††institutetext: The Mathematical Institute, University of Oxford, Woodstock
Road, Oxford OX2 6GG, United Kingdom

# Yang-Mills form factors on self-dual backgrounds

Giuseppe Bogna  Lionel Mason111On leave at the Institut des Haut Études
Scientifique, 35 Route de Chartres, 91440 Bures-Sur-Yvette, and the
Laboratoire de Physiques, ENS, Rue L’Homnond, Paris 75005, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The construction of perturbative quantities on non-linear backgrounds leads to
the possibility of incorporating strong field effects in perturbation theory.
We continue a programme to construct QFT observables on self-dual backgrounds.
The approach works with asymptotic data for fields defined at null infinity
$\mathscr{I}$, extending earlier work on Yang-Mills amplitudes on self-dual
backgrounds to form factors and incorporating supersymmetry. Since our
analysis is based on reconstruction from data at null infinity, it naturally
ties into work on celestial and twisted holography. We study form factors both
in pure Yang-Mills and their supersymmetric counterparts in $\mathcal{N}=4$
SYM, giving a full treatment of $\mathcal{N}=4$ super-Yang-Mills at null
infinity and their self-dual nonlinear backgrounds. We obtain tree-level MHV
form factors around these backgrounds using new formulae for lifting operators
to twistor space leading to simple dressings of the corresponding form factors
around the vacuum. We give brief indications on how to go beyond the MHV
sector by introducing dressed versions of the MHV diagram propagator. We
discuss generating functionals of the MHV all plus 1-loop amplitude in this
context together with its various dual conformal representations.

##  1 Introduction

Integrability is a powerful tool for the study of non-linear problems, and,
although it doesn’t apply directly to generic gauge and gravity theories, in
four dimensions such theories possess self-dual sectors that are integrable
Penrose:1976js ; Ward:1977ta ; Mason:1991rf . There is by now a long tradition
of exploiting the integrability of the self-dual sector to provide non-
perturbative results such as the construction of instantons and monopoles
Ward:1990vs . These structures are also intimately related to the rich
structures discovered in scattering amplitudes, from the famous Parke-Taylor
formula for the tree-level MHV scattering of gluons Parke:1986gb ; Nair:1988bq
, to the more general constructions of Witten:2003nn ; Berkovits:2004hg ;
Roiban:2004yf ; Cachazo:2004kj ; Britto:2005fq ; Brandhuber:2008pf ;
Bern:2010ue ; ArkaniHamed:2010kv ; Cachazo:2013hca ; Cachazo:2014xea ;
Geyer:2014fka , see Dixon:1996wi ; Bern:2007dw ; Elvang:2013cua ;
Travaglini:2022uwo ; Geyer:2022cey  for reviews. The role of integrability was
made explicit in studies of scattering amplitudes using twistor actions
defined on twistor space that allow the direct exploitation of the
integrability of the self-dual sectors of Yang-Mills and gravity theories.
Moreover, the non-self-dual theory can be formulated as a perturbation around
the self-dual sector Chalmers:1996rq ; Mason:2005zm ; Boels:2006ir ;
Boels:2007qn ; Mason:2010yk ; Adamo:2011cb  on twistor space, so many
interesting results for the full theory can be readily obtained via
perturbation theory; see the reviews Adamo:2011pv ; Adamo:2013cra  for further
details. From a different perspective, the existence of an integrable self-
dual sector was at the heart of recent, exciting developments in flat-space
holography, most notably celestial Strominger:2017zoo  and twisted
Costello:2022wso  holography. These sectors host chiral symmetry algebras –
whose existence is underpinned by infinitely many soft symmetries
Adamo:2019ipt ; Strominger:2021lvk ; Ball:2021tmb ; Strominger:2021mtt  – that
can be used to significantly constrain celestial correlators. The non-local
nature of twistor constructions means that they can be naturally formulated at
null infinity Newman:1976gc ; Hansen:1978jz ; Sparling:1990 ; Newman:1978ze ,
so these features become apparent if one adopts a twistorial description of
flat holography; for example, twistor methods give a nice way to understand
the symmetries of self-dual gravity Adamo:2021lrv  and can be used to derive
the gluon celestial OPE at all orders Adamo:2022wjo .

These approaches have by now been extended to obtain formulae for form
factors. Form factors are expectation values of local composite operators
between the vacuum and an $n$-particle on-shell state and therefore represent
intermediate observables between on-shell amplitudes and off-shell
correlators. They have important physical applications as well, for example,
arising as scattering amplitudes after additional fields have been integrated
out in effective field theories and, since form factors are only partially
off-shell, many amplitudes techniques have been extended to the construction
of these observables. This includes the MHV formalism, recursion relations,
and methods inspired by twistor theory in both $\mathcal{N}=4$ super-Yang-
Mills and pure Yang-Mills, both at tree and at loop level, see
Brandhuber:2010ad ; Brandhuber:2011tv ; Brandhuber:2012vm ; Penante:2014sza ;
Brandhuber:2016xue ; He:2016jdg ; Brandhuber:2017bkg ; Brandhuber:2018kqb ;
Brandhuber:2018xzk ; Bianchi:2018peu ; Bianchi:2018rrj ;
AccettulliHuber:2019abj . The twistor action approach has been pursued also
for form factors leading to many further results Koster:2016ebi ;
Koster:2016loo ; Koster:2016fna ; Chicherin:2016soh ; Chicherin:2016ybl ;
Chicherin:2016fac ; Chicherin:2016fbj ; Chicherin:2014uca , building on work
on correlation functions in twistor space Adamo:2011dq ; Adamo:2011pv ;
Eden:2017fow . Form factors have also played an interesting role in celestial
and twisted holography where they were recently used to resolve singularities
in the celestial amplitudes Casali:2022fro , to deform the soft symmetry
algebras Melton:2022fsf , as well as to compute both amplitudes and form
factors in terms of correlators of a 2d chiral algebra reminiscent of the
celestial soft algebra Costello:2022wso .

The above works concern observables around a flat background. In this paper,
we address the question of extending the computations of form factors around
non-trivial backgrounds. Although work already exists in this direction in the
case of scattering amplitudes Furry:1951zz ; DeWitt:1967ub ; tHooft:1975uxh ;
Abbott:1981ke , these face difficulties as soon as symmetries are lost and
explicit formulae for background-coupled fields are no longer easily
available; standard techniques and constraints based on momentum space and
hence the rationality of tree diagrams, such as BCFW recursion and unitarity,
cannot be applied. However, progress can be made on simple backgrounds: it is
for example possible to extract much information about scattering around plane
wave backgrounds Adamo:2017nia ; Adamo:2017sze ; Adamo:2018mpq ; Adamo:2020qru
; Adamo:2021hno . In this work, we work around _self-dual_ and _radiative_
backgrounds, continuing the programme established in Adamo:2020syc ;
Adamo:2020yzi . In these works, all-multiplicity expressions for tree-level
MHV gluon scattering amplitude around such backgrounds were obtained for the
first time, together with conjectural formulae for the NkMHV amplitudes.
Analogous gravitational formulae were found in Adamo:2022mev .

The class of backgrounds we consider are _radiative_ in the sense of being
determined by their data at $\mathscr{I}$, but they are otherwise generic.
They could be taken to be a sum of plane waves at infinity, to make contact
with higher-point formulae, but there is no particular reason to do so and one
can choose data for more general backgrounds, such as instantons. We use the
complete integrability of the self-dual sector to construct background-coupled
fields from their asymptotic data at null infinity and study their
interactions. Without local symmetries of the background, there will now be no
straightforward local definition of plane waves in the interior, but fields
can be taken to be plane waves at infinity; they can be characterized by the
same data at null infinity that leads to momentum eigenstates on the trivial
background. Complete integrability implies that these solutions do not
themselves scatter as they pass through the background field, so their values
from past null infinity $\mathscr{I}^{-}$ can be identified with their values
in the future at $\mathscr{I}^{+}$ with no ambiguity, thus preserving crossing
symmetry. Moreover, the backgrounds themselves are by assumption determined by
their data at null infinity, so the whole setting is intrinsically holographic
and fits well into the celestial holography programme.

As we explain below, the argument in Adamo:2020yzi  that leads to the MHV
amplitude is closely related to the construction of an MHV form factor: the
generating functional for the MHV amplitude can be viewed as the $q\to 0$
limit of the generating functional for the form factor of the operator
$\,\mathrm{tr}\,B^{2}$, where $B_{\alpha\beta}$ is the anti-self-dual
component of the field strength in the chiral Yang-Mills formulation of
Chalmers and Siegel Chalmers:1996rq ; Chalmers:1997sg  and $q$ is the momentum
associated to the local operator $\,\mathrm{tr}\,B^{2}$ in the expectation
value. Similar reasoning was also the basis of the twistor action approach of
Boels:2007qn  to the construction of the MHV formalism of Cachazo:2004kj , as
well as recent works in celestial holography on a trivial background
Costello:2022wso ; Costello:2022upu ; Bu:2022dis , but see also
Costello:2022jpg  for an intriguing example on Burns space. Even away from
$q=0$, the tree-level, colour-ordered MHV form factor for
$\,\mathrm{tr}\,B^{2}$ around a Cartan-valued self-dual radiative background
is extremely simple

$\mathscr{F}_{\,\mathrm{tr}\,B^{2}}(1^{+},\ldots,i^{-},\ldots,j^{-},\ldots
n^{+};q)=\frac{\langle ij\rangle^{4}}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,.$ (1)

Here $Q=k_{1}+\ldots+k_{n}$ is the sum of the gluon momenta measured at null
infinity, $g$ is a function that depends on the background field222For
example, for a self-dual plane wave background, $g$ is known as _Volkov
exponent_ Wolkow:1935zz ; Seipt:2017ckc . and $e_{j}$ is the set of charges of
each gluon with respect to the background, with values in the Cartan
subalgebra of the gauge group – relatively simple formulae are available for
generic backgrounds as well. The corresponding expression for the form factor
for the operator $\,\mathrm{tr}\,\tilde{F}^{2}$ is quite different due to the
chirality in our focus on MHV form factors. We show that the tree-level MHV
form factor around a Cartan-valued self-dual radiative background has the
compact form

$\mathscr{F}_{\,\mathrm{tr}\,\tilde{F}^{2}}(1^{+},\ldots,n^{+};q)=\frac{(q\cdot
Q)^{2}}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,.$ (2)

In both these examples, the space-time integral represents a simple dressing
of the form factor around the flat background; in the limit $g\to 0$
(corresponding to the flat-background limit) we recover momentum conservation
in the presence of the local operators, i.e. the momentum of either
$\,\mathrm{tr}\,B^{2}$ or $\,\mathrm{tr}\,\tilde{F}^{2}$ is constrained to be
equal to the sum of the gluon momenta, as a consequence of the translational
invariance of the trivial background. In that limit, the expression of the
$\,\mathrm{tr}\,\tilde{F}^{2}$ form factor reduces to the well-known formula
for the tree-level scattering of a massive Higgs and arbitrarily many
positive-helicity gluons Dixon:2004za . The form factor around a general
background thus retains much of the simplicity observed in the trivial
background case, although it involves a single residual space-time integral
because the background is not translation invariant, but the kinematical
prefactors coincide on the support of momentum conservation. Similar
considerations apply to other tree-level form factors in the MHV sector, see
Equations (102) and (93) for $k=3$ below for explicit examples of
$\,\mathrm{tr}\,\tilde{F}^{3}$ and $\,\mathrm{tr}\,B^{3}$, as well as for
tree-level MHV form factors in $\mathcal{N}=4$ SYM, see Equation (110).

This work is organized as follows. We review some elementary results in
twistor theory and develop the necessary tools to describe self-dual radiative
gauge fields in Section 2, as well as the extension to self-dual radiative
backgrounds in $\mathcal{N}=4$ SYM in §3, paying special regard to the
fermionic expansion for the asymptotic data near $\mathscr{I}$. The key
technique is introduced in §4, where we introduce new explicit integral
representations for the background fields in terms of their radiative data and
for the linear fields propagating around these backgrounds. These are then
used to lift for example $\,\mathrm{tr}\,\tilde{F}^{2}$ and
$\,\mathrm{tr}\,\tilde{F}^{3}$ to twistor space in Section 5. We use these to
show how expressions for the tree-level MHV form factors around non-trivial
backgrounds in pure Yang-Mills can be readily obtained by their expressions
around the trivial background. We conclude by showing that similar results
hold for tree-level MHV super form factors in $\mathcal{N}=4$ SYM around
gluonic self-dual radiative backgrounds. In the discussion section 6.1 we
briefly explain how the MHV-diagram propagator can be dressed; this can in
principle be used to compute higher MHV degree and loop-level expressions on
backgrounds. The machinery developed in the text is used in 6.2 to discuss
generating functions for the MHV all plus one-loop amplitude and its extension
to backgrounds, and we show how the formulation can be naturally used to
obtain dual conformal invariant region momentum formulae for the amplitude
such as those obtained by Henn:2019mvc ; Chicherin:2022bov . In Appendix A we
show an equivalence between the equations of motion for $\mathcal{N}=4$ SYM
and constraint equations for super-connections on chiral superspace. Finally,
we defer some more computational details of our construction to Appendix B.

##  2 Self-dual radiative backgrounds in pure Yang-Mills

In this section, we review self-dual radiative Yang-Mills backgrounds in four-
dimensional complexified Minkowski space-time and their twistor theory; see
also Penrose:1986uia ; Mason:1991rf ; Adamo:2017qyl ; Adamo:2020yzi  for
further details. Working on complex Minkowski space-time
$\mathbb{M}\cong\mathbb{C}^{4}$ with coordinates $x^{\mu}$, $\mu=0,1,2,3$,
it’s useful to recall the local isomorphism between
$\mathrm{SO}(4,\mathbb{C})$ and
$\mathrm{SL}(2,\mathbb{C})\times\mathrm{SL}(2,\mathbb{C})$ and to introduce
the 2-spinor notation, trading tensor indices with pairs of spinor indices
with opposite chirality

$x^{\alpha\dot{\alpha}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c
c}x^{0}+x^{3}&x^{1}-\mathrm{i}x^{2}\\\
x^{1}+\mathrm{i}x^{2}&x^{0}-x^{3}\end{array}\right)\,.$ (3)

The $\mathrm{SL}(2,\mathbb{C})$-invariant Levi-Civita symbols
$\varepsilon_{\alpha\beta}$ and $\varepsilon_{\dot{\alpha}\dot{\beta}}$ are
used to raise and lower spinor indices. Following standard spinor-helicity
notation, we denote the contractions between spinors by $\langle
ab\rangle\coloneqq
a^{\alpha}b_{\alpha}=\varepsilon^{\alpha\beta}a_{\beta}b_{\alpha}$ and
$[\tilde{a}\tilde{b}]\coloneqq\tilde{a}^{\dot{\alpha}}\tilde{b}_{\dot{\alpha}}=\varepsilon^{\dot{\alpha}\dot{\beta}}\tilde{a}_{\dot{\beta}}\tilde{b}_{\dot{\beta}}$.
Given a gauge field on $\mathbb{M}$, the field strength $F_{ab}$ admits the
decomposition

$F_{\alpha\dot{\alpha}\beta\dot{\beta}}=\varepsilon_{\alpha\beta}\tilde{F}_{\dot{\alpha}\dot{\beta}}+\varepsilon_{\dot{\alpha}\dot{\beta}}F_{\alpha\beta}\,,$
(4)

where $F_{\alpha\beta}$ and $\tilde{F}_{\dot{\alpha}\dot{\beta}}$ are
symmetric spinors and represent the anti-self-dual (ASD) and self-dual (SD)
parts of the field strength, respectively.

A source-free gauge field will be said to be radiative if it extends to null
infinity inside the conformal compactification of Minkowski space-time and is
completely determined by its free characteristic data at either past or future
null infinity. Here we will in fact assume that we are working with complex
fields on the conformal compactification of $\mathbb{M}$ that includes
$\mathscr{I}_{\mathbb{C}}=\mathscr{I}^{+}_{\mathbb{C}}\cup\mathscr{I}^{-}_{\mathbb{C}}$.
This is a partial complexification of standard real null infinity
$\mathscr{I}=\mathscr{I}^{+}\cup\mathscr{I}^{-}$ of $\mathbb{R}^{1,3}$ where
advanced and retarded fields are allowed to be complex. Focusing on future
null infinity, recall that in the real case,
$\mathscr{I}^{+}\cong\mathbb{R}\times S^{2}$ can be understood as the
inversion of the light-cone of the origin of $\mathbb{R}^{1,3}$;
$\mathscr{I}^{+}_{\mathbb{C}}$ is obtained by complexifying the $\mathbb{R}$
factor to $\mathbb{C}$ while keeping the $S^{2}$ base.333This will guarantee
that each $\alpha$-plane representing a twistor in $\mathbb{M}$ will intersect
$\mathscr{I}_{\mathbb{C}}^{+}$ in a unique point. In order to connect with
homogeneous coordinates on twistor space, we will use a homogeneous version
$(u,\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})$ of Bondi coordinates
subject to the equivalence relation

$(u,\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})\sim(b\bar{b}u,b\lambda_{\alpha},\bar{b}\bar{\lambda}_{\dot{\alpha}})\,,$
(5)

for any $b\in\mathbb{C}^{*}$. $u$ is a complexification of the standard Bondi
retarded time, while $(\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})$ are
homogeneous coordinates on the celestial sphere thought of as the complex
projective line $\mathbb{CP}^{1}$. The homogeneous coordinates allow us to
encode spin and conformal weights in terms of homogenous line bundles
$\mathcal{O}(p,q)\to\mathscr{I}^{+}_{\mathbb{C}}$, where a section of
$\mathcal{O}(p,q)$ is represented as a function of
$f_{p,q}(u,\lambda,\bar{\lambda})$ with weights $(p,q)$ under rescaling of the
homogeneous coordinates

$\displaystyle
f_{p,q}(|b|^{2}u,b\lambda,\bar{b}\bar{\lambda})=b^{p}\bar{b}^{q}f_{p,q}(u,\lambda,\bar{\lambda})\,.$
(6)

The conformal and spin weights $(h,s)$ are then given by $h=(p+q)/2$ and
$s=(p-q)/2$, respectively. A similar description can be set up on
$\mathscr{I}^{-}_{\mathbb{C}}$ by replacing the retarded time $u$ with the
advanced time $v$.

Within this projective formalism, the restriction of a gauge field $A$ on
$\mathbb{M}$ to $\mathscr{I}^{+}_{\mathbb{C}}$ in temporal gauge $A_{u}=0$ is
vanderBurg:1969 ; Newman:1978ze ; Strominger:2013lka ; Barnich:2013sxa

$\left.A\right|_{\mathscr{I}^{+}}=A_{-}(u,\lambda,\bar{\lambda})\mathrm{D}\lambda+A_{+}(u,\lambda,\bar{\lambda})\mathrm{D}\bar{\lambda}\,,$
(7)

where $\mathrm{D}\lambda\coloneqq\langle\lambda\,\mathrm{d}\lambda\rangle$,
$\mathrm{D}\bar{\lambda}\coloneqq[\bar{\lambda}\,\mathrm{d}\bar{\lambda}]$.
The restriction of the leading components of the SD and ASD parts of the
curvature are

$\displaystyle F^{0}_{+}$ $\displaystyle=$
$\displaystyle\partial_{u}A_{+}\,\mathrm{d}u\wedge\mathrm{D}\bar{\lambda}\,,$
(8a) $\displaystyle F^{0}_{-}$ $\displaystyle=$
$\displaystyle\partial_{u}A_{-}\,\mathrm{d}u\wedge\mathrm{D}\lambda\,,$ (8b)

thus $A_{+}$, $A_{-}$ are the free data for the SD and ASD components of the
field strength, respectively. In terms of the line bundles above, $A_{+}$ is a
section of $\mathcal{O}(-2,0)\otimes\mathfrak{g}$, while $A_{-}$ is a section
of $\mathcal{O}(0,-2)\otimes\mathfrak{g}$, where $\mathfrak{g}$ is the Lie
algebra of the gauge group. A self-dual, radiative gauge field is a gauge
field completely characterized by the free data
$\left.A\right|_{\mathscr{I}^{+}_{\mathbb{C}}}=A_{+}\,\mathrm{D}\bar{\lambda}$,
while $A_{-}=0$.

#### Radiative fields from their asymptotic data.

We can understand the previous discussion in terms of the peeling properties
of the gauge fields and by means of the Kirchhoff-d’Adhémar integral formula
Penrose:1962ij ; Penrose:1980yx ; Penrose:1984uia ; these can in turn be
regarded as twistor integral formulae using twistor representatives built from
asymptotic data at future null infinity. This perspective will also provide a
natural way to introduce radiative data for scalars and fermions in
$\mathcal{N}=4$ SYM.

Recall that the radiative data for a field
$\Phi_{\alpha_{1}\ldots\alpha_{2|h|}}$ of helicity $h\leq 0$ consist of a
function $\Phi_{2h}=\Phi_{2h}(u,\lambda,\bar{\lambda})$ of weight
$(-2|h|-1,-1))$ on (future) null infinity; it is well-known that the
characteristic data $\Phi_{2h}$ is the leading-order component of the field
that decays as $r^{-1}$ as $r\rightarrow\infty$, where $r$ is an affine
parameter; the $2|h|+1$ components of $\Phi_{\alpha_{1}\ldots\alpha_{2|h|}}$
_peel_ at different rates as $r\rightarrow\infty$ Penrose:1984uia . In terms
of $\Phi_{2h}$, the bulk field can be reconstructed using the
Kirchhoff-d’Adhémar formula

$\Phi_{\alpha_{1}\ldots\alpha_{2|h|}}(x)=\int_{\mathbb{CP}^{1}}\frac{\mathrm{D}\lambda\wedge\mathrm{D}\bar{\lambda}}{2\pi\mathrm{i}}\
\lambda_{\alpha_{1}}\ldots\lambda_{\alpha_{2|h|}}\left.\frac{\partial\Phi_{2h}}{\partial
u}\right|_{u=\langle\lambda|x|\bar{\lambda}]}\,,$ (9)

where the integral is evaluated on the _light-cone cut_ of $x$, that is the
intersection of the null cone with apex $x$ and future null infinity

$u=x^{\alpha\dot{\alpha}}\lambda_{\alpha}\bar{\lambda}_{\dot{\alpha}}\,.$ (10)

In particular, for $h=-1$, we identify the radiative data $\Phi_{-2}$ for an
ASD gauge field $\tilde{\mathcal{A}}$ by

$\Phi_{-2}(u,\lambda,\bar{\lambda})=\partial_{u}A_{-}(u,\lambda,\bar{\lambda})\,,$
(11)

that is, $\Phi_{-2}$ is precisely the leading part at
$\mathscr{I}^{+}_{\mathbb{C}}$ of the ASD curvature. For positive helicities,
we can take the conjugate of the above and the radiative data $\Phi_{2h}$ are
valued in $\mathcal{O}(-1,2h-1)$ (in particular, we identify
$\Phi_{2}=\partial_{u}A_{+}$), but the corresponding Kirchhoff-d’Adhémar will
have a less direct connection with twistor representatives.

#### The $J$ and $K$ potentials for self-dual gauge fields.

Self-dual gauge fields can be described also in terms of scalar second
potentials. Given a reference spinor $\iota^{\alpha}$, the vanishing of the
ASD curvature component $\iota^{\alpha}\iota^{\beta}F_{\alpha\beta}=0$ implies
flatness in the two-plane tangent to $\iota^{\alpha}\beta^{\dot{\alpha}}$ for
all $\beta^{\dot{\alpha}}$, so we can work with the ansatz

$A_{\alpha\dot{\alpha}}=\iota_{\alpha}A_{\dot{\alpha}}\,,$ (12)

for the gauge connection. In particular, the gauge field is in light-cone
gauge with respect to any null vector of the form
$n^{\alpha\dot{\alpha}}=\iota^{\alpha}\beta^{\dot{\alpha}}$. The equation
$\iota^{\alpha}F_{\alpha\beta}=0$ then implies the existence of a matrix-
valued scalar potential $K$, the $K$-matrix, so that

$A_{\dot{\alpha}}=\iota^{\alpha}\partial_{\alpha\dot{\alpha}}K\,.$ (13)

The gauge field is automatically in Lorentz gauge as well and the curvature
can now be written in terms of $K$ as

$\displaystyle F_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\iota_{\alpha}\iota_{\beta}(\square
K+\mathrm{i}[\tilde{\mathrm{d}}_{\dot{\alpha}}K,\tilde{\mathrm{d}}^{\dot{\alpha}}K])\,,$
(14a) $\displaystyle\tilde{F}_{\dot{\alpha}\dot{\beta}}$ $\displaystyle=$
$\displaystyle\mathrm{d}_{\dot{\alpha}}\mathrm{d}_{\dot{\beta}}K\,,$ (14b)

where we introduced the notation
$\mathrm{d}_{\dot{\alpha}}\coloneqq\iota^{\alpha}\partial_{\alpha\dot{\alpha}}$.
The self-duality equation is therefore

$\square
K+\mathrm{i}[\mathrm{d}_{\dot{\alpha}}K,\mathrm{d}^{\dot{\alpha}}K]=0\,.$ (15)

The $J$-matrix potential requires the choice of a second spinor $o_{\alpha}$,
which we normalize by $\langle\iota o\rangle=1$. Using
$o^{\alpha}o^{\beta}F_{\alpha\beta}=0$, we can deduce the existence of a
matrix function $J$ so that

$A_{\alpha\dot{\alpha}}=-\mathrm{i}\,\iota_{\alpha}J^{-1}o^{\beta}\partial_{\beta\dot{\alpha}}J\,.$
(16)

Defining $\tilde{\mathrm{d}}_{\dot{\alpha}}\coloneqq
o^{\alpha}\partial_{\alpha\dot{\alpha}}$, the curvatures become

$\displaystyle F_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-\mathrm{i}\,o_{(\alpha}\iota_{\beta)}\mathrm{d}^{\dot{\alpha}}(J^{-1}\tilde{\mathrm{d}}_{\dot{\alpha}}J)\,,$
(17a) $\displaystyle\tilde{F}_{\dot{\alpha}\dot{\beta}}$ $\displaystyle=$
$\displaystyle-\mathrm{i}\,\mathrm{d}_{(\dot{\alpha}}(J^{-1}\tilde{\mathrm{d}}_{\dot{\beta})}J)\,,$
(17b)

so that the self-dual Yang-Mills equations for this potential become

$\mathrm{d}^{\dot{\alpha}}(J^{-1}\tilde{\mathrm{d}}_{\dot{\alpha}}J)=0\,.$
(18)

###  2.1 Twistor-space description

To define twistor space, introduce homogeneous coordinates
$Z^{A}=(\mu^{\dot{\alpha}},\lambda_{\alpha})$ on $\mathbb{CP}^{3}$ subject to
the equivalence relation $Z^{A}\sim tZ^{A}$ for $t\in\mathbb{C}^{*}$. The
twistor space $\mathbb{PT}$ of $\mathbb{M}$ is the open subset of
$\mathbb{CP}^{3}$ given by

$\mathbb{PT}=\\{[Z^{A}]\in\mathbb{CP}^{3}:\lambda_{\alpha}\neq 0\\}\,,$ (19)

and can thus be described as the total space of the holomorphic bundle
$\mathcal{O}(1)\oplus\mathcal{O}(1)\to\mathbb{CP}^{1}$ over the Riemann
sphere. Its relationship with $\mathbb{M}$ is encoded in the incidence
relations

$\mu^{\dot{\alpha}}=x^{\alpha\dot{\alpha}}\lambda_{\alpha}\,.$ (20)

For fixed $x\in\mathbb{M}$, the incidence relations describe a _twistor line_
, that is a linearly and holomorphically embedded Riemann sphere
$X\subset\mathbb{PT}$, while for constant $Z^{A}\in\mathbb{PT}$ they give an
$\alpha$-plane in $\mathbb{M}$ Penrose:1967wn , i.e. a totally null 2-planes
with self-dual tangent bivectors. The definition (19) removes the twistor line
$I\subseteq\mathbb{CP}^{3}$ corresponding to spatial infinity $i^{0}$.

Radiative linear fields admit a natural description in twistor space
Mason:1986 . This can be obtained by pulling back asymptotic data on
$\mathscr{I}^{+}_{\mathbb{C}}$ to twistor space $\mathbb{PT}$ via the natural
projection $p$ to $\mathscr{I}^{+}_{\mathbb{C}}$ given by

$p:(\mu^{\dot{\alpha}},\lambda_{\alpha})\mapsto(u=\mu^{\dot{\alpha}}\bar{\lambda}_{\dot{\alpha}},\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}})\,.$
(21)

We can define line bundles $\mathcal{O}(n)\rightarrow\mathbb{PT}$ whose
sections can be represented by functions of homogeneity-degree $n$ in the
homogeneous coordinates. These line bundles are identified with both the pull-
backs $p^{*}\mathcal{O}(n,0)$ by $p$ from $\mathscr{I}^{+}_{\mathbb{C}}$ and
the pull-backs of the line bundles $\mathcal{O}(n)\to\mathbb{CP}^{1}$ by the
holomorphic projection $\mathbb{PT}\to\mathbb{CP}^{1}$. For fields of helicity
$h\leq 0$, we can pull back the characteristic data at
$\mathscr{I}^{+}_{\mathbb{C}}$ to twistor space to give $(0,1)$-forms of
holomorphic weight $2h-2$ that defines a Dolbeault cohomology class

$\omega_{2h}=p^{*}\left(\frac{\partial\Phi_{2h}}{\partial
u}\mathrm{D}\bar{\lambda}\right)\,\in H^{0,1}(\mathbb{PT},\mathcal{O}(2h-2)).$
(22)

Following Mason:1986 , the Kirchhoff-d’Adhémar integral formula (9) can now be
re-interpreted as a version of the twistor integral formula or Penrose
transform Penrose:1969ae ; Eastwood:1981jy  using Dolbeault cohomology

$\Phi_{\alpha_{1}\ldots\alpha_{2|h|}}(x)=\int_{X}\mathrm{D}\lambda\wedge\lambda_{\alpha_{1}}\ldots\lambda_{\alpha_{2|h|}}\left.\omega_{2h}\right|_{X}\,.$
(23)

Holomorphicity of $\omega_{2h}$ then implies that
$\Phi_{\alpha_{1}\dots\alpha_{2|h|}}$ solves the ZRM equation, so that it
correctly represent a helicity-$h$ field. For $h=1/2$, we take
$\omega_{1}=p^{*}(\Phi_{1}\mathrm{D}\bar{\lambda})\in
H^{0,1}(\mathbb{PT},\mathcal{O}(-1))$, but the integral formula now requires a
derivative

$\Phi_{\dot{\alpha}}(x)=\int_{X}\left.\mathrm{D}\lambda\wedge\frac{\partial\omega_{1}}{\partial\mu^{\dot{\alpha}}}\right|_{X}\,,$
(24)

and similarly for other positive-helicity linear fields.

In order to extend this construction to fully non-linear, non-abelian self-
dual gauge fields, we follow a strategy due to Sparling Sparling:1990 . We
first use the asymptotic gauge field at $\mathscr{I}^{+}_{\mathbb{C}}$ to
construct the $\mathfrak{g}$-valued $(0,1)$-form

$\mathsf{a}\coloneqq
p^{*}(A_{+}\,\mathrm{D}\bar{\lambda})=A_{+}(\mu^{\dot{\alpha}}\bar{\lambda}_{\dot{\alpha}},\lambda,\bar{\lambda})\mathrm{D}\bar{\lambda}\,,$
(25)

valued in $\Omega^{0,1}(\mathbb{PT},\mathcal{O}(0)\otimes\mathfrak{g})$ on
twistor space. Since $\mathsf{a}$ points only along the
$\mathrm{D}\bar{\lambda}$ direction and is holomorphic in
$\mu^{\dot{\alpha}}$, the $\bar{\partial}$ operator
$\bar{D}=\bar{\partial}+\mathsf{a}$ satisfies $\bar{D}^{2}=0$, so it defines a
holomorphic vector bundle $E$ on twistor space and gives a direct method to
construct the Ward transform of the self-dual gauge field from the
characteristic data at null infinity. More in detail, recall that self-dual
gauge fields are described on twistor space by the Ward correspondence
Ward:1977ta ; Newman:1978ze :

###### Theorem 1

There exists a one-to-one correspondence between

  * •

self-dual gauge fields on $\mathbb{M}$ with gauge group
$G=\mathrm{GL}(r,\mathbb{C})$,

  * •

holomorphic rank-$r$ vector bundles $E\to\mathbb{PT}$ such that
$\left.E\right|_{X}$ is trivial for each $x\in\mathbb{M}$.

We can use the reconstruction part of the Ward construction to solve the
characteristic data initial value problem. We give some details here as they
will be needed in the calculations that follow. If $X\subseteq\mathbb{PT}$ is
any line in twistor space, $\left.E\right|_{X}$ is topologically trivial by
assumption, and holomorphic with $\bar{\partial}$-operator
$\left.\bar{D}\right|_{X}$. For small $\mathsf{a}$ – that is, in perturbation
theory– this implies that $\left.E\right|_{X}$ is _holomorphically_ trivial,
so that there exists a frame
$\mathsf{H}\colon\left.E\right|_{X}\to\mathbb{C}^{r}$ satisfying the Sparling
equation Sparling:1990

$\left.\bar{D}\right|_{X}\mathsf{H}(x,\lambda,\bar{\lambda})\coloneqq\left.(\bar{\partial}+\mathsf{a})\right|_{X}\mathsf{H}=0\,.$
(26)

It is possible to understand this equation in both geometric and holographic
terms as follows Newman:1980fr : the image of $X$ under $p$ is the light-cone
cut of $x$ and it can be shown that the Sparling equation is satisfied when
$\mathsf{H}$ is taken to be the parallel propagator from the point $x$ to
$\mathscr{I}_{\mathbb{C}}^{+}$ along the light-cone. As we will see below,
this parallel propagator determines the bulk gauge field, giving a holographic
interpretation to the Ward correspondence. Note also that the frame is defined
up to a matrix-valued function $g(x)$ on $\mathbb{M}$,
$\mathsf{H}(x,\lambda)\to\mathsf{H}(x,\lambda)g(x)$, with the resulting
ambiguity identified with gauge transformations on $\mathbb{M}$. We can remove
the ambiguity by requiring that $\mathsf{H}(x,\iota)$ is the identity matrix
for some fixed spinor $\iota_{\alpha}$. The incidence relations and the chain
rule imply that
$\lambda^{\alpha}\partial_{\alpha\dot{\alpha}}\left.\mathsf{a}\right|_{X}=0$,
as $\left.\mathsf{a}\right|_{X}$ only depends on $x$ through
$\mu^{\dot{\alpha}}$, which is annihilated by
$\lambda^{\alpha}\partial_{\alpha\dot{\alpha}}$ on the support of the
incidence relations (20). Thus differentiating (26) along
$\lambda^{\alpha}\partial_{\alpha\dot{\alpha}}$ we quickly find

$\left.\bar{\partial}\right|_{X}(\mathsf{H}^{-1}\lambda^{\alpha}\partial_{\alpha\dot{\alpha}}\mathsf{H})=0\,,$
(27)

that is,
$\mathsf{H}^{-1}\lambda^{\alpha}\partial_{\alpha\dot{\alpha}}\mathsf{H}$ is a
holomorphic function on $X$ of weight $+1$ in $\lambda_{\alpha}$. Liouville’s
theorem ensures the existence of a $\mathfrak{g}$-valued function
$A_{\alpha\dot{\alpha}}$ on $\mathbb{M}$ such that

$\mathsf{H}^{-1}(x,\lambda)\lambda^{\alpha}\partial_{\alpha\dot{\alpha}}\mathsf{H}(x,\lambda)=-\mathrm{i}\lambda^{\alpha}A_{\alpha\dot{\alpha}}(x)\,.$
(28)

$A_{\alpha\dot{\alpha}}(x)$ is the desired self-dual gauge potential
transforming in the normal way under the gauge transformation $g$. Defining
the covariant derivative $\nabla=\mathrm{d}-\mathrm{i}A$, equation (28) can be
recast as

$\lambda^{\alpha}\nabla_{\alpha\dot{\alpha}}\mathsf{H}^{-1}\coloneqq\lambda^{\alpha}(\partial_{\alpha\dot{\alpha}}-iA_{\alpha\dot{\alpha}})\mathsf{H}^{-1}=0\,.$
(29)

This directly implies

$\lambda^{\alpha}\lambda^{\beta}F_{\alpha\dot{\alpha}\beta\dot{\beta}}\mathsf{H}^{-1}=[\lambda^{\alpha}\nabla_{\alpha\dot{\alpha}},\lambda^{\beta}\nabla_{\beta\dot{\beta}}]\mathsf{H}^{-1}=0\,.$
(30)

Since this equation holds for any value of $\lambda_{\alpha}$, we deduce that
$A_{\alpha\dot{\alpha}}$ is indeed self-dual. As promised, this equation
implies that $\mathsf{H}$ is parallel propagated along light-rays from $x$ to
infinity in the direction $\lambda_{\alpha}\bar{\lambda}_{\dot{\alpha}}$. If
we fix the gauge freedom with the choice $\mathsf{H}(x,\iota)=1$ for the
frame, (28) can be used to express the scalar potentials as well, namely

$\displaystyle J$ $\displaystyle=$ $\displaystyle\mathsf{H}(x,o)\,,$ (31a)
$\displaystyle K$ $\displaystyle=$
$\displaystyle\mathrm{i}\,o^{\alpha}\left.\frac{\partial}{\partial\lambda^{\alpha}}\mathsf{H}\right|_{\lambda=\iota}\,.$
(31b)

Finally, in the following, it will be important to know the Green’s function
for $\left.\bar{D}\right|_{X}$ acting on sections of $\mathcal{O}(-1)$, which
can be immediately found in terms of the holomorphic frame as

$\mathsf{U}_{X}(\lambda,\lambda^{\prime})=\frac{1}{2\pi\mathrm{i}}\frac{\mathsf{H}(x,\lambda)\mathsf{H}^{-1}(x,\lambda^{\prime})}{\langle\lambda\lambda^{\prime}\rangle}\,.$
(32)

##  3 Self-dual radiative backgrounds in $\mathcal{N}=4$ SYM

We now extend the discussion from the previous section to $\mathcal{N}=4$
super-Yang-Mills. In order to have a discussion adapted to twistor theory, we
consider the chiral formulation of the theory Chalmers:1996rq ;
Chalmers:1997sg , where the $\mathcal{N}=4$ supermultiplet can be described by
fields

$\\{A_{\alpha\dot{\alpha}},\tilde{\psi}_{a\dot{\alpha}},\phi_{ab},\psi^{a}_{\alpha},B_{\alpha\beta}\\}\,.$
(33)

In the full $\mathcal{N}=4$ theory, $B_{\alpha\beta}=B_{(\alpha\beta)}$ is an
ASD 2-form proportional to the ASD part of the curvature $F_{\alpha\beta}$,
while in the self-dual theory it is a linear field imposing the self-duality
condition. In both cases, we can make supersymmetry manifest by working with a
superspace description: the superspace that is best adapted to make contact
with twistor theory is chiral Minkowski superspace.

###  3.1 Chiral super-fields

We enlarge Minkowski space $\mathbb{M}$ to chiral Minkowski super-space
$\mathbb{M}^{4|8}$ with coordinates $x^{\alpha
A}\coloneqq(x^{\alpha\dot{\alpha}},\theta^{\alpha a})$ and also define the
corresponding coordinate derivatives $\partial_{\alpha
A}\coloneqq(\partial_{\alpha\dot{\alpha}},\partial_{\alpha a})$, where we used
the $2|4$ index $A=(\dot{\alpha},a)$. We can then introduce the super-
connection444We will underline super-fields on $\mathbb{M}^{4|8}$ to make the
distinction with space-time fields on $\mathbb{M}$ clear.
$\underline{\nabla}_{\alpha A}=\partial_{\alpha
A}-\mathrm{i}\underline{A}_{\alpha A}$, with $\underline{A}_{\alpha
A}(x,\theta)$ taking values in the Lie algebra of the gauge group.

On non-chiral superspace, the $\mathcal{N}=4$ equations of motion for the non-
chiral super-fields are equivalent to constraint equations for the super-
connection Witten:1978xx ; Harnad:1985bc . On chiral superspace, an analogue
statement can be derived by imposing the following constraints on the super-
connection

$[\underline{\nabla}_{a(\alpha},\underline{\nabla}_{\beta)A}\\}=0\,,\,.$ (34)

However, if we wish to ensure that the super-fields are a solution to the full
$\mathcal{N}=4$ SYM equations of motion, we must require the further
constraint on the ASD part of the bosonic supercurvature

$\underline{F}_{\alpha\beta}=\lambda\underline{B}_{\alpha\beta}\,\,,$ (35)

where the fields can be scaled so that $\lambda$ is the ’t Hooft coupling and
$\underline{B}_{\alpha\beta}$ is defined as a consequence of the first set of
constraints (34), see Equation (37b) below and appendix A for an extensive
discussion. The self-dual theory is recovered in the limit $\lambda\to 0$: in
this limit, it is well known that the constraint equations can be supplemented
with $[\underline{\nabla}_{A(\alpha},\underline{\nabla}_{\beta)B}\\}=0$ and
give the $\mathcal{N}=4$ self-dual SYM equations of motion for the super-
fields, both on non-chiral Witten:1978xx ; Witten:1985nt ; Harnad:1985bc  and
chiral Devchand:1996gv ; Adamo:2013cra  superspace.

In terms of the super-connection,
$\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}$ and
$\underline{F}_{\alpha\beta}$ are defined as usual. $\underline{\phi}_{ab}$
and $\underline{\tilde{\psi}}_{\dot{\alpha}a}$ are also defined as superspace
curvatures by

$\displaystyle-\varepsilon_{\alpha\beta}\underline{\phi}_{ab}$
$\displaystyle\coloneqq$ $\displaystyle\\{\underline{\nabla}_{\alpha
a},\underline{\nabla}_{\beta b}\\}\,,$ (36a)
$\displaystyle\varepsilon_{\alpha\beta}\underline{\tilde{\psi}}_{a\dot{\alpha}}$
$\displaystyle\coloneqq$ $\displaystyle[\underline{\nabla}_{\alpha
a},\underline{\nabla}_{\beta\dot{\alpha}}]\,,$ (36b)

The remaining super-fields $\underline{\psi}^{a}_{\alpha}$ and
$\underline{B}_{\alpha\beta}$ are given by consistency conditions following
from the constraints (34) and suitable Jacobi identities and are defined by

$\displaystyle\epsilon_{abcd}\underline{\psi}_{\alpha}^{d}$
$\displaystyle\coloneqq$ $\displaystyle\underline{\nabla}_{\alpha
a}\underline{\phi}_{bc}\,,$ (37a) $\displaystyle\underline{B}_{\alpha\beta}$
$\displaystyle\coloneqq$
$\displaystyle\frac{1}{4}\underline{\nabla}_{a(\alpha}\underline{\psi}_{\beta)}^{a}\,,$
(37b)

see appendix A for more details. In each case the super-field at
$\theta^{a\alpha}=0$ will be the corresponding $\mathcal{N}=4$ field on
$\mathbb{M}$, for example the scalar fields $\phi_{ab}$ are the lowest
components terms of $\underline{\phi}_{ab}$. The Jacobi identity for
$\underline{\nabla}_{a\alpha}$, $\underline{\nabla}_{b\beta}$ and
$\underline{\nabla}_{\gamma\dot{\gamma}}$ gives

$\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\phi}_{ab}=-\underline{\nabla}_{a\alpha}\underline{\tilde{\psi}}_{b\dot{\alpha}}\,.$
(38)

###  3.2 Chiral super-fields at $\mathscr{I}$

At null infinity, we take the tangent to the generators of
$\mathscr{I}^{+}_{\mathbb{C}}$ to be $\iota^{\alpha}\iota^{\dot{\alpha}}$ and
impose the gauge condition

$\displaystyle\underline{A}_{\alpha a}$ $\displaystyle=$
$\displaystyle\iota_{\alpha}\underline{A}_{a}\,,$ (39a)
$\displaystyle\underline{A}_{\alpha\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\iota_{\alpha}\underline{A}_{\dot{\alpha}}+\tilde{\iota}_{\dot{\alpha}}\underline{A}_{\alpha}\,.$
(39b)

In this gauge we can now investigate the $\theta$ dependence of the various
super-fields at null infinity. It’s useful to separate the fermionic variables
the variables

$\chi^{a}\coloneqq\theta^{a\alpha}o_{\alpha}\,,\qquad\tilde{\chi}^{a}\coloneqq-\theta^{a\alpha}\iota_{\alpha}\,.$
(40)

At null infinity, we focus on the $\chi^{a}$ dependence of the super-fields
and set $\tilde{\chi}^{a}=0$ accordingly. Peeling means that as one approaches
$\mathscr{I}$, fields align with $o_{\alpha}$ to leading order
$B_{\alpha\beta}\sim B_{2}o_{\alpha}o_{\beta}/r+O(1/r^{2})$. Equivalently, we
are considering a supersymmetrization
$\mathscr{I}_{\mathbb{C},\,\mathcal{N}=4}^{+}$ of
$\mathscr{I}^{+}_{\mathbb{C}}$ coordinatized by homogeneous coordinates
$(u,\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}},\chi^{a})$ defined up to the
equivalence relation

$(u,\lambda_{\alpha},\bar{\lambda}_{\dot{\alpha}},\chi^{a})\sim(b\bar{b}\,u,b\lambda_{\alpha},\bar{b}\bar{\lambda}_{\dot{\alpha}},b\chi^{a})\,,$
(41)

for any $b\in\mathbb{C}^{*}$. To obtain the $\chi^{a}$ dependence, we contract
the spinor indices in (37a), (37b), and (38) with $\iota^{\alpha}$ and, using
$\partial/\partial\chi^{a}=\iota^{\alpha}\partial_{a\alpha}$, integrate with
respect to $\chi^{a}$ to get

$\displaystyle\langle\iota\underline{\psi}^{a}\rangle$ $\displaystyle=$
$\displaystyle\langle\iota\psi^{a}\rangle+\chi^{a}B_{2}+\mathcal{O}(\chi^{2})\,,$
(42a) $\displaystyle\underline{\phi}_{ab}$ $\displaystyle=$
$\displaystyle\phi_{ab}+\epsilon_{abcd}\chi^{c}\langle\iota\psi^{d}\rangle+\frac{1}{2}\epsilon_{abcd}\chi^{c}\chi^{d}B_{2}+\mathcal{O}(\chi^{3})\,,$
(42b) $\displaystyle{[\tilde{\iota}\underline{\tilde{\psi}}_{a}]}$
$\displaystyle=$
$\displaystyle[\tilde{\iota}\tilde{\psi}_{a}]+\chi^{b}\partial_{u}\phi_{ab}+\frac{1}{2}\epsilon_{abcd}\chi^{b}\chi^{c}\langle\iota\psi^{d}\rangle+\chi_{a}^{3}B_{2}+\mathcal{O}(\chi^{4})$
(42c)

where

$\displaystyle B_{2}$ $\displaystyle\coloneqq$
$\displaystyle\iota^{\alpha}\iota^{\beta}B_{\alpha\beta}\,,$ (43a)
$\displaystyle\chi_{a}^{3}$ $\displaystyle\coloneqq$
$\displaystyle\frac{1}{3!}\epsilon_{abcd}\chi^{b}\chi^{c}\chi^{d}\,,$ (43b)

and where we have taken the integration constants to be the space-time field
associated with the corresponding super-field.

The same computation applied to (36a) and (36b) leads to the expansion of the
super-connection

$\displaystyle\underline{A}_{a}$ $\displaystyle=$
$\displaystyle\partial_{u}^{-1}[\tilde{\iota}\,\tilde{\psi}_{a}]+\phi_{ab}\chi^{b}+\langle\iota\psi^{d}\rangle\frac{1}{2}\epsilon_{abcd}\chi^{b}\chi^{c}+B_{2}\chi_{a}^{3}+O(\chi^{4})\,,$
(44a) $\displaystyle\tilde{\iota}^{\dot{\alpha}}\underline{A}_{\dot{\alpha}}$
$\displaystyle=$
$\displaystyle\tilde{\iota}^{\dot{\alpha}}A_{\dot{\alpha}}+[\tilde{\iota}\tilde{\psi}_{a}]\chi^{a}+\partial_{u}\phi_{ab}\frac{1}{2}\chi^{a}\chi^{b}+\langle\iota\,\partial_{u}\psi^{a}\rangle\chi_{a}^{3}+\partial_{u}B_{2}\chi^{1}\chi^{2}\chi^{3}\chi^{4}\,.$
(44b)

On restriction to $\mathscr{I}^{+}_{\mathbb{C}}$, the super-connection thus
determines the 1-form $\underline{\mathcal{A}}\,\mathrm{D}\bar{\lambda}$,
where we identify $\underline{\mathcal{A}}$ with
$\left.\underline{A}_{\dot{\alpha}}\right|_{\mathscr{I}_{\mathbb{C}}}$. This
means that we take

$\underline{\mathcal{A}}(u,\lambda,\bar{\lambda},\theta)=A_{+}+\Phi_{1,a}\chi^{a}+\partial_{u}\Phi_{0,ab}\chi^{a}\chi^{b}+\partial_{u}\Phi_{-1}^{a}\chi^{3}_{a}+\partial_{u}\Phi_{-2}\chi^{1}\chi^{2}\chi^{3}\chi^{4}\,,$
(45)

where $\chi^{a}\coloneqq\theta^{a\alpha}\lambda_{\alpha}$ and the coefficients
are the characteristic data at $\mathscr{I}^{+}_{\mathbb{C}}$ for the
$\mathcal{N}=4$ super Yang-Mills multiplet

$\\{\Phi_{2}=\partial_{u}A_{+},\Phi_{1,a},\Phi_{0,ab}=\Phi_{0,[ab]},\Phi_{-1}^{a},\Phi_{-2}\\}\,.$
(46)

Note that the definition of the radiative data doesn’t require any self-
duality condition to be valid, in complete analogy with (7).

On the other hand, if we consider $\mathcal{N}=4$ _self-dual_ super Yang-
Mills, the equations of motion are equivalent to the graded integrability
conditions Witten:1978xx ; Witten:1985nt ; Harnad:1985bc ; Devchand:1996gv

$[\underline{\nabla}_{A(\alpha},\underline{\nabla}_{\beta)B}\\}=0\,,$ (47)

and they imply that there exists a gauge for which $\underline{A}_{\alpha}=0$
as for the bosonic case. The integrability condition (47) implies the
existence of supersymmetrized versions of the scalar potentials
$\underline{J}(x,\theta)$ and $\underline{K}(x,\theta)$, so that

$\underline{A}_{\alpha A}=\iota_{\alpha}\iota^{\beta}\partial_{\beta
A}\underline{K}=-\mathrm{i}\,\iota_{\alpha}o^{\beta}\underline{J}^{-1}\partial_{\beta
A}\underline{J}\,.$ (48)

As before, the space-time fields $\tilde{F}_{\dot{\alpha}\dot{\beta}}$,
$\tilde{\psi}_{\dot{\alpha}a}$, and $\phi_{ab}$ can also be understood as the
lowest components of the self-dual super-curvature, defined by
$\varepsilon_{\alpha\beta}\underline{\mathcal{F}}_{AB}\coloneqq[\underline{\nabla}_{\alpha
A},\underline{\nabla}_{\beta B}\\}$. We now have, with the definitions
$\mathrm{d}_{A}\coloneqq\iota^{\beta}\partial_{\beta A}$ and
$\tilde{\mathrm{d}}_{A}\coloneqq o^{\beta}\partial_{\beta A}$

$\underline{\mathcal{F}}_{AB}=\mathrm{d}_{A}\mathrm{d}_{B}\underline{K}=-\mathrm{i}\,\mathrm{d}_{(A}(\underline{J}^{-1}\tilde{\mathrm{d}}_{B\\}}\underline{J})\,,$
(49)

in addition to (14b) and (17b). Similarly, the $\mathcal{N}=4$ self-duality
equations are the obvious super-symmetrizations of (15) and (18)

$\displaystyle
2\tilde{\mathrm{d}}_{[A}\mathrm{d}_{B\\}}\underline{K}+\mathrm{i}[\mathrm{d}_{A}\underline{K},\mathrm{d}_{B}\underline{K}\\}$
$\displaystyle=$ $\displaystyle 0\,,$ (50a)
$\displaystyle\mathrm{d}_{[A}(\underline{J}^{-1}\tilde{\mathrm{d}}_{B\\}}\underline{J})$
$\displaystyle=$ $\displaystyle 0\,.$ (50b)

In this gauge, the expansion of the $\underline{K}$ matrix at $\tilde{\chi}=0$
can be obtained by integrating the fermionic part of (48) using (44) to obtain

$\underline{K}=K-\partial_{u}^{-1}[\tilde{\iota}\tilde{\psi}_{a}]\chi^{a}+\frac{1}{2}\phi_{ab}\chi^{a}\chi^{b}-\langle\iota\psi^{a}\rangle\chi_{a}^{3}+\iota^{\alpha}\iota^{\beta}B_{\alpha\beta}\chi^{1}\chi^{2}\chi^{3}\chi^{4}\,.$
(51)

###  3.3 Super-twistor space description

As in the pure Yang-Mills case, solutions to the $\mathcal{N}=4$ self-dual SYM
equations are compactly obtained from a supersymmetric version of the Ward
correspondence. Introducing homogeneous coordinates
$Z^{A}=(\mu^{\dot{\alpha}},\lambda_{\alpha},\chi^{a})$, $a=1,\ldots,4$, on
$\mathbb{CP}^{3|4}$ subject to $Z^{A}\sim tZ^{A}$ for $t\in\mathbb{C}^{*}$,
the super-twistor space of $\mathbb{M}^{4|8}$ is defined to be Ferber:1977qx

$\mathbb{PT}=\\{[Z]\in\mathbb{CP}^{3|4}:\lambda_{\alpha}\neq 0\\}\,,$ (52)

and we still denote it as $\mathbb{PT}$. The incidence relations are

$\mu^{A}\coloneqq(\mu^{\dot{\alpha}},\chi^{a})=(x^{\alpha\dot{\alpha}}\lambda_{\alpha},\theta^{\alpha
a}\lambda_{\alpha})\,,$ (53)

and we denote the line in super-twistor space again by $X$, even though it now
depends on $(x,\theta)$. The Ward correspondence becomes Volovich:1983aa ;
Volovich:1983ii

###### Theorem 2

There exists a one-to-one correspondence between

  * •

solutions to the $\mathcal{N}=4$ self-dual Yang-Mills equations on
$\mathbb{M}^{4|8}$ with gauge group $G=\mathrm{GL}(r,\mathbb{C})$,

  * •

holomorphic rank-$r$ vector bundles $E$ over super-twistor space such that
$\left.E\right|_{X}$ is trivial for each $(x,\theta)\in\mathbb{M}^{4|8}$.

In the Dolbeault framework, these bundles are equipped with an integrable
super-connection $\underline{\mathsf{a}}$ and Dolbeault operator
$\bar{D}=\bar{\partial}+\underline{\mathsf{a}}$, which can be obtained using
the characteristic data built out of the null data. Together with the gluonic
background twistor connection $\mathsf{a}$ of the previous section, we can
construct the connection on supertwistor space

$\underline{\mathsf{a}}\coloneqq\mathsf{a}+\mathsf{a}_{a}\chi^{a}+\frac{1}{2}\mathsf{a}_{ab}\chi^{a}\chi^{b}+\mathsf{a}^{a}\chi^{3}_{a}+\tilde{\mathsf{a}}\chi^{1}\chi^{2}\chi^{3}\chi^{4}\,,$
(54)

valued in $\Omega^{0,1}(\mathbb{PT},\mathcal{O}(0)\otimes\mathfrak{g})$. Here
$\\{\mathsf{a},\mathsf{a}_{a},\mathsf{a}_{ab},\mathsf{a}^{a},\tilde{\mathsf{a}}\\}$
are $(0,1)$-forms of respective homogeneity $0,-1,-2,-3,-4$ in the bosonic
twistor variables and can be obtained as pullbacks of the radiative data for
the multiplet respectively
$p^{*}\\{A_{+},\Phi_{1,a},\partial_{u}\Phi_{0,ab},\partial_{u}\Phi_{-1}^{a},\partial_{u}\Phi_{-2}\\}$
on $\mathscr{I}^{+}_{\mathbb{C}}$.

The reconstruction part of the Ward correspondence is unaltered from the pure
Yang-Mills case, the only difference being the promotion of every field to a
super-field. In this way, at least for small data, there exists a holomorphic
frame $\underline{\mathsf{H}}(x,\theta,\lambda)$ satisfying

$\left.\bar{D}\right|_{X}\underline{\mathsf{H}}(x,\theta,\lambda)=0\,,$ (55)

in terms of which we can construct the super-connection $\underline{A}_{\alpha
A}$ on $\mathbb{M}^{4|8}$ as

$\underline{\mathsf{H}}^{-1}(x,\theta,\lambda)\lambda^{\alpha}\partial_{\alpha
A}\underline{\mathsf{H}}(x,\theta,\lambda)=-\mathrm{i}\lambda^{\alpha}\underline{A}_{\alpha
A}(x,\theta)\,.$ (56)

##  4 Integral formulae for the curvature and background coupled fields

In order to lift space-time formulae to twistor space, in this Section we
obtain explicit expressions for the space-time gauge field in terms of the
twistor connection $\mathsf{a}$ and the frame $\mathsf{H}$; although these are
in principle already determined by the previous Section, we will need more
explicit formulae. Similarly, it has been known for some time how to obtain
formulae for background-coupled linear fields and super-fields, but here we
give more detailed formulae for momentum eigenstates.

###  4.1 Connection and curvature formulae

We first introduce Green’s functions on the Riemann sphere for inverting the
$\bar{\partial}$ operators on different line bundles. Let
$\mathcal{O}(n)\to\mathbb{CP}^{1}$ be the line bundle of homogenous functions
of degree $n$ in $\lambda_{\alpha}$. Provided $n\geq-1$, we can invert

$\bar{\partial}g_{n}=f_{n}\,,$ (57)

for any $f_{n}\in\Omega^{0,1}(\mathbb{CP}^{1},\mathcal{O}(n))$ to find a
solution $g_{n}\in\Omega^{0,0}(\mathbb{CP}^{1},\mathcal{O}(k))$, namely

$g_{n}(\lambda)=\int\frac{\mathrm{D}\lambda^{\prime}}{2\pi\mathrm{i}}\;f_{n}(\lambda^{\prime})\frac{1}{\langle\lambda\lambda^{\prime}\rangle}\left(\frac{\langle\iota\lambda\rangle}{\langle\iota\lambda^{\prime}\rangle}\right)^{n+1}\,.$
(58)

The integral is over $\mathbb{CP}^{1}$, on which
$\lambda_{\alpha},\lambda^{\prime}_{\alpha}$ are homogeneous coordinates, and
the reference spinor $\iota_{\alpha}$ is used to fix the freedom in adding
polynomials of degree $n$ in $\lambda$ to $g_{n}$ by making it vanish to
$n$-th order at $\iota_{\alpha}$; note that for $n=-1$ the solution is unique,
while for $n\geq 0$ the ambiguity in $g_{n}$ is a consequence of
$H^{0}(\mathbb{CP}^{1},\mathcal{O}(n))\cong\mathbb{C}^{n-1}$.

We can now find an integral formula for $A_{\alpha\dot{\alpha}}$ by
differentiating the Sparling equation (26) and eliminating the twistor
connection via the definition of the holomorphic frame. In this way, we find

$\left.\bar{\partial}\right|_{X}(\partial_{\alpha\dot{\alpha}}\mathsf{H}^{-1}\,\mathsf{H})=\lambda_{\alpha}\mathsf{H}^{-1}\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\mathsf{H}\,,$
(59)

and using the Green’s functions (58)

$\partial_{\alpha\dot{\alpha}}\mathsf{H}^{-1}(x,\lambda)\,\mathsf{H}(x,\lambda)=\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\lambda\lambda^{\prime}\rangle}\frac{\langle\iota\lambda\rangle}{\langle\iota\lambda^{\prime}\rangle}\lambda^{\prime}_{\alpha}\mathsf{H}^{-1}(x,\lambda^{\prime})\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}\mathsf{H}(x,\lambda^{\prime})\,.$
(60)

The possible ambiguity in adding a constant to the right-hand side at
homogeneity degree zero is fixed by the vanishing of both sides of the
equation at $\lambda_{\alpha}=\iota_{\alpha}$ for the gauge
$\mathsf{H}(x,\iota)=I$ where
$A_{\alpha\dot{\alpha}}=\iota_{\alpha}A_{\dot{\alpha}}$. Contracting this
equation with $\lambda^{\alpha}$ using (29) yields

$A_{\alpha\dot{\alpha}}(x)=\frac{\iota_{\alpha}}{2\pi}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\iota\lambda^{\prime}\rangle}\mathsf{H}^{-1}(x,\lambda^{\prime})\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}\mathsf{H}(x,\lambda^{\prime})\,.$
(61)

Comparing this last equation with (13), we can identify the integral in (61)
with $\mathrm{d}_{\dot{\alpha}}K$. The associated field strength can be
straightforwardly checked to be self-dual, with SD component

$\tilde{F}_{\dot{\alpha}\dot{\beta}}=\int_{X}\frac{\mathrm{D}\lambda_{1}}{2\pi\mathrm{i}}\mathsf{H}_{1}^{-1}\partial_{{\dot{\alpha}}}\partial_{\dot{\beta}}\mathsf{a}_{1}\mathsf{H}_{1}-\int_{X^{2}}\frac{\mathrm{D}\lambda_{1}\mathrm{D}\lambda_{2}}{(2\pi\mathrm{i})^{2}\langle\lambda_{1}\lambda_{2}\rangle}[\mathsf{H}_{1}^{-1}\partial_{{\dot{\alpha}}}\mathsf{a}_{1}\mathsf{H}_{1},\mathsf{H}_{2}^{-1}\partial_{{\dot{\beta}}}\mathsf{a}_{2}\mathsf{H}_{2}]\,,$
(62)

where $\mathsf{H}_{i}=\mathsf{H}(x,\lambda_{i})$,
$\mathsf{a}_{i}=\left.\mathsf{a}\right|_{X}(x,\lambda_{i})$, and we denoted
$\mu^{\dot{\alpha}}$ derivatives as
$\partial_{{\dot{\alpha}}}\coloneqq\partial/\partial\mu^{\dot{\alpha}}$. The
expression for $\tilde{F}_{\dot{\alpha}\dot{\beta}}$ has the advantage of
being now both Lorentz and gauge invariant; the linear Penrose transform would
lead to the first term in (62) only, but for the fully non-linear field we
need also the second, double integral over $X$.

For $\mathcal{N}=4$ SYM, we similarly have

$\underline{A}_{\alpha
A}(x,\theta)=\frac{\iota_{\alpha}}{2\pi}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\iota\lambda^{\prime}\rangle}\underline{\mathsf{H}}^{-1}(x,\theta,\lambda)\left.\partial_{A}\underline{\mathsf{a}}\right|_{X}\underline{\mathsf{H}}(x,\theta,\lambda)\,.$
(63)

Here
$\partial_{A}\coloneqq(\partial/\partial\mu^{\dot{\alpha}},\partial/\partial\chi^{a})$.
The supersymmetrized $J$\- and $K$-matrices and the propagator on the line $X$
can be defined in this context as well by replacing the holomorphic frame with
its supersymmetrized version in (31a), (31b), (32). The covariant derivative
$\underline{\nabla}_{\alpha A}$ is self-dual by construction with
$[\nabla_{aA},\nabla_{\beta
B}\\}=\varepsilon_{\alpha\beta}\underline{F}_{AB}$; the gluon self-dual
curvature, the positive-helicity fermions and the scalar fields arise as the
lowest components of this self-dual super-curvature. Moreover, the
supersymmetrized Ward correspondence now gives gauge and Lorentz invariant
expressions for both the fermions and the scalars, in complete analogy with
the pure Yang-Mills case. Explicitly, the space-time fermion
$\tilde{\psi}_{a\dot{\alpha}}$ is the lowest component of the superfield

$\underline{\tilde{\psi}}_{a\dot{\alpha}}=\int_{X}\frac{\mathrm{D}\lambda_{1}}{2\pi\mathrm{i}}\underline{\mathsf{H}}_{1}^{-1}\partial_{{a}}\partial_{\dot{\alpha}}\underline{\mathsf{a}}_{1}\underline{\mathsf{H}}_{1}-\int_{X^{2}}\frac{\mathrm{D}\lambda_{1}\mathrm{D}\lambda_{2}}{(2\pi\mathrm{i})^{2}\langle\lambda_{1}\lambda_{2}\rangle}[\underline{\mathsf{H}}_{1}^{-1}\partial_{a}\underline{\mathsf{a}}_{1}\underline{\mathsf{H}}_{1},\underline{\mathsf{H}}_{2}^{-1}\partial_{\dot{\alpha}}\underline{\mathsf{a}}_{2}\underline{\mathsf{H}}_{2}]\,,$
(64)

whilst the scalar $\phi_{ab}$ is the lowest component of

$\underline{\phi}_{ab}=\int_{X}\frac{\mathrm{D}\lambda_{1}}{2\pi\mathrm{i}}\underline{\mathsf{H}}_{1}^{-1}\partial_{{a}}\partial_{b}\underline{\mathsf{a}}_{1}\underline{\mathsf{H}}_{1}-\int_{X^{2}}\frac{\mathrm{D}\lambda_{1}\mathrm{D}\lambda_{2}}{(2\pi\mathrm{i})^{2}\langle\lambda_{1}\lambda_{2}\rangle}\\{\underline{\mathsf{H}}_{1}^{-1}\partial_{a}\underline{\mathsf{a}}_{1}\underline{\mathsf{H}}_{1},\underline{\mathsf{H}}_{2}^{-1}\partial_{b}\underline{\mathsf{a}}_{2}\underline{\mathsf{H}}_{2}\\}\,.$
(65)

This is also the prescription given in Koster:2016ebi ; Koster:2016loo ;
Koster:2016fna  for the construction of vertices for composite operators in
$\mathcal{N}=4$ SYM.

###  4.2 Perturbations, linearized modes and momentum eigenstates

Massless fields of helicity $n/2$ on a self-dual background are well-known to
be given as first cohomology classes on twistor space with values in the
appropriate representation of the Ward bundle $E$ twisted by
$\mathcal{O}(n-2)$. In our radiative framework, for $n\leq 0$, these can be
represented by their $\mathscr{I}$ data
$f_{n-2}\coloneqq\partial_{u}\Phi_{n}\mathrm{D}\bar{\lambda}$. These are
$\bar{\partial}$ closed around the non-trivial self-dual background too,
because both $\mathsf{a}$ and and $f_{n-2}$ are pointing only along the
$\mathrm{D}\bar{\lambda}$ direction and $\Phi_{n}$ is holomorphic in
$\mu^{\dot{\alpha}}$. For $n\leq 0$, we can obtain a space-time linear field
via the standard integral representation of the Penrose transform; the
coupling with the background arises because the bundle $E$ must first be
trivialized before performing the twistor integral. Taking $f_{n-2}$ to be in
the adjoint, we obtain

$\Phi_{\alpha_{1}\ldots\alpha_{|n|}}=\frac{1}{2\pi\mathrm{i}}\int_{X}\lambda_{\alpha_{1}}\mathsf{H}^{-1}(x,\lambda)\left.f_{n-2}\right|_{X}\mathsf{H}(x,\lambda)\wedge\mathrm{D}\lambda\,.$
(66)

For concrete calculations, we will take our $f_{n-2}$ to be momentum
eigenstates with colour $T_{j}\in\mathfrak{g}$ and null momentum
$k^{\alpha\dot{\alpha}}_{j}=\kappa_{j}^{\alpha}\tilde{\kappa}_{j}^{\dot{\alpha}}$.

$f^{j}_{n-2}(Z)=T_{j}\int_{\mathbb{C}^{*}}\frac{\mathrm{d}s}{s^{n-1}}\bar{\delta}^{2}(\kappa_{j}-s\lambda)e^{\mathrm{i}s[\mu\tilde{\kappa}_{j}]}\,,$
(67)

where the holomorphic $\delta$-function is defined by

$\bar{\delta}^{2}(\kappa_{j}-s\lambda)\coloneqq\frac{1}{(2\pi\mathrm{i})^{2}}\bigwedge_{\alpha=1,2}\bar{\partial}\left(\frac{1}{\kappa_{j\,\alpha}-s\lambda_{\alpha}}\right)\,.$
(68)

#### Negative-helicity gluons.

The ASD linearized field strength itself is the case $n=-2$ where the integral
formula is immediately performed against the delta functions to give the ASD
field strength

$f^{j}_{\,\alpha\beta}(x)=\kappa_{j\alpha}\kappa_{j\beta}\mathsf{H}^{-1}_{j}T_{j}\mathsf{H}_{j}\,e^{\mathrm{i}k_{j}\cdot
x}\,,\qquad\qquad\mathsf{H}_{j}\coloneqq\mathsf{H}(x,\kappa_{j})\,.$ (69)

It is now easily checked that the spin-1 equation (70b) below follows using
(29) under the integral sign. We will see that this only gives the
perturbation $f_{j\,\alpha\beta}$ of the curvature, whilst the construction of
the corresponding gauge field perturbation $a^{j}_{\alpha\dot{\alpha}}$ is not
so straightforward as we now describe.

#### Backgound perturbations

The linearized equations of motion for a general perturbation
$a_{\alpha\dot{\alpha}}$ of a self-dual background gauge field read

$\displaystyle\nabla_{\dot{\alpha}(\alpha}a_{\beta)}^{\dot{\alpha}}$
$\displaystyle=$ $\displaystyle f_{\alpha\beta}\,,$ (70a)
$\displaystyle\nabla_{\alpha\dot{\alpha}}f^{\alpha\beta}$ $\displaystyle=$
$\displaystyle 0\,.$ (70b)

In a general background, it is no longer consistent to decompose a linear
field on the background into those whose perturbation of the curvature is
self-dual and anti-self-dual as (70b) would have an extra term from the
background ASD curvature Adamo:2021rfq . However, on a self-dual background,
it is still consistent to require that $f_{\alpha\beta}=0$ in which case the
perturbation preserves the self-duality condition. Such solutions are still
naturally identified with positive-helicity gluons. Conversely, non-trivial
solutions to (70b) are interpreted as negative-helicity gluons around the SD
background, but the corresponding potential that solves (70a) will generally
lead to a non-trivial self-dual curvature perturbation too: even if it is
imposed it to be zero asymptotically, it will develop a non-zero value as the
field is evolved through the space-time. This follows because the MHV tree
amplitude can be understood as being generated by the self-dual field at
$\mathscr{I}^{+}$ associated with a potential crossing space-time on an SD
background whose data at $\mathscr{I}^{-}$ is purely ASD, see Mason:2008jy ;
Adamo:2021rfq  for details.

#### Positive-helicity gluons.

In the following, we focus on MHV form factors, which contain arbitrarily many
positive-helicity external states. The Penrose transform relates these self-
dual gluon perturbations to the cohomology classes of weight 0 whose
representative $a_{j}$ with colour $T_{i}\in\mathfrak{g}$ and null momentum
$k^{\alpha\dot{\alpha}}_{j}=\kappa_{j}^{\alpha}\tilde{\kappa}_{j}^{\dot{\alpha}}$
is as above

$a_{j}(Z)=T_{j}\int_{\mathbb{C}^{*}}\frac{\mathrm{d}s}{s}\bar{\delta}^{2}(\kappa_{j}-s\lambda)e^{\mathrm{i}s[\mu\tilde{\kappa}_{j}]}\,.$
(71)

These will be used around a non-trivial background as well as in the radiative
framework, the construction provides a perturbation that is asymptotic to a
positive-helicity plane wave at $\mathscr{I}_{\mathbb{C}}$. The corresponding
space-time perturbation can be reconstructed by perturbing the formulae above.
Considering just one of the perturbations, we perturb
$\mathsf{a}\to\mathsf{a}+\epsilon_{j}a_{j}$ so that
$\mathsf{H}\to\mathsf{H}+\epsilon_{j}\,\delta_{j}\mathsf{H}$. The perturbation
of (26) to first order in $\epsilon_{j}$ yields

$\left.\bar{\partial}\right|_{X}(\mathsf{H}^{-1}\delta_{j}\mathsf{H})=\mathsf{H}^{-1}a_{j}\mathsf{H}\,,$
(72)

where we used the definition of $\mathsf{H}$ to eliminate the background
twistor connection. Using the Green’s function (58) and integrating against
the delta function then gives

$\mathsf{H}^{-1}\delta_{j}\mathsf{H}=\frac{\langle\iota\lambda\rangle}{\langle\iota
j\rangle\langle\lambda
j\rangle}\mathsf{H}^{-1}_{j}T_{j}\mathsf{H}_{j}e^{\mathrm{i}k_{j}\cdot x}\,.$
(73)

where insertions of $\kappa_{j}$ into angle-brackets are denoted just by $j$.
The variations of the $J$\- and $K$-matrices can be obtained via (31b) and
(31a) to give

$\displaystyle\delta_{j}K$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{\langle\iota
j\rangle^{2}}\mathsf{H}^{-1}_{j}T_{j}\mathsf{H}_{j}e^{\mathrm{i}k_{j}\cdot
x}\,,$ (74a) $\displaystyle J^{-1}\delta_{j}J$ $\displaystyle=$
$\displaystyle-\frac{1}{\langle\iota j\rangle\langle
oj\rangle}\mathsf{H}^{-1}_{j}T_{j}\mathsf{H}_{j}e^{\mathrm{i}k_{j}\cdot x}\,.$
(74b)

These then yield the space-time perturbation

$\displaystyle a_{j\,\alpha\dot{\alpha}}(x)=\frac{\iota_{\alpha}}{\langle\iota
j\rangle}\mathsf{H}^{-1}_{j}\left(\tilde{\kappa}_{j\,\dot{\alpha}}T_{j}+[g_{j\dot{\alpha}}(x),T_{j}]\right)\mathsf{H}_{j}e^{\mathrm{i}k_{j}\cdot
x}\,,$ (75)

where we have used (29) to define $g_{\dot{\alpha}}(x,\lambda)$ by

$\nabla_{\alpha\dot{\alpha}}\mathsf{H}^{-1}=\lambda_{\alpha}\mathsf{H}^{-1}g_{\dot{\alpha}}\,,$
(76)

and the subscript $j$ denotes evaluation at $\lambda=\kappa_{j}$. Similar
formulae for the variation of the curvature can be obtained. Here the usual
gauge dependent undotted spinor present in the vector polarization of a spin-1
momentum eigenstate is taken to be $\iota_{\alpha}$. Note in particular that
the reconstruction of the space-time gauge field, rather than of its
curvature, relies heavily on the existence of (perturbed) $J$\- and
$K$-matrices, in other words it’s possible only for positive-helicity
perturbations that preserve the self-duality condition. For negative-helicity
fields, we can at best construct the linearized field strength (69).

Perturbations corresponding to positive-helicity gluons also modify the
propagator (32). For a general perturbation $\delta a$ –that is, not
necessarily a momentum eigenstate– the variation $\delta\mathsf{U}_{X}$
follows again from (73)

$\displaystyle\delta\mathsf{U}_{X}(\lambda,\lambda^{\prime})$
$\displaystyle=-\int_{X}\mathrm{D}\lambda^{\prime\prime}\,\mathsf{U}_{X}(\lambda,\lambda^{\prime\prime})\left.\delta
a\right|_{X}\mathsf{U}_{X}(\lambda^{\prime\prime},\lambda^{\prime})\,.$ (77)

This variation can be iterated $n$ times to obtain the colour-ordered $n$th
perturbation of the propagator on $X$ as

$\mathsf{U}_{X}(\lambda,\lambda^{\prime})=\sum_{n=0}^{\infty}\left(\frac{-1}{2\pi\mathrm{i}}\right)^{n}\mathsf{H}(x,\lambda)\int_{X^{n}}\mathrm{D}\lambda_{1}\ldots\mathrm{D}\lambda_{n}\frac{\mathsf{H}_{1}^{-1}\delta
a_{1}\mathsf{H}_{1}\ldots\mathsf{H}_{n}^{-1}\delta
a_{n}\mathsf{H}_{n}}{\langle\lambda\lambda_{1}\rangle\langle\lambda_{1}\lambda_{2}\rangle\ldots\langle\lambda_{n}\lambda^{\prime}\rangle}\mathsf{H}^{-1}(x,\lambda^{\prime})\,,$
(78)

where $\delta a_{j}\coloneqq\left.\delta a\right|_{X}(x,\lambda_{j})$. The
frames in (78) are the frames for the background $\mathsf{a}$, and as before
each term is evaluated on the line $X$. If the perturbations are taken to be
momentum eigenstates, the integrations can be directly performed against the
delta functions.

####  $\mathcal{N}=4$ super-momentum eigenstates.

The Penrose transform provides momentum eigenstates for different helicities
as well. In the following, we will consider super form factors in
$\mathcal{N}=4$ where we super-symmetrize the external states (but not the
local operator), so we arrange the possible external states on space-time in
terms of Nair’s super-field Nair:1988bq , i.e. we consider the external state

$\Phi_{j}=g_{j}^{+}+\psi^{+}_{j\,a}\eta^{a}_{j}+\frac{1}{2}\phi_{j\,ab}\eta^{a}_{j}\eta^{b}_{j}+\frac{1}{3!}\varepsilon_{abcd}\psi^{-\,a}_{j}\eta_{j}^{b}\eta_{j}^{c}\eta_{j}^{d}+g^{-}_{j}\eta_{j}^{1}\eta_{j}^{2}\eta_{j}^{3}\eta_{j}^{4}\,,$
(79)

with super-momentum $k_{j\,\alpha
A}\coloneqq\kappa_{j\,\alpha}\tilde{\kappa}_{j\,A}=(\kappa_{j\,\alpha}\tilde{\kappa}_{j\,\dot{\alpha}},\kappa_{j\,\alpha}\eta_{j\,a})$.
The associated twistor representative around the flat background is

$\underline{a}_{i}(Z)=T_{i}\int_{\mathbb{C}^{*}}\frac{\mathrm{d}s}{s}\bar{\delta}^{2}(\kappa_{i}-s\lambda)e^{\mathrm{i}s[\mu\tilde{\kappa}_{i}]+\mathrm{i}s\\{\chi\eta_{i}\\}}\,,$
(80)

and since we are considering horizontal background fields on twistor space,
such a representative can be used around these backgrounds as well. The
corresponding linear perturbation on chiral superspace is

$\underline{a}_{j\,\alpha A}(x,\theta)=\frac{\iota_{\alpha}}{\langle\iota
j\rangle}\underline{\mathsf{H}}_{j}^{-1}(\tilde{\kappa}_{j\,A}+[\underline{g}_{j\,A}(x,\theta),T_{j}])\underline{\mathsf{H}}_{j}e^{\mathrm{i}k_{j}\cdot
x+\mathrm{i}\kappa_{j\,\beta}\eta_{j\,b}\theta^{b\beta}}\,,$ (81)

where $\underline{g}_{A}$ is the natural supersymmetrization of
$g_{\dot{\alpha}}$. Notice in particular that super-momentum eigenstates will
perturb the propagator on the line $X$ as well, the variation being given by
the supersymmetrization of (78).

##  5 MHV (super) form factors

Given a local operator $\mathscr{O}(x)$, its form factor
$\mathscr{F}_{\mathscr{O}}=\mathscr{F}_{\mathscr{O}}(1^{h_{1}},\ldots,n^{h_{n}};q)$
in presence of $n$ external gluons is defined as the Fourier transform of the
matrix element of $\mathscr{O}(x)$ between the vacuum and the $n$-gluon
multiparticle state

$\mathscr{F}_{\mathscr{O}}(1,\ldots,n;q)\coloneqq\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{-\mathrm{i}q\cdot
x}\langle 1^{h_{1}},\ldots,n^{h_{n}}|\mathcal{O}(x)|0\rangle\,,$ (82)

where we implicitly assumed to take our gluons to be outgoing plane waves.
Since we focus on MHV form factors, the helicities are almost all positive,
the number of negative-helicity gluons being equal to the number of $B$ fields
appearing in $\mathscr{O}$.

###  5.1 MHV form factors and Cartan backgrounds

If $\mathscr{O}$ is a composite operator depending only on
$\tilde{F}_{\dot{\alpha}\dot{\beta}}$ but not on $B_{\alpha\beta}$, it’s
straightforward to show that its tree-level MHV form factor can be readily
computed in self-dual Yang-Mills Chalmers:1996rq  by simply putting it on-
shell Boels:2013bi . More generally, in the MHV sector, we can treat form
factors involving $B_{\alpha\beta}$ as well by treating these ASD terms as
perturbations away from the self-dual sector of $S_{\text{SDYM}}$ below. The
generating functional for such a form factor is the path integral with action

$S_{\text{SDYM}}+\int_{\mathbb{M}}\mathrm{d}^{4}x\,\mathscr{J}\mathscr{O}\,,$
(83)

$S_{\text{SDYM}}$ being the action for self-dual Yang-Mills Chalmers:1996rq

$S_{\text{SDYM}}=\int_{\mathbb{M}}\mathrm{d}^{4}x\,\,\mathrm{tr}\,B_{\alpha\beta}F^{\alpha\beta}\,,$
(84)

and $\mathscr{J}$ being a source for $\mathscr{O}$. At tree-level, the
generating functional reduces to (the exponential of) the on-shell action in
the presence of the source, but even for non-trivial $\mathscr{J}$ the
existence of the $J$\- and $K$-matrices is not affected, as long as
$\mathscr{O}$ is a polynomial in $\tilde{F}_{\dot{\alpha}\dot{\beta}}$ and its
derivatives. The argument for the generating functional still holds for
operators involving the $B$ field, because in the MHV sector, the number of
negative-helicity gluons is then precisely the number of $B_{\alpha\beta}$’s
appearing in the operator under study, so for non-exceptional kinematical
configurations we can treat each $B_{\alpha\beta}$ as a linear perturbation
away from self-duality. Similar considerations hold for form factors in the
self-dual sector of $\mathcal{N}=4$ SYM. From the perspective of twisted
holography Costello:2022wso , this procedure can be interpreted as an explicit
realization Bu:2022dis  of the correspondence between local operators in four
dimensions and conformal blocks of a two-dimensional chiral algebra living on
the celestial sphere.

If the form factor is computed in the MHV sector, the set of external
positive-helicity gluons defines a self-dual background obtained as the
coherent state whose data is a sum of plane waves at
$\mathscr{I}_{\mathbb{C}}^{+}$, thus the on-shell expression of $\mathscr{O}$
around a general self-dual radiative background is the generating functional
for the MHV form factor of $\mathscr{O}$. In practice,
$\mathscr{F}_{\mathscr{O}}(1^{+},\ldots,n^{+})$ can be computed by considering
the bulk gauge field that reduces to
$\epsilon_{1}a_{1}+\ldots\epsilon_{n}a_{n}$ at $\mathscr{I}_{\mathbb{C}}$,
where $\\{\epsilon_{j}\\}$ are formal parameters, and extracting the term in
$\mathscr{O}$ proportional to $\epsilon_{1}\ldots\epsilon_{n}$ – see also
Rosly:1996vr ; Selivanov:1998hn  for a related approach. The twistor theory
developed in the previous sections reduces the task of finding such coherent
states from the non-linear equations of motion on $\mathbb{M}$ with boundary
condition at $\mathscr{I}^{+}_{\mathbb{C}}$ to a linear problem on
$\mathbb{PT}$, leading to the construction of all-multiplicity formulae for
the form factor from integrability rather than perturbation theory. The same
strategy can be used for form factors around _any_ self-dual radiative
background, as long as we include the twistor data for the radiative
background together with the momentum eigenstates representatives, and treat
the former non-perturbatively.

For the most part, we restrict ourselves to backgrounds valued in a Cartan
subalgebra $\mathfrak{h}\subseteq\mathfrak{g}$. Although our methods naturally
yield concrete formulae for more general backgrounds, this restriction leads
to simpler formulae in which the background is encoded into abelian factors
obtainable by quadratures, i.e., direct integral formulae. These formulae
still decompose an observable into colour-ordered components, with the
additional information of a set of charges $e^{i}$ of the positive helicity
gluons relative to the background. These are determined by the relation
$[t^{i},T_{j}]=e^{i}T_{j}$, where $\\{t^{i}\\}$ are a basis of $\mathfrak{h}$
and $T_{j}$ the colour of the $j$-th gluon. If we compute observables around
non-Cartan backgrounds, we need to introduce non-abelian $\mathsf{H}$ factors
associated with the background which, although determined as above, are no
longer expressible as quadratures and further interleave the colour-ordered
expression.

When the background is valued in the Cartan subalgebra, we can express the
holomorphic frame $\mathsf{H}=e^{-g}$ explicitly in terms of the Cartan-valued
function $g$ Ward:1977ta ; Adamo:2020yzi

$g(x,\lambda)=\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\lambda\lambda^{\prime}\rangle}\frac{\langle\iota\lambda\rangle}{\langle\iota\lambda^{\prime}\rangle}\left.\mathsf{a}\right|_{X}\,,$
(85)

and further use it to provide integral formulae for the background coupled
fields. For example, the $J$\- and $K$-matrices are given by

$\displaystyle\log J$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\iota\lambda^{\prime}\rangle\langle
o\lambda^{\prime}\rangle}\left.\mathsf{a}\right|_{X}\,,$ (86a) $\displaystyle
K$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\iota\lambda^{\prime}\rangle^{2}}\left.\mathsf{a}\right|_{X}\,.$
(86b)

Similarly, one can further simplify the field strength for linear
perturbations around a Cartan-valued background. Negative-helicity gluons have
a linearized ASD field strength

$b_{j\,\alpha\beta}(x)=\kappa_{j\alpha}\kappa_{j\beta}T_{j}e^{\mathrm{i}k_{j}\cdot
x+e_{j}g(x,\kappa_{j})}\,,$ (87)

whilst positive-helicity gluons have linearized potential

$a_{j\,\alpha\dot{\alpha}}(x)=\frac{\iota_{\alpha}}{\langle\iota
j\rangle}(\tilde{\kappa}_{j\dot{\alpha}}+e_{j}g_{\dot{\alpha}}(x,\kappa_{j}))T_{j}e^{\mathrm{i}k_{j}\cdot
x+e_{j}g(x,\kappa_{j})}\,.$ (88)

In particular, there is a factorization between the colour and kinematical
degrees of freedom (as opposed to the more general structure in (75)), the
background dresses the dotted component of the momentum as

$\kappa_{j\,\alpha}\tilde{K}_{j\,\dot{\alpha}}(x)=\kappa_{j\,\alpha}(\tilde{\kappa}_{j\,\dot{\alpha}}+e_{j}g_{\dot{\alpha}}(x,\kappa_{j}))\,,$
(89)

while leaving invariant the undotted component. This is expected as a
consequence of self-duality. The corresponding linearized SD field strength
for a positive-helicity gluon is

$\tilde{f}_{j\,\dot{\alpha}\dot{\beta}}(x)=\tilde{K}_{j\,\dot{\alpha}}(x)\tilde{K}_{k\,\dot{\beta}}(x)T_{j}e^{\mathrm{i}k_{j}\cdot
x+e_{j}g(x,\kappa_{j}))}-\frac{\mathrm{i}}{\langle\iota
j\rangle}\iota^{\alpha}\partial_{\alpha\dot{\alpha}}g_{\dot{\beta}}(x)T_{j}e^{\mathrm{i}k_{j}\cdot
x+e_{j}g(x,\kappa_{j})}\,.$ (90)

###  5.2 Form factors for powers of $B$ and $\tilde{F}$

We first consider the form factor for $\,\mathrm{tr}\,B^{2}$. At $q=0$, this
is the interaction term in the Chalmers-Siegel action that extends the SDYM
action to the full YM action. Its generating functional is

$\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(k_{i}+k_{j}-q)\cdot
x+e_{i}g(x,\kappa_{i})+e_{j}g(x,\kappa_{j})}\langle
ij\rangle^{4}\,\mathrm{tr}\,(\mathsf{U}_{X}(\kappa_{i},\kappa_{j})T_{i}\mathsf{U}_{X}(\kappa_{j},\kappa_{i})T_{j})\,,$
(91)

where we assumed that the negative-helicity gluons have momenta $k_{i},k_{j}$
and colours $T_{i},T_{j}$. Since the addition of $\,\mathrm{tr}\,B^{2}$ to
$S_{\text{SDYM}}$ yields an action perturbatively equivalent to the ordinary
full Yang-Mills action, the $q\to 0$ limit of the generating functional is
interpreted as the amplitude for the helicity flip of a single negative-
helicity gluon traversing an SD background Mason:2009afn . If the background
connection on twistor space is further taken to be in the form
$\mathsf{a}+\epsilon_{1}a_{1}+\ldots+\epsilon_{n-2}a_{n-2}$, where $a_{i}$ are
twistor representatives for positive-helicity momentum eigenstates, the
2-point amplitude expands into the $n$-point MHV amplitude around the
background defined by $\mathsf{a}$ Adamo:2020yzi , in particular the Parke-
Taylor denominator arises from the perturbation (78) of the propagator on the
line $X$, the integrals over $X$ being saturated by the holomorphic $\delta$
functions of the momentum eigenstates. More generally, the same expansion
allows us to obtain the $\,\mathrm{tr}\,B^{2}$ form factor away from $q=0$

$\mathscr{F}_{\,\mathrm{tr}\,B^{2}}(1^{+},\ldots,i^{-},\ldots,j^{-},\ldots
n;q)=\frac{\langle ij\rangle^{4}}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,.$ (92)

The generalization to the form factor of an arbitrary power
$\,\mathrm{tr}\,B^{k}\coloneqq\,\mathrm{tr}\,\tensor{B}{{}_{\alpha_{1}}^{\alpha_{2}}}\ldots\tensor{B}{{}_{\alpha_{k}}^{\alpha_{1}}}$
is straightforward

$\mathscr{F}_{\,\mathrm{tr}\,B^{k}}(1^{+},\ldots,i_{1}^{-}\ldots,i_{k}^{-},\ldots,n^{+};q)=\frac{(\langle
i_{1}i_{2}\rangle\ldots\langle i_{k}i_{1}\rangle)^{2}}{\langle
12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,.$ (93)

For generic operators containing $\tilde{F}_{\dot{\alpha}\dot{\beta}}$ as
well, the resulting formulae can still be considerably involved, so in the
present section we first consider the form factor for
$\,\mathrm{tr}\,\tilde{F}_{\dot{\alpha}\dot{\beta}}\tilde{F}^{\dot{\alpha}\dot{\beta}}$.
Around the flat background, the tree-level colour-ordered MHV form factor is
Dixon:2004za

$\mathscr{F}_{\,\mathrm{tr}\,\tilde{F}^{2}}(1^{+},\ldots,n^{+};q)=\frac{(q^{2})^{2}}{\langle
12\rangle\ldots\langle n1\rangle}\delta^{4}(Q-q)\,,$ (94)

where $Q=k_{1}+\ldots+k_{n}$ is the sum of the external gluon momenta. If one
interprets the form factor as the amplitude for a massive complex scalar
chirally coupled to the SD field strength, this beautiful formula can be
proved by Berends-Giele recursion Dixon:2004za ; alternatively, the parity-
conjugate form factor
$\mathscr{F}_{\,\mathrm{tr}\,B^{2}}(1^{-},\ldots,n^{-};q)$ can be computed as
the ”maximally-non-MHV” form factor using the MHV formalism Brandhuber:2011tv
. It’s remarkable that such a compact formula exists at all for a maximally
googly form factor. We now show that the simplicity of this form factor is a
consequence of the existence of the $K$-matrix. Expressing the SD field
strength in terms of the $K$-matrix and integrating by parts twice, it is
straightforward to rewrite the Fourier transform of
$\,\mathrm{tr}\,\tilde{F}^{2}$ as

$\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{-\mathrm{i}q\cdot
x}\iota_{\alpha}q^{\alpha\dot{\alpha}}\iota_{\beta}q^{\beta\dot{\beta}}\,\mathrm{tr}\,\mathrm{d}_{\dot{\alpha}}K\,\mathrm{d}_{\dot{\beta}}K\,,$
(95)

and the lifting of this expression to twistor space reads

$\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{-\mathrm{i}q\cdot
x}\iota_{\alpha}q^{\alpha\dot{\alpha}}\iota_{\beta}q^{\beta\dot{\beta}}\\\
\quad\int_{X^{2}}\frac{\mathrm{D}\lambda_{1}\mathrm{D}\lambda_{2}\,\langle\lambda_{1}\lambda_{2}\rangle^{2}}{\langle\iota\lambda_{1}\rangle\langle\iota\lambda_{2}\rangle}\,\mathrm{tr}\,\left(\left.\frac{\partial\mathsf{a}_{1}}{\partial\mu_{1}^{\dot{\alpha}}}\right|_{X}\mathsf{U}_{12}\left.\frac{\partial\mathsf{a}_{2}}{\partial\mu^{\dot{\beta}}_{2}}\right|_{X}\mathsf{U}_{21}\right)\,,$
(96)

where we defined
$\mathsf{U}_{ij}\coloneqq\mathsf{U}_{X}(\lambda_{i},\lambda_{j})$ for the
propagator. Since $\,\mathrm{tr}\,\tilde{F}^{2}$ coincides on-shell with the
topological term $\,\mathrm{tr}\,F\wedge F$, one naïvely expects to write
$\,\mathrm{tr}\,\tilde{F}^{2}$ as the divergence of the Chern-Simons current,
and indeed one can check that the latter simply takes the form

$\mathcal{J}^{\alpha\dot{\alpha}}=\iota^{\alpha}\,\mathrm{tr}\,(\mathrm{d}_{\dot{\beta}}K\,\mathrm{d}^{\dot{\alpha}}\mathrm{d}^{\dot{\beta}}K)\,.$
(97)

From this relation, it’s clear that expressing the gauge field via the
$K$-matrix allows to extract a further total derivative from
$\mathcal{J}^{\alpha\dot{\alpha}}$. Notice also that (96) seemingly depends on
the gauge spinor $\iota_{\alpha}$, but it is actually gauge invariant because
it coincides with the generating functional computed with (62); this latter
expression, although gauge invariant, is more involved and leads to less
compact formulae for the form factors.

We now evaluate the generating functional (96) on a perturbed background,
$\mathsf{a}\to\mathsf{a}+a$. The perturbation $a$ is assumed to be the sum of
$n$ momentum eigenstates (71), and the term in (96) containing precisely $n$
distinct momentum eigenstates is the form factor. Expanding the twistor
propagators with the aid of (78) and decomposing the form factor into colour-
ordered terms, we obtain the expression

$\displaystyle\mathscr{F}=$ $\displaystyle\frac{1}{\langle
12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\iota_{\alpha}q^{\alpha\dot{\alpha}}\iota_{\beta}q^{\beta\dot{\beta}}\sum_{i,j}\Biggl{\\{}\frac{\langle
ij\rangle^{2}}{\langle\iota i\rangle\langle\iota
j\rangle}\tilde{\kappa}_{i\dot{\alpha}}\tilde{\kappa}_{j\dot{\beta}}$ (98)
$\displaystyle-2\int_{X}\frac{\mathrm{D}\lambda}{2\pi\mathrm{i}}\frac{\langle
i-1,i\rangle\langle\lambda j\rangle^{2}\tilde{\kappa}_{j\dot{\beta}}}{\langle
i-1,\lambda\rangle\langle\lambda
i\rangle\langle\iota\lambda\rangle\langle\iota
j\rangle}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}$
$\displaystyle+\int_{X^{2}}\frac{\mathrm{D}\lambda\mathrm{D}\lambda^{\prime}}{(2\pi\mathrm{i})^{2}}\frac{\langle
i-1,i\rangle\langle
j-1,j\rangle\langle\lambda\lambda^{\prime}\rangle^{2}}{\langle
i-1,\lambda\rangle\langle\lambda i\rangle\langle
j-1,\lambda\rangle\langle\lambda
j\rangle\langle\iota\lambda\rangle\langle\iota\lambda^{\prime}\rangle}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}\left.\frac{\partial\mathsf{a}}{\partial{\mu^{\prime}}^{\dot{\beta}}}\right|_{X}\Biggr{\\}}\,.$

In principle, the background twistor connection can give derivative
contributions to the form factor, namely the second and third line in (98).
However, one can use the integral

$\displaystyle\int_{X}\frac{\mathrm{D}\lambda}{2\pi\mathrm{i}}\frac{\lambda_{\alpha}\lambda_{\beta}}{\langle\iota\lambda\rangle\langle\ell-1,\lambda\rangle\langle\lambda\ell\rangle}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}=-\mathrm{i}$
$\displaystyle\left(\frac{\kappa_{\ell-1\,\alpha}\kappa_{\ell-1\,\beta}}{\langle\ell-1,\iota\rangle\langle\ell-1,\ell\rangle}g_{\dot{\alpha}}(x,\kappa_{\ell-1})\right.$
(99)
$\displaystyle+\frac{\kappa_{\ell\,\alpha}\kappa_{\ell\,\beta}}{\langle\ell\iota\rangle\langle\ell,\ell-1\rangle}g_{\dot{\alpha}}(x,\kappa_{\ell})$
$\displaystyle\left.+\frac{\iota_{\alpha}\iota_{\beta}}{\langle\iota,\ell-1\rangle\langle\iota\ell\rangle}g_{\dot{\alpha}}(x,\iota)\right)\,,$

to realize that the terms containing derivatives of the background actually
give a vanishing contribution to the form factor. The final expression for the
colour-ordered form factor around a Cartan-valued self-dual radiative
background is

$\mathscr{F}_{\,\mathrm{tr}\,\tilde{F}^{2}}(1^{+},\ldots,n^{+};q)=\frac{(q\cdot
Q)^{2}}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,,$ (100)

see Appendix B for more details on the derivation. As previously anticipated,
the dependence on the gauge spinor $\iota_{\alpha}$ dropped out and the result
is fully gauge-invariant. Around a non-trivial background, we expect
translations to be broken, and therefore the ”momentum-conserving”
$\delta$-function in (94) is replaced by the residual integral over space-time
in (100). This integral cannot be performed analytically for a generic
background, but for specific, highly-symmetric examples, it is possible to
further simplify it. For example, three of the four integrals can be evaluated
around a self-dual plane wave background Adamo:2017nia ; Adamo:2018mpq ;
Adamo:2019zmk , and we recover three momentum-conserving $\delta$-functions on
the directions along which the gauge field is constant. Furthermore, we note
that for the form factor around the trivial background, $q^{2}=q\cdot Q$ on
the support of the momentum conserving $\delta$ function, so that the form
factor around a non-trivial background can be obtained simply by replacing the
$\delta$ function with the space-time integral in (100) while leaving the
prefactor containing the spinor brackets unaffected.

A similar analysis can be set up for form factors of other polynomials in
$\tilde{F}_{\dot{\alpha}\dot{\beta}}$ and its derivatives. Let us now consider
the cubic operator

$\,\mathrm{tr}\,\tilde{F}^{3}\coloneqq\,\mathrm{tr}\,\tensor{\tilde{F}}{{}_{\dot{\alpha}}^{\dot{\beta}}}\tensor{\tilde{F}}{{}_{\dot{\beta}}^{\dot{\gamma}}}\tensor{\tilde{F}}{{}_{\dot{\gamma}}^{\dot{\alpha}}}\,.$
(101)

Note that this is the unique cubic operator not involving derivatives of the
SD field strength, up to a sign. The lifting of this operator to twistor space
cannot be simplified using the $K$-matrix anymore but can be straightforwardly
obtained using (62) and again the generating functional is the Fourier
transform of such lifting. Since the expression of this lifting is rather
involved, we prefer to display it in Equation (165) in Appendix B. The
perturbative expansion proceeds as before: at arbitrary multiplicity, the
tree-level, colour-ordered MHV form factor around a self-dual, Cartan-valued,
radiative background is

$\displaystyle\mathscr{F}_{\,\mathrm{tr}\,\tilde{F}^{3}}=$
$\displaystyle\frac{1}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\times$ (102)
$\displaystyle\times\Biggl{(}\sum_{i,j,k}\langle ij\rangle\langle jk\rangle
ki\rangle[ij][jk][ki]$ $\displaystyle+3\sum_{i,j,k,\ell}\langle
jk\rangle\langle k\ell\rangle\langle\ell i\rangle[k\ell]([\ell
i][jk]+[ik][\ell j])$ $\displaystyle+3\sum_{i,j,k,\ell,m}\langle
jk\rangle\langle\ell m\rangle\langle mi\rangle\Bigl{(}[mi]([jk][\ell
m]+[j\ell][km])+(i\leftrightarrow j)\Bigr{)}$
$\displaystyle+\sum_{i,j,k,\ell,m,n}\langle jk\rangle\langle\ell
m\rangle\langle ni\rangle\Bigl{(}[jk]([ni][\ell m]+[mi][\ell
n])+(k\leftrightarrow\ell)$ $\displaystyle+(i\leftrightarrow
j)+(i\leftrightarrow j,k\leftrightarrow\ell)\Bigr{)}\Biggr{)}\,.$

In particular, the colour-ordered formulae obtained with our generating
functional match the first few expressions around the trivial background,
namely the minimal form factor

$\mathscr{F}_{\,\mathrm{tr}\,\tilde{F}^{3}}(1^{+},2^{+},3^{+};q)=[12][23][31]\,\delta^{4}(Q-q)\,,$
(103)

and the 4-point form factor Brandhuber:2017bkg ; Koster:2016loo

$\mathscr{F}_{\,\mathrm{tr}\,\tilde{F}^{3}}(1^{+},2^{+},3^{+},4^{+};q)=\frac{[12][23][34][41]}{\langle
12\rangle[21]}\left(1+\frac{[31][4|q|3\rangle}{\langle
23\rangle[32][41]}\right)\delta^{4}(Q-q)+\text{cyclic}\,.$ (104)

###  5.3 Generic MHV form factors

The most remarkable feature of the expressions (93), (100), and (102) is that
the form factor around a non-trivial background is obtained by a simple
dressing of the form factor around the trivial background, namely one only has
to replace the $\delta$ function with the space-time integral in the last line
of (102), while keeping intact the kinematical prefactor. This property is
generic of any form factor of a composite operator, as the lifting of any
polynomial in $\tilde{F}_{\dot{\alpha}\dot{\beta}}$ and its derivatives555The
possible presence of $B_{\alpha\beta}$ fields does not bring in any additional
issue, as we are working in the MHV sector. will be a sum of products of
integrals over $X$ of terms of the form
$\lambda^{\alpha_{1}}_{i}\ldots\lambda^{\alpha_{s}}_{i}\partial_{\mu_{i}^{\dot{\alpha}_{1}}}\ldots\partial_{\mu_{i}^{\dot{\alpha}_{s}}}\mathsf{a}$,
intertwined by propagators $\mathsf{U}_{X}(\lambda_{i},\lambda_{j})$ between
adjacent positions on the sphere $X$ and with some contraction of the spinor
indices. Once we expand such an expression in terms of momentum eigenstates,
the background connection can contribute to the form factor in two distinct
ways: it can either be present only in the holomorphic frames
$\mathsf{H}(x,\lambda_{i})$, or it can potentially give a contribution when
present in a $\mu^{\dot{\alpha}}$ derivative. The first type of contribution
gives a factor of $\exp(e_{j}g(x,\kappa_{j}))$ when the background acts by
conjugation on the $j$th external gluon. Conversely, the second type of
contribution is proportional to (possibly a spacetime derivative of)

$\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda}{\langle
j-1,\lambda\rangle\langle\lambda
j\rangle}\lambda_{\alpha}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}\,,$
(105)

when we consider a colour-ordered form factor. Such a term must be inserted in
any possible position inside the perturbative expansion, and for a colour-
ordered form factor, this means that we must sum over $j$. Moreover, any of
these contributions come together with a partial Parke-Taylor denominator
$1/(\langle 12\rangle\ldots\widehat{\langle j-1,j\rangle}\ldots\langle
n1\rangle)$ from which the factor $1/\langle j-1,j\rangle$ is removed. The
integral (105) can be evaluated for generic external momenta by considering
$\kappa_{j-1,\alpha}$ and $\kappa_{j\,\alpha}$ as a basis of undotted spinors,
and it’s equal to

$\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda}{\langle
j-1,\lambda\rangle\langle\lambda
j\rangle}\lambda_{\alpha}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}=\mathrm{i}\frac{\kappa_{j-1,\alpha}g_{\dot{\alpha}}(x,\kappa_{j-1})-\kappa_{j\,\alpha}g_{\dot{\alpha}}(x,\kappa_{j})}{\langle
j-1,j\rangle}\,.$ (106)

The denominator $\langle j-1,j\rangle$ is precisely the missing factor needed
to reconstruct the Parke-Taylor denominator, so that the residual sum over $j$
is telescopic

$\mathrm{i}\sum_{j=1}^{n}(\kappa_{j-1,\alpha}g_{\dot{\alpha}}(x,\kappa_{j-1})-\kappa_{j\,\alpha}g_{\dot{\alpha}}(x,\kappa_{j}))=0\,.$
(107)

Then the only non-vanishing contribution to the form factor around a non-
trivial background comes from terms where the background is present only in
the holomorphic frames, and this means that the desired form factor coincides
with the one around the trivial background, once we replace the $\delta$
function with the integral

$\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,.$ (108)

###  5.4 MHV super form factors in $\mathcal{N}=4$ SYM

The results from the previous section can be extended to super form factors in
$\mathcal{N}=4$ super-Yang-Mills as well, starting with the space-time
expression of a composite operator, lifting it to twistor space and expanding
around the desired background using the on-shell states (80). As in the pure
Yang-Mills case, any MHV super form factor around a Cartan-valued gluonic
background is obtained by replacing the momentum conserving $\delta$ function
with the by-now usual background-dependent integral. The simplest case is the
super form factor for $\frac{1}{2}\,\mathrm{tr}\,\phi_{ab}\phi_{ab}$, which
can be computed starting from the supersymmetrized $K$-matrix, that is using
the lifting formula

$\frac{1}{2}\,\mathrm{tr}\,\phi_{ab}\phi_{ab}=\frac{1}{2}\int\mathrm{d}^{8}\theta\,\iota_{\alpha}\iota_{\beta}\prod_{\begin{subarray}{c}\gamma\neq\alpha,\beta\\\
c\neq a,b\end{subarray}}\theta^{\gamma
c}\int_{X^{2}}\frac{\mathrm{D}\lambda_{1}\,\mathrm{D}\lambda_{2}\,\langle\lambda_{1}\lambda_{2}\rangle^{2}}{\langle\iota\lambda_{1}\rangle\langle\iota\lambda_{2}\rangle}\,\mathrm{tr}\,\left(\left.\frac{\partial\underline{\mathsf{a}}_{1}}{\partial\chi_{1}^{a}}\right|_{X}\underline{\mathsf{U}}_{12}\left.\frac{\partial\underline{\mathsf{a}}_{2}}{\chi^{b}_{2}}\right|_{X}\underline{\mathsf{U}}_{21}\right)\,,$
(109)

the fermionic integrals being necessary to extract the lowest component of
$\frac{1}{2}\,\mathrm{tr}\,\underline{\phi}_{ab}\underline{\phi}_{ab}$. The
corresponding MHV super form factor around a non-trivial background is given
by

$\mathscr{SF}_{\frac{1}{2}\,\mathrm{tr}\,\phi^{2}}(1,\ldots,n;q)=\frac{1}{4}\frac{(\mathcal{Q}^{2})^{2}}{\langle
12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,,$ (110)

where $\mathcal{Q}_{\alpha
a}=\kappa_{1\,\alpha}\eta_{1\,a}+\ldots+\kappa_{n\,\alpha}\eta_{n\,a}$, as we
show in Appendix B. In the flat-background limit, we recover the standard
expression for the super form factor Brandhuber:2011tv . For generic operators
around purely gluonic backgrounds, the same argument of Section 5.3 holds and
the corresponding form factor can obtained by dressing the form factor around
the trivial background with (108). Conversely, for backgrounds including
fermions and scalars, the grading in the fermionic coordinates prevents having
results as simple as in the purely gluonic case,666Presumably, one could
obtain interesting results by considering super form factors of
supersymmetrized operators, i.e. Fourier transforms over the entire chiral
superspace of matrix elements of a local, composite operator of superfields
between the vacuum and an on-shell superstate. but we can still obtain
reasonably compact formulae. For example, one can consider fermionic space-
time backgrounds: introducing the function

$g_{a}^{(f)}(x,\lambda)=\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\lambda\lambda^{\prime}\rangle}\left.\mathsf{a}_{a}\right|_{X}\,,$
(111)

the super form factor in the presence of a mixed gluonic and fermionic
background is

$\mathscr{SF}_{\frac{1}{2}\,\mathrm{tr}\,\phi^{2}}(1,\ldots,n;q)=\frac{1}{4\langle
12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,(\mathcal{Q}^{\alpha}_{a}\tilde{\mathcal{Q}}_{\alpha
a})^{2}e^{\mathrm{i}(Q-q)\cdot x+\sum_{j}e_{j}g(x,\kappa_{j})}\,,$ (112)

where $\tilde{\mathcal{Q}}_{\alpha
a}=\kappa_{1\,\alpha}\tilde{\eta}_{1\,a}+\ldots+\kappa_{n\,\alpha}\tilde{\eta}_{n\,a}$
is the dressed super-momentum defined in terms of the fermionic background by

$\kappa_{i\,\alpha}\tilde{\eta}_{i\,a}(x)=\kappa_{i\,\alpha}(\eta_{i\,a}+e_{i}g^{(f)}_{a}(x,\kappa_{i}))\,,$
(113)

in complete analogy with (89).

##  6 Discussion

We have extended the twistor framework for constructing form factors by
exploiting the representation of the Ward correspondence at null infinity
Sparling:1990 ; Newman:1978ze  and using asymptotic $\mathscr{I}$-data for
background fields to give formulae on such non-trivial self-dual radiative
backgrounds. We have further extended the $\mathscr{I}$ framework to
incorporate supersymmetry. This presentation links directly into the celestial
and twisted holography programmes. In particular, we gave a novel proof of the
MHV form factor for $\,\mathrm{tr}\,\tilde{F}^{2}$ extending it to the formula
(100) for the same form factor, but now evaluated on a general Cartan-valued,
self-dual, radiative background. The framework is set up so that, in
principle, we can compute the tree-level MHV form factor for an arbitrary
composite operator of the field strength and the MHV super form factor for an
arbitrary composite operator in $\mathcal{N}=4$ SYM, but for general operators
we don’t expect to have such simple expressions, as exemplified by the form
factor of $\,\mathrm{tr}\,\tilde{F}^{3}$. It’s nevertheless remarkable that
around any self-dual background, tree-level MHV form factors can be obtained
by a simple dressing via a single, residual space-time integral encoding the
details of the background, without any modification of the rational prefactor
depending on the spinor brackets. This is to be contrasted with the expected
complexity of the result, which should naïvely consist of $n-2$ space-time
integrals for an observables at $n$ points.

###  6.1 NkMHV observables

The main restriction in this work was the restriction to tree-level MHV form
factors. To go to higher MHV degrees or loop form factors on non-trivial
backgrounds, we must incorporate twistor space propagators on such non-trivial
backgrounds – recall that single insertion raises the MHV degree by one at
tree-level and each integration reduces the MHV degree by two. The main issue
is the absence of a known, compact expression for the propagator around a
radiative background: at the abstract level, twistor space propagators have
been studied for many years, see for example Atiyah:1981ey . More recently, in
the context of twistor actions expressions for the twistor space propagators
have been found in Mason:2010yk ; Adamo:2011cb ; Adamo:2011pv  in a gauge that
induces the axial gauge used in the MHV formalism Cachazo:2004kj ;
Boels:2007qn  when represented in momentum space. This propagator was used for
form factors and correlators in Chicherin:2014uca ; Koster:2016ebi ;
Koster:2016fna ; Koster:2016loo ; Chicherin:2016fac ; Chicherin:2016soh ;
Chicherin:2016fbj ; Chicherin:2016ybl . A twistor space propagator
$\Delta_{\mathsf{a}}(Z,Z^{\prime})$ on a background defined by a Lie algebra-
valued $(0,1)$-form $\mathsf{a}$ must satisfy the defining relation

$(\bar{\partial}_{0}+\mathsf{a})\Delta_{\mathsf{a}}(Z,Z^{\prime})=\bar{\delta}^{3}(Z,Z^{\prime})$
(114)

This can be solved in Euclidean signature by starting from the MHV propagator
$\Delta_{0}(Z,Z^{\prime})=\bar{\delta}^{2|4}(Z,Z^{\prime},Z_{*})$ around the
trivial background as in Mason:2010yk ; Adamo:2011cb ; Adamo:2011pv , where
$Z_{*}$ is the chosen reference twistor, and conjugating with the holomorphic
frame $\mathsf{H}(Z)$ for the line joining $(Z,Z^{\prime})$. This will give
the expression

$\Delta_{\mathsf{a}}(Z,Z^{\prime})=\mathsf{H}(Z,Z_{*})\bar{\delta}^{2|4}(Z,Z^{\prime},Z_{*})\mathsf{H}(Z^{\prime},Z_{*})^{-1}$
(115)

where $\mathsf{H}(z,Z_{*})$ satisfies
$(\bar{\partial}_{0}+\mathsf{a})\mathsf{H}(Z,Z_{*})=0$ on the line joining $Z$
to $Z_{*}$. This then satisfies the defining relations above as a consequence
of (55) on the support of the delta functions. The fact that
$\Delta_{\mathsf{a}}(Z,Z^{\prime})$ has four delta functions in its definition
should then lead to easy integrations at higher MHV degrees, and even to find
formulae for loop integrands Lipstein:2012vs , although carrying out the loop
integrals is possible but subtle, see Lipstein:2013xra . With these, one
should be able to compute both amplitudes and form factors at higher MHV
degrees on a background in pretty much the same way as in Adamo:2011cb .

###  6.2 The one-loop all plus amplitude

This framework can be used to give new insights into loop amplitudes. The
simplest case is the all-plus one-loop amplitude in pure Yang-Mills and we can
justify old generating function formulae, extending them to a nontrivial
background, and to understand the more recent dual-conformal invariant
formulae of Henn:2019mvc ; Chicherin:2022bov .

The vanishing of the tree-level amplitude ensures that the one-loop amplitude
around the trivial background is finite and a rational function of the spinor
brackets. There are a number of versions of the all-multiplicity formula for
this Bern:1993qk ; Mahlon:1993si ; Bern:1994ju , for example, ignoring
constant prefactors

$\mathcal{A}^{\text{1-loop}}=\sum_{i<j<s<t}\frac{\langle ij\rangle[js]\langle
st\rangle[ti]}{\langle 12\rangle\ldots\langle n1\rangle}\,.$ (116)

The following generating functional, originally due to one of us, was briefly
quoted without explanation in Boels:2007gv  as

$\int_{\mathbb{M}}\mathrm{d}^{4}x\int_{X^{2}}\mathrm{D}\lambda_{1}\mathrm{D}\lambda_{2}\,\mathrm{tr}\,(\partial_{\alpha}^{\dot{\alpha}}\mathsf{U}_{21}\partial^{\alpha}_{\dot{\beta}}\left.\mathsf{a}_{1}\right|_{X}\partial^{\dot{\beta}}_{\beta}\mathsf{U}_{12}\partial^{\beta}_{\dot{\alpha}}\left.\mathsf{a}_{2}\right|_{X})\,.$
(117)

We can express the derivative of the propagator as the variation under a
translation using (77)

$\partial_{\alpha\dot{\alpha}}\mathsf{U}_{X}(\lambda_{i},\lambda_{j})=-\int_{X}\mathrm{D}\lambda\mathsf{U}_{X}(\lambda_{i},\lambda)\partial_{\alpha\dot{\alpha}}\left.\mathsf{a}\right|_{X}\mathsf{U}_{X}(\lambda,\lambda_{j})\,.$
(118)

With this, the generating functional becomes

$\int_{\mathbb{M}}\mathrm{d}^{4}x\int_{X^{4}}\prod_{i=1}^{4}\mathrm{D}\lambda_{i}\,\langle\lambda_{1}\lambda_{2}\rangle\langle\lambda_{3}\lambda_{4}\rangle\,\mathrm{tr}\,(\partial_{{\dot{\alpha}}}\mathsf{a}_{1}\,\mathsf{U}_{12}\,\partial^{{\dot{\beta}}}\mathsf{a}_{2}\,\mathsf{U}_{23}\,\partial_{{\dot{\beta}}}\mathsf{a}_{3}\,\mathsf{U}_{34}\,\partial^{{\dot{\alpha}}}\mathsf{a}_{4}\,\mathsf{U}_{41})\,.$
(119)

This is more clearly a background-coupled 4-vertex generating amplitudes with
at least four external legs. As recently observed in Bu:2022dis , it clearly
gives (116) at any multiplicity when expanded around the trivial background,
since the angle brackets (respectively, $\mu^{\dot{\alpha}}$ derivatives) give
the correct angle brackets (square brackets) in the numerator, whilst the
expansion of the propagators as in (78) reproduces the Parke-Taylor
denominator. The same expansion around a Cartan-valued background yields

$\mathcal{A}^{\text{1-loop}}_{\text{background}}=\sum_{i<j<s<t}\frac{\langle
ij\rangle[js]\langle st\rangle[ti]}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,\exp\left(\sum_{j=1}^{n}(\mathrm{i}k_{j}\cdot
x+e_{j}g(x,\kappa_{j}))\right)\,,$ (120)

in agreement with our results on form factors, as well as with the previously
known results on gluon amplitudes Adamo:2020yzi .

In a different direction, the two-point generating function (117) neatly ties
into region momenta formulae, and, in particular, the observation of
Henn:2019mvc  that the 1-loop all plus rational amplitude can be derived from
the 1-loop maximally supersymmetric MHV 1-loop integrand in region momentum
space where the region momentum for the loop is placed at infinity. We can see
this in a slightly different way here. We first introduce the region momenta
$y_{i}$’s by

$y_{ij}=y_{i}-y_{j}=\sum_{i\leq s<j}k_{s}\,.$ (121)

We can then expand $\partial_{\alpha\dot{\alpha}}\mathsf{U}_{ij}$ in momentum
eigenstates, using (118) to obtain, for a given cyclic ordering, region
momenta appearing at the $(j-i)$-th order in the form

$\partial^{\alpha\dot{\alpha}}\mathsf{U}_{ij}=\frac{y_{i\,j+1}^{\alpha\dot{\alpha}}}{\langle
ii+1\rangle\ldots\langle
j-1j\rangle}\,,\qquad\partial^{\alpha\dot{\alpha}}\partial^{\beta\dot{\beta}}\mathsf{U}_{ij}=\frac{y_{i\,j+1}^{\alpha\dot{\alpha}}y_{i\,j+1}^{\beta\dot{\beta}}}{\langle
ii+1\rangle\ldots\langle j-1j\rangle}\,.$ (122)

Using the first, we see that our generating function (117) expands to give

$\sum_{i<j}\frac{\langle i|y_{i\,j+1}|j]\langle j|y_{j+1\,i}|i]}{\langle
12\rangle\langle 23\rangle\ldots\langle n1\rangle}\,.$ (123)

and, using, for example, §6 of Bullimore:2010pj , this can be recognised to be
the sum over $2$-mass easy and $1$-mass boxes Bern:1994zx  written in terms of
region momenta, with the loop insertion point $x_{0}$ –represented by the line
$(A,B)$ in Bullimore:2010pj – placed at infinity. Of course, this formula
could be seen directly by summing the $j$ and $t$ in (116) to give region
momenta.

An alternative version of the one-loop amplitude was observed in Henn:2019mvc

$\mathcal{A}^{\text{1-loop}}=\sum_{i<j}\frac{\langle
1|y_{1i}y_{ij}|1\rangle^{2}}{\langle 12\rangle\ldots\langle
n1\rangle}\frac{\langle i-1,i\rangle\langle j-1,j\rangle}{\langle
1,i-1\rangle\langle 1i\rangle\langle 1,j-1\rangle\langle 1j\rangle}\,.$ (124)

As explained in Chicherin:2022bov , this is the so-called _Kermit_ formula for
the supersymmetric MHV loop integrand, with the loop integrand region momentum
placed at infinity. This is the form obtained from the all-loop BCFW recursion
of ArkaniHamed:2010kv  and also the MHV-diagram version of Bullimore:2010pj ;
in the former, leg 1 is the direction of the BCFW shift, whereas, in the
latter, it can be taken to be an arbitrary reference twistor associated to the
reference spinor of the MHV formalism. In Bullimore:2010pj , a detailed
argument was given, based on the calculations in Brandhuber:2004yw  to relate
this to the more familiar 2-mass-easy boxes of Bern:1994ju . At the level of
the loop integrand (here again with the loop insertion point placed at
infinity) this follows from the algebraic relation

$\displaystyle\mathcal{A}^{\text{1-loop}}$
$\displaystyle=\sum_{i<j}\frac{1}{\langle 12\rangle\ldots\langle
n1\rangle}\left(\frac{\langle 1|y_{1j}y_{ji}|i\rangle\langle
1|y_{1i}y_{ij}|j\rangle}{\langle 1i\rangle\langle 1j\rangle}-\frac{\langle
1|y_{1j}y_{j,i-1}|i-1\rangle\langle 1|y_{1i}y_{ij}|j\rangle}{\langle
1,i-1\rangle\langle 1j\rangle}\right.$ (125)
$\displaystyle\quad\left.-\frac{\langle 1|y_{1j}y_{ji}|i\rangle\langle
1|y_{1i}y_{i,j-1}|j-1\rangle}{\langle 1i\rangle\langle
1,j-1\rangle}+\frac{\langle 1|y_{1j}y_{j,i-1}|i-1\rangle\langle
1|y_{1i}y_{i,j-1}|j-1\rangle}{\langle 1,i-1\rangle\langle
1,j-1\rangle}\right)$

so it should be possible to write a two-point generating functional for this
form of the amplitude, along the lines of (117). However, this generating
functional is unlikely to be local on space-time, as the summands for this
version of the one-loop amplitude contain spurious poles (the poles cancel in
the sum though, as they should Henn:2019mvc ).

An open question is how to construct these generating functionals as local
expressions in the self-dual Yang-Mills potentials. The central puzzle is that
(119) contains four derivatives of the twistor connection $\mathsf{a}$, so it
corresponds to a local space-time expression containing two derivatives on the
space-time gauge field. Gauge invariance and self-duality restrict the
possible terms to the integral of $\,\mathrm{tr}\,\tilde{F}^{2}$, but this
integral vanishes, as one can infer from the $q\to 0$ limit of the form factor
we presented in this work 777Alternatively, the radiative backgrounds we are
considering are topologically trivial, so the winding number vanishes.. A more
conservative approach towards a first-principle derivation of (119) could be a
computation of a one-loop determinant around the background gauge field, but
at the moment only the _variation_ of the determinant seems to be easy to
compute; it’s however worth mentioning that a one-loop determinant is
generically non-local, but its variation becomes local precisely around self-
dual backgrounds Brown:1978yj ; Corrigan:1979di . It would also be nice to
reproduce the chiral algebra computations of Costello:2022wso  directly from a
space-time perspective, without referring to the 2d conformal blocks; indeed,
the Green-Schwarz mechanism for the anomaly cancellation on twistor space
Bardeen:1995gk ; Costello:2021bah  suggests that the all-plus one-loop
amplitude could be computed using tree diagrams only, as showed in
Monteiro:2022nqt  for the parity-conserving terms of the amplitude. We hope to
return to these issues in future work.

#### Acknowledgements:

We thank Atul Sharma and Arthur Lipstein for useful discussions and Atul
Sharma for providing feedback on the draft. GB is supported by a joint
Clarendon Fund and Merton College Mathematics Scholarship. LJM is grateful to
the IHES and ENS Paris for hospitality while this work was progressing and to
the STFC for support under grant ST/T000864/1.

##  Appendix A $\mathcal{N}=4$ constraint equations on chiral superspace

It is well known that there exist an equivalence between the field equations
for the super-fields and constraint equations for the super-connections. The
precise statement depend on the superspace one is considering: the $d=10$,
$\mathcal{N}=1$ SYM equations of motion are equivalent to the constraint
equation Witten:1978xx ; Harnad:1985bc

$\\{\underline{\nabla}_{\mathcal{A}},\underline{\nabla}_{\mathcal{B}}\\}=2\gamma^{M}_{\mathcal{A}\mathcal{B}}\underline{\nabla}_{\mu}\,,$
(126)

where $(x^{M},\theta^{\mathcal{A}})$ are coordinates on $\mathbb{C}^{10|16}$.
When compactified to 4 dimensions, this equivalence becomes an equivalence
between the constraint equations and the $\mathcal{N}=4$ equations of motion
on non-chiral superspace $\mathbb{C}^{4|16}$ with coordinates
$(x^{\alpha\dot{\alpha}},\theta^{a\alpha},\bar{\theta}^{\dot{\alpha}}_{a})$.
If one further wants to reduce to chiral superspace, additional conditions are
required Devchand:1996gv ; Adamo:2013cra : in this Appendix, we show the
following result

###### Theorem 3

There is an equivalence between

  1. 1.

the constraint equations on chiral superspace

$\displaystyle[\underline{\nabla}_{a(\alpha},\underline{\nabla}_{\beta)B}\\}$
$\displaystyle=$ $\displaystyle 0\,,$ (127a)
$\displaystyle\underline{F}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\lambda\underline{B}_{\alpha\beta}\,,$ (127b)

where $\underline{F}_{\alpha\beta}$ is the ASD component of the supercurvature
$[\underline{\nabla}_{\alpha\dot{\alpha}},\underline{\nabla}_{\beta\dot{\beta}}]$,
$\underline{B}_{\alpha\beta}$ will be determined below by the constraint
equations (127a), and $\lambda$ is the ’t Hooft coupling,

  2. 2.

super-fields
$(\underline{A}_{\alpha\dot{\alpha}},\underline{\tilde{\psi}}_{a\dot{\alpha}},\underline{\phi}_{ab},\underline{\psi}^{\alpha}_{a},\underline{B}_{\alpha\beta})$
on chiral superspace satysfing

$\displaystyle\underline{\nabla}_{a\alpha}\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}$
$\displaystyle=$
$\displaystyle\underline{\nabla}_{\alpha(\dot{\alpha}}\underline{\tilde{\psi}}_{\dot{\beta})a}\,,$
(128a)
$\displaystyle\underline{\nabla}_{a\alpha}\underline{\tilde{\psi}}_{\dot{\alpha}b}$
$\displaystyle=$ $\displaystyle
2\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\phi}_{ab}\,,$ (128b)
$\displaystyle\underline{\nabla}_{a\alpha}\underline{\phi}_{bc}$
$\displaystyle=$
$\displaystyle\epsilon_{abcd}\underline{\psi}^{d}_{\alpha}\,,$ (128c)
$\displaystyle\underline{\nabla}_{a\alpha}\underline{\psi}^{b}_{\beta}$
$\displaystyle=$
$\displaystyle-\varepsilon_{\alpha\beta}[\underline{\phi}_{ac},\underline{\phi}^{bc}]+\delta^{b}_{a}\underline{B}_{\alpha\beta}\,,$
(128d) $\displaystyle\underline{\nabla}_{a\alpha}\underline{B}_{\alpha\beta}$
$\displaystyle=$
$\displaystyle-2\varepsilon_{\alpha(\beta}[\underline{\psi}^{b}_{\gamma)},\underline{\phi}_{ab}]\,,$
(128e) $\displaystyle\underline{F}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\lambda\underline{B}_{\alpha\beta}\,,$ (128f)

which in turn imply the $\mathcal{N}=4$ super-field equations of motion

$\displaystyle\underline{\nabla}_{\dot{\alpha}}^{\beta}\underline{B}_{\alpha\beta}$
$\displaystyle=$
$\displaystyle-\\{\underline{\tilde{\psi}}_{a\dot{\alpha}},\underline{\psi}^{a}_{\alpha}\\}-\frac{1}{2}[\underline{\phi}_{ab},\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\phi}^{ab}]\,,$
(129a)
$\displaystyle\underline{\nabla}_{\alpha}^{\dot{\alpha}}\underline{\tilde{\psi}}_{a\dot{\alpha}}$
$\displaystyle=$ $\displaystyle
2\lambda[\underline{\psi}^{b}_{\alpha},\underline{\phi}_{ab}]\,,$ (129b)
$\displaystyle\underline{\square}\underline{\phi}_{ab}$ $\displaystyle=$
$\displaystyle\\{\underline{\tilde{\psi}}_{\dot{\alpha}[b},\underline{\tilde{\psi}}^{\dot{\alpha}}_{a]}\\}+\lambda\epsilon_{abcd}\\{\underline{\psi}^{c}_{\alpha},\underline{\psi}^{d\alpha}\\}+2\lambda[\underline{\phi}_{c[a},[\underline{\phi}^{cd},\underline{\phi}_{b]d}]]\,,$
(129c)
$\displaystyle\underline{\nabla}_{\dot{\alpha}}^{\alpha}\underline{\psi}^{a}_{\alpha}$
$\displaystyle=$
$\displaystyle[\underline{\tilde{\psi}}_{b\dot{\alpha}},\underline{\phi}^{ab}]\,,$
(129d) $\displaystyle\underline{F}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\lambda\underline{B}_{\alpha\beta}\,,$ (129e)

  3. 3.

component fields
$(A_{\alpha\dot{\alpha}},\tilde{\psi}_{a\dot{\alpha}},\phi_{ab},\psi^{\alpha}_{a},B_{\alpha\beta})$
on space-time satisfying the $\mathcal{N}=4$ equations of motion – that is the
$\theta^{\alpha a}=0$ truncation of the super-field equations of motion–,
supplemented by the recursion relations (147a), (147b), (147c), (147d),
(147e), (147f), (147g), and (147h).

Our result is therefore a deformation away from self-duality of the results in
Devchand:1996gv  for $\mathcal{N}=4$ and reduces to that construction in the
$\lambda=0$ limit, and is more self-contained compared to the result of
Adamo:2013cra .

For the 1. $\Rightarrow$ 2\. implication, we begin with the construction of
the various super-fields appearing in the equations of motion, slightly
varying the normalizations from the main body of the paper. The super-
connection is taken to be $\underline{\nabla}_{\alpha A}=\partial_{\alpha
A}+\underline{A}_{\alpha A}$. We define the supercurvature components as
implied by the constraint equations

$\displaystyle[\underline{\nabla}_{\alpha\dot{\alpha}},\underline{\nabla}_{\beta\dot{\beta}}]$
$\displaystyle\coloneqq$
$\displaystyle\varepsilon_{\alpha\beta}\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}+\varepsilon_{\dot{\alpha}\dot{\beta}}\underline{F}_{\alpha\beta}\,,$
(130a) $\displaystyle{[\underline{\nabla}_{\alpha
a},\underline{\nabla}_{\beta\dot{\alpha}}]}$ $\displaystyle\coloneqq$
$\displaystyle\varepsilon_{\alpha\beta}\underline{\tilde{\psi}}_{\dot{\alpha}a}\,,$
(130b) $\displaystyle\\{\underline{\nabla}_{\alpha
a},\underline{\nabla}_{\beta b}\\}$ $\displaystyle\coloneqq$ $\displaystyle
2\varepsilon_{\alpha\beta}\underline{\phi}_{ab}\,,$ (130c)

as well as

$\underline{\phi}^{ab}\coloneqq\frac{1}{2}\epsilon^{abcd}\underline{\phi}_{cd}\,.$
(131)

The remaining super-fields are defined by

$\displaystyle\underline{\psi}^{a}_{\alpha}$ $\displaystyle\coloneqq$
$\displaystyle-\frac{1}{3!}\epsilon^{abcd}\underline{\nabla}_{\alpha
b}\underline{\phi}_{cd}\,,$ (132a) $\displaystyle\underline{B}_{\alpha\beta}$
$\displaystyle\coloneqq$
$\displaystyle\frac{1}{4}\underline{\nabla}_{a(\alpha}\psi^{a}_{\beta)}\,.$
(132b)

The definitions for $\underline{\psi}^{a}_{\alpha}$
$\underline{B}_{\alpha\beta}$ can be interpreted as arising from Jacobi
identities for the super-connection. For example, the identity for
$\underline{\nabla}_{\alpha a}$, $\underline{\nabla}_{\beta b}$, and
$\underline{\nabla}_{\gamma c}$ implies

$\underline{\nabla}_{\alpha
a}\underline{\phi}_{bc}=\underline{\nabla}_{\alpha[a}\phi_{bc]}\,.$ (133)

Similarly, the constraint equation (130c) implies that the super-field

$\underline{\nabla}_{a(\alpha}\underline{\nabla}_{\beta)b}\phi_{cd}\,,$ (134)

is totally skew in the $\,\mathrm{SU}(4)$ indices. We complete our definitions
by introducing the ’t Hooft coupling $\lambda$ and by requiring

$\underline{F}_{\alpha\beta}=\lambda\underline{B}_{\alpha\beta}\,,$ (135)

for the ASD part of the bosonic supercurvature.

With these definitions, we can derive the $\mathcal{N}=4$ equations of motion
as follows: we first consider the equations of motion for
$\underline{\psi}^{a}_{\alpha}$ and $\underline{B}_{\alpha\beta}$ and notice
that the Jacobi identity for $\underline{\nabla}_{\alpha\dot{\alpha}}$,
$\underline{\nabla}_{\beta b}$, and $\underline{\nabla}_{\gamma c}$ can be
written as

$2\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\phi}_{ab}=\underline{\nabla}_{\alpha
a}\underline{\tilde{\psi}}_{\dot{\alpha}b}\,,$ (136)

while the skew part of $\nabla_{\alpha
a}\underline{\tilde{\psi}}_{b\dot{\alpha}}$ in $a$, $b$ vanishes. Using
(130b), (130c) and (136), we reproduce (129d)

$\underline{\nabla}_{\dot{\alpha}}^{\alpha}\underline{\psi}_{\alpha}^{a}=[\underline{\tilde{\psi}}_{\dot{\alpha}b},\phi^{ab}]\,.$
(137)

In the same way, we can use (130c) to obtain for the fermionic derivative
acting of $\underline{\psi}^{a}_{\alpha}$

$\underline{\nabla}_{a\alpha}\underline{\psi}^{b}_{\beta}=-\varepsilon_{\alpha\beta}[\underline{\phi}_{ac},\underline{\phi}^{bc}]+\delta^{b}_{a}\underline{B}_{\alpha\beta}\,.$
(138)

From (130b), (136), and (138) one can straightforwardly show that

$\underline{\nabla}^{\beta}_{\dot{\alpha}}\underline{B}_{\alpha\beta}=-\\{\underline{\tilde{\psi}}_{a\dot{\alpha}},\underline{\psi}^{a}_{\alpha}\\}-\frac{1}{2}[\underline{\phi}_{ab},\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\phi}^{ab}]\,,$
(139)

thus reproducing (129a). The fermionic derivative of
$\underline{B}_{\alpha\beta}$ be obtained via (130c) and reads

$\underline{\nabla}_{a\alpha}\underline{B}_{\beta\gamma}=-2\varepsilon_{\alpha(\beta}[\underline{\psi}^{b}_{\gamma)},\underline{\phi}_{ab}]\,,$
(140)

We can finally derive the equations of motion for the positive-helicity
spinors and the scalars. Starting with the spinors, the Jacobi identity for
$\underline{\nabla}_{\alpha a}$, $\underline{\nabla}_{\beta\dot{\beta}}$ and
$\underline{\nabla}_{\gamma\dot{\gamma}}$ can be written as

$\displaystyle\underline{\nabla}_{a\alpha}\underline{F}_{\alpha\beta}$
$\displaystyle=$
$\displaystyle-\epsilon_{\alpha(\beta}\underline{\nabla}^{\dot{\alpha}}_{\gamma)}\underline{\tilde{\psi}}_{\dot{\alpha}a}\,,$
(141a)
$\displaystyle\underline{\nabla}_{a\alpha}\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}$
$\displaystyle=$
$\displaystyle\underline{\nabla}_{\alpha(\dot{\alpha}}\underline{\tilde{\psi}}_{\dot{\beta})a}\,,$
(141b)

so that (127b) and (140) directly give (129b)

$\underline{\nabla}_{\alpha}^{\dot{\alpha}}\underline{\tilde{\psi}}_{a\dot{\alpha}}=2\lambda[\underline{\psi}^{b}_{\alpha},\underline{\phi}_{ab}]\,.$
(142)

Similarly, acting with $\underline{\nabla}^{\alpha\dot{\alpha}}$ on (136) and
using (130b) we obtain

$\underline{\square}\phi_{ab}=\\{\underline{\tilde{\psi}}_{\dot{\alpha}[b},\underline{\tilde{\psi}}^{\dot{\alpha}}_{a]}\\}+\frac{1}{2}\underline{\nabla}_{\alpha
a}\underline{\nabla}^{\alpha\dot{\alpha}}\tilde{\psi}_{\dot{\alpha}b}\,,$
(143)

so that (138) gives

$\underline{\square}\underline{\phi}_{ab}=\\{\underline{\tilde{\psi}}_{\dot{\alpha}[b},\underline{\tilde{\psi}}^{\dot{\alpha}}_{a]}\\}+\lambda\epsilon_{abcd}\\{\underline{\psi}^{c}_{\alpha},\underline{\psi}^{d\alpha}\\}+2\lambda[\underline{\phi}_{c[a},[\underline{\phi}^{cd},\underline{\phi}_{b]d}]]\,.$
(144)

To prove the remaining implications, we adopt essentially the strategy
developed in Harnad:1985bc ; Devchand:1996gv , to which we refer for more
details. We partially fix the gauge on chiral superspace by going to the
radial gauge

$\mathcal{D}\underline{A}_{\alpha a}=0\,,$ (145)

where $\mathcal{D}$ is the Euler vector field along the fermionic directions

$\mathcal{D}\coloneqq\theta^{a\alpha}\partial_{a\alpha}\,.$ (146)

The residual gauge invariance corresponds to gauge transformations on chiral
superspace that are independent of the fermionic variables, i.e. to ordinary
gauge transformations on space-time. Moreover, in this gauge the equations
(128a), (128b), (128c), (128d), and (128e) readily imply the recursion
relations

$\displaystyle\mathcal{D}\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}$
$\displaystyle=$
$\displaystyle\theta^{a\alpha}\underline{\nabla}_{\alpha(\dot{\alpha}}\underline{\tilde{\psi}}_{\dot{\beta})a}\,,$
(147a) $\displaystyle\mathcal{D}\underline{\tilde{\psi}}_{a\dot{\alpha}}$
$\displaystyle=$
$\displaystyle-2\theta^{b\alpha}\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\phi}_{ab}\,,$
(147b) $\displaystyle\mathcal{D}\underline{\phi}_{ab}$ $\displaystyle=$
$\displaystyle\epsilon_{abcd}\theta^{c}_{\alpha}\underline{\psi}^{d\alpha}\,,$
(147c) $\displaystyle\mathcal{D}\underline{\psi}_{\alpha}^{a}$
$\displaystyle=$
$\displaystyle\theta^{a\beta}\underline{B}_{\alpha\beta}-\theta^{b}_{\alpha}[\underline{\phi}_{bc},\underline{\phi}^{ac}]\,,$
(147d) $\displaystyle\mathcal{D}\underline{B}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-2\theta^{a}_{(\alpha}[\underline{\psi}^{b}_{\beta)},\underline{\phi}_{ab}]\,,$
(147e) $\displaystyle\mathcal{D}\underline{F}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\lambda\mathcal{D}\underline{B}_{\alpha\beta}\,,$ (147f)
$\displaystyle\mathcal{D}\underline{A}_{\alpha\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\theta^{a}_{\alpha}\underline{\tilde{\psi}}_{a\dot{\alpha}}\,,$
(147g) $\displaystyle(1+\mathcal{D})\underline{A}_{a\alpha}$ $\displaystyle=$
$\displaystyle 2\theta^{b}_{\alpha}\underline{\phi}_{ba}\,.$ (147h)

These relations defines uniquely the super-fields in terms of their lowest
component fields, since the RHSs are all linear in the fermionic variables and
since a homogeneous polynomial in the fermionic variables is an eigenstate of
$\mathcal{D}$, the eigenvalue being the degree of the polynomial. In
particular, in this gauge the lowest terms in the $\theta$ expansion are

$\displaystyle\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}$
$\displaystyle=$
$\displaystyle\tilde{F}_{\dot{\alpha}\dot{\beta}}+\theta^{a\alpha}\nabla_{\alpha(\dot{\alpha}}\tilde{\psi}_{\dot{\beta})a}$
(148a)
$\displaystyle+\frac{1}{2}\theta^{a\alpha}\theta^{b\beta}(\nabla_{\alpha(\dot{\alpha}}\nabla_{\dot{\beta})\beta}\phi_{ab})+\varepsilon_{\beta\alpha}\\{\tilde{\psi}_{b(\dot{\alpha}},\tilde{\psi}_{\dot{\beta})a}\\})+\mathcal{O}(\theta^{3})\,,$
$\displaystyle\underline{\tilde{\psi}}_{a\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\tilde{\psi}_{a\dot{\alpha}}-2\theta^{b\alpha}\nabla_{\alpha\dot{\alpha}}\phi_{ab}-\theta^{b\beta}\theta^{c\gamma}(\epsilon_{abcd}\nabla_{\beta\dot{\alpha}}\psi^{d\beta}+\varepsilon_{\gamma\beta}[\tilde{\psi}_{c\dot{\alpha}},\phi_{ab}])+\mathcal{O}(\theta^{3})\,,$
(148b) $\displaystyle\underline{\phi}_{ab}$ $\displaystyle=$
$\displaystyle\phi_{ab}+\epsilon_{abcd}\theta^{c}_{\alpha}\psi^{d\alpha}+\frac{1}{2}\epsilon_{abcd}\theta^{c}_{\alpha}\theta^{e\beta}(\delta^{d}_{e}B^{\alpha}_{\beta}-\delta^{\alpha}_{\beta}[\phi_{ef},\phi^{df}])+\mathcal{O}(\theta^{3})\,,$
(148c) $\displaystyle\underline{\psi}^{a}_{\alpha}$ $\displaystyle=$
$\displaystyle\psi^{a}_{\alpha}+\theta^{b\beta}(\delta_{b}^{a}B_{\alpha\beta}-\varepsilon_{\beta\alpha}[\phi_{bc},\phi^{ac}])-\frac{1}{2}\theta^{a\beta}\theta^{b}_{(\alpha}[\psi^{c}_{\beta)},\phi_{bc}]$
(148d)
$\displaystyle-\frac{1}{2}\theta^{b}_{\alpha}\theta^{f}_{\beta}(2[\phi_{bc},\delta^{[a}_{f}\psi^{c]\beta}]-\epsilon_{bcfe}[\phi^{ac},\psi^{e\beta}])+\mathcal{O}(\theta^{3})\,,$
$\displaystyle\underline{B}_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
B_{\alpha\beta}-2\theta^{a}_{(\alpha}[\psi^{b}_{\beta)},\phi_{ab}]$ (148e)
$\displaystyle-\theta^{a}_{(\alpha}\theta^{c\gamma}(\delta^{b}_{c}[B_{\gamma\beta},\phi_{ab}]-\varepsilon_{\gamma\beta}[[\phi_{cd},\phi^{bc}],\phi_{ab}]-\\{\psi^{b}_{\beta)},\psi^{d}_{\gamma}\\})+\mathcal{O}(\theta^{3})\,,$
$\displaystyle\underline{A}_{\alpha\dot{\alpha}}$ $\displaystyle=$
$\displaystyle
A_{\alpha\dot{\alpha}}+\theta^{a}_{\alpha}\tilde{\psi}_{a\dot{\alpha}}-\theta^{a}_{\alpha}\theta^{b\beta}\nabla_{\beta\dot{\alpha}}\phi_{ab}+\mathcal{O}(\theta^{3})\,,$
(148f) $\displaystyle\underline{A}_{a\alpha}$ $\displaystyle=$
$\displaystyle\theta^{b}_{\alpha}\phi_{ba}+\frac{1}{3}\theta^{b}_{\alpha}\theta^{c}_{\beta}\epsilon_{abcd}\psi^{d\beta}+\mathcal{O}(\theta^{3})\,.$
(148g)

Conversely, if we assume that we have super-fields defined by the recursion
relations and the lowest-component fields, and if we assume that the
$\mathcal{N}=4$ equations of motion holds for the lowest-component fields,
then the super-fields relations (128a), (128b), (128c), (128d), (128e), (128f)
can be obtained by induction on the fermionic degree via the recursion
relations

$\displaystyle(1+\mathcal{D})(\underline{\nabla}_{a\alpha}\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}-\underline{\nabla}_{\alpha(\dot{\alpha}}\underline{\tilde{\psi}}_{\dot{\beta})a}$
$\displaystyle=$
$\displaystyle-\theta^{b\beta}\underline{\nabla}_{a\alpha}\underline{\nabla}_{\beta(\dot{\alpha}}\underline{\tilde{\psi}}_{\dot{\beta})a}+2\theta^{b}_{\alpha}[\underline{\phi}_{ba},\underline{\tilde{F}}_{\dot{\alpha}\dot{\beta}}]$
(149a)
$\displaystyle-\underline{\nabla}_{\alpha(\dot{\alpha}}(\theta^{b\beta}\underline{\nabla}_{\dot{\beta})\beta}\underline{\phi}_{ab})+\theta^{b}_{\alpha}\\{\underline{\tilde{\psi}}_{b(\dot{\alpha}},\underline{\tilde{\psi}}_{\dot{\beta})a}\\}\,,$
$\displaystyle(1+\mathcal{D})(\underline{\nabla}_{a\alpha}\underline{\tilde{\psi}}_{\dot{\alpha}b}-2\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\phi}_{ab})$
$\displaystyle=$ $\displaystyle
2\theta^{c\beta}\underline{\nabla}_{a\alpha}\underline{\nabla}_{\beta\dot{\alpha}}\underline{\phi}_{bc}+2\theta^{b}_{\alpha}[\underline{\phi}_{ba},\underline{\tilde{\psi}}_{\dot{\alpha}b}]$
(149b)
$\displaystyle-2\epsilon_{abcd}\theta^{c}_{\beta}\underline{\nabla}_{\alpha\dot{\alpha}}\underline{\psi}^{d\beta}+2\theta^{c}_{\alpha}[\underline{\tilde{\psi}}_{c\dot{\alpha}},\underline{\phi}_{ab}]\,,$
$\displaystyle(1+\mathcal{D})(\underline{\nabla}_{a\alpha}\underline{\phi}_{bc}-\epsilon_{abcd}\underline{\psi}^{d}_{\alpha})$
$\displaystyle=$
$\displaystyle-\epsilon_{bcde}\theta^{d}_{\beta}\underline{\nabla}_{a\alpha}\underline{\psi}^{e\beta}+2\theta^{d}_{\alpha}[\underline{\phi}_{da},\underline{\phi}_{bc}]$
(149c)
$\displaystyle-\epsilon_{abcd}(\theta^{d\beta}\underline{B}_{\alpha\beta}-\theta^{e}_{\alpha}[\underline{\phi}_{ef},\underline{\phi}^{df}])\,,$
$\displaystyle(1+\mathcal{D})(\underline{\nabla}_{a\alpha}\underline{\psi}^{b}_{\beta}+\varepsilon_{\alpha\beta}[\underline{\phi}_{ac},\underline{\phi}^{bc}]$
$\displaystyle=$
$\displaystyle\delta^{b}_{a}(1+\mathcal{D})\underline{B}_{\alpha\beta}-\theta^{b\gamma}\underline{\nabla}_{a\alpha}\underline{B}_{\beta\gamma}$
(149d)
$\displaystyle+\theta^{c}_{\beta}\underline{\nabla}_{a\alpha}[\underline{\phi}_{cd},\underline{\phi}^{bd}]+2\theta^{c}_{\alpha}[\underline{\phi}_{ca},\underline{\psi}^{b}_{\beta}]$
$\displaystyle+\varepsilon_{\alpha\beta}(\epsilon_{acde}\theta^{d}_{\gamma}[\underline{\psi}^{e\gamma},\underline{\phi}^{bc}]+\theta^{[b}_{\gamma}\delta^{c]}_{g}[\underline{\phi}_{ac},\underline{\psi}^{g\gamma}])$
$\displaystyle+2\delta^{b}_{a}\theta^{c}_{(\alpha}[\underline{\psi}^{d}_{\beta)},\underline{\phi}_{cd}],,$
$\displaystyle(1+\mathcal{D})(\underline{\nabla}_{a\alpha}\underline{B}_{\alpha\beta}+2\varepsilon_{\alpha(\beta}[\underline{\psi}^{b}_{\gamma)},\underline{\phi}_{ab}])$
$\displaystyle=$ $\displaystyle
2\theta^{b}_{(\beta}\underline{\nabla}_{|a\alpha|}[\underline{\psi}^{c}_{\gamma)},\underline{\phi}_{bc}]+2\theta^{b}_{\alpha}[\underline{\phi}_{ba},\underline{B}_{\beta\gamma}]$
(149e)
$\displaystyle+2\varepsilon_{\alpha(\beta}([\theta^{b\delta}\underline{B}_{\gamma)\delta}-\theta^{c}_{\gamma)}[\underline{\phi}_{cd},\underline{\phi}^{bd}],\underline{\phi}_{ab}])$
$\displaystyle+2\epsilon_{abcd}\varepsilon_{\alpha(\beta}[\underline{\psi}^{b}_{\gamma)},\theta^{c}_{\delta}\underline{\psi}^{d\delta}]\,,$

which follow directly from (147a), (147b), (147c), (147d), (147e), (147g), and
(147h).

Similarly, the constraint equations (127a) follows from induction on the
fermionic degree on the recursion relations

$\displaystyle(1+\mathcal{D})([\underline{\nabla}_{a\alpha},\underline{\nabla}_{\beta\dot{\alpha}}]-\varepsilon_{\alpha\beta}\underline{\tilde{\psi}}_{a\dot{\alpha}})$
$\displaystyle=$
$\displaystyle-\theta^{b}_{\beta}\underline{\nabla}_{a\alpha}\underline{\tilde{\psi}}_{b\dot{\beta}}+2\theta^{b}_{\alpha}\underline{\nabla}_{\beta\dot{\beta}}\underline{\phi}_{ba}$
(150a)
$\displaystyle+2\varepsilon_{\alpha\beta}\theta^{b\gamma}\underline{\nabla}_{\gamma\dot{\beta}}\underline{\phi}_{ab}\,,$
$\displaystyle(2+\mathcal{D})(\\{\underline{\nabla}_{a\alpha},\underline{\nabla}_{b\beta}\\}-2\varepsilon_{\alpha\beta}\underline{\phi}_{ab})$
$\displaystyle=$
$\displaystyle-2\theta^{c}_{\beta}\underline{\nabla}_{a\alpha}\underline{\phi}_{cb}-2\theta^{c}_{\alpha}\underline{\nabla}_{b\beta}\underline{\phi}_{ca}$
(150b)
$\displaystyle-2\varepsilon_{\alpha\beta}\epsilon_{abcd}\theta^{c}_{\gamma}\underline{\psi}^{d\gamma}\,,$

thus establishing the equivalence.

##  Appendix B Computational details

In this brief appendix, we show in more detail how to derive (100) from the
generating functional (96)

$\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{-\mathrm{i}q\cdot
x}\iota_{\alpha}q^{\alpha\dot{\alpha}}\iota_{\beta}q^{\beta\dot{\beta}}\int_{X^{2}}\frac{\mathrm{D}\lambda_{1}\mathrm{D}\lambda_{2}}{\langle\iota\lambda_{1}\rangle\langle\iota\lambda_{2}\rangle}\langle\lambda_{1}\lambda_{2}\rangle^{2}\,\mathrm{tr}\,\left(\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}_{1}}\right|_{X}\mathsf{U}_{X}(\lambda_{1},\lambda_{2})\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\beta}}_{2}}\right|_{X}\mathsf{U}_{X}(\lambda_{2},\lambda_{1})\right)\,,$
(151)

here rewritten for the sake of clarity. Its expansion at $n$ points evaluated
on momentum eigenstates reads

$\displaystyle\left(\frac{-1}{2\pi\mathrm{i}}\right)^{n-2}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x}\iota_{\alpha}q^{\alpha\dot{\alpha}}\iota_{\beta}q^{\beta\dot{\beta}}\sum_{i,j}\Biggl{\\{}\frac{\,\mathrm{tr}\,\hat{T}_{{p_{1}}}\ldots\hat{T}_{{p_{n}}}}{\langle
p_{1}p_{2}\rangle\ldots\langle p_{n}p_{1}\rangle}\frac{\langle
ij\rangle^{2}}{\langle\iota i\rangle\langle\iota
j\rangle}\tilde{\kappa}_{i\,\dot{\alpha}}\tilde{\kappa}_{j\,\dot{\beta}}$
(152)