Scaling dimensions in QED3 from the ϵ-expansion

Journal of High Energy Physics, Dec 2017

We study the fixed point that controls the IR dynamics of QED in d = 4 − 2ϵ dimensions. We derive the scaling dimensions of four-fermion and bilinear operators beyond leading order in the ϵ-expansion. For the four-fermion operators, this requires the computation of a two-loop mixing that was not known before. We then extrapolate these scaling dimensions to d = 3 to estimate their value at the IR fixed point of QED3 as function of the number of fermions N f . The next-to-leading order result for the four-fermion operators corrects significantly the leading one. Our best estimate at this order indicates that they do not cross marginality for any value of N f , which would imply that they cannot trigger a departure from the conformal phase. For the scaling dimensions of bilinear operators, we observe better convergence as we increase the order. In particular, the ϵ-expansion provides a convincing estimate for the dimension of the flavor-singlet scalar in the full range of N f .

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Scaling dimensions in QED3 from the ϵ-expansion

HJE QED3 from Lorenzo Di Pietro 0 1 3 Emmanuel Stamou 0 1 2 0 Chicago , IL 60637 , U.S.A 1 Waterloo , ON N2L 2Y5 , Canada 2 Enrico Fermi Institute, University of Chicago 3 Perimeter Institute for Theoretical Physics We study the xed point that controls the IR dynamics of QED in d = 4 dimensions. We derive the scaling dimensions of four-fermion and bilinear operators beyond leading order in the -expansion. For the four-fermion operators, this requires the computation of a two-loop mixing that was not known before. We then extrapolate these scaling dimensions to d = 3 to estimate their value at the IR xed point of QED3 as function of the number of fermions Nf . The next-to-leading order result for the four-fermion operators corrects signi cantly the leading one. Our best estimate at this order indicates that they do not cross marginality for any value of Nf , which would imply that they cannot trigger a departure from the conformal phase. For the scaling dimensions of bilinear operators, we observe better convergence as we increase the order. In particular, the -expansion provides a convincing estimate for the dimension of the avor-singlet scalar in the full range of Nf . Conformal Field Theory; Field Theories in Lower Dimensions; Renormaliza- - 2 tion Group 3.1 3.2 3.3 5.1 5.2 5.3 1 Introduction 2 QED in d = 4 2 2.1 Operator mixing 3 Four-fermion operators in d = 4 2 Operator basis Renormalizing Green's functions Evaluation of Feynman diagrams 3.4 Anomalous dimensions at the xed point 4 Bilinear operators in d = 4 2 5 Extrapolation to d = 3 Pade approximants Bilinears as d ! 3 6 Conclusions and future directions A Feynman rules B Renormalization constants B.1 Flavor-singlet four-fermion operators B.2 Flavor-nonsinglet four-fermion operators C Flavor-nonsinglet four-fermion operators persist beyond this large-Nf regime, but not much is known about it. Ref. [16] employed the conformal bootstrap approach to derive bounds on the scaling dimensions of some monopole operators. Another method to study the small-Nf CFT is the -expansion, which exploits the existence of a xed point of Wilson-Fisher type [17] in QED continued to d = 4 2 dimensions. When 1 we can access observables via a perturbative expansion in and subsequently attempt an extrapolation to . The -expansion of QED was employed to estimate scaling dimensions [18, 19], the free energy F [20], and the = 12 coe cients CT and CJ [14]. In particular, ref. [18] considered operators constructed out of gauge-invariant products of either four or two fermion elds. Four-fermion operators are interesting because of the dynamical role they can play in the transition from the conformal to a symmetry-breaking phase, which is conjectured to exist if Nf is smaller than a certain critical number Nfc [21{26]. In fact, the operators with the lowest UV dimension that are singlets under the symmetries of the theory are fourfermion operators. If for small Nf they are dangerously irrelevant, i.e., their anomalous dimension is large enough for them to ow to relevant operators in the IR, they may trigger the aforementioned transition [7, 27, 28].1 The one-loop result of ref. [18] led to the estimate N c f 2. Bilinear operators, i.e., operators with two fermion elds, are interesting because they are presumably among the operators with lowest dimension. For instance, when continued to d = 3, the two-form operators [ ] become the additional conserved currents of the SU(2Nf ) symmetry, of which only a SU(Nf ) subgroup is visible in d = 4 2 . This leads to the conjecture that their scaling dimension should approach the value = 2 as ! 2 1 which was tested at the one-loop level in ref. [18]. In order to assess the reliability of the -expansion in QED, and to improve the estimates from the one-loop extrapolations, it is desirable to extend the calculation of these anomalous dimensions beyond leading order in . This is the purpose of the present paper. Let us describe the computations we perform and the signi cance of the results. We rst consider four-fermion operators. In the UV theory in d = 4 2 , there are two such operators that, upon continuation to d = 3, match with the singlets of the SU(2Nf ) symmetry. We compute their anomalous dimension matrix (ADM) at two-loop level by renormalizing o -shell, amputated Green's functions of elementary elds with a single operator insertion. As we discuss in detail in a companion paper [32], knowing this two-by-two ADM is not su cient to obtain the O( 2) scaling dimensions at the IR xed point. We also need to take into account the full one-loop mixing with a family of in nitely many operators that have the same dimension in the free theory. These operators are of the form ( n 1::: n )2 ; (1.1) 1More precisely, when the four-fermion operators are slightly irrelevant in the IR, there is an additional nearby UV xed point. As these operators become marginal, the two xed points cross each other, and they can annihilate and disappear [29]. For a more detailed discussion, see section 5 of ref. [20]. Ref. [ 30 ] pointed out that in order to describe properly the conjectured transition, one cannot ignore higher-order terms in the four-fermion couplings. A study of the RG ow that employed -expansion and included four-fermion couplings appeared recently in ref. [31]. { 2 { HJEP12(07)54 where n is an odd integer, and n gamma matrices. All the operators in this family except for the rst two, i.e., n = 1; 3, vanish for the integer values d = 4 and d = 3, but are non-trivial for intermediate values 3 < d < 4. For this reason, they are called evanescent operators. Taking properly into account the contribution of the evanescent operators, via the approach described in ref. [32], we obtain the next-to-leading order (NLO) scaling dimension of the rst two operators. We then extrapolate to = 12 using a Pade approximant, leading to the result presented in subsection 5.2 and summarized in gure 2. The deviation from the leading order (LO) scaling dimension is considerable for small Nf , indicating that at this order we cannot yet obtain a precise estimate for this observable of the three-dimensional CFT. Taking, however, the NLO result at face value, we would conclude that the four-fermion operators are never dangerously irrelevant. This resonates with recent results that suggest that QED3 is conformal in the IR for any value of Nf . Namely, refs. [33{35] argued, based on 3d bosonization dualities [36{39], that for Nf = 1 the SU(2) U(1) symmetry is in fact enhanced to O(4) (this is related to the self-duality present in this theory [40]). Also, a recent lattice study [41] found no evidence for a symmetry-breaking condensate (for previous lattice studies see refs. [42{44]). We then consider the bilinear \tensor-current" operators of the form n crease the order. As mentioned above, in the limit d ! 3 the operators with n = 1; 2 approach conserved currents of the SU(2Nf ) symmetry. Indeed, we show in subsection 5.3 (see gure 4) that the extrapolated scaling dimension of the two-form operators approaches the value = 2 as we increase the order. As d ! 3, the operators with n = 0; 3 approach scalar bilinears, which are either in the adjoint representation of SU(2Nf ) or are singlets. For the singlet scalar, which is continued by a bilinear with n = 3, the results of various extrapolations that we perform are all close to each other (see gure 5), indicating that the expansion provides a good estimate for this scaling dimension in the full range of Nf . For the adjoint scalar, di erent components are continued by operators with either n = 0 or n = 3, giving two independent extrapolations at each order in . As expected, we nd that the two independent extrapolations approach each other as we increase the order (see gure 5). The rest of the paper is organized as follows: in section 2 we set up our notation and describe the xed point of QED in d = 4 2 ; in section 3 we present the computation of the two-loop ADM of the four-fermion operators, and then the result for their scaling dimension at the IR 2 ; in section 4 we present the same result for the bilinear operators; in section 5 we extrapolate the scaling dimensions to d = 3, and plot the resulting dimensions as a function of Nf for the various operators we consider; nally in section 6 we present our conclusions and discuss possible future directions. In the appendices we collect additional material and some useful intermediate results. { 3 { QED in d = 4 grangian is with the covariant derivative de ned as We consider QED with Nf Dirac fermions a, a = 1; : : : ; Nf , of unit charge. The La1 4 LQED = F F + ai D a ; Summation over repeated avor indices is implicit. We work in the R -gauge, de ned by adding the gauge- xing term We collect the Feynman rules in appendix A. The algebra of the gamma matrices is f ; g = 2 , with = and = d. We will employ some useful results on d-dimensional Cli ord algebras from ref. [46]. We normalize the traces by Tr[1] = 4, for any d. For d = 3, a decomposes as (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) is the (2.7) (2.8) giving 2Nf complex two-component 3d fermions Correspondingly, the gamma matrices decompose as i, i = 1; : : : ; 2Nf , all with charge 1. (3) where f g =1;2;3 are two-by-two 3d gamma matrices. In d = 4, the global symmetry preserved by the gauge-coupling is SU(Nf )L 2 , evanescent operators violate the conservation of the nonsinglet axial currents [47], so only the diagonal subgroup SU(Nf ) is preserved. In d = 3, this symmetry enhances to SU(2Nf ) U(1). We de ne coupling is given by D Lg.f. = 1 2 a d!!3 " a # a+Nf ; d!!3 " 0 (3) (3)# 0 ; 0 = Z ( ) 2 ; = 2 + ( ; ) ; d log Z d log : { 4 { where the renormalization constant Z ( ; ) absorbs the poles at = 0, and renormalization scale. The beta function reads where 2 e 16 2 and denote bare quantities with a subscript \0". The renormalized d d log ( ; ) where (n) is the Riemann zeta function. Our convention for renormalizing elds is = 3 4Nf + 27 16Nf2 To compute the anomalous dimension of local operators Oi, we add these operators to the LQED ! LQED + X(C0)i(O0)i ; i and compute their renormalized couplings Ci at linear level in the bare ones In Minimal Subtraction (MS), depends only on and not on . The MS QED function is known up to four-loop order for generic Nf [48, 49] ( ) = 8 3 Nf 2 + 8Nf 3 898 Nf2 + 4Nf 4 2244634 Nf3 + 16 27 2 the theory has a xed point at (C0)j = X CiZi j : i The Zi j are the mixing renormalization constants from which we obtain the ADM Like , does not depend on in the MS scheme. We introduce the following notation for the coe cients of the expansion in and (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) The most direct way to compute the mixing Zi j is to renormalize amputated oneparticle-irreducible Green's functions with zero-momentum operator insertions and elementary elds as external legs. Alternatively, one can renormalize the two-point functions of the composite operators. The former method has two main advantages. The rst is that to extract n-loop poles only n-loop diagrams need to be computed. The second is that we can insert the operators with zero momentum. This makes higher-loop computations more tractable. The disadvantage is that o -shell Green's functions with elementary elds as external legs are not gauge-invariant, so some results in the intermediate steps of the calculation are -dependent, which is why we need to include the -dependent wave-function renormalization of external fermions. In addition, operators that vanish via the equations of the four-fermion operators and use it to obtain the O( 2) IR scaling dimension at the xed point. Next, we employ the already existing results of the three-loop anomalous dimension of bilinear operators [45] to obtain their IR dimension to O( 3). 3 Four-fermion operators in d = 4 2 In this section, we present the computation of the ADM of the four-fermion operators En = ( a n a)2 + anQ1 + bnQ3 ; at the two-loop level. The antisymmetrization in ventional normalization factor n1! . In d = 4, the operators in eq. (3.1) are the only two operators with scaling dimension six at the free xed point that are singlets under the global symmetry SU(Nf )L a subset of the diagrams. We report the result for some nonsinglet operators in appendix C. 2 , insertions of Q1 and Q3 in loop diagrams generate additional structures that are linearly independent to the Feynman rules of Q1 and Q3. To renormalize the divergences proportional to such structures, we need to enlarge the operator basis. It is most convenient to de ne the complete basis by adding operators that vanish for and hence are called evanescent operators, as opposed to Q1 and Q3 that we refer to as physical operators. There is an in nite set of such evanescent operators. One choice of basis for them is 5. The terms proportional to the arbitrary constants an and bn are of the form times a physical operator; they parametrize di erent possible choices for the basis of evanescent operators.2 For the computation of the ADM we adopt the subtraction scheme introduced in refs. [50, 51]. Since this is the most commonly used scheme for applications in avor physics, we refer to it as the avor scheme. We label indices of the ADM using odd integers n 1, so that n = 1; 3 correspond to the physical operators, eq. (3.1), and n 5 to the evanescent operators, eq. (3.2). The ADM up to two-loop order is3 the avor scheme. No other two-loop entry enters the prediction of the O( 2) prediction of the scaling dimensions at the xed point. 2Adding terms with higher powers in have no e ect in the two-loop computation that we discuss here. 3The one-loop (Q1, Q3) block of the ADM can be found in ref. [52]; it is su cient to obtain the O( ) prediction of the scaling dimensions [18]. { 7 { > > > : 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : 8 " > > > > > > > > > > > > > > > > > > > > > > : 5 ; 5 ; Notice that the invariant (Q1; Q3) block of (2;0) depends on the coe cients a5 and b5, which parametrize our choice of basis. This dependence can be understood as a sign of scheme-dependence [53]. Clearly, this implies that the scaling dimensions at O( 2) are not simply obtained from the eigenvalues of this invariant block, as its eigenvalues depend on a5 and b5 too. The additional contribution that cancels this basis-dependence originates from the O( ) term (1; 1) in the one-loop ADM. Such O( ) terms are indeed induced in every scheme that contains nite renormalizations, such as the avor scheme. For a thorough discussion of the scheme/basis-dependence and its cancellation we refer to ref. [32]. There are a few non-trivial ways of partially testing the correctness of the two-loop results: i) We performed all computations in general R gauge. This allowed us to explicitly check that the mixing of gauge-invariant operators indeed does not depend on . ii) We veri ed that all the two-loop counterterms are local, i.e., the local counterterms from one-loop diagrams subtract all terms proportional to 1 log in two-loop diagrams. iii) We checked that the 12 poles of the two-loop mixing constants satisfy the relation Z (2;2) = 2 Z 1 (1;1) (1;1) Z 2 1 (1;0) (1;1) ; Z (3.7) where (1;0) is the one-loop coe cient of the beta-function. This is equivalent to the -independence of the anomalous dimension [54]. In the next two subsections, we discuss the renormalization of the one- and two-loop Green's functions from which we extract the relevant entries of the mixing matrix Z | and ultimately the ADM entries in eqs. (3.4), (3.5), and (3.6) | and some technical aspects of the two-loop computation. A reader more interested in the results for the scaling dimensions may proceed directly to section 3.4. 3.1 Operator basis As argued in section 2.1, in general we need to consider also EOM-vanishing operators when renormalizing o -shell Green's functions. Moreover, in our computation we adopt an IR regulator that breaks gauge-invariance, so we also need to take into account some gaugevariant operators. Below we list all operators that, together with (Q1; Q3) and fEngn 5, enter the renormalization of the two-loop Green's functions we consider. EOM-vanishing operators. There is a single EOM-vanishing operator, N1, that a ects the ADM at the one-loop level and another one, N2, that a ects it at the two-loop level. They read 1 e 1 e N1 = N2 = a) + 1 { 8 { N1 + Q1 ; N1 + N2 : (3.8) (3.9) HJEP12(07)54 In addition, there are EOM-vanishing operators that are only necessary to close the basis of independent Lorentz structures for certain Green's functions. For completeness, we list them here N3 = i a D*= D*= D*= a ; N4 = a( D= ( + * D= ) a F : Here D= D and the arrow indicates on which eld the derivative is acting. Gauge-variant operators. Renormalization constants subtract UV poles of Green's functions. It is thus essential to ensure that no IR poles are mistakenly included in the renormalization constants. In practice, this means that an energy scale must be present in dimensionally regularized integrals. Otherwise, UV and IR contributions cancel each other and the result of the loop integral is zero in dimensional regularization [47]. One possibility to introduce a scale is to keep the external momentum in the loop integral. However, i) such loop integrals are more involved than integrals obtained by expanding in powers of external momenta over loop momenta, and ii) keeping external momenta does not necessarily cure all the IR divergences, e.g., diagrams with gluonic snails in non-abelian gauge theories. Another possibility for QED would be to introduce a mass for the Dirac fermions. The drawback in this case is that we would have to consider many more EOM-vanishing operators. Instead, we apply the method of \Infrared Rearrangement" [55, 56]. This method consists in rewriting the massless propagators as a sum of a term with a reduced degree of divergence and a term depending on an arti cial mass, mIRA. Section 3.3 contains more details about the method. The caveat is that the method violates gauge invariance in intermediate steps of the computation. All breaking of gauge invariance is proportional to mI2RA and explicitly cancels in physical quantities. However, to restore gauge-invariance, also gaugevariant operators proportional to mI2RA need to be consistently included in the computation. Fortunately, due to the factor of mI2RA, at each dimension there are only a few of them. At the dimension-four level, a single operator is generated, i.e., the photon-mass operator: At the dimension-six level, there are more operators, but only one, P, enters our ADM computation because Q1 and Q3 mix into it at one-loop. It reads mI2RAA A : P = 1 e mI2RA X a a aA : 3.2 Renormalizing Green's functions In this subsection, we highlight the relevant aspects in the computation of the renormalization constants Zi j , from which we extracted the ADM presented above, via the renormalization of amputated one-particle irreducible Green's functions. For each Green's function we need to specify the operator we insert as well as the elementary elds on the external legs. In our case, the external legs are either four elementary fermions, or two fermions and a photon, or two photons. At tree-level, a Wick contraction with the elementary elds de nes a vertex structure for each operator. We { 9 { (3.10) (3.11) (3.12) (3.13) hN1i(2) ~ S A 2 4 . . . . . . ~ S 7 + 6 . . . 5 5 ~ S A 7 + 2Z(1) 6 4 Z(2) + Z(2) + Z(1)Z(1) 6 A 5 7 + ~ S 3 7 5 ~ S eq. (A.10). The squares denote operator insertions and the crosses counterterms. The rst parenthesis collects two-loop insertions of the operator N1, which is a linear combination of N1 and Q1. The second collects the one-loop insertions with counterterms on the propagators and the QED vertices. The third and fourth are one-loop insertions multiplied with the eld and charge renormalization of the elds and charges composing the N1, see eq. (3.15). The fth are the tree-level insertions multiplied with the two-loop eld and charge renormalization constant from the N1 . denote the ^ S. An additional subscript indicates the operator associated to a given structure. The representation in terms of Feynman diagrams is structures with S, the A ones with S~, and the A A one with = iCO S~ O We collect all structures that enter the computation in appendix A. In what follows, we refer to as a sum over a speci c subset of Feynman diagrams: i) All these diagrams have a single insertion of the operator O. ii) They are dressed with interactions such that they contribute at O( L). In particular, we include all counterterm diagrams proportional to eld and charge renormalization constants, but we do not include diagrams that contain mixing constants. We keep those separate to demonstrate how we extract them. iii) The subscript S indicates that out of this sum of diagrams we only take the part proportional to the structure S. In short, the notation of eq. (3.14) denotes the L-loop insertion of O projected on S, including contributions from eld and charge renormalization constants. To illustrate the notation we show in gure 1 a small subset of the Feynman diagrams for the non-trivial case of hN1i (2) ~, with S~ any of the structures in eq. (A.10). Note that, S since N1 is a linear combination of terms with di erent elds, see eq. (3.8), its eld and (3.14) hOi(L) S Green's function Depends on Constant(s) extracted One-loop Two-loop A A A A A A (1) ZON2 ZO(1N)1, ZO(1P) , ZON2 (1) (1) ZOO0, ZON1 (1) ZQ(2N)2, ZQ(1N) , ZQ(1;)P ZQ(2N) , ZQ(1O) , ZQ(1N) , ZQP (1) ZQ(2Q)0, ZQ(2N)1, ZQ(1O) , ZQ(1N) , ZQP (1) (1) ZON2 (1) ZO(1N)1, ZOP (1) ZOO0 ZQN2 (2) (2) ZQN1 (2) ZQQ0 the mixing renormalization constants that the given Green's function depends on. The last column contains the ones we extract in each case. charge renormalizations depend on the part we insert, namely 1 e (N1)0 = Z 1=2ZA1=2Z (3.15) Next we derive the conditions on the Green's functions that determine the mixing constants. For transparency we frame the constant(s) that we extract from a given condition. In table 1 we summarize which Green's functions we consider, on which mixing renormalization constants they depend, and which one we extract in each case. For brevity we use the following shorthand notation: Q; Q0 = Q1; Q3 ; E = En ; N = N1; N2 ; O; O0 = Q1; Q3; En : We collect the results for the renormalization constants in appendix B. A A At one-loop there is no insertion of any four-fermion operator that contributes to the Green's function with only two external photons. Thus (1) ZON2 = 0 : A at one-loop. Contrarily, one-loop insertions of four-fermion operator contribute to the Green's function. By expanding the diagram in the basis of S~ structures, we determine the mixing into operators with a tree-level projection onto A , namely N1 and P. For the physical operators the conditions are hQi(1) S~N1 (1) + ZQN1 hN1i(0) S~N1 (1) +ZQN2 hN2i(0) hQi(1) (1) ~ + ZQP hPi(0) SP S~N1 ~ SP In the rst line we use that ZQN2 = 0, as extracted from the A A Green's function. Similarly, we determine the mixing of En into N1 Notice that in this case the mixing constants subtract nite terms, as required by the avor scheme we adopt. (3.16) (3.17) (3.18) (3.19) Next, we compute the one-loop insertions in the Green's function. Firstly, we insert physical operators, i.e., Q, with the only non-vanishing hN1i(0) SO (1) ZQQ1 assumes knowledge of ZQ(1N)1 , which we have previously determined via the Green's function. Next, we insert evanescent operators. Again, the only di erence here is A that their mixing constants into physical operators subtract nite pieces being the one for O = Q1. We see that extracting This completes the computation of all one-loop constants required to determine the mixing of physical operators at the two-loop level. Next, we renormalize the same Green's functions at the two-loop level. A A A A at two-loop. At the two-loop order Q1 and Q3 insertions do contribute to the Green's function. They can thus mix into the operator N2. Even though N2 itself does not have a tree-level projection on physical operators, we need this mixing to extract the two-loop mixing of Q1 and Q3 into N1 in the next step. The projection onto the S^ (3.20) (3.21) (3.22) structure results in the condition level. We only need the two-loop mixing of physical operators into N1, because only N1 has a tree-level projection onto Q1. To unambiguously determine the projection on the structure S~N1 , we have to x a basis of linear independent structures, which correspond to linearly independent operators. At this loop order, we nd that apart from N1 we also need to include the operators N3 and N4 to project all generated structures. This projection is the only point in which these operators enter our computation. The niteness of the two-loop Green's function determines the two-loop mixing of physical operators into N1 via4 at two-loop. Finally, we have collected all results necessary to renormalize the two-loop Green's function. The renormalization conditions for the mixing in the 4Note that hN1i(1) S~N1 = hN1 i(1) S~N1 +hQ1i(1) S~N1 ; as N1 has two Feynman rules. physical sector read We see here explicitly that, because N1 has a tree-level projection onto Q1, we need ZQN1 Already at the two-loop level the number of Feynman diagrams entering the Green's functions is quite large. The present computation is thus performed in an automated setup. Firstly, the program QGRAF [57] generates all diagrams creating a symbolic output for each diagram. This output is converted to the algebraic structure of a loop diagram and subsequently computed using self-written routines in FORM [58]. The methods for the computation and extraction of the UV poles of two-loop diagrams are not novel and also widely used throughout the literature. Here, we shall only sketch the steps and mention parts speci c to our computation. One major simpli cation of the computation comes from the fact that we can always expand the integrand in powers of external momenta over loop-momenta and drop terms beyond the order we are interested in. For instance, for the Green's function all external momenta can be directly set to zero, while for the A one we need to keep the external momenta up to second order to obtain the mixing into N1 (see S~N1 in eq. (A.10)). After the expansion, all propagators are massless so the resulting loop-integrals vanish in dimensional regularization. To regularize the IR poles and perform the expansion in external momenta we implement the \Infrared Rearrangement" (IRA) procedure introduced in refs. [55, 56]. In IRA, an | in our case massless | propagator is replaced using the identity 1 (p + q)2 = p2 1 mI2RA q2 + 2p q + mI2RA p2 mI2RA 1 (p + q)2 ; (3.27) where p is the loop momentum, q is a linear combination of external momenta, and mIRA is an arti cial, unphysical mass. We see that the rst term in the decomposition contains the scale mIRA and carries no dependence on external momenta in its denominator. In the second term, the original propagator reappears, but thanks to the additional factor the overall degree of divergence of the diagram is reduced by one. When we apply the decomposition recursively, we obtain a sum of terms with only loop-momenta and mIRA 1 in the denominators plus terms proportional to (p+q)2 . These last terms, however, can be made to have an arbitrary small degree of divergence. Therefore, in a given diagram we can always perform the decomposition as many times as necessary until terms proportional 1 to (p+q)2 are nite and can thus be dropped if we are only interested in the UV poles. When applying IRA on photon propagators, the resulting coe cients of the poles are not gauge-invariant, because we drop the nite terms in the expansion of propagators. This is why some gauge-variant operators/counterterms enter in intermediate stages of the computation, for instance the operator P. Such operators are always proportional to mI2RA and so only a small number of them enters at each dimension. For more details on the prescription we refer to the original work [56]. The IRA procedure results in integrals with denominators that i) are independent of the external momenta, and ii) contain the arti cial mass mIRA. We can always reduce these integrals to scalar \vacuum" diagrams by contracting them with metric tensors and solving the resulting system of linear equations, e.g., see ref. [56]. This tensor reduction reduces all integrals to one- and two-loop scalar integrals of the form Z ddp (p2 m21)n1 and Z Z ddp1ddp2 (p21 m21)n1 (p22 m22)n2 (p1 p2)2n3 ; (3.28) with the integers n1, n2, n3 1, and m1 6= 0. The one-loop integral can be directly evaluated, whereas all two-loop integrals can be reduced to a few master integrals using the recursion relation in ref. [59]. In fact, in our case m1 = m2 = mIRA and the use of recursion relations is not required. In the evaluation of the Feynman diagrams, we use the Cli ord algebra in d dimensions for i) the evaluation of traces with gamma matrices when the diagram in question has closed fermion loops, and ii) the reduction of the Dirac structures to the operator structures S or S~ listed in appendix A. 3.4 Anomalous dimensions at the xed point By substituting the value of the coupling at the xed point, eq. (2.10), in the result of eq. (3.3), we obtain the ADM at the xed point as an expansion in >>>> + > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : > > : 1 24Nf2 (3.29) (3.30) (3.31) ( 2 )nm = < >>> N24f ( 1) n(n2 1) (n > > for m = 1; n 7:39 46:1 13:4 84:0 3:07 14:1 8:07 23:5 1:72 7:43 6:39 11:6 1:10 4:84 5:60 7:12 0:766 3:51 5:17 4:94 0:562 2:73 4:90 3:70 0:429 2:21 4:71 2:91 0:337 1:86 4:59 2:37 0:272 1:59 4:49 1:99 0:224 1:39 4:42 1:70 for Nf = 1; : : : ; 10. Only three signi cant digits are being displayed. Note that the physical-physical block is not invariant at order 2, because there are non-zero entries ( )n1 and ( )n3 for all n We are interested in nding the rst two eigenvalues of up to order 2 . They determine the scaling dimensions of the corresponding eigenoperators at the IR xed point. We denote these scaling dimensions by ( IR)i = UV( ) + ( 1)i + 2 ( 2)i + O( 3) ; (3.32) with i = 1; 2 and UV( ) = 6 4 . To compute the rst two eigenvalues we have truncated the problem to include a large but nite number of evanescent operators. Taking a su ciently large truncation, the scheme/basis-dependence of the approximated result can be made negligible at the level of precision we are interested in (for details see ref. [32]). In table 2, we list the values of ( 1)i and ( 2)i for Nf = 1; : : : ; 10 after we included enough evanescent operators such that the three signi cant digits listed remain unchanged. The table is the main result of this section. In section 5, we will use these results as a starting point to extrapolate the scaling dimensions to d = 3. 4 Bilinear operators in d = 4 2 In this section we consider operators that are bilinear in the fermionic elds. The most generic bilinear operators without derivatives are a n b ; a n 1::: n 5 b ; (4.1) with n prescription [60, 61]. The indices a; b = 1; : : : ; Nf are indices in the fundamental of the diagonal \vector" SU(Nf ) subgroup of the SU(Nf )L SU(Nf )R symmetry of the theory in d = 4. In d = 4 2 , the conservation of the nonsinglet axial currents is violated by evanescent operators [47], and thus only the diagonal SU(Nf ) is a symmetry. On the other hand, the CFT in d = 3 is expected to enjoy the full SU(Nf )L SU(Nf )R symmetry, which is actually enhanced to SU(2Nf ) U(1). Therefore, in continuing the operators of eq. (4.1) to d = 3, we nd that the ones with 5 are in the same multiplets of the avor symmetry as those without. So even though their scaling dimensions can di er as a function of , the enhanced symmetry entails that they should agree when Since the operators with 5 do not provide new information about the 3d CFT, and the = 12 't Hooft-Veltman prescription makes computations technically more involved, we restrict our discussion here to operators without 5. As a future direction, it would be interesting to test this prediction of the enhanced symmetry by comparing the scaling dimensions of operators with 5 after extrapolating to d = 3 at su ciently high order. We also restrict the discussion to operators with n 3, because the others are evanescent in d = 3. The anomalous dimension of bilinear operators without 5 has been computed for a generic gauge group at three-loop accuracy in ref. [45]. For our U(1) gauge theory we substitute CA = 0 and CF = TF = 1. Moreover, there is a di erence in the normalization convention for the anomalous dimension, so that operator decomposes into a singlet and an adjoint component, here = 2 there. Under SU(Nf ) each a n a n respectively. A priori, the two components can have di erent anomalous dimensions. The di erence between the singlet and the adjoint originates from diagrams in which the operator is inserted in a closed fermion loop. When the operator has an even number of gamma matrices, the closed loop gives a trace with an odd total number of gamma matrices, which vanishes. So for even n there is no di erence between the singlet and the adjoint, i.e., they have the same anomalous dimension. Below we collect the results for n Scalar: (Bs(i0n)g) = (Ba(0d)j) = 3 2 + 140Nf2 9 2Nf + 60Nf + 135 2 16Nf2 81Nf (16 (3) 5) 32Nf3 Vector: for n = 1 both operators are conserved currents, so they do cannot have an anomalous dimension, i.e., Two-form: (Bs(i2n)g) = (Ba(2d)j) = 3 Three-form: (Bs(i1n)g) = (Ba(1d)j) = 3 (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) 3078 3 + O( 4) : 3078 3 + O( 4) : In d = 4 these three-form operators are Hodge-dual to axial currents. Actually, the fact that they do not get an anomalous dimension at one-loop, as seen from the equations above, is related to this. However, Hodge-duality cannot be de ned in d = 4 2 and the anomalous dimensions start to di er from those of the axial current at the two-loop level. This exhausts the list of bilinears without 5 that ow to physical operators as d ! 3. In section 5.3 we discuss which operators of the CFT in d = 3 are continued by the operators above, and extrapolate the above results to obtain estimates for their scaling dimensions. 5 provides an approximation to the observable, e.g. the scaling dimension , in terms of a polynomial HJEP12(07)54 k i=1 = UV( ) + X i i + O( k+1) : Taking ! 12 in this polynomial gives the \ xed order" d = 3 prediction of the -expansion. Typically, the xed-order results show poor convergence as the order is increased. A standard resummation technique adopted for these kind of extrapolations is to replace the polynomial with a Pade approximant. The Pade approximant of order (k,l) is de ned as The coe cients ci and di are determined by matching the expansion of eq. (5.2) with eq. (5.1). k + l must equal the order at which we are computing. Another condition comes from the fact that we are interested in the result for to smoothly interpolate from = 0 to = 12 , an employable Pade approximant should not ! 12 . In order for the -expansion have poles for 2 [0; 12 ] for the values of Nf that we consider. In what follows, we show the predictions from a Pade approximation only if it does not contain any pole on the positive axis of for any value of Nf = 1; : : : ; 10. 5.2 Four-fermion operators as d ! 3 In d = 3, the two four-fermion operators in the UV can be rewritten as Q1 d!!3 ( i (3) i)2 ; Q3 d!!3 ( i i)2 ; where i = 1; : : : ; 2Nf . In this rewriting we see explicitly that these operators are singlets of SU(2Nf ). We now evaluate the scaling dimensions ( )1 and ( )2 of the two corresponding IR eigenoperators, at NLO. For the NLO prediction we employ the Pade approximation of (5.1) (5.2) (5.3) LO ( )2 LO NLO Pade (1,1) 4:12 4:23 4:27 4:27 4:26 4:24 4:23 4:21 4:20 4:19 four-fermion operators at d = 3 for various values of Nf . Only three signi cant digits are being to d = 3, as a function of Nf . In black (lower two lines) ( )1 and in red (upper two lines) ( )2. Dashed lines are the LO estimate and solid lines the NLO Pade (1,1). order (1,1). We list the values of the LO and NLO Pade (1,1) predictions for the values of Nf = 1; : : : ; 10 in table 3. We visualize the results in gure 2. The dashed lines are the result of the one-loop -expansion computation. Indeed, as discussed in ref. [18], the one-loop approximation predicts that the lowest eigenvalue becomes relevant for Nf < 3. The two-loop computation presented here changes this prediction. The two solid lines represent the NLO Pade (1,1) approximation to the two scaling dimensions. We observe that for no value of Nf does the lowest eigenvalue reach marginality. We also see that the corrections to the LO result are signi cant, especially for small Nf , i.e., Nf = 1; 2. This means that for such small values of Nf , NLO accuracy is not su cient to obtain a precise estimate for this scaling dimension. Nevertheless, at face value, the result of the two-loop -expansion suggests that QED3 is conformal in the IR for any value of Nf . Next, we comment on the relation of our result to the 1=Nf -expansion in d = 3. At large Nf , the gauged U(1) current, i (3) i, is set to zero by the EOM of the gauge eld, hence the operator Q1 is an EOM-vanishing operator. However, besides Q3, there still is another avor-singlet scalar operator of dimension 4 for Nf = 1, namely F 2 . Q3 and F 2 mix at order 1=Nf [11]. Looking at the -expansion result in gure 2 we see that indeed only the lowest eigenvalue ( )1 (black lines) approaches 4 for large Nf . The other scaling dimension (red lines) approaches 6 as Nf ! 1, implying that the two eigenoperators cannot mix at large Nf . This is consistent precisely because there is only one non-trivial singlet four-fermion operator at large Nf . Its mixing with F 2 cannot be captured within the -expansion, because the UV dimension of F 2 di ers from that of a four-fermion operator in d = 4 2 . We can, however, test whether for any value of eigenvalue ( )1, which starts o larger at level-crossing would require to revisit the extrapolation to estimate. The scaling dimension of F 2 in -expansion is = 0, crosses the dimension of F 2 . Such a = 12 and possibly a ect the = ( Ba(1d)j ) ij = i i (3) i (3) j ; 1 2Nf k (3) k j i : 2 d log = ; with given in eq. (2.10) up to O( 4). At three- and four-loop order the only Pade approximation without poles in the positive real axis of is the order (2,1) and (2,2), respectively. In gure 3 we plot ( )1;2 and (F 2) as a function of d for the representative cases of Nf = 1; 2; and 10. We observe that the only case in which ( )1 crosses before d = 3 is when Nf = 1 and when we employ N2LO Pade (2,1) to predict (F 2). The N3LO Pade (2,2) prediction for Nf = 1 does not cross ( )1 and the same holds for larger values of Nf . Therefore, at least at this order, F 2 should not play a signi cant role in obtaining the four-fermion scaling dimension. 5.3 Bilinears as d ! 3 derivatives, the possibilities are Next we consider bilinear operators in d = 3. In the UV, restricting to the ones without Bs(i0n)g = ( Ba(0d)j )ij = i i i j ; 1 2Nf k k j i : The subscript refers to the representation of SU(2Nf ). The singlet is parity-odd. We can combine parity with an element of the Cartan of SU(2Nf ), in such a way that one component of the adjoint scalar is parity-even. Since parity squares to the identity, this Cartan element can only have +1 and 1 along the diagonal, which up to permutations we can take to be the rst Nf , and the second Nf diagonal entries, respectively. With this choice, the parity-even bilinear is PaN=f1( a a This is the candidate to be the \chiral condensate" in QED3 [22]. a+Nf a+Nf ). (5.4) (F 2) (5.5) (5.6) (5.7) (5.8) 7 6 5 4 3 3 Nf = 1 Nf = 2 4 3 4 3 4 d d d (F 2) : N2LO Pade (2,1) (F 2) : N3LO Pade (2,2) HJEP12(07)54 fermion operators (black and red lines) and F 2 (blue lines) as a function of the dimension d, i.e., for Nf = 1; 2; and 10, respectively. We observe that the N3LO Pade (2,2) prediction of (F 2) never 2 [0; 12 ]. The left, center, and right panel show the result for the representative cases of crosses the NLO Pade (1,1) prediction of ( )1 in the extrapolation region. The singlet is the current of the gauged U(1). When the interaction is turned on, it recombines with the eld strength and does not ow to any primary operator of the IR CFT. The adjoint is the current that generates the SU(2Nf ) global symmetry. Therefore, we expect it to remain conserved along the RG and ow to a conserved current of dimension = 2 in the IR. We now identify which d = 4 2 bilinears from section 4 approach the d = 3 bilinears above. Substituting the decomposition of eqs. (2.4) and (2.5), and also using 3d Hodge duality, we nd that We denote by certain bilinear B (B) the scaling dimension of the operator in the IR CFT in d = 3 that a ows to. Therefore, we expect Bs(i3n)g Bs(i1n)g Ba(0d)j ; Ba(3d)j Ba(1d)j ; Ba(2d)j d!3 ! d!3 ! d!3 ! d!3 ! Bs(i0n)g ; Ba(0d)j ; Bs(i1n)g ; Ba(1d)j : (Ba(0d)j); (Bs(i3n)g) (Ba(3d)j) (Ba(2d)j) d!3 ! d!3 ! d!3 ! (Bs(i0n)g) ; (Ba(0d)j) ; 2 : (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) (5.15) (Bs(i0n)g) LO (Ba(2d)j) 2 2:65 2:49 2:51 1:32 0:238 2 0:500 1:69 1:99 2:75 1:58 1:58 2:00 2 2 2 2 2 2 2 2 2 2:48 2:38 2:32 2:27 2:24 2:21 2:19 2:17 2:16 Ba(0d)j, and the conserved current Ba(1d)j. In cases in which the NLO Pade (1,1) approximant is singular we list instead the values of the xed-order NLO prediction. Only three signi cant digits are being The last equation provides a test of the -expansion and the rst two provide estimates of the observables (Bs(i0n)g) and (Ba(0d)j). To this end, we employ the viable Pade approximants for Nf = 1; : : : ; 10. In table 4 we list the -expansion predictions at d = 3 for Nf = 1; : : : ; 10. For the cases in which the order (1,1) Pade approximant is singular, we list the xed-order NLO prediction. In gure 4 we plot the extrapolations for the scaling dimension of the conserved avornonsinglet current Ba(1d)j as a function of Nf . We observe that both N2LO Pade approximants are closer to 2 than the LO and NLO ones, and they remain close to 2 even for small values of Nf . We consider this to be a successful test of the -expansion, which supports its viability as a tool to study QED3. In gure 5 we plot the various extrapolations for the scaling dimension of the two scalar operators Bs(i0n)g and Ba(0d)j as a function of Nf . For Bs(i0n)g we nd good convergence behaviour between the NLO Pade (1,1) and the two N2LO Pade approximations. Therefore, for this observable we are able to provide a rather convincing estimate. We do stress, however, that the comparison of the various approximations does not provide rigorous error estimates, 3 3 (1) Ba(2d)j : LO Ba(2d)j : NLO Ba(2d)j : N2LO Pade (2,1) Ba(2d)j : N2LO Pade (1,2) 5 6 Nf 2 3 4 7 8 9 10 HJEP12(07)54 at d = 3. The operator is associated to the conserved avor-nonsinglet current of SU(2Nf ), thus its scaling dimension is expected to equal 2. We observe that the N2LO Pade approximations are indeed close to this expectation even for small values of Nf . (1) since the error due to the extrapolation is not under control. For Ba(0d)j we have two di erent operators that provide a continuation to d = 4 2 . It is encouraging that as the order increases, the two resulting estimates approach each other. Even so, we nd that for small Nf the N2LO Pade approximations are spread, so the -expansion at this order does not provide a de nite prediction. As Nf increases the situation improves, namely all NLO and N2LO approximations begin to converge. In table 4 we list the numerical values for the various estimates of the bilinear scaling dimensions for Nf = 1; : : : ; 10. Next, we compare to the large-Nf predictions for the scaling dimensions of the bilinears. The Pade approximants used to estimate the dimensions of Bs(i0n)g and Ba(0d)j do not develop a pole in the extrapolation region 0 consider the approximants evaluated at Nf = 1, i.e., 12 for any value of Nf 1. Therefore, we can = 12 , as a function of Nf , expand them around For Bs(i0n)g, the prediction from large-Nf is [5, 6] and compare the coe cient c(k;l) with its exact value obtained from the large-Nf expansion, clarge-Nf . In what follows we use ' to denote that we display only two signi cant digits. and the extrapolation obtained from the three-form singlet gives Nf + O(Nf 2) ; Bs(i0n)g: (5.16) (5.17) (5.18) 2:6 )32:4 (2:2 2:0 5 6 Nf Bs(i3n)g : LO Bs(i3n)g : NLO Pade (1,1) Bs(i3n)g : N2LO Pade (2,1) Bs(i3n)g : N2LO Pade (1,2) (0) (0) Badj 2:0 1:8 1:6 1:4 1:2 1:0 1 2 3 4 7 8 (left panel), and Ba(0d)j (right panel) at d = 3. The di erent colors for Ba(0d)j correspond to estimates from di erent continuations of the operator (see legend). For Ba(0d)j, the prediction from large-Nf is [4, 6] and the extrapolations obtained from the three-form and scalar adjoints give (0) Badj: Ba(3d)j: Ba(0d)j: 1:4 ; This suggests that the extrapolation of the three-form may provide a better estimate for the scaling dimension of the adjoint scalar at this order. 6 Conclusions and future directions We employed the -expansion to compute scaling dimensions of four-fermion and bilinear operators at the IR xed point of QED in d = 4 2 . We estimated the corresponding value for the physically interesting case of d = 3. The results seem to con rm the expectations from the enhancement of the global symmetry as d ! 3 (see gures 4 and 5). Therefore, going beyond the leading order gave us more con dence that the continuation is sensible. 5 6 Nf Ba(0d)j : LO Ba(0d)j : NLO Pade (1,1) Ba(0d)j : N2LO Pade (2,1) Ba(3d)j : LO Ba(3d)j : NLO Ba(3d)j : N2LO Pade (2,1) Ba(3d)j : N2LO Pade (1,2) (0) (5.19) (5.20) (5.21) At the same time, it appears that | with the exception of the scalar-singlet bilinear | to obtain precise estimates for the scaling dimensions for small values of Nf requires even higher-order computations and perhaps more sophisticated resummation techniques (see for instance chapter 16 of ref. [62] and references therein). The computation of such higher orders in via the standard techniques used in the present work would require hard Feynman-diagram calculations. In recent years, several authors exploited conformal symmetry to introduce a variety of novel techniques to compute observables of the xed point in -expansion. Ref. [63] proposed an approach based on multiplet recombination, further applied and developed in refs. [64{79]. Another approach is the analytic bootstrap, either together with the largespin expansion [80], or in its Mellin-space version [81{84]. Finally, ref. [85] aimed at directly computing the dilatation operator at the Wilson-Fisher xed point. It would be interesting to attempt to apply these techniques to QED in d = 4 On a di erent note, ref. [86] recently argued that QCD3 with massless quarks undergoes a transition from a conformal IR phase, which exists for su ciently large number of avors, to a symmetry-breaking phase when Nf f N c. This is analogous to the long-standing f conjecture for QED3, and so four-fermion operators may play the same role. Therefore, at least for the case of zero Chern-Simons level, -expansion can be employed in a similar manner to estimate N c. A LO estimate appeared in ref. [87]. In light of our results for QED3, it would be worth studying how this estimate is modi ed at NLO. Acknowledgments we thank Joachim Brod, Martin Gorbahn, John Gracey, Igor Klebanov, Zohar Komargodski, and David Stone for their interest and the many helpful discussions. We are also indebted to the Weizmann Institute of Science, in which this research began. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. LQED+g.f. = F F ai D a (1 q q q2 ; (A.1) (A.2) (A.3) (A.4) A Feynman rules From the QED Lagrangian in R -gauge, we obtain the Feynman rules Ψa Aμ Ψa Ψa q → q → 1 4 Ψa = Aν = 1 2 i q = ; i q2 Aμ = ie : There is one additional counterterm coupling that we need to specify. It is a relic of the procedure with which we regulate IR divergences (see section 3.3), which essentially breaks gauge invariance. For this reason to consistently renormalize Green's function we need to include a counterterm analogous to a mass for the photon, i.e., Only the one-loop value of mI2RA enters our computations. It reads Aμ Aν = i mI2RA : mI2RA = 4Nf mI2RA + O( 2) : To nd the EOM-vanishing operators at the non-renormalizable level we apply the EOM of the fermion and photon. They read D a = 0 ; D = 0 ; + e a a = 0 : (A.7) For brevity we use the shorthand notation D direction in which the derivative in D= acts, i.e. D(= and D= * D= . D= and use an arrow to indicate the We consider the Lagrangian with additional couplings proportional to the operators introduced in section 3.1 L = LQED + X CiOi : To compute the Green's function we need the Feynman rules of the operators we insert, as well as all the structures that we need to project the amplitude. For instance, to renormalize the Green's function of A with one-loop insertions of Q1 we need not only the Feynman rule of Q1, but also the A structure of all operators that Q1 generates In our case, the Feynman rules for the following three nal states su ce: Ψa where the structures SO , S~O , S^O depend on the inserted operator. For the set of operators Ψa Ψb Aμ q → O O Ψa Ψb O O p → Aμ O q ↓ Ψa O (A.5) (A.6) (A.8) HJEP12(07)54 relevant to our computation they read S O O Q1 : 2 O N2 : 2 4 q2q q : In this appendix we list the mixing-renormalization constants of four-fermion operators. First we list the constants we need to compute the ADM of avor-singlet four-fermion operators, which we discussed in the main text, and subsequently the constants entering the computation of the ADM of avor-nonsinglet four-fermion operators, which we discuss in appendix C. and (3.5) via B.1 Flavor-singlet four-fermion operators The divergent and nite pieces of the one-loop constants of the mixing between physical and evanescent operators are directly related to the one-loop anomalous dimension of eqs. (3.4) with O; O0 any physical or evanescent operator from section 3.1. To extract these constant from the Green's function we had to rst compute the one-loop mixing of the fourfermion operators into the EOM-vanishing operator N1. For the physical operators the corresponding constants are ZQ(11;1N) 1 = ZQ(13;1N) 1 = 4 3 8 ; (2Nf + 1) ; ZQ(11;0N) 1 = 0 ; ZQ(13;0N) 1 = 0 ; and for the evanescent operators they are ZE(1n;1N)1 = 0 ; ZE(1n;0N)1 = ( 1) 2 16(n 2)(n 5)! 4 3 (2Nf + 1)an 8bn ; (A.9) (A.10) (A.11) (B.1) (B.2) (B.3) (B.4) 5. To compute these constants for generic n we used Cli ordalgebra identities from ref. [46]. As explained in section 3.2, in the computation of the mixing at two-loop level more operators enter. The only one-loop mixings entering the computation, apart from those above, is the mixing of the physical four-fermion operators into the EOM-vanishing operator N2, and the gauge-variant operator P. The former vanish, i.e., and the latter read ZQ N2 ZQ(1;P0) = 0 ; with Q = Q1; Q3. Finally, the two-loop mixing constants of the two physical operators read ZQ(2;Q2)0 = " 29 24Nf2 + 20Nf + 103 23 (3Nf + 1)# ; ZQ(2;Q1)0 = 838 (3Nf + 1) " 514 (8Nf + 2275) 49 (107Nf + 253) " 1 2 + a5 23 (3Nf + 14) 1 22 19 (3Nf + 49)# 56 (8Nf + 9) 0 # " 0 + b5 44 + a7 12 0 0 0 # 1 ; 2 (103Nf + 86) 3 a5(2Nf + 1) 8b5 : # with Q = Q1; Q3. We do not list the corresponding constants for the evanescent operators because they do not enter the two-loop computation of the mixing of physical operators. In table 1 we summarised on which renormalization constants the Green's functions we computed depend on. We see that to determine the two-loop mixing of the four-fermion operators we rst need to determine the two-loop mixing of the physical operators into the two EOM-vanishing operators N1 and N2. The corresponding constants read (12Nf2 + 10Nf + 11) ; (Nf + 11) ; ZQ(21;2N) 2 = ZQ(21;2N) 1 = ZQ(23;2N) 2 = ZQ(23;2N) 1 = 4 9 8 3 8 9 Nf (2Nf + 1) ; 16 3 Nf ; (24Nf + 11) ; ZQ(21;1N) 2 = ZQ(21;1N) 1 = ZQ(23;1N) 2 = ZQ(23;1N) 1 = 8 9 Nf ; 4 27 32 9 8 9 4 Nf ; (B.5) (B.6) (B.7) (B.8) (B.9) (B.10) (B.11) (B.12) HJEP12(07)54 Flavor-nonsinglet four-fermion operators The renormalization of the Green's functions with insertions of avor-nonsinglet fourfermion operators is analogous to the one with avor-singlets but less involved. Their avor-o -diagonal structure forbids them to receive contributions from any EOM-vanishing or gauge-variant operator at two-loop order. Therefore, in this case we only need the mixing constants within the physical and evanescent sectors. As in the avor-singlet case, the one-loop mixing is directly related to the one-loop anomalous dimensions of eqs. (C.4) and (C.5) via with O; O0 any physical or evanescent avor-nonsinglet four-fermion operator; the oneloop anomalous dimensions above are given in appendix C. Finally, the two-loop mixing constants of the two physical operators read 24Nf 18 81 2 4(11Nf " + a5 + a7 12 0 9) 1 2 12) 1 19 (Nf + 63) # 16 (32Nf + 3) 0 # " 0 + b5 36 " 0 0 # 1 ; 2 # with Q = Q1; Q3. C Flavor-nonsinglet four-fermion operators In the main part of this work we investigated bilinear and avor-singlet four-fermion operators. There exist also four-fermion operators that are not singlets under avor. The ones we consider in this appendix are spanned by the basis Q1 = Tbadc( a Q3 = Tbadc( a 3 b)( c d) ; b)( c 3 En = Tbadc( a n 1::: n b)( c n d) ; d) + anQ1 + bnQ3 ; with Tdabc = Tbcda and Taadc = Tbadb = 0. The computation of their ADM at one- and two-loop order entails only a subset of the Feynman diagrams needed for avor-singlet case and is actually less involved as discussed in appendix B. In this appendix we present their ADM and their scaling dimensions at the IR xed point in d = 4 2 , and use this to estimate the corresponding d = 3 observables. (B.14) (B.15) (C.1) (C.2) (C.3) In the avor scheme, the full one-loop ADM of the physical and evanescent operators and the two-loop entries required read: (1;0) = < nm 8 2n(n 1)(n 5)(n 3) for m = n for 8 " (2;0) = < nm 176Nf 78 2 0 83 Nf 2 # + b5 72 49 Nf # + 634 Nf " 0 2 # 0 0 not required for n; m = 1; 3 for n 5 and m = 1; 3 The part of the one-loop result that does not depend on an and bn was rst computed in ref. [50]. Next we evaluate these ADMs at the xed point (C.5) (C.6) (C.7) (C.8) 8Nf2 324 + 792Nf >>>> + 8N3f2 a5 153 + 2Nf 351 + 96Nf # " 0 + 8N3f2 b5 108 4Nf 3 # 5)(n 3)an an+2 + 36bn) for m = 1; n 1 0 # 729 3 0 4Nf 3 1)(n 1)(n 1)(n 3)bn the scaling dimension of the avor-nonsinglet four-fermion operators for various cases of Nf . To obtain the two-loop ( 2)i values we implemented the algorithm to include the e ect of evanescent ( 1)1 ( 2)1 ( 1)2 ( 2)2 9:00 35:6 9:00 101 operators [32]. ( )1 LO ( )2 LO NLO 4:50 8:53 4:50 29:3 Nf NLO Pade (1,1) 3:00 3:63 3:00 14:9 2:25 1:95 2:25 9:40 1:80 1:19 1:80 6:67 1:50 0:782 1:50 5:09 1:29 0:544 1:29 4:08 1:12 0:393 1:12 3:38 1:00 0:292 1:00 2:87 0:900 0:221 0:900 2:49 1 0:500 3:26 8:50 16:7 2 3 4 5 6 7 8 1:75 2:50 2:88 3:10 3:25 3:36 3:44 3:50 3:55 3:17 3:22 3:30 3:37 3:43 3:49 3:53 3:57 3:60 6:25 5:50 5:12 4:90 4:75 4:64 4:56 4:50 4:45 1:09 1:78 2:78 3:23 3:48 3:62 3:72 3:78 3:83 avor-nonsinglet four-fermion operators at d = 3 for various values of Nf . Only three signi cant digits are being displayed. Following ref. [32] we shift to the scheme in which the physical-physical subblock forms an invariant subspace. In this scheme we are able to extract the scheme-independent O( 2) corrections to the scaling dimensions, i.e., the ( 2)is. 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Lorenzo Di Pietro, Emmanuel Stamou. Scaling dimensions in QED3 from the ϵ-expansion, Journal of High Energy Physics, 2017, 54, DOI: 10.1007/JHEP12(2017)054