# Next-to-MHV Yang-Mills kinematic algebra

Gang Chen b,c Henrik Johansson b,d Fei Teng e and Tianheng Wang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Centre for Research in String Theory, School
of Physics and Astronomy, Queen Mary University of London, Mile End Road,
London E1 4NS, U.K.Department of Physics and Astronomy, Uppsala University,
Box 516, 75120 Uppsala, SwedenNordita, Stockholm University and KTH Royal
Institute of Technology, Hannes Alfvéns väg 12, 10691 Stockholm,
SwedenInstitute for Gravitation and the Cosmos, Pennsylvania State University,
University Park, PA 16802, U.S.AInstitut für Physik und IRIS Adlershof,
Humboldt-Universität zu Berlin, Zum Großen Windkanal 6, 12489 Berlin, Germany

###### Abstract

Kinematic numerators of Yang-Mills scattering amplitudes possess a rich Lie
algebraic structure that suggest the existence of a hidden infinite-
dimensional kinematic algebra. Explicitly realizing such a kinematic algebra
is a longstanding open problem that only has had partial success for simple
helicity sectors. In past work, we introduced a framework using tensor
currents and fusion rules to generate BCJ numerators of a special subsector of
NMHV amplitudes in Yang-Mills theory. Here we enlarge the scope and explicitly
realize a kinematic algebra for all NMHV amplitudes. Master numerators are
obtained directly from the algebraic rules and through commutators and
kinematic Jacobi identities other numerators can be generated. Inspecting the
output of the algebra, we conjecture a closed-form expression for the master
BCJ numerator up to any multiplicity. We also introduce a new method, based on
group algebra of the permutation group, to solve for the generalized gauge
freedom of BCJ numerators. It uses the recently introduced binary BCJ
relations to provide a complete set of NMHV kinematic numerators that consist
of pure gauge.

††preprint: UUITP-21/21
NORDITA 2021-031
HU-EP-21/10
QMUL-PH-21-22

##  1 Introduction

Scattering amplitudes provide a rich source for better understanding of
quantum field theory and gravity by exposing hidden structures that are not
visible in a conventional Lagrangian approach. A structure that underlies many
new insights is the Bern-Carrasco-Johansson (BCJ) duality between color and
kinematics Bern:2008qj ; Bern:2010ue ; Bern:2019prr . The duality states that
scattering amplitudes in many gauge theories may be organized through cubic
diagrams, where each diagram consists of a kinematic numerator and a
corresponding color factor that satisfy isomorphic Lie-algebraic relations.
Since the color factors inherit their properties from the Lie algebra of the
gauge group, the duality implies the existence of a hidden kinematic Lie
algebra that builds up the kinematic numerators in a similar fashion.

For gauge theories that have only massless adjoint fields, such as pure Yang-
Mills (YM) theory, color-kinematics duality at tree level can be equivalently
phrased as the existence of BCJ amplitude relations Bern:2008qj ;
Stieberger:2009hq ; BjerrumBohr:2009rd ; Feng:2010my ; BjerrumBohr:2010hn .
Color-kinematics duality and BCJ amplitude relations were first identified for
pure YM theory Bern:2008qj  and related pure supersymmetric theories
Bern:2010ue ; Bern:2010yg ; Stieberger:2009hq ; BjerrumBohr:2009rd . Many
incarnations of the duality were later discovered in a variety of gauge
theories Bargheer:2012gv ; Huang:2012wr ; Broedel:2012rc ; Chiodaroli:2013upa
; Johansson:2014zca ; Chiodaroli:2014xia ; Johansson:2015oia ;
Chiodaroli:2015rdg ; Johansson:2017srf ; Chiodaroli:2018dbu ;
Johansson:2018ues ; Johansson:2019dnu ; Bautista:2019evw ; Plefka:2019wyg  as
well as scalar effective field theories Chen:2013fya ; Cheung:2016prv ;
Carrasco:2016ldy ; Mafra:2016mcc ; Carrasco:2016ygv ; Low:2019wuv ;
Cheung:2020qxc ; Rodina:2021isd . The duality has been generalized to loop-
level amplitudes Bern:2010ue ; Carrasco:2011mn ; Bern:2012uf ; Boels:2013bi ;
Bjerrum-Bohr:2013iza ; Bern:2013yya ; Nohle:2013bfa ; Mogull:2015adi ;
Mafra:2015mja ; He:2015wgf ; Johansson:2017bfl ; Hohenegger:2017kqy ;
Mafra:2017ioj ; Faller:2018vdz ; Kalin:2018thp ; Ben-Shahar:2018uie ;
Duhr:2019ywc ; Geyer:2019hnn ; Edison:2020uzf ; Casali:2020knc ;
DHoker:2020prr ; Carrasco:2020ywq ; Bridges:2021ebs  and form factors
Boels:2012ew ; Yang:2016ear ; Boels:2017ftb ; Lin:2020dyj , and it appears to
transpire to certain curved-space observables Adamo:2017nia ; Farrow:2018yni ;
Adamo:2018mpq ; Lipstein:2019mpu ; Prabhu:2020avf ; Armstrong:2020woi ;
Albayrak:2020fyp ; Adamo:2020qru ; Alday:2021odx . The existence of color-
kinematics duality and BCJ relations in massless gauge theories has been
understood from a variety of different perspectives, including string theory,
scattering equations, and positive geometry BjerrumBohr:2009rd ;
Stieberger:2009hq ; Cachazo:2012uq ; Arkani-Hamed:2017mur ; Mizera:2019blq .

A major insight from the color-kinematics duality is that gravitational
amplitudes have the same diagrammatic structure as gauge theory amplitudes,
except that the color factors are replaced by a second copy of kinematic
numerators Bern:2008qj ; Bern:2010ue . This construction of gravity amplitudes
is known as the double copy Bern:2010ue . It clarifies the existence of a
general connection between gauge and gravity theories, which first came to
light through the Kawai-Lewellen-Tye (KLT) relations Kawai:1985xq  between
open and closed strings. The double copy provides an efficient tool for loop
calculations in a large class of gravity theories Bern:2011rj ;
BoucherVeronneau:2011qv ; Bern:2013uka ; Bern:2014sna ; Chiodaroli:2015wal ;
Johansson:2017bfl ; Chiodaroli:2017ehv ; Bern:2018jmv ; Bern:2021ppb . Notable
recent results using the double copy includes applications to classical
solutions Monteiro:2014cda ; Luna:2015paa ; Luna:2016hge ; Bahjat-
Abbas:2017htu ; Carrillo-Gonzalez:2017iyj ; Berman:2018hwd ;
CarrilloGonzalez:2019gof ; Goldberger:2019xef ; Huang:2019cja ; Bahjat-
Abbas:2020cyb ; Easson:2020esh ; Emond:2020lwi ; Godazgar:2020zbv ;
Chacon:2021wbr ; Chacon:2020fmr ; Alfonsi:2020lub ; Monteiro:2020plf ;
White:2020sfn ; Elor:2020nqe ; Pasarin:2020qoa , black-hole scattering and
gravitational wave physics Luna:2016due ; Goldberger:2016iau ; Luna:2017dtq ;
Shen:2018ebu ; Plefka:2018dpa ; Bern:2019nnu ; Plefka:2019hmz ; Bern:2019crd ;
Bern:2020buy ; Almeida:2020mrg ; Bern:2021dqo .

In this paper we are interested in the mathematical consequences of color-
kinematics duality for pure YM theory. The duality-satisfying numerators, also
known as BCJ numerators, can by now be computed for pure YM theory in
principle up to any multiplicity at tree level BjerrumBohr:2010hn ;
Mafra:2011kj ; Fu:2012uy ; Mafra:2015vca ; Bjerrum-Bohr:2016axv ; Du:2017kpo ;
Chen:2017bug ; Fu:2018hpu ; Edison:2020ehu ; Hou:2021mvg . However, the
details of the underlying kinematic Lie algebra are still understood only at a
rudimentary level. General properties of it can be inferred from our knowledge
of the BCJ numerators: the algebra should be infinite dimensional, since the
structure constants must be parametrized by continuous momenta. It should not
be invariant under gauge transformations since the BCJ numerators transform
non-trivially. Indeed, gauge transformations may play a central role in the
details of the algebra. The structure constants of the algebra should be
isomorphic to interaction terms in a cubic Lagrangian that give BCJ numerators
through its Feynman rules. However, there are many unknowns regarding the
precise formulation of the algebra. Some details have been exposed by explicit
constructions in simple helicity sectors.

The first explicit kinematic-algebra construction was formulated in the self-
dual sector of YM theory, by Monteiro and O’Connell Monteiro:2011pc . They
found an explicit generator representation of the kinematic algebra in this
sector, and recognized it as corresponding to area-preserving diffeomorphisms.
A plane wave factor and a derivative operator built up a generator, and the
commutator of two generators closed in the Lie algebra with structure
constants identified as interaction terms of self-dual YM. Thus the tree-level
Feynman diagrams mapped to a Lie bracket structure, explicitly manifesting
color-kinematics duality. While these diagrams provide a self-dual solution to
the YM field equations, taken on shell they give tree-level scattering
amplitudes that vanish. Indeed, only one-loop amplitudes are non-vanishing in
self-dual YM theory Cangemi:1996rx  (see ref. Boels:2013bi  for color-
kinematics duality for these amplitudes). The maximally-helicity-violating
(MHV) sector of YM gives the simplest non-zero amplitudes in 4D, and while
ref. Monteiro:2011pc  considered it through a non-local gauge choice, a local
kinematic algebra of the MHV sector was unknown for quite some time.

A second example of an explicit construction of a kinematic algebra sub-sector
is by Cheung and Shen Cheung:2016prv . They realized a cubic Lagrangian for
the $SU(N)$ non-linear sigma model, which directly gave tree-level Feynman
rules that obey color-kinematics duality. A curious feature of their
construction was that they relied on vector fields for describing a scalar
field. Indeed, in later work it was made clear that this Lagrangian is a
truncation of YM via dimensional reduction operations Cheung:2017yef ;
Cheung:2017ems . And in ref. Chen:2019ywi  it was observed that the Cheung-
Shen Lagrangian indirectly computes local BCJ numerators for the MHV sector of
YM theory.

It is convenient to promote the 4D concept of (Next-to)k-MHV sectors to
general spacetime dimension by recognizing that this grading of YM translates
to independent structures in a BCJ numerator Chen:2019ywi . By the MHV sector
we mean the numerator terms with the fewest
$\varepsilon_{i}{\cdot}\varepsilon_{j}$ factors, which in YM implies the
schematic form
$\sim\varepsilon_{i}{\cdot}\varepsilon_{j}\prod\varepsilon_{k}{\cdot}p_{l}$.
Such numerator terms are sufficient for computing MHV amplitudes in 4D
Chen:2019ywi . It turns out that the BCJ numerator built from these terms is
unique at every multiplicity Chen:2019ywi . The Cheung-Shen Lagrangian
computes all these terms in the unique BCJ numerator, thus providing a local
realization of the kinematic algebra for the MHV sector of YM. Curiously, a
closely related kinematic algebra has recently been formulated as a non-
abelian generalization of the Navier-Stokes equation Cheung:2020djz  (see also
ref. Keeler:2020rcv ).

In earlier work, duality-satisfying effective Lagrangians were constructed
using non-local terms or auxiliary fields, which gave Feynman rules that
computed BCJ numerators up to five Bern:2010yg  and six points Tolotti:2013caa
. These constructions encountered a proliferation of ambiguities related to
the non-uniqueness of the BCJ numerators. We now understand that these
ambiguities start in the NMHV sector of the YM kinematic algebra Chen:2019ywi
, and they are a reflection of the generalized gauge freedom of the BCJ
numerators Bern:2008qj ; Bern:2010ue , which include the standard gauge
freedom. Because of these difficulties, refined frameworks are needed for
getting the kinematic algebra under control.

A first attempt at formulating a NMHV-sector kinematic algebra, for
$D$-dimensional local numerators, was made in our earlier work Chen:2019ywi .
The problem was distilled down to a sub-sector of the NMHV numerators, made
out of terms proportional to a fixed polarization product
$\varepsilon_{1}{\cdot}\varepsilon_{n}$, which effectively meant studying two
scalars interacting with the YM field. This bi-scalar sector retains much of
the NMHV sector details, and a new framework based of tensor currents and
fusion products realized the algebra in this sector. Interestingly, the tensor
currents both provided the algebra with generators of enlarged Lorentz tensor
structure, and encoded part of the generalized gauge freedom in an
enlightening way.

In the first part of this paper, we construct a realization of the kinematic
algebra for the complete NMHV sector of $D$-dimensional YM. We expand on the
framework of tensor currents and fusion products previously introduced
Chen:2019ywi . It is convenient to divide up the computational task into an
algebraic construction of a pre-numerator, and subsequently a BCJ numerator is
obtained through a nested set of commutators applied to the pre-numerator.
This mimics the construction of color factors by expressing them as nested
commutators of traces of the gauge group generators. In the bi-scalar
subsector, the nested commutator collapsed Chen:2019ywi  into an ordered
product of generators which meant that the distinction between pre-numerator
and numerator was not necessary in that subsector. Compared to the previous
work Chen:2019ywi , we have a larger set of tensor currents and somewhat more
involved fusion products that defines an algebraic structure in the NMHV
sector. After constructing the algebra, we note that the resulting pre-
numerator admits an all-multiplicity closed formula. The master numerators
(the half-ladder diagram numerators) as well as all the other numerators, are
obtained through $(2n-5)!!$ different nested commutators of the pre-numerator,
and enjoys a manifest $S_{n-1}$ crossing symmetry.

In the second part of the paper, we observe that the binary BCJ relations
introduced in ref. Chen:2019ywi  endows the local BCJ numerators with a
complete parametrization of the generalized gauge freedom for the NMHV sector.
This is an important observation that we expect to have natural
generalizations for higher NkMHV sectors, and thus may enable a full control
over the ambiguities in a local description of the kinematic algebra or
equivalent Lagrangian descriptions. In the present context, we analyse the
generalized gauge freedom in great detail by prescribing how to fully classify
the polynomials that can appear. They are given by invariant functions under a
specific element of the group algebra over the permutation group. This element
acts as a simple projector on monomials, which reduces the problem of
constructing pure-gauge BCJ numerators to a computationally straightforward
task. We explicitly count all the degrees of freedom that can appear in pure-
gauge BCJ numerators up to seven points, including the N2MHV sector.

The paper is organized as follows: In section 2, we introduce the general
setup that is based on the analogous structures of matrix products between Lie
algebra generators and fusion products of vector and tensor currents. In
section 3, we construct the fusion products necessary for describing the NMHV
sector pre-numerator. In section 4, we present a closed formula for the pre-
numerator as obtained from the algebraic construction. After completing the
particular kinematic algebra, we switch gears and study the more general
problem of pure gauge freedom in the NMHV sector BCJ numerators. We introduce
essential mathematical background on group algebra in section 5. We then show
in section 6 that the NMHV sector pure gauge terms with $S_{n-2}$ crossing
symmetry can be classified by the kinematic polynomials that are invariant
under certain group algebra action. In section 7, we discuss the
generalization of this construction to $S_{n-1}$ and $S_{n}$ crossing
symmetric pure gauge terms. Conclusion and outlook are in section 8.

##  2 Kinematic algebra from tensor currents

Here we review and extend some formal notations that were introduced in ref.
Chen:2019ywi . We denote a generator of a putative algebra as $J_{U}$, and
consider products of two such generators, $J_{U}\star J_{V}$, which we refer
to as a fusion product. The fusion product takes the form

$\displaystyle J_{U}\star J_{V}=\sum_{W}f_{UV}^{W}J_{W},$ (1)

where the $f$’s are coefficients that encode the algebraic structure. The
fusion product is required to be linear in the first argument,

$\displaystyle(c_{1}J_{U}+c_{2}J_{V})\star J_{W}$
$\displaystyle=c_{1}(J_{U}\star J_{W})+c_{2}(J_{V}\star J_{W})\,.$ (2)

In principe, it should also be linear in the second argument, but it will not
be necessary for the purpose of this paper.

Let us refine the details of the generators, so that the algebraic framework
can describe objects of a kinematic algebra for YM numerators. We upgrade the
generators to tensor (vector) currents $J_{U}\rightarrow J^{(w)}_{U}(p)$ that
are specified by three types of arguments:  (1) a Lorentz tensor structure
$U$;  (2) momentum argument $p$;  (3) additional degeneracy index $w$ to
distinguish otherwise identical generators.  The tensor structure is
constructed as a tensor product $U=a_{1}\otimes a_{2}\otimes\cdots\otimes
a_{r}$, where $a_{i}$’s are polarization vectors and momenta. We will refer to
the number of vectors in $U$ as the _tensor rank_ $r$, and a rank-one tensor
current is thus a vector current. The coefficients in the fusion product
$J^{(w)}_{U}(p)\star J^{(w^{\prime})}_{V}(p^{\prime})$ should be polynomials
of the Lorentz invariant products between the kinematic objects that appear in
the arguments of the generators. However for some tensor types we will need a
slight generalization, which we will discuss later.

The language of currents is borrowed from the old topic of current algebras,
but unlike that context the fusion products we consider will be strictly
_local_ , since we are interested in describing the kinematic algebra of local
BCJ numerators, which have no poles. In principle, it would be ideal to label
the generators with only the kinematic data of the external states of an
amplitude, however, there is no guarantee that these are one-to-one. As
already observed in ref. Chen:2019ywi , currents corresponding to internal
states of identical tensor and momentum structure may require additional
labels for the algebra to close. Thus an additional degeneracy index $w$ is
necessary to allow for a richer set of potential fusion products. These
considerations imply that we have to be careful to distinguish the vector
current associated with a (physical, on-shell) external gluon from an internal
vector current of a similar type. We denote the vector current associated with
a on-shell gluon as $J_{\varepsilon_{i}}(p_{i})$, where $\varepsilon_{i}$ and
$p_{i}$ are the polarization vector and on-shell momentum of the gluon, and
the $w$ index is omitted.

By analogy with the workings of a Lie algebra, we imagine that the generators
of the kinematic algebra transform in some particular representation that is
appropriate for spin-1/2 particles. This implies that we can try to realize an
on-shell gluon as a bilinear in the on-shell fermion wave function, as should
be very familiar from the spinor-helicity framework. However, we will only
require that one gluon, corresponding to leg $n$ of an $n$-point amplitude, is
obtained this way, whereas the other gluons $i\in\\{1,\ldots,n-1\\}$ are
obtained by writing down the corresponding generators
$J_{\varepsilon_{i}}(p_{i})$. That is, we formally split the polarization
vector $\varepsilon_{n}$ into a spinor $u_{n}(p_{n})=|n\rangle$ and a
conjugate spinor $\bar{v}_{q}(q)=\langle q|$, where the latter carries soft
(reference) momentum $q$. Then we use these spinors to sandwich the remaining
$n{-}1$ generators that are fused according to an ordering that corresponds to
a half-ladder (or multiperiferal) diagram:

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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$\@add@centering&\text{kinematic
factor}$\@add@centering\cr(T^{a_{1}}T^{a_{2}}\cdots
T^{a_{n-1}})_{i_{q}}^{\bar{\imath}_{n}}$\@add@centering&$\@add@centering&\langle
q|J_{\varepsilon_{1}}\star J_{\varepsilon_{2}}\star\cdots\star
J_{\varepsilon_{n-1}}|n\rangle$\@add@centering\end{array}\,.$ (5)

This is analogous to how a color factor of $n$ adjoint particles can be
constructed using the Lie algebra generators $T^{a_{i}}$, and where the color
of leg $n$ is only implicitly represented through the bi-fundamental indices
$(\bar{\imath}_{n},i_{q})$.111To obtain a physical state one needs to project
the bi-fundamental $(\bar{\imath}_{n},i_{q})$ onto the adjoint representation,
and likewise project the kinematical bi-spinor $\langle q|\cdots|n\rangle$
onto a gluon state. Note that the kinematic factor in eq. (5) cannot be
directly calculated, as we will not give an explicit representation for the
generators. Instead, we need to use the algebraic properties of the fusion
product to reduce it down to simpler objects involving fewer generators.

By convention, we always evaluate a fusion product from left to right,

$\displaystyle J_{\varepsilon_{1}}\star J_{\varepsilon_{2}}\star\cdots\star
J_{\varepsilon_{n-1}}\equiv\left(\cdots\left((J_{\varepsilon_{1}}\star
J_{\varepsilon_{2}})\star J_{\varepsilon_{3}}\right)\cdots\right)\star
J_{\varepsilon_{n-1}}\,,$ (6)

as this will reduce the number of fusion product coefficients that we need to
specify. At every step we encounter a two-to-one map $X\star
J_{\varepsilon_{i}}\rightarrow Y$, where one generator is a physical gluon
current. Evaluating such fusion products recursively will simplify the
expression down to terms with a single current sandwiched between the spinors,
which we can identify with an external state corresponding to leg $n$.

The kinematic factor in eq. (5) is not yet the BCJ numerator of a pure YM
diagram, since pictorially it resembles more a diagram with a massless
fermion. Indeed, we will call the result of the above ordered fusion product
the _pre-numerator_ , denoted by $\mathcal{N}$,

$\displaystyle\mathcal{N}(1,2,\cdots,n-1,n)\equiv\langle
q|J_{\varepsilon_{1}}\star J_{\varepsilon_{2}}\star\cdots\star
J_{\varepsilon_{n-1}}|n\rangle\Big{|}_{q~{}{\rm soft}}\,.$ (7)

Kinematically one should consider the momentum $q$ to be soft compared to
$p_{n}$, thus we can identify $p_{n}+q\approx p_{n}$, but $q$ will provide a
reference direction for the polarization vector
$\varepsilon_{n}=\varepsilon_{n}(p_{n},q)$. Note that we do not aim at
identifying the pre-numerator with a fermion diagram in full detail, the
figure in eq. (5) only serves as a motivation for understanding the general
properties of the pre-numerator.

Following the computational steps outlined, the pre-numerator becomes a linear
superposition of individual vector and tensor currents sandwiched between the
two spinors. To identify such objects with external states in the final step,
we make the choice to not distinguish between currents of different $w$ when
they are sandwiched between two spinors, hence we drop this index. Then, for a
vector current $J_{a}$, it is natural to evaluate it as

$\displaystyle\langle q|J_{a}|n\rangle\big{|}_{q\text{
soft}}=\bar{v}_{q}\gamma_{\mu}u_{n}a^{\mu}\big{|}_{q\text{ soft}}\equiv
a{\cdot}\varepsilon_{n}\,,$ (8)

and for a tensor current $J_{a_{1}\otimes a_{2}\otimes\cdots\otimes a_{r}}$,
we identify it as

$\displaystyle\langle q|J_{a_{1}\otimes a_{2}\otimes\cdots\otimes
a_{r}}|n\rangle\big{|}_{q~{}\rm
soft}=(\bar{v}_{q}\gamma_{\mu_{1}}\ldots\gamma_{\mu_{r}}u_{n})a_{1}^{\mu_{1}}\ldots
a_{r}^{\mu_{r}}\big{|}_{q~{}\rm soft}\,.$ (9)

This object is natural. If we antisymmetrize the Lorentz indices, then the
object
$(\bar{v}_{q}\gamma_{[\mu_{1}}\ldots\gamma_{\mu_{r}]}u_{n})\big{|}_{q~{}\rm
soft}$ is just the on-shell polarization tensor of an $r$-form field. It is
automatically transverse to both $p_{n}$ and $q$ hence it encodes the tensor
structure of the $\text{SO}(D-2)$ little group.

For completeness, we may consider an analogous decomposition of the color
factor in eq. (5) into representations of the Lie algbra, but we are primarily
interested in the adjoint representation. This is obtained through the
projection

$\displaystyle\mathcal{C}(1,2,\ldots,n)\equiv(T^{a_{1}}T^{a_{2}}\cdots
T^{a_{n-1}})_{i_{q}}^{\bar{\imath}_{n}}(T^{a_{n}})_{i_{n}}^{\bar{\imath}_{q}}=\text{tr}(T^{a_{1}}T^{a_{2}}\cdots
T^{a_{n}})\,,$ (10)

which gives a standard trace factor. Thus, by analogy, one may be tempted to
think of the pre-numerator as a kinematic analog of the trace of $n$
generators. However, a slight obstruction is that the cyclic symmetry of the
pre-numerator is not guaranteed, and we will not assume it in general, thus we
keep leg $n$ in a fixed position in $\mathcal{N}(1,2,\ldots,n)$. See ref.
Bern:2011ia , where an alternative interpretation of kinematic traces were
given.

Next, we use the analogy between $\mathcal{C}(1,2,\ldots,n)$ and
$\mathcal{N}(1,2,\ldots,n)$ to realize the color-kinematics duality of pure YM
theory. As is by now well known Bern:2008qj , one can write color-dressed pure
YM tree amplitudes as a sum over $(2n-5)!!$ cubic graphs,

$\displaystyle\mathcal{A}_{n}=\sum_{\Gamma\in\text{cubic}}\frac{C_{\Gamma}\,N_{\Gamma}}{D_{\Gamma}}\,,$
(11)

where $C_{\Gamma}$, $N_{\Gamma}$ and $1/D_{\Gamma}$ are respectively the color
factor, kinematic numerator factor and propagator factor of the graph
$\Gamma$. Throughout this paper we set gauge coupling constants, and other
overall numerical factors in the amplitude, to unity.

It is convenient to represent an $n$-point cubic graph $\Gamma$ as a nested
commutator of the elements $1,\ldots,n{-}1$, where the number of commutators
used is $n{-}2$. For example, it can be recursively constructed starting from
the seed $[1,2]$ and then, for $i=3,\ldots,n-1$, substitute any Lie-valued
object (particle label or commutator) using the rule $*\rightarrow[*,i]$.
Consider the next step, there are three Lie-valued objects in $[1,2]$, namely
$\\{1,2,[1,2]\\}$. Applying rule for $i=3$ then gives:
$\\{[[1,3],2],[1,[2,3]],[[1,2],3]\\}$. At multiplicity $n$, one can count that
there are $1\times 3\times 5\times\ldots\times(2n-5)=(2n-5)!!$ nested
commutator expressions generated by this rule, precisely matching the number
of cubic graphs. Note that the nested commutator representation has the
feature that leg $1$ always appears in the first position, legs
$2\,\ldots,n{-}1$ can appear in any order, whereas leg $n$ does not appear at
all. For a graph $\Gamma$, we identify its nested commutator representation
with the graph itself; e.g. $\Gamma=[\ldots[[[1,2],3],4],\ldots,n{-}1]$.

We can use this notation to define the inverse propagator factor more
precisely. We have $D_{\Gamma}=\prod_{\gamma\in\Gamma}s_{\gamma}$, where
$\gamma$ is a nested commutator subgraph of $\Gamma$ containing at least two
particle labels, and $s_{\gamma}=\sum_{i,j\in\gamma}p_{i}{\cdot}p_{j}$ gives
the invariant momentum square of this subgraph. Similarly, using the color
traces $\mathcal{C}(1,2,\ldots,n)$ defined in eq. (10) one can define the
color factor of a cubic graph as $C_{\Gamma}$,

$\displaystyle C_{\Gamma}=\mathcal{C}(\Gamma,n)\,.$ (12)

where the commutators are expanded according to the natural rule
$\mathcal{C}(\ldots[A,B]\ldots,n)=\mathcal{C}(\ldots
A,B\ldots,n)-\mathcal{C}(\ldots B,A\ldots,n)$. As a consequence of this
commutator rule, the color factors will satisfy Jacobi identities. For
example, at $n{=}4$ the color factor for the $s$-channel diagram can be
written as

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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.21802pt}{-13.93153pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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$\displaystyle=\mathcal{C}\big{(}\overbrace{[[1,2],3]}^{\Gamma},4\big{)}$
$\displaystyle=\mathcal{C}(1,2,3,4)-\mathcal{C}(2,1,3,4)-\mathcal{C}(3,1,2,4)+\mathcal{C}(3,2,1,4)$
$\displaystyle=f^{a_{1}a_{2}b}f^{ba_{3}a_{4}}\,.$ (13)

Similarly for the other two channels, we have

$\displaystyle
C_{t}=\mathcal{C}\big{(}[1,[2,3]],4\big{)}=\mathcal{C}(1,2,3,4)-\mathcal{C}(1,3,2,4)-\mathcal{C}(2,3,1,4)+\mathcal{C}(3,2,1,4)\,,$
$\displaystyle
C_{u}=\mathcal{C}\big{(}[[1,3],2],4\big{)}=\mathcal{C}(1,3,2,4)-\mathcal{C}(3,1,2,4)-\mathcal{C}(2,1,3,4)+\mathcal{C}(2,3,1,4)\,,$
(14)

and the Jacobi identity $C_{s}-C_{u}=C_{t}$ is automatic.

Analogously, the color-kinematics duality will be manifest by construction if
the numerators are obtained from the pre-numerators through the same nested
commutator expression,

$\displaystyle N_{\Gamma}=\mathcal{N}(\Gamma,n)\,.$ (15)

Of particular importance are the numerators of half-ladder diagrams with leg
$1$ and $n$ fixed, which are obtained through the left-nested commutators

$\displaystyle N(1,2,3,\ldots,n-1,n)$ $\displaystyle\equiv
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$\displaystyle\equiv\mathcal{N}\big{(}[\ldots[[1,2],3],\ldots,n-1],n\big{)}\,.$
(16)

The relabelings of this half-ladder (or master) numerator,
$N(1,\sigma_{2},\sigma_{3},\cdots,\sigma_{n-1},n)$, form a $(n{-}2)!$ basis
under Jacobi identities, also known as the Del-Duca-Dixon-Maltoni (DDM) basis
DelDuca:1999rs . Therefore, the half-ladder numerator $N(1,2,\ldots,n-1,n)$,
as well as the pre-numerator $\mathcal{N}(1,2,\ldots,n-1,n)$, are the central
objects that we study in this paper. Other BCJ numerators can be inferred by
Jacobi relations and permutations.

Consider the same examples as for the color factors at multiplicity $n{=}4$.
The DDM basis is given by the $s$\- and $u$-channel numerators,

$\displaystyle N_{s}=N(1,2,3,4)$
$\displaystyle=\mathcal{N}\big{(}[[1,2],3],4\big{)}$
$\displaystyle=\mathcal{N}(1,2,3,4)-\mathcal{N}(2,1,3,4)-\mathcal{N}(3,1,2,4)+\mathcal{N}(3,2,1,4)\,,$
$\displaystyle N_{u}=N(1,3,2,4)$
$\displaystyle=\mathcal{N}\big{(}[[1,3],2],4\big{)}$
$\displaystyle=\mathcal{N}(1,3,2,4)-\mathcal{N}(3,1,2,4)-\mathcal{N}(2,1,3,4)+\mathcal{N}(2,3,1,4)\,.$
(17)

The $t$-channel numerator is not a basis element, it is written as
$N_{t}=\mathcal{N}\big{(}[1,[2,3]],4\big{)}$, which automatically implies the
kinematic Jacobi identity $N_{s}-N_{u}=N_{t}$.

Using the BCJ numerators and color factors in the DDM basis, we can re-write
the color-dressed YM amplitude as Bern:2008qj ; Vaman:2010ez

$\displaystyle\mathcal{A}_{n}=\sum_{\Gamma\in\text{cubic}}\frac{C_{\Gamma}\,N_{\Gamma}}{D_{\Gamma}}=\sum_{\sigma,\rho\in
S_{n-2}}C(1,\sigma_{2},\cdots,\sigma_{n-1},n)m(\sigma|\rho)N(1,\rho_{2}\cdots,\rho_{n-1},n)\,,$
(18)

where the $(n{-}2)!$-by-$(n{-}2)!$ matrix $m(\sigma|\rho)$ is built out of
linear combinations of the scalar-type propagators $1/D_{\Gamma}$, as given by
the decomposition of the BCJ numerators and color factors into the DDM basis.
It goes by many names in the literature, it is called the “propagator matrix”
Vaman:2010ez , the “inverse of the KLT kernel” Kawai:1985xq ; Cachazo:2013iea
, or the “bi-adjoint scalar amplitude” Cachazo:2013iea . It may also be
identified as the double-partial amplitudes of “dual-scalar theory”
Bern:2010yg ; BjerrumBohr:2012mg , “color-scalar theory” Du:2011js  or “scalar
$\phi^{3}$ theory” Bern:1999bx ; Chiodaroli:2014xia .

The color-ordered partial amplitudes follow from eq. (18) as the kinematic
factor multiplying each independent color factor, which gives a (non-
invertible) map between BCJ numerators and partial amplitudes

$\displaystyle A(1,\sigma,n)=$ $\displaystyle\sum_{\rho\in
S_{n-2}}m(\sigma|\rho)N(1,\rho,n)\,.$ (19)

The propagator matrix is not invertible for on-shell momenta and hence it has
a kernel (or null space). This implies that one can find contributions to BCJ
numerators that live in this kernel, and do not feed into the partial
amplitudes. Hence, BCJ numerators are in general not unique. The ambiguity is
called _generalized gauge freedom_ Bern:2008qj ; Bern:2010ue  and it
corresponds to shifting the existing numerators by what we call _pure gauge_
numerators

$\displaystyle N(1,2,\ldots,n-1,n)\sim
N(1,2,\ldots,n-1,n)+N^{\text{gauge}}(1,2,\ldots,n-1,n)\,,$ (20)

where the pure gauge numerators are annihilated by the propagator matrix,

$\displaystyle\sum_{\rho\in
S_{n-2}}m(\sigma|\rho)N^{\text{gauge}}(1,\rho,n)=0\,.$ (21)

The generalized gauge freedom subsume the standard gauge freedom for vector
fields, and it generalizes it to also include any other operations, such as
field redefinitions, that changes the cubic diagram numerators but leaves the
amplitude invariant.

Finally, we note that the BCJ numerators constructed from pre-numerators will
automatically satisfy crossing symmetry relations in the legs
$1,\ldots,n{-}1$, because these legs are on equal footing in the pre-
numerator, meaning that any permutations of them are allowed to appear. We
will refer to this property as exhibiting (manifest) $S_{n-1}$ crossing
symmetry, and we note that this symmetry is larger than what is naively
obtained from a standard DDM basis of numerators, which naturally admits a
manifest $S_{n-2}$ crossing symmetry. See, e.g., ref. Edison:2020ehu  for an
all-multiplicity form of BCJ numerators for YM that exhibit $S_{n-2}$ crossing
symmetry.

##  3 Fusion product of currents in the NMHV sector

In this section, we explicitly construct the fusion product rules which are
used to compute the duality-satisfying kinematic numerators up to the
quadratic order in $\varepsilon_{i}{\cdot}\varepsilon_{j}$ factors. Before we
get into the details, let us briefly clarify the decomposition of the
kinematic numerator into different sectors based on the structure of the
variables that it contains.

As previously mentioned, we consider only local kinematic numerators which are
polynomials of Lorentz products of momenta and polarization vectors. We
classify the terms in the numerators according to the number of
$\varepsilon_{i}{\cdot}\varepsilon_{j}$ factors, which we call _polarization
power_. Terms of different polarization power splits into independent sectors
in the BCJ numerator; that is, the sectors do not mix under: permutations of
labels, momentum identities, or Jacobi identities. Since these sectors are in
a one-to-one correspondence with the 4D helicity sectors, denoted by NkMHV,
via the gauge choices for the polarization vectors explained in ref.
Chen:2019ywi , we will often use the 4D language when referring to these
sectors.

In this paper, we will mostly focus our attention to terms of polarization
power two (NMHV), as terms of polarization power one (MHV) are already
uniquely known Chen:2019ywi , and terms of polarization power $k{+}1$ (NkMHV)
are beyond our scope. Schematically, the two simplest sectors look as

$\begin{array}[]{ c c c c c }\\!\\!\\!\\!\text{polarization power
one:}&(\varepsilon_{i}{\cdot}\varepsilon_{j})\prod(\varepsilon_{k}{\cdot}p_{l})\phantom{\Big{|}}&\\!\\!\rightarrow&\text{MHV
sector}\\\ \\!\\!\\!\\!\text{polarization power
two:}&\\!\\!(\varepsilon_{i_{1}}{\cdot}\varepsilon_{j_{1}})(\varepsilon_{i_{2}}{\cdot}\varepsilon_{j_{2}})(p_{i_{3}}{\cdot}p_{j_{3}})\prod(\varepsilon_{k}{\cdot}p_{l})\phantom{\Big{|}}&\\!\\!\rightarrow&\text{NMHV
sector}\,.\end{array}$ (22)

Recall that the mass dimension of kinematic numerators in YM must be $n-2$.
Thus the requirement of locality induces a correspondence between the number
of $\varepsilon_{i}{\cdot}\varepsilon_{j}$ factors and $p_{i}{\cdot}p_{j}$
factors. The polarization power one terms contain no $p_{i}{\cdot}p_{j}$
factors whereas those of polarization power two are linear in the Mandelstam
variables. This explains why BCJ numerators of polarization power two are not
unique, since the $p_{i}{\cdot}p_{j}$ factors may conspire with the propagator
denominators and produce a contact interaction, which cannot be uniquely
attributed to a cubic diagram.

###  3.1 Currents in NMHV kinematic algebra

Ref. Chen:2019ywi  introduced the fusion products necessary to compute the
kinematic numerators in the so-called bi-scalar sector of YM, in which the
terms have polarization power up to two, and a common fixed factor
$\varepsilon_{1}{\cdot}\varepsilon_{n}$. In order to generalize to the full
NMHV sector, we need additional fusion rules that compute the
$\varepsilon_{n}{\cdot}p_{i}$ and $\varepsilon_{i}{\cdot}\varepsilon_{n}$
terms. As we restrict our discussion to terms of polarization power one and
two, we only need to consider currents whose tensor ranks are at most three. A
consideration based on mass dimension of the tensor currents indicates that we
can have at most two momenta in the tensor labels Chen:2019ywi . In hindsight,
we claim that the relevant tensor currents in the NMHV sector are as follows,

$\displaystyle J_{\varepsilon_{i}}(p)\,,\qquad J_{p_{i}}(p)\,,\qquad
J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}(p)\,,\qquad
J_{p_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}(p)\,,$ $\displaystyle
J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes p}(p)\,,\qquad
J_{p_{i}\otimes\varepsilon_{j}\otimes p}(p)\,.$ (23)

where we use $p$ without subscript to denote the momentum carried by the
tensor. When computing the ordered fusion products in the pre-numerator (7),
we only need those tensors with $i{<}j{<}k$. The tensor currents also depend
on the momentum of the polarization vectors in the tensor label. The
superscripts that label the vector and tensor types are suppressed in eq.
(3.1), which will be later worked out based on their behavior in fusion
products. The current $J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes p}(p)$
and $J_{p_{i}\otimes\varepsilon_{j}\otimes p}(p)$, where the last tensor label
agrees with its momentum, will play a special role and provide a simple
realization of the fusion products.

In the on-shell limit, we impose a Clifford-algebra like relation among
tensors,

$\displaystyle J_{a_{i}\otimes a_{j}\otimes a_{k}}+J_{a_{i}\otimes
a_{k}\otimes a_{j}}=2a_{j}{\cdot}a_{k}J_{a_{i}}\,.$ (24)

Note that the on-shell representations of vectors and tensors are agnostic to
types. One can use it to put tensor currents into a minimal basis. Different
basis choices may lead to different vector currents. As we will see later, on-
shell vector currents correspond to the kinematic numerators, while the
difference caused by the tensor basis choice is just part of the generalized
gauge freedom. This is our motivation of introducing the relation (24) to the
tensors.

###  3.2 Method to determine the fusion product

Here we try to construct the fusion products through an ansatz approach. More
precisely, we make an ansatz for every fusion product that contributes to eq.
(7). We then use eq. (15) to convert the resultant pre-numerators into the
kinematic numerator associated with a graph $\Gamma$. The ansatz can be solved
by matching with the amplitude on the support of maximal factorization,
namely, all the propagators of $\Gamma$ are on-shell,

$\displaystyle\mathcal{C}(\Gamma,n)\mathcal{N}(\Gamma,n)\Big{|}_{\begin{subarray}{c}\text{maximal}\\\
\text{factorization}\end{subarray}}=\text{Res}_{\Gamma}\mathcal{A}_{n}\,,$
(25)

where $\mathcal{A}_{n}$ is the full color-dressed amplitude and
$\text{Res}_{\Gamma}\mathcal{A}_{n}$ is the residue when all the propagators
of the graph $\Gamma$ are taken on-shell. We note that eq. (25) is a necessary
condition and sufficient only up to polarization power two. For higher
polarization powers, we need to consider such conditions for non-maximal
factorization channels due to higher powers of Mandelstam variables.

In practice, we impose additional constraints on the fusion products to reduce
the number of free parameters and simplify the solution. First, we require
that the fusion products relevant to eq. (7) do not reduce the tensor rank.
Since only vectors and rank-three tensors are present in the NMHV sector, this
means that the fusion between two vectors can lead to both vectors and rank-
three tensors, while the fusion between a rank-three tensor and a vector can
only give rank-three tensors. Then following a similar consideration in
Chen:2019ywi , we fix one particular fusion product as

$\displaystyle J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes p}(p)\star
J_{\varepsilon_{k}}(p_{k})$
$\displaystyle=\frac{1}{2}p^{2}J^{(1)}_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}(p+p_{k})\,,$
$\displaystyle J_{p_{i}\otimes\varepsilon_{j}\otimes p}(p)\star
J_{\varepsilon_{k}}(p_{k})$
$\displaystyle=\frac{1}{2}p^{2}J^{(1)}_{p_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}(p+p_{k})\,,$
(26)

We also assume that there is only one type of the tensor
$J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes p}$ and
$J_{p_{i}\otimes\varepsilon_{j}\otimes p}$ so that we drop the superscript.
Since this is the first time that the tensor
$J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}$ and
$J_{p_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}$ appear, we can freely
assign a type label to them, as have been done above. For other currents, the
tensor type is determined in an iterative way. At a given multiplicity $n$,
suppose we have two currents $J^{(w_{1})}_{U}$ and $J^{(w_{2})}_{U}$ in the
output of $J_{\varepsilon_{1}}\star J_{\varepsilon_{2}}\star\cdots\star
J_{\varepsilon_{n-2}}$ that have identical tensor labels, we say they are of
the same type, namely, $w_{1}{=}w_{2}$, if their fusion products with
$J_{\varepsilon_{n-1}}(p_{n-1})$ are identical when evaluated on-shell,

$\displaystyle\langle q|J_{U}^{(w_{1})}\star
J_{\varepsilon_{n-1}}(p_{n-1})|n\rangle\Big{|}_{q\,{\rm soft}}=\langle
q|J_{U}^{(w_{2})}\star J_{\varepsilon_{n-1}}(p_{n-1})|n\rangle\Big{|}_{q\,{\rm
soft}}\,.$ (27)

In other words, we ignore the possibility that the fusion products above may
differ off-shell. Such difference includes, for example, some tensors in the
results are of different types, and thus have different fusion products at
higher multiplicity. Under this simplification, new types of currents only
appear in the last stage of the consecutive fusion products of eq. (7) at each
multiplicity. This choice keeps the number of current types minimal and
reduces the complexity of our ansatz to a manageable level.

###  3.3 Constructing NMHV kinematic algebra

Now we start to construct the fusion products relevant for the pre-numerator
(7). The first type of currents we encounter is $J_{\varepsilon_{i}}(p_{i})$,
which corresponds to the on-shell particle $i$. As already discussed in
section 2, we omit the type label for these special currents. The fusion
products that will appear in the process (7) are of the form

$\displaystyle J^{(w)}_{a_{i}}(p)\star J_{\varepsilon_{i}}(p_{i})\,,\quad
J^{(w)}_{a_{i}\otimes a_{k}\otimes a_{l}}(p)\star
J_{\varepsilon_{i}}(p_{i})\,,$ (28)

where the subscripts $a_{i}$ follow those in eq. (3.1), and we will further
determine the tensor type $w$. At three points, we encounter only one fusion
product, and we make the following ansatz for it,

$\displaystyle J_{\varepsilon_{1}}(p_{1})\star J_{\varepsilon_{2}}(p_{2})$
$\displaystyle=\varepsilon_{2}{\cdot}p_{1}J_{\varepsilon_{1}}(p_{12})+x_{1}\varepsilon_{1}{\cdot}p_{2}J_{\varepsilon_{2}}^{(w_{0})}(p_{12})$
$\displaystyle\quad+x_{2}\varepsilon_{1}{\cdot}\varepsilon_{2}J_{p_{2}}^{(1)}(p_{12})-\frac{x_{0}}{2}J_{\varepsilon_{1}\otimes\varepsilon_{2}\otimes
p_{12}}(p_{12})\,,$ (29)

where $p_{12}{=}p_{1}{+}p_{2}$. It is an extension of the one in the bi-scalar
sector Chen:2019ywi . As we have preluded in eq. (3.2), we only use one type
of the tensor $J_{\varepsilon_{1}\otimes\varepsilon_{2}\otimes p_{12}}$ such
that we drop its type label. We expect that the $J_{\varepsilon_{1}}(p_{12})$
on the right hand side of eq. (3.3) is of the same type as the
$J_{\varepsilon_{i}}$’s on the left hand side, as inferred from the bi-scalar
sector algebra Chen:2019ywi . On the other hand, $J_{p_{2}}^{(1)}(p_{12})$ is
the first current of this type that appears in the algebra so that we can
freely assign its type. Thus $J_{\varepsilon_{2}}^{(w_{0})}(p_{12})$ is the
only current with undetermined type. The three-point numerator is then
computed by

$\displaystyle\mathcal{N}([1,2],3)=\mathcal{N}(1,2,3)-\mathcal{N}(2,1,3)=\langle
q|J_{\varepsilon_{1}}(p_{1})\star
J_{\varepsilon_{2}}(p_{2})|3\rangle\Big{|}_{q\,{\rm soft}}-(1\leftrightarrow
2)\,,$ (30)

where the vector and tensor currents are evaluated according to eq. (8) and
(9). In particular, the tensor vanishes due to momentum conservation and Dirac
equation,

$\displaystyle\langle q|J_{\varepsilon_{1}\otimes\varepsilon_{2}\otimes
p_{12}}|3\rangle\Big{|}_{q\,{\rm
soft}}=\bar{v}_{q}\not{\varepsilon}_{1}\not{\varepsilon}_{2}\not{p}_{12}u_{3}=-\bar{v}_{q}\not{\varepsilon}_{1}\not{\varepsilon}_{2}\not{p}_{3}u_{3}=0\,.$
(31)

At three points, the numerator (30) should match directly with the amplitude

$\displaystyle\mathcal{A}_{3}=f^{a_{1}a_{2}a_{3}}(\varepsilon_{2}{\cdot}p_{1}\varepsilon_{1}{\cdot}\varepsilon_{3}-\varepsilon_{1}{\cdot}p_{2}\varepsilon_{2}{\cdot}\varepsilon_{3}+\varepsilon_{1}{\cdot}\varepsilon_{2}p_{2}{\cdot}\varepsilon_{3})\,,$
(32)

which gives $x_{1}=0$ and $x_{2}=\frac{1}{2}$ while $x_{0}$ remains free. Thus
$J_{\varepsilon_{2}}^{(w_{0})}$ drops out so that we do not need to fix the
type.

At four points, the $s$ channel numerator is given by

$\displaystyle\mathcal{N}([[1,2],3],4)=\mathcal{N}(1,2,3,4)-\mathcal{N}(2,1,3,4)-\mathcal{N}(3,1,2,4)+\mathcal{N}(3,2,1,4)\,.$
(33)

We compute $\mathcal{N}(1,2,3,4)=\langle q|J_{\varepsilon_{1}}\star
J_{\varepsilon_{2}}\star J_{\varepsilon_{3}}|4\rangle\big{|}_{q\,{\rm soft}}$
from the algebra and obtain the other three by permuting indices. While
$J_{\varepsilon_{1}}\star J_{\varepsilon_{2}}$ is known from eq. (3.3), we
just need to compute the fusion with $J_{\varepsilon_{3}}$ from the right.
This leaves us to determine222We use the somewhat unusual convention:
$s_{ij}=p_{i}{\cdot}p_{j}$ and $s_{ij\ldots
k}=\frac{1}{2}(p_{i}{+}p_{j}{+}\ldots{+}p_{k})^{2}$.

vector-vector fusion: $\displaystyle J_{\varepsilon_{1}}(p_{12})\star
J_{\varepsilon_{3}}(p_{3})\,,\quad J^{(1)}_{p_{2}}(p_{12})\star
J_{\varepsilon_{3}}(p_{3})\,,$ tensor-vector fusion: $\displaystyle
J_{\varepsilon_{1}\otimes\varepsilon_{2}\otimes p_{12}}(p_{12})\star
J_{\varepsilon_{3}}(p_{3})=s_{12}J_{\varepsilon_{1}\otimes\varepsilon_{2}\otimes\varepsilon_{3}}^{(1)}(p_{123})\,,$
(34)

where $p_{123}{=}p_{1}{+}p_{2}{+}p_{3}$. The tensor-vector fusion here follows
immediately our assumption (3.2). We can write down the result of
$J_{\varepsilon_{1}}(p_{12})\star J_{\varepsilon_{3}}(p_{3})$ by simply
assuming that eq. (3.3) still holds when the momentum of $J_{\varepsilon_{1}}$
is off-shell. Thus we only need to make an ansatz for the last vector-vector
fusion,

$\displaystyle J^{(1)}_{p_{2}}(p_{12})\star J_{\varepsilon_{3}}(p_{3})$
$\displaystyle=x_{3}\varepsilon_{3}{\cdot}p_{12}J^{(w_{1})}_{p_{2}}(p_{123})+x_{4}s_{23}J^{(w_{2})}_{\varepsilon_{3}}(p_{123})+x_{5}\varepsilon_{3}{\cdot}p_{2}J^{(w_{3})}_{p_{3}}(p_{123})$
$\displaystyle\quad+x_{6}J_{p_{2}\otimes\varepsilon_{3}\otimes
p_{123}}(p_{123})\,.$ (35)

To solve the unknowns, we evaluate eq. (33) in the limit $s_{12}\rightarrow 0$
and then equate it to the residue of the four-point amplitude on the $s_{12}$
pole, following eq. (25). After a proper rearrangement using the relation
(24), the only tensor that appears in eq. (33) in the on-shell limit is
$J_{\varepsilon_{1}\otimes\varepsilon_{2}\otimes\varepsilon_{3}}$, which then
drops out when $s_{12}\rightarrow 0$ since it is multiplied by $s_{12}$. It
can be verified that this tensor term is in the kernel of the propagator
matrix. This will be the case for all tensor terms at higher multiplicities as
well, although the maximal factorization condition does not seem to directly
imply this property. More details will be discussed later in the paper. The
$s_{12}$ factorization channel constraint gives

$\displaystyle x_{3}=1,$ $\displaystyle x_{4}=x_{0}-1,$ $\displaystyle
x_{5}=1\,,$ (36)

while $x_{6}$ remains undetermined. We cannot determine the superscript
$w_{1,2,3}$ at four points, since in the on-shell limit these vector currents
are indistinguishable. Instead, they will be retrospectively determined at
five points according to their fusion products with
$J_{\varepsilon_{4}}(p_{4})$. Here $x_{0}$ is still a free parameter, and it
will remain free even at higher multiplicities. In the following, we choose
$x_{0}=\frac{1}{4}$. This choice keeps the fusion products and the form of
pre-numerator the simplest. We will leave the more general expressions with a
free $x_{0}$ to appendix B.

At five points, we need to set up ansatze for the fusion of
$J_{p_{2}}^{(w_{1})}$, $J_{\varepsilon_{3}}^{(w_{2})}$, $J_{p_{3}}^{(w_{3})}$
and $J_{\varepsilon_{1}\otimes\varepsilon_{2}\otimes\varepsilon_{3}}$ with
$J_{\varepsilon_{4}}$, and solve them by imposing the “maximal factorization”
condition (25). This gives $x_{6}=-\frac{1}{4}$ and fixes the tensor type

$\displaystyle J_{p_{2}}^{(w_{1})}=J_{p_{2}}^{(1)}\,,\quad
J_{p_{3}}^{(w_{3})}=J_{p_{3}}^{(1)}\,,\quad
J_{\varepsilon_{3}}^{(w_{2})}=J_{\varepsilon_{3}}^{(2)}\,,$ (37)

along with solutions to new parameters introduced at five points. Here the
identification is made under the assumption of eq. (27). We iterate this
process at each multiplicity and observe the closure of the algebra at seven
points. That is, assuming no new currents are generated, we verify that the
set of fusion products determined up to seven points computes the correct
numerator at higher multiplicities. We have checked explicitly that the
algebra gives the correct amplitude at eight points up to polarization power
two terms. At nine points, we have checked that the BCJ numerators generate
the correct maximal factorization for each cubic graph.

To summarize, there are in all 13 relevant vector and tensor currents,

vector: $\displaystyle J_{\varepsilon_{i}}\,,\quad
J_{\varepsilon_{i}}^{(2)}\,,\quad J_{p_{i}}^{(1)}\,,\quad J_{p_{i}}^{(2)}$
tensor: $\displaystyle J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes
p}\,,\quad J_{p_{i}\otimes\varepsilon_{j}\otimes p}\,,\quad
J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}^{(1)}\,,\quad
J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}^{(2)}\,,$
$\displaystyle
J_{p_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}^{(1)}\,,\ldots\,,J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}^{(5)}\,.$
(38)

The fusion products involving vector currents are

$\displaystyle J_{\varepsilon_{i}}(p)\star J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{\varepsilon_{i}}(p{+}p_{j})+\frac{1}{2}\varepsilon_{i}{\cdot}\varepsilon_{j}J^{(1)}_{p_{j}}(p{+}p_{j})-\frac{1}{8}J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes(p{+}p_{j})}(p{+}p_{j}),$
$\displaystyle J^{(1)}_{p_{i}}(p)\star J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ^{(1)}_{p_{i}}(p{+}p_{j})+\varepsilon_{j}{\cdot}p_{i}J^{(1)}_{p_{j}}(p{+}p_{j})-\frac{3}{4}s_{ij}J_{\varepsilon_{j}}^{(2)}(p{+}p_{j})$
$\displaystyle\quad-\frac{1}{4}J_{p_{i}\otimes\varepsilon_{j}\otimes(p+p_{j})}(p{+}p_{j}),$
$\displaystyle J_{\varepsilon_{i}}^{(2)}(p)\star J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{\varepsilon_{i}}^{(2)}(p{+}p_{j})-\varepsilon_{i}{\cdot}p_{j}J_{\varepsilon_{j}}^{(2)}(p{+}p_{j})+\frac{2}{3}\varepsilon_{i}{\cdot}\varepsilon_{j}J_{p_{j}}^{(2)}(p{+}p_{j}),$
$\displaystyle J_{p_{i}}^{(2)}(p)\star J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{p_{i}}^{(2)}(p{+}p_{j})+\varepsilon_{j}{\cdot}p_{i}J_{p_{j}}^{(2)}(p{+}p_{j}).$
(39)

The fusion rules involving $J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes
p}$ and $J_{p_{i}\otimes\varepsilon_{j}\otimes p}$ are assumed as in eq.
(3.2),

$\displaystyle J_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes p}(p)\star
J_{\varepsilon_{k}}(p_{k})$
$\displaystyle=\frac{1}{2}p^{2}J^{(1)}_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}(p{+}p_{k})\,,$
$\displaystyle J_{p_{i}\otimes\varepsilon_{j}\otimes p}(p)\star
J_{\varepsilon_{k}}(p_{k})$
$\displaystyle=\frac{1}{2}p^{2}J^{(1)}_{p_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}(p{+}p_{k})\,,$
(40)

The fusion rules involving
$J^{(1)}_{\varepsilon_{i}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}$
and $J^{(1)}_{p_{i}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}$ are
similar such that they can be combined into

$\displaystyle
J^{(1)}_{a\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}(p)\star
J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{a\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(2)}(p{+}p_{j})+\varepsilon_{j}{\cdot}aJ_{p_{j}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(4)}(p{+}p_{j})$
$\displaystyle\quad+\varepsilon_{i_{2}}{\cdot}aJ_{p_{i_{2}}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{j}}^{(3)}(p{+}p_{j})-\varepsilon_{i_{1}}{\cdot}aJ_{p_{i_{1}}\otimes\varepsilon_{i_{2}}\otimes\varepsilon_{j}}^{(3)}(p{+}p_{j})$
$\displaystyle\quad+\varepsilon_{i_{2}}{\cdot}pJ_{a\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{j}}^{(1)}(p{+}p_{j})-\varepsilon_{i_{1}}{\cdot}pJ_{a\otimes\varepsilon_{i_{2}}\otimes\varepsilon_{j}}^{(1)}(p{+}p_{j})\,,$
(41)

where the $a=\varepsilon_{i}$ or $p_{i}$. There are new types of tensors
generated on the right hand side. The tensor
$J^{(2)}_{\varepsilon_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}$ and
$J^{(2)}_{p_{i}\otimes\varepsilon_{j}\otimes\varepsilon_{k}}$ again have
similar fusion rules,

$\displaystyle
J^{(2)}_{a\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}(p)\star
J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{a\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(2)}(p{+}p_{j})+\varepsilon_{j}{\cdot}aJ_{p_{j}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(4)}(p{+}p_{j})$
$\displaystyle\quad+\varepsilon_{i_{2}}{\cdot}aJ_{p_{i_{2}}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{j}}^{(3)}(p{+}p_{j})-\varepsilon_{i_{1}}{\cdot}aJ_{p_{i_{1}}\otimes\varepsilon_{i_{2}}\otimes\varepsilon_{j}}^{(3)}(p{+}p_{j})\,,$
(42)

where the $a=\varepsilon_{i}$ or $p_{i}$. The fusion rule for
$J^{(3)}_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}$ gives
one new type of tensor,

$\displaystyle
J^{(3)}_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}(p)\star
J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(5)}(p{+}p_{j})+\varepsilon_{j}{\cdot}p_{r}J_{p_{j}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(4)}(p{+}p_{j})$
$\displaystyle\quad+\varepsilon_{i_{2}}{\cdot}p_{r}J_{p_{i_{2}}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{j}}^{(3)}(p{+}p_{j})-\varepsilon_{i_{1}}{\cdot}pJ_{p_{r}\otimes\varepsilon_{i_{2}}\otimes\varepsilon_{j}}^{(1)}(p{+}p_{j})\,,$
(43)

while $J^{(4)}_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}$ is
self-closed under the fusion with $J_{\varepsilon_{j}}$,

$\displaystyle
J^{(4)}_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}(p)\star
J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(4)}(p{+}p_{j})+\varepsilon_{j}{\cdot}p_{r}J_{p_{j}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(4)}(p{+}p_{j})\,.$
(44)

Finally, the fusion rule of
$J^{(5)}_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}$ does not
have new type of tensors,

$\displaystyle
J^{(5)}_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}(p)\star
J_{\varepsilon_{j}}(p_{j})$
$\displaystyle=\varepsilon_{j}{\cdot}pJ_{p_{r}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(5)}(p{+}p_{j})+\varepsilon_{j}{\cdot}p_{r}J_{p_{j}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{i_{2}}}^{(4)}(p{+}p_{j})$
$\displaystyle\quad+\varepsilon_{i_{2}}{\cdot}p_{r}J_{p_{i_{2}}\otimes\varepsilon_{i_{1}}\otimes\varepsilon_{j}}^{(3)}(p{+}p_{j})\,,$
(45)

which closes our algebra for the NMHV sector pre-numerators.

##  4 BCJ numerators from the fusion product

The final result for the NMHV sector algebra contains vector currents
$J_{a_{i}}$ and rank-three tensor currents $J_{a_{i_{1}}\otimes
a_{i_{2}}\otimes a_{i_{2}}}$, where $a_{i}$’s are momenta or polarization
vectors. Although we have different types of vector and tensor currents
participating in the fusion products, we consider their on-shell
representations to only depend on their Lorentz structure. Thus we evaluate
them in the on-shell limit as eq. (8) for vectors and eq. (9) for tensors. In
particular, for rank-three tensors, we have

$\displaystyle\langle q|J_{a_{i_{1}}\otimes a_{i_{2}}\otimes
a_{i_{2}}}|n\rangle\Big{|}_{q\,{\rm soft}}$
$\displaystyle=\bar{v}_{q}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\gamma_{\mu_{3}}u_{n}a^{\mu_{1}}_{1}a^{\mu_{2}}_{2}a^{\mu_{3}}_{3}\Big{|}_{q\,{\rm
soft}}\equiv\varepsilon_{n}(a_{1},a_{2},a_{3})\,.$ (46)

The tensor $\varepsilon_{n}(a_{1},a_{2},a_{3})$ satisfies

$\displaystyle\varepsilon_{n}(a_{1},a_{2},a_{3})+\varepsilon_{n}(a_{2},a_{1},a_{3})=2a_{1}{\cdot}a_{2}\,a_{3}{\cdot}\varepsilon_{n}\,,$
$\displaystyle\varepsilon_{n}(a_{1},a_{2},a_{3})+\varepsilon_{n}(a_{1},a_{3},a_{2})=2a_{2}{\cdot}a_{3}\,a_{1}{\cdot}\varepsilon_{n}\,,$
(47)

which is the on-shell version of the Clifford algebra relation (24). We can
use these relations to bring tensors into an irreducible basis. Once in this
irreducible basis, they must reside in the null space of the propagator matrix
since they are not physical states of YM, and thus they can be removed by
hand. However, even if the tensors are not in an irreducible basis, their
contribution must cancel out of the amplitude through the propagator matrix.
This implies that there are certain vector contributions that live in the null
space of the propagator matrix, and they can be exposed by changing the tensor
basis. Therefore, as shown in ref. Chen:2019ywi , the basis choice encodes a
certain subset of the generalized gauge freedom for numerators of physical
gluon amplitudes.

We will now study the output of the kinematic algebra constructed in the
previous section. Remarkably, the fusion products that we found lead to a
closed all-multiplicity formula for the NMHV pre-numerators,

$\displaystyle\mathcal{N}(1,2,\ldots,n)=\Big{(}1+\frac{1}{2}Q_{n}\Big{)}\mathcal{N}^{(1)}_{V}+\frac{1}{4}(1+Q_{n})\mathcal{N}_{T}^{(2)}+\mathcal{N}^{(2)}_{V}\,,$
(48)

where $\mathcal{N}^{(1)}_{V}$ and $\mathcal{N}_{T}^{(2)}$ are given by

$\displaystyle\mathcal{N}^{(1)}_{V}$
$\displaystyle=(\varepsilon_{1}{\cdot}\varepsilon_{n})\prod_{j=2}^{n-1}\varepsilon_{j}{\cdot}X_{j}\,,$
(49a) $\displaystyle\mathcal{N}_{T}^{(2)}$
$\displaystyle=-\frac{1}{4}\sum_{i=2}^{n-2}\sum_{\begin{subarray}{c}\ell,m=i\\\
m>\ell\end{subarray}}^{n-1}(-1)^{\ell-i}X^{2}_{i+1}\Bigg{[}\prod_{j\notin\\{1,i,i+1,\ldots,\ell,m,n\\}}\varepsilon_{j}{\cdot}X_{j}\Bigg{]}$
(49b)
$\displaystyle\quad\times(\varepsilon{\cdot}X)^{i,i+1,i+2\ldots,\ell-1}_{[\ldots[[i+2,i+3],i+4]\ldots,\ell+1]}\varepsilon_{n}(\varepsilon_{1},\varepsilon_{\ell},\varepsilon_{m})\,.$

Here $X_{j}=\sum_{i=1}^{j-1}p_{i}$ denotes the region momentum
Stieberger:2016lng ; Nandan:2016pya ; Chiodaroli:2017ngp , and we define

$\displaystyle(\varepsilon{\cdot}X)^{i,i+1,i+2\ldots,\ell-1}_{[\ldots[[i+2,i+3],i+4]\ldots,\ell+1]}=\varepsilon_{i,\mu}\varepsilon_{i+1,\nu}\varepsilon_{i+2,\rho}\ldots\varepsilon_{\ell-1,\lambda}X^{\mu}_{[\ldots[[i+2}X^{\nu\vphantom{\mu}}_{\ldots
i+3]}X^{\rho\vphantom{\mu}}_{i+4]}\ldots
X^{\lambda\vphantom{\mu}}_{\ldots\ell+1]},$ (50)

where the indices of $X$ are in the left-nested commutator. For example, the
following term appears at $n{=}7$ and beyond,

$\displaystyle(\varepsilon{\cdot}X)^{2,3,4}_{[[4,5],6]}$
$\displaystyle=\varepsilon_{2}{\cdot}X_{4}\varepsilon_{3}{\cdot}X_{5}\varepsilon_{4}{\cdot}X_{6}-\varepsilon_{2}{\cdot}X_{5}\varepsilon_{3}{\cdot}X_{4}\varepsilon_{4}{\cdot}X_{6}$
$\displaystyle\quad-\varepsilon_{2}{\cdot}X_{6}\varepsilon_{3}{\cdot}X_{4}\varepsilon_{4}{\cdot}X_{5}+\varepsilon_{2}{\cdot}X_{6}\varepsilon_{3}{\cdot}X_{5}\varepsilon_{4}{\cdot}X_{4}\,.$
(51)

We note that when $\ell{=}i$ the commutator does not exist and we set it to
unity. When $\ell{=}i{+}1$, it simply becomes $\varepsilon_{i}{\cdot}X_{i+2}$.
The superscripts in $\mathcal{N}_{V}$ and $\mathcal{N}_{T}$ denote the
polarization power. We note that $\mathcal{N}^{(1)}_{V}$ and
$\mathcal{N}^{(2)}_{T}$ are exactly the vector and tensor part of the bi-
scalar-YM sector numerators obtained in a previous work Chen:2019ywi , whereas
the notations therein are slightly different. The connection between the pre-
numerator here and the bi-scalar sector numerator in Chen:2019ywi  will be
discussed in appendix B. Next, $Q_{n}$ is a differential operator,

$\displaystyle Q_{n}$
$\displaystyle=\varepsilon_{1}{\cdot}\mathcal{D}_{2}{\cdot}\mathcal{D}_{3}\cdots\mathcal{D}_{n-1}{\cdot}\frac{\partial}{\partial\varepsilon_{1}}-1$
$\displaystyle=\sum_{r=1}^{n-2}\sum_{2\leqslant
i_{1}<i_{2}<\ldots<i_{r}\leqslant
n-1}(\varepsilon_{1}{\cdot}\varepsilon_{i_{1}})\Bigg{[}\prod_{\ell=1}^{r-1}(p_{i_{\ell}}{\cdot}\varepsilon_{i_{\ell+1}})\frac{\partial}{\partial\varepsilon_{i_{\ell+1}}{\cdot}p_{1}}\Bigg{]}\frac{\partial}{\partial\varepsilon_{i_{1}}{\cdot}p_{1}}\left(p_{i_{r}}{\cdot}\frac{\partial}{\partial\epsilon_{1}}\right)\,,$
(52)

where
$(\mathcal{D}_{i})_{\mu\nu}=\varepsilon_{i,\mu}p_{i,\nu}\frac{\partial}{\partial\varepsilon_{i}{\cdot}p_{1}}+\eta_{\mu\nu}$.
The action of $Q_{n}$ on $\mathcal{N}^{(1)}_{V}$ and $\mathcal{N}_{T}^{(2)}$
does not change the polarization power, and it can be evaluated directly as,

$\displaystyle\Big{(}1+\frac{1}{2}Q_{n}\Big{)}\mathcal{N}^{(1)}_{V}$
$\displaystyle=\frac{1}{2}\Bigg{[}(\varepsilon_{1}{\cdot}\varepsilon_{n})\prod_{j=2}^{n-1}\varepsilon_{j}{\cdot}X_{j}+\varepsilon_{1}{\cdot}G_{2,3,\ldots,n-1}{\cdot}\varepsilon_{n}\Bigg{]},$
(53) $\displaystyle(1+Q_{n})\mathcal{N}_{T}^{(2)}$
$\displaystyle=-\frac{1}{4}\sum_{i=2}^{n-2}\sum_{\begin{subarray}{c}\ell,m=i\\\
m>\ell\end{subarray}}^{n-1}(-1)^{\ell-i}X^{2}_{i+1}$ (54)
$\displaystyle\quad\times\varepsilon_{n}\big{(}\varepsilon_{1}{\cdot}G_{2,\ldots,i-1}{\cdot}G^{i,i+1,\ldots,\ell-1}_{[\ldots[i+2,i+3],\ldots,\ell+1]}{\cdot}G_{\ell+1,\ldots,\hat{m},\ldots,n-1},\varepsilon_{\ell},\varepsilon_{m}\big{)}\,.$

We first introduce
$(G^{i}_{j})_{\mu\nu}=\varepsilon_{i,\mu}p_{i,\nu}+\varepsilon_{i}{\cdot}X_{j}\,\eta_{\mu\nu}$
and $G_{i}\equiv G^{i}_{i}$. Then in the second line of eq. (54), we have the
shorthand notation

$\displaystyle G_{i,\ldots,k}=G_{i}{\cdot}G_{i+1}\cdots
G_{k-1}{\cdot}G_{k}\,,$ (55)

and a hatted index should be skipped in this product. We also have

$\displaystyle
G^{i,\ldots,\ell-1}_{[\ldots[i+2,i+3],\ldots,\ell+1]}=G^{i}_{[\ldots[{i+2}}{\cdot}G^{i+1}_{{i+3}]}\cdots
G^{\ell-1}_{{\ell+1}]}\,.$ (56)

The subscripts of $G$ are in the same left-nested commutator as in eq. (50).
Finally, $\mathcal{N}^{(2)}_{V}$ contains additional polarization power two
terms that are generated from the fusion products of the vector currents,

$\displaystyle\mathcal{N}^{(2)}_{V}$
$\displaystyle=-\frac{1}{4}\Bigg{[}\sum_{j=2}^{n-3}(\varepsilon_{1}{\cdot}G_{2,\ldots,j}{\cdot}p_{j+1})\sum_{\ell=j+1}^{n-2}(\varepsilon_{j+1}{\cdot}\tilde{G}_{j+2,\ldots,\ell}{\cdot}\varepsilon_{\ell+1})(p_{\ell+1}{\cdot}G_{\ell+2,\ldots,n-1}{\cdot}\varepsilon_{n})\Bigg{]}_{(\varepsilon{\cdot}\varepsilon)^{2}}$
$\displaystyle\quad-\frac{3}{8}\Bigg{[}\sum_{j=2}^{n-2}(\varepsilon_{1}{\cdot}G_{2,\ldots,j}{\cdot}p_{j+1})(\varepsilon_{j+1}{\cdot}\tilde{G}_{j+2,\ldots,n-1}{\cdot}\varepsilon_{n})\Bigg{]}_{(\varepsilon{\cdot}\varepsilon)^{2}}\,,$
(57)

where
$\tilde{G}_{i,\ldots,k}=\tilde{G}_{i}{\cdot}\tilde{G}_{i+1}\cdots\tilde{G}_{k-1}{\cdot}\tilde{G}_{k}$
and
$(\tilde{G}_{i})_{\mu\nu}=-p_{i,\mu}\varepsilon_{i,\nu}+\varepsilon_{i}{\cdot}X_{i}\,\eta_{\mu\nu}$.
Note that we only need to keep the polarization power two terms in the above
equation.

As discussed in section 2, the BCJ numerators obtained from the pre-numerators
satisfy the $S_{n-1}$ crossing symmetry. At this point, the numerators still
contain on-shell tensors, and the crossing symmetry is realized even with
these extra contributions. To obtain the conventional on-shell numerators, we
need to expand the tensors in a minimal basis and then remove them using BCJ
amplitude relations. This process will contribute additional terms due to the
relation (4). Such contributions resulted from the tensor basis choice are
part of the generalized gauge freedom Chen:2019ywi . More specifically, the
tensors will be removed by the (generalized) binary BCJ relations Chen:2019ywi
. Certain basis choices can retain the crossing symmetry in the conventional
on-shell numerators. One particularly convenient choice is to antisymmetrize
the labels of polarization vectors in the on-shell tensor (46). We will
discuss pure gauge terms in more details in section 6.

We have checked up to eight points that the tensors are indeed in the null
space of the propagator matrix when put into a basis. We have also checked
that the resulting amplitude is gauge invariant through the NMHV sector.
Namely, after the replacement $\varepsilon_{i}\rightarrow p_{i}$, the
polarization power zero and one terms are in the null space of the propagator
matrix. Finally, we present the explicit pre-numerators up to five points,

$\displaystyle\mathcal{N}(1,2,3,4)$
$\displaystyle=\frac{1}{2}\Big{(}\varepsilon_{1}{\cdot}\varepsilon_{4}\,\varepsilon_{2}{\cdot}X_{2}\,\varepsilon_{3}{\cdot}X_{3}+\varepsilon_{1}{\cdot}G_{2}{\cdot}G_{3}{\cdot}\varepsilon_{4}\Big{)}-\frac{1}{2}s_{12}\varepsilon_{4}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})$
(58) $\displaystyle\mathcal{N}(1,2,3,4,5)$
$\displaystyle=\frac{1}{2}\Big{[}(\varepsilon_{1}{\cdot}\varepsilon_{5})\varepsilon_{2}{\cdot}X_{2}\varepsilon_{3}{\cdot}X_{3}\varepsilon_{4}{\cdot}X_{4}+\varepsilon_{1}{\cdot}G_{2}{\cdot}G_{3}{\cdot}G_{4}{\cdot}\varepsilon_{5}\Big{]}$
$\displaystyle\quad-\frac{1}{8}s_{12}\Big{[}\varepsilon_{5}\big{(}\varepsilon_{1}{\cdot}G_{4},\varepsilon_{2},\varepsilon_{3}\big{)}+\varepsilon_{5}\big{(}\varepsilon_{1}{\cdot}G_{3},\varepsilon_{2},\varepsilon_{4}\big{)}-\varepsilon_{5}\big{(}\varepsilon_{1}{\cdot}G^{2}_{4},\varepsilon_{3},\varepsilon_{4}\big{)}\Big{]}$
$\displaystyle\quad-\frac{1}{8}s_{123}\varepsilon_{5}\big{(}\varepsilon_{1}{\cdot}G_{2},\varepsilon_{3},\varepsilon_{4}\big{)}-\left[\frac{1}{4}(\varepsilon_{1}{\cdot}G_{2}{\cdot}p_{3})(\varepsilon_{3}{\cdot}\varepsilon_{4})(p_{4}{\cdot}\varepsilon_{5})\right]_{(\varepsilon{\cdot}\varepsilon)^{2}}$
$\displaystyle\quad-\frac{3}{8}\left[\vphantom{\frac{1}{4}}(\varepsilon_{1}{\cdot}G_{2}{\cdot}p_{3})(\varepsilon_{3}{\cdot}\tilde{G}_{4}{\cdot}\varepsilon_{5})+(\varepsilon_{1}{\cdot}G_{2}{\cdot}G_{3}{\cdot}p_{4})(\varepsilon_{4}{\cdot}\varepsilon_{5})\right]_{(\varepsilon{\cdot}\varepsilon)^{2}}\,.$
(59)

In the five-point pre-numerator, the tensors can be expanded by, for example,

$\displaystyle\varepsilon_{5}\big{(}\varepsilon_{1}{\cdot}G_{4},\varepsilon_{2},\varepsilon_{3}\big{)}=\varepsilon_{1}{\cdot}\varepsilon_{4}\varepsilon_{5}(p_{4},\varepsilon_{2},\varepsilon_{3})+\varepsilon_{4}{\cdot}X_{4}\varepsilon_{5}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})\,,$
$\displaystyle\varepsilon_{5}\big{(}\varepsilon_{1}{\cdot}G^{2}_{4},\varepsilon_{3},\varepsilon_{4}\big{)}=\varepsilon_{1}{\cdot}\varepsilon_{2}\varepsilon_{5}(p_{2},\varepsilon_{3},\varepsilon_{4})+\varepsilon_{2}{\cdot}X_{4}\varepsilon_{5}(\varepsilon_{1},\varepsilon_{3},\varepsilon_{4})\,.$
(60)

The $(\varepsilon{\cdot}\varepsilon)^{2}$ contribution from the last two
brackets are given by

$\displaystyle-\left[\frac{1}{4}(\varepsilon_{1}{\cdot}G_{2}{\cdot}p_{3})(\varepsilon_{3}{\cdot}\varepsilon_{4})(p_{4}{\cdot}\varepsilon_{5})+\frac{3}{8}(\varepsilon_{1}{\cdot}G_{2}{\cdot}p_{3})(\varepsilon_{3}{\cdot}\tilde{G}_{4}{\cdot}\varepsilon_{5})+\frac{3}{8}(\varepsilon_{1}{\cdot}G_{2}{\cdot}G_{3}{\cdot}p_{4})(\varepsilon_{4}{\cdot}\varepsilon_{5})\right]_{(\varepsilon{\cdot}\varepsilon)^{2}}$
$\displaystyle=-\frac{1}{4}s_{23}\varepsilon_{1}{\cdot}\varepsilon_{2}\Big{[}\varepsilon_{3}{\cdot}\varepsilon_{4}\varepsilon_{5}{\cdot}p_{4}+\frac{3}{2}(\varepsilon_{3}{\cdot}\varepsilon_{5}\varepsilon_{4}{\cdot}X_{4}-\varepsilon_{4}{\cdot}\varepsilon_{5}\varepsilon_{3}{\cdot}p_{4})\Big{]}-\frac{3}{8}s_{24}\varepsilon_{1}{\cdot}\varepsilon_{2}\varepsilon_{4}{\cdot}\varepsilon_{5}\varepsilon_{3}{\cdot}X_{3}$
$\displaystyle\quad-\frac{3}{8}s_{34}\varepsilon_{4}{\cdot}\varepsilon_{5}\Big{[}\varepsilon_{1}{\cdot}\varepsilon_{2}\varepsilon_{3}{\cdot}p_{2}+\varepsilon_{1}{\cdot}\varepsilon_{3}\varepsilon_{2}{\cdot}X_{2}\Big{]}\,.$
(61)

##  5 Group algebra actions for numerators

In this section, we briefly review the mathematical topic called group algebra
(or group ring), which provides a convenient formal language for operating on
numerators and polynomial building blocks for kinematics. As we will see in
the next section, this language provides an essential tool for studying
generalized gauge freedom in local BCJ numerators.

A _group algebra_ $Z[S]$ (see e.g. refs. algebra ) is defined over a
commutative ring $Z$ and a group $S$ with two binary operations (addition and
multiplication). In a representation space, the action of an group algebra
element $\mathbb{V}\in Z[S]$ is given by

$\displaystyle\mathbb{V}=\sum_{\sigma\in S}a^{(\sigma)}\mathbb{P}_{\sigma}\,,$
(62)

where $a^{(\sigma)}\in Z$ and $\mathbb{P}_{\sigma}$ is the action of the group
element $\sigma$. The identity $\mathbb{I}$ of the group $S$ is also the
identity of the group algebra $Z[S]$. The sum and product are inherited from
the ring $Z$ and group $S$,

$\displaystyle\mathbb{V}+\mathbb{U}=\sum_{\sigma\in
S}(a^{(\sigma)}+b^{(\sigma)})\mathbb{P}_{\sigma}\,,$ (63a)
$\displaystyle\mathbb{V}\mathbb{U}=\sum_{\sigma,\gamma\in
S}a^{(\sigma)}b^{(\gamma)}\mathbb{P}_{\sigma}\mathbb{P}_{\gamma}=\sum_{\sigma,\gamma\in
S}a^{(\sigma\gamma^{-1})}b^{(\gamma)}\mathbb{P}_{\sigma}\,.$ (63b)

The multiplication is associative and distributive over the sum. The
multiplication by a scalar $c\in Z$ can be defined as
$c\mathbb{V}\equiv\sum_{\sigma\in S}(ca^{(\sigma)})\mathbb{P}_{\sigma}$ and
thus is compatible with the multiplication (63b). If there exist two nonzero
elements $\mathbb{V},\mathbb{U}\in Z[S]$ such that $\mathbb{VU}=0$, then
$\mathbb{V}$ is a _left zero divisor_ of $\mathbb{U}$ (and $\mathbb{U}$ is a
_right zero divisor_ of $\mathbb{V}$). We will focus on the cases that $S$ is
a permutation group and $Z$ the integer ring. The action of a permutation
group element $\sigma\in S_{n}$ on a function $f=f(1,2,3,\ldots,n)$ is defined
as

$\displaystyle\mathbb{P}_{\sigma}\circ
f(1,2,3,\cdots,n)=f(\sigma_{1},\sigma_{2},\sigma_{3},\cdots,\sigma_{n})\,,$
(64)

and products of functions obey
$\mathbb{P}_{\sigma}\circ(fg)=(\mathbb{P}_{\sigma}\circ
f)(\mathbb{P}_{\sigma}\circ g)$.

We now give some group algebra elements that will be useful for later
discussions. First, due to the crossing symmetry, we can obtain the generic
DDM basis numerator $N(1,\beta_{2},\ldots,\beta_{n-1},n)$ by the action of
$\mathbb{P}_{\beta}$,

$\displaystyle N(1,\beta_{2},\ldots,\beta_{n-1},n)=\mathbb{P}_{\beta}\circ
N(1,2,\ldots,n-1,n)\,,\quad~{}~{}\beta\in S_{n-2}\,.$ (65)

Next, nested commutators can be generated by the action of333We use the cycle
decomposition for the permutations appearing in the group action. For example,
the cycle $(a_{1}a_{2}\ldots a_{m})$ corresponds to the permutation
$a_{1}\rightarrow a_{2},a_{2}\rightarrow a_{3},\ldots,a_{m}\rightarrow a_{1}$.

$\displaystyle\mathbb{L}(i_{1},i_{2},\ldots,i_{m})\equiv\Big{[}\mathbb{I}-\mathbb{P}_{(i_{2}i_{1})}\Big{]}\Big{[}\mathbb{I}-\mathbb{P}_{(i_{3}i_{2}i_{1})}\Big{]}\cdots\Big{[}\mathbb{I}-\mathbb{P}_{(i_{m}\cdots
i_{2}i_{1})}\Big{]}\,.$ (66)

Indeed, one can check with explicit computations that

$\displaystyle\mathbb{L}(i_{1},i_{2},\ldots,i_{m})\circ
f(i_{1},i_{2},\ldots,i_{m})$
$\displaystyle=f([\ldots[[i_{1},i_{2}],i_{3}]\ldots,i_{m}])$ (67)
$\displaystyle=f(i_{1},i_{2},i_{3}\cdots,i_{m})-f(i_{2},i_{1},i_{3}\cdots,i_{m})$
$\displaystyle\quad-f(i_{3},i_{1},i_{2}\cdots,i_{m})+f(i_{3},i_{2},i_{1}\cdots,i_{m})+\cdots\,.$

Therefore, we can use it to construct BCJ numerators from pre-numerators,

$\displaystyle
N(1,2,\ldots,n-1,n)=\mathbb{L}(1,2\ldots,n-1)\circ\mathcal{N}(1,2,\ldots,n-1,n)\,.$
(68)

We note that the operator $\frac{1}{m}\mathbb{L}(i_{1},i_{2},\cdots,i_{m})$ is
satisfies

$\displaystyle\left[\frac{1}{m}\mathbb{L}(i_{1},i_{2},\cdots,i_{m})\right]^{2}=\frac{1}{m}\mathbb{L}(i_{1},i_{2},\cdots,i_{m})\,,$
(69)

which implies that it is a projector. It is also known as a Lie idempotent in
the topic of free Lie algebras reutenauer2003free . Using the unit operator
$\mathbb{I}$, we can construct a mutually orthogonal operator

$\displaystyle\left[\mathbb{I}-\frac{1}{m}\mathbb{L}(i_{1},\ldots,i_{m})\right]\mathbb{L}(i_{1},\ldots,i_{m})=\mathbb{L}(i_{1},\ldots,i_{m})\left[\mathbb{I}-\frac{1}{m}\mathbb{L}(i_{1},\ldots,i_{m})\right]=0\,.$
(70)

From this orthogonality, it is clear that the operator
$\mathbb{I}-\frac{1}{m}\mathbb{L}(i_{1},\ldots,i_{m})$ generates the principal
ideals of both the left and right zero divisors of
$\mathbb{L}(i_{1},\ldots,i_{m})$. And, conversely,
$\mathbb{L}(i_{1},\ldots,i_{m})$ generates both the left and right zero
divisors of $\mathbb{I}-\frac{1}{m}\mathbb{L}(i_{1},\ldots,i_{m})$.

Another interesting group algebra element is

$\displaystyle\mathbb{B}(i_{1},i_{2},\ldots,i_{m})\equiv\Big{[}\mathbb{I}-\mathbb{P}_{(i_{1}i_{2}\ldots
i_{m})}\Big{]}\Big{[}\mathbb{I}-\mathbb{P}_{(i_{1}i_{2}\ldots
i_{m-1})}\Big{]}\cdots\Big{[}\mathbb{I}-\mathbb{P}_{(i_{1}i_{2})}\Big{]}\,,$
(71)

which can be used to generate the so-called binary BCJ relations Chen:2019ywi
. These BCJ amplitude relations are equivalent to other versions of the BCJ
relations, but they are nicely organized into $2^{m}$ terms, as is clear from
the iterated product of two “binary” terms. In ref. Chen:2019ywi , the binary
BCJ relation was expressed as the following sum

$\displaystyle\mathcal{B}_{[i,\ldots,n-1]}(1,2,\ldots,n)\equiv\sum_{j=i}^{n-1}\sum_{\begin{subarray}{c}\gamma\in\\{j-1,\ldots,i\\}\\\
\shuffle\\{j+1,\ldots,n-1\\}\end{subarray}}\sum_{\begin{subarray}{c}\sigma\in\\{2,\ldots,i-1\\}\\\
\shuffle\\{j,\gamma\\}\end{subarray}}\\!\\!\\!(-1)^{j-i}p_{j}{\cdot}X_{j}(\sigma)A(1,\sigma,n)\cong
0\,,$ (72)

where $X_{i}(\sigma)=\sum_{\sigma_{k}<i}p_{\sigma_{k}}$ is the region momenta.
The “$\cong 0$” means that the expression should be equal to zero for physical
amplitudes evaluated for on-shell kinematics. Here we only care about the
abstract functional form and hence “BCJ relation” is used to denote a
polynomial in kinematic variables and $A(1,\sigma,n)$. We mostly drop the
“$\cong 0$” in the remainder of the paper. Using the group algebra the above
binary BCJ relation can be simplified to

$\displaystyle\mathcal{B}_{[i,\ldots,n-1]}(1,2,\ldots,n)=\mathbb{B}(i,\ldots,n-1)\circ\sum_{\begin{subarray}{c}\sigma\in\\{2,\ldots,i-1\\}\\\
\shuffle\\{i,\ldots,n-1\\}\end{subarray}}p_{i}{\cdot}X_{i}(\sigma)A(1,\sigma,n)\,.$
(73)

Relations with more general arguments can be obtained by acting by an
arbitrary permutation $\beta\in S_{n-2}$ onto the above relation,

$\displaystyle\mathcal{B}_{[\beta_{i},\ldots,\beta_{n-1}]}(1,\beta_{2},\ldots,\beta_{n-1},n)=\mathbb{P}_{\beta}\circ\mathcal{B}_{[i,\ldots,n-1]}(1,2,\ldots,n)\,.$
(74)

In appendix A, we will give an independent basis for the binary BCJ relations
over local coefficients that are linear in $s_{ij}$. Binary BCJ relations are
important for parametrizing all the local pure gauge freedom in the BCJ
numerators of the NMHV sector, as we will get to shortly.

We can expand the product in $\mathbb{B}(i_{1},i_{2},\ldots,i_{m})$ and arrive
at a form that follows the definition (62),

$\displaystyle\mathbb{B}(i_{1},i_{2},\ldots,i_{m})=\sum_{a=1}^{m}\sum_{\rho\in\\{i_{a},\\{i_{a-1},\ldots,i_{1}\\}\shuffle\\{i_{a+1}\ldots
i_{m}\\}\\}}(-1)^{a-1}\mathbb{P}_{\rho}\,.$ (75)

Interestingly, the operator $\mathbb{L}(i_{1},i_{2},\ldots,i_{m})$ has an
identical expansion after making the replacement
$\mathbb{P}_{\rho}\rightarrow\mathbb{P}_{\rho^{-1}}$ in the above equation,

$\displaystyle\mathbb{L}(i_{1},i_{2},\ldots,i_{m})=\sum_{a=1}^{m}\sum_{\rho\in\\{i_{a},\\{i_{a-1},\ldots,i_{1}\\}\shuffle\\{i_{a+1}\ldots
i_{m}\\}\\}}(-1)^{a-1}\mathbb{P}_{\rho^{-1}}\,.$ (76)

Here, $\rho^{-1}$ is the inverse of $\rho$ such that $\rho^{-1}_{i}=a$ if
$\rho_{a}=i$. For convenience, we abbreviate the two equations as

$\displaystyle\mathbb{B}(i_{1},i_{2},\ldots,i_{m})=\hskip-14.22636pt\sum_{\sigma\in
K[i_{1},\ldots,i_{m}]}\hskip-8.5359pt\mu^{(\sigma)}\mathbb{P}_{\sigma}\,,$
$\displaystyle\hskip
14.22636pt\mathbb{L}(i_{1},i_{2},\ldots,i_{m})=\hskip-14.22636pt\sum_{\sigma\in
K[i_{1},\ldots,i_{m}]}\hskip-8.5359pt\mu^{(\sigma)}\mathbb{P}_{\sigma^{-1}}\,,$
(77)

where $K[i_{1},\ldots,i_{m}]$ is the set of relevant permutations and
$\mu^{(\sigma)}=\pm 1$ is the coefficient associated to each permutation. As
we will see later, the connection between $\mathbb{B}$ and $\mathbb{L}$ is the
key to understand the pure gauge terms in a BCJ numerator.

Generic numerators consist of a linear combination of _kinematic monomials_
$\mathsf{m}$, defined as a product of $\varepsilon{\cdot}\varepsilon$,
$\varepsilon{\cdot}p$ and $p{\cdot}p$. In addition, our algebraic construction
detailed in section 3 also gives tensors in the numerators. Without losing
generality, we use tensors in the basis that the $p$ labels and $\varepsilon$
labels are antisymmetrized respectively,

$\displaystyle\hat{\varepsilon}_{n}(p_{i_{1}},{\ldots},p_{i_{k}},\varepsilon_{j_{k+1}},{\ldots},\varepsilon_{j_{2m+1}})\equiv\varepsilon_{n}(p_{[i_{1}},{\ldots},p_{i_{k}]},\varepsilon_{[j_{k+1}},{\ldots},\varepsilon_{j_{2m+1}]})$
(78)

as additional building blocks of the kinematic monomials. For example, in the
NMHV sector we have
$\hat{\varepsilon}_{n}(p_{i_{1}},\varepsilon_{i_{2}},\varepsilon_{i_{3}})$ and
$\hat{\varepsilon}_{n}(\varepsilon_{i_{1}},\varepsilon_{i_{2}},\varepsilon_{i_{3}})$.

We classify monomials by the orbit of the $S_{n-2}$ group action. Two
monomials belong to the same _monomial equivalence class_ (MEC) if they are
related by an $S_{n-2}$ permutation.444MECs can be classified by the
topologies of certain diagrams, a terminology used in ref. Chen:2019ywi .
Therefore, each MEC forms a representation space of the permutation groups.
For example, the following two monomials belong to the same MEC,

$\displaystyle
p_{3}{\cdot}p_{7}\varepsilon_{3}{\cdot}\varepsilon_{4}\varepsilon_{5}{\cdot}p_{4}\hat{\varepsilon}_{8}(\varepsilon_{1},\varepsilon_{6},\varepsilon_{7})\quad\text{and}\quad
p_{6}{\cdot}p_{7}\varepsilon_{6}{\cdot}\varepsilon_{4}\varepsilon_{5}{\cdot}p_{4}\hat{\varepsilon}_{8}(\varepsilon_{1},\varepsilon_{3},\varepsilon_{7})\,,$
(79)

since they are related by the permutation $3\leftrightarrow 6$.

More specifically, we now consider the independent building blocks for the
kinematic monomials in $S_{n-2}$ crossing symmetric numerators. It is obvious
that all the $\varepsilon_{i}{\cdot}\varepsilon_{j}$ are independent as there
are no relations among them. As to $\varepsilon_{i}{\cdot}p_{j}$, momentum
conservation and transversality together allow us to take
$\varepsilon_{i}{\cdot}p_{j}$ with $j\neq i$ and $n$ as independent variables.
For $\varepsilon_{n}{\cdot}p_{j}$, we instead require
$2{\leqslant}j{\leqslant}n-1$ to accommodate the $S_{n-2}$ crossing symmetry.
We choose those Mandelstam variables $p_{i}{\cdot}p_{j}$ that do not contain
leg $p_{n}$ as independent variables. This leaves us with
$\frac{(n-1)(n-2)}{2}$ Mandelstam variables. We do not use the on-shell
condition $p_{n}^{2}=0$ to reduce the number to $\frac{n(n-3)}{2}$ since this
minimal basis does not respect the $S_{n-2}$ crossing symmetry. Finally, we
simply require that $p_{n}$ does not appear in the arguments of the tensor
$\hat{\varepsilon}_{n}$.

Numerators are just polynomials composed of a linear combination of the
kinematic monomials. Then, in general, we can expand the numerators as

$\displaystyle
N(1,2,\ldots,n-1,n)=\sum_{\mathsf{C}\in\text{MECs}}\sum_{\sigma\in
S_{n-2}}a^{(\sigma)}_{\mathsf{C}}\mathbb{P}_{\sigma}\circ\mathsf{m}_{\mathsf{C}}\,.$
(80)

Here $\mathsf{m}_{\mathsf{C}}$ is a representative monomial of the MEC
$\mathsf{C}$, and the group algebra element $\sum_{\sigma\in
S_{n-2}}a^{(\sigma)}_{\mathsf{C}}\mathbb{P}_{\sigma}$ generates a polynomial
that is solely composed of the monomials in $\mathsf{C}$.

##  6 Pure gauge degrees of freedom for the NMHV sector

Recall that a pure gauge numerator, called $N^{\text{gauge}}$ in eq. (20),
resides in the null space of the propagator matrix and thus does not
contribute to the amplitude. Equivalently, the pure gauge terms should give
zero when fed into some version of a double copy formula. They must thus
satisfy the local constraint

$\displaystyle\sum_{\beta\in
S_{n-2}}A(1,\beta_{2},\ldots,\beta_{n-1},n)N^{\text{gauge}}(1,\beta_{2},\ldots,\beta_{n-1},n)=\sum_{i}c_{i}\mathcal{B}_{i}\cong
0\,,$ (81)

where $\mathcal{B}_{i}$ are a set of independent BCJ relations and $c_{i}$ are
some local kinematic coefficients. If we treat the partial amplitudes on the
left hand side as formal objects, then the above double copy formula
indirectly suggests a local parametrization of the pure gauge terms using the
BCJ relations $\mathcal{B}_{i}$.

This brings us to the central question that we will answer in this and the
next section: How do we classify and construct the pure gauge terms that can
reside in local numerators that respect color-kinematics duality?
Specifically, can we count the number of free parameters corresponding to pure
gauge degrees of freedom, and how does the count change when we impose
stronger manifest crossing symmetries on the numerators? A keyword of this
section is “local”, as we are only interested in studying numerators that are
polynomials of the kinematic Lorentz invariants, and thus contain no physical
or spurious poles.

Due to our objective of finding local numerators, we need to distinguish
between independent BCJ relations belonging to the vector space over:
polynomials in $s_{ij}$ versus rational functions in $s_{ij}$. Recall that in
ref. Bern:2008qj , a complete solution to all BCJ relations was constructed,
which gave a basis of $(n-3)!$ partial amplitudes. Since the Kleiss-Kuijf
basis of partial amplitudes is of size $(n-2)!$ Kleiss:1988ne , one is
justified to say that there exist $(n-3)(n-3)!=(n-2)!-(n-3)!$ independent BCJ
relations. More precisely the BCJ relations live in a
$(n-3)(n-3)!$-dimensional vector space over the rational functions in
$s_{ij}$. This vector space is spanned by the explicit solution in ref.
Bern:2008qj , or alternatively, by the permutations of the simplest such
relation at each multiplicity. The latter were dubbed the “fundamental BCJ
relations” Feng:2010my , because of their ability to span the full vector
space.

With eq. (81) in mind, it is important that we consider all BCJ relations that
have local coefficients, that is

$\sum_{\sigma\in
S_{2}}a_{\sigma}\,A(1,\sigma,n)=0\,,~{}~{}~{}~{}~{}a_{\sigma}\in{\rm
Poly}[s_{ij}]$ (82)

While it is still true that this set of local BCJ relations live in the
$(n-3)(n-3)!$-dimensional vector space over the rational functions in
$s_{ij}$, it is not a useful characterization for identifying them. Instead we
need to consider a basis of local BCJ relations that spans a vector space over
the polynomials in $s_{ij}$. This vector space has a larger dimension than the
corresponding vector space over the rational functions in $s_{ij}$. In
appendix A, we argue that the binary BCJ relations give such a basis for all
linear-in-$s_{ij}$ BCJ relations, and its dimension is
$(n-2)!\sum_{i=2}^{n-2}(n-i)^{-1}$ (the generalized Stirling numbers). For BCJ
relations with polynomial coefficients of higher degrees, there is as of yet
no established basis for a vector space over the polynomials.

The binary BCJ relations are important because they provide exactly the basis
needed for the local expansion (81) in the NMHV sector. With these at our
disposal, we can proceed as follows to simplify the construction of pure gauge
numerators: pure gauge terms are split according to the independent MEC
sectors, and then kinematic polynomials are constructed such that they are
(1) composed only of monomials from the MEC;  (2) are invariant under the
$\mathbb{L}$ operator action defined in eq. (67).  We will show that when
those $\mathbb{L}$-invariant polynomials are appropriately assembled and feed
into eq. (81) they will reassemble into a superposition of the binary BCJ
relations, and hence they correspond to pure gauge terms in the numerator, and
will not contribute to the amplitude.

This section will focus on numerators exhibiting $S_{n-2}$ crossing symmetry,
and subsequently we discuss imposing manifest $S_{n-1}$ and $S_{n}$ crossing
symmetry in the next section. The case study is the NMHV sector of YM, where
we have good control over the building blocks, and we will only briefly touch
on the N2MHV in the next section. We remind the reader of the important result
from ref. Chen:2019ywi : the MHV sector does not contain any local generalized
gauge freedom, and the local numerators of this sector are thus unique.

###  6.1 Pure gauge terms and group algebra invariants

We start with proving that the following sum contains only pure gauge degrees
of freedom in the sense of eq. (81),

$\displaystyle N_{[i,\ldots,n-1]}^{\text{gauge}}(1,2,\ldots,n-1,n)$
$\displaystyle=\sum_{\begin{subarray}{c}\sigma^{-1}\in\\{2,\ldots,i-1\\}\\\
\shuffle\\{i,\ldots,n-1\\}\end{subarray}}p_{\sigma_{i}}{\cdot}X_{\sigma_{i}}(\mathbb{I})\,\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}(1,\sigma,n)\,,$
(83)

where $2\leqslant i\leqslant n-2$ and
$X_{\sigma_{i}}(\mathbb{I})=\sum_{j<\sigma_{i}}p_{j}$. The summation is over
all the permutations whose inverse are in the shuffle
$\\{2,\ldots,i-1\\}\shuffle\\{i,\ldots,n-1\\}$. For example, at $n=5$ eq. (83)
gives

$\displaystyle N^{\text{gauge}}_{[2,3,4]}(1,2,3,4,5)$
$\displaystyle=p_{2}{\cdot}p_{1}\,\mathcal{I}_{\mathbb{L}(2,3,4)}(1,2,3,4,5)\,,$
(84a) $\displaystyle N^{\text{gauge}}_{[3,4]}(1,2,3,4,5)$
$\displaystyle=p_{3}{\cdot}(p_{1}+p_{2})\,\mathcal{I}_{\mathbb{L}(3,4)}(1,2,3,4,5)+p_{2}{\cdot}p_{1}\,\mathcal{I}_{\mathbb{L}(3,4)}(1,3,2,4,5)$
$\displaystyle\quad+p_{2}{\cdot}p_{1}\,\mathcal{I}_{\mathbb{L}(3,4)}(1,4,2,3,5)\,.$
(84b)

In particular, the second equation comes from summing over

$\displaystyle\sigma^{-1}\in\\{2\\}\shuffle\\{3,4\\}=\big{\\{}\\{2,3,4\\},\\{3,2,4\\},\\{3,4,2\\}\big{\\}}\,,$
(85)

which implies that
$\sigma\in\big{\\{}\\{2,3,4\\},\\{3,2,4\\},\\{4,2,3\\}\big{\\}}$. The function
$\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}(1,\sigma,n)$ is obtained from a
permutation

$\displaystyle\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}(1,\sigma,n)=\mathbb{P}_{\sigma}\circ\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}\,,$
(86)

where the canonically labeled object $\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}$
is an eigenfunction, or group algebra invariant, of the $\mathbb{L}$ operator.
Specifically, we take it to be a kinematic polynomial multilinear in the
polarizations of the external states and of mass dimension $(n{-}4)$, and it
solves the eigenfunction equation

$\displaystyle\frac{1}{n-i}\mathbb{L}(i,\ldots,n-1)\circ\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}=\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}\,.$
(87)

Using the orthogonality relation (70), it is straightforward to see that any
solution to eq. (87) can be written as

$\displaystyle\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}=\mathbb{L}(i,\ldots,n-1)\circ\mathsf{m}\,,$
(88)

where $\mathsf{m}$ is a generic kinematic monomial, which we take to be of
mass dimension $(n{-}4)$ and multilinear in the polarizations.

Let us show that the numerator given by eq. (83), with the general solution
(88), indeed satisfies the pure gauge condition (81). We start with the
definition (83) and obtain the full DDM basis of numerators through
permutations,

$\displaystyle\sum_{\beta\in
S_{n-2}}A(1,\beta,n)\,N^{\text{gauge}}_{[\beta_{i},\ldots,\beta_{n-1}]}(1,\beta,n)$
$\displaystyle=\sum_{\beta\in S_{n-2}}A(1,\beta,n)\,\mathbb{P}_{\beta}\circ
N^{\text{gauge}}_{[i,\ldots,n-1]}(1,2,\ldots,n-1,n)$
$\displaystyle=\sum_{\beta\in
S_{n-2}}\sum_{\begin{subarray}{c}\sigma^{-1}\in\\{2,\ldots,i-1\\}\\\
\shuffle\\{i,\ldots,n-1\\}\end{subarray}}A(1,\beta,n)\,p_{\beta\sigma_{i}}{\cdot}X_{\beta\sigma_{i}}(\beta)\,\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}(1,\beta\sigma,n)\,.$
(89)

Using the form of $\mathbb{L}$ given in eq. (77), we have

$\displaystyle\mathcal{I}_{\mathbb{L}(i,\ldots,n-1)}(1,\beta\sigma,n)$
$\displaystyle=\mathbb{P}_{\beta\sigma}\circ\mathbb{L}(i,\ldots,n-1)\circ\mathsf{m}=\sum_{\gamma\in
K[i,\ldots,n-1]}\mu^{(\gamma)}\,\mathbb{P}_{\beta\sigma\gamma^{-1}}\circ\mathsf{m}\,.$
(90)

We plug it back into eq. (6.1), and redefine dummy variables as
$\beta\sigma\gamma^{-1}\rightarrow\beta$, then we obtain

$\displaystyle\sum_{\beta\in
S_{n-2}}A(1,\beta,n)\,N^{\text{gauge}}_{[\beta_{i},\ldots,\beta_{n-1}]}(1,\beta,n)$
$\displaystyle=\sum_{\gamma\in K[i,\ldots,n-1]}\sum_{\beta\in
S_{n-2}}\sum_{\begin{subarray}{c}\sigma^{-1}\in\\{2,\ldots,i-1\\}\\\
\shuffle\\{i,\ldots,n-1\\}\end{subarray}}A(1,\beta\gamma\sigma^{-1},n)\,p_{\beta\gamma_{i}}{\cdot}X_{\beta\gamma_{i}}(\beta\gamma\sigma^{-1})\,\mu^{(\gamma)}\,\mathbb{P}_{\beta}\circ\mathsf{m}$
$\displaystyle=\sum_{\beta\in
S_{n-2}}\Big{[}\mathbb{P}_{\beta}\circ\mathbb{B}(i,\ldots,n-1)\,\circ\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\sigma\in\\{2,\ldots,i-1\\}\\\
\shuffle\\{i,\ldots,n-1\\}\end{subarray}}p_{i}{\cdot}X_{i}(\sigma)A(1,\sigma,n)\Big{]}\mathbb{P}_{\beta}\circ\mathsf{m}$
$\displaystyle=\sum_{\beta\in
S_{n-2}}\big{[}\mathbb{P}_{\beta}\circ\mathcal{B}_{[i,\ldots,n-1]}(1,2,\ldots,n-1,n)\big{]}\,(\mathbb{P}_{\beta}\circ\mathsf{m})$
(91)

To obtain the third line, we have again used the abbreviated form of
$\mathbb{B}$ given in eq. (77), and renamed the dummy variable
$\sigma^{-1}\rightarrow\sigma$. We see that eq. (6.1) indeed vanishes on-shell
due to the binary BCJ relations
$\mathcal{B}_{[i,\ldots,n-1]}(1,2,\ldots,n{-}1,n)\cong 0$.

In principle, the above construction is not restricted to the NMHV sector, it
should apply more generally to pure gauge terms in any sector of YM. However,
there is a slight complication when going beyond NMHV since we do not yet have
a complete basis of local BCJ relations over non-linear polynomials.
Nevertheless, if we restrict our attention to the NMHV sector, one can argue
that eq. (83) forms a complete basis for the pure gauge terms in this sector.
We have explicitly and independently checked this statement up to seven points
using a direct Ansatz construction of the pure gauge numerator. Such a brute
force calculation of pure gauge terms becomes computationally intractable at
eight points and beyond.

The argument that eq. (83) captures all NMHV pure gauge freedom goes as
follows. We defined the NMHV sector numerators to have at most one power of
the $p_{i}{\cdot}p_{j}$ factors. All such factors are explicit in eq. (83),
and hence the invariant function $\mathcal{I}_{\mathbb{L}}$ cannot contain
such factors. This implies that when plugging in the numerators into the pure-
gauge constraint (81) the only BCJ relations that are expected to show up on
the right-hand side are those that have coefficients linear in
$p_{i}{\cdot}p_{j}$. Since we know the complete basis of local BCJ relations
linear in in $p_{i}{\cdot}p_{j}$, namely the binary BCJ relations, one can
conclude that we have fully taken into account the local freedom associated
with the constraint (81).

A caveat to this argument is that we do not yet have a mathematical proof that
the binary BCJ relations span the vector space of local BCJ relations with
linear coefficients. What we have checked up to multiplicity nine is that the
binary BCJ relations span the space of all currently known such BCJ relations.
This includes all permutations of the fundamental BCJ relations, and
generalized BCJ relations Bern:2008qj ; BjerrumBohr:2009rd ; Chen:2011jxa .
The fact that the dimension of this basis happens to non-trivially coincide
with the generalized Stirling numbers, gives support to the presumption that
the basis is complete.

As our next step, we explore the basis structure of pure gauge terms in the
NMHV sector. One can consider a fixed monomial $\mathsf{m}$ which belongs to a
certain MEC, and then we count the number of independent
$\mathbb{L}$-invariant polynomials for this MEC.

###  6.2 Independent group algebra invariants

Given a MEC denoted by $\mathsf{C}$, the group algebra invariants
$\mathcal{I}_{\mathbb{L}}=\mathbb{L}\circ\mathsf{m}$, for different
$\mathbb{L}$’s, will constitute a subset of the polynomial functions that can
be built out of the monomials $\mathsf{m}\in\mathsf{C}$. As discussed,
$\mathbb{L}$ has right zero divisors, or, equivalently, $\mathbb{L}$ is a
projector. This implies that the number of independent
$\mathcal{I}_{\mathbb{L}}$ should be equal to the _rank of $\mathbb{L}$ in the
representation space $\mathsf{C}$._

We start with a few examples. At $n=4$, we consider
$\mathbb{L}(2,3)=\mathbb{I}-\mathbb{P}_{(23)}$. We can find three MECs that
are of mass dimension zero,

$\displaystyle\mathsf{C}_{4,1}=\Big{[}\varepsilon_{2}{\cdot}\varepsilon_{3}\,\varepsilon_{1}{\cdot}\varepsilon_{4}\Big{]}\,,$
$\displaystyle\mathsf{C}_{4,1^{\prime}}=\Big{[}\hat{\varepsilon}_{4}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})\Big{]},$
$\displaystyle\mathsf{C}_{4,2}=\left[\begin{array}[]{c}\varepsilon_{1}{\cdot}\varepsilon_{2}\,\varepsilon_{3}{\cdot}\varepsilon_{4}\\\
\varepsilon_{1}{\cdot}\varepsilon_{3}\,\varepsilon_{2}{\cdot}\varepsilon_{4}\end{array}\right].$
(94)

The subscripts of $\mathsf{C}$ indicate the number of particles and the
dimension of the representation, respectively. Extra labels may be used to
distinguish representations of the same dimension, and a prime is used to
signal the presence of tensors. The permutations have the following
representation matrices: