compat=1.1.0 /tikzfeynman/warn luatex = false aainstitutetext: Department of
Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
bbinstitutetext:  Institut fur Theoretische Physik, ETH Zurich, 8093 Zurich,
Switzerland S6pt L7pt

#  Double-copy towards supergravity inflation
with $\bm{\alpha}$-attractor models

John Joseph M. Carrasco, a,b Matthew Lewandowski, a and Nicolas H. Pavao
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Key to the simplicity of supergravity $\alpha$-attractor models of inflation
are Volkov-Akulov fermions, often in the form of nilpotent superfields. Here
we explore the possibility of using the double-copy to construct theories of
Dirac-Born-Infeld-Volkov-Akulov (DBIVA) coupled to supergravity. A color-dual
bootstrap admits scattering amplitudes involving pions and vectors through
five-point tree-level order by order in mass-dimension, but requires the
introduction of a $\text{Tr}(F^{3})$ operator. Gauge theories with this
operator were recently found to require a tower of higher-derivative operators
to be compatible with color-kinematics duality. Adjoint-type double-copy
construction at its most conservative seems to require the UV completion of
DBIVA \+ pure Poincaré supergravity scattering amplitudes to a family of
theories involving DBIVA-like particles coupled to Weyl-Einstein supergravity.
We also point out an alternative solution to color-dual gauged pions that
allows adjoint double-copy without a tower of higher derivative corrections
but at the cost of exchange symmetry between scalars.

##  1 Introduction

Inflation is a framework involving dynamic spacetime. Can we begin to
understand questions asked of an inflationary universe by considering much
simpler questions asked of a gauge theory evolving in flat spacetime? The
existence of color-dual double-copy structure BCJ ; BCJLoop  for both quantum
field theory scattering amplitudes in Einstein-Hilbert gravity, as well as for
classical solutions, is incredibly suggestive. For recent reviews of the
double-copy literature see refs. White:2017mwc ; Borsten:2020bgv ; BCJreview .
Here we make some first steps towards a double-copy construction of scattering
amplitudes in theories that could support supergravity $\alpha$-attractor
models of inflation.

Supergravity $\alpha$-attractor models are among the set of simple yet
phenomenologically viable inflationary models consistent with Planck and BICEP
(see e.g. BICEP:2021xfz ; Kallosh:2021mnu  and references therein). The
simplicity of these models relies on a construction involving nilpotent
superfields, which tame the proliferation of unwanted scalar modes during
inflation. We have long known that these seemingly esoteric superfields can be
understood111See also ref. Bansal:2020krz  for connections made to de Sitter
solutions from a unimodular approach to supergravity Nagy:2019ywi . as Volkov-
Akulov (VA) fermions Rocek:1978nb ; Lindstrom:1979kq ; Casalbuoni:1988xh ;
Komargodski:2009rz ; Antoniadis:2014oya . The equivalence of various nonlinear
field redefinitions that relate these representations have been explicitly
verified through direct calculation of tree-level scattering amplitudes
Karlsson:2017bfv . So, with eyes on eventually exploring these inflationary
models, we can start by introducing double-copy descriptions of Volkov-Akulov
fermions with gravity.

To set the stage, we recall a few well-known double-copy constructions. The
first is that dynamical gravity222To be precise $\mathcal{N}=0$ supergravity
is the result of this straightforward double-copy. One can recover Einstein-
Hilbert gravity by including additional wrong-sign matter Johansson:2014zca ;
Luna:2017dtq  or by explicitly projecting out unwanted states Carrasco:2021bmu
. can be described by double-copying two Yang-Mills theories, schematically
denoted by

$\mathcal{M}^{\rm gravity}=\mathcal{A}^{\rm YM}\otimes\mathcal{A}^{\rm YM}\ .$
(1)

We will explain this construction in detail in section 2, but the main point
is that we construct scattering amplitudes in a theory of spin-2 particles by
combining simpler building blocks from theories of spin-1 particles. It is
also known that Volkov-Akulov fermions have a double-copy description in terms
of nonlinear sigma model (NLSM) scalar pions (denoted here and elsewhere as
‘$\pi$’ for short) and fermions through

$\mathcal{A}^{\rm VA}=\mathcal{A}^{\pi}\otimes\mathcal{A}^{\rm fermions}\ .$
(2)

The above VA fermion amplitudes can be imbedded in a maximally supersymmetric
theory by double-copying NLSM pions with the full $\mathcal{N}=4$ super Yang-
Mills (sYM) multiplet, giving a theory of Dirac scalars, VA fermions, and
Born-Infeld photons, referred to as Dirac-Born-Infeld-Volkov-Akulov (DBIVA)
Cachazo2014xea ; He:2016vfi ; Cachazo2016njl ; Carrasco2016ldy ,

$\mathcal{A}^{\text{DBIVA}}=\mathcal{A}^{\pi}\otimes\mathcal{A}^{\rm sYM}\ .$
(3)

We will build toward a double-copy construction of $\alpha$-attractor models
by addressing the possible double-copy structure of scattering amplitudes
involving Volkov-Akulov fermions, gravitons, and their superpartners. We do so
by conjecturing the existence of a special color-dual gauge theory of NLSM
pions interacting with vectors and attempt to bootstrap it, adding higher-
derivative operators as necessary for double-copy consistency f3Ladder . When
double-copied with $\mathcal{N}=4$ super Yang-Mills (sYM) theory, this color-
dual vector-pion theory’s predictions would presumably generate scattering
amplitudes in $\mathcal{N}=4$ Dirac-Born-Infeld-Volkov-Akulov-Supergravity
($\text{DBIVA}+\text{SG}$), schematically

$\mathcal{A}^{S^{2}=0}\sim\mathcal{A}^{\text{DBIVA}+\text{SG}}\equiv\mathcal{A}^{\text{YM}+\pi}\otimes\mathcal{A}^{\text{sYM}}\
,$ (4)

where $\mathcal{A}^{S^{2}=0}$ are S-matrix elements describing nilpotent
superfields coupled to dynamic spacetime. In our approach, employing the
double-copy construction, the spectrum of the $\mathcal{N}=4$ supergravity
theory is realized as a tensor product between the fields of the constituent
two gauge theories: a single vector field coupled to NLSM pions and the states
belonging to an ${{\cal N}=4}$ vector multiplet,

$(A^{+},\lambda^{+}_{ABC},s_{AB},\lambda^{-}_{A},A^{-})\otimes(A^{+},A^{-},\pi)=(h^{++},\psi^{+}_{ABC},A^{+}_{AB},\chi^{+}_{A},{{\bar{\rm
t}}})\\\ \oplus({{\rm
t}},\chi^{-}_{ABC},A^{-}_{AB},\psi^{-}_{A},h^{--})\oplus(A^{\text{BI}}{}^{+},\lambda^{\text{VA}}{}^{+}{}_{ABC},s^{\text{DBI}}{}_{AB},\lambda^{\text{VA}}{}^{-}{}_{A},A^{\text{BI}}{}^{-})\
.$ (5)

All supersymmetry relations in the supergravity theory are inherited from the
supersymmetry of the $\mathcal{N}=4$ sYM factors. Following the conventions of
e.g. ref. Carrasco:2013ypa , we use $({{\rm t}},{{\bar{\rm t}}})$ to refer to
the complex field that labels the external scalar states in the supergravity
scattering amplitudes. In terms of the two vector fields, these scalar fields
are defined by,

${{\rm t}}=A^{+}_{{{\cal N}=4}}\otimes
A^{-}_{\text{YM}+\pi}\quad,\qquad{{\bar{\rm t}}}=A^{-}_{{{\cal N}=4}}\otimes
A^{+}_{\text{YM}+\pi}\ .$ (6)

Similarly the double-copy of our conjectured color-dual vector-pion theory
with any supersymmetric Yang-Mills theory, including e.g. $\mathcal{N}=1$ sYM,
should land on an associated reduced supergravity theory coupled to a likewise
reduced DBIVA multiplet. Indeed many reduced symmetry supergravity theories
can be considered by orbifolding the supersymmetric gauge theory
Carrasco:2012ca ; Chiodaroli:2013upa  in combination with explicit projecting
out of unwanted states Johansson:2014zca .

In this paper we demonstrate how a color-dual bootstrap allows us to build
amplitudes in two models of double-copy consistent gauge theories of vectors
and pions. In the most conservative approach, we discover that additional
higher-derivative operators must be added to the action of a simple
covariantized nonlinear sigma model to preserve the color-dual structure
beyond four-points. As was found in f3Ladder , including these additional
operators induces a tower of color-dual contact terms required by the
principle of double-copy consistency. By collecting terms at each order in the
pion decay width, $\Lambda$, we conjecture that the amplitudes can be resummed
to a dimensional reduction of $(DF)^{2}+\text{YM}$ JohanssonConformal . We
then use the resulting color-dual vector-pion amplitudes in the double-copy
construction of Einstein-Weyl-DBIVA theory. There is a natural question here
as to whether the modified Adler’s zero, induced by the required higher
derivative operators, has allowed us to achieve our aims of actually building
nilpotent superfields coupled to supergravity. This motivates us to identify
an alternative approach to an adjoint color-dual gauged non-linear sigma model
with unmodified soft-behavior. This comes at a cost, however, which is to
break manifest exchange symmetry among pions.

The paper is organized as follows. In section 2, we review the amplitude
relations required for double-copy construction and the procedure for
constructing double-copy consistent theories using color-dual numerators. In
section 3, we use an on-shell bootstrap to identify additional operators
required for our candidate vector-pion theory to obey color-kinematics
duality. In section 4, we observe that these additional operators induce a
tower of higher-derivative corrections, which leads us to conjecture that our
desired color-dual vector-pion theory is a dimensional reduction of
$(DF)^{2}+\text{YM}$ theory. In section 5, we identify virtual gluon
corrections to the six-point pure-pion amplitudes, required by double-copy
consistency in the $(DF)^{2}+\text{YM}+\pi$ theory, and comment on their
implications for the expected Adler’s zero of pure Volkov-Akulov observables.
In section 6 we present a second model for color-dual gauged pions that leaves
Adler’s zero unmodified at the cost of scalar exchange symmetry. Our
conclusions and discussion of future directions are in section 7.

##  2 Review of amplitude relations and the double-copy

In this work, we are interested in combining the two theories written
schematically on the right-hand side of eq. (4). The key to our procedure for
calculating the desired scattering amplitudes of Volkov-Akulov (VA) fermions
coupled to gravity is the double-copy construction BCJ . Here we provide a
brief review of how to implement the double-copy, along with a general
overview of on-shell methods. For more details we encourage the reader to
consult recent dedicated review papers BCJreview ; Adamo:2022dcm .

We start by considering a pure $SU(N)$ Yang-Mills theory, whose action is
given by

$\mathcal{L}^{\rm YM}=-\frac{1}{4}\,{\rm Tr}[F_{\mu\nu}F^{\mu\nu}]\ ,$ (7)

where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig[A_{\mu},A_{\nu}]$
is the field strength, $A_{\mu}=A_{\mu}^{a}T^{a}$ is the massless gluon field,
$T^{a}$ are the generators of $SU(N)$ (in the adjoint representation), and the
trace is over color indices. The field strength can also be written in terms
of the covariant derivative $D_{\mu}$ as

$F_{\mu\nu}=\frac{i}{g}[D_{\mu},D_{\nu}]\ ,\quad\text{for}\quad
D_{\mu}=\partial_{\mu}-igA_{\mu}\ ,$ (8)

where $g$ is the gauge coupling. The generators satisfy the normalization,

$[T^{a},T^{b}]=f^{abc}T^{c}\ ,\quad\text{and}\quad\,{\rm
Tr}[T^{a}T^{b}]=\delta^{ab}\ ,$ (9)

where $f^{abc}$ are the totally antisymmetric structure constants that satisfy
the Jacobi identity,

$f^{abe}f^{ecd}-f^{dae}f^{ebc}-f^{dbe}f^{eca}=0\ .$ (10)

To compute scattering amplitudes in this theory, one option is to fix a gauge
and use the action of eq. (7) to define Feynman rules. This has the advantage
of making locality manifest and attaching factors of $f^{abc}$ to each three-
point vertex, but in turn requires ghosts to remove gauge redundancy at loop
level, and involves generally laborious calculation. Off-shell Feynman rules
are verbose, whereas physical on-shell quantities can be relatively compact.

Two of the lessons from the on-shell community over the past few decades has
been to recycle on-shell information whenever possible through unitarity
methods UnitarityMethod ; Fusing , as well as to minimize the work being done
at any given time by considering color-ordered or color-stripped quantities in
gauge theories. We will discuss both, focusing first on color-decomposition.

###  2.1 Cubic representations

Gauge theory scattering amplitudes at tree-level can be expressed in terms of
cubic (trivalent) graphs, as in the scaffolding of $\phi^{3}$ Feynman
diagrams,

$\mathcal{A}_{m}=g^{m-2}\sum_{g\in\Gamma_{3}^{m}}\frac{c_{g}n_{g}}{d_{g}}\,.$
(11)

Here $\mathcal{A}_{m}$ is referred to in the amplitudes community as a full or
color-dressed $m$-point or $m$-particle amplitude. The sum is over $(2m-5)!!$
distinct, connected, cubic tree-graphs with $m$ external legs. The color-
weight $c_{g}$ of each graph $g$ is calculated by giving each vertex a
structure constant $f^{abc}$, and every internal edge a delta function
$\delta^{ab}$. The external edges for these graphs are always labeled,
$1,\ldots,m$ and their color indices are in concordance with their labeling,
$a_{1},\ldots,a_{m}$. The propagators or denominator weights, $d_{g}$ are
dressed as normal for massless $\phi^{3}$ Feynman diagrams, and the kinematic
numerator weights $n_{g}$ are dressed with whatever product of Lorentz
invariants (including polarizations for external gluons) are required to
correctly reproduce the scattering amplitude.

The famous Yang-Mills quartic vertex familiar from Feynman rules is absorbed
into this cubic representation by appropriate factors of inverse propagators
appearing in the kinematic numerators. Such assignments are not at all unique,
and so there are many distinct representations that are all sufficient. Indeed
one way of generating such a representation is to begin by drawing all cubic
Feynman diagrams, dressing them as normal, and then drawing all quartic
diagrams, and expanding them out to cubic diagrams consistent with their
color-dressings. This would not be efficient but is certainly a consistent way
of generating a cubic representation.

For the three-point amplitude (non-vanishing only for complex momenta), with
$k_{1}+k_{2}+k_{3}=0$ and $k_{i}^{2}=0$, there is only one distinct graph with
no internal propagators,

$\mathcal{A}^{\text{YM}}_{3}=g\,n_{3}\,c_{3}$ (12)

with $c_{3}=f^{a_{1}a_{2}a_{3}}$. Due to the adjoint structure of $c_{3}$, the
kinematic weight must also be antisymmetric in exchange of particle labels,
otherwise the full amplitude would vanish. Furthermore, there must be one
polarization per external leg per term, and the amplitude must be gauge
invariant, i.e., $\mathcal{A}|_{\varepsilon_{i}\to k_{i}}=0$. As all
Mandelstam invariants $s_{ij}\equiv(k_{i}+k_{j})^{2}$ vanish at three-points,
there are only two possible mass-dimensions one can write down. The lowest
order is associated with Yang-Mills, $\text{Tr}(F^{2})$:

$n_{3}=(\varepsilon_{1}\cdot\varepsilon_{2})\left((k_{1}-k_{2})\cdot\varepsilon_{3}\right)+\text{cyclic}\,,$
(13)

the other is associated with the $\text{Tr}(F^{3})$ operator, which it turns
out we will have use for in this paper, so we quote its functional form here:

$n^{F^{3}}_{3}=\left((k_{1}-k_{2})\cdot\varepsilon_{3}\right)\left((k_{2}-k_{3})\cdot\varepsilon_{1}\right)\left((k_{3}-k_{1})\cdot\varepsilon_{2}\right)\,.$
(14)

Note that the $n^{F^{3}}_{3}$ kinematic weight, when used, must come in with a
dimensionful coupling relative to $n_{3}$. Again, up to the overall Wilson
coefficient of the $\,{\rm Tr}(F^{3})$ operator, the kinematic weight of
$n^{F^{3}}_{3}$ is entirely fixed by mass-dimension and anti-symmetry. As
there are no additional graphs, each of these numerators must be gauge-
invariant independently, which indeed they are.

Let us look at a slightly more interesting example, four-points, in Yang-
Mills. Here the scattering amplitude is given by,

$g^{-2}\,\mathcal{A}^{\text{YM}}_{4}=\frac{c_{s}n_{s}}{s_{12}}+\frac{c_{t}n_{t}}{s_{14}}+\frac{c_{u}n_{u}}{s_{13}}\,,$
(15)

where $s_{ij}$ is used to mean $(k_{i}+k_{j})^{2}$. The color factors are
given as follows:

$\displaystyle c_{s}$ $\displaystyle=f^{a_{1}a_{2}e}f^{ea_{3}a_{4}}\,,$ (16)
$\displaystyle\,c_{t}$ $\displaystyle=f^{a_{4}a_{1}e}f^{ea_{2}a_{3}}\,,$ (17)
$\displaystyle c_{u}$ $\displaystyle=f^{a_{1}a_{3}e}f^{ea_{2}a_{4}}\,.$ (18)

The traditional contact term has been absorbed to the cubic dressings by
appropriately multiplying $c_{s}$, $c_{t}$, or $c_{u}$ within the contact term
by $s_{12}/s_{12}$, $s_{14}/s_{14}$, and $s_{13}/s_{13}$ respectively. As the
color-weights satisfy the Jacobi identity $c_{s}=c_{t}+c_{u}$ this assignment
is by no means unique.

One such assignment results in the following functional dressing BCJreview
for the kinematic numerators:

$n(abcd)=\Bigl{\\{}\Big{[}(\varepsilon_{a}\cdot\varepsilon_{b})k_{a}^{\mu}+2(\varepsilon_{a}\cdot
k_{b})\varepsilon_{b}^{\mu}-(a\leftrightarrow
b)\Big{]}\Big{[}(\varepsilon_{c}\cdot\varepsilon_{d})k_{c\,\mu}+2(\varepsilon_{c}\cdot
k_{d})\varepsilon_{d\,\mu}-(c\leftrightarrow d)\Big{]}\\\
+s_{ab}\Bigl{[}(\varepsilon_{a}\cdot\varepsilon_{c})(\varepsilon_{b}\cdot\varepsilon_{d})-(\varepsilon_{a}\cdot\varepsilon_{d})(\varepsilon_{b}\cdot\varepsilon_{c})\Bigr{]}\Bigr{\\}}\,,$
(19)

with

$\displaystyle n_{s}$ $\displaystyle=n(1234)\,,$ (20) $\displaystyle n_{t}$
$\displaystyle=n(4123)\,,$ (21) $\displaystyle n_{u}$
$\displaystyle=n(1324)\,.$ (22)

In particular it is worth noticing that the coefficient of $s_{ab}$ in eq.
(19) is exactly the piece of the contact term absorbed by each cubic graph.

###  2.2 Color decomposition

At four-points, as well as higher multiplicity, local dressings of individual
graphs are no longer individually gauge invariant. Why? There can be
cancellations between graphs depending upon the redundancy between color
factors. Generically the color-factors satisfy the Jacobi identity,

$f^{abe}f^{ecd}=f^{dae}f^{ebc}+f^{cae}f^{edb}$ (23)

with an implied sum over the repeated indices. Because of this redundancy in
color-factors, any representation that makes use of all $(2m-5)!!$ cubic
graphs and their individual color-dressings are not in a minimal color basis.
One can always use the Jacobi identities to re-express the color dressings in
a minimal color basis of $(m-2)!$ color weights DixonMaltoni , and the
kinematic coefficients of those color weights will be gauge invariant.

Let us look at the four-point example above to see a concrete realization. One
can rewrite eq. (15) using $c_{t}=c_{s}-c_{u}$ to achieve,

$g^{-2}\,\mathcal{A}^{\text{YM}}_{4}=c_{s}\left(A^{\text{YM}}_{4}(1234)\equiv\frac{n_{s}}{s_{12}}+\frac{n_{t}}{s_{14}}\right)+c_{u}\left(A^{\text{Y}M}_{4}\equiv\frac{n_{u}}{s_{13}}-\frac{n_{t}}{s_{14}}\right)\,.$
(24)

The color-ordered or partial amplitudes $A^{\text{YM}}_{4,\text{tree}}(1234)$
and $A^{\text{YM}}_{4,\text{tree}}(1324)$ are gauge invariant and only depend
on an exponential number of cubic graphs as multiplicity increases BCJreview .
They are related functionally and satisfy various field theory relations
between them.

$1$$\sigma(2)$$\sigma(3)$$\sigma(n-1)$$n$ Figure 1:  The basis graphs for any
tree-level kinematics that satisfy Jacobi-like relations can be expressed in
terms of half-ladder diagrams where two legs, say $1$ and $n$, are held fixed,
and all other leg labels $\sigma(i)$ are permuted. See, e.g., ref.
DixonMaltoni  for the relevant argument applied to color-factors.

Generically, we can write $m$-multiplicity gauge theory scattering amplitudes
in the following form:

$\displaystyle\mathcal{A}^{\rm YM}_{n,\text{ tree}}$
$\displaystyle=g^{n-2}\sum_{g_{i}\in\Gamma_{3}^{(n)}}\frac{c_{g_{i}}\,n_{g_{i}}}{d_{g_{i}}}$
(25) $\displaystyle=g^{n-2}\sum_{\sigma\in
S_{n-2}}c(1,\sigma(2),\dots,\sigma(n-1),n)A_{n}^{\rm
YM}(1,\sigma(2),\dots,\sigma(n-1),n)\ .$ (26)

where the $c(1,\sigma(2),\dots,\sigma(n-1),n)$ corresponded precisely to a
product of color factors dressing a half-ladder or multiperipheral graph,

$c(1,\sigma(2),\dots,\sigma(n-1),n)\equiv
f^{a_{1}a_{\sigma(2)}b_{1}}f^{b_{1}a_{\sigma(3)}b_{2}}\cdots
f^{b_{n-3}a_{\sigma(n-1)}a_{n}}\ .$ (27)

Given this color structure, we call eq. (26) the half-ladder representation,
as it restructures the color-dressed amplitude in terms of a sum over color
factors associated with “half-ladder” cubic graphs, see e.g. Fig. 1. In the
case of eq. (27), root legs $1$ and $n$ are held fixed, and all other legs
take on permutations of labels $\sigma$. The remaining color dressings are
then related to this choice of $c(1,\sigma(2),\dots,\sigma(n-1),n)$ by
iterated application of Jacobi identities on the internal edges, or
antisymmetry of individual vertices DixonMaltoni ,

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(mid2)--[thick](a2), (mid2)--[thick](a3), (mid1)--[thick,hred](mid2) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(mid1) at (-0.5,
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at (-.5, 1) {$a_{2}$}; \vertex(a3) at (.5, 1) {$a_{3}$}; \vertex(a4) at (1.5,
0) {$a_{4}$}; \diagram{ (mid1)--[thick](a1), (mid1)--[thick](a2),
(mid2)--[thick](a4), (mid2)--[thick](a3), (mid1)--[thick,hred](mid2) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}-{\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{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(mid1) at (-0.5,
0); \vertex(mid2) at (0.5,0); \vertex(a1) at (-1.5, 0) {$a_{1}$}; \vertex(a2)
at (-.5, 1) {$a_{3}$}; \vertex(a3) at (.5, 1) {$a_{2}$}; \vertex(a4) at (1.5,
0) {$a_{4}$}; \diagram{ (mid1)--[thick](a1), (mid1)--[thick](a2),
(mid2)--[thick](a4), (mid2)--[thick](a3), (mid1)--[thick,hred](mid2) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,$ (28)

where $a_{i}$ are labels of internal or external edges. The half-ladder basis
of eq. (26) is expressed in terms of a minimal color basis, meaning each
kinematic partial amplitude contributes to the overall amplitude weighted by a
completely distinct direction in color space. Before proceeding, it is worth
taking a moment to appreciate the utility of reframing gauge invariant
amplitudes in this way. While the local behavior manifest in the Feynman rules
generated eq. (7) has been obscured (at least for now), the local gauge
symmetry and Bose symmetry is now absolutely manifest.

One can choose alternative color-bases depending on which external legs are
held fixed as the half-ladder root legs. Their coefficients are also gauge
invariant and indeed the partial amplitudes of one basis can be linearly
expressed in terms of partial amplitudes of other bases. This is due to a set
of linear relations known as the Kleiss-Kuijf relations Kleiss:1988ne , which
follow simply from the fact that kinematic dressings of graphs obey
antisymmetry around vertices, correspondent with the color-weights, to
preserve Bose symmetry. The partial amplitudes also inherit reflection and
cyclic identities,

$A_{n,\text{tree}}^{\rm YM}(1,2,\dots,n)=A_{n,\text{tree}}^{\rm
YM}(2,\dots,n,1)\ ,$ (29)

and

$A_{n,\text{tree}}^{\rm YM}(1,2,\dots,n)=(-1)^{n}A_{n,\text{tree}}^{\rm
YM}(n,\dots,2,1)\ ,$ (30)

and also obey what is called the photon decoupling identity,

$\sum_{\sigma\in\text{cyclic}}A_{n,\text{tree}}^{\rm
YM}({\sigma(1),\sigma(2),\dots,\sigma(n-1)},n)=0\ .$ (31)

As we will now show, the partial amplitudes respect an additional set of
relations that underly the local properties of the kinematic numerators hidden
in each $A_{n,\text{tree}}^{\rm YM}$.

###  2.3 Color-dual numerators

Let us spend a second revisiting the generalized gauge choices that allow us
to make different assignments of contact terms to cubic graphs. As we
mentioned earlier, when a higher-point contact interaction, $X_{g}$, weighted
by a color factor $c_{g}$, is produced by the Feynman rules, it can be made
compatible with the form of eq. (25) by multiplying by an overall factor of
unity, taking $c_{g}X_{g}\rightarrow\frac{c_{g}}{d_{g}}(d_{g}X_{g})$. In which
case, $n_{g}\rightarrow n_{g}+d_{g}X_{g}$. Take for example any three graphs
$\\{g_{1},g_{2},g_{3}\\}$ in $\Gamma_{3}^{(m)}$ whose color factors are
related by the Jacobi identity

$c_{g_{1}}=c_{g_{2}}+c_{g_{3}}\ .$ (32)

Then the amplitude is invariant under shifting the numerator factors by the
following _generalized gauge transformation_

$n_{g_{1}}\rightarrow n_{g_{1}}+d_{g_{1}}\Delta_{\\{g_{1},g_{2},g_{3}\\}}\
,\quad n_{g_{2}}\rightarrow
n_{g_{2}}-d_{g_{2}}\Delta_{\\{g_{1},g_{2},g_{3}\\}}\ ,\quad
n_{g_{3}}\rightarrow n_{g_{3}}-d_{g_{3}}\Delta_{\\{g_{1},g_{2},g_{3}\\}}\ ,$
(33)

leading to an overall shift in the amplitude of,

$\mathcal{A}^{\rm YM}_{n,\text{ tree}}\rightarrow\mathcal{A}^{\rm
YM}_{n,\text{
tree}}+(c_{g_{1}}-c_{g_{2}}-c_{g_{3}})\Delta_{\\{g_{1},g_{2},g_{3}\\}}=\mathcal{A}^{\rm
YM}_{n,\text{ tree}}\,$ (34)

for any function $\Delta_{\\{g_{1},g_{2},g_{3}\\}}$ of the kinematic
invariants. This is equivalent to the process of absorbing contact terms into
the trivalent graph representation of the full amplitude.

In the case of Yang-Mills amplitudes, for all graphs whose color factors are
related by the Jacobi identity, these $\Delta$ can be chosen such that the
$n_{g}$ satisfy the same relationships as the color factors $c_{g}$, i.e.

$\displaystyle\begin{split}c_{g_{i}}+c_{\bar{g}_{i}}=0\quad&\Rightarrow\quad
n_{g_{i}}+n_{\bar{g}_{i}}=0\ ,\\\
c_{g_{i}}-c_{g_{j}}-c_{g_{k}}=0\quad&\Rightarrow\quad
n_{g_{i}}-n_{g_{j}}-n_{g_{k}}=0\ ,\end{split}$ (35)

where $\bar{g}$ is a cubic graph with an odd number of vertex flips relative
to $g$. In this gauge, the interplay between the algebraic relations of the
local numerators and the color factors now make Bose symmetry and gauge
invariance manifest. Given the parallel structure between the color and
kinematic factors, we say that Yang-Mills theory obeys the duality between
color and kinematics, or simply, that it is color dual.

A wide variety of theories have been shown in the literature to be color dual,
as recently reviewed in refs. BCJreview ; Carrasco:2020ywq ; Adamo:2022dcm .
Such theories permit this special representation of the numerators if and only
if the partial amplitudes $A_{n,\text{tree}}^{\rm YM}$ satisfy a further set
of relations, identified by Bern, Johansson and one of the authors BCJ ,

$\sum_{i=2}^{n-1}k_{1}\cdot\left(k_{2}+\dots+k_{i}\right)A_{n,\text{tree}}^{\rm
YM}(2,\dots,i,1,i+1,\dots,n)=0\ ,$ (36)

sometimes called the fundamental BCJ relations Feng:2010my . These additional
kinematic identities further reduce the basis of partial amplitudes from
$(n-2)!$ to $(n-3)!$ elements. This leads to an associated reduction in
complexity of constructing both tree-level amplitudes and multi-loop
integrands via cut-construction.

###  2.4 The double-copy

A reduction in computational complexity is not the only advantage of
performing calculations in color-dual theories. Consider the replacement of
color-weight with color-dual vector kinematic weight, $c_{g}\rightarrow
n_{g}$, yielding,

$\mathcal{M}_{n,\text{tree}}=\sum_{g_{i}\in\Gamma_{3}^{(n)}}\frac{n_{g_{i}}n_{g_{i}}}{d_{g_{i}}}\
.$ (37)

The resulting theory is local by construction and remains Bose symmetric due
to the algebraic properties of the numerators and the explicit sum over all
permutations of graph labels. But now the amplitudes are colorless with states
determined by the tensor product
$\varepsilon_{i}^{\mu}\otimes\varepsilon_{i}^{\nu}$. Furthermore, the new
amplitude is invariant under _linear diffeomorphisms_ , the “square” of gauge
invariance. In other words, two factors of Yang-Mills theory numerators yields
a theory of axion-dilaton-gravity. We call this procedure the double-copy
construction.

More generally, we can be agnostic to the origin of the numerators, and
construct a general class of theories of the form

$\mathcal{M}_{n,\text{tree}}=\sum_{g_{i}\in\Gamma_{3}^{(n)}}\frac{a_{g_{i}}b_{g_{i}}}{d_{g_{i}}}\
.$ (38)

where the numerator weights $a_{g_{i}}$ and $b_{g_{i}}$ are local functions of
color and kinematics that obey the same algebraic relations of eq. (35),
subject to a suitable gauge choice. As long as both the $a$-stripped ordered
amplitudes and $b$-stripped ordered amplitudes are consistent across
multiplicities via factorization, we call such theories double-copy
consistent. By construction they are local and invariant under generalized
gauge transformations acting on each set of numerators.

It is worth also briefly mentioning how this process continues to loop level.
In Yang-Mills, at $L$ loop level, the $n$-point amplitude in $D$ dimensions is
given by,

$\mathcal{A}^{\rm
YM}_{n,L\text{-loop}}=i^{L-1}g^{n-2+2L}\prod_{l=1}^{L}\int\frac{d^{D}\ell_{l}}{(2\pi)^{D}}\sum_{i}\frac{1}{S_{i}}\frac{c_{i}n_{i}}{d_{i}}$
(39)

where $\ell_{l}$ are the loop momenta, the sum is over all cubic diagrams that
can be built from Feynman rules with distinct color structure $c_{i}$, the
$d_{i}$ are products of propagators for each internal line. The $n_{i}$ depend
on contractions of the various kinematic quantities (including loop momenta),
and $S_{i}$ are symmetry factors. Then, the corresponding gravitational theory
has loop amplitudes BCJ ; BCJLoop  of the form,

$\mathcal{M}_{n,L\text{-loop}}=\mathcal{A}^{\rm
YM}_{n,L\text{-loop}}\Big{|}_{c_{i}\rightarrow\tilde{n}_{i},g\rightarrow\kappa/2}=i^{L-1}\left(\frac{\kappa}{2}\right)^{n-2+2L}\prod_{l=1}^{L}\int\frac{d^{D}\ell_{l}}{(2\pi)^{LD}}\sum_{i}\frac{1}{S_{i}}\frac{\tilde{n}_{i}n_{i}}{d_{i}}\
.$ (40)

As noted previously, the above considerations naturally extend to theories
other than Yang-Mills. The key to the double-copy in all of these cases is
finding color-dual kinematic numerators $n_{g}$, i.e. that satisfy the Jacobi
relations eq. (35). The goal in this paper is to double-copy sYM with a color-
dual theory of vectors coupled to pions, which we will build up using an on-
shell bootstrap. Once we have done so, we can replace adjoint color factors
with color-dual sYM numerators, thereby constructing the desired observables
in a putative theory of VA fermions coupled to supergravity.

###  2.5 Unitarity methods

We presented gauge-invariant building blocks of gauge theories in terms of
easier to compute, purely kinematic partial amplitudes, and we saw how it was
possible to have algebraic constraints on the building blocks of such partial
amplitudes by imposing color-kinematics duality on the numerator factors. How
do we know that these partial amplitudes are consistent with the theory we
mean to be talking about? Generally they must satisfy what are called
unitarity cuts. At tree-level, this is simply factorization to lower-
multiplicity on-shell amplitudes, summed over states in the cut. The
requirement of such consistency allows for verification, and verification is
sufficient for construction, as we will shortly discuss. Furthermore, by
considering ordered amplitudes that contribute to ordered-cuts, far fewer
graphs will be needed for any given cut.

To proceed, we note that the sum of scalar states is trivial, and that the sum
over massless vector states is given by the following completeness relation in
terms of formal polarizations on a cut leg with momenta $k$:

$\displaystyle\sum_{s\in\text{states}}\varepsilon_{s}^{\mu}(k)\varepsilon_{\bar{s}}^{\nu}(-k)=\eta^{\mu\nu}-\frac{k^{\mu}q^{\nu}+q^{\mu}k^{\nu}}{k\cdot
q}\,,$ (41)

where $q$ is an arbitrary null reference momentum that must cancel out in any
physical quantities, such as ordered amplitudes.

As an example, consider the $s_{12}$-channel cut of the partial amplitude
$A^{\text{YM}}(1234)=n_{s}/s_{12}+n_{t}/s_{14}$. We are looking to verify,

$\lim_{s_{12}\to
0}s_{12}A(1234)=\sum_{s\in\text{states}}A(1,2,l_{s})A(-l_{\bar{s}},3,4)$ (42)

where $A(abc)=n_{3}(abc)$ is given in eq. (13). From the LHS of our cut
equation we have:

$\displaystyle\lim_{s_{12}\to 0}s_{12}A(1234)$
$\displaystyle=\left.n_{s}\right|_{s_{12}\to 0}$
$\displaystyle=\left.n(1234)\right|_{s_{12}\to 0}$
$\displaystyle=\left.A(1,2,l)A(-l,3,4)\right|_{\varepsilon(l)^{\mu}\varepsilon(-l)^{\nu}\to\eta^{\mu\nu}}\,.$
(43)

In the last line we have exploited the property that — on the cut — the
contact term drops out of eq. (19), and the remaining inner product in the
numerator can be written as a product between two on-shell three-point
amplitudes if the cut-polarizations are interpreted as a spacetime metric.
This is manifestly equal to the RHS of our cut equation – since in eq. (41),
only the metric contributes in our sum over states, as $k_{i}\cdot l=0$ for
all external legs $i$ due to three-point on-shell kinematics
Kosmopoulos:2020pcd .

###  2.6 Unitarity based bootstrap

One can invert verification to construction in the form of unitarity-based
bootstrap methods UnitarityMethod ; Fusing . This is a longstanding approach
applied to both tree amplitudes and loop-level integrands. We will focus here
on the idea of requiring the duality between color and kinematics in
conjunction with unitarity based factorization.

Let us consider as an example a bootstrap through four-point for the
amplitudes associated with the covariantized free scalar:
$\mathcal{L}=(D\phi)^{2}+\text{Tr}(F^{2})$. We have already fixed the purely
gluonic sector, so we are left to consider amplitudes involving scalars.
Consider the three-point amplitude between adjoint scalars and a massless
vector. The color weight will be of the form $f^{abc}$ and so the kinematic
weight is incredibly restricted. As all dot products between on-shell external
momenta vanish for three-point kinematics, the only three-point amplitude one
can write down is:

$\mathcal{A}^{{(D\phi)^{2}}}_{3}(1,2,3_{A})=gf^{abc}(k_{1}-k_{2})\cdot\varepsilon_{3}\,.$
(44)

Strictly speaking we could have given a new coupling to the three-point
amplitude at this stage, but consistency at four-point will force the same
coupling as the three-point gluon.

At four-points, the first and simplest amplitude to consider is one with all-
scalar external states. The full four-point amplitude will be of the form,

$\mathcal{A}^{{(D\phi)^{2}}}_{4}(1,2,3,4)=g^{2}\left(\frac{n_{s}c_{s}}{s_{12}}+\frac{n_{t}c_{t}}{s_{14}}+\frac{n_{u}c_{u}}{s_{13}}\right)\,.$
(45)

The kinematic weight will be linear in Mandelstam invariants if it is to
factorize to eq. (44). Requiring the duality between color and kinematics
lands us uniquely on the kinematic weight up to a constant of proprtionality,
$\xi$,

$n^{(D\phi)^{2}}(ijkl)=\xi\,(s_{ik}-s_{jk})\,.$ (46)

Here $n_{s}=n^{(D\phi)^{2}}(1234)$ and others following standard relabeling as
for the gluonic numerators previously constructed in eq. (19). Factorization
on the $s_{12}$-channel requires

$\displaystyle n^{(D\phi)^{2}}(1234)|_{s_{12}\to 0}$
$\displaystyle=\left.\xi(s_{13}-s_{23})\right|_{s_{12}\to 0}$ (47)
$\displaystyle=2\,\xi\,s_{13}$ (48)
$\displaystyle=\sum_{s}A(1,2,l_{A,s})A(-l_{A,\bar{s}},3,4)$ (49)
$\displaystyle=\left.(k_{1}-k_{2})\cdot(k_{3}-k_{4})\right|_{s_{12}\to 0}$
(50) $\displaystyle=4\,s_{13}\,$ (51)

fixing the constant of proportionality $\xi=2$. The only remaining amplitude
to consider through four-points is the two-gluon two-scalar amplitude. In this
case we actually have two distinct graph topologies – one where each of the
external gluons interacts with a separate vertex connected by a scalar
propagator, $n_{\text{I}}$, and one where the two gluons interact at the same
vertex, $n_{\text{II}}$. When legs labeled 1 and 4 are vectors, we have,

$\displaystyle\mathcal{A}^{{(D\phi)^{2}}}(1_{A},2,3,4_{A})$
$\displaystyle=g^{2}\left(\frac{c_{s}n_{\text{I}}(1_{A},2,3,4_{A})}{s_{12}}+\frac{c_{t}n_{\text{II}}(1_{A},4_{A},3,2)}{s_{14}}+\frac{c_{u}n_{\text{I}}(1_{A},3,2,4_{A})}{s_{23}}\right)$
(52)
$\displaystyle=g^{2}\left(c_{s}\left[A(1234)\equiv\frac{n_{\text{I}}(1_{A},2,3,4_{A})}{s_{12}}+\frac{n_{\text{II}}(1_{A},4_{A},3,2)}{s_{14}}\right]\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+\,\,c_{u}\left[A(1324)\equiv\frac{n_{\text{I}}(1_{A},3,2,4_{A})}{s_{12}}+\frac{n_{\text{II}}(1_{A},4_{A},2,3)}{s_{14}}\right]\right)\,.$
(53)

The duality between color and kinematics requires that:

$n_{\text{II}}(1_{A},4_{A},3,2)=n_{\text{I}}(1_{A},2,3,4_{A})-n_{\text{I}}(1_{A},3,2,4_{A})\,,$
(54)

which is manifestly antisymmetric under exchange $j\leftrightarrow k$. Thus,
we need only give an ansatz to $n_{\text{I}}$. At this mass-dimension, a
spanning set of independent Lorentz invariants is simply:

$\left\\{\varepsilon_{1}\cdot\varepsilon_{4},\varepsilon_{1}\cdot
k_{2},\varepsilon_{1}\cdot k_{4},\varepsilon_{4}\cdot
k_{1},\varepsilon_{4}\cdot k_{2},s_{13},s_{14}\right\\}\,$ (55)

and our initial ansatz is thus:

$n_{\text{I}}(1_{A},2,3,4_{A})=\varepsilon_{1}\cdot\varepsilon_{4}(\xi_{1}s_{13}+\xi_{2}s_{14})+(\varepsilon_{1}\cdot
k_{2})\left(\xi_{3}(\varepsilon_{4}\cdot k_{1})+\xi_{4}(\varepsilon_{4}\cdot
k_{2})\right)\\\ +(\varepsilon_{1}\cdot
k_{4})\left(\xi_{5}(\varepsilon_{4}\cdot k_{1})+\xi_{6}(\varepsilon_{4}\cdot
k_{2})\right)$ (56)

Fixing on the $s$-channel ordered cut of $A(1234)$,

$n_{\text{I}}(1_{A},2,3,4_{A})|_{s_{12}\to
0}=A(1_{A},2,l)A(-l,3,4_{A})=-4(\varepsilon_{1}\cdot
k_{2})(\varepsilon_{4}\cdot k_{2})\,,$ (57)

yields $\xi_{1}=\xi_{2}$, $\xi_{4}=4$, and $\xi_{3}=\xi_{5}=\xi_{6}=0$. One
can fix the remaining parameters by either considering the $t$-channel cut,

$n_{\text{II}}(1_{A},4_{A},3,2)|_{s_{14}\to
0}=\sum_{s}A(1_{A},4_{A},l_{A,s})A(-l_{A,\bar{s}},3,4)\,,$ (58)

or demanding gauge invariance of $A(1234)$. Either of these constraints fixes
$\xi_{2}=1$. This yields ultimately,

$n_{\text{I}}(1_{A},2,3,4_{A})=-4(k_{2}\cdot\varepsilon_{1})(k_{3}\cdot\varepsilon_{4})-s_{12}(\varepsilon_{1}\cdot\varepsilon_{4})\,,$
(59)

and we are done through four-points. Note that the presence of $s_{12}$ means
that our bootstrap yields the contact-term required for the gauge-invariance
of $(D\phi)^{2}$. We will employ a similar strategy to attempt to couple pions
to vectors in a color-dual fashion.

###  2.7 Vector tower coming from $\,{\rm Tr}(F^{3})$

We will see that the $\,{\rm Tr}(F^{3})$ operator must be included in our
first candidate color-dual vector-pion theory, and this has consequences.
Here, we briefly review the relevant results of f3Ladder  which described the
color-dual bootstrap for a pure-vector Yang-Mills theory deformed by $\,{\rm
Tr}(F^{3})$, which ultimately must be a part of the vector-scalar theory that
we consider in this work. It was shown in f3Ladder  that an action of the
form333To compare the notation of f3Ladder  to the current paper, we note that
f3Ladder  had $g=1$. Restoring powers of $g$, the coupling of $\,{\rm
Tr}(F^{3})$ should be $g\alpha^{\prime}$ as in eq. (60), which allows us to
match the two notations with the identification
$\alpha^{\prime}=-\Lambda/g^{2}$.

$\mathcal{L}_{\text{YM}+F^{3}}=-\frac{1}{4}\,{\rm
Tr}(F^{2})+\frac{g\alpha^{\prime}}{3}\,{\rm Tr}(F^{3})\ ,$ (60)

while obviously gauge invariant, is not color dual to all tree-level
multiplicity at all orders in $\alpha^{\prime}$. Instead, it appears that in
order to make the theory double-copy consistent, one must add an infinite
tower of higher-derivative operators with increasing powers of
$\alpha^{\prime}$. This is required even at low multiplicity.

$\begin{split}A^{(n)}{(12345)}\Big{|}_{s_{45}-\text{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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey1](mid1) at (-1,0) {$\,\alpha^{\prime n-1}\,$};
\vertex[white](mid3) at (-1,0) {$\,\alpha^{\prime n-1}\,$};
\vertex[blob,hred](mid2) at (0.7,0) {$\,\alpha^{\prime 1}\,$};
\vertex[white](mid4) at (0.7,0) {$\,\alpha^{\prime 1}\,$}; \vertex(a3) at
(1.6, 1.2) {$4$}; \vertex(a4) at (1.6, -1.2) {$5$}; \vertex(a7) at (0, -0.9);
\vertex(a6) at (0, 0.9); \diagram{ (mid2) --[gluon, thick](a3), (mid2)
--[gluon, thick](a4), (mid2) -- [gluon, thick](mid1), (a1)--[gluon,
thick](mid1), (a2) -- [gluon, thick](mid1), (mid1) --[gluon, thick](a5),
(a6)--[scalar, thick,cutred](a7) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+{\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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey2](mid1) at (-1,0) {$\,\alpha^{\prime}\,{}^{n}\,$};
\vertex[white](mid3) at (-1,0) {$\,\alpha^{\prime}\,{}^{n}\,$};
\vertex[blob,hblue](mid2) at (0.7,0) {$\,\alpha^{\prime 0}\,$};
\vertex[white](mid4) at (0.7,0) {$\,\alpha^{\prime 0}\,$}; \vertex(a3) at
(1.6, 1.2) {$4$}; \vertex(a4) at (1.6, -1.2) {$5$}; \vertex(a7) at (0, -0.9);
\vertex(a6) at (0, 0.9); \diagram{ (mid2) --[gluon, thick](a3), (mid2)
--[gluon, thick](a4), (mid2) -- [gluon, thick](mid1), (a1)--[gluon,
thick](mid1), (a2) -- [gluon, thick](mid1), (mid1) --[gluon, thick](a5),
(a6)--[scalar, thick,cutred](a7) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\mathcal{O}(\alpha^{\prime
n})}\end{split}$

$\begin{split}A^{(n+1)}{(12345)}\Big{|}_{s_{45}-\text{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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey2](mid1) at (-1,0) {$\,\alpha^{\prime}\,{}^{n}\,$};
\vertex[white](mid3) at (-1,0) {$\,\alpha^{\prime}\,{}^{n}\,$};
\vertex[blob,hred](mid2) at (0.7,0) {$\,\alpha^{\prime 1}\,$};
\vertex[white](mid4) at (0.7,0) {$\,\alpha^{\prime 1}\,$}; \vertex(a3) at
(1.6, 1.2) {$4$}; \vertex(a4) at (1.6, -1.2) {$5$}; \vertex(a7) at (0, -0.9);
\vertex(a6) at (0, 0.9); \diagram{ (mid2) --[gluon, thick](a3), (mid2)
--[gluon, thick](a4), (mid2) -- [gluon, thick](mid1), (a1)--[gluon,
thick](mid1), (a2) -- [gluon, thick](mid1), (mid1) --[gluon, thick](a5),
(a6)--[scalar, thick,cutred](a7) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+{\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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey3](mid1) at (-1,0) {$\,\alpha^{\prime n+1}\,$};
\vertex[white](mid3) at (-1,0) {$\,\alpha^{\prime n+1}\,$};
\vertex[blob,hblue](mid2) at (0.7,0) {$\,\alpha^{\prime 0}\,$};
\vertex[white](mid4) at (0.7,0) {$\,\alpha^{\prime 0}\,$}; \vertex(a3) at
(1.6, 1.2) {$4$}; \vertex(a4) at (1.6, -1.2) {$5$}; \vertex(a7) at (0, -0.9);
\vertex(a6) at (0, 0.9); \diagram{ (mid2) --[gluon, thick](a3), (mid2)
--[gluon, thick](a4), (mid2) -- [gluon, thick](mid1), (a1)--[gluon,
thick](mid1), (a2) -- [gluon, thick](mid1), (mid1) --[gluon, thick](a5),
(a6)--[scalar, thick,cutred](a7) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\\!\\!\\!\\!{\mathcal{O}(\alpha^{\prime
n+1})}\end{split}$

Figure 2:  Inductive ladder of higher-derivative contacts for the pure-vector
theory generated by cuts of the form
$A^{(k)}_{5}|_{\text{cut}}\sim(A_{4}^{(k)}A_{3}^{\text{YM}}+A_{4}^{(k-1)}A_{3}^{F^{3}})$
and considering double-copy consistency, where $A_{n}^{(k)}$ denotes $n$-point
amplitudes appearing at mass-dimension $\alpha^{\prime k}$.

Consider the four-point amplitude. The action in eq. (60) gives a four-point
amplitude at $\mathcal{O}(\alpha^{\prime}{}^{2})$ by combining two
$\alpha^{\prime}\,{\rm Tr}(F^{3})$ vertices. This amplitude is not color dual
by itself, but can be made so by adding a term proportional to
$\alpha^{\prime}{}^{2}\,{\rm Tr}(F^{4})$ to the theory Broedel2012rc . The new
$\alpha^{\prime}{}^{2}\,{\rm Tr}(F^{4})$ term then contributes to the five-
point amplitude. In particular, there is an
$\mathcal{O}(\alpha^{\prime}{}^{3})$ contribution at five points coming from
combining $\alpha^{\prime}{}^{2}\,{\rm Tr}(F^{4})$ with $\alpha^{\prime}\,{\rm
Tr}(F^{3})$. This contribution was also shown to not be color dual by itself.
To make it color dual, one has to add a new operator at four points,
schematically $\alpha^{\prime}{}^{3}\,{\rm Tr}(\partial^{2}F^{4})$, which
combines with the $\mathcal{O}(\alpha^{\prime}{}^{0})$ three-point vertex in
eq. (60) to give an $\mathcal{O}(\alpha^{\prime}{}^{3})$ contribution at five
points and make the amplitude color dual at that order. However, the new
$\alpha^{\prime}{}^{3}\,{\rm Tr}(\partial^{2}F^{4})$ can now combine with
$\alpha^{\prime}\,{\rm Tr}(F^{3})$ to give an
$\mathcal{O}(\alpha^{\prime}{}^{4})$ contribution at five points, which again
is not color dual, but can be fixed by a term schematically like
$\alpha^{\prime}{}^{4}\,{\rm Tr}(\partial^{4}F^{4})$. The essence of this cut-
consistency is outlined in Fig. 2. Thus, by just considering the four- and
five-point amplitudes, this tower was conjectured to continue indefinitely,
requiring an infinite number of higher-derivative operators to be color dual
and satisfy the correct factorization.

##  3 Bootstrapping to a color-dual vector-pion theory

One might expect that a color-dual theory involving vectors and pions would be
an inevitability. After all, both pure NLSM amplitudes Chen2013fya ;
Cachazo2014xea  and pure gluonic amplitudes are double-copy consistent
KiermaierTalk ; BjerrumMomKernel  to all multiplicity. While one can be
inspired by color-dual all-vector amplitudes and all-pion amplitudes, the
existence of a double-copy consistent theory admitting mixed pion and vector
amplitudes is an open question that we will now begin to address.

###  3.1 Gauged nonlinear sigma models

A natural first step would be to couple Yang-Mills to the NLSM via simple
covariantization, by promoting $\partial_{\mu}\rightarrow D_{\mu}$. Consider
then the covariantized NLSM Lagrangian444Starting from the form of the pion
action given in e.g. Cachazo2014xea . for $SU(N)$ adjoint scalars
$\pi=\pi^{a}T^{a}$ coupled to gluons $A_{\mu}=A_{\mu}^{a}T^{a}$,

$\mathcal{L}^{\rm{cov.}\pi}=-\frac{1}{4}\,{\rm Tr}(F^{2})+\frac{1}{2}\,{\rm
Tr}\left[\left(1-\Lambda\pi^{2}\right)^{-1}D_{\mu}\pi\left(1-\Lambda\pi^{2}\right)^{-1}D^{\mu}\pi\right]\,,$
(61)

where $\Lambda$ is related to the pion decay constant $f_{\pi}$ by
$\Lambda\sim 1/f_{\pi}^{2}$, and the covariant derivative is given as

$D_{\mu}\pi=(\partial_{\mu}\pi^{a}-igf^{abc}A_{\mu}^{b}\pi^{c})T^{a}\,.$ (62)

Such theories, also with additional operators, have a venerable history in the
literature, see refs. Brezin:1980ms ; Gates:1984nk  for some earlier examples,
and are referred to generally as gauged nonlinear sigma models (e.g.
Hull:1989jk ).

Our approach is as follows. We take eq. (61) as our starting point for a
theory of vectors and scalars. We then look at amplitudes of increasing
multiplicity and check whether they are color dual. Finding that at five-
points they are not, we will modify eq. (61) using the color-dual bootstrap,
essentially adding new operators to eq. (61), to ensure that the resulting
theory is double-copy consistent. In the end, we find towers of higher-
derivative operators that must be added to eq. (61) in order for the
amplitudes to be color dual-and satisfy consistent factorization to lower-
point amplitudes.

All tree-level amplitudes in this theory through four-points are already known
to be color dual. The all-vector amplitudes belong to the predictions of Yang-
Mills theory. The $\mathcal{O}(\Lambda)$ contribution to the all-scalar four-
point amplitude belongs to the predictions of NLSM. The remaining
contributions through four point belong to the scattering amplitudes of the
covariantized free-scalar theory, sometimes referred to as the simple-scalar,
i.e. $\Lambda\to 0$ in eq. (61). The first opportunity for an amplitude
original to the theory eq. (61) is at five points, which we discuss in the
following section.

###  3.2 $\,{\rm Tr}(F^{3})$ from double-copy consistency

To begin, we look at the $\mathcal{O}(g\Lambda)$ behavior of
$\mathcal{A}^{\rm{cov.}\pi}(\pi\pi\pi\pi A)$, with a factorization channel
involving the characteristic four-pion amplitude as well as the three-point
scalar-vector interaction. Through explicit calculation from eq. (61) we find
that the tree-level five-point amplitude
$\mathcal{A}^{\rm{cov.}\pi}(\pi\pi\pi\pi A)$, fails to satisfy the consistency
relations of ref. BCJ  required to participate in standard double-copy
construction. Recall, however, that we only care about the theory of eq. (61)
as a starting point. There are any number of gauge-invariant operators one
might introduce, and indeed may be required, for the full theory to be double-
copy consistent, as described in f3Ladder .

There are two approaches to resolving this issue that we follow. A potentially
more radical solution is to admit the possibility that pions may belong to
possibly distinct fields – breaking exchange symmetry between scalars. We
leave the discussion of such a model to section 6. The more conservative
approach, which we consider now, is to simply admit higher derivative
corrections to the fields we have already present in our theory. To identify
the requisite operators, we follow the color-dual bootstrap approach of
starting with an ansatz for the amplitude of the appropriate kinematic mass-
dimension and little-group scaling. We then establish whether or not it is
possible to fix parameters so that our desired scattering amplitude is color
dual and satisfies the generalized unitarity cuts we value. Previous examples
in the literature of a similar approach include bootstrapping NLSM pion
amplitudes in ref. BCJreview  and massive scalar QCD in ref. Carrasco:2020ywq
. In these cases, at a given kinematic mass-dimension, satisfactying color-
kinematics duality and factorization alone entirely constrain the amplitudes
of the theory.

In the current problem of identifying the existence and nature of a not-
completely specified theory, we recognize that we might not have included at
lower multiplicity all the interactions required to admit both color-
kinematics duality as well as healthy factorization at five-points. So as far
as factorization is concerned, we will at first only insist that our five-
point amplitude of the correct mass-dimension must be color dual, and satisfy
the factorization to four-pion scattering

$\mathcal{A}_{5}(\pi\pi\pi\pi
A)|_{(k_{4}+k_{5})\text{-cut}}=\mathcal{A}_{4}(\pi\pi\pi\pi)\times\mathcal{A}_{3}(\pi\pi
A)\,.$ (63)

We will learn what was missing from the simple covariantization of eq. (61) by
probing the analytic structure of this new color-dual amplitude.

Five-point amplitudes with four external scalars can be expressed in terms of
kinematic and color dressings of the fifteen distinctly labeled cubic
(trivalent) graphs, $\Gamma^{(5)}_{3}$,

$\mathcal{A}_{5}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})=\sum_{g\in\Gamma^{(5)}_{3}}\frac{c_{g}n_{g}}{d_{g}}\,.$
(64)

The color-weights $c_{g}$ are given by dressing each vertex with adjoint
structure constants, $f^{abc}$, the propagators $d_{g}$ are dressed as normal
for massless particles, and any remaining kinematics and coupling constants
are absorbed into the kinematic numerators, $n_{g}$.

The fifteen graphs contributing to the five-particle amplitude can be
described as relabelings of only two graph topologies when one encodes
external particle type in the structure of the graph. In our case the four
equivalent particles are the pions, and the lone distinct fifth particle is
the vector. The basis topology graph, with external gluon on a terminal
vertex, we will refer to as $g_{1}$,

$g_{1}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})={\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, -0.9)
{$1$}; \vertex(a2) at (-1, 0.9) {$2$}; \vertex(mid1) at (-1,0); \vertex(mid2)
at (0.3,0); \vertex(mid3) at (1.6, 0); \vertex(a5) at (0.3, 0.9) {$3$};
\vertex(a3) at (1.6, 0.9) {$4$}; \vertex(a4) at (1.6, -0.9) {$5$}; \diagram{
(mid2) --[ thick](a5), (mid2) -- [ thick,scalar](mid3), (a1)--[ thick](mid1),
(a2) -- [ thick](mid1), (mid1)--[ thick,scalar](mid2), (mid3)--[ thick](a3),
(a4) -- [ thick,gluon](mid3) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,.$ (65)

The other topology, $g_{2}$, has the external vector on the central vertex,

$g_{2}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})={\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, -0.9)
{$3$}; \vertex(a2) at (-1, 0.9) {$4$}; \vertex(mid1) at (-1,0); \vertex(mid2)
at (0.3,0); \vertex(mid3) at (1.6, 0); \vertex(a5) at (0.3, 0.9) {$5$};
\vertex(a3) at (1.6, 0.9) {$1$}; \vertex(a4) at (1.6, -0.9) {$2$}; \diagram{
(mid2) --[ thick,gluon](a5), (mid2) -- [ thick,scalar](mid3), (a1)--[
thick](mid1), (a2) -- [ thick](mid1), (mid1)--[ thick,scalar](mid2), (mid3)--[
thick](a3), (a4) -- [ thick](mid3) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,.$ (66)

Note in neither topology do we bother assigning a particle type to the
internal legs (represented as dashed edges here). All propagating particles
are massless and it does not affect dressing the graph with kinematic weights,
color-weights, or its propagators. Contact terms are included by allowing for
terms proportional to inverse propagators in the kinematic numerators of the
cubic graphs.

A color-dual kinematic Jacobi relation gives the kinematic weight of the
second topology, $n(g_{2})$, in terms of two labelings of the first,
$n(g_{1})$,

$n(g_{2}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A}))=n(g_{1}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A}))-(3\leftrightarrow
4)\,.$ (67)

All fifteen graphs in eq. (64) are thus given by the kinematic weight of
various relabelings of the basis graph $g_{1}$, or linear combinations
thereof. Thus, we need only give an ansatz of the correct mass-dimension and
little-group scaling to $n(g_{1})$. Little group scaling demands a
polarization for the gluon, $\varepsilon_{5}$, in each term. For the
$\mathcal{O}(g\Lambda)$ contribution to $\mathcal{A}_{5}(\pi\pi\pi\pi A)$, we
require $n(g_{1})$ to be third-order in Lorentz-invariants, $(k_{i}\cdot
k_{j})(k_{l}\cdot k_{m})(k_{n}\cdot\varepsilon_{5})$. In terms of a minimal
basis of on-shell kinematics, the ansatz starts with 45 free parameters. We
enforce color-kinematics by imposing antisymmetry of each vertex as well as
kinematic Jacobi on every edge of both graphs. All remaining parameters are
constrained uniquely via the four-pion cut eq. (63) and the vector Ward
identity. In contrast to the amplitudes constructed from the
$\mathcal{L}^{\rm{cov.}\pi}$ of eq. (61), we can now build the partial
amplitudes that satisfy the color-dual BCJ relations at five-point,

$\displaystyle\begin{split}A_{5}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})&=g\Lambda\Bigg{[}\frac{s_{35}\kappa_{2}^{(5)}-s_{25}\kappa_{3}^{(5)}}{s_{23}}+\frac{s_{35}s_{25}\kappa_{12}^{(5)}}{s_{12}s_{34}}+\frac{s_{25}\kappa_{3}^{(5)}}{s_{34}}-\frac{s_{35}\kappa_{2}^{(5)}}{s_{12}}\\\
&\hskip
130.08621pt+3\left(\frac{s_{24}\kappa_{1}^{(5)}}{s_{15}}-\frac{s_{13}\kappa_{4}^{(5)}}{s_{45}}+\kappa_{24}^{(5)}\right)\Bigg{]},\end{split}$
(68)

where $s_{ij}=(k_{i}+k_{j})^{2}$, and $\kappa_{i\dots
j}^{(5)}\equiv(k_{i}+\dots+k_{j})\cdot\varepsilon_{5}$.

Now we can see what was missing from the simple gauged NLSM in order to
manifest the duality between color and kinematics. Had we just considered the
contribution from $\mathcal{L}^{\rm{cov.}\pi}$, only the final term in eq.
(68) would have been generated. As can be seen in the first line, the newly
constructed $A(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})$ has a two-propagator
residue, admitting a non-vanishing maximal-cut proportional to $g\Lambda$
contributing to the secondary $g_{2}$ topology,

$\mathcal{A}_{5}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})|_{g_{2}\text{
Max.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, -0.8)
{$3$}; \vertex(a2) at (-1, 0.8) {$4$}; \vertex(mid1) at (-1,0); \vertex(midAb)
at (0,0) ; \vertex(midAt) at (0,.8) {$5$}; \vertex(mid2) at (1,0); \vertex(a3)
at (1, 0.8) {$1$}; \vertex(a4) at (1,- 0.8) {$2$}; \vertex(a5) at (0.5, -0.7);
\vertex(a6) at (0.5, 0.7); \vertex(a7) at (-0.5, -0.7); \vertex(a8) at (-0.5,
0.7); \diagram{ (midAb) --[ thick,gluon](midAt), (mid2) --[ thick](a3), (mid2)
--[ thick](a4), (mid2) -- [ thick,scalar](midAb), (midAb) --[
thick,scalar](mid1), (a1) --[ thick](mid1), (a2) --[ thick](mid1),
(a5)--[scalar, thick,cutred](a6), (a7)--[scalar, thick,cutred](a8) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,.$

If we had required that the above channels vanish (as they do for the theory
of eq. (61)) then we would not have found a color-dual solution. This cut can
only be satisfied at this mass-dimension (in a non-abelian manner) by the
inclusion of amplitudes associated with the $\,{\rm Tr}(F^{3})$
operator555Additional higher-derivative contact interactions may be required
by double-copy consistency. We also note that the $\,{\rm Tr}(F^{3})$ operator
that we consider here is the only other independent all-vector three-point
amplitude besides the one contained in pure YM. In our notation, it is given
explicitly by $\,{\rm Tr}(F^{3})=\,{\rm
Tr}(F_{\mu}{}^{\nu}F_{\nu}{}^{\rho}F_{\rho}{}^{\mu})$. whose coupling is fixed
to be $g_{F^{3}}=\Lambda/g$. Intriguingly, $\,{\rm Tr}(F^{3})$ was the first
higher-derivative vector operator found to be compatible Broedel2012rc  with
adjoint-type double-copy, at least for all amplitudes admitting only single-
insertions SchlottererBGCurrent . This leads us to the following starting
point for the double-copy consistent theory for YM + $\pi$,

$\mathcal{L}^{\rm
dcc}_{\text{YM}+\pi}\supset\mathcal{L}^{\text{cov.}\pi}-\frac{g_{F^{3}}}{3}\,{\rm
Tr}(F^{3})+\dots\ .$ (69)

In the next section we will demonstrate that this is just the beginning. Once
$\,{\rm Tr}(F^{3})$ forces itself into the conversation, we find for double
copy consistency that an infinite tower of scalar-vector operators are needed
at fixed multiplicity, thereby pushing both scalar and vector sectors of the
theory into concordance with particular ultraviolet completions.

##  4 Completing the vector-pion theory under double-copy consistency

###  4.1 New contributions for vector amplitudes with zero scalars

The first set of amplitudes to consider are the all-vector observables
resulting from the inclusion of $\,{\rm Tr}(F^{3})$ to the Lagrangian of eq.
(61). As reviewed in section 2.7, these contributions were identified in
f3Ladder , where the interplay between pure Yang-Mills theory and $\,{\rm
Tr}(F^{3})$ induces a ladder of higher-derivative four-point local contact
terms. The coefficients for a subset of the available color-dual contacts are
completely fixed by the gauge coupling, $g$, and the dimensionful $\,{\rm
Tr}(F^{3})$ coefficient, $g_{F^{3}}=\Lambda/g$, when we demand that the theory
is double-copy consistent. By setting any additional higher-derivative freedom
to zero, we find that for $k\geq 2$, the required color-dual contacts take the
form:

$A_{4}^{(k)}(1_{A},2_{A},3_{A},4_{A})=g^{2}\left(\frac{\Lambda}{g^{2}}\right)^{k}u\times\Bigg{[}\frac{\,{\rm
tr}[F_{1}F_{2}]\,{\rm
tr}[F_{3}F_{4}]}{s_{12}^{2}}s_{12}^{k-1}+\text{cyc}(2,3,4)\Bigg{]},$ (70)

where

$F_{i}^{\mu\nu}=k_{i}^{\mu}{\varepsilon}_{i}^{\nu}-k_{i}^{\nu}\varepsilon_{i}^{\mu}\,,$
(71)

and

$A^{(F^{3})^{2}+F^{4}}_{4}\equiv A_{4}^{(2)},\qquad A_{4}^{D^{2k}F^{4}}\equiv
A^{(k+2)}_{4}.$ (72)

It will be helpful to note that $A^{(1)}_{4}$ is intimately related to the
$A^{(DF)^{2}}$ amplitude of Johansson:2017srf  as written in $D$-dimensions
Azevedo2018dgo ,

$A^{(1)}_{4}=A^{F^{3}}_{4}-A^{(DF)^{2}}_{4}\,.$ (73)

As stated above, these are precisely the building blocks that appear in the
inductive ladder described in section 2.7. By considering pure-vector contacts
that contribute to two-scalar three-vector five-point amplitudes via
factorization, we have now been able to explicitly verify eq. (70) through
$\mathcal{O}(\Lambda^{5})$.

In ref. f3Ladder  it was conjectured that the resulting two-parameter
effective field theory can be resummed to $(DF)^{2}+\text{YM}$ theory, first
identified in Johansson:2017srf  when considering whether conformal gravity is
double-copy constructible. The amplitudes generated by the
$(DF)^{2}+\text{YM}$ Lagrangian have been used in the double-copy construction
of open bosonic string and heterotic string amplitudes at tree-level. The
conjectured resummation of the inductive ladder in ref. f3Ladder  is due to
the observation that if one collects $A_{4}^{(k)}$ to all orders in mass-
dimension, this is equivalent to a low energy expansion of the four-point
ordered $(DF)^{2}+\text{YM}$ amplitude:

$\begin{split}g^{-2}A^{(DF)^{2}+\text{YM}}(1234)&=A^{\text{YM}}_{(1234)}+\alpha^{\prime}A^{(DF)^{2}}_{(1234)}+\alpha^{\prime}u\Bigg{[}\frac{\,{\rm
tr}[F_{1}F_{2}]\,{\rm
tr}[F_{3}F_{4}]}{s^{2}_{12}(1+\alpha^{\prime}s_{12})}+\text{cyc}(2,3,4)\Bigg{]}\\\
&=A^{\text{YM}}_{(1234)}+\alpha^{\prime}A^{F^{3}}_{(1234)}+\sum_{k=2}^{\infty}\alpha^{\prime\,k}A^{(k)}_{(1234)}\,.\end{split}$
(74)

Recall, as stated in section 2.7, the dimensionful coupling
$\alpha^{\prime}=-\Lambda/g^{2}$. The additional freedom alluded to above was
then described in ref. f3Ladder  by the following double-copy consistent
higher-derivative vector amplitude:

$A^{\text{dcc}}_{(1234)}=A^{(DF)^{2}+\text{YM}}(1234)\left[1+\sum_{x\geq
1,y}c_{(x,y)}\sigma_{3}^{x}\sigma_{2}^{y}\right]$ (75)

where $\sigma_{3}=(stu)/8$ and $\sigma_{3}=(s^{2}+t^{2}+u^{2})/8$ are
permutation invariants of four-point kinematics. Again, by considering two-
scalar three-vector factorization, we have verified that the double-copy
consistent building blocks in our tower are reproduced by the low energy
expansion of eq. (75) up through $\mathcal{O}(\Lambda^{5})$, at which point
there are two additional free parameters, $c_{(1,0)}$ and $c_{(1,1)}$. Thus,
it would appear that in order for our candidate vector-pion theory to satisfy
the duality between color and kinematics to all-multiplicity, we are
interested in a Lagrangian of the form, $(DF)^{2}+\text{YM}+\pi+\ldots$, where
the ellipses denote the potential of additional higher-derivative operators
that are unconstrained by double-copy consistency.

For the remainder of this section, we will focus our attention on the
operators whose Wilson coefficients are fixed completely in terms of $\Lambda$
and $g$, and thus required by double-copy consistency, and allow all
unconstrained freedom in the free parameters, $c_{(x,y)}$, to vanish.

###  4.2 New contributions for vector amplitudes with two scalars

Now, in our vector-scalar theory, the inclusion of $\,{\rm Tr}(F^{3})$ creates
a new $\mathcal{O}(\Lambda)$ contribution at four points to the two-pion, two-
vector scattering amplitude, $\mathcal{A}_{4}(\pi\pi AA)$. Consider the four-
point scattering where legs one and two are pions, and legs three and four are
vectors. Demanding factorization to the correct $\,{\rm Tr}(F^{3})$ vertex and
double-copy consistency, there is a new $\Lambda$ contribution to the partial
amplitude,

$A_{4}^{(1)}(1_{\pi},2_{\pi},3_{A},4_{A})=\,{\Lambda}\frac{u}{s}\Big{[}(k_{3}\cdot\varepsilon_{4})(k_{4}\cdot\varepsilon_{3})-(k_{3}\cdot
k_{4})(\varepsilon_{3}\cdot\varepsilon_{4})\Big{]}\,,$ (76)

where $s=(k_{1}+k_{2})^{2}$, $u=(k_{1}+k_{3})^{2}=-s-t$, and for the remainder
of the paper, the superscript (k) denotes $k$-th order in $\Lambda$.

We also find that demanding factorization and double-copy consistency has
forced an additional two-pion-two-vector contact operator. This can most
easily be seen by considering the other ordered amplitudes given by the BCJ
relations. At four-point these are

$\frac{A_{4}(1_{\pi},2_{\pi},3_{A},4_{A})}{u}=\frac{A_{4}(1_{\pi},2_{\pi},4_{A},3_{A})}{t}=\frac{A_{4}(1_{\pi},3_{A},2_{\pi},4_{A})}{s}\
,$ (77)

which is true for our amplitudes by construction. Then, we can see from eq.
(76) that the $s$-channel pole in $A_{4}(1_{\pi},3_{A},2_{\pi},4_{A})$
cancels, making it a pure contact term. For completeness, the full color-
dressed amplitude is given by

$\mathcal{A}_{4}(\pi\pi
AA)=\left(c_{s}u+c_{u}s\right)\frac{A_{4}(1_{\pi},2_{\pi},3_{A},4_{A})}{u}.$
(78)

$\begin{split}A^{(n)}{(1^{\pi}2^{A}3^{\pi}4^{A}5^{A})}\Big{|}_{s_{45}-\text{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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey1](mid1) at (-1,0) {$\,\Lambda^{n-1}\,$};
\vertex[white](mid3) at (-1,0) {$\,\Lambda^{n-1}\,$}; \vertex[blob,hred](mid2)
at (0.7,0) {$\,\Lambda^{1}\,$}; \vertex[white](mid4) at (0.7,0)
{$\,\Lambda^{1}\,$}; \vertex(a3) at (1.6, 1.2) {$4$}; \vertex(a4) at (1.6,
-1.2) {$5$}; \vertex(a7) at (0, -0.9); \vertex(a6) at (0, 0.9); \diagram{
(mid2) --[gluon, thick](a3), (mid2) --[gluon, thick](a4), (mid2) -- [gluon,
thick](mid1), (a1)--[ thick](mid1), (a2) -- [gluon, thick](mid1), (mid1) --[
thick](a5), (a6)--[scalar, thick,cutred](a7) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+{\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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey2](mid1) at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[white](mid3)
at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[blob,hblue](mid2) at (0.7,0)
{$\,\Lambda^{0}\,$}; \vertex[white](mid4) at (0.7,0) {$\,\Lambda^{0}\,$};
\vertex(a3) at (1.6, 1.2) {$4$}; \vertex(a4) at (1.6, -1.2) {$5$}; \vertex(a7)
at (0, -0.9); \vertex(a6) at (0, 0.9); \diagram{ (mid2) --[gluon, thick](a3),
(mid2) --[gluon, thick](a4), (mid2) -- [gluon, thick](mid1), (a1)--[
thick](mid1), (a2) -- [gluon, thick](mid1), (mid1) --[ thick](a5),
(a6)--[scalar, thick,cutred](a7) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\qquad{\mathcal{O}(\Lambda^{n})}\end{split}$

$\begin{split}A^{(n+1)}{(1^{\pi}2^{A}3^{\pi}4^{A}5^{A})}\Big{|}_{s_{45}-\text{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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey2](mid1) at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[white](mid3)
at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[blob,hred](mid2) at (0.7,0)
{$\,\Lambda^{1}\,$}; \vertex[white](mid4) at (0.7,0) {$\,\Lambda^{1}\,$};
\vertex(a3) at (1.6, 1.2) {$4$}; \vertex(a4) at (1.6, -1.2) {$5$}; \vertex(a7)
at (0, -0.9); \vertex(a6) at (0, 0.9); \diagram{ (mid2) --[gluon, thick](a3),
(mid2) --[gluon, thick](a4), (mid2) -- [gluon, thick](mid1), (a1)--[
thick](mid1), (a2) -- [gluon, thick](mid1), (mid1) --[ thick](a5),
(a6)--[scalar, thick,cutred](a7) }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+{\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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey3](mid1) at (-1,0) {$\,\Lambda^{n+1}\,$};
\vertex[white](mid3) at (-1,0) {$\,\Lambda^{n+1}\,$};
\vertex[blob,hblue](mid2) at (0.7,0) {$\,\Lambda^{0}\,$}; \vertex[white](mid4)
at (0.7,0) {$\,\Lambda^{0}\,$}; \vertex(a3) at (1.6, 1.2) {$4$}; \vertex(a4)
at (1.6, -1.2) {$5$}; \vertex(a7) at (0, -0.9); \vertex(a6) at (0, 0.9);
\diagram{ (mid2) --[gluon, thick](a3), (mid2) --[gluon, thick](a4), (mid2) --
[gluon, thick](mid1), (a1)--[ thick](mid1), (a2) -- [gluon, thick](mid1),
(mid1) --[ thick](a5), (a6)--[scalar, thick,cutred](a7) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\qquad{\mathcal{O}(\Lambda^{n+1})}\end{split}$

Figure 3: Inductive ladder of higher-derivative contacts generated by
consideration of color-dual consistency and cuts of the form $A(\pi\pi
AA)\times(A_{3}^{\text{YM}}+A_{3}^{F^{3}})$.

With the inclusion of $\,{\rm Tr}(F^{3})$ and an $\mathcal{O}(\Lambda)$
contribution to $A_{4}(\pi\pi AA)$ required by double-copy consistency, we are
now forced to consider contributions to $\mathcal{A}_{5}(\pi\pi AAA)$ at
$\mathcal{O}(\Lambda^{2})$. We proceed with the same procedure as the previous
section, constructing an ansatz that respects mass-dimension and little-group
scaling, and then impose BCJ relations, and we will find an inductive process,
in the same spirit to that identified in f3Ladder  and reviewed in section
2.7, that generates a tower of higher derivative operators with Wilson
coefficients fixed in terms of $\Lambda$ and $g$.

After requiring that our kinematic weights are color-dual, we can fix the
remaining freedom on consistent factorization to the available lower
multiplicity building blocks. At this mass-dimension, the
$\mathcal{O}(\Lambda)$ partial amplitude of eq. (76) will contribute to a
three-to-two-particle generalized unitarity cut with $\,{\rm Tr}(F^{3})$. This
will leave the $\,{\rm Tr}(F^{2})$ contribution to the two-particle cut
available to interact with a potential $\mathcal{O}(\Lambda^{2})$ contact on
the other side of the cut, if required by double-copy consistency (shown in
Fig. 3 for general mass-dimension). Indeed, we find that an additional four-
point contact term is required of the form,

$A_{4}^{(2)}(1_{\pi},2_{\pi},3_{A},4_{A})=\frac{\Lambda^{2}}{g^{2}}\,u\Big{[}(k_{3}\cdot\varepsilon_{4})(k_{4}\cdot\varepsilon_{3})-(k_{3}\cdot
k_{4})(\varepsilon_{3}\cdot\varepsilon_{4})\Big{]}\,,$ (79)

for our candidate vector pion theory to continue to respect double-copy
consistency. By the same logic, this now requires us to further consider
contributions at the next order in $\Lambda$ – yielding yet another four-point
contact at $\mathcal{O}(\Lambda^{3})$. In fact, we observe that in order for
$\mathcal{A}_{5}(\pi\pi AAA)$ to be consistent to all orders in $\Lambda$,
contact terms contributing to the partial amplitude, $A_{4}^{(k)}(\pi\pi AA)$,
at $\mathcal{O}(\Lambda^{k})$ must be completed by an additional contact
$A_{4}^{(k+1)}(\pi\pi AA)$ at $\mathcal{O}(\Lambda^{k+1})$, as required by
double-copy consistency. This inductive process is sketched in Fig. 3. It
turns out that the general form of these color-dual four-point contributions
at each additional order in mass-dimension can evidently be written as,

$A_{4}^{(k)}(1_{\pi},2_{\pi},3_{A},4_{A})=g^{2}\left(\frac{\Lambda}{g^{2}}\right)^{k}u\,s^{k-1}\frac{\,{\rm
tr}[F_{3}F_{4}]}{s}\,,$ (80)

where the superscript $k$ counts additional orders in mass-dimension, and
$F_{i}^{\mu\nu}$ is given in eq. (71). The resulting tensor,

$\,{\rm tr}[F_{i}F_{j}]\equiv
F_{i}^{\mu\nu}F_{j}^{\rho\sigma}\eta_{\mu\rho}\eta_{\nu\sigma}=-2\Big{[}(k_{i}\cdot\varepsilon_{j})(k_{j}\cdot\varepsilon_{i})-(k_{i}\cdot
k_{j})(\varepsilon_{i}\cdot\varepsilon_{j})\Big{]}\ ,$ (81)

is the same tensor structure found in eq. (76) and eq. (79), for example. We
have explicitly verified eq. (80) up to $\mathcal{O}(\Lambda^{5})$ when
unconstrained freedom in higher-derivative Wilson coefficients is set to zero.

###  4.3 New contributions for vector amplitudes with four scalars

$\begin{split}A{(1^{\pi}2^{\pi}3^{\pi}4^{\pi}5^{A})}\supset{\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 (-2.2,
-1.2) {$3$}; \vertex(a2) at (-2.6, 0) {$4$}; \vertex(a5) at (-2.2, 1.2) {$5$};
\vertex[blob,hgrey](mid1) at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[white](mid3)
at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[blob,hblue](mid2) at (0.7,0)
{$\,\Lambda^{0}\,$}; \vertex[white](mid4) at (0.7,0) {$\,\Lambda^{0}\,$};
\vertex(a3) at (1.6, 1.2) {$1$}; \vertex(a4) at (1.6, -1.2) {$2$}; \vertex(a7)
at (0, -0.9); \vertex(a6) at (0, 0.9); \diagram{ (mid2) --[ thick](a3), (mid2)
--[ thick](a4), (mid2) -- [gluon, thick](mid1), (a1)--[ thick](mid1), (a2) --[
thick](mid1), (mid1) --[gluon, thick](a5), (a6)--[scalar, thick,cutred](a7) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}+{\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 (-2.2,
-1.2) {$1$}; \vertex(a2) at (-2.6, 0) {$2$}; \vertex(a5) at (-2.2, 1.2) {$3$};
\vertex[blob,hgrey](mid1) at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[white](mid3)
at (-1,0) {$\,\Lambda^{n}\,$}; \vertex[blob,hblue](mid2) at (0.7,0)
{$\,\Lambda^{0}\,$}; \vertex[white](mid4) at (0.7,0) {$\,\Lambda^{0}\,$};
\vertex(a3) at (1.6, 1.2) {$4$}; \vertex(a4) at (1.6, -1.2) {$5$}; \vertex(a7)
at (0, -0.9); \vertex(a6) at (0, 0.9); \diagram{ (mid2) --[ thick](a3), (mid2)
--[gluon, thick](a4), (mid2) --[ thick](mid1), (a1)--[ thick](mid1), (a2) --[
thick](mid1), (mid1) --[ thick](a5), (a6)--[scalar, thick,cutred](a7) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\end{split}$ (82)
Figure 4: Generalized unitarity cuts used to constrain BCJ satisfying partial
amplitudes contributing to $\mathcal{A}_{5}(\pi\pi\pi\pi A)$. Given the
existence of $A_{4}^{(n)}(\pi\pi AA)$ at any order $\mathcal{O}(\Lambda^{n})$,
an additional four-point all-pion higher-derivative contact is required to
maintain double-copy consistency at five-point, such that $A_{4}^{(n)}(\pi\pi
AA)\iff A_{4}^{(n)}(\pi\pi\pi\pi)$.

With this tower of contacts contributing to $\mathcal{A}_{4}(\pi\pi AA)$ in
hand, we can immediately see that additional higher derivative corrections
must be added to the all-pion amplitude, $\mathcal{A}_{4}(\pi\pi\pi\pi)$, in
order for $\mathcal{A}_{5}(\pi\pi\pi\pi A)$ to be double-copy consistent to
all orders in mass-dimension.

Imposing the factorization constraints in Fig. 4, and setting any remaining
freedom to zero, we find that the four-point ordered amplitude contributing to
the all-pion cuts at a given order in $\Lambda$ takes the form,

$A_{4}^{(k)}(1_{\pi},2_{\pi},3_{\pi},4_{\pi})=g^{2}\left(\frac{\Lambda}{g^{2}}\right)^{k}u\,\Big{[}s^{k-1}+t^{k-1}+u^{k-1}\Big{]}\
.$ (83)

We have verified through $\mathcal{O}(\Lambda^{5})$ that four-point amplitudes
of the form stated in eq. (80) and eq. (83) are required for double-copy
consistency at five points.

$\mathcal{O}(\Lambda^{n})$ $||\pi||=2k$ | $k=0$ | $k=1$
---|---|---
$n=0$ |  |
$n=1$ |  $\,F^{3}\,$$\,F^{3}\,$ |
Table 1: Three-point amplitudes for $2k$ pions – Green cells contain
amplitudes generated by eq. (61). The additional $\,{\rm Tr}(F^{3})$
interaction highlighted in yellow was needed in eq. (68) for double-copy
consistency.

##  5 Analysis of the double-copy consistent vector-pion theory

In this section we will identify out some novel features of this candidate
vector-pion theory, and some directions for future studies. To begin, we
provide a set of tables that display the additional color-dual operators that
are required for our conjectured vector-pion theory. All cells that appear
with a diagram indicate that the amplitude has been constructed, and is
available in the ancillary file.

###  5.1 Summary of new contributions

$\mathcal{O}(\Lambda^{n})$ $||\pi||=2k$ | $k=0$ | $k=1$ | $k=2$
---|---|---|---
$n=0$ |  0 |  0 |  0
$n=1$ |  $\,F^{3}\,$$\,F^{3}\,$ 0 |  0 |  0
$n=2$ |  $\,F^{4}\,$$\,F^{4}\,$ 0 |  0 |  0
$n=3$ |  $\,D^{2}F^{4}\,$$\,D^{2}F^{4}\,$ 1 |  1 |  1
$n=4$ |  $\,D^{4}F^{4}\,$$\,D^{4}F^{4}\,$ 1 |  1 |  1
$n=5$ |  $\,D^{6}F^{4}\,$$\,D^{6}F^{4}\,$ 2 |  2 |  2
Table 2: Four-point amplitudes for $k$ pairs of pions – Green cells contain amplitudes generated by eq. (61). The additional interactions highlighted in yellow were needed in eq. (68) for double-copy consistency. $\mathcal{O}(\Lambda^{n})$ $||\pi||=2k$ | $k=0$ | $k=1$ | $k=2$
---|---|---|---
$n=0$ |  0 |  0 |  0
$n=1$ |  0 |  0 |  $\,\text{cov.}\pi\,$$\,\text{cov.}\pi\,$ 0
$n=2$ |  0 |  0 |  0
$n=3$ |  1 |  1 |  1
$n=4$ |  1 |  1 |  1
$n=5$ |  |  |  3
Table 3: Five-point amplitudes for $k$ pairs of pions – Green cells contain
amplitudes generated by eq. (61). The amplitude highlighted in red violated
BCJ relations when only considering interactions of eq. (61). This amplitude,
$A^{\text{cov.}\pi}$, needed additional interactions in eq. (68) to be double-
copy consistent.

Cells in green correspond to amplitudes that are generated by the Feynman
rules of eq. (61), our starting point. The four-pion five-point amplitude in
Table 3 is highlighted in red, to indicate that it is the first amplitude that
forced us to consider additional operators; these additional operators, that
served as the base-steps of our inductive tower, are highlighted in yellow in
Tables 1 and 2. Finally, the bold-face numbers in each cell correspond to the
number of free Wilson-coefficients generated by our bootstrap. These
additional parameters are unconstrained by double-copy consistency and five-
point factorization; however there could in principle be additional
constraints at higher multiplicity.

As outlined in the review of unitarity methods in section 2.5, these
amplitudes were constrained by (1) color-kinematics (2) factorization and (3)
gauge invariance. To elucidate the requisite four-point contacts through
$\mathcal{O}(\Lambda^{5})$ and the available freedom, we only considered
color-kinematic and factorization constraints on the same mass-dimension five-
point amplitude; and did not perform the computationally onerous process of
further imposing gauge-invariance. For this reason, the zero- and two-scalar
five-point amplitudes at $\mathcal{O}(\Lambda^{5})$ are left blank.

In addition to the three-, four- and five-point amplitudes, we have
constructed the all-scalar six-point amplitude at NLSM mass-dimension,
$\mathcal{O}(\Lambda^{2})$. Due to the inclusion of a five-point amplitude at
$\mathcal{O}(\Lambda^{2}/g)$ required by double-copy consistency, there is a
novel correction to the pure NLSM six-point amplitude, shown in Table 4. We
will discuss the implications of this in more detail in the next subsection.

$\mathcal{O}(\Lambda^{n})$ $||\pi||=2k$ | $k=3$
---|---
$n=2$ |  | \+ $\,A^{\text{new}}_{6}\,$$\,A^{\text{new}}_{6}\,$
Table 4: Six-point amplitude for 3 pairs of pions – Green cells contain
amplitudes generated by eq. (61). The additional contribution at
$\mathcal{O}(\Lambda^{2})$ is required for double-copy consistency of the full
vector-pion theory.

###  5.2 Modifying NLSM and thus DBIVA behavior

As suggested in Table 4, the existence of five-point contributions at
$\mathcal{O}(\Lambda^{2}/g)$ to $\mathcal{A}(\pi\pi\pi\pi A)$ mandates
corrections to the already color-dual six-point pion amplitude for our theory.
This additional $\mathcal{O}(\Lambda^{2}/g)$ term leads to non-trivial
corrections to the pure pion six-point amplitude at
$\mathcal{O}(\Lambda^{2})$, such that

$A_{6}^{(DF)^{2}+\text{YM}+\pi}(123456)\Big{|}_{\Lambda^{2}}=A_{6}^{\pi}(123456)+A_{6}^{\text{new}}(123456)$
(84)

where the first term is the usual six-point NLSM partial amplitude:

$A_{6}^{\pi}(123456)=9\Lambda^{2}\left[\frac{s_{13}s_{46}}{s_{123}}+\frac{s_{24}s_{51}}{s_{234}}+\frac{s_{35}s_{62}}{s_{345}}-s_{135}\right]$
(85)

and the novel contribution appearing at $\mathcal{O}(\Lambda^{2})$ takes the
form:

$A_{6}^{\text{new}}(123456)=\Lambda^{2}\Bigg{[}\frac{1}{2}\left(\frac{(s_{15}+s_{25})\tau_{12|34}}{s_{12}s_{34}}+\frac{s_{36}\tau_{12|45}}{s_{12}s_{45}}+\frac{(s_{35}+s_{36})\tau_{12|56}}{s_{12}s_{56}}\right)\\\
+\frac{\tau_{12|56}-\tau_{12|45}}{s_{12}}-(s_{14}+s_{23})+\text{cyc}(1,2,3,4,5,6)\Bigg{]}$
(86)

where we have introduced the notation,
$\tau_{ij|kl}=s_{ik}s_{jl}-s_{il}s_{jk}$. The novel contribution,
$A^{\text{new}}_{6}$, has non-vanishing residues on cuts that expose
individual three-point amplitudes,

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[blob,hgrey](mid1)
at (0,0) {$\,A_{6}|_{\Lambda^{2}}\,\,$}; \vertex[white](mid2) at (0,0)
{$\,A_{6}|_{\Lambda^{2}}\,\,$}; \vertex(a1) at (1, -.8) {}; \vertex(a2) at (1,
.8) {}; \vertex(a3) at (-1, 0.8) {}; \vertex(a4) at (-1,- 0.8) {}; \vertex(a5)
at (0,1.24) {}; \vertex(a6) at (0,-1.24) {}; \diagram{ (mid1) --[, thick](a1),
(mid1) --[, thick](a2), (mid1) -- [, thick](a3), (mid1) -- [, thick](a4),
(mid1) -- [, thick](a5), (mid1) -- [, thick](a6), };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\supset{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex[blob,hgrey](mid1)
at (0,0) {$\Lambda^{2}/g$}; \vertex[white](mid2) at (0,0) {$\Lambda^{2}/g$};
\vertex(a3) at (-1, 0.8) {}; \vertex(a4) at (-1,- 0.8) {}; \vertex(a5) at
(0,1.24) {}; \vertex(a6) at (0,-1.24) {}; \vertex[blob,hgrey](mid3) at (2,0)
{$g$}; \vertex[white](mid4) at (2,0) {$g$}; \vertex(a1) at (3, -.8) {};
\vertex(a2) at (3, .8) {}; \vertex(b2) at (1.1,-0.8){}; \vertex(b1) at
(1.1,0.8){}; \diagram{ (mid3) --[ thick](a1), (mid3) --[thick](a2), (mid3)
--[gluon, thick](mid1), (mid1) -- [thick](a3), (mid1) -- [thick](a4), (mid1)
-- [thick](a5), (mid1) -- [thick](a6), (b1) -- [scalar,thick, cutred](b2) };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,.$ (87)

This new contribution alters the Addler’s zero behavior of the theory. Note
that $A^{\text{new}}_{6}$ does not contribute to typical factorization channel
available to six-point NLSM pion amplitudes, the sewing of two four-point
trees. It is particularly interesting to note that this novel contribution to
the six-point amplitude independently satisfies the BCJ relations, unlike the
additional term in eq. (68) that was required to make
$\mathcal{A}(\pi\pi\pi\pi A)$ color-dual. The identification of the coupling
with $\Lambda^{2}$ is only required by double-copy consistency with the lower-
multiplicity results found thus far. The gauge theory cut, depicted above,
will then lead to a non-vanishing factorization channel in the resulting
double-copy to $\text{DBIVA}+$Weyl-Einstein supergravity:

$\mathcal{A}_{6}(\lambda^{\text{VA}}\cdots\lambda^{\text{VA}})\Big{|}_{l^{2}\text{-cut}}\\!\\!\\!\\!\sim\sum_{\text{states}}\mathcal{A}_{5}(\lambda^{\text{VA}}\cdots\lambda^{\text{VA}},l^{\bar{s}})\mathcal{A}_{3}(-l^{s},\lambda^{\text{VA}}\lambda^{\text{VA}})$
(88)

This is obviously in contrast to the usual behavior of pure DBIVA, that
mandates factorization to three-point trees vanish in the scattering. As both
contributions above in eq. (84) appear at $\mathcal{O}(\Lambda^{2})$, it is
not possible to tune the interaction to recover pure DBIVA theory from the
resulting double-copy.

We must admit this is a somewhat disturbing aspect of our construction – we
wished to build Volkov-Akulov coupled to supergravity, but we seem to have
altered the very soft-limit behavior we identify with Volkov-Akulov scattering
to begin with. Does this destroy the nilpotence associated with the fermionic
field in DBIVA uncoupled to supergravity? The answer to that question awaits
future work. We note that the more radical construction presented in section 6
does leave DBIVA behavior unmodified to all multiplicity at the cost of
manifest exchange symmetry between scalars in the constituent gauged NLSM
theory.

###  5.3 Resumming to dimensionally reduced $(DF)^{2}+\text{YM}$

Was it in some sense inevitable that an effective field theory of identical
scalars, NLSM, could only be combined with an effective field theory of
vectors, $(DF)^{2}+\text{YM}$, in a double-copy consistent way? We are
prepared to conjecture an answer by considering the suggestive field theory
question: what is the dimensional reduction of $(d+1)$-dimensional
$(DF)^{2}+\text{YM}$, a known color dual theory, to $d$-dimensions?

We can perform a dimensional reduction of this form directly at the level of
amplitudes. To do so, we will define the $(d+1)$-dimensional momenta,
$\mathcal{K}_{a}$, and polarizations, $\mathcal{E}_{a}$, as follows:

$\mathcal{K}_{a}=(k_{\mu},0)_{a}\qquad\mathcal{E}_{a}=\begin{cases}1&a=d\\\
\varepsilon_{\mu}&a\neq d\end{cases}$ (89)

where $\mu$ takes on values in the lower dimensional subspace,
$\mu=0,...,d-1$. When we identify the pions with the extra-dimensional
polarization mode, $\mathcal{E}_{a=d}$, one finds that the building blocks in
eq. (74) exactly reproduce the scalar amplitudes in eq. (80) and eq. (83). The
same goes the for available five-point amplitudes through
$\mathcal{O}(\Lambda^{4})$.

Furthermore, this dimensional reduction procedure is in one-to-one
correspondence with the application of transmutation operators,
$\mathcal{T}_{ij}=\frac{\partial}{\partial(\varepsilon_{i}\varepsilon_{j})}$,
of Cheung, Shen, and Wen Cheung:2017ems ; Cheung:2017yef , related to the
‘generalized dimensional reductions’ of Cachazo, He, and Yuan Cachazo2014xea .
So the observation that the all-vector building blocks in eq. (70), generated
by $\mathcal{L}_{(DF)^{2}+\text{YM}}$, are secretly related to the mixed
vector-pion color-dual building blocks identified in eq. (80) and eq. (83) can
be neatly captured by the following set of relations:

$A_{4}^{(n)}(1_{\pi},2_{\pi},3_{A},4_{A})=\mathcal{T}_{12}\circ
A^{({n})}_{4}(1_{A},2_{A},3_{A},4_{A})$ (90)
$A_{4}^{(n)}(1_{\pi},2_{\pi},3_{\pi},4_{\pi})=\left(\mathcal{T}_{12}\mathcal{T}_{34}+\text{cyc}(2,3,4)\right)\circ
A^{({n})}_{4}(1_{A},2_{A},3_{A},4_{A})$ (91)

Dimensional reduction does not seem to affect the double-copy consistency
BCJreview , but does offer consistently interacting scalars. Likewise,
performing this dimensional reduction on the ${(DF)^{2}+\text{YM}}$ Lagrangian
yields the type of interaction terms required to build the color-dual mixed
amplitudes we have presented here through six-points. So adding appropriately
weighted higher derivative vector operators to the covariantized NLSM seems
exactly the right approach to build a theory compatible with the double-copy
web of theories through the interactions we have yet explored.

The double-copy of sYM with ${(DF)^{2}+\text{YM}}$ is known to yield a Weyl-
Einstein type conformal supergravity (CSG) Johansson:2017srf ,

$A^{\text{CSG}}=A^{\text{sYM}}\otimes A^{(DF)^{2}+\text{YM}}\,.$ (92)

This theory is related to the gravitational amplitudes in the heterotic string
by a particular choice of Wilson coefficients for an additional tower of
double-copy consistent higher derivative operators. Our conjectured vector-
pion theory double-copied with sYM is then related to this family of Weyl-
Einstein type conformal supergravities, by what appears to be a sort of
dimensional reduction – but one that only applies to one factor in the double-
copy construction. Our model appears consistent with the behavior of modified
Volkov-Akulov fermions dynamically coupled to spacetime of interest in
$\alpha$-attractor models, which naturally reside in gravity theories where
the super Poincaré symmetry is upgraded to the super-conformal group
Kallosh:2013yoa . This arises from a candidate vector-pion theory that is
related to the color-dual $(DF)^{2}+\text{YM}$ via dimensional-reduction. All-
order in $\Lambda$ predictions in this theory could be extracted directly from
any multiplicity $(DF)^{2}+\text{YM}$ theory amplitudes.

Moreover, constructing the Lagrangian for our vector-pion theory, with all
residual higher-derivative freedom set to zero, is just a matter of redefining
the gauge fields, $A_{a}\rightarrow(A_{\mu},\pi)_{a}$ and projecting the
derivatives down to the lower dimensional subspace,
$\partial_{a}\rightarrow(\partial_{\mu},0)_{a}$. This gives us the following
conjecture for the Lagrangian that reproduces the amplitudes constructed in
this paper:

$\mathcal{L}_{(DF)^{2}+\text{YM}+\pi}\equiv\mathcal{L}_{(DF)^{2}+\text{YM}}\Big{|}^{\partial_{a}\rightarrow(\partial_{\mu},0)_{a}}_{A_{a}\rightarrow(A_{\mu},\pi)_{a}}\,.$
(93)

The observation that a double-copy consistent vector-pion theory could be
equivalent to the dimensional reduction of $(DF)^{2}+\text{YM}+\text{HD}$ also
sheds some light on the question of why we cannot recover the usual NLSM
Adler’s zero condition from taking limits of coupling in our theory. Since the
$\alpha^{\prime}$ in $(DF)^{2}+\text{YM}$-amplitudes is related to the pion
decay-width, $\alpha^{\prime}=-\Lambda/g^{2}$, the limit $g\rightarrow 0$ is
equivalent to the $\alpha^{\prime}\rightarrow\infty$ limit, where the theory
becomes a dimensional reduction of the dimension six theory,
$\mathcal{L}^{(DF)^{2}}$Johansson:2017srf . In this limit, dimensional
reduction is one-to-one with dimensional truncation and all pionic amplitudes
vanish.

###  5.4 Helicity selection rule from new contributions

The tensor structure of eq. (76) is different from the standard $\,{\rm
Tr}(F^{2})$ contribution, which has particularly sharp implications for the
helicity structure of gluon scattering in four dimensions. The standard
$\,{\rm Tr}(F^{2})$ term only contributes to this amplitude for mixed-helicity
gluons, while the $\,{\rm Tr}(F^{3})$ operator only contributes to this
amplitude for same-helicity gluons. Using the four-dimensional spinor-helicity
formalism, when the gluons are both plus helicity the amplitude is simply

$A_{4}(1_{\pi},2_{\pi},3_{A}^{+},4_{A}^{+})=\Lambda\frac{u}{s}{\left[34\right]}^{2}\
,$ (94)

which starts at $\mathcal{O}(\Lambda)$ because the standard YM vertex does not
contribute. It is worth spending a few words considering the implications this
has for the double-copy theory we are attempting to build. Aiming for
$\text{DBIVA}+\text{supergravity}$, double-copying these amplitudes with
$\mathcal{N}\leq 4$ super-Yang-Mills amplitudes, we now find in our
constructed gravity theory non-vanishing four-point interactions such as,
e.g.:

$\mathcal{A}_{4}(\chi^{\text{VA}}{}^{+}\chi^{\text{VA}}{}^{-}{}h^{++}{{\bar{\rm
t}}})=\mathcal{A}_{4}(\lambda^{+}\lambda^{-}A^{+}A^{-})\otimes\mathcal{A}_{4}(\pi\pi
A^{+}A^{+})\,.$ (95)

This is quantitatively different than the amplitude arising from the double-
copy with only the covariantized NLSM of eq. (61).

###  5.5 Double-copy consistency of massive Yang-Mills

The structure of the hidden tower of corrections also lends itself to studying
the double-copy consistency of massive gauge theory. By starting with the
Proca Lagrangian,

$\mathcal{L}=-\frac{1}{4}\text{Tr}(F^{2})+\frac{1}{2}m^{2}\text{Tr}\left[A_{\mu}A^{\mu}\right]$
(96)

we can recover the starting point of this paper, (61), by performing a
Stuckelberg transformation on the gauge fields,

$A_{\mu}\rightarrow\frac{i}{g}U^{\dagger}D_{\mu}U$ (97)

where the $U\in SU(N)$ group elements are expressed in terms of the Cayley
parameterization as in (61), and $\Lambda=m^{2}/g^{2}$. The color-dual nature
of this action was previously studied in Momeni:2020vvr  with application to
the double-copy construction of dRGT gravity amplitudes. As we have outlined
throughout the text, many of the irrelevant operators that one could add to
(61), included in the tables of section 5.1, will be required for double-copy
consistency in higher-multiplicity tree amplitudes.

##  6 A flavorful approach to a color-dual vector-pion theory

Here we demonstrate a more radical possibility for building a color-dual
gauged pion theory. A recent study by one of the authors Pavao:2022kog
demonstrated that $D$-dimensional gauge theory amplitudes can be decomposed
into building blocks that preserve all the partial amplitude relations of the
parent vector theory. This expansion, referred to as the reducible-amplitude
block-decomposition, takes the following form for Yang-Mills,

$A^{\text{YM}}_{(\sigma)}=\sum_{k=1}^{\lfloor|\sigma|/2\rfloor}\sum_{\rho\in
S^{2|k}_{\sigma}}\epsilon_{(\rho)}\Delta^{(\rho)}_{(\sigma)}$ (98)

where $\rho=(a_{1}b_{1})\dots(a_{k}b_{k})$ is a list of paired legs that
appear as external labels in the ordered list, $\sigma$, and
$\epsilon_{(\rho)}=(\epsilon_{a_{1}}\epsilon_{b_{1}})\cdots(\epsilon_{a_{k}}\epsilon_{b_{k}})$
are dot products of $D$-dimensional polarizations. The sum starts at $k=1$
since all kinematic monomials for Yang-Mills at tree-level contain at least
one factor of $\epsilon_{(ab)}$. The observation relevant for the following
discussion is that $\Delta^{(\rho)}_{(\sigma)}$ are kinematic functions that
independently satisfy BCJ relations with a flavor structure determined by
$\rho$. Furthermore, the building blocks, $\Delta^{(\rho)}_{(\sigma)}$, can be
consistently combined to produce pion amplitudes via the dimensional reduction
of ref. Cheung:2017yef ,

$\mathcal{E}^{Z}_{a}=\frac{1}{gf_{\pi}}(k^{\mu}_{a},0,ik^{\mu}_{a}),\qquad\mathcal{E}^{Y}_{a}=(\vec{0},1,\vec{0}),$
(99)

where momenta $\mathcal{K}_{a}$ live in the $d$-dimensional scattering
subspace:

$\mathcal{K}_{a}=(k^{\mu}_{a},0,0).$ (100)

Applying this dimensional reduction to pure gluon amplitudes in $D=2d+1$
dimensions will yield NLSM tree-level amplitudes when two particle $(ij)$ are
projected along $Y$-directions, and the remaining particles take on
$Z$-polarizations Cheung:2017yef :

$A^{\text{NLSM}}_{(12...n)}=A^{\text{YM}}(Z_{1},...,Y_{i},...,Y_{j},...,Z_{n})\equiv\Delta^{(ij)}_{(12...n)}\Big{|}_{\epsilon\rightarrow
k}.$ (101)

These amplitudes can also be generated using Feynman rules from a Lagrangian
that is perturbatively equivalent to NLSM given a particular weight of
$Z$-particles Cheung2016prv . Since they are equivalent to a special
dimensional reduction from Yang-Mills Cheung:2017yef , they can be expressed
in terms of $\Delta^{(ij)}_{(12...n)}$ by considering how eq. (99) acts on the
block decomposition Pavao:2022kog .

Now let’s return to the starting point of this paper. In pursuit of
constructing amplitudes for DBIVA dynamically coupled to spacetime via the
double-copy, the goal was to identify a double-copy consistent theory of NLSM
pion interacting with gluons. While the construction of eq. (101) introduces
different particle flavors that obscures the scalar-exchange symmetry of the
NLSM Lagrangian, it does suggest an alternative approach to higher derivative
model analyzed in previous sections. Namely, one could augment the flavored
dimensional reduction in eq. (99) by introducing an additional gluon
polarization that is restricted to the $d$-dimensional subspace:

$\mathcal{E}^{A}_{a}=(\epsilon^{\mu}_{a},0,0).$ (102)

Truncating gluons from $D=2d+1$ down to $d$-dimensions provides a potential
path towards consistently coupling the longitudinal pions with gluons; albeit
in a way that violates exchange symmetry between the external pions. Take for
example, the following four-scalar five-point amplitudes at pion mass
dimension:

$\displaystyle A^{\text{YM}}_{5}(1_{Z},2_{Y},3_{Y},4_{Z},5_{A})$
$\displaystyle=\frac{g}{f_{\pi}^{2}}\left(\frac{s_{24}\kappa_{1}^{(5)}}{s_{15}}-\frac{s_{13}\kappa_{4}^{(5)}}{s_{45}}+\kappa_{24}^{(5)}\right)\frac{s_{14}}{s_{23}}\sim
s_{14}\Delta^{(14)(23)}_{(12345)},$ (103) $\displaystyle
A^{\text{YM}}_{5}(1_{Y},2_{Z},3_{Z},4_{Y},5_{A})$
$\displaystyle=\frac{g}{f_{\pi}^{2}}\left(\frac{s_{24}\kappa_{1}^{(5)}}{s_{15}}-\frac{s_{13}\kappa_{4}^{(5)}}{s_{45}}+\kappa_{24}^{(5)}\right)\sim
s_{23}\Delta^{(14)(23)}_{(12345)},$ (104) $\displaystyle
A^{\text{YM}}_{5}(1_{Y},2_{Z},3_{Y},4_{Z},5_{A})$
$\displaystyle=\frac{g}{f_{\pi}^{2}}\left(\frac{s_{24}\kappa_{1}^{(5)}}{s_{15}}-\frac{s_{24}\kappa_{4}^{(5)}}{s_{45}}\right)\sim
s_{24}\Delta^{(13)(24)}_{(12345)}.$ (105)

where the relationship of these amplitudes to $\Delta^{(\rho)}_{(\sigma)}$ is
due to the Ward identities derived in Pavao:2022kog . It is worth noting that
$A^{\text{YM}}_{5}(1_{Y},2_{Z},3_{Z},4_{Y},5_{A})$ happens to be identical to
the last term in eq. (68). We remind the reader that this term is precisely
the five-point amplitude, $A_{5}^{\text{cov.}\pi}$, generated by the gauged
NLSM Lagrangian of eq. (61):

$A_{5}^{\text{cov.}\pi}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})\equiv
A^{\text{YM}}_{5}(1_{Y},2_{Z},3_{Z},4_{Y},5_{A}).$ (106)

However, given this definition, one can check that the four-pion five-point
amplitude of $A^{\text{cov}.\pi}_{5}$ does not satisfy BCJ relations when the
pions are taken to be identical bosons. This observation led earlier to
introducing an additional $\text{Tr}(F^{3})$ operator with coupling
$\Lambda/g$ in section 3.2. Indeed, one can check that the following BCJ
relation, eq. (36), does not hold when applied to $A_{5}^{\text{cov.}\pi}$:

$\begin{split}s_{1|2}A_{5}^{\text{cov.}\pi}(1_{\pi},2_{\pi},3_{\pi},4_{\pi},5_{A})&+s_{13|2}A^{\text{cov.}\pi}_{5}(1_{\pi},3_{\pi},2_{\pi},4_{\pi},5_{A})\\\
&+s_{134|2}A^{\text{cov.}\pi}_{5}(1_{\pi},3_{\pi},4_{\pi},2_{\pi},5_{A})\neq
0,\end{split}$ (107)

where $s_{i|j_{1}...j_{n}}=k_{i}\cdot(k_{j_{1}}+\cdots+k_{j_{n}})$.

In contrast, when we keep track of particle flavor for the dimensionally
reduced pionic amplitudes of eq. (104) and eq. (105), with $Y$\- and
$Z$-particles coupled to gluons, we find that the usual BCJ relation does
hold:

$\begin{split}s_{1|2}A^{\text{YM}}_{5}(1_{Y},2_{Z},3_{Z},4_{Y},5_{A})&+s_{13|2}A^{\text{YM}}_{5}(1_{Y},3_{Z},2_{Z},4_{Y},5_{A})\\\
&+s_{134|2}A^{\text{YM}}_{5}(1_{Y},3_{Z},4_{Y},2_{Z},5_{A})=0.\end{split}$
(108)

This is simply a consequence of Yang-Mills satisfying BCJ relations in
$D$-dimensions, and the pion-vector amplitudes above just being a special
choice of $D$-dimensional polarizations. Thus, the amplitudes generated using
the dimensional reduction of eq. (99) and eq. (102) are manifestly color-dual.
Indeed an important feature of this model is that any $m$-point tree-level
amplitude with $2(m-1)$ $Z$ particles and 2 $Y$ particles will match exactly
the same scattering amplitudes and have the same soft behavior as the NLSM.

Since we require no additional interactions beyond those generated by the
dimensional reduction from $D=2d+1$ dimensions, the six-point pure-scalar NLSM
amplitude is unaffected, and consequently the behavior of six-point DBIVA
amplitudes in the double-copy theory coupled to supergravity is likewise
unaffected. It is, however, worth noting that eq. (103) has additional poles
not generated by the gauged NLSM of eq. (61). In this case, unlike the
approach studied in previous sections, these additional residues avoid the
introduction of $\text{tr}(F^{3})$ that forced a tower of higher-derivative
pion-vector interactions. These additional poles are generated by a
combination of the pure-scalar three-point vertices derived in Cheung:2017yef
, and additional vector-pion interactions. The following are the off-shell
Feynman rules for the novel interactions between gluons and $Y$-particles,

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(mid1) at (0,0);
\vertex(a1) at (-1*1.3, -.8*1.3) {$\,Y_{1}^{a}\,$}; \vertex(a2) at (-1*1.3,
.8*1.3) {$\,Y_{2}^{b}\,$}; \vertex(a3) at (1.3*1.3, 0*1.3)
{$\,A_{3}^{c,\mu}\,$}; \diagram{ (mid1) --[scalar, thick](a1), (mid1)
--[scalar, thick](a2), (mid1) -- [gluon, thick](a3), };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\equiv
V_{abc}^{\mu}=g\,f^{abc}(k_{1}-k_{2})^{\mu},$ (109)

and gluons with the $Z$-particle,

${\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(mid1) at (0,0);
\vertex(a1) at (-1*1.3, -.8*1.3) {$\,A_{1}^{a,\mu}\,$}; \vertex(a2) at
(-1*1.3, .8*1.3) {$\,A_{2}^{b,\nu}\,$}; \vertex(a3) at (1.3*1.3, 0*1.3)
{$\,Z_{3}^{c}\,$}; \diagram{ (mid1) --[gluon, thick](a1), (mid1) --[gluon,
thick](a2), (mid1) -- [, thick](a3), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\equiv
V_{abc}^{\mu\nu}=f_{\pi}^{-1}\,f^{abc}\big{[}\eta^{\mu\nu}\,k_{3}\\!\cdot\\!(k_{1}-k_{2})+(k_{3}^{\nu}k_{2}^{\mu}-k_{3}^{\mu}k_{1}^{\nu})\big{]}.\\\
\ $ (110)

Both of these vertex rules can be constructed by applying the dimensional
reduction of eq. (99) to the off-shell Feynman rules of the Yang-Mills three-
point function in eq. (13). While the vertex $V_{abc}^{\mu}$ of eq. (109)
produces the familiar three-point amplitude of eq. (44), the vertex
$V_{abc}^{\mu\nu}$ of eq. (110) is novel. A consequence is that any tree-level
amplitudes involving external $Y$ and $A$ external states will simply be
amplitudes shared with the covariantized free scalar.

Furthermore, while $V_{abc}^{\mu\nu}$ is needed for consistent factorization
of the $A^{\text{YM}}_{(AZZA)}$ four-point amplitude, we can see that it must
vanish on-shell666The $A^{\text{YM}}_{(AAZ)}$ amplitude is equivalent on-shell
to three-point function $A^{(\chi)}_{(gg\pi)}$ of Pavao:2022kog , in that they
both vanish. However as we note, $A^{\text{YM}}_{(AAZ)}$ vanishes due to on-
shell three-point kinematics, whereas the three-point function
$A^{\chi}_{(gg\pi)}$ vanished due to the kinematic factor being symmetric
under color ordering. when external polarizations are transverse and momenta
are light-like. In addition to $V_{abc}^{\mu\nu}$ at three-point, there will
also be novel four-point vertices that are required for
$A^{\text{YM}}_{(AZZA)}$ to be gauge invariant.

In summary, while this approach violates exchange symmetry between external
scalars, it has merit in that we can smoothly interpolate between pure Yang-
Mills amplitudes, when taking $f_{\pi}\rightarrow\infty$, and pure NLSM
amplitudes, when taking the gauge coupling $g\rightarrow 0$. Furthermore,
since this flavored formulation of gauged pions avoids the
$\mathcal{O}(\Lambda^{2}/g)$ five-point contribution described in section 5.2,
the Adler’s zero of pure pions at six-point is preserved. In other words, any
pure scalar amplitudes at pion mass dimension are identical to those generated
by the NLSM Lagrangian, free of corrections from higher-dimension contacts
needed for double-copy consistency.

##  7 Conclusions

In this paper we have begun to establish whether a color-dual theory of
vectors coupled to pions can exist. We have demonstrated the existence of two
distinct candidate theories through six-point interactions. We include
expressions for all basis kinematic weights $n(g)$ and associated amplitudes
in a machine readable auxiliary file. One fascinating promise of such a color-
dual theory is its role in the double-copy description of a Dirac-Born-Infeld-
Volkov-Akulov-type theory coupled to dynamic spacetime – a necessary
ingredient if we aspire to double-copy descriptions of many phenomenological
$\alpha$-attractor models involving supergravity. As the literature provides
color-dual maximally supersymmetric Yang-Mills theory representations through
four-loops Bern:2012uf , we have reduced multi-loop calculations in
$\text{DBIVA}+\text{CSG}$ to carrying out cut-construction in our candidate
color-dual vector-pion theory.

The possibilities for future study are rather large, with non-trivial payoff,
and these are just first steps. To craft arbitrary inflationary potentials we
will want to investigate double-copy construction of higher-derivative
operators targeting the inflaton sector. It will be intriguing to explore the
consequences of these choices especially considering the good UV behavior of
conformal supergravity theories, which seem a necessary feature of our more
conservative double-copy construction of section 3. This model, however,
suffers from modified Adler’s zero behavior which could call into question
whether the fermions in the double-copy supergravity theory can rightly be
associated with nilpotent superfields – a question that awaits future study.
Our second model described in section 6 neatly evades this issue at the cost
of manifest exchange symmetry between scalars in the color-dual gauged pion
theory.

Towards constructing the type of ultra-soft UV behavior potentially offered by
string theory one should, in both models, consider questions of attempting to
climb the ladder of higher-derivative operators towards closed-string
scattering with the abelianized open-string. Furthermore, it will be
fascinating to see what type of interpretation one can give to other double-
copy constructions associated with these color-dual vector-pion theories and
to explore their potential phenomenology. Consider, for example, the double-
copy of each color-dual gauged-pion model with itself. By identifying color-
dual gauged pion theories, we have effectively constructed color dual theories
describing a decoupling limit of massive vectors – theories whose consistent
double-copy should describe a decoupling limit of massive gravity. Indeed, the
resulting double-copy spectrum should describe Goldstone modes of broken time-
translation generators in a theory of $\mathcal{N}=0$ gravity + Born-Infeld +
Galileons, arguably a theory of dark energy.

#### Acknowledgements:

We gratefully acknowledge Alex Edison, Renata Kallosh, Ian Low, James Mangan,
Radu Roiban, Aslan Seifi and Suna Zekioğlu for related collaboration, useful
conversations, feedback on earlier drafts, and encouragement at various
stages. This work was supported by the DOE under contracts DE-SC0021485 and
DE-SC0015910, and by the Alfred P. Sloan Foundation.

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