3 Perturbative c-theorem in d-dimensions 1 0 2 n a Kazuya Yonekura J 5 1 School of Natural Sciences, Institute for Advanced Study, ] Princeton, NJ 08540, USA h t - p e h Abstract [ 2 We study perturbative behavior of free energies on a d-dimensional sphere Sd v for theories with marginal interactions. The free energies are interpreted as the 8 2 “dilaton effective action” with the dilaton having a nontrivial background vacuum 0 expectation value. We compute the dependence of the free energies on the radius of 3 the sphere by using dimensional regularization. It is shown that the first (second) . 2 derivative of the freeenergies in odd (even) dimensions with respect to the radius of 1 2 thesphereareproportionaltothesquareofthebetafunctionsofcouplingconstants. 1 The result is consistent with the c, F and a-theorems in two, three, four and six : v dimensions. Theresultisalsousedtoruleoutalargeclassofscaleinvarianttheories i which are not conformally invariant. X r a 1 Introduction In two dimensional quantum field theory, two elegant theorems are known. Zamolod- chikov showed [1] that there exists a function c(r) of a length scale r which monotonically decreases as r is increased, and becomes constant only on conformal fixed points. Roughly speaking, this result indicates that a number of “degrees of freedom” monotonically de- creases along renormalization group (RG) flows. This is the famous Zamolodchikov’s c-theorem. Then, Polchinski proved [2] that all scale invariant theories (with discrete spectrum of scaling dimensions) are also conformally invariant. The result of Ref. [1] played a crucial role in the proof of Ref. [2]. There have also been significant developments in the study of monotonically decreas- ing quantities in higher dimensions. In even dimensional CFT, the trace of the energy- momentum tensor has anomaly when the CFT is coupled to external gravitational back- ground. It is given as [3] Tµµ = (−1)d2+1aEd +···, (1) where Ed ∝ ǫµ1µ2···µd−1µdǫν1ν2···νd−1νdRµ1µ2ν1ν2···Rµd−1µdνd−1νd is the Euler density, and the dots indicate terms which vanish in conformally flat background. In two dimensional CFTs, thecoefficient of theEuler density E in thetrace anomaly(written as ain Eq. (1)) d coincides with the Zamolodchikov’s c function. In general even dimensional field theories, it was conjectured [4] that a decreases along RG flows in theories which interpolate UV and IR CFTs, that is, a < a . IR UV The quantity a may be extracted in the following way. Let us put a CFT on a d- dimensional sphere Sd with radius r and consider the partition function on the sphere, Z = [Dϕ]e S, (2) − Z where ϕ denotes dynamical fields of the theory, S is the action on the sphere, and [Dϕ] is the path integral. Using the fact that the change of the radius r as r eσrRfor a → constant σ is equivalent to the Weyl rescaling of the metric as g e2σg , the free µν µν → energy F = logZ satisfies − dF = dd√gTµ ( 1)d2a, (3) dlogr − µ ∝ − (cid:28)Z (cid:29) 2 where we have used the fact that the metric of the sphere is conformally flat in projective coordinates, ds2 = (2r2/(x2+r2))2dx2, andhence the terms denoted by thedots inEq. (1) do not contribute. Therefore, the above conjecture may be interpreted as the conjecture d that the function ( 1)2dF/dlogr decreases as r is increased. − In odd dimensions, there is a similar conjecture about the free energy F [5, 6]. In d+1 this case, it is ( 1) 2 F that is conjectured to decrease. Therefore, in both even and odd − dimensions, the free energy on the sphere, F = logZ, plays an important role in the − study of monotonically decreasing quantities. Recently, aproofthatasatisfies a < a wasgiveninfourdimensions [7]andfurther IR UV discussed in Refs. [8, 9]. Also, a monotonically decreasing quantity was constructed in three dimensions [10] which coincides with F in CFT [11]. Completely different methods were used in the proofs in two [1], three [10] and four [7] dimensions. There is still no proof in general space-time dimensions, although holography suggests the existence of a monotonically decreasing function in arbitrary dimensions [12, 13, 14]. See Refs. [15, 16, 17] for recent works in six and higher dimensions. Progress has also been made regarding the equivalence of scale and conformal invari- anceinfourdimensions. Aproofoftheequivalencewasgiveninperturbationtheory[9,18] (see also Refs. [19, 2, 20, 21, 22]).1 Ref. [9] also gave a non-perturbative argument in favor of the equivalence. In that proof, the existence of a monotonically decreasing quantity a (or more precisely the dilaton forward scattering amplitude) is essential. This is similar to the proof of the equivalence in two dimensions [1, 2]. However, much less is known in other dimensions.2 In particular, there are many perturbative fieldtheoriesinthreedimensions, andthereisapossibility thatsome ofthem could be scale invariant without conformal invariance by the same mechanism discussed in Refs. [23, 24, 25, 26]. As discussed above, one of the ways to generalize the two dimensional theorems to arbitrary dimensions may be to use the free energy on the sphere. The flow of the free 1 Although the existence ofexplicit counterexamples arediscussed [23, 24, 25, 26], they are arguedto be conformally invariant [27, 28, 18] based on the results of Refs. [29, 30]. 2 Thereexistfreefieldtheorycounterexamplesind>4[31,32]. Butthereisnolocalcurrentoperator forscalingsymmetryandonlythechargeiswell-definedinthosetheories. Therestillremainsapossibility that every scale invariant theory with a scaling current is conformally invariant. 3 Dimensions Lagrangians d = 2 G (φ)(∂ φ)2 +G (φ)ψ∂/ψ +H(φ)ψ4 b µ f d = 3 (A∂A+ 2A3)+(D φ)2 +ψD/ψ +φ6 +φ2ψ2 3 µ d = 4 (F )2 +(D φ)2 +ψD/ψ +φ4 +φψ2 µν µ d = 6 (∂ φ)2 +φ3 µ Table 1: Schematic forms of the Lagrangians of perturbative theories with marginal in- teractions. The fields φ are bosons, ψ are fermions, A are gauge bosons, F are gauge field strengths, and possible indices specifying these fields are suppressed. G (φ), G (φ) and b f H(φ) are arbitrary functions of scalar fields φ. All the coupling constants are dimension- less in these Lagrangians. energy was studied when a CFT is deformed by adding slightly marginal operators to O the Lagrangian [4, 6]. The operators were assumed to have scaling dimensions d y with − y 1. ≪ Inthispaper, westudythefreeenergyforgeneralweakly interactingfieldtheorieswith d+1 marginal interactions. A list of such theories is given in Table. 1. We show that ( 1) 2 F − d (in odd dimensions) or ( 1)2dF/dlogr (in even dimensions) decreases monotonically in − these theories. Furthermore, following Ref. [9], we arguethat scale invarianceisequivalent to conformal invariance in these theories. Therest ofthepaperisorganizedasfollows. Insection2wegivearelationbetween the free energy on the sphere and the “dilaton effective action” which was used in the proof of the a-theorem in four dimensions [7, 8, 9]. It enables us to compute perturbative flows of the free energy by using the method of Refs. [8, 9]. We obtain the dilaton effective action in dimensional regularization. In section 3, we compute the flow of the free energy using the dilaton effective action. We check our result in two, three, four and six dimensions. Using the result, we argue the equivalence of scale and conformal invariance. Section 4 is devoted to conclusions. 2 Dilaton effective action We define the free energy of a theory on a d-dimensional sphere as a dilaton effective action in the following way. We first consider the partition function as a functional of a background metric gˆ . (The hat is used on the metric following the notation of µν 4 Refs. [7, 8, 9].) It is given as Z = [Dϕ]exp( S[ϕ,gˆ ] S [gˆ ]) µν c.t. µν − − Z = Z exp( S [gˆ ] S [gˆ ]), (4) 0 eff,0 µν c.t. µν − − where ϕ denotes dynamical fields of the theory, and S[ϕ,gˆ ] is the action of the fields µν ϕ coupled to the metric gˆ . The factor Z is the contribution to the partition function µν 0 which does not depend on the background metric, and S is the (bare) effective action eff,0 of the metric obtained as a result of the path integral. The counterterm S is taken so c.t. that the functional S [gˆ ] = S [gˆ ]+S [gˆ ], (5) eff µν eff,0 µν c.t. µν becomes finite. We will impose further condition on the counterterms S later. c.t. We introduce a dilaton field τ anda new metric g asgˆ = e 2τg . Then the dilaton µν µν − µν effective action is defined as S[τ,g ] = S [gˆ = e 2τg ]. (6) µν eff µν − µν This definition of the dilaton effective action is emphasized in Ref. [9]. When g = η , it µν µν givesthedilatoneffectiveactioninflatspace, andthisdefinitionmakescleartheinvariance ofthedilationeffectiveactionunderconformaltransformations. Thisisbecauseconformal transformations are just the subgroup of the diffeomorphism of the original metric gˆ µν which preserves the form dsˆ2 = e 2τdx2. − The metric of the sphere with radius r can be written using the projective coordinates as dsˆ2 = gˆ dxµdxν = [2r2/(x2 +r2)]2dx2. However, we may also interpret this as a flat µν metric g = η with a nontrivial background for the dilaton, e τ = 2r2/(x2+r2). Then, µν µν − the free energy of the theory on the sphere, F = logZ, is given as − 2r2 F(r) = logZ +S e τ = ,η . (7) − 0 " − x2 +r2 µν# By this interpretation, we can use the results of Refs. [8, 9] for the dilaton effective action to compute the free energy on the sphere. Itisclearthatthedependence ofF(r)ontheradiusofthespherer shouldbecontained in the second term of Eq. (7). In this paper we attempt to calculate only the derivatives 5 of F(r) with respect to r. Then we may neglect the term logZ , and focus on the dilaton 0 effective action. The above definition still has an ambiguity regarding the choice of the counterterms in S . Although the divergent part of S is determined uniquely so that it makes c.t. c.t. the metric effective action S finite, the finite part of S is not fixed. We impose eff c.t. the following requirement on the finite part. In this paper we only consider massless theories which do not contain dimensionful parameters (see Table. 1). Furthermore, we always use dimensional regularization as a regularization method. Then, by using mass- independent renormalization scheme (such as minimal subtraction), counterterms which contain dimensionful coefficients are not necessary (see e.g. Ref. [33]). That is, we can set all the counterterms to zero aside from counterterms with dimensionless coefficients which are schematically given as Sc.t.[gˆµν] ddx gˆ(Rµνρσ(gˆ))d2, (8) ∼ Z q where R is the Riemann tensor and indices are contracted in arbitrary ways. There- µνρσ fore, we only introduce counterterms of the form (8). This is our criterion for choosing the counterterms. In the case of odd dimensions, terms like Eq. (8) do not exist and hence we need no counterterms at all. Therefore F(r) is uniquely determined by our criterion. In even dimensions, finite counterterms of the form (8) are allowed,3 and hence the ambiguity in defining F(r) remains. However, one can see that the contributions coming from these finite counterterms disappear if we take the derivative of the free energy, dF/dr.4 As discussed in the introduction, the important quantity in even dimensions is dF/dr rather than F itself, and hence the remaining ambiguity in choosing the counterterms does not matter. The above requirement on the counterterms is a little technical. More physical re- quirement may be that the free energy F (in odd dimensions) or its derivative dF/dr (in 3The finite counterterms are in fact necessary in order for the effective action S (gˆ ) to be RG eff µν invariant. Even if they are set to zero at some RG scale, they are generated along RG flows. 4Strictly speaking, these contributions are not precisely zero in dimensional regularization. They are suppressedby ǫ, where the space-time dimensions is givenby d=(integer) 2ǫ. Then, the contributions − of the finite part of the counterterms become zero when we take ǫ 0, but the contributions from → divergent part of the counterters are important. 6 even dimensions) becomes constant on UV/IR fixed points. This physical requirement will be satisfied by the above choice of the counterterms. Now let us study the dilaton effective action S[τ] in flat (Euclidean) space-time. The most important part of S[τ] in perturbation theory has been given in Refs. [8, 9]. We use dimensional regularization where we work in d = d 2ǫ dimensions with d an integer. 0 0 − We expand the action as S[ϕ,gˆ = e 2τη] = S[ϕ,η ]+ ddxτTµ +O(τ2), (9) µν − µν µ Z where Tµν is the energy-momentum tensor. The linear term in τ is proportional to the trace anomaly. We assume that interaction terms are present in the Lagrangian as λi , where λi are bare couplings and are bare operators. For example, in a four- 0Oi 0 Oi dimensional scalar φ4 theory, we may define = 1φ4. Then, if the energy-momentum O 4! tensor is improved appropriately, the trace anomaly may be given as Tµ = i[ ], (10) µ − B Oi i X where [ ] are the renormalized operators corresponding to the bare operator , and i i i O O B are the beta functions of the renormalized coupling constants λi. In cases where there are many flavors of matter fields, there are ambiguities in the definition of usual beta functions βi, while the beta functions i appearing in Eq. (10) is unambiguous [29, 30]. B Following the notation of Refs. [29, 30], we denote this unambiguous beta functions as i B rather than βi. The improvement of the energy-momentum tensor is related to the term Rφ2 in the Lagrangian,whereRistheRicciscalarandφarescalarfieldsofthetheory. Foramoment, let us assume that this term is chosen so that Eq. (10) holds. We will revisit this point at the end of this section. The higher orderterms ofτ inEq. (9)areaccompanied byadditional powers ofǫorthe coupling constants [9]. The reason is the following. In theories with only dimensionless parameters which are listed in Table. 1, the dilaton appears in the combination ǫτ in the bare Lagrangian after performing appropriate Weyl rescaling of matter fields. For example, in the case of a four dimensional φ4 theory, the bare Lagrangian of the theory 7 is given as 5 1 d 2 λ λ gˆ gˆµν∂ φˆ∂ φˆ+ − R(gˆ)φˆ2 + 0φˆ4 = ηµν∂ φ∂ φ+e 2ǫτ 0φ4 (11) µ ν µ ν − 2 8(d 1) 4! ! 4! ! q − where φˆ is the bare field, gˆ = e 2τη and φ = e (d 2)τ/2φˆ. Loop calculations give µν − µν − − divergences which may cancel the factor ǫ in ǫτ. However, whenever ǫ is cancelled, there is always an additional loop suppression factor L (e.g., L = λ/16π2 in the φ4 theory). Therefore, τ appears only in the combination ǫτ or Lτ. Then, neglecting the higher order terms, the leading order term in the dilaton effective action is given by 1 S [gˆ = e 2τη ] = ddxddyτ(x)τ(y) i j [ (x)][ (y)] + (12) eff,0 µν − µν i i −2 B B h O O i ··· Z i,j X where dots denote higher order terms in ǫ or loop factors. At the leading order of perturbation theory, correlation functions of [ ] are given as i O µ2(d di)c δ − i ij [ (x)][ (y)] = , (13) i i h O O i x y 2di | − | where c are dimensionless constants (e.g., c = 1(Γ(d/2 1)/4πd/2)4 for = 1φ4 ), µ i 4! − O 4! is the unit of mass of dimensional regularization (or in other words the RG scale), and d is the classical scaling dimension of . Although we are considering only marginal i i O interactions, d differs from d by order ǫ (e.g., d = 2(d 2) = 4 4ǫ for = 1φ4). The i i − − O 4! operators [ ] are assumed to be normalized so that [ (x)][ (y)] is proportional to δ i i i ij O h O O i at the leading order. The constants c are ensured to be positive by reflection positivity. i Then the dilaton effective action becomes 1 µ2(d di)c 2 S [gˆ = e2τη ] = ddxddyτ(x)τ(y) − iBi + eff,0 µν µν −2 x y 2di ··· Z Xi | − | = ddk τ˜(k) 2 πd22d−2diΓ(d/2−di)c 2µ2(d di)k2di d + − (2π)d| | 2Γ(d ) iBi − − ··· Z i i X (14) 5Here we pretend as if the term Rφ2 is chosen as the conformal coupling of a free scalar, (d−2) Rφ2. 8(d−1) This is not correct at higher orders of perturbation theory [34], but the corrections occur at sufficiently higher orders so that the following discussion is not violated. 8 where we have Fourier-transformed the dilaton as τ˜(k) = ddxe ikxτ(x). − In odd dimensions, the above dilaton effective action isRfinite in the limit ǫ 0. This → is consistent with the fact that we need no counterterms in odd dimensions as discussed above. In even dimensions, there is a divergence coming from the factor Γ(d/2 d ) and i − we have to renormalize it. The counterterm should be local and is given as S [gˆ = e 2τη ] = ddk τ˜(k) 2 a + πd22d−2diΓ(d/2−di)c 2 µd d0kd0 + c.t. µν − µν (2π)d| |  0 2Γ(d ) iBi − ··· Z i i X = ddxτ(x)(−∂2)d20τ(x)a0 + π2d2d−22diΓΓ((dd/)2−di)ciBi2µd−d0 +··· Z i i X   (15) where a is a constant which is finite in the limit ǫ 0. This counterterm makes the 0 → dilaton effective action finite. Although it is not immediately evident whether the counterterm (15) can be obtained from counterterms for the metric S [gˆ ] by replacing the metric as gˆ e 2τη , it c.t. µν µν − µν → is known to be possible [15, 16]. It may be instructive to see it explicitly in the simplest case where the space-time dimension is d = 2 2ǫ. There is only one candidate for the − counterterm which is given by S [gˆ ] ddx gˆR(gˆ). (16) c.t. µν ∝ Z q Then, the dilaton counterterm is obtained as S [gˆ = e 2τg ] ddx√ge2ǫτ R(g) 2ǫ(1 2ǫ)( τ)2 c.t. µν − µν ∝ − − ∇ Z h i = ddx√g R(g)+2ǫ τR(g) ( τ)2 +O(ǫ2) . (17) − ∇ Z h (cid:16) (cid:17) i Thus, by taking g η , the counterterm of the form 1τ∂2τ (a single pole term in ǫ) µν → µν ǫ is obtained from 1√gˆR(gˆ) (a double pole term in ǫ). ǫ2 One should also notice that the finite term in the dilaton counterterm S [gˆ = c.t. µν e 2τη ] actually comes from the divergent term in the metric counterterm S [gˆ ]. In − µν c.t. µν fact, this is how the Wess-Zumino action for the dilaton [35, 7] arises in dimensional regularization. The integral of the Euler density E is a topological quantity in d - d0 0 dimensions. In the case of d = d 2ǫ dimensional space-time, this topological property 0 − 9 is broken by ǫ, and the change of the metric gˆ = e 2τg gives µν − µν ddx gˆE (gˆ) = ddx√gE (g)+2ǫS +O(ǫ2), (18) d0 d0 WZ Z q Z where S = ddx√g(τE (g)+ ) (19) WZ d0 ··· Z is the Wess-Zumino action for the dilaton. See Eq. (17) for the case of d = 2. In CFTs, 0 we need a counterterm of the form ddx(a/ǫ)E to make the energy-momentum tensor d0 finite [36]. This counterterm leads toRthe trace anomaly Tµ aE . One can see that the µ ∼ d0 presence of this counterterm gives the finite Wess-Zumino action aS for the dilaton by WZ using Eq. (18). Let us return to the computation of the dilaton effective action. We have neglected higher order terms in ǫτ and loop factors. We continue to neglect the higher order corrections of the loop factors. However, for our purposes it is important to recover the higher order terms of ǫτ. Actually, there are divergences when we compute the free energy by substituting e τ = 2r2/(x2 + r2). It turns out that terms containing extra − powers of τ = log((x2 + r2)/2r2), τk (k = 0,1,2, ), give additional divergences 1 . ··· ǫk Therefore it is necessary to retain higher order terms in ǫτ. To recover the dependence on ǫτ, we use the conformal invariance of the dilaton effective action. As discussed above, the dilaton effective action should be conformally invariant since the conformal transformations are just the subgroup of the diffeomorphism of the original effective action for the metric. We can make Eqs. (14) and (15) conformally invariant by replacing them as 1 e (d di)τ(x) 1 e (d di)τ(y) ddxτ(x) τ(y) ddxddy − − − − , (20) Z |x−y|2di → Z d−di ! |x−y|2di d−di ! and ddxτ(x)( ∂2)d20τ(y) ddx e−(d−2d0)τ(x) ( ∂2)d20 e−(d−2d0)τ(x) . (21) − → (d d )/2 − (d d )/2 Z Z − 0 − 0     Byexpandinginτ, onecancheckthatthelineartermsinτ areabsentduetotheproperties of dimensional regularization. The quadratic terms in τ just reproduce the original ones. 10