# Hidden Amplitude Zeros From Double Copy

Christoph Bartsch◇ Taro V. Brown♠ Karol Kampf$\,{}^{\Diamond}$ Umut Oktem♠
Shruti Paranjape♠ Jaroslav Trnka♠ ♠Center for Quantum Mathematics and Physics
(QMAP), University of California, Davis, CA, USA  ◇Institute for Particle and
Nuclear Physics, Charles University, Prague, Czech Republic

###### Abstract

Recently, Arkani-Hamed et al. proposed the existence of zeros in scattering
amplitudes in certain quantum field theories including the cubic adjoint
scalar theory Tr($\phi^{3}$), the $SU(N)$ non-linear sigma model (NLSM) and
Yang-Mills (YM) theory. These hidden zeros are special kinematic points where
the amplitude vanishes and factorizes into a product of lower-point
amplitudes, similar to factorization near poles. In this letter, we show a
close connection between the existence of such zeros and color-kinematics
duality. In fact, all zeros can be derived from the Bern-Carrasco-Johansson
(BCJ) relations. We also show that these zeros extend via the Kawai-Lewellen-
Tye (KLT) relations to special Galileon amplitudes and their corrections,
evincing that these hidden zeros are also present in permutation-invariant
amplitudes.

##  I Introduction

Scattering amplitudes in quantum field theory are constrained by physical
principles. Tree-level amplitudes have poles which are well-understood from
the local nature of interactions. Unitarity dictates that amplitudes factorize
on these poles into products of lower-point amplitudes and this can be used to
reconstruct the amplitude recursively Britto  _et al._ (2005); Cohen  _et al._
(2011); Cheung  _et al._ (2015a). In specific theories there are even more
kinematic properties that scattering amplitudes can have, one very interesting
set is the presence of _zeros_. The most famous example is the _Adler zero_
Adler (1965); Kampf  _et al._ (2013a), i.e. vanishing of the amplitude in the
soft limit. This is a consequence of the shift symmetry of the underlying
Lagrangian, present in theories with a spontaneously broken symmetry. In fact,
just factorization properties and the Adler zero are enough to completely fix
all $n$-point amplitudes in the NLSM theory. Motivated by that, one can impose
the behavior in the soft limit as a kinematic constraint and search for new
(or already known) theories. Dubbed the _soft bootstrap_ Cheung  _et al._
(2015b, 2016, 2017); Elvang  _et al._ (2019), this has led to many interesting
insights and singled out certain special theories such as the NLSM, Born-
Infeld and Galileon models as unique theories with specified soft behavior.
These theories also play a special role in different contexts such as in the
Cachazo-He-Yuan (CHY) scattering equations, ambitwistor strings or color-
kinematics duality Cachazo  _et al._ (2015); Casali  _et al._ (2015); Bern
_et al._ (2008) and were also classified using algebraic methods Bogers and
Brauner (2018); Roest  _et al._ (2019). One particular constraint in the
bootstrap is an _enhanced soft limit_ Cheung  _et al._ (2015b) i.e. scaling
$p_{i}\rightarrow tp_{i}$ and $t\rightarrow 0$, requires

$\lim_{t\rightarrow 0}\mathcal{A}_{n}={\cal O}(t^{\sigma})\,.$ (1)

In multi-field theories, we discuss _ordered_ amplitudes $A_{n}$, where the
full amplitude is decomposed into cyclic sectors with fixed orderings of
external legs. For NLSM amplitudes such a decomposition is known as _flavor
ordering_ ,

${\cal A}_{n}=\sum_{a\in S_{n}/Z_{n}}{\rm Tr}\,(T^{a_{1}}\dots
T^{a_{n}})\,A_{n}[a_{1}a_{2}\dots a_{n}]\,,$ (2)

and the Adler zero ($\sigma=1$ soft limit) is then enjoyed by the full
amplitude ${\cal A}_{n}$ as well as the ordered amplitudes $A_{n}$ Kampf  _et
al._ (2013b, a). In theories without internal group structure, there are
_unordered_ amplitudes. One notable example is the special Galileon,
discovered from the bootstrap method as the unique theory with $\sigma=3$ soft
behavior Cheung  _et al._ (2015b), later explained by the presence of an
enhanced symmetry of the Lagrangian Hinterbichler and Joyce (2015). The same
amplitude also naturally appears in the CHY formalism Cachazo  _et al._
(2015). Using the double copy relations, two copies of NLSM produce the
special Galileon amplitudes Cachazo  _et al._ (2015).

In recent remarkable work Arkani-Hamed  _et al._ (2023a, 2024a); Arkani-Hamed
and Figueiredo (2024), Arkani-Hamed, Cao, Dong, Figueiredo and He discovered
that amplitudes in certain theories exhibit very different types of zeros:
they vanish when certain Mandelstam invariants $s_{ij}=(p_{i}+p_{j})^{2}$ go
to zero. A similar zero has also been noted to exist in vector boson pair
production Dixon  _et al._ (1999). More generally, these _hidden zeros_
originate from the kinematic mesh picture Arkani-Hamed  _et al._ (2022), and
exist in amplitudes of ${\rm Tr}(\phi^{3})$ theory and the NLSM, but also to
YM theory (where the zeros involve polarization vectors). At low multiplicity,
the four-point amplitude vanishes if $s_{13}=0$, and the five-point if
$s_{13}=s_{14}=0$. The six-point amplitude has two types of zeros, we refer to
them as _skinny_ and _rectangle_ :

$\displaystyle\begin{split}C_{1}&=\\{s_{13}=s_{14}=s_{15}=0\\},\\\
C_{2}&=\\{s_{14}=s_{15}=s_{24}=s_{25}=0\\},\end{split}$ (3)

and similarly for all cyclic permutations of labels. More generally, at $n$
points the cyclically inequivalent zeros are

$\displaystyle
C_{m}=\begin{Bmatrix}s_{1m+2}\,=\,s_{1m+3}\,=\ldots=s_{1n-1}\,\,=0\\\
s_{2m+2}\,=\,s_{2m+3}\,=\ldots=s_{2n-1}\,\,=0\\\ \vdots\hskip
39.83368pt\vdots\hskip 62.59596pt\vdots\\\ \hskip
8.5359pts_{mm+2}=s_{mm+3}=\ldots=s_{mn-1}=0\end{Bmatrix},$ (4)

where $m=1,\dots,\lfloor\frac{n}{2}\rfloor-1$. Furthermore, the amplitudes
factorize in the neighborhood of these zeros into a four-point prefactor (not
necessarily an amplitude in the same theory) and a “top” and “bottom”
amplitude,

$\displaystyle A_{n}\big{\rvert}_{C_{m}}\xrightarrow{s^{\ast}\neq
0}A_{4}\times A_{m+2,B}\times A_{n-m,T}\,,$ (5)

as can also be read off directly from the kinematic mesh picture, with
$s_{i\cdots n}\equiv(p_{i}+\cdots+p_{n})^{2}$,

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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ (6)

Note that the skinny zero actually implies the Adler zero (since setting
$s_{1n}=0$ further sends $p_{1}=0$). The goal of this letter is to find the
relationship between these hidden zeros and the BCJ relations, as both
Tr($\phi^{3}$) and NLSM satisfy both constraints. We find that all zeros can
be derived from BCJ relations for any number of points. Analogous to the soft
bootstrap, we use hidden zeros as a constraint, construct higher-derivative
corrections to NLSM and compare it to the Adler zero and BCJ relations.
Finally, we show that via the KLT relations the hidden zeros carry over to
permutationally invariant theories. We show this explicitly in the case of the
special Galileon and check both the existence of hidden zeros (now for all
permutations) as well as factorizations near these zeros. This raises the
question of whether the special Galileon (and other theories produced by the
KLT relations) have their origin in a surfacehedron picture Arkani-Hamed  _et
al._ (2023b, c, 2024b).

While this work was in its final stages, the preprint Cao  _et al._ (2024)
appeared with some overlapping ideas.

##  II Zeros from BCJ relations

Let us investigate how the NLSM amplitude zeros are related to the BCJ
relations Bern  _et al._ (2008, 2019). At six points the relations can be
written in the following form:

$\displaystyle
A_{6}[123456]=\frac{1}{s_{12}s_{123}s_{56}}\big{[}A_{6}[162543]\
s_{13}s_{25}(s_{56}-s_{24})$ $\displaystyle+A_{6}[162345]\
s_{15}(s_{12}+s_{23})(s_{14}-s_{56})$ $\displaystyle-A_{6}[162354]\
s_{14}(s_{12}+s_{23})(s_{25}+s_{35})$ (7) $\displaystyle+A_{6}[162435]\
s_{13}s_{15}s_{24}+A_{6}[162453]\ s_{13}s_{24}(s_{15}+s_{35})$ $\displaystyle-
A_{6}[162534]\ s_{14}s_{25}(s_{12}+s_{23})\big{]}\,.$

Since the NLSM has no three-point interactions, its amplitudes have no two-
particle poles. Looking at the RHS of (7), it vanishes on both the $C_{1}$ and
$C_{2}$ zeros defined in (4). Thus at six points, the zeros of the NLSM
amplitude are a direct consequence of their BCJ-compatibility.

For Tr($\phi^{3}$)-theory the relation (7) holds when replacing
$A_{6}[\alpha]\to m_{6}[\alpha|\beta]$ where $m_{6}$ is a bi-adjoint scalar
six-point amplitude and $\beta$ is any ordering such that none of the BCJ
basis elements $m_{6}[162\sigma(345)|\beta]$ vanish. On choosing
$\beta\\!=\\!\\{123456\\}$ for example on the LHS, none of the amplitudes on
the RHS contain poles at the locations of the zeros and so once again the RHS
vanishes due to the Mandelstam coefficients. Other orderings vanish trivially
through appropriately relabeled BCJ relations.

The proof that BCJ-compatibility implies the presence of zeros at $n$ points
is involved but conceptually straightforward. It requires careful
consideration of which terms in the BCJ relation vanish on the support of a
generic zero $C_{m}$ (4). Below, we present the details of the proof, starting
with a convenient representation of the BCJ relation Bern  _et al._ (2008),

$\displaystyle A_{n}[123\cdots n]=(-1)^{n}\\!\\!\\!\\!\\!\sum_{\sigma(3\ldots
n-1)}\\!\\!\\!\\!A_{n}[1n2\sigma]\,\prod_{k=3}^{n-1}\frac{\mathcal{F}_{k}[2\sigma
1]}{s_{kk+1\ldots n}}.$ (8)

The factors $\mathcal{F}_{k}$ are given by

$\mathcal{F}_{k}[\rho]=\begin{cases}\sum\limits_{l=t_{k}}^{n-1}\mathcal{S}_{k,\rho_{l}}+\theta(t_{k-1},t_{k})s_{kk+1\ldots
n}&\text{if }t_{k}>t_{k+1},\\!\\\
-\\!\\!\sum\limits_{l=1}^{t_{k}}\mathcal{S}_{k,\rho_{l}}\\!-\theta(t_{k},t_{k-1})s_{kk+1\ldots
n}&\text{if }t_{k}<t_{k+1},\\!\end{cases}$ (9)

where $t_{k}$ is the position of leg $k$ in the ordered list
$\rho\\!=\\!\\{2\sigma 1\\}$, $\rho_{l}$ denotes the $l$-th element of
$\\{2\sigma 1\\}$ and we always set $t_{2}\equiv 0$ and $t_{n}\equiv t_{n-2}$.
The function $\theta(t_{i},t_{j})=1$ if $t_{i}>t_{j}$ and 0 otherwise. The
factors $\mathcal{S}_{i,j}$ are defined as

$\displaystyle\mathcal{S}_{i,j}=\begin{cases}s_{ij}&\text{if }i>j\text{ or
}j=1,2,\\\ 0&\text{else}.\end{cases}$ (10)

Our objective is now to show that for each of the $(n-3)!$ permutations
$\\{2\sigma 1\\}$ where $\sigma\\!=\\!\sigma(34\ldots n\\!-\\!1)$ there is at
least one index $k\\!\in\\!\\{34\ldots n\\!-\\!1\\}$ such that on the zero
constraint $C_{m}$ (4) the factor $\mathcal{F}_{k}[2\sigma
1]\big{\rvert}_{C_{m}}\\!\\!\\!\\!=0$. First we define the set of labels

$\displaystyle I_{m}=\\{m\\!+\\!2\ldots n\\!-\\!1\\}.$ (11)

For any term in the BCJ relation corresponding to a particular permutation
$\\{2\sigma 1\\}$ we proceed as follows:

  * •

Find the position $t_{m+1}$ of the label $m\\!+\\!1$ in $\\{2\sigma 1\\}$.

  * •

Identify the label in $I_{m}$ which appears as the left-most in $\\{2\sigma
1\\}$. Denote it $L$ and its position $t_{L}$. Note that by definition
$t_{L}<t_{i}$ for any $i\in
I_{m}\setminus\\{L\\}\\!=\\!\\{m\\!+\\!2\ldots\cancel{L}\ldots n\\!-\\!1\\}$.

  * •

Identify the label in $I_{m}$ which appears as the right-most in $\\{2\sigma
1\\}$. Denote it $R$ and its position $t_{R}$. Note that by definition
$t_{R}>t_{i}$ for any $i\in
I_{m}\setminus\\{R\\}\\!=\\!\\{m\\!+\\!2\ldots\cancel{R}\ldots n\\!-\\!1\\}$.

Having carried out this procedure, the permutation $\\{2\sigma 1\\}$
necessarily falls into one of three categories:

  1. (a)

$\\{2\sigma 1\\}=\\{2\ldots m\\!+\\!1\ldots L\ldots R\ldots 1\\}$, where
$t_{R}>t_{m+1}$ and $t_{L}>t_{m+1}$,

  2. (b)

$\\{2\sigma 1\\}=\\{2\ldots L\ldots R\ldots m\\!+\\!1\ldots 1\\}$, where
$t_{R}<t_{m+1}$ and $t_{L}<t_{m+1}$.

  3. (c)

$\\{2\sigma 1\\}=\\{2\ldots L\ldots m\\!+\\!1\ldots R\ldots 1\\}$, where
$t_{R}>t_{m+1}$ and $t_{L}<t_{m+1}$.

For the case (a) we set $k\\!=\\!R$. Then by definition of $R$, all elements
in $I_{m}\setminus\\{R\\}$ appear to the left of $R$ in $\\{2\sigma 1\\}$ and
$m+1$ appears to the left of $R$. This guarantees that $t_{R}>t_{R+1}$ and
$t_{R}>t_{R-1}$ for any $R\in\\!I_{m}$, leading to

$\displaystyle(\text{a})\\!:\,\,\,\mathcal{F}_{R}[2\sigma
1]=\sum_{i=t_{R}}^{n-1}\mathcal{S}_{R,\rho_{i}}=\\!\\!\sum_{R>\rho_{i}}s_{R\rho_{i}}.$
(12)

Since we are summing over all labels to the right of $R$, the range for
$\rho_{i}$ can only include $\rho_{i}\subset\\{134\ldots m\\}$. But for any
choice of $R\in\\!I_{m}$ and $\rho_{i}\in\\{134\ldots m\\}$, the invariant
$s_{R\rho_{i}}$ will be part of the constraint $C_{m}$ (4). Hence

$\displaystyle(\text{a})\\!:\,\,\,\mathcal{F}_{R}[2\sigma
1]\big{\rvert}_{C_{m}}\\!\\!\\!\\!=0.$ (13)

Setting $k=L$ for case (b) and either $k=L$ or $k=R$ for case (c), one can
show by analogous arguments that

$\displaystyle(\text{b})\\!:\,\,\,\mathcal{F}_{L}[2\sigma
1]\big{\rvert}_{C_{m}}\\!\\!\\!\\!=0,\hskip
14.22636pt(\text{c})\\!:\,\,\,\mathcal{F}_{R/L}[2\sigma
1]\big{\rvert}_{C_{m}}\\!\\!\\!\\!=0.$ (14)

Thus for any permutation $\\{2\sigma 1\\}$ appearing in the BCJ relation (8),
there exists at least one $k\\!\in\\!I_{m}$ such that $\mathcal{F}_{k}[2\sigma
1]\big{\rvert}_{C_{m}}\\!\\!\\!\\!=0$ holds. This in turn proves that

$\displaystyle A_{n}[12\ldots n]\big{\rvert}_{C_{m}}\\!\\!\\!\\!=0,$ (15)

for any $C_{m}$ of the form (4).

##  III Zeros of higher-derivative theories

Following the logic of the soft and BCJ bootstraps Low and Yin (2019); Brown
_et al._ (2023); Elvang  _et al._ (2019); Kampf  _et al._ (2021); Kampf (2021)
we can now ask about the existence of zeros for higher-derivative corrections
to NLSM theory, and compare the constraints they impose to the Adler zero
conditions and BCJ relations. Let us look at the first generic case of six-
point amplitudes. At the two-derivative or ${\cal O}(p^{2})$ level, we find
that the NLSM amplitude is the unique solution to all three questions: it is
completely fixed by the Adler zero, it is the only BCJ-satisfying amplitude,
and it is the unique solution to the hidden zero problem. We start with a
general ansatz for the six-point amplitude in ${\cal O}(p^{2k})$ power-
counting,

$A_{6}=\sum_{i}\sum_{\rm
cycl}\frac{a_{i}P_{i}}{s_{123}}+\sum_{j}b_{j}Q_{j}\,,$ (16)

where $P_{i}\sim s^{k{+}1}$, $Q_{j}\sim s^{k}$ are polynomials in $s_{ij}$.
Imposing the zero conditions $C_{1}$ and $C_{2}$ allows us to constrain the
coefficients $a_{i}$, $b_{j}$. For $k=1$ all coefficients get fixed and we
find a unique solution – the NLSM amplitude,

$A_{6}^{\rm
NLSM}=-\frac{1}{2}\frac{(s_{12}+s_{23})(s_{45}+s_{56})}{s_{123}}+s_{12}+\text{cyc}.$
(17)

For $k{=}2$, i.e. at ${\cal O}(p^{4})$ order, there are no BCJ-satisfying six-
point (and higher-point) amplitudes, but starting with (16) and applying the
zero conditions (3), we find one solution. It can be identified as the
amplitude of a particular operator in the chiral Lagrangian (with coefficient
$L_{0}$ in the conventions of Bijnens  _et al._ (2019)). Explicitly, it reads

$\displaystyle A_{6}^{(4)}$
$\displaystyle=s_{12}(s_{12}+s_{34}+2s_{45})-s_{123}(s_{345}-2s_{61})$
$\displaystyle\hskip
56.9055pt-\frac{(s_{12}+s_{23})(s_{45}+s_{56})^{2}}{s_{123}}+\text{cyc}.$ (18)

On poles, this consistently factorizes into an NLSM amplitude and a higher-
derivative four-point amplitude,

$\displaystyle A_{4}^{(4)}=(s_{12}+s_{23})^{2}=s_{13}^{2},$ (19)

which itself satisfies the four-point zero conditions.

Interestingly, the BCJ-incompatible amplitude (18) factorizes close to the
zeros, similar to the leading order NLSM. Let us first consider the skinny
zero $C_{1}$ in (3). Turning on $s^{*}=s_{15}\neq 0$ in (5) we find,

$\displaystyle A_{6}^{(4)}\big{\rvert}_{C_{1}}\xrightarrow{s_{15}\neq
0}A_{4}^{(2)}{\times}m_{3,B}^{(0)}{\times}A_{5,T}^{(2)}+A_{4}^{(4)}{\times}m_{3,B}^{(0)}{\times}A_{5,T}^{(0)}\,,$
(20)

where $m_{3,B}^{(0)}=1$ is the three-point Tr($\phi^{3}$) amplitude and we
identify (as functions of $s_{ij}$) the pre-factors

$\displaystyle\begin{split}A_{4}^{(2)}=-s_{15},\hskip
14.22636ptA_{4}^{(4)}=s_{15}^{2},\end{split}$ (21)

as the NLSM amplitude and its hidden zero-compliant higher-derivative
correction (19). The five-point “top” amplitudes in (20) involve pions and
Tr($\phi^{3}$) scalars,

$\displaystyle
A_{5,T}^{(0)}(\phi\pi\pi\phi\phi)=1-\frac{s_{23}+s_{34}}{s_{234}}-\frac{s_{34}+s_{45}}{s_{345}},$
(22)

as well as the higher-derivative correction,

$\displaystyle
A_{5,T}^{(2)}(\phi\pi\pi\phi\phi){=}s_{23}{+}2s_{34}{+}s_{45}{-}\frac{(s_{23}{+}s_{34})^{2}}{s_{234}}{-}\frac{(s_{34}{+}s_{45})^{2}}{s_{345}}.$
(23)

In particular the higher-derivative amplitude retains the Adler zero for pions
and correctly factorizes on poles into (19) and the three-scalar amplitude
$m_{3,B}$. The factorization (20) presents a natural generalization of the
known factorization theorem for NLSM amplitudes Arkani-Hamed  _et al._
(2023a). In an EFT expansion, we get a sum over products of NLSM and mixed
amplitudes (with higher-derivative corrections) such that their combined mass
dimension matches the given power counting. Setting $s^{*}$ to a different
$s_{ij}$ in (20) leads to the appearance of different mixed amplitudes and
their higher-derivative corrections. The distribution of pions and scalars
follows the pattern described for the leading order NLSM in Arkani-Hamed  _et
al._ (2023a).

The even-even factorization for higher-derivative amplitudes close to the zero
$C_{2}$ in (3) also generalizes the behavior of NLSM amplitudes. For
$s^{*}=s_{25}$,

$\displaystyle A_{6}^{(4)}\big{\rvert}_{C_{2}}\xrightarrow{s_{25}\neq
0}m_{4}\times\big{\\{}A_{4,B}^{(4)}\\!\times
A_{4,T}^{(2)}{+}A_{4,B}^{(2)}\times A_{4,T}^{(4)}\big{\\}},$ (24)

where the prefactor $m_{4}$ takes the form of a four-point Tr($\phi^{3}$)
amplitude

$\displaystyle m_{4}=\frac{1}{s_{123}}+\frac{1}{s_{345}},$ (25)

and the “bottom” and “top” amplitudes only involve (higher-derivative) pions.

Returning to the bootstrap analysis, we note that while the amplitude (18)
satisfies the Adler zero condition, we want to emphasize that the constraints
from the hidden zeros are in fact _more_ stringent. At $\mathcal{O}(p^{4})$
there are two independent operators in the chiral Lagrangian. Both generate
amplitudes with the Adler zero, but only (18) also has the hidden zeros. We
can continue to higher ${\cal O}(p^{2k})$ orders and find a general pattern,

$(\mbox{BCJ satisfying})\subset(\mbox{Hidden zeros})\subset(\mbox{Adler
zero})$ (26)

which can be supported by the explicit counting of solutions to higher $k$
found in table 1.

Table 1: Number of six-point amplitudes from various constraints in NLSM ${\cal O}(p^{\\#})$ | 2 | 4 | 6 | 8 | 10 | 12 | 14
---|---|---|---|---|---|---|---
Alder zero | 1 | 2 | 10 | 29 | 78 | 203 | 461
Hidden zeros | 1 | 1 | 5 | 13 | 41 | 112 | 282
BCJ satisfying | 1 | 0 | 1 | 1 | 2 | 4 | 7

As a fun side check, one can repeat the exercise imposing only skinny zeros
$C_{1}$ (ignoring the rectangle zero $C_{2}$). Here we get more solutions. In
particular, at ${\cal O}(p^{8})$ order, we get 14 solutions which reduce to 13
when we impose the rectangle zero. This suggests a pattern that the presence
of skinny zeros are stronger conditions than that of Adler zeros, and
rectangle zeros provide further constraints when fixing coefficients in the
ansatz (16).

##  IV KLT relations and special Galileon

The geometric origin of hidden zeros Arkani-Hamed  _et al._ (2023a) suggests
that their presence requires an ordered set of external states. On the other
hand, since the BCJ relations imply their existence (assuming the absence of
two-particle poles), a natural question is whether or not these zeros persist
in permutation-invariant double copy theories. In particular: do the hidden
zeros in the NLSM double copy to zeros of tree amplitudes in special Galileon
theory? Note that Tr($\phi^{3}$) is not a BCJ-compatible theory, nor is it the
product of a double copy and so the only scalar theory of interest here will
be the special Galileon theory Bern  _et al._ (2019).

Let us start with the six-point amplitude,

$\displaystyle
M_{6}=\frac{s_{12}s_{23}s_{13}s_{45}s_{46}s_{56}}{s_{123}}+\text{perms}-\frac{1}{2}G(12345)\,,$
(27)

where $G$ denotes the Gram determinant. On the zero $C_{1}$, all terms but the
following vanish

$\displaystyle\frac{s_{12}s_{26}s_{16}s_{34}s_{45}s_{35}}{s_{126}}+s_{12}^{2}s_{34}s_{35}s_{45}\,.$
(28)

On the support of $C_{1}$, $s_{12}=-s_{16}$ and we see that the above
expression must vanish – $C_{1}$ is indeed a zero of the special Galileon
amplitude! Indeed the amplitude (27) vanishes on $C_{2}$ as well and the
permutation symmetry of the amplitude means that it vanishes at many more
locations than its single copy.

Moving on to eight points, we find that the Galileon amplitude vanishes on all
three expected zeros $C_{1}$, $C_{2}$ and $C_{3}$. In the next section, we
will see that these zeros are not a low-point accident, but rather their
presence is guaranteed by the KLT double copy formula.

In light of the known factorization theorem (5) for ordered theories, let us
now study the behavior of special Galileon amplitudes close to the hidden
zeroes. The structure of the known theorem is determined by the color-ordering
of external states, making it seem unlikely that permutation-invariant double
copy products inherit such properties. Contrary to this expectation, we find
that special Galileon amplitudes indeed do factorize near their zeros. Take
for example the six-point amplitude (27) near the zero $C_{2}$,

$\displaystyle M_{6}\big{\rvert}_{C_{2}}\xrightarrow{s_{25}\neq 0}m_{4}\times
M_{4,B}\times M_{4,T}\,,$ (29)

where $m_{4}$ is as in (25) and the “top”- and “bottom”-amplitude can be
functionally identified as the four-point special Galileon,

$\displaystyle M_{4,B}=s_{61}s_{12}(s_{61}+s_{12}),\hskip
8.5359ptM_{4,T}=s_{34}s_{45}(s_{34}+s_{45}).$ (30)

Thus, despite the permutation invariance of Galileon theory, it still
satisfies the same factorization theorem as NLSM and Tr($\phi^{3}$). We
checked that this factorization near zeros persists at eight points.

We end this section with a short discussion on factorization into odd-point
subamplitudes. Already in the case of NLSM, this involves the introduction of
mixed pion-scalar amplitudes in a theory of NLSM coupled to a Tr($\phi^{3}$)
scalar $\phi$. Due to its analogous role in the soft limit and double copy, a
natural guess for odd-point factorization in Galileon theory is subamplitudes
in a theory of a Galileon coupled to pions and scalars, first studied in
Cachazo  _et al._ (2016). For example near $C_{1}$ with $s_{15}\neq 0$,

$\displaystyle M_{6}\big{\rvert}_{C_{1}}\xrightarrow{s_{15}\neq
0}M_{4}\times\tilde{M}_{5,T}$ (31)

where $\tilde{M}_{5,T}\\!=\\!\tilde{M}_{5}(\phi\,\Gamma\,\Gamma\,\phi\,\phi)$
involves special Galileons and scalars. Limited checks at six-point verify
this guess, but a deeper discussion is left for future work.

##  V KLT preserves zeros

Encouraged by the explicit checks of the previous section at six and eight
points, we investigate how the special Galileon amplitudes inherit the zeros
through the double copy procedure. To see how this works, we can write the KLT
kernel in a basis where each element vanishes at these zeros, which implies
that the KLT formula,

$M_{n}=\sum_{\sigma\gamma}S[\sigma|\gamma]A_{n}(1\gamma
m{+}1n)A_{n}(1m{+}1n\sigma)\,,$ (32)

preserves the zeros $C_{m}$ if the double copied amplitude has no two-particle
poles.

For a given zero $C_{m}$ as in (4), let us consider the two sets $I_{m}$, as
defined in (11), and $J_{m}\\!=\\!\\{2,3,...,m\\}$ such that $s_{zf}=0$ for
any $z\\!\in\\!I_{m}$ and $f\\!\in\\!J_{m}$. The form of the KLT kernel Kawai
_et al._ (1986); Bjerrum-Bohr  _et al._ (2011) we will be using is given by

$\displaystyle S[a_{1},\cdots a_{n-3}|b_{1}\cdots b_{n-3}]$ (33)
$\displaystyle\hskip
14.22636pt=\left(\frac{1}{2}\right)^{-(n-3)}\prod_{t=1}^{n-3}\left(p_{1}\cdot
p_{a_{t}}+\sum_{q>t}^{n-3}\theta(a_{t},a_{q})p_{a_{t}}\cdot
p_{a_{q}}\right)\\!,$

and is evaluated in the KLT formula on different permutations i.e.
$S[\sigma(a_{1},\cdots a_{n-3})|\gamma(b_{1},\cdots,b_{n-3})]$. It is
convenient to choose a basis for the kernel where $\gamma$ and $\sigma$
contain only elements that live in $J_{m}$ or $I_{m}$. This corresponds to
fixing the positions of $\\{1,m{+}1,n\\}$ as in (32), instead of the usual
choice of $\\{1,n{-}1,n\\}$ Kawai  _et al._ (1986); Bjerrum-Bohr  _et al._
(2011). Note that if $s_{ij}=0$ is a zero of $M(12...m{+}1...n{-}1\ n)$, it is
also a zero of $M(12...n{-}1...m{+}1\ n)$ due to the permutational invariance
of $M$, allowing a $m{+}1\leftrightarrow n{-}1$ swap in (32).

The KLT kernel is proportional to $s_{1z_{0}}\\!=\\!0$ for any
$z_{0}\\!\in\\!I_{m}$ if the permutation $\sigma$ ends with $z_{0}$.
Similarly, if $\gamma$ starts with any $z_{1}\in I_{m}$, the kernel is zero.
This is because as soon as we consider $i_{t}=z_{1}$ in the kernel (33),
anything that follows $z_{1}$ in the $\sigma$ ordering necessarily follows it
in the $\gamma$ ordering. This preserves the order in both sets and sets the
KLT kernel proportional to $s_{1z_{1}}=0$. Thus the only terms of interest in
the KLT formula are of the following form:

$M=\sum_{\sigma\gamma}S[\sigma f_{0}|f_{1}\gamma]A_{n}(1f_{1}\gamma
m+1n)A_{n}(1f_{0}\bar{\sigma}nm+1)$ (34)

where $f_{0},f_{1}\in J_{m}$ and $\bar{\sigma}$ denotes the reversed $\sigma$
ordering. The non-zero contributions to $M$ are then orderings where $\gamma$
starts with an element of $J_{m}$ and $\sigma$ ends with one. Denoting the
rightmost label in $\sigma$ that lives in $I_{m}$ as $z_{0}$, the kernel is
proportional to $s_{1z_{0}}+\sum_{f>z_{0}}s_{z_{0}f}$. Here the sum is over
all labels $f$ in $\sigma$ to the right of $z_{0}$, such that $f$ is to the
left of $z_{0}$ in $\gamma$. However, if $z_{0}\in I_{m}$ and $f\in J_{m}$,
$s_{z_{0}f}=0$, meaning the kernel vanishes. As a result, all the terms
contributing to $M$ vanish, and the KLT formula preserves the zeros $C_{m}$ as
long as the amplitude does not have any two-particle poles.

Unlike the presence of zeros in double copy theories, factorization near them
is less apparent from the KLT formula. We turned instead to a different
formulation of Galileon amplitudes, the CHY formula Cachazo  _et al._ (2015).
Though less straightforward, it is possible to see that the factorization of
Galileon tree amplitudes into lower-point ones near “even” zeros $C_{2m}$ is a
consequence of the CHY formula, as we will discuss further in upcoming work
Bartsch  _et al._ .

##  VI Zeros in special Galileon EFT

In this section, we investigate the hidden zeros from the point of view of
higher-derivative corrections to the leading order NLSM and special Galileon
amplitudes. It was shown in Elvang  _et al._ (2019); Brown  _et al._ (2023);
Carrillo González  _et al._ (2020) that certain special higher-derivative
corrections to NLSM satisfy BCJ relations and results were provided up to
${\cal O}(p^{18})$ order for six-point amplitudes. In general, the BCJ
conditions on higher-point amplitudes provided constraints on lower-point
amplitudes, and it is an open question how many BCJ satisfying theories there
are to all points.

The BCJ-satisfying ordered amplitudes can be plugged into the KLT formula,
producing amplitudes in a higher-derivative single scalar theory. In the last
section, we saw that the KLT formula preserves all zeros of the ordered
amplitude – in fact, the new amplitude has hidden zeros at all permutations of
the kinematic conditions (4). In Brown  _et al._ (2023) we inspected these
amplitudes from the point of view of soft limits. The BCJ amplitudes have
${\cal O}(t)$ Adler zeros Adler (1965) and the KLT relations enhance this
behavior to ${\cal O}(t^{3})$, as evident from the NLSM - special Galileon
examples. Though this works more generally, we find that not all ${\cal
O}(t^{3})$ amplitudes are generated from KLT (the first counter example
appears at 16-derivative order), only a special subset is. Here we explore
this question in the context of hidden zeros. In the schematic KLT formula,

$(\mbox{KLT amp})=(\mbox{BCJ amp})\otimes(\mbox{BCJ amp})$ (35)

we can count the number of possible terms on the left hand side and compare
them to the scalar bootstrap construction of all amplitudes with hidden zeros
at a given derivative order. Because the hidden zero conditions are trivial
for four-point amplitudes, we begin the analysis at six points. In particular,
we find:

Table 2: Number of six-point amplitudes from various constraints in special Galileon theory ${\cal O}(p^{\\#})$ | 10 | 12 | 14 | 16 | 18 | 20 | 22
---|---|---|---|---|---|---|---
${\cal O}(t^{3})$ soft behavior | 1 | 0 | 1 | 3 | 10 | 23 | 49
Hidden zeros | 1 | 0 | 1 | 1 | 4 | 6 | 14
Generated from KLT | 1 | 0 | 1 | 1 | 3 | 5 | 10

At ${\cal O}(p^{10})$ order, we identify one solution for each of the
constraints and that is the special Galileon. There is no solution at the next
${\cal O}(p^{12})$ order to any of the constraints, and one for each at the
${\cal O}(p^{14})$ order (all corresponding to the same theory). The first
difference appears at ${\cal O}(p^{16})$, where the ${\cal O}(t^{3})$ soft
limit constraints produce three solutions while only one of them is generated
from the KLT relations and also enjoys the hidden zeros. Note that up to this
derivative order one can identify all solutions to the ${\cal O}(t^{3})$
constraint as higher-derivative corrections to the special Galileon theory. At
the next level, ${\cal O}(p^{18})$ we find that out of the four solutions to
the hidden zeros constraint, only three of them are generated by the KLT
relations. This leaves us with the following hierarchy of constraints,

$(\mbox{From KLT})\subset(\mbox{Hidden zeros})\subset({\cal
O}(t^{3})\,\mbox{behavior})$

Note that here we only imposed the presence of hidden zeros as a constraint,
not factorization near these zeros. To do that one would need to provide a
precise prescription for the higher-derivative theories. It is possible that
on doing so, this restricted set of “zeros+factorization” theories is the same
as the set of all “KLT theories”.

Here it is important to note that the factorization theorem for EFT amplitudes
is modified when compared to the leading order NLSM or Galileon theories. They
now involve a higher-derivative Tr($\phi^{3}$) amplitude, leading to

$\displaystyle
M^{\text{h.d.}}_{n}\big{\rvert}_{C_{m}}\xrightarrow[]{s^{\ast}\neq
0}A^{\text{Tr}\phi^{3}\text{+h.d.}}_{4}\times M^{\text{h.d.}}_{B}\times
M^{\text{h.d.}}_{T}\,.$ (36)

The Tr($\phi^{3}$) amplitude with higher-derivative corrections obtained from
the near-zero factorization of both BCJ-compatible NLSM and special Galileon
theory are equal,

$\displaystyle
A^{\text{Tr}\phi^{3}\text{+h.d.}}_{4}\\!=\\!\Lambda\\!\left(\frac{1}{s}{+}\frac{1}{t}\right)\\!{-}\frac{t}{\Lambda}{+}\frac{t^{2}}{\Lambda^{2}}{+}\frac{t}{\Lambda^{3}}(7t^{2}{-}us)+\cdots$
(37)

where $\Lambda$ is the EFT scale. Tr($\phi^{3}$) amplitudes coincide with
diagonal bi-adjoint scalar amplitudes, whose higher-derivative corrections
have been studied in the context of modified KLT kernels Chi  _et al._ (2022);
Chen  _et al._ (2023, 2022); Carrasco  _et al._ (2020, 2021) and (37) is one
with a specific choice of Wilson coefficients. What type of KLT kernel this
leads to and whether we can use a modified kernel to make the factorization
theorem for leading and subleading orders coincide are interesting questions
left for future work.

##  VII Outlook

The relationship between hidden zeros and the double copy discussed above
opens many avenues of investigation. Some, such as the possible relation
between higher-derivative KLT kernels and factorization near zeros of EFT
amplitudes, as well as a detailed study of Galileon-pion-scalar mixed
amplitudes were discussed in the text above. The most interesting one relates
to the presence of zeros and factorization in gravitational amplitudes
obtained as a double copy from YM amplitudes. The YM zeros as described in
Arkani-Hamed  _et al._ (2023a) include conditions involving the polarization
vectors as well as momenta. Additionally the amplitudes contain 2-particle
poles, voiding the KLT argument. Nonetheless, we find that special properties
of YM ensure that gravity amplitudes do indeed vanish on these zeros via the
CHY formula, which we discuss in an upcoming work Bartsch  _et al._ .

The original context of the hidden zeros in NLSM and Tr($\phi^{3}$) was that
of the associahedron Arkani-Hamed  _et al._ (2024c, 2022, 2018a), but an
independent geometric description of NLSM, similar to the amplituhedron
Arkani-Hamed and Trnka (2014a, b); Arkani-Hamed  _et al._ (2018b); Damgaard
_et al._ (2019); He  _et al._ (2023) for planar ${\cal N}=4$ SYM or the
Tr($\phi^{3}$) associahedron itself is yet-to-be found. Furthermore, nothing
is known about the geometric description of permutation-invariant amplitudes
and any insight into the geometric origin of Galileon zeros is absent.

A final interesting question about these hidden zeros is that of uniqueness of
tree-level amplitudes Arkani-Hamed  _et al._ (2018c); Rodina (2019a, b) and
also loop integrands Arkani-Hamed  _et al._ (2011); Bartsch  _et al._ (2022,
2024). For the NLSM, one can write down a recursion relation similar to soft
subtracted recursion Kampf  _et al._ (2013b); Cheung  _et al._ (2016); Luo and
Wen (2016); Cheung  _et al._ (2017); Kampf  _et al._ (2020). To what extent
double copy theories are fixed by the presence of their zeros, especially in
an EFT expansion is an open problem.

Acknowledgements: We thank Nima Arkani-Hamed, Carolina Figueiredo, Song He,
Alfredo Guevara, Callum Jones and Jiri Novotny for very useful discussions.
This work is supported by GAČR 24-11722S, GAUK-327422, MEYS LUAUS23126, DOE
grant No. SC0009999 and the funds of the University of California.

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