Quantum equivalence of f (R) gravity and scalar–tensor theories in the Jordan and Einstein frames

Quantum equivalence of f (R) gravity and scalar–tensor theories in the Jordan and Einstein frames Prog. Theor. Exp. Phys. 2018, 033B02 (13 pages) DOI: 10.1093/ptep/pty008 Quantum equivalence of f (R) gravity and scalar–tensor theories in the Jordan and Einstein frames 1,2,∗ Nobuyoshi Ohta Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502, Japan Maskawa Institute for Science and Culture, Kyoto Sangyo University, Kyoto 603-8555, Japan E-mail: ohtan@phys.kindai.ac.jp Received December 18, 2017; Accepted January 16, 2018; Published March 5, 2018 ................................................................................................................... The f (R) gravity and scalar–tensor theory are known to be equivalent at the classical level. We study if this equivalence is valid at the quantum level. There are two descriptions of the scalar– tensor theory in the Jordan and Einstein frames. It is shown that these three formulations of the theories give the same determinant or effective action on shell, and thus they are equivalent at the quantum one-loop level on shell in arbitrary dimensions. We also compute the one-loop divergence in f (R) gravity on an Einstein space. ................................................................................................................... Subject Index B32, B39, E03, E05 1. Introduction It is always of great interest to consider various modifications of Einstein gravity for phenomeno- logical applications. Among others, what is called f (R) gravity attracts much attention, especially in the context of the inflationary scenario in cosmology. For early attempts, see Refs. [1–3] and [4,5] for reviews. This class of theories has a nice feature that even though the theory involves higher derivatives, there is no ghost introduced. In addition to the massless spin-two graviton, the theory involves an additional scalar degree of freedom. The simplest way to see this is to rewrite the theory using a scalar field coupled to the Einstein theory, leading to scalar–tensor theory. It has been known for a long time that the equivalence is valid at the classical level on shell (see, e.g., Refs. [4,6,7]), but it has not been much discussed at the quantum level. One possible way to understand the quantum properties of gravity is the asymptotic safety sce- nario [8–12]. The idea is that the theory of quantum gravity is searched for within a large class of theories, and one single theory is chosen by the condition that it corresponds to a fixed point of the renormalization group flow. The use of a functional renormalization group equation has given considerable evidence in support of the existence of a nontrivial fixed point. The asymptotic safety scenario is discussed for f (R) gravity in Refs. [13–25], and for scalar–tensor theory in Refs. [26–31]. In fact, Benedetti and Guarnieri considered the problem of the equivalence by using the functional renormalization group approach [32]. They rewrite the f (R) gravity in the scalar–tensor theory in the form without a kinetic term for the scalar, and then introduced a kinetic term for the scalar with constant coefficient ω. After deriving the functional renormalization group equation in the Feynman and Landau gauges, they try to find fixed points in these gauges, in particular for the ω = 0 case. © The Author 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 Funded by SCOAP by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta They found that the results disagree with each other, and argued that this is evidence against the equivalence to f (R) theory. However, this might be just a gauge artifact. While this work was in progress, a paper appeared in which one-loop divergence in f (R) gravity on arbitrary background is computed in a specific gauge [33]. Based on this result, it is argued that f (R) gravity and classically equivalent scalar–tensor theory are also equivalent on shell at the quantum level [34]. It would be interesting to further check the equivalence at the level of effective action and/or functional renormalization group satisfied by the effective average action. As the equivalence holds on shell classically, it is expected that the equivalence only holds on shell also at the quantum level. It is a general belief that the effective potential is gauge independent on shell [35], so it is expected that the equivalence holds on shell in any gauge. We also expect that the result is independent of the parametrization of the metric. In this paper, we discuss this problem by studying the effective actions or equivalent determinants obtained in the path integral formulation in arbitrary dimensions. There are two equivalent (at the classical level) formulations of scalar–tensor theories related by conformal transformation. They are known as scalar–tensor theories in the Jordan and Einstein frames. There is still ongoing debate about the quantum equivalence of these theories in the different frames [36–43]. We also study the relation in this paper. We find that all these formulations are equivalent on shell at the quantum level. This paper is organized as follows. In Sect. 2, we first review how the f (R) theory is rewritten into a theory of a scalar field coupled to the Einstein theory in the Jordan frame. Then we make a conformal transformation to map the theory in the Einstein frame. In Sect. 3, we start studying the effective actions in these theories in the background field formalism. In Sect. 3.1, we derive the Hessian for the metric fluctuation h in f (R) gravity on an Einstein background, which is μν assumed throughout this paper. Using the exponential parametrization of the metric, which has the nice feature of giving results rather close to on-shell [23,24,30,44–48], we calculate the determinant with a general linear gauge with two gauge parameters α and β. We show that the resulting effective action or determinant after the path integral does not depend on the gauge parameters (if we use partial gauge fixing h = 0). For completeness, we also give the one-loop divergent part of the effective action and the resulting functional renormalization group equation. The first agrees with the recent calculation [33]. In Sect. 3.2, we repeat the calculation in the scalar–tensor theory in the Jordan frame. The Hessian has a matrix structure but after taking the determinant, we find that the result precisely agrees with that in the f (R) theory if we use field equations for the background. In Sect. 3.3, we go on to the scalar–tensor theory in the Einstein frame, and find that the resulting determinant is different off shell but becomes the same on shell. This is to be expected because classically the theory is equivalent only on shell. We take these facts as evidence of the quantum equivalence of these theories. In Sect. 4, we give conclusions and discussions. In the appendix, we give a formula for the conformal transformation. 2. Classical equivalence Let us consider the Euclidean theory S = d x gf (R), (2.1) where g = det(g ). Classically it is known that this theory is equivalent to a scalar field φ coupled μν to the Einstein gravity. 2/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta To move to such a formulation, let us first consider the theory S = d x g f (χ )(R − χ) + f (χ ) , (2.2) μν where χ is a new scalar field. If we take the variation with respect to χ and g ,weget f (χ )(R − χ) = 0, f (χ )R − [f (χ )(R − χ) + f (χ )]g −∇ ∇ f (χ ) + g ∇ f (χ ) = 0. (2.3) μν μν μ ν μν We assume that f (χ ) = 0, and then we get χ = R. Substituting this into Eq. (2.2) or the second equation in Eq. (2.3), we find that it reduces to f (R)R − f (R)g −∇ ∇ f (R) + g ∇ f (R) = 0, (2.4) μν μν μ ν μν which is nothing but the field equation obtained from the action (2.1). Thus the theory (2.2)is classically equivalent to Eq. (2.1). We can now rewrite the theory further using another scalar field φ. We set Z φ =−f (χ ), (2.5) where Z = with G being the Newton constant. It should be understood that we solve Eq. 16πG (2.5) for the field χ in terms of φ. Then Eq. (2.2) takes the form S = d x g [Z φ{χ(φ) − R}+ f (χ (φ))]. (2.6) The field equations following from this action are δφ :Z (χ (φ) − R + φχ (φ)) + f (χ (φ))χ (φ) = 0, μν 2 δg :Z (−φR +∇ ∇ φ − g ∇ φ) − [Z φ{χ(φ) − R}+ f (χ (φ))] g = 0. (2.7) N μν μ ν μν N μν Here and in what follows, the primes should be understood as differentiations with respect to the df (χ ) dχ(φ) arguments, so f (χ ) = and χ (φ) = , and they should not be confused. Using Eq. (2.5)in dχ dφ the first equation, we find χ(φ) = R. Together with Eq. (2.5) again, the second equation in Eq. (2.7) then recovers Eq. (2.4). So this theory is also classically equivalent to Eq. (2.1). We define a potential by V (φ) = Z φχ (φ) + f (χ (φ)). (2.8) Using Eq. (2.5), the derivatives of the potential are found to be V (φ) = Z χ(φ), V (φ) = Z χ (φ). (2.9) We also have χ (φ) =− . (2.10) f (χ ) The action (2.6) is what is known as a theory of a scalar field coupled to gravity in the Jordan frame. We refer to this theory as scalar–tensor theory in the Jordan frame. 3/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta We can go to the Einstein frame by setting −2/(d−2) g = φ g ˜ . (2.11) μν μν With the help of the formula in the appendix, the action (2.6) is transformed into d − 1 (∂ φ) d −2/(d−2) −d/(d−2) S = d x g ˜ Z −R + + φ χ(φ) + φ f (χ (φ)) . (2.12) d − 2 φ We change the scalar kinetic term by setting d − 2 ln φ = ϕ, (2.13) d − 1 and get 2 d √ √ − ϕ − ϕ d 2 ˜ (d−1)(d−2) (d−1)(d−2) S = d x g ˜ Z −R + (∂ ϕ) + e χ(ϕ) + e f (χ (ϕ)) . (2.14) EF N μ We also refer to this theory as scalar–tensor theory in the Einstein frame. This should again be equivalent to Eq. (2.1). Thus we have two equivalent formulations of the theory (2.1) at the classical level. Note that this equivalence is valid on shell, i.e., when we use the field equations. The question that we would like to address is whether these descriptions are also equivalent at the quantum level. We expect that the equivalence is also valid only on shell. From Eq. (2.13), we have d − 1 −f (χ ) ϕ = ln , (2.15) d − 2 Z and d − 2 f (χ ) d − 2 f (χ ) f (χ ) − f (χ )f (χ ) χ (ϕ) = , χ (ϕ) = etc. (2.16) d − 1 f (χ ) d − 1 f (χ ) f (χ ) (n) We can get all the equations for χ (ϕ) in terms of f and its derivatives. Then if we define the potential by d d−2 − ϕ ϕ d−1 (d−1)(d−2) U (ϕ) = e Z e χ(ϕ) + f (χ (ϕ)) , (2.17) we can express the condition of a minimum of the potential in terms of f (χ ): d−2 −f (χ ) 2χf (χ ) − df (χ ) U (ϕ) = √ . (2.18) (d − 2)(d − 1) In addition we also have that d−2 −f (χ ) U (ϕ) = f (χ ) − χf (χ ) . (2.19) The Einstein equation for g ˜ is μν ˜ ˜ − Z R − g ˜ −Z R + Z (∂ ϕ) + U (ϕ) + Z ∂ ϕ∂ ϕ = 0. (2.20) N μν μν N N ρ N μ ν 4/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta For constant backgrounds, this gives Z R = U (ϕ). (2.21) d − 2 On the other hand, by the transformation (2.11), we also have that −2 d−2 −f (χ ) R = R. (2.22) At the minimum of the potential, we have d−2 −f (χ ) d − 2 U (ϕ) = Z χ (2.23) min N Z d and we get the Einstein equation −2 −2 d−2 d−2 −f (χ ) −f (χ ) R = χ, (2.24) Z Z N N or R = χ. (2.25) 3. Quantum equivalence In order to discuss quantum theory, we use the background field method and expand the metric and the scalar fields as h ρ ¯ ˜ g =¯ g (e ) , φ = φ + φ, ϕ =¯ ϕ +˜ ϕ. (3.1) μν μρ ν We will consider constant background φ and ϕ ¯. Note that we use the exponential parametrization for the metric. This is because this parametrization has various virtues like least gauge dependence. For the one-loop calculation, we have to derive the Hessian. Henceforth we assume that the background space is an Einstein space with ¯ ¯ R = g , R = const. (3.2) μν μν 3.1. f (R) gravity For the f (R) gravity, we have the quadratic term [24] 1 2 (2) TT TT μν ¯ ¯ I =− f (R)h − R h μν 2 f (R) 4 d ¯ ¯ d − 1 2(d − 1) R d − 2 R ¯ ¯ + σ f (R) − + f (R) − σ L0 L L0 4d d d − 1 d d − 1 1 2(d − 1) R + h f (R) − L0 4 d d − 1 (d − 1)(d − 2) 2 1 ¯ ¯ ¯ + f (R) − R + f (R) h d d − 2 2 ¯ ¯ d − 1 2(d − 1) R d − 2 R ¯ ¯ + h f (R) − + f (R) − σ , (3.3) L0 L0 L0 2d d d − 1 d d − 1 5/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta where we have suppressed the overall factor g ¯ and , , and are the Lichnerowicz L2 L1 L0 Laplacians on the symmetric tensor, vector, and scalar, respectively, defined by 2 ρ ρ ρσ σρ T =−∇ T + R T + R T − R T − R T , L μν μν μ ρν ν μρ μρνσ μρνσ 2 ρ V =−∇ V + R V , L μ μ μ ρ S =−∇ S. (3.4) L0 We have also used the York decomposition defined by 1 1 TT 2 h = h +∇ ξ +∇ ξ +∇ ∇ σ − g ¯ ∇ σ + g ¯ h, (3.5) μν μ ν ν μ μ ν μν μν μν d d where TT μν TT μ ∇ h =¯ g h =∇ ξ = 0. (3.6) μ μ μν μν The above formula agrees with Ref. [49]. In terms of s = σ + h, Eq. (3.3) is put into L0 1 2 (2) TT TT μν ¯ ¯ I =− f (R)h − R h L2 μν f (R) 4 d ¯ ¯ ¯ ¯ (d − 1) (d − 2)f (R) − 2Rf (R) R + s f (R) + − s L L 0 0 2d 2(d − 1) d − 1 ¯ ¯ ¯ df (R) − 2Rf (R) + h . (3.7) 8d We then consider the gauge fixing term d μν S = d x g ¯ g ¯ F F , (3.8) GF μ ν 2α with β + 1 F =∇ h − ∇ h. (3.9) μ ρ μ μ Following Ref. [35], it is convenient to reparametrize the scalar sector in terms of the gauge-invariant variable s and a new degree of freedom u defined as [(d − 1) − R]σ + βh L0 s = h + σ , u = . (3.10) L0 (d − 1 − β) − R The gauge fixing action then becomes 2 2 ¯ ¯ 1 2R (d − 1 − β) R S = dx g ¯ ξ − ξ + u − u . (3.11) GF μ L1 L0 L0 2α d d d−1−β On shell, the last term in Eq. (3.7) is zero, so the quadratic part of the action is written entirely in TT terms of the physical degrees of freedom h and s, and the gauge fixing entirely in terms of the gauge degrees of freedom ξ and u. The ghost action for this gauge fixing contains a non-minimal operator β + 1 μ ν 2 ν ν ¯ ¯ S = dx g ¯ C δ ∇ + 1 − 2 ∇ ∇ + R C . (3.12) gh μ μ ν 6/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta Upon decomposing the ghost into transverse and longitudinal parts T L T L C = C +∇ C = C +∇ √ C , (3.13) ν ν ν ν ν L0 and the same for C, the ghost action splits in two terms: ¯ ¯ 2R d − 1 − β R T μ T L L ¯ ¯ S = dx g ¯ −C − C − 2 C − C . (3.14) gh L1 L0 d d d − 1 − β Now if we make a partial gauge fixing to set h = 0, which can be done by sending β →∞,we get the following one-loop determinants: −1/2 −1/2 ¯ ¯ ¯ ¯ 2 (d − 2)f (R) − 2Rf (R) R ¯ ¯ Det − R Det f (R) + − , L2 L0 L0 d 2(d − 1) d − 1 (3.15) from Eq. (3.7), −1 −1 ¯ ¯ 2R R −1/2 Det − Det [ ] Det − , (3.16) L1 L0 L0 d d − 1 − β from the gauge fixing term (3.11), and ¯ ¯ 2R R Det − Det − , (3.17) L1 L0 d d − 1 − β from the ghost terms (3.14). The York decomposition has the Jacobian 1/2 1/2 2 R 1/2 Det − R Det[ ] Det − (3.18) L L L 1 0 0 d d − 1 whereas the subsequent transformation (σ , h) → (s, u) has a unit Jacobian. We see that many of these cancel and we are left with 1/2 Det − R L1 . (3.19) 1/2 1/2 ¯ ¯ ¯ (d−2)f (R)−2Rf (R) 2R Det − Det + L L 2 0 2(d−1)f (R) As observed in Ref. [24] and confirmed in Ref. [48], this result does not depend on the gauge parameters α and β, and the result is close to on-shell once the partial gauge choice h = 0 is made. This is the advantage of the exponential parametrization [23,24,30,44–48]. The above determinant is what governs the quantum theory at the one-loop level, in particular effective action. Given the above result, we can evaluate the effective action, which is related to the partition function −(g ¯ ) by Z (g ¯ ) = e . Neglecting field-independent terms, we find ¯ ¯ 1 2R 1 2R (g ¯ ) = log Det − − log Det − L2 L1 2 d 2 d ¯ ¯ 1 R (d − 2)f (R) + log Det − + . (3.20) 2 d − 1 2(d − 1)f (R) 7/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta The divergent part of the effective action can be computed by standard heat kernel methods [50]. On an Einstein background in four dimensions, the logarithmically divergent part is 2  2 ¯ ¯ ¯ 1  71 433 f (R) Rf (R) 4 2 2 ¯ ¯ (g ¯ ) = d x g ¯ log − R + R − + , log μνρσ 2 2 ¯ ¯ 12(4π) μ 10 120 3f (R) f (R) (3.21) where  stands for a cutoff and we have introduced a reference mass scale μ. On shell, we have ¯ ¯ 2f (R) ¯ ¯ f (R) = , and R = for maximally symmetric space; this reduces to μνρσ ¯ 6 2 2 ¯ ¯ 1  97 8f (R) 4f (R) 4 2 (g ¯ ) = d x g ¯ log R − + , (3.22) log 2 2 2  2 ¯ ¯ ¯ 24(4π) μ 20 3R f (R) f (R) in agreement with Ref. [33]. The flow equation for the f (R) theory on the 4-sphere was derived in Refs. [23,24]. The result using the spectral sum is 32π ( − 2r + 4) ˙ ˙ d d 2r − 2 −  d  − 2r + d  d 2 3 4 1 5 = − + + , 6 + (6α + 1)r  (3 + (3β − 1)r) +  4 + (4γ − 1)r (3.23) where 5[6 + (6α − 1)r][12 + (12α − 1)r] d = , 5[6 + (6α − 1)r][3 + (3α − 2)r] d = , [2 + (2β + 3)r][3 + (3β − 1)r][6 + (6β − 5)r] d = , [2 + (2β − 1)r][12 + (12β + 11)r] d = , 2 2 −72 − 18r(1 + 8γ) + r 19 − 18γ − 72γ d = , (3.24) −2 −4 ¯ ¯ and we have used the dimensionless quantities r = Rk and (r) = k f (R), and α, β, and γ are the parameters of endomorphism, not to be confused with the gauge parameters. 3.2. Scalar–tensor theory in the Jordan frame Next we discuss the one-loop determinant in the scalar–tensor theory in the Jordan frame. We find from Eq. (2.9) that the quadratic terms in the fluctuations in the scalar field are √ Z 1 2 2 ¯ ˜ ˜ ¯ gV (φ) g ¯ χ (φ)(φ + hφ) + V (φ)h . (3.25) 2 8 Together with the contribution from the rest of the terms, we find, using theYork decomposition (3.5), ¯ ¯ 1 2R (d − 2)(d − 2) R d − 2 (2) TT TT μν 2 ¯ ¯ I =−Z φ − h − h + s − s + Rh N L L 2 0 μν 4 d 4d d − 1 8d ¯ ¯ R d − 1 R Z 1 2 2 ˜ ˜ ¯ ˜ ˜ ¯ − Z hφ + φ − s + χ (φ)(φ + hφ) + V (φ)h . (3.26) N L0 2 d d − 1 2 8 8/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta We employ the same gauge fixing as in the preceding subsection. Here we also use the partial gauge fixing h = 0. Then we again find that the quadratic part of the action is written entirely in terms of TT the physical degrees of freedom h , s, and φ and the gauge fixing entirely in terms of the gauge degrees of freedom ξ and u. The one-loop determinant from the scalar sector (s, φ) is ⎛     ⎞ −1/2 ¯ ¯ (d−1)(d−2) R d−1 R ¯ ¯ Z φ − − Z φ − 2 N L0 N L0 d−1 d d−1 2d ⎝   ⎠ Det d−1 R ¯ ¯ − Z φ − Z χ (φ) N L N d d−1 −1/2 ¯ ¯ (d − 1) R R d − 2 ¯ ¯ = Det Z − − + φχ (φ) . (3.27) L0 L0 d d − 1 d − 1 d − 1 It follows from Eqs. (2.5) and (2.10) that f (χ ) φχ (φ) = . (3.28) f (χ ) So, writing out the resulting whole one-loop determinant, we get 1/2 Det − R L1 . (3.29) 1/2 1/2 ¯ ¯ (d−2)f (χ) ¯ 2R R Det − Det + − L2 L0 d 2(d−1)f (χ) ¯ d−1 If we use χ ¯ = R, this precisely agrees with Eq. (3.19), the result for the f (R) theory. Thus we conclude that this formulation by scalar–tensor theory is equivalent to the original f (R) theory at the quantum (at least) one-loop level on shell. 3.3. Scalar–tensor theory in the Einstein frame Next we discuss the Hessian and one-loop determinant in the scalar–tensor theory in the Einstein frame. If we take one more step in the discussion in Sect. 2, we find that the second derivative of the potential is given by d−2 −f (χ ) (d − 2)f − 2χf d U (ϕ) = Z − √ U (ϕ), (3.30) Z (d − 1)f (d − 2)(d − 1) so the Hessian for the fluctuation ϕ ˜ is (d − 2)f − 2χf d (2) I = Z + − √ U (ϕ) ¯ . (3.31) N L ϕϕ 0 2(d − 1)f 2 (d − 2)(d − 1) Note that if we exploit the equations of motion for the background U (ϕ) ¯ = 0 and χ ¯ = R,wefind that this is proportional to that of the field s in Eq. (3.7): (2) (2) S ∝ S . (3.32) ss ϕϕ on-shell There are also some mixing terms between the graviton h and the scalar ϕ, but these drop out in the gauge h = 0 in the exponential parametrization. 9/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta The rest of the theory is the usual Einstein theory. The Hessian for this theory can be obtained from the result in the preceding subsection by setting f (R) =−Z R. We thus find 1 2 (d − 1)(d − 2) R d − 2 (2) TT TT μν 2 ¯ ¯ I =−Z − h − R h + s − s + Rh . N L2 L0 μν 4 d 4d d − 1 8d (3.33) The gauge fixings can be taken as in the preceding subsections. With the partial gauge fixing h = 0, we again find the separation of the physical degrees of freedom and the gauge fixing terms. It is now straightforward to derive the one-loop determinant: 1/2 Det − R L1 . (3.34) 1/2 1/2 (d−2)f −2χf 2R d Det − Det Z + − U (ϕ) L2 N L0 d 2(d−1)f 2 (d−2)(d−1) As noted above, on shell, this is equivalent to the result of f (R) gravity. However, before concluding that the theory is equivalent at the quantum level, we have to take the Jacobian from the transformation (2.13) into account. The path integral would produce divergences in the form δ(0) times the volume from this change of variable. However, such terms would affect the power-law divergence coefficients, which are subject to a regularization scheme. The coefficients of logarithmic divergence are not affected and are universal. It is true that we have to take into account the difference in the definition of the scales in different frames, but that can be easily incorporated since the forms of the effective action are the same. One may worry that the conformal transformation introduces a change in the path integral measure and hence leads to a trace anomaly. However, this will be taken into account in the form of a logarithmic ultraviolet (UV) cutoff dependence when the determinant is regularized with a UV cutoff, which is dependent on the conformal transformation in terms of the scalar field [42]. We will discuss this problem of the difference in scale in the next section. All the results for one-loop divergence, effective action, and the flow equation can be derived from these determinants. The scale dependence in different frames would introduce a formal difference in the resulting functional renormalization group equations, but should not affect the physical results. We thus conclude that the theory is also equivalent to the f (R) theory at quantum level on shell. 4. Conclusions and discussions In this paper, we first summarized the relation of the f (R) theory and the reformulations of the theory in the form of a scalar field coupled to the Einstein theory in the Jordan and Einstein frames, and then calculated determinants, which correspond to the effective actions, after path integral over the fluctuation fields in the background field formalism. It turns out that all three formulations give the same determinant on shell. If we evaluate the determinant with a suitable cutoff, this gives the divergences in the theory. After renormalization, this produces an effective action. We have also given the one-loop divergent term in the effective action. One could also try to derive the functional renormalization group equation by introducing a suitable cutoff function. The fact that the determinants, from which these are all derived, are the same is strong evidence that these theories are equivalent at the quantum level, at least at one loop. One possible caveat is that the transformation into the Einstein frame involves conformal trans- formation. This transformation would produce δ(0) type divergence, which could be removed by a local counterterm. However, this also produces a difference in the scales in different frames. In a 10/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta given frame, a short-distance cutoff  may be defined by 2 μ ν = g x x . (4.1) μν When the metric is transformed as Eq. (2.11), the cutoff lengths in the Jordan and Einstein frames are related by 2 J μ ν −2/(d−2) E μ ν −2/(d−2) 2 ¯ ¯ = g x x = (φ) g x x = (φ)  . (4.2) J μν μν E The UV cutoff is then related by 2 2/(d−2) 2 = (φ)  . (4.3) J E This would result in slightly different-looking functional renormalization group equations in the two frames due to the different cutoffs. If this difference is dealt with suitably, the physical result should not depend on the difference because the effective action is the same. For related discussions, see Ref. [42]. To summarize, we have found strong evidence that the f (R) theory and the scalar–tensor theories are equivalent on shell in arbitrary dimensions. As a byproduct, we have also found evidence that the theories in the Jordan and Einstein frames are equivalent. Note that our discussions are based on the Einstein space with the curvature (3.2). It would be interesting to try to extend our result to more general spacetime. Acknowledgements We would like to thank Kevin Falls for valuable discussions at the early stage of this work, and Roberto Percacci for valuable comments. This work was supported in part by a Grant-in-Aid for the Scientific Research Fund of the Japan Society for the Promotion of Science (C) No. 16K05331. Funding Open Access funding: SCOAP . Appendix. Conformal transformation We give a formula relevant in the text here. Under the transformation −2ρ g = e g ˜ , (A.1) μν μν the Einstein term changes as (2−d)ρ 2 2 gR = ge ˜ R + 2(d − 1)∇ ρ − (d − 1)(d − 2)(∂ ρ) . (A.2) Note that the contraction is made on the right-hand side by g ˜ . References [1] J. Hwang and H. Noh, Phys. Lett. B 506, 13 (2001) [arXiv:astro-ph/0102423][Search INSPIRE]. [2] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, and S. Zerbini, J. Cosmol. Astropart. Phys. 02, 010 (2005) [arXiv:hep-th/0501096][Search INSPIRE]. [3] S. Capozziello, V. F. Cardone, and A. Troisi, Phys. Rev. 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Percacci, An Introduction to Covariant Quantum Gravity and Asymptotic Safety (World Scientific, Singapore, 2017). 13/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Progress of Theoretical and Experimental Physics Oxford University Press

Quantum equivalence of f (R) gravity and scalar–tensor theories in the Jordan and Einstein frames

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Prog. Theor. Exp. Phys. 2018, 033B02 (13 pages) DOI: 10.1093/ptep/pty008 Quantum equivalence of f (R) gravity and scalar–tensor theories in the Jordan and Einstein frames 1,2,∗ Nobuyoshi Ohta Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502, Japan Maskawa Institute for Science and Culture, Kyoto Sangyo University, Kyoto 603-8555, Japan E-mail: ohtan@phys.kindai.ac.jp Received December 18, 2017; Accepted January 16, 2018; Published March 5, 2018 ................................................................................................................... The f (R) gravity and scalar–tensor theory are known to be equivalent at the classical level. We study if this equivalence is valid at the quantum level. There are two descriptions of the scalar– tensor theory in the Jordan and Einstein frames. It is shown that these three formulations of the theories give the same determinant or effective action on shell, and thus they are equivalent at the quantum one-loop level on shell in arbitrary dimensions. We also compute the one-loop divergence in f (R) gravity on an Einstein space. ................................................................................................................... Subject Index B32, B39, E03, E05 1. Introduction It is always of great interest to consider various modifications of Einstein gravity for phenomeno- logical applications. Among others, what is called f (R) gravity attracts much attention, especially in the context of the inflationary scenario in cosmology. For early attempts, see Refs. [1–3] and [4,5] for reviews. This class of theories has a nice feature that even though the theory involves higher derivatives, there is no ghost introduced. In addition to the massless spin-two graviton, the theory involves an additional scalar degree of freedom. The simplest way to see this is to rewrite the theory using a scalar field coupled to the Einstein theory, leading to scalar–tensor theory. It has been known for a long time that the equivalence is valid at the classical level on shell (see, e.g., Refs. [4,6,7]), but it has not been much discussed at the quantum level. One possible way to understand the quantum properties of gravity is the asymptotic safety sce- nario [8–12]. The idea is that the theory of quantum gravity is searched for within a large class of theories, and one single theory is chosen by the condition that it corresponds to a fixed point of the renormalization group flow. The use of a functional renormalization group equation has given considerable evidence in support of the existence of a nontrivial fixed point. The asymptotic safety scenario is discussed for f (R) gravity in Refs. [13–25], and for scalar–tensor theory in Refs. [26–31]. In fact, Benedetti and Guarnieri considered the problem of the equivalence by using the functional renormalization group approach [32]. They rewrite the f (R) gravity in the scalar–tensor theory in the form without a kinetic term for the scalar, and then introduced a kinetic term for the scalar with constant coefficient ω. After deriving the functional renormalization group equation in the Feynman and Landau gauges, they try to find fixed points in these gauges, in particular for the ω = 0 case. © The Author 2018. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 Funded by SCOAP by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta They found that the results disagree with each other, and argued that this is evidence against the equivalence to f (R) theory. However, this might be just a gauge artifact. While this work was in progress, a paper appeared in which one-loop divergence in f (R) gravity on arbitrary background is computed in a specific gauge [33]. Based on this result, it is argued that f (R) gravity and classically equivalent scalar–tensor theory are also equivalent on shell at the quantum level [34]. It would be interesting to further check the equivalence at the level of effective action and/or functional renormalization group satisfied by the effective average action. As the equivalence holds on shell classically, it is expected that the equivalence only holds on shell also at the quantum level. It is a general belief that the effective potential is gauge independent on shell [35], so it is expected that the equivalence holds on shell in any gauge. We also expect that the result is independent of the parametrization of the metric. In this paper, we discuss this problem by studying the effective actions or equivalent determinants obtained in the path integral formulation in arbitrary dimensions. There are two equivalent (at the classical level) formulations of scalar–tensor theories related by conformal transformation. They are known as scalar–tensor theories in the Jordan and Einstein frames. There is still ongoing debate about the quantum equivalence of these theories in the different frames [36–43]. We also study the relation in this paper. We find that all these formulations are equivalent on shell at the quantum level. This paper is organized as follows. In Sect. 2, we first review how the f (R) theory is rewritten into a theory of a scalar field coupled to the Einstein theory in the Jordan frame. Then we make a conformal transformation to map the theory in the Einstein frame. In Sect. 3, we start studying the effective actions in these theories in the background field formalism. In Sect. 3.1, we derive the Hessian for the metric fluctuation h in f (R) gravity on an Einstein background, which is μν assumed throughout this paper. Using the exponential parametrization of the metric, which has the nice feature of giving results rather close to on-shell [23,24,30,44–48], we calculate the determinant with a general linear gauge with two gauge parameters α and β. We show that the resulting effective action or determinant after the path integral does not depend on the gauge parameters (if we use partial gauge fixing h = 0). For completeness, we also give the one-loop divergent part of the effective action and the resulting functional renormalization group equation. The first agrees with the recent calculation [33]. In Sect. 3.2, we repeat the calculation in the scalar–tensor theory in the Jordan frame. The Hessian has a matrix structure but after taking the determinant, we find that the result precisely agrees with that in the f (R) theory if we use field equations for the background. In Sect. 3.3, we go on to the scalar–tensor theory in the Einstein frame, and find that the resulting determinant is different off shell but becomes the same on shell. This is to be expected because classically the theory is equivalent only on shell. We take these facts as evidence of the quantum equivalence of these theories. In Sect. 4, we give conclusions and discussions. In the appendix, we give a formula for the conformal transformation. 2. Classical equivalence Let us consider the Euclidean theory S = d x gf (R), (2.1) where g = det(g ). Classically it is known that this theory is equivalent to a scalar field φ coupled μν to the Einstein gravity. 2/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta To move to such a formulation, let us first consider the theory S = d x g f (χ )(R − χ) + f (χ ) , (2.2) μν where χ is a new scalar field. If we take the variation with respect to χ and g ,weget f (χ )(R − χ) = 0, f (χ )R − [f (χ )(R − χ) + f (χ )]g −∇ ∇ f (χ ) + g ∇ f (χ ) = 0. (2.3) μν μν μ ν μν We assume that f (χ ) = 0, and then we get χ = R. Substituting this into Eq. (2.2) or the second equation in Eq. (2.3), we find that it reduces to f (R)R − f (R)g −∇ ∇ f (R) + g ∇ f (R) = 0, (2.4) μν μν μ ν μν which is nothing but the field equation obtained from the action (2.1). Thus the theory (2.2)is classically equivalent to Eq. (2.1). We can now rewrite the theory further using another scalar field φ. We set Z φ =−f (χ ), (2.5) where Z = with G being the Newton constant. It should be understood that we solve Eq. 16πG (2.5) for the field χ in terms of φ. Then Eq. (2.2) takes the form S = d x g [Z φ{χ(φ) − R}+ f (χ (φ))]. (2.6) The field equations following from this action are δφ :Z (χ (φ) − R + φχ (φ)) + f (χ (φ))χ (φ) = 0, μν 2 δg :Z (−φR +∇ ∇ φ − g ∇ φ) − [Z φ{χ(φ) − R}+ f (χ (φ))] g = 0. (2.7) N μν μ ν μν N μν Here and in what follows, the primes should be understood as differentiations with respect to the df (χ ) dχ(φ) arguments, so f (χ ) = and χ (φ) = , and they should not be confused. Using Eq. (2.5)in dχ dφ the first equation, we find χ(φ) = R. Together with Eq. (2.5) again, the second equation in Eq. (2.7) then recovers Eq. (2.4). So this theory is also classically equivalent to Eq. (2.1). We define a potential by V (φ) = Z φχ (φ) + f (χ (φ)). (2.8) Using Eq. (2.5), the derivatives of the potential are found to be V (φ) = Z χ(φ), V (φ) = Z χ (φ). (2.9) We also have χ (φ) =− . (2.10) f (χ ) The action (2.6) is what is known as a theory of a scalar field coupled to gravity in the Jordan frame. We refer to this theory as scalar–tensor theory in the Jordan frame. 3/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta We can go to the Einstein frame by setting −2/(d−2) g = φ g ˜ . (2.11) μν μν With the help of the formula in the appendix, the action (2.6) is transformed into d − 1 (∂ φ) d −2/(d−2) −d/(d−2) S = d x g ˜ Z −R + + φ χ(φ) + φ f (χ (φ)) . (2.12) d − 2 φ We change the scalar kinetic term by setting d − 2 ln φ = ϕ, (2.13) d − 1 and get 2 d √ √ − ϕ − ϕ d 2 ˜ (d−1)(d−2) (d−1)(d−2) S = d x g ˜ Z −R + (∂ ϕ) + e χ(ϕ) + e f (χ (ϕ)) . (2.14) EF N μ We also refer to this theory as scalar–tensor theory in the Einstein frame. This should again be equivalent to Eq. (2.1). Thus we have two equivalent formulations of the theory (2.1) at the classical level. Note that this equivalence is valid on shell, i.e., when we use the field equations. The question that we would like to address is whether these descriptions are also equivalent at the quantum level. We expect that the equivalence is also valid only on shell. From Eq. (2.13), we have d − 1 −f (χ ) ϕ = ln , (2.15) d − 2 Z and d − 2 f (χ ) d − 2 f (χ ) f (χ ) − f (χ )f (χ ) χ (ϕ) = , χ (ϕ) = etc. (2.16) d − 1 f (χ ) d − 1 f (χ ) f (χ ) (n) We can get all the equations for χ (ϕ) in terms of f and its derivatives. Then if we define the potential by d d−2 − ϕ ϕ d−1 (d−1)(d−2) U (ϕ) = e Z e χ(ϕ) + f (χ (ϕ)) , (2.17) we can express the condition of a minimum of the potential in terms of f (χ ): d−2 −f (χ ) 2χf (χ ) − df (χ ) U (ϕ) = √ . (2.18) (d − 2)(d − 1) In addition we also have that d−2 −f (χ ) U (ϕ) = f (χ ) − χf (χ ) . (2.19) The Einstein equation for g ˜ is μν ˜ ˜ − Z R − g ˜ −Z R + Z (∂ ϕ) + U (ϕ) + Z ∂ ϕ∂ ϕ = 0. (2.20) N μν μν N N ρ N μ ν 4/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta For constant backgrounds, this gives Z R = U (ϕ). (2.21) d − 2 On the other hand, by the transformation (2.11), we also have that −2 d−2 −f (χ ) R = R. (2.22) At the minimum of the potential, we have d−2 −f (χ ) d − 2 U (ϕ) = Z χ (2.23) min N Z d and we get the Einstein equation −2 −2 d−2 d−2 −f (χ ) −f (χ ) R = χ, (2.24) Z Z N N or R = χ. (2.25) 3. Quantum equivalence In order to discuss quantum theory, we use the background field method and expand the metric and the scalar fields as h ρ ¯ ˜ g =¯ g (e ) , φ = φ + φ, ϕ =¯ ϕ +˜ ϕ. (3.1) μν μρ ν We will consider constant background φ and ϕ ¯. Note that we use the exponential parametrization for the metric. This is because this parametrization has various virtues like least gauge dependence. For the one-loop calculation, we have to derive the Hessian. Henceforth we assume that the background space is an Einstein space with ¯ ¯ R = g , R = const. (3.2) μν μν 3.1. f (R) gravity For the f (R) gravity, we have the quadratic term [24] 1 2 (2) TT TT μν ¯ ¯ I =− f (R)h − R h μν 2 f (R) 4 d ¯ ¯ d − 1 2(d − 1) R d − 2 R ¯ ¯ + σ f (R) − + f (R) − σ L0 L L0 4d d d − 1 d d − 1 1 2(d − 1) R + h f (R) − L0 4 d d − 1 (d − 1)(d − 2) 2 1 ¯ ¯ ¯ + f (R) − R + f (R) h d d − 2 2 ¯ ¯ d − 1 2(d − 1) R d − 2 R ¯ ¯ + h f (R) − + f (R) − σ , (3.3) L0 L0 L0 2d d d − 1 d d − 1 5/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta where we have suppressed the overall factor g ¯ and , , and are the Lichnerowicz L2 L1 L0 Laplacians on the symmetric tensor, vector, and scalar, respectively, defined by 2 ρ ρ ρσ σρ T =−∇ T + R T + R T − R T − R T , L μν μν μ ρν ν μρ μρνσ μρνσ 2 ρ V =−∇ V + R V , L μ μ μ ρ S =−∇ S. (3.4) L0 We have also used the York decomposition defined by 1 1 TT 2 h = h +∇ ξ +∇ ξ +∇ ∇ σ − g ¯ ∇ σ + g ¯ h, (3.5) μν μ ν ν μ μ ν μν μν μν d d where TT μν TT μ ∇ h =¯ g h =∇ ξ = 0. (3.6) μ μ μν μν The above formula agrees with Ref. [49]. In terms of s = σ + h, Eq. (3.3) is put into L0 1 2 (2) TT TT μν ¯ ¯ I =− f (R)h − R h L2 μν f (R) 4 d ¯ ¯ ¯ ¯ (d − 1) (d − 2)f (R) − 2Rf (R) R + s f (R) + − s L L 0 0 2d 2(d − 1) d − 1 ¯ ¯ ¯ df (R) − 2Rf (R) + h . (3.7) 8d We then consider the gauge fixing term d μν S = d x g ¯ g ¯ F F , (3.8) GF μ ν 2α with β + 1 F =∇ h − ∇ h. (3.9) μ ρ μ μ Following Ref. [35], it is convenient to reparametrize the scalar sector in terms of the gauge-invariant variable s and a new degree of freedom u defined as [(d − 1) − R]σ + βh L0 s = h + σ , u = . (3.10) L0 (d − 1 − β) − R The gauge fixing action then becomes 2 2 ¯ ¯ 1 2R (d − 1 − β) R S = dx g ¯ ξ − ξ + u − u . (3.11) GF μ L1 L0 L0 2α d d d−1−β On shell, the last term in Eq. (3.7) is zero, so the quadratic part of the action is written entirely in TT terms of the physical degrees of freedom h and s, and the gauge fixing entirely in terms of the gauge degrees of freedom ξ and u. The ghost action for this gauge fixing contains a non-minimal operator β + 1 μ ν 2 ν ν ¯ ¯ S = dx g ¯ C δ ∇ + 1 − 2 ∇ ∇ + R C . (3.12) gh μ μ ν 6/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta Upon decomposing the ghost into transverse and longitudinal parts T L T L C = C +∇ C = C +∇ √ C , (3.13) ν ν ν ν ν L0 and the same for C, the ghost action splits in two terms: ¯ ¯ 2R d − 1 − β R T μ T L L ¯ ¯ S = dx g ¯ −C − C − 2 C − C . (3.14) gh L1 L0 d d d − 1 − β Now if we make a partial gauge fixing to set h = 0, which can be done by sending β →∞,we get the following one-loop determinants: −1/2 −1/2 ¯ ¯ ¯ ¯ 2 (d − 2)f (R) − 2Rf (R) R ¯ ¯ Det − R Det f (R) + − , L2 L0 L0 d 2(d − 1) d − 1 (3.15) from Eq. (3.7), −1 −1 ¯ ¯ 2R R −1/2 Det − Det [ ] Det − , (3.16) L1 L0 L0 d d − 1 − β from the gauge fixing term (3.11), and ¯ ¯ 2R R Det − Det − , (3.17) L1 L0 d d − 1 − β from the ghost terms (3.14). The York decomposition has the Jacobian 1/2 1/2 2 R 1/2 Det − R Det[ ] Det − (3.18) L L L 1 0 0 d d − 1 whereas the subsequent transformation (σ , h) → (s, u) has a unit Jacobian. We see that many of these cancel and we are left with 1/2 Det − R L1 . (3.19) 1/2 1/2 ¯ ¯ ¯ (d−2)f (R)−2Rf (R) 2R Det − Det + L L 2 0 2(d−1)f (R) As observed in Ref. [24] and confirmed in Ref. [48], this result does not depend on the gauge parameters α and β, and the result is close to on-shell once the partial gauge choice h = 0 is made. This is the advantage of the exponential parametrization [23,24,30,44–48]. The above determinant is what governs the quantum theory at the one-loop level, in particular effective action. Given the above result, we can evaluate the effective action, which is related to the partition function −(g ¯ ) by Z (g ¯ ) = e . Neglecting field-independent terms, we find ¯ ¯ 1 2R 1 2R (g ¯ ) = log Det − − log Det − L2 L1 2 d 2 d ¯ ¯ 1 R (d − 2)f (R) + log Det − + . (3.20) 2 d − 1 2(d − 1)f (R) 7/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta The divergent part of the effective action can be computed by standard heat kernel methods [50]. On an Einstein background in four dimensions, the logarithmically divergent part is 2  2 ¯ ¯ ¯ 1  71 433 f (R) Rf (R) 4 2 2 ¯ ¯ (g ¯ ) = d x g ¯ log − R + R − + , log μνρσ 2 2 ¯ ¯ 12(4π) μ 10 120 3f (R) f (R) (3.21) where  stands for a cutoff and we have introduced a reference mass scale μ. On shell, we have ¯ ¯ 2f (R) ¯ ¯ f (R) = , and R = for maximally symmetric space; this reduces to μνρσ ¯ 6 2 2 ¯ ¯ 1  97 8f (R) 4f (R) 4 2 (g ¯ ) = d x g ¯ log R − + , (3.22) log 2 2 2  2 ¯ ¯ ¯ 24(4π) μ 20 3R f (R) f (R) in agreement with Ref. [33]. The flow equation for the f (R) theory on the 4-sphere was derived in Refs. [23,24]. The result using the spectral sum is 32π ( − 2r + 4) ˙ ˙ d d 2r − 2 −  d  − 2r + d  d 2 3 4 1 5 = − + + , 6 + (6α + 1)r  (3 + (3β − 1)r) +  4 + (4γ − 1)r (3.23) where 5[6 + (6α − 1)r][12 + (12α − 1)r] d = , 5[6 + (6α − 1)r][3 + (3α − 2)r] d = , [2 + (2β + 3)r][3 + (3β − 1)r][6 + (6β − 5)r] d = , [2 + (2β − 1)r][12 + (12β + 11)r] d = , 2 2 −72 − 18r(1 + 8γ) + r 19 − 18γ − 72γ d = , (3.24) −2 −4 ¯ ¯ and we have used the dimensionless quantities r = Rk and (r) = k f (R), and α, β, and γ are the parameters of endomorphism, not to be confused with the gauge parameters. 3.2. Scalar–tensor theory in the Jordan frame Next we discuss the one-loop determinant in the scalar–tensor theory in the Jordan frame. We find from Eq. (2.9) that the quadratic terms in the fluctuations in the scalar field are √ Z 1 2 2 ¯ ˜ ˜ ¯ gV (φ) g ¯ χ (φ)(φ + hφ) + V (φ)h . (3.25) 2 8 Together with the contribution from the rest of the terms, we find, using theYork decomposition (3.5), ¯ ¯ 1 2R (d − 2)(d − 2) R d − 2 (2) TT TT μν 2 ¯ ¯ I =−Z φ − h − h + s − s + Rh N L L 2 0 μν 4 d 4d d − 1 8d ¯ ¯ R d − 1 R Z 1 2 2 ˜ ˜ ¯ ˜ ˜ ¯ − Z hφ + φ − s + χ (φ)(φ + hφ) + V (φ)h . (3.26) N L0 2 d d − 1 2 8 8/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta We employ the same gauge fixing as in the preceding subsection. Here we also use the partial gauge fixing h = 0. Then we again find that the quadratic part of the action is written entirely in terms of TT the physical degrees of freedom h , s, and φ and the gauge fixing entirely in terms of the gauge degrees of freedom ξ and u. The one-loop determinant from the scalar sector (s, φ) is ⎛     ⎞ −1/2 ¯ ¯ (d−1)(d−2) R d−1 R ¯ ¯ Z φ − − Z φ − 2 N L0 N L0 d−1 d d−1 2d ⎝   ⎠ Det d−1 R ¯ ¯ − Z φ − Z χ (φ) N L N d d−1 −1/2 ¯ ¯ (d − 1) R R d − 2 ¯ ¯ = Det Z − − + φχ (φ) . (3.27) L0 L0 d d − 1 d − 1 d − 1 It follows from Eqs. (2.5) and (2.10) that f (χ ) φχ (φ) = . (3.28) f (χ ) So, writing out the resulting whole one-loop determinant, we get 1/2 Det − R L1 . (3.29) 1/2 1/2 ¯ ¯ (d−2)f (χ) ¯ 2R R Det − Det + − L2 L0 d 2(d−1)f (χ) ¯ d−1 If we use χ ¯ = R, this precisely agrees with Eq. (3.19), the result for the f (R) theory. Thus we conclude that this formulation by scalar–tensor theory is equivalent to the original f (R) theory at the quantum (at least) one-loop level on shell. 3.3. Scalar–tensor theory in the Einstein frame Next we discuss the Hessian and one-loop determinant in the scalar–tensor theory in the Einstein frame. If we take one more step in the discussion in Sect. 2, we find that the second derivative of the potential is given by d−2 −f (χ ) (d − 2)f − 2χf d U (ϕ) = Z − √ U (ϕ), (3.30) Z (d − 1)f (d − 2)(d − 1) so the Hessian for the fluctuation ϕ ˜ is (d − 2)f − 2χf d (2) I = Z + − √ U (ϕ) ¯ . (3.31) N L ϕϕ 0 2(d − 1)f 2 (d − 2)(d − 1) Note that if we exploit the equations of motion for the background U (ϕ) ¯ = 0 and χ ¯ = R,wefind that this is proportional to that of the field s in Eq. (3.7): (2) (2) S ∝ S . (3.32) ss ϕϕ on-shell There are also some mixing terms between the graviton h and the scalar ϕ, but these drop out in the gauge h = 0 in the exponential parametrization. 9/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta The rest of the theory is the usual Einstein theory. The Hessian for this theory can be obtained from the result in the preceding subsection by setting f (R) =−Z R. We thus find 1 2 (d − 1)(d − 2) R d − 2 (2) TT TT μν 2 ¯ ¯ I =−Z − h − R h + s − s + Rh . N L2 L0 μν 4 d 4d d − 1 8d (3.33) The gauge fixings can be taken as in the preceding subsections. With the partial gauge fixing h = 0, we again find the separation of the physical degrees of freedom and the gauge fixing terms. It is now straightforward to derive the one-loop determinant: 1/2 Det − R L1 . (3.34) 1/2 1/2 (d−2)f −2χf 2R d Det − Det Z + − U (ϕ) L2 N L0 d 2(d−1)f 2 (d−2)(d−1) As noted above, on shell, this is equivalent to the result of f (R) gravity. However, before concluding that the theory is equivalent at the quantum level, we have to take the Jacobian from the transformation (2.13) into account. The path integral would produce divergences in the form δ(0) times the volume from this change of variable. However, such terms would affect the power-law divergence coefficients, which are subject to a regularization scheme. The coefficients of logarithmic divergence are not affected and are universal. It is true that we have to take into account the difference in the definition of the scales in different frames, but that can be easily incorporated since the forms of the effective action are the same. One may worry that the conformal transformation introduces a change in the path integral measure and hence leads to a trace anomaly. However, this will be taken into account in the form of a logarithmic ultraviolet (UV) cutoff dependence when the determinant is regularized with a UV cutoff, which is dependent on the conformal transformation in terms of the scalar field [42]. We will discuss this problem of the difference in scale in the next section. All the results for one-loop divergence, effective action, and the flow equation can be derived from these determinants. The scale dependence in different frames would introduce a formal difference in the resulting functional renormalization group equations, but should not affect the physical results. We thus conclude that the theory is also equivalent to the f (R) theory at quantum level on shell. 4. Conclusions and discussions In this paper, we first summarized the relation of the f (R) theory and the reformulations of the theory in the form of a scalar field coupled to the Einstein theory in the Jordan and Einstein frames, and then calculated determinants, which correspond to the effective actions, after path integral over the fluctuation fields in the background field formalism. It turns out that all three formulations give the same determinant on shell. If we evaluate the determinant with a suitable cutoff, this gives the divergences in the theory. After renormalization, this produces an effective action. We have also given the one-loop divergent term in the effective action. One could also try to derive the functional renormalization group equation by introducing a suitable cutoff function. The fact that the determinants, from which these are all derived, are the same is strong evidence that these theories are equivalent at the quantum level, at least at one loop. One possible caveat is that the transformation into the Einstein frame involves conformal trans- formation. This transformation would produce δ(0) type divergence, which could be removed by a local counterterm. However, this also produces a difference in the scales in different frames. In a 10/13 Downloaded from https://academic.oup.com/ptep/article-abstract/2018/3/033B02/4922028 by Ed 'DeepDyve' Gillespie user on 16 March 2018 PTEP 2018, 033B02 N. Ohta given frame, a short-distance cutoff  may be defined by 2 μ ν = g x x . (4.1) μν When the metric is transformed as Eq. (2.11), the cutoff lengths in the Jordan and Einstein frames are related by 2 J μ ν −2/(d−2) E μ ν −2/(d−2) 2 ¯ ¯ = g x x = (φ) g x x = (φ)  . (4.2) J μν μν E The UV cutoff is then related by 2 2/(d−2) 2 = (φ)  . (4.3) J E This would result in slightly different-looking functional renormalization group equations in the two frames due to the different cutoffs. If this difference is dealt with suitably, the physical result should not depend on the difference because the effective action is the same. For related discussions, see Ref. [42]. To summarize, we have found strong evidence that the f (R) theory and the scalar–tensor theories are equivalent on shell in arbitrary dimensions. As a byproduct, we have also found evidence that the theories in the Jordan and Einstein frames are equivalent. Note that our discussions are based on the Einstein space with the curvature (3.2). It would be interesting to try to extend our result to more general spacetime. 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