On $C_{J}$ and $C_{T}$ in Conformal QED

QED with a large number $N$ of massless fermionic degrees of freedom has a conformal phase in a range of space-time dimensions. We use a large $N$ diagrammatic approach to calculate the leading corrections to $C_T$, the coefficient of the two-point function of the stress-energy tensor, and $C_J$, the coefficient of the two-point function of the global symmetry current. We present explicit formulae as a function of $d$ and check them versus the expectations in 2 and $4-\epsilon$ dimensions. Using our results in higher even dimensions we find a concise formula for $C_T$ of the conformal Maxwell theory with higher derivative action $F_{\mu \nu} (-\nabla^2)^{\frac{d}{2}-2} F^{\mu \nu}$. In $d=3$, QED has a topological symmetry current, and we calculate the correction to its two-point function coefficient, $C^{\textrm{top}}_{J}$. We also show that some RG flows involving QED in $d=3$ obey $C_T^{\rm UV}>C_T^{\rm IR}$ and discuss possible implications of this inequality for the symmetry breaking at small values of $N$.


Introduction and Summary
One of the important observables in Conformal Field Theory (CFT) is C T , the coefficient of the two-point function of the stress-energy tensor T µν , defined via [1] T µν (x 1 )T λρ (x 2 ) = C T I µν,λρ (x 12 ) (x 2 12 ) d , where I µ ν,λρ (x) ≡ 1 2 (I µλ (x)I νρ (x) + I µρ (x)I νλ (x)) − 1 d δ µν δ λρ , If the CFT has a global symmetry generated by conserved currents J a µ , then another interesting observable is C J , the coefficient of their two-point functions: In CFTs with a large number of degrees of freedom, N , these observables typically admit 1/N expansions of the form The values of C J1 and C T 1 have been calculated in a variety of models. Petkou [2] has used large N methods and operator products expansions to calculate them as a function of d in the scalar O(N ) model. Very recently, these results were reproduced using the large N diagrammatic approach in [3], where the same technique was also used to calculate C J1 and C T 1 as a function of d in the conformal Gross-Neveu model. An important feature of the diagrammatic approach, which was uncovered in [3], is the necessity, in the commonly used regularization scheme [4][5][6][7][8], of a divergent multiplicative "renormalization" Z T for the stressenergy tensor. This factor is required by the conformal Ward identities in the regularized theory.
In this paper we extend the methods of [3]  Our main results are where ψ(x) = Γ (x)/Γ(x). Here η m1 (d) encodes the electron mass anomalous dimension; it . (1.7) In the physically interesting case of d = 3 we find Let us compare our results with the earlier diagrammatic calculations [10,11], which were carried out in d = 3 using a regulator different from ours. Our result for C J1 (3) agrees with that given by Huh and Strack in [11]. 2 However, our value of C T 1 (3) does not agree with that given in [11], which after translating to our convention for N is C H&S The source of the disagreement is the effect of Z T , which was not included in [11].
A nontrivial check of our results (1.5) and (1.6) comes from comparing them with the known exact values in d = 2 and the 4 − expansions, see sections 3 and 5. Had we not included Z T , there would be no agreement with the 4 − expansion. In higher even d, the conformal QED reduces to a free theory of N fermions and a conformal higher-derivative Maxwell theory with the action (see e.g. [12]) (1.9) Using the value of C T 1 in general even dimensions, we extract the C T of this conformal Maxwell theory 11) where N = 4N f is twice the number of two-components Dirac fermions. The leading order term is in agreement with [10,13].
The QED 3 Lagrangian also has an enhanced SU (2N f ) global symmetry, and for small N f this symmetry may be broken spontaneously to [14,15]. In section 6 we present a new estimate for the critical value of N f above which the symmetry breaking cannot occur by using the RG inequality C UV T > C IR T . It implies that the chiral symmetry cannot be broken for N f > 1 + √ 2. The status of this conclusion is uncertain, since there are known violations of the inequality in some supersymmetric RG flows [16]. Nevertheless, it is interesting that the critical value of N f it yields is close to other available estimates [12,[17][18][19][20] and is consistent with the results available from lattice gauge theory [21,22].

Large N Expansion for Conformal QED d
The action for Maxwell theory coupled to N f massless charged fermions in flat Euclidean Here the fermions ψ i are taken to be four-component complex spinors. We define the dimensional continuation of the theory by keeping the number of fermion components fixed.
One may develop the 1/N expansion of the theory by integrating out the fermions [15,23].
This produces an effective action for the gauge field of the form is the conserved U (1) current. Using the bare fermion propagator where / p ≡ γ µ p µ , the current two-point function in the free fermion theory is found to be Thus, when d < 4, one sees that the non-local kinetic term in (2.2) is dominant in the low momentum (IR) limit compared to the two-derivative Maxwell term. Hence, the latter can be dropped at low energies, and one may develop the 1/N expansion of the critical theory by using the induced quadratic term Note that this effective action is gauge invariant as it should, due to conservation of the current.
The induced photon propagator is obtained by inverting the non-local kinetic term in (2.6). As usual, this requires gauge-fixing. Working in a generalized Feynman gauge, the propagator is where ξ is an arbitrary gauge parameter (ξ = 0 corresponds to Landau gauge ∂ µ A µ = 0).
The normalization constant C A is given by and in (2.7) we have introduced, as in [3], a regulator ∆ to handle divergences [4][5][6][7][8], which should be sent to zero at the end of the calculation. This makes the interaction vertex in (2.6) dimensionful, and one should introduce a renormalization scale µ so that The Feynman rules of the model are summarized in figure 1. In what follows we calculate the two-point function of the SU (N f ) current and stress-energy tensor, which are given by 4 Note that there is no Maxwell term contribution in T µν , as this term was dropped in (2.6) in the critical limit.
We will work in flat Euclidean d-dimensional metric and introduce a null vector z µ , which satisfies z 2 = z µ z ν δ µν = 0 .
where we have introduced the notation x z ≡ z µ x µ . It will be also useful to report the form of these two-point functions in momentum space, which may be obtained by Fourier transform and reads where p z ≡ z µ p µ .
For the stress-tensor of conformal QED, we may write T = T ψ + T A , where the two terms are given in momentum space by The diagrammatic representation is shown in figure 2. 3 Calculation of C J1 and C T 1 The diagrams contributing to JJ up to order 1/N are shown in figure 3. Their expressions in momentum space and explicit results are listed in Appendix B. Putting together the results, we find where C J1 (d) is given in (1.5), and is the free fermion contribution. A plot of C J1 as a function of d is given in figure 4. The value in d = 3 was given in (1.8) above. One may also extract the following -expansions In d = 3 the leading correction is quite small even for small N ; for N = 4, corresponding to N f = 1, it makes C J around 7% bigger than the free fermion result.
Let us now turn to the calculation of C T . Up to order N 0 , the stress-tensor two-point function receives contribution from the diagrams shown in figure 5. Note that for some topologies we did not draw explicitly diagrams with the opposite fermion loop direction, but they have to be included. We list the integrands and results for these diagrams in Appendix B. We have where we have introduced a "Z T -factor" [3], which is computed in Appendix A from the Ward identity. It reads Z T = 1 , where η m1 is given in (1.7). Putting together the results for the diagrams given in Appendix T A (p) Figure 5: Diagrams contributing to C T up to N 0 order.

B, we obtain
where C T 1 (d) is given in (1.6), and the free fermion contribution is As a check of our calculation, we note that the final result does not depend on the gauge parameter ξ.
A plot of C T 1 (d) in 2 < d < 4 is given in figure 6. We see that C T 1 is negative for 2 < d < 2.79. This means that the inequality C UV T > C IR T is violated for the flow from conformal QED d (which may be thought of as the UV fixed point of the Thirring model) to the free fermion theory for 2 < d < 2.79. However, it holds for 2.79 < d < 4, including in particular d = 3.
Near some even dimensions we find Note that in d = 2 we get This result is precisely as expected, since the conformal QED 2 corresponds to the multiflavor Schwinger model with 2N f Dirac fermions, which is described by a CFT with central charge [26,27]. Normalizing (3.10) by the free scalar contribution C sc , and recalling N = 4N f , we obtain precisely this central charge. In section 5 we will see that Figure 6: Plot of C T 1 .
Near even dimensions the QED d theory is expected to be described by the free fermions weakly coupled to a U (1) gauge theory with the local kinetic term (1.9). For example, in d = 6 this higher-derivative theory was explored in [12,[28][29][30][31][32][33]. We may use (1.6) to extract the C T coefficient for the conformal Maxwell theory (1.9). From (1.6) it follows that Recalling that the contribution of the free massless fermions is given by (3.8), we find that the C T of the conformal Maxwell theory is  In d = 3, it is interesting to compute C top J for the "topological" U (1) current where the factor of i arises because we are working in Euclidean signature, and the normalization is such that the associated charges are integers. The diagrams contributing to the current two-point function up to order 1/N 2 are shown in figure 7. The diagrams D 1 and D 2 have the same structure as the corresponding ones in fig. 4 for the SU (N f ) current, 5 with the difference that at the external points we now have the gauge U (1) current, to which we attach the two induced photon propagators. Thus, using the results from Appendix B, we where C A and C J1 (d) are given in (2.8) and (1.5), which yield C A | d=3 = 32 and the value of C J1 (3) given in (3.4). Therefore, we finally get Comparing with the momentum space normalization in (2.12), we find the result given in eq. (1.11). We note that this is related to C J in (1.4)-(1.8) by an inversion, C A ∼ 1/C J . This essentially follows from the fact that in the large N critical QED, A µ and J µ are related by a Legendre transformation [35,36], see eq. (2.6).
The conformal bootstrap constraints on the values of C J , C T and C top J in QED 3 for N f = 1, 2, 3 were recently discussed in [13]. In Table 1 we summarize our results for these coefficients in d = 3 and for different values of N f (the number of 4-component fermions).
These results appear to fall within the regions allowed by the bootstrap for N f = 1, 2, 3. The renormalization of the electric charge is well-known, and in minimal subtraction scheme it reads [37]: where e is the renormalized coupling, and the corresponding beta function is Then, one finds an IR stable perturbative fixed point at Computing the diagrams in figure 3 with the photon propagator (5.1), taking a Fourier transform to coordinate space, and setting e = e * at the end, we obtain in d = 4 − To calculate the 4 − expansion of C T to order , we will use as a shortcut the fact that in d = 4 the C T coefficient may be obtained as (see e.g. [38,39]) where β a is the beta function for the Weyl-squared term, which is known to be [40,41] β The first term corresponds to the contributions of the free fermions and of the Maxwell field, while the second one encodes the leading interaction corrections. The second term, when evaluated at the IR fixed point (5.4) in d = 4 − , gives 7 6(16π) 2 . However, this is not the only contribution of order because the free field contributions need to be evaluated in 4 − dimensions. The contribution of free massless fermions is given in (3.8). The contribution of the Maxwell field is more subtle, since this theory is scale invariant but not conformal away from four dimensions [42]. However, defining the projected stress-tensor T Maxwell = z µ z ν F µα F α ν (this selects the traceless part of T µν ), and using the field strength two-point function [42] we find that T Maxwell (x)T Maxwell (0) takes the form (2.11), just as in a conformal field theory, with the normalization given by This serves as the natural definition of C T for the Maxwell theory (in d = 4, it agrees with the well-known result [1]). Putting these results together we find which exactly agrees with (3.9). This gives a highly non-trivial test of the dimension dependence of C T 1 .

A New Estimate for Symmetry Breaking in QED 3
In d = 3, the QED Lagrangian has SU (2N f ) global symmetry. For N f < N f,crit it may be broken via the generation of vacuum expectation value of the operator N f j=1ψ j ψ j (this is written using the 4-component spinors ψ i and gamma-matrices) [14,15]. This operator preserves the 3-d time reversal symmetry, but it breaks the global symmetry to SU (N f ) × In an earlier paper [12], using the F -theorem inequality F UV > F IR [43][44][45][46][47] we showed that theories with N f = 5 and higher must be in the conformal phase. The F -theorem method is inconclusive, however, for theories with N f ≤ 4. There is lattice evidence that theories with N f = 1, 2 are not conformal [21,22], 6 but little is known about theories with Let us now consider a different RG inequality: which is sometimes called "the C T theorem". While there is a known d = 3 counter-example to this inequality [16], which involves theories with N = 2 supersymmetry, many known RG flows appear to obey (6.1). For example, it is obeyed for flows involving the scalar O(N ) [2,3] and the Gross-Neveu model [3]. If we think of the conformal QED 3 theory as the UV fixed point of the Thirring model, then the inequality (6.1) is obeyed by the flow to the free fermion theory because C T 1 (3) > 0. We may also test this inequality for the flow from the QED theory in the extreme UV, which consists of the free decoupled Maxwell field and N f 4-component fermions, to the conformal QED 3 . For the former we find using (5.9) and (3.8) For the interacting conformal phase, using our result (1.8), we have We see that at large N f (6.1) is obeyed to order N 0 f because 9 > 3 4192 45π 2 − 8 ≈ 4.32. Let us now try applying (6.1) to the d = 3 flow from QED in the extreme UV to the broken symmetry phase. For the former we have (6.2). The latter is a free conformal field theory of 2N 2 f + 1 scalar fields; therefore, it has We find that the two expressions are equal for N f = N f,crit = 1 + √ 2 ≈ 2.414. This suggests that theories with N f = 3 and higher are in the conformal phase. The inequality (6.1), however, does not require the N f = 1, 2 theories to be conformal, and indeed there is lattice evidence that they are not [21,22] where ψ i with i = 1, .., N f are the quark fields belonging to the fundamental representation of the colour group G, A a µ is the gluon field and c a andc a are the ghost fields in the adjoint 7 A more stringent value N f,crit = 3/2 follows from the RG inequality based on the coefficient of the thermal free energy [49]. This appears to be in contradiction with the lattice gauge theory work [22] claiming that the N f = 2 theory is not conformal. However, both N f,crit = 3/2 and N f,crit = 1 + √ 2 ≈ 2.414 are consistent with the recent paper [48] claiming that the symmetry breaking does not take place even for N f = 1.
representation of the colour group. We will use the following notation for the Casimirs of the Lie group generators t a ([t a , t b ] = if abc t c ): tr(t a t b ) = C(r)δ ab , t a t a = C 2 (r) · I, f acd f bcd = C 2 (G) · I (7.2) and also tr(I) = d(r) and δ ab δ ab = d(G). The stress-energy tensor is (2.9) with D µ = ∂ µ + iA a µ t a , and as we mentioned above we can omit the gauge fixing and ghost parts of T µν when computing correlation functions of gauge invariant operators. The diagrams contributing to C T to order 1/N f are the same as in the QED case (see figure 5). It is not hard to show that the relations between QED and QCD diagrams are where for some diagrams we used the identity d(r)C 2 (r) = d(G)C(r). Therefore, we find The subtraction of the second term is due to the gauging of the SU (N c ) Kac-Moody algebra with level k. Since there are 2N f 2-d Dirac flavors in the fundamental representation of SU (N c ), we have k = 2N f . This theory may be described by a SU (2N f ) Nc × U (1) WZW model [26,27]. Its central charge is which is in agreement with (7.5) evaluated in d = 2. For a general gauge group G we have . . , (7.8) which agrees with (7.4). Analogously, one can easily see that we have the same relation between C J in QCD and QED: where C J0 and C J1 are the results for QED given in (3.3) and (1.5).
Note Added: After the first version of this paper appeared, the value of C T for the d = 6 conformal Maxwell theory was calculated directly in [58]. The result is in agreement with our (3.12), providing a check of our methods.

Acknowledgments
We thank L.

A Calculation of Z T
In this appendix we present the computation of the Z T factor for the stress-energy tensor in the theory of Critical QED. As we show below, a non-trivial Z T is required for the Ward identity to hold. We define the "renormalized" stress-energy tensor T ren µν by , and T µν is the "bare" stress-tensor. To find Z T we will use the three-point function T ren where O ren m = Z Om O m is the electron mass operator, Z Om is its renormalization constant and the bare operator is This three point function is gauge invariant. So using conformal invariance and conservation of the stress-tensor, one has the general expression for the three-point function The conformal Ward identity gives where C Om and ∆ Om are two-point constant and anomalous dimension of the operator O m in coordinate space: Taking the Fourier transform of (A.3) and setting the momentum of the stress-energy tensor to zero for simplicity, one finds in terms of the projected stress tensor T = z µ z ν T µν whereC Computing these diagrams one finds   figure   9, and the explicit results are listed in eq. (A.14) below. Putting these diagrams together and equating the expression (A.7) required by conformal symmetry with the diagrammatic result for (A.13), we find that the required Z T factor is the one given in (3.6). As a check of our calculation, we note that dependence on the gauge parameter ξ drops out from the final result.
Let us end this section by listing the results for the diagrams in figure 9. They are given by , , , η m1 is given in (A.11). We notice that