From Twistor-Particle Models to Massive Amplitudes

From Twistor-Particle Models to Massive Amplitudes††This paper is a
contribution to the Special Issue on Twistors from Geometry to Physics in
honour of Roger Penrose. The full collection is available at
https://www.emis.de/journals/SIGMA/Penrose.html

Giulia ALBONICO a, Yvonne GEYER b and Lionel MASON a

G. Albonico, Y. Geyer and L. Mason

a) The Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>
b) Department of Physics, Faculty of Science, Chulalongkorn University,
b) Thanon Phayathai, Pathumwan, Bangkok 10330, Thailand<EMAIL_ADDRESS>
Received March 19, 2022, in final form June 08, 2022; Published online June
19, 2022

In his twistor-particle programme of the 1970’s, Roger Penrose introduced a
representation of the massive particle phase space in terms of a pair of
twistors subject to an internal symmetry group. Here we use this
representation to introduce a chiral string whose target is a complexification
of this space, extended so as to incorporate supersymmetry. We show that the
gauge anomalies associated to the internal symmetry group vanish only for
maximal supersymmetry, and that correlators in these string models describe
amplitudes involving massive particles with manifest supersymmetry. The models
and amplitude formulae exhibit a double copy structure from gauge theory on
the Coulomb branch to gravity, although the graviton remains massless. The
formulae are closely related to those obtained earlier by the authors
expressed in terms of the polarised scattering equations.

twistor theory; scattering amplitudes; ambitwistor string

81U20; 83C60; 32L25; 81T30; 81T13

Dedicated to Roger Penrose on the occasion of his 90th birthday and the recent
award of his Nobel prize in physics, who has been a constant source of
inspiration to us.

##  1 Introduction

The twistor-strings of Witten [66], Berkovits [4] and Skinner [63] led to
remarkably compact amplitude formulae for tree amplitudes for both super
Yang–Mills [58] and supergravity theories [12, 16, 17, 29]. These twistor-
strings theories and formulae, including the worldsheet formulae of Cachazo,
He and Yuan [13, 14, 15], can all be understood under the umbrella of
ambitwistor-strings [45]; a family of quantum field theories of holomorphic
maps from a Riemann surface to the complexification of phase spaces of
massless particles. These can come in various different twistorial or non-
twistorial representations, with and without spin or supersymmetry and in
different numbers of space-time dimensions. They exhibit a number of beautiful
and perhaps unexpected features, such as a uniform simple structure in terms
of residues in the moduli space $\mathfrak{M}_{0,n}$ of $n$-points on
$\mathbb{CP}^{1}$ modulo Möbius transformations, and one of the most direct
realizations [10, 14, 46, 47] of the double-copy between gauge and gravity
amplitudes [6, 7].

In their original form, all these ambitwistor-string theories and associated
worldsheet formulae appear to be tightly restricted to theories and amplitudes
involving only massless particles. However the underlying approach suggests
that, if one wishes to compute amplitudes for theories with massive particles,
one should consider quantum field theories of holomorphic strings whose target
is the complexified phase space of massive particles. The massless case has
also shown that, in order to incorporate fermions simply, we should use a
twistor representations of the phase space.

Nearly 50 years ago, Roger Penrose, followed by Zoltan Perjes, gave a twistor
description of massive particles in terms of a set of two or more twistors up
to an internal symmetry group [51, 54]. He proposed the _twistor-particle
programme_ based on the twistor quantization of this description. In
particular, it was hoped that the representation theory of the internal
symmetry group should classify elementary particles; see for example [36, 37,
52, 55] and references therein. Although this programme has not been pursued
further by the twistor community, the framework was taken up by other authors
in the particle physics community. For these authors, quantization via a
worldline Lagrangian approach was used leading to studies of the spectrum of
such twistor particle models, often incorporating supersymmetry. Two-twistor
particle models include [8, 9, 22], see also [23, 24] for more recent studies
that have a good number of references to the evolution of the subject,
including [26, 27].111Note also a two-twistor model along the lines of an
ambitwistor string [40, 41], but focussed on massless particles. Such
worldline actions are a stepping stone to ambitwistor-string formulations.
These are holomorphic strings whose target is a complexified phase space; the
action being built directly from the holomorphic symplectic potential on such
a complexified phase space [45].

A particular advantage of the two-twistor representation of the phase space of
massive particles is that it reduces to the nonlinear massive phase space via
a symplectic quotient from the vector space of a pair of twistors. Such a
symplectic quotient can be done via BRST in the quantum field theory, and all
computations can be performed in a linear free-field quantum field theory on
the Riemann surface. However, a key lesson from the massless cases is that,
even if one is only interested in bosonic Yang–Mills or gravity, fermionic
symmetries are needed on the worldsheet and supersymmetries on space-time to
obtain simple uniform formulae incorporating all relevant helicities. We will
see that these supersymmetries can also be introduced in the massive case,
leading to simple compact formulae for amplitudes in otherwise complicated,
non-linear gauge and gravity theories.

In its simplest approach, massive particles were understood in terms of a pair
of 4d twistors $\big{(}Z_{a},\bar{Z}_{a}\big{)}\in\mathbb{T}\times\mathbb{T}$,
$a=1,2$. Each twistor $Z\in\mathbb{T}$ has four complex components, that
according to more recent (and less Penrosian) conventions are written as
$Z=\big{(}\lambda_{\alpha},\mu^{\dot{\alpha}}\big{)}$, i.e., as a pair of
2-component spinors; $\bar{Z}$ is the ${\rm SU}(2,2)$ complex conjugate of
$Z$, defining a dual twistor by
$\bar{Z}=\big{(}\bar{\lambda}_{\dot{\alpha}},\bar{\mu}^{\alpha}\big{)}$. This
description of massive particles was defined up to an internal symmetry group
${\rm SU}(2)\times\mathbb{C}$, where the ${\rm SU}(2)$ acts conformally
invariantly on the $a$ index. It can be understood in more conventional terms
as the stabilizer of the massive momentum in the Lorentz group, the little
group, see for example the massive spinor-helicity framework of [3].222More
generally, $n$-twistor descriptions were considered with symmetry groups
containing ${\rm SU}(n)$; in the twistor particle programme, particle
multiplets were to be understood via the representation theory of such
internal symmetry groups. The quantization of massive worldline models based
on these descriptions has been studied by a number of authors, see [23, 50]
and references therein. The factor of $\mathbb{C}$ in the symmetry group
breaks conformal invariance and determines the particle masses.

Here we complexify twistors so that the complex conjugate twistor $\bar{Z}$
becomes a dual twistor $\tilde{Z}$ independent of $\bar{Z}$ giving the pair
$Y=\big{(}Z,\tilde{Z}\big{)}\in\mathbb{T}_{\mathbb{C}}:=\mathbb{T}\times\mathbb{T}^{*}$.
We can also think of such a complexified twistor as a Dirac twistor
$Y=\big{(}\lambda_{A},\mu^{A}\big{)}$, given as a pair of 4-component Dirac
spinors
$\lambda_{A}=\big{(}\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}}\big{)}$
and $\mu^{A}=\big{(}\tilde{\mu}_{\alpha},\mu^{\dot{\alpha}}\big{)}$. We will
also incorporate supersymmetry extending
$Y\rightarrow\mathcal{Y}=\big{(}\lambda_{A},\mu^{A},\eta^{I}\big{)}$ with
$\mathcal{N}$ additional fermionic components $\eta^{I}$. This description
gives a natural inner product
$\mathcal{Y}\cdot\mathcal{Y}:=Z\cdot\tilde{Z}+\eta_{I}\eta^{I}$ from the
duality between $\mathbb{T}$ and $\mathbb{T}^{*}$ and skew form $\Omega_{IJ}$.

Our 4d massive twistorial models are given by holomorphic maps from the
Riemann surface $\Sigma$ to the complexified two-twistor description of
massive particles. They consist of a pair of complexified twistor fields
$\mathcal{Y}_{a}(\sigma)=\big{(}\mathcal{Z}_{a}(\sigma),\widetilde{\mathcal{Z}}_{a}(\sigma)\big{)}$,
$a=1,2$ taking values in worldsheet spinors $\sqrt{\Omega^{1,0}_{\Sigma}}$ .
To reduce to the twistor representation of the massive particle phase space,
we also gauge the currents
$\big{(}\mathcal{Y}_{a}\cdot\mathcal{Y}_{b},\lambda^{2},\tilde{\lambda}^{2}\big{)}$
that generate the (complexified) internal symmetry group ${\rm
SL}(2)\times\mathbb{C}\times\tilde{\mathbb{C}}$. Here
$\lambda^{2}:=\det(\lambda)$ and its conjugate $\tilde{\lambda}^{2}$ determine
the squared mass of the massive momentum
$P_{\alpha\dot{\alpha}}=\lambda_{\alpha}^{a}\tilde{\lambda}_{\dot{\alpha}}^{b}\epsilon_{ab}$.
Thus we arrive at the model

$\displaystyle
S_{4d}=\int_{\Sigma}\mathcal{Y}^{a}\cdot\bar{\partial}\mathcal{Y}_{a}+A_{ab}\mathcal{Y}^{a}\cdot\mathcal{Y}^{b}+A\big{(}\lambda^{2}-j^{H}\big{)}+\tilde{A}\big{(}\tilde{\lambda}^{2}-j^{H}\big{)}+S_{m}.$

Here $S_{m}$ is some theory dependent additional worldsheet matter that in
particular can give rise to a current $j^{H}$ associated to some symmetry
generator $H$. The $A_{ab}=A_{(ab)}$ are gauge fields for the ${\rm SL}(2)$
little group, and $\big{(}A,\tilde{A}\big{)}$ gauge the
$\mathbb{C}\times\tilde{\mathbb{C}}$ part of the internal symmetry group; they
are also Lagrange multipliers relating the values of the particle masses to
their charges under $H$.

Although this two-twistor massive model is a string whose target is the
complexified two-twistor description of massive particles of [54], it can be
identified with the dimensional reduction of the 6d and 5d ambitwistor strings
in [34]. The contractions of the massive spinor helicity variables correspond
to two components of the internal momentum when embedding the massive
variables in a six dimensional massless momentum. The two-twistor string
produces correlators that localize on delta functions that fix the values of
internal momenta in terms of charges under $H$ and $\tilde{H}$ for all
particles involved. In addition to this, the correlators are further localized
by delta functions imposing a polarised version of the scattering equations as
in [1, 30].

The models above not only allow us to derive formulae involving any number of
massive particles, but also give an alternative formulation of the massless
models in [29]. This is of particular importance as it presents a framework in
which a massless field can be _deformed_ to go off-shell, which is a necessary
prerequisite for defining a gluing operator in the four dimensional twistorial
model and producing loop amplitudes.

Paper summary. In the next section we introduce the two-twistor geometry of
the massive particle phase space. We then briefly present the Penrose
transform and its complexification. This can be used in a two-twistor string
that computes amplitudes for theories with massive particles in four
dimensions. Such formulae incorporate fermions and supersymmetry, generalizing
the massless case of [29]. They are based on the polarised scattering
equations that have already been introduced and studied in six and five
dimensions [1, 31, 34]; these can also be related [62] to the formulae of
[11]. We focus on a model adapted to the Coulomb branch of $\mathcal{N}=4$
SYM; this contains a gauge field, fermions and scalars that have a vacuum
expectation value to give masses to some of the particles, analogous to the
standard model. Nevertheless, as in the models of [33, 34], we can write down
a full range of models for particles of spin-0, spin-1 and spin-2 following
the double copy, although the scope for introducing masses into the gravity
models are limited.

##  2 Massive particles

We first review the twistor description of massive particles in terms of a
pair of twistors with redundancy described by the two-twistor internal
symmetry group ${\rm SU}(2)\times\mathbb{C}$. This framework ties in directly
with the (more recent) spinor-helicity formalism for expressing polarization
data for massive particles. Anticipating the string model we then complexify
the two-twistor description, and introduce the Penrose transform for massive
momentum eigenstates.

Conventions and notation. We will work with spinors in the complex, so that
the Lorentz group in $4$ dimensions is double-covered by ${\rm
SL}(2,\mathbb{C})\times{\rm SL}(2,\mathbb{C})$ with each factor acting
separately on the positive and negative chirality spinors. The positive and
negative chirality spinors carry indices $\alpha=1,2$,
$\dot{\alpha}=\dot{1},\dot{2}$, raised and lowered with
$\varepsilon_{\alpha\beta}=\varepsilon_{[\alpha\beta]}$, $\varepsilon_{12}=1$
and its inverse, as well as the corresponding dotted version. We denote spinor
inner products by the conventional bracket notation,

$\displaystyle\langle\lambda_{1}\lambda_{2}\rangle=\lambda_{1\alpha}\lambda_{2}^{\alpha},\qquad[\lambda_{1}\lambda_{2}]=\tilde{\lambda}_{1\dot{\alpha}}\tilde{\lambda}_{2}^{\dot{\alpha}},\qquad\tilde{\lambda}_{\dot{\alpha}}=\tilde{\lambda}^{\dot{\beta}}\varepsilon_{\dot{\beta}\dot{\alpha}},\qquad\tilde{\lambda}^{\dot{\alpha}}=\varepsilon^{\dot{\alpha}\dot{\beta}}\tilde{\lambda}_{\dot{\beta}}.$

###  2.1 Review of twistor internal symmetry groups for massive particles

Massless particles. As described in [51, 53, 54, 55], a general twistor
$Z^{\mathcal{A}}=(\lambda_{\alpha},\mu^{\dot{\alpha}})\in\mathbb{T}$
determines a massless particle whose momentum $P_{\alpha\dot{\alpha}}$ and
angular momentum
$M^{\mu\nu}=M^{\alpha\beta}\varepsilon^{\dot{\alpha}\dot{\beta}}+\text{c.c.}$
about the origin, can be assembled into the _angular momentum twistor_ given
by

$\displaystyle
L^{\mathcal{A}\mathcal{B}}:=\begin{pmatrix}0&P_{\alpha}^{\dot{\beta}}\\\
P_{\beta}^{\dot{\alpha}}&M^{\dot{\alpha}\dot{\beta}}\end{pmatrix}=\begin{pmatrix}0&\lambda_{\alpha}\bar{\lambda}^{\dot{\beta}}\\\
\lambda_{\beta}\bar{\lambda}^{\dot{\alpha}}&\bar{\lambda}^{(\dot{\alpha}}\mu^{\dot{\beta})}\end{pmatrix}=Z^{(\mathcal{A}}I^{\mathcal{B})\mathcal{C}}\bar{Z}_{\mathcal{C}}.$

In this formula, the _infinity twistor_ breaks conformal invariance and is
defined by

$\displaystyle
I^{\mathcal{A}\mathcal{B}}\bar{Z}_{\mathcal{B}}=\big{(}0,\bar{\lambda}^{\dot{\alpha}}\big{)},\qquad
I_{\mathcal{A}\mathcal{B}}Z^{\mathcal{B}}=(0,\lambda^{\alpha}),$

extending the spinor contractions to degenerate inner products $\langle
Z_{1}Z_{2}\rangle:=I_{\mathcal{A}\mathcal{B}}Z^{\mathcal{A}}_{1}Z^{\mathcal{B}}_{2}=\langle\lambda_{1}\lambda_{2}\rangle$
on twistor space. The angular momentum twistor is invariant under the internal
symmetry transformation $Z\rightarrow\mathrm{e}^{{\rm i}\theta}Z$, which we
can identify as the little group rotating the phase of the constituent spinors
of the massless momentum
$P_{\alpha\dot{\alpha}}=\lambda_{\alpha}\bar{\lambda}_{\dot{\alpha}}$.

Massive particles. In order to describe massive particles, we introduce a sum
over two twistors $Z_{a}^{\mathcal{A}}$, $a=1,2$ with complex conjugates
$\bar{Z}_{\mathcal{A}}^{a}$. These yield the angular momentum twistor

$\displaystyle
L^{\mathcal{A}\mathcal{B}}=Z^{(\mathcal{A}}_{a}I^{\mathcal{B})\mathcal{C}}\bar{Z}_{\mathcal{C}}^{a}.$

In particular, the momentum is given by

$\displaystyle
P_{\alpha\dot{\alpha}}=\lambda_{a\alpha}\bar{\lambda}^{a}_{\dot{\alpha}},$

and so we can identify the indices $a$, $b$ as the ${\rm SU}(2)$ little-group
indices that stabilizes the massive momentum $P_{\alpha\dot{\alpha}}$ inside
the Lorentz group.

Penrose [51] and Perjes [54] define the two-twistor internal symmetry group to
be the Poincaré invariant transformations that preserve the angular momentum
twistor. This group is ${\rm SU}(2)\times\mathbb{C}$, where the ${\rm SU}(2)$
acts as the massive little group, and the factor of $\mathbb{C}$ is given by
the complex transformations

$\displaystyle\delta Z^{\mathcal{A}}_{a}\propto
I^{\mathcal{A}\mathcal{B}}\bar{Z}_{\mathcal{B}}^{b}\epsilon_{ba}.$

These symmetries all preserve the symplectic form and potential [64]

$\displaystyle\Omega_{m}:={\rm d}\Theta_{m},\qquad 2\Theta_{m}:={\rm
i}Z^{\mathcal{A}}_{a}{\rm d}\bar{Z}_{\mathcal{A}}^{a}-{\rm
i}\bar{Z}_{\mathcal{A}}^{a}{\rm d}Z^{\mathcal{A}}_{a}.$

The internal symmetry group action with respect to the potential $\Theta_{m}$
is generated by the Hamiltonians

$\displaystyle
Z^{\mathcal{A}}_{(a}\cdot\bar{Z}_{b)\mathcal{A}},\qquad\lambda^{2}:=\frac{1}{2}\lambda_{\alpha}^{a}\lambda_{a}^{\alpha}=\frac{1}{2}\langle
Z_{a},Z^{a}\rangle=\frac{1}{2}I_{\mathcal{A}\mathcal{B}}Z^{\mathcal{A}}_{a}Z^{\mathcal{B}a},$

for the factors of ${\rm SU}(2)$ and $\mathbb{C}$ respectively. We can
therefore define the phase space $\mathcal{P}_{m}$ for particles of mass $m$
as the symplectic quotient

$\displaystyle\mathcal{P}_{m}=\big{\\{}Z_{a}\in\mathbb{T}\times\mathbb{T}\mid
Z_{(a}\cdot\bar{Z}_{b)}=0,\langle Z_{a}Z^{a}\rangle=m\big{\\}}/\\{{\rm
SU}(2)\times\mathbb{C}\\}.$ (2.1)

It is easy to see that this is a 6 real-dimensional symplectic manifold with
symplectic potential $\Theta_{m}$.

###  2.2 Dirac spinors and spinor-helicity for massive particles

As remarked above, the ${\rm SU}(2)$ of the internal symmetry group is the
massive particle “little group”, the subgroup of the spin double cover of the
Lorentz group that preserves a time-like momentum $P_{\alpha\dot{\alpha}}$;
the representations of this little group are naturally identified with the
polarization states of massive particles as follows. For a massive particle of
momentum $k_{\alpha\dot{\alpha}}$ we write as above

$\displaystyle
k_{\alpha\dot{\alpha}}=\kappa_{a\alpha}\tilde{\kappa}_{\dot{\alpha}}^{a},$

where $a=1,2$ is an ${\rm SU}(2)$ little group index raised and lowered by
$\varepsilon_{ab}=\varepsilon_{[ab]}$, $\varepsilon_{12}=1$. In the real case
$\tilde{\kappa}^{a}_{\dot{\alpha}}$ can be taken to be the complex conjugate
of $\kappa_{\alpha a}$ reducing the little group to ${\rm SU}(2)$. We denote
little group contractions by

$\displaystyle(v_{1}v_{2}):=v_{1a}v_{2b}\varepsilon^{ab}.$

The mass $m$ is given by
$k^{2}=m^{2}=\det(k_{\alpha\dot{\alpha}})=\det\kappa\det\tilde{\kappa}$; so
defining

$\displaystyle\det\kappa=M,\qquad\det\tilde{\kappa}=\tilde{M},$

we have $M\tilde{M}=m^{2}$ and although we can fix the phases of $\kappa$ and
$\tilde{\kappa}$ so that $M=\tilde{M}=\pm m$, later we will want to keep them
independent before they are fixed by the model.

Massive particles are not chiral, and two-component spinors necessarily double
up with their conjugates. For a more compact notation, we introduce Dirac
4-component spinors with indices denoted by capital Roman letters from the
beginning of the alphabet as

$\displaystyle\psi_{A}=\big{(}\psi_{\alpha},\tilde{\psi}^{\dot{\alpha}}\big{)},\qquad\psi^{A}=\varepsilon^{AB}\psi_{A}:=\big{(}\psi^{\alpha},\tilde{\psi}_{\dot{\alpha}}\big{)},\qquad\psi_{1A}\psi_{2}^{A}=\psi_{1\alpha}\psi_{2}^{\alpha}+\tilde{\psi}_{1}^{\dot{\alpha}}\tilde{\psi}_{2\dot{\alpha}},$

and we will raise and lower indices with $\varepsilon^{AB}$,
$\varepsilon_{AB}$, $\varepsilon^{AB}\varepsilon_{AC}=\delta^{B}_{C}$. Also
note the $\gamma_{5}$ matrix defined by

$\displaystyle\gamma_{5A}^{B}\psi_{B}={\rm
i}\big{(}\psi_{\alpha},-\tilde{\psi}^{\dot{\alpha}}\big{)}.$

The mass-$m$ Dirac operator $D^{m}_{AB}=D^{m}_{[AB]}$ in this notation is

$\displaystyle D^{m}_{AB}:=-{\rm i}\nabla_{AB}+m\varepsilon_{AB}=-{\rm
i}\begin{pmatrix}0&\nabla_{\alpha}^{\;\dot{\beta}}\\\
-\nabla^{\dot{\alpha}}_{\;\beta}&0\end{pmatrix}+m\varepsilon_{AB}.$

The spin $s$ massive field equations for $\Psi^{A_{1}\cdots
A_{2}s}=\Psi^{(A_{1}\cdots A_{2}s)}$ becomes

$\displaystyle D^{m}_{BA_{1}}\Psi^{A_{1}\cdots A_{2}s}=0.$

At spin $s=1$, we obtain $F_{AB}=F_{(AB)}$ whose $2\times 2$ block-
decomposition contains the 2-form curvature spinors along the diagonal and
$mA_{\alpha\dot{\beta}}$ on the off-diagonal, where $A_{\alpha\dot{\alpha}}$
is the one-form potential.

Introducing a little group spinor $\epsilon_{a}$, the general plane wave on
Minkowski space of spin-$s$ can be decomposed into Dirac spinor wave functions
as

$\displaystyle\Psi_{A_{1}\cdots
A_{2s}}=\epsilon_{A_{1}}\cdots\epsilon_{A_{2s}}\mathrm{e}^{{\rm i}k\cdot
x},\qquad\epsilon_{A}=\epsilon_{a}\kappa^{a}_{A}=:(\epsilon\kappa_{A}),\qquad\kappa_{Aa}=\big{(}\kappa_{\alpha
a},\tilde{\kappa}^{\dot{\alpha}}_{a}\big{)}.$ (2.2)

For spin $1/2$, this is an ordinary massive Dirac field momentum eigenstate
with polarization $\epsilon_{a}$; for spin $s=1$ this describes a massive
field with potential
$A_{\alpha\dot{\alpha}}=\frac{1}{m}\epsilon_{ab}\,\kappa^{a}_{\alpha}\tilde{\kappa}^{b}_{\dot{\alpha}}\,\mathrm{e}^{{\rm
i}k\cdot x}$, with polarization $\epsilon_{(ab)}=\epsilon_{a}\epsilon_{b}$. In
general, spin-$s$ massive particles transform as the symmetric part of rank
$2s$ tensors of the massive little group ${\rm SU}(2)$, with polarization data
$\epsilon_{a_{1}\cdots a_{2s}}=\epsilon_{(a_{1}\cdots a_{2s})}$. Note that the
polarization in (2.2) is taken to be simple to tie in with later
supersymmetric expressions, corresponding to a null polarization vector. We
refer to [3, 19, 20] for more extended recent discussions of spinor-helicity
for massive particles.

To reduce to the massless case, we can take half the spinor components to
vanish $\kappa_{1\alpha}=0=\tilde{\kappa}_{0}^{\dot{\alpha}}$, whereupon the
little group spinor components $\epsilon_{0}$ and $\epsilon_{1}$ parametrize
the positive and negative helicity states respectively.

###  2.3 The complexified particle phase space and Penrose transform

In all (massless) ambitwistor strings [45], the target space is the
complexification of the massless particle phase space, often referred to as
ambitwistor space and denoted by $\mathbb{A}$. To define this we first
introduce complexified twistor space $\mathbb{T}_{\mathbb{C}}$ by

$\displaystyle
Y=\big{(}Z,\tilde{Z}\big{)}\in\mathbb{T}_{\mathbb{C}}:=\mathbb{T}\times\mathbb{T}^{*}.$

Then the complexified phase space of massless particles in four dimensions has
become known as ambitwistor space $\mathbb{A}$, defined non-projectively as
the holomorphic symplectic quotient

$\displaystyle\mathbb{A}=\\{Y\in\mathbb{T}_{\mathbb{C}}\mid Y\cdot
Y:=Z\cdot\tilde{Z}=0\\}/\big{\\{}Z\cdot\partial_{Z}-\tilde{Z}\cdot\partial_{\tilde{Z}}\big{\\}},$
(2.3)

with respect to the symplectic structure

$\displaystyle\Omega_{\mathbb{A}}={\rm
d}\Theta_{\mathbb{A}},\qquad\Theta_{\mathbb{A}}:={\rm i}Z\cdot{\rm
d}\tilde{Z}-{\rm i}\tilde{Z}\cdot{\rm d}Z.$

This is the target of the original twistor strings [4, 63, 66] and the closely
related ambitwistor strings [29].

In analogy with the massless case, here we take the target space to be
$\mathcal{P}_{m}^{\mathbb{C}}$, the complexification of the massive particle
phase space $\mathcal{P}_{m}$. We represent $\mathcal{P}^{\mathbb{C}}_{m}$ as
the holomorphic symplectic quotient analogue of (2.1) as

$\displaystyle\mathcal{P}^{\mathbb{C}}_{m}:=\big{\\{}Y_{a}\in\mathbb{T}^{\mathbb{C}}\times\mathbb{T}^{\mathbb{C}}\mid
Z_{(a}\cdot\tilde{Z}_{b)}=0,\,\langle
Z_{a}Z^{a}\rangle=\big{[}\tilde{Z}_{a}\tilde{Z}^{a}\big{]}=m\big{\\}}/\mathrm{SL}(2,\mathbb{C})\times\mathbb{C}\times\tilde{\mathbb{C}}.$

One of the oldest applications of twistor theory has been to provide solutions
to the free field equations. In the massless case this is achieved via the
Penrose transform, which represents zero-rest mass helicity-$h$ fields as
twistor cohomology classes $H^{1}(\mathbb{P}\mathbb{T},\mathcal{O}(2h-2))$.
Using the identifications $\mathbb{A}=T^{*}\mathbb{PT}=T^{*}\mathbb{PT}^{*}$,
representatives of these cohomology classes can also be pulled back to
ambitwistor space. While two-twistor descriptions in the literature [37] lead
to $H^{2}$ representatives by building on the real massive particle phase
space, we use the complexification $\mathcal{P}^{\mathbb{C}}_{m}$ to obtain
representatives in (Dolbeault) cohomology classes
$H^{1}\big{(}\mathcal{P}^{\mathbb{C}}_{m},\mathcal{O}(2s-2)\big{)}$ that
couple naturally to the worldsheet. Here we will focus on the scalar case
$s=0$, the extension to spinning particles can be achieved most
straightforwardly via supersymmetry and is discussed in Section 2.4.

To represent the plane wave (2.2) with momentum
$k_{\alpha\dot{\alpha}}=\kappa_{\alpha a}\tilde{\kappa}_{\dot{\alpha}}^{a}$ on
$\mathbb{T}^{\mathbb{C}}\times\mathbb{T}^{\mathbb{C}}$ it will be convenient
to reorganise the spinor constituents of $Y_{a}$ as a “Dirac twistor”

$\displaystyle
Y_{a}=\big{(}\lambda_{aA},\mu^{A}_{a}\big{)},\qquad\lambda_{aA}:=\big{(}\lambda_{a\alpha},\tilde{\lambda}_{a}^{\dot{\alpha}}\big{)},\qquad\mu^{A}_{a}:=\big{(}\mu^{\dot{\alpha}}_{a},\tilde{\mu}_{\alpha
a}\big{)}.$

Writing $\kappa_{Aa}=\big{(}\kappa_{\alpha
a},\tilde{\kappa}^{\dot{\alpha}}_{a}\big{)}$, we define the corresponding
cohomology representative in
$H^{1}\big{(}\mathcal{P}^{\mathbb{C}}_{m},\mathcal{O}(-2)\big{)}$ by
introducing four auxiliary complex variables $u_{a}$, $v_{a}$;

$\displaystyle\Phi_{\kappa}(Y_{a})=\int{\rm d}^{2}u\,{\rm
d}^{2}v\bar{\delta}^{4}((u\lambda_{A})-(v\,\kappa_{A}))\bar{\delta}((v,\epsilon)-1)\exp\big{(}\big{(}u\mu^{A}\big{)}\epsilon_{A}\big{)}.$
(2.4)

Here the line bundle $\mathcal{O}(n)$ is the bundle of homogeneity degree $n$
in the $Y_{a}$, and for a complex variable $z$ we define $\bar{\delta}(z)$ to
be the distributional $(0,1)$-form

$\displaystyle\bar{\delta}(z)=\bar{\partial}\frac{1}{2\pi{\rm
i}z}=\delta(\operatorname{Re}z)\delta(\operatorname{Im}z)\,{\rm d}\bar{z}.$

After the $u_{a}$, $v_{a}$ integrals have been performed,
$\Phi_{\kappa}(Y_{a})\in
H^{1}\big{(}\mathcal{P}^{\mathbb{C}}_{m},\mathcal{O}(-2)\big{)}$ is indeed a
$(0,1)$-form as desired, and the integration over $u$ ensures invariance under
the ${\rm SL}(2,\mathbb{C})$ little group. For the Penrose transform, we take
$\det(\kappa)$ and $\det(\tilde{\kappa})$ to be unconstrained; in the two-
twistor string these quantities will be constrained to agree with the particle
masses determined by the underlying theory.

To see that this indeed corresponds to a plane-wave on space-time, we impose
the incidence relations $\mu^{\dot{\alpha}}_{a}={\rm
i}x^{\alpha\dot{\alpha}}\lambda_{a\alpha}$ and $\tilde{\mu}^{\alpha}_{a}=-{\rm
i}x^{\alpha\dot{\alpha}}\tilde{\lambda}_{a\dot{\alpha}}$. Then on the support
of the delta functions we have $(u\lambda_{A})=(v\kappa_{A})$ and
$(v\epsilon)=1$, giving333This Penrose transform is closely related to the
(indirect) 6d Penrose transform [44, 60]. The twistor space for 6d is
$\mathbb{T}_{\mathbb{C}}|_{Y\cdot Y=0}$ and the plane wave (2.2) is
represented by  $\displaystyle\Psi_{\kappa}(Y)=\int(\epsilon v)^{n}\frac{{\rm
d}s}{s^{n-1}}(v{\rm
d}v)\bar{\delta}^{4}(s\lambda_{A}-(v\kappa_{A}))\exp\big{(}s\mu^{A}\epsilon_{A}/(\epsilon
v)\big{)}\in H^{2}(\mathbb{P}\mathbb{A},\mathcal{O}(n-2)).$ Following [34, 43,
56, 61], the massive $\Phi_{\kappa}(Y_{a})$ in (2.4) can then be constructed
via $\Phi_{\kappa}(Y_{a})=\int\Psi_{\kappa}((u,Y))(u{\rm d}u)$.

$\displaystyle\big{(}u\mu^{A}\big{)}\epsilon_{A}={\rm
i}x^{\alpha\dot{\alpha}}(v\kappa_{\alpha})(\epsilon\tilde{\kappa}_{\dot{\alpha}})-{\rm
i}x^{\alpha\dot{\alpha}}(v\tilde{\kappa}_{\dot{\alpha}})(\epsilon\kappa_{\alpha})={\rm
i}x\cdot k.$ (2.5)

The parameters $(u_{a},v_{a})$ can be integrated against the delta functions
to yield the single delta function $\bar{\delta}\big{(}k\cdot
P-m\lambda^{2}-m\tilde{\lambda}^{2}\big{)}$, where
$P_{\alpha\dot{\alpha}}=\lambda_{a\alpha}\tilde{\lambda}_{\dot{\alpha}}^{a}$;
this gives the Dolbeault representation for a simple pole. This delta function
imposes a massive analogue of the _scattering equations_ that play a key role
in the CHY formulae for massless amplitudes [12, 16, 17, 29]. A version of
these _massive scattering equations_ has been studied for massive amplitudes
in the CHY representation [48], see the discussion in Section 5 for more
details. The delta-functions in $\Phi_{\kappa}$ have become known as the
_polarized scattering equations_ due to their dependence on a choice of
polarization spinor.444The polarization spinor will play a more prominent role
in amplitude formulae based on the polarized scattering equations, because the
path integrals introduce $\epsilon$-dependence in
$\lambda_{\scalebox{0.6}{$A$}}(\sigma)$. These serve to define additional and
unique parameters $u_{a}$, $v_{a}$ on the support of the massive scattering
equations that will play a key role later.

###  2.4 Supersymmetric extension

We will aim to generate supersymmetric formulae for two reasons. Amplitudes
for theories with a variety of spins become drastically simpler in the
supersymmetric case because many or all the particles can be expressed as one
multiplet. This leads to uniform formulae from which different sectors with
particles of different spins can be read off. Moreover, amplitudes for non-
supersymmetric theories can be extracted from these superamplitudes at tree-
level and at one-loop [33, 38]. A more structural reason is that all (ambi-)
twistor string models that describe gauge theory and gravity require space-
time supersymmetry to be anomaly free – the supersymmetric extension of
twistor space includes additional fermionic variables that cancel anomalies
from the bosonic variables. Thus we introduce a supersymmetric extension of
$\mathcal{P}_{m}^{\mathbb{C}}$, as well as the plane wave $\Phi_{\kappa}$.

On the Coulomb branch of $\mathcal{N}=4$ super Yang–Mills, some scalars
acquire a vacuum expectation value, effectively breaking the gauge group from
${\rm SU}(N+M)$ down to ${\rm SU}(N)\times{\rm SU}(M)$. The states can then be
organised into two types of multiplets; a massless vector multiplet
transforming in the adjoint of the residual gauge group, and a massive vector
multiplet in the bifundamental of ${\rm SU}(N)\times{\rm SU}(M)$. Massive
multiplets are in the so-called $1/2$-BPS, ultrashort massive representations
of $\mathcal{N}=4$ with central extension $Z_{IJ}=2M\Omega_{IJ}$, with
Sp$(\mathcal{N}/2)$ $R$-symmetry, with skew form $\Omega_{IJ}$ and indices
$I,J=1,\dots,\mathcal{N}=4$. For massless multiplets on the other hand,
$R$-symmetry is enhanced to a full ${\rm SU}(4)$ by the vanishing of the
central extension as $m\rightarrow 0$. Employing the notation of the previous
section, we can combine the supercharges into a Dirac spinor
$Q_{I}^{A}=\big{(}Q_{\alpha I},Q^{\dagger\dot{\alpha}}_{I}\big{)}$, such that
the supersymmetry algebra takes the compact form

$\displaystyle\\{Q_{AI},Q_{BJ}\\}=2\Omega_{IJ}D^{m}_{AB}.$

In the massless case, the structure of the supersymmetry algebra greatly
simplifies as the only non vanishing component of the Dirac operator is
$D_{\alpha\dot{\alpha}}^{0}=\nabla_{\alpha\dot{\alpha}}$.

The action of the supercharges arranges the states in multiplets as follows.
The massive multiplet is composed of a massive spin one field $F_{AB}$, five
massive scalars $\phi_{IJ}$ and four massive Weyl–Majorana spinors
$\Psi_{A}^{I}$:

$\displaystyle\mathscr{F}^{m}=\big{(}\phi_{IJ}=\phi_{[IJ]},\Psi_{I}^{A},F^{AB}=F^{(AB)}\big{)},\qquad\phi_{IJ}\Omega^{IJ}=0.$
(2.6)

The massless multiplet is

$\displaystyle\mathscr{F}^{0}=\big{(}\phi_{IJ}=\phi_{[IJ]},\Psi^{I}_{\alpha},\tilde{\Psi}_{I\dot{\alpha}},F_{\alpha\beta},F_{\dot{\alpha}\dot{\beta}}\big{)},$
(2.7)

where the $R$-symmetry indices now label the fundamental of ${\rm SU}(4)$ and
can therefore no longer be raised and lowered. It contains the two familiar
${\pm 1}$ helicity states of the massless spin-1, six real massless scalars
$\phi_{IJ}$ and eight massless gluino states via the chiral parts of
$\Psi_{\alpha}^{I}$, $\tilde{\Psi}_{I\dot{\alpha}}$. We note that the massless
scalars $\phi_{IJ}$ are no longer trace-free; the extra 6th component arises
from the loss of one of the polarization degrees of freedom going from the
massive spin-1 field $F_{AB}$ to the massless case.

For momentum eigenstates with space-time dependence $\phi=\exp({\rm i}k\cdot
x)$, the supersymmetry generators reduce to the massive little group as

$\displaystyle
Q_{AI}=\kappa_{A}^{a}Q_{aI},\qquad\\{Q_{aI},Q_{bJ}\\}=2\Omega_{IJ}\varepsilon_{ab},$

where $\kappa_{a}^{A}$ is defined by (2.2), because the Dirac operator reduces
as $D_{AB}^{m}\phi=(\kappa_{A}\kappa_{B})\phi$. In the massless limit we have
the natural embedding of the little group via
$\kappa_{1\alpha}=0=\tilde{\kappa}_{0\dot{\alpha}}$,
$\kappa_{0\alpha}=\kappa_{\alpha}$,
$\tilde{\kappa}_{1\dot{\alpha}}=\tilde{\kappa}_{\dot{\alpha}}$.

Both the massive and massless multiplets are annihilated by half of the
supercharges so that their $8$ bosonic and $8$ fermionic states can all be
encoded into the exterior powers of $\mathcal{N}=4$ fermionic supermomenta
$q_{I}$, $I=1,\dots,4$. These are defined to be the eigenvalues of an
anticommuting subset of the $Q_{Ia}$. To define this subset, we introduce a
basis $(\epsilon_{a},\xi_{a})$ of the fundamental representation of ${\rm
SL}(2)$ so that the supermomenta are defined by the action of the supercharges
on functions on on-shell superspace via

$\displaystyle
Q_{aI}\tilde{\mathscr{F}}(\kappa,q)=\left(\xi_{a}q_{I}+\epsilon_{a}\Omega_{IJ}\frac{\partial}{\partial
q_{J}}\right)\tilde{\mathscr{F}}(\kappa,q).$ (2.8)

The massive and massless multiplets are expanded on on-shell superspace as
follows

$\displaystyle\tilde{\mathscr{F}}_{(\kappa,q)}^{(m)}=F^{\epsilon\epsilon}(\kappa)+q_{I}\Psi^{\epsilon
I}(\kappa)+q^{2}F^{\epsilon\xi}(\kappa)+\frac{1}{2}q_{I}q_{J}\Phi^{IJ}(\kappa)+q^{2}q_{I}\Psi^{\xi
I}(\kappa)+q^{4}F^{\xi\xi}(\kappa),$
$\displaystyle\tilde{\mathscr{F}}_{(\kappa,q)}^{(0)}=g^{h}(\kappa)+q_{I}\Psi^{\epsilon
I}(\kappa)+\frac{1}{2}q_{I}q_{J}\varphi^{IJ}(\kappa)+q^{2}q_{I}\Psi^{\xi
I}(\kappa)+q^{4}g^{-h}(\kappa),$ (2.9)

with $q^{4}=(q_{I}q_{J})\big{(}q^{I}q^{J}\big{)}$ and
$\big{(}q^{3}\big{)}^{I}_{a}=\partial q^{4}/\partial q^{a}_{I}$.

It is then standard procedure to encode such multiplets in superfields on a
supersymmetric extension of Minkowski space satisfying (2.8) and to derive a
supersymmetric Penrose transform by establishing a supergeometric
correspondence with super-twistor space. We can bypass some of this by
studying the action of supersymmetry on super-twistors.

Supertwistors and the Penrose transform. We extend the bosonic complexified
twistor $Y\in\mathbb{T}^{\mathbb{C}}$ with $\mathcal{N}$ fermionic coordinates
$\eta^{I}$, $I=1,\dots,\mathcal{N}$ to give
$\mathcal{Y}=\big{(}\lambda_{A},\mu^{A},\eta^{I}\big{)}\in\mathcal{T}^{\mathbb{C}}$,
using Dirac-spinor notation. These fermionic coordinates allow the
supersymmetry to act geometrically as

$\displaystyle
Q_{AI}=\lambda_{A}\frac{\partial}{\partial\eta^{I}}+\eta^{J}\Omega_{JI}\frac{\partial}{\partial\mu^{A}},\qquad\\{Q_{AI},Q_{BJ}\\}=2\Omega_{IJ}\lambda_{[A}\frac{\partial}{\partial\mu^{B]}},$

where the anticommutator now generates the action of translations on
$\mathcal{T}^{\mathbb{C}}$. This extends in the obvious way to the two-twistor
description of supersymmetric massive particles in terms of $\mathcal{Y}_{a}$
with sums over the $a$-index in each term. Again, the supersymmetric extension
for the $\mathcal{Y}_{a}$ becomes:

$\displaystyle\mathcal{Y}_{a}=\big{(}\lambda_{aA},\mu^{A}_{a},\eta^{I}_{a}\big{)},$

with again $I=1,\dots,\mathcal{N}$. The plane wave representative for
particles with spinor helicity data $\kappa_{Aa}=\big{(}\kappa_{\alpha
a},\tilde{\kappa}^{\dot{\alpha}}_{a}\big{)}$, supermomentum $q$ and
polarization data $\epsilon_{a}$ will take the form:

$\displaystyle\Phi_{(\kappa,q)}(\mu,\lambda)=\int{\rm d}^{2}u\,{\rm
d}^{2}vw\bar{\delta}^{4}((u\lambda_{A})-(v\kappa_{A}))\bar{\delta}((\epsilon
v)-1){\rm e}^{{\rm
i}u_{a}(\mu^{Aa}\epsilon_{A}+q_{I}\eta^{Ia})-\frac{1}{2}(\xi v)q^{2}}.$ (2.10)

Here $w$ is a function of weight $2$ in $\mathcal{Y}_{a}$(or $-2$ in $u$); as
far as the Penrose transform is concerned, this can be taken to be
$w=\big{(}\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}\big{)}{\rm
e}^{\alpha\dot{\alpha}}$, where ${\rm e}^{\alpha\dot{\alpha}}$ is a
polarization vector for the spin-1 1-form. In the string model however, $w$
plays an important role in the vertex operators, and will be constructed
differently. This representative indeed satisfies

$\displaystyle
Q_{AI}\Phi_{\kappa,q}(\mathcal{Y}_{a})=\bigg{(}(\kappa_{A}\xi)q_{I}+\epsilon_{A}\Omega_{IJ}\frac{\partial}{\partial
q_{I}}\bigg{)}\Phi_{\kappa,q}(\mathcal{Y}_{a}),$

and we can then read off the Penrose transform for the component fields from
the action of the supersymmetry generators.

##  3 Massive two-twistor string

The significance of the twistor representations of spaces of massless and
massive particles is that they are represented as symplectic quotients of
vector spaces. This means that in order to construct a theory of maps from a
Riemann surface $\Sigma\rightarrow\mathcal{P}^{\mathbb{C}}_{m}$, we can start
with a quantum field theory of maps $\Sigma\rightarrow\mathbb{T}^{\mathbb{C}}$
in the massless case and
$\Sigma\rightarrow\mathbb{T}^{\mathbb{C}}\times\mathbb{T}^{\mathbb{C}}$ in the
massive case; in both cases, by virtue of the twistor representations, these
are free field theories on the worldsheet $\Sigma$. We then realize the
symplectic quotient in the Lagrangian framework by gauging the Hamiltonian
symmetries as we shall describe below. These gauge symmetries are then dealt
with via BRST in the quantum field theory. In both cases, the free field
theory action is based on the restriction of the symplectic potential $\Theta$
to $T^{0,1}_{\Sigma}$. This has the consequence that the worldsheet
commutators and OPEs encode the symplectic structure $\Omega_{m}$ on
$\mathcal{P}_{m}^{\mathbb{C}}$.

We first briefly review the massless case; although the construction for the
massive two-twistor string will be analogous, but with target
$\mathcal{P}_{m}^{\mathbb{C}}$ and the different massive supersymmetry
representation. In the next section we explain how the models allow us to
construct amplitudes as correlation functions of vertex operators in these
models.

The massless case. The twistor strings of Witten [66] and Berkovits [4] and
the 4d ambitwistor string of [29] are theories of holomorphic maps
$\mathcal{Y}=\big{(}\mathcal{Z},\tilde{\mathcal{Z}}\big{)}\colon\Sigma\rightarrow\mathbb{T}^{\mathbb{C}}$
gauged by $\mathbb{C}^{*}$, where we now use supertwistors
$\mathcal{Z}=\big{(}\lambda_{\alpha},\mu^{\dot{\alpha}},\eta^{I}\big{)}\in\mathbb{T}=\mathbb{C}^{4|\mathcal{N}}$,
$I=1,\dots,\mathcal{N}$ and their complexification
$\mathbb{T}^{\mathbb{C}}=\mathbb{T}\times\mathbb{T}^{*}$; this is the
complexification of the four-dimensional massless Ferber superparticle [28].

The four-dimensional ambitwistor string of [29] is closest to the massive
case, being with worldsheet fields twisted to take values in
$\Omega^{\nicefrac{{1}}{{2}},0}_{\Sigma}$ and so we briefly review it here. It
is a theory of holomorphic maps from a Riemann surface
$\mathcal{Y}\colon\Sigma\rightarrow\mathbb{T}_{\mathbb{C}}\otimes\Omega^{\nicefrac{{1}}{{2}},0}_{\Sigma}$,
so that the coordinates $\mathcal{Y}$ are worldsheet spinors. The reduction to
ambitwistor space is enforced by gauging the little-group Hamiltonian
$\mathcal{Y}\cdot\mathcal{Y}:=\mathcal{Z}\cdot\tilde{\mathcal{Z}}$ with the
worldsheet gauge field $A\in\Omega^{0,1}_{\Sigma}$. The basic bosonic 4d
ambitwistor action in conformal gauge555The full action would start with a
term $eT$, where $e\in T^{1,0}_{\Sigma}\otimes\Omega^{0,1}_{\Sigma}$ is a
Beltrami differential thought of as a gauge field parametrizing complex
structures on $\Sigma$ up to coordinate transformations and
$T\in\big{(}\Omega^{1,0}_{\Sigma}\big{)}^{2}$ is the holomorphic stress energy
tensor; this is then gauge fixed, giving rise to ghosts
$(b,c)\in\big{(}\big{(}\Omega^{1,0}_{\Sigma}\big{)}^{2},T_{\Sigma}^{1,0}\big{)}$
and BRST operator $Q=\oint cT+bc\partial c/2$.  is based on the symplectic
potental

$\displaystyle
S^{0}_{4d}=\int_{\Sigma}\widetilde{\mathcal{Z}}\cdot\bar{\partial}\mathcal{Z}-\mathcal{Z}\cdot\bar{\partial}\widetilde{\mathcal{Z}}+A\,\mathcal{Y}\cdot\mathcal{Y}.$
(3.1)

Classically, $A$ is a Lagrange multiplier that enforces the constraint
$\mathcal{Y}\cdot\mathcal{Y}=0$ and the quotient by its Hamiltonian vector
field arises because $\big{(}Z,\tilde{Z}\big{)}\rightarrow\big{(}\alpha
Z,\alpha^{-1}\tilde{Z}\big{)}$ are gauge symmetries of the action when
accompanied by the gauge transformations $A\rightarrow
A+\bar{\partial}\log\alpha$. Thus the holomorphic symplectic quotient to
$\mathbb{A}$ in (2.3) is realized in this Lagrangian framework by the gauge
field $A$. In the QFT this is implemented via BRST quantization. The models of
[29] also include additional worldsheet matter fields but these are much as
described for the massive case below.

Massive models. In order to have target space $\mathcal{P}_{m}^{\mathbb{C}}$,
we start with maps
$\mathcal{Y}_{a}\colon\Sigma\rightarrow\mathbb{T}_{\mathbb{C}}\times\mathbb{T}_{\mathbb{C}}$,
with the reduction to $\mathcal{P}^{\mathbb{C}}_{m}$ obtained by gauging the
complexified two-twistor massive internal symmetry group. Thus, our theory is
one of maps
$\mathcal{Y}_{a}\colon\Sigma\rightarrow\mathbb{T}_{\mathbb{C}}\otimes\mathbb{C}^{2}\otimes\Omega^{\nicefrac{{1}}{{2}},0}_{\Sigma}$
with action (again in conformal gauge)

$\displaystyle
S_{4d}=\int_{\Sigma}\mathcal{Y}^{a}\bar{\partial}\mathcal{Y}_{a}+A_{ab}\mathcal{Y}^{a}\cdot\mathcal{Y}^{b}+A\big{(}\lambda^{2}-j^{H}\big{)}+\tilde{A}\big{(}\tilde{\lambda}^{2}-j^{H}\big{)}+S_{m}.$

Here $a=1,2$ is the little group index, and
$\big{(}A_{ab}=A_{(ab)},A,\tilde{A}\big{)}$ are worldsheet $(0,1)$-forms that
act as Lagrange multipliers for the constraints, and as gauge fields for the
internal two-twistor symmetry group. With this symmetry, we no longer have the
freedom to allow worldsheet fields of different degrees as we did for the
twistor-string. In order to describe specific space-time theories, the basic
action must be supplemented by further worldsheet fields such as a current
algebra for gauge theory and some analogue of worldsheet supergravity for
gravity with details given below. Here we assume that it contains a current-
algebra that gives rise to a $(1,0)$-form $j_{H}$ on the worldsheet that
generates some symmetry.

To be more explicit, in quantizing the fields
$\mathcal{Y}_{a}(\sigma)=\big{(}\mathcal{Z}_{a}(\sigma),\tilde{\mathcal{Z}}_{a}(\sigma)\big{)}$,
for $\sigma$ a coordinate on $\Sigma$, the only non-trivial OPEs are

$\displaystyle\mathcal{Z}^{\mathcal{A}}_{a}(\sigma)\tilde{\mathcal{Z}}_{b\mathcal{B}}(0)=\frac{\delta^{\mathcal{A}}_{\mathcal{B}}}{\sigma}\varepsilon_{ab}+\cdots$

reflecting the Poisson brackets. These OPEs can lead to anomalies for the
little group ${\rm SL}(2,\mathbb{C})$ generated by
$J^{ab}=\mathcal{Y}^{a}\cdot\mathcal{Y}^{b}$. For a consistent model these
anomalies have to vanish, which requires judicious choices for the worldsheet
matter $S_{m}$.

The fields $a$, $\tilde{a}$ gauge the constraints
$\lambda^{2}-j^{H}=0=\tilde{\lambda}^{2}-j^{H}$. These equations constrain the
mass operators

$\displaystyle\lambda^{2}:=\frac{1}{2}\lambda^{a}_{\alpha}\lambda_{a}^{\alpha}=\det(\lambda_{\alpha}^{a}),\qquad\tilde{\lambda}^{2}:=\frac{1}{2}\tilde{\lambda}_{\dot{\alpha}}^{a}\tilde{\lambda}_{a}^{\dot{\alpha}}=\det\big{(}\tilde{\lambda}_{a}^{\dot{\alpha}}\big{)},$

to be given by a $(1,0)$-form $j^{H}$ on the worldsheet $\Sigma$. We write
$j^{H}$ to indicate that this will be taken to be the current associated to
the element $h\in\mathfrak{g}$, living in the Cartan subalgebra of some
symmetry of the system. This $j^{H}$ will be constructed from the matter
fields and, through the constraints above, will determine the masses of the
particles. For a given matter content, different choices of $j^{H}$ correspond
to different distributions of masses within the models. The massless models
(3.1) are recovered from these massive ones when $j^{H}=0$ by reducing the
path integral.

Worldsheet matter. A variety of physically interesting models can be
constructed from different choices of $S_{m}$. These will be made up of
current algebras, whose action will be denoted by $S_{C}$, and worldsheet
fermions providing a supersymmetric extension of the worldsheet gauge algebra,
denoted by $S_{\rho}$. The latter will play a similar role to worldsheet
supergravity in the superstring, and is requried for models describing gauge
theory and supergravity.

A worldsheet current algebra is a theory on the worldsheet from which one can
construct worldsheet currents
$j^{a}\in\Omega^{1,0}_{\Sigma}\otimes\mathfrak{g}$ for some Lie algebra
$\mathfrak{g}$, satisfying the OPE

$\displaystyle
j^{\mathfrak{a}}(\sigma)j^{\mathfrak{b}}(0)\sim\frac{l\,\delta^{\mathfrak{ab}}}{\sigma^{2}}+\frac{f^{\mathfrak{ab}}_{\mathfrak{c}}j^{\mathfrak{c}}}{\sigma},$

where $\mathfrak{a}$, $\mathfrak{b}$ are Lie-algebra indices, $l\in\mathbb{Z}$
is the level and $f^{\mathfrak{ab}}_{\mathfrak{c}}$ the structure constants of
$\mathfrak{g}$. Such current algebras can be constructed in a number of ways,
most easily for SO$(n)$ and ${\rm SU}(n)$ by “real” or “complex” free fermions
on the worldsheet. See also [33] for a construction referred to as a comb-
system, with level zero and novel properties that allow the construction of
Einstein-Yang–Mills amplitudes. We will not specify the action $S_{C}$
explicitly, but merely assume that we have the currents $j^{\mathfrak{a}}$ in
the theory.

For gauge and gravity theories, we need a supersymmetric extension of the
worldsheet gauge algebra. This plays a similar role to the worldsheet
supergravity of the conventional RNS models, see also [45] for the
ambitwistor-string version. The supersymmetric extension of the bosonic gauge
algebra $\mathfrak{sl}_{2}\times\mathbb{C}^{2}$ is constructed by introducing
the worldsheet fermions
$\big{(}\rho_{A},\tilde{\rho}^{A}\big{)}\in\Omega^{0}\big{(}\Sigma,K_{\Sigma}^{1/2}\big{)}$
with action

$\displaystyle
S_{\rho}=\int_{\Sigma}\tilde{\rho}^{\scalebox{0.6}{$A$}}\bar{\partial}\rho_{\scalebox{0.6}{$A$}}+b_{a}\big{(}\gamma_{5}^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}\lambda_{\scalebox{0.6}{$A$}}^{a}\rho_{\scalebox{0.6}{$B$}}\big{)}+\tilde{b}_{a}\lambda_{{\scalebox{0.6}{$A$}}}^{a}\tilde{\rho}^{\scalebox{0.6}{$A$}}.$
(3.2)

Here the $\big{(}b^{a},\tilde{b}^{a}\big{)}$ are fermionic gauge fields and so
are $(0,1)$-forms on the worldsheet. They are Lagrange multipliers that impose
the constraints
$\gamma_{5}^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}\lambda_{\scalebox{0.6}{$A$}}^{a}\rho_{\scalebox{0.6}{$B$}}=\lambda_{{\scalebox{0.6}{$A$}}}^{a}\tilde{\rho}^{\scalebox{0.6}{$A$}}=0$
and their gauge transformations translate $\mu^{A}_{a}$ in the direction of
$\big{(}\rho^{A},\tilde{\rho}^{A}\big{)}$. The only non-trivial OPE’s of the
constraints are given by

$\displaystyle\big{(}\gamma_{5}^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}\lambda_{\scalebox{0.6}{$A$}}^{a}\rho_{\scalebox{0.6}{$B$}}\big{)}(z)\;\big{(}\lambda_{\scalebox{0.6}{$B$}}^{b}\tilde{\rho}^{\scalebox{0.6}{$B$}}\big{)}(w)\sim\frac{\varepsilon_{ab}}{z-w}\big{(}\lambda^{2}-\tilde{\lambda}^{2}\big{)}.$
(3.3)

These symmetries thus give a supersymmetric extension of the two-twistor
internal symmetry group $\mathbb{C}\times\mathfrak{sl}_{2}\ltimes H(0,4)$,
where $H$ denotes the Heisenberg Lie superalgebra.666The Heisenberg
superalgebra $H(m_{b},m_{f})$ has a central element $z$, as well as $2m_{b}$
even and $m_{f}$ odd generators, $H=\langle
x_{1},\dots,x_{2m_{b}},z\rangle\oplus\langle\psi_{1},\dots,\psi_{m_{f}}\rangle$.
The generators satisfy the “usual” commutation relations
$[x_{i},x_{2i}]=z,\qquad\\{\psi_{r},\psi_{s}\\}=2\delta_{rs}z.$

Models. With these ingredients, models without ${\rm
SL}(2,\mathbb{C})$-anomalies can be constructed by combining a pair of
worldsheet matter systems, much along the lines of the double copy for the RNS
ambitwistor strings as in [33] as follows:

massive bi-adjoint scalar $\displaystyle
S^{\scalebox{0.7}{BAS}}=S_{4d}+S_{C}+S_{\tilde{C}},$ super Yang–Mills on the
Coulomb branch $\displaystyle S^{\scalebox{0.7}{CB}}\hskip
4.0pt=S_{4d}+S_{\rho}+S_{C},$ super-gravity $\displaystyle
S^{\scalebox{0.7}{sugra}}=S_{4d}+S_{\rho_{1}}+S_{\rho_{2}}.$

In this construction two points are worth highlighting:

  * $(i)$

The closure of the constraint algebra requires that both constraints
$\lambda^{2}-j^{H}=0=\tilde{\lambda}^{2}-j^{H}$ involve the _same_ current
$j^{H}$ for super Yang–Mills, whereas a more general construction is possible
for the bi-adjoint scalar.

  * $(ii)$

Unlike the twistor- and ambitwistor models for 4d massless theories, these
models fit neatly into the double copy format [7] expressed directly in the
CHY formulae [12] and in the corresponding RNS ambitwistor strings [45].
However, it is harder to find a $j^{H}$ to endow our particles with mass in
the gravitational case because there is no additional current algebra, and
with $j^{H}=0$ our models are massless. We also note that as in [5, 34, 45],
both $S^{\scalebox{0.7}{CB}}$ and $S^{\scalebox{0.7}{BAS}}$ also contain a
gravity sector, but it is of higher order and remains massless.

BRST and anomalies. Gauge fixing the action via BRST generates ghost systems,
the well-known $(b,c)\in\big{(}\Omega^{1}_{\Sigma}\big{)}^{2}\times
T_{\Sigma}$ for worldsheet diffeomorphisms, as well as additional fermionic
ghosts associated to internal two-twistor symmetry group, and bosonic ghosts
for the fermionic currents in $S_{\rho}$. The BRST operator takes the usual
form:

$\displaystyle Q=\oint c^{i}\bigg{(}T^{m}_{i}+\frac{1}{2}T^{g}_{i}\bigg{)},$
(3.4)

where the sum runs over all sets of ghosts, and $T^{m}$ and $T^{g}$ are the
matter and ghost parts of the currents respectively. By construction $Q^{2}=0$
classically, but in the QFT double contractions (or worldsheet bubble diagrams
with two external gauge fields) can lead to anomalies so that $Q^{2}\neq 0$
with a potential obstruction arising from any of the gauged symmetries. Here
we briefly summarize the results of such calculations.

The models above only have a vanishing ${\rm SL}(2,\mathbb{C})$ anomaly
(corresponding to the two-twistor internal symmetry group) for maximal space-
time supersymmetry, as evident from the anomaly coefficient

$\displaystyle\mathfrak{a}_{\scalebox{0.7}{$\mathrm{SL}(2)$}}=\sum_{i}(-1)^{F_{i}}\mathrm{tr}_{R_{i}}\big{(}t^{k}t^{k}\big{)}=\begin{cases}4\,\mathrm{tr}_{\scalebox{0.7}{F}}\big{(}t^{k}t^{k}\big{)}-\mathrm{tr}_{\scalebox{0.7}{adj}}\big{(}t^{k}t^{k}\big{)}=0&\text{bi-
adjoint scalar},\\\ \frac{3}{4}(4-\mathcal{N})&\text{Coulomb branch},\\\
\frac{3}{4}(8-\mathcal{N})&\text{supergravity}.\end{cases}$

The anomaly coefficient vanishes trivially for the bi-adjoint scalar, and for
maximal supersymmetry in the case of gauge theory and gravity. The sum here
runs over all fields that transform non-trivially under the internal two-
twistor symmetry group $\mathrm{SL}(2,\mathbb{C})$. Similarly, the Virasoro
central charge can be calculated for all models, giving

$\displaystyle\mathfrak{c}^{\scalebox{0.6}{BAS}}=-40+\mathfrak{c}_{j},\qquad\mathfrak{c}^{\scalebox{0.6}{CB}}=-32+\mathcal{N}+\mathfrak{c}_{j},\qquad\mathfrak{c}^{\scalebox{0.6}{sugra}}=-20+\mathcal{N},$

where $\mathfrak{c}_{j}$ denotes the central charge of the internal current
algebra. The conformal anomaly thus vanishes for suitable choice of $S_{j}$,
with $\mathcal{N}=4$ and $\mathfrak{c}_{j}=28$ for Yang–Mills theory on the
Coulomb branch, and $\mathfrak{c}_{j}=40$ for the bi-adjoint scalar. The
Virsaoro anomaly for the supergravity model also vanishes if we include a
central charge term $\mathfrak{c}_{6d}=12$ arising from six compactified
dimensions. After BRST gauge-fixing, all such models are free worldsheet
theories with vanishing anomalies.777Strictly speaking, the Lie algebra
element $H$ should also be null or the current algebra should have level $l=0$
as in the comb systems of [18]. We now explain how to obtain $n$-point
amplitudes from these models.

##  4 Amplitudes and vertex operators

In string theory, amplitudes are constructed as correlation functions of
_vertex operators_ , one for each external particle. Vertex operators come in
various forms that depend on how residual gauge freedom is fixed after initial
gauge fixing, see for example [65]. The easiest to understand are the
_integrated vertex operators_ that require integration over $\Sigma$; these
are the generic case and arise as the perturbations of the action
corresponding to infinitesimal background plane-wave fields. However, one also
needs a small number of _fixed vertex operators_ that fix residual worldsheet
diffeomorphisms and other symmetries such as those associated to the $a$,
$\tilde{a}$ and $b$, $\tilde{b}$ fields. We here just give brief details of
the basic recipe required for the amplitude formulae and refer to further work
[2] for full details of the derivation.

###  4.1 Vertex operators

Just as for the original twistor string, vertex operators for the two-twistor
string are constructed from the Penrose-transform of plane waves
$\Phi_{\kappa}(Y_{a})\in
H^{1}\big{(}\mathcal{P}^{\mathbb{C}}_{m},\mathcal{O}(-2)\big{)}$ (2.4), or its
supersymmetric extension $\Phi_{\kappa,q}(\mathcal{Y}_{a})$ (2.10) with an
appropriate choice of $w$. The identification of the $Y_{a}$ or
$\mathcal{Y}_{a}$ as spinors on the worldsheet implies that
$\Phi_{\kappa}(Y_{a})$ restricts to $\Sigma$ to define an element of
$H^{0,1}(\Sigma,T_{\Sigma})$. Similar considerations apply to
$\Phi_{\kappa,q}(\mathcal{Y}_{a})$ but the line bundle on $\Sigma$ depends on
the choice of $w$. We will therefore choose $w$ so as to take values in
$\big{(}\Omega^{1,0}\big{)}^{2}$ so that $w\circ\Phi$ defines a $(1,1)$-form
on $\Sigma$ and can be integrated; here the notation $\circ$ indicates
schematically that $w$ may depend on $u$, and should be included under the
integral in $\Phi_{\kappa}$. Thus fixed and integrated vertex operator can be
constructed by including a state- and theory dependent factor
$w\in\Omega^{0}\big{(}\Sigma,K_{\Sigma}^{2}\big{)}$ respectively as

$\displaystyle V=c\;w\circ\Phi_{\kappa}(\sigma),\qquad\mathcal{V}=\int{\rm
d}\sigma\,w\circ\Phi_{\kappa}(\sigma).$ (4.1)

Here $c$ is the fermionic ghost with values in $T^{1,0}\Sigma$ associated to
worldsheet diffeomorphisms.

Vertex operators are required to be BRST $Q$-closed which implies invariance
under the symmetries. For example, the presence of the $c$-ghost in $V$ gives
$Q$-closure trivially for that part of $Q$ containing $c$ which is associated
to worldsheet diffeomorphisms, whereas, on the other hand, the integration of
the $(1,1)$-form on $\Sigma$ that defines $\mathcal{V}$ is manifestly
diffeomorphism invariant. We also expect to see similar invariance of both
fixed and integrated vertex operators under the gauged two-twistor internal
symmetry and the gauged fermionic symmetries in $S_{\rho}$ if present. BRST-
closed vertex operators can thus be constructed by including ghost factors for
the remaining generators, including fermionic ghosts $t$, $\tilde{t}$
associated to the mass constraints gauged by $A$, $\tilde{A}$, and bosonic
ghosts $\gamma^{a}$, $\tilde{\gamma}^{a}$ corresponding to $b^{a}$,
$\tilde{b}^{a}$ respectively.

It is this $Q$-invariance that will fix the masses of our external particles.
In the sum over ghosts in (3.4), $Q$ contains the term for the ghosts $t$ and
$\tilde{t}$ that arose from gauge fixing the $A$, $\tilde{A}$:

$\displaystyle Q_{m}:=\oint
t\big{(}\lambda^{2}-j^{H}\big{)}+\tilde{t}\big{(}\tilde{\lambda}^{2}-j_{H}\big{)}.$

When acting on a vertex operator built as in (4.1), the operator $\lambda^{2}$
becomes the mass operator, yielding the $Q$-invariance condition that requires
support on $\kappa^{2}=M^{H}$, where $M^{H}$ is the eigenvalue of the
$H$-action on the vertex operator under $j^{H}$. Thus $Q$-invariance requires
that the signed mass is given by the action of $j^{H}$. This action is theory-
dependent as follows.

Biadjoint scalars. This theory involves two Lie algebras $\mathfrak{g}$,
$\tilde{\mathfrak{g}}$ with fields $\phi^{\mathfrak{a}\tilde{\mathfrak{a}}}$,
where $\mathfrak{a}$, $\tilde{\mathfrak{a}}$ are the respective Lie-algebra
indices. The action is $\int{\rm
d}^{4}x\big{(}(\partial\phi)^{2}+f_{\mathfrak{a}\mathfrak{b}\mathfrak{c}}\tilde{f}_{\tilde{\mathfrak{a}}\tilde{\mathfrak{b}}\tilde{\mathfrak{c}}}\phi^{\mathfrak{a}\tilde{\mathfrak{a}}}\phi^{\mathfrak{b}\tilde{\mathfrak{b}}}\phi^{\mathfrak{c}\tilde{c}}\big{)}$,
where the $f_{\mathfrak{a}\mathfrak{b}\mathfrak{c}}$,
$\tilde{f}_{\tilde{\mathfrak{a}}\tilde{\mathfrak{b}}\tilde{\mathfrak{c}}}$ are
the structure constants of the respective Lie algebras. We will also
incorporate a mass term $\mathrm{tr}([\phi,H])^{2}$, where
$H\in\mathfrak{g}\oplus\tilde{\mathfrak{g}}$. Each particle then has a pair of
“colours”
$\big{(}\mathfrak{t}_{\mathfrak{a}},\tilde{\mathfrak{t}}_{\tilde{\mathfrak{a}}}\big{)}\in\mathfrak{g}\oplus\tilde{\mathfrak{g}}$
and momentum $k$ with mass determined by the eigenvalues of the action of $H$
in the form of the plane wave
$\phi^{\mathfrak{a}\tilde{\mathfrak{a}}}=\mathfrak{t}^{\mathfrak{a}}\tilde{\mathfrak{t}}^{\tilde{\mathfrak{a}}}\mathrm{e}^{{\rm
i}k\cdot x}$. These theories don’t admit supersymmetry so we will simply use
(2.4) and the two colours
$\mathfrak{t}^{\mathfrak{a}}\tilde{\mathfrak{t}}^{\tilde{\mathfrak{a}}}$ to
build the corresponding vertex operators, so that the mass is assigned via

$\displaystyle
j^{H}(z)\;\mathfrak{t}_{\mathfrak{a}}j^{\mathfrak{a}}(w)\sim\frac{M^{H}_{\mathfrak{t}}}{z-w}\mathfrak{t}_{\mathfrak{a}}j^{\mathfrak{a}}.$
(4.2)

Coulomb branch of maximal super-Yang–Mills. Here one takes the standard
$\mathcal{N}=4$ super-Yang–Mills with Lie algebra $\mathfrak{g}$, and assumes
that the scalars of the massless sector $\phi_{IJ}$ are expanded around the
constant $H\Omega_{IJ}$, where $H\in\mathfrak{g}$ is taken to be semi-simple.
Decomposing $\mathfrak{g}$ into eigenspaces under the adjoint action of $H$,
those with eigenvalue zero give rise to massless multiplets (2.7), but those
with non-zero eigenvalue $M^{H}$ acquire a mass $|M^{H}|$ and become massive
multiplets as in (2.6), where the scalar that is a multiple of $\Omega_{IJ}$
becomes a polarization state of the massive spin-1 field via the Higgs
mechanism.

To be more explicit, consider the symmetry breaking $U(N+M)\rightarrow
U(N)\times U(M)$, obtained by choosing
$H\sim\mathrm{diag}(\mathds{1}_{N},0_{M})$. This leaves two types of states in
the reduced theories, the first corresponding to
$\mathfrak{t}\in\mathfrak{u}_{N}\times\mathfrak{u}_{M}$, the diagonal blocks
that commute with $H$, and the second the off-diagonal blocks consisting of
$\mathfrak{m}\in\mathbb{C}^{N}\otimes\big{(}\mathbb{C}^{M}\big{)}^{*}\oplus\mathbb{C}^{M}\otimes\big{(}\mathbb{C}^{N}\big{)}^{*}$
for which $[H,\mathfrak{m}]=M^{H}_{\mathfrak{m}}\mathfrak{m}$. The first type
then corresponds to massless states, and the second will have mass
$\big{|}M^{H}_{\mathfrak{m}}\big{|}$, following from the $Q$-invariance of the
vertex operators as above. As a consequence, the OPEs (4.2) of the respective
currents take the following form:

$\displaystyle j^{H}(z)\;\mathfrak{t}\cdot j(w)\sim 0,\qquad
j^{H}(z)\;\mathfrak{m}\cdot
j(w)\sim\frac{M^{H}_{\mathfrak{m}}}{z-w}\mathfrak{m}\cdot j.$ (4.3)

We can thus identify vertex operators built from currents $\mathfrak{t}\cdot
j$ with the massless vector multiplet transforming in the adjoint of the
residual $U(N)\times U(M)$ gauge group, whereas vertex operators built from
$\mathfrak{m}\cdot j$ describe the massive vector multiplet.888The massive
states are $1/2$-BPS, ultrashort massive representations of $\mathcal{N}=4$
with central extension $|Z_{IJ}|=2m$.

Supergravity. We are not able to introduce massive spin-2 fields with this
mechanism. However, we can introduce masses elsewhere into the supermultiplet
by choosing for $j^{H}$ a generator of the $R$-symmetry. While we will not
develop this theme in any detail, a short discussion is included in Section 5.
The graviton states remain in a massless supermutliplet with $\mathcal{N}=4$,
formed as spin-2 part of the tensor product of two copies of the gauge
mutiplet (2.7), supplemented by an ultrashort matter multiplet, again with
$\mathcal{N}=4$.

Because both the $A$, $\tilde{A}$ fields and $b^{a}$, $\tilde{b}^{a}$ fields
have moduli that need to be fixed when computing correlators, we need to
adjust some vertex operators to fix this residual gauge freedom. Here we will
suppress these details and give only the equivalent of integrated vertex
operators for these symmetries, and refer to the analogous calculations in the
5d ambitwistor string [34] or [2] for details.

The main new ingredient here required for BRST-invariance of the vertex
operators are delta functions fixing the value of $\lambda^{2}$ and
$\tilde{\lambda}^{2}$ to the relevant masses $M_{i}^{H}$ arising from the
eigenvalues of $j^{H}$. These arise from integrating out the moduli of the
associated gauge fields in the presence of vertex operators. Thus the generic
integrated vertex operators for these theories can be summarized as

$\displaystyle\begin{split}w_{\scalebox{0.6}{BAS}}(\sigma_{i})={}&\delta\big{(}\mathrm{Res}_{\sigma_{i}}\big{(}\lambda^{2}-j^{H}\big{)}\big{)}\delta\big{(}\mathrm{Res}_{\sigma_{i}}\big{(}\tilde{\lambda}^{2}-j^{H}\big{)}\big{)}\mathfrak{t}_{\mathfrak{a}}j^{\mathfrak{a}}\tilde{\mathfrak{t}}_{\mathfrak{a}}\tilde{j}^{\mathfrak{a}},\\\
w_{\scalebox{0.6}{CB}}(\sigma_{i})={}&\delta\big{(}\mathrm{Res}_{\sigma_{i}}\big{(}\lambda^{2}-j^{H}\big{)}\big{)}\delta\big{(}\mathrm{Res}_{\sigma_{i}}\big{(}\tilde{\lambda}^{2}-j^{H}\big{)}\big{)}\big{(}e_{i}^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}(\lambda_{\scalebox{0.6}{$B$}}\lambda_{\scalebox{0.6}{$A$}})+\epsilon_{i}^{\scalebox{0.6}{$A$}}\epsilon_{i{\scalebox{0.6}{$B$}}}\rho_{\scalebox{0.6}{$A$}}\tilde{\rho}^{\scalebox{0.6}{$B$}}\big{)}\mathfrak{t}_{\mathfrak{a}}j^{\mathfrak{a}},\\\
w_{\scalebox{0.6}{grav}}(\sigma_{i})={}&\delta\big{(}\mathrm{Res}_{\sigma_{i}}\big{(}\lambda^{2}-j^{H}\big{)}\big{)}\delta\big{(}\mathrm{Res}_{\sigma_{i}}\big{(}\tilde{\lambda}^{2}-j^{H}\big{)}\big{)}\big{(}e_{i}^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}(\lambda_{\scalebox{0.6}{$A$}}\lambda_{\scalebox{0.6}{$B$}})-\epsilon_{i}^{\scalebox{0.6}{$A$}}\epsilon_{i{\scalebox{0.6}{$B$}}}\rho_{\scalebox{0.6}{$A$}}\tilde{\rho}^{\scalebox{0.6}{$B$}}\big{)}\\\
&\times\big{(}\tilde{e}_{i}^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}(\lambda_{\scalebox{0.6}{$A$}}\lambda_{\scalebox{0.6}{$B$}})-\tilde{\epsilon}_{i}^{\scalebox{0.6}{$A$}}\epsilon_{i{\scalebox{0.6}{$B$}}}\rho_{\scalebox{0.6}{$A$}}\tilde{\rho}^{\scalebox{0.6}{$B$}}\big{)},\end{split}$
(4.4)

where $e_{i}$ and $\tilde{e}_{i}$ are polarisation vectors. When evaluating
correlators including these vertex operators, we will see that the delta-
functions enforce the mass-shell condition. One can see from (4.2)–(4.3) that
taking the residue of $j^{H}$ acting on a $H$-eigenstate extracts the mass
$M^{H}_{i}$. Note that in the gravitational case the $M^{H}_{i}$ will be zero
for any multiplet containing the gravitational spin two field.

###  4.2 Massive amplitudes as two-twistor string correlators

Amplitudes are calculated as correlators in the ambitwistor string models. Due
to ghost zero modes and residual symmetry after gauge fixing, a non-trivial
correlator with $n$ vertex operators must contain three fixed vertex operators
$V_{1}$, $V_{2}$, $V_{3}$, not necessarily of the same type, whose details we
suppress:

$\displaystyle\mathcal{A}_{n}=\bigg{\langle}V_{1}V_{2}V_{3}\prod_{i=4}^{n}\mathcal{V}_{i}\bigg{\rangle}.$

After gauge fixing, all the fields are free and these correlation functions
can be straightforwardly evaluated explicitly for any $n$ to give formulae for
scattering amplitudes.

Each vertex operator contains an exponential from the $\Phi_{\kappa}$ factor
as in the classes (2.10) embedded in the definition of the vertex operators
(4.1). Directly computing their correlation functions would be awkward, but in
the expression of the correlation function as a path integral, the
exponentials from the vertex operators can be absorbed as source terms into
the action for the $\mathcal{Y}_{a}$-path integral. Since the correlator is
independent of $\mu^{\scalebox{0.6}{$A$}}_{a}$, and the action is linear in
$\mu^{\scalebox{0.6}{$A$}}_{a}$, the path integrals over these fields can be
performed directly, localizing the correlator onto the classical equations of
motion, but now with sources arising from the vertex operators

$\displaystyle\bar{\partial}\lambda_{A}^{a}=\sum_{i=1}^{n}u_{i}^{a}\epsilon_{iA}\,\bar{\delta}(\sigma-\sigma_{i}),\qquad\bar{\partial}\eta_{I}^{a}=\frac{1}{2}\sum_{i=1}^{n}u_{i}^{a}\,q_{iI}\,\bar{\delta}(\sigma-\sigma_{i}).$

On the Riemann sphere, these are solved uniquely by

$\displaystyle\lambda_{A}^{a}(\sigma)=\sum_{i=1}^{n}\frac{u_{i}^{a}\epsilon_{iA}}{\sigma-\sigma_{i}},\qquad\eta_{I}^{a}(\sigma)=\frac{1}{2}\sum_{i=1}^{n}\frac{u_{i}^{a}\,q_{iI}}{\sigma-\sigma_{i}}.$
(4.5)

Using this, the remainder of the correlator can be evaluated as follows:

  * •

_Polarized scattering equations and measure._ Evaluating the ghost path
integrals and collecting the various measure factors as well as the delta-
functions from $\Phi_{\kappa}$ gives the following measure:

$\displaystyle{\rm d}\mu_{n}^{\mathrm{pol}}:=\frac{\prod_{j}{\rm
d}\sigma_{j}\,{\rm d}^{2}u_{j}\,{\rm
d}^{2}v_{j}}{\operatorname{vol}\mathrm{SL}(2,\mathbb{C})_{\sigma}\times\mathrm{SL}(2,\mathbb{C})_{u}}\;\prod_{i=1}^{n}\bar{\delta}^{4}\big{(}(u_{i}\lambda_{A}(\sigma_{i}))-(v_{i}\kappa_{iA})\big{)}.$
(4.6)

A short counting reveals that the integrals are fully localized on the
constraints enforced by the delta-functions, known as the _polarised
scattering equations_ $\mathcal{E}_{i{\scalebox{0.6}{$A$}}}=0$, where

$\displaystyle\mathcal{E}_{i{\scalebox{0.6}{$A$}}}:=(u_{i}\lambda_{A}(\sigma_{i}))-(v_{i}\kappa_{iA})=\sum_{j\neq
i}\frac{(u_{i}u_{j})\epsilon_{jA}}{\sigma_{i}-\sigma_{j}}-(v_{i}\kappa_{iA}).$
(4.7)

These equations and corresponding measure are by now well studied in [1, 30];
the polarized scattering equations imply a massive analogue of the original
scattering equations for the $\sigma_{i}$ described in Section 5, and the
measure reduces to the CHY measure [14]. After performing the integrals, the
amplitudes still contain six residual delta-functions, whose form can be
extracted from

$\displaystyle\epsilon_{i[A}\mathcal{E}_{iB]}=\sum_{i}K_{iAB}=0,\qquad\text{where}\quad
K_{iAB}:=(\kappa_{iA}\kappa_{iB})=\begin{pmatrix}\kappa_{i}^{2}\varepsilon_{\alpha\beta}&k_{i\alpha}{}^{\dot{\beta}}\\\
-k_{i\beta}^{\dot{\alpha}}&\tilde{\kappa}_{i}^{2}\varepsilon^{\dot{\alpha}\dot{\beta}}\end{pmatrix}\\!.$
(4.8)

The residual delta-functions thus encode conservation of momentum
$k_{\alpha\dot{\alpha}}$, as well as restrictions on
$\kappa^{2}=\det(\kappa_{\alpha}^{a})$ and $\tilde{\kappa}^{2}$:
$\sum_{i}\kappa_{i}^{2}=0=\sum_{i}\tilde{\kappa}_{i}^{2}$. Notice that the
closure of the algebra of constraints (3.3) imposes that
$\kappa_{i}^{2}=\tilde{\kappa}_{i}^{2}$ so that these two conditions are
actually equivalent. One can show that, because each $M_{i}^{H}$ is the charge
under a symmetry on a compact worldsheet, the amplitude vanishes unless
$\sum_{i}M_{i}^{H}=0$. Then this condition, together with conservation of
$\kappa^{2}$ and the $n-1$ mass assigning delta functions, entails that also
for the remaining particle $\kappa_{1}^{2}=M_{1}^{H}$.

  * •

_Mass fixing._ The action of the operators $\lambda^{2}$ and
$\tilde{\lambda}^{2}$ reduces to the residues of their zero-modes (4.5) at
$\sigma_{i}$. From the above formula, on the support of the delta functions in
(4.6), this can be seen to yield $\kappa^{2}_{i}$ and
$\tilde{\kappa}_{i}^{2}$. Thus the mass parameters for each particle
$\kappa_{i}^{2}$ and $\tilde{\kappa}_{i}^{2}$ are fixed to the appropriate
value $M_{i}^{H}$ via the delta function in that vertex operator in (4.4):

$\displaystyle\prod_{i=2}^{n}\delta\big{(}\mathrm{Res}_{\sigma_{i}}\lambda^{2}-M^{H}_{i}\big{)}\delta\big{(}\mathrm{Res}_{\sigma_{i}}\tilde{\lambda}^{2}-M^{H}_{i}\big{)}=\prod_{i=2}^{n}\delta\big{(}\kappa_{i}^{2}-M_{i}^{H}\big{)}\delta\big{(}\tilde{\kappa}_{i}^{2}-M_{i}^{H}\big{)},$

now clearly enforcing the mass-shell condition for $n-1$ particles. That for
the $1$st particle is omitted in one of the fixed vertex operators (to fix a
residual degree of gauge freedom) but arises instead as a consequence of the
residual delta functions that gives
$\sum_{i}\kappa_{i}^{2}=0=\sum_{i}\tilde{\kappa}_{i}^{2}$ as discussed above.

  * •

_Supersymmetry._ In the supersymmetric cases, the vertex operators contain
exponential factors in the $\eta^{a}_{I}$s; these localize onto the solution
to (4.5), to yield:

$\displaystyle{\rm
e}^{F_{\mathcal{N}}}:=\exp\Bigg{(}\sum_{j<k}\frac{(u_{j}u_{k})q_{j}\cdot
q_{k}}{\sigma_{j}-\sigma_{k}}-\frac{1}{2}\sum_{j=1}^{n}(\xi_{j}v_{j})q_{j}^{2}\Bigg{)}.$

This factor carries all the dependence on the supermomenta and can be expanded
to yield the various component-amplitudes according to the supermomentum
expansion of the supermultiplet (2.9). The form of this factor moreover
guarantees that the amplitudes are supersymmetrically invariant [1, 30].

Combining these factors leads to the following expression for massive
amplitudes for super Yang–Mills on the Coulomb branch and massive bi-adjoint
scalar theory;

$\displaystyle\mathcal{A}_{n}=\int{\rm
d}\mu_{n}^{\mathrm{pol}}\,\mathcal{I}_{n}{\rm e}^{F_{\mathcal{N}}}.$ (4.9)

Here, $\mathcal{I}_{n}$ – the “integrand” of the amplitude expression –
contains all the remaining correlator factors. Since the current algebra and
Heisenberg superalgebra do not interact, $\mathcal{I}_{n}$ can be split
further into two “half-integrand” factors that can be calculated individually.
The current algebra correlator at leading trace gives the omnipresent
Parke–Taylor factor,

$\displaystyle\langle\mathfrak{t}_{\mathfrak{a}_{1}}j^{\mathfrak{a}_{1}}(\sigma_{1})\cdots\mathfrak{t}_{\mathfrak{a}_{n}}j^{\mathfrak{a}_{n}}(\sigma_{n})\rangle=\sum_{\alpha\in
S_{n}/D_{n}}\frac{\operatorname{tr}\left(\mathfrak{t}_{\alpha(1)}\cdots\mathfrak{t}_{\alpha(n)}\right)}{\sigma_{\alpha(1)\alpha(2)}\cdots\sigma_{\alpha(n)\alpha(1)}}=:\sum_{\alpha\in
S_{n}/D_{n}}\operatorname{PT}(\alpha),$

with the sum running over dihedrally inequivalent orderings. The evaluation of
the $\rho\tilde{\rho}$ half-integrand is also standard (cf. [34]), giving the
reduced determinant,

$\displaystyle\det{}^{\prime}H:=\frac{1}{(u_{1}u_{2})}\det H^{[12]}_{[12]},$

where, the $n\times n$ matrix $H$ is defined by

$\displaystyle H_{ij}=\frac{\epsilon_{iA}\epsilon_{j}^{A}}{\sigma_{ij}},\qquad
H_{ii}=-e_{i}^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}(\lambda_{\scalebox{0.6}{$A$}}\lambda_{\scalebox{0.6}{$B$}})(\sigma_{i}),$

and the sub- and superscripts indicate that both the rows and the columns 1
and 2 have been removed. It can be shown that the reduced determinant
$\det{}^{\prime}H$ is invariant under permutation of _all_ particle labels;
this can be seen directly [1, 30], but also follows from its correlator
origin. We thus arrive at the following integrands;

$\displaystyle\mathcal{I}_{n}^{\scalebox{0.6}{BAS}}=\operatorname{PT}(\alpha)\operatorname{PT}(\beta),\qquad\mathcal{I}_{n}^{\scalebox{0.6}{CB}}=\operatorname{PT}(\alpha)\det{}^{\prime}H,\qquad\mathcal{I}_{n}^{\rm
Grav}=\det{}^{\prime}H\det{}^{\prime}\tilde{H}.$ (4.10)

This completes the specification of the massive amplitude formulae (4.9). For
worked out examples at three and four points, we refer the reader to the
computations in six dimensions presented in [1]. They reproduce the formulae
present in the literature for two or four particles in the massive
supermultiplet and the remaining particles massless.

##  5 Summary and discussion

We have therefore seen that a chiral string whose target is the
complexification of Penrose’s two-twistor representation of the massive
particle phase space yields theories of massive particles in four dimensions.
The spectrum of these models includes massive particles, and correlators give
amplitude formulae for super Yang–Mills on the Coulomb branch among other
theories. These string models represent the confluence of two separate
developments: the twistor-particle program of the 70’s describing massive
particles, and the more recent ambitwistor string models describing scattering
amplitudes for massless particles. In the latter approach a chiral or
holomorphic string whose target is the complexification of the space of
massless particles yields amplitudes for theories of massless fields. Here we
have seen that the logic extends naturally to massive particles. The
significance of Penrose’s twistor description is that it provides a canonical
representation of the space as the symplectic quotient of a vector space
modulo a Hamiltonian group action allowing the BRST quantization of a free
quantum string. The twistorial description furthermore facilitates the
incorporation of fermions and supersymmetry.

From the path integral of these models, we have arrived at the compact
formulae (4.9), supplemented by (4.6) and (4.10) supported on a massive
version (4.7) of the polarised scattering equations with manifest
supersymmetry for appropriate gauge and gravity theories including massive
particles. Like all twistor-string [17, 59], CHY [13] and ambitwistor-string
amplitude formulae, all the integrations are saturated against delta functions
so that these are really residue formulae summing contributions from the
$(n-3)!$ solutions to a massive extension of the scattering equations
discussed further below. As shown in [1], the extra data in the polarised
extension is uniquely obtained by linear equations on the support of these
scattering equations and the formulae are linear in the polarization data.

The models described in this paper, and related ones, can also be derived via
a symmetry reduction of the higher-dimensional ambitwistor string models [2].
This alternative interpretation of the two-twistor string highlights that
these worldsheet models describe a subset of massive models, where the
particle masses are related to their (higher-dimensional) charges under a
symmetry. While this may appear restrictive, it includes many theories of
immediate interest; in particular, all massive particles that we encounter in
the standard model arise from the Higgs mechanism that can be obtained by
symmetry reduction. The upcoming paper [2] will also contain full details of
the fixed vertex operators and picture changing operators that we omitted for
brevity. Mirroring the close relation of the models, the resulting amplitude
formulae for massive particles are also closely related to those obtained
previously by dimensional reduction from six and five dimensions [1, 30, 34].
These can further be related to the formulae of [11] by a change of gauge
choice of an embedding inside a Lagrangian grassmannian [62]. At low point
orders, these expressions match the results obtained in [3, 21, 35, 39, 42,
49] by BCFW recursion.

Contrary to the massless four-dimensional formulae of [29], in which the
double copy properties are hidden in the measure, the expressions derived here
present the standard structure with two half integrands that can be combined
to form amplitudes for scalars, spin-1 and spin-2 particles as in the CHY
formulae and corresponding RNS models of [18, 45]. The embedding of the
massless models in these novel massive ones give a new outlook on the
corresponding massless formulae.

We conclude by discussing further features and connections to related formulae
for massive amplitudes and future directions that we plan to address in [2].

Massive scattering equations. The massive polarised scattering equations (4.7)
are closely related to the massive scattering equations in the CHY framework
proposed by Dolan & Goddard and by Naculich [25, 48]. We first note that as
remarked after (2.5) that the delta functions imply:

$\displaystyle
K^{AB}_{i}P_{AB}(\sigma_{i}):=k_{i}^{\alpha\dot{\alpha}}P_{\alpha\dot{\alpha}}-m_{i}\lambda^{2}(\sigma_{i})-m_{i}\tilde{\lambda}^{2}(\sigma_{i})=0.$

After the path-integral we have that $\lambda_{Aa}$ is given in (4.5) and we
can compute [1, 30]

$\displaystyle
P_{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}(\sigma):=(\lambda_{A}\lambda_{B})=\sum_{i}\frac{K_{i{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}}{\sigma-\sigma_{i}}\,{\rm
d}\sigma,$ (5.1)

where $K_{i}$ is defined as in (4.8). When the momenta $K_{i}$ are null,
$P_{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}$ plays a crucial role in the
vector representation of the ambitwistor string [45], where a similar path
integral calculation to the one in Section 4 localizes $P$ onto (5.1).
Naculich then showed that certain massive amplitudes localize on a massive
extension of the CHY scattering equations,

$\displaystyle
0=K_{i{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}P^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}(\sigma_{i})=\sum_{j\neq
i}\frac{k_{i}\cdot k_{j}-M_{i}M_{j}}{\sigma_{i}-\sigma_{j}}.$

The polarised scattering equations $\mathcal{E}_{i{\scalebox{0.6}{$A$}}}$ then
clearly imply Naculich’s massive scattering equations via
$K_{i{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}P^{{\scalebox{0.6}{$A$}}{\scalebox{0.6}{$B$}}}(\sigma_{i})=\det\big{(}\kappa_{i{\scalebox{0.6}{$A$}}}^{a},\lambda^{b{\scalebox{0.6}{$A$}}}(\sigma_{i})\big{)}=0$,
where the last equality holds on the support of
$\mathcal{E}_{i{\scalebox{0.6}{$A$}}}$. Vector ambitwistor models that
directly yield these equations will be presented in [2].

Masses from $\boldsymbol{R}$-symmetry. Novel gauge and gravitational models
with massive degrees of freedom – albeit massless gluons and gravitons – can
be formulated by choosing $j^{H}$ from the Cartan of the $R$-symmetry. In
particular, we may take
$j^{H}=m_{1}\tilde{\eta}^{1}_{a}\eta_{1}^{a}+m_{2}\tilde{\eta}^{2}_{a}\eta_{2}^{a}$
in the two-twistor models $S^{\scalebox{0.7}{CB}}$ and
$S^{\scalebox{0.7}{sugra}}$. For generic parameters $m_{1}$ and $m_{2}$, this
results in non-supersymmetric models with massive scalars and fermions (and
spin-1 in the supergravity model); no mass can be assigned to the spin-2 field
since it transforms trivially under the $R$-symmetry. It is also possible to
preserve a residual $\mathcal{N}=2$ supersymmetry for gauge theory (and
$\mathcal{N}=4$ for supergravity) by constructing the current $j^{H}$ from
only one Cartan generator with $m_{2}=0$.

Loops from a gluing operator. This two-twistor string provides an alternative
formulation of the massless ambitwistor string [29], but in a framework in
which a massless field can be _deformed_ to go off-shell. This allows us to
adapt the elegant method of deriving loop amplitudes in [57] via a _gluing
operator_ but now to theories with fermions and supersymmetry such as super
Yang–Mills theory. This construction arises from the vector model of the
ambitwistor string, where it has been shown that loop integrands can be
localized on a _nodal sphere_ rather than the torus that more usually arises
in string theory [32, 33]. At the level of the worldsheet model, this nodal
structure of loop correlators is realized in the worldsheet CFT as the _gluing
operator_ $\Delta$, which encodes the propagator of the target-space field
theory. One-loop amplitudes then have two equivalent descriptions; a string-
inspired one as correlators on a torus, and an alternative representation as
$g=0$ correlators in the presence of a gluing operator.

### Acknowledgements

We are grateful to Alexander Ochirov for discussions and comments. GA is
supported by the EPSRC under grant EP/R513295/1. YG gratefully acknowledges
support from the CUniverse research promotion project “Toward World-class
Fundamental Physics” of Chulalongkorn University (grant reference CUAASC). LJM
is grateful to the STFC for support under grant ST/T000864/1. We would also
like to thank the SIGMA team for their courage and dedication, working through
difficult circumstances in Kyiv in the face of the Russian invasion and
aggression.

## References

  * [1] Albonico G., Geyer Y., Mason L., Recursion and worldsheet formulae for 6d superamplitudes, J. High Energy Phys. 2020 (2020), no. 8, 066, 90 pages, arXiv:2001.05928.
  * [2] Albonico G., Geyer Y., Mason L., Massive ambitwistor string models, Parts I & II, in preparation.
  * [3] Arkani-Hamed N., Huang T.C., Huang Y., Scattering amplitudes for all masses and spins, J. High Energy Phys. 2021 (2021), no. 11, 070, 75 pages, arXiv:1709.04891.
  * [4] Berkovits N., Alternative string theory in twistor space for $N=4$ super-Yang–Mills theory, Phys. Rev. Lett. 93 (2004), 011601, 3 pages, arXiv:hep-th/0402045.
  * [5] Berkovits N., Lize M., Field theory actions for ambitwistor string and superstring, J. High Energy Phys. 2018 (2018), no. 9, 097, 16 pages, arXiv:1807.07661.
  * [6] Bern Z., Carrasco J.J.M., Johansson H., New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008), 085011, 19 pages, arXiv:0805.3993.
  * [7] Bern Z., Carrasco J.J.M., Johansson H., Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010), 061602, 4 pages, arXiv:1004.0476.
  * [8] Bette A., de Azcárraga J.A., Lukierski J., Miquel-Espanya C., Massive relativistic free fields with Lorentz spins and electric charges, Phys. Lett. B 595 (2004), 491–497, arXiv:hep-th/0405166.
  * [9] Bette A., Lukierski J., Miquel-Espanya C., Two-twistor space, commuting composite Minkowski coordinates and particle dynamics, in Fundamental Interactions and Twistor-Like Methods, AIP Conf. Proc., Vol. 767, Amer. Inst. Phys., Melville, NY, 2005, 44–56, arXiv:hep-th/0503134.
  * [10] Bjerrum-Bohr N.E.J., Bourjaily J.L., Damgaard P.H., Feng B., Manifesting color-kinematics duality in the scattering equation formalism, J. High Energy Phys. 2016 (2016), no. 9, 094, 17 pages, arXiv:1608.00006.
  * [11] Cachazo F., Guevara A., Heydeman M., Mizera S., Schwarz J.H., Wen C., The S matrix of 6D super Yang–Mills and maximal supergravity from rational maps, J. High Energy Phys. 2018 (2018), no. 9, 125, 85 pages, arXiv:1805.11111.
  * [12] Cachazo F., He S., Yuan E.Y., Scattering in three dimensions from rational maps, J. High Energy Phys. 2013 (2013), no. 10, 141, 20 pages, arXiv:1306.2962.
  * [13] Cachazo F., He S., Yuan E.Y., Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014), 171601, 4 pages, arXiv:1307.2199.
  * [14] Cachazo F., He S., Yuan E.Y., Scattering of massless particles: scalars, gluons and gravitons, J. High Energy Phys. 2014 (2014), no. 7, 033, 20 pages, arXiv:1309.0885.
  * [15] Cachazo F., He S., Yuan E.Y., Scattering equations and matrices: from Einstein to Yang–Mills, DBI and NLSM, J. High Energy Phys. 2015 (2015), no. 7, 149, 43 pages, arXiv:1412.3479.
  * [16] Cachazo F., Mason L., Skinner D., Gravity in twistor space and its Grassmannian formulation, SIGMA 10 (2014), 051, 28 pages, arXiv:1207.4712.
  * [17] Cachazo F., Skinner D., Gravity from rational curves in twistor space, Phys. Rev. Lett. 110 (2013), 161301, 4 pages, arXiv:1207.0741.
  * [18] Casali E., Geyer Y., Mason L., Monteiro R., Roehrig K.A., New ambitwistor string theories, J. High Energy Phys. 2015 (2015), no. 11, 038, 29 pages, arXiv:1506.08771.
  * [19] Conde E., Joung E., Mkrtchyan K., Spinor-helicity three-point amplitudes from local cubic interactions, J. High Energy Phys. 2016 (2016), no. 8, 040, 29 pages, arXiv:1605.07402.
  * [20] Conde E., Marzolla A., Lorentz constraints on massive three-point amplitudes, J. High Energy Phys. 2016 (2016), no. 9, 041, 42 pages, arXiv:1601.08113.
  * [21] Craig N., Elvang H., Kiermaier M., Slatyer T.R., Massive amplitudes on the Coulomb branch of $\mathcal{N}=4$ SYM, J. High Energy Phys. 2011 (2011), no. 12, 097, 38 pages, arXiv:1104.2050.
  * [22] de Azcarraga J.A., Fedoruk S., Izquierdo J.M., Lukierski J., Two-twistor particle models and free massive higher spin fields, J. High Energy Phys. 2015 (2015), no. 4, 010, 39 pages, arXiv:1409.7169.
  * [23] Deguchi S., Okano S., Gauged twistor formulation of a massive spinning particle in four dimensions, Phys. Rev. D 93 (2016), 045016, 22 pages, Erratum, Phys. Rev. D 93 (2016), 1 page, arXiv:1512.07740.
  * [24] Deguchi S., Suzuki T., Twistor formulation of a massive particle with rigidity, Nuclear Phys. B 932 (2018), 385–424, arXiv:1707.04713.
  * [25] Dolan L., Goddard P., Proof of the formula of Cachazo, He and Yuan for Yang–Mills tree amplitudes in arbitrary dimension, J. High Energy Phys. 2014 (2014), no. 4, 010, 24 pages, arXiv:1311.5200.
  * [26] Fedoruk S., Lukierski J., Massive twistor particle with spin generated by Souriau–Wess–Zumino term and its quantization, Phys. Lett. B 733 (2014), 309–315, arXiv:1403.4127.
  * [27] Fedoruk S., Zima V.G., Bitwistor formulation of massive spinning particle, arXiv:hep-th/0308154.
  * [28] Ferber A., Supertwistors and conformal supersymmetry, Nuclear Phys. B 132 (1978), 55–64.
  * [29] Geyer Y., Lipstein A.E., Mason L., Ambitwistor strings in four dimensions, Phys. Rev. Lett. 113 (2014), 081602, 5 pages, arXiv:1404.6219.
  * [30] Geyer Y., Mason L., Polarized scattering equations for 6D superamplitudes, Phys. Rev. Lett. 122 (2019), 101601, 6 pages, arXiv:1812.05548.
  * [31] Geyer Y., Mason L., Supersymmetric S-matrices from the worldsheet in 10 & 11d, Phys. Lett. B 804 (2020), 135361, 6 pages, arXiv:1901.00134.
  * [32] Geyer Y., Mason L., Monteiro R., Tourkine P., Loop integrands for scattering amplitudes from the Riemann sphere, Phys. Rev. Lett. 115 (2015), 121603, 5 pages, arXiv:1507.00321.
  * [33] Geyer Y., Mason L., Monteiro R., Tourkine P., One-loop amplitudes on the Riemann sphere, J. High Energy Phys. 2016 (2016), no. 3, 114, 53 pages, arXiv:1511.06315.
  * [34] Geyer Y., Mason L., Skinner D., Ambitwistor strings in six and five dimensions, J. High Energy Phys. 2021 (2021), no. 8, 153, 34 pages, arXiv:2012.15172.
  * [35] Herderschee A., Koren S., Trott T., Constructing $\mathcal{N}=4$ Coulomb branch superamplitudes, J. High Energy Phys. 2019 (2019), no. 8, 107, 50 pages, arXiv:1902.07205.
  * [36] Hughston L.P., Twistors and particles, Lecture Notes in Physics, Vol. 97, Springer-Verlag, Berlin – New York, 1979.
  * [37] Hughston L.P., Hurd T.R., A cohomological description of massive fields, Proc. Roy. Soc. London Ser. A 378 (1981), 141–154.
  * [38] Johansson H., Ochirov A., Pure gravities via color-kinematics duality for fundamental matter, J. High Energy Phys. 2015 (2015), no. 11, 046, 52 pages, arXiv:1407.4772.
  * [39] Johansson H., Ochirov A., Double copy for massive quantum particles with spin, J. High Energy Phys. 2019 (2019), no. 9, 040, 44 pages, arXiv:1906.12292.
  * [40] Kunz C., Four dimensional anomaly-free twistor string, arXiv:2004.04842.
  * [41] Kunz C., A note on classical aspects of the four dimensional anomaly-free twistor string, arXiv:2104.06584.
  * [42] Lazopoulos A., Ochirov A., Shi C., All-multiplicity amplitudes with four massive quarks and identical-helicity gluons, J. High Energy Phys. 2022 (2022), no. 3, 009, 38 pages, arXiv:2111.06847.
  * [43] Mason L.J., Reid-Edwards R.A., The supersymmetric Penrose transform in six dimensions, arXiv:1212.6173.
  * [44] Mason L.J., Reid-Edwards R.A., Taghavi-Chabert A., Conformal field theories in six-dimensional twistor space, J. Geom. Phys. 62 (2012), 2353–2375, arXiv:1111.2585.
  * [45] Mason L.J., Skinner D., Ambitwistor strings and the scattering equations, J. High Energy Phys. 2014 (2014), no. 7, 048, 34 pages, arXiv:1311.2564.
  * [46] Monteiro R., O’Connell D., The kinematic algebras from the scattering equations, J. High Energy Phys. 2014 (2014), no. 3, 110, 28 pages, arXiv:1311.1151.
  * [47] Naculich S.G., Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, J. High Energy Phys. 2014 (2014), no. 9, 029, 21 pages, arXiv:1407.7836.
  * [48] Naculich S.G., CHY representations for gauge theory and gravity amplitudes with up to three massive particles, J. High Energy Phys. 2015 (2015), no. 5, 050, 19 pages, arXiv:1501.03500.
  * [49] Ochirov A., Helicity amplitudes for QCD with massive quarks, J. High Energy Phys. 2018 (2018), no. 4, 089, 22 pages, arXiv:1802.06730.
  * [50] Okano S., Deguchi S., A no-go theorem for the $n$-twistor description of a massive particle, J. Math. Phys. 58 (2017), 031701, 6 pages, arXiv:1606.01339.
  * [51] Penrose R., Twistors and particles: an outline, in Quantum Theory and the Structure of Space-Time (Feldafing Conference of the Max-Planck Institute), Carl Hanser Verlag, Münich, 1975, 129–145.
  * [52] Penrose R., The twistor programme, Rep. Math. Phys. 12 (1977), 65–76.
  * [53] Penrose R., Rindler W., Spinors and space-time. Vol. 2. Spinor and twistor methods in space-time geometry, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1986.
  * [54] Perjés Z., Twistor variables of relativistic mechanics, Phys. Rev. D 11 (1975), 2031–2041.
  * [55] Perjés Z., Perspectives of Penrose theory in particle physics, Rep. Math. Phys. 12 (1977), 193–211.
  * [56] Reid-Edwards R., On closed twistor string theory, arXiv:1212.6047.
  * [57] Roehrig K.A., Skinner D., A gluing operator for the ambitwistor string, J. High Energy Phys. 2018 (2018), no. 1, 069, 33 pages, arXiv:1709.03262.
  * [58] Roiban R., Spradlin M., Volovich A., Tree-level $S$ matrix of Yang–Mills theory, Phys. Rev. D 70 (2004), 026009, 10 pages, arXiv:hep-th/0403190.
  * [59] Roiban R., Volovich A., All conjugate-maximal-helicity-violating amplitudes from topological open string theory in twistor space, Phys. Rev. Lett. 93 (2004), 131602, 4 pages, arXiv:hep-th/0402121.
  * [60] Sämann C., Wolf M., On twistors and conformal field theories from six dimensions, J. Math. Phys. 54 (2013), 013507, 44 pages, arXiv:1111.2539.
  * [61] Sämann C., Wolf M., Non-abelian tensor multiplet equations from twistor space, Comm. Math. Phys. 328 (2014), 527–544, arXiv:1205.3108.
  * [62] Schwarz J.H., Wen C., Unified formalism for 6D superamplitudes based on a symplectic Grassmannian, J. High Energy Phys. 2019 (2019), no. 8, 125, 26 pages, arXiv:1907.03485.
  * [63] Skinner D., Twistor strings for $\mathcal{N}=8$ supergravity, J. High Energy Phys. 2020 (2020), no. 4, 047, 50 pages, arXiv:1301.0868.
  * [64] Tod P., Some symplectic forms arising in twistor theory, Rep. Math. Phys. 11 (1977), 339–346.
  * [65] Witten E., Superstring perturbation theory revisited, arXiv:1209.5461.
  * [66] Witten E., Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys. 252 (2004), 189–258, arXiv:hep-th/0312171.