Perturbative linearization of supersymmetric Yang-Mills theory

Supersymmetric gauge theories are characterized by the existence of a transformation of the bosonic fields (Nicolai map) such that the Jacobi determinant of the transformation equals the product of the Matthews-Salam-Seiler and Faddeev-Popov determinants. This transformation had been worked out to second order in the coupling constant. In this paper, we extend this result (and the framework itself) to third order in the coupling constant. A diagrammatic approach in terms of tree diagrams, aiming to extend this map to arbitrary orders, is outlined. This formalism bypasses entirely the use of anti-commuting variables, as well as issues concerning the (non-)existence of off-shell formulations for these theories. It thus offers a fresh perspective on supersymmetric gauge theories and, in particular, the ubiquitous $\mathcal N{=}\,4$ theory.


Introduction and summary
A key consequence of supersymmetry is the dramatic improvement it produces in the ultra-violet behavior of quantum field theories. The importance of anti-commuting variables in formulating and quantizing supersymmetric field theories is well known. These variables are, however, not the easiest to work with, especially when it comes to setting up an off-shell formulation, which often leads to a proliferation of auxiliary variables and unphysical degrees of freedom (and for the most interesting theories is not believed to exist). The idea that supersymmetric theories could be formulated without anti-commuting variables, and thus characterized in a more economical fashion in terms of a purely bosonic functional measure was first proposed in [1,2] and further developed in [3][4][5][6]. 1 This approach to supersymmetric fermion-boson models, referred to as a 'Nicolai map', is designed to cancel out the fermion determinant while simultaneously reducing the boson measure to a free one (see [7] for a pedagogical introduction). Supersymmetric gauge theories, the focus of this paper, can be characterized by the existence of such a functional map T g -a transformation of the bosonic fields such that the Jacobi determinant of the transformation exactly equals the product of the Matthews-Salam-Seiler (MSS) [8] and Faddeev-Popov (FP) [9] determinants. This was explicitly shown to second order in the coupling constant in [1]. In this paper, we extend these results and the framework itself to third order in the coupling constant by presenting an explicit formula for T g , in equation (2.1). In addition, we provide a Feynman-like graphical approach using tree diagrams that in principle allows one to extend this construction to all orders.
This novel approach sidesteps entirely the use of abstractly defined anti-commuting objects and hence offers a fresh perspective on the quantization of supersymmetric gauge theories. Given the central role quantum gauge theories play in describing the real world, a new window into their workings is invaluable. This framework, for example, allows us to re-derive [10] the classical result [11] that interacting supersymmetric Yang-Mills theories exist only in space-time dimensions D = 3, 4, 6 and 10 (together with the extended supersymmetric Yang-Mills theories obtained from these by dimensional reduction). Specifically, for the N = 4 theory in D = 4, all known results for scalar correlators may also be obtained within this formalism, at least up to the order for which the map had previously been worked out [12]; the computational efforts involved compare well with the more standard techniques of perturbative quantum field theory. An understanding of scattering amplitudes in this approach is likely to yield new insights into the symmetries that underlie these simple structures. A related long-term goal is to move beyond perturbation theory and establish a link between the map and the integrable properties of the N = 4 model (see e.g. [13]). There also exists earlier work which focussed primarily on the search for, and the exploitation of, variables providing a local realization of the map T g [14][15][16][17] (the precise relation between these older results and the non-local map T g will be left for future study), as well as an alternative construction of T g in [18].
We start by stating the main theorem from [1,2].

Main theorem
Supersymmetric gauge theories are characterized by the existence of a non-linear and non-local transformation T g of the Yang-Mills fields which is invertible at least in the sense of a formal power series such that 1. The Yang-Mills action without gauge-fixing terms is mapped to the bosonic abelian action, , at least order by order in perturbation theory.

The gauge fixing function G a [A] is a fixed point of T g .
A general proof of this theorem is presented in appendix A, and is largely based on existing work [2][3][4][5][6]. However, the present proof is more general in that it takes into account recent insights from [10] that the theorem is actually valid for all supersymmetric Yang-Mills theories in space-time dimensions D = 3, 4, 6, 10. These include the corresponding extended theories obtained by reduction (like N = 4 Yang-Mills theory from the D = 10 theory).
Even with the general proof and explicit expressions for A ′ a µ (x) at hand it is by no means obvious that the transformed field A ′ a µ (x) satisfies all three statements in the main theorem above, at every order in g. Our main goal with this paper is to work out the transformation at O(g 3 ) and to explain in detail how all the necessary conditions are satisfied. We hope that the explicit expressions, derived in section 3, illustrate the non-triviality of this result. With these findings one can now proceed to compute quantum correlators by means of the formula (where · · · g denotes the expectation value for coupling g) in terms of the free functional measure for the gauge field, and thus extend the results in [12] to the next order.
Let us also emphasize that, at this point, all the statements in the above theorem are to be understood to hold in the sense of formal power series. Non-perturbatively, we will have to worry about zero modes of the relevant determinants [19]: on the mass shell, we have , (1.2) directly relating the Jacobian of the transformation to the second-order fluctuation operator around a given background solution of the Yang-Mills field equations, such as an instanton solution (the divergence term on the r.h.s. of (1.2) can be ignored on the Landau gauge hypersurface ∂ µ A a µ = ∂ µ A ′ a µ = 0). Since the eigenvalues of the fluctuation operator are known to be related to the ones of the MSS determinant, matching zero modes imposes an extra restriction on the theory. As already pointed out in [19] the match works only for the maximally extended N = 4 theory in D = 4, which is additionally singled out for this reason.
The work presented below is in Euclidean space. However, the Euclidean metric is by no means crucial to our discussion, as all of these results can be derived in spacetimes with a Lorentzian signature as in [3][4][5]. We also restrict ourselves (mostly) to the Landau gauge, in order to avoid complicating the presentation. Other choices for the gauge function G a [A], in particular the axial and light-cone gauges, will be discussed elsewhere. The light-cone Hamiltonians for pure and maximally supersymmetric Yang-Mills theories are quadratic forms [20]. This "complete square" structure, highly reminiscent of the map, is likely related to a light-cone realization of T g .

Notation and conventions
We restrict our attention in this paper to super Yang-Mills theories and work in Euclidean space, rendering upper and lower indices equivalent. With the admissible dimensions D = 3, 4, 6 and 10, the corresponding numbers of fermionic degrees of freedom are In section 3 and the appendix A.2 we will rederive this relation in a novel manner. We shall employ the generator R of the inverse transformation T −1 g (see below). A key role in the R operator is played by the fermionic propagator S ab (x, y; A) in a gauge-field dependent background characterized by A a µ (x), with (1.4) The limit g = 0 gives us the free fermionic propagator S ab 0 (x) which obeys Here C(x) is the free scalar propagator (with the Laplacian ∂ µ ∂ µ ≡ ✷) It satisfies ✷ C(x) = −δ(x). We use the convention that the derivative always acts on the first argument, i.e. ∂ ρ C(x−y) ≡ (∂/∂x ρ )C(x−y) ≡ ∂ x ρ C(x−y). Thus, we need to be careful with sign flips when using ∂ . In general, the R operator works for a number of gauge choices. However, special elements of the present calculation do depend on the chosen gauge. Henceforth, for simplicity we will only consider the Landau gauge function Finally, we require the implementation of the ghost propagator G ab (x, y; A), obeying Hence, the free ghost propagator satisfies It is important to note that not only G ab (x, y; A) depends on g and the background field A a µ (x) but that (D µ G) ab (x, y; A) does so as well, viz. 3 with the abelian transversal projector where ≃ means equality in the sense of a distribution. We will later see that the terms of O(g) in (1.10) become relevant for the map T g from order g 3 onwards.

The R operator
To determine the map T g one first constructs its inverse T −1 g via its infinitesimal generator R, a non-local and non-linear functional differential operator first introduced for the N = 1, D = 4 theory in [3][4][5]. The image (T g A) a µ (x) of the map is then obtained order by order in g by formally inverting the power series where R is the infinitesimal generator of the inverse map. Details of the construction of the R operator are provided in appendix A.1, thus generalizing the original proof given in [3] for D = 4. We use the background field dependent propagators λ a (x)λ b (y) ≡ S ab (x, y; A) and C a (x)C b (y) ≡ G ab (x, y; A) (1.13) 3 We note that the corresponding formula (1.18) in [10] is incomplete in that it missed out on the g-dependence of (DµG) ab . However, this correction kicks in only at O(g 3 ) and beyond, hence does not affect the results up to second order in [1,10].
to rewrite (A. 19). Here λ a (x) are the gaugino fields (prior to their elimination via the MSS determinant), and C a (x) andC a (x) denote the ghost and anti-ghost fields. For the Landau gauge the R operator is then represented by the functional differential operator (1.14) where we introduced the 'covariant' (or 'non-abelian') transversal projector (1.15) Since the bosonic background field A a µ (x) does not depend on g, the first application of R to A a µ (x) is straightforward. For all higher orders we also need and as well as These equations are obtained from (1.4) and (1.8). After iteratively computing R n for any desired n, we set g = 0, which in particular maps S ab (x, y) to the free propagator S ab 0 (x−y) and G ab (x, y) to the free propagator G ab 0 (x−y), and finally obtain ( The actual map T g is then obtained by power series inversion. Let (1.20) Expanding T −1 g T g = 1 in powers of g and matching coefficients we readily obtain (1.21) The explicit expression for (T g A) a µ (x) up to O(g 3 ) is provided in the following section.

Result and discussion
We now present the main new result which is the explicit formula for T g to cubic order O(g 3 ) 4 (2.1) 4 As usual, all anti-symmetrizations are with strength one, such that e.g.
The first two lines above correspond to the result obtained in [1] and extended to dimensions D = 4 in [10]. The last two lines are the new terms arising from the g-dependence of (D µ G) ab in (1.8); they are crucial for the fulfillment of the conditions in the main theorem. While the result up to O(g 2 ) was originally obtained by trial and error in [1], this becomes tricky at higher orders because the number of terms is significantly larger at O(g 3 ) than below. In addition, from the last term we see that new structures appear. In the following section we will verify that this result indeed satisfies all three statements of the main theorem (subsection 1.1) simultaneously, providing a highly non-trivial test.

Tests
A general proof that the statements in the main theorem in subsection 1.1 are true at any order of g is given in appendix A. A detailed explanation of how this works up to O(g 2 ) can be found in [10]. Thus, we only present here the calculations at third order in g.

Gauge condition
We first verify that Applying ∂ µ to the terms of order g 3 in (2.1) and removing all terms that are manifestly anti-symmetric under the exchange of indices µ and ρ yields (3.1) The first two terms cancel each other. In the third term we use ✷ C(x − y) = −δ(x − y). It is then easy to see that

Free action
By the first statement in the main theorem the transformed gauge field must satisfy We stress that, unlike the matching of determinants, the fulfillment of this condition will turn out not to depend on the dimension D. Because of the invariance of the Landau gauge function, we can ignore the second term on the l.h.s. and the corresponding term on the r.h.s. of this equation. The calculation, up to O(g 2 ), is presented in detail in [10]. At third order, (3.3) has two contributions which must cancel each other: To check this we collect all the terms to obtain (3.5) The general procedure to simplify this expression is rather straightforward. However, there are a few terms which require additional attention. The first step is to integrate each term by parts such that the Laplacian acts on the C(x − y), which then simplifies to a δ-function and we obtain We notice that we can replace any if the full expression is symmetric under simultaneous exchange a ↔ c and µ ↔ λ. This allows us to integrate the respective terms by parts again. After the integration the index contractions must be spelled out and most terms cancel. Subsequently, we obtain (3.7) The first term vanishes by the Jacobi identity, i.e.
The second, third and fourth term in (3.7) can be integrated by parts and after removing the terms that are anti-symmetric under the exchange of two indices, we get Thus, the condition (3.3) holds up to and including O(g 3 ). It is worthwhile to point out here that the very existence of a non-local field transformation mapping one local action to another local action is a remarkable fact in itself, independently of supersymmetry (but in the absence of supersymmetry, locality would be spoilt by the Jacobian).

Jacobians, fermion and ghost determinants
Finally, we need to perturbatively show that on the gauge surface the Jacobian determinant is equal to the product of the MSS and FP determinants. This is done order by order in g by considering the logarithms of the determinants rather than the determinants themselves; since the relevant checks up to O(g 2 ) were already performed in [1,10], we can here concentrate on the third order, viz. . (3.10) Of the three statements in subsection 1.1 this is the most complicated condition to verify. Moreover, it is the only condition that depends on the dimension of our field theory and will impose the constraint (1.3) on the latter.
The ghost determinant is computed from the functional matrix using the well-known equation (3.13) For the MSS determinant we have (3.14) After accounting for the Majorana (or Weyl) condition by including an extra factor of 1 2 we get (

3.15)
For both determinants there is no contribution at O(g) and we have simplified the results at O(g 2 ) by using f abc f abd = n δ cd . Taking the trace in (3.15) and multiplying the two determinants subsequently yields the right hand side of (3.10) (3.16) We thus end up with a total of five independent structures; we use color coding to help us identify the corresponding terms in the Jacobian determinant.
At O(g 3 ) the logarithm of the Jacobian determinant schematically consists of three terms (3.17) and the final trace is done by setting µ = ν, a = b, x = y and integrating over x. The computation is straightforward and we find (3.18) The second term gives Finally, the first term gives (3.20) There are two special features about this part of the Jacobian determinant. First, we have to use the gauge condition G a [A] ≡ ∂ µ A a µ = 0 to eliminate two terms. Secondly, we find terms which do not match any of the five structures from the fermion and ghost determinants and hence must cancel among themselves. However, before addressing those terms, let us first analyze the color coded terms. Imposing the equality (3.10) yields the following conditions thus extending the result of [10] to cubic order. It remains to be shown that the remaining (black) terms from (3.20) vanish. Using the Jacobi identity in the first term and f abc f abd = n δ cd in the latter two yields (3.23) The second term is rewritten using the identity with a formally divergent piece C(0) which can be appropriately regulated. This simplifies the expression above to (3.25) Subsequent integration by parts produces (3.26) Thus, (3.10) is satisfied. Let us note that, unlike for the O(g 2 ) computation, we had to make use of the Landau gauge condition (1.7) to achieve this equality. This feature, which arises only from O(g 3 ) onwards, is entirely due to the g-dependence of the ghost propagator in (1.8).

Beyond the third order: graphical representation
It is not difficult to present the full perturbative expansion of the R operator. To streamline the notation, we suppress the color indices and position variables and integrations, but understand all objects to be multiplied as color matrices or vectors and by convoluting integration kernels with A insertions. Concretely, abbreviating and expanding where derivatives act on everything on their right, we may write where we now let the variation with respect to g and A act to the left in order to conform with the implicit color and position ordering in these expressions. In the last line, we have expanded the abelian transversal projector Π and simplified The individual terms of the expansion in powers of g allow for a simple representation in terms of Feynman diagrams, built from free fermion and ghost propagators dressed with photon insertions. They have the graphical structure of linear trees mostly starting with a modified double ghost line 2δ αβ µν ∂ α G 0 ∂ ν A β G 0 and ending in a "double photon emission" 1 2 / A 2 . In between, one finds k further ghost lines (k = 0, 1, 2, . . .) followed by ℓ fermion lines (ℓ = 0, 1, 2, . . .), separated by the appropriate photon emission insertions. Since the initial A-variation reduces the power of A by one, such a term contributes to O(g k+ℓ+1 A k+ℓ+2 ). In addition, there are the ghostfree linear trees, which are rooted in a modified fermion line γ µν ∂ ν G 0 , comprise ℓ additional fermion lines and also terminate in The Feynman-like rules for these graphs are as follows: Furthermore, a trace has to be performed in spinor space. Since for each fermion line, gamma matrices from the free fermion propagators S 0 and from the photon insertions / A are multiplied along the linear tree, the trace short-circuits the tree in spinor space and contracts the Lorentz indices of the partial derivatives on C and of the photon emissions A in every possible fashion.
The perturbative map is obtained by iterating the R operation to build (T −1 g A) a µ (x) according to (1.12) and inverting the power series (see (1.21)). In terms of the power-series components R n defined above, the n-th order contribution to T g A is given by [18]  and with coefficients c {n} = (−1) s n! n 1 · (n 1 +n 2 ) · (n 1 +n 2 +n 3 ) · · · (n 1 + · · · +n s ) . (4.8) To order g 4 , this expansion reads (4.9) The repeated action of R on itself grafts linear trees onto linear trees. This produces binary trees of all kinds with double leaves / A 2 , where multiple gamma-matrix traces run over all possible parts of those trees, and any part of a tree may have fermion lines replaced by ghost lines. Excluded only are length-one ghost lines and ghost lines ending in a double leaf. After perfoming the gamma-matrix traces, all lines become scalar propagators C dressed with a partial derivative, whose Lorentz indices get contracted in almost all possible ways. However, there appear partial cancellations of gamma-matrix traces between trees of the same topology. The combinatorial factors and Feynman rules for these trees will be given elsewhere.
Finally, one may raise the question of the uniqueness of T g A. As the map is constructed iteratively from the R operator, a non-uniqueness will originate from an ambiguity in R. In D=4, such an ambiguity arises from the freedom to add a topological term ∼ F ∧ F to the bosonic action, which allows for the modification in the R operator and, hence, in the Feynman rules. As a consequence, novel terms carrying the epsilon tensor appear in the expansion of T g A. This offers, for κ = ±1, the option of a chiral Nicolai map [2,6]. The possibility of such a chiral projection may be explored for the other critical dimensions as well.

A Proof of the main theorem
In this appendix we firstly construct the R operator from the response of the vacuum expectation value of an arbitrary product of bosonic operators to changes in the coupling constant. This generalizes the argument first presented in [3] for the D=4, N =1 theory to all other pure supersymmetric Yang-Mills theories. Secondly, we prove the distributivity of the R operation. Thirdly, we recall the argument that R annihilates the gauge-invariant bosonic Yang-Mills action as well as the gauge-fixing function. These properties imply the determinant matching and suffice to establish the main theorem.

A.1 Construction of the R operator
In this section we show how to construct the R operator for any pure super Yang-Mills theory.
The action consists of a gauge invariant part and a gauge fixing part (with gauge parameter ξ) For an arbitrary product of bosonic operators X the linear response of its vacuum expectation value to a change in the coupling constant is given by Making use of supersymmetry, we want to rewrite the right-hand side in terms of a derivational operator R. To this end we introduce and use to compute Notice that δ α anticommutes with other anticommuting operators. With and the gauge condition ∂ µ A b µ = 0 we arrive at d dg We want to rewrite using the supersymmetry Ward identity Employing the Slavnov variations one finds that Thus, the Ward identity becomes (A.14) By (A.10) we need to apply this to Y = ∆ α X, and from s(∆ α X) = s(∆ α )X − ∆ α s(X) we also require the Slavnov variation of ∆ α . Making use of the Jacobi identity we get Subsequently we can put everything back together, In the next subsection we will show that the multiplicative contribution vanishes, only in the critical dimensions D = 3, 4, 6 and 10, where r = 2(D−2) indeed. Thus, by integrating over all fermionic degrees of freedom we obtain R X = dX dg + δ α X · ∆ α + dxC a (x)δ α ∂ µ A a µ (x) ∆ α s(X) . (A.19) For N = 1 super Yang-Mills theory this result has first been derived in [3], see also [2].
A.3 R annihilates the bosonic action and the gauge-fixing function Statement [2.] in the main theorem, about determinant matching, follows from the other two parts and from the equality valid for any string X of bosonic operators, where there subscript '0' on the right-hand side indicates an expectation value in the free theory (g=0). This statement can be seen to be equivalent [18] to a fixed-point property of T g under the coupling constant flow, from which in fact T g can be constructed directly [18]. For completeness, let us recall the proof that R annihilates the bosonic invariant action [2]. Recall the form (1.14) of the R operator with the covariant transversal projector (1.15). From dx dz dy D σ F σµ a (x) P ae µν (x, z) tr γ ν S eb (z, y) γ ρλ f bcd A c ρ (y)A d λ (y) .

(A.29)
We can take advantage of the fact that S g is gauge invariant: Since dx D σ F σµ a (x) D µ G ae (x, z) ∂ ν = − dx (D µ D σ F σµ a (x) G ae (x, z) ∂ ν = 0 , (A.30) the projector P µν in R can be replaced by the identity. Then the second term in (A.29) becomes where we partially integrated D σ , employed γ µν γ σ = 2γ [µ δ ν]σ + γ µνσ and observed the Bianchi identity γ σµν D σ F µν = 0. In this form, the first term (A.33) is seen to cancel the second term (A.31), which proves the assertion.
Statement [3.] in the main theorem is almost trivial, given the form (1.15) of the transversal projector inside R. The fixed-point feature of the gauge fixing function, is built into the formalism, since the projector by construction annihilates G a . For the Landau gauge, by the definition of the ghost propagator. This is also apparent from the form (A.19) of the R operator, where for X = G a [A] the second and third terms cancel each other. In fact, this consideration generalizes to an arbitrary gauge. 5 The main theorem is herewith proved, for the Landau gauge.

B Explicit expression for R 3
In this section we give the explicit expression for R 3 A a µ (x) . Applying the R-operator three times and repeatedly using (1.14) as well as (1.16) and (1.17) we arrive at