Nicolai maps and uniqueness in the light-cone gauge
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Nipun Bhave and Saurabh Pant
Indian Institute of Science Education and Research
Pune 411008, India
We compute the Nicolai map for the supersymmetric Yang-Mills theory, in the light-cone gauge, to the second order in the coupling constant for all critical dimensions ($d=3,4,6,10$). The process of integrating out unphysical degrees of freedom in this gauge, produces a four fermion interaction term. We show that, to the order investigated here, this term is harmless.  We demonstrate the existence of a particularly `simple'  map in $d=4$ in the light-cone gauge and address the issue of uniqueness in the context of the map. We also investigate the map in the light-cone superspace in $d=4$.

§ INTRODUCTION

Supersymmetric gauge theories have been studied extensively because of their interesting ultraviolet properties. The flagship theory, the maximally supersymmetric $\mathcal{N}=4$ Yang-Mills theory, for example, is a perturbatively finite quantum theory in four dimensions. An alternative perspective on supersymmetric gauge theories is offered by the Nicolai map [2, 1, 3] which makes the formulation of these theories possible without the use of anti-commuting variables [5, 4].

While this map has been investigated in the Landau gauge [6, 7, 8], its study in other gauges has been limited [9, 11, 10]. Not surprisingly, the map appears more complicated in the axial gauge (and in the light-cone gauge) than in the Landau gauge [9]. In this paper, we work with the $LC_2$ light-cone gauge approach which means that only the physical degrees of freedom of the theory are retained. This means that the role of the little group is explicit while manifest covariance is sacrificed. The close link between the light-cone gauge and spinor-helicity variables [12], implies that studying the map in this language could prove useful in the study of scattering amplitudes.

One aim of this paper is to demonstrate the existence of a particularly `simple'  map in $d=4$ in the light-cone gauge and consequently in terms of the helicity variables. We also find a map that works in $d=3,4,6$ and $10$. When we write this map explicitly in $d=4$ in terms of the helicity variables, we find that this map is distinct from the `simple' $d=4$ map. This raises the question of the uniqueness of the map which we attempt to address in this paper [8, 13].

The second part of this paper focuses on superspace. The first all-order proof of finiteness for the $\mathcal{N}=4$ Yang-Mills theory was provided using light-cone superspace [14, 15]. It is therefore of interest to ask how this proof of finiteness may be achieved within the framework of the Nicolai map. With this goal in mind, we find a non-linear and non-local transformation for the $\mathcal{N}=1$ superfield with a trivial Jacobian such that the full $\mathcal{N}=1$ Lagrangian is mapped to a free Lagrangian in light-cone superspace.

A future direction of interest would be to ask whether the issues addressed in this paper could be extended to supergravity formulated in light-cone superspace.

The paper is organized as follows. In section 2, we start with the $\mathcal{N}=1$ Yang-Mills theory in the light-cone gauge to write a map in $d=4$. In section 3, we generalize the map to all critical dimensions ($d=3,4,6,10$). We also establish its connection with the map derived in general gauges [9]. In section 4, we comment on the uniqueness of the light-cone Nicolai map in $d=4$. In section 5, we compute the Nicolai map in light-cone superspace in four dimensions. In the last section, we discuss the possible connection between the Nicolai map and the quadratic form structure in the light-cone Hamiltonian for pure Yang-Mills theory.
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Note: While working on the manuscript, we became aware of [16], which contains some overlapping results.

§ LIGHT-CONE NICOLAI MAP IN $D=4$

We work with the light-cone coordinates given by
x^±=x^0 ± x^3/√(2) ,  x=x^1+ix^2/√(2) ,x̅=x^* ,

and their derivatives $\del_{\pm}\,(-\del^{\mp})\,,\,\bar{\del}\,,\,\del$ respectively$\,$[We define the `inverse' derivative using the step function: $\frac{1}{\del^+}\,f(x^-)\,\equiv\,-\int dy^-\,\,\theta(x^--y^-)f(y^-)$.].

We start with the $\mathcal{N}=1$ supersymmetric Yang-Mills Lagrangian in the light-cone gauge written purely in terms of the physical degrees of freedom [17]. Details of the procedure to obtain this Lagrangian are presented in Appendix A. The Lagrangian written entirely in terms of physical fields - the gauge fields and fermion fields ($A,\bar{A},\chi,\bar{\chi}$) - is

ℒ   =   A̅^a A^a - 2g f^abc (∂̅/∂^+A^a∂^+A̅^b A^c+∂/∂^+A̅^a∂^+A^bA̅^c)

-2g^2 f^abc f^ade 1/∂^+(∂^+A^bA̅^c)1/∂^+(∂^+A̅^d A^e)

+ i/√(2)χ̅^a(/^+δ^ac-2gf^abc1/^+(∂A̅^̅b̅+∂̅A^b) +2gf^abcA̅^b∂/^+)χ^c + i√(2) gf^abc χ̅^a∂̅/^+(A^b χ^c)

+ i√(2) g^2 f^abcf^bde χ̅^a1/^+2(A^d^+A̅^e+A̅^d^+A^e)χ^c-i√(2) g^2 f^abdf^becχ̅^a A̅^d1/^+(A^eχ^c)

+g^2f^abcf^ade1/^+(χ̅^bχ^c)1/^+(χ̅^dχ^e) .

where $\Box=(-2\del^+\del^-+2\del\bar{\del})$ and the $f^{abc}$ are the structure constants of the gauge group.

The statement of the Nicolai map is the following: there exists a non-linear and non-local transformation $\mathcal{T}_g(A)$ which satisfies the following three properties:

1. The transformation $\mathcal{T}_g(A)$ when substituted in the free bosonic Lagrangian (Maxwell theory) yields the full interacting bosonic Lagrangian (Yang-Mills theory).

2. The Jacobian of the transformation is equal to the fermion determinant (or the product of the fermion and ghost determinants in cases where the gauge choice does not eliminate all unphysical degrees of freedom).

3. The transformation preserves the gauge choice.

Essentially, this means that one works with a free bosonic theory to compute correlators in a supersymmetric gauge theory - through the inverse transformations $\mathcal{T}_g^{-1}(A^{'})$ [18].

§.§ The transformation

We now write down a field transformation for the physical fields $A^a$ and $\bar{A}^a$ by trial and error so the Yang-Mills Lagrangian may be written as a purely kinetic term in the new (primed) variables: $\bar{A}^{'\,a}\Box A^{'\,a}$. We introduce a Green's function through $\Box C(x-y)=-\delta^{(4)}(x-y)$ to write such an ansatz upto $\mathcal{O}(g^2)$

A^' a(x,g;A,A̅)   =   A^a(x)+2gf^abc ∫dy   ^+C(x-y) ∂̅/^+ A^b(y)A^c(y)

-g^2 f^abcf^bde ∫dy   ^+ C(x-y) A^c(y) 1/^+ 2(^+A^d(y)A̅^e(y))

- 2g^2 f^abcf^bde  ∫dy  dz   ( C(x-y) A̅^c(y) - ^+ C(x-y)∂/^+ A̅^c(y) )

×  ^+  C(y-z)  ∂̅/^+ A^d(z)A^e(z) .
Here $dy\,,\,dz$ denote the four dimensional space-time measure. In this section, all measures and delta functions will be assumed to be four dimensional and the dimension will be suppressed henceforth. The transformation for $\bar{A}^{'\,a}$ is just the complex conjugate of the above.

In a covariant approach, the map at order $g^2$, contains terms of the form $\del\, C\, A \,\del\, C \,A \,A$ (with space-time indices and color indices suppressed), all of which contribute to the Jacobi determinant. In the light-cone Nicolai map at order $g^2$, we find that there is a term with a single Green's function (line 2 of eq. (<ref>)). This term produces the pure Yang-Mills quartic vertex (line 2 of eq. (<ref>)) but does not contribute to the Jacobian at order $g^2$ as we show below.

The functional variation of the fields are
δA^a(x)/δA^b(w)=δA̅^a(x)/δA̅^b(w)=δ^abδ(x-w) ,     δA^a(x)/δA̅^b(w)=δA̅^a(x)/δA^b(w)=0 .

The Jacobi matrix[The matrix elements $\cfrac{\delta A^{'\,a}(x)}{\delta\bar{A}^m(w)}$ and $\cfrac{\delta \bar{A}^{'\,a}(x)}{\delta A^m(w)}$ do not contribute to the trace at order $g^2$] of the above transformation (<ref>) is

δA^' a(x)/δA^m(w)   =   δ^amδ(x-w)+2gf^abc ∫dy   {^+C(x-y) ∂̅/^+ δ^bmδ(y-w)A^c(y)

+ ^+C(x-y) ∂̅/^+ A^b(y) δ^cmδ(y-w)}

-2g^2 f^abcf^bde ∫dy  dz (  C(x-y) A̅^c(y) - ^+ C(x-y)∂/^+ A̅^c(y) )

×  { ^+  C(y-z) ∂̅/^+δ^dm δ(z-w)A^e(z) + ^+  C(y-z)  ∂̅/^+ A^d(z) δ^em δ(z-w)} ,

where we have dropped all terms that vanish after taking the trace as they are proportional to $\del_{\mu} C(0)$. The Jacobi determinant of the map can be computed using the relation

log(1+X)=Tr log(1+X) =Tr  X- 1/2Tr  X^2 ±..

After partial integrations, taking the trace (by setting $a=m$ and $x=w$), using $f^{abc}f^{abd} = n\delta^{cd}$ and integrating over $x$, the Jacobi determinant upto $O(g^2)$ reads

log(δA_i'^a(x)/δA_j^m(w) )   =   2ng^2 ∫dx  dy  {   ∂̅ C(x-y) A^b(y)  ∂̅  C(y-x)  A^b(x)

+ ^+ C(x-y)∂̅/^+ A^b(y) ^+ C(y-x)∂̅/^+ A^b(x)

- 2 ∂̅ C(x-y) A^b(y) ^+ C(y-x)∂̅/^+ A^b(x)

+   ∂/^+ C(x-y) A̅^b(y)  ^+ C(y-x) A^c(x)

-  ∂ C(x-y) A̅^b(y) ^+ C(y-x)∂̅/^+ A^b(x)

-  ∂̅ C(x-y) A^b(y) ^+ C(y-x)∂/^+ A̅^b(x)

+  ^+ C(x-y)∂/^+ A̅^b(x) ^+ C(y-x)∂̅/^+ A^b(x)} + c.c. .

Here $i,j$ run over the transverse variables $x,\bar{x}$.

§.§ The fermion determinant

Computing the fermion determinant is complicated by the presence of a four fermion interaction term in (<ref>). It was shown in [19] that the construction of Nicolai maps can also be extended to supersymmetric theories with four fermion interaction terms. In Appendix B, we explain why this term does not contribute to the fermion determinant at order $g^2$. The quadratic operator[Note that the zero modes of the operator $\del^+$ can be removed using the appropriate boundary conditions [20]] $Q^{ac}$ relevant to order $g^2$ (again dropping terms which vanish after `trace-ing'), is
Q^ac(x;A)   =   /^+δ^ac-2gf^abc1/^+(∂A̅^̅b̅+∂̅A^b)+2gf^abc∂̅/^+(A^b +2gf^abcA̅^b∂/^+ ,
which may be written as
Q^ac(x,y;A)   =   /^+(δ^ac+2gf^abc∫dy  ^+ C(x-y)1/^+(∂A̅^b(y)+∂̅A^b(y))

-2gf^abc∫dy  ∂̅C(x-y)A^b(y) -2gf^abc∫dy ^+ C(x-y)A̅^b(y)∂^(y)/^+) .

As $\det (Q)=\det(\Box/\del^+)\times \det(1+\textbf{Y})$, we use (<ref>) to compute the fermion determinant order by order in $g$ upto an overall constant $\det(\Box/\del^+)$.

The fermion determinant to order $g^2$ is

log(1+Y)   =   2ng^2 ∫dx  dy  {   ∂̅ C(x-y) A^b(y)  ∂̅  C(y-x)  A^b(x)

+ ^+ C(x-y)∂̅/^+ A^b(y) ^+ C(y-x)∂̅/^+ A^b(x)

- 2 ∂̅ C(x-y) A^b(y) ^+ C(y-x)∂̅/^+ A^b(x)

+   ∂/^+ C(x-y) A̅^b(y)  ^+ C(y-x) A^c(x)

-  ∂ C(x-y) A̅^b(y) ^+ C(y-x)∂̅/^+ A^b(x)

-  ∂̅ C(x-y) A^b(y) ^+ C(y-x)∂/^+ A̅^b(x)

+  ^+ C(x-y)∂/^+ A̅^b(x) ^+ C(y-x)∂̅/^+ A^b(x)} + c.c. .
Thus, we find that the Jacobi determinant of the bosonic transformation (<ref>) exactly matches the fermion determinant (<ref>) upto $O(g^2)$.

§ EXTENSION OF THE LIGHT-CONE MAP TO ALL CRITICAL DIMENSIONS

We now illustrate how this light-cone realization of the Nicolai map extends nicely to all critical dimensions (<ref>). One perhaps obvious observation is that such a result cannot begin from a Lagrangian in a helicity basis, which is closely tied to four dimensions (and the little group $SO(2)$).

§.§ The move away from a helicity basis

We start instead from the Lagrangian (<ref>) in Appendix A. Note that the free bosonic and the free fermionic degrees of freedom match only in $d=3,4,6,10$. We show that when interactions are switched on, the fermion determinant matches the Jacobian only in the critical dimensions, hence confirming the existence of the supersymmetric Yang-Mills theories in these dimensions.

We identify a field transformation for the physical fields $A_i$ so that we can write the pure Yang-Mills theory (first two lines of equation (<ref>)) as $\frac{1}{2} {A_i}^{\prime\,a}\Box A_i^{\prime\,a}$. In this section, all measures and delta functions will be $d$-dimensional (dimensions will get fixed using the determinant matching). Again, we introduce $\Box C(x-y)=-\delta^{(d)}(x-y)$ to write the map to order $g^2$

A_i^'  a(x)   =   A_i^a(x)+ gf^abc ∫dy   (^+ C(x-y) _j/^+ A_j^b(y)A_i^c(y) - _j C(x-y) A_j^b(y) A_i^c(y) )

-   g^2/2 f^abcf^bde  ∫dy   ^+ C(x-y) A_i^c(y) 1/^+  2(^+ A_j^d(y) A_j^e(y))

+   g^2/2 f^abcf^bde  ∫dy  dz   {  _j C(x-y)  A_k^c(y)

×  ( _i C(y-z) A_k^d(z)A_j^e(z)+ _k C(y-z) A_j^d(z)A_i^e(z))

-  _i C(x-y) A_j^c(y)  ^+ C(y-z) _k/^+ A_k^d(z) A_j^e(z)

+   ^+ C(x-y) _j/^+ A_j^c(y)  ^+ C(y-z) _k/^+ A_k^d(z) A_i^e(z)

+   2   _i C(x-y)  A_j^c(y)  _k C(y-z) A_k^d(z)A_j^e(z)

-   2 ^+ C(x-y) _j/^+ A_j^c(y)  _k C(y-z) A_k^d(z)A_i^e(z)

+   ^- C(x-y) A_k^c(y) ^+C(y-z) A_k^d(z) A_i^e(z)

+   ^+ C(x-y) A_k^c(y)^-C(y-z) A_k^d(z) A_i^e(z)

-   _j C(x-y)  A_k^c(y)  _j C(y-z) A_k^d(z)A_i^e(z)
} .

§.§ Jacobian

We calculate below the Jacobian of the transformation (<ref>).
δA_i^' a (x)/δA_m^ p (w)   =   δ_i^m δ^apδ(x-w)+gf^abc ∫dy  { ^+ C(x-y)  δ(y-w) ( _j/^+ δ_j^m δ^bpA_i^c(y)

+   _j/^+ A_j^b(y)  δ_i^m δ^cp ) -_j C(x-y)  δ(y-w) ( δ_j^m δ^bp A_i^c(y) +  A_j^b(y)  δ_i^m δ^cp )  }

+   g^2/2 f^abcf^bde  ∫dy  dz  { _j C(x-y)  A_k^c(y)  δ(z-w) {_i C(y-z) δ_k^m δ^dp A_j^e(z)

+   _i C(y-z) A_k^d(z) δ_j^m δ^ep +   _k C(y-z) δ_j^m δ^dp A_i^e(z)

+   _k C(y-z) A_j^d(z) δ_i^m δ^ep  } -    _i C(x-y) A_j^c(y) δ(z-w)

×   (  ^+ C(y-z) _k/^+ δ_k^m δ^dp A_j^e(z) +  ^+ C(y-z) _k/^+ A_k^d(z) δ_j^m δ^ep )

+   ^+ C(x-y) _j/^+ A_j^c(y)  δ(z-w) ^+ C(y-z)

×   (  _k/^+ δ_k^m δ^dp  A_i^e(z) + _k/^+ A_k^d(z) δ_i^m δ^ep  ) + 2   _i C(x-y)  A_j^c(y)  δ(z-w)

×   ( _k C(y-z) δ_k^m δ^dp A_j^e(z) + _k C(y-z) A_k^d(z)  δ_j^m δ^ep )

-   2 ^+ C(x-y) _j/^+ A_j^c(y)  δ(z-w)

×   ( _k C(y-z) δ_k^m δ^dp A_i^e(z) + _k C(y-z) A_k^d(z)δ_i^m δ^ep )

+  ^- C(x-y) A_k^c(y)   δ(z-w)

×  (^+ C(y-z) A_k^d(z) δ_i^m δ^ep +  ^+ C(y-z) A_i^e(z) δ_k^m δ^dp )

+   ^+ C(x-y) A_k^c(y)  δ(z-w)

×  ( ^- C(y-z) A_k^d(z) δ_i^m δ^ep +  ^- C(y-z) A_i^e(z) δ_k^m δ^dp )

-   _j C(x-y)  A_k^c(y)   δ(z-w)

×  ( _j C(y-z)  δ_k^m δ^dp A_i^e(z) +  _j C(y-z)   A_k^d(z) δ_i^m δ^ep ) } ,

where we have only written the non-trivial terms relevant till order $g^2$.
We take the trace by setting $a=m$, $w=x$ and integrating over $x$. We also use the $SU(n)$ identity $f^{abc}f^{abd} = n\delta^{cd}$ to obtain the Jacobi determinant upto $O(g^2)$ using (<ref>)

log(δA_i'^a(x)/δA_j^m(w) )   =    ng^2 ∫dx  dy   (d-2){    ∂_i  C(x-y) A_i^b(y)  ∂_j  C(y-x)  A_j^b(x)

+   ∂^+C(x-y)  _j/^+ A_j^b(y) ∂^+C(y-x) _i/^+A_i^b(x)

- 2  ∂_i  C(x-y) A_i^b(y) ∂^+C(y-x) _j/^+A_j^b(x)

- 1/2  ∂_i  C(x-y) A_j^b(y)  ∂_i  C(y-x)  A_j^b(x)

+   1/2 _i^2/^+ C(x-y) A_j^b(y)  ^+ C(y-x) A_j^b(y)  } ,

where we have used the relation $2\,\del^-C(x-y) = \frac{\del_i^2}{\del^+} C(x-y)+ \frac{1}{\del^+} \delta(x-y)$.

§.§ Fermion Determinant

As noted in section 2, the presence of the four fermion interaction term in (<ref>) makes the computation of the fermion determinant slightly involved. But the contribution from such a term to the fermion determinant is trivial at order $g^2$ as shown in Appendix B.

We simplify the quadratic operator in (<ref>) by expanding the covariant derivatives and using the constraint equation (<ref>). We get
Δ={1/2 /∂^+δ^ac- 1/2gf^abc  γ^iγ^j ∂_i/∂^+ (A_j^b -  1/2gf^abc γ^iγ^j A_i^b ∂_j/∂^+   -  gf^abc ∂_i/∂^+A_i^b

-   g^2f^abcf^bde  1/∂^+ 2(A_i^d∂^+A_i^e) -1/2g^2f^adef^bcdγ^iγ^j A_i^e 1/∂^+A_j^b} .
The non-trivial part of the quadratic operator relevant to order $g^2$ is

Δ=(1/2 /∂^+)·{δ^ac+gf^abcγ^iγ^j ∫dy   ^+C(x-y)A_i^b(y) _j/^+

+ gf^abcγ^iγ^j ∫dy  _iC(x-y)  A_j^b(y)  +2 gf^abc   ∫dy ^+ C(x-y)_i/^+ A_i^b(y) } .

We now compute the fermion determinant  pertubatively using the (<ref>). We use the trace relation satisfied by the gamma matrices given in appendix A (<ref>) to obtain

log(1+Y)   =    ng^2 ∫dx  dy   r/4{   2  ∂_i  C(x-y) A_i^b(y)  ∂_j  C(y-x)  A_j^b(x)

+ 2  ∂^+C(x-y)  _j/^+ A_j^b(y) ∂^+C(y-x) _i/^+A_i^b(x)

- 4  ∂_i  C(x-y) A_i^b(y) ∂^+C(y-x) _j/^+A_j^b(x)

-  ∂_i  C(x-y) A_j^b(y)  ∂_i  C(y-x)  A_j^b(x)

+   _i^2/^+ C(x-y) A_j^b(y)  ^+ C(y-x) A_j^b(y)  } ,

where $r={\rm Tr}\,{\bf 1}\;  ({\bf 1}$ is the identity matrix) and it counts the number of off-shell fermionic degrees of freedom.

§.§ Existence of map in critical dimensions

We see now that the Jacobian determinant (<ref>) matches against the fermion determinant (<ref>) if and only if
r/2 = d-2  ,

r/2 = d-2    ,

-r = - 2(d-2)  ,

-r/4 = - d-2/2  ,

+r/4 = d-2/2   , all implying that $r=2(d-2)$ which happens for $d=3,4,6$ and $10$ [6, 7, 9]. We find that the matching of determinants depends on the dimension of our field theory, which imposes a constraint on the allowed values of space-time dimensions. We recover the old result using this approach that the pure supersymmetric Yang-Mills theories can only exist in $d=3,4,6,10$ dimensions. This result was first obtained in [21] using the closure of supersymmetry transformations and requiring the use of specific Fierz identity. Hence supersymmetric Yang-Mills theories can be formulated in the light-cone gauge in all critical dimensions without using anti-commuting variables.

Note that the constraints on the fermion fields are dimension dependent. For example in $d=3,4$, the fermion fields are Majorana spinors, in $d=6$  Weyl spinors, while in $d=10$, they are Majorana-Weyl spinors.

Further we also note that we can recover our map for the physical fields $A_i$ (<ref>) (to order $g$) from the one derived in [9] in general gauges ($n^{\mu}A_{\mu}=0$). Note that in ref. [9], the map is written covariantly. In our case, we have lost manifest covariance due to the elimination of unphysical degrees of freedom. Hence, we can only match the transverse part of the map in [9] by employing the following conditions in equation (4.1) of ref. [9] : $n^{\mu}$ is chosen to be null. In the light cone coordinates, this means setting the components $n^+=n^i=0$ and $n^-=1$. The ghost propagator (in the light-cone gauge) is  $\frac{1}{\del^+}$. Using the above conditions and the constraint equation ((<ref>) from the appendix) without the fermion term, the map in ref.[9] matches with our map (<ref>) to the cubic order.

§ ON THE ISSUE OF UNIQUENESS

The map obtained in the preceding section (<ref>) can be written in $d=4$ in terms of the helicity variables and fields. This is distinct from the map (<ref>) obtained in section 2. One can understand this distinction at the level of Lagrangian. The cubic terms (purely involving the gauge fields) in (<ref>) and (<ref>) are related by partial integrations in $d=4$. However, the maps (<ref>) and (<ref>) to the cubic order cannot be connected by any partial integrations. Structurally the map (<ref>) is of the form $\del\,C\,A\,A$ while the map (<ref>) is of the form $\del\,C\,(\,A\,A\,+\,A\,\bar{A})$ to the cubic order ($\del$ collectively represents the space-time derivatives). Hence, these two maps are distinct.

We can, in principle, write distinct maps by writing the Lagrangian in different ways by performing partial integrations. The non-trivial check required for each map is determinant matching. In our case, we find that there are two maps at order $g^2$ that satisfy all the conditions of the Main theorem (listed in section 2). The determinant matching condition is about the equality of the derivative (Jacobian) of the map with the fermion determinant and not about the map itself, hence the non-uniqueness. Moving to higher orders may fix this uniqueness issue. It is useful to point out that when one works to any finite order in a perturbation theory, one can always find the simplest map relevant to that order, which will simplify the computation of correlation functions (scattering amplitudes).

We find that the maps can be related  at the level of the Jacobian[The two six dimensional maps in [7, 8] can also be related to each other at the level of the Jacobi determinant.  To do this, we simply need to add terms proportional to $(d-6)$ to the Jacobian of the six dimension map (obtained by trial and error) in [8].]. The existence of more than one map at a particular order in the coupling constant is due to the freedom in writing the Lagrangian. In this paper, we found two distinct four dimensional maps (<ref>) and (<ref>).  We can relate the map (<ref>)  that  works in all critical dimensions to the simple four dimensional map (<ref>) at the level of the Jacobi determinant by adding and subtracting a term of the form  $4ng^2 \del \,C(x-y)\,\bar{A}^a\bar{\del}\,C(y-x)\,A^a$ in (<ref>). This implies that when the Jacobian (<ref>) is written in $d=4$ in the helicity basis, some terms get canceled. The Jacobian of the simpler map (<ref>) in $d=4$ only produces the non-trivial terms. Thus, to any given order one can find different maps which satisfy all the conditions of the main theorem.

§.§ Towards $\mathcal{N}=1$ light-cone superspace in $d=4$

In the light-cone gauge, we found that the Lagrangian (<ref>) contains a four fermion interaction term. This might complicate the construction of Nicolai map at higher orders in the coupling. An alternative way out is to study this approach in the language of superspace.  In the following section, we find a superspace map and comment on its properties.

§ THE MAP IN SUPERSPACE

In the light-cone superspace, the Lagrangian (<ref>) takes a very simple form. The main ingredient in superspace is the superfield given by
ϕ(y,θ)=iA(y)+θχ̅(y) .

where $y=(x^+,x^--\frac{i}{\sqrt{2}}\theta\bar{\theta},x,\bar{x})$ and $\theta,\bar{\theta}$ are the anti-commuting variables. We also have the covariant derivatives $\mathrm{d}\,,\bar{\mathrm{d}}$ given by
d=-∂/∂θ̅-i/√(2)θ∂^+ ,  d̅=∂/∂θ+i/√(2)θ̅ ∂^+ ,

which satisfy $\{\mathrm{d},\bar{\mathrm{d}}\,\}=-i\sqrt{2}\partial^+$. The superfield and its conjugate satisfy $\mathrm{d}\phi=0$ and  $\bar{\mathrm{d}}\bar{\phi}=0$. The action is then

𝒮=∫d^4x  d^2θ  {-i/√(2)ϕ̅^a/∂^+ϕ^a +√(2)gf^abc(∂̅/∂^+ϕ^aϕ^bϕ̅^c+∂/∂^+ϕ̅^aϕ̅^bϕ^c)

-g^2f^abcf^ade1/∂^+(ϕ^b  dϕ̅^c)1/∂^+(ϕ̅^d d̅ϕ^e)} .

In the following, we adopt the notation : $\textbf{x}=(y,\theta)$ and $\widetilde{\textbf{x}}=(x,\theta,\bar{\theta)}$. The former will be referred to as `chiral coordinates' while the latter as `super coordinates'. The chiral superfield in the Lagrangian thus depends on the chiral coordinates as $\phi^a(\textbf{x})$. The integration measure is written as $d\widetilde{\textbf{x}}$ . We find a transformation to order $g^2$ such that it maps the full $\mathcal{N}=1$ Lagrangian to a free Lagrangian in superspace.  The Jacobi determinant of the transformation is one and it preserves the chirality of the superfield. In order to write such a transformation, we use
G(x-y)=dd̅/-i√(2)^+δ^(6)(x-y) .
Note that $G(\widetilde{\textbf{x}}-\widetilde{\textbf{y}})$ is chiral w.r.t $\widetilde{\textbf{x}}$ while it is anti-chiral w.r.t. $\widetilde{\textbf{y}}$.

We find the following transformation

ϕ^' a(x)=      ϕ^a(x)+2igf^abc∫dy  ∂^+G(x-y)∂̅/∂^+ϕ^b(y)ϕ^c(y)

+i/√(2) g^2 f^abcf^bde∫dy  ∂^+G(x-y)  d̅ϕ^c(y)1/∂^+ 2[ϕ^d(y)  dϕ̅^̅e̅(y̅)]

+ 2g^2f^abcf^bde∫dy  dz  ∂^+G(x-y)∂/∂^+ϕ̅^̅c̅(y̅) ∂^+G(y-z)∂̅/∂^+ϕ^d(z)ϕ^e(z) ,

where $\textbf{x}=(y,\theta)$, $\textbf{y}=(y^{'},\theta^{'})$, $\textbf{z}=(y^{''},\theta^{''})$ are the chiral coordinates and $\widetilde{\textbf{y}}=(x^{'},\theta^{'},\bar{\theta}^{'})\,,\,\widetilde{\textbf{z}}=(x^{''},\theta^{''},\bar{\theta}^{\,''})$ are the super coordinates. The map for $\bar{\phi}^{\,'\,a}$ is obtained by complex conjugation.

It is easy to check that we can now write the Lagrangian as $-\frac{i}{\sqrt{2}}\,\bar{\phi}^{'\,a}\frac{\Box}{\partial^+}\phi^{'\,a}$. We compute the Jacobian of the transformation using the fact that
δϕ^a(x)/δϕ^b(y)=δ_b^a  δ(x-y) ,  δϕ^a(x)/δϕ̅^b(y̅)=0 .
The delta functions of the chiral coordinates are $5$-dimensional while those of the super coordinates are $6$-dimensional.  Note that the above functional variation is chiral.  We can write the delta function in terms of $\widetilde{\textbf{x}}$ and $\widetilde{\textbf{y}}$ as

δ(x-y)=dδ(x-y) .

The non-trivial contributions to the Jacobian relevant at order $g^2$ are
δϕ^' a(x)/δϕ^ m(w)   =   δ_m^a δ(x-w)

+ 2igf^abc∫dy  ∂^+G(x-y){δ_m^b ∂̅/∂^+(δ(y-w))ϕ^c(y)

+∂̅/∂^+ϕ^b(y) δ_m^c  δ(y-w)}

+ 2g^2f^abcf^bde∫dy  dz  ∂^+G(x-y)∂/∂^+ϕ̅^c(y) ∂^+G(y-z)

×  { δ_m^d∂̅/∂^+ (δ(z-w))ϕ^e(z)+∂̅/∂^+ϕ^d(z) δ_m^e δ(z-w)} .

Using (<ref>) along with (<ref>) and the fact that $\mathrm{d} G(\widetilde{\textbf{x}}-\widetilde{\textbf{y}})=0$ yields
( δΦ_i^' a(x)/δΦ_j^ m(w))=1+ 𝒪(g^3) ,
where $i,j=1,2$ and $\Phi_1\equiv\phi,\;\Phi_2\equiv\bar{\phi}$. Thus we find that by using the chirality constraint in superspace, the trace vanishes till order $g^2$. We expect this to hold at higher orders in the coupling as well. Hence, the superspace map offers an alternative way to compute correlation functions using a free theory in superspace. Finally, the goal would be to find such a map for the case of the maximally supersymmetric Yang-Mills theory and see what it can teach us about the mathematical properties of the theory.

§  DISCUSSION ON NICOLAI MAP AND QUADRATIC FORMS

The light-cone Hamiltonians for  the pure and the maximally supersymmetric theories in $d=4$ can be expressed as quadratic forms [22, 23, 24]. Here, we focus on the quadratic form structure in the pure Yang-Mills theory. The light-cone Hamiltonian for this theory may be written as
ℋ=2∫d^3x   𝒟A̅^a𝒟̅A^a ,

𝒟̅A^a=∂̅A^a-gf^abc1/(A̅^bA^c) ,
and $\mathcal{D}\bar{A}^a$ is obtained by complex conjugation.

We now propose an alternative way to express this Hamiltonian. We introduce new variables $A'^{a}\,,\bar{A}'^{a}$ given by

A'^a(x)=A^a(x)-2gf^abc∫d^2y ∂C^T(x-y)1/[A̅^b(x')A^c(x')] ,
where $x'=(x^+,x^-,y,\bar{y})$ and $2\,\partial\bar{\partial}C^T(x-y)=-\delta^2(x-y)$. In these new variables, the Hamiltonian takes the form
ℋ=-2∫d^3x  A̅'̅^a ∂∂̅ A'^a .

Thus, (<ref>) is a bosonic transformation that maps the Yang-Mills Hamiltonian to a free Hamiltonian. It is therefore of interest to explore whether this transformation (<ref>) has a connection with the Nicolai map (<ref>). We believe that the expression (<ref>) represents a good place to begin an investigation of possible links between the Nicolai map and quadratic form.

§ ACKNOWLEDGMENTS

We thank Sudarshan Ananth for many helpful discussions. SP and NB acknowledge support from the Prime Minister’s Research Fellowship (PMRF) and CSIR-NET fellowship respectively.

§ APPENDIX

0.2 cm

§ NOTATIONS AND CONVENTIONS

We start with a gauge theory involving bosonic ($A_{\mu}$) and fermionic ($\psi$) degrees of freedom in the adjoint representation. We work here in the light-cone gauge and do not explicitly specify the dimension.   The allowed dimensions for the supersymmetric Yang-Mills theories to exist will get fixed using the Nicolai map (particularly the matching of fermion and the Jacobi determinants). These dimensions are referred as the `critical dimensions' [7].

The light cone coordinates are given by
x^±=(x^0 ± x^d-1)/√(2)   .

The transverse coordinates are given by $x_i$ where $i=1,...d-2$. The derivatives with respect to the light-cone coordinates are denoted by $\del_{\pm}$ ($-\del^{\mp}$) while those with respect to the transverse coordinates are denoted by $\del_i$.

Gamma matrices satisfy $\left\{\gamma^{\m}\, ,\,\gamma^{\nu}\right\}=-2\eta^{\mu \nu}$ where $\eta_{\mu\nu}$ is the light-cone metric. The $\gamma^{\pm}$ are defined as
γ^±=1/√(2)(γ^0±γ^d-1) .
They satisfy
γ^±  2=0 , γ^+ ^ †=γ^-  ,  {γ^± ,  γ^i}=0 ,  {γ^+,γ^-}=2 .
The gamma matrices satisfy the following trace identities

Tr ( γ^ γ^ν)   =    - r  η^μν

Tr ( γ^ γ^ν γ^ρ γ^σ)   =   -r(η^μν η^ρσ-η^μρ η^νσ+η^μσ η^ρν)
where $r=2^\frac{[D]}{2} $ and it counts the number of off-shell fermionic degrees of freedom.

We introduce two hermitian projection operators
P_+=1/2 γ^-γ^+  ,      P_-=1/2 γ^+γ^-     which satisfy   P_±^ 2=P_±   ,   P_+P_-=P_-P_+=0.

We start with the action

S= ∫d^d x  (-1/4F_μν^ aF^μν a+ i/2ψ̅^a γ^μ(D_μ ψ)^a ) ,
where $F_{\mu\nu}^{\,a}=\partial_{\mu}A_{\nu}^{\,a}-\partial_{\nu}A_{\mu}^{\,a}+gf^{abc}A_{\mu}^{\,b}A_{\nu}^{\,c}$ , $\bar{\psi}=\psi^{\dagger}\gamma^0$ and $D_\mu=\partial_\m\delta^{ac}+gf^{abc}A_\mu^b$ . The $f^{abc}$ are the structure constants of the gauge group $SU(n)$.

Here, we will not explicitly distinguish between Majorana, Weyl, and Majorana-Weyl spinors to keep notations simple. Note that this is justified because our calculations require only basic Clifford algebra and the trace relation (<ref>).

The equation of motion corresponding to the gauge field is

D_μ F^μν a-i/2gf^abcψ̅^b γ^νψ^c=0 .

We make the gauge choice $A_-^{a}=0$ which renders $\nu=+$ in (<ref>) as a constraint equation and we get

A_+^a=-1/^+(∂_i A^i a)-gf^abc1/^+ 2(A_i^b^+ A^i c)-i/2gf^abc1/^+(ψ̅_+^aγ^+ψ_+^a) ,

where the operation $\frac{1}{\del^+}$ is defined as

1/^+f(x^-)=-∫dy^-   θ(x^- - y^-) f(y^-) ,
where $\theta(x^- - y^-)$ is the step function.

The equation of motion for the fermion field is given by $\gamma^\m D_\m^{ac} \psi^c=0.$  The fermion field $\psi$ can be decomposed into $\psi_{\pm}$ using the projection operators
ψ_±= P_± ψ   ,    ψ̅_±= ψ̅P_∓   and   ψ=ψ_+ +ψ_-   ,     ψ̅ =ψ̅_+ +ψ̅_- .
Acting $P_+$ and $ P_-$ on the equation of motion, we get the following two equations

D_-^ac ψ_-^c=-1/2γ^+γ^i  D_i^ac ψ_+^c        i=1,.....d-2   ,

D_+^ac ψ_+^c=-1/2γ^-γ^i  D_i^ac ψ_-^c       i=1,.....d-2 .
Since  (<ref>) is a constraint so we solve for $\psi_-^a$ and we obtain

ψ_-^a=1/2γ^+γ^i1/∂^+ D_i^ac ψ_+^c .

Expanding the fermion term in terms of $\psi_{\pm}$ (and its conjugate) and substituting the constraint equations (<ref>),(<ref>) in (<ref>), we obtain the Lagrangian in the light-cone gauge

ℒ   =   1/2 A_i^a A_i^a - g f^abc (_i /^+A_i^a  ^+ A_j^b A_j^c +_i A_j^a A_i^b A_j^c )

-g^2 f^abc f^ade (  1/4 A_i^b A_j^c A_i^d A_j^e + 1/21/∂^+(∂^+A_i^b A_i^c)1/∂^+(∂^+A_j^d A_j^e)  )

+ i/2 ψ̅_+^a  γ^+(D_+^ ad-1/2 γ^i D_i^ ac1/∂^+γ^j D_j^ cd)ψ_+^d

-1/8  g^2 f^abc f^ade 1/∂^+ (ψ̅_+^b γ^+ ψ_+^c ) 1/∂^+ (ψ̅_+^d γ^+ ψ_+^e ) .

The last term of (<ref>) is a four fermion interaction term which was absent in the original action. This is a feature which distinguishes the light-cone formulation of the theory from other gauges.

We now restrict to $d=4$ and  introduce the transverse coordinates
x=(x^1+ix^2)/√(2) , x̅=x^* ,
and their derivatives $\bar{\del}\,,\,\del$ respectively. We also introduce the helicity field
A^a=A^a_1+iA^a_2/√(2) ,
and its conjugate $\bar{A}^a$.

The $4 \times 4$ gamma matrices are
0      1

1      0

0      σ^i

-σ^i      0

where $\sigma^i$ are the standard Pauli matrices.

The fermion fields in $d=4$ satisfy the Majorana condition $\psi = C \bar{\psi}^T$ where the charge conjugation matrix $C$ is
iσ^2      0

0      -iσ^2

The fermion field $\psi^a(x)$ takes the form

The Lagrangian (<ref>) can now be simplified and written purely in terms of the physical fields ($A^a,\bar{A}^a,\chi^a,\bar{\chi}^a$) as

ℒ   =   A̅^a A^a - 2g f^abc (∂̅/∂^+A^a∂^+A̅^b A^c+∂/∂^+A̅^a∂^+A^bA̅^c)

-2g^2 f^abc f^ade 1/∂^+(∂^+A^bA̅^c)1/∂^+(∂^+A̅^d A^e)

+ i/√(2)χ̅^a(/^+δ^ac-2gf^abc1/^+(∂A̅^̅b̅+∂̅A^b) +2gf^abcA̅^b∂/^+)χ^c + i√(2) gf^abc χ̅^a∂̅/^+(A^b χ^c)

+ i√(2) g^2 f^abcf^bde χ̅^a1/^+2(A^d^+A̅^e+A̅^d^+A^e)χ^c-i√(2) g^2 f^abdf^becχ̅^a A̅^d1/^+(A^eχ^c)

+g^2f^abcf^ade1/^+(χ̅^bχ^c)1/^+(χ̅^dχ^e) .

§ QUARTIC FERMION TERM

For simplicity we study the $d=4$ case. The analysis however holds true in any $d$. The path integral takes the form
Z=∫DA^a DA̅^a Dχ^a Dχ̅^a   exp[i∫d^4x ℒ_1+B] ,
where $\mathcal{L}_1$ contains all terms in (<ref>) except the four fermion interaction term and $B$ is the four fermionic interaction. We now expand the exponent of $B$ to the linear order (since we are working to order $g^2$ and $B$ is exactly of that order)

Z=∫DA^a DA̅^a Dχ^a Dχ̅^a  (1+i∫d^4x B)  exp[i∫d^4x ℒ_1] ,
Z=Z_0+Z_1 .
The fermion determinant can be evaluated in the term $Z_0$. Let us denote this determinant by $\Delta_F(A,\bar{A},g)$. The form of $Z_0$ is then
Z_0=∫DA^a DA̅^a   Δ_F(A,A̅,g)   exp[i∫d^4x ℒ_YM] ,

where $\mathcal{L}_{YM}$ denotes the Yang-Mills Lagrangian. The term $Z_1$ is the path integral $Z_0$ with the four fermion interaction term as an insertion. The term $Z_1$ may be computed using the standard technique of introducing sources by considering

Z_0[J]   =   ∫DA^a DA̅^a Dχ^a Dχ̅^a

exp[i∫d^4x ℒ_YM+i/√(2)((χ̅^a+J̅^b Q^-1 ba)Q^ac(χ^c+Q^-1 cdJ^d)-J̅^a Q^-1 acJ^c)] .

Here $Q^{ac}$ denotes the quadratic operator in the fermionic part of the Lagrangian while $Q^{-1\,ac}$ denotes the fermion propagator in the presence of gauge field. A change of variables in the path integral from $\chi^a$ to $\chi^{'\,a}=\chi^a+Q^{-1\,ac}J^c$ and similarly for its complex conjugate allows us to integrate the fermion fields and get a factor of $\Delta_F(A,\bar{A},g)$. By differentiating $Z_0[J]$ with respect to the sources and putting them to zero we find the form of $Z_1$ to be

Z_1=∫DA^a DA̅^a  Δ_F(A,A̅,g)G_4(A,A̅,g)  exp[i∫d^4x ℒ_YM] .
Z=∫DA^a DA̅^a  Δ_F(A,A̅,g)[1+G_4(A,A̅,g)]  exp[i∫d^4x ℒ_YM] .

We can define $\Delta'_F(A,\bar{A},g)=\Delta_F(A,\bar{A},g)[1+G_4(A,\bar{A},g)]$ as the effective fermion determinant. Note that $G_4$ is a product of two fermion propagators. The fermion propagator in the presence of a gauge field is an infinite series in the coupling $g$. Since $Z_1$ is itself at order $g^2$, we must only consider the leading term in the series which is nothing but the free fermion propagator given by $\del^+C(x-y)$. But since all the four fermion fields are at the same space-time point, $G_4$ vanishes as $\del^+C(0)=0$.

Thus four fermion term  starts contributing to the fermion determinant from order $g^4$.

[1]
H. Nicolai,
“Supersymmetry and Functional Integration Measures”,
Nucl. Phys. B 176, 419-428 (1980)

[2]
H. Nicolai,
“On a New Characterization of Scalar Supersymmetric Theories”,
Phys. Lett. B 89, 341 (1980)

[3]
H. Nicolai,
“Supersymmetric functional integration measures,”

[4]
O. Lechtenfeld,
“Construction of the Nicolai mapping in supersymmetric field theories,”

[5]
O. Lechtenfeld,
“The Nicolai-map approach to supersymmetry,”
J. Phys. Conf. Ser. 2667, no.1, 012001 (2023)
[arXiv:2309.00481 [hep-th]].

[6]
S. Ananth, H. Nicolai, C. Pandey and S. Pant,
“Supersymmetric Yang–Mills theories:not quite the usual perspective”,
J. Phys. A 53, no.17, 17 (2020)
[arXiv:2001.02768 [hep-th]].

[7]
S. Ananth, O. Lechtenfeld, H. Malcha, H. Nicolai, C. Pandey and S. Pant,
“Perturbative linearization of supersymmetric Yang-Mills theory,”
JHEP 10, 199 (2020)
[arXiv:2005.12324 [hep-th]]

[8]
S. Ananth, H. Malcha, C. Pandey and S. Pant,
“Supersymmetric Yang-Mills theory in $D=6$ without anticommuting variables,”
Phys. Rev. D 103, no.2, 025010 (2021)
[arXiv:2006.02457 [hep-th]].

[9]
H. Malcha and H. Nicolai,
“Perturbative linearization of super-Yang-Mills theories in general gauges,”
JHEP 06, 001 (2021)
[arXiv:2104.06017 [hep-th]].

[10]
V. de Alfaro, S. Fubini and G. Furlan,

Phys. Lett. B 163 (1985), 176-180

[11]
O. Lechtenfeld and M. Rupprecht,
“Construction method for the Nicolai map in supersymmetric Yang–Mills theories,”
Phys. Lett. B 819 (2021), 136413
[arXiv:2104.09654 [hep-th]].

[12]
S. Ananth,
“Spinor helicity structures in higher spin theories,”
JHEP 11, 089 (2012),
[arXiv:1209.4960 [hep-th]].

[13]
O. Lechtenfeld and M. Rupprecht,
“Is the Nicolai map unique?,”
JHEP 09, 139 (2022)
[arXiv:2207.09471 [hep-th]].

[14]
S. Mandelstam,
“Light Cone Superspace and the Ultraviolet Finiteness of the N=4 Model,”
Nucl. Phys. B 213, 149-168 (1983).

[15]
L. Brink, O. Lindgren and B. E. W. Nilsson,
“The Ultraviolet Finiteness of the N=4 Yang-Mills Theory,”
Phys. Lett. B 123, 323-328 (1983).

[16]
O. Lechtenfeld,
“Nicolai maps for super Yang-Mills on the light cone,”
[arXiv:2406.04406 [hep-th]].

[17]
L. Brink, O. Lindgren and B. E. W. Nilsson,
“N=4 Yang-Mills Theory on the Light Cone”,
Nucl. Phys. B 212, 401-412 (1983)

[18]
H. Nicolai and J. Plefka,
“$N=4$ super-Yang-Mills correlators without anticommuting variables,”
Phys. Rev. D 101, no.12, 125013 (2020)
[arXiv:2003.14325 [hep-th]].

[19]
L. Casarin, O. Lechtenfeld and M. Rupprecht,
“Nicolai maps with four-fermion interactions,”
JHEP 12, 132 (2023)
[arXiv:2310.19946 [hep-th]].

[20]
G. Leibbrandt,
“Introduction to Noncovariant Gauges,”
Rev. Mod. Phys. 59, 1067 (1987)

[21]
L. Brink, J. H. Schwarz and J. Scherk,
“Supersymmetric Yang-Mills Theories,”
Nucl. Phys. B 121 (1977), 77-92

[22]
S. Ananth, L. Brink, S. S. Kim and P. Ramond,
“Non-linear realization of PSU(2, 2|4) on the Light-Cone”,
Nucl. Phys. B 722, 166-190 (2005)
[arXiv:hep-th/0505234 [hep-th]]

[23]
S. Ananth, L. Brink and M. Mali,
“Yang-Mills theories and quadratic forms”,
JHEP 08, 153 (2015)
[arXiv:1507.01068 [hep-th]].

[24]
S. Ananth, C. Pandey and S. Pant,

JHEP 07 (2020) no.07, 100
[arXiv:2005.10376 [hep-th]].

§ THE MAP IN SUPERSPACE

A convenient way to do quantum calculations in a supersymmetric theory is using the superspace formalism. In the light-cone superspace, the Lagrangian (<ref>) takes a very simple form. The main ingredient in superspace is the superfield given by

and its conjugate where $y=(x^+,x^--\frac{i}{\sqrt{2}}\theta\bar{\theta},x,\bar{x})$. We also have the covariant derivatives $\mathrm{d}\,,\bar{\mathrm{d}}$ given by

d=-∂/∂θ̅-i/√(2)θ∂^+  d̅=∂/∂θ+i/√(2)θ̅ ∂^+

which satisfy $\{\mathrm{d},\bar{\mathrm{d}}\,\}=-i\sqrt{2}\partial^+$. The superfields satisfy $\mathrm{d}\phi=0$ and $\bar{\mathrm{d}}\bar{\phi}=0$. The action is then

𝒮=∫d^4x  d^2θ  {-i/√(2)ϕ̅^a/∂^+ϕ^a +√(2)gf^abc(∂̅/∂^+ϕ^aϕ^bϕ̅^c+∂/∂^+ϕ̅^aϕ̅^bϕ^c)

-g^2f^abcf^ade1/∂^+(ϕ^b  dϕ̅^c)1/∂^+(ϕ̅^d d̅ϕ^e)}

where $d^2\theta=d\theta\,d\bar{\theta}$. In the following, we adopt the notation : $\textbf{x}=(y,\theta)$ and $\widetilde{\textbf{x}}=(x^{\mu},\theta,\bar{\theta)}$ where $x^{\mu}=(x^+,x^-,x,\bar{x})$. Thus the chiral fields in the Lagrangian can be written as $\phi^a(\textbf{x})$ while the anti-chiral fields as $\bar{\phi}^a(\bar{\textbf{x}})$. The integration measure may be written as $d\widetilde{\textbf{x}}$ . We now find a Jacobian one transformation which preserves the chirality such that it maps the full $\mathcal{N}=1$ Lagrangian to a free Lagrangian in superspace. We guess the following transformation

ϕ^' a(x)=      ϕ^a(x)+2igf^abc∫dy  ∂^+G(x-y)∂̅/∂^+ϕ^b(y)ϕ^c(y)

+i/√(2) g^2 f^abcf^bde∫dy  ∂^+G(x-y)  d̅ϕ^c(y)1/∂^+ 2[ϕ^d(y)  dϕ̅^̅e̅(y̅)]

+2g^2f^abcf^bde∫dy  dz  ∂^+G(x-y)∂/∂^+ϕ̅^̅c̅(y̅) ∂^+G(y-z)∂̅/∂^+ϕ^d(z)ϕ^e(z)

where $\textbf{x}=(y,\theta)$, $\textbf{y}=(y^{'},\theta^{'})$, $\textbf{z}=(y^{''},\theta^{''})$ and $y,y^{'},y^{''}$ are the shifted space-time coordinates. Also $\widetilde{\textbf{y}}=(x^{'},\theta^{'},\bar{\theta}^{'}),\widetilde{\textbf{z}}=(x^{''},\theta^{''},\bar{\theta}^{\,''})$ represent the superspace-coordinates integrated over in the above map. We also have
Note that $G(\widetilde{\textbf{x}}-\widetilde{\textbf{y}})$ is chiral w.r.t $\widetilde{\textbf{x}}$ while it is anti-chiral w.r.t. $\widetilde{\textbf{y}}$. Thus $\del^+G(\widetilde{\textbf{x}}-\widetilde{\textbf{y}})$ is the propagator given by $<\phi(\textbf{x})\,\,\bar{\phi}(\bar{\textbf{y}})>$. The transformation for $\bar{\phi}^{'\,a}(x)$ is given by the complex conjugate of (<ref>).

It is easy to check that we can now write the Lagrangian as $\bar{\phi}^{'\,a}\frac{\Box}{\partial^+}\phi^{'\,a}$. Now we compute the Jacobian of the transformation using the fact that
δϕ^a(x)/δϕ^b(y)=δ_b^a  δ^(5)(x-y)
Note that this variation is chiral since the delta function which appears on the RHS depends on the shifted coordinates. We can write this delta function in terms of $\widetilde{\textbf{x}}$ and $\widetilde{\textbf{y}}$ as

We supress the dimension of the delta function with the convention that the delta functions of the bold variables are $5$-dimensional while those of the bold tilde variables are $6$-dimensional. We now write the non-trivial contributions to the Jacobian relevant at order $g^2$

δϕ^' a(x)/δϕ^ m(w)   =   δ_m^a δ(x-w)

+ 2igf^abc∫dy  ∂^+G(x-y){δ_m^b ∂̅/∂^+(δ(y-w))ϕ^c(y)

+∂̅/∂^+ϕ^b(y) δ_m^c  δ(y-w)}

+ 2g^2f^abcf^bde∫dy  dz  ∂^+G(x-y)∂/∂^+ϕ̅^c(y) ∂^+G(y-z)

×  { δ_m^d∂̅/∂^+ (δ(z-w))ϕ^e(z)+∂̅/∂^+ϕ^d(z) δ_m^e δ(z-w)}

Using (<ref>) along with (<ref>) and the fact that $\mathrm{d} G(\widetilde{\textbf{x}}-\widetilde{\textbf{y}})=\bar{\mathrm{d}}\overline{G}(\widetilde{\textbf{x}}-\widetilde{\textbf{y}})=0$ yields
log[( δϕ^' a(x)/δϕ^ m(w))]=𝒪(g^3)
Thus we find that by using the chirality constraints in superspace, the trace vanishes till order $g^2$. We expect this to hold at higher orders in the coupling as well. In this way we can map the action (<ref>) to a free action by a Jacobian one transformation.

§.§ A note on quadratic form structure in $\mathcal{N}=1$ theory

An interesting feature of the light-cone Hamiltonians for the pure and the maximally supersymmetric theories (in superspace) is that they may be written as quadratic forms[22, 23]. We now show that this feature extends to the $\mathcal{N}=1$ Hamiltonian in light-cone superspace. The Hamiltonian may be written as
ℋ=∫d^3x d^2θ  𝒲^ a𝒲^ a
𝒲^ a=∂/∂^+dϕ̅^a-igf^abc1/∂^+(ϕ^b dϕ̅^c)
and $\overline{\mathcal{W}}^{\,a}$ is obtained by complex conjugation. Note that $\mathcal{W}^{\,a}$ is chiral fermionic while $\overline{\mathcal{W}}^{\,a}$ is anti-chiral fermionic superfield. Thus this quadratic form structure suggests a change of variables from $\phi^a$ to $\mathcal{W}^a$ hinting at some possible connection with the Nicolai map. In the last section we speculate over possible link between the two.


# Longitudinal boost-invariance of charge balance function in hadron-hadron
and nucleus-nucleus collisions

Li Na, Li Zhiming, and Wu Yuanfang  Key Laboratory of Quark and Lepton Physics
(Huazhong Normal University),Ministry of Education Institute of Particle
Physics, Huazhong Normal University, Wuhan 430079, China

###### Abstract

Using Monte Carlo generators of the PYTHIA model for hadron-hadron collisions
and a multi-phase transport (AMPT) model for nucleus-nucleus collisions, the
longitudinal boost-invariance of charge balance function and its transverse
momentum dependence are carefully studied. It shows that the charge balance
function is boost-invariant in both p+p and Au+Au collisions in these two
models, consistent with experimental data. The balance function properly
scaled by the width of the pseudorapidity window is independent of the
position or the size of the window and is corresponding to the balance
function of the whole pseudorapidity range. This longitudinal property of
balance function also holds for particles in small transverse momentum ranges
in the PYTHIA and the AMPT default models, but is violated in the AMPT with
string melting. The physical origin of the results are discussed.

###### pacs:

25.75.Gz,25.75.Ld

##  I Introduction

Charge balance function (BF) has been widely used as an effective exploring
for the hadronization scheme in hadron-hadron collisions at the ISR energies
oldbf1  and $e^{+}+e^{-}$ annihilations at PETRA energies petra . Recently,
the charge BF gains special attentions in clocking hadronization at
relativistic heavy-ion collisions. A narrowing of the BF is suggested as a
signature for delayed hadronization bf1 ; bf2 .

The dependence of the BF on centrality and system size has been reported by
several relativistic heavy-ion experiments star130 ; na49 . However, most of
the current heavy-ion experiments are limited by the pseudorapidity range
star130 ; na49 ; phenix , it is impossible to quantitatively compare the
results from the experiments with the coverage at different pseudorapidity
ranges. The dependence of the BF on the pseudorapidity window is essential for
understanding the physics of the BF star130 ; na49 ; na22bf ; tom1 , and has
been carefully studied by NA22 na22bf  and STAR experiments for hadron-hadron
and relativistic heavy ion collisions, respectively.

The NA22 experiment has full 4$\pi$ acceptance and excellent momentum
resolution na22bf . It is found in the experiment that the BF in
$\pi^{+}{\mathrm{p}}$ and ${\mathrm{K}}^{+}{\mathrm{p}}$ Collisions at 22 GeV
is invariant under longitudinal boost over the whole rapidity range of
produced particles, in spite of the non-boost-invariance of the single-
particle density. Moreover, the BF of the whole rapidity range can be deduced
from the BF properly scaled by the width of rapidity windows na22bf .

The STAR experiment covers a finite but relative wide pseudorapidity range.
The scaling property of the BF in Au +Au collisions at 200 GeV is further
observed in the experiment star200 . This scaling property of the balance
function is also found in different $p_{\rm T}$ ranges of final state
particles.

These results from both hadron-hadron and nuclear collisions indicate that
charge balance of produced particles in strong interactions is boost-
invariance in longitudinal phase-space, in contrary with the single particle
density.

Therefore, it is interesting to see if those properties are taken into account
in the models which are successfully described hadron-hadron and nuclear
collisions, and how they associate with the mechanisms of particle production
in the models.

##  II Charge balance function and implement models

Charge balance function measures how the conserved electric charge compensate
in the phase space, i.e., how the surrounding net charge are rearranged if the
charge of a selected point changesoldbf1 . In high energy collisions, the
production of charged particles are constrained by charge balance in the phase
space. The BF therefore provides a direct access to collision dynamics.

The BF has been originally defined in terms of a combination of four kinds of
charge-related conditional densities in pseudorapidity oldbf1

$\displaystyle B(\eta_{1}|\eta_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\rho(+,\eta_{1}|-,\eta_{2})-\rho(+,\eta_{1}|+,\eta_{2})\right.$
(1)
$\displaystyle\quad\left.+\rho(-,\eta_{1}|+,\eta_{2})-\rho(-,\eta_{1}|-,\eta_{2})\right],$

where the notation $\rho(a,\eta_{a}|b,\eta_{b})$ represents the ratio
$\rho_{ab}(\eta_{a},\eta_{b})/\rho_{b}(\eta_{b})=\langle
n_{ab}(\eta_{a},\eta_{b})\rangle/\langle n_{b}(\eta_{b})\rangle$ with $a,b$
standing for $+$ or $-$ charged particles. Projecting to pseudorapidity
difference $\delta\eta=\eta_{1}-\eta_{2}$ in an pseudorapidity window
$\eta_{w}$, it becomes bf1 ; star130

$\displaystyle B(\delta\eta|\eta_{\rm w})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{\langle n_{+-}(\delta\eta)\rangle-\langle
n_{++}(\delta\eta)\rangle}{\langle n_{+}\rangle}\right.$ (2)
$\displaystyle\quad\left.+\frac{\langle n_{-+}(\delta\eta)\rangle-\langle
n_{--}(\delta\eta)\rangle}{\langle n_{-}\rangle}\right]$

where $n_{ab}(\delta\eta)$ is the total number of pairs of opposite charged
particles with pseudorapidity difference $\delta\eta$ in the pseudorapidity
window $\eta_{\rm w}$. $n_{+}$ and $n_{-}$ are the number of positively and
negatively charged particles in the window $\eta_{\rm w}$, respectively.
$\langle\cdots\rangle$ is the average over the whole event sample.

From the findings of the BF at NA22 na22bf  and STAR experiments star200 , the
BF is boost-invariant in the whole rapidity range in hadron-hadron collisions
and may be in nuclear collisions as well. In the case, the properly scaled BF
is corresponding to the BF of the whole pseudorapidity range and is deduced by

$\displaystyle B_{s}(\delta\eta)=\frac{B(\delta\eta|\eta_{\rm
w})}{1-\frac{\delta\eta}{|\eta_{\rm w}|}}$ (3)

where $|\eta_{\rm w}|$ is the width of pseudorapidity window.

The PYTHIA 5.720 pythia  is well set up for p+p collisions. It is a standard
Monte Carlo generator with string fragmentation as hadronization scheme. Two
versions of a multi-phase transport (AMPT) model ampt  are used to study Au+Au
collisions. One is the AMPT default and the other one is the AMPT with string
melting. In both versions, the initial conditions are obtained from the HIJING
model, and then the scattering among partons are given by ZPC. In the AMPT
default model, the partons recombined with their parent strings when they stop
interacting, and the resulting strings are converted to hadrons using the Lund
string fragmentation model, while in the AMPT model with string melting, quark
coalescence is used in combining partons into hadrons. The dynamics of the
hadronic matter is described by ART model.

It is commonly believed that in relativistic heavy ion collisions, the charge
ordering during the string fragmentation in elementary collisions is no longer
valid, and it should be replaced by the quark-coalescence mechanism in
hadronization recombination . So it is interesting to see whether the boost-
invariance of the BF is sensitive to the mechanisms of hadronization.

In this paper, we firstly study the boost-invariance of the BF for p+p
collisions at $\sqrt{s}=22$ GeV and $\sqrt{s}=200$ GeV using the PYTHIA, and
for Au+Au collisions at $\sqrt{s}=200$ GeV using two versions of the AMPT. The
transverse momentum dependence of longitudinal scaling property of the BF is
then examined in the models. The obtained results are compared with
corresponding experimental data and discussed.

##  III Boost-invariance and longitudinal scaling of the BF

Figure 1:  Upper panel: the $B(\delta\eta|\eta_{\rm w})$ in four
pseudorapidity windows with equal size $|\eta_{\rm w}|=3$ at the different
positions for p+p collisions at (a) $\sqrt{s}=22$ GeV and (b) $\sqrt{s}=200$
GeV by PYTHIA model. Lower panel: the scaled balance function,
$B_{s}(\delta\eta)$, deduced from the directly measured BF at six different
sizes and positions of pseudorapidity windows for p+p collisions at (c)
$\sqrt{s}=22$ GeV and (d) $\sqrt{s}=200$ GeV by PYTHIA model. The solid down
triangle is the BF of the whole $\eta$ range.

In order to demonstrate directly whether the BF is invariant under a
longitudinal Lorentz transformation over the whole rapidity in hadron-hadron
collisions, we choose four equal size ($|\eta_{\rm w}|=3$) pseudorapidity
windows locating at different positions ($-3,0$), ($-2,1$), ($-1,2$) and
($0,3$). The results for p+p collisions at $\sqrt{s}=22$ GeV and
$\sqrt{s}=200$ GeV are shown in Fig. 1(a) and (b) respectively. The statistic
errors are smaller than the markers. It is clear that the BF measured in four
windows are approximately identical to each other at two incident energies.
This indicates that the charge compensation is essentially the same in any
longitudinally-Lorentz-transformed frame for p+p collisions in the PYTHIA
model, consistent with the data from NA22 experiment. These results show that
the string fragmentation mechanism implemented in PYTHIA well describes the
production mechanisms of charge particles and their charge balance in
longitudinal phase space.

Fig. 1(c) and (d) are the scaled balance function $B_{s}(\delta\eta)$ at two
incident energies. They are deduced from directly measured
$B(\delta\eta|\eta_{\rm w})$ at six different pseudorapidity windows,
($-0.8,0.8$) (open circles), ($1,3$) (open triangles), ($-3,1$) (open
squares), ($-2.4,2.4$) (open diamonds), ($0,3$) (open crosses), and ($-2,-1$)
(open stars). From the figures we can see that all the $B_{s}(\delta\eta)$
deduced from different windows are coincide with each other within errors, as
expected from boost-invariance of the BF bf2 . The solid down triangles in the
same figures are the BF of the whole pseudorapidity range,
$B(\delta\eta|\eta_{\infty})$. It is close to the scaled balance function
$B_{s}(\delta\eta)$. These results indicate that the scaled BF is in fact
corresponding to the BF of the whole pseudorapidity range
$B(\delta\eta|\infty)$ bf2 .

Figure 2:  Upper panel: the $B(\delta\eta|\eta_{\rm w})$ in five
pseudorapidity windows with equal size $|\eta_{\rm w}|=2$ at the different
positions for Au+Au collisions at $\sqrt{s}=200$ GeV by (a) the AMPT default
and (b) the AMPT with string melting. Lower panel: the scaled balance
function, $B_{s}(\delta\eta)$, deduced from the directly measured BF at
various pseudorapidity windows with different sizes and positions for Au+Au
collisions at $\sqrt{s}=200$ GeV by (c) the AMPT default and (d) the AMPT with
sting melting.

It is then interesting to see whether the boost-invariance of the BF is held
in nucleus-nucleus collisions. STAR experiment only observe the boost-
invariance of BF in cental pseudorapidity range $-1<\eta<1$ star200 , where
the single particle distribution is almost flat, or boost-invariance. Now in
model investigation, we can carefully examine the property in the whole
pseudorapidity range.

The upper panel of Fig. 2 is the BF in five pseudorapidity windows with equal
size $\eta_{\rm w}=2$ at different positions ($-3,-1$), ($-2,0$), ($-1,1$),
($0,2$) and ($1,3$). Where the Fig. 2(a) and (b) are the results from the AMPT
default (v1.11) and the AMPT with string melting (v2.11), respectively. Both
figures show that the BF is boost-invariance in pseudorapidity range (-3, 3)
in two versions of the AMPT.

The lower panel of Fig. 2 is the scaled balance functions, which are obtained
from directly measured BF at six different windows as indicated at legend of
the figure, where the solid down triangles are the BF in pseudorapidity range
(-4, 4). It shows that the scaled BF does not depend on the size and position
of the windows, and corresponds to the BF of the whole pseudorapidity in two
versions of the AMPT, consistent with the results of p+p collisions in the
PYTHIA model.

##  IV The transverse-momentum dependence of the boost-invariance of the BF

The longitudinal property of boost invariance of BF comes from the special
longitudinal interaction of charged particles under the constraint of global
electric charge balance. Global electric charge conservation not only applies
to all final-state charged particles, but also constrains particles which are
produced at the same proper time of evolution. It is argued that the
transverse-momentum of final-state particles may be roughly used as a scale of
the proper time of their production in the expansion of nuclear collisions
rudy ; bmuller ; hama1 ; ptscale . Examining the $p_{\rm T}$ dependence of
longitudinal property of the BF will provide direct access on whether
particles in specified $p_{\rm T}$ range are consistent to be produced
simultaneously with well balanced electric charge.

So we turn to check whether the boost-invariant of BF holds for particles in
different $p_{\rm T}$ ranges. Fig. 3 shows the BF for p+p collisions at
$\sqrt{s}=22$ GeV and $\sqrt{s}=200$ GeV from PYTHIA in three transverse
momentum bins ($0<p_{\rm T}<0.2$), ($0.2<p_{\rm T}<0.4$), and ($p_{\rm
T}>0.2$) GeV/$c$, respectively. These $p_{\rm T}$ bins are selected to make
the multiplicity in each bin comparable. The result shows that the points at a
given $\delta\eta$ in a restricted $p_{\rm T}$ interval are approximately
coincide with each other, i.e., the boost-invariance of the BF hold in small
$p_{\rm T}$ ranges. It indicates that particles produced at different $p_{\rm
T}$ ranges are also boost-invariant for hadron-hadron collisions in the PYTHIA
model.

Figure 3: For each of three $p_{\rm T}$ ranges, the $B(\delta\eta|\eta_{\rm
w})$ in four pseudorapidity windows with equal size $|\eta_{\rm w}|=3$ at the
different positions for p+p collisions at $\sqrt{s}=22$ GeV and $\sqrt{s}=200$
GeV in upper and lower panels, respectively. Figure 4:  For each of four
$p_{\rm T}$ ranges, the $B(\delta\eta|\eta_{\rm w})$ in five pseudorapidity
windows with equal size $|\eta_{\rm w}|=2$ at the different positions for
Au+Au collisions at $\sqrt{s}=200$ GeV from the AMPT default (in upper panel)
and the AMPT with string melting (in lower panel).

The same study for Au+Au 200 GeV collisions from the two versions of the AMPT
are presented in the upper and lower panels of Fig. 4, respectively. Where
four $p_{\rm T}$ bins are, ($0.15,0.4$), ($0.4,0.7$), ($0.7,1$) and ($1,2$)
GeV/$c$. From the upper panel of the figure, we can see that the BF of
different pseudorapidity windows in each $p_{\rm T}$ bin are close to each
other, in consistent with the data from STAR experiment star200 . However, in
the AMPT with string melting, as shown in the lower panel of the figure, where
the BF of different pseudorapidity windows are not as close to each other as
those in the upper panel.

This is because in the AMPT with string melting, each parton in the evolution
of nuclear collision has its own freeze-out time, which last a very long
period after the interaction of two nucleus liu-yu . The particles in the same
transverse-momentum range are not freezed-out simultaneously with well
balanced charge, and therefore the longitudinal boost-invariance of the BF in
small $p_{\rm T}$ ranges is violated. In the AMPT default, the partons
recombined with their parent strings immediately after they stop interacting,
and converted to hadrons. So the charge balance of the produced particles in
the same $p_{\rm T}$ ranges is preserved and boost-invariance of the BF keeps.

##  V Summary

In the paper, we systematically study the longitudinal boost-invariance of
charge balance function and its $p_{\rm T}$ dependence for p+p and Au+Au
collisions using PYTHIA the AMPT models. It shows that charge balance function
is boost-invariance in both hadron-hadron and nuclear interactions, in
contrary to the single particle density. As expected, this boost-invariance of
the BF make the BF properly scaled by window size is independent of window and
corresponds to the BF of the whole (pseudo)rapidity range. Therefore, the BF
is a good measure free from the restriction of finite longitudinal acceptance.

It is further show that the boost invariance of the BF in specified $p_{\rm
T}$ range is valid in PYTHIA for hadron-hadron collisions and the AMPT default
for Au+Au collisions. While the AMPT with string melting fails to reproduce
this property due to the different schemes at hadronization. So the $p_{\rm
T}$ dependence of the longitudinal property of the BF may be served as a
sensitive probe for charge balance in hadronization mechanism.

##  VI Acknowledgments

We thank Prof. Liu Lianshou and Dr. Yu Meiling for valuable discussions and
remarks. This work is supported in part by the NSFC of China with project No.
10835005 and No. 10647124, and the MOE of China with project No. IRT0624 and
No. B08033.

## References

  * (1) D. Drijard et al., Nucl. Phys. B 155 269 (1979); B 166, 233 (1980); I. V. Ajinenko et al., ibid. C 43, 37 (1989).
  * (2) R. Brandelik et al., Phys. Lett. B 100, 357 (1981); M. Althoff et al., Z. Phys. C 17, 5 (1983); H. Aihara et al., Phys. Rev. Lett. 53, 2199 (1984); 57, 3140 (1986) ; P. D. Acton et al., Phys. Lett. B 305, 415 (1993).
  * (3) S. A. Bass, P. Danielewicz and S. Pratt, Phys. Rev. Lett. 85, 2689 (2000).
  * (4) S. Jeon and S. Pratt, Phys. Rev. C 65, 044902 (2002).
  * (5) J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 90, 172301 (2003); Gary. D. Westfall (for STAR Collaboration), J. Phys. G. 30, S345 (2004).
  * (6) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 71, 034903 (2005); C. Alt et al. (NA49 Collaboration), arXiv: 0705.1122.
  * (7) K. Adcox et al. (PHENIX Collaboration), Phys. Rev. Lett. 89, 082301 (2002).
  * (8) T. A. Trainor, hep-ph/0301122.
  * (9) M. R. Atayan et al. (NA22 Collaboration), Phys. Lett. B 637, 39 (2006).
  * (10) Li Zhiming, Li Na, Liu Lianshou and Wu Yuanfang, Int. J Mod. Phys. E 16, 3347 (2007).
  * (11) T. Sjöstrand, Comp. Phys. Commun. 82, 74 (1994).
  * (12) Z. W. Lin, C. M. Ko, B. A. Li, B. Zhang and S. Pal, Phys. Rev. C 72, 064901 (2005).
  * (13) A. Bialas, Phys. Lett. B 579, 31 (2004); R. C. Hwa and C. B. Yang, Phys. Rev. C 70, 024904 (2004).
  * (14) R. C. Hwa and Y. Wu, Phys. Rev. C 60, 054904 (1999).
  * (15) M. Asakawa, S. A. Bass, B. Müller and C. Nonaka, Phys. Rev. Lett. 101, 122302 (2008).
  * (16) F. Grassi, Y. Hama, and T. Kodama, Phys. Lett. B 355, 9 (1995); Y. M. Sinyukov, S. V. Akkelin, and Y. Hama, Phys. Rev. Lett. 89, 052301 (2002);
  * (17) M. Asakawa, S. A. Bass, B. Müller and C. Nonaka, Phys. Rev. Lett. 101, 122302 (2008).
  * (18) Yu Meiling, Du Jiaxin, and Liu Lianshou, Phys. Rev. C (Rapid communication) 74, 044906 (2006).