Yukawa CFTs and Emergent Supersymmetry

We study conformal field theories with Yukawa interactions in dimensions between 2 and 4; they provide UV completions of the Nambu-Jona-Lasinio and Gross-Neveu models which have four-fermion interactions. We compute the sphere free energy and certain operator scaling dimensions using dimensional continuation. In the Gross-Neveu CFT with $N$ fermion degrees of freedom we obtain the first few terms in the $4-\epsilon$ expansion using the Gross-Neveu-Yukawa model, and the first few terms in the $2+\epsilon$ expansion using the four-fermion interaction. We then apply Pade approximants to produce estimates in $d=3$. For $N=1$, which corresponds to one 2-component Majorana fermion, it has been suggested that the Yukawa theory flows to a ${\cal N}=1$ supersymmetric CFT. We provide new evidence that the $4-\epsilon$ expansion of the $N=1$ Gross-Neveu-Yukawa model respects the supersymmetry. Our extrapolations to $d=3$ appear to be in good agreement with the available results obtained using the numerical conformal bootstrap. Continuation of this CFT to $d=2$ provides evidence that the Yukawa theory flows to the tri-critical Ising model. We apply a similar approach to calculate the sphere free energy and operator scaling dimensions in the Nambu-Jona-Lasinio-Yukawa model, which has an additional $U(1)$ global symmetry. For $N=2$, which corresponds to one 2-component Dirac fermion, this theory has an emergent supersymmetry with 4 supercharges, and we provide new evidence for this.


Introduction and Summary
Physical applications of relativistic quantum field theories with four-fermion interactions date back to Fermi's theory of beta decay. The first application to strong interactions was the seminal Nambu and Jona-Lasinio (NJL) model. In the original paper [1] they considered the model in 3 + 1 dimensions with a single 4-component Dirac fermion and Lagrangian In addition to the U (1) symmetry ψ → e iβ ψ, this Lagrangian possesses the U (1) chiral symmetry under ψ → e iαγ 5 ψ. Using the gap equation it was shown that the chiral U (1) can be broken spontaneously, giving rise to the massless Nambu-Goldstone boson. This was the discovery of the crucial role of chiral symmetry breaking in the physics of strong interactions.
One of the goals of this paper is to study a generalization of (1.1) to N f 4-component Dirac fermions ψ j , j = 1, . . . N f : 2) and its continuation to dimensions below 4. We define N = 4N f , so that N is the number of 2-component Majorana fermions in d = 3. In addition to the chiral U (1) symmetry ψ j → e iαγ 5 ψ j this multi-flavor NJL model possesses a U (N f ) symmetry. 1 When considered in 2 < d < 4 this model gives rise to an interacting conformal field theory which describes the second-order phase transition separating the phases where the U (1) chiral symmetry is broken and restored.
In addition to studying the NJL model with the U (1) chiral symmetry, we will present new results for the Gross-Neveu (GN) model [2], which has a simpler quartic interaction Instead of the continuous chiral symmetry it possesses the discrete chiral symmetry ψ j → γ 5 ψ j . As discovered in [2], in d = 2 this theory is asymptotically free for N > 2. When considered in 2 < d < 4 this is believed to be an interacting conformal field theory which describes the second-order phase transition where the discrete chiral symmetry is broken.
In d > 2 the four-fermion interactions (1.2) and (1.3) are non-renormalizable. While they are renormalizable in the sense of the 1/N expansion [5], at finite N it is important to know the UV completion of these theories. In [6,7] it was suggested that the UV completion This theory, known as the Gross-Neveu-Yukawa (GNY) model, will be discussed in section 2.
There is a large body of literature on the GN and NJL CFTs in d = 3 and their applications; see, for example, [6,7,[9][10][11][12][13][14][15][16][17][18][19][20]. We will carry out further studies of these CFTs using the 4 − and 2 + expansions followed by Padé extrapolations. In addition to studying the scaling dimensions of some low-lying operators, we will calculate the sphere free energy F . The latter determines the universal entanglement entropy across a circle [21], and is the quantity that enters the F -theorem [22][23][24][25]. We will also discuss C T , the normalization of the correlation function of two stress-energy tensors. For the GN model, the 1/N and expansions of C T were studied in [20]; in this paper we extend these results to the NJL model and also provide the numerical estimates in d = 3 for various values of N .
When considered for N f = 1/2, i.e. the single 4-component Majorana fermion (which is equivalent to one Dirac fermion in d = 3), the NJLY model is expected to flow to the well-known supersymmetric Wess-Zumino model with 4 supercharges. In d < 4 this theory defines a CFT with "emergent supersymmetry" [26,27], in the sense that the RG flow drives the interactions to a supersymmetric IR stable fixed point, where the global U (1) symmetry becomes the U (1) R symmetry (see figure 1). We will provide additional evidence for this using the 4 − expansion of the NJLY model with N f = 1/2 to two loops, both for certain scaling dimensions and for the sphere free energy. A three loop calculation of scaling dimensions, which supports the emergent supersymmetry, was carried out recently [28].
Even more intriguingly, when the GNY model is continued to N f = 1/4, which corresponds to a single 2-component Majorana fermion in d = 3, it appears to flow to a CFT with 2 supercharges [19,26,[29][30][31]. We will show that the O( 2 ) corrections to scaling dimensions 2 In d = 3, one may express the Lagrangian (1.2) in terms of 2N f 2-component Dirac spinors χ i , χ i+N f by writing ψ i = (χ i , χ i+N f ), i = 1, . . . , N f . See for instance [8] ? ?  [19]. We also estimate C T and F for this theory. These results will be presented in section 4.

The Gross-Neveu-Yukawa Model
The β-functions for the GNY model with action (1.4) in d = 4 − , up to two-loop order, where N = N f tr1 = 4N f . The model possesses an IR stable fixed point at the critical couplings g * i given by Of course, there is also a fixed point g * 1 = 0, g * 2 = g Ising 2 = 16π 2 3 + O( 2 ) which corresponds to the decoupled product of the single-scalar Wilson-Fisher fixed point and N f free fermions.
By looking at the derivative of the beta functions at the fixed points, one can verify that (2.2) is attractive for all N f , so one can flow to it from the "Ising" fixed point along a relevant direction. Let us mention that there is formally also a third fixed point obtained from (2.2) by changing the sign of √ N 2 + 132N + 36. This fixed point is unstable in d = 4 − due to the negative quartic coupling, g * 2 < 0, but its dimensional continuation may produce a CFT in d = 3.
The scaling dimensions of σ and ψ are found to be [14] At the IR stable fixed point (2.2), one gets 5 These dimensions agree with [12] after correcting some typos in eq. (11) of that paper (in particular, the coefficient 33 should be changed to 3). Our O( ) term in ∆ ψ corrects a typo in eq. (4.44) of [16].
3), we may also recover the result at the Ising fixed point ∆ Ising . One can then see that the Yukawa operator σψψ is relevant at this decoupled fixed point, and can trigger a flow to the IR stable fixed point (2.2). We expect this to be true in d = 3 as well, since it is known that ∆ 3d Ising σ ≈ 0.518 [38], and so ∆ 3d Ising σψψ ≈ 2.518 < 3.
The anomalous dimension of the operator σ 2 , which determines the critical exponent ν −1 = 2 − γ σ 2 , may be read off from eq. (18) of [14] At the fixed point (2.2) we find We have also calculated the one-loop anomalous dimensions of the operatorsψψ and σ 3 : At higher orders these operator will mix, and one has to find the eigenvalues of their mixing 5 Throughout the paper, one can obtain the corresponding expansions at the unstable fixed point by changing the sign of the square root. matrix. At the fixed point (2.2), we find The first of these dimensions corresponds to a descendant of σ, as can be seen from the fact that it equals 2 + ∆ σ .
Let us also review the known result for the 4 − expansion of C T in the GNY model, which was discussed in [20]; the diagrams contributing to the term ∼ g 2 1 are shown in fig.  4.7 there. Evaluation of these diagrams yields , (2.9) where S d = 2π d/2 /Γ(d/2), and we used the values of C T for free scalar and fermion theories: where R is the scalar curvature, W 2 is the square of the Weyl tensor, and E the Euler density The parameters η 0 , a 0 , b 0 , c 0 are bare curvature couplings whose renormalization can be fixed order by order in perturbation theory. On a sphere, the Weyl square term drops out, and to the order we work below we will only need the renormalization of the Euler coupling b 0 (the R 2 term and the renormalization of conformal coupling are expected to play a role at higher orders [36,39,40]). The corresponding beta function can be extracted from the results of [36,41], and we find (2.13) where b is the renormalized coupling, and its the relation to the bare 360(4π) 2 + . . .) can be inferred from the above beta function. The coupling independent term is related to the a-anomaly of the free fermions and scalar.
The calculation of the sphere free energy now proceeds as in [42,43]. Keeping terms that contribute up to order 2 , we have (2.14) where gE is the contribution of the curvature term, and F f , F s are the sphere free energies of free fermion and scalar, which can be found in [44]. Starting from the flat space propagators , and then Weyl transforming to the sphere, we find Here I 2 (∆) denotes the integrated 2-point function of an operator of dimension ∆, which is given by [24,45,46] .
For the 4-point function, we find where we used a shorthand notation for the chordal distance s xy ≡ s(x, y). The integral of this 4-point function over the sphere cannot be calculated explicitly, but one can find it as a series in d = 4 − . For this we used the Mellin-Barnes approach, which is described in [42,43]. The result for the integral reads Putting everything together, we find for the free energy in d = 4 − 1,0 N (N + 6) 2(4π) 4 + N N − 6 + 6(N + 6)(3 + γ + log(4πR 2 )) 12(4π) 4 (2.20) Now replacing the bare couplings with the renormalized ones we find that all pole cancels, and the free energy is a finite function of the renormalized couplings g 1 , g 2 , b. 6 As explained in [42], in order to calculate the free energy at the critical point we should 6 Note that the coupling dependent part in the renormalization of b is necessary to cancel poles coming from diagrams at the next order. However, we still have to carefully include the Euler term, as in [42,43], as it affects the free energy at the fixed point to order 2 . now tune all couplings, including b, to their fixed point values. Using (2.2), we get From the curvature beta function (2.13), we find at the critical point and, using (2.2) and d d x √ gE = 64π 2 + O( ), we find that the Euler term contributes (2.24) Substituting this into (2.22), and writing the result in terms ofF = − sin(πd/2)F , we find (2.25)

2 + expansions
In this section we review the known results for operator dimensions at the UV fixed point of the Gross-Neveu model in d = 2 + , and then compute its sphere free energy to order 3 .
The action for the Gross-Neveu model [2] in Euclidean space in terms of bare fields and coupling reads where i = 1, . . . 2N f , and we have included the Euler term which is needed for the calculation of the sphere free energy below.
The β-function for the renormalized coupling constant g in d = 2 + is known to be [16,47,48] Therefore, one can see that there is a perturbative UV fixed point at a critical coupling g * given by 7 The scaling dimensions are found to be [48] where σ ∼ψψ. At the UV fixed point this gives It is also not hard to determine the dimension Let us now turn to the calculation of the free energy on S 2+ . To order g 4 , we have where F f is the free fermion contribution, derived as a function of d in [44], and Using the flat space fermion propagator in (2.15), and then performing a Weyl transformation to the sphere, we find where the integral I 2 (∆) is given in (2.17), and I 3 (∆) denotes the integrated 3-point function [24,45,46] For the integrated 4-point function we find where s mn ≡ s(x m , x n ) is a chordal distance on a sphere. Using the methods described in [42,43], we find After expressing the bare coupling g 0 in terms of the renormalized one we find a surviving pole in F of order g 4 / . This pole can be cancelled provided we renor-malize the Euler density parameter as where the coupling independent term is due to the trace anomaly of the free fermion field, In order to obtain the correct expression for F at the 41) and putting this together with the contributions of S 2 and S 3 , 8 we find 42) or, in terms ofF = − sin(πd/2)F : This result agrees with the one obtained in [42] using a conformal perturbation theory approach; this provides a non-trivial check on the procedure used here which involves the curvature terms.

Large N expansions
In this section we test the 4 − and 2 + expansions by comparing them with the known 1/N expansions [9,49,50]. The general form of the large N expansions of scaling dimensions in the GN model is 9 and In these equations we defined where ψ(x) = Γ (x)/Γ(x) denotes the digamma function. 9 The 1/N 3 term in ∆ ψ may be found in [51,52].
In d = 3, the above large N expansion of the scaling dimensions read (2.49) ForF , the O(N 0 ) result can be obtained from the general formula [44,53] for the change where O ∆ is a scalar primary operator of dimension ∆. In terms ofF = − sin(πd/2)F , the result is In the present case O ∆ =ψψ, and so ∆ = d − 1. Therefore, The 4 − expansion of this agrees with the large N expansion of (2.25), i.e.
correction to the answer for free feermions is given in [20]: . (2.54) Its expansion in d = 4 − can be seen to match (2.9), in particular reproducing the extra propagating scalar present in the GNY description.

Padé approximants
To obtain estimates for the CFT observables in d = 3, we will use "two-sided" Padé approximants that combine information from the 4− and 2+ expansions. Namely, we consider the is the spacetime dimension, and we fix the coefficients a i , b j so that its Taylor expansion near d = 4 and d = 2 agrees with the available perturbative results. Clearly, the "degree" n + m of the approximant is bound by how many terms in the -expansion are known. Such approximants may be derived for any finite N , and it is useful to compare their large N behavior as a function of d to the 1/N -expansion results listed in the previous section. When several approximants Pade [m,n] are possible for the same quantity, we use such comparisons to large N results to choose the one which appears to work best.
For ∆ ψ , ∆ σ and ∆ σ 2 we know the 4 − expansion to order 2 , and the 2 + expansion to order 3 . This allows to use Padé approximants with m + n = 6. For ∆ ψ and ∆ σ , we find that Pade [4,2] has no poles in 2 < d < 4 and for large N is in good agreement with the results (2.45) and (2.46). For ∆ σ 2 , we perform the Padé on the critical exponent ν −1 = d − ∆ σ 2 , and then translate to ∆ σ 2 at the end. In this case, we find that Pade [1,5] is the only approximant with no poles, and it matches well to the large N result. The resulting estimates in d = 3 for these scaling dimensions are given in Table 1. We note that the scaling dimensions for N = 8 are in good agreement with the Monte Carlo results from [14]. In figure 2, we plot our 3d estimates for ∆ ψ and ∆ σ , compared to the large N curve obtained from (2.49) by eliminating N to express ∆ σ as a function of ∆ ψ . The N = 1 values, which correspond to the N = 1 SUSY fixed point, are obtained in Section 4. For the sphere free energy of the interacting CFT,F , we find it convenient to perform the Padé approximation on the quantity Large N SUSY which is essentially the interacting part of the free energy in the GN description, but it includes the contribution of a free scalar from the GNY point of view. Using the results (2.25) and (2.43), we can use Padé approximants with n + m = 8, and we find that Pade [4,4] has no poles and agrees well with the large N result (2.50). The resulting d = 3 estimates for F , normalized by the free fermion contribution N F f , are given in Table 1   We can use our estimates to make some tests of the F -theorem. In the GNY description, we can flow to the critical theory from the free UV fixed point of N fermions plus a scalar, while in the GN description one can flow to the critical theory to the free fermions. Then, the F -theorem implies the inequalities We verified that our estimates satisfy these inequalities for all values of N . As an example, we also see that we can flow to the critical GN point from the decoupled product of the Ising CFT and N free fermions. This implies Using the estimates for F Ising derived in [42], we have checked that this inequality indeed holds. Using the Padé approximants as a function of 2 < d < 4, we can also verify that both (2.56) and (2.57) are satisfied, in terms ofF , in the whole range of d. This is in agreement with the "generalized F -theorem" [42].
Finally, we discuss N = 2, which is a special case where the β-function in d = 2 vanishes exactly; therefore, the theory has a line of fixed points. For N = 2 we cannot apply the and for the sphere free energy in agreement with the F -theorem.

The Nambu-Jona-Lasinio-Yukawa Model
Using the results of [33][34][35][36][37], we have found the following β-functions for the NJLY model (1.5) up to two loops: (3.1) The one-loop terms above agree with the N b = 1 case of the results in [13]. Solving (3.1), we find the following fixed point: The scaling dimensions of the fields are found to be At the fixed point (3.2) these give: The NJLY has two types of operators quadratic in the scalar fields: the U (1) invariant operator φφ, and the charged operators φ 2 andφ 2 . The one-loop scaling dimension of φφ was determined in [13]. Using [37], we find up to two loops At the fixed point this gives and The 4 − expansion of C T in the NJLY model proceeds similarly to that for the GNY model presented in [20] and reviewed in section 2. In the NJLY model there are two real scalar fields, and we need to replace T σ by T σ 1 + T σ 2 . It is not hard to check that each diagram contributing to the term ∼ g 2 1 picks up a factor of 2 compared to the GNY model, since each of the internal scalar lines can be either φ 1 or φ 2 . Thus, we find . (3.8)

Free energy on S 4−
The calculation of F for the NJLY model follows the same steps as the one for the GNY model discussed earlier. The integrals are also nearly identical, except for combinatorics factors due to the fact that we have two scalar fields, one of which has iγ 5 coupling.
The perturbative expansion of the free energy is given by where we defined the operators O 1 =ψ j (φ 1 + iγ 5 φ 2 )ψ j and O 2 = (φ 2 1 + φ 2 2 ) 2 , and δF b = b 0 d d xE is the Euler term. Evaluating the correlation functions above, we find where I 2 is given in (2.17) and I 4 corresponds to the integrated 4-point function of O 1 , for which we find Now replacing the bare couplings with the renormalized ones g 1,0 = µ 2 g 1 + N + 4 32π 2 g 3 1 + . . . , g 2,0 = µ g 2 + . . . (3.12) we find that all poles cancel as they should. As explained in the GNY and GN calculations, in order to obtain the correct expression for F at the IR fixed point in d = 4 − we need to include the effect of the Euler term. Using an improved version of the result from [36,41], adapted to the presence of γ 5 in the vertices, we get: (3N − 7)) . Putting this together with the integrals in (3.10), we finally find in terms ofF :

2 + expansions
In this section we consider the theory in 2 + dimensions with 4-fermion interactions which respect the U (1) chiral symmetry. We begin with the action in Euclidean space of the form [4] where i = 1, . . . 2N f is the number of two-component spinors, and γ 0 = σ 1 , γ 1 = σ 2 and γ 5 = −iγ 0 γ 1 = σ 3 . For g V = 0 and g S = −g P this reduces to the well-known chiral Gross-Neveu model [2] in d = 2. However, as we will see below, for our purposes it is not consistent to set g V = 0 -the corresponding operator respects the U (1) chiral symmetry and gets induced.
The one-loop beta-functions and anomalous dimension of the ψ field were found in [4] using MS scheme and read β S = g S − 1 π 1 2 (N − 2)g 2 S − g P g S − 2g V (g S + g P ) + . . . , β P = g P + 1 π 1 2 (N − 2)g 2 P − g P g S + 2g V (g S + g P ) + . . . , (3.17) (3. 18) We note that at the leading order in the 2 + expansion the evanescent operators do not appear [4,54,55]. One of the UV fixed points of (3.17) is 10 which corresponds to the SU (2N f ) Thirring model [4,56]. Indeed using the relation for the SU (2N f ) generators (T a ) i j (T a ) k l = 1 2 (δ i l δ k j − 2 N δ i j δ k l ) and the Fierz identity in d = 2 (γ µ ) αβ (γ µ ) γδ = δ αδ δ βγ − (γ 5 ) αδ (γ 5 ) βγ one finds So the action for this model is with the beta-function and anomalous dimension It is plausible that this model is the continuation of the NJLY model (1.5) to d = 2 + . One finds that the UV fixed point is g * = 2π /N , and the critical anomalous dimension reads Also one finds that the dimension of the quartic operator is We can also calculate the free energy of this model. To order g 2 , we have: where F f is the free fermion contribution and where, under the Thirring description (3.20), O = −2(ψγ µ T a ψ) 2 . Going through the combinatorics, we find Evaluating this in d = 2 + and plugging in the fixed point value (3.19) we finally find (3.28)

Large N expansions
For the NJL model, the 1/N expansions of operator dimensions again assume the general form (2.44), where now [10,57] and There is a considerable similarity between these results and the corresponding results for the (3.33) Turning to the sphere free energy, we can obtain the O(N 0 ) result forF by simply doubling the δF from eq. (3.8) of [44]. Therefore, we havẽ The 4 − expansion of this agrees with the large N expansion of (3.15), i.e. (3.35) and its 2 + expansion agrees with the large N expansion of (3.28), which yields For C T , the presense of the extra scalar field compared to the GN case [20] again poses no difficulty. After some simple excercise commuting γ 5 , we conclude that all the diagrams in [44] contributing to C T in the GN model should receive a factor of 2 due to the presence of two scalar fields. Hence, we find for the NJL model , (3.37) i.e. the correction C T 1 is just twice the corresponding term in the GN model (note that γ ψ,1 in the NJL model is twice that of the GN model). This result is, in particular, consistent with the fact that in the limit d → 4 we expect C T 1 to reproduce the contribution of two free scalar fields. Its 4 − expansion can be also seen to agree with (3.8).

Padé approximants
Following the same methods as described in Section 2.4, we now use the 4 − and 2 + expansions 12 derived in the previous sections to obtain rational approximants of scaling dimensions and sphere free energy at the NJL fixed point in 2 < d < 4, for N > 2 (the case N = 2, which displays the emergent supersymmetry, is treated separately in Section 4). The results in d = 3 are given in Table 2, indicating which approximants was chosen in each case.
In figure 4, we also plot our 3d estimated for ∆ ψ and ∆ φ , compared to the large N results. 12 For ∆ φ , we use the boundary condition ∆ φ = 1 + O( ) in d = 2 + .  For the sphere free energyF , we again find it convenient to perform the Padé approximation on the quantity f (d) =F − NF f , which corresponds to the interacting part of the NJLY free energy plus the contribution of two free scalars. In Figure 3, we plot the resulting Padé approximants as a function of 2 < d < 4 for a few values of N , showing that they approach well the analytical large N formula (3.34).
We can again use our estimates to test the d = 3 F -theorem and its proposed generalization in 2 < d < 4 in terms ofF . We find that in the whole range of dimensions, the inequalities NF f + 2F s >F > NF f (3.38) hold, in accordance with the conjectured generalized F -theorem [42,44]. Using the results in [42] for the free energy of the O(2) Wilson-Fisher model, we have also checked that which is consistent with the fact that the theory can flow from the fixed point consisting of free fermions decoupled from the O(2) model, to the IR stable NJL fixed point.

Theory with 4 supercharges
A well-known supersymmetric theory with 4 supercharges is the Wess-Zumino model of a single chiral superfield Φ with superpotential W ∼ λΦ 3 . In d = 4 the model is classically conformally invariant, but it has a non-vanishing beta function and is expected to be trivial in the IR. Continuation of this model to lower dimensions (defined such that the number of supercharges is fixed in 2 ≤ d ≤ 4) was discussed in [44,58,59]. In d = 3, one finds the N = 2 theory of a single chiral superfield (a complex scalar and a 2-component Dirac fermion) with cubic superpotential, which flows to a non-trivial CFT in the IR [60]. In d = 2, the model matches onto the (2, 2) supersymmetric CFT with c = 1, which is the k = 1 member of the superconformal discrete series with c = 3k k+2 . The component Lagrangian of the WZ model in 4d reads where ψ is a 4-component Majorana fermion and φ = φ 1 + iφ 2 . The beta function of the from which one finds an IR fixed point with λ 2 * = 16π 2 3 + O( 2 ). The dimension of the chiral operator φ at the fixed point is determined by its R-charge to be One also has the exact result ∆ φ 2 = ∆ φ + 1, since the operator φ 2 is obtained from φ by acting twice with the supercharges (this is because, due to cubic superpotential, one has the relation Φ 2 = 0 in the chiral ring). It also follows from supersymmetry [26,28] that the dimension of the operator φφ at the fixed point is given by which agrees with the explicit three loop calculation of [28]. In performing Padé extrapolation of this result to d = 3, we have found that Padé [1,2] and Padé [2,1] give answers close to each other. Their average is ≈ 1.909, which is very close to the value ≈ 1.91 reported using numerical bootstrap studies [58,59]. We can also take into account the fact that in d = 2 the dimension of φφ should approach 2. Since it also approaches 2 in d = 4, it is not a monotonic function of d, which makes Padé extrapolation difficult. If we instead perform a "two-sided" extrapolation of ν −1 = d − ∆ φφ , and then return to ∆ φφ in d = 3, then we find ≈ 1.94. This is somewhat further from the numerical bootstrap estimate.
Since in d = 4 the WZ model includes a complex scalar and a 4-component Majorana fermion (i.e. one half of a Dirac fermion), one would expect it to correspond to N f = 1/2 and N = 2 in the NJLY model [26,27]. Note that the NJLY Lagrangian (1.5), specialized to the case of a single Majorana fermion, coincides with the WZ Lagrangian (4.1) provided 13 (4.5) 13 One should rescale ψ → ψ/ √ 2 in (1.5) to get a canonical kinetic term when the fermion is Majorana.
Indeed, setting N = 2 in the result for the fixed point couplings (2.2), we find: in agreement with the supersymmetry. In particular, the fact that the O( 2 ) terms vanish is consistent with the exact result (4.3), and we also see ∆ φ 2 = ∆ φ + 1 as discussed above.  This precisely agrees with the expansion of (5.23) in [44], which was derived using a proposal for supersymmetric localization in continuous dimension. We note that the curvature term (3.14) contributes at O( 3 ) order toF : δF b = − π 3 1296 . This contribution is crucial for agreement with [44]. Thus, (4.8) provides a nice perturbative test of the exact formula for F as a function of d proposed in [44] for the Wess-Zumino model.
In d = 3, the result obtained from localization [63] yields F W =Φ 3 ≈ 0.290791. In [42], the value of F for the O(2) Wilson-Fisher fixed point in d = 3 was estimated to be F O(2) ≈ 0.124.
Finally, we discuss C T of the N = 2 SCFT with superpotential W ∼ λΦ 3 . Its exact value in d = 3 has been determined using the supersymmetric localization [64,65]: where C UV T = 4C T,s is the value for the free UV theory of two scalars and one two-component Dirac fermion. Let us compare this with a Padé extrapolation of the ratio C T /C T,s using the boundary conditions where the 4 − expansion was obtained by setting N = 2 in (3.8). The Pade [1,1] approximant with these boundary conditions is

Theory with 2 supercharges
It has been suggested that in d = 3 there exists a minimal N = 1 superconformal theory containing a 2-component Majorana fermion ψ [19,26,[29][30][31]. This theory must also contain a pseudoscalar operator σ, whose scaling dimension is related to that of ψ by the supersymmetry, Some evidence for the existence of this N = 1 supersymmetric CFT was found using the conformal bootstrap [19].
To describe the theory in d = 3, one can write down the Lagrangian [19,26,29,30] (4.14) This model has N = 1 supersymmetry in d = 3; the field content can be packaged in the real superfield Σ = σ +θψ + 1 2θ θf , and the interactions follow from the cubic superpotential W ∼ λΣ 3 . 15 It is natural to expect that this model flows to a non-trivial N = 1 SCFT in the IR. Note that the theory cannot be described as the UV fixed point of a lagrangian for a Majorana fermion with quartic interaction, because the term (ψψ) 2 vanishes for a 2-component spinor.
The theory (4.14) is super-renormalizable in d = 3, and one may attempt its 4 − expansion [26]. To formulate a Yukawa theory in d = 4, one strictly speaking needs a 4component Majorana fermion, which corresponds to the GNY model with N = 2. However, the GNY description may be formally continued to N = 1. A sign of the simplification that occurs for this value is that √ N 2 + 132N + 36, which appears in the 4 − expansions (2.4), (2.6), equals 13 for N = 1. For this value of N , we find that the fixed point couplings in In [26] it was found using one-loop calculations that ∆ σ = 1 − 3 7 = ∆ ψ − 1 2 . Let us check that the supersymmetry relation (4.13) continues to hold at order 2 . Using (2.4) with N = 1, we indeed find The dimensions of operators σ 2 and σψ should also be related by the supersymmetry, ∆ σ 2 = ∆ σψ − 1 2 . Since σψ is a descendant, we also have ∆ σψ = ∆ ψ + 1. Thus, the supersymmetry relation assumes the form [30] ∆ σ 2 = ∆ ψ + 1 2 = ∆ σ + 1 . (4.18) Substituting N = 1 into (2.6) we find so that the supersymmetry relation (4.18) holds to order 2 . These non-trivial checks provide strong evidence that the continuation of the GNY model to N = 1 flows to a superconformal theory to all orders in the 4 − expansion.
It is also important to know how the theory behaves when continued to d = 2. It is plausible that the d = 2 theory has N = 1 superconformal symmetry, and the obvious candidate is the tri-critical Ising model [29,31], which is the simplest supersymmetric minimal model [66,67]. The Padé extrapolation (4.20) gives ∆ σ ≈ 0.217, which is quite close to the dimension 1/5 of the energy operator in the tri-critical Ising model. This provides new evidence that the GNY model with N = 1 extrapolates to the tri-critical Ising model in d = 2; in figure 6 we show how the operator spectrum matches with the exact results in d = 2. Imposing the boundary condition that ∆ σ = 1/5 in d = 2 enables us to perform a "two-sided" Padé estimate. The resulting value in d = 3 is ≈ 0.588, which is very close to that following from (4.20). The agreement with the bootstrap result ∆ σ ≈ 0.582 is excellent.
Since ∆ 1 corresponds to the θθ component of the superfield Σ 3 , we know that As a check, we may set N = 1 in (2.8) and see that the order term in ∆ σ 3 indeed vanishes.
It is further possible to argue that this primary operator, when continued to d = 2, matches onto the dimension 3 operator in the tri-critical Ising model, i.e. the product of left and right supercurrents. This implies that ∆ σ 3 is not monotonic as a function of d and is likely to be somewhat smaller than 3 in d = 3. 17  Performing a Padé approximation of the quantity f (d) =F −F f , we find that the average of the standard Padé [2,1] and Padé [1,2] approximants yieldsF /F s ≈ 0.68, which is quite close to c = 7/10. Therefore, in order to get a better estimate in d = 3, it makes sense to impose it as an exact boundary condition in d = 2. Following this procedure, and taking an average of the Padé approximants with n + m = 4, we find the d = 3 estimate In the UV, we have the free CFT of one scalar and one Majorana fermion, which has F UV = 1 4 log 2 ≈ 0.173, and therefore we find F IR /F UV ≈ 0.91. This is a check of the Ftheorem for the flow from the free to the interacting N = 1 SCFT. It is also interesting to compare the value of F at the SUSY fixed point to the decoupled Ising fixed point in Figure   1. A plot comparingF −F f withF Ising , which was obtained in [42], is given in Figure 7. It shows thatF <F f +F Ising in the whole range 2 < d < 4, in agreement with the generalized F -theorem [42,44] and the expectation that the SUSY fixed point is IR stable.
Finally we consider C T . Its 4 − -expansion for general N is given in eq. (2.9), and we can use it to estimate the C T value at the N = 1 SUSY fixed point in d = 3. We can perform a Padé approximation on the ratio C T /C T,s , where C T,s is the free scalar value, using the boundary conditions where we have imposed the d = 2 value corresponding to the tri-critical Ising model. A Pade [1,1] approximant with these boundary conditions is Let us also include a brief discussion of the non-supersymmetric fixed point of the N = 1 GNY model, marked by the red triangle in figure 1. While it has g * 2 < 0 in d = 4 − , it may become stable for sufficiently small d. Changing the sign of the square root in (2.25), we find the 4 − expansion of the sphere free energy at this fixed point: that could correspond to ψ in the Yukawa theory (such a field is not present in the standard modular invariants that retain fields of integer spin only). 18 In d = 3 our extrapolation gives 18 Another possibility is that the continuation of the non-supersymmetric fixed point to d = 2 gives the F non−SUSY ≈ 0.16. The latter quantity is bigger than (4.24); therefore, the non-SUSY fixed point, if it is stable, can flow to the SUSY one in d = 3.
It is also interesting to look at the scaling dimensions of σ, ψ and σ 2 at the non- These numbers are not far from the corresponding operator dimensions in either the (5,6) or the (6, 7) minimal models. For example, in the (5, 6) interpretation the exact scaling dimension of ψ in d = 2 should be 11/20, while in the (6, 7) it should be 19/28. The Padé value we find lies between the two, but the accuracy of the extrapolation all the way to d = 2 is hard to assess.