momentum proportional to $k_{L}^{\mu}\equiv(1,-1)$. On-shell ZM amplitudes
naturally divide into sectors corresponding to the configuration of left and
right movers, where the scattering in many sectors is rather subtle. Only
color-ordered amplitudes can be defined unambiguously from the naive Feynman
rules. At four-point, only the alternating sector is free of subtleties and
the amplitude for this process vanishes. In equations, $A[LRLR]=0$ where $R$
corresponds to a right mover and $L$ corresponds to a left mover. In every
other configuration of right and left movers, such as $A[RRRR]$, one of the
internal propagators is accidentally on-shell since $k_{R}^{2}=k_{L}^{2}=0$.

Figure 2: The two physical cut topologies shared by NLSM, sGal, BI, and DBIVA

It is tempting to try to match the pion four-point two-loop integrand to ZM on
the cuts, but given that the cuts only probe the on-shell four-point
amplitude, which is either subtle or vanishes for ZM, it is better to compare
the off-shell numerators directly. The numerators of the two basis graphs are

$\displaystyle N^{\text{ZM}}\left[\scalebox{0.75}{ { \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.5,1){1}; \vertex(a2) at (1.5,-1){3}; \vertex(a3) at (1.5,1){4};
\vertex(a4) at (-1.5,-1){2}; \vertex(mid1) at (0,-.5); \vertex(mid2) at
(0,.5); \vertex(mid3) at (1,.5); \vertex(mid4) at (1,-.5) ; \vertex(mid5) at
(-1,.5); \vertex(mid6) at (-1,-.5) ; \vertex(l1) at (.5,.85){$\ell_{1}$};
\vertex(l2) at (-.5,.85){$\ell_{2}$}; \diagram{ (mid6) --[ultra
thick,nhpBlue](a4), (mid3) --[ultra thick,nhpBlue](a3), (mid4) --[ultra
thick,nhpBlue](a2), (mid5) --[ultra thick,nhpBlue](a1), (mid1) --[ultra
thick,nhpBlue](mid2), (mid3) --[ultra thick, fermion2,nhpBlue](mid2), (mid1)
--[ultra thick,nhpBlue](mid6), (mid1) --[ultra thick,nhpBlue](mid4), (mid5)
--[ultra thick, fermion2,nhpBlue](mid2), (mid5) --[ultra thick,nhpBlue](mid6),
(mid3) --[ultra thick,nhpBlue](mid4), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} } }\right]$
$\displaystyle=\begin{aligned} &\langle\ell_{1}\ell_{2}\rangle\langle
k_{4}\ell_{1}\rangle\langle\ell_{2}k_{1}\rangle\langle
k_{3}(\ell_{1}+k_{4})\rangle\\\
&\times\langle(k_{1}+\ell_{2})k_{2}\rangle\langle(\ell_{2}+k_{12})(\ell_{1}-k_{12})\rangle\end{aligned}$
(61) $\displaystyle N^{\text{ZM}}\left[\scalebox{0.75}{ { \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.5,1){4}; \vertex(a2) at (.5,-1){2}; \vertex(a3) at (.5,1){3}; \vertex(a4)
at (-1.5,-1){1}; \vertex(mid1) at (-.5,-.5); \vertex(mid2) at (-1,0);
\vertex(mid3) at (0,.5) ; \vertex(mid4) at (0,-.5) ; \vertex(mid5) at (-1,.5);
\vertex(mid6) at (-1,-.5) ; \vertex(l1) at (-1.35,.3){$\ell_{1}$}; \vertex(l2)
at (-1.35,-.3){$\ell_{2}$}; \diagram{ (a4) --[ultra thick,nhpBlue](mid6), (a3)
--[ultra thick,nhpBlue](mid3), (a2) --[ultra thick,nhpBlue](mid4), (a1)
--[ultra thick,nhpBlue](mid5), (mid1) --[ultra thick,nhpBlue](mid2), (mid6)
--[ultra thick, fermion2,nhpBlue](mid2), (mid1) --[ultra thick,nhpBlue](mid6),
(mid1) --[ultra thick,nhpBlue](mid4), (mid5) --[ultra thick,nhpBlue](mid3),
(mid5) --[ultra thick, fermion2,nhpBlue](mid2), (mid3) --[ultra
thick,nhpBlue](mid4), }; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} } }\right]$
$\displaystyle=\begin{aligned} &\langle\ell_{1}\ell_{2}\rangle\langle
k_{4}\ell_{1}\rangle\langle\ell_{2}k_{1}\rangle\langle
k_{3}(\ell_{1}+k_{4})\rangle\\\
&\times\langle(k_{1}-\ell_{1})k_{2}\rangle\langle(\ell_{2}+k_{1})(\ell_{1}-k_{1})\rangle\end{aligned}$
(62)

where $k_{12}=k_{1}+k_{2}$ and, again, $\langle ab\rangle\equiv
p_{a}^{\mu}\varepsilon_{\mu\nu}p_{b}^{\nu}$ is the color-stripped ZM vertex.
As local functions, the numerators never suffer from any of the subtleties of
on-shell 2D kinematics. We introduce a proportionality constant $z$ between
the numerators of the two theories, $N_{\text{NLSM}}|_{2D}=z~{}N_{\text{ZM}}$,
because we are only interested in how the kinematical structure of
$N_{\text{ZM}}$ can be used to eliminate gauge freedom in
$N_{\text{NSLM}}$.888Another reason for introducing the parameter $z$ is
because the ZM and NLSM are believed to differ at loop level even though they
are dual classically in 2D Nappi:1979ig . Once all of the constraints from
this section are imposed, $z$ is fixed to $216/565$ where the four-point pion
tree amplitude is normalized to $-k_{1}\cdot k_{2}$ and the ZM vertex is
normalized to $k_{1}^{\mu}\varepsilon_{\mu\nu}k_{2}^{\nu}$. Mechanically, the
pion numerator is matched to ZM by first taking every possible assignment of
right and left movers for the external particles and then restricting the pion
numerator to 2D. The loop momenta are restricted to 2D but left off-shell.
After performing the 2D matching, there are only 365 parameters of generalized
gauge freedom.

The loop momentum structure of ZM theory provides one final hint for
simplifying the pion numerators. A generic term in the ZM numerator (of either
basis graph) looks schematically like

$\ell_{1}^{m}\ell_{2}^{n}k^{12-(m+n)}\text{ where }5\leq m+n\leq 8,$ (63)

whereas a generic set of local, cubic Feynman rules could have produced terms
with up to $m+n=12$ powers of loop momenta. When the pion numerators are
forced to have the loop power counting structure in eq. 63 of ZM theory, the
number of generalized gauge freedom parameters reduces to 58. The pion
numerators appearing in the ancillary files make exactly this choice.

###  4.2 Double-copy verification

With a cubic color-dual pion representation in hand, we can perform double
copies with many theories to extract colorless gravity-like amplitudes. In
particular, we produce numerators for special Galileons (sGal), Born-Infeld
(BI) theory, and Dirac-Born-Infeld-Volkov-Akulov (DBIVA) by double-copying
against pions, pure YM, and $\mathcal{N}=4$ super-Yang-Mills (sYM)
respectively Cachazo:2014xea ; Cheung:2015ota ; Cheung:2016drk . Such double-
copy constructions are important nontrivial checks on the underlying single-
copy theories as the Jacobi relations in the single copy conspire to produce
the double-copy theory’s version of linear diffeomorphism invariance: enhanced
shift symmetry for special Galileons and gauge invariance of the BI photon and
DBIVA supermultiplet Hinterbichler:2015pqa . All three of these double-copy
theories were recently studied extensively by one of the authors and Carrasco
from the perspective of direct unitarity cut construction Carrasco:2023qgz .
One of the shared features of these three theories is that they all have the
same physical cut topologies in 4D: the two diagrams shown in fig. 2, which
are only composed of four-point amplitudes.

To construct the double-copy theories, we source the cubic pure YM numerators
from Ref. Bern:2015ooa  (which additionally satisfies a relaxed form of color-
kinematics duality), and the cubic $\mathcal{N}=4$ sYM representation from the
well-known $n_{2\text{box}}=n_{\text{cross-box}}=s^{2}t\,A^{\text{tree}}$
Bern:1997nh . On the other hand, we use the methods of Ref. Carrasco:2023qgz
to directly compute the needed basis cuts in all three theories without
relying on a color-dual pion representation. We find exact agreement in each
of the three theories for both physical cuts. Since Ref. Carrasco:2023qgz  has
already exhaustively explored the properties of four-point loop amplitudes in
these theories, we direct interested readers there for more information.

###  4.3 Two-loop Yang-Mills revisited

Given the close ties between pion and gluon scattering for trees and at one
loop, the existence of the two-loop pion numerator prompts us to investigate
the most general local numerator for YM, without any of the loop power
counting assumptions of Ref. Bern:2015ooa . We follow the same general
procedure as in section 4.1, again identifying the double-box and penta-
triangle as the Jacobi basis graphs and building the most general ansatz for
each of their numerators compatible with the assumptions in section 2.3.
Because we are now considering pure Yang-Mills, each monomial in the local
ansatz must consist of five Lorentz scalar dot products instead of the six for
NLSM, and each monomial must be linear in each of the four external gluon
polarizations. Thus, the numerator ansatz for both diagrams will be built from
terms of the form

$N^{\text{YM}}\left[\scalebox{0.75}{ { \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.5,1){1}; \vertex(a2) at (1.5,-1){3}; \vertex(a3) at (1.5,1){4};
\vertex(a4) at (-1.5,-1){2}; \vertex(mid1) at (0,-.5); \vertex(mid2) at
(0,.5); \vertex(mid3) at (1,.5); \vertex(mid4) at (1,-.5) ; \vertex(mid5) at
(-1,.5); \vertex(mid6) at (-1,-.5) ; \vertex(l1) at (.5,.85){$\ell_{1}$};
\vertex(l2) at (-.5,.85){$\ell_{2}$}; \diagram{ (mid6) --[ultra thick,
fermion2](a4), (mid3) --[ultra thick, fermion2](a3), (mid4) --[ultra thick,
fermion2](a2), (mid5) --[ultra thick, fermion2](a1), (mid1) --[ultra
thick,](mid2), (mid3) --[ultra thick, fermion2](mid2), (mid1) --[ultra
thick,](mid6), (mid1) --[ultra thick,](mid4), (mid5) --[ultra thick,
fermion2](mid2), (mid5) --[ultra thick,](mid6), (mid3) --[ultra thick,](mid4),
}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} } }\right]\text{ and
}N^{\text{YM}}\left[\scalebox{0.75}{ { \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.5,1){4}; \vertex(a2) at (.5,-1){2}; \vertex(a3) at (.5,1){3}; \vertex(a4)
at (-1.5,-1){1}; \vertex(mid1) at (-.5,-.5); \vertex(mid2) at (-1,0);
\vertex(mid3) at (0,.5) ; \vertex(mid4) at (0,-.5) ; \vertex(mid5) at (-1,.5);
\vertex(mid6) at (-1,-.5) ; \vertex(l1) at (-1.35,.3){$\ell_{1}$}; \vertex(l2)
at (-1.35,-.3){$\ell_{2}$}; \diagram{ (a4) --[ultra thick,](mid6), (a3)
--[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1) --[ultra
thick,](mid5), (mid1) --[ultra thick,](mid2), (mid6) --[ultra thick,
fermion2](mid2), (mid1) --[ultra thick,](mid6), (mid1) --[ultra thick,](mid4),
(mid5) --[ultra thick,](mid3), (mid5) --[ultra thick, fermion2](mid2), (mid3)
--[ultra thick,](mid4), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }
}\right]\in\text{span}\left\\{(\varepsilon_{i}\varepsilon_{j}),(k_{i}\varepsilon_{j}),(k_{i}k_{j})\right\\}$
(64)

where the $k_{i}$ take the same definition as described near eq. 59. Without
any power counting restrictions imposed, both the double-box and the penta-
triangle have an ansatz with 10,010 terms each. Imposing diagram automorphisms
and maximal-cut gauge invariance on the two basis diagrams, we reduce the
number of terms to 2,235 for the double-box, and 4,133 for the penta-triangle.
The minimal set of spanning physical cuts is shown in fig. 3, but we choose to
work within the framework of the method of maximal cuts Bern:2007ct  in order
to identify the simplest cut that is in tension with the kinematic Jacobi
relations. Maximal cuts and symmetries are then imposed on the 14 non-
pathological cubic diagrams shown above in fig. 1. Doing so, we are left with
596 total parameters in the ansatz.

$(mid3)+(0.8,-0.6)$$(mid3)+(.8,0.6)$

Figure 3: The three spanning physical unitarity cuts of pure Yang-Mills at two
loops

Proceeding to the next-to-maximal cuts, we continue to avoid pathological
diagrams including the newly-appearing type discussed in eq. 27. After
discarding these types of cuts as well as those involving cuts of external
Mandelstams, there are 8 one-particle-irreducible cut topologies (see fig. 4).
Seven of these next-to-maximal cuts are consistent with the kinematic Jacobi
relations and symmetries. Critically, the “bowtie” cut,

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.25,.85){}; \vertex(a2) at (1.25,-.85){}; \vertex(a3) at (1.25,.85){};
\vertex(a4) at (-1.25,-.85){}; \vertex(mid1) at (0,0); \vertex(mid3) at
(.75,.4) ; \vertex(mid4) at (.75,-.4) ; \vertex(mid5) at (-.75,.4);
\vertex(mid6) at (-.75,-.4) ; \diagram{ (a4) --[ultra thick,](mid6), (a3)
--[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1) --[ultra
thick,](mid5), (mid3) --[ultra thick,](mid1), (mid1) --[ultra thick,](mid6),
(mid1) --[ultra thick,](mid4), (mid5) --[ultra thick,](mid1), (mid5) --[ultra
thick,](mid6), (mid3) --[ultra thick,](mid4), }; \vertex[dot,scale=2.4](midC)
at (0,0){}; \vertex[dot,scale=1.8,hgrey0](midC) at (0,0){};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,$

cannot be satisfied by the ansatz once Jacobi relations and symmetries are
applied.

,  ,  ,  ,
,  ,  ,  ,

Figure 4: The 8 non-pathological 1-particle-irreducible next-to-maximal-cut
diagrams used in our cut constraints on the Yang-Mills integrand.

In fact, we can make the failure extremely precise. First, start with the
diagrams shown in the Jacobi relation from eq. 19, which includes the double-
box, crossed-box, and penta-triangle. Then _without_ defining the crossed-box
in terms of the other two diagrams, write down the most generic parity-even
local ansatz involving four polarizations and six momenta (any combination of
external or loop) for each of the three diagrams. Each ansatz will have 10,010
terms initially. Next impose the symmetry constraints on each of the three
diagrams, leaving 2,761 free parameters on the double-box, 2,576 free
parameters on the crossed-box, and 5,040 free parameters on the penta-
triangle. Finally impose the “bowtie” next-to-maximal cut, which in the _non-
planar_ $t$-$u$ color channel only receives contributions from the double-box

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.25,.85){}; \vertex(a2) at (1.25,-.85){}; \vertex(a3) at (1.25,.85){};
\vertex(a4) at (-1.25,-.85){}; \vertex(mid1) at (0,0); \vertex(mid3) at
(.75,.4) ; \vertex(mid4) at (.75,-.4) ; \vertex(mid5) at (-.75,.4);
\vertex(mid6) at (-.75,-.4) ; \diagram{ (a4) --[ultra thick,](mid6), (a3)
--[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1) --[ultra
thick,](mid5), (mid3) --[ultra thick,](mid1), (mid1) --[ultra thick,](mid6),
(mid1) --[ultra thick,](mid4), (mid5) --[ultra thick,](mid1), (mid5) --[ultra
thick,](mid6), (mid3) --[ultra thick,](mid4), }; \vertex[dot,scale=2.4](midC)
at (0,0){}; \vertex[dot,scale=1.8,hgrey0](midC) at (0,0){};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{1}{{\color[rgb]{0,0.390625,0.56640625}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.390625,0.56640625}\ell_{1}^{2}}}\
N\left[{\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at
(-1.5,1){4}; \vertex(a2) at (1.5,-1){2}; \vertex(a3) at (1.5,1){3};
\vertex(a4) at (-1.5,-1){1}; \vertex(mid1) at (0,-.5); \vertex(mid2) at
(0,.5); \vertex(mid3) at (1,.5) ; \vertex(mid4) at (1,-.5) ; \vertex(mid5) at
(-1,.5); \vertex(mid6) at (-1,-.5) ; \vertex(m1) at ($(mid5)!0.5!(mid6)$);
\vertex(m2) at ($(mid1)!0.5!(mid6)$); \vertex(m3) at ($(mid5)!0.5!(mid2)$);
\vertex(clo) at ($(m2)!0.5!(m3)$) {}; \vertex(c1) at ($(clo)+(90:0.8)$);
\vertex(c2) at ($(clo)+(180:0.8)$); \vertex(c3) at ($(clo)+(270:0.8)$);
\vertex(m4) at ($(mid3)!0.5!(mid4)$); \vertex(m5) at ($(mid3)!0.5!(mid2)$);
\vertex(m6) at ($(mid4)!0.5!(mid1)$); \vertex(cro) at ($(m5)!0.5!(m6)$) {};
\vertex(c4) at ($(cro)+(90:0.8)$); \vertex(c5) at ($(cro)+(0:0.8)$);
\vertex(c6) at ($(cro)+(270:0.8)$); \par\par\diagram{ (a4) --[ultra
thick,](mid6), (a3) --[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1)
--[ultra thick,](mid5), (mid1) --[ultra thick,ucp-color](mid2), (mid3)
--[ultra thick,](mid2), (mid1) --[ultra thick,](mid6), (mid1) --[ultra
thick,](mid4), (mid5) --[ultra thick,](mid2), (mid5) --[ultra thick,](mid6),
(mid3) --[ultra thick,](mid4), (clo)--[ultra thick,dashed,cut-color](c1),
(clo)--[ultra thick,dashed,cut-color](c2), (clo)--[ultra thick,dashed,cut-
color](c3), (cro)--[ultra thick,dashed,cut-color](c4), (cro)--[ultra
thick,dashed,cut-color](c5), (cro)--[ultra thick,dashed,cut-color](c6) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right]+\frac{1}{{\color[rgb]{0,0.390625,0.56640625}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.390625,0.56640625}\ell_{2}^{2}}}\
N\left[{\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1) at
(-1.5,1){4}; \vertex(a2) at (1.5,-1){3}; \vertex(a3) at (1.5,1){2};
\vertex(a4) at (-1.5,-1){1}; \vertex(mid1) at (0,-.5); \vertex(mid2) at
(0,.5); \vertex(mid3) at (1,.5) ; \vertex(mid4) at (1,-.5) ; \vertex(mid5) at
(-1,.5); \vertex(mid6) at (-1,-.5) ; \vertex(m1) at ($(mid5)!0.5!(mid6)$);
\vertex(m2) at ($(mid1)!0.5!(mid6)$); \vertex(m3) at ($(mid5)!0.5!(mid2)$);
\vertex(clo) at ($(m2)!0.5!(m3)$) {}; \vertex(c1) at ($(clo)+(90:0.8)$);
\vertex(c2) at ($(clo)+(180:0.8)$); \vertex(c3) at ($(clo)+(270:0.8)$);
\vertex(m4) at ($(mid3)!0.5!(mid4)$); \vertex(m5) at ($(mid3)!0.5!(mid2)$);
\vertex(m6) at ($(mid4)!0.5!(mid1)$); \vertex(cro) at ($(m5)!0.5!(m6)$) {};
\vertex(c4) at ($(cro)+(90:0.8)$); \vertex(c5) at ($(cro)+(0:0.8)$);
\vertex(c6) at ($(cro)+(270:0.8)$); \par\par\diagram{ (a4) --[ultra
thick,](mid6), (a3) --[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1)
--[ultra thick,](mid5), (mid1) --[ultra thick,ucp-color](mid2), (mid3)
--[ultra thick,](mid2), (mid1) --[ultra thick,](mid6), (mid1) --[ultra
thick,](mid4), (mid5) --[ultra thick,](mid2), (mid5) --[ultra thick,](mid6),
(mid3) --[ultra thick,](mid4), (clo)--[ultra thick,dashed,cut-color](c1),
(clo)--[ultra thick,dashed,cut-color](c2), (clo)--[ultra thick,dashed,cut-
color](c3), (cro)--[ultra thick,dashed,cut-color](c4), (cro)--[ultra
thick,dashed,cut-color](c5), (cro)--[ultra thick,dashed,cut-color](c6) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right]\,.$ (65)

After imposing this cut, we find that the original Jacobi relation, eq. 19,
can no longer be satisfied. In other words, the constraints imposed by the
single Jacobi relation and the three sets of diagram symmetries conspire in a
way that is inconsistent with the “bowtie” cut. The contrapositive to this
surprising result can be summarized in the following diagrammatic statement,
which prioritizes the physical unitarity cut:

$\scalebox{0.9}{ { \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.25,.85){}; \vertex(a2) at (1.25,-.85){}; \vertex(a3) at (1.25,.85){};
\vertex(a4) at (-1.25,-.85){}; \vertex(mid1) at (0,0); \vertex(mid3) at
(.75,.4) ; \vertex(mid4) at (.75,-.4) ; \vertex(mid5) at (-.75,.4);
\vertex(mid6) at (-.75,-.4) ; \diagram{ (a4) --[ultra thick,](mid6), (a3)
--[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1) --[ultra
thick,](mid5), (mid3) --[ultra thick,](mid1), (mid1) --[ultra thick,](mid6),
(mid1) --[ultra thick,](mid4), (mid5) --[ultra thick,](mid1), (mid5) --[ultra
thick,](mid6), (mid3) --[ultra thick,](mid4), }; \vertex[dot,scale=2.4](midC)
at (0,0){}; \vertex[dot,scale=1.8,hgrey0](midC) at (0,0){};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }
}\\!\\!\Rightarrow\\!\\!\scalebox{0.85}{ { \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.5,1){4}; \vertex(a2) at (1.5,-1){2}; \vertex(a3) at (1.5,1){3};
\vertex(a4) at (-1.5,-1){1}; \vertex(mid1) at (0,-.5); \vertex(mid2) at
(0,.5); \vertex(mid3) at (1,.5) ; \vertex(mid4) at (1,-.5) ; \vertex(mid5) at
(-1,.5); \vertex(mid6) at (-1,-.5) ; \diagram{ (a4) --[ultra thick,](mid6),
(a3) --[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1) --[ultra
thick,](mid5), (mid1) --[ultra thick,](mid2), (mid3) --[ultra thick,](mid2),
(mid1) --[ultra thick,](mid6), (mid1) --[ultra thick,](mid4), (mid5) --[ultra
thick,jacobi-color](mid2), (mid5) --[ultra thick,](mid6), (mid3) --[ultra
thick,](mid4), }; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} } }+\scalebox{0.85}{
{ \leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.5,1){4}; \vertex(a2) at (1.5,-1){2}; \vertex(a3) at (1.5,1){3};
\vertex(a4) at (-1.5,-1){1}; \vertex(mid1) at (0,-.5); \vertex(mid2) at
(0,.5); \vertex(mid3) at (1,.5); \vertex(mid4) at (1,-.5); \vertex(mid5) at
(-1,.5); \vertex(mid6) at (-1,-.5); \vertex(mid7) at (-.5,0) {}; \diagram{
(a4) --[ultra thick,](mid6), (a3) --[ultra thick,](mid3), (a2) --[ultra
thick,](mid4), (a1) --[ultra thick,](mid5), (mid1) --[ultra thick,](mid6),
(mid3) --[ultra thick,](mid2), (mid1) --[ultra thick,](mid4), (mid5) --[ultra
thick,](mid7), (mid7) --[ultra thick,](mid1), (mid2) --[ultra thick,](mid6),
(mid5) --[ultra thick,jacobi-color](mid2), (mid3) --[ultra thick,](mid4), };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} } }+\scalebox{0.85}{
{ \leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at
(-1.5,1){4}; \vertex(a2) at (.5,-1){2}; \vertex(a3) at (.5,1){3}; \vertex(a4)
at (-1.5,-1){1}; \vertex(mid1) at (-.5,-.5); \vertex(mid2) at (-1,0);
\vertex(mid3) at (0,.5) ; \vertex(mid4) at (0,-.5) ; \vertex(mid5) at (-1,.5);
\vertex(mid6) at (-1,-.5) ; \diagram{ (a4) --[ultra thick,](mid6), (a3)
--[ultra thick,](mid3), (a2) --[ultra thick,](mid4), (a1) --[ultra
thick,](mid5), (mid1) --[ultra thick,](mid2), (mid6) --[ultra thick,](mid2),
(mid1) --[ultra thick,](mid6), (mid1) --[ultra thick,](mid4), (mid5) --[ultra
thick,](mid3), (mid5) --[ultra thick,jacobi-color](mid2), (mid3) --[ultra
thick,](mid4), }; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} } }\neq 0\,.$ (66)

Thus we have identified the minimal failure state for a globally-color-dual
two-loop Yang-Mills representation: off-shell locality, symmetry, and the
kinematic Jacobi relations for just three diagrams are incompatible with the
“bowtie” cut.

##  5 Conclusions and Outlook

In summary, we have found that there is a tension between color-kinematics
duality and an off-shell local construction of the two-loop four-point pure
Yang-Mills integrand in $D$-dimensions. At tree level, we found that the
duality can be manifested by breaking manifest Bose symmetry at the Lagrangian
level, either by explicitly picking different gluon states (as in SDYM) or by
picking states that only restore Bose symmetry in the final amplitude (as in
NLSM). Both of these theories can be encapsulated in the semi-abelian YM
theory presented in eq. 41 which obscures the Bose symmetry of the external
states due to the presence of the two gauge fields $A_{\mu}$ and
$\bar{A}_{\mu}$.

In addition to the tree-level construction, we found that kinetic mixing
between the two gauge fields permits globally color-dual one-loop integrands.
After plugging in suitable on-shell states, the integrands produced by semi-
abelian YM correspond to those of SDYM in eq. 53 and NLSM in eq. 55, along
with Chern-Simons theory, which we discuss in section A.5. Due to the kinetic
mixing that stems from the broken Bose symmetry, eq. 57 shows why it is
impossible to construct a two-loop integrand for semi-abelian YM theory. In
section 4, we overcame this obstruction by successfully constructing two-loop
NLSM numerators with an ansatz-based bootstrap. However, we found that the
same approach fails for two-loop pure Yang-Mills, and pin-pointed the failure
to a particular unitarity cut captured by eq. 66. Thus, obtaining a globally
color-dual integrand beyond one-loop then requires sacrificing more than just
Bose symmetry, and in this paper we have argued that off-shell locality999An
example of this can already be found in the literature, where locality is
relaxed in Ref. Mogull:2015adi  in order to obtain color-dual two-loop
integrands with on-shell 4D states. must be abandoned for color-dual
constructions of multi-loop Yang-Mills. To expand the available function
space, we propose that one should consider including rational functions of
kinematics, rather than just polynomials.

We concede building loop-level numerators from rational functions of the
kinematics is rather unnatural from the perspective of point-like quantum
field theories. After all, operators that produce rational functions are
typically non-local in their construction. The archetypal example of non-local
quantum operators are string vertex operators with $\alpha^{\prime}$
corrections. These operators promote local tree-level amplitudes to stringy
extended objects by integrating over disc integrals of the open string
worldsheet

$\mathcal{A}^{\text{YM}}\stackrel{{\scriptstyle\alpha^{\prime}}}{{\longrightarrow}}\mathcal{A}^{\text{OS}}\supset
t_{8}F^{4}\int_{0}^{1}dx\,\frac{x^{\alpha^{\prime}s_{12}}(1-x)^{\alpha^{\prime}s_{23}}}{x(1-x)(s_{12}+s_{23})}\,.$
(67)

The resulting Veneziano factor of Gamma functions produces rational functions
of kinematics, while preserving color-kinematics duality MafraBCJAmplString ;
Broedel:2013tta ; Mafra:2016mcc ; Carrasco:2019yyn . While introducing some
type of worldsheet formulation of color-dual numerators might seem unjustified
given the results of this work, our findings certainly suggest that we must do
something to relax the constraint of off-shell locality, and realize locality
only in the limit of on-shell kinematics. This could be achieved by either
modifying the kinetic term or potentially something new and more exotic. Below
we provide some simple examples at tree-level of what one might consider for
implementing such a construction.

###  5.1 Non-local construction of scattering amplitudes

As an exemplar of an off-shell non-local structure, consider the simple four-
point example of a color-dual representation of Yang-Mills theory. We can
define a functional numerator, $N_{(12|34)}^{\text{YM}}$, as follows,

$N_{(12|34)}^{\text{YM}}=\frac{t_{8}F^{4}}{3}\frac{s_{13}-s_{12}}{s_{23}{s_{13}}}$
(68)

where $N_{s}=N_{(12|34)}$, $N_{t}=N_{(14|23)}$ and $N_{u}=N_{(13|42)}$. By
construction, this functionally symmetric numerator is antisymmetric and obeys
the Jacobi identity

$N_{(12|34)}^{\text{YM}}+N_{(13|42)}^{\text{YM}}+N_{(14|23)}^{\text{YM}}=0.$
(69)

However, it must also factorize to kinematics that are consistent with the
local Feynman rules of eq. 28. Evaluating the residue on the $s=s_{12}\to 0$
pole we find

$\displaystyle\text{Res}\left[\frac{N_{s}^{\text{YM}}}{s}\right]^{s=0}$
$\displaystyle=\frac{2}{3}A_{(12|l)}^{\text{YM}}A_{(-l|34)}^{\text{YM}}$ (70)
$\displaystyle\text{Res}\left[\frac{N_{t}^{\text{YM}}}{t}\right]^{s=0}$
$\displaystyle=-\frac{1}{3}A_{(12|l)}^{\text{YM}}A_{(-l|34)}^{\text{YM}}$ (71)
$\displaystyle\text{Res}\left[\frac{N_{u}^{\text{YM}}}{u}\right]^{s=0}$
$\displaystyle=-\frac{1}{3}A_{(12|l)}^{\text{YM}}A_{(-l|34)}^{\text{YM}}\,,$
(72)

with $t=s_{23}$ and $u=s_{13}$. Plugging these into a cubic graph
representation of Yang-Mills, the tree level version of eq. 2, we find

$\text{Res}\left[\mathcal{A}_{4}^{\text{YM}}\right]^{s=0}=\frac{1}{3}A_{(12|l)}^{\text{YM}}A_{(-l|34)}^{\text{YM}}(2c_{s}-c_{t}-c_{u})=\mathcal{A}_{(12|l)}^{\text{YM}}\mathcal{A}_{(-l|34)}^{\text{YM}}\,,$
(73)

where we have applied the color structure Jacobi identity,
$c_{s}+c_{t}+c_{u}=0$. In light of our findings in eq. 66, building color-dual
numerators in this way where locality is only realized on-shell might be a
more natural approach. While we can absolutely apply a generalized gauge
transformation BCJ  to restore field theoretic locality to the cubic
numerators, this could merely be an aesthetic choice that is only permissible
at tree-level.

Moreover, by abandoning traditional notions of off-shell locality, we
incidentally have gained enough functional freedom to massage the color-dual
numerators into a form that is manifestly gauge invariant for all particles.
As an organizational principle, pulling out overall gauge-invariant factors
comes with the added advantage of possibly simplifying a loop-level
construction of color-dual numerators for Yang-Mills.

###  5.2 Future Directions

With an eye towards generalizing the non-local construction above to future
multi-loop studies, we note that the four-point half-ladder of Yang-Mills
secretly makes use of the color-dual structure of NLSM. At four-point, we can
construct permutation invariants from the single BCJ basis amplitude of both
NLSM and pure YM theory as follows

$stA^{\text{YM}}_{(s,t)}=t_{8}F^{4}\qquad stA^{\text{NLSM}}_{(s,t)}=stu\,.$
(74)

Using this, we can redefine the non-local numerators of eq. 69 so that the
vector structure of YM is captured in a permutation invariant prefactor and
the kinematic Jacobi identity is entirely due to the NLSM numerators

$N_{(12|34)}^{\text{YM}}\equiv
t_{8}F^{4}\,\frac{N_{(12|34)}^{\text{NLSM}}}{stu}\,.$ (75)

where four-point pion numerator, $N_{s}^{\text{NLSM}}$, is given in eq. 5. One
advantage of this construction is that it reduces some of the $D$-dimensional
complexity of vector theories that arises due to the mixing between external
polarization and internal loop momenta. But maybe more importantly, it puts
all the heavy lifting of functional Jacobi relations on the scalar kinematic
numerator, $N_{(12|34)}^{\text{NLSM}}$. Thus, rather than building an ansatz
from the irreducible scalar products of eq. 64, one might instead consider the
following construction

$N^{\text{YM}}_{m,L}=\sum_{i}\mathcal{O}_{i}\mathcal{R}^{(i)}_{m,L}$ (76)

where $\mathcal{O}_{i}$ are the on-shell gauge invariant tensor basis elements
of Bern:2017tuc ; Carrasco:2019yyn , and $\mathcal{R}^{(i)}_{m,L}$ are
rational functions of irreducible scalar products of eq. 59. It would be
interesting if a similar construction as our tree-level example could be
uplifted to two-loop using the globally color-dual NLSM integrand that we have
computed in this work. We see this as a natural future direction worth
investigating that is now made possible by our findings.

### Acknowledgments

The authors would like to thank John Joseph Carrasco, Sasank Chava, Clifford
Cheung, Kezhu Guo, Nia Robles, Aslan Seifi, Fei Teng, and Suna Zekioğlu for
insightful conversations, feedback on earlier drafts, and encouragement
throughout the completion of this work. This work was supported by the DOE
under contract DE-SC0015910 and by the Alfred P. Sloan Foundation.
Additionally we would like to acknowledge the Northwestern University
Amplitudes and Insight group, the Department of Physics and Astronomy, and
Weinberg College for their generous support. Feynman diagrams were typeset
using TikZ-Feynman Ellis:2016jkw .

##  Appendix A Catalog of color-dual theories

Below we review the five manifestly color-dual theories that we mention in the
text, self-dual Yang-Mills (SDYM) Monteiro2011pc , two formulations of NLSM
Cheung:2016prv ; Cheung:2020djz ; Cheung:2021zvb , ZM Zakharov:1973pp ;
Cheung:2022mix , and Chern-Simons theory Ben-Shahar:2021zww . All of these
theories are closely tied together by the semi-abelian YM theory in eq. 41,
which gives the cubic sector of pure YM. The manifestly color-dual equations
of motion for SDYM and NLSM are derived by picking a gauge and then placing a
constraint directly on the field strength. In almost all of these theories,
Bose symmetry is broken or obscured at the Lagrangian level. Since Bose
symmetry is broken, the ensuing kinetic mixing prevents color-dual
representations beyond one-loop without the introduction of additional
interactions.

###  A.1 Self-Dual Yang-Mills

The first example in the literature of color-dual Feynman rules is that of
Yang-Mills theory in the self-dual sector (SDYM). We summarize the approach of
Ref. Monteiro2011pc  for extracting the kinematic algebra. To obtain the
Lagrangian for self-dual Yang-Mills from standard Yang-Mills, we apply the
self-duality condition that equates the non-abelian $SU(N)$ field strength,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+g[A_{\mu},A_{\nu}]$,
to its dual two-form,

$F_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}\,.$ (77)

Solutions to the self-dual condition automatically satisfy the Yang-Mills
equation of motion due to the Bianchi identity,

$D^{\mu}F^{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}D^{\mu}F^{\rho\sigma}=0\,.$
(78)

Furthermore, field configurations that satisfy eq. 77 correspond to instantons
which are probed by the path integral in the full non-perturbative description
of Yang-Mills. To construct a manifestly perturbative action, one begins by
transforming from Cartesian coordinates to light cone coordinates,
$(t,x,y,z)\rightarrow(u,v,w,\bar{w})$, via

$u=t+x,\quad v=t-x,\quad w=y+iz,\quad\bar{w}=y-iz\,,$ (79)

which yields the invariant line element,

$ds^{2}=2(dudv-dwd\bar{w})=dt^{2}-d\vec{x}\cdot d\vec{x}\,.$ (80)

Applying this coordinate transformation to the self-dual condition of eq. 77,
produces three constraint equations on the field strength,

$\displaystyle F_{uw}$ $\displaystyle=0$ (81) $\displaystyle F_{uv}$
$\displaystyle=F_{w\bar{w}}$ (82) $\displaystyle F_{v\bar{w}}$
$\displaystyle=0\,.$ (83)

In light-cone gauge, $A_{u}=0$, we the self-duality condition restricts the
gauge field as follows:

$\displaystyle A_{u}=0\text{ and }(\ref{SDLightCone1})\quad$
$\displaystyle\Rightarrow\quad A_{w}=0$ (84) $\displaystyle
A_{w}=0\,,\,A_{u}=0\text{ and }(\ref{SDLightCone2})\quad$
$\displaystyle\Rightarrow\quad\partial_{u}A_{v}=\partial_{w}A_{\bar{w}}\,.$
(85)

The last constraint is satisfied by writing the gauge field in terms of a
single scalar field $\Psi$ via

$A_{v}=\frac{1}{2}\partial_{w}\Psi~{}~{}~{}\text{ and
}~{}~{}~{}A_{\bar{w}}=\frac{1}{2}\partial_{u}\Psi\,.$ (86)

Using this definition on the first self-dual constraint equation yields the
following equation of motion for the dynamical scalar field:

$(\ref{SDLightCone1})\text{ and
}(\ref{SDDef})\quad\Rightarrow\quad\Box\Psi+ig[\partial_{u}\Psi,\partial_{w}\Psi]=0\,.$
(87)

Adding back in the anti-holomorphic field $\bar{\Psi}$ as a Lagrange
multiplier, we obtain the following Lagrangian for self-dual Yang-Mills
theory:

$\mathcal{L}^{\text{SDYM}}=(\partial\bar{\Psi})(\partial\Psi)-ig\bar{\Psi}[\partial_{u}\Psi,\partial_{w}\Psi]\,.$
(88)

The color-ordered cubic Feynman rule for this theory can be immediately read
off as

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \diagram{ (mid1)
--[ultra thick,fermion2,](a3), (mid1) --[ultra thick,fermion2,](a2), (a1)
--[ultra thick,fermion2,](mid1), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=X(k_{1},k_{2})\equiv
k_{u,1}k_{w,2}-k_{w,1}k_{u,2}\,.$ (89)

In this form, the Lagrangian and Feynman rules are manifestly color-dual off-
shell. The effect of eq. 77 on the full YM Lagrangian is to decouple the anti-
MHV three-point vertex from the theory, leaving only the MHV three-point
vertex. Before proceeding, it is important to note that the mass-dimension of
the theory appears to differ from that of Yang-Mills – we comment on this in
detail in appendix B.

###  A.2 YZ-theory

Self-dual Yang-Mills involves explicit reliance on the spacetime dimension,
which makes the theory poorly suited for constructing dimensionally-regulated
loop-level amplitudes. On the other hand, $YZ$-theory of Cheung:2016prv  is an
honest $D$-dimensional theory that captures the classical physics of NLSM
while manifesting the duality between color and kinematics.

One can understand $YZ$-theory as a particular dimensional reduction of Yang-
Mills in $D=2d+1$ dimensions Cheung:2017yef , down to $d$-dimensions. Starting
with the Yang-Mills Lagrangian of eq. 28, one can redefine the gauge fields,
$A_{M}$, in terms of $X$, $Y$ and $Z$ fields:

$\displaystyle X_{M}$ $\displaystyle=(X_{\mu},0,-iX_{\mu})$ (90)
$\displaystyle Y_{M}$ $\displaystyle=(0,Y,0)$ (91) $\displaystyle Z_{M}$
$\displaystyle=(Z_{\mu},0,iZ_{\mu})$ (92)

where $X$ and $Z$ are $d$-vectors and $Y$ is a scalar field. Considering the
conjugate nature of the $XZ$ propagator, in this work we make the replacement
$X\rightarrow\bar{Z}$. Plugging in this redefinition of the gauge fields
produces the following Lagrangian up to cubic order in the interactions

$\mathcal{L}^{\text{YZ}}=\frac{1}{2}(\partial Y)^{2}+(\partial
Z)(\partial\bar{Z})-gf^{abc}\left(\bar{Z}_{\mu\nu}Z^{\mu}Z^{\nu}+[Y,\partial_{\mu}Y]Z^{\mu}\right)\,.$
(93)

The Feynman rules for this Lagrangian are simply

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \diagram{ (a3) --[ultra
thick,,nhpRed](mid1), (a2) --[ultra thick,,nhpRed](mid1), (mid1) --[ultra
thick,fermion2,](a1), }; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=i(\varepsilon_{3}p_{2})\qquad{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \diagram{ (mid1)
--[ultra thick,fermion2,](a3), (mid1) --[ultra thick,fermion2,](a2), (a1)
--[ultra thick,fermion2,](mid1), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=i(\varepsilon_{1}p_{3})(\varepsilon_{2}\bar{\varepsilon}_{3})-i(\varepsilon_{2}p_{3})(\varepsilon_{1}\bar{\varepsilon}_{3})\,.$
(94)

Note that the pure vector vertex on the right is exactly what was obtain from
semi-abelian YM in eq. 41. We now review of the construction of NLSM
numerators at tree-level from $YZ$-theory. In this construction, we can define
$D$-dimensional generators of the kinematic algebra as follows,

$T^{a}_{ij}=i\varepsilon_{a}(p_{i}-p_{j})\,,$ (95)

where momentum conservation requires,

$p_{a}+p_{i}+p_{j}=0\,.$ (96)

The kinematic half-ladder diagrams then take on the following concise form

$n^{\text{NLSM}}_{(i|a_{1}a_{2}...a_{n}|j)}={}_{i}\langle
T^{a_{1}}T^{a_{2}}\cdots T^{a_{n}}\rangle_{j}\,.$ (97)

Since there are no pole cancelling factors of $s_{ij}=(p_{i}+p_{j})^{2}$, this
definition of the kinematic algebra is manifestly cubic. Thus, the kinematic
structure constants defined in terms of these generators are invariant under
generalized gauge freedom because there are no quartic vertices. They can be
defined implicitly through

$[T^{a},T^{b}]_{ij}=F^{a}_{\,b|c}T^{c}_{ij}\,.$ (98)

Given this definition, the Feynman rule associated with kinematic structure
constant is

$iF^{a}_{\,b|c}=(\varepsilon_{b}p_{ab})(\varepsilon_{a}\bar{\varepsilon}_{c})-(\varepsilon_{a}p_{ab})(\varepsilon_{b}\bar{\varepsilon}_{c})\,,$
(99)

where $p_{ab}=p_{a}+p_{b}$ and $\varepsilon$ and $\bar{\varepsilon}$ are the
polarizations of the $Z$-vectors particle and its conjugate field,
respectively. The gauge-fixed state sum for this theory is simply

$\sum_{\text{states}}\varepsilon^{\,\mu}_{(p)}\bar{\varepsilon}^{\,\nu}_{(-p)}=\eta^{\mu\nu}\,.$
(100)

Notice that the vector state sum is gauge fixed since the $YZ$ model
explicitly chooses Lorenz gauge for the $Z$ particles,
$\partial_{\mu}Z^{\mu}=0$. Tree-level NLSM amplitudes are recovered from the
kinematic structure constants by plugging in

$\varepsilon^{\mu}_{(p)}=p^{\mu}\qquad\bar{\varepsilon}^{\mu}_{(p)}=\frac{q^{\mu}}{pq}$
(101)

for the on-shell polarizations for $Z$ and $\bar{Z}$, respectively, where
$q^{2}=0$ is some null reference momentum. The tree-level pion amplitude can
then be defined in two equivalent ways,

$A^{\text{NLSM}}=A(...,Y,...,Y,...)~{}~{}~{}\text{ and
}~{}~{}~{}A^{\text{NLSM}}=A(...,\bar{Z},...)\,,$ (102)

where the ellipses denote additional on-shell $Z$-particles. In a suitable
gauge, the kinematic numerators in the latter definition for pion scattering
are equivalent to those of $J$-theory, where $\bar{Z}$ corresponds to the root
leg of $J$-theory Cheung:2021zvb .

It is instructive to see how both of the constructions in eq. 102 produce
valid tree-level amplitudes for the pion. First we will start with $Y$
particles on legs 1 and 4. Applying the Feynman rules above, and plugging in
on-shell states for the $Z$-particles produces the following $s$\- and
$t$-channel numerators:

$\displaystyle n^{YY}_{s}$ $\displaystyle=(T^{2}T^{3})_{14}=s_{12}^{2}$ (103)
$\displaystyle n^{YY}_{t}$
$\displaystyle=F^{3}_{\,2|X}T^{X}_{14}=s_{14}(s_{13}-s_{12})\,.$ (104)

Plugging these numerators into the ordered amplitudes $A(s,t)$ yields the
desired result,

$A^{YY}_{(s,t)}=\frac{n^{YY}_{s}}{s_{12}}+\frac{n^{YY}_{t}}{s_{14}}=s_{13}\,.$
(105)

For the $Z$ and $\bar{Z}$ configuration, the numerators are

$\displaystyle n^{\bar{Z}Z}_{s}$ $\displaystyle={}_{4}\langle
F^{3}F^{2}\rangle_{1}=s_{12}(s_{14}-s_{13})p_{2}^{\mu_{1}}+s_{12}^{2}(p_{3}-p_{4})^{\mu_{1}}$
(106) $\displaystyle n^{\bar{Z}Z}_{t}$ $\displaystyle={}_{2}\langle
F^{3}F^{4}\rangle_{1}=s_{14}(s_{12}-s_{13})p_{4}^{\mu_{1}}+s_{14}^{2}(p_{3}-p_{2})^{\mu_{1}}\,,$
(107)

where we have used the shorthand

${}_{x}\langle F^{a_{1}}F^{a_{2}}\cdots F^{a_{n}}\rangle_{y}\equiv
F^{a_{1}}_{\,x|b_{2}}F^{a_{2}}_{\,b_{2}|b_{3}}\cdots F^{a_{n}}_{\,b_{n}|y}\,.$
(108)

The polarization vector of the $\bar{Z}$ particle has not been contracted into
the numerators above, resulting in the free $\mu_{1}$ index. These numerators
produce the partial amplitude

$A^{\bar{Z}Z}_{(s,t)}=\frac{n^{\bar{Z}Z}_{s}}{s_{12}}+\frac{n^{\bar{Z}Z}_{t}}{s_{14}}=-s_{13}(p_{2}+p_{3}+p_{4})^{\mu_{1}}=s_{13}\,p_{1}^{\mu_{1}}\,.$
(109)

Plugging in the on-shell polarization of the conjugate field in eq. 101
produces precisely the desired result of eq. 105. Indeed, this construction is
valid to all multiplicity at tree-level. One can see this by considering the
only two possible factorization channels that contribute to each of these
amplitudes, the $YY$ cut and the $\bar{Z}Z$ cut,

$\displaystyle A(...,Y,...,Y,...)$ $\displaystyle\rightarrow
A(...,Y,...,Y)A(Y,...,Y,...)$ (110) $\displaystyle\rightarrow
A(...,Y,...,Y,...,Z)A(\bar{Z},...)\,.$ (111)

Both factorization channels are valid descriptions of NLSM amplitudes when
plugging in the $Z$ and $\bar{Z}$ on-shell states. As long as we plug in valid
on-shell states for $Z$ and $\bar{Z}$ from eq. 101, these numerators will
produce NLSM ordered amplitudes. Given this special property of $YZ$-theory
(namely that the $Z\bar{Z}$ and $YY$ constructions are equivalent), the $Y$
particle need not be considered when constructing NLSM amplitudes on shell.
This is essentially what $J$-theory does, as we will show below.

###  A.3 $J$-theory

$J$-theory is obtained through a first order reformulation of NLSM in terms of
the chiral current Cheung:2021zvb .101010For related formulations of NLSM that
do not treat color-kinematics duality see Freedman:1980us ; Slavnov:1971mz .
Two constraints are placed on the chiral current. First, the field strength
associated with $J_{\mu}$ should vanish,

$F_{\mu\nu}(J)=\partial_{\mu}J_{\nu}-\partial_{\nu}J_{\mu}+g[J_{\mu},J_{\nu}]=0\,,$
(112)

which then implies that $J^{\mu}$ is pure gauge
$J_{\mu}=U\partial_{\mu}U^{-1}$. The second condition is that the chiral
current should be in Lorenz gauge,

$\partial_{\mu}J^{\mu}=0\,,$ (113)

which then yields the NLSM equation of motion
$\partial_{\mu}(U\partial^{\mu}U^{-1})=0$. Taking a linear combination of eq.
112 and eq. 113 yields

$\Box J^{\mu}+gf^{abc}J^{\nu}\partial_{\nu}J^{\mu}=0.$ (114)

Integrating in an auxiliary field $\bar{J}^{\mu}$ as a Lagrange multiplier
trivially produces the Lagrangian of semi-abelian YM eq. 41 but with the gauge
fields renamed to $J_{\mu}$ and $\bar{J}_{\mu}$. Since both theories are in
Lorenz gauge this means that _$J$ -theory and semi-abelian YM are identical_.
Furthermore, this theory is one-to-one with the amplitudes of $YZ$-theory when
only considering on-shell $Z$$\bar{Z}$ states. As such, the single vertex in
this theory is the same as the pure vector vertex of $YZ$ theory in Lorenz
gauge,

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \diagram{ (mid1)
--[ultra thick,fermion2,](a3), (mid1) --[ultra thick,fermion2,](a2), (a1)
--[ultra thick,fermion2,](mid1), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=i(\varepsilon_{1}p_{2})(\varepsilon_{2}\bar{\varepsilon}_{3})-i(\varepsilon_{2}p_{1})(\varepsilon_{1}\bar{\varepsilon}_{3})\,.$
(115)

By calculating an off-shell four-point correlation function it is possible to
check that the Lagrangians of $J$-theory and semi-YM theory automatically
satisfies color-kinematics duality, as described in section 2.2. As might be
expected, the kinematic algebra is the diffeomorphism algebra Cheung:2021zvb .
Note that the different external particles break manifest Bose symmetry.

###  A.4 Zakharov-Mikhailov theory

While $J$-theory manifests color-kinematics duality, it only does so at tree
level because $J$-theory breaks manifest Bose symmetry. Manifest color-
kinematics duality can be achieved at all loop orders by restricting
$J$-theory to two spacetime dimensions where the chiral current can be
dualized to a scalar $J^{\mu}=\varepsilon^{\mu\nu}\partial_{\nu}\varphi$,
which trivially enforces Lorenz gauge. After plugging $\varphi$ into eq. 114
and rearranging, another copy of $\varphi$ (rather than some new field) can be
integrated in to obtain the Zakharov-Mikhailov (ZM) Lagrangian,

$\mathcal{L}^{\text{ZM}}=\frac{1}{2}(\partial\varphi)^{2}+gf^{abc}\varphi^{a}\varepsilon^{\mu\nu}(\partial_{\mu}\varphi^{b})(\partial_{\nu}\varphi^{c}).$
(116)

This procedure makes it clear that ZM and NLSM encode the same classical
physics, since their equations of motion are dual to each other. Furthermore,
ZM theory can also be motivated from SDYM theory. The power counting of the
interaction term in the SDYM equation of motion eq. 87 makes it clear that the
theory is critical in 2D. This suggests that the two lightcone coordinates $u$
and $w$ should be identified as the coordinates of the 2D space. The 4D
d’Alembertian in lightcone coordinates,
$\Box=4(\partial_{u}\partial_{v}-\partial_{w}\partial_{\bar{w}})$, cannot be
interpreted as the 2D equivalent, so the kinetic term of SDYM must be replaced
with the correct 2D version. However, changing the kinetic term in eq. 14 will
have no effect on the Jacobi identity of eq. 17, and thus does not interfere
with color-kinematics duality.

The color-stripped Feynman rule for the three-point vertex can be read off
from the Lagrangian eq. 116, giving

$\langle ab\rangle\equiv p_{a}^{\mu}\varepsilon_{\mu\nu}p_{b}^{\nu}\,.$ (117)

Unlike $J$-theory and $YZ$-theory, the color-stripped vertex is functionally
antisymmetric on all legs

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \diagram{ (mid1)
--[ultra thick,,nhpBlue](a3), (mid1) --[ultra thick,,nhpBlue](a2), (a1)
--[ultra thick,,nhpBlue](mid1), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\langle
12\rangle=\langle 23\rangle=\langle 31\rangle=-\langle 21\rangle\,.$ (118)

This key property is due to momentum conservation alone. To prove off-shell
color kinematics duality for any multiplicity and loop order it is enough to
observe that the sum of the off-shell $s$-, $t$-, and $u$-channel numerators
sum to zero

$\langle ab\rangle\langle cd\rangle+\langle ac\rangle\langle db\rangle+\langle
ad\rangle\langle bc\rangle=0$ (119)

by virtue of the 2D Levi-Civita identity
$\varepsilon_{\mu\nu}\varepsilon_{\rho\sigma}=\eta_{\mu\rho}\eta_{\nu\sigma}-\eta_{\mu\sigma}\eta_{\nu\rho}$.
Amusingly, eq. 119 is simply a manifestation of the Schouten identity. Since
ZM theory manifests color-kinematics duality without any reference to the on-
shell condition, a mass term can be added to ZM theory without spoiling color-
kinematics duality Cheung:2022mix . In general, if the on-shell conditions are
not used, color-kinematics duality is unaffected by the details of the
propagator structure.

Of course, dramatically altering the pole structure could spoil locality in
the double-copied theory. As already mentioned, the only difference
classically between ZM and SDYM is the propagator structure. While both
theories are color dual for the same reason, altering the propagators has
profound consequences since SDYM tree amplitudes vanish and manifest color-
kinematics does not persist to all loop orders. Color dual theories with the
same interactions but different kinetic terms have appeared in the literature
before. For example, $J$-theory and the theory of non-Abelian fluids presented
in Cheung:2020djz  differ only in their propagators. Specifically, taking the
static limit of the fluid’s $(D+1)$-dimensional kinetic term
$\partial_{t}-\nabla^{2}$ and Wick rotating one of the coordinates yields the
relativistic d’Alembertian of $D$-dimensional $J$-theory.

While the color-dual nature of ZM theory is rather elegant, the loop-level
construction introduces complications when applying dimensional
regularization. Similar to the difficulties of renormalizing chiral fermions
tHooft:1972tcz , there is ambiguity in promoting the integrands to formal
$D$-dimensional expressions. As such, in our two-loop bootstrap of NLSM we
only used the ZM integrands as a mechanism for reducing the residual gauge
freedom in our final solution.

###  A.5 Chern-Simons theory

The final theory that is unified – through one loop – by the semi-abelian
Yang-Mills construction is Chern-Simons (CS) theory. As we will show below,
plugging in appropriate on-shell states to the semi-YM numerators that we
constructed in the text will produce precisely the kinematic numerators for CS
theory. Much like SDYM and ZM theory, the CS action has a preferred integer
dimension of $D=3$,

$S_{\text{CS}}=\int
d^{3}x\,\text{tr}\left[\epsilon^{\mu\nu\rho}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2}{3}A_{\mu}A_{\nu}A_{\rho}\right)\right]$
(120)

where $\epsilon^{\mu\nu\rho}$ is a 3D Levi-Civita symbol. The color-dual
properties of this theory were previously studied in Ref. Ben-Shahar:2021zww ,
which demonstrated that color-kinematics duality is manifest off-shell. At the
classical level, CS is clearly related to $J$-theory (and thus semi-YM) since
the CS equation of motion,

$\epsilon^{\rho\mu\nu}F_{\mu\nu}=0\Leftrightarrow F_{\mu\nu}=0\,,$ (121)

and the Lorenz gauge choice,

$\partial_{\mu}A^{\mu}=0\,,$ (122)

are the same as the $J$-theory conditions in eqs. 112 and 113, respectively.

The correspondence between CS and semi-YM can be made more precise at the
level of the Feynman rules. The color-stripped three-point vertex and
propagator can be read off from the CS action in eq. 120 producing

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \diagram{ (mid1)
--[ultra thick,nhpRed,](a3), (mid1) --[ultra thick,nhpRed,](a2), (a1) --[ultra
thick,nhpRed,](mid1), }; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\epsilon^{\mu\nu\rho}\varepsilon^{\mu}_{1}\varepsilon^{\nu}_{2}\varepsilon^{\rho}_{3}\qquad\qquad{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a2) at
(1,0){$A^{\mu}$}; \vertex(a3) at (-1,0){$A^{\nu}$}; \vertex(mid1) at
(0,.3){$k\rightarrow$}; \diagram{ (a3) --[ultra thick,nhpRed](a2), };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{i}{k^{2}}\epsilon^{\mu\nu\rho}k^{\rho}\,.$
(123)

At first glance, this seems far removed from the semi-abelian Yang-Mills
analogs in eq. 44 and eq. 45. After all, semi-YM makes no reference to an
antisymmetric 3-tensor and the mass dimensions of the vertices do not agree.
However, to construct the full on-shell amplitudes of CS theory, we must plug
in the Fourier transforms of the on-shell currents, $\bar{\varepsilon}(p)$, as
described in Ref. Ben-Shahar:2021zww . These are related to the polarizations,
$\varepsilon_{i}$, of the Feynman rules above as follows:

$\varepsilon^{\mu}(k)=\epsilon^{\mu\nu\rho}k^{\nu}\bar{\varepsilon}^{\rho}(k)\,.$
(124)

With this definition of the on-shell states, it is clear how the Feynman rules
of eq. 123 are related to those of eq. 44 and eq. 45 – we must simply
canonicalize the CS kinetic term by absorbing $\epsilon^{\mu\nu\rho}k^{\rho}$
into the definition of the three-point vertex:

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a2) at
(1,0){$A^{\mu}$}; \vertex(a3) at (-1,0){$A^{\nu}$}; \vertex(mid1) at
(0,.3){$k\rightarrow$}; \diagram{ (a3) --[ultra thick,nhpRed](a2), };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{i}{k^{2}}\epsilon^{\mu\nu\rho}k^{\rho}\quad\longrightarrow\quad{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex(a2) at
(1,0){$\bar{A}^{\mu}$}; \vertex(a3) at (-1,0){$A^{\nu}$}; \vertex(mid1) at
(0,.3){$k\rightarrow$}; \diagram{ (a3) --[ultra thick,fermion2](a2), };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{i}{k^{2}}\eta^{\mu\nu}\,.$
(125)

This has the effect of orienting the three-point vertex of eq. 123 by making
the replacement
$\varepsilon_{i}\rightarrow\epsilon^{\mu\nu\rho}k_{i}^{\nu}\bar{\varepsilon}^{\rho}_{i}$
on one of the legs,

$\displaystyle{\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \diagram{ (mid1)
--[ultra thick,nhpRed,](a3), (mid1) --[ultra thick,nhpRed,](a2), (a1) --[ultra
thick,nhpRed,](mid1), }; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\rightarrow{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\feynman \vertex[dot,
scale=2.6](mid1) at (0,0){}; \vertex[dot, scale=2.6](mid2) at (0,0){};
\vertex[dot, scale=2,hgrey0](mid3) at (0,0){}; \vertex(a1) at (0,1){3};
\vertex(a2) at (.85,-.55){2}; \vertex(a3) at (-.85,-.55){1}; \diagram{ (mid1)
--[ultra thick,,fermion2](a3), (mid1) --[ultra thick,,fermion2](a2), (a1)
--[ultra thick,,fermion2](mid1), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\epsilon^{\mu\nu\rho}\varepsilon^{\mu}_{1}\varepsilon^{\nu}_{2}(\epsilon^{\rho\alpha\beta}k_{3}^{\alpha}\bar{\varepsilon}^{\beta}_{3})=(\varepsilon_{1}\bar{\varepsilon}_{3})(\varepsilon_{2}k_{3})-(\varepsilon_{2}\bar{\varepsilon}_{3})(\varepsilon_{1}k_{3})\,,$
(126)

where we have used the identity
$\epsilon^{\mu\nu\rho}\epsilon^{\rho\alpha\beta}=\eta^{\mu\alpha}\eta^{\nu\beta}-\eta^{\mu\beta}\eta^{\nu\alpha}$.
As desired, the redefined CS vertex is precisely the three-point vertex of
semi-YM in eq. 44. Clearly the $D$-dimensional amplitudes of semi-YM can be
mapped to the 3D observables of CS theory by making the following replacement
on all of the external states of semi-abelian YM:

$\varepsilon^{\mu}_{i}\rightarrow\epsilon^{\mu\nu\rho}k^{\nu}_{i}\bar{\varepsilon}^{\rho}_{i}\qquad\qquad\bar{\varepsilon}_{i}^{\mu}\rightarrow\bar{\varepsilon}^{\mu}_{i}\,.$
(127)

Upon making this replacement, the amplitudes of $D$-dimensional semi-YM are
mapped onto CS theory in $D=3$, $\mathcal{A}^{\text{semi-
YM}}\rightarrow\mathcal{A}^{\text{CS}}$. While this may seem like a marginal
gain at tree-level, by using our $D$-dimensional construction at one-loop, the
manifestly color-dual integrands of eq. 53 can resolve regularization
ambiguity of CS theory due to its explicit dependence on $D=3$. For more
background on CS theory, we refer the reader to Ref. Ben-Shahar:2021zww .

##  Appendix B Spinor-helicity and conventions

We use the same conventions as Ref. jjmcTASI2014 , which we quote now. The
component-wise definitions of the spinor brackets are

$\displaystyle\langle ab\rangle$
$\displaystyle=\frac{(a_{1}+ia_{2})(b_{0}+b_{3})-(b_{1}+ib_{2})(a_{0}+a_{3})}{\sqrt{(a_{0}+a_{3})(b_{0}+b_{3})}}\,,$
(128) $\displaystyle[ab]$
$\displaystyle=\frac{(b_{1}-ib_{2})(a_{0}+a_{3})-(a_{1}-ia_{2})(b_{0}+b_{3})}{\sqrt{(a_{0}+a_{3})(b_{0}+b_{3})}}\,,$
(129)

where the $a_{i}$ are components of the four-vector,
$k^{\mu}_{a}=(a_{0},a_{1},a_{2},a_{3})$. For massless momenta, $k_{a}$ and
$k_{b}$, one can apply the on-shell condition, $k_{a}^{2}=k_{b}^{2}=0$, to
show that

$s_{ab}=(k_{a}+k_{b})^{2}=\langle ab\rangle[ba]\,.$ (130)

Four-dimensional polarization dot products with fixed helicity states are
mapped to

$\displaystyle k_{a}\cdot\varepsilon_{b}^{(+)}$ $\displaystyle=\frac{\langle
qa\rangle[ab]}{\sqrt{2}\langle qb\rangle}\,,\qquad\quad\qquad
k_{a}\cdot\varepsilon_{b}^{(-)}=-\frac{[qa]\langle
ab\rangle}{\sqrt{2}[qb]}\,,$ (131)
$\displaystyle\varepsilon_{a}^{(-)}\cdot\varepsilon_{b}^{(+)}$
$\displaystyle=-\frac{\langle qa\rangle[qb]}{[qa]\langle
qb\rangle}\,,\qquad\qquad\varepsilon_{a}^{(\pm)}\cdot\varepsilon_{b}^{(\pm)}=0\,,$

Note that given the above definition, the spinor helicity variables carry the
same mass dimension for both the angle and square brackets. In terms of the
light-cone coordinates that we employed in appendix A, the angles and squares
can be rewritten as

$\displaystyle\langle ab\rangle$
$\displaystyle=\frac{a_{w}b_{u}-a_{u}b_{w}}{\sqrt{a_{u}b_{u}}}\,,$ (132)
$\displaystyle[ab]$
$\displaystyle=\frac{b_{\bar{w}}a_{u}-a_{\bar{w}}b_{u}}{\sqrt{a_{u}b_{u}}}\,.$
(133)

The conventions we use in the text where the holomorphic spinor, $X(p,k)$,
carries extra mass dimension amounts to shifting the bracket by an overall
factor,

$\displaystyle\langle ab\rangle$ $\displaystyle\rightarrow
a_{w}b_{u}-a_{u}b_{w}\equiv X(a,b)\,,$ (134) $\displaystyle[ab]$
$\displaystyle\rightarrow\frac{b_{\bar{w}}a_{u}-a_{\bar{w}}b_{u}}{a_{u}b_{u}}\equiv
Q(a,b)\,.$ (135)

With this choice we still obtain the same completeness relation for the SDYM
spinors,

$X(a,b)Q(b,a)=s_{ab}\,.$ (136)

##  Appendix C Regulating BEL integrals

This appendix covers the bubble-on-external-leg (BEL) integrals that result
from the $n$-gon numerator of $Y\\!Z$ theory. Before covering the BEL
integrals themselves, we will need the appropriate numerator. Weight counting
tells us that the $n$-gon master numerator must have $n$ on-shell
$Z$-particles. Unitarity requires that there are three distinct contributions
to the $n$-gon: one from an off-shell $Y$-loop particle and then two more from
different orientations of a $\bar{Z}Z$-loop. Thus, $YZ$ theory gives us the
following one-loop $n$-gon numerator:

$N^{n\text{-gon},YZ}_{(12...n)}=\langle T^{1}T^{2}\cdots T^{n}\rangle+\langle
F^{1}F^{2}\cdots F^{n}\rangle$ (137)

where $\langle\,\cdots\rangle$ indicates an internal contraction over the $YY$
and $\bar{Z}Z$ loops and $T$ and $F$ were given in eq. 95 and eq. 99,
respectively. In terms of the external momenta $k_{i}$ and the loop momenta
$\ell_{i}$, where $\ell_{i}$ flows into $k_{i}$ and out of $k_{i-1}$, the NLSM
numerator is

$N_{(12...n)}^{\text{NLSM}}=\langle T^{1}T^{2}\cdots
T^{n}\rangle=2^{n}(\ell_{1}k_{1})(\ell_{2}k_{2})\cdots(\ell_{n}k_{n})$ (138)

which is the same as eq. 55 after using
$2(k_{i}\cdot\ell_{i})=\ell_{i}^{2}-\ell^{2}_{i-1}$. Similarly, there is a
pure vector contribution that is identical to the semi-abelian YM $n$-gon
numerator that we constructed in eq. 53 after projecting external states along
longitudinal modes $\varepsilon\rightarrow k$,

$\langle F^{1}F^{2}\cdots
F^{n}\rangle=D(\ell_{1}k_{1})(\ell_{2}k_{2})\cdots(\ell_{n}k_{n})+\mathcal{O}(D^{0})\,.$
(139)

The dimension-dependent factor essentially counts that number of internal
vector states. While this $n$-gon is manifestly color-dual, it does not
produce the right cuts for NLSM. However, the scalar contribution, that comes
dressed with an overall factor $D$ does manifest the duality globally, and
satisfies all the desired pion cuts due to the factorization of eq. 110 and
eq. 111. In order for the Feynman rules for $Y\\!Z$ theory compute one-loop
color-dual numerators consistent with NLSM cuts, one would have to add
additional states to cancel off the spurious poles, while preserving color-
kinematics duality. We leave this as a direction of future work.

While the $\bar{Z}Z$ vector loop spoils color-kinematics off-shell, the
$\bar{Y}Y$ loop alone gives us the desired expression for the $n$-gon.
However, as we noted in the text, the $n$-gon numerator has the strange
property that it produces non-vanishing bubble-on-external-leg (BEL) graphs.
However, the BEL diagram integrates to zero after integral reduction. As an
example, the four-point BEL diagram can be reconstructed from the $n$-gon,

$N^{\text{BEL}}_{1|2,34}=N^{\text{NLSM}}_{(1234|\ell)}-N^{\text{NLSM}}_{(1243|\ell)}-N^{\text{NLSM}}_{(1342|\ell)}+N^{\text{NLSM}}_{(1432|\ell)}\,,$
(140)

where we define the loop momentum to be in between the left most and right
most leg on the box. Plugging in particular values for $\ell_{i}$, the BEL
reduces to

$N^{\text{BEL}}_{1|2,34}=s_{12}(\ell+k_{1})^{2}\ell^{\mu}k_{1}^{\nu}k_{2}^{[\mu}k_{[34]}^{\nu]}\,.$
(141)

Notice there is an overall factor that cancels one of the propagators.
Plugging this numerator into the amplitude produces an integral of the form

$\mathcal{I}^{\text{BEL}}_{1|2,34}=s_{12}k_{1}^{\nu}k_{2}^{[\mu}k_{[34]}^{\nu]}\int\frac{d^{D}\ell}{i\pi^{D/2}}\frac{\ell^{\mu}}{\ell^{2}-\mu^{2}}\sim
s_{12}(s_{13}-s_{14})(\mu^{2})^{D/2}\,,$ (142)

where we have introduced a mass regulator that will be proportional to the on-
shell momentum inside the BEL, $\mu^{2}\equiv k_{1}^{2}$. Thus, in
sufficiently large dimension, $D>2$, this integral suppresses the $\mu^{-2}$
divergence appearing in the denominator of the BEL diagram.

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