}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\!\\!$
$\displaystyle=\frac{1}{s_{123}}\left[{\scalebox{0.85}{ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue4](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{0}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue1](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{9}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\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=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue3](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{3}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue2](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{6}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\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=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue2](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{6}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue3](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{3}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\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=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue1](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{9}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue4](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{0}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right]\Rightarrow
c_{(3,0)}=\frac{c_{(1,0)}^{3}}{3!}$ (80) $\displaystyle{\lim_{s_{123}\to
0}}\,\,{\scalebox{0.85}{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F^{4}$}; \vertex(b1) at (-2.8,.6) {3};
\vertex(b2) at (-2.8,0) {2}; \vertex(b3) at (-2.8,-.6) {1}; \vertex(b4) at
(-.2,.6) {4}; \vertex(b6) at (-.2,0) {5}; \vertex(b5) at (-.2,-.6) {6};
\vertex[dot, scale=1,hgrey0](mid1a) at (-1.5,0) {$D^{2n+}F^{4}$};
\vertex(mid1b) at (-1.5,0) {\large$A_{6}^{(12)}$}; \diagram{ (mid1a) --[ultra
thick](b1), (mid1a) --[ultra thick](b2), (mid1a) --[ultra thick](b3), (mid1a)
--[ultra thick](b4), (mid1a) --[ultra thick](b6), (mid1a) --[ultra thick](b5),
}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\!\\!$
$\displaystyle=\frac{1}{s_{123}}\left[{\scalebox{0.85}{ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue4](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{0}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue5](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{12}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\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=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue3](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{3}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue1](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{9}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\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=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue2](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{6}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue2](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{6}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\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=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue1](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{9}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue3](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{3}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>\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=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue5](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {$\alpha^{12}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue4](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {$\alpha^{0}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right]\Rightarrow
c_{(4,0)}=\frac{c_{(1,0)}^{4}}{4!}$ (81)

In general, we expect to find the following set of factorization relations for
all orders in $\alpha^{\prime n}$:

$\mathcal{O}(\alpha^{\prime 3n}):\quad\sum_{k=0}^{n}c_{(n,0)}{n\choose
k}\sigma_{(123)}^{k}\sigma_{(456)}^{n-k}=\sum_{k=0}^{n}c_{(k,0)}c_{(n-k,0)}\sigma_{(123)}^{k}\sigma_{(456)}^{n-k}$
(82)

Which implies that the Wilson coefficients, $c_{(n,0)}$ at four-point that
label matrix elements of the form $\sigma_{3}^{n}A_{4}$, should precisely
exponentiate:

$c_{(n,0)}=\frac{c_{(1,0)}^{n}}{n!}$ (83)

We will now generalize this construction towards the all-multiplicity
structure and argue that one should expect double-copy consistency to
introduce exponentiation whenever a new contact is added to a color-dual EFT.

###  V.3 All-multiplicity Structure

We believe that this exponential behavior is likely a universal property of
higher-derivative contact terms in double-copy consistent theories, and not
just a special property of four-point permutation invariants. In the previous
section we just considered how $\sigma_{3}^{n}A_{4}^{\text{YM}}$ type contacts
exponentiate. Instead, consider a color-dual $n$-point contact, $X_{n}$, with
dimensionful coupling at $\mathcal{O}(\Lambda)$, that is constructed from
transformation matrix $\Omega_{n}$ acting on an $(n-3)!$ basis of $n$-point
vector amplitude at $\mathcal{O}(\Lambda^{0})$. Then we can define a general
family of degree $k$ contact operators, $X^{(k)}_{n}$, as follows,

${\scalebox{1.0}{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F$}; \vertex[dot,
scale=1,hgrey0](mid1a) at (-1.5,0) {$D^{2n+}F$}; \vertex(mid1b) at (-1.5,0)
{$X_{n}^{(k)}$}; \vertex(b1) at (-2.6,.8) {$2$}; \vertex(b2) at (-2.8,0)
{$1$}; \vertex(b3) at (-2.6,-.8) {$n$}; \vertex(b6) at (-.6,.1) {$\vdots$};
\diagram{ (mid1a) --[ultra thick](b1), (mid1a) --[ultra thick](b2), (mid1a)
--[ultra thick](b3), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=c_{(k)}\Omega_{n}^{k}A_{n}^{(0)}$
(84)

where $c_{(k)}$ is an unconstrained Wilson coefficient fo the $X^{k}_{n}$
operator. Now we can promote the product structure of these contacts to higher
orders in $\alpha^{\prime}$ by taking a series of $\text{N}^{k}\text{Max}$
cuts at successively higher point amplitudes, and imposing color-kinematics at
each step:

$\mathcal{O}(\Lambda^{k})\begin{cases}\begin{split}{\textit{color-
dual}\atop\textit{constraints}}\,\,+&\,\,{\scalebox{1.0}{ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F$}; \vertex[dot,
scale=1,gray!10](mid1a) at (-1.5,0) {$D^{2n+}F$}; \vertex[dot,
opacity=.3,scale=1,nhpBlue5](mid1a) at (-1.5,0) {$\qquad\quad\,$};
\vertex(mid1b) at (-1.5,0) {$A_{n+1}^{(k)}$}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\Bigg{|}^{\text{MC}}\,=\,{\scalebox{1.0}{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.4,0) {$D^{2n}\,F$}; \vertex[dot,
scale=1,hgrey0](mid1a) at (-1.4,0) {$D^{2n}\,F$}; \vertex(mid1b) at (-1.4,0)
{$\,X^{(k)}_{n}$}; \vertex[dot, scale=1.08](mid2) at (.3,0) {$\alpha^{n+}$};
\vertex(a1) at (-.45,-.4) ; \vertex(a2) at (-.45,.44) ; \vertex(mid3) at (0,0)
{} ; \diagram{ (mid1) --[ thick](mid2), (a1) --[thick,scalar,nhpRed](a2), };
\vertex[dot, scale=1,white](mid2a) at (.3,0) {$\alpha^{n+}$};
\vertex[dot,opacity=0.4, scale=5.2,nhpBlue4](mid2c) at (.3,0) {};
\vertex(mid2b) at (.3,0) {$A_{3}$}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad\Rightarrow\quad
A_{n+1}^{(k)}=c_{(k)}\Omega^{k}_{n+1}A^{(0)}_{n+1}\\\ {\textit{color-
dual}\atop\textit{constraints}}\,\,+&\,\,{\scalebox{1.0}{ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F$}; \vertex[dot,
scale=1,gray!10](mid1a) at (-1.5,0) {$D^{2n+}F$}; \vertex[dot,
opacity=.45,scale=1,nhpBlue5](mid1a) at (-1.5,0) {$\qquad\quad\,$};
\vertex(mid1b) at (-1.5,0) {$A_{n+2}^{(k)}$}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\Bigg{|}^{\text{N}^{1}\text{MC}}\\!\\!\\!\\!=\,{\scalebox{1.0}{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.4,0) {$D^{2n}\,F$}; \vertex[dot,
scale=1,gray!10](mid1a) at (-1.4,0) {$D^{2n}\,F$}; \vertex[dot,
opacity=.3,scale=1,nhpBlue5](mid1a) at (-1.4,0) {$\qquad\,\,\,\,\,$};
\vertex(mid1b) at (-1.4,0) {$A_{n+1}^{(k)}$}; \vertex[dot, scale=1.08](mid2)
at (.3,0) {$\alpha^{n+}$}; \vertex(a1) at (-.45,-.4) ; \vertex(a2) at
(-.45,.44) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[ thick](mid2),
(a1) --[thick,scalar,nhpRed](a2), }; \vertex[dot, scale=1,white](mid2a) at
(.3,0) {$\alpha^{n+}$}; \vertex[dot,opacity=0.4, scale=5.2,nhpBlue4](mid2c) at
(.3,0) {}; \vertex(mid2b) at (.3,0) {$A_{3}$};
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad\Rightarrow\quad
A_{n+2}^{(k)}=c_{(k)}\Omega^{k}_{n+2}A^{(0)}_{n+2}\\\
\downarrow\qquad\quad&\qquad\qquad\,\qquad\qquad\qquad\downarrow\qquad\qquad\qquad\qquad\quad\downarrow\\\
{\textit{color-dual}\atop\textit{constraints}}\,\,+&\,\,{\scalebox{1.0}{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F$}; \vertex[dot,
scale=1,gray!10](mid1a) at (-1.5,0) {$D^{2n+}F$}; \vertex[dot,
opacity=.75,scale=1,nhpBlue5](mid1a) at (-1.5,0) {$\qquad\quad\,$};
\vertex(mid1b) at (-1.5,0) {$A_{2n-2}^{(k)}$};
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\Bigg{|}^{\text{N}^{n-2}\text{MC}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\,{\scalebox{1.0}{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.4,0) {$D^{2n}\,F$}; \vertex[dot,
scale=1,gray!10](mid1a) at (-1.4,0) {$D^{2n}\,F$}; \vertex[dot,
opacity=.6,scale=1,nhpBlue5](mid1a) at (-1.4,0) {$\qquad\,\,\,\,\,$};
\vertex(mid1b) at (-1.4,0) {$A_{2n-3}^{(k)}$}; \vertex[dot, scale=1.08](mid2)
at (.3,0) {$\alpha^{n+}$}; \vertex(a1) at (-.45,-.4) ; \vertex(a2) at
(-.45,.44) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[ thick](mid2),
(a1) --[thick,scalar,nhpRed](a2), }; \vertex[dot, scale=1,white](mid2a) at
(.3,0) {$\alpha^{n+}$}; \vertex[dot,opacity=0.4, scale=5.2,nhpBlue4](mid2c) at
(.3,0) {}; \vertex(mid2b) at (.3,0) {$A_{3}$};
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\quad\Rightarrow\quad
A_{2n-2}^{(k)}=c_{(k)}\Omega^{k}_{2n-2}A^{(0)}_{2n-2}\end{split}\end{cases}$
(85)

The blue interaction regions are three-point vertices at
$\mathcal{O}(\alpha^{\prime\,0})$, which do not contribute to the mixing of
Wilson coefficients. This is parallel to the first step in checking six-point
factorization above, where consistent factorization down to five-point
promoted the product structure to the six-point factorization channel.
Carrying out double-copy consistency up to $\text{N}^{n-2}\text{MCut}$ level,
allows us to now impose a split factorization constraint on the
$s_{12...n\\!-\\!1}$ pole. At this level, the $A^{k}_{2n-2}$ amplitude would
factor into two contact diagrams that set the seed of our recursion:

$\displaystyle{\textit{color-dual}\atop\textit{constraints}}\,$
$\displaystyle{\scalebox{0.85}{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F^{4}$}; \vertex(b1) at (-2.8,.6)
{$\,\,$}; \vertex(b1a) at (-3.1,.6) {$n\\!-\\!1$}; \vertex(b2) at (-2.8,.1)
{$\vdots$}; \vertex(b3) at (-2.8,-.6) {$1$}; \vertex(b4) at (-.2,.6) {$n$};
\vertex(b6) at (-.2,.1) {$\vdots$}; \vertex(b5) at (-.2,-.6) {$\,\,\,$};
\vertex(b7) at (.2,-.6) {$2n\\!-\\!2$}; \vertex[dot, scale=1,hgrey0](mid1a) at
(-1.5,0) {$D^{2n+}F^{4}$}; \vertex(mid1b) at (-1.5,0)
{\large$A_{2n-2}^{(k)}$}; \diagram{ (mid1a) --[ultra thick](b1), (mid1a)
--[ultra thick](b3), (mid1a) --[ultra thick](b4), (mid1a) --[ultra thick](b5),
}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\!\\!\\!\\!\\!\\!=\frac{1}{s_{12...n\\!-\\!1}}\sum_{j=0}^{k}c_{(k)}{k\choose
j}\Omega^{\,j}_{(12...n\\!-\\!1)}\Omega^{k-j}_{(n...2n\\!-\\!2)}\,\left[{\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[dot,
scale=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue4](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {\large$A_{n}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue4](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {\large$A_{n}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right]$ (86)
$\displaystyle{\textit{split-channel}\atop\textit{factorization}}\,$
$\displaystyle{\scalebox{0.85}{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F^{4}$}; \vertex(b1) at (-2.8,.6)
{$\,\,$}; \vertex(b1a) at (-3.1,.6) {$n\\!-\\!1$}; \vertex(b2) at (-2.8,.1)
{$\vdots$}; \vertex(b3) at (-2.8,-.6) {$1$}; \vertex(b4) at (-.2,.6) {$n$};
\vertex(b6) at (-.2,.1) {$\vdots$}; \vertex(b5) at (-.2,-.6) {$\,\,\,$};
\vertex(b7) at (.2,-.6) {$2n\\!-\\!2$}; \vertex[dot, scale=1,hgrey0](mid1a) at
(-1.5,0) {$D^{2n+}F^{4}$}; \vertex(mid1b) at (-1.5,0)
{\large$A_{2n-2}^{(k)}$}; \diagram{ (mid1a) --[ultra thick](b1), (mid1a)
--[ultra thick](b3), (mid1a) --[ultra thick](b4), (mid1a) --[ultra thick](b5),
}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\!\\!\\!\\!\\!\\!=\frac{1}{s_{12...n\\!-\\!1}}\sum_{j=0}^{k}c_{(j)}c_{(k-j)}\Omega^{\,j}_{(12...n\\!-\\!1)}\Omega_{(n...2n\\!-\\!2)}^{k-j}\left[{\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[dot,
scale=1.08](mid1) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=1,white](mid1a) at (0,-.6) {$\alpha^{n+}$}; \vertex[dot,
scale=5.3,opacity=0.4,nhpBlue4](mid1a) at (0,-.6) {$$}; \vertex(mid1b) at
(0,-.6) {\large$A_{n}$}; \vertex[dot, scale=1.08](mid2) at (0,.6)
{$\alpha^{n+}$}; \vertex[dot, scale=1,white](mid2a) at (0,.6) {$\alpha^{n+}$};
\vertex[dot, scale=5.3,opacity=0.4,nhpBlue4](mid2a) at (0,.6) {$$};
\vertex(mid2b) at (0,.6) {\large$A_{n}$}; \vertex(a1) at (-.4,0) ; \vertex(a2)
at (.4,0) ; \vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (a1)
--[thick,scalar,nhpRed](a2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\right]$ (87)

Much like at four-point, all-order color-kinematics and factorization would
then require that the Wilson coefficients, $c_{(k)}$, are related to each
other by the binomial expansion of
$(\Omega_{12...n-1}+\Omega_{n+1...2n-2})^{k}$ appearing in eq. (86) – thus
giving following set of relations:

$c_{(k)}=\frac{1}{k!}c_{(1)}^{k}$ (88)

Since we have made no assumptions about the structure of the color-dual
contact beyond it being related to a BCJ basis amplitude,
$X_{n}=\Omega_{n}A_{n}$, we state here an all-order ansatz for the all-
multiplicity structure of double-copy consistent theories:

$A^{\text{UV}}_{n}=\prod_{k}\exp\left[c_{k}\,\Omega^{\text{DCC}}_{k}\right]A^{\text{IR}}_{n}$
(89)

where $\Omega^{\text{DCC}}_{k}$ are $(n-3)!\times(n-3)!$ matrices sourcing
color-dual contact operators that act on an $(n-3)!$ basis of vector
amplitudes at leading order in the EFT expansion. While this remains a
conjecture, motivated by structure uncovered in the four-point $\sigma_{3}$
sector, directly analyzed at five and six-point, we find it exciting to
consider that by making no assumptions about the high energy behavior of an
EFT, exponential factors that are compatible with massive modes encoding UV
soft behavior could be required at all-multiplicity through double-copy
consistency. This is entirely compatible with the position that double-copy
consistency as a physical principle is well situated to bootstrap UV theories
directly from low energy effective operators that live in the IR, which we now
turn to.

##  VI Consistent Massive Resonance Models

Effective field theories allow a consistent parameterization of our ignorance
beyond data relevant to the scales at hand. We, especially in the field of
scattering amplitudes, are accustomed to considering EFT perturbatively,
recognizing a finite bound of validity in relevant energy scales. In the
context of double-copy consistent theories, we have shown how to propagate
Wilson-coefficients from four-points to higher multiplicity within the EFT
framework. Both $\text{DF}^{2}+\text{YM}$ and the SV-map that appears in
closed string and heterotic amplitudes provide evidence for a compelling idea:
Wilson coefficients that encode massive resonance order by order in
$\alpha^{\prime}$, when constrained by double-copy and factorization, should
be compatible with (and in some cases could even be sufficient for) the
interpretation of a consistent massive spectrum to propagate from multiplicity
to multiplicity solely via Wilson coefficients. Such a conjecture allows
massive-resonance model building in terms of massless kinematics starting at
directly from four-points.

Before we continue with this story let us imagine where such a conjecture
could fail. Factorization alone is insufficient to fix contact terms
generically. However, double-copy consistency as we have seen in the emergence
of $\text{DF}^{2}+\text{YM}$ theory establishes uniquely a minimum set of
higher-point contact Wilson coefficients in terms of lower-point Wilson
coefficients as a bootstrap from Yang-Mills deformed by $F^{3}$. In that case,
factorization and the duality between color and kinematics is sufficient to
establish the propagation of massive modes $M$ established as a four-point
global prefactor of simply:

$M^{\text{DF}^{2}+\text{YM}}=\frac{1}{(1-s\alpha^{\prime})(1-t\alpha^{\prime})(1-u\alpha^{\prime})}$
(90)

against a IR theory that at four-points is color-dual.

One might ask if factorization and the requirement of color-kinematics duality
to all multiplicity may be sufficient to entirely fix the $\alpha^{\prime}$
expansion of Chan-Paton dressed Z-theory at five-points. This possibility,
while attractive, is at least not obvious nor necessary. Starting at
$\alpha^{\prime 5}$ in the five-point amplitude there are local color-dual
Wilson-Coefficients required by Z-theory unfixed by factorization to four-
points. The presence of multiple zeta values starting at five-points is a
clear sign that any such constraints imposed by all-multiplicity factorization
would have to be incredibly non-trivial and seem intimately tied to how
massive resonances must interact. It would be an interesting exercise to learn
which Wilson coefficients are ultimately fixed by factorization of color-dual
higher-points amplitudes, and what additional constraints are required for
consistent interaction of mass-spectra multiplicity by multiplicity. That of
course does not rule out a bootstrap, but suggests that additional principles
will be relevant as the multiplicity increases. Given the factorization
consistency of color-dual amplitudes with an exponential structure, we are
confident a resonance color-dual bootstrap starting from four-points exists,
and it will be fascinating to learn the minimal amount of additional
principles required to extract Z-theory.

Of course one can imprint massive resonances of some desired spectrum (from
some hierarchically encompassing EFT) into Wilson-coefficients of color-dual
IR four-point amplitudes. We will discuss how straightforward such a program
is, with some words of caution. We then discuss the much more fascinating
knowledge-discovery problem of backing out information about massive
resonances, directly from a finite set of Wilson coefficients using
resurgence.

###  VI.1 From Masses to Wilson Coefficients

To go from a finite spectrum of masses to Wilson coefficients of our EFT local
operators at four-points, we simply expand in the large effective cutoff
limit. If the spectrum of masses is given in terms of $\mu_{k}$ with mass
scale $\Lambda$,

$M=\prod_{k=1}\frac{P_{k}(\sigma_{2},\sigma_{3})}{(s/\Lambda-\mu_{k})(t/\Lambda-\mu_{k})(u/\Lambda-\mu_{k})}$
(91)

then we simply series expand $\Lambda\to\infty$ and read off the Wilson
coefficients as functions of scaleless $\mu_{k}$ order by order in some
$\alpha^{\prime}=1/\Lambda$. Of course, generic multiplication by terms of
order:

$M=\prod_{k=1}\frac{f_{k}}{(s/\Lambda-\mu_{k})(t/\Lambda-\mu_{k})(u/\Lambda-\mu_{k})}$
(92)

where $f_{k}$ is independent of kinematics, certainly establishes a tower of
massive modes, but are subject to whatever massless factorization properties
are present in the IR theory. One might worry about contact terms of the form
$\sigma_{2}^{m}/s$, which suggest the presence of higher-spin massless
exchange in the EFT. In this paper, we postpone judgement on the matter for
the following reasons:

  * •

Such dubious channels can be disposed of by the other theory appearing in the
double-copy with, e.g. the nonlinear sigma model (NLSM) is famous for
absorbing channels in the double-copy. We will use NLSM as an exemplar of
color-dual model building with massive resonance in the next section.

  * •

If $\sigma_{2}^{m}/s$ issues are resolved via double-copy with all theories of
interest, this can be suggestive of applying adjoint-type double-copy
construction where symmetric $d^{abc}$ type double-copy results in identical
well-behaved double-copy theories but with more physical looking ingredients
Carrasco and Pavao (2023a, b). This strikes us as similar to using a gauge
with spurious poles that cancel in the end — perhaps aesthetically
unappealing, but not actually problematic unless one has real issues in the
final product.

That said, $\sigma_{2}$, glitches can be entirely disposed of if one is
willing to start with an IR theory that is proportional to $\sigma_{3}$.
Another choice of polynomial residue sufficient for canceling unwanted higher
spin modes at four-point would be,
$P_{n}(\sigma_{2},\sigma_{3})=\mu_{n}^{3}-\mu_{n}\sigma_{2}+\sigma_{3}$. Given
such a choice of numerator, the massive spectra are related to the Wilson
coefficients of the exponential factors at four-point as follows:

$c_{(1,m)}=\sum_{n=1}^{\infty}\frac{1}{\mu^{2m+3}_{n}}$ (93)

Clearly, when $\mu_{n}=n$, the Wilson coefficients that we have shown
exponentiate at four-point, just reproduce the zeta values of the single-value
promotion, $c_{(1,m)}=\zeta_{2m+3}$.

We should note that simply engaging with a finite spectrum of massive modes,
while it may lead to incredibly well behaved UV scaling, could introduce
problematic behavior like violating positivity bounds Adams  _et al._
(2006)555See for example refs. de Rham  _et al._ (2017); Bellazzini  _et al._
(2019, 2021); Bern  _et al._ (2021); Creminelli  _et al._ (2022) for recent
applications of imposing positivity constraints., or disturbing partial wave
unitarity upon resummation666A venerable consistency constraint Martin (1963);
Goddard and Thorn (1972); Soldate (1987); Aoki  _et al._ (1990) but see also
e.g. refs. Caron-Huot  _et al._ (2017); Green and Wen (2019); Arkani-Hamed
_et al._ (2022) for recent discussions.. Of course, one is free to keep adding
and tuning modes ad-infinitum until one lands on an EFT that looks like a
healthy UV theory, or be content with an EFT that allows one to describe the
relevant physics at hand to desired accuracy. Either way, empirically there is
a much more pressing problem.

The mass parameters are non-perturbative, requiring information of the full
theory at energy scales beyond the UV cutoff scale,
$\Lambda=1/\alpha^{\prime}$. However, often we only have empirical access to
the Wilson coefficients, $c_{k}$, that are valid below the cutoff scale.
Solving for the mass spectrum analytically runs into an inverse problem where
we must determine an infinite number of degrees of freedom, $\mu_{n}$, from
finite number of outputs, $c_{(1,m)}$ (up to some fixed order in
$\alpha^{\prime}$), and thus may seem forever out of reach from an amplitudes-
based perturbative bootstrap. Fortunately, there are incredibly well-developed
mathematical tools that we can use to extract non-perturbative information
about the spectrum, directly from the Wilson coefficients of color-dual
operators appearing in eq. (89). We turn to these methods now.

###  VI.2 From Wilson Coefficients to Masses

In the remainder of this section we will introduce Padé approximants as an
analytic tool for probing the non-perturbative spectra of double-copy
consistent amplitudes with generic Wilson coefficients. A Padé approximant is
a rational function of the form,

$\mathcal{R}_{[m,n]}(x)\equiv\frac{A_{m}(x)}{B_{n}(x)}\,,$ (94)

where $[m,n]$ is the degree of the polynomials, $A_{m}$ and $B_{n}$,
respectively:

$A_{m}(x)=\sum_{j=0}^{m}a_{j}x^{j},\quad
B_{n}(x)=1+\sum_{j=0}^{n}b_{j}x^{j}\,.$ (95)

The coefficients of $\mathcal{R}_{[m,n]}$ are defined in order to match the
first $m+n$ derivatives when expanded around a point, $x=x_{0}$,

$f^{(k)}(x_{0})=\mathcal{R}^{(k)}_{[m,n]}(x_{0})\qquad k\leq m+n\,.$ (96)

Padé approximants have been used widely in the resurgence literature for
extracting information about non-perturbative physics, like instantons, when
only pertubative information is available. For background, we direct the
reader to refs. Costin and Dunne (2020, 2022); Dunne and Harris (2021, 2023).
For our purposes, we will use them to study the behavior of mass spectra in
generic classes of consistent mass resonance double-copy models, even when the
closed form resummation of the amplitude is not directly computable.

As straightforward example that demonstrates the power of these ideas in the
context of effective field theory, let us consider a scalar effective field
theory that looks like a chiral perturbation theory Weinberg (1979), with two
resonant masses beyond the $\alpha^{\prime}=1$ scale of the pion decay width.
To demonstrate the reach of these non-perturbative methods, we introduce an
order of magnitude separation between resonances with $\mu_{1}=1$ and
$\mu_{2}=10$. In our framework the purely IR amplitude,
$\mathcal{A}_{\text{IR}}$, is just that of the nonlinear sigma model (NLSM), a
known color-dual theory Chen and Du (2014); Cachazo  _et al._ (2015); Cheung
and Shen (2017); Cheung  _et al._ (2018), which is the leading order
contribution in chiral Lagrangian Weinberg (1979). The four-point amplitude is
simply:

$\mathcal{A}_{\text{IR}}=f^{a_{1}a_{2}b}f^{ba_{3}a_{4}}(s_{23}-s_{13})+f^{a_{4}a_{1}b}f^{ba_{2}a_{3}}(s_{12}-s_{13})+f^{a_{3}a_{1}b}f^{ba_{4}a_{2}}(s_{23}-s_{12})\,.$
(97)

To recover the pions of the Standard Model, one simply treats the structure
constants as describing some flavor symmetry, speficying $SU(2)$ for the
desired isospin symmetry of low energy QCD. We will embed the UV mass spectrum
encoded by,

$M^{\text{2-mode}}=\frac{1\times
10^{3}}{(s\alpha^{\prime}-1)(t\alpha^{\prime}-1)(u\alpha^{\prime}-1)(s\alpha^{\prime}-10)(t\alpha^{\prime}-10)(u\alpha^{\prime}-10)}\,.$
(98)

Again, $\alpha^{\prime}=1/\Lambda$, where $\Lambda$ is the cutoff scale and
$\alpha^{\prime}$ is the pion decay width for the NLSM. For the purposes of
this exercise, we take ourselves as living deep in the IR and so only have
experimental access to the Wilson coefficients measured in some fixed-angle
scattering, taking $u=-s-t$ and $t=-s/2$, up to some finite precision.

Let us consider the difference between what we can infer with Padé
approximants if we have access to 6 vs 10 Wilson coefficients, to a precision
of four significant digits777Perhaps we should emphasize that this does not
come for free. In order to measure the Wilson coefficients to such precision
at $\mathcal{O}(\alpha^{\prime})^{10}$ deep in the IR requires astoundingly
accurate experiments.. The relevant expansions are subsets of the series
expansion,

$\displaystyle\mathcal{A}$
$\displaystyle=\mathcal{A}^{\text{IR}}\times\left[1.000+1.010(\alpha^{\prime})^{2}\sigma_{2}+1.001(\alpha^{\prime})^{3}\sigma_{3}+1.010(\alpha^{\prime})^{4}\sigma_{2}^{2}+2.01(\alpha^{\prime})^{5}\sigma_{2}\sigma_{3}\right.$
$\displaystyle~{}+(\alpha^{\prime})^{6}\left(1.010\sigma_{2}^{3}+1.001\sigma_{3}^{2}\right)+3.02(\alpha^{\prime})^{7}\sigma_{2}^{2}\sigma_{3}+(\alpha^{\prime})^{8}\left(1.010\sigma_{2}^{4}+3.01\sigma_{2}\sigma_{3}^{2}\right)$
$\displaystyle~{}\left.+(\alpha^{\prime})^{9}\left(4.03\sigma_{2}^{3}\sigma_{3}+1.001\sigma_{3}^{3}\right)+(\alpha^{\prime})^{10}\left(1.01\sigma_{2}^{5}+6.03\sigma_{2}^{2}\sigma_{3}^{2}\right)+\mathcal{O}(\alpha^{\prime})^{11}\right]\,.$
(99)

Of course any perturbative expansion will fail to catch even the first
resonance. Even with access to only four-significant digits, the
$\mathcal{R}_{[3,3]}$ approximant to the $\mathcal{O}(\alpha^{\prime})^{6}$
expansion captures the first resonance, and the $\mathcal{R}_{[5,5]}$
approximant to the $\mathcal{O}(\alpha^{\prime})^{10}$ expansion manages to
encode both resonances. We plot the fixed-angle predictions for the $A(1234)$
ordered amplitude888The coefficient of $f^{a_{1}a_{2}b}f^{ba_{3}a_{4}}$ after
using the Jacobi identity to rewrite
$f^{a_{4}a_{1}b}f^{ba_{2}a_{3}}=f^{a_{1}a_{2}b}f^{ba_{3}a_{4}}-f^{a_{3}a_{1}b}f^{ba_{4}a_{2}}$
in eq. (VI.2) as per section II. in Fig. 2.

As we discuss in the conclusion, leveraging non-perturbative data from such a
small number of Wilson-coefficients has a number of potential opportunities
both for empirical expectations for new signals as well as formal insight
relating non-perturbative behavior of theories participating in the double-
copy web of theories.

Figure 2: Expanded series expansions, $\mathcal{O}(\alpha^{\prime})^{6}$ and
$\mathcal{O}(\alpha^{\prime})^{10}$, as well as Padé approximant
$\mathcal{R}_{[3,3]}$ to the $\mathcal{O}(\alpha^{\prime})^{6}$ expansion
(left) plotted against the fully resummed UV function. No matter how high the
perturbative expansion, it will never capture even the first emergent
resonance, but the behavior is spectacularly captured in the
$\mathcal{R}_{[3,3]}$ approximant. On the second panel (right), we show the
behavior of Padé approximants $\mathcal{R}_{[3,3]}$ to the
$\mathcal{O}(\alpha^{\prime})^{6}$ series expansion and $\mathcal{R}_{[5,5]}$
to the $\mathcal{O}(\alpha^{\prime})^{10}$ series expansion near the domain of
the second resonance. We find that a relatively small number of Wilson
coefficients allow us in $\mathcal{R}_{[5,5]}$ to recover the full spectrum of
massive modes.

##  VII Conclusions

In this work we have demonstrated two mechanisms for understanding UV behavior
of scattering amplitudes directly from perturbative information in the IR –
double-copy consistency of effective field theory, and Padé approximants for
fixed-angle 2-to-2 scattering amplitudes. To properly frame these approaches,
we provide below a brief outline of the main results of the paper and future
directions made possible by this work.

###  VII.1 Summary

#### Menu of Massive Resonance

We began with an overview of the interplay between massive resonance and
double-copy constructible theories studied in the literature from
$\text{YM}+\text{F}^{3}$ theory in section III, to string amplitudes and
$Z$-theory in section IV. We provided an argument for why one might expect
color-kinematics duality to impose all-order constraints on effective
operators that in principle could resum to non-local kinetic operators in the
UV. In Fig. 1, we introduced the color-dual factorization pyramid, where gauge
field operators at fixed orders in $\alpha^{\prime\,n}$,

$\mathcal{O}(\alpha^{\prime n}):\quad
A^{2n+4}\leftrightarrow\partial^{2}A^{2n+2}\leftrightarrow\partial^{4}A^{2n}\leftrightarrow\cdots\leftrightarrow\partial^{2n+2}A^{2}\,,$
(100)

are mixed via color-dual constraints on kinematic numerators (or partial
amplitudes). Then, factorization bundles together remaining operators higher
up on the pyramid at all orders $\alpha^{\prime m<n}$. Through this bundling,
one might expect massive resonances to appear due to modified kinetic terms,
$\mathcal{L}_{\text{kin}}\supset\prod_{k}(\partial^{2}-\mu_{k}^{2})A^{2}$. We
thus proposed a simple paradigm for encoding massive resonance through all
order towers of higher derivatives that live in massless color-dual numerators
of the form,

$\mathcal{M}^{\text{mass}}=\sum_{g}\frac{n_{g}N^{\text{HD}}_{g}}{d_{g}}\,.$
(101)

Motivated by the higher derivative bootstrap of $\text{YM}+\text{F}^{3}$
studied Carrasco  _et al._ (2023a), we then pursued a bootstrap to determine
to what extent the Wilson coefficients of higher-derivative operators are
constrained by double-copy consistency.

#### Color-Dual Bootstrap

In section V, we began by studying the promotion of a color-dual four-point
contact, $\sigma_{3}^{n}A_{4}$, with Wilson coefficients, $c_{(n,0)}$, to
five-point amplitudes, $A_{5}^{(n)}$, under double-copy consistency. We found
that through third order, that the amplitude is completely determined, and
exhibits an iterative structure in terms of a higher-derivative transformation
matrix, $A_{5}^{(n)}=\Omega^{n}A_{5}$. Requiring that the six-point amplitude
is color-dual and factorizes to $\Omega^{n}A_{5}$, imposes a constraint on the
split factorization channel of eq. (78). Double-copy consistency thus requires
that the four-point contact exponentiates:

$c_{(n,0)}=\frac{1}{n!}c_{(1,0)}^{n}$ (102)

Guided by the structure of this bootstrap, in eq. (84) through eq. (87), we
provided an all-order procedure for generalizing this exponential behavior to
all-multiplicity by imposing a series of $\text{N}^{k}\text{MCut}$ conditions
– finding that IR color-dual amplitudes must in general exponentiate any
higher-derivative contact operator consistent with the duality:

$A^{\text{UV}}_{n}=\prod_{k}\exp\left[c_{k}\Omega^{\text{DCC}}_{k}\right]A^{\text{IR}}_{n}\,,$
(103)

where $\Omega^{\text{DCC}}_{k}$ are double-copy consistent higher derivative
matrices that map color-dual amplitudes to color-dual contacts at higher
orders in the effective expansion. In doing so, we have demonstrated that
double-copy consistency alone could be sufficient to introduce kinematic
factors that soften the UV behavior of gravity, among all other theories that
are double-copy constructible.

#### UV Spectra via Padé

Finally, to resolve the inverse problem posed by the expression of eq. (93),
we invoked Padé approximants used widely in the resurgence literature Costin
and Dunne (2022, 2020); Dunne and Harris (2021, 2023),

$R^{(k)}_{[m,n]}(x_{0})=f^{(k)}(x_{0})\qquad k\leq m+n\,,$ (104)

where $f^{(k)}$ is the $k$-th derivative of $f(x)$. This analytic continuation
can be used as a mechanism for recovering non-perturbative information about
the mass spectrum, directly from the Wilson coefficients of eq. (89). In doing
so we show how increasing orders in Wilson coefficients allow us to gain
further reach in the spectrum of the UV theory as per Fig. 2, directly from
Wilson coefficients in the IR.

Before discussing some specific future directions, we will comment about the
potential implications of the color-dual factorization pyramid for novel
representations of color-dual loop-level integrands.

###  VII.2 Pyramid Implications for Loop Corrections

If we take seriously the possibility that a $D$-dimensional classification of
the kinematic algebra requires one to introduce to non-local field theory
operators, then restricting ourselves to just consider $F^{2}$ operators at
Yang-Mills mass-dimension might prove too restrictive for constructing all-
loop color-dual integrands. Here we sketch an argument for why one might
expect higher dimensional operators to force themselves into the conversation
starting at two-loop.

Suppose we have a color-dual three-point operator that manifests Jacobi-like
relations at all multiplicity in arbitrary dimensions, $\mathbb{V}_{(abc)}$.
As we argued above, consistency conditions between higher derivative contact
diagrams could introduce non-local propagators with modified kinetic terms.
These would appear as corrections to the propagators – yielding the following
sum over physical states:

$\mathbb{V}_{(ab|k)}\cdot\mathbb{P}_{(k)}\cdot\mathbb{V}_{(-k|cd)}$ (105)

where in general, our state-projector operator could be some string-like
vertex operator with a tower of massive-resonant modes, like those appearing
in the amplitudes of $Z$-theory:

$\mathbb{P}_{(k)}\sim\frac{\eta^{\mu\nu}}{k^{2}}+\sum_{n=0}^{\infty}\sum_{j=1}^{n}\frac{P^{\mu_{1}\mu_{2}...\mu_{j}}_{\nu_{1}\nu_{2}...\nu_{j}}}{(k^{2}-\mu_{n})}$
(106)

However, as we have argued in the text, color-dual theories with massive
resonances should be captured as higher derivative corrections in the cubic
graphs. After absorbing these massive residues into the definition of the
vertices of kinematic numerators, we could expect to see interactions
resembling,

$\mathbb{V}_{(123)}\rightarrow\mathbb{V}^{\text{YM}}_{(123)}+\sum_{n_{1},n_{2},n_{3}}\frac{B_{1}^{\otimes({n_{1}})}\cdot
B_{2}^{\otimes({n_{2}})}\cdot
B_{3}^{\otimes({n_{3}})}}{(k_{1}^{2}-\mu_{n_{1}})(k_{2}^{2}-\mu_{n_{2}})(k_{3}^{2}-\mu_{n_{3}})}$
(107)

where $B^{\otimes(n)}$ is an on-shell state tensor valued operator with upto
spin-$n$ modes. When inserting only on-shell vector states of Yang-Mills, in
the $\alpha^{\prime}\rightarrow 0$ limit, we know that this must recover the
normal on-shell Yang-Mills vertex:

$\lim_{\alpha^{\prime}\to
0}\mathbb{V}_{(123)}\cdot\varepsilon_{i}\rightarrow\mathbb{V}^{\text{YM}}_{(123)}\cdot\varepsilon_{i}$
(108)

At both tree-level and one-loop, the basis diagrams for color-dual theory
expose every internal vertex to at least one on-shell state:

$\cdots$$n$$n-1$

$n\\!-\\!1$$n$$1$$2$

(109)

However, at two-loop, the off-shell vertices become relevant in the basis
diagrams of a color-dual gauge theory:

(110)

The red vertices are completely off-shell, and thus there would be no external
state to temper any non-local denominator introduced by $D$-dimensional double
copy consistency. Since purely non-vector modes would be allowed to contribute
to the state-sum of the two-loop basis graph, color-dual numerators at two-
loop and beyond could be constructed from non-local kinematic numerators of
rational functions, rather than of polynomial kinematics that one would expect
from local Feynman rules. Off-shell local color-dual numerators of minimal
powercounting was excluded for Yang-Millls at two-loops and four-points in
arbitrary dimensions in ref. Bern  _et al._ (2016). This exclusion was
recently confirmed and extended by relaxing all power-counting assumptions by
recent work in collaboration with one of the authors Edison  _et al._ (2023).
A by now venerable example of non-local representations at two-loop can be
found in Mogull and O’Connell (2015), which studied five-point Yang-Mills
integrands for a specific choice of 4D external states.

###  VII.3 Future Directions

We see many future directions and applications of this work. Possibly the most
exciting is the prospect for phenomenological model building. The results of
our bootstrap suggest that the space of double-copy consistent amplitudes is
necessarily softened at high energy, while providing freedom to tune Wilson
coefficients beyond the zeta values of string theory. In section VI.1, we
shared our framework for using Padé approximant to study massive spectra above
the cutoff scale. Better understanding the sensitivity of UV physics on the IR
operators would allow the construction of bespoke UV completions. It would be
worth understanding how our double-copy consistent operators,
$\Omega_{k}^{\text{DCC}}$, when appropriately tuned match the spectral
constructions of Cheung and Remmen (2023).

To this end, a natural application of our results towards model building would
be in the low energy confining regime of QCD. One could use the massive
resonance framework of eq. (37) to construct pionic scattering models that
incorporate excited meson resonances, and then promote the spectrum to higher-
multiplicity through exponentiated contacts of eq. (89). Indeed, a large class
models in the web of color-dual theories Bern  _et al._ (2019), not just low
energy QCD, could be uplifted to consistent higher-multiplicity massive
resonance scattering in this way.

Beyond the possibility of model building, we should emphasize that the
exponential structure we have identified only takes into account constraints
placed by color-kinematics and split factorization. However, these constraints
do not preclude the possibility of additional relations between Wilson
coefficients from more factorization channels at higher multiplicity. Consider
for example an eight-point factorization channel that mixes four-point
contacts at different orders:

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0}}}\,\,{\scalebox{0.85}{ \leavevmode\hbox to0pt{\vbox
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\vertex[dot, scale=5.3,opacity=0.4,nhpBlue1](mid2a) at (.6,0) {$$};
\vertex(mid2b) at (.6,0) {$\sigma_{7}$}; \vertex(a1) at (0,.-.4) ; \vertex(a2)
at (0,.4) ; \vertex[dot, scale=1.08](mid4) at (1.8,0) {$\alpha^{n+}$};
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\vertex(mid3) at (0,0) {} ; \diagram{ (mid1) --[thick](mid2), (mid4)
--[thick](mid2), (a1) --[thick,scalar,nhpRed](a2), (a3)
--[thick,scalar,nhpRed](a4), }; \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{split}+\cdots\right]$
(111)

where $\sigma_{k}=c_{(k)}A^{(0)}_{4}(s^{k}+t^{k}+u^{k})$. By considering
additional all-order constraints, our exponentiated contacts could further
conspire at overlapping orders in the effective $\alpha^{\prime}$ expansion.
Better understanding whether there are additional relations between the Wilson
coefficients of eq. (89) we see as an important direction of future study,
that could give further understanding about the uniqueness of stringy
dynamics.

Absent additional constraints, our bootstrap at the very least offers an IR
perspective on why certain terms are excluded from the low energy effective
expansions in the closed string. For example, consider the multiple zeta
value, $\zeta_{3,5}$, that appears in the open string expansion, but is absent
in the closed string Broedel  _et al._ (2013a). From a UV perspective,
$\zeta_{3,5}$ is not single-valued and thus cannot be constructed from the
spherical integration of closed string amplitudes. Fascinatingly, we can now
identify an IR reason from our bootstrap for why this term must be excluded
from the closed string expansion.

Consider, by way of example, the $\mathcal{O}(\alpha^{\prime 8})$ sector at
five-point, the order at which one would expect $\zeta_{3,5}$ to appear. At
this order in the open string expansion, $\zeta_{3,5}$ weights a contact of
the form,
$\Omega^{\text{OS}}_{5,8}\sim\zeta_{3,5}\left(\Omega^{(1,0)}\Omega^{(1,1)}-\Omega^{(1,1)}\Omega^{(1,0)}\right)$,
where $\Omega^{(m,n)}$ are the five-point higher-derivative matrices that we
defined in eq. (68), and $\Omega^{\text{OS}}_{5,8}$ just indicates the open
string five-point contact that appears at $\mathcal{O}(\alpha^{\prime 8})$.
The only permissible factorization channel at this order is the following:

${\lim_{s_{45}\to 0}}\,\,{\scalebox{0.85}{ \leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.5,0) {$D^{2n+}F^{4}$}; \vertex(b1) at (-2.8,.6) {3};
\vertex(b2) at (-2.8,0) {2}; \vertex(b3) at (-2.8,-.6) {1}; \vertex(b4) at
(-.2,.6) {4}; \vertex(b5) at (-.2,-.6) {5}; \vertex[dot,
scale=1,hgrey0](mid1a) at (-1.5,0) {$D^{2n+}F^{4}$}; \vertex(mid1b) at
(-1.5,0) {\large$A_{5}^{(8)}$}; \diagram{ (mid1a) --[ultra thick](b1), (mid1a)
--[ultra thick](b2), (mid1a) --[ultra thick](b3), (mid1a) --[ultra thick](b4),
(mid1a) --[ultra thick](b5), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\!\\!=c_{(2,1)}\frac{\sigma_{3}^{2}\sigma_{2}}{s_{45}}\,{\scalebox{0.85}{
\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[dot,
scale=1.05](mid1) at (-1.4,0) {$D^{2n+}F^{4}$}; \vertex(b1) at (-2.65,.6) {3};
\vertex(b2) at (-2.65,0) {2}; \vertex(b3) at (-2.65,-.6) {1}; \vertex(b4) at
(1.5,.6) {4}; \vertex(b5) at (1.5,-.6) {5}; \vertex[dot,
scale=1,hgrey0](mid1a) at (-1.4,0) {$D^{2n+}F^{4}$}; \vertex(mid1b) at
(-1.4,0) {\large$A_{4}$}; \vertex[dot, scale=1.08](mid2) at (.6,0)
{$\alpha^{n++}$}; \vertex(a1) at (-.3,-.48) ; \vertex(a2) at (-.3,.52) ;
\vertex(mid3) at (0,0) {} ; \diagram{ (mid1a) --[ultra thick](b1), (mid1a)
--[ultra thick](b2), (mid1a) --[ultra thick](b3), (mid2) --[ultra thick](b4),
(mid2) --[ultra thick](b5), (mid1) --[ultra thick](mid2), (a1) --[ultra
thick,scalar,nhpRed](a2), }; \vertex[dot, scale=1,white](mid2a) at (.6,0)
{$\alpha^{n++}$}; \vertex[dot,opacity=0.5, scale=6.6,nhpBlue4](mid2c) at
(.6,0) {}; \vertex(mid2b) at (.6,0) {\large$A_{3}$};
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ (112)

After imposing color-kinematic constraints at five-point, and demand
consistent factorization down to four-point, we find that there are only two
additional color-dual five-point contacts. Let us call these contacts
$\Omega_{5,8}^{1}$ and $\Omega_{5,8}^{2}$, where $\Omega_{m,n}$ is a $m$-point
contact at $\mathcal{O}(\alpha^{\prime n})$. Intriguingly, one finds that
$\Omega^{\text{OS}}$ is not compatible with any linear combination of
$\Omega_{5,8}^{1}$ and $\Omega_{5,8}^{2}$. Thus we find that the $\zeta_{3,5}$
Wilson coefficient is restricted from the closed string expansion because its
operator is excluded by double-copy consistency. An obviously compelling
future direction is to build a detailed understanding of which IR operators in
the closed and open string expansions are prohibited by double-copy
consistency, thereby understanding entirely IR constraints on effective
operators .

Finally, it would be worth further studying the extent to which resurgence can
be leveraged to make UV predictions from IR physics. One could start by
understanding the relationship between partial wave unitarity Caron-Huot  _et
al._ (2017); Arkani-Hamed  _et al._ (2022) applied to massive residues of Padé
approximants, and positivity bounds on Wilson coefficients from the optical
theorem Adams  _et al._ (2006). Indeed, utilizing analytic continuations of
effective field theory beyond the perturbative sector could be fruitful in
further bounding the UV directly from IR data. Furthermore, it is clear that
leveraging resurgence methods in an EFT context should serve as a mechanism
for identifying potential energy scales for new physics directly from a finite
number of Wilson coefficients. It does not escape us that applying resurgence
methods to Wilson coefficients gleaned from loop-level effective predictions,
directly constructed via on-shell methods, could represent a rather novel
amplitudes path towards probing non-perturbative physics in theories both of
phenomenological and formal interest. This has the potential to expose shared
non-perturbative structure within the double-copy web of theories.

###### Acknowledgements.

The authors would like to thank Rafael Aoude, Sasank Chava, Lance Dixon, André
de Gouvêa, Gerald Dunne, Alex Edison, Kays Haddad, Karol Kampf, Ian Low, James
Mangan, Frank Petriello, Paolo Pichini, Radu Roiban, Oliver Schlotterer, Aslan
Seifi, Jaroslav Trnka, and Suna Zekioğlu for insightful conversations, related
collaboration, and encouragement throughout the completion of this work. We
would especially like to thank Gerald Dunne, Oliver Schlotterer, and Radu
Roiban for very valuable feedback on earlier versions of the manuscript. The
completion of this manuscript benefited from the hospitality of NORDITA during
the program “Amplifying Gravity at All Scales”, and ICTP-SAIFR during the
program “Gravitational Waves meet Amplitudes in the Southern Hemisphere.” This
work was supported by the DOE under contract DE-SC0015910 and by the Alfred P.
Sloan Foundation. N.H.P. acknowledges the DOE Office of Science Graduate
Student Research (SCGSR) program under contract DE‐SC0014664, the Northwestern
University Amplitudes and Insight group, the Department of Physics and
Astronomy, and Weinberg College for their generous support.

##  Appendix A Hypergeometric Residues

Below we provide some identities that we use to recover the residues of the
hypergeometric functions appearing in five-point $Z$-theory amplitudes. The
${}_{3}F_{2}$ hypergeometric function can be expressed as an infinite sum over
Pochhammer symbols,

${}_{3}F_{2}\left[{a_{1},\,a_{2},\,a_{3}\atop
a_{4},\,a_{5}};z\right]=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}(a_{2})_{n}(a_{3})_{n}}{(a_{4})_{n}(a_{5})_{n}}\frac{z^{n}}{n!}\,,$
(113)

where $(x)_{n}=\Gamma(x+n)/\Gamma(x)$. In the text we found that the fully
resummed $Z$-theory $(n-3)!$ basis amplitudes can be expressed in terms of a
single functional with different kinematic arguments, depending on whether the
color ordering is planar or non-planar with respect to the Chan-Paton factors:

$F^{2}$$F^{2}$

$\displaystyle=Z_{5}(-s_{34},-s_{45},-s_{15},-s_{12},-s_{23})\,,$ (114)

$F^{2}$$F^{2}$

$\displaystyle=-Z_{5}(1-s_{34},-s_{45},-s_{15},1-s_{12},-s_{23})\,.$ (115)

Here we have defined the five-point $Z_{5}$-function using a particular
ordering of ${}_{3}F_{2}$ as follows,

$Z_{5}(a_{1},a_{2},a_{3},a_{4},a_{5})\equiv{}_{3}F_{2}\left[{\,a_{1},\,a_{4},\,a_{23}-a_{5}\atop
a_{12},\,a_{34}};1\right]\frac{\Gamma(a_{1})\Gamma(a_{2})\Gamma(a_{3})\Gamma(a_{4})}{\Gamma(a_{12})\Gamma(a_{34})}\,,$
(116)

and employed the shorthand notation
$a_{i_{1}i_{2}...i_{n}}=a_{i_{1}}+a_{i_{2}}+\cdots+a_{i_{n}}$. Since
${}_{3}F_{2}\rightarrow 1$ when anyone of the top arguments vanishes, it is
clear that $Z_{5}$ factorizes to the appropriate four-point Veneziano
amplitudes in the $a_{1},a_{4}\rightarrow 0$ limits. However, as noted in the
text, the other factorization channels are obscured when defined in this way.
To elucidate the remaining pole structure, one can apply a Thomae
transformation to the $Z_{5}$ function,

$\frac{(x_{123})!}{(x_{13})!(x_{23})!}{}_{3}F_{2}\left[{1+x_{123},\,-x_{4},\,-x_{5}\atop
1+x_{13},\,1+x_{23}};1\right]=\frac{(x_{345})!}{(x_{34})!(x_{35})!}{}_{3}F_{2}\left[{1+x_{345},\,-x_{1},\,-x_{2}\atop
1+x_{34},\,1+x_{35}};1\right]\,.$ (117)

By making the following variable replacements, one can show that the $Z_{5}$
is cyclically invariant in its arguments:

$\displaystyle 1+x_{123}=a_{1}\quad$ $\displaystyle\Rightarrow\quad
Z_{5}(a_{1},a_{2},a_{3},a_{4},a_{5})=Z_{5}(a_{2},a_{3},a_{4},a_{5},a_{1})$
(118) $\displaystyle 1+x_{123}=a_{4}\quad$ $\displaystyle\Rightarrow\quad
Z_{5}(a_{1},a_{2},a_{3},a_{4},a_{5})=Z_{5}(a_{5},a_{1},a_{2},a_{3},a_{4})$
(119)

As such, we can use this relationship between different cyclic orderings of
$Z_{5}$ do derive the $a_{i}\rightarrow 0$ limits of the hypergeometric
function appearing in Z-theory amplitudes:

$\displaystyle\lim_{a_{1}\rightarrow
0}{}_{3}F_{2}\left[{\,a_{1},\,a_{4},\,a_{23}-a_{5}\atop
a_{12},\,a_{34}};1\right]$ $\displaystyle=1$ (120)
$\displaystyle\lim_{a_{2}\rightarrow
0}{}_{3}F_{2}\left[{\,a_{1},\,a_{4},\,a_{23}-a_{5}\atop
a_{12},\,a_{34}};1\right]$
$\displaystyle={}_{2}F_{1}\left[{a_{4},\,a_{3}-a_{5}\atop
a_{34}};1\right]=\frac{\Gamma(a_{5})\Gamma(a_{34})}{\Gamma(a_{3})\Gamma(a_{45})}$
(121) $\displaystyle\lim_{a_{3}\rightarrow
0}{}_{3}F_{2}\left[{\,a_{1},\,a_{4},\,a_{23}-a_{5}\atop
a_{12},\,a_{34}};1\right]$
$\displaystyle={}_{2}F_{1}\left[{a_{1},\,a_{2}-a_{5}\atop
a_{12}};1\right]=\frac{\Gamma(a_{5})\Gamma(a_{12})}{\Gamma(a_{2})\Gamma(a_{15})}$
(122) $\displaystyle\lim_{a_{4}\rightarrow
0}{}_{3}F_{2}\left[{\,a_{1},\,a_{4},\,a_{23}-a_{5}\atop
a_{12},\,a_{34}};1\right]$ $\displaystyle=1$ (123)
$\displaystyle\lim_{a_{5}\rightarrow
0}{}_{3}F_{2}\left[{\,a_{1},\,a_{4},\,a_{23}-a_{5}\atop
a_{12},\,a_{34}};1\right]$
$\displaystyle=\frac{1}{a_{5}}\frac{\Gamma(a_{12})\Gamma(a_{34})}{\Gamma(a_{1})\Gamma(a_{23})\Gamma(a_{4})}+\mathcal{O}(1)$
(124)

Together these limits of the ${}_{3}F_{2}$ hypergeometric function give the
desired factorization properties endowed in the fully resummed $Z$-theory
amplitudes studied in the text. While in the text we have just studied the
$s_{12}$ and $s_{34}$ factorization channels at five-point – the remaining
residues can simply be recovered through functional relabelings due to the
cyclicity of $Z_{5}$.

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