UUITP-37/21

10D Super-Yang-Mills Scattering Amplitudes From Its Pure Spinor Action

Maor Ben-Shahar††† † email<EMAIL_ADDRESS>and Max Guillen‡†† ‡
email<EMAIL_ADDRESS>

Department of Physics and Astronomy, 75108 Uppsala, Sweden

Using the pure spinor master action for 10D super-Yang-Mills in the gauge
$b_{0}V=Q\Xi$, tree-level scattering amplitudes are calculated through the
perturbiner method, and shown to match those obtained from pure spinor CFT
techniques. We find kinematic numerators made of nested $b$-ghost operators,
and show that the Siegel gauge condition $b_{0}V=0$ gives rise to color-
kinematics duality satisfying numerators whose Jacobi identity follows from
the Jacobi identity of a kinematic algebra.

August 2021

1\. Introduction

It has been known for a long time that maximally supersymmetric theories lack
a manifestly Lorentz covariant Lagrangian formulation in ordinary superspace
[[1]1,,[2]2,,[3]3], making it more difficult to exploit supersymmetry in
amplitudes computations. However, things change when superspace is extended
[4]. Indeed, non-minimal pure spinor variables have been used in
[[5]5,,[6]6,,[7]7,,[8]8] for constructing manifestly supersymmetric actions
for several maximally supersymmetric theories like 10D super-Yang-Mills, 10D
super-Born-Infeld and 11D supergravity. This makes pure spinor quantum field
theory a promising approach for evaluating scattering amplitudes in an elegant
way. This should not be a surprise at all. In fact, the pure spinor formalism
for superstrings [9] is nowadays arguably the most powerful and efficient
framework for computing string scattering amplitudes as compared to the
traditional Ramond-Neveu-Schwarz and Green-Schwarz formalisms
[[10]10,,[11]11,,[12]12,,[13]13,,[14]14].

Such pure spinor field theory actions have been constructed from a single pure
spinor superfield, which exhibits the field-antifield symmetry of the
corresponding Batalin-Vilkovisky descriptions of the theories in study. Since
standard gauge-fixing fermions are incompatible with this symmetry,
alternative gauge-fixing conditions are necessary for the computation of
scattering amplitudes. Using inspiration from string field theory, the Siegel
gauge condition $b_{0}V=0$ [[15]15,,[16]16], has been proposed in
[[8]8,,[17]17] as a natural gauge choice. Although no explicit computations
have been done so far in this pure field theory setting, power counting
arguments have been used to discuss the ultraviolet behaviour of 10D super-
Yang-Mills and 11D supergravity [[18]18,,[19]19,,[20]20,,[21]21].

On the other hand, recent progress in the study of scattering amplitudes has
revealed new structures previously hidden from Lagrangian formulations of
field theories. The BCJ duality between color-and kinematics
[[22]22,,[23]23,,[24]24] states that given an amplitude formulated as a sum
over cubic diagrams,

$\eqalignno{A&=\sum_{i\in\Gamma_{i}}{c_{i}n_{i}\over D_{i}}\ ,&({1.1})}$

it is possible to find [[25]25,,[26]26] representations of the kinematic
numerators such that they obey the same relations as the color factors

$\eqalignno{c_{i}+c_{j}+c_{k}&=0\Rightarrow n_{i}+n_{j}+n_{k}=0\ .&({1.2})}$

Having obtained such numerators, it is possible to “double-copy” them by
replacing $c_{i}\to n_{i}$, obtaining amplitudes from a theory of gravity,
dramatically simplifying the computation of gravity observables
[[27]27,,[28]28]. Since the identities obeyed by the color factors are
inherited from the Jacobi identity of the color algebra, it is tempting to ask
if a similar algebra exists for kinematics. Indeed, algebraic origins of this
duality have been observed in certain sectors of Yang-Mills theory
[[29]29,,[30]30,,[31]31,,[32]32], but the general case is still not fully
understood, appearing to require an infinite number of auxiliary fields
[[33]33,,[34]34,,[35]35,,[36]36,,[37]37] whose job is to modify the numerators
by generalized gauge transformations.

In this paper we study the 10D super-Yang-Mills pure spinor action subject to
two different gauge-fixing conditions. The first gauge choice is a relaxation
of the standard Siegel gauge condition, namely, $b_{0}V=Q\Xi$, where $b_{0}$
is the quantum-mechanical operator version of the pure spinor $b$-ghost, and
$\Xi$ is some superfield. The usual unintegrated vertex operators
$V=\lambda^{\alpha}A_{\alpha}$, widely used in superstring scattering
amplitudes, are explicitly shown to satisfy such a condition. We then compute
the corresponding equations of motion in pure spinor superspace, and apply the
perturbiner method to compute tree-level scattering amplitudes from simple
contractions of Berends-Giele currents. Our numerators take the form of nested
$b$-ghosts acting on external states, for example the half-ladder numerator at
five points is $\langle b_{0}(b_{0}(V_{1}V_{2})V_{3})V_{4}V_{5}\rangle$. We
explicitly calculate such expressions for 4-point and 5-point amplitudes by
making use of the so-called physical operators introduced in [7], in the
process collecting useful identities for these operators. The amplitudes thus
obtained are shown to coincide with those found from open pure spinor
superstrings in the particle-limit [[10]10,,[38]38,,[39]39]. We also study the
Siegel gauge, $b_{0}V=0$, in which the vertex operators are dependant on non-
minimal variables. In this scenario, nilpotency of the $b$-ghost is shown to
imply color-kinematics duality off-shell. The nested $b$-ghosts define a
Poisson bracket, whose Jacobi identity is the kinematic Jacobi identity. The
dual Lie algebra to this Poisson algebra is an algebra of diffeomorphisms that
preserve the Siegel gauge. Such a gauge appears to be the only one that allows
for both manifest crossing symmetry and color-kinematics duality.

The paper is organized as follows. In section 2, we discuss the 10D pure
spinor superparticle and its BRST-cohomology. Non-minimal pure spinors are
then introduced in order to construct both well-defined action principles, and
the $b$-ghost which satisfies $\\{Q,b\\}={P^{2}\over 2}$. In section 3, we
discuss two approaches for computing tree amplitudes, namely, Feynman diagrams
and Berends-Giele recursion relations. In section 4, we apply the systematics
of the latter and compute N-point scattering amplitudes in 10D super-Yang-
Mills when external states satisfy $b_{0}V=Q\Xi$. The stronger condition
$b_{0}V=0$ is then discussed and shown to manifestly reproduce color-
kinematics duality. Calculations of different expressions involving nested
$b$-ghosts are explicitly carried out in Appendix A. Finally, we close with
discussions and further research directions in section 5.

2\. Pure Spinor Action Of 10D Super-Yang-Mills

In this section we review the minimal 10D pure spinor superparticle. After
introducing non-minimal variables, a well-defined pure spinor measure is
constructed as well as a composite operator $b$, the so-called $b$-ghost,
satisfying $\\{Q,b\\}={P^{2}\over 2}$.

2.1. 10D Pure Spinor Superparticle

The 10D pure spinor superparticle [40] is defined by the action

$\eqalignno{S&=\int
d\tau\,\bigg{[}P_{m}\partial_{\tau}X^{m}+p_{\alpha}\partial_{\tau}\theta^{\alpha}+w_{\alpha}\partial_{\tau}\lambda^{\alpha}-{1\over
2}P^{2}\bigg{]}\ ,&({2.1})}$

and the BRST operator

$\eqalignno{Q&=\lambda^{\alpha}d_{\alpha}\ ,&({2.2})}$

where we are using letters from the beginning/middle of the Greek/Latin
alphabet to denote $SO(1,9)$ spinor/vector indices. Furthermore,
$(X^{m},\theta^{\alpha})$ stand for the superspace coordinates, and
$(P_{m},p_{\alpha})$ denote their respective conjugate momenta. The variable
$\lambda^{\alpha}$ is a pure spinor satisfying $\lambda\gamma^{m}\lambda=0$,
and $w_{\alpha}$ is its respective conjugate momentum, which is defined up to
the gauge transformation $\delta
w_{\alpha}=(\gamma^{m}\lambda)_{\alpha}\rho_{m}$, for any $\rho_{m}$. The
Green-Schwarz fermionic constraint $d_{\alpha}$ in (2.2) is defined as usual:
$d_{\alpha}=p_{\alpha}+{1\over 2}(\gamma^{m}\theta)_{\alpha}P_{m}$, and it
commutes with the supersymmetry generators: $q_{\alpha}=p_{\alpha}-{1\over
2}(\gamma^{m}\theta)_{\alpha}P_{m}$. Finally, $(\gamma^{m})_{\alpha\beta}$,
$(\gamma^{m})^{\alpha\beta}$ denote the familiar $SO(1,9)$ Pauli matrices
satisfying
$(\gamma^{(m})_{\alpha\beta}(\gamma^{n)})^{\beta\delta}=\eta^{mn}\delta^{\delta}_{\alpha}$.

The action (2.1) is invariant under Super-Poincare transformations as well as
under the global symmetry generated by $J=-\lambda^{\alpha}w_{\alpha}$. The
charge associated to the current $J$ will be referred to as ghost number, and
thus $\lambda^{\alpha}$, $w_{\alpha}$ carry ghost numbers $1$, $-1$,
respectively. As is well known, the Hilbert space of the superparticle (2.1)
will be described by the cohomology of the BRST operator (2.2), which can be
conveniently separated into different ghost number sectors. Indeed, it has
been computed from different methods that there only exists non-trivial
cohomology up to ghost number 3. Schematically,

$\eqalignno{\Psi(x,\theta,\lambda)&=\Psi^{(0)}(x,\theta,\lambda)+\Psi^{(1)}(x,\theta,\lambda)+\Psi^{(2)}(x,\theta,\lambda)+\Psi^{(3)}(x,\theta,\lambda)\
,&({2.3})}$

where the superscript in $\Psi^{(i)}$ stands for the ghost number sector which
$\Psi^{(i)}$ belongs to. In this manner, the pure spinor superfields
$\Psi^{(0)}$, $\Psi^{(1)}$, $\Psi^{(2)}$, $\Psi^{(3)}$ were found to describe
the gauge symmetry ghosts, physical fields, antifields, and ghost antifields
of 10D super-Yang-Mills, respectively. Let us illustrate this with the ghost
number one sector
$\Psi^{(1)}(x,\theta,\lambda)=\lambda^{\alpha}A_{\alpha}(x,\theta)$. This
superfield is subject to the physical state conditions

$\eqalignno{Q\Psi^{(1)}&=0\ ,\ \ \ \ \ \delta\Psi^{(1)}=Q\Lambda\ ,&({2.4})}$

where $\Lambda$ is an arbitrary gauge superfield. The implications of (2.4) on
$A_{\alpha}(x,\theta)$ read

$\eqalignno{(\gamma^{mnpqr})^{\alpha\beta}D_{\alpha}A_{\beta}&=0\ ,\ \ \ \ \
\delta A_{\alpha}=D_{\alpha}\Lambda\ ,&({2.5})}$

where $D_{\alpha}=\partial_{\alpha}+{1\over
2}(\gamma^{m}\theta)_{\alpha}\partial_{m}$ is the usual supersymmetric
derivative. Eqns. (2.5) are nothing but the superspace equations of motion of
linearized 10D super-Yang-Mills. They can easily be solved in the so-called
Harnad-Shnider gauge [41], which imposes $\theta^{\alpha}A_{\alpha}=0$. In
this gauge, $A_{\alpha}(x,\theta)$ takes the form [42]

$\eqalignno{A_{\alpha}(x,\theta)&={1\over
2}(\gamma^{m}\theta)_{\alpha}a_{m}(x)-{1\over
3}(\gamma^{m}\theta)_{\alpha}(\theta\gamma_{m}\chi(x))-{1\over
16}(\gamma_{p}\theta)_{\alpha}(\theta\gamma^{mnp}\theta)\partial_{m}a_{n}(x)\cr&+{1\over
60}(\gamma_{p}\theta)_{\alpha}(\theta\gamma^{mnp}\theta)(\theta\gamma_{m}\partial_{n}\chi(x))+\ldots&({2.6})}$

where $\ldots$ stands for higher derivative terms of $a_{m}$, $\chi^{\alpha}$.
The field $a_{m}$ in (2.6) satisfies the relations $\partial^{m}a_{m}=0$,
$\delta a_{m}=\partial_{m}s$ for any $s$, and $\chi^{\alpha}$ satisfies the
equation $(\gamma^{m})_{\alpha\beta}\partial_{m}\chi^{\beta}=0$, so that
$a_{m}$, $\chi^{\alpha}$ are identified with the 10D super-Yang-Mills gluon
and gluino fields, respectively. Similar arguments apply to the other ghost
sectors, and so (2.3) indeed describes the Batalin-Vilkovisky formulation of
linearized 10D super-Yang-Mills.

The top scalar cohomology of $Q$ can be used to define manifestly
supersymmetric correlators [9]. Such a measure has played a fundamental role
in the computation of superstring scattering amplitudes. Its explicit form
reads

$\eqalignno{\langle(\lambda\gamma^{m}\theta)(\lambda\gamma^{n}\theta)(\lambda\gamma^{p}\theta)(\theta\gamma_{mnp}\theta)\rangle&=1\
.&({2.7})}$

One might then naively use this measure for construcing a manifestly
supersymmetric action reproducing eqns. (2.4). However, the measure (2.7) is
degenerate [[8]8,,[43]43] and so it is not adequate for such a purpose. This
problem is solved by introducing non-minimal pure spinor variables. This is
what we do next.

2.2. Non-Minimal Variables

The pair of conjugate variables
$(\overline{\lambda}_{\alpha},\overline{w}^{\alpha})$,
$(r_{\alpha},s^{\alpha})$, where $\overline{\lambda}_{\alpha}$ is a pure
spinor satisfying $\overline{\lambda}\gamma^{m}\overline{\lambda}=0$ and
$r_{\alpha}$ is a fermionic variable satisfying
$\overline{\lambda}\gamma^{m}r=0$, can be introduced into the model (2.1)
through the quartet argument, that is

$\eqalignno{S&=\int
d\tau\,\bigg{[}P_{m}\partial_{\tau}X^{m}+p_{\alpha}\partial_{\tau}\theta^{\alpha}+w_{\alpha}\partial_{\tau}\lambda^{\alpha}+\overline{w}^{\alpha}\partial_{\tau}\overline{\lambda}_{\alpha}+s^{\alpha}\partial_{\tau}r_{\alpha}-{1\over
2}P^{2}\bigg{]}\ ,&({2.8})}$

and,

$\eqalignno{Q&=Q_{0}+r_{\alpha}\overline{w}^{\alpha}\ ,&({2.9})}$

so that the BRST-cohomology does not change [44]. Indeed, it is not hard to
see it is always possible to find a representative element in the cohomology
of (2.9) which is independent of the non-minimal variables. These variables
allow us to write down a well-defined measure for formulating action
principles in pure spinor superspace. Such a measure takes the form
$[dZ]=[d\lambda][d\overline{\lambda}][dr]$, where

$\eqalignno{[d\lambda]\lambda^{\beta}\lambda^{\delta}\lambda^{\gamma}&=(\epsilon
T^{-1})^{\beta\delta\gamma}_{\alpha_{1}\ldots\alpha_{11}}d\lambda^{\alpha_{1}}\ldots
d\lambda^{\alpha_{11}}&({2.10})\cr[d\overline{\lambda}]\overline{\lambda}_{\beta}\overline{\lambda}_{\delta}\overline{\lambda}_{\gamma}&=(\epsilon
T)_{\beta\delta\gamma}^{\alpha_{1}\ldots\alpha_{11}}d\overline{\lambda}_{\alpha_{1}}\ldots
d\overline{\lambda}_{\alpha_{11}}&({2.11})\cr[dr]&=(\epsilon
T^{-1})^{\beta\delta\gamma}\overline{\lambda}_{\beta}\overline{\lambda}_{\delta}\overline{\lambda}_{\gamma}\big{(}{\partial\over\partial
r_{\alpha_{1}}}\big{)}\ldots\big{(}{\partial\over\partial
r_{\alpha_{11}}}\big{)}&({2.12})}$

and the Lorentz invariant tensors $(\epsilon
T^{-1})^{\beta\gamma\delta}_{\alpha_{1}\ldots\alpha_{11}}$, $(\epsilon
T)_{\beta\gamma\delta}^{\alpha_{1}\ldots\alpha_{11}}$ are symmetric and gamma-
traceless in $(\beta,\delta,\gamma)$ and antisymmetric in
$[\alpha_{1},\ldots,\alpha_{11}]$. Their explicit forms can be found in
[[9]9,,[44]44,,[45]45]. For instance, up to an overall normalization factor

$\eqalignno{(\epsilon
T)^{\alpha_{1}\ldots\alpha_{11}}_{\beta\gamma\delta}&=\epsilon^{\alpha_{1}\ldots\alpha_{16}}(\gamma^{m})_{\alpha_{12}\eta}(\gamma^{n})_{\alpha_{13}\epsilon}(\gamma^{p})_{\alpha_{14}\kappa}(\gamma_{mnp})_{\alpha_{15}\alpha_{16}}\bigg{[}\delta^{\eta}_{(\beta}\delta^{\epsilon}_{\gamma}\delta^{\kappa}_{\delta)}-{1\over
40}(\gamma^{q})_{(\beta\gamma}\delta^{\eta}_{\delta)}(\gamma_{q})^{\epsilon\kappa}\bigg{]}\cr&&({2.13})}$

The linearized 10D super-Yang-Mills action then reads

$\eqalignno{S&=\int d^{10}xd^{16}\theta[dZ]{\cal{N}}\bigg{(}{1\over 2}\Psi
Q\Psi\bigg{)}\ ,&({2.14})}$

where ${\cal{N}}=e^{-\\{Q,\xi\\}}$ with $\xi=\overline{\lambda}\theta$, is a
regularization factor which prevents the appearance of undetermined
expressions as a result of the zero mode integrations of non-compact bosonic
and fermionic variables, and $\Psi$ is a pure spinor superfield which
generically depends on non-minimal variables. Due to the existence of the
operator
$\tilde{\xi}={\overline{\lambda}\theta\over\lambda\overline{\lambda}+r\theta}$
satisfying $Q\tilde{\xi}=1$, pure spinor integrands as that in (2.14) will be
restricted to diverge slower than $\lambda^{-8}\overline{\lambda}^{-11}$ [44].
Integrands diverging faster than $\lambda^{-8}\overline{\lambda}^{-11}$
require a different regularization scheme [46], which will not be discussed in
this paper. It is straightforward to show that the pure spinor action (2.14)
indeed reproduces eqns. (2.4). In order to introduce interactions, the field-
antifield structure, inherently described by the pure spinor superfield
$\Psi$, can be exploited by introducing the so-called pure spinor antibracket,
defined via

$\eqalignno{(A,B)&=\int{\delta_{R}A\over\delta\Psi(Z)}[dZ]{\delta_{L}B\over\delta\Psi(Z)}\
,&({2.15})}$

where $L$, $R$ denote left, right derivatives, respectively. Then, a pure
spinor master action must satisfy

$\eqalignno{(S,S)&=0\ .&({2.16})}$

As can easily be demonstrated, the action (2.14) satisfies the master equation
(2.16). The simplest deformation of (2.14) which does not explicitly involve
non-minimal variables and satisfies (2.16) reads

$\eqalignno{S&=\int\,d^{10}xd^{16}\theta[dZ]{\cal{N}}\,{\tenrm
Tr}\bigg{(}{1\over 2}\Psi Q\Psi+{g\over 3}\Psi\Psi\Psi\bigg{)}\ ,&({2.17})}$

where ${\tenrm Tr}$ is the trace taken over gauge group generator matrices,
and $g$ is the coupling constant. The equations of motion and gauge
transformations following from (2.17) can readily be computed to be

$\eqalignno{Q\Psi+g\Psi\Psi&=0\ ,\ \ \ \ \ \
\delta\Psi=Q\Lambda+g[\Psi,\Lambda]\ ,&({2.18})}$

where $\Lambda$ is an arbitrary gauge superfield. Eqns. (2.18) imply that the
ghost number one sector $\Psi^{(1)}=\lambda^{\alpha}A_{\alpha}$ satisfies

$\eqalignno{(\gamma^{mnpqr})^{\alpha\beta}(D_{\alpha}A_{\beta}+gA_{\alpha}A_{\beta})&=0\
,\ \ \ \ \ \ \delta A_{\alpha}=D_{\alpha}\Lambda+g[A_{\alpha},\Lambda]\
.&({2.19})}$

These equations are nothing but the superspace constraints describing 10D non-
abelian super-Yang-Mills. Therefore, (2.18) can be viewed as the superspace
equations of motion describing full 10D super-Yang-Mills in its antifield
formulation.

2.3. The $b$-ghost

In the non-minimal pure spinor worldline formalism, there exists a composite
operator $b$, also known as the $b$-ghost, which obeys the relation

$\eqalignno{\\{Q,b\\}&={P^{2}\over 2}\ .&({2.20})}$

Its explicit form reads

$\eqalignno{b={(\overline{\lambda}\gamma^{m}d)\over
2(\lambda\overline{\lambda})}P_{m}+&{(\overline{\lambda}\gamma^{mnp}r)[-(d\gamma_{mnp}d)+24N_{mn}P_{p}]\over
192(\lambda\overline{\lambda})^{2}}-{(r\gamma^{mnp}r)(\overline{\lambda}\gamma_{m}d)N_{np}\over
16(\lambda\overline{\lambda})^{3}}\cr&-{(r\gamma^{mnp}r)(\overline{\lambda}\gamma^{pqr}r)N_{mn}N_{qr}\over
128(\lambda\overline{\lambda})^{4}}\ ,&({2.21})}$

where $N^{mn}={1\over 2}(\lambda\gamma^{mn}w)$. This operator has been notably
simplified in [47] to a quadratic polynomial in a fermionic vector
$\overline{\Gamma}^{m}$, which is linear in $d_{\alpha}$ and $N^{mn}$.
Although this simplified form is useful to check several properties satisfied
by the $b$-ghost, we will use an alternative expression which makes use of the
so-called physical operators, introduced in [7] in the search for deformations
of the 10D super-Yang-Mills action. These operators are defined by the
relations

$\eqalignno{[Q,{\tenbf{A}}_{\alpha}]&=-d_{\alpha}+(\gamma^{m}\lambda)_{\alpha}{\tenbf{A}}_{m}\
,&({2.22})\cr\\{Q,{\tenbf{A}}_{m}\\}&=P_{m}+(\lambda\gamma_{m}{\tenbf{W}})\
,&({2.23})\cr[Q,{\tenbf{W}}^{\alpha}]&={1\over
4}(\gamma^{mn})_{\alpha}{}^{\beta}\lambda^{\alpha}{\tenbf{F}}_{mn}\
,&({2.24})\cr\\{Q,{\tenbf{F}}_{mn}\\}&=-2(\lambda\gamma_{[m}\partial_{n]}{\tenbf{W}})\
,&({2.25})}$

which are reminiscent of the standard 10D super-Yang-Mills equations of motion

$\eqalignno{D_{\alpha}A_{\beta}+D_{\beta}A_{\alpha}=(\gamma^{m})_{\alpha\beta}A_{m}&,\
\ \ \ D_{\alpha}A_{m}=\partial_{m}A_{\alpha}+(\gamma_{m}W)_{\alpha}\ ,\cr
D_{\alpha}W^{\beta}={1\over 4}(\gamma^{mn})_{\alpha}{}^{\beta}F_{mn}&,\ \ \ \
D_{\alpha}F_{mn}=-2(\gamma_{[m}\partial_{n]}W)_{\alpha}\ .&({2.26})}$

Since the physical operators carry negative ghost number, the solution to
eqns. (2.22)-(2.25) will necessarily make use of non-minimal variables. Such a
solution is given by

$\eqalignno{{\tenbf{A}}_{\alpha}&={1\over
4(\lambda\overline{\lambda})}\bigg{[}N^{mn}(\gamma_{mn}\overline{\lambda})_{\alpha}+J\overline{\lambda}_{\alpha}\bigg{]}\
,&({2.27})\cr{\tenbf{A}}_{m}&={(\overline{\lambda}\gamma_{m}d)\over
2(\lambda\overline{\lambda})}+{(\overline{\lambda}\gamma_{mnp}r)\over
8(\lambda\overline{\lambda})^{2}}N^{np}\
,&({2.28})\cr{\tenbf{W}}^{\alpha}&={(\gamma^{m}\overline{\lambda})^{\alpha}\over
2(\lambda\overline{\lambda})}\tenbf{\Delta}_{m}\
,&({2.29})\cr{\tenbf{F}}_{mn}&=-{(r\gamma_{mn}{\tenbf{W}})\over
2(\lambda\overline{\lambda})}={(\overline{\lambda}\gamma_{mn}{}^{p}r)\over
4(\lambda\overline{\lambda})}\tenbf{\Delta}_{p}\ ,&({2.30})}$

where,

$\eqalignno{\tenbf{\Delta}_{m}&=-P_{m}+{(r\gamma_{m}d)\over
2(\lambda\overline{\lambda})}+{(r\gamma_{mnp}r)\over
8(\lambda\overline{\lambda})^{2}}N^{np}\ .&({2.31})}$

After a few algebraic manipulations, one can show these operators act on the
on-shell ghost number one superfield $\Psi^{(1)}=\lambda^{\alpha}A_{\alpha}$
as follows

$\eqalignno{\hat{{\tenbf{A}}}_{\alpha}\Psi^{(1)}&=A_{\alpha}+(\lambda\gamma_{m})_{\alpha}\sigma_{m}\
,&({2.32})\cr\hat{{\tenbf{A}}}_{m}\Psi^{(1)}&=A_{m}-(\lambda\gamma_{m}\rho)+Q\sigma_{m}\
,&({2.33})\cr\hat{{\tenbf{W}}}^{\alpha}\Psi^{(1)}&=W^{\alpha}-Q\rho^{\alpha}+(\gamma^{mn}\lambda)^{\alpha}s_{mn}+\lambda^{\alpha}s\
,&({2.34})\cr\hat{{\tenbf{F}}}_{mn}\Psi^{(1)}&=F_{mn}-4Qs_{mn}+(\lambda\gamma_{[m}g_{n]})+(\lambda\gamma_{mn}g)\
,&({2.35})}$

where,

$\eqalignno{\sigma_{m}&=-{(\overline{\lambda}\gamma_{m}A)\over
2(\lambda\overline{\lambda})}\ ,\ \ \ \
\rho^{\alpha}={(\gamma^{p}\overline{\lambda})^{\alpha}\over
2(\lambda\overline{\lambda})}(A_{p}+Q\sigma_{p})\ ,\ \ \ \
\xi^{\alpha}=W^{\alpha}-Q\rho^{\alpha}\ ,\cr
s_{mn}&={(\overline{\lambda}\gamma_{mn}\xi)\over
8(\lambda\overline{\lambda})}\ ,\ \ \ \ s={(\overline{\lambda}\xi)\over
4(\lambda\overline{\lambda})}\ ,\ \ \ \ r_{mn}=-F_{mn}+4Qs_{mn}\ ,&({2.36})\cr
g_{\alpha}&={(\gamma^{mn}\overline{\lambda})_{\alpha}\over
8(\lambda\overline{\lambda})}r_{mn}-{\overline{\lambda}_{\alpha}\over
2(\lambda\overline{\lambda})}Qs\ ,\ \ \ \
g_{m}^{\alpha}={(\gamma^{n}\overline{\lambda})^{\alpha}\over(\lambda\overline{\lambda})}r_{nm}\
,}$

and we are using hatted symbols to denote the operator version of their
corresponding unhatted symbols, that is,
$(\hat{\tenbf{A}}_{\alpha},\hat{\tenbf{A}}_{m},\hat{\tenbf{W}}^{\alpha},\hat{\tenbf{F}}_{mn})$
are obtained from
$({\tenbf{A}}_{\alpha},{\tenbf{A}}_{m},{\tenbf{W}}^{\alpha},{\tenbf{F}}_{mn})$
after performing the replacements:

$\eqalignno{&P_{m}\rightarrow\partial_{m}\ ,\ \ d_{\alpha}\rightarrow
D_{\alpha}\ ,\ \ w_{\alpha}\rightarrow-\partial_{\lambda^{\alpha}}\
.&({2.37})}$

Using the physical operators (2.27)-(2.30), the $b$-ghost (2.21) can then be
rewritten in the form

$\eqalignno{b&={1\over
2}\bigg{[}P^{m}{\tenbf{A}}_{m}-d_{\alpha}{\tenbf{W}}^{\alpha}-{1\over
2}N^{mn}{\tenbf{F}}_{mn}\bigg{]}\ ,&({2.38})}$

as can easily be demonstrated by directly expanding the right-hand side of
(2.38). As a check, the use of eqns. (2.22)-(2.25) allows us to show that
(2.38) indeed obeys $\\{Q,b\\}={P^{2}\over 2}$, as expected. Next, writing the
$b$-ghost as a first order differential operator we observe the identity

$\eqalignno{P^{m}\hat{\tenbf{A}}_{m}-D_{\alpha}\hat{\tenbf{W}}_{\alpha}-{1\over
2}N^{mn}\hat{\tenbf{F}}_{mn}&=\partial^{m}{\tenbf{A}}_{m}-d_{\alpha}{\tenbf{W}}^{\alpha}+{1\over
4}(\lambda\gamma^{mn}\partial_{\lambda}){\tenbf{F}}_{mn}\ ,&({2.39})}$

which is a direct consequence of eqns. (2.27)-(2.30). Then, eqns.
(2.32)-(2.35) allow us to show that

$\eqalignno{\\{b,\Psi^{(1)}\\}&=P^{m}A_{m}-D_{\alpha}W^{\alpha}-{1\over
2}N^{mn}F_{mn}+Q\Sigma\ ,&({2.40})}$

where $\Sigma$ is given by

$\eqalignno{\Sigma&=P^{m}\sigma_{m}-d_{\alpha}\rho^{\alpha}+(\lambda\gamma^{mn}w)s_{mn}-(\lambda
w)s\ ,&({2.41})}$

which is consistent with the particle-limit of the stringy relation
$\\{b,V\\}=U$, where $U$ is the usual ghost number zero vertex operator.

3\. Amplitudes In Pure Spinor Superspace

In this section we review Feynman rules in pure spinor superspace, discussing
the gauge-fixing needed to invert the propagator. We then show that these
Feynman rules are equivalent to Berends-Giele currents and use the latter to
compute tree-level scattering amplitudes. Since Feynman rules are derived from
an action, they are manifestly crossing-symmetric, while Berends-Giele
currents follow from equations of motion and permit modifications that may
break manifest crossing symmetry.

3.1. Feynman Rules

Since the pure spinor superfield $\Psi$ in (2.17) contains both fields and
antifields in its definition, we have to employ a non-standard gauge-fixing
procedure. Using inspiration from string field theory, the so-called Siegel
gauge $b_{0}\Psi=0$ has been proposed [[8]8,,[17]17] as a candidate for such a
purpose, where $b_{0}$ can be viewed as the second order differential operator
obtained from (2.38) after using the correspondence principle (2.37). In this
section, we will be less restrictive and require that $b_{0}\Psi=Q\Xi$
instead, where $\Xi$ is a ghost number -1 superfield depending on non-minimal
variables. As we will see, such a weaker gauge fixing is enough for defining
propagators. The gauge-fixed action (2.17) then takes the form

$\eqalignno{S&=\int\,d^{10}xd^{16}\theta[dZ]{\cal{N}}\,Tr\bigg{[}{1\over
2}\Psi Q\Psi+{g\over 3}\Psi\Psi\Psi+e(b_{0}\Psi-Q\Xi)\bigg{]}\ ,&({3.1})}$

where $e$ is a Lagrange multiplier enforcing the gauge condition. The off-
shell fields are Lie-algebra valued, defined as $\Psi=\Psi^{a}T^{a}$ where
$T^{a}$ are the generators of the color group. We take these generators to
satisfy $[T^{a},T^{b}]=f^{abc}T^{c}$, where $f^{abc}$ are the totally
antisymmetric structure constants and the generators are normalized such that
${\tenrm Tr}(T^{a}T^{b})=\delta^{ab}$. Note that the usual unintegrated vertex
operator $\Psi^{(1)}(x,\theta,\lambda)=\lambda^{\alpha}A_{\alpha}(x,\theta)$,
satisfies the weaker gauge condition

$\eqalignno{b_{0}\Psi^{(1)}&=Q\bigg{[}\partial^{m}\sigma_{m}-D_{\alpha}\rho^{\alpha}-(\lambda\gamma^{mn}\partial_{\lambda})s_{mn}+7(\lambda\partial_{\lambda})s\bigg{]}\
,&({3.2})}$

as shown in Appendix A.1. For the purpose of obtaining the color-dressed
Feynman rules, (3.1) can be written as

$\eqalignno{S&=\int\,d^{10}xd^{16}\theta[dZ]{\cal{N}}\,\bigg{[}{1\over
2}\Psi^{a}Q\Psi^{a}+{g\over
6}f^{abc}\Psi^{a}\Psi^{b}\Psi^{c}+e^{a}(b_{0}\Psi^{a}-Q\Xi^{a})\bigg{]}\
,&({3.3})}$

by evaluating the trace. The Feynman rules following from (3.3) are then given
by

The Propagator: Using the relation $\\{Q,b_{0}\\}=\hbox{$\sqcup$}\hbox
to0.0pt{\hss\hbox{$\sqcap$}}$ on $\Psi^{a}$, which satisfies
$b_{0}\Psi^{a}=Q\Xi^{a}$, one gets that $b_{0}Q=\hbox{$\sqcup$}\hbox
to0.0pt{\hss\hbox{$\sqcap$}}$, and so the propagator is defined by

$\eqalignno{{\cal
G}^{ab}(Z,Z^{\prime})&=\delta^{ab}{2b_{0}\over\hbox{$\sqcup$}\hbox
to0.0pt{\hss\hbox{$\sqcap$}}}\delta(Z-Z^{\prime})\ .&({3.4})}$

The 3-point vertex: The Chern-Simons like term in (3.3) gives rises to the
3-point vertex

$\eqalignno{{\cal V}^{abc}&=gf^{abc}\int d^{10}xd^{16}\theta[dZ]{\cal N}\
.&({3.5})}$

The full amplitude is then obtained by a sum over all Feynman graphs,

$\eqalignno{{\cal A}_{n}&=\sum_{i\in\Gamma_{n}}{c_{i}n_{i}\over D_{i}}\
,&({3.6})}$

where the color-factors $c_{i}$ are given by contractions of structure
constants, the denominators $D_{i}$ are products of Mandelstam variables, and
$n_{i}$ are the numerators and encode the remaining kinematic dependence,
including pure spinor integrations.

The color factors of the amplitudes can be expanded in sums over traces of
generator matrices $T^{a}$, and the full amplitude can then be written as

$\eqalignno{{\cal A}_{n}&=\sum_{\sigma\in
S_{n-1}}A_{\sigma(1,\ldots,n-1),n}{\tenrm Tr}(T^{\sigma_{1}}\ldots
T^{\sigma_{n-1}}T^{n})\ ,&({3.7})}$

where $S_{k}$ is the set of permutations of $k$ external-state labels, and the
objects $A_{\ldots}$ are called color-ordered amplitudes, each contributing to
one trace structure in the full amplitude. They can be obtained directly from
the color-ordered Feynman rules which are given by:

The Propagator:

$\eqalignno{{\cal G}(Z,Z^{\prime})&={2b_{0}\over\hbox{$\sqcup$}\hbox
to0.0pt{\hss\hbox{$\sqcap$}}}\delta(Z-Z^{\prime})\ .&({3.8})}$

The color-ordered 3-point vertex: It is given by the part of the color-dressed
three-point vertex that is proportional to ${\tenrm Tr}(T^{a}T^{b}T^{c})$,

$\eqalignno{{\cal V}&=g\int d^{10}xd^{16}\theta[dZ]{\cal N}\ .&({3.9})}$

The color-ordered amplitudes are then computed by sums over all planar Feynman
diagrams. Throughout this work we focus on the color-ordered amplitudes due to
their relative computational simplicity.

3.2. Berends-Giele Currents

The color-ordered amplitudes can equivalently be obtained from Berends-Giele
currents [48]. Starting from eqn. (3.1), the equations of motion are

$\eqalignno{Q\Psi+\Psi\Psi-gb_{0}(e)&=0\ ,&({3.10})\cr b_{0}\Psi-Q\Xi&=0\
,&({3.11})}$

where we have rescaled $\Psi\rightarrow{\Psi\over g}$. Applying $b_{0}$ on
both sides of (3.10), and using the nilpotency of the $b$-ghost, one finds
that

$\eqalignno{{1\over 2}\hbox{$\sqcup$}\hbox
to0.0pt{\hss\hbox{$\sqcap$}}\Psi+b_{0}(\Psi\Psi)&=0\ ,&({3.12})}$

where one has to use eqn. (3.11), which renders the kinetic term invertible.
We now introduce the $n$-particle perturbiner expansion [[49]49,,[39]39]

$\eqalignno{\Psi&=\sum_{P}\Psi_{P}T^{P}e^{k_{P}.x}=\sum_{i}\Psi_{i}T^{a_{i}}e^{k_{i}.x}+\sum_{i,j}\Psi_{ij}T^{a_{i}}T^{a_{j}}e^{k_{ij}.x}+\ldots&({3.13})}$

where $i=1,\ldots,n$ and $P$ stands for non-empty words in particle labels.
Lie algebra generators and momenta carrying multi-indices are defined as
$T^{P}=T^{a_{1}}\ldots T^{a_{n}}$, and $k_{P}=k_{1}+\ldots+k_{n}$. Mandelstam
variables are defined by $s_{P}={1\over 2}k_{P}^{2}$. After plugging (3.13)
into the equation of motion (3.12), and collecting all terms with the same Lie
generator products, one obtains

$\eqalignno{QV_{i}&=0\ ,\ \ \ \ \Psi_{P}=-{1\over
s_{P}}\sum_{QR=P}b_{0}(\Psi_{Q}\Psi_{R})\ ,\ \ \ \ b_{0}V_{i}=Q\Xi_{i}\
.&({3.14})}$

where $\Xi_{i}$ is the single-particle superfield in the perturbiner expansion
of $\Xi$, $V_{i}=\Psi_{i}$, for $i=1,\ldots,n$, and the sum runs over all the
possible deconcatenations of $P$ into the non-empty ordered words $Q$, $R$.
For example, the deconcatenations of $12345$ are $(1234,5),(123,45),(12,345)$
and $(1,2345)$. The $n$-point color-ordered amplitude then reads

$\eqalignno{{{A}}_{1\ldots n}&=(-1)^{n}\sum_{PQ=(1\ldots
n-1)}\langle\Psi_{P}\Psi_{Q}V_{n}\rangle\ ,&({3.15})}$

where ${\Psi}_{1\ldots i}$ is given by the recursive formula (3.14), see [38]
for a similar construction. The product $\Psi_{P}\Psi_{Q}$ is just
$\Psi_{1...n-1}$ with the outer propagator $b_{0}/\hbox{$\sqcup$}\hbox
to0.0pt{\hss\hbox{$\sqcap$}}$ stripped off. The prescription to remove the
last propagator can be seen as the LSZ reduction in pure spinor superspace;
the BRST operator is to be applied from the left in order to cancel the
external propagator, then the limit of on-shell momentum is taken. The angle
brackets $\langle\ldots\rangle$ represent the application of the pure spinor
measure studied in section 2.2. Integration over $x$ then imposes the momentum
conserving delta function, which we ignore in what follows, while the
integration over pure spinors and $\theta$ is responsible for imposing various
contractions between on-shell states.

The Berends-Giele currents precisely reproduce the amplitudes from the color-
ordered Feynman rules. This follows from evaluating the pure spinor integrals
in the latter until all delta functions from propagators are localized, and
then observing that terms in the sum over deconcatenations in (3.15) are in
one-to-one correspondence with the planar diagrams Feynman diagrams.

3.3. Examples

In what follows we compute a few lower-point amplitudes and show that they
reproduce those obtained in [[50]50,,[38]38,,[51]51]. We start from the three-
point amplitude, using eqn. (3.15) we have

$\eqalignno{A_{123}&=\langle V_{1}V_{2}V_{3}\rangle\ ,&({3.16})}$

which is exactly the three-particle amplitude in pure spinor superspace.
Notice that we implicitly used the prescription in (3.15), and removed the
outer propagator from $\Psi_{12}$.

For the four-point amplitude it is convenient to introduce a diagrammatic
representation of the currents,

$\eqalignno{\Psi_{123}&=\matrix{\epsfbox[0153050]{fig1.eps}\cr}\cr&=-{b_{0}\over
s_{123}}\big{(}\Psi_{12}V_{3}+V_{1}\Psi_{23}\big{)}\cr&={b_{0}\over
s_{123}}\bigg{(}{b_{0}\over
s_{12}}\big{(}V_{1}V_{2}\big{)}V_{3}+V_{1}{b_{0}\over
s_{23}}\big{(}V_{2}V_{3}\big{)}\bigg{)}\ ,&({3.17})}$

where each diagram has a natural Feynman-rule interpretation as compared to
those in [52]. After removing the last propagator $b_{0}/s_{123}$ and
multiplying with an external state $V_{4}$, we have

$\eqalignno{A_{1234}&={1\over s_{12}}\langle
b_{0}(V_{1}V_{2})V_{3}V_{4}\rangle+{1\over s_{23}}\langle
V_{1}b_{0}(V_{2}V_{3})V_{4}\rangle\ .&({3.18})}$

As discussed in A.2 , the action of $b_{0}$ on two single-particle superfields
can be demonstrated to take the form

$\eqalignno{b_{0}(V_{1}V_{2})&=V_{12}+Q\Lambda_{12}\ ,&({3.19})}$

where $V_{12}=\lambda^{\alpha}A_{12\,\alpha}$, and $A_{12\,\alpha}$ is the
2-particle superfield introduced in [52]. After dropping BRST-exact terms,
eqn. (3.18) becomes

$\eqalignno{{A}_{1234}&={1\over s_{12}}\langle V_{12}V_{3}V_{4}\rangle+{1\over
s_{23}}\langle V_{1}V_{23}V_{4}\rangle\ ,&({3.20})}$

giving the expected result.

The five point amplitude can be obtained along similar lines. The four-
particle current is given by five planar diagrams,

$\eqalignno{\Psi_{1234}&=\matrix{\epsfbox[15103050]{fig2.eps}\cr}\cr&=-{b_{0}\over
s_{1234}}\bigg{[}{b_{0}(b_{0}(V_{1}V_{2})V_{3})V_{4}\over
s_{123}s_{12}}+{b_{0}(V_{1}b_{0}(V_{2}V_{3}))V_{4}\over
s_{123}s_{23}}+{b_{0}(V_{1}V_{2})b_{0}(V_{3}V_{4})\over s_{12}s_{34}}\cr&\ \ \
\ \ \ \ \ \ \ \ \ \ +{V_{1}b_{0}(V_{2}b_{0}(V_{3}V_{4}))\over
s_{234}s_{34}}+{V_{1}b_{0}(b_{0}(V_{2}V_{3})V_{4})\over
s_{234}s_{23}}\bigg{]}\ .&({3.21})}$

And from this, the five point amplitude reads

$\eqalignno{A_{12345}&=-{\langle
V_{1}b_{0}(V_{2}(b_{0}(V_{3}V_{4}))V_{5}\rangle\over s_{234}s_{34}}-{\langle
V_{1}b_{0}(b_{0}(V_{2}V_{3})V_{4})V_{5}\rangle\over s_{234}s_{23}}-{\langle
b_{0}(V_{1}V_{2})b_{0}(V_{3}V_{4})V_{5}\rangle\over s_{12}s_{34}}\cr&-{\langle
b_{0}(V_{1}(b_{0}(V_{2}V_{3}))V_{4}V_{5}\rangle\over s_{123}s_{23}}-{\langle
b_{0}(b_{0}(V_{1}V_{2})V_{3})V_{4}V_{5}\rangle\over s_{123}s_{12}}\
.&({3.22})}$

As shown in A.3, the expressions involving two nested $b_{0}$ can be
calculated to be

$\eqalignno{b_{0}(b_{0}(V_{i}V_{j})V_{k})&=V_{ijk}+s_{ij}T_{ijk}+s_{ijk}\Lambda_{ij}V_{k}+Q(\Lambda_{[ij]k})&({3.23})}$

where $V_{ijk}=\lambda^{\alpha}A_{ijk\,\alpha}$ is the 3-particle lowest
dimensional superfield studied in [52], $T_{ijk}$ is given by

$\eqalignno{T_{ijk}&=\Lambda_{i}V_{j}V_{k}-\Lambda_{ij}V_{k}+{\tenrm
cyclic}(ijk)\ ,&({3.24})}$

and $\Lambda_{[ij]k}$ can be found in (A.46). The amplitude (3.22) then takes
the form

$\eqalignno{A_{12345}&={\langle[V_{342}+s_{34}T_{342}+s_{342}\Lambda_{34}V_{2}]V_{1}V_{5}\rangle\over
s_{234}s_{34}}-{\langle[V_{234}+s_{23}T_{234}+s_{123}\Lambda_{23}V_{4}]V_{1}V_{5}\rangle\over
s_{234}s_{23}}\cr&+{\langle(V_{12}+Q\Lambda_{12})(V_{34}+Q\Lambda_{34})V_{5}\rangle\over
s_{12}s_{34}}-{\langle[V_{231}+s_{23}T_{231}+s_{231}\Lambda_{23}V_{1}]V_{4}V_{5}\rangle\over
s_{123}s_{23}}\cr&+{\langle[V_{123}+s_{12}T_{123}+s_{123}\Lambda_{12}V_{3}]V_{4}V_{5}\rangle\over
s_{123}s_{12}}&({3.25})}$

Cancellations between terms in this expression must happen over common poles,
since all objects appearing in the numerator are local. For example, we can
focus on terms with the propagator $s_{123}^{-1}$. There are two such
contributions

$\eqalignno{A_{12345}\bigg{|}_{s_{123}^{-1}}&=\langle
T_{123}V_{4}V_{5}\rangle-\langle T_{231}V_{4}V_{5}\rangle\ .&({3.26})}$

But the $T_{ijk}$ have a cyclic symmetry, and hence these terms do not
contribute to the amplitude. Let us also examine the terms with the propagator
$s_{34}$, once again there are exactly two such contributions,

$\eqalignno{A_{12345}\bigg{|}_{s_{34}^{-1}}&=\langle\Lambda_{34}V_{2}V_{1}V_{5}\rangle+{\langle(V_{12}+Q\Lambda_{12})Q\Lambda_{34}V_{5}\rangle\over
s_{12}}\cr&=\langle\Lambda_{34}V_{2}V_{1}V_{5}\rangle+{\langle\Lambda_{34}Q(V_{12}+Q\Lambda_{12})V_{5}\rangle\over
s_{12}}\cr&=\langle\Lambda_{34}V_{2}V_{1}V_{5}\rangle+\langle\Lambda_{34}V_{1}V_{2}V_{5}\rangle\
,&({3.27})}$

where in the last equality we used that $Qb_{0}(V_{1}V_{2})=s_{12}V_{1}V_{2}$,
or equivalently that $QV_{12}=s_{12}V_{1}V_{2}$. A similar analysis follows
for the other terms, giving the expected result

$\eqalignno{A_{12345}&=\langle V_{1}M_{234}V_{5}\rangle+\langle
M_{12}M_{34}V_{5}\rangle+\langle M_{123}V_{4}V_{5}\rangle\ ,&({3.28})}$

where

$\eqalignno{M_{ij}={V_{ij}\over s_{ij}}\ ,\ \ \ \ {\tenrm and}\ \ \ \
M_{ijk}&={1\over s_{ijk}}\bigg{(}{V_{ijk}\over s_{ij}}-{V_{jki}\over
s_{jk}}\bigg{)}\ .&({3.29})}$

Once the non-minimal variables are decoupled in this way, it is
straightforward to evaluate the pure spinor integrals remaining in the
expressions [53]. In the following section we will argue that our formalism
should generate the correct amplitudes to any multiplicity by relating our
numerators to those in [10].

4\. Properties Of The Numerators

The operators used in the previous sections were required to satisfy
$b_{0}V=Q\Xi$. We saw in eqn. (3.2) that the usual unintegrated vertex
operator $V=\lambda^{\alpha}A_{\alpha}$, which only depends on minimal
variables, satisfies such a gauge condition. In this section we study the
properties of the numerators in such a gauge as well as in the Siegel gauge
$b_{0}V=0$. Unlike the former, the latter implies that $V$ is a function of
non-minimal variables with singularities in $\lambda\overline{\lambda}$.
Interestingly though, this gauge will be shown to realize color-kinematics
duality, via a mechanism reminiscent of the one in [37]. We will restrict our
study to the algebraic properties of such a gauge choice, and leave the task
of explicit calculations for future work.

4.1. Generalized BRST Blocks And Gauge Invariance

We will first consider the action of the BRST operator on a generic numerator.
Each time $Q$ anti-commutes with a $b_{0}$ we are left with a Mandelstam
invariant. In general we have

$\eqalignno{Qb_{0}(b_{0}(\Psi_{\alpha}\Psi_{\beta})\Psi_{\gamma})&=s_{\alpha\beta\gamma}b_{0}(\Psi_{\alpha}\Psi_{\beta})\Psi_{\gamma}-s_{\alpha\beta}b_{0}(\Psi_{\alpha}\Psi_{\beta}\Psi_{\gamma})+\ldots\
,&({4.1})}$

where the Greek letters label some tree diagram, and the ellipses stand for
additional terms obtained by propagating the BRST operator through more
$b$-ghosts. Focusing on the $s_{\alpha\beta}$ part only, we can represent this
diagrammatically,

$\eqalignno{&\matrix{\epsfbox[0020050]{fig3.eps}\cr}&({4.2})}$

where the blobs are place-holders for the unspecified incoming states
$\Psi_{\alpha}$, $\Psi_{\beta}$, $\Psi_{\gamma}$. The right-hand-side is
proportional to $s_{\alpha\beta}$, hence the corresponding propagator has been
contracted in the diagram. In the full amplitude, this term can cancel only
with other Feynman diagrams that have the same propagator structure, meaning
there must be some other diagram whose BRST variation matches the right-hand-
side of the equation above. Indeed, this is

$\eqalignno{&\matrix{\epsfbox[0020050]{fig4.eps}\cr}&({4.3})}$

where the minus sign comes form commuting the BRST operator past the $\alpha$
leg. We notice that diagrams cancel in pairs, where each pair of diagrams
differ by exactly one propagator. This observation holds even if we embed our
$\alpha\beta\gamma$ sub-diagram in any other larger diagram. Let us see an
example of this, take the BRST variation of the four-particle half-ladder
numerator,

$\eqalignno{Qb_{0}(b_{0}(b_{0}(V_{1}V_{2})V_{3})V_{4})&=s_{1234}b_{0}(b_{0}(V_{1}V_{2})V_{3})V_{4}-s_{123}b_{0}(b_{0}(V_{1}V_{2})V_{3}V_{4})\cr&\
\ \ +s_{12}b_{0}(b_{0}(V_{1}V_{2}V_{3})V_{4})\ .&({4.4})}$

Clearly the $s_{12}$ part has a cyclic symmetry in $(1,2,3)$ which is needed
for the cancellation of this term with the $s_{23}$ part coming from the BRST
operator acting on $b_{0}(b_{0}(V_{1}b_{0}(V_{2}V_{3}))V_{4})$. Similarly, the
term proportional to $s_{123}$ has a cyclic symmetry in $(12,3,4)$, where
labels $12$ are treated as a single unit. This is again needed in order to
cancel the $s_{34}$ contribution from the variation of
$b_{0}(b_{0}(V_{1}V_{2})b_{0}(V_{3}V_{4}))$.

The transformations of the numerators add up and give the expected equation of
motion for Berends-Giele currents [38]

$\eqalignno{Q\Psi_{Z}&=-\sum_{XY=Z}\Psi_{X}\Psi_{Y}\ .&({4.5})}$

without the gauge-fixing terms. From this it follows that the amplitudes are
BRST-closed, and gauge invariant under transformations of the last leg $\delta
V_{n}=Q\omega$. To see this, consider a gauge transformation of the particle
$n$ in an $n-$point amplitude, $V_{n}\to Q\omega$, it can be written as

$\eqalignno{\delta A_{12...n}&=\sum_{XY=(1\ldots
n-1)}\langle\Psi_{X}\Psi_{Y}Q\omega\rangle\cr&=\sum_{XY=(1\ldots n-1)}\langle
Q(\Psi_{X}\Psi_{Y})\omega\rangle\cr&=\sum_{XY=(1\ldots
n-1)}\bigg{(}-\sum_{AB=X}\langle\Psi_{A}\Psi_{B}\Psi_{Y}\omega\rangle+\sum_{AB=Y}\langle\Psi_{X}\Psi_{A}\Psi_{B}\omega\rangle\bigg{)}\cr&=0\
.&({4.6})}$

Notice that a similar argument has been worked out for superstring amplitudes
in [54]. To see that the amplitude is invariant under gauge transformations of
any external leg, we first discuss the four-particle numerator as an example.
Under the BRST variation $V_{1}\to Q\omega$, the same method of anti-commuting
the BRST operator with $b$-ghosts gives

$\eqalignno{b_{0}(b_{0}(b_{0}(Q\omega
V_{2})V_{3})V_{4})&=s_{1234}b_{0}(b_{0}(\omega
V_{2})V_{3})V_{4}-s_{123}b_{0}(b_{0}(\omega V_{2})V_{3}V_{4})\cr&\ \ \
+s_{12}b_{0}(b_{0}(\omega V_{2}V_{3})V_{4})\ ,&({4.7})}$

up to BRST-exact terms. Once again the $s_{12}$ part above cancels with the
$s_{23}$ contribution form $b_{0}(b_{0}(V_{1}b_{0}(V_{2}V_{3}))V_{4})$. The
variation of (4.7) is identical to (4.4) under the replacement
$V_{1}\to\omega$. In this manner, it is not hard to see that under one
replacement of $V_{i}\to Q\omega$ in a numerator, the general transformation
rule is

$\eqalignno{b_{0}(b_{0}(...)V_{n})\big{|}_{V_{i}\to
Q\omega}&=\pm\big{[}Qb_{0}(b_{0}(...)V_{n})\big{]}\big{|}_{V_{i}\to\omega}\
,&({4.8})}$

where the overall sign is determined by the location of the leg being
transformed. Gauge invariance of the full amplitude can now be shown by
substituting this transformation rule into the variation of the amplitude, in
a similar fashion to (4.6).

Our discussion here mirrors the discussion for the cancellation of contact
terms in (3.23), and for good reason; they are both related to gauge
invariance. So far we only dealt with linearized gauge invariance of external
states, but we actually learned that if we modify our numerators by contact
terms that have the same symmetry properties as the ones we discovered in the
gauge transformations, they will decouple in the amplitude. For example, one
could add to all numerators containing $b_{0}(b_{0}(V_{i}V_{j})V_{k})$ a term
proportional to $s_{ij}$ so long as it is cyclic in $(ijk)$. These kinds of
modifications are known as generalized gauge transformations in the amplitudes
literature [24], and reflect the fact that the numerators, which are gauge
dependent, are not unique††1 It is also possible to apply generalized gauge
transformations that are polynomial in Mandelstams. These may have
cancellations between more than just pairs of diagrams, but they do not play a
major role in our analysis..

In [[10]10,,[52]52], gauge invariance of the amplitudes followed from a multi-
particle equation of motion,

$\eqalignno{QV_{1...n}&=\sum_{j=2}^{n}s_{1...j}\bigg{(}\sum_{\alpha\in
P(\beta_{j})}V_{1...(j-1)\alpha}V_{j(\beta_{j}-\alpha)}\cr&\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ +\sum_{\tilde{\alpha}\in
P(\beta_{(j+1)})}V_{1...(j)\tilde{\alpha}}V_{j+1(\beta_{(j-1)}-\tilde{\alpha})}\bigg{)}\
,&({4.9})}$

where $\beta_{j}$ is the range $(j+1,...,n)$ and $P(\beta_{j})$ is its power
set. To unpack this a little, we return to the four-particle example, in which
case we have

$\eqalignno{QV_{1234}&=s_{1234}V_{123}V_{4}-s_{123}(V_{123}V_{4}-V_{12}V_{34}-V_{124}V_{3})\cr&+s_{12}(V_{1}V_{234}-V_{12}V_{34}+V_{13}V_{24}+V_{14}V_{23}-V_{124}V_{3}+V_{134}V_{2})\
.&({4.10})}$

The important observation for us is that the term proportional to $s_{12}$ has
a cyclic symmetry in $(1,2,3)$, just as we observed was necessary for gauge
invariance. Now the fact that both our numerators and those in [[52]52]
compute the same amplitudes implies that they should be related by some
generalized gauge transformation. To prove this statement in the general case,
we make use of Appendices A.2-A.3, where we explicitly show that
$b_{0}(V_{1}V_{2})$ and $b_{0}(b_{0}(V_{1}V_{2})V_{3})$ differ from $V_{12}$
and $V_{123}$ by such generalized gauge transformations, and assume that this
same pattern holds up to some multiplicity $n$, that is

$\eqalignno{b_{0}(b_{0}(\ldots)V_{n})&=V_{12...n}+{\cal C}_{n}\ ,&({4.11})}$

where ${\cal C}_{n}$ is a combination of contact terms possessing the right
symmetry to decouple from the Berends-Giele currents at multiplicity $n$, or
BRST exact terms that decouple irrespective of their symmetry properties.
Given that ${\cal C}_{n}$ decouples at multiplicity $n$ immediately implies it
decouples from all higher multiplicities due to the recursive definition of
the Berends-Giele currents (wrapping the contact terms by an additional
$b_{0}$ does not affect their cancellations). Therefore we simply have to show
that

$\eqalignno{b_{0}(V_{12...n}V_{n+1})&=V_{12...n+1}+{\cal C}\ ,&({4.12})}$

where ${\cal C}$ decouples from currents of order $n+1$. This follows from
observing that the multi-particle equations of motion of [52], for example
(4.9), are equivalent to the single particle equations of motion up to contact
terms which have the right symmetries to decouple from amplitudes. We have
shown in the appendix that $b_{0}(V_{1}V_{2})=V_{12}+Q\Lambda_{12}$, and so
the multi-particle generalization of this simply includes additional contact
terms that decouple.

Under the operation of $Q$, our numerators transform into a sum of cubic
diagrams with one quartic vertex, corresponding to a cancelled propagator.
Naively this seems different to the transformation rules in eqn. (4.9), but in
fact they are related, with ours generalizing (4.9) by the addition of more
Mandelstam variables. We will return to this at the end of the next
subsection, after we develop a better understanding of how the $b$-ghost acts
on products of fields.

4.2. Color-Kinematics Duality

To start with, it is convenient to write down the $b$-ghost in the simplified
form

$\eqalignno{b&=\bigg{[}P^{m}+{(\lambda\gamma^{mn}r)\over
4(\lambda\overline{\lambda})}{\tenbf{A}}_{n}\bigg{]}{\tenbf{A}}_{m}\
,&({4.13})}$

where $\tenbf{A}_{m}$ was defined in (2.28). Up to shift-symmetry terms, the
expression inside the square brackets is nothing but $-\Delta_{m}$ defined in
(2.31). Eqn. (4.13) can then be compactly written as

$\eqalignno{b&=-\Delta^{m}{\tenbf{A}}_{m}\ .&({4.14})}$

In this manner, when $b_{0}$ acts on the general vertices $V_{1}$, $V_{2}$,
one gets

$\eqalignno{b_{0}(V_{1}V_{2})&=(b_{0}V_{1})V_{2}+\hat{\Delta}^{m}V_{1}\hat{\tenbf{A}}_{m}V_{2}-\hat{\tenbf{A}}^{m}V_{1}\hat{\Delta}_{m}V_{2}-V_{1}(b_{0}V_{2})\
,&({4.15})}$

where, as before, we are using hatted symbols to represent the operator
version of the corresponding fields. Similarly, using the Leibniz rule, one
can show the action of $b_{0}$ on three general superfields $V_{1}$, $V_{2}$,
$V_{3}$ is given by

$\eqalignno{b_{0}(V_{1}V_{2}V_{3})=&\
b_{0}(V_{1}V_{2})V_{3}+b_{0}(V_{2}V_{3})V_{1}+b_{0}(V_{3}V_{1})V_{2}\cr&-(b_{0}V_{1})V_{2}V_{3}+V_{1}(b_{0}V_{2})V_{3}-V_{1}V_{2}(b_{0}V_{3})\
.&({4.16})}$

When the Siegel gauge condition is imposed on external vertex operators, eqns.
(4.15), (4.16) take the simple form

$\eqalignno{b_{0}({\cal{V}}_{1}{\cal{V}}_{2})&=\hat{\Delta}^{m}{\cal{V}}_{1}\hat{\tenbf{A}}_{m}{\cal{V}}_{2}-\hat{\tenbf{A}}^{m}{\cal{V}}_{1}\hat{\Delta}_{m}{\cal{V}}_{2}&({4.17})\cr
b_{0}({\cal{V}}_{1}{\cal{V}}_{2}{\cal{V}}_{3})&=b_{0}({\cal{V}}_{1}{\cal{V}}_{2}){\cal{V}}_{3}+b_{0}({\cal{V}}_{2}{\cal{V}}_{3}){\cal{V}}_{1}+b_{0}({\cal{V}}_{3}{\cal{V}}_{1}){\cal{V}}_{2}&({4.18})}$

where we are using calligraphic letters to denote Siegel gauge operators. The
definition of the Poisson bracket

$\eqalignno{\\{X,Y\\}&=\hat{\Delta}^{m}X\hat{\tenbf{A}}_{m}Y-\hat{\tenbf{A}}^{m}X\hat{\Delta}_{m}Y\
,&({4.19})}$

allows us to rewrite eqn. (4.17) in the suggestive form

$\eqalignno{b_{0}({\cal{V}}_{1}{\cal{V}}_{2})&=\\{{\cal{V}}_{1},{\cal{V}}_{2}\\}\
.&({4.20})}$

In addition, the application of $b_{0}$ on both sides of (4.18) gives

$\eqalignno{b_{0}(b_{0}({\cal{V}}_{1}{\cal{V}}_{2}){\cal{V}}_{3})+b_{0}(b_{0}({\cal{V}}_{2}{\cal{V}}_{3}){\cal{V}}_{1})+b_{0}(b_{0}({\cal{V}}_{3}{\cal{V}}_{1}){\cal{V}}_{2})&=0\
.&({4.21})}$

This cancellation is precisely all that is needed in order to prove that the
Siegel gauge numerators obey the color-kinematics duality††2 Some subtleties
might arise after applying the regularization scheme developed in [46]. We
elaborate more on this in the Discussions section.. For external states the
identity follows from the Siegel gauge, while internal states are always
dressed with a $b$-ghost, and due to the fact that $b_{0}^{2}=0$, they are
effectively in the Siegel gauge too. Let us show this by an example, and also
explain how integration by parts within the pure spinor measure allows to deal
with the case when one of the particles is the root leg of the diagram.
Consider the six-point ladder numerator

$\eqalignno{n_{123456}&=&\matrix{\epsfbox[0022560]{fig5.eps}\cr}\cr&=\langle
b_{0}(b_{0}(b_{0}({\cal{V}}_{1}{\cal{V}}_{2}){\cal{V}}_{3}){\cal{V}}_{4}){\cal{V}}_{5}{\cal{V}}_{6}\rangle\
.&({4.22})}$

The Jacobi identity surrounding the $s_{12}$ propagator, that is, the cyclic
sum on particles $\\{1,2,3\\}$ has already been discussed. The Jacobi identity
on the $s_{123}$ propagator follows from the cyclic sum on (multi-)particle
labels $\\{(12),3,4\\}$, where $b_{0}({\cal{V}}_{1}{\cal{V}}_{2})$ is treated
as a single particle, which is clearly in the Siegel gauge. Finally, we may
also take the cyclic sum over the labels $\\{4,5,6\\}$. It is possible, by use
of integration by parts of the $b$-ghost, to show that this is equivalent by
relabelling to the Jacobi identity on legs $\\{4,5,6\\}$. Alternatively, one
can use eqn. (4.18) followed by integration by parts to see that this is
indeed zero. That integration by parts is possible follows from the fact that
$b_{0}$ commutes with the regularization factor
${\cal{N}}=e^{-\lambda\overline{\lambda}-r\theta}$, which can be shown by
observing that
$[{\tenbf{A}}_{m},\lambda\overline{\lambda}]=[{\tenbf{A}}_{m},r\theta]=0$††3
We thank Carlos Mafra for pointing out a different argument, based on
representation theory, which also establishes the validity of this property..

The kinematic Jacobi identity is related to the Jacobi identity of the Poisson
algebra (4.19),

$\eqalignno{\\{\\{{\cal{V}}_{1},{\cal{V}}_{2}\\},{\cal{V}}_{3}\\}+\\{\\{{\cal{V}}_{2},{\cal{V}}_{3}\\},{\cal{V}}_{1}\\}+\\{\\{{\cal{V}}_{3},{\cal{V}}_{1}\\},{\cal{V}}_{2}\\}&=0\
.&({4.23})}$

That this bracket to obey a Jacobi identity not only on functions in the
Siegel gauge it is required that
$[\hat{\Delta}^{m},\hat{\Delta}^{n}]=\\{\hat{\tenbf{A}}_{m},\hat{\tenbf{A}}_{m}\\}=[\hat{\Delta}^{n},\hat{\tenbf{A}}_{m}]=0$,
which indeed hold. Note that these are also the constraints needed to show
that $b_{0}^{2}=0$. This Poisson algebra has a dual Lie algebra with
generators

$\eqalignno{L_{\psi}:&={\hat{\Delta}}^{m}(\psi){\hat{\tenbf
A}}_{m}-{\hat{\tenbf A}}_{m}(\psi)\hat{\Delta}^{m}\ ,&({4.24})}$

which obey

$\eqalignno{[L_{\psi},L_{\phi}]&=L_{b_{0}(\psi\phi)}\ .&({4.25})}$

For generators defined from functions $\psi$ that satisfy $b_{0}\psi=0$, this
becomes an algebra of infinitesimal diffeomorphisms in pure spinor superspace
that preserve the Siegel gauge.

For numerators with particles in the Siegel gauge, there is no difference if
one replaces $b_{0}\to\\{\ ,\ \\}$. But Siegel gauge external states must be
related to any other external states by BRST-exact deformations, ${\cal
V}=V+Q(\zeta)$, with $\zeta$ being some superfield depending on non-minimal
variables. Since $Q(\zeta)$ is a gauge transformation it decouples from
amplitudes, see section 4.1, and we find that the numerators obtained by
nested Poisson brackets generate the correct amplitudes with any external
states. However, what is lost by this procedure is crossing symmetry, since we
cannot integrate by parts the Poisson brackets. Therefore we learn that the
Siegel gauge is special in that it allows for both manifest color-kinematics
duality and crossing symmetry at the level of individual numerators.

We have one last observation to make regarding the multi-particle equations of
motion. We noted earlier in the section that our numerators obey slightly
different equations of motion than (4.9), yet still had all the right
transformation properties to ensure that amplitudes are gauge invariant. It
can be shown that taking a half-ladder numerator $b_{0}((...)V_{n})$ and
operating with $Q$ on the left would reproduce the equation of motion (4.9) if
the external states are in the Siegel gauge. For this, one needs to use the
identity (4.21), and identify $V_{i...jk}$ with the nested $b$-ghost
numerators, $b_{0}((...){\cal V}_{k})$. So if external states are not in the
Siegel gauge we find that (4.9) is contaminated by terms with $b_{0}(V_{i})$
which contribute more Mandelstam variables. These generalize eqn. (4.9) by
adding to the right-hand side terms that are polynomial in Mandelstams and are
functions of $\Lambda_{i}$ as well as $V_{i}$.

5\. Discussions and Future Research Directions

In this work we studied amplitudes obtained directly from the 10D pure spinor
action for super-Yang-Mills theory. We employed Berends-Giele currents, which
streamline computations for tree-level amplitudes, and showed that they are
equivalent to the amplitudes obtained from the particle-limit of pure spinor
open superstrings. We have made extensive use of physical operators, finding
that they dramatically simplify computations. In the study of properties of
individual numerators we found that the color-kinematics duality emerged from
the second-order Leibniz rule obeyed by the $b$-ghost, which is the
propagator-numerator for our Feynman rules. Therefore color-kinematics duality
emerges directly from the Feynman rules. This is in contrast to other
constructions where no mention of an action is made [55], or where the color-
kinematics duality is enforced by taking gauge-transformations of the
kinematic numerators [[52]52,,[26]26,,[56]56,,[37]37].

Individual numerators are harder to obtain in this formalism than complete
amplitudes, since each numerator is given by a pure spinor superspace integral
of nested $b$-ghosts acting on on-shell states, and one cannot decouple the
non-minimal variables from it. As discussed in section 2.2, when the
numerator’s integrand diverges slower than
$\lambda^{-8}\overline{\lambda}^{-11}$, the integral is well-defined. Since
$b$-ghosts carry poles in $\lambda{\overline{\lambda}}$, the poles of the
numerators grow with multiplicity and eventually exceed
$\lambda^{-8}\overline{\lambda}^{-11}$. Fortunately, these divergences do not
contribute to amplitudes, meaning they completely decouple. However, to define
individual numerators these divergences need to be regularized. It was shown
in [46] that in principle the $b$-ghost can be regularized in a BRST-invariant
manner, so that $b^{\prime}(z)=b(z)+\\{Q,\chi_{\epsilon}(z)\\}$, for some
$\chi_{\epsilon}(z)$ which depends on the regularization parameter $\epsilon$.
If such a $\chi$ can be shown to satisfy $\chi(z)=\\{b_{0},{\cal{K}}(z)\\}$,
for some conformal weight zero ${\cal{K}}(z)$, then
$b_{0}^{\prime}(z)=e^{-\\{Q,{\cal{K}}\\}}b_{0}(z)e^{\\{Q,{\cal{K}}\\}}$, and
thus nilpotency of $b_{0}$ is preserved after regularization. Such an idea has
been used in [57] for instance. This idea has been used in [57] for studying
the UV behavior of 11D supergravity scattering amplitudes. In such a scenario,
the color-kinematics duality realized by the Siegel gauge operators is
maintained after regularization, even though explicit computations of the
numerators might still be a challenging problem.

There are many additional avenues for further exploration. Color-kinematics
duality followed directly from the second order nature of the $b$-ghost,
without kinematic structure constants appearing in the action, so it would be
interesting to find other theories with a similar action and propagator-
numerator. Since it has been observed that the color-kinematics duality has
connections to supersymmetry [58], exploring pure spinor type formulations of
theories in lower dimensions [[5]5,,[59]59,,[60]60] seems promising. Perhaps
in lower dimensions numerators would be more tractable for direct
computations. In fact, an analogous discussion has been carried out for Chern-
Simons theory [61], which has an action formulation similar to the pure spinor
action (2.17).

On the other hand, it would also be interesting to apply the ideas developed
in this work to other pure spinor field theories. For instance, the abelian
Born-Infeld action in pure spinor superspace [7] involves the operator
$\Delta_{m}$ defined in eqn. (2.31), so the perturbiner method implies that
tree-level Born-Infeld scattering amplitudes are determined from the knowledge
of the algebraic properties of the $b$-ghost and $\Delta_{m}$, and their
respective actions on single-particle superfields $V$. Likewise, the 11D
supergravity pure spinor action [6] contains 11D physical operators whose
actions on single-particle superfields have recently been calculated in
[[62]62,,[63]63]. The perturbiner method then tells us that tree-level 11D
supergravity scattering amplitudes can be directly obtained from the
properties satisfied by these operators and the 11D $b$-ghost. We plan to
investigate this in the near future.

Finally, the pure spinor Feynman rules of section 3.1 can, in principle, be
used for computing loop-level amplitudes. Since loop-level Feynman diagrams
involve ghost particles running in the loop, the calculation of loop-level
processes seems to require the use of the other ghost sectors of the pure
spinor superfield $\Psi$. It would be interesting to understand such a
mechanism in a more transparent way, as well as exploring the realization of
on-shell techniques in pure spinor superspace, which might dramatically
simplify loop-level computations.

Acknowledgements: We are grateful to Oliver Schlotterer, Carlos Mafra and
Henrik Johansson for providing valuable comments on an earlier version of this
work. MG would like to thank Nathan Berkovits for useful discussions. MBS is
thankful to Marco Chiodaroli and Henrik Johansson for collaboration on related
topics. The research of MG is supported by the European Research Council under
ERC-STG-804286 UNISCAMP. The research of MBS is supported by the Knut and
Alice Wallenberg Foundation under grants KAW 2018.0116 (From Scattering
Amplitudes to Gravitational Waves) and KAW2018.0162 (Exploring a Web of
Gravitational Theories through Gauge-Theory Methods).

Appendix A. The $b$-ghost As A Second-Order Differential Operator

A.1. The $b$-ghost On A Single Superfield

The action of the $b$-ghost on the physical sector of the pure spinor
superfield $V=\lambda^{\alpha}A_{\alpha}$, displayed in (3.2), can be computed
by using eqn. (2.38) after implementing the usual correspondence principle,
namely $P_{m}\rightarrow\partial_{m}$, $d_{\alpha}\rightarrow D_{\alpha}$,
$w_{\alpha}\rightarrow-\partial_{\lambda^{\alpha}}$. In this manner, one has

$\eqalignno{b_{0}V&=\bigg{(}\partial^{m}\hat{\tenbf{A}}_{m}-D_{\alpha}\hat{\tenbf{W}}^{\alpha}+{1\over
4}(\lambda\gamma^{mn}\partial_{\lambda})\hat{\tenbf{F}}_{mn}\bigg{)}V&({\hbox{A.}1})}$

Using eqns. (2.32)-(2.35), eqn. (A.1) becomes

$\eqalignno{b_{0}V&=Q\bigg{[}\partial^{m}\sigma_{m}-D_{\alpha}\rho^{\alpha}-(\lambda\gamma^{mn}\partial_{\lambda})s_{mn}\bigg{]}-(\lambda
D)s+{9\over 4}(\lambda\gamma^{n}g_{n})+{1\over
4}\lambda^{\alpha}(\lambda\gamma^{m})_{\beta}\partial_{\lambda^{\alpha}}g_{m}^{\beta}\cr&-{45\over
2}(\lambda g)-{5\over
2}\lambda^{\alpha}\lambda^{\beta}\partial_{\lambda^{\alpha}}g_{\beta}\cr&=Q\bigg{[}\partial^{m}\sigma_{m}-D_{\alpha}\rho^{\alpha}-(\lambda\gamma^{mn}\partial_{\lambda})s_{mn}\bigg{]}-(\lambda
D)s+2(\lambda\gamma^{n}g_{n})-20(\lambda g)+{5\over 4}(\lambda D)s-{5\over
4}Qs\cr&-{(\lambda\gamma^{mn}\overline{\lambda})\over
4(\lambda\overline{\lambda})}(\lambda
D)s_{mn}+{(\lambda\gamma^{mn}\overline{\lambda})\over
4(\lambda\overline{\lambda})}Qs_{mn}&({\hbox{A.}2})}$

where we used the identities

$\eqalignno{\partial_{\lambda^{\alpha}}g_{m}^{\beta}=&-{\overline{\lambda}_{\alpha}\over(\lambda\overline{\lambda})}g_{m}^{\beta}+{4(\gamma^{n}\overline{\lambda})^{\beta}\over(\lambda\overline{\lambda})}\bigg{[}D_{\alpha}s_{mn}-Q({\overline{\lambda}_{\alpha}\over(\lambda\overline{\lambda})}s_{mn})\bigg{]}&({\hbox{A.}3})\cr\partial_{\lambda^{\alpha}}g_{\beta}=&-{\overline{\lambda}_{\alpha}\over(\lambda\overline{\lambda})}g_{\beta}-{\overline{\lambda}_{\alpha}\over
2(\lambda\overline{\lambda})}\bigg{[}D_{\beta}s-Q({\overline{\lambda}_{\beta}\over(\lambda\overline{\lambda})}s)\bigg{]}\cr&+{(\gamma^{mn}\overline{\lambda})^{\beta}\over
2(\lambda\overline{\lambda})}\bigg{[}D_{\beta}s_{mn}-Q({\overline{\lambda}_{\beta}\over(\lambda\overline{\lambda})}s_{mn})\bigg{]}&({\hbox{A.}4})}$

Likewise, the use of eqns. (2.36) allows us to state that

$\eqalignno{2(\lambda\gamma^{n}g_{n})-20(\lambda
g)&={(\lambda\gamma^{mn}\overline{\lambda})\over
2(\lambda\overline{\lambda})}F_{mn}-2{(\lambda\gamma^{mn}\overline{\lambda})\over(\lambda\overline{\lambda})}Qs_{mn}+10Qs&({\hbox{A.}5})}$

Therefore,

$\eqalignno{b_{0}V&=Q\bigg{[}\partial^{m}\sigma_{m}-D_{\alpha}\rho^{\alpha}-(\lambda\gamma^{mn}\partial_{\lambda})s_{mn}\bigg{]}+{1\over
4}(\lambda D)s-{5\over 4}Qs+{(\lambda\gamma^{mn}\overline{\lambda})\over
2(\lambda\overline{\lambda})}F_{mn}\cr&-{(\lambda\gamma^{mn}\overline{\lambda})\over
4(\lambda\overline{\lambda})}(\lambda
D)s_{mn}+{(\lambda\gamma^{mn}\overline{\lambda})\over
4(\lambda\overline{\lambda})}Qs_{mn}-2{(\lambda\gamma^{mn}\overline{\lambda})\over(\lambda\overline{\lambda})}Qs_{mn}+10Qs&({\hbox{A.}6})}$

This expression can be simplified further by making use of the pure spinor
constraints. Indeed, one can show that

$\eqalignno{{(\lambda\gamma^{mn}\overline{\lambda})\over(\lambda\overline{\lambda})}Qs_{mn}&={\overline{\lambda}Q\xi\over(\lambda\overline{\lambda})}+Qs&({\hbox{A.}7})}$

and so, eqn. (A.6) takes the form

$\eqalignno{b_{0}V&=Q\bigg{[}\partial^{m}\sigma_{m}-D_{\alpha}\rho^{\alpha}-(\lambda\gamma^{mn}\partial_{\lambda})s_{mn}\bigg{]}+{1\over
4}(\lambda D)s-{5\over 4}Qs+8(\lambda D)s-{5\over 4}(\lambda D)s\cr&-7(\lambda
D)s-{7\over
4}Qs+10Qs\cr&=Q\bigg{[}\partial^{m}\sigma_{m}-D_{\alpha}\rho^{\alpha}-(\lambda\gamma^{mn}\partial_{\lambda})s_{mn}+7s\bigg{]}&({\hbox{A.}8})}$

After plugging (2.36) into (A.8), the expression inside the square brackets
can be written in the more compact way

$\eqalignno{\Lambda&=-{\partial^{m}(\overline{\lambda}\gamma_{m}A)\over(\lambda\overline{\lambda})}+2{(\overline{\lambda}W)\over(\lambda\overline{\lambda})}+{(\overline{\lambda}\gamma^{m}D)\over
4(\lambda\overline{\lambda})^{2}}(r\gamma_{m}A)&({\hbox{A.}9})}$

As a check, one can easily show this result yields the same expression found
by using the Y-formalism $b$-ghost in the minimal pure spinor framework
($r=0$), defined as $b_{Y}=(Y\gamma^{m}D)\partial_{m}$, with
$Y_{\alpha}\rightarrow{\overline{\lambda}_{\alpha}\over(\lambda\overline{\lambda})}$.

A.2. The $b$-ghost On Two Superfields

The action of the $b$-ghost on two single-particle superfields, say
$V_{1}=\lambda^{\alpha}A_{1\,\alpha}$, $V_{2}=\lambda^{\alpha}A_{2\,\alpha}$,
can be calculated from the following ansatz

$\eqalignno{b_{0}(V_{1}V_{2})&=V_{12}+Q\Lambda_{12}&({\hbox{A.}10})}$

where $V_{12}=\lambda^{\alpha}A_{12\,\alpha}$, and $A_{12\,\alpha}$ is given
by

$\eqalignno{A_{12\,\alpha}&={1\over 2}\bigg{[}(k_{2}\cdot
A_{1})A_{2\,\alpha}-(k_{1}\cdot
A_{2})A_{1\,\alpha}+(\gamma^{p}W_{1})_{\alpha}A_{2\,p}-(\gamma^{p}W_{2})_{\alpha}A_{1\,p}\bigg{]}&({\hbox{A.}11})}$

which, interestingly, matches the lowest-dimensional multiparticle superfield
introduced in [52]. Notice that the proposal (A.10) is reasonable since both
sides of the equality behave in the same way under the action of the BRST
charge. Indeed, $Q(b_{0}(V_{1}V_{2}))=s_{12}V_{1}V_{2}=QV_{12}$. To compute
$\Lambda_{12}$, one can let the $b$-ghost act on both-sides of (A.10) to get

$\eqalignno{\Lambda_{12}&=-{b_{0}(V_{12})\over s_{12}}&({\hbox{A.}12})}$

Thus, $\Lambda_{12}$ is defined by eqn. (A.12) up to BRST-exact terms. Let us
compute then $b_{0}V_{12}$. For this purpose, it is convenient to first list
the equations of motion satisfied by the two-particle superfields

$\eqalignno{D_{\alpha}A_{12\,\beta}+D_{\beta}A_{12\,\alpha}&=(\gamma^{m})_{\alpha\beta}A_{12\,m}+(k_{1}\cdot
k_{2})(A_{1\,\alpha}A_{2\,\beta}+A_{1\,\beta}A_{2\,\alpha})&({\hbox{A.}13})\cr
D_{\alpha}A_{12\,m}&=(\gamma_{m}W_{12})_{\alpha}+k_{12\,m}A_{12\,\alpha}+(k_{1}\cdot
k_{2})(A_{1\,\alpha}A_{2\,m}-A_{2\,\alpha}A_{1\,m})&({\hbox{A.}14})\cr
D_{\alpha}W_{12}^{\beta}&={1\over
4}(\gamma^{mn})_{\alpha}{}^{\beta}F_{12\,mn}+(k_{1}\cdot
k_{2})(A_{1\,\alpha}W^{\beta}_{2}-A_{2\,\alpha}W^{\beta}_{1})&({\hbox{A.}15})\cr
D_{\alpha}F_{12\,mn}&=k_{12\,m}(\gamma_{n}W_{12})_{\alpha}-k_{12\,n}(\gamma_{m}W_{12})_{\alpha}+(k_{1}\cdot
k_{2})\bigg{[}A_{1\,\alpha}F_{2\,mn}-A_{2\,\alpha}F_{1\,mn}\cr&+A_{1\,n}(\gamma_{m}W_{2})_{\alpha}-A_{2\,n}(\gamma_{m}W_{1})_{\alpha}-A_{1\,m}(\gamma_{n}W_{2})_{\alpha}+A_{2\,m}(\gamma_{n}W_{1})_{\alpha}\bigg{]}\cr&&({\hbox{A.}16})}$

where,

$\eqalignno{A_{12\,m}&={1\over 2}\bigg{[}A_{2\,m}(k_{2}\cdot
A_{1})-A_{1\,m}(k_{1}\cdot A_{2})+(k_{2\,m}-k_{1\,m})(A_{1}\cdot
A_{2})+2(W_{1}\gamma_{m}W_{2})\bigg{]}\cr&&({\hbox{A.}17})\cr
W_{12}^{\alpha}&={1\over
4}(\gamma^{mn}W_{2})^{\alpha}F_{1\,mn}+W_{2}^{\alpha}(k_{2}\cdot
A_{1})-{1\over
4}(\gamma^{mn}W_{1})^{\alpha}F_{2\,mn}-W_{1}^{\alpha}(k_{1}\cdot
A_{2})&({\hbox{A.}18})\cr
F_{12\,mn}&=k_{12\,m}A_{12\,n}-k_{12\,n}A_{12\,m}-(k_{1}\cdot
k_{2})(A_{1\,m}A_{2\,n}-A_{1\,n}A_{2\,m})&({\hbox{A.}19})}$

In this manner, the action of $\tenbf{A}_{\alpha}$ given in (2.27) is readily
computed to be

$\eqalignno{\hat{\tenbf{A}}_{\alpha}V_{12}&=A_{12\,\alpha}+(\lambda\gamma^{m})_{\alpha}\sigma_{12\,m}&({\hbox{A.}20})}$

with,

$\eqalignno{\sigma_{12\,m}&=-{(\overline{\lambda}\gamma_{m}A_{12})\over
2(\lambda\overline{\lambda})}&({\hbox{A.}21})}$

The action of $\tenbf{A}_{m}$ in (2.28) can be written as

$\eqalignno{\hat{\tenbf{A}}_{m}V_{12}&={(\gamma^{m}\overline{\lambda})^{\alpha}\lambda^{\beta}\over
2(\lambda\overline{\lambda})}\bigg{[}D_{\alpha}A_{12\,\beta}+D_{\beta}A_{12\,\alpha}\bigg{]}+{(\overline{\lambda}\gamma^{m}\gamma^{p}\lambda)\over
2(\lambda\overline{\lambda})}Q\bigg{[}-{(\overline{\lambda}\gamma_{p}A_{12})\over
2(\lambda\overline{\lambda})}\bigg{]}&({\hbox{A.}22})}$

which, after using eqn. (A.13), becomes

$\eqalignno{\hat{\tenbf{A}}_{m}V_{12}&=A_{12\,m}-(\lambda\gamma_{m}\rho_{12})+Q(\sigma_{12\,m})+(k_{1}\cdot
k_{2})\bigg{[}{1\over
2(\lambda\overline{\lambda})}\bigg{(}V_{1}(\overline{\lambda}\gamma_{m}A_{2})-V_{2}(\overline{\lambda}\gamma_{m}A_{1})\bigg{)}\bigg{]}\cr&&({\hbox{A.}23})}$

where $\rho^{\alpha}_{12}$ is defined by

$\eqalignno{\rho_{12}^{\alpha}&={(\gamma^{p}\overline{\lambda})^{\alpha}\over
2(\lambda\overline{\lambda})}\bigg{[}A_{12\,p}+Q(\sigma_{12\,p})\bigg{]}&({\hbox{A.}24})}$

Likewise, the action of $\tenbf{W}^{\alpha}$ in (2.29) takes the form

$\eqalignno{\hat{\tenbf{W}}^{\alpha}V_{12}&={(\gamma^{m}\overline{\lambda})^{\alpha}\over
2(\lambda\overline{\lambda})}\bigg{[}-\partial_{m}V_{12}+QA_{12\,m}-(\lambda\gamma_{m}Q\rho_{12})\cr&+(k_{1}\cdot
k_{2})Q\bigg{[}{1\over
2(\lambda\overline{\lambda})}\bigg{(}V_{1}(\overline{\lambda}\gamma_{m}A_{2})-V_{2}(\overline{\lambda}\gamma_{m}A_{1})\bigg{)}\bigg{]}\bigg{]}&({\hbox{A.}25})}$

The use of eqn. (A.14) allows us to write (A.25) in the form

$\eqalignno{\hat{\tenbf{W}}^{\alpha}V_{12}=&\xi_{12}^{\alpha}+(\gamma^{mn}\lambda)^{\alpha}s_{12\,mn}+\lambda^{\alpha}s_{12}+(k_{1}\cdot
k_{2})\bigg{[}{(\gamma^{m}\overline{\lambda})^{\alpha}\over
2(\lambda\overline{\lambda})}\bigg{(}V_{1}A_{2\,m}-V_{2}A_{1\,m}\bigg{)}\cr&+{(\gamma^{m}\overline{\lambda})^{\alpha}\over
4(\lambda\overline{\lambda})^{2}}\bigg{(}-V_{1}(r\gamma_{m}A_{2})+V_{2}(r\gamma_{m}A_{1})\bigg{)}\bigg{]}&({\hbox{A.}26})}$

where

$\eqalignno{\xi^{\alpha}_{12}&=W_{12}^{\alpha}-Q\rho_{12}^{\alpha}\ \ \ \
s_{12\,mn}={(\overline{\lambda}\gamma_{mn}\xi_{12})\over
8(\lambda\overline{\lambda})}\ ,\ \ \
s_{12}={(\overline{\lambda}\xi_{12})\over
4(\lambda\overline{\lambda})}&({\hbox{A.}27})}$

Furthermore, the action ${\tenbf{F}}_{mn}$ in (2.30) can be calculated to be

$\eqalignno{\hat{\tenbf{F}}_{mn}V_{12}=&F_{12\,mn}-4Qs_{12\,mn}-(\lambda\gamma_{[m}g_{12\,n]})+(\lambda\gamma_{mn}g_{12})+{(k_{1}\cdot
k_{2})\over
2(\lambda\overline{\lambda})}\bigg{[}V_{1}(\overline{\lambda}\gamma_{mn}W_{2})\cr&-V_{2}(\overline{\lambda}\gamma_{mn}W_{1})+{(\overline{\lambda}\gamma^{mnp}r)\over
2(\lambda\overline{\lambda})}(V_{1}A_{2\,m}-V_{2}A_{1\,m})\cr&+{(\overline{\lambda}\gamma^{mnp}r)\over
4(\lambda\overline{\lambda})^{2}}\bigg{(}-V_{1}(r\gamma_{p}A_{2})\\}+V_{2}(r\gamma_{m}A_{1})\bigg{)}\bigg{]}&({\hbox{A.}28})}$

where eqn. (A.16) was used, and

$\eqalignno{r_{12\,mn}=-F_{12\,mn}+4Qs_{12\,mn}&,\ \ \
g_{12\,\alpha}={(\gamma^{mn}\overline{\lambda})_{\alpha}\over
8(\lambda\overline{\lambda})}r_{12\,mn}-{\overline{\lambda}_{\alpha}\over
2(\lambda\overline{\lambda})}Qs_{12}\cr
g^{\alpha}_{12\,m}&={(\gamma^{n}\overline{\lambda})^{\alpha}\over(\lambda\overline{\lambda})}r_{12\,mn}&({\hbox{A.}29})}$

Therefore, one finds that

$\eqalignno{-\Lambda_{12}&={b_{0}(V_{12})\over
s_{12}}\cr&=-(A_{1}W^{2}-A_{2}W^{1})-{2\over(\lambda\overline{\lambda})}\bigg{[}V_{1}(\overline{\lambda}W_{2})-V_{2}(\overline{\lambda}W_{1})\bigg{]}+{k_{12}^{m}\over
2(\lambda\overline{\lambda})}\bigg{[}V_{1}(\overline{\lambda}\gamma_{m}A_{2})-V_{2}(\overline{\lambda}\gamma_{m}A_{1})\bigg{]}\cr&-{(\overline{\lambda}\gamma^{m}D)\over
2(\lambda\overline{\lambda})}\bigg{[}V_{1}A_{2\,m}-V_{2}A_{1\,m}\bigg{]}-{(\overline{\lambda}\gamma^{m}D)\over
4(\lambda\overline{\lambda})^{2}}\bigg{[}-V_{1}(r\gamma_{m}A_{2})+V_{2}(r\gamma_{m}A_{1})\bigg{]}\cr&+{1\over
4}(\lambda\gamma^{mn}\partial_{\lambda})\bigg{[}{1\over
2(\lambda\overline{\lambda})}\bigg{(}V_{1}(\overline{\lambda}\gamma_{mn}W_{2})-V_{2}(\overline{\lambda}\gamma_{mn}W_{1})\bigg{)}+{(\overline{\lambda}\gamma^{mnp}r)\over
4(\lambda\overline{\lambda})^{2}}\bigg{(}V_{1}A_{2\,p}-V_{2}A_{1\,p}\bigg{)}\cr&+{(\overline{\lambda}\gamma_{mnp}r)\over
8(\lambda\overline{\lambda})^{3}}\bigg{(}-V_{1}(r\gamma^{p}A_{2})+V_{2}(r\gamma^{p}A_{1})\bigg{)}\bigg{]}&({\hbox{A.}30})}$

where we have ignored BRST-exact terms. This expression can be put in a
simpler form through the use of pure spinor identities. Indeed, the
$r$-independent part of (A.30) can be cast as

$\eqalignno{-\Lambda_{12}&=k_{12\,m}V_{1}(\overline{\lambda}\gamma^{m}A_{2})-k_{12\,m}V_{2}(\overline{\lambda}\gamma^{m}A_{1})+2V_{1}(\overline{\lambda}W_{2})-2V_{2}(\overline{\lambda}W_{1})\cr&-{1\over(\lambda\overline{\lambda})}(\lambda\gamma^{mp}\overline{\lambda})A_{1\,m}A_{2\,p}+Q_{0}\bigg{[}{1\over
2(\lambda\overline{\lambda})}\bigg{[}(\overline{\lambda}\gamma^{m}A_{1})A_{2\,m}-(\overline{\lambda}\gamma^{m}A_{2})A_{1\,m}\bigg{]}\bigg{]}+O(r)\cr&&({\hbox{A.}31})}$

where we used that

$\eqalignno{-{1\over
2(\lambda\overline{\lambda})}(\overline{\lambda}\gamma^{m}D)\bigg{[}V_{1}A_{2\,m}-V_{2}A_{1\,m}\bigg{]}&=Q_{0}\bigg{[}{1\over
2(\lambda\overline{\lambda})}\bigg{[}(\overline{\lambda}\gamma^{m}A_{1})A_{2\,m}-(\overline{\lambda}\gamma^{m}A_{2})A_{1\,m}\bigg{]}\bigg{]}\cr&+{k_{12\,m}\over
2(\lambda\overline{\lambda})}(\overline{\lambda}\gamma^{m}A_{1})V_{2}-{k_{12\,m}\over
2(\lambda\overline{\lambda})}(\overline{\lambda}\gamma^{m}A_{2})V_{1}\cr&+{(\lambda\gamma^{mp}\overline{\lambda})\over(\lambda\overline{\lambda})}A_{1\,p}A_{2\,m}+5V_{1}(\overline{\lambda}W_{2})-5V_{2}(\overline{\lambda}W_{1})\cr&+{(\overline{\lambda}\gamma^{m}A_{1})\over
2(\lambda\overline{\lambda})}(\lambda\gamma_{m}W_{2})-{(\overline{\lambda}\gamma^{m}A_{2})\over
2(\lambda\overline{\lambda})}(\lambda\gamma_{m}W_{1})&({\hbox{A.}32})}$

and also,

$\eqalignno{{1\over 4}(\lambda\gamma^{mn}\partial_{\lambda})\bigg{[}{1\over
2(\lambda\overline{\lambda})}\bigg{(}V_{1}(\overline{\lambda}\gamma_{mn}W_{2})-&V_{2}(\overline{\lambda}\gamma_{mn}W_{1})\bigg{)}\bigg{]}={5\over
4(\lambda\overline{\lambda})^{2}}\bigg{[}V_{2}(\overline{\lambda}W_{1})-V_{1}(\overline{\lambda}W_{2})\bigg{]}\cr&+{1\over
2(\lambda\overline{\lambda})}\bigg{[}(\lambda\gamma^{m}W_{2})(\overline{\lambda}\gamma_{m}A_{1})-(\lambda\gamma^{m}W_{1})(\overline{\lambda}\gamma_{m}A_{2})\bigg{]}\cr&+{1\over
4(\lambda\overline{\lambda})}\bigg{[}V_{1}(\overline{\lambda}W_{2})-V_{2}(\overline{\lambda}W_{1})\bigg{]}+A_{1}W_{2}-A_{2}W_{1}\cr&&({\hbox{A.}33})}$

On the other hand, the $r$-dependent part can be computed from the following
results:

$\eqalignno{{(\overline{\lambda}\gamma^{mnp}r)\over
16(\lambda\overline{\lambda})^{2}}\bigg{[}(\lambda\gamma_{mn}A_{1})A_{2\,p}-(\lambda\gamma_{mn}A_{2})A_{1\,p}\bigg{]}&={1\over
2(\lambda\overline{\lambda})}(r\gamma^{p}A_{1})A_{2\,p}-{(\lambda r)\over
2(\lambda\overline{\lambda})^{2}}(\overline{\lambda}\gamma^{p}A_{1})A_{2\,p}\cr&-{1\over
4(\lambda\overline{\lambda})^{2}}(\overline{\lambda}\gamma^{m}A_{1})(\lambda\gamma^{p}\gamma_{m}r)A_{2\,p}\cr&-{1\over
2(\lambda\overline{\lambda})}(r\gamma^{p}A_{2})A_{1\,p}+{(\lambda r)\over
2(\lambda\overline{\lambda})^{2}}(\overline{\lambda}\gamma^{p}A_{2})A_{1\,p}\cr&+{1\over
4(\lambda\overline{\lambda})^{2}}(\overline{\lambda}\gamma^{m}A_{2})(\lambda\gamma^{p}\gamma_{m}r)A_{1\,p}&({\hbox{A.}34})}$
$\eqalignno{{(\overline{\lambda}\gamma^{mnp}r)\over
32(\lambda\overline{\lambda})^{2}}\bigg{[}-&(\lambda\gamma_{mn}A_{1})(r\gamma^{p}A_{2})+(\lambda\gamma_{mn}A_{2})(r\gamma^{p}A_{1})\bigg{]}=-{(r\gamma^{p}A_{1})\over
4(\lambda\overline{\lambda})^{2}}(r\gamma_{p}A_{2})\cr&+{(\lambda r)\over
2(\lambda\overline{\lambda})^{3}}(\overline{\lambda}\gamma^{p}A_{1})(r\gamma_{p}A_{2})-{(\overline{\lambda}\gamma^{m}A_{1})\over
8(\lambda\overline{\lambda})^{3}}(\lambda\gamma_{m}\gamma^{p}r)(r\gamma_{p}A_{2})\cr&+{(r\gamma^{p}A_{2})\over
4(\lambda\overline{\lambda})^{2}}(r\gamma_{p}A_{1})-{(\lambda r)\over
2(\lambda\overline{\lambda})^{3}}(\overline{\lambda}\gamma^{p}A_{2})(r\gamma_{p}A_{1})\cr&+{(\overline{\lambda}\gamma^{m}A_{2})\over
8(\lambda\overline{\lambda})^{3}}(\lambda\gamma_{m}\gamma^{p}r)(r\gamma_{p}A_{1})&({\hbox{A.}35})}$
$\eqalignno{{1\over
4(\lambda\overline{\lambda})^{2}}(\overline{\lambda}\gamma^{m}D)\bigg{[}V_{1}(r\gamma_{m}A_{2})-&V_{2}(r\gamma_{m}A_{1})\bigg{]}=Q_{0}\bigg{[}-{(\overline{\lambda}\gamma^{m}A_{1})\over
4(\lambda\overline{\lambda})^{2}}(r\gamma_{m}A_{2})+{(\overline{\lambda}\gamma^{m}A_{2})\over
4(\lambda\overline{\lambda})^{2}}(r\gamma_{m}A_{1})\bigg{]}\cr&+{(r\gamma^{m}D)\over
4(\lambda\overline{\lambda})^{2}}\bigg{[}V_{2}(\overline{\lambda}\gamma_{m}A_{1})-V_{1}(\overline{\lambda}\gamma_{m}A_{2})\bigg{]}\cr&+{1\over
4(\lambda\overline{\lambda})^{2}}\bigg{[}(\overline{\lambda}\gamma^{m}A_{1})(r\gamma_{m}\gamma^{s}\lambda)A_{2\,s}-(\overline{\lambda}\gamma^{m}A_{2})(r\gamma_{m}\gamma^{s}\lambda)A_{1\,s}\cr&+(\overline{\lambda}\gamma^{m}\gamma^{s}\lambda)A_{1\,s}(r\gamma_{m}A_{2})-(\overline{\lambda}\gamma^{m}\gamma^{s}\lambda)A_{2\,s}(r\gamma_{m}A_{1})\bigg{]}&({\hbox{A.}36})}$

In this manner, one learns that

$\eqalignno{\Lambda_{12}&=k_{12\,m}V_{1}(\overline{\lambda}\gamma^{m}A_{2})-k_{12\,m}V_{2}(\overline{\lambda}\gamma^{m}A_{1})+2V_{1}(\overline{\lambda}W_{2})-2V_{2}(\overline{\lambda}W_{1})-{(\lambda\gamma^{mp}\overline{\lambda})\over(\lambda\overline{\lambda})}A_{1\,m}A_{2\,p}\cr&+{(\lambda\gamma^{p}\gamma_{m}r)\over
4(\lambda\overline{\lambda})^{2}}(\overline{\lambda}\gamma^{m}A_{1})A_{2\,p}-{(\lambda\gamma^{p}\gamma_{m}r)\over
4(\lambda\overline{\lambda})^{2}}(\overline{\lambda}\gamma^{m}A_{2})A_{1\,p}+{(\lambda\gamma_{m}\gamma^{p}r)\over
8(\lambda\overline{\lambda})^{3}}(\overline{\lambda}\gamma^{m}A_{1})(r\gamma_{p}A_{2})\cr&-{(\lambda\gamma_{m}\gamma^{p}r)\over
8(\lambda\overline{\lambda})^{3}}(\overline{\lambda}\gamma^{m}A_{2})(r\gamma_{p}A_{1})+{1\over
4(\lambda\overline{\lambda})^{2}}(r\gamma^{m}D)\bigg{[}V_{2}(\overline{\lambda}\gamma_{m}A_{1})-V_{1}(\overline{\lambda}\gamma_{m}A_{2})\bigg{]}\cr&+Q\bigg{[}{1\over
2(\lambda\overline{\lambda})}\bigg{[}(\overline{\lambda}\gamma^{m}A_{1})A_{2\,m}-(\overline{\lambda}\gamma^{m}A_{2})A_{1\,m}\bigg{]}\bigg{]}&({\hbox{A.}37})}$

As before, one can check this expression coincides with that one obtained by
using the Y-formalism $b$-ghost in the minimal pure spinor framework ($r=0$),
with
$Y_{\alpha}\rightarrow{\overline{\lambda}_{\alpha}\over(\lambda\overline{\lambda})}$.

A.3. Two $b$-ghosts On Three Superfields

The simplest and non-trival expression involving nested $b$-ghost is that one
containing two $b$-ghosts and three superfields, which was relevant for the
5-point amplitude computation (3.22). Such an expression reads

$\eqalignno{&b_{0}(b_{0}(V_{1}V_{2})V_{3})&({\hbox{A.}38})}$

One way of computing (A.38) is by using the Y-formalism $b$-ghost, which also
satisfies $\\{Q,b_{Y}\\}=\hbox{$\sqcup$}\hbox to0.0pt{\hss\hbox{$\sqcap$}}$.
Then, the $r$-dependent piece is computed by requiring consistency under the
action of the BRST charge on both sides of the equality. This computation
makes a heavy use of pure spinor and gamma-matrix identities, as well as the
10D super-Yang-Mills equations of motion (2.26), (A.13)-(A.16), and thus we
will not reproduce it here. An alternative and more elegant way to calculate
(A.38), which can then be generalized to higher-points, is via the 3-particle
equation of motion

$\eqalignno{D_{\alpha}A_{123\,\beta}+D_{\beta}A_{123\,\alpha}=&(\gamma^{m})_{\alpha\beta}A_{123\,m}+(k_{1}\cdot
k_{2})[A_{1\,\alpha}A_{23\,\beta}+A_{2\,\alpha}A_{31\,\beta}+(\alpha\leftrightarrow\beta)]\cr&+(k_{12}\cdot
k_{3})[A_{12\,\alpha}A_{3\,\beta}-(12\leftrightarrow 3)]&({\hbox{A.}39})}$

where $A_{123\,\alpha}$ is the 3-particle superfield $\hat{A}_{123\,\alpha}$
of [52], defined as

$\eqalignno{A_{123\,\alpha}&={1\over 2}\bigg{[}(k_{3}\cdot
A_{12})A_{3\,\alpha}-(k_{12}\cdot
A_{3})A_{12\,\alpha}+(\gamma^{p}W_{12})_{\alpha}A_{3\,p}-(\gamma^{p}W_{3})_{\alpha}A_{12\,p}\bigg{]}&({\hbox{A.}40})}$

After contracting with pure spinor variables, eqn. (A.39) becomes

$\eqalignno{QV_{123}&=s_{123}V_{12}V_{3}+s_{12}\bigg{[}V_{1}V_{23}+V_{2}V_{31}+V_{3}V_{12}\bigg{]}\
,&({\hbox{A.}41})}$

Applying $b_{0}$ on both sides of (A.41) yields

$\eqalignno{s_{123}V_{123}-Q(b_{0}(V_{123}))&=s_{123}b_{0}(V_{12}V_{3})+s_{12}b_{0}\bigg{[}V_{1}V_{23}+V_{2}V_{31}+V_{3}V_{12}\bigg{]}&({\hbox{A.}42})}$

One can now use eqn. (A.10) to get

$\eqalignno{s_{123}V_{123}-Q(b_{0}(V_{123}))&=s_{123}\bigg{[}b_{0}(b_{0}(V_{1}V_{2})V_{3})+s_{12}\bigg{(}V_{1}\Lambda_{23}+V_{2}\Lambda_{31}+V_{3}\Lambda_{12}\bigg{)}-s_{123}\Lambda_{12}V_{3}\bigg{]}\cr&+s_{12}b_{0}(V_{1}b_{0}(V_{2}V_{3}))+s_{12}b_{0}(V_{2}b_{0}(V_{3}V_{1}))+s_{12}b_{0}(V_{3}b_{0}(V_{1}V_{2}))\cr&+Q\bigg{[}s_{123}b_{0}(\Lambda_{12}V_{3})-s_{12}\bigg{(}b_{0}(V_{1}\Lambda_{23})+b_{0}(V_{2}\Lambda_{31})+b_{0}(V_{3}\Lambda_{12})\bigg{)}\bigg{]}\cr&&({\hbox{A.}43})}$

The use of the identity then tells us that

$\eqalignno{V_{123}&=\bigg{[}b_{0}(b_{0}(V_{1}V_{2})V_{3})+s_{12}\bigg{(}V_{1}\Lambda_{23}+V_{2}\Lambda_{31}+V_{3}\Lambda_{12}\bigg{)}-s_{123}\Lambda_{12}V_{3}\bigg{]}\cr&-s_{12}\bigg{(}\Lambda_{1}V_{2}V_{3}+V_{1}\Lambda_{2}V_{3}+V_{1}V_{2}\Lambda_{3}\bigg{)}+Q\bigg{[}b_{0}(\Lambda_{12}V_{3})-{s_{12}\over
s_{123}}\bigg{[}b_{0}\bigg{(}V_{1}\Lambda_{23}\cr&+V_{2}\Lambda_{31}+V_{3}\Lambda_{12}-\Lambda_{1}V_{2}V_{3}-V_{1}\Lambda_{2}V_{3}-V_{1}V_{2}\Lambda_{3}\bigg{)}\bigg{]}+{b_{0}(V_{123})\over
s_{123}}\bigg{]}\cr&&({\hbox{A.}44})}$

Therefore,

$\eqalignno{b_{0}(b_{0}(V_{1}V_{2})V_{3})&=V_{123}+s_{12}T_{123}+s_{123}\Lambda_{12}V_{3}+Q\Lambda_{[12]3}&({\hbox{A.}45})}$

where $T_{123}$ was defined in (3.24), and $\Lambda_{[12]3}$ is given by

$\eqalignno{\Lambda_{[12]3}&=-b_{0}\bigg{[}{s_{12}\over
s_{123}}T_{123}+{V_{123}\over
s_{123}}+\Lambda_{12}V_{3}\bigg{]}&({\hbox{A.}46})}$

Although this expression appears to not be manifestly local, we have
explicitly checked that the $1/s_{123}$ pole cancels in the $r$-independent
sector, and $\Lambda_{[12]3}$ just reduces to the multi-particle
generalization of $\Lambda_{12}$.

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