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# The color-dual fate of $\mathcal{N}=4$ supergravity

John Joseph M. Carrasco  Department of Physics and Astronomy, Northwestern
University, Evanston, Illinois 60208, USA  Institut de Physique Théorique,
Universite Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France  Matthew
Lewandowski  Department of Physics and Astronomy, Northwestern University,
Evanston, Illinois 60208, USA  Nicolas H. Pavao  Department of Physics and
Astronomy, Northwestern University, Evanston, Illinois 60208, USA

###### Abstract

We conjecture that the massless gauge theory of Yang-Mills deformed by a
higher-derivative $\,{\rm Tr}(F^{3})$ operator will not be tree-level color-
dual without an infinite number of additional counterterms. The requirement of
color-dual kinematics and consistent factorization between four- and five-
points induces a tower of increasingly higher-dimensional operators. We find
through explicit calculation that their amplitudes are consistent with the
$\alpha^{\prime}$ expansion of those generated by the $(DF)^{2}+\text{YM}$
theory, a known color-dual theory where the $\text{Tr}(F^{2})$ term has been
given a mass squared proportional to $1/\alpha^{\prime}$. Considering
consistent double-copy construction as a physical principle would suggest that
a $\,{\rm Tr}(F^{3})$-based attempt at a color-dual resolution of the UV
divergence of $\mathcal{N}=4$ supergravity may be able to tame the bad
ultraviolet behavior of the theory, but at the cost of field-theoretic
locality.

††preprint:

##  I Introduction

Recent years have demonstrated that perturbative calculation in quantum
gravity theories is not as prohibitive as Feynman diagram approaches, in
generic gauges, might suggest. The duality between color and kinematics Bern
_et al._ (2008), and associated double-copy construction Kawai  _et al._
(1986); Bern  _et al._ (2008, 2010), reduces the complexity of calculations in
many gravity theories to understanding predictions in much simpler gauge
theories.

On the other hand, identifying consistent high-energy, or ultraviolet (UV),
completions of quantum gravity theories still remains challenging from a
point-like quantum field theory perspective. The only proven UV completion to
quantum gravity, the closed string, requires an infinite number of higher-
derivative corrections from the QFT perspective–arguably rendering the theory
non-local. The best candidate for a perturbatively finite local quantum field
theory of gravity is the maximally supersymmetric theory Cremmer  _et al._
(1978), $\mathcal{N}=8$ supergravity (SG), which remains finite in four
dimensions at least through the five-loop correction Bern  _et al._ (2007,
2009, 2018a). Counterterms compatible with known linearly-realized symmetries
have been identified which would be relevant starting at seven loops Beisert
_et al._ (2011); Bossard  _et al._ (2011), although their coefficients have
not been determined and could vanish in four dimensions. The ultimate fate of
$\mathcal{N}=8$ SG awaits explicit calculation.

The pure half-maximal $\mathcal{N}=4$ SG survives Bern  _et al._ (2012) the
analogous test at three-loops Bossard  _et al._ (2011), only to diverge at
four loops Bern  _et al._ (2013). The double-copy structure of pure
$\mathcal{N}=4$ SG and its corresponding spectrum follows from

$\left(\mathcal{N}=4\,\text{SG}\right)\,=\,\left(\mathcal{N}=4\,\,\text{sYM}\right)\otimes\text{YM}\,.$
(1)

One copy is the maximally supersymmetric $\mathcal{N}=4$ super Yang-Mills
theory (sYM) in four dimensions, and the other pure non-supersymmetric Yang-
Mills. The observed divergence at four loops has been linked to the $U(1)$
anomalous behavior Marcus (1985); Carrasco  _et al._ (2013); Bern  _et al._
(2017) of $\mathcal{N}=4$ SG amplitudes. Such anomalous behavior at one loop
can be removed with a simple local counterterm whose double-copy description
involves adding the $\,{\rm Tr}(F^{3})$ operator to the pure Yang-Mills
theory. Does including this counterterm render the theory finite? This too
awaits explicit calculation–but investigation at one and two loops Bern  _et
al._ (2018b, 2019) has verified that the addition of this counterterm,
appropriately tuned, does indeed remove the identified anomalous behavior.

Let us now consider a notion of double-copy consistency that applies equally
well to gauge theories and gravity theories. Often gauge theories are called
color-dual if all color-ordered tree-level amplitudes can be made compatible
with the duality between color and kinematics. Gravity theories are called
double-copy constructable if their tree-level amplitudes can be constructed
from the product of kinematics of color-dual gauge theories. We will describe
a quantum field theory as _double-copy consistent_ if every tree-level
amplitude can be written as a generalized product of factors that obey the
same algebraic relations, and the separate factors can be organized into
ordered amplitudes which satisfy all unitarity constraints, i.e. factorizing
on physical poles to lower-multiplicity ordered amplitudes from the same
theory. We will show in this Letter that the consequence of requiring double-
copy consistency of $\text{YM}+F^{3}$ appears to be an infinite tower of
counterterms. The double-copy of this tower, with $\mathcal{N}=4$ sYM, may
well be able to tame the UV behavior of the $\mathcal{N}=4$ SG theory, but at
the ultimate cost of field-theory locality.

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

The idea of double-copy consistency gets to the heart of an open question
regarding double-copy construction. Should we regard the double copy as a
technical trick to be employed piecemeal, amplitude by amplitude as necessary,
or rather as a physical principle pointing to the presence of an as-yet-
unrecognized physical mechanism braiding together factors of two otherwise
consistent theories? In the context of appreciating excellent UV behavior, we
note that the scattering amplitudes of both open and closed tree-level string
theory, for external massless states, demand field-theoretic double-copy
consistency Mafra  _et al._ (2013); Broedel  _et al._ (2013); Carrasco  _et
al._ (2017a); Mafra and Schlotterer (2017); Carrasco  _et al._ (2017b).

As a precursor to the more complicated gauge theory involving $\,{\rm
Tr}(F^{3})$, let us consider a simple effective field theory that requires an
infinite number of counterterms to be double-copy consistent. Consider a
theory of massless scalars with only the following four-field interaction
term,

$\mathcal{L}_{4}=\Lambda
f^{abe}f^{ecd}(\partial_{\mu}\phi_{a})\phi_{b}\phi_{c}(\partial^{\mu}\phi_{d})\,.$
(2)

All odd-multiplicity amplitudes vanish. The adjoint color-weights (sometimes
called flavor) satisfy Jacobi and antisymmetry at every even multiplicity. The
color-stripped amplitudes for this theory, however, admit adjoint color-dual
representations only at four-points. This theory requires including an
additional six-field two-derivative operator in order for the theory’s six-
point amplitude to satisfy the adjoint-type duality between color and
kinematics. The form of the operator and its coefficient are entirely fixed by
the duality between color and kinematics as well as unitarity via consistent
factorization to lower multiplicity Carrasco and Rodina (2019). Indeed
requiring double-copy consistency of this theory involves adding an infinite
sequence of even-field two-derivative operators, which resums to the venerable
nonlinear sigma model–a theory whose kinematic weights are known to be color-
dual to all multiplicity at tree-level Cheung and Shen (2017). Demanding
double-copy consistency for this massless scalar theory involving only a two-
derivative four-point interaction encodes the same physical Nambu-Goldstone
symmetry as imposing the constraint of the famous Adler’s zero.

Let us now consider Yang-Mills theory with a straightforward inclusion of the
simple $\,{\rm Tr}(F^{3})$ operator. For this theory,

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

all three-point amplitudes satisfy the duality between color and kinematics,
as does the four-point tree-level amplitude through
$\mathcal{O}(\alpha^{\prime})$ Broedel and Dixon (2012). As noted in the
introduction, this $\mathcal{O}(\alpha^{\prime})$ contribution is particularly
important: when double-copied with $\mathcal{N}=4$ sYM, it describes the
local-counterterm required to remove the one-loop $U(1)$ anomalous behavior
from $\mathcal{N}=4$ SG. In contrast, the $\mathcal{O}(\alpha^{\prime\,2})$
contribution to the four-point amplitude, while gauge invariant, does not
satisfy the duality between color and kinematics. One must modify
$\mathcal{L}_{\text{YM}+F^{3}}$ with an additional $\,{\rm Tr}(F^{4})$
operator Broedel and Dixon (2012) if one wishes for the four-point amplitudes
generated at $\mathcal{O}(\alpha^{\prime\,2})$ to be color-dual. We would say
that the unmodified $\mathcal{L}_{\text{YM}+F^{3}}$ theory is not double-copy
consistent.

In fact, it appears that no finite number of local operators is sufficient to
render $\mathcal{L}_{\text{YM}+F^{3}}$ double-copy consistent. The situation
may be even worse than the scalar example of eq. (2), which at least requires
only a finite number of operators to allow any particular multiplicity
amplitude to be color-dual. For the theory of $\text{YM}+F^{3}$ we may need to
climb to all orders in mass-dimension even at relatively small multiplicity.
Indeed, we need not look any further than factorization consistency between
four-points and five-points in the theory.

Consider the generalized unitarity cut of a five-point tree-level amplitude
into four-points sewn with three-points as depicted in fig. 1 and expressed
as,

$A_{5}{(12345)}|_{(k_{4}+k_{5})^{2}\text{-cut}}=\sum_{\text{states}}A_{4}(123l^{s})A_{3}(-l^{\overline{s}}45)\,.$
(4)

Considering the decomposition of the three-point amplitude into Yang-Mills and
$\,{\rm Tr}(F^{3})$ contributions, $A_{3}\equiv
A_{3}^{\text{YM}}+\alpha^{\prime}A_{3}^{F^{3}}$, the presence of a purely
local four-point contribution at order $\mathcal{O}(\alpha^{\prime\,n})$ means
the existence of a non-vanishing and factorizing five-point amplitude of
$\mathcal{O}(\alpha^{\prime\,n+1})$. An $\mathcal{O}(\alpha^{\prime\,n+1})$
contribution at five-points may not itself be compatible with the duality
between color-and kinematics without the additional factorizing contribution
from some specific four-point contact term of order
$\mathcal{O}(\alpha^{\prime\,n+1})$ contracted with the $\text{Tr}(F^{2})$
terms. Contraction of any such new four-point contact with the
$\alpha^{\prime}\,{\rm Tr}(F^{3})$ terms now forces consideration of the
color-dual requirements of a non-vanishing $\mathcal{O}(\alpha^{\prime\,n+2})$
contribution at five-points and so forth.

Figure 1: Generalized unitarity cut on five-point tree-level amplitude at
$\mathcal{O}(\alpha^{\prime\,n+1})$. The three-point contribution is
decomposed into the $F^{2}$ and $\alpha^{\prime}F^{3}$ building blocks.

Indeed, this is precisely what we find, by explicit calculation, through
$\mathcal{O}(\alpha^{\prime\,4})$. To tease out this inductive ladder, we
start with four-points where adjoint color-dual amplitudes can always be
written in terms of cubic, or trivalent, graphs:

$\mathcal{A}_{4}=\frac{n_{s}c_{s}}{s}+\frac{n_{t}c_{t}}{t}+\frac{n_{u}c_{u}}{u}.$
(5)

We label the graphs by their propagators, using $s$, $t$ and $u$ to refer to
the following Mandelstam invariants:

$\displaystyle s$ $\displaystyle=(k_{1}+k_{2})^{2}\,=(k_{3}+k_{4})^{2},$ (6)
$\displaystyle t$ $\displaystyle=(k_{2}+k_{3})^{2}=(k_{1}+k_{4})^{2}\,,$ (7)
$\displaystyle u$ $\displaystyle=(k_{1}+k_{3})^{2}=(k_{2}+k_{4})^{2}=-s-t\,.$
(8)

Each graph has a color weight arising from dressing every vertex with a color-
structure constant, e.g. $c_{s}=c(1,2,3,4)=f^{a_{1}a_{2}b}f^{ba_{3}a_{4}}$.
Similarly, keeping polarization vectors formal, we require a functional map
from the labeled graph to kinematic weight such that, e.g.,

$\displaystyle n_{s}$ $\displaystyle=n(1,2,3,4)\ ,$ (9) $\displaystyle n_{t}$
$\displaystyle=n(1,4,3,2)\ ,$ (10) $\displaystyle n_{u}$
$\displaystyle=n(1,3,2,4)\ .$ (11)

We can give this color-dual numerator a generic ansatz through some mass-
dimension. We impose the requisite antisymmetry and Jacobi-type relation,
$n_{s}=n_{t}+n_{u}$, and finally impose factorization to three-point vertices
of the form $A^{\text{YM}}_{3}+\alpha^{\prime}A^{F^{3}}_{3}$. Because there is
a maximum mass-dimension associated with the kinematics of a three-point
contribution in our theory, every contribution to $\mathcal{A}_{4}$ above
$\mathcal{O}(\alpha^{\prime\,2})$ must be purely local. Any purely local
color-dual terms will be entirely unconstrained, but any terms that require
factorization channels so as to satisfy color-kinematics duality are fixed to
be related to coupling constants appearing in the three-point amplitude.

We therefore find the following contributions through kinematic weights of
mass-dimension $\alpha^{\prime 4}$ over Yang-Mills in the functional color-
dual numerator of the $s$-channel graph,

${n}^{\text{dcc}}_{s}=n_{s}^{\text{YM}}+\alpha^{\prime}n_{s}^{\text{YM}+F^{3}}+\alpha^{\prime\,2}\,n_{s}^{(F^{3})^{2}+F^{4}}+\\\
\alpha^{\prime\,3}\left[a_{3}\,\left(n_{s}^{D^{2}F^{4}}+\sigma_{2}\,n_{s}^{\text{YM}+F^{3}}\right)+a_{3,\text{YM}}\,\sigma_{3}\,n_{s}^{\text{YM}}\right]+\\\
\alpha^{\prime\,4}\left[a_{4,1}\,\left(n_{s}^{(DF)^{4}_{1}}+\sigma_{2}\,n_{s}^{(F^{3})^{2}+F^{4}}\right)+\right.\\\
\left.a_{4,2}\,n_{s}^{(DF)^{4}_{2}}+a_{4,F^{3}}\,\sigma_{3}\,n_{s}^{\text{YM}+F^{3}}\right]+\mathcal{O}(\alpha^{\prime
5})\,.$ (12)

We have scaled out mass-dimension using the $\,{\rm Tr}(F^{3})$ dimensionful
coupling $\alpha^{\prime}$, leaving all unconstrained ansatz parameters,
$a_{i}$, dimensionless. We introduce the notation $\sigma_{i}$ to refer to the
normalized scalar permutation invariants $\sigma_{2}=(s^{2}+t^{2}+u^{2})/8$
and $\sigma_{3}=(stu)/8$. Here the numerator is given in terms of six of the
eight spanning color-dual four-point vector building blocks
$n_{s}^{\mathcal{O}_{i}}$ of Carrasco  _et al._ (2020, 2021), with explicit
definitions given in an ancillary machine-readable file. We have dropped any
contributions that are pure generalized gauge and must cancel out in all
physical amplitudes. The factors of $a_{i}$ are free numeric parameters
unconstrained by the factorization of four-point to three-point amplitudes,
and appear next to terms that go as $\mathcal{O}(\alpha^{\prime}{}^{i})$. If
nothing in our analysis at higher points fixed the $a_{i}$ to take on any
particular non-vanishing value we would not be forced to admit a tower of
higher-derivative operators. On the contrary, however, requiring that color-
dual five-point amplitudes factorize correctly, entirely fixes $a_{3}$,
$a_{4,1}$, and $a_{4,2}$, and relates $a_{4,F^{3}}$ to $a_{3,\text{YM}}$.

Since we know there are only eight distinct color-dual gauge-invariant vector
building blocks at four-points, the most efficient four-point ansatz to arrive
at eq. (12) is small–exposing how powers of scalar permutation invariants can
map the vector basis to span all relevant mass-dimensions. No such vector
basis has yet been clarified at five-points. At five-points we must build an
ansatz in terms of bare Lorentz invariants so as to identify the constraints
from manifesting color-dual kinematics and factorization.

At five-points there are up to 30 distinct Lorentz products to contend with,
and like four-points, there is only one cubic topology to dress. To complete
the calculation involves consideration of a 58,923 parameter ansatz spanning
from order four in dot products for the Yang-Mills numerator to order eight
for a mass-dimension $\alpha^{\prime\,4}$ above Yang-Mills. Imposing the
duality between color and kinematics without consideration of factorization
simply relates these parameters to each other, while setting some to vanish.
Generalized unitarity then fixes the coefficients of monomials to appear in a
double-copy consistent theory in terms of couplings introduced at lower
multiplicity.

We find that requiring color-dual amplitudes at five-points places the
following constraints on the free parameters in eq. (12),

$\displaystyle a_{4,F^{3}}$ $\displaystyle=1+a_{3,\text{YM}}\,,$ (13)
$\displaystyle a_{3}$ $\displaystyle=1\,,$ (14) $\displaystyle a_{4,1}$
$\displaystyle=a_{4,2}=1\,.$ (15)

In summary, the five-point vector amplitude requires consideration of
kinematics at $\mathcal{O}(\alpha^{\prime\,3})$, simply by virtue of three
insertions of the $\,{\rm Tr}(F^{3})$ operator. Evidently such a factorizing
vector amplitude at five-point can only be color-dual if it also includes
terms from a now-required order $\mathcal{O}(\alpha^{\prime\,3})$ contact term
at four-points, eq. (14). This new contact at four-points means we must
consider five-points through order $\mathcal{O}(\alpha^{\prime\,4})$.
Requiring a factorizable amplitude of that kinematic mass-dimension to be
color-dual in turn demands inclusion of the order
$\mathcal{O}(\alpha^{\prime\,4})$ contact term at four-points, eq. (15). We
conjecture that if we are to have a double-copy consistent theory, this tower
will continue to all orders in higher-derivative corrections.

A natural question is if the results found so far through five-points and
$\mathcal{O}(\alpha^{\prime\,4})$ can resum to a known theory. We apparently
have the freedom to set $a_{3,\text{YM}}=0$. If we do so, both our four-point
and five-point amplitudes through $\mathcal{O}(\alpha^{\prime\,4})$ are
consistent with the $\alpha^{\prime}$ expansion of the $B(1,\ldots,n)$
amplitudes of Huang  _et al._ (2016). These $B$ amplitudes Azevedo  _et al._
(2018) belong to the $(DF)^{2}+\text{YM}$ theory of ref. Johansson and Nohle
(2017), where the $(DF)^{2}$ has been deformed by a massive gauge-theory, with
mass scale set by $1/\alpha^{\prime}$. Indeed, in ref. Carrasco  _et al._
(2020), the four-point amplitude of $(DF)^{2}+\text{YM}$ theory was recovered
in terms of the above identified color-dual building blocks:

$n_{s}^{(DF)^{2}+\text{YM}}=n_{s}^{\text{YM}}+\\\
\frac{\alpha^{\prime}n_{s}^{F^{3}}+\alpha^{\prime\,2}n_{s}^{(F^{3})^{2}+F^{4}}+\alpha^{\prime\,3}n_{s}^{D^{2}F^{4}}+\alpha^{\prime\,4}n_{s}^{(DF)^{4}}}{1-\alpha^{\prime}{}^{2}\,\sigma_{2}-\alpha^{\prime}{}^{3}\,\sigma_{3}}\
,$ (16)

with $n_{s}^{(DF)^{4}}\equiv n_{s}^{(DF)^{4}_{1}}+n_{s}^{(DF)^{4}_{2}}$.

The $(DF)^{2}+\text{YM}$ theory that generates the $B$ amplitudes is a
fascinating color-dual dimension-six theory involving the $\,{\rm Tr}(F^{3})$
operator with higher-order propagators. It was first written down by Johansson
and Nohle Johansson and Nohle (2017) with the explicit aim of finding a
double-copy description of conformal supergravity.

Here we come to the most important consequence of our analysis. If one regards
double-copy construction as a mathematical trick to simplify the calculation
of the kinematic form of specific amplitudes of interest, there is no
motivation to introduce a tower of higher-derivative operators. If, on the
other hand, we require double-copy consistency as a matter of principle, and
we wish to grapple with the UV behavior of $\mathcal{N}=4$ supergravity by
adding the $\,{\rm Tr}(F^{3})$ operator to the Yang-Mills copy, we find
ourselves lifting the Poincaré supergravity theory to Berkovits-Witten
conformal supergravity Berkovits and Witten (2004); Johansson  _et al._
(2018),

$(\mathcal{N}=4\,\,\text{Berkovits-Witten}+\ldots)=\\\
\left(\mathcal{N}=4\,\,\text{sYM}\right)\otimes\left((DF)^{2}+\text{YM}+\ldots\right)\,.$
(17)

We include ellipses to emphasize the potential addition of higher-derivative
operators unfixed by solely requiring the double-copy consistency of
$\text{YM}+F^{3}$. Such freedom is exposed through $\mathcal{O}(k^{10})$ by
the entirely unconstrained parameter $a_{3,\text{YM}}$. This freedom can be
fixed with particular single-valued multiple zeta values (MZV) Stieberger and
Taylor (2014); Schlotterer and Schnetz (2019) so that
$((DF)^{2}+\text{YM}+\ldots)\to[(DF)^{2}+\text{YM}]^{\text{sv}}$, where the
superscript denotes a particular mapping to all orders in $\alpha^{\prime}$
that we will discuss. Such a choice will promote $\mathcal{N}=4$ SG to the
gravitational heterotic string at tree-level.

Using eq. (16) to rewrite our constrained ansatz in eq. (12), without
requiring that $a_{3,\text{YM}}$ vanishes offers a revised form of the four-
point numerator for our double-copy consistent theory:

$n^{\text{dcc}}=n^{(DF)^{2}+\text{YM}}+\mathcal{O}(\alpha^{\prime 5})\\\
+a_{3,\text{YM}}\,\alpha^{\prime}{}^{3}\,\sigma_{3}\left(n^{\text{YM}}+\alpha^{\prime}n^{F^{3}}+\mathcal{O}(\alpha^{\prime\,2})\right)\,.$
(18)

Note that the terms from the second line,
$n^{\text{YM}}+\alpha^{\prime}n^{F^{3}}$, are the first two terms of the
$\alpha^{\prime}$ expansion of $n^{(DF)^{2}+\text{YM}}$ given in eq. (16).
This suggests that color-dual consistent amplitudes can be promoted to higher-
order contact terms via a product of their color-dual numerators with scalar
permutation invariants, and that this information can be consistently
propagated to higher multiplicity color-dual amplitudes. Indeed, as we shall
discuss shortly, there is a known map from string-theory considerations which
offers not only a proof of concept but a prescriptive understanding of how
such higher multiplicity color-dual amplitudes may be consistently included.
In light of this behavior, it is not hard to imagine that the following
resummation spans the full set of double-copy consistent four-point
amplitudes, $A^{\text{dcc}}_{4}$, for $\,{\rm Tr}(F^{3})$ theories:

$A^{\text{dcc}}_{4}=B(1,2,3,4)\left[1+\\!\\!\sum_{x\geq
1,y}c_{(x,y)}\sigma_{3}^{x}\sigma_{2}^{y}\alpha^{\prime\,3x+2y}\right]\,,$
(19)

where $\sigma_{3}$ and $\sigma_{2}$ are the four-point scalar permutation
invariants, and all remaining freedom is given in terms of numeric ansatz
parameters $c_{(x,y)}$, which encode the Wilson coefficients of higher-
derivative corrections.

##  III Heterotic string and the SV promotion

We will now show that the additional UV freedom to add operators to the
$(DF)^{2}+\text{YM}$ theory will allow us to lift the $\mathcal{N}=4$
Berkovits-Witten supergravity theories to the tree-level graviton amplitudes
of the heterotic string. First, let us recall the field-theoretic double-copy
structures appearing in tree-level string-theory amplitudes. The most
important clarifying example is the double-copy construction of ordered open
superstring amplitudes from ordered Yang-Mills amplitudes and doubly-ordered
$Z$-theory amplitudes Mafra  _et al._ (2013); Broedel  _et al._ (2013);
Carrasco  _et al._ (2017a); Mafra and Schlotterer (2017); Carrasco  _et al._
(2017b):

$A^{\text{OSS}}_{A}=A^{\text{sYM}}_{a}\otimes^{ab}Z_{Ab}\,,$ (20)

where the indices, $a,b$ and $A$, refer to various orderings of kinematic
labels, and the outer product is taken to mean a field-theoretic double-copy –
expressed particularly with a field-theory KLT kernel acting on ordered
amplitudes, and an implied sum over all repeated indices. The doubly-ordered
$Z$-theory disc amplitudes encode the string-theoretic scalar corrections at
each order in $\alpha^{\prime}$.

For $n$-point amplitudes, we use capital $A$ to refer to orderings of external
legs that satisfy string-theoretic monodromy relations Bjerrum-Bohr  _et al._
(2009); Stieberger (2009), and the lowercase $a$ to index orderings that
satisfy field-theoretic amplitude relations Kleiss and Kuijf (1989); Bern  _et
al._ (2008). So dressing the $A$ ordering of $Z_{Aa}$ with Chan-Paton color-
weights and the $a$ ordering with field-theory color-weights yields color-
dressed $n$-point amplitudes in $Z$-theory. This higher-derivative bi-colored
scalar theory has the property that its $a$-stripped amplitudes satisfies
adjoint-type color-kinematics duality order by order in $\alpha^{\prime}$
Carrasco  _et al._ (2020, 2021). A similar double-copy structure exists for
the open bosonic string Huang  _et al._ (2016); Azevedo  _et al._ (2018):

$A^{\text{OBS}}_{A}=B_{a}\otimes^{ab}Z_{Ab}\,,$ (21)

where the so-called “$B$-theory” amplitudes on the right-hand-side are
generated by the $(DF)^{2}+\text{YM}$ Lagrangian.

Field-theoretic double-copy construction, as per eq. (20) and eq. (21), is
equally applicable to closed superstring construction. This can be seen by
first noting the construction of closed supersymmetric string amplitudes via
the string KLT kernel Kawai  _et al._ (1986), represented here by
$\otimes_{\alpha^{\prime}}$,

$\text{CSS}=A^{\text{OSS}}_{A}\otimes^{AB}_{\alpha^{\prime}}A^{\text{OSS}}_{B}\
.$ (22)

Plugging in the expressions for open superstring amplitudes, given in eq.
(20), reveals a field-theory double-copy,

CSS
$\displaystyle=(A^{\text{sYM}}_{a}\otimes^{ab}Z_{Ab})\otimes^{AB}_{\alpha^{\prime}}(Z_{Bc}\otimes^{cd}A^{\text{sYM}}_{d})\,$
(23)
$\displaystyle=A^{\text{sYM}}_{a}\otimes^{ab}(Z_{Ab}\otimes^{AB}_{\alpha^{\prime}}Z_{Bc}\otimes^{cd}A^{\text{sYM}}_{d})\,$
(24)
$\displaystyle=A^{\text{sYM}}_{a}\otimes^{ab}(A^{\text{sYM}})^{\text{sv}}_{b}\
,$ (25)

where in the last line we introduced the single-valued promotion of field-
theory amplitudes,

$\displaystyle(Y)^{\text{sv}}_{a}\equiv(Z_{Aa}\otimes^{AB}_{\alpha^{\prime}}Z_{Bb}\otimes^{bc}Y_{c}).$
(26)

This operation is called ‘single-valued’ because all the coefficients of
$\alpha^{\prime}$ introduced by the promotion come with only single-valued
multiple zeta values. All resulting amplitudes satisfy color-kinematics as
well as factorization. At four-points this can be understood as multiplying
the $Y$ theory color-dual numerators by scalar permutation invariants at each
order in $\alpha^{\prime}$. The existence of such a map that is consistent to
all multiplicity means that we are free to conjecture the most general double-
copy consistent UV completion of $B(1,2,3,4)$ to be that of eq. (19).

It was pointed out in ref. Azevedo  _et al._ (2018) that the $B$-theory
amplitudes generated by the $(DF)^{2}+\text{YM}$ Lagrangian also play a
critical role in the field-theoretic construction of the gravitational
heterotic string amplitude,

$\text{HS}=B_{a}\otimes^{ab}\left(A^{\text{sYM}}\right)^{\text{sv}}_{b}\,.$
(27)

It is clear from the above construction that one could equally well describe
the hetoretic string as

$\text{HS}=\left(B\right)^{\text{sv}}_{a}\otimes^{ab}A^{\text{sYM}}_{b}\,.$
(28)

We see that the remaining freedom in the set of consistent double-copy
completions to $\text{YM}+F^{3}$ must allow for the single-valued promotion of
$(DF)^{2}+\text{YM}$. Indeed such is the remaining freedom we find through
$\mathcal{O}(\alpha^{\prime}{}^{4})$ in eq. (18), and posit in our conjectured
resummation of eq. (19). We realize the low-energy expansion of
$\left[(DF)^{2}+\text{YM}\right]^{\text{sv}}$ by setting
$a_{3,\text{YM}}=c_{(1,0)}=\zeta(3)$ in eqs. (18) and (19) rather than zero.
This means that in the sea of potential double-copy consistent attempts to
tame the UV behavior of $\mathcal{N}=4$ SG by engaging with its anomalous
behavior via $\,{\rm Tr}(F^{3})$, one of them, with particular single-valued
MZV coefficients of additional higher-derivative operators, results in the
gravitational amplitudes of the heterotic string.

##  IV Next steps

We have presented evidence that demanding double-copy consistency of a gauge
theory with the $\,{\rm Tr}(F^{3})$ operator induces an all-order tower of
$\alpha^{\prime}$ corrections, which seems to require at a minimum all higher-
derivative corrections associated with $(DF)^{2}+\text{YM}$. There exists a
small basis of color-dual vector building blocks, up to trivial scalar
permutation invariants, at four-points Carrasco  _et al._ (2020). Using this
basis reduces the complexity of four-point color-dual vector amplitudes to
simple considerations of what permutation invariant scalars are required for a
given mass-dimension. Developing a similar basis for vector building-blocks at
five-points, as has already been done for higher-derivative color-weights
Carrasco  _et al._ (2021), could allow a simple derivation that
$\text{YM}+F^{3}$ must close to $(DF)^{2}+\text{YM}$ under double-copy
consistency. If we require double-copy consistency of an $\mathcal{N}=4$ SG
theory modified by $(\mathcal{N}=4\text{ sYM})\otimes\,{\rm Tr}(F^{3})$ it
appears that the fate of the theory lies in a family of Berkovitz-Witten
conformal supergravity theories with freedom to add an additional tower of
higher-derivative corrections. A particular choice of their Wilson
coefficients takes the theory to the graviton amplitudes of the heterotic
string. We look forward to verifying the interplay of such higher-derivative
operators on the UV behavior of the theory through explicit calculation.

### Acknowledgements:

We gratefully acknowledge related collaboration and insightful conversations
with Zvi Bern, James Mangen, Laurentiu Rodina, Radu Roiban, Aslan Seifi, and
Suna Zekioğlu. We would like to thank Henrik Johansson, James Mangen, Olver
Schlotterer, and Suna Zekioğlu for very helpful feedback regarding earlier
drafts. This work was supported by the DOE under contract DE-SC0021485 and by
the Alfred P. Sloan Foundation. M.L. and N.H.P. would additionally like to
acknowledge the Northwestern University Amplitudes and Insight group,
Department of Physics and Astronomy, and Weinberg College for support.

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