Easy-plane QED$_3$'s in the large $N_f$ limit

We consider Quantum Electrodynamics in $2{+}1$ dimensions with $N_f$ fermionic or bosonic flavors, allowing for interactions that respect the global symmetry $U(N_f/2)^2$. There are four bosonic and four fermionic fixed points, which we analyze using the large $N_f$ expansion. We systematically compute, at order $O(1/N_f)$, the scaling dimensions of quadratic and quartic mesonic operators. We also consider Quantum Electrodynamics with minimal supersymmetry. In this case the large $N_f$ scaling dimensions, extrapolated at $N_f{=}2$, agree quite well with the scaling dimensions of a dual supersymmetric Gross-Neveu-Yukawa model. This provides a quantitative check of the conjectured duality.

Quantum Electrodynamics (QED) in 2+1 dimensions, with fermionic and/or bosonic flavors, is a prime example of interacting Quantum Field Theory, with both theoretical and experimental relevance. In this paper we study QED's in the limit of large number of flavors, the large N f limit, where perturbation theory allows to find quantitative results.
Our goal is to define and study models that admit a tractable large N f expansion but at the same time might be realistic when the number of flavor is small. For this reason we consider an even number of flavors and allow for interactions that respect only U (N f /2) 2 global symmetry, instead of the usual U (N f ). Most of the results derived in this paper have been presented in [1]. We use the name "easy plane" QED's because for N f =2, one of the bosonic fixed points is the "easy-plane" CP 1 model. Together with SU (2)-CP 1 model it describes the Nèel -Valence Bond Solid quantum phase transition in the SU (2) and XY magnets [2][3][4].
We find four bosonic and four fermionic fixed points. The various models differ by the form of the quartic interactions, which in the large N f limit are modeled introducing one or two Hubbard-Stratonovich scalar fields. In each of the 8 models we systematically compute the anomalous dimensions of all the scalar operators that at infinite-N f have small scaling dimension (∆=1 or ∆=2). Some operators are quadratic or quartic in the charged fields, some are linear or quadratic in the Hubbard-Stratonovich fields. We work at the leading non trivial order in the large N f expansion, O(1/N f ), providing many details of the computations, including results for all individual Feynman diagrams.
The results of this paper were used in [1] to argue for the following physical picture. If the number of flavors N f is large enough, it is well known that the Renormalization Group (RG) flows in the infrared to a Conformal Field Theory (CFT) describing a second order phase transition. However, when N f is decreased, the large N f anomalous dimensions suggest that below a certain critical value N * f (which is a model dependent quantity), the various fixed points merge pairwise and become complex Conformal Field Theories. This is also called annihilation and merging scenario: varying a parameter, in this case the number of flavors, two real fixed points of the RG flow (real CFT's) annihilate and become a pair of complex conjugate CFT's. The RG flow preserves unitarity and doesn't hit those complex fixed points, instead it slows down while passing between the complex fixed points (walking) [5]. For N f < N * f , the infrared physics is not described by a second order phase transition, but by a weakly first order phase transition.
Studying quantum field theories in the large N f limit has been proved to be useful in different circumstances. In 2+1d the large N f limit has recently been applied to calculate scaling dimensions of monopole operators, S 3 partition functions and central charges [52][53][54][55][56][57][58][59]. We think that it would be interesting to generalize those computations to the "easy-plane" models described in this paper.
After discussing QED's with bosonic flavors in section 2 and QED's with fermionic flavors in section 3 we, move to QED with minimal supersymmetry, N = 1. In section 4 we compute the scaling dimensions of bilinear and quartic mesonic operators. We also include the large N f dimensions of monopole operators from [57]. In this case, lowering N f there is no evidence of merging or symmetry breaking. Instead, even for N f =2 the large N f results are physical. N = 1 QED with N f = 2 is supposed to be dual to a supersymmetric Gross-Neveu-Yukawa model [60,61], which can be studied quantitatively in the 4 − ε expansion [61]. We compare the large N f results on the gauge theory side of the duality with the 4 − ε results on the Gross-Neveu-Yukawa side of the duality, and we find good quantitative agreement, providing a check of the conjectured N = 1 duality.
2 Four bosonic QED fixed points in the large N f limit In this section we study bosonic QED with large N f complex scalar fields, imposing at least U (N f /2) 2 global symmetry. There are four different fixed points, two fixed points have U (N f ) global symmetry, two fixed points have U (N f /2) 2 global symmetry.
We start by considering the following UV (Euclidean) lagrangian ., N f /2) carry charge +1 under the gauge group. The conformal gauge fixing is defined by the last term in (2.1). Choosing the gauge fixing parameter to be zero (ξ = 0) simplifies the calculations a lot, however we prefer to keep ξ arbitrary (notice that in this parametrization ξ = 1 is the Landau gauge). Calculating correlation functions of gauge invariant operators, we will see that some Feynman graphs depend on ξ, but the sum (at a given order in 1/N f ) doesn't as expected. This is a useful check of the calculations. In the following, we will always assume conformal gauge fixing for all the QED actions, but will not write it explicitly. The quartic potential in (2.1) is a relevant deformation of the free theory. As explained in [1], depending on the form of the quartic couplings {λ ep , λ} there are four different fixed points 2 : • bQED (tricritical), defined by vanishing quartic potential, In appendix (A) we study the RG flow diagram and the fixed points of the model (2.1) using the epsilon expansion technique. The zeros of the beta functions support the existence of precisely these four RG fixed points. See [1,85] for discussions about the ungauged fixed points and RG flow.
We study the critical behaviour of the fixed points in the large N f limit. For this purpose we engineer the quartic interactions in terms of cubic and quadratic interactions via the Hubbard-Stratonovich trick. Introducing two Hubbard-Stratonovich (HS) fields σ andσ, we get an expression equivalent to (2.1) Integrating out σ andσ, one recovers the quartic potential in (2.1) with couplings {λ ep , λ} expressed in terms of {η 1 , η 2 }: It is sometimes convenient to work with the following HS fields With the choice (2.5) there is no mixed quadratic term between σ + and σ − .

bQED (tricritical QED)
The bQED is reached tuning to zero both the mass terms and the quartic interactions. For this reason another name for it is tricritical. The large N f effective action is described by N f copies of complex scalars Φ i (we collected all the scalars (Φ,Φ) into a single field and denoted it by Φ) minimally coupled to the effective photon   Figure 2: bQED Feynman rules.
Effective photon propagator is obtained by summing geometric series of bubble diagrams such as (1).
The feynman rules for the bQED action (2.7) are summarised in the table (2). The faithful global symmetry is Where Z N f is the center of SU (N f ), generated by e 2πi/N f I ∈ SU (N f ), which is a gauge transformation, so the symmetry is P SU There is also parity symmetry. Using the Feynman rules (2), we compute anomalous dimensions of gauge-invariant operators at order O(1/N f ). For this purpose, first we calculate the 2-point correlation function for a given operator, then using it we extract anomalous contribution to the scaling. It might happen that for a given model there are several gauge invariant operators that have the same scaling dimensions at the order O(N 0 f ) and carry the same quantum numbers. These operators can mix by quantum corrections at order O(1/N f ) and one needs to study the matrix of mixed 2-point correlation functions in order to correctly identify the eigenbasis of mixed operators and their anomalous dimensions.

Scaling dimension of low-lying scalar operators
Bilinear mesonic operators At the quadratic level, there are N 2 f operators of the form Φ * i Φ j . They transform in the adjoint plus singlet representations of SU (N f ): The 2-point correlation function for the adjoint operator is the sum of the graphs A,B,C (3). For the adjoint operator the graphs D and E are of order O(1/N 2 f ) because each photon brings a factor 1/N f and the two blue loops are of order O(N 0 f ). All the divergent graphs are regularized by putting an UV cutoff Λ on the momentum integrals. Check the appendix (C) for more details of the loop calculations.
adj . (2.12) Where we defined anomalous dimension of adjoint operator ∆ (1) adj , so ∆ (1) adj = − 64 3π 2 N f . We extract the anomalous dimension for the singlet operator in a similar way. Notice that for the singlet operator there is an additional order O(1/N f ) contribution coming from the graph D (3) (as opposed to the adjoint bilinear, in the singlet case each loop in (D,E) gives a factor N f ). Below we give the final results Quartic mesonic operators Next we consider scalar quartic operators T ij kl is a gauge invariant operator, symmetric in its upper and lower indices. The following decomposition of T into irreducible representations under the SU (N f ) group is useful for discussion of their scaling dimensions The first, second and third terms in the right hand side of (2.16) are correspondingly singlet, adjoint and adjoint-2 (Dynkin labels [2, 0, . . . , 0, 2]) quartic operators. All of them have scaling dimension 2 at leading order, it remains to calculate order O(1/N f ) corrections.
Let us consider quartic adjoint-2 operator defined by the last term of (2.16). It is enough to study the two-point correlation function for only one component of the adjoint-2 representation, which we choose to be All the relevant graphs for extracting the anomalous dimension of the operator (2.17) are collected in the table (4) (the last graph comes at sub-leading order). It receives contribution from the anomalous dimensions of the Φ i fields (there are 4 such graphs) plus graphs with a photon connecting two different legs ("kite"-graphs, there are 6 "kite"-graphs). In 2 "kite"-graphs the photon connects the scalar propagators with arrows going in the same direction, while in the other 4 "kite"-graphs the photon connects propagators with arrows going in the opposite direction. The contribution of a "kite"-graph where the photon connects arrows going in the same direction is equal to minus the contribution of a "kite"-graph where the photon connects arrows going in the opposite direction. So effectively we are left with the contribution of 2 such "kite"-graphs. 4 For the quartic adjoint operator the last graph in (4) contributes at order O(1/N f ). For the singlet quartic operator the last graph contributes twice as much as for the 4 One can consider degree-2k operators which transform in the adjoint-k representation (Dynkin labels [k, 0, . . . , 0, k]). These operators do not mix with other operators. The anomalous dimension of a degree-2k adjoint-k operator, at order O(1/N f ), receives contribution from the anomalous dimensions of the Φ i fields (there are 2k such graphs) plus the contribution of "kite" graphs (there are 2k 2 = 2k 2 −k "kite"-graphs). In 2 · k 2 = k 2 − k "kite"-graphs the photon connects fields with arrows going in the same direction, while in the other k 2 "kite"-graphs the photon connects fields with arrows going in the opposite direction. These two groups of "kite"-graphs contribute with opposite signs, so effectively we are left with the contribution of k 2 − (k 2 − k) = k such "kite"-graphs. Therefore the scaling dimension of the degree-2k adjoint-k operator is quartic adjoint operator. We list the quartic operators and their scaling dimensions and by tuning the mass term to zero. In the literature this model is also known as Abelian Higgs model or CP N f −1 model. The large N f effective action is described by N f copies of complex scalars Φ i (we collected all the scalars (Φ,Φ) into a single field and denoted it by Φ), minimally coupled to an effective photon and interacting with a single Hubbard-Stratonovich field σ + via a cubic interaction: The effective photon propagator is the same as in (2.8) and the effective propagator for the HS field is obtained from summing geometric series of the bubble diagrams in (5).
The global symmetry is the same as in bQED (2.9).

Scaling dimension of low-lying scalar operators
The N 2 f gauge invariant operators Φ * i Φ j transform in the adjoint plus singlet of SU (N f ). The singlet operator is set to zero by the equation of motion of the Hubbard-Stratonovich field σ + . 5 So we consider the scaling dimension of σ + instead. The scaling dimensions of these operators can be read from the table (6) As a simple check of this statement, one can explicitly check that the two point function The 1-loop diagram cancels with a 2-loop diagram given by two bubbles connected by a σ + propagator (normalizing the singlet operator as 1 √ both such graphs are of order 1 at large N f ). We thank to Silviu Pufu for clarifying this point.
Next we discuss scaling dimension of the quartic adjoint-2 operator (with Dynkin labels [2, 0, ..., , 0, 2]). This operator is in the spectrum and has scaling dimension 2 at order O(N 0 f ). The graphs that contribute to its 2-point correlation function at the order  (7). There are 4 graphs with HS field connecting a leg with itself and 6 kite graphs with the HS field joining two different legs. Summing all the contributions we can extract anomalous dimension of the adjoint-2 quartic operator

ep-bQED ("easy-plane" QED)
The ep-bQED fixed point is reached with the quartic potential V ∼ ( 2 and by tuning the mass terms to zero. The large N f effective action is described by complex scalar fields (Φ i ,Φ i ) minimally coupled to the effective photon and interacting with two HS fields via cubic interactions (2.28) The effective propagator for the photon is the same as in (2.8), and the effective propagators for the HS fields are (2.29) The photon "sees" all the N f flavors, σ andσ only "see" N f /2 flavors. In the Feynman graphs, we use red dashed (double dashed) line for σ (σ) and blue (double blue) line for Φ (Φ).
The global symmetry of the effective action (2.28) is There is also parity invariance.

Scaling dimension of low-lying scalar operators
The N 2 f quadratic gauge invariant operators transform as two adjoints, two singlets and two bifundamentals of SU (N f /2) 2 . More precisely, in the reducible representation where by F we denoted the fundamental representation of SU (N f /2). Feynman graphs that contribute to the anomalous scaling dimension of |Φ| 2 adj are the graphs in the right column of the table (6). One has to keep in mind that the photon "sees" all the flavors, while each sigma field "sees" only half of them, therefore the contribution of graphs that involve an HS propagator is twice as big as the contribution of the corresponding graphs in bQED + . For the adjoint operator |Φ| 2 adj one has the same set of graphs, but the blue lines are exchanged by blue double lines and red dashed lines are exchanged by red dashed double lines. On the other hand, the scaling dimension of the bifundamental operators (Φ iΦ * j , Φ * iΦ j ) is corrected by graphs similar to those in the right column of (6), except that the last graph is absent. The two scalar-bilinear singlets are set to zero by the equations of motion of the HS fields σ andσ.
The 2-point correlation function σ(x)σ(0) is corrected by the left column graphs (6), and similar graphs stand for σ(x)σ(0) . It is preferable to denote by U the effective propagator of the HS field σ (2.29), then the graphs involving single photon contribute as in bQED + , the graphs involving HS propagator contribute 2 times the corresponding graphs in bQED + , the graph involving two photons contributes twice less than the same graph in bQED + . So we conclude that the O(1/N f ) corrected propagator for the HS σ field is (2.32) It turns out that already at order O(1/N f ) there is a mixing between HS fields σ and σ (8). Figure 9: Renormalization of the (sym, sym) quartic operator.
The HS fields σ ± defined in (2.5) are the eigenvectors of the mixing matrix. Using (2.32, 2.33) one readily extracts anomalous dimensions of those fields.
The N 4 f quartic gauge invariant operators transform as reducible representation of SU (N f /2) 2 with the following decomposition into irreducible blocks Where by R we denote the representation of SU (N f /2) with Dynkin labels [2, 0, ..., 0, 1]. All the irreducible blocks in the third row of (2.34) contain a singlet quadratic factor and therefore they are out of spectrum. In the table (9) we collected all the relevant graphs for extracting the anomalous scaling dimension of the operator (sym, sym) = Φ * i Φ * jΦ kΦl . One can make similar tables for the other quartic operators which are in the spectrum.
The scaling dimensions are as follows

bQED −
The bQED − is reached with quartic deformation V ∼ ( |Φ i | 2 − |Φ i | 2 ) 2 and by tuning mass terms to zero. Large N f effective action is described by complex scalar fields (Φ i ,Φ i ) minimally coupled to the effective photon and interacting with single HS field via cubic interaction.
Effective propagator for the photon is the same as in (2.8), and the effective propagators for the HS field σ − is as follows In the Feynman graphs we will use red dashed line for the effective propagator of σ − . The global symmetry of the bQED − action is the same as for ep-bQED (2.30).

Scaling dimension of low-lying scalar operators
The N 2 f quadratic gauge invariant operators are decomposed into irreducible representations of SU (N f /2) 2 as in (2.31). Feynman graphs that contribute to the scaling dimensions of the operators {|Φ| 2 adj , |Φ| 2 adj } are those in the left column of (6). The same graphs can be used to calculate scaling dimension of the bifundamental operators {Φ iΦ * j , Φ * iΦ j }, however the graph with HS field σ − joining propagators Φ andΦ contributes with the opposite sign compared to the similar graph in the bQED + . This is because the cubic vertices with HS field coupled to the scalar flavors (Φ,Φ) have different signs as it follows from the effective action (2.44). Notice that EOM of the HS field σ − sets to zero the operator ( |Φ i | 2 − |Φ i | 2 ). Therefore that operator is out of the spectrum, while the plus combination is in the spectrum and has a dimension 1 at leading order.
The N 4 f quartic gauge invariant operators are decomposed into irreducible representations of SU (N f /2) 2 as in (2.34). Notice that in the last line of (2.34) not all the operators are excluded from the spectrum: the quartic operators which are a product of a quadratic operator ( |Φ i | 2 + |Φ i | 2 ) and a quadratic adjoint or bifundamental operator, as well as the quartic operator ( |Φ i | 2 + |Φ i | 2 ) 2 are in the spectrum and have scaling dimension equal to 2 in the leading order. In the table (10) we collected all the graphs that contribute to the anomalous scaling dimension of the quartic bifundamental operator . Similar computations can be done for the other operators. Below we give the list of operators and their scaling dimensions.
3 Four fermionic QED fixed points in the large N f limit In this section we study fermionic QED, with large N f complex fermionic flavors, imposing at least U (N f /2) 2 global symmetry. There are four different fixed points, two fixed points have U (N f ) global symmetry, two fixed points have U (N f /2) 2 global symmetry.
As explained in [1], depending on the form of the Yukawa interactions, there are four different fixed points: • fQED, both HS fields are massive and the Yukawa interactions are absent, • QED-GN + , the Yukawa interaction involving HS field ρ + is turned on and the HS field ρ − is massive, • QED-NJL (gauged Nambu-Jona-Lasinio), both HS fields ρ ± are massless, and both Yukawa interactions are turned on, • QED-GN − , the Yukawa interaction involving HS field ρ − is turned on and the HS field ρ + is massive.

fQED
In fQED both HS fields massive and therefore decoupled from the IR spectrum. The large N f effective action for the fQED fixed point is described by N f copies of Dirac fermions Ψ i (we collected all the fermions (Ψ,Ψ) into a single field and denoted it by Ψ) minimally coupled to the effective photon The effective photon propagator is obtained summing geometric series of bubble diagrams (1), where all the scalar (blue) loops are exchanged with fermion (green) loops.
We notice that the effective photon propagator in the fQED coincides with the effective photon propagator in the bosonic QED's. This is because the fermion and boson loops that appear in the geometric sums are equal to each other. Feynman rules for the vertices and for the propagators are given in the table (11).
The faithful global symmetry is Where Z N f is generated by e 2πi/N f I, −1 ∈ SU (N f ) × U (1) top (this fact comes from a careful treatment of the monopoles operators, which are dressed with fermionic zeromodes). Z C 2 is the charge-conjugation symmetry. There is also symmetry under parity. 7 Scaling dimension of low-lying scalar operators 8 The N 2 f gauge invariant operatorsΨ i Ψ j transform in the adjoint plus singlet of SU (N f ). Their scaling dimensions at large N f can be read from (12) and have been already discussed in [72][73][74]

QED-GN +
In the QED-GN + fixed point the action is (3.1), with Yukawa interaction involving HS field ρ + , while the HS field ρ − is massive and is decoupled from the IR spectrum. The large N f effective action is described by N f copies of Dirac fermions Ψ i (we collected all the fermions (Ψ,Ψ) into a single field and denoted it by Ψ) minimally coupled to the effective photon and interacting with HS field ρ + via Yukawa interaction The effective propagator for the photon is the same as in the fQED (3.3). The effective propagator for the HS field ρ + follows from summing geometric series of bubble diagrams as in (5) with all the scalar (blue) loops exchanged with fermion(green) loops (3.8) In the Feynman graphs we use a red dashed line in order to represent the ρ + propagator. The global symmetry is the same as in fQED (3.4). There is also parity symmetry (ρ + is parity-odd).

Scaling dimension of low-lying scalar operators
As in fQED, the N 2 f gauge invariant operatorsΨ i Ψ j transform in the adjoint plus singlet representation of SU (N f ). However, the singlet operator is set to zero by the equation of motion of the HS field ρ + . Order O(1/N f ) scaling dimensions for the adjoint operator and for ρ + can be read using table (13), for ρ 2 + using table (14) The scaling dimension of the order parameter ρ 2 + is related to the critical exponent ν: Note added: soon after we presented these results in [1], also [75] computed the scaling dimensions (3.9, 3.10, 3.11). Their results agree with ours.

QED-NJL
In the QED-NJL fixed point, the action is (3.1). It involves Yukawa interactions and the masses of the HS fields are tuned to zero. The large N f effective action is described by N f Dirac fermions (Ψ i ,Ψ i ) minimally coupled to the effective photon and interacting with the HS fields (ρ,ρ) via Yukawa interactions The last Feynman graph is vanishing because the triangle subgraphs made by fermion propagators are identically zero.
where ρ = ρ + + ρ − (3.14) The photon "sees" all the flavors, therefore effective photon propagator is the same as in fQED (3.3). The effective propagators for the HS fields are (3.16) The continuous global symmetry is: Parity is preserved, provided (ρ,ρ) and ρ ± are odd under parity transformation. The other global symmetries act as follows.

Scaling dimension of low-lying scalar operators
The gauge invariant fermion bilinear operators are classified as irreducible representations (2.31) under SU (N f /2) 2 symmetry group. The calculation of the scaling dimensions for the adjoint and the bifundamental operators is parallel to the calculation of the scaling dimensions of the similar operators in the ep-bQED and can be done using the graphs (13).
The quadratic singlet operators are out of the spectrum, they are set to zero by the EOM of the HS fields ρ,ρ. The two-point correlation function for the ρ field can be calculated using the left column diagrams of the table (15). Taking into account the necessary changes we get (3.18) Notice that at order O(1/N f ) there is a mixing between ρ andρ (15).
Instead, the fields ρ + , ρ − do not mix, they are the eigenvectors of the mixing matrix. Using (3.18, 3.19) one can calculate anomalous dimensions of these fields.
In the table (16) we collected all the graphs that contribute to the mixing of operators quadratic in HS fields: {ρ 2 (x),ρ 2 (x), √ 2ρρ(x)}. We get the following mixing matrix (3.20) WhereZ is defined in (16). Using (3.20) it is straightforward to pass to the eigenbasis and find the scaling dimension for each of the eigenbasis operators. Below we give the list of operators and their scaling dimensions A similar model with two HS scalars was studied in [76]. Their model seems to be different from QED-NJL we discuss, in particular the anomalous dimensions of HS fields are different from ours.  Figure 16: (QED-NJL) mixing of quadratic in HS field operators.

QED-GN −
In the QED-GN − fixed point the action is (3.1), with Yukawa interaction involving the HS field ρ − , while the HS field ρ + is massive and is decoupled from the IR spectrum.
The large N f effective action is described by Dirac fermions (Ψ i ,Ψ i ) minimally coupled to the effective photon and interacting with the HS field ρ − via Yukawa interaction The effective photon propagator is as in (3.3) because the photon "sees" all the flavors. The effective propagator for the HS field ρ − is (3.29) The continuous global symmetry is the same as that of QED-NJL (3.17). Parity is preserved provided HS field ρ − is odd under parity transformation. The discrete symmetry

Scaling dimension of low-lying scalar operators
The fermion bilinear operators are classified according to irreducible representations of SU (N f /2) 2 , like in (2.31). Notice that ρ − takes the operator out from the spectrum, while the plus combination remains in the spectrum and has dimension 2 at the leading order. The scaling dimension of the HS field ρ − is calculated using the graphs in the left column of the table (13). The contributions of the first 5 graphs remain unchanged, while there are 4 three-loop graphs and they are canceling each other. This is due to the fact that ρ − field couples to the fermion flavors (Ψ,Ψ) with different signs and therefore the three loop graph which has fermions Ψ running in one of its loops and fermionsΨ running in the other loop comes with an opposite sign with respect to the three loop graph made solely by fermions Ψ (orΨ). The scaling dimensions for the other operators can be calculated easily.
Some of these results have been obtained in [77][78][79], which also include some scaling dimensions at order O(1/N 2 f ).

Super-QED in the large N f limit
In this section we compute the large N f scaling dimension of mesonic operators in QED with minimal supersymmetry, and then compare the results, at N f = 2, with a dual Gross-Neveu-Yukawa model. At the end we also consider an N = 2 superQED. The UV action of 2 + 1d QED with minimal supersymmetry, N = 1 (i.e. 2 supercharges), N f flavors and zero superpotential has the form The action (4.2) is written in the Minkowski metric. Our convention for the Minkowski metric is (+, −, −). The kinetic terms for the photon and for the gaugino are non canonically normalised, the covariant derivative is D µ = ∂ µ + iA µ . We have N f flavors of Dirac fermions and complex scalars: Ψ j , Φ j , j = 1, ..., N f . Our conventions for the gamma matrices are: γ 0 = σ 2 , γ 1 = iσ 1 , γ 2 = iσ 3 , where σ i are the Pauli matrices. We defineΨ = Ψ † γ 0 . Notice that gaugino is a Majorana fermion, with our conventions for the gamma matrices it has two real components. The action (4.2) is written in the Wess-Zumino gauge, which explicitly breaks supersymmetry, the remaining gauge symmetry is fixed by adding the conformal gauge fixing term in the action. The N = 1 supersymmetry of the action (4.2) becomes obvious when one constructs it using superspace integrals and superfields, for more details check [80]. The fields are organized in N = 1 super-multiplets: a vector multiplet {λ, A µ } and N f scalar matter multiplets {Φ i , Ψ i , F i }. Going on-shell one sets F i = 0. The global symmetry of the action is Additionally there is parity invariance. These symmetries prevent the generation of additional interactions (quadratic or quartic superpotential interactions would break parity invariance), therefore there is no need of tuning interactions to zero.
The large N f effective action of the N = 1 SQED is described by N f scalar and N f fermion flavors minimally coupled to the effective photon and interacting with the effective gaugino via a Yukawa interaction The effective photon propagator is obtained by summing a geometric series of bubble diagrams with fermion and scalar loops. We give the effective photon propagator after Wick rotation x µ x ν |x| 2 . (4.5) The effective gaugino propagator is obtained by summing a geometric series of bubble diagrams with each bubble made by one fermion and one boson propagators, after Wick rotation we have following expression We use red dotted line to represent effective gaugino propagator in the Feynman graphs.

Scaling dimension of low-lying mesonic operators
The following three quadratic operators sit inside the same N = 1 supermultiplet where α is a spinor index. Depending how the flavor indices are contracted we can construct a singlet and an adjoint representation of the global symmetry SU (N f ). Let us first discuss the adjoint supermultiplet. Using the graphs A,B,C,D from the table (17) we can extract the scaling dimension of the adjoint scalar-bilinear operator adj . (4.8) The anomalous dimension of the adjoint operator is ∆ (1) adj = − 24 3π 2 N f . Due to supersymmetry the scaling dimensions of the components in (4.7) are related to each other.
It is also possible to construct another scalar-fermion bilinear We checked that anomalous dimension of (4.12) is vanishing at order O(1/N f ). This is not surprising since operator (4.12) sits in the same supermultiplet with the gauge invariant flavor current operator Ψ γ µ Ψ + i(Φ * D µ Φ − D µ Φ · Φ) adj which is conserved and has a scaling dimension exactly equal to 2 for any N f . In order to compute the anomalous dimension of the singlet scalar-bilinear operator, we use all the graphs in the table (17), since for this operator all of them contribute at order O(1/N f ). It turns out that the anomalous scaling dimension vanishes at that order (there seems to be no reason to think that at higher orders in the 1/N f expansion the anomalous corrections are going to be absent). Also the singlet supermultiplet (4.7) has the dimensions of its components related to each other: Figure 18: (N = 1 SQED) quartic adjoint-2 operator renormalization.
Notice that the singlet counterpart of (4.12) is out of spectrum. This is precisely the operator that couples to gaugino in the effective action (4.4) and it is set to zero by the EOM of the gaugino field. Next we consider scalar quartic operators (2.15). In the equation (2.16) we decomposed this operator into irreducible representations of SU (N f ) group: singlet, adjoint, adjoint-2. For the quartic adjoint-2 operator we need the Feynman graphs of table (18) to extract the scaling dimension. Similar calculations can be done for the other two operators. We skip the details and give the final result below The N = 1 super-QED with two flavors (N f = 2) has been argued to be dual to a cubic N = 1 supersymmetric Wess-Zumino model with SU (2) × U (1) global symmetry( [60,61]). The field content of the WZ model is given by 7 real N = 1 supermultiplets: a real triplet µ I and a complex doublet M α . The superpotential of the WZ model is dictated by the SU (2) × U (1) global symmetry and by parity invariance: The real fields µ I map to the quadratic mesons on the gauge theory side. The complex fields M α map to the monopoles with minimal topological charge. This duality can be obtained starting from the N = 4 mirror symmetry ( [81,82]), which in the IR relates abelian gauge theory with one hypermultiplet flavor to a free massless hypermultiplet. The N = 1 duality also has a description in terms of S-duality of Type IIB brane setups [83].
In table (19) we collect the basic gauge invariant operators. On the left side we list the operators which belong to the spectrum of N = 1 SQED, their approximate scaling dimensions are calculated using the large N f formulas obtained in the previous two sections. We also include the scaling dimension of the monopole operators M ±1 in the large N f limit (∆[M ±1 ] = 0.3619N f + O(1)), which we extract from the results of [57] in appendix D. On the right side we list the operators of the dual WZ model, their scaling dimensions are calculated using 4 − expansion in [61]. Using the map discussed in detail in [61], on each row the two operators map into each other under the duality. We notice a quite good agreement between the dimensions of the corresponding operators, providing a nice quantitative check of the N = 1 duality. Where the Φ is a complex scalar, and Ψ is a two-component Dirac fermion and F is an auxiliary complex field. The vector multiplet has the field content A µ σ λ 1 λ 2 D .
The N = 2 SQED action can be written as a N = 1 SQED action (4.2), plus a kinetic term for H and interaction from the superpotential Written in Lorentzian metric, the full action in components become where the first term in the right hand side of (4.24) is defined in (4.2). The gaugino λ in (4.2) is replaced by λ 1 . We have a quartic term in the second line because we have integrated out auxiliary fields F j : Performing a 1/N f expansion with quartic vertix is usually more involved task than working with the cubic vertex, therefore one usually doesn't integrate out F j (4.25 Due to supersymmetry, these dimensions are exact in 1/N f expansion. This follows from the fact that the dimension of the gauge field A µ is exactly 1 and the fields (A µ , σ, λ 1 , λ 2 , D) sit in the same vector multiplet. The operator (Φ * Φ adj ) has a scaling dimension exactly equal to 1, since it sits in the same N = 2 supermultiplet of the flavor SU (N f ) currents. These observations allow us to check our results for the singlet and adjoint operator dimensions obtained in N = 1 SQED. For this purpose first we notice that the action (4.24) without the first term is the N = 1 supersymmetric O(N ) sigma model [84].
The large N f scaling dimensions of the field σ and of the bilinear adjoint operator for this model model have been computed in [84] (see [39] for a finite-N f study in the 4 − expansion): . Therefore the sum of this contributions should be such that anomalous scaling dimensions for |Φ| 2 adj and σ are exactly zero. As one can see from (4.27, 4.28) and (4.9, 4.15) this is true.
Finally, for the completeness, we compute the scaling dimensions of the scalar mesonic operators in N = 2 SQED, which at leading order have dimension 2, but are not protected. One such operator is the quartic adjoint-2: |Φ| 4 adj−2 , to calculate its scaling dimension one uses graphs in the tables (18,20). The operator σ 2 has its scaling dimension twice the scaling dimension of σ field (which is exactly equal to 1) plus the contributions of the last two graphs in the table (14).  Figure 20: (N = 2 SQED) quartic adjoint-2 operator renormalization. Dotted red line stands for the effective gaugino λ 2 propagator. Thick red line stands for effective D-field propagator.
Where the D µ = ∂ µ + ieA µ , and the kinetic term for the photon is canonically normalized. The beta functions of the gauge and quartic couplings at one loop order are The beta function of the gauge coupling has two zeroes. One trivial zero is when the gauge coupling vanishes, then we have the ungauged O(N f ) × O(N f ) vector model. See [1,85] for discussions about the ungauged fixed points and RG flow. The other zero is when Plugging this value into the beta functions (A.3, A.4) we generically expect to find four fixed points. There are two fixed points (which we identify with bQED and bQED + discussed in the main text) for which λ = 2λ ep with a global symmetry group SU (N f ). The values of the quartic couplings at those fixed points are It follows from (A.6) that for N f < 182.95 the quartic couplings become complex. Similarly one writes solutions for the remaining two fixed points (which we identify with ep-bQED and bQED − discussed in the main text), for which global symmetry is Where we defined the discriminant It follows from (A.9) that at these two fixed points the quartic couplings become complex when N f < 196. 22. We provide the expansions of solutions (A.6, A.7, A.8) in the large N f limit ( = 1) bQED (tricritical): .13) Notice that at the tricritical point: λ ∼ 1/N 2 f , while in bQED + : λ ∼ 1/N f . Similarly, in bQED − : λ, λ ep ∼ 1/N f , while in ep-bQED: λ ep ∼ 1/N f , λ ∼ 1/N 2 f . This justifies our identification with the four fixed points discussed at large N f in the main text.
The figure (22) is an example of RG flow and fixed points in the space of quartic couplings.

C Feynman graphs
In this appendix we give some examples of computation of the Feynman diagrams, using an approach similar for instance to [71]. We read the graphs using position space Feynman rules, then we identify the region from where UV logarithmic divergences appear. 8δ µα N f π 2 |y − z| 2 8δ νβ N f π 2 |y − w| 2 where the 4 is the symmetry factor of the graph (23). Each blue loop in the graph gives a factor N f , and since we normalized singlet bilinear operator as follows 1 √ N f N f i=1 |Φ i | 2 we also get a factor 1/N f . After cancelation we obtain the factor N f in (C.1). Also we choose to work in the ξ = 0 gauge since the graph (23) turns to be independent from the choice of the gauge parameter.
The logarithmic divergences come from the region where y, z, w are close to 0.