Anomalous dimensions of scalar operators in $QED_3$

The infrared dynamics of $2+1$ dimensional quantum electrodynamics (QED$_3$) with a large number $N$ of fermion flavors is governed by an interacting CFT that can be studied in the $1/N$ expansion. We use the $1/N$ expansion to calculate the scaling dimensions of all the lowest three scalar operators that transform under the $SU(N)$ flavor symmetry as a Young diagram with two columns of not necessarily equal heights and that have vanishing topological charge. In the case of $SU(N)$ singlets, we study the mixing of $(\bar \psi_i \psi^i)(\bar \psi_j \psi^j)$ and $F_{\mu\nu} F^{\mu\nu}$, which are the lowest dimension parity-even singlets. Our results suggest that these operators are irrelevant for all $N>1$.


Introduction and summary
Quantum electrodynamics in 2+1 dimensions (QED 3 ) with N (two-component complex) charged fermions can be shown to flow to an interacting CFT perturbatively in the 1/N expansion [1,2]. In this CFT, there are many quantities that have been computed to various orders in 1/N . Examples include: the scaling dimensions of the lowest SU (N ) singlet scalars [3][4][5][6], adjoints [7,8], and a couple of other scalar operators [6]; the scaling dimensions of monopole operators [9][10][11]; the two-point functions of the canonically-normalized stressenergy tensor and of the conserved currents [12][13][14]; the S 3 free energy [15]; as well as various finite temperature quantities [8]. 1 Our goal here is to add to this list the scaling dimensions of many more operators: For each SU (N ) irreducible representation with two columns of fixed lengths, we identify the 3 lowest-lying scalar operators with zero monopole charge, and we compute their scaling dimensions to order 1/N .
Our interest in the scaling dimensions of scalar operators transforming in non-trivial representations of SU (N ) comes in part from the recent conformal bootstrap study [19] of where (λ ν 1 1 , λ ν 2 2 , . . . ) denotes a Young tableau with ν i rows of length λ i , and the n = 1 case is the adjoint. In tensor notation, the irrep (1) can be represented as a traceless tensor with n antisymmetric fundamental indices and n antisymmetric anti-fundamental indices.
While the bootstrap study [19] only examined relatively small values of N (namely N = 2, 4, and 6), future studies may be able to access larger values of N , and in order to assess their accuracy, one would benefit from more large N analytical approximations than those currently available in the literature. We thus develop the large N expansion for the scaling dimensions of scalar operators transforming as (1) under SU (N ). Of course, such large N expansions could also be useful independently of the conformal bootstrap program, for 1 See also [16][17][18] for estimates of some of the same quantities coming from the 4 − expansion.
instance if one engineers a new material that exhibits critical behavior described by QED 3 with a sufficiently large number of flavors.
An additional motivation exists for computing the scaling dimensions of lowest-lying parity even SU (N ) singlet operators. As we explain below, the lowest such operator has scaling dimension approximately equal to 4 at N = ∞, with negative 1/N corrections. If this operator becomes relevant at some finite value of N , it may completely change the IR physics if no tuning is performed. It is conceivable that for N ≤ N crit , the deep IR corresponds to a chiral symmetry breaking phase and that N crit can be estimated from when the scaling dimension of the lowest lying SU (N ) singlet approaches 3 [6,16,17]. Computing this scaling dimensions as a function of N would allow us to estimate N crit .
Let us present a summary of our results. For any n ≥ 0, for which the SU (N ) irrep is given by (1), we denote the lowest dimension operator by O n . As we show in Section 3.1, SU (N ) group theory requires that for n > 0, O n must be constructed from a product of precisely n distinct fermions anti-symmetrized in their SU (N ) indices and symmetrized in their spinor indices and a product of n distinct anti-fermions with the same property.
Furthermore, only a single operator can be built in this way, namely 2 where α m = 1, 2 are Lorentz spinor indices and i m = 1, . . . , N are flavor indices-see Section 2 for our conventions. This operator is parity even (odd) depending on whether n is even (odd).
We provide a formula for the scaling dimension ∆ n of this operator to order 1/N in Eq. (56) for all n > 0. This formula is rather complicated, so we record the scaling dimensions here only for the first several cases: 3 Next, we consider the lowest dimension operator in the same SU (N ) irrep as O n but with opposite parity. For that purpose, we must consider an operator constructed with one more 2 The construction of this operator requires n ≤ N/2. The regime where n is comparable to N is outside of the validity range of our approximation-we first fix n and then take N to be large. 3 The scaling dimension ∆ 1 was already computed in [7,8]. ∆ 2 agrees with the result of Sections II.B and II.C of [6]. The other operator in Section II.C of [6], with scaling dimension 4 + 64 , is a four-fermion operator transforming under SU (N ) as the irrep (2 N −1 , 4 1 ). ψ andψ each than O n . As we will show, for 0 < 2n < N there are two linearly independent such operators, which can be taken to be 4 By considering the mixing of these two operators (4), we calculate the scaling dimensions ∆ n,± to order 1/N for all n > 0. Since the final expression (Eq. (69)) is rather complicated, we will only record here the scaling dimensions for the first several cases: Lastly, the case n = 0 (SU (N ) singlet) requires special treatment. The lowest dimension Its scaling dimension is [5] ∆ 0 = 2 + 128 The two lowest dimension parity even operators are mixtures of the operators ψ i ψ i ψ j ψ j and F µν F µν . We find that the scaling dimensions are This result agrees with that of Ref. [6] that was obtained through a different method. 5 4 The construction of O n requires n ≤ N/2 − 1, and the construction of O n requires n ≤ N/2. The regime where n is comparable to N is outside of the validity range of our approximation-we first fix n and then take N to be large. 5 Ref. [6] studied the mixing of the operators ψ i ψ i ψ j ψ j and ψ i γ µ ψ i ψ j γ µ ψ j by adding these operators to the action and studying the renormalization of their couplings as one integrates out momentum shells. In our approach, we extract the scaling dimensions from the matrix of two-point functions, and in doing so we can make use of the equations of motion. The gauge field equation of motion,ψ i γ µ ψ i = 0, implies that the two-point function of ψ i γ µ ψ i ψ j γ µ ψ j vanishes at separated points. Instead of considering the Extrapolating (7) to finite N , one finds that all parity-even SU (N ) singlets are irrelevant for all values of N > 1. This result might suggest that the interacting CFT obtained in the 1/N expansion extends to all values of N > 1, in agreement with the recent lattice simulations of [20]. 6 It is worth mentioning that the scaling dimension of the four-fermion parity-even singlet was also estimated from the 4 − expansion in [16], where it was found that this operator is irrelevant only for N > 2 [16]. It would be interesting to understand how the mixing between the four-fermion operator and F 2 µν studied here affects the 4 − expansion estimates.
The rest of this paper is organized as follows. In Section 2, we set up our conventions and Feynman rules for QED 3 . Sections 3 and 4 contain the bulk of our computations.

Setup and conventions
Before we delve in the computations of the various scaling dimensions mentioned above, let us describe our conventions and the setup of our computation. The Euclidean signature Lagrangian of QED 3 with N fermion flavors is where e is the gauge coupling. The gamma matrices obey the Clifford algebra {γ µ , γ ν } = 2δ µν I and can be taken to be equal to the Pauli matrices γ µ = σ µ , for µ = 1, 2, 3. We choose to write fundamental spinor indices as lower and fundamental SU (N ) indices as upper, as in ψ i α , with i = 1, . . . , N and α = 1, 2. Anti-fundamental indices have the opposite index placement, as inψ α i . In the following we try to avoid as much as possible writing down explicit spinor indices, but we do write down the SU (N ) flavor indices explicitly. Repeated indices are always summed over.
As will become clear shortly, the gauge coupling e drops out of all computations in the IR CFT. Therefore, one can think of the fermions in (8) as carrying any gauge charge, and not necessarily the smallest unit of charge allowed by the U (1) gauge symmetry. The results of this paper are thus independent of the gauge charge of the fermions. mixing of ψ i ψ i ψ j ψ j and ψ i γ µ ψ i ψ j γ µ ψ j , we consider the mixing of ψ i ψ i ψ j ψ j and F µν F µν , as we do in Section 4. Despite the different methods, we obtain the same result as Ref. [6]. It would be interesting to perform a similar computation to the one in this paper in the case of an SU (2) gauge theory and compare with the results of [6]. 6 See, however, [21] where it was observed that for N = 2 there is spontaneous chiral symmetry breaking. Also, the F -theorem [17] implies that chiral symmetry breaking is ruled out for N ≥ 10.
In momentum space, the fermion propagator G(p) and the gauge field propagator D µν (p) are The gauge-fermion vertex factor is simply iγ µ . See Figure 1. Computing diagrams using these rules is then straightforward. The IR CFT behavior can be extracted by taking the limit of small external momenta. This limit is equivalent to taking e 2 → ∞ in all correlation functions because by dimensional analysis e 2 always appears as e 2 / |p|, where p is one of the external momenta.
Using the Feynman rules in Figure 1 is cumbersome for two reasons. The first reason is that at the CFT fixed point the Maxwell term is irrelevant, so there should be a way of performing the computation such that e 2 never appears and no limit needs to be taken at the end. In other words, there should be a way of performing the computation where the e 2 → ∞ limit is taken from the very beginning. The second reason is that at each order in 1/N there is an infinite number of fermion bubble diagrams that always get resummed in the same way, so one should resum them once and for all.
Let us address the second concern first. In order to avoid resumming the same bubble diagrams every time, one can define an effective gauge field propagator obtained after the resummation. See Figure 2. In order to obtain an explicit expression for the effective  (10): Each fermion bubble is nothing but the two-point function of the gauge current j µ =ψ i γ µ ψ i ; as follows from performing the required Wick contraction and using (11). Passing to momentum space, one has as follows from the formulas given in (94). As defined above, the effective gauge field propagator is just the sum of the fermion bubbles and takes the form of a geometric series: One can thus replace the gauge propagator (10) with (14) in order to not have to resum the bubble diagrams every time, and otherwise compute Feynman diagrams as usual.

Feynman rules with non-standard gauge fixing
As already mentioned, it would be nice to have a way of performing computations at the CFT fixed point without having to carry around e 2 and to take the limit e 2 → ∞ at the end of the computation. Unfortunately, the Maxwell propagator (10) and the effective propagator (14) do not generally have finite limits as e 2 → ∞, so this limit cannot in general be taken at the beginning of the computation.
An exception occurs in the gauge ξ = 0, where the effective gauge propagator (14) does have a finite limit as e 2 → ∞ and one can indeed take e 2 → ∞ from the beginning. As we now show, it is also possible to modify the gauge fixing term (9) so as to have a one-parameter family of gauge-fixing terms, not just that for ξ = 0, for which one can take e 2 → ∞ from the beginning.
Instead of (9), one can consider the non-local gauge-fixing term where ζ is a gauge-fixing parameter. Using (15) instead of (9), the Maxwell gauge field propagator in (10) gets replaced bỹ and the effective gauge propagator in (14) gets replaced bỹ As advertised, this expression has a finite limit as e 2 → ∞ for any ζ. Gauge invariant observables should of course be independent of ζ.

Summary of Feynman rules
To summarize, the momentum and position space Feynman rules we will work with are: where x 12 ≡ x 1 − x 2 and the position space expression for D µν is derived in (97). The vertex factor is iγ µ . See Figure 3.
In working with the effective gauge field propagator D µν one should keep in mind that this propagator stands for the sum of the bubble diagrams in Figure 2, so one should not Figure 3: Feynman rules used in this paper.
count the same Feynman diagram multiple times. In particular, one should not consider any effective gauge propagators renormalized by fermion bubbles, for instance as on the RHS of Figure 2 if the dotted lines were replaced by wavy lines.

General strategy for anomalous dimension computation
In this paper, we compute anomalous dimensions from the matrix of two point functions in We expect the following x dependence of the first two coefficients: This expression serves as a definition of the r × r matrices N and M. Here, Λ is the UV cutoff, which is required in order to make the argument of the logarithm dimenisonless. At order 1/N the anomalous dimensions are the eigenvalues ∆ (1) a of the matrix (see for instance [22]). The total scaling dimensions are thus In the examples below, we compute the matrices N and M and use this procedure to extract ∆ a .

Previous results
In the following we will use the previously computed results for the leading 1/N corrections to the scaling dimensions of the fermion field ψ and that of the 2-fermion

Correction to fermion propagator
Because the gauge fixing term (15) is conformally invariant, the two point function of ψ has powerlaw decay for any ζ. However, the corresponding scaling dimension ∆ ψ will depend on ζ and does not have to obey the unitarity bound for a spin-1/2 operator. We have The correction coefficient ∆ (1) ψ can be found from the diagram in Figure 4. It is found to be [5] ∆ (1)

2-fermion singlet
The We exhibit the diagrams that were used in evaluating ∆ (1) 0 in Figure 5.

2-fermion adjoint
Similarly, the dimension of the 2-fermion adjoint The diagrams that contribute to ∆

Number of operators
The scalar operators in 1 N −2n , 2 n , being gauge invariant, must be constructed from an equal number of ψ's andψ's. Let us count how many linearly independent operators we can construct out of m ψ's and mψ's and determine the smallest value of m that is necessary in order to be able to construct at least one such operator.
where {x} denotes the fractional part of x and (2j) denotes the spin-j irrep of SU (2). See Each term in the sum (30) can be further decomposed as a sum of irreducible representations of SU (N ). Performing this decomposition is a straightforward group theory exercise, and one can then count how many times the irrep 1 N −2n , 2 n we are interested in appears in this decomposition. The result is that if m < n, the irrep 1 N −2n , 2 n does not appear at all: we need at least n ψ's and nψ's in order to construct an operator transforming in 1 N −2n , 2 n . If m = n, the irrep 1 N −2n , 2 n appears in the decomposition of (30) only once, and it comes from the term j = n/2; the corresponding operator can be written explicitly as where we symmetrize and anti-symmetrized with unit weight, and the traces are over SU (N ) indices. This operator is non-zero only for 2n ≤ N . When m = n + 1, the irrep 1 N −2n , 2 n appears in (30) twice, once coming from j = m/2 and once form j = m/2 − 1. The corresponding linearly independent operators can be taken to be where O n corresponds to j = m/2 and O n is a linear combination of an operator from j = m/2 − 1 and j = m/2 that is easy to write down. Note that O n is non-zero only if 2n < N and O n is non-zero only for 2n ≤ N . It is straightforward to use the same method to also count the multiplicity of the irrep 1 N −2n , 2 n when m ≥ n + 2, but we will not be concerned with those cases here.

Scaling dimension of O n
We consider a particular operator representing (31) by taking i k = k: where the trace term in (31) does not contribute because all the i k are distinct. This operator can be rewritten as where the spinor indices are contracted between adjacent fermions, and sig(σ) is the signature of the permutation σ ∈ S n . The conjugate of O We would like to express the two-point function of O n as in Section 2.3. Directly from the definition (34), we can write For any permutation τ ∈ S n , we can perform the relabeling ψ i → ψ τ (i) , which shows that . We can freely apply such a transformation to each term in (36) separately. Taking τ = (σ ) −1 and denoting σ τ = σ, we see that (36) reduces where I is the identity permutation. Noticing that each permutation in a given conjugacy class gives an equal contribution to the two-point function, we can express (37) as a sum over conjugacy classes C n,i of the symmetric group S n : Since conjugacy classes of the symmetric group S n will appear several times in this section, let us briefly review their properties. Conjugacy classes of S n are in one-to-one correspondence with integer partitions of n. Suppose we write such an integer partition corresponding to a conjugacy class C n,i as for some positive integers a ij . All permutations in C n,i have a ij cycles of length j. In terms of this data, the size and signature of C n,i can be expressed as , sig(C n,i ) = (−1) n j=1 a ij (j−1) .

Leading order
At leading order at large N , we can evaluate O   where we used (40) and the fermion propagator in (18). Then, using (38), we find The sum n j=1 a ij gives the number of cycles in conjugacy class C n,i . Explicitly, and so on.

Next-to-leading order
For the next order in 1/N , we should consider diagrams with one photon line. In there are several possibilities for where to draw the photon line: . . .
x 0 • the photon line can connect a fermion line to itself. Each such diagram gives There are 2j such diagrams for a permutation cycle of length j, for a total of 2n diagrams. See Figure 7 for an example.
• the photon line can connect fermion lines belonging to different cycles of C n,i . These diagrams cancel in pairs-See Figure 8.
• the photon line can connect distinct fermion lines of opposite types (one G(x, 0) and one G(0, x)) within the same cycle of C n,i . See the lefthand diagram in Figure 9. Let this cycle have length k. In position space, such a diagram is where the first term in parentheses comes from the cycles without photon lines, and the contribution we exhibited is that coming from the photon line. The number of fermion propagators between those containing photon lines is k 1 and k 2 , with k 1 + k 2 = k − 1.
• the photon line can connect distinct fermion lines of the same type (either both G(x, 0) or both G(0, x)) within in the same cycle of C n,i . See the righthand diagram in Figure 9.   contribution equal to where again the first term in parentheses comes from the cycles without photon lines, and the contribution we exhibited is that coming from the photon line. Here, From D k and E k we have to extract the logarithmic divergence. While these are very complicated diagrams and their full evaluation would be an onerous task, the extraction of the logarithmic divergence is quite easy, because it comes either from when z and w are both close to x or to 0. Both limits give the same answer, so we can just take the limit where both z and w are close to 0 and multiply the answer by 2. For D k , Eq. (46) thus becomes where we used G(x, 0)G(0, x) = − 1 (4π) 2 x 4 I. The position space integral can be written in Fourier space as an integral over a Fourier momentum q: This expression can be seen to evaluate to −8(3 − ζ) log Λ 2 /(N π 2 ) after performing the required gamma matrix algebra and using the gauge field propagator in (18). Here, Λ is the UV cutoff and it must appear inside the logarithm in the combination Λ |x|. Thus, D k (x) evaluates to A similar strategy works for evaluating the logarithmic divergence in E k . Taking the limits when z, w go to x or 0, one obtains The momentum space integral now gives 8(1 + ζ) log Λ 2 /(N π 2 ), so in the end Due to the various ways of placing the gauge propagator, there are k 2 diagrams that give D k and k(k − 1) diagrams that give E k . Along with the leg contributions, we have Quite nicely, after plugging in (45), (50), and (52) into (53) one obtains an expression independent of ζ: Using (37) and (42), we can write the ratio between the 1/N correction to the two-point function and the leading result as The results of Section 2.3 then imply that the scaling dimension of O n is This expression can be evaluated for any n using the data for the conjugacy classes of the permutation group. When n = 1, one has only one conjugacy class C 1 of size |C 1 | = 1 and a 11 = 1; it is easy to see that (56) reduces to (27).

Scaling dimension of O n and O n
We consider particular operators representing (32) by choosing i k = k: with O n as in (33) and O 0 as in Section 2.4.2. Since we have two operators that mix together, we must consider the matrix of 2-point functions as in Section 2.3 and expand it in 1/N .

Note that we can write
where σ is a permutation of the set {1, . . . , n + 1}, and σ k = π k • σ, π k being the map π k (i) = i for i = 1, . . . , n and π k (n + 1) = k. This expression is somewhat similar to that for O n+1 , which is an observation that will simplify some of our computations.
The two-point function O n (x)Ō n (0) is the simplest to calculate because it factorizes not just at leading order in 1/N , but also at the first subleading order: The factorization at next-to-leading order is because the diagrams formed by photon lines between O n and O 0 all cancel in pairs. From (55) and (26), we have the following ratio of subleading to leading orders where I is the identity permutation, andĨ is the transposition that flips n and n + 1.
At sub-leading order, the two-point functions (63) have the same diagrams as O n+1Ōn+1 (1) , but also differ from O n+1Ōn+1 (1) due to the occurrence of additional diagrams where the ψ k andψ k belonging to either O n or O n are joined together by a fermion line. See Figure 10.
These additional diagrams are similar to the last two diagrams in Figure 5. We thus obtain The n-dependent prefactors in the second lines of (64) and (65) can be understood as follows. In the case where ψ k andψ k belong to O n , we either have that ψ k can be contracted withψ k , or ψ k andψ k can be part of a bigger cycle. Out of (n + 1)! total possibilities, the first case occurs n! times, while the latter occurs n!n times and has an extra factor of −1/2 relative to the first, because there is one fewer trace and permutation for this diagram.
Summing both cases we find that whenever O n is involved we must include a factor of n! − n!n/2 (n + 1)! = 1 − n/2 n + 1 (66) relative to the n = 0 case of the last two diagrams in Figure 5. For O n we do not need any extra factors. Thus (64) contains one power of (66) and (65) contains two powers of (66).
Gathering the previous results we can write down the M and N matrices defined in (20): where From this expression and (22), we can extract the anomalous dimensions by diagonalizing Particular cases are given in (5) in the Introduction.

The mixing of lowest parity-even SU (N ) singlets
We now consider the parity-even SU (N ) singlets. At large N , all these operators are irrele-  For instance, at order N 0 , the diagrams in Figure 11 can be seen to cancel exactly. 7 Let us thus focus on the matrix of two-point functions of O i , with i = 1, 2 and write it at large N in the form given in Section 2.3. In particular, let us compute the matrices M and N defined in (20).
At leading order in N , we have Rewriting O 2 as 7 We thank Mark Mezei for discussions on this issue. and using the gauge field propagator D µν in (18) gives Since at order N 0 , the two-point function O 1 (x)O 2 (0) (0) vanishes, the matrix N defined in (20) is In order to compute M, it is natural to think of each O 1 as a composite betweenψ i ψ i andψ j ψ j , and of O 2 as the composite between F µν and F µν . There are many diagrams that contribute to M but they can be split into diagrams referred to as leg corrections coming from each of the factors of the composites as well as diagrams referred to as vertex corrections that mix together the two factors.
The leg correction diagrams have already been computed. Since the operatorψ i ψ i ac-  quires an anomalous dimension given by 128/(3π 2 N ) + O(1/N 2 ) (see (6)) and F µν has no anomalous dimension, we have that the leg contribution to M is Let's start with O 1 (x)Ō 2 (0) (bottom diagram in Figure 12). In position space, this diagram can be written as As in the previous sections, we are content with extracting only the logarithmic divergence of this diagram, leaving its full evaluation to future work. The log divergences come from regions where z, w, y, t are either close to x or to 0. The first one is This is just a one-loop diagram, written in Fourier space as This expression evaluates to To evaluate the contribution from when z, w, y, t are close to 0 in (76) one has to be more careful. To obtain the log divergence, one has to expand the D's in the first line of (76) to linear order in z and w as these quantities tend to zero: where (The terms proportional to z α z β and w α w β in the expansion give vanishing contribution to the final answer and can be dropped.) In (81), the x dependence appears only implicitly as Λ |x|, Λ being the UV cutoff. This is a 3-loop diagram
The expressions for H 2 and H 3 differ from (90) only in the placement of the derivatives ∂ σ and ∂ τ . It is tedious but straightforward to perform the r integral first and then the q integral using the formulas in [23]. The result is that these integrals do not have logarithmic divergences.
Putting everything together, we have that Combining with (74) with (91) and (75), we can write down the anomalous dimension matrix From the eigenvalues of this matrix, we conclude that the two parity even SU (N ) singlet operators have scaling dimensions given in (7).