my dot/.style= /tikzfeynman/dot, /tikz/minimum size=15pt, aainstitutetext:
Department of Physics and Astronomy, Northwestern University, Evanston,
Illinois, 60208, USA bbinstitutetext: Department of Physics and Astronomy,
Uppsala University, Box 516, 75120 Uppsala, Sweden

# Color-kinematics dual representations of one-loop matrix elements in the
open-superstring effective action

Alex Edison b and Micah Tegevi<EMAIL_ADDRESS>
###### Abstract

The $\alpha^{\prime}$-expansion of string theory provides a rich set of
higher-dimension operators, indexed by $\zeta$ values, which can be used to
study color-kinematics duality and the double copy. These two powerful
properties, actually first noticed in tree-level string amplitudes, simplify
the construction of both gauge and gravity amplitudes. However, their
applicability and limitations are not fully understood. We attempt to
construct a set of color-kinematics dual numerators at one loop and four
points for insertions of operator combinations corresponding to the lowest
four $\zeta_{2}$-free operator insertions from the open superstring:
$\zeta_{3}$, $\zeta_{5}$, $\zeta_{3}^{2}$, and $\zeta_{7}$. We succeed in
finding a representation for the first three in terms of box, triangle, and
bubble numerators. In the case of $\zeta_{7}$ we find an obstruction to a
fully color-dual representation related to the bubble-on-external-leg type
diagrams. We discuss two paths around this obstruction, both of which signal
an incompatability between color-kinematics duality and manifesting certain
desired properties. Using the constructed color-dual numerators, we find two
different Bern-Carrasco-Johansson double copies that produce candidate closed-
string-insertion numerators. Both approaches to the double copy match the
kinematics of the cuts, with relative normalization set by either summing over
both double copies including degeneracy or by including an explicit prefactor
on the double-copy numerator definitions.

††preprint: UUITP-47/22

##  1 Introduction

The study of scattering amplitudes has long built on the interplay between
generating new theoretical data, manifesting different properties of the data,
and finding new ways of encoding the results. One significant achievement of
this progression is the discovery of double-copy relations between gauge
theory and gravity amplitudes. At tree level, the connections between three
different manifestations of the double copy: Kawai-Lewellen-Tye
(KLT)Kawai:1985xq , Bern-Carrasco-Johansson (BCJ)Bern:2010ue , and Cachazo-He-
Yuan (CHY)Cachazo:2013iea , have opened up a wide range of methods for
studying gauge and gravity amplitudes. All three representations can also be
useful in studying loop-level amplitudes. The KLT and CHY formulations are
effective at generating on-shell data for both gauge and gravity theories. The
loop level incarnation of the BCJ double copy allows calculating gravitational
diagram numerators directly from gauge theory numerators.

The tree-level BCJ double copy starts from the observation that when writing
the four-point Yang–Mills (YM) amplitude in terms of cubic diagrams, the group
Jacobi relations between the color factors of the $s$, $t$ and $u$ channels

$c_{s}-c_{t}-c_{u}=0$ (1)

are a necessary relation to ensure gauge invariance of the amplitude. If we
then suppose that we can write a gravitational amplitude in terms of the
square of some numerator factor $N_{g}$, linearized diffeomorphism invariance
requires that

$N_{s}-N_{t}-N_{u}=0\,.$ (2)

The algebraic duality between the color Jacobi relations, eq. 1, and eq. 2 is
referred to as color-kinematics (CK) duality. The full BCJ double copy states
that if one can write an amplitude in terms of numerators that obey eq. 2 for
any $s$, $t$, $u$ triplet of cubic graphs whose color obeys eq. 1, then a
“gravitational” amplitude can be obtained by replacing all color factors with
their corresponding numerators. This process can even work at loop level, with
the caveat that in the color Jacobi relations one should only sum over the
color contractions within the triplet. It has also been shown to work for a
wide range of theories. Of interest in the current work is that when $N_{g}$
is a set of numerators for an $n$-point $\mathcal{N}=4$ super-Yang–Mills (sYM)
amplitude obeying eq. 2, then $N_{g}^{2}$ is an $n$-point numerator for
$\mathcal{N}=8$ supergravity (SG). See Ref. Bern:2019prr  for a comprehensive
review of the subject.

Recent decades have also seen a growing effort to explore ways that scattering
amplitudes and string theoretic methods can inform each other. Even just
limiting the scope to the $\alpha^{\prime}$-expansion of the open superstring,
we find a rich playground to: study higher-dimension operators, generalize the
notion of double copies Chi:2021mio , find interesting representations of
theories Carrasco:2016ldy ; Carrasco:2016ygv , and even inform number theory
Schlotterer:2012ny ; Stieberger:2013wea ; Stieberger:2014hba ;
Schlotterer:2018zce ; Vanhove:2018elu ; Brown:2019wna . Working in the other
direction, amplitudes methods were recently applied to calculating one-loop
matrix elements in both the open- and closed-string effective actions
Edison:2021ebi . These methods were able to verify pieces of the closed-string
results of D’Hoker and Green DHoker:2019blr , produce new predictions for the
open string, and target terms currently unreachable by standard string
methods.

The goal of this paper is to continue building the groundwork for exploring
loop-level BCJ with higher-dimension operators. We will attempt to construct
CK dual numerators for open-superstring matrix elements up through
$\alpha^{\prime 7}$. The paper is structured as follows. We begin by reviewing
the structure of amplitudes related to color-kinematics duality in section 2,
including discussion on loop-level considerations and tree-level constructions
in sYM and the open superstring. Next, in section 3 we demonstrate the
simplicity of constructing one-loop unitarity cuts from well-structured tree
amplitudes, even with the higher dimension operators of the open string low
energy effective action. Finally, in section 4 we apply the method of maximal
cuts Bern:2007ct ; Bern:2008pv ; Bern:2010tq ; Bern:2012uf ; Bern:2018jmv  in
conjunction with CK duality to construct color-dual representations for
insertions of open string operators, and double copy the numerators to find
representations for closed-string operator insertions.

##  2 Review

We begin with a review of the structure of tree amplitudes in
(super-)Yang–Mills, with a particular eye towards the duality between color
and kinematics. A tree-level gauge amplitude can be written

$\mathcal{A}^{\mathrm{tree}}_{n}=-ig_{*}^{n-2}\sum_{g\in\Gamma_{3,n}}\frac{c_{g}N_{g}}{\prod_{i}p_{g_{i}}^{2}.},$
(3)

where the sum runs over the set of all cubic graphs with $n$ external legs,
$\Gamma_{3,n}$, including permutations of external legs. The index $n$ is the
number of external particles, and $g_{*}$ is the coupling constant and other
overall normalizations for the theory. The kinematic denominator is the
product of all propagators in the graph, $p_{g_{i}}$. The numerators are
decomposed into color factors $c_{g}$, and kinematic factors $N_{g}$. The
color factors are gauge group structure constants. The kinematic contain
Lorentz products of polarization vectors for the external particles,
$\varepsilon_{i}$, as well as the momenta for the particles, $p_{i}$. In this
paper, we will use all-outgoing external momenta.

We work in a normalization where the Mandelstam variables of massless momenta
carry a factor of $2$ and no explicit factors of $\alpha^{\prime}$, i.e.

$s_{ij}\equiv 2p_{i}\cdot p_{j}=(p_{i}+p_{j})^{2}|_{p^{2}=0}$ (4)

with a natural extension to multi-labled $s$

$s_{ij\dots}\equiv(p_{i}+p_{j}+\dots)^{2}|_{p^{2}=0}=\sum_{a,b\in\\{i,j,\dots\\}}2p_{a}\cdot
p_{b}\,.$ (5)

When off-shell momenta, like loop momenta, are included we use the sum-of-
products definition

$s_{\ell i\dots}=s_{i\dots}+\sum_{a\in\\{i,\dots\\}}2\ell\cdot p_{a}\,.$ (6)

The color factors $c_{g}$ come from dressing every cubic-vertex in the graph
with group theory structure constants
$f^{abc}=\mathrm{Tr}([T^{a},T^{b}]T^{c})$, where the $T^{a}$’s are generators
of the gauge group. We assume that the structure constants belong to a Lie
group so that they satisfy the Jacobi relations

$f^{12\alpha}f^{\alpha 34}-f^{41\alpha}f^{\alpha 23}-f^{31\alpha}f^{\alpha
42}=0.$ (7)

Identifying the three products of structure constants in eq. 7 with the color
factors for the $s$-, $t$-, and $u$-channel cubic graphs, fig. 1, the Jacobi
relation can be simplified to

$c_{s}-c_{t}-c_{u}=0\,.$ (8)

The full color-dressed amplitude $\mathcal{A}_{n}^{\mathrm{tree}}$ can be
decomposed into an $(n-2)!$ basis of color-ordered amplitudes through
judicious application of cyclicity, reflection (anti-)symmetry, the photon
decoupling identity, and the Kleiss-Kuijf relations Mangano:1990by ;
Zeppenfeld:1988bz ; Mangano:1988kk ; Kosower:1987ic ; Berends:1987cv ;
Cvitanovic:1980bu ; Kleiss:1988ne . At four points, the Jacobi relation tells
us that the color coefficients of the color-ordered amplitudes can be chosen
as any two of $c_{s},c_{t},$ and $c_{u}$.

[scale=0.6,mode=buildnew]figures/sDiag [scale=0.6,mode=buildnew]figures/tDiag
[scale=0.6,mode=buildnew]figures/uDiag

Figure 1: The s-, t-, and u-channel cubic 4-point tree level graphs

Bern and collaborators Bern:2008qj ; Bern:2010ue ; Bern:2010yg  discovered
that one can arrange the kinematic numerators such that they obey the same
algebraic relations as the color factors, e.g.

$N\left(\begin{gathered}\includestandalone[scale=0.7,mode=buildnew]{figures/sBCJ}\end{gathered}\right)-N\left(\begin{gathered}\includestandalone[scale=0.7,mode=buildnew]{figures/tBCJ}\end{gathered}\right)-N\left(\begin{gathered}\includestandalone[scale=0.7,mode=buildnew]{figures/uBCJ}\end{gathered}\right)=0\,,$
(9)

for all diagrams that only differ on the four-point sub diagram. Further, once
such numerators are found, gravity amplitudes can be constructed via a “BCJ
double copy” Bern:2010ue  by simply replacing the $c_{g}$ in eq. 3 with a
second copy of $N_{g}$

$\mathcal{A}_{n}^{\mathrm{tree}}(1,2,\ldots,n)=\sum_{g\in\Gamma_{3,n}}\frac{c_{g}N_{g}}{D_{g}}\Rightarrow\mathcal{M}_{n}^{\mathrm{tree}}(1,2,\ldots,n)=\sum_{g\in\Gamma_{3,n}}\frac{N_{g}^{2}}{D_{g}}\,.$
(10)

Since the numerator relations, eq. 9, are algebraically the same as the group
theory Jacobi relations, they are referred to as the kinematic Jacobi
relations. A set of numerators which obey all possible kinematic Jacobi
relations are referred to as color-kinematics (CK) dual numerators. Asymmetric
double copies are also possible between two distinct theories, as long as they
share a basis of cubic graphs and at least one of theories is written in terms
of CK dual numerators. There are a wide variety of methods for calculating
manifestly CK dual numerators at tree level Cachazo:2013hca ; Cachazo:2013iea
; Lam:2016tlk ; Du:2017gnh ; Fu:2017uzt ; Du:2017kpo ; Edison:2020ehu ;
Cheung:2021zvb ; Ben-Shahar:2021zww ; Carrasco:2021ptp ; Carrasco:2019yyn ,
one of which we will recap below in section 2.2. Significant work has also
gone into finding double copy representations of a wide range of theories; we
direct interested readers to Ref. Bern:2019prr , a recent comprehensive review
of the topic.

###  2.1 CK duality at one loop

The general structure of a loop level amplitude is a straightforward extension
of the tree-level structure (dropping the coupling for simplicity)

$\mathcal{A}^{L\text{-loop}}_{n}=\sum_{g\in\Gamma^{(L)}_{3,n}}\int\left(\frac{d^{D}\ell}{(2\pi)^{D}}\right)^{L}\frac{1}{S_{g}}\frac{c_{g}N_{g}}{\prod_{i}p_{g_{i}}^{2}.}\,,$
(11)

where the diagram sum is now over diagrams with exactly $L$ loops, and the
symmetry factors $S_{g}$ are the same ones that appear in Feynman diagrams and
take care of any over-count due to external orderings or internal
automorphisms. Since the amplitude is again written in terms of pairings
between color and kinematic structures, color kinematics duality can still be
explored and exploited to simplify both gauge theory and gravity calculations.
The gauge theory simplifications come from the fact that the kinematic Jacobi
relations can be used to express all cubic loop diagram numerators in terms of
a small number of _basis_ diagrams. This can be useful even in the one-loop
four-point case. The relevant graph topologies that will enter our
calculations are the box, the triangle, and the 2-2 bubble

$\Gamma^{(1)}_{3,4}=\left\\{\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/box}\end{gathered},\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/tri}\end{gathered},\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/bub}\end{gathered}\right\\}+\text{relabelings}\,.$
(12)

Through the kinematic Jacobi relations, we can relate the triangle and bubble
numerators to different labelings of the box numerator. We start by applying
the Jacobi relation on the edge connecting externals 1 and 2 of the box-
diagram with color-order 1234, giving the following relation between diagrams

$\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/boxBCJ}\end{gathered}-\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/box2134BCJ}\end{gathered}-\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triAltBCJ}\end{gathered}=0.$
(13)

Expressed as the numerators of the graphs, eq. 13 fixes the numerator for the
triangle graph, $N_{\bigtriangleup}(12|3|4)$, in terms of the two box
numerators

$\begin{split}&N_{\square}(1|2|3|4)-N_{\square}(1|2|4|3)-N_{\bigtriangleup}(1|2|3,4)=0\\\
\Rightarrow&N_{\bigtriangleup}(1|2|3,4)=N_{\square}(1|2|3|4)-N_{\square}(1|2|4|3).\end{split}$
(14)

When referring to functions of loop diagrams (both numerators and cuts), we
use vertical bars to separate vertices in the loop, commas to separate
external legs which meet at a three-point vertex, and concatenated labels for
when external legs meet at a higher–point vertex. We can get another relation
involving the 2-2 bubble diagram by applying the Jacobi relation on the loop
edge connecting externals 3 and 4 of the triangle

$\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triAltBCJ2}\end{gathered}-\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triALTBCJ1243}\end{gathered}-\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/bubBCJ}\end{gathered}=0.$
(15)

This relates the bubble numerator, $N_{\bigcirc}(1,2|3,4)$, to the triangle
and, using the relation we just found in eq. 14, the box numerators

$\begin{split}&N_{\bigtriangleup}(1|2|3,4)-N_{\bigtriangleup}(2|1|3,4)-N_{\bigcirc}(1,2|3,4)=0\\\
\Rightarrow&N_{\bigcirc}(1,2|3,4)=N_{\bigtriangleup}(1|2|3,4)-N_{\bigtriangleup}(2|1|3,4)\\\
&\qquad=N_{\square}(1|2|3|4)-N_{\square}(1|2|4|3)-N_{\square}(2|1|3|4)+N_{\square}(2|1|4|3).\end{split}$
(16)

Since eqs. 14 and 16 are used to define triangle and bubble numerators in
terms of the box numerator, we refer to them as the _defining_ kinematic
Jacobi relations. In addition to these defining relations, there are also
_boundary_ kinematic Jacobi relations that relate a difference of two non-zero
diagrams to bubble-on-external-leg (BEL) or tadpole diagrams, whose numerators
we often set to $0$. The main boundary relation stems from applying the Jacobi
relation on one of the triangle edges only adjacent to one external leg, which
yields

$\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triBoundBCJ}\end{gathered}-\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triBoundBCJ2}\end{gathered}-\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/belBoundBCJ}\end{gathered}=0\,.$
(17)

In the “snail” regularization of Ref. Bern:2012uf , the third diagram in the
relation is either exactly zero or carries a numerator factor of
$k_{1}^{2}\sim 0$ meaning that the kinematic Jacobi relation reduces to a
consistency condition on the triangle numerators. Alternatively, one can use
“Minahaning” regularization Minahan:1987ha ; Edison:2022jln , in which the BEL
numerators are not definied to be $0$ (and can thus be defined via eq. 17),
but are regulated by relaxing momentum conservation from $n$-point to
$n+1$-point and requiring that they conspire between themselves on the cuts
and at the level of the integrand to cancel the unphysical $k_{1}^{-2}$ pole.
We adopt the “snail” regularization in this work, and will generally assume
that the BEL numerator is identically $0$ unless explicitly stated otherwise.

While the four-point one-loop Jacobi relations are rather simple, the same
basic method can become quite complicated yet powerful for higher loops and
legs. For example, the four-loop four-point integrand of maximal sYM can be
expressed in terms of 85 diagrams. The numerators for 83 of these diagrams can
be expressed in terms of only two planar diagrams, or one non-planar diagram
Bern:2012uf . Beyond four loops there are problems manifesting CK duality
Bern:2017ucb , and effort has been put into finding more-accessible examples
for testing the assumptions that go into finding CK numerators Bern:2013yya ;
Bern:2015ooa ; Mogull:2015adi ; Johansson:2017bfl ; Bridges:2021ebs .

###  2.2 Tree amplitudes with manifest CK duality in sYM and the open
superstring

A systematic and efficient method of constructing tree amplitudes that
manifests their double copy structures is using the Cachazo, He, and Yuan
(CHY) formalism Cachazo:2013hca . In this form, tree amplitudes are given by
an integral over $n$ punctures on a sphere against a product of theory-
dependent half-integrands $\mathcal{I}_{L}(\sigma)$, $\mathcal{I}_{R}(\sigma)$

$\mathcal{A}_{n}=\int\frac{d^{n}\sigma}{\mathrm{Vol}\left(SL(2,\mathbb{C})\right)}\prod_{a}{}^{\prime}\delta\left(\sum_{b\neq
a}\frac{s_{ab}}{\sigma_{a}-\sigma_{b}}\right)\mathcal{I}_{L}(\sigma)\mathcal{I}_{R}(\sigma)\,.$
(18)

The $\prod{}^{\prime}\delta(\dots)$ fully localizes the integrals, and the
arguments of the $\delta(\dots)$ are known as the scattering equations. For
(super-)Yang–Mills, one of the half-integrands encodes the color information
and the other half-integrand encodes the kinematics, while for (super)gravity
both half-integrands carry kinematics. In both cases, the integrals can be
resolved in terms of bi-adjoint scalar amplitudes $m(\alpha|\beta)$
Vaman:2010ez ; Du:2011js ; Bjerrum-Bohr:2012kaa ; Cachazo:2013iea  as

$\displaystyle\mathcal{A}_{\text{sYM}}^{\text{tree}}$
$\displaystyle=\sum_{\sigma,\rho\in
S_{n-2}}c\left(\begin{minipage}{86.72267pt}\includestandalone[width=mode=buildnew]{figures/half-
ladder}\end{minipage}\right)m(1,\rho,n|1,\sigma,n)N\left(\begin{minipage}{86.72267pt}\includestandalone[width=mode=buildnew]{figures/half-
ladder-sigma}\end{minipage}\right)$ $\displaystyle=\sum_{\sigma,\rho\in
S_{n-2}}c(1,\rho,n)m(1,\rho,n|1,\sigma,n)N(1,\sigma,n)$ (19)
$\displaystyle\mathcal{M}_{\text{GR}}^{\text{tree}}$
$\displaystyle=\sum_{\sigma,\rho\in
S_{n-2}}\tilde{N}(1,\rho,n)m(1,\rho,n|1,\sigma,n)N(1,\sigma,n)$ (20)

where the sums run over permutations of $2,3,\dots,n{-}1$, the $c(1,\rho,n)$
are DDM half-ladder color structures DelDuca:1999rs , and the $N$, $\tilde{N}$
are corresponding kinematic half-ladder numerators. These two equations
manifest the double-copy relations between gauge theory and gravity: a
gravitational amplitude can be obtained by simply replacing the color
structures of eq. 19 with a second copy of the kinematic numerator. The (s)YM
representation also manifests the decomposition into an $(n-2)!$ basis of
color-ordered amplitudes DelDuca:1999rs ; Kleiss:1988ne , by simply
restricting to a single element in the sum over $\rho$ and stripping the color
factor

$A_{\text{sYM}}^{\text{tree}}(1,\rho,n)=\sum_{\sigma\in
S_{n-2}}m(1,\rho,n|1,\sigma,n)N(1,\sigma,n)\,.$ (21)

The $N(\beta)$ are functions of external polarizations and momenta, and are
half-ladder _basis numerators_ for the BCJ relations: all non-half-ladder
graph numerators can be obtained from them via repeated application of the
kinematic Jacobi relations, eq. 9. For example, at four points we have

$\displaystyle A_{\text{sYM}}^{\mathrm{tree}}(1,2,3,4)$
$\displaystyle=\left(\frac{1}{s}+\frac{1}{t}\right)N(1,2,3,4)-\frac{1}{t}N(1,3,2,4)$
$\displaystyle=\frac{N(1,2,3,4)}{s}+\frac{N(1,2,3,4)-N(1,3,2,4)}{t},$ (22)

so that the $N(1,\sigma,4)$ are equivalent to the numerators in the standard
presentation of the amplitude of eq. 3

$\displaystyle N(1,2,3,4)\equiv N_{s}\quad N(1,3,2,4)\equiv N_{u}\quad
N_{t}\equiv N(1,2,3,4)-N(1,3,2,4)\,.$ (23)

The composition of $N(\dots)$ is dependent on the theory under consideration.
Since the original presentation by CHY, significant work has gone into finding
efficient calculations and compact representations of $N(\beta)$ for
(super-)Yang–Mills Mafra:2015vca ; Bjerrum-Bohr:2016axv ; Lam:2016tlk ;
Du:2017gnh ; Fu:2017uzt ; Du:2017kpo ; Edison:2020ehu ; Cheung:2021zvb . Most
convenient for the current work is the construction developed by one of the
current authors and Teng Edison:2020ehu , which calculates the numerators via
auxiliary combinatoric structures known as _increasing trees_. The resulting
numerators for external gluons can be rearranged as

$\displaystyle N(1,\sigma,n)$ $\displaystyle=\sum_{\sigma=A\cup
B}\left(\varepsilon_{1}\cdot f_{b_{1}}\cdot f_{b_{2}}\ldots
f_{b_{|B|}}\cdot\varepsilon_{n}\right)\mathfrak{N}_{\sigma}(1;A;n)$ (24)

with linearized field strengths
$f_{i}^{mn}=p_{i}^{m}\varepsilon_{i}^{n}-p_{i}^{n}\varepsilon_{i}^{m}$, and
where $\mathfrak{N}_{\sigma}(1;A;n)$ is independent of $n$ and dependent on
the momenta of $1$ and $A$, but only the polarizations of $A$. The exact
content of $\mathfrak{N}_{\sigma}(\dots)$ will not be relevant for the current
discussion; we will only encounter $\mathfrak{N}_{\sigma}(a;\emptyset;b)=1$
here. There is also a natural extension when particles $1$ and $n$ are
fermions

$N(1_{f},\sigma,n_{f})=\sum_{\sigma=A\cup
B}\left(\chi_{1}\not{f}_{b_{1}}\ldots\not{f}_{b_{|B|}}\xi_{n}\right)\mathfrak{N}_{\sigma}(1;A;n)$
(25)

with fermion polarizations

$\not{p}_{i}\cdot\chi_{i}=0,\qquad\not{p}_{n}\cdot\xi_{n}=\chi_{n}$ (26)

and $\gamma$-slash conventions

$\not{p}\equiv
p^{\mu}\gamma_{\mu},\qquad\not{f}_{i}\equiv\frac{1}{8}f_{i}^{\mu\nu}\left[\gamma_{\mu},\gamma_{\nu}\right]=\frac{1}{2}\not{p}_{i}\not{\varepsilon}_{i}$
(27)

which is convenient for computing loops in sYM Edison:2020uzf ; Edison:2021ebi
. With an eye towards maximal sYM in $D=10$ and its dimensional reductions, it
is most straightforward to work in the $16\times 16$ Weyl representation of
the $10$D Dirac matrices, in which case the fermion polarizations will be
16-component $\text{SO}(1,9)$ spinors. We will shorthand the products of $f$s
appearing in eqs. 24 and 25 as

$\displaystyle W_{2g}(1,B,n)=\varepsilon_{1}\cdot f_{B_{1}}\ldots\cdot
f_{B_{|B|}}\cdot\varepsilon_{n},$ (28) $\displaystyle
W_{2f}(1_{f},B,n_{f})=\chi_{1}\not{f}_{B_{1}}\ldots\not{f}_{B_{|B|}}\xi_{n}.$
(29)

The particular usefulness of this representation for the task at hand stems
from the fact that the polarizations for $1$ and $n$ are structurally
distinguished from the other particles, and thus behave well when used as the
sewn legs in the construction of unitarity cuts.

The CHY integral representation for sYM, eq. 18, also has an interpretation as
the $\alpha^{\prime}\to 0$ limit of superstring tree amplitudes computed via
disk integrals. We limit ourselves to a brief presentation here, and refer the
reader to Ref. Edison:2021ebi  and references therein for a deeper discussion
of the subject. From the disk integral representation of open-superstring
amplitudes

$A_{\mathrm{super}}^{\mathrm{tree}}(1,2,\ldots,n)=\int\limits_{0<z_{2}<\ldots<z_{n-2}}\frac{dz_{2}dz_{3}\ldots
dz_{n-2}}{\mathrm{Vol}(SL(2,\mathbb{R}))}\left\langle V_{1}(z_{1})\ldots
V_{n}(z_{n})\right\rangle$ (30)

the worldsheet correlator $\langle V_{1}(z_{1})\dots V_{n}(z_{n})\rangle$ can
be rearranged with the help of total $z$ derivatives to collect all
$\alpha^{\prime}$ dependence into the Koba-Nielsen factor

$\displaystyle A_{\mathrm{super}}^{\mathrm{tree}}(1,2,\ldots,n)$
$\displaystyle=\int\limits_{0<z_{2}<\ldots<z_{n-2}}\frac{dz_{2}dz_{3}\ldots
dz_{n-2}}{\mathrm{Vol}\left(SL(2,\mathbb{R})\right)}\prod_{1\leq
i<j}^{n-1}|z_{i}-z_{j}|^{2\alpha^{\prime}k_{i}\cdot k_{j}}$
$\displaystyle\times\sum_{\rho,\tau\in
S_{n-3}}A_{\mathrm{sYM}}^{\mathrm{tree}}(1,\rho,n-1,n)\frac{S_{0}\left[\rho|\tau\right]_{1}}{(z_{1}-z_{\tau_{1}})\dots(z_{n}-z_{n-1})}\,,$
(31)

where $S_{0}[\rho|\tau]_{1}$ is the field-theory KLT matrix, and is related to
the matrix inverse of $m(\alpha|\beta)$ with legs $1,n-1,n$ at fixed positions
in $\alpha,\beta$ Bern:1998sv ; Bjerrum-Bohr:2010pnr . The
$A_{\text{sYM}}^{\text{tree}}$ and $S_{0}[\rho|\tau]_{1}$ are independent of
the $z_{i}$. As such, all of the $z_{i}$ and $\alpha^{\prime}$ dependent terms
can be collected into a single object Mafra:2011nv ; Broedel:2013tta

$Z(a_{1},a_{2},\ldots,a_{n}|b_{1},b_{2},\ldots,b_{n})=\\!\\!\int\limits_{z_{a_{1}}<z_{a_{2}}<\ldots<z_{a_{n}}}\\!\\!\\!\frac{dz_{1}dz_{2}\ldots
dz_{n}}{\mathrm{Vol}(SL_{2}(\mathbb{R}))}\frac{\prod_{1\leq
i<j}^{n-1}|z_{i}-z_{j}|^{2\alpha^{\prime}k_{i}\cdot
k_{j}}}{(z_{b_{1}}-z_{b_{2}})\dots(z_{b_{n}}-z_{b_{1}})}\,,$ (32)

known as $Z$ functions or $Z$-theory amplitudes. Explicit expressions for the
$Z$-theory amplitudes can be computed using the BGap package BGap  or
extracted from the $F$ matrix representation Schlotterer:2012ny ; MZVWebsite ;
Broedel:2013aza . In the limit $\alpha^{\prime}\to 0$, $Z$-theory amplitudes
become bi-adjoint amplitudes

$\lim_{\alpha^{\prime}\to 0}Z(\rho|\sigma)=m(\rho|\sigma)\,.$ (33)

Using $Z$ functions, the open-superstring tree amplitude can be written as

$A_{\mathrm{super}}^{\mathrm{tree}}(a_{1},\ldots,a_{n})=\sum_{\rho,\tau\in
S_{n-3}}Z(a_{1},\dots|1,\rho,n,n-1)S_{0}[\rho|\tau]_{1}A_{\mathrm{sYM}}(1,\tau,n-1,n)\,.$
(34)

The $\alpha^{\prime}$ expansion of the matrix $Z\cdot S_{0}$ can be further
decomposed into kinematic matrices $\mathbb{P}$, $\mathbf{M}_{2n+1}$, and
$\mathbb{Q}$ Schlotterer:2012ny ; MZVWebsite ; Broedel:2013aza

$\displaystyle Z\cdot S_{0}$
$\displaystyle=\mathbb{P}\,\mathbb{Q}:\exp\left(\sum_{i\geq
1}\zeta_{2i+1}\mathbf{M}_{2i+1}\right):$
$\displaystyle=\mathbb{P}\left(1+\zeta_{3}\mathbf{M}_{3}+\zeta_{5}\mathbf{M}_{5}+\frac{1}{2}\zeta_{3}^{2}\mathbf{M}_{3}^{2}+\zeta_{3}\zeta_{5}\mathbf{M}_{5}\mathbf{M}_{3}+\zeta_{3,5}[\mathbf{M}_{5},\mathbf{M}_{3}]+\dots\right)$
(35)

which contain the kinematic information paired with $\zeta$ values. The
matrices $\mathbb{P}$ and $\mathbb{Q}$ are infinite series containing all of
the $\zeta_{2k}$ and the higher-depth multiple-$\zeta$ values, respectively.
The entries of $\mathbf{M}_{w}$ are homogenous polynomials of degree $w$ in
$s_{ij}$ with rational coefficients. The ordering colons prescribe that all
products of $\mathbf{M}_{i}$ resulting from the expansion of the exponential
only appear in descending order of $i$, i.e. we only have
$\mathbf{M}_{i_{1}}\mathbf{M}_{i_{2}}\dots$ when $i_{1}>i_{2}>\dots$.

Through the use of eq. 21 and the relation between $S_{0}$ and
$m(\alpha|\beta)$ we can further rewrite eq. 34 as

$\displaystyle A_{\mathrm{super}}^{\mathrm{tree}}(a_{1},\ldots,a_{n})$
$\displaystyle=\\!\sum_{\rho,\tau\in
S_{n-3}}\\!Z(a_{1}\dots|1,\rho,n,n{-}1)S[\rho|\tau]_{1}\\!\sum_{\sigma\in
S_{n-2}}\\!\\!m(1,\tau,n{-}1,n|1,\sigma,n)N(1,\sigma,n)$
$\displaystyle=\sum_{\rho\in S_{n-2}}Z(a_{1}\dots|1,\rho,n)N(1,\rho,n)\,.$
(36)

Thus we see that sYM tree amplitudes can be promoted to open-superstring tree
amplitudes by simply upgrading bi-adjoint amplitudes to $Z$-theory amplitudes.
Further, we can interpret eq. 34 as writing the open superstring as a double
copy between the amplitudes of $Z$-theory and super-Yang–Mills. Expanding both
sides of the double copy in $\alpha^{\prime}$ leads to the schematic
identification of operators between the respective effective actions
Mafra:2016mcc

$\begin{split}(\alpha^{\prime})^{0}:F^{2}&=\mathrm{sYM}\otimes\phi^{3},\\\
\zeta_{2}(\alpha^{\prime})^{2}:\
F^{4}&=\mathrm{sYM}\otimes(\partial^{2}\phi^{4}+\phi^{5}),\\\
\zeta_{3}(\alpha^{\prime})^{3}:\
D^{2}F^{4}+F^{5}&=\mathrm{sYM}\otimes(\partial^{4}\phi^{4}+\partial^{2}\phi^{5}+\phi^{6}),\\\
\zeta_{4}(\alpha^{\prime})^{4}:\
D^{4}F^{4}+D^{2}F^{5}+F^{6}&=\mathrm{sYM}\otimes(\partial^{6}\phi^{4}+\partial^{4}\phi^{5}+\partial^{2}\phi^{6}+\phi^{7}).\end{split}$
(37)

For the first four orders in $\alpha^{\prime}$, there is only one combination
of operators that appears. Starting at $\alpha^{\prime 5}$ multiple
combinations appear. However, these operators can be grouped according to
their $\zeta$-value coefficients Kitazawa:1987xj ; Barreiro:2005hv ;
Barreiro:2012aw ; Koerber:2002zb ; Oprisa:2005wu . For instance,
$\alpha^{\prime 5}$ dresses two different operator combinations, one with
coefficient $\zeta_{2}\zeta_{3}$ and another with $\zeta_{5}$
Schlotterer:2012ny . Throughout the rest of this work, we will often refer to
terms in the expansion of $Z$-theory and their corresponding operator
insertions via their $\zeta$-value coefficient, e.g. we will talk about
$\zeta_{3}$ insertions as a shorthand for insertions of the $D^{2}F^{4}+F^{5}$
operator combination appearing in $Z$ theory at $\alpha^{\prime 3}$.
Similarly, we will often index objects by the $\zeta$ value of the operator
they correspond to, as in $Z_{\zeta_{3}}$ or $N_{g}^{\zeta_{5}}$.

Closed-string amplitudes can be expanded in a very similar manner using the
KLT relation Kawai:1985xq  which equates closed-string amplitudes to products
of open-string amplitudes via the $\alpha^{\prime}$ uplift of the kinematic
kernel $S_{0}\to S_{\alpha^{\prime}}$

$\mathcal{M}_{\text{closed}}^{\text{tree}}=\left(A_{\text{open}}^{\text{tree}}\right)^{T}\cdot
S_{\alpha^{\prime}}\cdot A_{\text{open}}^{\text{tree}}\,.$ (38)

Inserting eq. 35, noting that the transposition operation reverses the color
ordering prescription, and with the help of
$\mathbb{P}^{T}S_{\alpha^{\prime}}\mathbb{P}=S_{0}$ and
$\mathbf{M}_{i}^{T}S_{0}=S_{0}\mathbf{M}_{i}$ we get Schlotterer:2012ny

$\displaystyle\mathcal{M}_{\text{closed}}^{\text{tree}}$
$\displaystyle=(A_{\text{sYM}}^{\text{tree}})^{T}S_{0}\left(1+2\zeta_{3}\mathbf{M}_{3}\dots+2\zeta_{3}\zeta_{5}\left(\mathbf{M}_{5}\mathbf{M}_{3}+\mathbf{M}_{3}\mathbf{M}_{5}\right)+\dots\right)A_{\text{sYM}}^{\text{tree}}\,.$
(39)

Making use of eq. 21, we find a multiplication of two sYM numerator factors
via the matrix kernel $m\cdot
S_{0}\cdot(1+2\zeta_{3}\mathbf{M}_{3}+\dots)\cdot m$ (suppressing the various
permutation labels), which can be identified as the result of applying the
single-valued map “sv” Schnetz:2013hqa ; Brown:2013gia 111Since we only deal
with single $\zeta$s in this work, we will omit discussion of the subtleties
related to the “sv” map and multiple $\zeta$ values.

$\operatorname{sv}1=1\qquad\operatorname{sv}\zeta_{2k}=0\qquad\operatorname{sv}\zeta_{2k+1}=2\zeta_{2k+1}\qquad\text{for
}k\in\mathbb{N}$ (40)

order-by-order in $\alpha^{\prime}$ on the $Z$-functions Schlotterer:2012ny ;
Stieberger:2013wea ; Stieberger:2014hba ; Schlotterer:2018zce ;
Vanhove:2018elu ; Brown:2019wna

$\displaystyle m(\gamma,|\dots)\cdot
S_{0}\cdot(1+2\zeta_{3}\mathbf{M}_{3}+\dots)\cdot m(\dots|\rho)$
$\displaystyle=\operatorname{sv}Z(\gamma|\rho)\,,$ (41)

leading to a compact representation of the closed-string tree amplitude
Broedel:2013tta

$\mathcal{M}_{\text{closed}}^{\text{tree}}=\sum_{\sigma,\rho\in
S_{n-2}}\tilde{N}(1,\rho,n)\left[\operatorname{sv}Z(1,\rho,n|1,\sigma,n)\right]N(1,\sigma,n)\,.$
(42)

##  3 Unitarity Cuts in bi-adjoint and Z-theory decompositions

###  3.1 Open String

As alluded to above, the packaging of tree amplitudes into a propagator-matrix
decomposition via eqs. 21 and 36 and the further arrangement of the numerator
terms is particularly conducive to calculating one-loop unitarity cuts, which
we now demonstrate. We restrict out attention to four points and only external
gluons in the current work.

Generically, the unitarity cut for a graph G with vertices $V(G)$ and edge
$E(G)$ is given by

$\mathcal{C}(G)=\sum_{\begin{subarray}{c}\text{states}\\\ \text{over
}E(G)\end{subarray}}\prod_{v\in V(G)}A^{\text{tree}}(v)\,.$ (43)

Immediately restricting to one-loop four-point, one of the most useful cuts is
the two-particle (2P) cut, which is given as

$\displaystyle\begin{gathered}\includestandalone[scale=0.7,mode=buildnew]{figures/bub22cut}\end{gathered}$
$\displaystyle\to\begin{gathered}\includestandalone[scale=0.7,mode=buildnew]{figures/bub22cutTrees}\end{gathered}$
(46) $\displaystyle\mathcal{C}(12|34)$
$\displaystyle=\sum_{\mathrm{states}}A^{\text{tree}}(\ell_{1},1,2,-\ell_{2})A^{\text{tree}}(\ell_{2},3,4,-\ell_{1})$
(47)

in the planar color ordering. The gluonic state sums are resolved as
Roehrig:2017gbt ; Kosmopoulos:2020pcd ; Edison:2020uzf

$\displaystyle\sum_{\text{states of
}\ell_{i}}\mathcal{C}_{\mu\nu}\varepsilon_{\ell_{i}}^{\mu}\varepsilon_{-\ell_{i}}^{\nu}+\mathcal{C}\left(\varepsilon_{\ell_{i}}\cdot\varepsilon_{-\ell_{i}}\right)\to\mathcal{C}_{\mu\nu}\eta^{\mu\nu}+(D-2)\mathcal{C}$
(48)

where $\mathcal{C}_{\mu\nu}$ is the component of $\mathcal{C}(12|34)$ in which
$\varepsilon_{\ell_{i}}^{\mu}\varepsilon_{-\ell_{i}}^{\nu}$ are not contracted
with each other, and $\mathcal{C}$ the one in which they are. Note that during
the sewing of the first leg, $\mathcal{C}=0$ since $\varepsilon_{\ell_{i}}$
and $\varepsilon_{-\ell_{i}}$ come from separate tree amplitudes and thus
cannot begin contracted. However, after sewing the first leg subsequent legs
may have been contracted by previous sewings. The fermionic state sums are
similarly Roehrig:2017gbt ; Edison:2020uzf 222This is not strictly true, but
along with the other prescritions below properly captures the parity even part
of the state sewing. See Edison:2022jln  for a more comprehensive treatment.

$\sum_{\text{states of
}\ell_{i}}\chi_{\ell_{i}}^{\alpha}(\xi_{\ell_{i}})_{\beta}=\delta_{\beta}^{\alpha}.$
(49)

Amplitudes expressed via eqs. 24 and 25 are particularly well-suited to
applying state sums on legs $1$ and $n$, with the state sums closing chains of
$W$s into appropriate traces. In the case of 2P cuts, we find

$\sum_{\mathrm{states}}W_{2\text{g}}(\ell_{1},X,-\ell_{2})W_{2\text{g}}(\ell_{2},Y,-\ell_{1})=\begin{cases}D-2&|X|=|Y|=0\\\
\mathrm{tr}_{v}(X,Y)&\text{otherwise}\end{cases}$ (50)

where

$\mathrm{tr}_{v}(X,Y)=f_{x_{1}}^{\mu}{}_{\nu}f_{x_{2}}^{\nu}{}_{\rho}\dots
f_{y_{|Y|}}^{\sigma}{}_{\mu}\,,$ (51)

for sewing gluons, while for fermions

$\sum_{\mathrm{states}}W_{2f}(\ell_{1},X,-\ell_{2})W_{2f}(\ell_{2},Y,-\ell_{1})=\begin{cases}\mathrm{tr}_{s}(\mathbf{I})=2^{\frac{D}{2}-1}&|X|=|Y|=0\\\
\mathrm{tr}_{s}(X,Y)&\text{otherwise}\end{cases}$ (52)

with

$\mathrm{tr}_{s}(X,Y)=\left(\not{f}_{x_{1}}\dots\not{f}_{y_{1}}\dots\right)^{\alpha}_{\beta}\delta^{\beta}_{\alpha}\,,$
(53)

following our Dirac matrix convetions eq. 27. The numerator factors
$\mathfrak{N}$ are unaffected other than renaming the legs.

These trace structures were studied in detail in Ref. Edison:2020uzf , and are
particularly well-behaved in the case of maximal SUSY 333For ease of
discussion we focus on $\mathcal{N}=1$ sYM in $D=10$, but as demonstrated in
Edison:2020uzf  the various dimensional reductions with maximal SUSY behave
identically, with a minor change in the way in which the cancellations occur
for $|X|+|Y|=0$.

$\sum_{\begin{subarray}{c}\text{states}\\\
s\in\\{2\text{g},2f\\}\end{subarray}}W_{s}(\ell_{1},X,-\ell_{2})W_{s}(\ell_{2},Y,-\ell_{1})=\begin{cases}0&|X|+|Y|<4\\\
\mathrm{tr}_{v}(X,Y)-\frac{1}{2}\mathrm{tr}_{s}(X,Y)&\text{otherwise}.\end{cases}$
(54)

The relative factor of $-\frac{1}{2}$ between the two traces is discussed in
more detail in Refs. Edison:2020uzf ; Edison:2022jln . Further, when
$|X|+|Y|=4$ the combination of traces is the permutation invariant contraction
of four field strengths via the $t_{8}$-tensor:

$\mathrm{tr}(1234)-\frac{1}{2}\mathrm{tr}_{s}(1234)=\frac{1}{2}\left(\mathrm{tr}(1234)+\frac{1}{4}\mathrm{tr}(12)\mathrm{tr}(34)+\mathrm{cyc}(234)\right)=\frac{1}{2}t_{8}(1,2,3,4)\,.$
(55)

We will ignore the overall normalization factor of $\frac{1}{2}$ from now on,
as it will appear on all cuts calculated in this manner so can be absorbed
into the definition of the coupling constants.

Thus, we have all of the components needed to evaluate the four-point 2P cut
from eq. 47 in maximal sYM

$\displaystyle\mathcal{C}_{\text{sYM}}(12|34)$
$\displaystyle=\sum_{\mathrm{states}}A(\ell_{1},1,2,-\ell_{2})A(\ell_{2},3,4,-\ell_{1})$
$\displaystyle=\sum_{\begin{subarray}{c}\rho\in S_{\\{1,2\\}}\\\ \sigma\in
S_{\\{3,4\\}}\end{subarray}}m(\ell_{1},1,2,-\ell_{2}|\ell_{1},\rho,-\ell_{2})m(\ell_{2},3,4,-\ell_{1}|\ell_{2},\sigma,-\ell_{1})$
$\displaystyle\qquad\times\mathfrak{N}_{\rho}(\ell_{1};\emptyset;-\ell_{2})\mathfrak{N}_{\sigma}(\ell_{2};\emptyset;-\ell_{1})\sum_{\begin{subarray}{c}\text{states}\\\
s\in\\{2\text{g},2f\\}\end{subarray}}W_{s}(\ell_{1},\rho,-\ell_{2})W_{s}(\ell_{2},\sigma,-\ell_{1})$
(56)

From Ref. Edison:2020ehu  we identify $\mathfrak{N}(a;\emptyset;b)=1$.
Additionally, the state sum is resolved to $t_{8}(1,2,3,4)$ as discussed
above, giving the well-known result

$\displaystyle\mathcal{C}_{\text{sYM}}(12|34)$
$\displaystyle=t_{8}(1,2,3,4)\sum_{\begin{subarray}{c}\rho\in S_{\\{1,2\\}}\\\
\sigma\in
S_{\\{3,4\\}}\end{subarray}}m(\ell_{1},1,2,-\ell_{2}|\ell_{1},\rho,-\ell_{2})m(\ell_{2},3,4,-\ell_{1}|\ell_{2},\sigma,-\ell_{1})$
$\displaystyle=t_{8}(1,2,3,4)\left[\frac{1}{s_{\ell_{1}1}}+\frac{1}{s_{12}}-\frac{1}{s_{12}}\right]\times\left[\frac{1}{s_{\ell_{2}3}}+\frac{1}{s_{34}}-\frac{1}{s_{34}}\right]=\frac{t_{8}(1,2,3,4)}{s_{\ell_{1}1}s_{\ell_{2}3}}.$
(57)

This construction is straightforward to extend from sYM trees to superstring
higher-dimension-operator insertions by noting that the doubly-color-ordered
amplitudes $m(\cdot|\cdot)$ are given by the $\alpha^{\prime}\to 0$ limit of
$Z(\cdot|\cdot)$. Thus, to calculate a cut for an open-superstring matrix
element, all the steps in eq. 57 remain the same save the evaluation of the
$m(\cdot|\cdot)$ which is replaced by the relevant $Z(\cdot|\cdot)$
expansions. Importantly, the operator identification via series expansion and
$\zeta$ value coefficient matching must be taken at the level of the entire
cut, e.g. for $\zeta_{4}\propto\zeta_{2}^{2}\leftrightarrow D^{4}F^{4}+\dots$
we must sum over $Z_{1}(L)Z_{\zeta_{4}}(R)$,
$Z_{\zeta_{2}}(L)Z_{\zeta_{2}}(R)$ and $Z_{\zeta_{4}}(L)Z_{1}(R)$ to get the
correct cut.

For a simple example, we compute the $\zeta_{2}\leftrightarrow t_{8}F^{4}$ 2P
cut via the above method

$\displaystyle\mathcal{C}^{\text{open}}_{\zeta_{2}}(12|34)$
$\displaystyle=t_{8}(1,2,3,4)\sum_{\rho\in S_{\\{1,2\\}}\atop\sigma\in
S_{\\{3,4\\}}}Z(\ell_{1},1,2,-\ell_{2}|\ell_{1},\rho,-\ell_{2})Z(\ell_{2},3,4,-\ell_{1}|\ell_{2},\sigma,-\ell_{1})\Big{|}_{\zeta_{2}}$
$\displaystyle=t_{8}(1,2,3,4)\sum_{\rho\in S_{\\{1,2\\}}\atop\sigma\in
S_{\\{3,4\\}}}\Big{(}m(\ell_{1},1,2,-\ell_{2}|\ell_{1},\rho,-\ell_{2})Z_{\zeta_{2}}(\ell_{2},3,4,-\ell_{1}|\ell_{2},\sigma,-\ell_{1})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+Z_{\zeta_{2}}(\ell_{1},1,2,-\ell_{2}|\ell_{1},\rho,-\ell_{2})m(\ell_{2},3,4,-\ell_{1}|\ell_{2},\sigma,-\ell_{1})\Big{)}$
$\displaystyle=-t_{8}(1,2,3,4)s_{12}\left(\frac{1}{s_{\ell_{1}1}}+\frac{1}{s_{\ell_{2}3}}\right)$
(58)

As expected, we find that this two-particle cut supports additional residues
corresponding to triangle topology cuts, but not the two simultaneous residues
to have non-zero contribution to the box topology.

The triangle and box cuts can be extracted by taking additional residues on
the 2P cut. The 1-3 bubble cut and one-particle cut can be calculated in a
similar manner via sewing together chains of $W_{s}$. However, special care
needs to be given to correct application of momentum conservation for these
cuts, first using momentum conservation on each vertex separately to cancel
potentially-singular terms before applying the overall four-point momentum
conservation Edison:2020uzf ; Edison:2021ebi ; Edison:2022jln .

###  3.2 Closed string

Gravitational cuts at one loop are nearly as simple as gauge theory cuts. The
abstract definition of unitarity cuts via products of trees is the same,
except with gravity tree amplitudes as the constituent objects, e.g. in the 2P
case

$\mathfrak{C}_{\text{grav}}(12|34)=\sum_{\text{states}}\mathcal{M}^{\text{tree}}(\ell_{1},1,2,-\ell_{2})\mathcal{M}^{\text{tree}}(\ell_{2},3,4,-\ell_{1})\,.$
(59)

Since gravitons are not colored, the tree amplitudes are fully permutation
invariant, and the cut is invariant under $1\leftrightarrow 2$ and
$3\leftrightarrow 4$. As spin-two particles, the state sum for gravitons is
more complicated than the one for gauge bosons Kosmopoulos:2020pcd

$\displaystyle\sum_{\text{states of
}\ell_{i}}\mathfrak{C}_{\mu\nu\rho\sigma}\varepsilon_{\ell_{i}}^{\mu\nu}\varepsilon_{-\ell_{i}}^{\rho\sigma}+\mathfrak{C}_{\mu\rho}\varepsilon_{\ell_{i}}^{\mu\nu}\varepsilon_{-\ell_{i},\nu}^{\rho}+\mathfrak{C}\varepsilon_{\ell_{i}}^{\mu\nu}\varepsilon_{-\ell_{i},\mu\nu}$
$\displaystyle\rightarrow\frac{1}{2}\left(\mathfrak{C}^{\mu\nu}{}_{\mu\nu}+\mathfrak{C}^{\mu\nu}{}_{\nu\mu}-\frac{2}{D-2}\mathfrak{C}^{\mu}{}_{\mu}{}^{\nu}{}_{\nu}\right)+\frac{D(D-3)}{2(D-2)}\mathfrak{C}^{\mu}{}_{\mu}+\frac{D(D-3)}{2}\mathfrak{C}\,,$
(60)

with $\varepsilon^{\mu\nu}$ the graviton polarizations and the various
$\mathfrak{C}$ are pieces of the cut with the specified polarization
contractions. Using eq. 20 for the tree amplitudes and evaluating the state
sums in eq. 59 for maximal supergravity yields444Showing this result via
direct state sewing in maximal supergravity is somewhat tedious. However, for
maximal supersymmetry no information is lost or corrupted by performing the
state sums separately on each of the two sYM numerators $\tilde{N}$ and $N$
and then identifying
$\tilde{\varepsilon}_{i}^{\mu}\varepsilon_{i}^{\nu}=\varepsilon_{i}^{\mu\nu}$
as the external graviton polarizations Elvang:2015rqa .

$\displaystyle\mathfrak{C}_{\text{max SG}}(12|34)$
$\displaystyle=t_{8}(1,2,3,4)^{2}\\!\\!\\!\sum_{\alpha\in
S_{\\{1,2\\}}\atop\beta\in S_{\\{3,4\\}}}\\!\\!\sum_{\rho\in
S_{\\{1,2\\}}\atop\sigma\in
S_{\\{3,4\\}}}\\!\\!m(\ell_{1},\alpha,-\ell_{2}|\ell_{0},\rho,-\ell_{2})m(\ell_{2},\beta,-\ell_{0}|\ell_{2},\sigma,-\ell_{0})$
$\displaystyle=t_{8}(1,2,3,4)^{2}\left(\frac{1}{s_{\ell_{1},1}s_{\ell_{2}3}}+\frac{1}{s_{\ell_{1},2}s_{\ell_{2}3}}+\frac{1}{s_{\ell_{1},1}s_{\ell_{2}4}}+\frac{1}{s_{\ell_{1},2}s_{\ell_{2}4}}\right)$
(61)

where we identify
$t_{8}(1,2,3,4)^{2}=(stA_{\text{sYM}}^{\text{tree}}(1,2,3,4))^{2}=stuM_{\text{GR}}^{\text{tree}}(1,2,3,4)$.
Similar to lifting sYM$\to$open superstring, gravity cuts also have a natural
uplift to string-insertion cuts via replacing the gravity tree amplitudes with
closed-string ones, eq. 42,

$\mathfrak{C}^{\text{closed}}(12|34)=t_{8}(1,2,3,4)^{2}\\!\\!\\!\sum_{\alpha\in
S_{\\{1,2\\}}\atop\beta\in S_{\\{3,4\\}}}\\!\\!\sum_{\rho\in
S_{\\{1,2\\}}\atop\sigma\in
S_{\\{3,4\\}}}\\!\\!\text{sv}\Big{[}Z(\ell_{1},\alpha,-\ell_{2}|\ell_{1},\rho,-\ell_{2})Z(\ell_{2},\beta,-\ell_{1}|\ell_{2},\sigma,-\ell_{1})\Big{]}\,.$
(62)

A direct calculation of closed-string-insertion cuts is invaluable for
verifying the double-copy behavior of our CK-dual numerators. In particular,
it is a priori unclear exactly how to apply the BCJ double copy to get the
desired closed-string insertion. Some natural choices are

$\displaystyle N^{\text{closed}}_{\zeta_{i}\zeta_{j}}(g)$
$\displaystyle\overset{?}{\propto}N^{\text{sYM}}(g)N^{\text{open}}_{\zeta_{i}\zeta_{j}}(g)$
(63)
$\displaystyle\overset{?}{\propto}N^{\text{open}}_{\zeta_{i}}(g)N^{\text{open}}_{\zeta_{j}}(g)$
(64)

or a linear combination of the two. The first is calculationally desirable, as
at four points we have

$N^{\text{sYM}}(g)=\begin{cases}1&g=\text{perms of }1|2|3|4\\\
0&\text{otherwise}\end{cases}\Rightarrow
N^{\text{closed}}_{\zeta_{i}\zeta_{j}}(g)\overset{?}{\propto}\begin{cases}N^{\text{open}}_{\zeta_{i}\zeta_{j}}(g)&g=\text{perms
of }1|2|3|4\\\ 0&\text{otherwise,}\end{cases}$ (65)

i.e. the closed-string insertion numerator would only be written in terms of
box diagrams. An important thing to note is that both eq. 63 and eq. 64 can be
valid double copies, while eq. 65 implies that

$N^{\text{sYM}}(g)N^{\text{open}}_{\zeta_{i}\zeta_{j}}(g)\not\propto
N^{\text{open}}_{\zeta_{i}}(g)N^{\text{open}}_{\zeta_{j}}(g)\,;$ (66)

they would only need to agree on the cuts such that the actual amplitudes
would only differ by a normalization.

We use “$\propto$” in all of these relations because the CK construction does
not care about the overall normalization of the numerators. A more-complete
treatment may include at least a prefactor of
$\operatorname{sv}(\zeta_{i}\zeta_{j})$ as part of the definition of
$N^{\text{closed}}_{\zeta_{i}\zeta_{j}}$, but then we would still be
interested if the double copy choices produce different normalizations.
Comparing cuts constructed from eqs. 63 and 64 against those constructed
directly via eq. 62 can identify which combinations actually produce the
desired physical observables. We will explore this in more detail in section
4.3, with explicit CK numerators in hand.

##  4 Construction of Color-Dual Representations

With the foundations of the theory and cut construction established, we turn
our attention to the assembly of color-kinematics-dual representations for the
odd-$\zeta$-indexed matrix elements. From eqs. 14 and 16 we know that four-
point one-loop color-kinematics-dual numerators can be written in terms of a
single basis numerator: that of the cubic box.

We will construct our color-dual representations following the method of
maximal cuts Bern:2007ct ; Bern:2008pv ; Bern:2010tq ; Bern:2012uf ;
Bern:2018jmv , which can be briefly summarized in the current context as:

#### Method of maximal cuts:

  1. 1.

Assemble an ansatz for the box diagram as a homogeneous polynomial of momentum
products. A quick analysis of how the $Z$-functions enter the cuts (or the
contact-diagram representation of Edison:2021ebi ) shows that the degree of
the box numerator must be equal to the desired $\alpha^{\prime}$ order.

  2. 2.

Impose power-counting constraints on the triangle and bubble numerators (as
constructed from the box numerator via kinematic Jacobi relations).

  3. 3.

Impose _boundary Jacobi relations_ on the ansatz. These are the kinematic
Jacobi relations which would relate triangles and bubbles to bubble-on-
external-leg and tadpole diagrams. They generally impose that an
antisymmetrization around an internal edge is zero.

  4. 4.

Impose diagram symmetries on the ansatz. For instance, the triangle diagram
has a $\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$ symmetry corresponding to

$N_{\triangle}(1,2|3|4)=-N_{\triangle}(2,1|3|4)\qquad\text{and}\qquad
N_{\triangle}(1,2|3|4)=-N_{\triangle}(1,2|4|3)\big{|}_{\ell\to\ell+k_{1}+k_{2}}$

In the current construction, the symmetries are always a subset of the
kinematic Jacobi relations.

  5. 5.

Match the cuts, as calculated via section 3. We impose the matching in order
of “distance from the box”: first matching the box itself, then the triangle
cut, followed by the 2-2 (product of two four-point amplitudes) and 1-3
(product of three-point and five-point amplitude) bubbles, and finally the
one-particle cut.

Let us now turn our attention to the details of ansatz construction. As we saw
in section 3, the external state data factors out of all cuts as an overall
$t_{8}(1,2,3,4)$. We require that this factorization holds for the full
integrand as well, so will leave the $t_{8}$ factor implicit from now on. We
will assume that the numerators are not secretly rational functions
themselves, and thus will be polynomials of the external Mandelstams and
$k\cdot\ell$. We use

$\mathcal{P}=a_{1}s_{12}+a_{2}s_{23}+a_{3}s_{1\ell}+a_{4}s_{2\ell}+a_{5}s_{3\ell}+a_{6}\ell^{2}$
(67)

to denote a degree one arbitrary polynomial in the variables, and
$\mathcal{P}_{n}$ to refer to a degree-$n$ arbitrary polynomial. For example
the degree-two ansatz $\mathcal{P}_{2}$ is given by

$\begin{split}\mathcal{P}_{2}=&a_{1}s_{12}^{2}+a_{2}s_{12}s_{\text{1$\ell$}}+a_{3}s_{12}s_{23}+a_{4}s_{12}s_{\text{2$\ell$}}+a_{5}s_{12}s_{\text{3$\ell$}}+a_{6}s_{12}\ell^{2}+a_{7}s_{\text{1$\ell$}}^{2}\\\
&+a_{8}s_{\text{1$\ell$}}s_{23}+a_{9}s_{\text{1$\ell$}}s_{\text{2$\ell$}}+a_{10}s_{\text{1$\ell$}}s_{\text{3$\ell$}}+a_{11}s_{\text{1$\ell$}}\ell^{2}+a_{12}s_{23}^{2}+a_{13}s_{23}s_{\text{2$\ell$}}\\\
&+a_{14}s_{23}s_{\text{3$\ell$}}+a_{15}s_{23}\ell^{2}+a_{16}s_{\text{2$\ell$}}^{2}+a_{17}s_{\text{2$\ell$}}s_{\text{3$\ell$}}+a_{18}s_{\text{2$\ell$}}\ell^{2}+a_{19}s_{\text{3$\ell$}}^{2}\\\
&+a_{20}s_{\text{3$\ell$}}\ell^{2}+a_{21}\ell^{4}.\end{split}$ (68)

From the explicit representations of the $Z$-functions BGap ;
Schlotterer:2012ny ; MZVWebsite , it is clear that only sYM supports a maximal
(box) cut. Using this observation, we can refine the initial ansatz for the
box with degree $w$ as

$N_{\square}^{w}=\ell_{1}^{2}\,\mathcal{P}_{w-1}+\text{cyc}(1,2,3,4)$ (69)

where $+\text{cyc}(1,2,3,4)$ sums the various relabelings of
$\mathcal{P}_{w-1}$, as well as the four possible inverse propagators

$\ell_{1}^{2}=(\ell+k_{1})^{2}\quad\ell_{2}^{2}=(\ell+k_{12})^{2}\quad\ell_{3}^{2}=(\ell+k_{123})^{2}\quad\ell_{4}^{2}=\ell^{2}\,.$
(70)

Note that this type of ansatz is in general overcomplete, as terms like
$\ell_{1}^{2}\ell_{2}^{4}$ can be sourced from the explicit $\ell_{1}^{2}$
prefactor multiplying against the linear combination of terms from
$\mathcal{P}_{2}$ that build $\ell_{2}^{4}$, or from the explicit
$\ell_{2}^{2}$ multiplying a different linear combination from
$\mathcal{P}_{2}$ generating $\ell_{1}^{2}\ell_{2}^{2}$. However, this ansatz
trivializes the vanishing of the box cut as well as the cyclic symmetry
properties of the box diagram. We will additionally impose that the triangle
and bubble ansatz have one and two powers of $s_{12}$ respectively

$N_{\bigtriangleup}(1|2|3,4)\propto s_{12}\qquad N_{\bigcirc}(1,2|3,4)\propto
s_{12}^{2}$ (71)

so that the loop amplitude has no explicit poles in $s_{ij}$. These conditions
are nontrivial to impose _a priori_ on $N_{\square}$, and enter as constraints
on the $a_{i}$.

We proceed to construct color-dual representations for the four lowest odd-
only $\zeta$ values: $\zeta_{3},\zeta_{5},\zeta_{3}^{2},$ and $\zeta_{7}$. For
all insertions with a factor of $\zeta_{2}$ it should be impossible to
construct a color-dual representation due to non-trivial monodromy relations
in the open superstring Bjerrum-Bohr:2009ulz ; Stieberger:2009hq ;
Broedel:2012rc . We have explicitly verified the construction failure for
$\zeta_{2}^{2}\sim\zeta_{4}$, $\zeta_{2}\zeta_{3}$, $\zeta_{2}^{3}$,
$\zeta_{2}\zeta_{5}$, and $\zeta_{2}^{2}\zeta_{3}$. These constructions fail
by being unable to match $\mathcal{C}_{w}(1|2|34)$ while also satisfying the
color-kinematics relations.

###  4.1 Detailed construction of $\boldsymbol{\zeta_{3}\leftrightarrow
D^{2}F^{4}+F^{5}}$

We begin with a detailed account of the construction of the $\zeta_{3}$
representation. Examining the color-ordered cuts we find

$\begin{split}\mathcal{C}_{\zeta_{3}}(1|2|34)&=s_{12}\left(s_{12}+s_{\text{1$\ell$}}+s_{\text{2$\ell$}}+s_{\text{3$\ell$}}\right),\\\
\mathcal{C}_{\zeta_{3}}(12|34)&=\frac{s_{12}(s_{12}+s_{123\ell})}{s_{1\ell}}+\frac{s_{12}(s_{12}+s_{1\ell})}{s_{123\ell}},\\\
\mathcal{C}_{\zeta_{3}}(1|234)&=s_{12}+s_{23}+\frac{s_{12}(s_{12}+s_{123\ell})}{s_{12\ell}}+\frac{s_{23}(s_{23}+s_{12\ell})}{s_{123\ell}}.\end{split}$
(72)

and that the one-particle cut is purely rational, thus containing no
additional information. We can diagrammatically expand these cuts onto cubic
diagrams as

$\displaystyle\mathcal{C}(1|2|34)$
$\displaystyle=\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/contactCuts}\end{gathered}$
$\displaystyle=\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/boxCut}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/tri34Cuts}\end{gathered},$
(76) $\displaystyle\mathcal{C}(12|34)$
$\displaystyle=\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/bub22}\end{gathered}$
$\displaystyle=\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triAltBubCuts}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triBubCuts}\end{gathered}$
(80)
$\displaystyle\qquad+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/boxBubCuts}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/bubBubCuts}\end{gathered},$
(83) $\displaystyle\mathcal{C}(1|234)$
$\displaystyle=\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/contactBebCuts}\end{gathered}$
$\displaystyle=\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triBelCuts}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/tri4123BelCuts}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/boxBelCuts}\end{gathered}.$
(88)

Here the dotted lines indicate cut propagators.

Noting that the explicit expression for the triangle cut in eq. 72 is a
degree-2 polynomial, we see that the ansatz for $N^{\zeta_{3}}_{\square}$ is
covered by

$N_{\square}^{\zeta_{3}}(1|2|3|4)=\ell_{1}^{2}\mathcal{P}_{2}+\text{cyc}(1,2,3,4)$
(89)

exactly as in eq. 68 with 21 free parameters, $a_{i}$. Through the Jacobi
relations, eqs. 13 and 15, we get the numerators for the cubic triangle and
2-2 bubble, $N_{\bigtriangleup}^{\zeta_{3}}$ and $N_{\bigcirc}^{\zeta_{3}}$.
Imposing the power counting of eq. 71 fixes 10 of the free parameters, with
the remaining 11 parameters only appearing in three distinct combinations. For
this case the overlap of the defining Jacobi relations, eqs. 14 and 16, and
power counting, eq. 71, turn out to be a superset of the symmetry conditions
and boundary Jacobi relations; both are trivially satisfied by the resulting
numerators. With the numerators defined we can start imposing the cuts.

Inserting the numerator definitions into the triangle cut, eq. 76, we get

$\displaystyle\mathcal{C}_{\zeta_{3}}(1|2|34)$
$\displaystyle=\operatorname*{Cut}_{1;1;2}\left[\frac{N_{\square}^{\zeta_{3}}(1|2|3|4)}{\ell_{3}^{2}}+\frac{N_{\bigtriangleup}^{\zeta_{3}}(1|2|3,4)}{s_{34}}\right]$
$\displaystyle=\operatorname*{Cut}_{1;1;2}\left[\frac{N_{\square}^{\zeta_{3}}(1|2|3|4)}{\ell_{3}^{2}}+\frac{N_{\square}^{\zeta_{3}}(1|2|3|4){-}N_{\square}^{\zeta_{3}}(1|2|4|3)}{s_{34}}\right]\,,$
(90)

where $\operatorname*{Cut}_{1;1;2}$ imposes
$\ell_{1}^{2}=\ell_{2}^{2}=\ell_{4}^{2}=0$. Performing the explicit matching
between the left and right hand sides of eq. 90 fixes one of the remaining
$a_{i}$. Continuing to the bubbles, we have

$\mathcal{C}_{\zeta_{3}}(12|34)=\operatorname*{Cut}_{2;2}\left[\frac{N_{\bigtriangleup}^{\zeta_{3}}(1,2|3|4)}{s_{12}\ell_{1}^{2}}+\frac{N_{\bigtriangleup}^{\zeta_{3}}(1|2|3,4)}{s_{12}\ell_{3}^{2}}+\frac{N_{\square}^{\zeta_{3}}(1|2|3|4)}{\ell_{1}^{2}\ell_{3}^{2}}+\frac{N_{\bigcirc}^{\zeta_{3}}(1,2|3,4)}{s_{12}^{2}}\right]$
(91)

and

$\mathcal{C}_{\zeta_{3}}(1|234)=\operatorname*{Cut}_{1;3}\left[\frac{N_{\bigtriangleup}^{\zeta_{3}}(1|2|3,4)}{s_{12}\ell_{2}^{2}}+\frac{N_{\bigtriangleup}^{\zeta_{3}}(1|2,3|4)}{s_{23}\ell_{3}^{2}}+\frac{N_{\square}^{\zeta_{3}}(1|2|3|4)}{\ell_{2}^{2}\ell_{3}^{2}}\right]$
(92)

which can of course be written exclusively in terms of labelings of the box
numerator via eqs. 14 and 16. The matching of the numerators against the cuts
from eq. 72 fixes the last two $a_{i}$ and we have unique color-kinematics-
dual numerators for the $\zeta_{3}$ insertion

$\displaystyle N_{\square}^{\zeta_{3}}(1|2|3|4)=$
$\displaystyle-4s_{12}^{2}s_{\text{1$\ell$}}+3s_{12}^{2}s_{23}-s_{12}s_{\text{1$\ell$}}s_{23}+2s_{\text{1$\ell$}}^{2}s_{23}+2s_{12}s_{23}^{2}+2s_{\text{1$\ell$}}s_{23}^{2}$
$\displaystyle-2s_{12}^{2}s_{\text{2$\ell$}}-4s_{12}s_{\text{1$\ell$}}s_{\text{2$\ell$}}+2s_{12}s_{23}s_{\text{2$\ell$}}+2s_{23}^{2}s_{\text{2$\ell$}}-2s_{12}s_{\text{2$\ell$}}^{2}-2s_{23}s_{\text{2$\ell$}}^{2}$
$\displaystyle-2s_{12}^{2}s_{\text{3$\ell$}}+4s_{12}s_{\text{1$\ell$}}s_{\text{3$\ell$}}-3s_{12}s_{23}s_{\text{3$\ell$}}+4s_{\text{1$\ell$}}s_{23}s_{\text{3$\ell$}}-4s_{23}s_{\text{2$\ell$}}s_{\text{3$\ell$}}$
(93)
$\displaystyle+2s_{12}s_{\text{3$\ell$}}^{2}+4s_{12}^{2}\ell^{2}+4s_{12}s_{23}\ell^{2}+4s_{23}^{2}\ell^{2}\,,$
$\displaystyle N_{\bigtriangleup}^{\zeta_{3}}(1|2|3,4)=$ $\displaystyle
s_{12}^{3}-2s_{12}^{2}s_{\text{1$\ell$}}+2s_{12}^{2}s_{23}-3s_{12}s_{\text{1$\ell$}}s_{23}+s_{12}^{2}s_{\text{2$\ell$}}+3s_{12}s_{23}s_{\text{2$\ell$}}$
$\displaystyle-s_{12}^{2}s_{\text{3$\ell$}}\,,$ (94) $\displaystyle
N_{\bigcirc}^{\zeta_{3}}(1,2|3,4)=$ $\displaystyle
2s_{12}^{2}(s_{23}-s_{13}).$ (95)

These numerators are also provided in the attached auxiliary file.

###  4.2 Higher orders

The construction procedure for the higher $\zeta$ values remains nearly
identical to the $\zeta_{3}$ case, just applied to larger initial ansatze.
However, there is one major difference. For $\zeta_{3}$, we only required
bubble cuts. With the higher $\zeta$s, the $Z$-theory power counting is such
that one-particle cuts contain polynomials of $s_{i\dots}$ in addition to
rational functions, which signals a potential contribution that is
undetectable on other cuts. Thus, we must also match the cubic numerators
against this class of cuts. The color structure of the contact tadpole is such
that the two external orderings, (1,2,3,4) and (4,3,2,1) contribute to the
same color ordering555This can be seen by considering the six-point half-
ladder color structures DelDuca:1999rs  $c(a,1,2,3,4,b)$ and $c(a,4,3,2,1,b)$.
Under normal circumstances these two color structures are distinct; attempting
to convert one into the other eventually results in a color Jacobi identity
involving a commutator of $a$ and $b$. For the tadpole color structure, $a$
and $b$ are identified and thus commute, removing the obstruction to mapping
them into each other. so to get the complete color-ordered cut we need the
expansion:

$\begin{split}\mathcal{C}(1234)=&\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/tad1234}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/tad4321}\end{gathered}=\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triTadCut}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/tri2TadCut}\end{gathered}\\\
&+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/triAltTadCut}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6]{figures/bubTadCut}\end{gathered}+\begin{gathered}\includestandalone[scale=0.6,mode=buildnew]{figures/boxTadCut}\end{gathered}+(\ell\leftrightarrow-\ell)\,.\end{split}$
(96)

At $(\alpha^{\prime})^{\geq 5}$ the boundary Jacobi relations are no longer a
subset of the defining Jacobi relations and thus we need to impose their
constraints on the numerator. We find that the symmetries are automatically
satisfied after application of all kinematic Jacobi relations including the
boundary relations.

Beyond the need to actually impose boundary Jacobi relations, the $\zeta_{5}$
and $\zeta_{3}^{2}$ constructions follow exactly the same process as the
$\zeta_{3}$ construction. The triangle and bubble cuts impose a small number
of conditions on the ansatz, after which the one-particle cuts are satisfied
without imposing additional constraints. Unlike the $\zeta_{3}$ numerator, the
$\zeta_{5}$ and $\zeta_{3}^{2}$ numerators are not unique. After all
conditions are imposed, the $\zeta_{5}$ numerator has two free parameters
remaining, while the $\zeta_{3}^{2}$ numerator has only one.

At $\zeta_{7}$ our construction fails. The first point of trouble occurs in
the matching of $\mathcal{C}_{\zeta_{7}}(12|34)$ and
$\mathcal{C}_{\zeta_{7}}(1|234)$. Each cut can be matched separately, but the
two cannot be matched simultaneously. This difficulty can be surmounted by
introducing a non-trivial numerator for the bubble-on-external-leg (BEL) cubic
diagram (the third diagram in eq. 17). In the “snail” regularization
Bern:2012uf  the singular $\frac{1}{k_{1}^{2}}$ propagator that would be
expected in such a diagram is allowed to be formally non-zero at the level of
the integrand, and is then cancelled by introducing an overall factor of
$k_{1}^{2}$ in the diagram’s numerator. This procedure effectively introduces
a contact correction with the problematic propagator collapsed, carrying only
the color structure of the cubic diagram. After double copying to
gravitational theories, these new diagrams will drop out from the
representation as the double-copied numerator will contain a $k_{1}^{4}$ while
there will only be one factor of $\frac{1}{k_{1}^{2}}$ as a propagator,
leaving a $k_{1}^{2}$ which can be safely set to zero using on-shell
conditions.

While this process solves the BEL obstruction, problems continue to arise when
attempting to match the one-particle cut. Unlike in the three previous cases,
the local terms of the one-particle cut are not automatically matched by the
constrained numerators. By itself this is not a problem. Unfortunately, we
further find that the matching cannot be completed with the remaining freedom
in the ansatz. As with the BEL numerators, one might hope to introduce only
appropriately-regularized cubic tadpole contributions that will drop out of
the resulting double-copied theory. Our attempts to absorb the necessary local
contributions into symmetry-obeying cubic tadpoles have failed. However, there
are significantly more subtleties with introducing cubic tadpole numerators to
absorb this mismatch, so we cannot claim that finding appropriate cubic
tadpoles is impossible. Even though the single-copy numerators do not match
all cuts, since they match the triangle and bubble cuts and obey color-
kinematics duality they will compute the triangle and bubble cuts correctly in
the double-copy theory.

$\boldsymbol{\zeta}$ weight |  Initial ansatz |  After powercounting |  After boundary Jacobies |  Free params after cuts
---|---|---|---|---
$\zeta_{5}$ |  126 |  43 |  25 |  2
$\zeta_{3}^{2}$ |  252 |  135 |  45 |  1
$\zeta_{7}$ |  462 (+30) |  223 (+30) |  118 (+30) |  -
462 |  352 $\left(\begin{subarray}{c}\text{relaxed}\\\ \text{bubble}\end{subarray}\right)$ |  280 |  4
Table 1: Number of free parameters for $\zeta_{5}$, $\zeta_{3}^{2}$ and
$\zeta_{7}$ at each step in the construction. The exact number of parameters
at each step depends on the representation of the initial ansatz. The final
number of free parameters is independent on such choices. For $\zeta_{7}$, we
report on two methods. First we use an additional 30 parameter ansatz for the
bubble-on-external-leg numerator which is unaffected by powercounting and
Jacobi relations. Since this construction of $\zeta_{7}$ is incomplete, we do
not present the final free parameter count. Second, we relax the bubble
powercounting constraint which allows a complete solution.

Note that the above problems only arise when trying to force the
representation to contain strictly cubic diagrams. The $\zeta_{7}$
representation from Ref. Edison:2021ebi  does not suffer from any of these
problems, but is a purely contact representation with no cubic diagrams. It is
possible that a full “Minahaning” Minahan:1987ha ; Edison:2022jln  treatment
of the BEL numerator would alleviate the problems, but the required
computational overhead for that approach is prohibitive for the current work.

On the other hand, rather than modifying or lifting the boundary Jacobi
conditions, we could interrogate the necessity of the power-counting
constraints, eq. 71. These constraints are imposed to manifest the property
that the amplitude has no poles in external Mandelstams, but strictly speaking
this only needs to hold after integration. Since the construction obstacle
occurs at the level of the bubble cuts, we loosen the powercounting
restrictions on the bubble to

$N_{\bigcirc}(1,2|3,4)\propto s_{12}\,.$ (97)

Repeating the construction with the weaker powercounting succeeds, with four
free parameters left over after all conditions are imposed.

The number of free parameters at each step in the construction for each of
$\zeta_{5},\ \zeta_{3}^{2},\ \zeta_{7}$ are summarized in table 1. Since the
degree of the box numerator is the same as the $\alpha^{\prime}$ order, the
resulting numerators are sufficiently large that they are not worth presenting
in text. They may be found in the auxiliary file for the arXiv posting, with
the remaining free parameters left unfixed.

Color-kinematics representations of numerators are often also manifest loop
power-counting representations, isolating the leading ultraviolet divergences
to particular diagram classes. For instance, there are two different
representations for the four-loop four-point intergrand in maximal super-
Yang–Mills: one without CK Bern:2010tq , and one with manifest CK Bern:2012uf
. The one with manifest CK numerators also manifests the UV structure of the
theory by pushing the bad power-counting terms from the box-like topolgies
into the bubble-like ones. The UV critical dimension and leading divergence
can be extracted from only these bubble-like numerators.

We can ask the same questions of our new representations (only using the
$\zeta_{7}$ with the relaxed bubble power counting). Namely, how does the CK
representation assign loop momentum power counting, and does the CK dual
representation generate the leading UV divergence from only the bubble? In
table 2 we present the relevant data. We find that the $\zeta_{3}$,
$\zeta_{5}$, and $\zeta_{7}$ representations _do_ have manifest UV properties.
The scalar bubble integral first develops a UV divergence in $D_{C}=4$, and
the bubble numerator for $\zeta_{5}$ ($\zeta_{7}$) carries two (four)
additional powers of $\ell$ to produce its critical dimension $D_{C}=2\,(0)$.
Interestingly, while the $\zeta_{3}^{2}$ representation does not have manifest
UV behavior, the two powers of $\ell$ enter the bubble numerator in such a way
that the leading UV divergence cancels at the level of the individual diagram.
Note that while this is not the best imaginable power counting, it is
significantly closer than the contact representation from Ref. Edison:2021ebi
. That representation also carries $\ell^{2}$ on bubble diagrams, but in
addition it contains explicit tadpole numerators, thus requiring a careful
interplay between multiple diagrams to see the UV cancellations.

$\boldsymbol{\zeta}$ weight |  Box $\ell$ |  Triangle $\ell$ |  2-2 Bubble $\ell$ |  UV Crit. Dim.
---|---|---|---|---
$\zeta_{3}$ |  $\ell^{2}$ |  $\ell$ |  1 |  4
$\zeta_{5}$ |  $\ell^{4}$ |  $\ell^{3}$ |  $\ell^{2}$ |  2
$\zeta_{3}^{2}$ |  $\ell^{4}$ |  $\ell^{3}$ |  $\ell^{2}$ |  4
$\zeta_{7}\,\left(\begin{subarray}{c}\text{relaxed}\\\ \text{bubble}\end{subarray}\right)$ |  $\ell^{6}$ |  $\ell^{5}$ |  $\ell^{4}$ |  0
Table 2: The loop momentum power counting for each of the topologies, and the
lowest dimension in which the representation develops a UV divergence. The
critical dimensions are in agreement with those reported in Ref.
Edison:2021ebi .

###  4.3 Double-copy cut comparison

With color-dual numerators in hand, we can easily construct a large number of
closed-string-insertion matrix elements. In particular, the double copy gives
easy access to all closed-string matrix elements in
$\\{\text{sYM},\zeta_{3},\zeta_{5},\zeta_{3}^{2},\zeta_{7}\\}\otimes\\{\text{sYM},\zeta_{3},\zeta_{5},\zeta_{3}^{2},\zeta_{7}\\}$.
Using the direct cut construction method from section 3.2, we can verify the
double-copy integrands and explore the different methods of arriving at
representations for products of $\zeta$ values. In table 3, we present the
various ratios between cuts constructed by taking double copies of our
representations and cuts calculated directly from eq. 62, without adopting any
shifts in normalization.

Double-copy form | $\frac{\text{double-copy cuts}}{\text{direct cuts}}$ | Analogous tree-level structure
---|---|---
$\text{sYM}\otimes\zeta_{3}$ | $\frac{1}{2}$ |  $\mathbf{M}_{3,L}+\mathbf{M}_{3,R}$
$\text{sYM}\otimes\zeta_{5}$ | $\frac{1}{2}$ |  $\mathbf{M}_{5,L}+\mathbf{M}_{5,R}$
$\text{sYM}\otimes\zeta_{7}$ | $\frac{1}{2}$ |  $\mathbf{M}_{7,L}+\mathbf{M}_{7,R}$
$\zeta_{3}\otimes\zeta_{3}$ | $\frac{1}{2}$ |  $\frac{1}{2}(2\mathbf{M}_{3,L}\mathbf{M}_{3,R})$
$\text{sYM}\otimes\zeta_{3}^{2}$ | $\frac{1}{4}$ |  $\frac{1}{2}(\mathbf{M}_{3,L}^{2}+\mathbf{M}_{3,R}^{2})$
$\zeta_{3}\otimes\zeta_{5}$ | $\frac{1}{4}$ |  $\mathbf{M}_{3,L}\mathbf{M}_{5,R}+\mathbf{M}_{5,L}\mathbf{M}_{3,R}$
$\text{sYM}\otimes\zeta_{3}\zeta_{5}$ | conjecture: $\frac{1}{4}$ |  $\mathbf{M}_{5,L}\mathbf{M}_{3,L}+\mathbf{M}_{3,R}\mathbf{M}_{5,R}$
$\zeta_{5}\otimes\zeta_{5}$ | $\frac{1}{2}$ |  $\frac{1}{2}(2\mathbf{M}_{5,L}\mathbf{M}_{5,R})$
$\text{sYM}\otimes\zeta_{5}^{2}$ | conjecture: $\frac{1}{4}$ |  $\frac{1}{2}(\mathbf{M}_{5,L}^{2}+\mathbf{M}_{5,R}^{2})$
$\zeta_{3}\otimes\zeta_{7}$ | $\frac{1}{4}^{*}$ |  $\mathbf{M}_{3,L}\mathbf{M}_{7,R}+\mathbf{M}_{7,L}\mathbf{M}_{3,R}$
$\text{sYM}\otimes\zeta_{3}\zeta_{7}$ | conjecture: $\frac{1}{4}$ |  $\frac{1}{2}(\mathbf{M}_{7,L}\mathbf{M}_{3,L}+\mathbf{M}_{3,R}\mathbf{M}_{7,R})$
$\zeta_{5}\otimes\zeta_{7}$ | $\frac{1}{4}^{*}$ |  $\mathbf{M}_{5,L}\mathbf{M}_{7,R}+\mathbf{M}_{7,L}\mathbf{M}_{5,R}$
$\text{sYM}\otimes\zeta_{5}\zeta_{7}$ | conjecture: $\frac{1}{4}$ |  $\frac{1}{2}(\mathbf{M}_{7,L}\mathbf{M}_{5,L}+\mathbf{M}_{5,R}\mathbf{M}_{7,R})$
$\zeta_{7}\otimes\zeta_{7}$ | $\frac{1}{2}^{*}$ |  $\frac{1}{2}(2\mathbf{M}_{7,L}\mathbf{M}_{7,R})$
$\text{sYM}\otimes\zeta_{7}^{2}$ | conjecture: $\frac{1}{4}$ |  $\frac{1}{2}(\mathbf{M}_{7,L}^{2}+\mathbf{M}_{7,R}^{2})$
Table 3: A comparison between the various BCJ double-copy constructions
buildable from sYM, $\zeta_{3}$, $\zeta_{5}$, $\zeta_{3}^{2}$, and $\zeta_{7}$
representations. The first column lists the possible constructions, and
$\otimes$ should be understood as commutative. The second column highlights
the differences between the symmetric and asymemtric constructions. While we
do not have access to both forms of double-copy numerators for
$\alpha{}^{\prime\geq 8}$, we conjecture that if such numerators exist then
the normalization pattern we observe in the asymmetric double copy continues.
The third column recalls analogous tree-level kinematic structures appearing
in the KLT representation of closed-string amplitudes, eq. 39. The $\zeta_{7}$
entries used the relaxed-bubble-powercounting representation. The starred
entries in the second column were only checked on the triangle cut,
$\mathfrak{C}(1|2|34)$.

We find that both double copy prescriptions, eqs. 63 and 64, reproduce the
correct kinematic structure of the cuts but lead to the wrong overall
normalization.

To account for the normalization mismatch, we need to dig a bit deeper into
what the various double-copy constructions are doing. First, note that the BCJ
double copy, eq. 10, is generally commutative: for a product between two
theories it does not matter which is assigned to $N$ and which to $\tilde{N}$.
As such the notation $X\otimes Y$ in the table and double-copy numerator
definitions in eqs. 63 and 64 do not distinguish between $N_{X}\tilde{N}_{Y}$
or $N_{Y}\tilde{N}_{X}$, and so have no reason to sum over the contributions.
On the other hand, the KLT double copy used to define the direct cut
construction, eqs. 38, 39, 42 and 62, sum over contributions to each $\zeta$
value from both the “left” theory and the “right” theory. We can make this
clear by artificially breaking the symmetry between the “left” and “right”
copies of the open string by putting “L” and “R” tag labels on eq. 38,

$\mathcal{M}^{\text{closed}}=(A_{\text{open},L})^{T}S_{\alpha^{\prime}}A_{\text{open},R}$
(98)

and following the same expansion and commutation process, yielding

$\mathcal{M}^{\text{closed}}=A_{\text{sYM}}S_{0}\left(1+\zeta_{3}(\mathbf{M}_{3,L}+\mathbf{M}_{3,R})+\dots\right)A_{\text{sYM}}\,.$
(99)

Clearly the closed-string tree amplitude sees contributions from both the
“left” and “right” theories, even though they are identical. Thus, in matching
the cuts using BCJ double-copy numerators, we should really be summing over
all relevant combinations of $N$ and $\tilde{N}$, even those that look
degenerate. For example, consider the $\zeta_{3}^{2}$ contributions, which on
all cuts behave like

$\mathfrak{C}_{\zeta_{3}^{2}\otimes\text{sYM}}+\mathfrak{C}_{\text{sYM}\otimes\zeta_{3}^{2}}+\mathfrak{C}_{\zeta_{3}\otimes\zeta_{3}}=\mathfrak{C}_{\zeta_{3}^{2}}\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{2}\right)=\mathfrak{C}_{\zeta_{3}^{2}}\,.$
(100)

We list the relevant terms in the expansion of eq. 99 in the right-most column
of table 3. Through this lens, the relative combinatorics is mundane while the
amazing thing is that both of the double copy forms correctly reproduce the
kinematics of the cuts individually. Cut-matching could have instead required
cancellations between the different double-copy numerators. We conjecture that
this normalization pattern continues for higher $\alpha{}^{\prime}$ operators.
Due to the difficulties with the $\zeta_{7}$ numerators mentioned above, we
limit ourselves to the more-symmetric double copy numerators which do continue
to follow the observed pattern. It is also interesting that adopting an
additional normalization on $N^{\text{closed}}_{\zeta_{i}\zeta_{j}}$ of
$\operatorname{sv}(\zeta_{i}\zeta_{j})$ leads to an exact match for all of the
asymmetric double-copy numerators, and a degeneracy overcount for the
symmetric numerators.

##  5 Conclusions

In this work, we constructed new color-dual representations at one loop and
four points that contain operator insertions from the open-string low-energy
effective action. The three lowest mass dimension operators for which this was
possible, $D^{2}F^{4}+F^{5}$ (with coefficient $\zeta_{3}\alpha^{\prime 3}$),
$D^{6}F^{4}+\dots$(with coefficient $\alpha^{\prime 5}\zeta_{5}$), and
$D^{8}F^{4}+\dots$(with coefficient $\alpha^{\prime 6}\zeta_{3}^{2}$) , are
represented in terms of box, triangle, and 2-2 bubble numerators. Via the
kinematic Jacobi relations, the latter two numerators are expressed purely in
terms of linear combinations of box numerators with different labelings. For
the next mass dimension operator, the $D^{10}F^{4}+\dots$ combination linked
with $\alpha^{\prime 7}\zeta_{7}$, we introduced a bubble-on-external-leg
contribution via “snail” regularization to match both bubble cuts, and could
not fully match the one-particle cuts. Interestingly, when double copied with
sYM this operator becomes a $D^{8}R^{4}$ type operator, the same type which
was found to cause a deviation between the $\mathcal{N}=4$ sYM and
$\mathcal{N}=8$ SUGRA UV behavior at five loops Bern:2018jmv . The known five-
loop sYM representations also have unresolved tension between color-kinematics
duality and unitarity constraints Bern:2017ucb ; Bern:2018jmv . We saw that it
was possible to overcome the $\zeta_{7}$ construction obtacle using a relaxed
powercounting condition on the bubble. It is also possible that the failure of
the $\zeta_{7}$ construction may be overcome by adopting the less-restrictive
“Minahan” Minahan:1987ha ; Edison:2022jln  regularization for BEL diagrams.
Both approaches provide guidance for a re-examination of the five loop
construction with considerations of relaxing either the “snail” scheme or
diagrammatic powercounting restrictions related to forced powers of external
mandelstams.

The $\zeta_{3}$, $\zeta_{5}$, and $\zeta_{7}$ representations have manfiest UV
behavior, with the critical dimension and actual divergence coming from only
the bubble numerators. On the other hand, the $\zeta_{3}^{2}$ representation
does not have manifest power counting. The UV critical dimension of $D_{C}=4$
suggests that the numerators should at worst be a tensor triangle or scalar
bubble. Instead, the CK dual numerator picks an $\ell^{2}$ bubble. However,
the cancellation of the potential $D_{C}=2$ UV divergence happens at the level
of an individual bubble diagram, and does not require summing over separate
diagrams or channels, in contrast to the contact representation of Ref.
Edison:2021ebi .

We hope that the representations we constructed will be useful for studying
loop-level color-kinematics duality and BCJ relations in a simpler context
than five-loop sYM or two- and three-loop $\mathcal{N}=0,1,2$ YM. In
particular, it would be interesting to investigate if the tree-level kinematic
composition construction of Carrasco, Rodina, Yin, and Zekioglu
Carrasco:2021ptp ; Carrasco:2019yyn  can be lifted to loop level, for which
our representations could serve as seed data.

As an additional bonus to constructing color-dual numerators, the BCJ double
copy allowed us to construct closed-string-insertion numerators from pairings
of the open-string-insertion numerators. We found that regardless of the
prescription used, the double-copy numerators produced the correct kinematic
structures to match against directly calculated cuts, but not the correct
normalziation. In fact, the mismatch in normalization is explained by
observaing that the sum over all allowed prescriptions (with degeneracy)
exactly matches the directly calculated cuts. This sum including degeneracy is
in direct analogy with terms and normalizations in the KLT representation of
the tree-level closed-string amplitudes, and may suggest a similar
decomposition for the one-loop numerators.

## Acknowledgements

We give significant thanks Henrik Johansson and Oliver Schlotterer for
discussion and feedback throughout the project. We appreciate Oliver
Schlotterer and Fei Teng for comments on the manuscript. This work is an
extension of MT’s master’s project at Uppsala University. AE is supported in
part by the Knut and Alice Wallenberg Foundation under KAW 2018.0116, From
Scattering Amplitudes to Gravitational Waves, by the USDOE under contract DE-
SC0015910, and by Northwestern University via the Amplitudes and Insight
Group, Department of Physics and Astronomy, and Weinberg College of Arts and
Sciences. The diagrams in this manuscript were made using TikZ-Feynman
Ellis:2016jkw .

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