Higher-derivative deformations of the ModMax theory

We present higher-derivative deformations of the ModMax theory which preserve both $\mathsf{U}(1)$ duality symmetry and Weyl invariance. In particular, we single out a class of deformations expected to describe a low-energy effective action for the ModMax theory. We also elaborate on (higher-derivative) deformations of the $\mathcal{N}=1$ super ModMax theory.


Introduction
Recently much interest has been attracted to the unique conformal U(1) duality-invariant nonlinear electrodynamics proposed by Bandos, Lechner, Sorokin and Townsend [1].It is described by the Lagrangian where γ is a non-negative real coupling constant, 1 and ω is defined by with F ab = 1 2 ε abcd F cd the Hodge dual of the field strength F ab .This model for nonlinear electrodynamics was called the ModMax theory, that is modified Maxwell electrodynamics.Originally it was derived using the Hamiltonian methods for nonlinear electrodynamics advocated by Bialynicki-Birula [2].Ten days after [1] had been released in the arXiv, the theory was re-derived in [3] using the Gaillard-Zumino-Gibbons-Rasheed (GZGR) formalism of duality rotations for nonlinear electrodynamics [4][5][6][7][8].The ModMax theory was also shown to be a unique conformal solution [9] within the Ivanov-Zupnik (IZ) approach to U(1) duality-invariant nonlinear electrodynamics [10][11][12].
There is an interesting historical curiosity concerning the ModMax theory.In the late 1990s and early 2000s, much work was done to describe general U(1) duality-invariant Lagrangians for nonlinear electrodynamics [5,7,8,[11][12][13] of which the most prominent example is the Born-Infeld theory [14] L BI = 1 g 2 1 − − det(η ab + gF ab ) = with g the coupling constant.A natural question is the following: Why wasn't the ModMax theory discovered at that time?One of the reasons is that all self-dual models for nonlinear electrodynamics which were studied before [1] were assumed to possess a well defined weak-field limit, and the existence of such a limit is an intrinsic feature of low-energy effective actions.
As is clear from (1.1), the ModMax theory does not have a weak-field limit. 2 To make this 1 The parameter γ is restricted to be non-negative since superluminal propagation is possible for γ < 0 [1]. 2 The N = 1 supersymmetric extension of the ModMax theory [9,15] also does not have a weak-field limit.
This is in contrast with the superconformal U(1) duality-invariant model for the N = 2 vector multiplet proposed in [9,16].The latter theory is described in terms of the N = 2 reduced chiral superfield strength, W (x, θ) = ϕ(x)+θ α i θ βi F (αβ) (x)+. . ., and its conjugate.The action functional is not analytic with respect to the physical scalars ϕ and φ, which are required to possess non-zero VEVs, but it is analytic in the electromagnetic field strength F αβ and and its conjugate F α β related to F ab according to (2.1).
point clearer, we remind the reader that every U(1) duality-invariant nonlinear electrodynamics is described by a Lorentz-invariant Lagrangian L(F ab ) which is a solution to the self-duality equation [2,5,7] where The self-duality equation guarantees invariance under U(1) duality rotations on the mass shell.If the Lagrangian L(F ab ) is expressed in terms of the electromagnetic invariants ω and ω, eq.(1.2), then the self-duality equation (1.4) turns into where Λ(ω, ω) is a real function related to the Lagrangian L(ω, ω) by the rule see [17] for the technical details. 3 For the nonlinear theory to possess a weak-field limit, the self-coupling Λ(ω, ω) must be a real analytic function in a neighbourhood of ω = 0, Here the reality of the Taylor coefficients follows from (1.7), see [17] for more comments.However, the ModMax theory (1.1) is characterised by the function [9] Λ which is evidently not of the form (1.9).
3 For any U(1) duality-invariant model for nonlinear electrodynamics, its compact duality group U(1) can be enhanced to the non-compact SL(2, R) group by coupling the electromagnetic field to the dilaton and axion fields [6][7][8].In the case of the ModMax theory, this was discussed in [9,18].It is also worth pointing out a Galilean cousin of the ModMax theory introduced in [19]; it is invariant under Galilean conformal transformations.
In accordance with the above discussion, it appears that the ModMax theory cannot arise as a perturbative low-energy effective action.However, its origin as a non-perturbative quantum correction is not excluded.It is also natural to wonder whether a quantum version of the ModMax theory may be defined and, if so, what is the explicit structure of the corresponding counterterms, Weyl anomalies etc.? While this theory clearly does not possess a Poincaréinvariant vacuum state, one may nevertheless still try to carry out a path-integral analysis.One-loop logarithmic divergences should respect the Weyl and duality symmetries (related formal arguments may be found in [20,21]), and the only possible functional structure of the form d4 x e L(F ab ) is given by (1.1).Such quantum corrections are absent [22]. 4Higherderivative quantum corrections are possible, and thus it becomes important to address the problem of finding consistent higher-derivative deformations of the ModMax theory.
A generalisation of the GZGR formalism to the case of U(1) duality-invariant theories with higher derivatives was sketched in [17]. 5Two modifications are required.Firstly, the definition of G is replaced with Secondly, the self-duality equation (1.4) is replaced with It is assumed in (1.11) and (1.12) that S[F ] is unambiguously defined as a functional of an unconstrained two-form F ab , i.e. no dependence on ∂ b F ab is present.In equation (1.12), F ab is considered to be an unconstrained bivector.Duality-invariant theories with higher derivatives naturally occur in N = 2 supersymmetry [16,17].Further aspects of duality-invariant theories with higher derivatives were studied in, e.g., [24][25][26][27].
Within the GZGR approach or its supersymmetric extensions, a consistent nonlinear deformation of a given U(1) duality-invariant theory (say, Maxwell's theory) is typically derived in perturbation theory and requires an infinite number of terms for the deformed action to satisfy the self-duality equation (1.4) or its supersymmetric extensions [16,17,25,28,29].Hence, a closed-form expression for the deformed action is difficult to obtain in such a setting.However, within the IZ approach, which makes use of auxiliary variables V ab = −V ba , all information about the given duality-invariant theory is encoded in its U(1)-invariant interaction Lagrangian L int (V ab ).Its U(1)-invariance is equivalent to the self-duality equation (1.4).A consistent deformation of the theory amounts to deforming L int (V ab ) → Lint (V ab ) in such a way that Lint (V ab ) is also U(1) invariant.In the presence of the auxiliary variables, the deformed theory is given in closed form.The IZ approach has also been extended to U(1) duality-invariant theories with higher derivatives [30,31]. 6his paper is organised as follows.In section 2 we describe several families of higherderivative deformations of the ModMax electrodynamics, including special classes expected to appear in loop quantum corrections to the theory.Such deformations are determined within the auxiliary variable formulation of [10][11][12].Thus, in section 3 we consider two important models and eliminate such variables in perturbation theory.Section 4 is devoted to higher-derivative deformations of the N = 1 super ModMax theory as an extension of the non-supersymmetric analysis of section 2. Concluding comments are provided in section 5.

Higher-derivative deformations of ModMax
A natural framework to generate U(1) duality-invariant models for nonlinear electrodynamics is the IZ approach [10][11][12].In the case of theories without higher derivatives, it is a reformulation of the GZGR formalism [4][5][6][7][8] which is obtained by replacing L(F ab ) → L(F ab , V ab ), where V ab = −V ba is an auxiliary unconstrained bivector.The latter is equivalent to a pair of symmetric rank-2 spinors, V αβ = V βα and its conjugate V α β , which are defined by (2.1) Our two-component spinor notation and conventions, including the definition of the relativistic Pauli matrices (σ a ) α α, follow [33][34][35].
The new Lagrangian L is at most quadratic in the electromagnetic field strength F ab , while the self-interaction is described by a nonlinear function of the auxiliary variables, L int (V ab ), The original theory L(F ab ) is derived from L(F ab , V ab ) by integrating out the auxiliary variables using their algebraic equations of motion.In terms of L(F ab , V ab ), the condition of U(1) duality as a systematic procedure to generate duality-invariant theories.However, it was demonstrated [30] that the construction of [25,28,32] naturally originates within the IZ approach proposed a decade earlier.Specifically, the twisted self-duality constraint corresponds to an equation of motion in the approach of [11,12].
invariance was shown [11,12] to be equivalent to the requirement that the self-interaction be invariant under linear U(1) transformations ν → e 2iϕ ν, with ϕ ∈ R, therefore The ModMax theory corresponds to the choice [9] (2.5) In the case of U(1) duality-invariant theories with higher derivatives, the self-coupling L int becomes a multivariable function, , and it becomes more economical to work with the action functional.The IZ reformulation is obtained by replacing the action functional S[F ] with a first-order action7 such that imposing the equation of motion In the case that the interaction has the form We are interested in those higher-derivative deformations of the ModMax theory which may contribute to a low-energy effective action of the theory.An important insight is obtained by considering the in-out vacuum amplitude for the ModMax theory where A a is the gauge potential, F ab = ∇ a A b − ∇ b A a , and ξ(x) is a background scalar field. 9n accordance with (2.2) and (2.5), the functional Formally, the effective action is expected to possess such a scale symmetry.Thus it is natural to assume that (a local part of) the effective action has the form and possesses the following properties:

8). 11
A solution to these requirements is given by where g n is a dimensionless numerical factor, and On the other hand, if we are only interested in a Weyl-invariant functional obeying the condition (2.8), then a more general action is allowed for some real function F(x) of a real argument.
The specific feature of the functional (2.13), including its special case (2.12), is that the auxiliary field appears in the deformation term only via the combination ν ν.Another choice is to replace F in (2.13) with a multivariable function of the form Here ∆ 0 denotes the Fradkin-Tseytlin operator [38] ∆ which is conformal when acting on the space of Weyl-neutral scalar fields. 13It should be pointed out that the structures Ξ n in (2.14) are independent for n = 1, 2, 3, 4. Specifically, it may be shown that 1 where we have introduced the following conformally primary structures: ) Each of these structures may be written in a different form by making use of So far we have considered only those primary deformation structures which do not contain the primary vector fields When considering deformations of Maxwell's theory, the structures containing χ a and χa would lead to contributions involving the classical equations of motion.This is the reason for avoiding such structures in the above discussion.However, making use of the vector fields (2.17) allows us to generate new deformations, including the following: which may originate at the one-loop level.

Elimination of auxiliary variables
Given a duality-invariant theory described by the first-order action (2.6), in which the self-coupling S int [V ] = S int [ν, ν] obeys the condition (2.9), the final goal is to derive its reformulation S[F ] which is obtained by imposing the equation of motion (2.7).This equation is equivalent to and its conjugate.The latter equations can be solved in perturbation theory, say, within the loop expansion.
Let us consider two examples.We start with a one-loop deformation in (2.12), and for simplicity we set = 1, The equation of motion (3.1) takes the form (ν ν) Eliminating the auxiliary fields gives where we have defined Our second example is defined by The corresponding equation of motion for the auxiliary variable V αβ is . (3.7) Eliminating the auxiliary fields gives Both models (3.4) and (3.8) involve one and the same composite field Ω, eq.(3.5).What is the significance of Ω? Let S[F, V ; g] be the action corresponding to the self-coupling (3.2), and S[F ; g] the U(1) duality-invariant model which is obtained upon elimination of the auxiliary field V ab .Since the parameter g is inert under the U(1) duality transformations, the functional is duality invariant for any value of g [7,8,17]. 14In particular, Υ(g = 0) is a duality-invariant functional in the ModMax theory.As demonstrated in [44], any two duality-invariant local observables H 1 (F ; γ) and H 2 (F ; γ) are functionally dependent, and ∂L MM /∂γ is such an observable. 15Thus it is natural to expect that the duality-invariant functional Υ(0) should be constructed in terms of Ω. Analogous considerations apply to the U(1) duality-invariant model generated by (3.6).
In accordance with [44], every duality-invariant scalar observable O(F ) in the ModMax theory can be expressed as a function of Ω.However, this is no longer the case if we allow for functionals involving derivatives of the field strength F ab .Let us consider an infinitesimal duality transformation in the ModMax theory
(3.12) 14 This property implies that, in perturbation theory, the leading contribution to the deformation of any selfdual theory must be invariant under the duality transformations of the original theory.We emphasise that this does not extend beyond first order; the sectors of (3.4) and (3.8) quadratic in are not duality invariant. 15The energy-momentum tensor T ab in every U(1) duality-invariant theory is duality invariant [5,7,8], and therefore every scalar duality-invariant observable H(F ) may be expressed as a function of T ab .For the Mod-Max theory it holds that ∂L MM /∂γ = 1 2 T ab T ab , see [41][42][43][44] for the technical details and earlier references. 16This transformation law follows from the auxiliary variable formulation.Specifically, the equation of motion for V ab in ModMax theory implies that (1 + cosh γ) √ ν = I.Since δ ϕ ν = 2iν under U(1) duality transformations, we obtain equation (3.12).
This result immediately implies that Ω is duality invariant.Moreover, it also implies the existence of new primary and duality-invariant observables, such as I( c √ Ī) 2 , which are functionally independent of Ω.

Higher-derivative deformations of super ModMax
It is also of interest to study consistent higher-derivative deformations of the N = 1 supersymmetric extension of the ModMax theory which was discovered in [15] and independently re-derived in [9] where E denotes the chiral integration measure.This theory is a unique representative in the family of U(1) duality-invariant nonlinear theories for the N = 1 vector multiplet of the general form [16,17,45,46] where W 2 = W α W α and W 2 = W α W α, the variable u is defined by the compensator Υ is a nowhere vanishing real scalar with the super-Weyl transformation and Λ(ω, ω) obeys the self-duality equation (1.7). 17We recall that the infinitesimal super-Weyl transformation of the spinor covariant derivatives [47] is given by see [35] for a review.The transformation law (4.4)implies that (4.2) is super-Weyl invariant.
We also remind the reader that the super-Weyl transformation law of the chiral field strength W α is and therefore the following composite antichiral scalar is super-Weyl invariant.What singles out the super ModMax theory (4.1), is that the corresponding Λ(ω, ω) is given by (1.10) and thus the Υ-dependence drops out.As a consequence, the action is locally superconformal. 18iven a U(1) duality-invariant vector multiplet model with action S[W, W ; Υ], there exists its reformulation with auxiliary variables [48,49] where the auxiliary spinor η α is only constrained to be covariantly chiral, D β η α = 0. We assume that η α and its conjugate η α are auxiliary superfields in the sense that the equation of motion for η α , and its conjugate may be solved to express η α as a functional of the field strength W α and its conjugate, As a result, we end up with the action which describes the dynamics of the vector multiplet.The action (4.8) defines a U(1) dualityinvariant system provided S int [η, η; Υ] is invariant under rigid U(1) transformations, This implies that the action S[W, W ; Υ] obeys the self-duality equation [16,17,45] where we have introduced In the relations (4.12), W α is chosen to be a general chiral spinor superfield.
It is worth pointing out that the self-coupling S int [η, η; Υ] corresponding to the dualityinvariant nonlinear supersymmetric electrodynamics (4.2) is This action is super-Weyl invariant provided η α transforms as The model is U(1) duality invariant if F(v, v) = F(vv), see [48] for the technical details.In the super ModMax case, it holds [9] that where κ is related to the ModMax coupling constant γ as in (2.5).In accordance with (4.13), this functional has no dependence on the compensator Υ.
Let us consider the in-out vacuum amplitude for the super ModMax theory where V = V is the gauge superfield which determines W α as the following descendant and κ(V ) is the gauge fixing function with ξ being a background chiral superfield.Finally, H denotes the Faddeev-Popov operator which acts on the space of chiral (φ) and antichiral ( φ) pairs.More details about the covariant quantisation of the vector multiplet in a supergravity background can be found, e.g., in [34].
In accordance with (4.8) and (4.15), the functional −1 S SMM [W, W , η, η] is invariant under rigid re-scalings Modulo quantum anomalies, the effective action is expected to respect this scale symmetry.We are interested in a local deformation of the super ModMax theory To understand the structure of allowed contributions to ∆Γ SMM [η, η], techniques are required to generate primary descendants of η α and η α.Such methods are available only in the presence of a compensator, for instance (D 2 − 4 R) η 2 Υ −2 is a primary dimension-zero antichiral scalar.However, since we are interested in locally superconformal deformations, no supergravity compensator can be present in the deformed action.The only remaining option is to allow for Υ being a descendant of the vector multiplet, which means that 19Υ = Dη Dη . ( However, in the ModMax theory the θ-independent component of Dη vanishes on-shell [15].Thus we are allowed to take into account only those structures which involve Υ in the numerator.
Let us describe two families of functionals that can be used to generate consistent deformations of the super ModMax theory.One of them involves the supersymmetric Fradkin-Tseytlin operator [53,54], ∆ which is defined by and has the super-Weyl transformation law provided Φ is inert under the super-Weyl transformations, δ σ Φ = 0.
Using η α and η α allows us to construct a primary dimension-zero antichiral scalar and its conjugate v. Now we can generate primary dimension-three chiral scalars of the form v−n ∆v −n , n > 0 , each of which can be integrated over the chiral subspace to result in a super-Weyl invariant This functional is obviously invariant under rigid U(1) transformations (4.11).Another type of consistent deformation is realised as an integral over the full superspace Above we have described two families of higher-derivative deformations of the super Mod-Max theory enjoying both U(1) duality invariance and super-Weyl symmetry.These are determined by the functionals (4.27) and (4.28).They are of particular interest since they are also symmetric under the re-scaling transformation (4.20), hence they may appear in the low-energy effective action for super ModMax theory.It should be noted, however, that these deformations do not describe supersymmetric extensions of those presented in section 2 due to their dependence on Dη.
It is possible to construct a supersymmetric generalisation of the primary field Ψ introduced in (2.14).It makes use of the primary dimensionless scalar which may admit a well-defined Dη → 0 limit in its bosonic sector.To probe this further, we take the flat-space limit, D A → D A , and find that w takes the form

.30)
Taking the θ-independent component of w and setting all fermionic fields to zero, the resulting expression is simply where we have made the definitions20 Finally, taking the legal limit τ → 0 in eq.(4.31a), we find that w| θ=0 = Ψ, see eq. (2.14).This analysis indicates that a functional of the form defines a superconformal and duality-invariant deformation of the super ModMax action.However, a separate analysis is required to investigate whether this action is non-singular in the bosonic sector, since so far we have only analysed the component structure of w.
where g 1 ∈ C and g 2 , g 3 , g 4 ∈ R. Eliminating the auxiliary fields to leading order in via the equation of motion (3.1) leads to the higher-derivative Lagrangian

2
+ O( 2) . (5.3) In accordance with footnote 14, the sector of L linear in must be duality invariant.This property immediately follows from the transformation law (3.12) in conjunction with the duality invariance of Ω, eq.(3.11).All structures in (5.3) may contribute to the logarithmically divergent part of the one-loop effective action for the ModMax theory.However, explicit calculation of one-loop divergences in the ModMax theory remain to be completed [55].
The main constructions of this paper can be extended to the conformal chiral two-form field theory in six dimensions [56] within the approach of [57], which combined the virtues of the six-dimensional Pasti-Sorokin-Tonin formulation [58,59] with the four-dimensional formalism developed by Ivanov, Nurmagambetov and Zupnik [60].Higher-derivative deformations of the conformal chiral two-form field theory in six dimensions can also be studied using the approach by Mkrtchyan et al. [61][62][63].
In section 4 we have studied higher-derivative deformations of the super ModMax theory.Their structure is more restrictive than in the non-supersymmetric case.Moreover, they do not look "natural" due to built-in singularities.In this sense explicit calculations of loop quantum corrections may be revealing.
Unlike the purely bosonic case, there exist consistent deformations of the super ModMax theory without higher derivatives that preserve both the U(1) duality invariance and super-Weyl symmetry, such as the following: (5.5) Upon elimination of the auxiliary field V ab in the perturbation theory, the equation of motion for the auxiliary field D of the vector multiplet has a solution D = 0, as in the ModMax theory [1].