Einstein equations are given by,3 H = κ (cid:18) − φ − k k ¯ φ ˙¯ φ + m k ¯ φ + λ k ¯ φ (cid:19) ;2 ˙ H + 3 H = κ (cid:18) − ˙¯ φ − k k ¯ φ ˙¯ φ + m k ¯ φ + λ k ¯ φ (cid:19) , (72)from which we obtain in the de-Sitter limit ( ˙ H ∼ ˙ φ ∼ H = κ (cid:18) m k ¯ φ + λ k ¯ φ (cid:19) . (73)The field equation for ¯ φ reads,( a + k k ¯ φ ) ¨¯ φ + k k ¯ φ ˙¯ φ + (3 aH + 2 k k H ¯ φ ) ˙¯ φ − m k ¯ φ − λ k ¯ φ = 0 . (74)22pplying the de-Sitter conditions, Eq. (74) yields the de-Sitter value of ¯ φ ,¯ φ = − k m λ . (75)Using Eq. (75) in (73), we find the de-Sitter value of Hubble parameter H : H = − κ m λ . (76)Clearly, the condition for existence of de-Sitter solutions is λ <
0. Demanding this conditionin Eq. (6), along with m > φ < f , leads to a constraint on the parameter α of theoriginal theory (2): 0 . < α .
1. Following the results of [25], we choose 0 . . α . . ∼O (1). Near this value of α , f ∼ M p = 1 /κ and Λ ∼ GeV . Substituting these inEqs. (6), we find m ∼ Λ /f ∼ − M p ; λ ∼ Λ /f ∼ − . Similarly, k ∼ k ∼ M − p . This also implies that in the low energy limit where momenta k ≪ GeV ≪ M p , k k /k ≪ λ/k , i.e. the derivative coupling term is suppressed.From the above, we can estimate the order of magnitude contributions of terms in A and B at O ( ¯ φ ) and O ( ¯ φ ) respectively. We estimate the magnitude of each type of termpresent at both orders. At quadratic order in background field, we find, κ m k ∼ λm k ∼ k m k ∼ GeV . (77)Similarly, at quartic order in background field, κ m k k ∼ κ m λk ∼ λ k ∼ m k k ∼ k m λk ∼ − . (78)Quite an interesting observation here is that the magnitudes of gravitational (terms witha factor of κ ) and non-gravitational (terms without κ ) corrections turn out to be exactlythe same for both quadratic and quartic order contributions. However, the correspondingquantum corrections are expectedly smaller by an order of 10 − compared to m and λ , ascan also be checked using the loop counting parameter for de Sitter inflation H /M P l with H ∼ GeV and M P l ∼ GeV . VI. CONCLUSION
The nonminimal natural inflation model in consideration here is approximately describedby a massive scalar field model with quartic self interaction and a derivative coupling in theregion where φ/f <
1. We study one-loop corrections to this theory, about a Minkowski23ackground, using a covariant effective action approach developed by DeWitt-Vilkovisky.The one-loop divergences and corresponding counterterms have been obtained. Along theway, we also recover several past results, both gauge-invariant non-gauge-invariant, for sim-ilar theories. In one such exercise, we obtain the φ coupling correction in a theory withnonminimal coupling of scalar field to gravity, originally considered in Ref. [46] and therebyextend their result.Finite corrections have been taken into account for the calculation of effective potential,where we assume that the background field changes sufficiently slowly so that all derivativesof background field(s) can be ignored. Although cosmologically relevant inferences are notfeasible as long as the metric background is Minkowski and not FRW, we can still estimateapproximately the magnitudes of quantum corrections. Using the range of parameters ap-plicable to our model, we find that the gravitational and non-gravitational corrections areof same order of magnitudes, while still being expectedly small compared to m and λ .This is quite an interesting observation, since one would naively assume that gravita-tional corrections are κ suppressed and thus would necessarily be small. There is thusenough motivation to go a step further, and calculate gravitational corrections in the FRWbackground so that cosmologically relevant inferences can be derived. ACKNOWLEDGMENTS
This work was partially funded by DST (Govt. of India), Grant No. SERB/PHY/2017041.
Appendix A: Loop Integrals
Most of the loop integrals are calculated using the well known PV reduction method [54].Some integrals, namely (A10,A11) are calculated the general method outlined in Ref. [55].Finite parts have been calculated for integrals needed for evaluating the effective potential.24ntegrals in h ˜ S i , h ˜ S i : Z d k (2 π ) k µ k ν k + m k = g µν π (cid:16) m k − m (cid:0) − − ǫ + γ + log( π ) + log( m k µ ) (cid:1) k (cid:17) (A1) Z d k (2 π ) k µ k + m k = 0 (A2) Z d k (2 π ) k + m k = 116 π m (cid:0) − − ǫ + γ + log( π ) + log( m k µ ) (cid:1) k (A3)Integrals in h ˜ S ˜ S i , h ˜ S ˜ S i : Z d k ′ (2 π ) k ′ µ k ′ ν k ′ (( k ′ − k ) + m k ) = 116 π (cid:18) g µν (cid:0) ǫ − γ − log( π ) (cid:1) + k µ k ν (cid:16) (cid:0) ǫ − γ − log( π ) (cid:1) + (cid:0) − ǫ + γ + log( π ) (cid:1)(cid:17) k (cid:19) (A4) Z d k ′ (2 π ) k ′ µ k ′ ν k ′ (( k ′ − k ) + m k ) = 116 π k µ k ν (cid:0) ǫ − γ − log( π ) (cid:1) − g µν (cid:0) ǫ − γ − log( π ) (cid:1)(cid:0) m k + k (cid:1) (A5) Z d k ′ (2 π ) k ′ µ k ′ (( k ′ − k ) + m k ) = 116 π k µ ǫ − γ − log( π )) (A6) Z d k ′ (2 π ) k ′ (( k ′ − k ) + m k ) = 116 π " (cid:16) m k − m (cid:0) − − ǫ + γ + log( π ) + log( m k µ ) (cid:1) k (cid:17) + m (cid:0) − − ǫ + γ + log( π ) + log( m k µ ) (cid:1) ( k ) k (A7) Z d k ′ (2 π ) k ′ (( k ′ − k ) + m k ) = 116 π (cid:16) ǫ − γ − log( π ) − log( m k µ ) − log (cid:0) k ( k ) m (cid:1)(cid:0) m k ( k ) (cid:1)(cid:17) (A8) Z d k ′ (2 π ) k ′ − k ) + m k ) = 116 π m (cid:0) − − ǫ + γ + log( π ) + log( m k µ ) (cid:1) k (A9)25ntegrals in h ˜ S ˜ S i : Z d k ′ (2 π ) k ′ ( k ′ + m k )(( k ′ − k ) + m k ) = 116 π m k + 3 m ǫk − m γ k − m log( π )8 k − m log( m k µ )8 k + 7 m ( k )8 k + 7 m ( k )4 ǫk − m γ ( k )8 k − m log( π )( k )8 k − m log( m k µ )( k )8 k + ( k ) + ( k ) ǫ − γ ( k ) − log( π )( k ) − log( m k µ )( k ) − arctan (cid:16) k / ( k ) / (cid:0) m − k ( k ) (cid:1) / (cid:17) ( k ) / (cid:0) m − k ( k ) (cid:1) / k / ! (A10) Z d k ′ (2 π ) k ′ k ′ µ ( k ′ + m k )(( k ′ − k ) + m k ) = − π ( 2 ǫ − γ − log( π )) 3 m k k µ (A11) Z d k ′ (2 π ) k ′ µ k ′ ν ( k ′ + m k )(( k ′ − k ) + m k ) = 116 π g µν (cid:18) m (cid:0) − − ǫ + γ + log( π ) + log( m k µ ) (cid:1) k + (cid:0) − m k − ( k ) (cid:1) − (cid:0) ǫ − γ − log( π ) (cid:1)(cid:16) m k + ( k ) − (cid:0) m k − m ( k ) k (cid:1)(cid:17) k ) (cid:19) + k x k x (cid:18) m (cid:0) − − ǫ + γ + log( π ) + log( m k µ ) (cid:1) k ( k ) + m k + ( k )18( k )+ (cid:0) ǫ − γ − log( π ) (cid:1)(cid:16) m k − m ( k ) k + ( k ) − (cid:0) m k − m ( k ) k (cid:1)(cid:17) k ) (cid:19) (A12) Z d k ′ (2 π ) k ′ µ ( k ′ + m k )(( k ′ − k ) + m k ) = − π k µ (cid:16) − ǫ + γ + log( π ) (cid:17) (A13) Z d k ′ (2 π ) k ′ + m k )(( k ′ − k ) + m k ) = 116 π (cid:16) ǫ − γ − log( π ) − log( m k µ ) − log (cid:16) (cid:0) m k ( k ) (cid:1) / − (cid:0) m k ( k ) (cid:1) / (cid:17)(cid:0) m k ( k ) (cid:1) / (cid:17) (A14)26ntegrals of type (A12,A13,A14) are also present in h ˜ S ˜ S ˜ S i . The rest of the integrals are, Z d k ′ (2 π ) k ′ µ k ′ ν k ′ ρ d d d = 116 π (cid:16) ǫ − γ − log( π ) (cid:17) × ( g νρ (2 k µ + p µ ) + g ρµ (2 k ν + p ν ) + g µν (2 k ρ + p ρ ) (A15) Z d k ′ (2 π ) k ′ µ k ′ ν d d d = 116 π g µν (cid:16) ǫ − γ − log( π ) (cid:17) (A16) [1] R. P. Woodard, Reports on Progress in Physics , 126002 (2009).[2] J. F. Donoghue, Phys. Rev. D , 3874 (1994).[3] J. F. Donoghue and B. R. Holstein, J. Phys. G , 103102 (2015).[4] J. C. Fabris, P. L. C. de Oliveira, D. C. Rodrigues, A. M. Velasquez-Toribio,and I. L. Shapiro, Proceedings, 10th Conference on Quantum field theory under theinfluence of external conditions (QFEXT 11): Benasque, Spain, September 18-24,2011 , Int. J. Mod. Phys.
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