Operator mixing in $\boldsymbol{\epsilon}$-expansion: scheme and evanescent (in)dependence

We consider theories with fermionic degrees of freedom that have a fixed point of Wilson-Fisher type in non-integer dimension $d = 4-2\epsilon$. Due to the presence of evanescent operators, i.e., operators that vanish in integer dimensions, these theories contain families of infinitely many operators that can mix with each other under renormalization. We clarify the dependence of the corresponding anomalous-dimension matrix on the choice of renormalization scheme beyond leading order in $\epsilon$-expansion. In standard choices of scheme, we find that eigenvalues at the fixed point cannot be extracted from a finite-dimensional block. We illustrate in examples a truncation approach to compute the eigenvalues. These are observable scaling dimensions, and, indeed, we find that the dependence on the choice of scheme cancels. As an application, we obtain the IR scaling dimension of four-fermion operators in QED in $d=4-2\epsilon$ at order $\mathcal{O}(\epsilon^2)$.


Introduction
One of the tools to study conformal eld theories (CFTs) is to realize them as the endpoint of a renormalization group (RG) ow. Starting from a description in terms of a weakly-coupled UV Lagrangian deformed by a relevant coupling, the dimension d of space(-time) can be continued close to the upper-critical value d c , in which the IR and the free UV xed points coalesce. When d = d c − 2 with 1, the observables of the IR CFT admit a systematic expansion in the parameter [1,2]. Eventually, an extrapolation to of O(1) is a empted in order to estimate observables of the original strongly-coupled CFT. 1 A known property of the dimensional continuation is that the spectrum of operators is enlarged, i.e., in d = d c − 2 there exist so-called evanescent operators that become redundant when → 0. ese operators are more than a mere curiosity: as a consequence of their existence, the xed points in non-integer d have qualitatively new features compared to the standard, integer-dimensional CFTs that they continue. For instance, it was shown in refs. [8,9] that in theories of free bosons and φ 4 -theories evanescent operators lead to negative-norm states in radial quantization, implying that the xed point in non-integer dimension is not unitary. ese negative norm states have large scaling dimensions, so in the example considered in ref. [8,9] the evanescent operators do not a ect the properties of the light spectrum. e departure from standard CFTs is even more pronounced in theories with fermionic degrees of freedom, due to the fact that theories with free fermions in non-integer dimension contain in nitely many evanescent operators with the same scaling dimension and spin. One way to construct them is by antisymmetrizing n gamma matrices, where n runs over the positive integers, such that they vanish in integer d < n. e simplest example is that of the scalar four-fermion operators O n = (ΨΓ n µ 1 ...µn Ψ) 2 , with Γ n µ 1 ...µn ≡ γ [µ 1 . . . γ µn] , (1.1) where the square brackets denote antisymmetrization. When interactions are turned on, all these operators can mix with each other. ese mixings result in an anomalous-dimension matrix (ADM) of in nite size, which makes the computation of the eigenvalues a considerably more involved problem than in the bosonic case.
At leading order (LO), there is a drastic simpli cation, because the operators that are not evanescent -the so-called physical operators 2 -form a nite-dimensional invariant subspace under mixing, i.e., the evanescent-physical entries of the LO ADM vanish. erefore, the IR scaling dimensions of the physical operators at LO in can be obtained by diagonalizing a nitesize matrix. At next-to-leading order (NLO) and beyond, the ADM is scheme-dependent. In the context of d = 4 computations within dimensional regularization, refs. [10,11] introduced a scheme choice with the a ractive property that the block form of the LO ADM is preserved at all orders. e same scheme was proposed in refs. [12,13] in the context of d = 2 Gross-Neveu/ irring models, though formulated in a di erent language.
In this paper we investigate the problem of obtaining the IR scaling dimension of physical operators beyond LO in when there is mixing with in nitely many evanescent operators. Naively, the scheme choice of ref. [10][11][12][13] seems to trivialize the problem by restricting it to the nite-dimensional invariant subspace spanned by physical operators. However, this leads to an apparent paradox, because the entries of the nite-size block of the ADM do not transform properly under a change of basis. For instance, they are a ected by rede nitions of the evanescent operators by a multiple of ×the physical operators [14]. is can be interpreted as a sign of renormalization-scheme dependence.
We resolve this issue by studying the transformation rules of the ADM under a change of scheme, keeping = 0. e general transformation rule turns out to depend on . We demonstrate how this dependence is important to ensure that the eigenvalues at the xed point are invariant under a change of scheme. In particular, we show that going from the minimal subtraction (MS) scheme to the scheme used for evanescent operators in ref. [10][11][12][13] introduces terms of order in the one-loop ADM. 3 At the xed point, these terms spoil the block structure of the ADM at O( 2 ). erefore, the nite-size physical block of the two-loop ADM is not sufcient by itself to extract the scaling dimensions at NLO. e additional input required is the full one-loop mixing of the in nite tower of evanescent operators into the physical operators. Once this is known, one can nally compute the nitely many eigenvalues associated to the physical operators. is requires a rotation or, equivalently, a further change of scheme that completely xes the aforementioned ambiguity in the choice of basis for the evanescent operators. We illustrate the procedure by carrying it out explicitly in the example of four-fermion operators in QED in d = 4 − 2 . A er computing all the relevant entries of the ADM up to order 2 , we approximate the rst two eigenvalues by truncating to a nite number of evanescent operators. We test that the approximations converge as we increase the truncation.
ere are several examples of CFTs with fermionic degrees of freedom that can be studied in -expansion and to which the method we describe here is applicable, see for instance the recent works [15][16][17][18][19][20][21][22][23][24][25][26] and references therein. In our companion paper [27], we focus on 3d QED and use the NLO eigenvalues obtained here to estimate the scaling dimensions of four-fermion operators in d = 3. e rest of the paper is organized as follows: in section 2 we review the general setup of the -expansion, x our notation, and relate the CFT scaling dimensions at NLO to renormalization constants; in section 3 we discuss the transformation rules of the beta function and the ADM under a change of renormalization scheme, rst to all orders in perturbation theory and then more explicitly at the two-loop order, illustrating the scheme-independence of the scaling dimensions; in section 4 we explain the block structure of the mixing between evanescent and physical operators and show that neither in MS nor in the scheme of ref. [10][11][12][13] does the ADM at the xed point have an invariant subspace at NLO; in section 5 we work out the example of four-fermion operators in QED and introduce the truncation algorithm that allows us to compute the scaling dimensions at the xed point. Supplementary material and formulas for the anomalous dimensions computed in ref. [27] are collected in the appendices.
2 Fixed points and scaling dimensions in d = 4 − 2 Consider a theory in d = 4 that admits a perturbative expansion in a classically marginal coupling α. For α = 0 the theory is free; its local operators are products of the elds and their derivatives, and their correlators are given by Wick contractions. When the interaction is turned on, we can compute corrections to the correlators in a perturbative expansion in α. Each order in this expansion can be continued to a non-integer value of the dimension d [2,28]. Upon continuation to d = 4, the coupling acquires a nonzero mass dimension. For de niteness we keep in mind the example of gauge theories, where α = g 2 16π 2 , and take this acquired mass dimension to be 4 − d ≡ 2 .
Correlators of local operators have poles at = 0, which can be subtracted by de ning the renormalized coupling and the renormalized operators as e subscript "0" labels bare quantities. Z α and (Z −1 ) j i are the renormalization constants that subtract the divergences. 4 We stress that here we are interested in the dynamics for = 0. erefore, the procedure of absorbing the divergences in Z α and (Z −1 ) j i is just an e cient way to keep track of the leading behavior of correlators for 1. is observation also appeared recently in ref. [29], see also section 1.35 of ref. [30].
In the perturbative expansion of the renormalization constants in α, each term admits an additional Laurent expansion in , i.e., Di erent choices of the terms that are nite for → 0 correspond to di erent (mass-independent) renormalization schemes. A standard choice is the MS scheme, 5 in which these nite terms are set to zero. When evanescent operators are present, it is more convenient to use a di erent scheme that includes some speci c nite terms Z (L,0) We discuss this in detail below. For the moment, we keep the scheme generic.
From the renormalization constants one obtains the RG functions, namely the beta function and the ADM. e beta function determines the running of the coupling α A convenient way to de ne the ADM is to consider the theory deformed by adding new couplings proportional to composite operators e ADM, γ, is de ned as the running of the couplings C i to linear order in them, namely It then follows from eq. (2.2) that the renormalized couplings C i are related to the bare ones via From the fact that bare quantities do not depend on the renormalization scale µ, we obtain via eqs. (2.5) and (2.7) the standard formulas (2.10) 5 We do not distinguish between MS and MS or any MS-like scheme, in which the same constant term is always subtracted together with a pole [31]. In the generic mass-independent schemes we consider, the nite terms in the renormalization constants are totally arbitrary, i.e., they cannot be reabsorbed with an overall rescaling of µ as in MS-like schemes.
In schemes in which nite terms Z (L,M ) α and Z (L,M ) with M ≤ 0 are present in the renormalization constants, β and γ contain terms with positive powers of . We shall keep track of them in order to discuss the scheme independence of observables in the next section. We de ne them via the expansions We are interested in studying non-trivial xed points of the RG in d = 4 − 2 with > 0. ese are de ned by the condition e solution of the above condition for 1, up to second order in is (2.14) By requiring α * > 0, we nd that an IR xed point exists only if β (1,0) < 0, i.e., when the coupling is marginally irrelevant in d = 4. ( is is, of course, because we assumed the mass dimension of α to be positive in d < 4; alternatively, one could have a marginally relevant coupling, which acquires a negative mass dimension, and nd a perturbative UV xed point.) In the free UV theory, the scaling dimensions of operators are just the sum of the canonical dimensions of the free elds that compose them; we denote these UV scaling dimensions by ∆ UV . e ADM has a block form, in the sense that only operators with the same spin and the same value of ∆ UV can mix. Within each block, the scaling dimensions of operators at the IR xed point are given by the eigenvalues of where γ * is the ADM evaluated at the xed point. is can be derived by applying the RG equation to the two-point correlation function [32][33][34]. See appendix A for a derivation. Up to second order in we have that where All terms on the right hand side of eqs. (2.17) and (2.18) are xed in terms of renormalization constants. We collect these relations in appendix B. e relevant aspect is that, while γ * 1 does not depend on nite renormalization constants, i.e., it is scheme independent, both terms of γ * 2 do depend on such nite constants.
∆ UV is just a constant shi within each block, so to obtain the IR scaling dimension up to O( 2 ) it is su cient to perturbatively diagonalize 1 γ * = γ * 1 + γ * 2 + O( 2 ). At this order, the corresponding set of eigenvalues are where (γ * 1 ) i denotes the i-th eigenvalue of γ * 1 , U is the rotation to the basis of eigenvectors of γ * 1 , i.e., U γ * in the rotated basis. e IR scaling dimension of the i-th operator with UV dimension ∆ UV then equals with the de nitions 3 Scheme independence of scaling dimensions e scaling dimensions are observables. erefore, they cannot depend on the subtraction scheme that we use to compute the renormalization constants. is is not evident from eqs. (2.20) and (2.21), because both the ADM and the beta function, which are used to de ne γ * , do depend on the scheme. We now explain how the scheme dependence cancels in the eigenvalues. Even though this is a well-known result (see for instance section 1.40 of ref. [30]), it is useful for us to review it here, because we shall make use of it later to identify a convenient scheme for the case in which evanescent operators are present. We rst review the general argument at all orders in perturbation theory and then present the explicit formulas for the change of scheme up to two-loop order.
Renormalization schemes are parametrized by the coe cients of the nite terms, Z e rst line de nes the renormalized coupling in the new scheme,α, as a function of α and . Since the divergent terms agree, i.e., Z for M > 0, the Laurent expansion of α =α(α, ) cannot contain negative powers of . We can then de ne the change of scheme to all orders in perturbation theory via functions that depend solely on and α, i.e., with the normalizations f (0, ) = 1, and F (0, ) = 1 and both functions regular at = 0.
the xed point in α gets mapped to the xed point inα, i.e.,α * = f (α * , )α * . As for the anomalous dimension, we have that In the evaluation of eq. (3.6) atα * , the second term drops out, and we see that the matrix at the xed point is a ected by the change of scheme only through a similarity transformation with the matrix F . is does not a ect the eigenvalues, thus proving that the scaling dimensions are scheme-independent.
Next, we show which terms enter the cancellation of the scheme dependence in perturbation theory, up to two-loop order. To this end, we must rst relate the one-and two-loop coe cients of the beta function and ADM in the two schemes; we list these relations in appendix B. Using them, we evaluate the anomalous dimensions at the xed point via eqs. (2.17) and (2.18) to obtainγ * ese equations can be understood as the perturbative expansion of the result in eq. (3.6) evaluated at the xed point. Since the di erence, δγ * 2 , is a commutator with γ * 1 , one readily derives that (U δγ * 2 U −1 ) i i = 0, which means that the IR scaling dimension of eq. (2.20) does not change with the scheme shi . is is the NLO manifestation of the scheme independence of the scaling dimensions.
We stress that the NLO scheme independence of the scaling dimension requires that we include the term γ (1,−1) in γ *

Evanescent operators
In the free theory at α = 0 local operators can be de ned by (gauge-invariant) products of the free elds and their derivatives. ese composite operators o en satisfy linear relations that reduce the number of independent monomials in the elds. However, many relations (in fact, in nitely many) are satis ed when = 0 but are violated by positive powers of . More generally, many relations hold only if d is integer. is implies that in non-integer dimension there are additional independent operators, which are called evanescent operators because they vanish when → 0.
For instance, any operator de ned through the antisymmetrization of n indices, such as the four-fermion operator .) However, O n is a non-trivial operator when d is non-integer, as can be seen by considering the associated Feynman rule, i.e., the contraction with two Ψ and two Ψ elds, which reads Using the standard rules for the Cli ord algebra in d dimensions which is non-zero except if d is an integer smaller than n. is demonstrates that, in general, also the structure in eq. (4.2) is non-trivial. For our purposes we consider an expansion around d = 4 and thus we call evanescent the operators in d = 4 − 2 that vanish when → 0, such as O n for n > 5. Similarly, one can de ne evanescent operators relative to other integer values of d . When interactions are turned on, physical and evanescent operators can mix. In fact, evanescent operators were rst introduced for the computation of anomalous dimensions in d = 4 in dimensional regularization in refs. [10,11], because this mixing a ects the result for the physical operators. (Here, by mixing between operators we mean a corresponding non-zero entry in the renormalization constant Z. In particular, the expression "O i mixes into O j " means that Z j i = 0.) Due to this mixing, as we ow to the IR xed point in d < 4, the eigenoperators, i.e., operators with de nite scaling dimensions, become linear combinations of physical and evanescent operators. Evanescent operators at the Wilson-Fisher xed point of a scalar eld theory in d < 4 dimensions were recently studied in ref. [9], where it was shown that, in general, they lead to loss of unitarity if d is not integer. ere is an important di erence between the scalar theory considered in ref. [9] and a theory with fermions. In the scalar theory, for any xed ∆ UV , there is a nite number of evanescent operators with this value of ∆ UV . In the fermionic theory, an in nite number of them may be present, as illustrated by the operators {O (n) } n∈N that all have ∆ UV = 2(d − 1) for α = 0.

Block structure of the anomalous-dimension matrix
Even before specifying the interactions, it is possible to draw some general conclusions on the form of the mixing between physical and evanescent operators. Consider a set of operators with the same UV dimension, and let us split them into physical and evanescent components, denoted collectively by Q and E , respectively, i.e., We add these operators to the Lagrangian with bare couplings ( and compute the mixing matrix Z by renormalizing these couplings.
To this end, consider the interaction vertices between the elementary elds that are propor- . Each coupling has a particular vertex structure associated to it at the tree level e structures S Ea vanish in the limit → 0. For instance, in a theory of a Dirac fermion, the four-fermion operators in eq. (4.1) give rise to the four-fermion vertices where S n αβγδ ∝ (Γ n µ 1 ...µn ) αβ (Γ n µ 1 ...µn ) γδ . Perturbative corrections to the vertex, order by order in the coupling α, can again be expressed as a linear combination of the structures (S Q i , S Ea ). For this step, it is important that (S Q i , S Ea ) form a complete basis of structures. e L-loop correction to the vertex then is where the coe cients A (L) contain poles when → 0. 6 ese are subtracted by the renormalization constants Z that de ne the renormalized couplings via eq. (2.8). Typically, the L-loop coe cients A (L) have a leading −L pole and also subleading ones. However, in the corrections to the evanescent vertices, i.e., the terms proportional to (C 0 ) a E , the projection to the physical structures S Q j are always accompanied with additional positive powers of [11,14]. is has important consequences for the structure of the matrix Z.
At one-loop, A E Q is nite due to the additional factor of coming from the projection. is results in a block form for the one-loop renormalization constant Z (1,1) and consequently for the one-loop ADM, with zero entries in the E Q block (4.10) At two-loop order, the choice of scheme begins to a ect the mixing constants and thus the ADM. A convenient choice is to use a slightly modi ed MS prescription that subtracts the nite terms in the E Q block [11]. In this scheme, the mixing constant Z E Q is chosen to cancel the nite term A (1) E Q . e nite, one-loop terms then are e motivation for choosing this scheme is that it simpli es the structure of the two-loop ADM, as we shall explain next.
At two-loop order, A QQ , A QE , and A E E can contain both 1 2 and 1 poles. e coe cient of the 1 2 divergence is xed by the one-loop result, i.e., the renormalization constants satisfy the RG identity which ensures that the ADM is free of divergences. e subleading 1 divergences, Z (2,1) , determine the two-loop ADM. In the A E Q block, the divergences are still down by a factor of due to the projection, but now this does not mean that the mixing constant is nite, because a 1 2 divergence from a loop integral can be multiplied with an from the projection resulting in 1 poles. erefore, in the Z (2,1) mixing matrix all the blocks are non-trivial (4.14) It is precisely because Z (2,1)

E Q and Z
(1,0) E Q originate from 1 2 and 1 loop-integral divergences, respectively, that these constants are related by an analogue of the RG identity of eq. (4.13), 6 In ref. [27] we use the notation (A namely, E Q , and for this reason we can write the above identity solely in terms of renormalization constants. We can now appreciate the motivation for this choice of nite terms: by inspection of formula (B.2) for the two-loop anomalous dimension, we see that eq. (4.15) implies that γ (2,0) E Q = 0 ! erefore, in this scheme the block structure of the one-loop ADM (eq. (4.10) persists also at two-loop order, i.e., For applications to d = 4 physics, this scheme has the advantage of enabling us to solve the RG ow without specifying the actual values of C E when d → 4 [14]. Indeed, the nite subtraction in eq. (4.12) was rst introduced for the calculation of the QCD NLO anomalous dimension of four-fermion operators in refs. [10,11], and in a di erent but equivalent language in the context of d = 2 Gross-Neveu/ irring models in refs. [12,13]. In the following, we shall refer to this scheme as the " avor scheme".
In d = 4−2 on the other hand, the one-loop nite renormalization introduces an additional term linear in in the one-loop ADM, namely As we discussed in the previous section, the term γ (1,−1) plays a role in cancelling the scheme dependence of the scaling dimension at the xed point. Recall from eq. (2.16) that γ * 2 depends also on γ (1,−1) , and it thus inherits a non-zero o -diagonal E Q block. erefore, as far as scaling dimensions are concerned the simpli ed block structure of eq. (4.16) in γ (2,0) is not particularly helpful because it does not persist in γ * 2 . Had we, instead, adopted the pure MS scheme, i.e., Z (1,0) = 0, γ (1,−1) would be zero, but the two-loop ADM γ (2,0) would itself have a non-zero E Q block.
Summarizing, we have shown that in d = 4 − 2 the ADM at the xed point has an invariant QQ block at order , i.e., (γ * 1 ) E Q = 0, but the block is no longer invariant when we include also 2 terms, i.e., (γ * 2 ) E Q = 0, neither in pure MS nor in the avor scheme. As such, the O( 2 ) corrections of the scaling dimensions cannot be computed solely from the QQ entries. is is particularly problematic in cases with in nitely many evanescent operators, as in the example of four-fermion operators in eq. (4.1). e computation of the eigenvalues in this case is the topic of the next section.

The evanescent tower
In this section, we show how to obtain the NLO IR scaling dimensions of physical operators in the presence of mixing with an in nite tower of evanescent operators. For concreteness, we demonstrate the method for a speci c example, which, however, should make clear how to apply it more generally.
We consider the example of four-fermion operators in QED in d = 4 − 2 , with N f avors of four-component Dirac fermions Ψ a , a = 1, . . . , N f , namely where the sum over repeated avor indices is implicit. For the application of this example to the dynamics of QED in d = 3, see ref. [27]. e tensor T ac bd = T ca db speci es the avor structure. In particular, we consider the " avor-nonsinglet" case, for which T ac ad = 0 and T ab bd = 0, and the " avor-singlet" case, for which T ac bd = δ a b δ c d . Since the interaction is avor blind, mixing does not spoil neither conditions on T ac bd . e gauge coupling α = e 2 16π 2 induces a mixing of the physical operators (Q 1 , Q 3 ) with the evanescent operators with n running over all odd positive integers ≥ 5. We have included terms proportional to with arbitrary coe cients a n , b n as in ref. [14]. ese terms re ect an intrinsic ambiguity in the de nition of the evanescent operators. e nal result for the scaling dimensions should not depend on these coe cients. We shall use this as a check of our computation. We do not include pieces of the form 2 × a physical operator because they have no e ect in the two-loop computation presented here. Since the expressions for the mixing matrices in this general basis are rather involved we set a n = b n = 0 in the rest of this section. We give the results in the more general basis in appendix C. In the following we use pairs of odd integers (n, m) as indices for matrices: the indices 1 and 3 refer to the physical operators Q 1 and Q 3 , respectively, while indices n ≥ 5 refer to the associated evanescent operators E n .

Flavor-nonsinglet operators
Let us consider rst the avor-nonsinglet case, i.e. T ac ad = 0 = T ab bd . First, we present the results in the avor scheme and subsequently perform a change of scheme. e one-and twoloop ADM are given in appendix C. Furthermore, for QED we have that β (1,0) = − 4 3 N f and β (2,0) = −4N f . Using eq. (2.17) we obtain that the O( ) anomalous dimension at the xed point is for m = n + 2 0 otherwise .

(5.4)
All the E Q entries in γ * 1 vanish, in agreement with the argument of the previous section. e result of the one-loop ADM for this nonsinglet case is also found in ref. [11].
At NLO in , we obtain via eq. (2.18) that the QQ block of the anomalous dimension at the xed point is is result derives from a two-loop computation of the corresponding renormalization constants. For details on the computation we refer to ref. [27]. In the E Q block there is a single non-vanishing entry in the nite renormalization Z It leads to a corresponding non-vanishing entry in the NLO ADM at the xed point which, as we explained, hinders us from extracting the scaling dimensions of physical operators solely from the QQ block.
To reduce the problem to a nite-dimensional one, we need to set the E Q entries of the ADM to zero at NLO. is can be achieved either by a change of basis or equivalently by a change of scheme. We adopt the la er approach. We denote the nite renormalization in the new scheme byZ . From the scheme shi in eq. (3.8) we obtain the expression for the E Q entries of γ * 2 in the new scheme. Requiring = 0, which means that we do not introduce any nite renormalization in the QQ block. As a second boundary condition, we require that (Z n3 ) do not grow too fast as n → ∞, where "too fast" will be speci ed in a moment.
In the new scheme, the computation of the physical eigenvalues is reduced to the diagonalization of the invariant QQ block. e NLO QQ block reads (5.14) By substituting the values of (Z In practice, we use the following algorithm to solve the recurrence relation: 1. We truncate the recurrence relation by se ing an upper cuto n tr to the index n, i.e., we only consider the equations with n < n tr ; 2. We solve the resulting system of linear equations, treating (Z where A 51 (n tr ) and A 53 (n tr ) are constants that depend on the truncation point n tr but not onZ is is the precise sense in which we requireZ   Figure 1: For the avor-nonsinglet four-fermion operators we compute the n tr and the n tr + 2 approximation to the two (∆ 2 ) i 's. We plot the change between two neighbouring approxima- , as function of the truncation point n tr for the case N f = 1. e le , right gure shows the truncation dependence of (∆ 2 ) 1 and (∆ 2 ) 2 , respectively.
In a nutshell, this algorithm simply consists in truncating the in nite-dimensional matrix to a nite size, nding the rst two eigenvalues for the truncated matrix, and then taking the limit in which the truncation is removed.
We implemented this algorithm for di erent values of the parameter N f . Figure 1 shows how the NLO contribution to the scaling dimension (∆ 2 ) i relaxes as we increase the point of truncation. To demonstrate this we plot the change in the approximation of (∆ 2 ) i when the n tr is increased by 2, i.e, 1− ( , as a function of n tr for the case of N f = 1. e behavior for larger N f is analogous. e plots show that as n tr increases the solution approaches a constant value, indicating that the limit in eq. (5.17) indeed exists. In table 1 we list the values of (∆ 2 ) 1,2 for N f = 1, . . . , 10 for a truncation point so large that the signi cant digits displayed are stable. For comparison, we also show the LO values (∆ 1 ) 1,2 . Note, that with this choice of basis, i.e., a n = b n = 0, diagonalizing only the physical-physical block in the avor scheme, i.e., not accounting for evanescent operators, amounts to a sizable numerical error. For instance, for (∆ 2 ) i = 51%, −21% for i = 1, 2, respectively. In appendix D we also include the arbitrary coe cients a n and b n of eq. (5.3). While the truncated solutions depend linearly on a n and b n , we show that the coe cients of the terms proportional to a n and b n decrease to zero as we increase n tr . is is an important check that the answer we obtain is indeed a physical observable, independent of the choice of basis and renormalization scheme.   Table 1: ree signi cant digits of the one-loop, (∆ 1 ) i , and the two-loop, (∆ 2 ) i , contributions to the scaling dimension of the avor-nonsinglet operators for various cases of N f . To obtain the two-loop (∆ 2 ) i values we implemented the algorithm to include the e ect of evanescent operators. Higher truncation of the procedure does not a ect the three signi cant digits displayed here.

Flavor-singlet operators
We use the same approach to compute the NLO scaling dimensions for the avor-singlet fourfermion operators for which T ac bd = δ a b δ c d , i.e., Since the traces of T ac bd are not zero, there are more diagrams contributing. As a result the ADM is not the same as in the avor-nonsinglet case, and there more non-zero entries of the mixing matrix compared to the avor-nonsinglet case. In particular, while the avor-nonsinglet case had a single non-zero E Q entry at NLO (see eq. (5.7)), there are in nitely many non-zero entries in the avor-singlet case. In addition to the (5, 3) entry of eq. (5.7), we nd that where n runs over all odd positive integers ≥ 5. In terms of the general avor tensor, this contribution is proportional to T ab bd , which explains why it vanishes in the avor-nonsinglet case. To compute eq. (5.20) we use the identity [35] Γ n µ 1 ...µn γ ν Γ n µ 1 ...µn = (−1) We collect the results for the ADM in appendix C. e LO ADM at the xed point then follows from them; it reads  Figure 2: For the avor-singlet four-fermion operators we compute the n tr and the n tr + 2 approximation to the two (∆ 2 ) i 's. We plot the change between two neighbouring approximations, , as function of the truncation point n tr for the case N f = 1. e le , right gure shows the truncation dependence of (∆ 2 ) 1 and (∆ 2 ) 2 , respectively. and the physical-physical block of the NLO ADM at the xed point is Analogously to the previous section we perform a change of scheme and x the nite renormalization constants by requiring that the E Q entries of γ * 2 vanish in the new scheme. is requirement de nes a recurrence relation analogous to the one of eqs. (5.9) and (5.10), which we solve with the same algorithm as above. Note that the presence of in nitely many odiagonal entries does not change qualitatively the procedure. Also in this case we see that the solution converges to a constant as we increase the size of the truncation, as shown in gure 2, which is the analogue of gure 1 for the avor-singlet case. We list the values of the O( 2 ) corrections, (∆ 2 ) i , for N f = 1, . . . , 10 in table 2. For comparison we also list the LO values, (∆ 1 ) i , for the respective N f values. Note that with this choice of basis, i.e., a n = b n = 0, diagonalizing only the physical-physical block, amounts for N f = 1 to a numerical error of (∆ 2 ) i = −14%, 140% for i = 1, 2, respectively. Also for this avor-singlet case, we demonstrate in appendix D the independence of the scaling dimension on the parameters a n and b n from eq. (5.3).   Table 2: ree signi cant digits of the one-loop, (∆ 1 ) i , and the two-loop, (∆ 2 ) i , contributions to the scaling dimension of the avor-singlet operators for various cases of N f . To obtain the twoloop (∆ 2 ) i values we implemented the algorithm to include the e ect of evanescent operators. Higher truncation of the procedure does not a ect the three signi cant digits displayed here.

Conclusions and future directions
and renormalization scheme. We explicitly computed the O( 2 ) corrections in the example of four-fermion operators in QED.
In light of our ndings, it would be interesting to revisit the O( 4 ) computation of the scaling dimension of the four-fermion interaction in the Gross-Neveu model from ref. [15]. Since in this case the evanescent operators are rst generated at three loops, we expect the whole evanescent tower to a ect the O( 4 ) term of the scaling dimension.
In the present work we only applied our method to extract the rst few eigenvalues of the ADM, whose eigenoperators approach the physical operators for → 0. e same procedure can also be applied to obtain additional eigenvalues. It would be interesting to study whether the additional eigenoperators also approach the physical operators as → 0, or whether they are evanescent. e rst case would mean that there exist multiple continuations of the physical operators to non-integer dimension and correspondingly multiple functions that continue their scaling dimensions. Another aspect that we have not addressed in this work and that deserves further investigation is the loss of unitarity of the d-dimensional CFT. In analogy to refs. [8,9], we expect that among the tower of evanescent operators one may nd states of negative norm, and operators of complex scaling dimensions.
Recall that in d = 4−2 we have a dimensionful coupling α 0 of dimension 2 . When |k| 2 α 0 with k the momentum of the operator insertion, we can expand the bare two-point function as We are interested in the IR limit of this two-point function, namely the limit of large α 0 |k| −2 . We keep 1 and xed. To constrain the two-point function we use input from the renormalized theory. More precisely, we use the fact that there exist renormalized variables α and O j de ned as such that the renormalized two-point function, as a function of the renormalized coupling, has a smooth → 0 limit. e renormalized two-point function also has a perturbative expansion, i.e., where α(µ) is the renormalized coupling, and µ the arbitrary renormalization scale. e negative powers of in eq. (A.1) were chosen to match the powers of log(k 2 /µ 2 ) in eq. (A.4) as we take → 0. From eqs. (2.9) and (2.10) we have that . e dimensionless parameter that becomes large in the IR limit is (A.9) erefore, taking this IR limit while maintaining µ = |k|, implies that on the right-hand side Z α α(µ) must become large. As α grows continuously on the positive real axis, starting from the UV value α = 0, the constant Z α can become large if α approaches a pole of the integrand of eq. (A.7), i.e., a non-trivial xed point α * . Close to the solution we expand −2 α + β(α, ) ∼ C(α − α * ). Substituting this in the integral, we nd the leading behavior of α 0 |k| −2 as α approaches the xed point Given that 1 we can see perturbatively that the xed point exists when β (1,0) < 0 and that C > 0. e la er inequality ensures that indeed α 0 |k| −2 grows as we approach the xed point.
Similarly to Z α , eq. (A.8) implies that as α → α * the leading behavior of Z j i is Using eq. (A.10), we nd that in the IR limit Z j i becomes a power-law in |k|, namely where we introduced the crossover scale Λ, whose leading behavior as a function of for 1 is (A.13) Recalling that and using eqs. (A.12) and (A.4) for µ = |k| we see that in the IR limit a new scaling behavior emerges, i.e., corresponding to the IR scaling dimension ∆ IR = ∆ UV + γ(α * , ). We also see that more precisely the crossover to the IR scaling happens when |k| ∼ Λ, with Λ given in eq. (A.13). As observed in Ref. [16], Λ is exponentially enhanced for 1 compared to the naive crossover scale α 1 2 0 .

B Beta functions and anomalous dimensions
In this appendix we collect: i) the one-and two-loop formulas for the beta function and ADM from eqs. (2.9) and (2.10), respectively. In terms of the renormalization-constant expansions from eqs. (2.3) and (2.4) they read ii) the relations between the one-and two-loop beta function and ADM in two di erent, mass-independent schemes distinguished by the superscript " ". Substituting the expansion of eqs. (2.9) and (2.11) in eq. (3.5) we nd that Similarly, the expansion of eqs. (2.10) and (2.12) in eq. (3.6) leads tõ C Anomalous dimensions of four-fermion operators in QED C.1 Flavor-nonsinglet operators e results for the one-loop anomalous dimension, and the one-loop nite renormalization in the avor scheme can be found in ref. [11]. Including also the dependence on a n , b n the result is for m = n + 2 0 otherwise , We computed also the (Q 1 , Q 3 ) entries of the two-loop anomalous dimension in the same scheme, nding Moreover, as explained in section 4, in this scheme γ for m = n + 2 0 otherwise . C.2 Flavor-singlet operators e one-loop anomalous dimension in the physical sector can be found in refs. [16,36]. e one-loop E Q nite terms and the two-loop QQ ADM in the avor scheme are computed in ref. [27]. e results are otherwise , 3 (2N f + 1)δ n1 − 2(n − 1)(n − 3) for m = n 1 for m = n + 2 0 otherwise , for n, m = 1, 3 ,
(C.10) D a n and b n independence In appendix C we collected the results for the ADM for both the avor-singlet and avornonsinglet operators computed in the generic basis of eq. (5.3). We observe that both the E Q nite one-loop mixing and the two-loop QQ mixing depends linearly on the coe cients a n and b n . As a result, in this generic basis, the entries of γ * 2 , which we use to compute the NLO scaling dimension, also depend on a n and b n . However, the parameters a n and b n are just a parametrization of our freedom to choose the basis of operators. erefore, the observables, i.e., the scaling dimensions, cannot depend on them. In this appendix, we demonstrate using the algorithm from section 5 how this unphysical dependence indeed cancels in the observables.
is has to be contrasted with the wrong procedure of naively diagonalizing the QQ block of the two-loop ADM in the avor scheme, which would lead to eigenvalues that depend on a n and b n .
To this end, we rst generalize eqs. (5.13) and (5.14) to the case in which the basis includes a n and b n . e generalizations for the two considered cases follow:  Figure 3: For the case of avor-nonsinglet operators, we plot the coe cient of the basis-dependent parameters a n and b n in the truncated result for the NLO scaling dimension, as a function of the truncation number n tr , for N f = 1. We see that as we increase the number of evanescent operators included, the dependence drops from the observable. See eq. (D.7) for more details.
Flavor-nonsinglet case:  Figure 4: For the case of avor-singlet operators, we plot the coe cient of the basis-dependent parameters a n and b n in the truncated result for the NLO scaling dimension, as a function of the truncation number n tr , for N f = 1. We see that as we increase the number of evanescent operators included, the dependence drops from the observable. See eq. (D.7) for more details.
terms proportional to a ntr−2 , a ntr , a ntr+2 , b ntr−2 , b ntr , and b ntr+2 all relax to zero as n tr → ∞, i.e., that We demonstrate this in gure 3 and gure 4 for the avor-nonsinglet and avor-singlet case, respectively, in which we plot the absolute value of these coe cients as a function of n tr for the case N f = 1. In both cases, we observe that even a er a few steps of the algorithm the coe cients have already relaxed to small values. e behavior for larger values of N f is analogous.