Generalized F-theorem and the ϵ expansion

Some known constraints on Renormalization Group flow take the form of inequalities: in even dimensions they refer to the coefficient a of the Weyl anomaly, while in odd dimensions to the sphere free energy F. In recent work [1] it was suggested that the a- and F-theorems may be viewed as special cases of a Generalized F -Theorem valid in continuous dimension. This conjecture states that, for any RG flow from one conformal fixed point to another, F˜UV>F˜IR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{F}}_{\mathrm{UV}}>{\tilde{F}}_{\mathrm{IR}} $$\end{document}, where F˜=sinπd/2logZSd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{F}= \sin \left(\pi d/2\right) \log {Z}_{S^d} $$\end{document}. Here we provide additional evidence in favor of the Generalized F-Theorem. We show that it holds in conformal perturbation theory, i.e. for RG flows produced by weakly relevant operators. We also study a specific example of the Wilson-Fisher O(N) model and define this CFT on the sphere S4−ϵ, paying careful attention to the beta functions for the coefficients of curvature terms. This allows us to develop the ϵ expansion of F˜\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{F} $$\end{document} up to order ϵ5. Padé extrapolation of this series to d = 3 gives results that are around 2–3% below the free field values for small N. We also study RG flows which include an anisotropic perturbation breaking the O(N) symmetry; we again find that the results are consistent with F˜UV>F˜IR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{F}}_{\mathrm{UV}}>{\tilde{F}}_{\mathrm{IR}} $$\end{document}.


Introduction and summary
A well-known set of constraints on Renormalization Group in d-dimensional relativistic Quantum Field Theory (QFT) takes the form of inequalities: if an RG trajectory leads from a short-distance unitary Conformal Field Theory (CFT) to a long-distance one, then a certain positive quantity defined on the space of CFTs decreases. In even dimensions d, using the modern terminology such theorems may be called the a-theorems. The Weyl anomaly equation takes the form where E d is the Euler density term, which is present in all even d ≥ 2, and c i are the coefficients of other Weyl invariant curvature terms. The universal Weyl anomaly coefficient a may be extracted from the free energy on a sphere of radius R: F = − log Z S d = (−1) d/2 a log R. In even d, the RG inequalities take the form [2][3][4][5][6][7] a UV > a IR . (1.2) In d = 2 this is well-known as the c-theorem [2], because there is only one Weyl anomaly coefficient, and it is standard to define c = 3a. In d = 4 a non-perturbative proof of (1.2) was found in [4,5], long after the early work of [3,8]. Very recently [6,7], there was major progress towards establishing (1.2) for supersymmetric flows in d = 6, building on the earlier work including [9,10].

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In odd dimensions the situation is quite different because there are no Weyl anomalies. This actually has some advantages: for a CFT, the sphere free energy F = − log Z S d is a finite, radius independent quantity, which has no ambiguities because there are no Weyl invariant terms constructed purely out of the curvature tensor. In d = 3 it was conjectured [11] that the RG inequality takes the form F UV > F IR . This F -theorem can be equivalently formulated in terms of the entanglement entropy across a circle [12,13]. A proof of the three-dimensional F -theorem has been found using properties of the entanglement entropy in relativistic theories [14] (see also [15]). For other odd dimensions, it was conjectured [16] that the RG inequality takes the formF UV >F IR , whereF = (−1) (d−1)/2 log Z S d . 1 Supporting evidence for this conjecture included a calculation of the change inF for the RG flow produced by a weakly relevant operator of dimension d− [16]. Such calculations, as well as their analogues in even dimensions which apply to the change in a [2,3,5], can be carried out perturbatively in .
The similarity between the RG inequalities in even and odd dimensions (both of them can be phrased in terms of the free energy on S d or, equivalently, in terms of the entanglement entropy across S d−2 [12]) has led to the idea that they are special cases of the RG inequality valid in continuous dimension [1]: 2 F UV >F IR ,F = sin(πd/2) log Z S d = − sin(πd/2)F .
(1. 3) In even dimensions, the factor sin(πd/2) cancels the pole present in F , and we haveF = πa/2; therefore (1.3) reduces to the a-theorem. In odd dimensions, this definition reduces to that proposed in [16]. Thus, the "Generalized F -Theorem" (1.3) smoothly interpolates between the a-theorems in even d and the F -theorems in odd d.
In [1] several pieces of evidence were provided in favor of the Generalized F -Theorem. 3 In free conformal scalar and fermion theories,F is positive for all d. For example, for a conformally coupled scalar it is which is a smooth, monotonically decreasing function of d. Therefore, (1.3) is valid for the RG flows produced by the scalar or fermion mass terms -such flows lead to trivial theories whereF = 0. Other tractable RG flows include those produced by double-trace operators in large N theories. As is well-known [19,20], turning on a relevant perturbation O 2 ∆ , which is the square of of a primary scalar operator of dimension ∆ < d/2, makes a large N CFT flow to another CFT where the corresponding operator has dimension d − ∆ + O(1/N ). This produces the following change inF [1,21]: (1.5)

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This is negative in the entire range (d − 2)/2 < ∆ < d/2 where the UV and IR CFTs are unitary, establishing consistency with the Generalized F -Theorem. However, when ∆ is sufficiently far below the unitarity bound (d − 2)/2, then the inequality (1.3) is violated (for d = 3 this was discussed in [22]). This provides a simple explicit example of how (1.3) may be violated in non-unitary theories. 4 Explicit applications of the a-theorem in d > 2 have been mostly to supersymmetric RG flows (see, for instance, [6,7,[24][25][26][27][28]), because few interacting non-supersymmetric CFTs are known in d = 4 and none in d = 6. On the other hand, in 2n − dimensions there is a multitude of non-supersymmetric RG flows connecting perturbative fixed points [29][30][31][32][33][34][35]. This opens the possibility of many tests of the Generalized F -Theorem in 2n− dimensions, and some of them were carried out in [1]. In this paper we extend these calculations, focusing mostly on the Wilson-Fisher fixed point in 4 − dimensions [29], which is the O(N ) symmetric theory of N real scalar fields φ i , i = 1, . . . , N , with interaction λ 4 (φ i φ i ) 2 . For large N , this is a double-trace operator with ∆ = d − 2. Furthermore, the fact that the coupling constant at the IR fixed point is of order allows one to develop the expansions for the critical exponents [29]; this works well for all values of N including N = 1, i.e. for the Ising model [36]. Fortunately,F is also amenable to expansion using renormalization of the O(N ) model on the sphere S 4− . In [1] the contributions of interactions toF were determined up to O( 4 ). The leading interaction correction, which is of order 3 , did not require renormalization and was quite straightforward. However, at the next order one needs to renormalize the theory including the effects of the terms quadratic in the curvature. In section 2 we elucidate the definition of the Wilson-Fisher fixed point on S 4− following [8,[37][38][39][40]. We find that it is important to define the IR theory by setting all the beta functions to zero, including the beta functions for the coefficients of the curvature terms. Applying this procedure up to order 5 we find As a byproduct of our analysis of the O(N ) model on S 4− , we extend the previous results [8,[38][39][40] and determine the beta function for the Euler density term to order λ 5 :

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As far as we know, the O(λ 5 ) term has not appeared in the previous literature. We also present a calculation of the sphere free energy and curvature beta functions in the most general quartic scalar field theory in appendix A. A constrained Padé approximation of the series (1.6) for N = 1, using the boundary condition that the 2-d Ising model hasF = π/12, corresponding to c = 1/2, gives that in 3-dimensions F Ising /F s ≈ 0.976. This is consistent with the F -theorem, but the surprise is how close F Ising is to the free field value. 5 For comparison, we note that the scaling dimension of φ i , ∆ φ ≈ 0.5182 is around 3.6% above the free field value. The coefficient of the stress tensor 2-point function, c T , is also known to be close to the free field value for the 3-d Ising model: c 3d Ising T /c s T ≈ 0.9466 [41,42]. In section 4 we also provide a check of the Generalized F -Theorem for weakly relevant perturbations. When an RG flow in d dimensions is sourced by an operator O of dimension d − then there is a nearby IR fixed point, and the change inF is where C 2 and C 3 are the coefficients of the two-and three-point functions of O in the UV CFT, (4.2). In odd d this result agrees with [16], and in even d with the change in a-anomaly computed in [5]. For a unitary CFT, C 2 > 0 and C 3 is real. So we find that the Generalized F -Theorem holds to leading order in conformal perturbation theory for all d. We have also extended the conformal perturbation theory approach to determine the term of order 4 in specific models. For example, for the O(N ) model it reproduces this term in (1.6). In section 4.2 we generalize the conformal perturbation approach to the case of several weakly relevant operators. We obtain a concise formula (4.39) for the leading order change inF . As a specific application, we study the O(N ) model with an extra "cubic anisotropy" operator i φ 4 i which breaks the O(N ) symmetry. The conformal perturbation approach allows us to calculate theF to order 3 at the different fixed points of this theory with two coupling constants. The results are found to be in agreement with the direct approach of section 2 which uses renormalization of the theory on S 4− .

The Wilson-Fisher fixed points in curved space
The renormalization of interacting scalar field theory in d = 4 − on a curved space was studied in [38,39] for the single scalar theory, and in [8,40] for the more general multicomponent theory (see appendix A). In the case of N real scalar fields with O(N ) invariant interaction, the full bare action on a curved manifold is

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where R is the Ricci scalar, H = R d−1 , and Here W 2 is the square of the Weyl tensor, and E is the Gauss-Bonnet term which is the topological Euler density in d = 4 (it is not topological in d = 4 − , but we will still refer to it as the Euler density in what follows). In (2.1), we have separated an arbitrary coupling η 0 to the scalar curvature from the conformal coupling term d−2 4(d−1) R. The action (2.1) contains all possible terms which are marginal in d = 4, which have to be included in order to consistently renormalize the theory in d = 4 − . Note that in d = 4 − the coupling λ 0 has dimension , a 0 , b 0 , c 0 dimension − , and η 0 is dimensionless in any d. We have omitted a mass term for the scalar field since we will be interested in the conformal theory, and the mass can be consistently set to zero in dimensional regularization.
Since the theory is renormalizable, the divergences in the free energy F = − log Z on a general manifold can be cancelled by expressing the bare couplings λ 0 , a 0 , b 0 , c 0 , η 0 in terms of renormalized ones λ, a, b, c, η. The renormalization of the quartic coupling is well known from flat space and reads Requiring dλ 0 d log µ = 0 this yields the beta function [43] The renormalization of the curvature couplings takes the form [38,39] where a, b, c, η are dimensionless renormalized parameters, and L a , L b , L c , L κ , L Λ , L η are functions of λ alone with pure poles in , 6 namely and similarly for c, η, κ, Λ. In the case of the quartic O(N ) theory (2.1), these functions are given explicitly by

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where the values of the known coefficients are [38][39][40] a 01 = N 120(4π) 2 , (2.8) We will be able to determine the coefficient b 51 by a direct perturbative calculation on the sphere in the next section, and the result is given in eq. (3.9). 7 The functions L κ and L Λ will not play an important role in what follows, but they have the structure Finally, in the η renormalization, Z 2 is the Z-factor in the flat space renormalization of the φ 2 operator, which is related to its anomalous dimension by [43] β and the function L η starts at order λ 4 [38,39] The coefficient η 41 may be read off from [39,40], but we will not need its explicit form here. Note that the λ independent terms in L a , L b and L Λ , which are proportional to N , are those required to cancel the divergencies of the free field determinant, F free = N 2 log det −∇ 2 + (d−2) 4(d−1) R + η d−1 R , which may be extracted for an arbitrary manifold using heat kernel methods.
After expressing the bare couplings in terms of renormalized ones, the free energy is finite in the limit → 0 for any values of the renormalized parameters λ, a, b, c, η. The renormalization process implies that the curvature parameters a, b, c, η acquire a scale dependence, just like λ, and have their own beta functions. Requiring that the bare couplings are independent of µ, from (2.5) one obtains the beta functions 7 Note that, when working on the sphere, we cannot distinguish the divergences associated with the Euler term from those associated with the R 2 term. Therefore, in order to determine b51, we have to assume the result for c51 obtained in [39,40].

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where the hatted beta functions depend on λ only (and not on ) and are given bŷ b , . . . , denote the residue of the simple pole as defined in (2.6). As usual, the beta functions depend only on the coefficients of the simple pole, but there is a tower of consistency conditions ('t Hooft identities) relating higher poles to lower ones.
Since the bare couplings are independent of the renormalization scale, the free energy satisfies the identity which implies that when F is expressed in terms of the renormalized couplings, it obeys the Callan-Symanzik equation Note that the dependence of F on the renormalized parameters is simply linear in a, b, c, since the corresponding curvature terms are field independent additive terms, therefore we may split where F φ (λ, η, µ) is finite for any λ, η. So far we have been working on a general curved manifold. Let us now specialize to the case of a round sphere S d of radius R. In this case, we have On the sphere, the radius dependence of the free energy is constrained to be F = F (λ, a, b, c, η, µR), and so we see that Note that ∂F ∂λ and ∂F ∂η are related respectively to the integrated one-point functions of the renormalized φ 4 and Rφ 2 operators. In fact, an unintegrated version of this equation (up to equations of motion and total derivative terms, and including the Weyl square term as well) also holds as an operator equation on a general manifold. 8 In order to have a scale invariant theory in curved space in d = 4 − , we see therefore that we should set to zero not only the usual beta function for λ, but also all curvature coupling beta functions 9 (2.20) Using the expressions for the beta functions (2.4), (2.12), these conditions can be solved in where λ * = 8π 2 N +8 + . . . is the value solving β λ (λ * ) = 0 which defines the Wilson-Fisher fixed point in flat space. More explicitly, using (2.8) and (2.13)

(2.22)
Tuning all couplings to their fixed point values, we then get the sphere free energy at the IR fixed point We emphasize that the resulting F IR (d) is just a function of d = 4 − , the radius and µ dependence drop out as a consequence of setting all beta functions to zero. We will verify this explicitly by our perturbative calculation below. eq. (2.12) is our definition of the Wilson-Fisher fixed point on a curved space. Note that it is important that we work in d = 4 − with a non-zero . If we take the → 0 limit first, then the curvature beta functions become independent of a, b, c and it is not possible to set them all to zero. Indeed, if we take d → 4 first, we get (since λ * = 0 in d = 4):

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where we used S 4 d 4 x √ gE = 64π 2 . This is the expected conformal a-anomaly of N free massless scalars. 10 A related fact is that if we dial to the fixed point (2.12) in d = 4 − , and then take → 0, the free energy diverges: it has a simple pole whose residue is related to the d = 4 anomaly. It follows that the functionF IR (d) = − sin( πd 2 )F IR (d) has a smooth limit as d → 4 proportional to the a-anomaly. Indeed, since the first term in (2.16) is finite as → 0, we have While here we are considering the scalar field theory which has a trivial free fixed point in d = 4, the fact that lim d→4FIR (d) reproduces the 4d a-anomaly coefficient is general and should apply also to cases where there is a non-trivial fixed point in d = 4.

Perturbative calculation of F for the O(N ) model in d = 4 −
In this section we perform a calculation of F at the IR fixed point of the O(N ) model in d = 4 − to order 4 , i.e. we determineF to order 5 (the case of most general quartic coupling is worked out in appendix A). Because of the structure of (2.22), this requires knowing the beta functions for the curvature terms to order λ 5 . Let us first argue that to this order we can neglect the effect of the renormalization of the conformal coupling parameter η. Note that once F is expressed in terms of renormalized parameters, it is finite for any value of η. Since the curvature counterterms L b , L c in eq. (2.5) are independent of η, we can fix them by working at η = 0. This means that we can set the bare parameter . The leading effect of this parameter is a divergence in F of order λ 6 , coming from a two point function diagram (G 2 in figure 1) with a η 0 insertion in one of the propagators (the 0-point and 1-point functions contribute to higher orders, respectively order λ 8 and λ 8 2 ). Hence, it does not affect the calculation of L b and L c to the order λ 5 that we consider below. After the η-independent part of the counterterms is fixed, we can reintroduce the η dependence perturbatively. Recall that at the fixed point we set η = η * = O( 3 ), see eq. (2.22). The leading contribution to F due to the η * is then a finite term of order η * λ 2 * ∼ O( 5 ), coming from the two point function with a "mass" insertion on one propagator. In addition, the value of c * at the fixed point is shifted by the η dependent terms in (2.21) only starting at order 6 . We conclude that, to the order we will be working on, we can ignore renormalization of the conformal coupling parameter η and work with the action Introducing the integrated n-point functions of the free theory 10 On a general manifold the term proportional to the square of the Weyl tensor a01 S 4 d 4 x √ gW 2 will of course also appear on the right hand side. where

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, the free energy up to fifth order is given by where F free is the contribution of the free field determinant [1] The divergence of this determinant fixes b 01 = − N 360(4π) 2 as given in (2.8). One can obtain the -expansion of F free to any desired order.
Using the two-point function where s(x, y) is the chordal distance we find where all diagrams are represented in figure 1. The integrals I 2 (∆) and I 3 (∆) are given in appendix B, also using the method described in this appendix we find for the four point and for the five point functions we get This allows to find the Euler beta function β b to order λ 5 , given in eq. (1.7), and hence determine the fixed point value b * to order 4 by (2.22). Using the fixed point coupling In this equation NF s (4− ) = − sin( πd 2 )F free . Note that the dependence on µR has dropped out, consistently with the conformal invariance. As a partial check of this result, we note that at large N it yields (3.11) which precisely agrees with the result derived in [1] using the Hubbard-Stratonovich transformation; it follows from (1.5) with ∆ = d − 2.
We can also obtain a result for F in d = 4 at a generic value of the renormalized coupling constant λ. Taking the limit → 0 of (3.3) (after expressing all bare couplings in JHEP12 (2015)155 terms of renormalized ones), we find For the integrated trace of the energy-momentum tensor due to interactions we get As a consistency check of this result, we note that it satisfies the Callan-Symanzik equation (2.15) to order λ 4 (without the β η term that we have not included here).

Padé approximation ofF
For any quantity f (d) known in the = 4 − d expansion up to a given order, we can construct a Padé approximant where the coefficients A i , B i are fixed by requiring that the expansion at small agrees with the known terms in f (4 − ). If a quantity is known in the -expansion to order k 0 , one can construct Padé approximants of total order m + n = k 0 . An improved version of the Padé approximant may be obtained if one knows a few terms of the -expansion of the quantity f (d) near two values of dimension: Using the known expansions near the two boundaries, one can fix more coefficients in (3.14) and hence use Padé approximants of higher order, which should provide a better resummation of the quantity f (d). For the Ising model (N = 1) in 2 ≤ d ≤ 4, we know the expansion ofF in d = 4 − to order 5 , eq. (1.6). In addition, we know that for d = 2 the IR fixed point of the quartic scalar field theory should coincide with the 2d Ising CFT, which is known to have central charge c = 1/2. This corresponds to the valueF Ising (d = 2) = π/12. In other words, we have:F Imposing the d = 2 value as boundary condition then allows us to use Padé [m,n] approximants with total order m + n = 6. All approximants Padé [1,5] , Padé [2,4] , Padé [3,3] , Padé [4,2] and Padé [5,1] appear to give very close results in the whole range 2 ≤ d ≤ 4. In d = 3, we getF with Padé [1,5] , 0.97383 with Padé [2,4] , 0.977106 with Padé [3,3] , 0.977248 with Padé [4,2] , 0.975379 with Padé [5,1] .
Notice that all these values are lower than one, consistently with the F -theorem in d = 3.
Given that these approximants are very close to each other, it seems reasonable to average over them. In this way, we obtain the estimate in d = 3: Using instead Padé approximants of total order m + n = 5, which are insensitive to the order 5 inF in d = 4 − , we find a very close estimateF Ising /F s ≈ 0.974, which indicates that the Padé resummation appears to be reliable. 11 In figure 2, we plotF Ising normalized by the free scalar resultF s , comparing the result of the two-sided Padé approximants and the unresummed -expansion.
In the general O(N ) case for N > 2, the IR fixed point of the quartic theory is known to have an equivalent description as the UV fixed point of the non-linear O(N ) sigma model, which is weakly coupled in d = 2 + . For d = 2, one has a free theory of N − 1 massless scalar, which has c = N − 1, orF = (N − 1)π/6. The same d = 2 value applies to the case N = 2, where one has a compact scalar with c = 1 (in this case, though, the behavior of the model in d = 2 + is not known).
For N > 2, the expansion ofF O(N) in d = 2 + is expected to take the form where the coefficients a 1 , a 2 , . . . may in principle be fixed by doing a perturbative calculation on S 2+ . This suggests that it may be more natural to apply the Padé resummation procedure not to the fullF O(N ) = NF s +F int , but to the quantity As a test of our results, we note that in the large N limit, our Padé extrapolation with d = 2 boundary condition yields, in d = 3: The term of order N 0 is within one percent of the exact value −ζ(3)/(8π 2 ) ≈ −0.0152242, which may be obtained from eq. (1.5) by setting ∆ = 1, d = 3. We have verified a similar good agreement with (1.5) over the whole range 2 ≤ d ≤ 4. The term of order 1/N is not known exactly in d = 3, and it would be interesting to compute it directly by large N methods and compare with our estimate (3.23). We note that that the ratioF O(N ) /(NF s ) in d = 3, given in table 1, has a minimum around N = 2 using the constrained two-sided Padé, while it has a minimum close to N = 3 using the unconstrained Padé. 13 We observe that a similar behavior, with a mimimum close to N = 2, was found in the conformal bootstrap approach [44] for the ratio c 12 Note that the expansion of the free scalar answerFs by itself contains log( ) terms in d = 2 + . 13 The latter result is closer to what was observed in [1] using the unresummed expansion.  N = 1 (b.c.) the entry is the average of several Padé approximants, eq. (3.20). For N ≥ 2, the constrained Padé is carried out on the quantity f (d) defined in (3.22), as explained in the text. The "free" Padé approximant is carried out on the fullF O(N ) , eq. (1.6), using Padé [3,2] .
The estimates for F given in table 1 imply in particular that, in d = 3 for N 4 .
The F-theorem then requires that For large N , this inequality was shown to hold in [16]. Using our results obtained from the expansion and Padé extrapolation, we have verified that it in fact holds for all N .

4F in conformal perturbation theory
Let us consider a conformal field theory CFT 0 in dimension d perturbed by a weakly relevant primary scalar operator

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The renormalization of the coupling constant can be deduced from a perturbative calculation of the two-point function in flat space. We have We can use the integral so we find, after plugging ∆ = d − , . (4.5) In the limit → 0, we get We can introduce a dimensionless renormalized coupling g by where µ is the renormalization scale. This corresponds to the renormalization of the local operator (4.8) We can fix z 1 by requiring that the two-point function of the renormalized operator is finite in the limit → 0. From (4.6) we find This gives (4.10) From (4.7) we then find the beta function [3,16] Then there is a IR stable fixed point given by

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The dimension of the renormalized operator O ren (x) at the fixed point is given by Note that the operator has become irrelevant at the IR fixed point. Setting g = g * in (4.9), we find that the two point function of the renormalized operator takes the form, to leading order (4.14) Note that the leading correction to the two-point function normalization is expected to be of order 2 .

The sphere free-energy
We now conformally map the CFT (4.1) to the sphere S d . The two and three point functions of the bare operator are the same as in (4.2) with the replacements |x − y| → s(x, y), etc., where s(x, y) is the chordal distance and the metric of the sphere is ds 2 = 4R 2 dx µ dx µ (1+x 2 ) 2 . Working in perturbation theory, we can compute the change in the free energy in the perturbed CFT to be [16] where in the second line we have used the relation (4.7), (4.10) between bare and renormalized coupling, and we have used the two and three point integrals on the sphere I 2 (∆) and I 3 (∆), which are given in appendix B. Expanding at small with fixed d, we find To leading order in and g we then find, using (4.17)

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Finally, in terms ofF = − sin( πd 2 )F , after some simplification of the d-dependent prefactor we obtain Note that to this order we find and soF is stationary at the fixed point. To evaluate the change inF from UV to IR, we can plug in the value of the fixed point coupling (4.12), and we get In odd d this result agrees with [16], and in even d with the change in a-anomaly computed in [5]. For a unitary CFT, C 2 > 0 and C 3 is real. So we find that for all d,F UV >F IR to leading order in conformal perturbation theory. This gives a perturbative proof of the Generalized F -theorem in the framework of conformal perturbation theory. Note that using (4.21) we may rewrite the leading order change inF as (4.23) It is natural to ask whether such a relation continues to hold at higher orders, provided we replace C 2 by a coupling dependent two-point function coefficient C 2 (g).

Several coupling constants
More generally, we can consider a perturbation by several primary operators where g i b are bare coupling constants. For simplicity, we take the bare operators O i to have the same dimension where as before we assume 1. The two and three point functions in flat space take the form Here C 2,ij is a symmetric matrix (positive in unitary theories). We could choose a basis of operators so that C 2,ij = δ ij , but we will not do this here.

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Let us consider the calculation of the two-point function in the perturbed CFT. We have Using the integral (4.4), and expanding at small , we find We can introduce the dimensionless renormalized couplings g i by The corresponding local operators renormalization is Plugging this into (4.28) and requiring that the renormalized two-point function has no poles as → 0, we get the condition This is solved by where C il 2 is the inverse of the matrix C 2,il . The beta functions are then given by The computation of the sphere free energy proceeds as in the single coupling case described above. In terms of the integrals I 2 and I 3 we have

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where in the second line we have used the small expansion of I 2 and I 3 , and we have plugged in the relation between bare and renormalized couplings. In terms ofF , we have Analogously to the one-coupling case, we then find To write the final result more explicitly, we note that the vanishing of the beta functions implies and so we find In a unitary theory, the matrix C 2,ij is positive definite and the coupling constants are real, so we find as expectedF UV >F IR . Note that (4.36) implies that the leading order change inF from UV to IR can be written as the line integral where the integral is along a path connecting the origin in coupling space to the point g * = (g 1 * , g 2 * , . . .). The integral is independent on the choice of path due to (4.36), which implies We can view the quartic interaction as a perturbation of the form (4.1), where the unperturbed CFT is the free theory, and the perturbing operator O(x) = 1 4 (φ i φ i ) 2 has dimension ∆ = 2d − 4 = d − in the UV. Then we can formally apply the results of the previous section to the present case. Note that in the above calculations we have assumed a CFT defined in dimension d with an operator of dimension d − , and in principle unrelated to d. Therefore, we are expanding in first with d fixed, and at the end set d = 4 − (this JHEP12(2015)155 corresponds to a different order of limits compared to the approach in section 3). In the free theory at dimension d, we have From which we can read off, in the notation of the previous section so that the conformal perturbation theory β-function reads (4.44) Note that setting d = 4, this agrees with the well-known result for the beta function in the quartic scalar field theory computed in the standard field theory approach, β = − g + N +8 8π 2 g 2 + O(g 3 ). To obtain the leading order change inF on the sphere, we just need to apply the general result (4.22). Using (4.43), and setting d = 4 − , we find This agrees precisely with the leading order term in (1.6), first obtained in [1].
In the conformal perturbation theory approach, it is not easy to deduce the next order correction in the beta function (4.11), since it depends on the 4-point function of the perturbing operator. However, in the present case we know that for d = 4 it should reduce to the known beta function computed in the minimal subtraction scheme, eq. (2.4), namely we should have Assuming that the relation of the form (4.23) holds to higher orders, and using the fact that C 2 (g) = C 2 + O(g 2 ) (see (4.14)), we then find (4.47) Remarkably, this agrees with the result (1.6) obtained by the direct method based on the renormalization of the field theory on curved space, provided one includes the effect of the Euler density counterterm which affects the order 4 inF .

The O(N ) model with anisotropic perturbation
Let us consider the O(N ) scalar field theory in d = 4 − perturbed by an operator that breaks O(N ) to S N × Z N 2 , which are the symmetries of the N -dimensional hypercube. This is why it is sometimes called the model with "cubic anisotropy" [43]: Both operators in the action, (φ i φ i ) 2 and ( i φ 4 i ), have dimension ∆ = 2d − 4 = d − in the free UV CFT, and so we can apply the results of section 4.2 to this model. In this basis of operators the two-point function matrix C 2,ij is not diagonal. While it is not necessary to do so, we find it convenient to change the basis to In terms of the operators The one-loop beta functions computed with the standard dimensional regularization and minimal subtraction scheme are known to be [43] (see A.1 for the higher order terms) (4.51) Besides the free UV fixed point, there are three IR fixed points, given to the one-loop order by , Anisotropic fixed point   figure 3. The value N c = 4 is the leading order one-loop result. Higher order corrections make N c decrease as we move away from d = 4. In d = 3, it is expected that N c 3 [43]. We now compute the leading order correction toF at each fixed point and check that the structure of the RG flows described above is consistent with the Generalized F -theorem. The beta functions (4.51) differ from the conformal perturbation theory beta functions (4.33) by the d-dependent factors in the quadratic terms. However, this difference does not affect the result forF at leading order in , so we can apply directly our final result (4.38). Using (4.50) and setting d = 4 − , this yields It is straightforward to check that this agrees with the result obtained by the approach of [1], where the leading contribution to δF just comes from the integrated 2-point function on the sphere. Using (4.52), we then get for the three fixed points (4.55) Of course, the leading correction toF is negative in all cases, consistently with the fact that we can always flow from the UV free theory to any of the fixed points. It is interesting to examine the constraints imposed by the Generalized F -theorem on the possible flows between them. For instance, from the above result we see that which forbids flows from the tensor product of N Ising models to the O(N ) fixed point for N > 10. This is consistent with the structure of the one-loop RG flows depicted in figure 3, even though the constraint imposed byF appears to be slightly weaker. The result (4.57) just follows from the leading order term inF ; as discussed in section 3.1, including higher orders and doing a Padé extrapolation yields (3.24) in d = 3. We also see that which is consistent with the flows between N -Ising and anisotropic fixed point for N > 4 (again, the bound fromF appears to be slightly weaker). We also note that at large N the anisotropic fixed point gets very close to the N -Ising point. Indeed we have The flow from the N decoupled Ising models to the anisotropic fixed point is caused by the "double-trace" operator i<j φ 2 i φ 2 j ; this is why the correction toF is of order N 0 . As done in eq. (4.47) for the O(N ) model, it is interesting to compute the next order correction toF at the anisotropic fixed point using the conformal perturbation theory approach, under the assumption that the line integral formula (4.39) holds at higher orders with C 2,ij (g) = C 2,ij + O(g 2 ). The β functions computed in conformal perturbation theory take the form (1 + O( )) . (4.60) Choosing a piecewise straight path from the origin to the fixed point g * 1 , g * 2 , we obtaiñ where the value of g * 1 , g * 2 to the needed order can be obtained from the β-functions (4.60). Carrying out the integrations, we find

The Gross-Neveu model in 2 + dimensions
The techniques of conformal perturbation theory may be also applied when the perturbing operator is weakly irrelevant and one finds a nearby UV fixed point. As an example, we consider the Gross-Neveu model [45] in d = 2 + : where ψ i is a collection ofÑ Dirac fermions (we will denote N =Ñ tr1). The perturbing operator O(x) = 1 2 (ψ i ψ i ) 2 has dimension ∆ = 2(d − 1) in the free IR theory. In d = 2 + , we have ∆ = d + , and the operator is weakly irrelevant.
The two-and three-point functions of the perturbing operator O(x) in the free theory are

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We can formally apply the result (4.11) with → − and obtain the beta function where we have included the subleading term by requiring that it matches the known beta function of the Gross-Neveu model in d = 2 [46]. Applying (4.23), and again assuming that C 2 (g) = C 2 + O(g 2 ), we obtain As shown in [47], this precisely agrees with the direct field theory calculation in curved space in d = 2 + , where the Euler curvature beta function is set to its zero, following the same logic as outlined in section 2.

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and the ellipses mean all inequivalent permutations of i, j, k, l . 14 One can recover the usual O(N ) theory using the tensor Using β-functions (A.2) one can find how bare couplings are connected with renormalized ones: due to cumbersome expressions we only write the first term. For the connected correlators we find  Demanding that the expression for the sphere free energy have no poles in cannot fix the counter terms completely, because E and H 2 are proportional to each other on a round sphere. However, using also the result for c 51 from [8], 14 For instance β (2) ijkl = λ abij λ abkl + λ abik λ abjl + λ abil λ abjk and β (3,1) ijkl = λ abcd λ aijk λ lbcd + λ abcd λ aijl λ kbcd + λ abcd λ aikl λ jbcd + λ abcd λ ajkl λ ibcd .

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where b 01 = − N 360(4π) 2 is the free field contribution determined by (3.4), which just depends on the total number of fields. For the curvature β-functions we have therefore All these results are in agreement with the results for the O(N ) model and the first terms in β b and β c agree with [8,40]. The term of order λ 5 in β b is, as far as we know, a new result.

A.1 The O(N ) model with anisotropy
As an interesting special case, let us consider the O(N ) model with anisotropy, whose action is given in (4.49). In this case our λ ijkl tensor is given by where δ ijkl = 1 if i = j = k = l and δ ijkl = 0 in another case. The relation between bare and renormalized couplings reads  (3))g 4 2 + . . . , (3)

B Feynman integrals on the sphere
In this appendix we describe a method for computing Feynman integrals on a sphere. We start from the well known integrals, which can be calculated explicitly [3,16,48]: , Working in stereographic coordinates all integrals on a sphere have the general form 15 Using rotational invariance we can always put one point to zero and multiply our integral by the volume of the sphere (one can always choose the point which simplifies the subsequent calculations). Then making iversion x µ i → x µ i /|x i | 2 we obtain (we choose x n = 0) Thus we can always reduce the number of integrals by one. We want to represent each integral I in the Mellin-Barnes form. After this is done we can use special Mathematica packages [49,50] to calculate the value of the integral as a series in [51]. All integrals I on a sphere can be represented diagramatically with the help of the external line and propagator (figure 4). Using Feynman parameters and the Mellin-Barnes formula, 16 we derive the following basic relations depicted in figure 5. where , , 15 In stereographical coordintates the chordal distance is s(x, y) = 2R|x−y| (1+x 2 ) 1/2 (1+y 2 ) 1/2 and the metric is 16 Here we mean the formula  = Γ 0 (a 1 , a 2 , b) x y z w × Γ(b 1 +z 1 +z 4 +z 5 )Γ(b 2 +z 2 +z 4 +z 6 )Γ(b 3 +z 3 +z 5 +z 6 ) Using these relations one can easily write any integral on a sphere in the Mellin-Barnes form. For example let us consider the diagram G 4 , which contributes to the free energy of φ 4 -theory at λ 4 order. This diagram is depicted on figure 6. In this graph each line represents the propagator C φ /s(x, y) d−2 and we don't include the symmetry factor, so the integral reads G (3) s(x, y) 2(d−2) s(x, z) d−2 s(x, w) d−2 s(y, z) d−2 s(y, w) d−2 s(z, w) 2(d−2) . (B.5)