compat=1.1.0 graviton/.style=circle, draw=green!60, fill=green!5, very thick,
minimum size=7mm codot/.style=/tikz/shape=circle, /tikz/fill=red,/tikz/minimum
size=0.1cm,/tikz/inner sep=1.8pt sb/.style= /tikz/shape=circle,
/tikz/fill=black, /tikz/minimum size=0.3cm,/tikz/inner sep=1.pt myblob/.style=
/tikz/shape=ellipse, /tikz/fill=red, /tikz/minimum width=0.5cm,
myblobFF/.style= /tikz/shape=circle, /tikz/fill=black, /tikz/minimum
width=0.25cm, myblob2/.style= /tikz/shape=rectangle, /tikz/fill=black,
/tikz/minimum width=0.1cm,/tikz/inner sep=1.8pt fp/.style=
/tikz/shape=rectangle, /tikz/fill=red, /tikz/minimum width=0.1cm,/tikz/inner
sep=1.8pt myvertex/.style= /tikz/shape=circle, /tikz/fill=black, /tikz/minimum
width=0.05cm, HV/.style= /tikz/shape=circle, /tikz/fill=rgb:black,1;white,2,
/tikz/minimum size=0.01cm,/tikz/inner sep=1.8pt

QMUL-PH-23-22

# Kinematic Hopf algebra for amplitudes from higher-derivative operators

Gang Chen$\mbox{}^{a}$ Laurentiu Rodina$\mbox{}^{b,c}$ Congkao
Wen$\mbox{}^{b}$

###### Abstract

Recently it has been shown that Bern-Carrasco-Johansson (BCJ) numerators of
colour-kinematic duality for tree-level scattering amplitudes in Yang-Mills
theory (coupled with scalars) can be determined using a quasi-shuffle Hopf
algebra. In this paper we consider the same theory, but with higher-derivative
corrections of the forms $\alpha^{\prime}F^{3}$ and $\alpha^{\prime 2}F^{4}$,
where $F$ is the field strength. In the heavy mass limit of the scalars, we
show that the BCJ numerators of these higher-derivative theories are governed
by the same Hopf algebra. In particular, the kinematic algebraic structure is
unaltered and the derivative corrections only arise when mapping the abstract
algebraic generators to physical BCJ numerators. The underlying kinematic Hopf
algebra enables us to obtain a compact expression for the BCJ numerators of
any number of gluons and two heavy scalars for amplitudes with higher-
derivative operators. The pure gluon BCJ numerators can also be obtained from
our results by a simple factorisation limit where the massive particles
decouple.

###### Contents

  1. 1 Introduction
  2. 2 Higher-derivative corrections and BCJ numerators
  3. 3 Review of Kinematic Hopf algebra
    1. 3.1 Review of quasi-shuffle Hopf algebra
    2. 3.2 The evaluation maps
  4. 4 Evaluation maps with higher-derivative corrections
    1. 4.1 Factorisation behaviour
    2. 4.2 Crossing symmetry
    3. 4.3 Summary and pure gluon amplitudes
  5. 5 Conclusions and outlook
  6. 6 Acknowledgements
  7. A Six-point pre-numerator

##  1 Introduction

The Bern-Carrasco-Johansson (BCJ) duality reveals deep connections between
Yang-Mills theory and gravity [1, 2], and provides remarkably powerful
computational tools for calculating scattering amplitudes including at loop
level. See [3, 4] for recent reviews and the references therein. A central
aspect of the duality is the fact that BCJ kinematic numerators of scattering
amplitudes satisfy non-trivial relations, in particular the Jacobi identity
associated with the dual colour factors. This extremely intriguing fact has
not been understood in full generality. Given both the practical and
theoretical implications of the duality, understanding the kinematic algebra
that is responsible for the duality remains an important open question.

As an abstract algebra, it is generally characterised in terms of fusion
products for a basis of generators. The generators can be realised from
several different points of view, such as differential operators [5, 6, 7, 8,
9, 10], auxiliary fields [11, 12, 13], string worldsheet operators [14, 15,
16], or BRST formulas [17, 18, 19]. Recently a systematic framework for the
kinematic algebra was proposed in [20, 21, 22], where the generator is taken
as the QCD current, or its heavy-mass limit. Especially in the heavy-mass
effective field theory [23, 24, 25, 26, 27, 22, 28], which had a wide
phenomenological applications in heavy quark and black hole physics [28, 29,
30, 31, 32, 33, 34, 35, 36, 37], the kinematic algebra was found to be
isomorphic to an infinite dimensional combinatorial algebra [38, 39], i.e. a
(generalised) quasi-shuffle Hopf algebra [40, 41, 42, 43, 44]. This underlying
kinematic algebra, which is called the kinematic Hopf algebra, was further
extended to scalar Yang-Mills theory with an arbitrary finite mass in [45]. A
geometrical perspective of this structure was also proposed in [46].

A natural question is to find out whether all theories that admit a colour-
kinematic representation also admit such a Hopf algebra structure. In this
paper we target higher derivative corrections to Yang-Mills theory and find
the answer is positive. Specifically, we consider the subset of higher-
derivative operators of the forms $\alpha^{\prime}F^{3}$ and $\alpha^{\prime
2}F^{4}$ that are compatible with adjoint type colour-kinematic duality, as
discussed in [47]. Such higher-derivative operators are relevant for the
bosonic string amplitude in the low energy expansion.

We find the algebra structure (the fusion product rules, etc.) is in fact
identical to the previously studied cases, and the only modification required
is to the map which turns the algebraic generators into concrete functions of
momenta and polarisation vectors – such map is called the evaluation map [45].
We provide explicit expressions for the $\alpha^{\prime}$ corrections to the
evaluation maps due to $\alpha^{\prime}F^{3},\alpha^{\prime 2}F^{4}$ terms.
The underlying Hopf algebra structure enables us to write down extremely
compact expressions for the BCJ numerators of the higher-derivative theory for
the amplitudes with any number of gluons and two heavy scalars. The BCJ
numerators of pure gluons can be also obtained from our results by an
appropriate factorisation limit in which the scalars decouple [38]. We have
explicitly verified the BCJ numerators obtained from the proposed kinematic
algebra up to eight external particles.

The rest of the paper is organised as follows. In Section 2 we introduce the
relevant higher derivative corrections to Yang-Mills with
$\alpha^{\prime}F^{3}$ and $\alpha^{\prime 2}F^{4}$ terms, and describe the
procedure resulting in the corresponding BCJ numerators with two heavy
scalars. In Section 3 we briefly review the kinematic Hopf algebra
construction of BCJ numerators by introducing fusion product rules of Hopf
algebra and the concept of evaluation maps. In Section 4 we propose an
explicit expression for the evaluation maps that compute BCJ numerators of all
multiplicity up to the order $\alpha^{\prime 2}$. We conclude in Section 5
with a summary and possible future directions. Finally, Appendix A contains
the explicit expression for the numerators with six particles, and higher-
point expressions can be found in the linked repository [48].

##  2 Higher-derivative corrections and BCJ numerators

In [47] it was pointed out that up to order $\alpha^{\prime 2}$ there exists a
unique combination of higher derivative operators in Yang-Mills theory that is
compatible with adjoint type colour-kinematic duality. In fact, it was
recently understood that the entire series of higher derivative corrections is
tightly constrained by demanding “double copy consistency" [49]. The
Lagrangian that is consistent with the adjoint type colour-kinematic duality
up to order $\alpha^{\prime 2}$ takes the following form,

$S_{\mathrm{YM}+\alpha^{\prime}F^{3}+\alpha^{\prime
2}F^{4}}=\int\mathrm{d}^{D}x\operatorname{Tr}\left\\{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{2\alpha^{\prime}}{3}F_{\mu}^{\nu}F_{\nu}^{\lambda}F_{\lambda}^{\mu}+\frac{\alpha^{\prime
2}}{4}\left[F_{\mu\nu},F_{\lambda\rho}\right]\left[F^{\mu\nu},F^{\lambda\rho}\right]\right\\}\,,$
(2.1)

and this is the theory we will study in this paper.

An efficient method to compute the corresponding amplitudes based on Berends-
Giele recursion [50] was proposed in [51]. We will be interested in scattering
amplitudes of gluons coupled to two heavy scalars, which have been shown to
play very important roles in the study of gravitational wave emission and
black hole physics [28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. One way to obtain
these gluon-scalar amplitudes is by performing a dimensional reduction of two
gluons to two massive scalars at the level of Lagrangian, starting from (2.1).
Effectively, one can directly perform the dimensional reduction at the level
of scattering amplitudes using the transmutation operator
$\partial_{e_{i}}{\cdot}\partial_{e_{j}}$ following [52] (see also [53]),
which turns two gluons into adjoint scalars. We then take the infinite mass
limit for these scalars, which have the momentum $mv^{\mu}$, with $m$ and
$v^{\mu}$ being the mass and velocity of the heavy particle.

The corresponding BCJ numerators of these amplitudes can be obtained through
the KLT procedure [54]:

$\mathcal{N}_{\mathrm{BCJ}}(1\alpha,v)=\sum_{\beta\in
S_{n-3}}\mathcal{S}(1\alpha\mid 1\beta)A(1\beta,v)\,,$ (2.2)

where $A(1\beta,v)$ is the amplitude of $n-2$ gluons and two heavy scalars
which can be obtained from the procedure outlined above, and
$\mathcal{S}(1\alpha\mid 1\beta)$ is the KLT matrix [55]. For example, using
(2.2), and taking the heavy mass limit, we find the numerators for the three
and four-point amplitudes determined by the Lagrangian (2.1) can be expressed
as

$\displaystyle\mathcal{N}_{\mathrm{BCJ}}(1,v)=\varepsilon_{1}{\cdot}v\,,$
(2.3)

which cannot receive any high-derivative corrections when two of external
particles are scalars, and

$\displaystyle\mathcal{N}_{\mathrm{BCJ}}(12,v)=$
$\displaystyle-\frac{v{\cdot}F_{12}{\cdot}v}{p_{1}{\cdot}v}+\alpha^{\prime}\text{tr}(F_{12})p_{1}{\cdot}v+\alpha^{\prime
2}p_{12}^{2}\text{tr}(F_{12})p_{1}{\cdot}v\,,$ (2.4)

where
$F^{\mu\nu}_{i}=p_{i}^{\mu}\varepsilon_{i}^{\nu}-\varepsilon_{i}^{\mu}p_{i}^{\nu}$,
and we have defined,

$\displaystyle F_{\sigma}=F_{i_{1}}\cdot F_{i_{2}}\cdots
F_{i_{r}}\,,\,p_{\sigma}=p_{i_{1}}+p_{i_{2}}+\cdots+p_{i_{r}},\quad{\rm
for}\quad\\{i_{1},i_{2},\ldots,i_{r}\\}=\sigma\,,$ (2.5)

in particular $F_{12}=F_{1}{\cdot}F_{2},p_{12}=p_{1}+p_{2}$. The higher
derivative corrections are particularly simple here, but naively they grow
quickly in complexity at higher multiplicity. In the next section, we will
present a Hopf algebra that underlies the mathematical structures of these BCJ
numerators with higher derivative corrections. The understanding of the
algebraic structures allows us to write compact expressions for the numerators
with arbitrary multiplicity.

It is worth mentioning that once the numerators are obtained, applying the
double copy procedure leads to potential higher derivative corrections in
gravity [47, 56, 51]. In particular, by squaring the four-point numerator
given in (2.4), we find the contribution arising from $F^{3}\times F^{3}$ is
given by

$\displaystyle\alpha^{\prime
2}\frac{(\text{tr}(F_{1}{\cdot}F_{2})\,p_{1}{\cdot}v)^{2}}{p^{2}_{12}}\,,$
(2.6)

which agrees with the result given [57] for the gravitational amplitude of two
gravitons and two heavy scalars with one insertion of $R^{3}$. The double copy
of $F^{3}\times F^{3}$ should also include the contribution from the
insertions of two $R^{2}\phi$, which however is proportional to that of
$R^{3}$ at four points, which explains the agreement with the amplitude of a
single $R^{3}$ insertion.

##  3 Review of Kinematic Hopf algebra

In this section we will give a brief review of constructing BCJ numerators
based on the Kinematic Hopf algebra [38, 45, 39]. The construction consists of
two steps. The first step is to introduce the quasi-shuffle Hopf algebra of
algebraic generators, whereas the second step is to determine the evaluation
map that translates algebraic generators into functions of kinematics (i.e.
momenta and polarization vectors).

###  3.1 Review of quasi-shuffle Hopf algebra

We begin by introducing the quai-shuffle Hopf algebra relevant for
constructing BCJ numerators, following the references [38, 45, 39]. More
details can also be found in [40, 41, 42, 43, 44]. We first construct an
abstract algebra numerator which is obtained directly from the fusion product
of the algebraic generators $T_{(i)}$ as follows,

$\displaystyle\widehat{\mathcal{N}}(12\ldots n{-}2,v)=T_{(1)}\star
T_{(2)}\cdots\star T_{(n{-}2)}\,,$ (3.1)

with the fusion product defined as the standard quasi-shuffle product. For
instance, for the cases with two, three, and four gluons, we have

$\displaystyle\widehat{\mathcal{N}}(12,v)=$ $\displaystyle\,\,T_{(1)}\star
T_{(2)}=-T_{(12)}\,,$ (3.2) $\displaystyle\widehat{\mathcal{N}}(123,v)=\,\,$
$\displaystyle T_{(1)}\star T_{(2)}\star
T_{(3)}=T_{(123)}-T_{(12),(3)}-T_{(13),(2)}\,,$ (3.3)
$\displaystyle\widehat{\mathcal{N}}(1234,v)=$ $\displaystyle-
T_{(1234)}+T_{(123),(4)}+T_{(14),(23)}+T_{(124),(3)}+T_{(12),(34)}-T_{(12),(3),(4)}$
$\displaystyle-
T_{(12),(4),(3)}-T_{(14),(2),(3)}+T_{(134),(2)}+T_{(13),(24)}-T_{(13),(2),(4)}$
$\displaystyle-T_{(13),(4),(2)}-T_{(14),(3),(2)}\,.$ (3.4)

In general, the fusion product is captured by the following formula,

$\displaystyle T_{(1\tau_{1}),\ldots,(\tau_{r})}\star
T_{(j)}=\hskip-14.22636pt\sum_{\sigma\in\\{(\tau_{1}),\ldots,(\tau_{r})\\}\shuffle\\{(j)\\}}T_{(1\sigma_{1}),\ldots,(\sigma_{r+1})}-\sum_{i=1}^{r}T_{(1\tau_{1}),\ldots,(\tau_{i-1}),(\tau_{i}j),(\tau_{i+1}),\ldots,(\tau_{r})}\,.$
(3.5)

It is important to stress that the fusion products, or more generally the
algebra, are identical to that given in the references [38, 45, 39], which was
introduced to describe BCJ numerators of Yang-Mills theory (i.e. without
higher-derivative corrections).

Following [38, 45, 39], once $\widehat{\mathcal{N}}(12\ldots n{-}2;v)$ is
constructed in terms of the algebraic objects
$T_{(1\tau_{1}),\ldots,(\tau_{r})}$ using the fusion product rules given in
(3.5), to obtain the BCJ numerators, we further introduce the evaluation maps
which translate abstract generators into functions of physical variables (such
as momenta and polarisation vectors). The evaluation maps for the BCJ
numerators of scattering amplitudes in Yang-Mills theory (with two heavy
massive scalars) were written down and proved in [38], and further extended to
more general cases (with arbitrary numbers of scalars with general masses)
[45]. As we emphasised previously that the fusion rules of the Hopf algebra we
propose for the BCJ numerators of the higher-derivative theory (2.1) is the
same as that for the Yang-Mills theory [38, 45, 39], therefore the higher-
derivative corrections can only arise in the evaluation maps, which we will
discuss in the next subsection.

###  3.2 The evaluation maps

We denote the evaluation maps of the generator $T$ as $\langle T\rangle$ [38,
45, 39]. To expose the pole structure of massive propagators, we propose that
$\langle T\rangle$ takes the following form

$\displaystyle\langle
T_{(1\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})}\rangle:=\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a)[myblob]{}; \vertex[left=0.8cm of a] (a0)[myblob]{};
\vertex[right=0.8cm of a] (a2)[myblob]{}; \vertex[right=0.8cm of a2]
(a3)[myblob]{}; \vertex[right=0.8cm of a3] (a4)[myblob]{}; \vertex[above=0.8cm
of a] (b1){$\tau_{1}~{}~{}~{}~{}$}; \vertex[above=0.8cm of a2]
(b2){$\tau_{2}$}; \vertex[above=0.8cm of a3] (b3){$\cdots$};
\vertex[above=0.8cm of a4] (b4){$\tau_{r}$}; \vertex[above=0.8cm of a0]
(b0){$1~{}~{}$}; \vertex[above=0.8cm of a0](j1){$$}; \vertex[right=0.2cm of
j1](j2){$$}; \vertex[right=0.6cm of j2](j3){$$}; \vertex[right=0.4cm of
j3](j4){$$}; \vertex[right=0.8cm of j4](j5){$$}; \vertex[right=0.2cm of
j5](j6){$$}; \vertex[right=0.6cm of j6](j7){$$}; \vertex[right=0.0cm of
j7](j8){$$}; \vertex[right=0.8cm of j8](j9){$$}; \diagram*{(a)--[very
thick](a0),(a)--[very thick](a2),(a2)--[very thick](a3), (a3)--[very
thick](a4),(a0) -- [thick] (j1),(a) -- [thick] (j2),(a)--[thick](j3),(a2) --
[thick] (j4),(a2)--[thick](j5),(a4) -- [thick] (j8),(a4)--[thick](j9)};
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle={1\over n-2}{G_{1\tau_{1}}(v)\over
v{\cdot}p_{1}}{G_{\tau_{2}}(p_{\Theta(\tau_{2})})\over
v{\cdot}p_{1\tau_{1}}}\,\cdots{G_{\tau_{r}}(p_{\Theta(\tau_{r})})\over
v{\cdot}p_{1\tau_{1}\ldots\tau_{r-1}}}\,,$ (3.6)

where the lines between red blobs are massive propagators, and the external
lines are gluons, which are grouped into sets, denoted as
$1,\tau_{1},\tau_{2},\ldots,\tau_{r}$. The $\tau_{i}$s are ordered non-empty
sets such that
$\tau_{1}\cup\tau_{2}\cup\cdots\cup\tau_{r}=\\{2,3,\ldots,n{-}2\\}$ and
$\tau_{i}\cap\tau_{j}=\emptyset$, i.e. they constitute a partition. The set
$\Theta(\tau_{i})$ consists of all indices to the left of $\tau_{i}$ and
smaller than the first index in $\tau_{i}$:

$\displaystyle\Theta(\tau_{i})=(\\{1\\}\cup\tau_{1}\cup\cdots\cup\tau_{i-1})\cap\\{1,\ldots,\tau_{i[1]}\\}\,.$
(3.7)

The expression on the second line of (3.2) manifests the massive propagators
in the large-mass limit, which are given by $v\cdot p_{1\tau_{1}\ldots}$.

The BCJ numerator is determined from the pre-numerators via the following
relation,

$\displaystyle\mathcal{N}_{\mathrm{BCJ}}(1\alpha,v)=\mathcal{N}([1\alpha],v)\,,$
(3.8)

where $[\bullet]$ stands for the left-nested commutator, defined as

$\displaystyle[i_{1}i_{2}\cdots
i_{r}]=[\ldots[[i_{1},i_{2}],i_{3}],\ldots,i_{r}]\,.$ (3.9)

For instance,
$\mathcal{N}_{\mathrm{BCJ}}(12,v)=\mathcal{N}(12,v)-\mathcal{N}(21,v)$. This
can also be expressed as [21]

$\displaystyle\mathcal{N}([1\ldots n{-}2],v)=\mathbb{L}(1\ldots
n-2)\circ\mathcal{N}(1\ldots n{-}2,v)\,,$ (3.10)

where the left-nested operator is defined as

$\displaystyle\mathbb{L}(i_{1},\ldots,i_{m})\equiv\Big{[}\mathbb{I}-\mathbb{P}_{(i_{2}i_{1})}\Big{]}\Big{[}\mathbb{I}-\mathbb{P}_{(i_{3}i_{2}i_{1})}\Big{]}\cdots\Big{[}\mathbb{I}-\mathbb{P}_{(i_{m}\cdots
i_{2}i_{1})}\Big{]}\,$ (3.11)

and $\mathbb{P}_{(i_{m}\cdots i_{2}i_{1})}$ denotes the cyclic permutations,
i.e. $\mathbb{P}_{(i_{2}i_{1})}$ is $(i_{2}\rightarrow i_{1},i_{1}\rightarrow
i_{2})$, $\mathbb{P}_{(i_{3}i_{2}i_{1})}$ is $(i_{3}\rightarrow
i_{2},i_{2}\rightarrow i_{1},i_{1}\rightarrow i_{3})$. Then
$\mathcal{N}([1\alpha],v)$ is obtained through the evaluation map (3.2),

$\displaystyle\mathcal{N}(1\alpha,v)=\langle\widehat{\mathcal{N}}(1\alpha,v)\rangle\,,$
(3.12)

with $\widehat{\mathcal{N}}(1\alpha,v)$ defined by the fusion product (3.1).

The kinematic algebra and the structure (3.2) together automatically induce a
recursive relation of the pre-numerator

$\displaystyle\mathcal{N}(1\alpha,v)$
$\displaystyle=(-1)^{n-3}\Big{(}{G_{1\alpha}(v)\over(n-2)v{\cdot}p_{1}}\Big{)}$
$\displaystyle+\sum_{\tau_{L}\subset\alpha}(-1)^{|\tau_{R}|}\Big{(}(n-2-|\tau_{R}|){\mathcal{N}(1\tau_{L},v)G_{\tau_{R}}(p_{\Theta_{L}(\tau_{R})})\over(n-2)v{\cdot}p_{1\tau_{L}}}\Big{)}\,,$
(3.13)

where again $\\{1\tau_{L}\\}\cup\\{\tau_{R}\\}=\\{1\alpha\\}=\\{12\ldots
n-2\\}$ and $|\tau_{R}|$ denotes the number of gluons in $\tau_{R}$. It is
straightforward to see that the number of terms of
$\mathcal{N}(1,2,\ldots,n{-}2;v)$ is nothing but the Fubini numbers [38]

$\displaystyle\mathsf{F}_{n{-}3}=\sum_{i=0}^{n-4}\binom{n{-}3}{i}\mathsf{F}_{i}\,,$
(3.14)

where $\mathsf{F}_{0}=1,\mathsf{F}_{1}=1$.

##  4 Evaluation maps with higher-derivative corrections

In this section, we will study the evaluation maps in the expression (3.2),
namely the $G$-function. We will constrain it using a variety of physical
properties such as factorisation behaviour, power counting, gauge invariance,
and crossing symmetry, and then propose our solution.

###  4.1 Factorisation behaviour

We first note, as for the Yang-Mills theory without higher-derivative
corrections, for certain special cases, the generators are mapped to $0$. In
particular, we have

$\displaystyle\langle T_{(1),\ldots}\rangle=0\,,\qquad\langle
T_{\ldots,(1\ldots),\ldots}\rangle=0\,.$ (4.1)

In general, the $G$ function is non-trivial. For a given ordered set of
gluons, say $\tau$, we propose it takes the following universal form,

$\displaystyle
G_{\tau}(x)=\Big{(}x{\cdot}F_{\tau}{\cdot}v+\sum_{\sigma_{1}i\sigma_{2}j\sigma_{3}=\tau}x{\cdot}F_{\sigma_{1}}{\cdot}p_{i}W(i\sigma_{2}j)p_{j}{\cdot}F_{\sigma_{3}}{\cdot}v\Big{)}\,,$
(4.2)

where the first term $x{\cdot}F_{\tau}{\cdot}v$ is precisely the result of the
heavy-mass effective field theory (HEFT) [38] without higher-derivative
operators. $W(i\sigma_{2}j)$ contains the information of higher-derivative
corrections, and is given by

$\displaystyle W(i\sigma
j)=\alpha^{\prime}\text{tr}(F_{[i\sigma]}F_{j})+\alpha^{\prime
2}\sum_{\rho\in[i\sigma]j}\sum_{\rho_{1}j_{2}\rho_{2}j_{1}={\rm
cyc}(\rho)}p_{j_{1}}{\cdot}F_{\rho_{1}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\rho_{2}]}F_{j_{1}})$
$\displaystyle+\alpha^{\prime
2}\sum_{\sigma_{1}j_{1}\sigma_{2}j_{2}\sigma_{3}=\sigma}\text{tr}(F_{[i\sigma_{1}]}F_{j_{1}})p_{j_{1}}{\cdot}F_{\sigma_{2}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\sigma_{3}]}F_{j})\,.$
(4.3)

The summation in (4.2) is over all the partitions of ordered $\tau$ in terms
of three ordered subsets $\sigma_{1},\\{i\sigma_{2}j\\},\sigma_{3}$ (including
empty $\sigma_{i}$). For example,

$\displaystyle G_{12}(x)$
$\displaystyle=x{\cdot}F_{12}{\cdot}v+x{\cdot}p_{1}W(12)p_{2}{\cdot}v$ (4.4)
$\displaystyle G_{123}(x)$
$\displaystyle=x{\cdot}F_{123}{\cdot}v+x{\cdot}p_{1}W(123)p_{2}{\cdot}v+x{\cdot}F_{1}{\cdot}p_{2}W(23)p_{3}{\cdot}v+x{\cdot}p_{1}W(12)p_{2}{\cdot}F_{3}{\cdot}v\,.$

Below we will motivate the structure of $G_{\tau}(x)$, especially from the
factorisation behaviour on the massive poles, also which imposes strong
constraints on the function $W(i\sigma_{2}j)$.

First, the term $x{\cdot}F_{\tau}{\cdot}v$, corresponding to Yang-Mills
interactions with no higher derivatives, is determined by mass dimension
counting, gauge invariance, linearity in $v$ (except for the case $x=v$, when
it is quadratic in $v$), and the ordering of gluons in the set $\tau$. Next,
the term corresponding to the higher derivative corrections is the most
natural way to generalise the original term $x{\cdot}F_{\tau}{\cdot}v$ by
including extra powers of momenta, in a way consistent with the ordering in
$\tau$. Corrections of order $\alpha^{\prime}$ require two extra powers of
momenta, and the simplest way to add two momenta is by inserting them inside
the chain of $F_{\tau}$. This can be either adjacent to each other,
$x{\cdot}F_{\sigma_{1}}{\cdot}p_{X}p_{Y}{\cdot}F_{\sigma_{2}}{\cdot}v$, with
$\sigma_{1}\sigma_{2}=\tau$, or more generally,
$x{\cdot}F_{\sigma_{1}}{\cdot}p_{X}W(F_{\sigma_{2}})p_{Y}{\cdot}F_{\sigma_{3}}{\cdot}v$,
with $\sigma_{1}\sigma_{2}\sigma_{3}=\tau$. Finally, the labels $X$ and $Y$
should be related to the location of the insertions. At order
$\alpha^{\prime}$ $W$ is only a function of field strengths, while at
$\alpha^{\prime 2}$ it can include another two powers of extra momenta.
Comparing this general expectation to explicit low point examples, we indeed
find they satisfy (4.2).

We will now further show that $G_{\tau}(x)$ defined in (4.2) gives the
correction factorisation behaviours of the numerators on the heavy mass
propagators,

$\displaystyle\mathcal{N}(1\ldots
n{-}2,v)\rightarrow{(n-2-|i\tau_{R}|)|i\tau_{R}|\over
n-2}{p_{\Theta_{L}(i\tau_{R})}{\cdot}p_{i}\over
v{\cdot}p_{1\tau_{L}}}\mathcal{N}(1\tau_{L},v)\mathcal{N}(i\tau_{R},v)\,.$
(4.5)

Let us begin with the four-point case. The pre-numerator takes the following
form,

$\displaystyle\mathcal{N}(12,v)=-{G_{12}(v)\over
2v{\cdot}p_{1}}=-{v{\cdot}F_{12}{\cdot}v+v{\cdot}p_{1}W(12)p_{2}{\cdot}v\over
2v{\cdot}p_{1}}\,.$ (4.6)

We see that the higher-derivative correction term has no pole, therefore the
factorisation is identical to that of Yang-Mills theory without corrections
[38].

Terms containing $W$’s in the $G$-function become relevant for higher-point
numerators. The first such example is the five-point pre-numerator, which is
given as

$\displaystyle\mathcal{N}(123,v)$ $\displaystyle={1\over
3}\left({G_{123}(v)\over v{\cdot}p_{1}}-{G_{12}(v)\over
v{\cdot}p_{1}}{G_{3}(p_{12})\over v{\cdot}p_{12}}-{G_{13}(v)\over
v{\cdot}p_{1}}{G_{2}(p_{1})\over v{\cdot}p_{13}}\right)$ (4.7)
$\displaystyle={v{\cdot}F_{123}{\cdot}v+v{\cdot}p_{1}W(123)p_{3}{\cdot}v+v{\cdot}p_{1}W(12)p_{2}{\cdot}F_{3}{\cdot}v+v{\cdot}F_{1}{\cdot}p_{2}W(23)p_{3}{\cdot}v\over
v{\cdot}p_{1}}$
$\displaystyle-\frac{\left(v{\cdot}F_{12}{\cdot}v+W(12)p_{1}{\cdot}vp_{2}{\cdot}v\right)p_{12}{\cdot}F_{3}{\cdot}v}{p_{1}{\cdot}vp_{12}{\cdot}v}-\frac{\left(v{\cdot}F_{13}{\cdot}v+W(13)p_{1}{\cdot}vp_{3}{\cdot}v\right)p_{1}{\cdot}F_{2}{\cdot}v}{p_{1}{\cdot}vp_{13}{\cdot}v}\,.$

The factorisation behaviour can be sketched as below,

$v$$1$$2$$3$ $\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(p1) {$v$}; \vertex[right=0.8cm of p1] (b1) [HV]{H};
\vertex[right=0.5cm of b1] (b2) []{}; \vertex[right=0.5cm of b2] (b3) [dot]{};
\vertex[above =1.2cm of p1] (u0) []{};\vertex[right =0.2cm of u0] (u1)
[]{$1$}; \vertex[above=1.2cm of b3] (u3) []{};\vertex[left=0.5cm of u3] (u3a)
[]{$3$};\vertex[right=0.0cm of u3] (u3b) []{$2$}; \vertex[right=0.8cm of
b3](p4){}; \vertex[right=0.5cm of b1] (cut1); \vertex[above=0.2cm of cut1]
(cut1u); \vertex[below=0.2cm of cut1] (cut1b); \vertex[right=0.25cm of b2]
(cut2); \vertex[above=0.2cm of cut2] (cut2u); \vertex[below=0.2cm of cut2]
(cut2b); \diagram* { (b1)--[gluon, thick](u1), (b3)-- [gluon, thick]
(u3b),(b1)-- [gluon, thick] (u3a), (p1) -- [thick] (b1)-- [thick] (b3)--
[thick] (p4), (cut1u)--[ red,thick] (cut1b), };
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(4.8)

Since the factorisation of the leading-order terms in $\alpha^{\prime}$ (i.e.
terms without $W$) has already been shown in [38], we will only focus on the
terms beyond the leading order, It is straightforward to see that from the
explicit expression of $\mathcal{N}(123,v)$, at the three poles,
$p_{1}{\cdot}v\to 0,p_{12}{\cdot}v\to 0$, and $p_{123}{\cdot}v\to 0$ (as shown
in (4.8)), $\mathcal{N}(123,v)$ factorises as

$\displaystyle\Big{(}{p_{1}{\cdot}p_{2}\over
p_{1}{\cdot}v}\Big{)}\Big{(}v{\cdot}\varepsilon_{1}\Big{)}\Big{(}-{p_{2}{\cdot}vW(23)p_{3}{\cdot}v\over
p_{2}{\cdot}v}\Big{)}\,,$ $\displaystyle\Big{(}{p_{12}{\cdot}p_{3}\over
p_{12}{\cdot}v}\Big{)}\Big{(}-{p_{1}{\cdot}vW(12)p_{2}{\cdot}v\over
p_{1}{\cdot}v}\Big{)}\Big{(}v{\cdot}\varepsilon_{3}\Big{)}\,,$
$\displaystyle\Big{(}{p_{1}{\cdot}p_{3}\over
p_{13}{\cdot}v}\Big{)}\Big{(}-{p_{1}{\cdot}vW(13)p_{3}{\cdot}v\over
p_{1}{\cdot}v}\Big{)}\Big{(}v{\cdot}\varepsilon_{2}\Big{)}\,.$ (4.9)

The above expressions agree with the general factorisation behaviour (4.5).
Once again importantly the term $W(123)$ does not contribute in the
factorisation limit. The factorisation behaviour of six-point numerators can
be analysed in a similar fashion. The explicit expression for the six-point
numerator is given in (A) of the appendix A. The factorisation behaviour on
most of the heavy massive channels is similar. The only non-trivial one is on
the pole $1\over p_{1}{\cdot}v$, for which we find (for the terms with
$\alpha^{\prime}$ corrections)

$\displaystyle{p_{1}{\cdot}p_{2}v{\cdot}\varepsilon_{1}\over p_{1}{\cdot}v}$
$\displaystyle\Big{(}W(23)p_{3}{\cdot}F_{4}{\cdot}v-W(234)p_{23}{\cdot}v$
(4.10)
$\displaystyle-\frac{W(23)p_{3}{\cdot}vp_{23}{\cdot}F_{4}{\cdot}v}{p_{23}{\cdot}v}-\frac{W(34)p_{23}{\cdot}vv{\cdot}F_{2}{\cdot}p_{3}}{p_{2}{\cdot}v}-\frac{W(24)p_{23}{\cdot}vp_{2}{\cdot}F_{3}{\cdot}v}{p_{3}{\cdot}v}\Big{)}\,.$

The result again agrees with the factorisation behaviour (4.5). These examples
show that the evaluation maps (3.2) and (4.2) are sufficient to ensure the
factorisation behaviours even without knowing the explicit form of the
$W$-function, and the analysis further confirms that the function
$G_{\tau}(x)$ has the correct form.

###  4.2 Crossing symmetry

In the following we will constrain the function $W(i\sigma_{2}j)$. The general
properties of $W(i\sigma_{2}j)$ are

  * •

It is manifestly gauge invariant without any poles, therefore it should
composed of the product of the factor $\text{tr}(F_{\sigma})$ or
$p_{X}{\cdot}F_{\sigma}{\cdot}p_{Y}$

  * •

By power counting, it should have mass dimension $|\sigma_{2}|$, where
$|\sigma_{2}|$ denotes the number of the gluon labels in $\sigma_{2}$.

  * •

It is constrained by the full crossing symmetry of the BCJ numerators.

We will now analyse in details the constraints on $W(i\sigma_{2}j)$ due to the
crossing symmetry on the BCJ numerator, as in the case of no higher-derivative
corrections [38, 39]. The crossing symmetry results in a novel relation
between the pre-numerator and BCJ numerator

$\displaystyle\mathbb{L}(1\ldots n-2)\circ\mathcal{N}(1\ldots
n{-}2,v)=(n{-}2)\mathcal{N}(1\ldots n{-}2,v)\,,$ (4.11)

and we will refer this as the crossing symmetry relation.

At four points, the crossing symmetry relation requires that

$\displaystyle\mathbb{L}(1,2)\circ\mathcal{N}(12,v)=2\mathcal{N}(12,v)\,.$
(4.12)

From (4.6), this implies that $W(12)$ is a symmetric function under the
swapping of label $1$ and $2$. The relative coefficient between these terms
may be fixed using the factorisation property of the amplitude for the
massless poles, and we find the $W(12)$ can only be a linear combination of
the following two terms

$\displaystyle\alpha^{\prime}\text{tr}(F_{12}),$ $\displaystyle\alpha^{\prime
2}p_{12}^{2}\text{tr}(F_{12})\,,$ (4.13)

for the first two orders in the $\alpha^{\prime}$-expansion.111Other
possibilities such as $\alpha^{\prime
2}p_{2}{\cdot}F_{1}{\cdot}F_{2}{\cdot}p_{1}$ are not independent. For example,
$\alpha^{\prime 2}p_{2}{\cdot}F_{1}{\cdot}F_{2}{\cdot}p_{1}$ is proportional
to $\alpha^{\prime 2}p_{12}^{2}\text{tr}(F_{12})$, so we do not include it.
The coefficients of these terms may be fixed using the factorisation property
of the amplitude for the massless poles, which turn out to be simply $1$, so
we find

$\displaystyle W(12)=\alpha^{\prime}\text{tr}(F_{12})+\alpha^{\prime
2}p_{12}^{2}\text{tr}(F_{12})\,.$ (4.14)

At five points, the crossing symmetry relation is

$\displaystyle\mathbb{L}(1,2,3)\circ\mathcal{N}(123,v)$
$\displaystyle=\mathcal{N}(123,v)-\mathcal{N}(213,v)-\mathcal{N}(312,v)+\mathcal{N}(321,v)$
$\displaystyle=3\mathcal{N}(123,v)\,.$ (4.15)

After using the crossing symmetry property for the $W(i_{1}i_{2})$, the above
relation implies the following relation on $W(123)$,

$\displaystyle\mathbb{L}(1,2,3)\circ(W(123)p_{3}{\cdot}v)=3W(123)p_{3}{\cdot}v\,,$
(4.16)

under the on-shell condition $p_{123}{\cdot}v=0$. Using the explicit
definition of the operator $\mathbb{L}(1,2,3)$, we find (4.16) yields
following conditions

$\displaystyle 2W(123)+W(213)+W(321)=0\,,$ $\displaystyle W(312)+W(321)=0\,.$
(4.17)

From power counting as well as manifest gauge invariance, we find $W(123)$
should be consisted of the following terms,

$\displaystyle\alpha^{\prime}:$ $\displaystyle\qquad\text{tr}(F_{123})\,,$
(4.18) $\displaystyle\alpha^{\prime 2}:$ $\displaystyle\qquad
p_{1}{\cdot}p_{2}\text{tr}(F_{123})\,,\quad
p_{1}{\cdot}p_{3}\text{tr}(F_{123})\,,\quad
p_{2}{\cdot}p_{3}\text{tr}(F_{123})\,,$ $\displaystyle\qquad
p_{1}{\cdot}F_{3}{\cdot}p_{2}\text{tr}(F_{12})\,,\quad
p_{1}{\cdot}F_{2}{\cdot}p_{3}\text{tr}(F_{13})\,,\quad
p_{2}{\cdot}F_{1}{\cdot}p_{3}\text{tr}(F_{23})\,.$ (4.19)

We then find that there are only three solutions, which are consistent with
the crossing symmetric constraints (4.17),222We write terms in this way to
manifest the symmetries. For example, in fact
$\text{tr}(F_{213})=-\text{tr}(F_{123})$, so one may write
$\text{tr}(F_{123})-\text{tr}(F_{213})=2\,\text{tr}(F_{123})$.

$\displaystyle(1):~{}~{}$
$\displaystyle\text{tr}(F_{123})-\text{tr}(F_{213})\,,$ (4.20)
$\displaystyle(2):~{}~{}$ $\displaystyle
p_{123}^{2}(\text{tr}(F_{123})-\text{tr}(F_{213}))\,,$
$\displaystyle(3):~{}~{}$
$\displaystyle\text{tr}(F_{12})p_{2}{\cdot}F_{3}{\cdot}p_{1}+\text{tr}(F_{23})p_{3}{\cdot}F_{1}{\cdot}p_{2}+\text{tr}(F_{31})p_{1}{\cdot}F_{2}{\cdot}p_{3}$
$\displaystyle-\text{tr}(F_{13})p_{3}{\cdot}F_{2}{\cdot}p_{1}-\text{tr}(F_{21})p_{1}{\cdot}F_{3}{\cdot}p_{2}-\text{tr}(F_{32})p_{2}{\cdot}F_{1}{\cdot}p_{3}\,.$

Interestingly, all the three solutions have another extra common crossing
symmetry under reversing the indices $\\{1,2,3\\}\rightarrow\\{3,2,1\\}$. So
we also have

$\displaystyle W(123)=-W(321).$ (4.21)

Under this relation, the first relation in (4.17) is not an independent one
anymore and the independent relations are

$\displaystyle W(312)+W(321)=0\,,$ $\displaystyle W(123)=-W(321)\,.$ (4.22)

By further comparing the factorisations on massless poles (or even the
amplitude), we fix all the coefficients for the terms in (4.20). Once again,
we find all the coefficients are simply $1$, and so

$\displaystyle W(123)=\alpha^{\prime}\text{tr}(F_{[12]}F_{3})+\alpha^{\prime
2}\sum_{\rho\in[12]3}\sum_{\rho_{1}j_{2}\rho_{2}j_{1}={\rm
cyc}(\rho)}p_{j_{1}}{\cdot}F_{\rho_{1}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\rho_{2}]}F_{j_{1}})\,,$
(4.23)

where $F_{[12]}=F_{1}{\cdot}F_{2}-F_{2}{\cdot}F_{1}$. Note when $\rho_{1}$ is
an empty set, the above expression leads to the term $(2)$ in (4.20).

At six points, the crossing symmetric condition implies

$\displaystyle\mathbb{L}(1,2,3,4)\circ\Bigg{(}\Big{(}W(1234)-W(12)W(34)p_{2}{\cdot}p_{3}\Big{)}p_{4}{\cdot}v\Bigg{)}$
$\displaystyle=4\Big{(}W(1234)-W(12)W(34)p_{2}{\cdot}p_{3}\Big{)}p_{4}{\cdot}v\,,$
(4.24)

where we have used the on-shell condition $p_{1234}{\cdot}v=0$ and the
symmetry property of $W(i_{1}i_{2})$ and $W(i_{1}i_{2}i_{3})$. For
convenience, we define

$\displaystyle W^{\prime}(1234)=W(1234)-W(12)W(34)p_{2}{\cdot}p_{3}\,,$ (4.25)

where the second term, which we subtract, can be viewed as a product of lower-
point order-$\alpha^{\prime}$ terms. Then we have the relations

$\displaystyle W^{\prime}(4312)+W^{\prime}(4321)=0\,,$ (4.26) $\displaystyle
W^{\prime}(4123)-W^{\prime}(4213)-W^{\prime}(4321)=0\,,$ (4.27) $\displaystyle
3W^{\prime}(1234)+W^{\prime}(2134)+W^{\prime}(3124)-W^{\prime}(3214)-W^{\prime}(4321)=0\,.$
(4.28)

Below are the terms that are consistent with power counting and manifest gauge
invariance at six points: at the $\alpha^{\prime}$ order, we have,333We have
omitted double-trace terms (i.e. terms proportional to
$\text{tr}(F_{12})\text{tr}(F_{34})$) in writing down possible terms for
$W^{\prime}(1234)$, since they are already contained in $W(12)W(34)$, and
furthermore they are inconsistent with the relations (4.26).

$\displaystyle\text{tr}(F_{1234}),$ $\displaystyle\text{tr}(F_{2134}),$ (4.29)

and for the $\alpha^{\prime 2}$ order

$\displaystyle p_{i}{\cdot}p_{j}\text{tr}(F_{1234}),$ $\displaystyle
p_{i}{\cdot}p_{j}\text{tr}(F_{2134}),$ $\displaystyle
p_{i}{\cdot}F_{4}{\cdot}p_{j}\text{tr}(F_{123}),$ $\displaystyle
p_{i}{\cdot}F_{3}{\cdot}p_{j}\text{tr}(F_{124})\,,$ $\displaystyle
p_{i}{\cdot}F_{2}{\cdot}p_{j}\text{tr}(F_{134})\,,$ $\displaystyle
p_{i}{\cdot}F_{1}{\cdot}p_{j}\text{tr}(F_{234})\,,$ $\displaystyle
p_{i}{\cdot}F_{34}{\cdot}p_{j}\text{tr}(F_{12})\,,$ $\displaystyle
p_{i}{\cdot}F_{24}{\cdot}p_{j}\text{tr}(F_{13}),$ $\displaystyle
p_{i}{\cdot}F_{23}{\cdot}p_{j}\text{tr}(F_{14})\,,$ $\displaystyle
p_{i}{\cdot}F_{14}{\cdot}p_{j}\text{tr}(F_{23})\,,$ $\displaystyle
p_{i}{\cdot}F_{13}{\cdot}p_{j}\text{tr}(F_{24})\,,$ $\displaystyle
p_{i}{\cdot}F_{12}{\cdot}p_{j}\text{tr}(F_{34})\,.$ (4.30)

Then the solutions of (4.26) are

$\displaystyle(1):~{}$
$\displaystyle\text{tr}(F_{1234})-\text{tr}(F_{2134})-\text{tr}(F_{3124})+\text{tr}(F_{3214})\,,$
(4.31) $\displaystyle(2):~{}$ $\displaystyle
p_{1234}^{2}\text{tr}(F_{1234})-p_{1234}^{2}\text{tr}(F_{2134})-p_{1234}^{2}\text{tr}(F_{3124})+p_{1234}^{2}\text{tr}(F_{3214})\,,$
$\displaystyle(3):~{}$
$\displaystyle\text{tr}(F_{[12]4})p_{1}{\cdot}F_{3}{\cdot}p_{4}-\text{tr}(F_{[13]2})p_{1}{\cdot}F_{4}{\cdot}p_{2}+\text{tr}(F_{[13]4})p_{1}{\cdot}F_{2}{\cdot}p_{4}-\text{tr}(F_{[14]2})p_{1}{\cdot}F_{3}{\cdot}p_{2}$
$\displaystyle+\text{tr}(F_{[21]3})p_{2}{\cdot}F_{4}{\cdot}p_{3}+\text{tr}(F_{[23]1})p_{1}{\cdot}F_{4}{\cdot}p_{2}+\text{tr}(F_{[24]1})p_{1}{\cdot}F_{3}{\cdot}p_{2}+\text{tr}(F_{[24]3})p_{2}{\cdot}F_{1}{\cdot}p_{3}$
$\displaystyle-\text{tr}(F_{[31]2})p_{2}{\cdot}F_{4}{\cdot}p_{3}-\text{tr}(F_{[31]4})p_{3}{\cdot}F_{2}{\cdot}p_{4}-\text{tr}(F_{[32]4})p_{3}{\cdot}F_{1}{\cdot}p_{4}-\text{tr}(F_{[34]2})p_{2}{\cdot}F_{1}{\cdot}p_{3}$
$\displaystyle+\text{tr}(F_{[41]3})p_{3}{\cdot}F_{2}{\cdot}p_{4}-\text{tr}(F_{[42]1})p_{1}{\cdot}F_{3}{\cdot}p_{4}+\text{tr}(F_{[42]3})p_{3}{\cdot}F_{1}{\cdot}p_{4}-\text{tr}(F_{[43]1})p_{1}{\cdot}F_{2}{\cdot}p_{4}\,,$
$\displaystyle(4):~{}$
$\displaystyle\text{tr}(F_{12})p_{1}{\cdot}F_{43}{\cdot}p_{2}-\text{tr}(F_{13})p_{1}{\cdot}F_{24}{\cdot}p_{3}+\text{tr}(F_{13})p_{1}{\cdot}F_{42}{\cdot}p_{3}-\text{tr}(F_{14})p_{1}{\cdot}F_{23}{\cdot}p_{4}$
$\displaystyle-\text{tr}(F_{21})p_{1}{\cdot}F_{34}{\cdot}p_{2}-\text{tr}(F_{23})p_{2}{\cdot}F_{14}{\cdot}p_{3}-\text{tr}(F_{24})p_{2}{\cdot}F_{13}{\cdot}p_{4}+\text{tr}(F_{24})p_{2}{\cdot}F_{31}{\cdot}p_{4}$
$\displaystyle-\text{tr}(F_{31})p_{1}{\cdot}F_{24}{\cdot}p_{3}+\text{tr}(F_{31})p_{1}{\cdot}F_{42}{\cdot}p_{3}+\text{tr}(F_{32})p_{2}{\cdot}F_{41}{\cdot}p_{3}+\text{tr}(F_{34})p_{3}{\cdot}F_{21}{\cdot}p_{4}$
$\displaystyle+\text{tr}(F_{41})p_{1}{\cdot}F_{32}{\cdot}p_{4}-\text{tr}(F_{42})p_{2}{\cdot}F_{13}{\cdot}p_{4}+\text{tr}(F_{42})p_{2}{\cdot}F_{31}{\cdot}p_{4}-\text{tr}(F_{43})p_{3}{\cdot}F_{12}{\cdot}p_{4}\,,$
$\displaystyle\cdots~{}$ $\displaystyle\text{three other solutions}\cdots\,.$

The coefficients of these terms are fixed by massless poles (or directly the
amplitude). It turns out, the terms of $(1-4)$ that are shown in the above
have coefficient $1$, while “three other solutions" that we did not show
explicitly have vanishing coefficients. In conclusion, we find,

$\displaystyle W^{\prime}(1234)$
$\displaystyle=\alpha^{\prime}\text{tr}(F_{[123]}F_{4})+\alpha^{\prime
2}p_{1234}^{2}\text{tr}(F_{[123]}F_{4})$ $\displaystyle+\alpha^{\prime
2}\sum_{\rho=[123]4}\Big{(}p_{\rho_{4}}{\cdot}F_{\rho_{1}}{\cdot}p_{\rho_{2}}\text{tr}(F_{[\rho_{2}\rho_{3}]}F_{\rho_{4}})+p_{\rho_{1}}{\cdot}F_{\rho_{2}}{\cdot}p_{\rho_{3}}\text{tr}(F_{[\rho_{3}\rho_{4}]}F_{\rho_{1}})$
$\displaystyle+p_{\rho_{2}}{\cdot}F_{\rho_{3}}{\cdot}p_{\rho_{4}}\text{tr}(F_{[\rho_{4}\rho_{1}]}F_{\rho_{2}})+p_{\rho_{3}}{\cdot}F_{\rho_{4}}{\cdot}p_{\rho_{1}}\text{tr}(F_{[\rho_{1}\rho_{2}]}F_{\rho_{3}})\Big{)}$
$\displaystyle+\alpha^{\prime
2}\sum_{\rho=[123]4}\Big{(}p_{\rho_{4}}{\cdot}F_{\rho_{1}\rho_{2}}{\cdot}p_{\rho_{3}}\text{tr}(F_{\rho_{3}}F_{\rho_{4}})+p_{\rho_{1}}{\cdot}F_{\rho_{2}\rho_{3}}{\cdot}p_{\rho_{4}}\text{tr}(F_{\rho_{4}}F_{\rho_{1}})$
$\displaystyle+p_{\rho_{2}}{\cdot}F_{\rho_{3}\rho_{4}}{\cdot}p_{\rho_{1}}\text{tr}(F_{\rho_{1}}F_{\rho_{2}})+p_{\rho_{3}}{\cdot}F_{\rho_{4}\rho_{1}}{\cdot}p_{\rho_{2}}\text{tr}(F_{\rho_{2}}F_{\rho_{3}})\Big{)}\,,$
(4.32)

which can be recast nicely as

$\displaystyle
W^{\prime}(1234)=\alpha^{\prime}\text{tr}(F_{[123]}F_{4})+\alpha^{\prime
2}\sum_{\rho\in[123]4}\sum_{\rho_{1}j_{2}\rho_{2}j_{1}={\rm
cyc}(\rho)}p_{j_{1}}{\cdot}F_{\rho_{1}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\rho_{2}]}F_{j_{1}})\,,$
(4.33)

where $\sum_{\rho=[123]4}f(\rho)\equiv f(1234)-f(2134)-f(3124)+f(3214)$ and
$F_{[123]}\equiv F_{123}-F_{213}-F_{312}+F_{321}$.

Interestingly, from the explicit expression of $W^{\prime}(1234)$, we note it
obeys an extra relation by reversing the indices

$\displaystyle W^{\prime}(1234)-W^{\prime}(4321)=0\,.$ (4.34)

Combined with this additional relation, the relations in (4.26) for
$W^{\prime}$-function become

$\displaystyle W^{\prime}(4312)+W^{\prime}(4321)=0\,,$ $\displaystyle
W^{\prime}(4123)-W^{\prime}(4213)-W^{\prime}(4321)=0\,,$ $\displaystyle
W^{\prime}(1234)-W^{\prime}(4321)=0\,.$ (4.35)

From the $W^{\prime}$, we get the form of $W$ function

$\displaystyle W(1234)=W^{\prime}(1234)+W(12)p_{2}{\cdot}p_{3}W(34)\,.$ (4.36)

The first term can be understood as the primary contact terms which are
constrained from the simple crossing symmetry. The contact terms contain two
factors of $W$ functions arise from lower-point BCJ numerators, which can be
viewed as the descendant contact terms.

Let us now consider the general case. Up to the $\alpha^{\prime 2}$ order, we
find the $W(12\ldots r)$ function has the following structure

$\displaystyle W(12\ldots r)\Big{|}_{\alpha^{\prime
2}}=\Big{(}W^{\prime}(12\ldots r)+\\!\\!\sum_{2\leq i<j\leq r-1}W(12\ldots
i)p_{i}{\cdot}F_{i+1\ldots j-1}{\cdot}p_{j}W(j\ldots
r)\Big{)}\Big{|}_{\alpha^{\prime 2}}\,,$ (4.37)

where the contact terms with two $W$-functions are constructed to preserve the
ordering of gluons. The primary contact term (i.e. $W^{\prime}(12\ldots r)$)
satisfies the crossing symmetry

$\displaystyle\mathbb{L}(1,2,\ldots,r)\circ\Big{(}W^{\prime}(12\ldots
r)\,p_{r}{\cdot}v\Big{)}=r\,W^{\prime}(12\ldots r)\,p_{r}{\cdot}v\,,$ (4.38)

valid under the on-shell condition $p_{12\ldots r}{\cdot}v=0$. This leads to
the following symmetry properties of the $W^{\prime}$-function

$\displaystyle\sum_{\sigma\in[\rho]}W^{\prime}(\sigma
i_{r})-(-1)^{|\rho|}W^{\prime}(i_{r}\rho^{\rm
rev})=r\,W^{\prime}(i_{1}i_{2}\ldots i_{r})\,,$ (4.39)
$\displaystyle\sum_{\sigma\in[\rho]}W^{\prime}(i_{r}\ldots i_{r-j-1}\sigma
i_{r-j})-(-1)^{|\rho|}W^{\prime}(i_{r}\ldots i_{r-j-1}i_{r-j}\rho^{\rm
rev})=0\,,$ (4.40)

where $[\rho]$ denotes left-nested commutators, and $|\rho|$ is the size of
$\rho$ and $\rho^{\rm rev}$ denotes the reversing of $\rho$. The second
relation is valid for $j\in\\{2,3,\ldots,r-1\\}$.

Following the patterns of five- and six-point examples, given in (4.23) and
(4.33), respectively, it is straightforward to show that the following general
solution is consistent with (4.38),

$\displaystyle W^{\prime}(123\ldots r)=\alpha^{\prime}\text{tr}(F_{[1\ldots
r-1]}F_{r})+\alpha^{\prime 2}\sum_{\rho\in[1\ldots
r-1]r}\sum_{\rho_{1}j_{2}\rho_{2}j_{1}={\rm
cyc}(\rho)}p_{j_{1}}{\cdot}F_{\rho_{1}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\rho_{2}]}F_{j_{1}})\,.$
(4.41)

Once again, the explicit expression for $W^{\prime}(123\ldots r)$ given in
(4.41) has an extra reversing symmetry

$\displaystyle W^{\prime}(\rho)-(-1)^{|\rho|}W^{\prime}(\rho^{\rm rev})=0\,.$
(4.42)

Combining all the discussion above, we obtain a closed formula for the
$W$-function

$\displaystyle W(i\sigma
j)=\alpha^{\prime}\text{tr}(F_{[i\sigma]}F_{j})+\alpha^{\prime
2}\sum_{\rho\in[i\sigma]j}\sum_{\rho_{1}j_{2}\rho_{2}j_{1}={\rm
cyc}(\rho)}p_{j_{1}}{\cdot}F_{\rho_{1}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\rho_{2}]}F_{j_{1}})$
$\displaystyle+\alpha^{\prime
2}\sum_{\sigma_{1}j_{1}\sigma_{2}j_{2}\sigma_{3}=\sigma}\text{tr}(F_{[i\sigma_{1}]}F_{j_{1}})p_{j_{1}}{\cdot}F_{\sigma_{2}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\sigma_{3}]}F_{j})\,.$
(4.43)

We have further checked explicitly the correctness of the BCJ numerators up to
eight points against the amplitudes computed from the action (2.1) using
Berends-Giele recursion [51]. This completes our proposal of the kinematic
Hopf algebra for the BCJ numerators with higher-derivative corrections,
$\alpha^{\prime}F^{3}$ and $\alpha^{\prime 2}F^{4}$.

The leading correction (the term proportional to $\alpha^{\prime}$) is due to
the $F^{3}$ term in the action
$S_{\mathrm{YM}+\alpha^{\prime}F^{3}+\alpha^{\prime 2}F^{4}}$ as defined in
(2.1). By power counting and the gauge invariance condition, the linear
$\alpha^{\prime}$ term can only be the trace of the strength tensors. The
particular combination in the nested commutator is related to the full
crossing symmetry for the BCJ numerators. The subleading $\alpha^{\prime 2}$
corrections contain two distinct structures. This is indeed expected, since
this order receives contributions from the $\alpha^{\prime 2}F^{4}$ operator,
and also from an operator of the type $(\alpha^{\prime}F^{3})^{2}$.

###  4.3 Summary and pure gluon amplitudes

Here we briefly summarise our final results of constructing BCJ numerators
using the quasi-Hopf algebra:

$\displaystyle\mathcal{N}_{\mathrm{BCJ}}(1\alpha,v)=\mathcal{N}([1\alpha],v)\,,$
(4.44)

where $[\bullet]$ denotes left-nested commutators, and
$\mathcal{N}([1\alpha],v)$ is obtained through the evaluation map,

$\displaystyle\mathcal{N}(1\alpha,v)=\langle\widehat{\mathcal{N}}(1\alpha,v)\rangle\,.$
(4.45)

The algebraic numerator $\widehat{\mathcal{N}}$ is defined as,

$\displaystyle\widehat{\mathcal{N}}(12\ldots n{-}2,v)=T_{(1)}\star
T_{(2)}\cdots\star T_{(n{-}2)}\,,$ (4.46)

with fusion product rules given in (3.5). The evaluation map is defined as

$\displaystyle\langle
T_{(1\tau_{1}),(\tau_{2}),\ldots,(\tau_{r})}\rangle={1\over
n-2}{G_{1\tau_{1}}(v)\over
v{\cdot}p_{1}}{G_{\tau_{2}}(p_{\Theta(\tau_{2})})\over
v{\cdot}p_{1\tau_{1}}}\,\cdots{G_{\tau_{r}}(p_{\Theta(\tau_{r})})\over
v{\cdot}p_{1\tau_{1}\ldots\tau_{r-1}}}\,,$ (4.47)

where

$\displaystyle
G_{\tau}(x)=\Big{(}x{\cdot}F_{\tau}{\cdot}v+\sum_{\sigma_{1}i\sigma_{2}j\sigma_{3}=\tau}x{\cdot}F_{\sigma_{1}}{\cdot}p_{i}W(i\sigma_{2}j)p_{j}{\cdot}F_{\sigma_{3}}{\cdot}v\Big{)}\,,$
(4.48)

and the $W(i\sigma j)$ term gives $\alpha^{\prime}$ corrections and is given
by

$\displaystyle W(i\sigma
j)=\alpha^{\prime}\text{tr}(F_{[i\sigma]}F_{j})+\alpha^{\prime
2}\sum_{\rho\in[i\sigma]j}\sum_{\rho_{1}j_{2}\rho_{2}j_{1}={\rm
cyc}(\rho)}p_{j_{1}}{\cdot}F_{\rho_{1}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\rho_{2}]}F_{j_{1}})$
$\displaystyle+\alpha^{\prime
2}\sum_{\sigma_{1}j_{1}\sigma_{2}j_{2}\sigma_{3}=\sigma}\text{tr}(F_{[i\sigma_{1}]}F_{j_{1}})p_{j_{1}}{\cdot}F_{\sigma_{2}}{\cdot}p_{j_{2}}\text{tr}(F_{[j_{2}\sigma_{3}]}F_{j})\,.$
(4.49)

Finally, as argued in [38], by a simple factorisation limit as shown below,

$\bm{\times}$$1$$2\cdots$$~{}~{}n{-}2$ $\displaystyle\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a){$\longrightarrow$}; \diagram*{}; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}~{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman
\vertex(a){$n{-}1$}; \vertex[above=0.6cm of a](b)[dot]{}; \vertex[left=0.6cm
of b](c); \vertex[left=0.22cm of b](c23); \vertex[above=0.13cm of
c23](v23)[dot]{}; \vertex[above=.4cm of c](j1){$1$}; \vertex[right=.7cm of
j1](j2){$2\cdots$}; \vertex[right=0.5cm of j2](j3){$~{}~{}n{-}2$};
\diagram*{(a) -- [thick] (b),(b) -- [thick] (j1),(v23) -- [thick]
(j2),(b)--[thick](j3)}; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (4.50)

one can simply decouple the heavy-massive scalars and obtain the BCJ
numerators of pure gluons from $\mathcal{N}([1\alpha],v)$. Explicitly, the BCJ
numerators of $(n{-}1)$ gluons can be deduced from the numerators with scalars
as a limit,

$\displaystyle\mathcal{N}^{\rm gluon}([1\ldots n{-}1])$
$\displaystyle=\mathcal{N}([1\ldots
n{-}2],v)\big{|}^{v\rightarrow\epsilon_{n{-}1}}_{p_{1\ldots
n{-}2}^{2}\rightarrow 0}\,,$ (4.51)

where $p_{1\ldots n{-}2}^{2}\rightarrow 0$ is used to impose the on-shell
condition for the $(n{-}1)$-th gluon that arises through the factorisation
limit, as shown in (4.50).

##  5 Conclusions and outlook

We have demonstrated that higher-derivative operators that are compatible with
colour-kinematics also admit a Hopf algebra representation. To be specific, we
have considered higher-derivative operators of the form $\alpha^{\prime}F^{3}$
and $\alpha^{\prime 2}F^{4}$ that were first studied in [47] in the context of
colour-kinematics duality. Understanding the underlying algebraic structure
leads to a very compact formula for the BCJ numerators of any number of
particles. Our results strongly suggest that the Hopf algebra may in fact
underlie all instances of the colour-kinematic duality, which is known to
apply to a wide range of theories. Our construction also hints on the
possibility of observing the kinematic algebra at the Lagrangian level. In
particular, the structure of the evaluation maps suggests that there is a
direct connection between higher-derivative operators and terms in the
evaluation maps.

There are two other natural research directions we could investigate in the
future. The first one is to understand the evaluation maps for the theories
with higher-derivative operators beyond the $\alpha^{\prime 2}$ order.
Particularly, our ansatz for the $G$-function may be valid even in the higher
orders of $\alpha^{\prime}$, and the structure of the $W$-function is very
suggestive hinting on possible higher-order generalisations. Such
generalisations could then complement studies on higher derivative operators
from positivity conditions on EFT’s that satisfy double copy (such as
gravity), or monodromy relations (in the case of string theory) [58, 59, 60,
61, 62, 63, 64, 65]. It would be fascinating if the Hopf algebra reveals any
further structure or relations between these operators. We may also obtain BCJ
numerators of other theories, in particular for scalars, from our results
using the transmutation procedure [52] and compare with other recent bootstrap
approaches (e.g. [66, 67, 68]).

The second one is to generalise the construction beyond purely kinematic
numerators following [69, 70]. By allowing colour factors (for example, colour
traces) to non-trivially combine with kinematic factors, it was found that
that all operators present in the low energy expansion of string theory can be
written in colour-kinematic dual form. It would be interesting to see if the
kinematic Hopf algebra is still present in this case, and to understand the
modification of the evaluation maps to incorporate more general colour
structures. If this is possible, it would allow re-writing the complete string
EFT in terms of this novel algebra. Similarly, one can attempt to modify the
Hopf algebra even beyond adjoint type numerators, as the colour-kinematic
duality is not necessarily tied to adjoint type constructions, as recently
explored in [71].

The BCJ numerators encoding higher dimensional corrections to Yang-Mills can
be double-copied to obtain corresponding corrections to gravity [47, 56],
which could be relevant to black hole scattering [57, 72]. Finally, given that
the close connection between the Hopf algebra and the geometrical perspective
described in [46], it is interesting to see if a similar permutohedra
construction also exists for the higher-derivative operators discussed in this
paper.

##  6 Acknowledgements

We would like to thank Andreas Brandhuber and Gabriele Travaglini for
stimulating discussions. GC has received funding from the European Union’s
Horizon 2020 research and innovation program under the Marie Skłodowska-Curie
grant agreement No. 847523 “INTERACTIONS”. LR is supported by the European
Union Horizon 2020 Marie Skłodowska-Curie Individual Fellowship, Grant No.
101025095 “BUFFS”. CW is supported by a Royal Society University Research
Fellowship URF\R\221015 and a STFC Consolidated Grant, ST\T000686\1
“Amplitudes, strings & duality".

##  Appendix A Six-point pre-numerator

At six points, the pre-numerator is given by

$\displaystyle\mathcal{N}(1234,v)={1\over 4}\Big{[}$
$\displaystyle-\frac{p_{1}{\cdot}F_{2}{\cdot}vp_{132}{\cdot}F_{4}{\cdot}v\left(v{\cdot}F_{13}{\cdot}v+p_{1}{\cdot}vp_{3}{\cdot}vW(13)\right)}{p_{1}{\cdot}vp_{13}{\cdot}vp_{132}{\cdot}v}-\frac{p_{1}{\cdot}F_{2}{\cdot}vp_{13}{\cdot}F_{4}{\cdot}v\left(v{\cdot}F_{13}{\cdot}v+p_{1}{\cdot}vp_{3}{\cdot}vW(13)\right)}{p_{1}{\cdot}vp_{13}{\cdot}vp_{134}{\cdot}v}$
$\displaystyle-\frac{p_{1}{\cdot}F_{2}{\cdot}vp_{12}{\cdot}F_{3}{\cdot}v\left(v{\cdot}F_{14}{\cdot}v+p_{1}{\cdot}vp_{4}{\cdot}vW(14)\right)}{p_{1}{\cdot}vp_{14}{\cdot}vp_{142}{\cdot}v}-\frac{p_{1}{\cdot}F_{2}{\cdot}vp_{1}{\cdot}F_{3}{\cdot}v\left(v{\cdot}F_{14}{\cdot}v+p_{1}{\cdot}vp_{4}{\cdot}vW(14)\right)}{p_{1}{\cdot}vp_{14}{\cdot}vp_{143}{\cdot}v}$
$\displaystyle-\frac{p_{12}{\cdot}F_{3}{\cdot}vp_{123}{\cdot}F_{4}{\cdot}v\left(v{\cdot}F_{12}{\cdot}v+p_{1}{\cdot}vp_{2}{\cdot}vW(12)\right)}{p_{1}{\cdot}vp_{12}{\cdot}vp_{123}{\cdot}v}-\frac{p_{12}{\cdot}F_{3}{\cdot}vp_{12}{\cdot}F_{4}{\cdot}v\left(v{\cdot}F_{12}{\cdot}v+p_{1}{\cdot}vp_{2}{\cdot}vW(12)\right)}{p_{1}{\cdot}vp_{12}{\cdot}vp_{124}{\cdot}v}$
$\displaystyle+\frac{\left(p_{12}{\cdot}F_{34}{\cdot}v+p_{3}{\cdot}p_{12}p_{4}{\cdot}vW(34)\right)\left(v{\cdot}F_{12}{\cdot}v+p_{1}{\cdot}vp_{2}{\cdot}vW(12)\right)}{p_{1}{\cdot}vp_{12}{\cdot}v}$
$\displaystyle+\frac{\left(v{\cdot}F_{14}{\cdot}v+p_{1}{\cdot}vp_{4}{\cdot}vW(14)\right)\left(p_{1}{\cdot}F_{23}{\cdot}v+p_{1}{\cdot}p_{2}p_{3}{\cdot}vW(23)\right)}{p_{1}{\cdot}vp_{14}{\cdot}v}$
$\displaystyle+\frac{\left(v{\cdot}F_{13}{\cdot}v+p_{1}{\cdot}vp_{3}{\cdot}vW(13)\right)\left(p_{1}{\cdot}F_{24}{\cdot}v+p_{1}{\cdot}p_{2}p_{4}{\cdot}vW(24)\right)}{p_{1}{\cdot}vp_{13}{\cdot}v}$
$\displaystyle+\frac{p_{123}{\cdot}F_{4}{\cdot}v\left(v{\cdot}F_{123}{\cdot}v+p_{1}{\cdot}vp_{2}{\cdot}F_{3}{\cdot}vW(12)+p_{3}{\cdot}vv{\cdot}F_{1}{\cdot}p_{2}W(23)+p_{1}{\cdot}vp_{3}{\cdot}vW(123)\right)}{p_{1}{\cdot}vp_{123}{\cdot}v}$
$\displaystyle+\frac{p_{12}{\cdot}F_{3}{\cdot}v\left(v{\cdot}F_{124}{\cdot}v+p_{1}{\cdot}vp_{2}{\cdot}F_{4}{\cdot}vW(12)+p_{4}{\cdot}vv{\cdot}F_{1}{\cdot}p_{2}W(24)+p_{1}{\cdot}vp_{4}{\cdot}vW(124)\right)}{p_{1}{\cdot}vp_{124}{\cdot}v}$
$\displaystyle+\frac{p_{1}{\cdot}F_{2}{\cdot}v\left(v{\cdot}F_{134}{\cdot}v+p_{1}{\cdot}vp_{3}{\cdot}F_{4}{\cdot}vW(13)+p_{4}{\cdot}vv{\cdot}F_{1}{\cdot}p_{3}W(34)+p_{1}{\cdot}vp_{4}{\cdot}vW(134)\right)}{p_{1}{\cdot}vp_{134}{\cdot}v}$
$\displaystyle-\frac{1}{p_{1}{\cdot}v}\Big{(}v{\cdot}F_{1234}{\cdot}v+p_{1}{\cdot}vp_{2}{\cdot}F_{34}{\cdot}vW(12)+v{\cdot}F_{1}{\cdot}p_{2}p_{3}{\cdot}F_{4}{\cdot}vW(23)+p_{4}{\cdot}vv{\cdot}F_{12}{\cdot}p_{3}W(34)$
$\displaystyle+p_{1}{\cdot}vp_{3}{\cdot}F_{4}{\cdot}vW(123)+p_{4}{\cdot}vv{\cdot}F_{1}{\cdot}p_{2}W(234)+p_{1}{\cdot}vp_{4}{\cdot}vW(1234)\Big{)}\Big{]}\,.$

One can generate more examples of the BCJ numerators and explicit form of the
$W$ functions in kinematicHopfAlgebra GitHub repository [48].

## References

  * [1] Z. Bern, J.J.M. Carrasco and H. Johansson, _New Relations for Gauge-Theory Amplitudes_ , _Phys. Rev._ D78 (2008) 085011 [0805.3993].
  * [2] Z. Bern, J.J.M. Carrasco and H. Johansson, _Perturbative Quantum Gravity as a Double Copy of Gauge Theory_ , _Phys. Rev. Lett._ 105 (2010) 061602 [1004.0476].
  * [3] Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, _The Duality Between Color and Kinematics and its Applications_ , 1909.01358.
  * [4] Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, _The SAGEX review on scattering amplitudes Chapter 2: An invitation to color-kinematics duality and the double copy_ , _J. Phys. A_ 55 (2022) 443003 [2203.13013].
  * [5] R. Monteiro and D. O’Connell, _The Kinematic Algebra From the Self-Dual Sector_ , _JHEP_ 07 (2011) 007 [1105.2565].
  * [6] M. Ben-Shahar and H. Johansson, _Off-shell color-kinematics duality for Chern-Simons_ , _JHEP_ 08 (2022) 035 [2112.11452].
  * [7] C. Cheung and J. Mangan, _Covariant color-kinematics duality_ , _JHEP_ 11 (2021) 069 [2108.02276].
  * [8] R. Monteiro, R. Stark-Muchão and S. Wikeley, _Anomaly and double copy in quantum self-dual Yang-Mills and gravity_ , _JHEP_ 09 (2023) 030 [2211.12407].
  * [9] R. Monteiro, _Celestial chiral algebras, colour-kinematics duality and integrability_ , _JHEP_ 01 (2023) 092 [2208.11179].
  * [10] A. Lipstein and S. Nagy, _Self-Dual Gravity and Color-Kinematics Duality in AdS4_ , _Phys. Rev. Lett._ 131 (2023) 081501 [2304.07141].
  * [11] C. Cheung and C.-H. Shen, _Symmetry for Flavor-Kinematics Duality from an Action_ , _Phys. Rev. Lett._ 118 (2017) 121601 [1612.00868].
  * [12] P. Ferrero and D. Francia, _On the Lagrangian formulation of the double copy to cubic order_ , _JHEP_ 02 (2021) 213 [2012.00713].
  * [13] M. Ben-Shahar, L. Garozzo and H. Johansson, _Lagrangians manifesting color-kinematics duality in the NMHV sector of Yang-Mills_ , _JHEP_ 08 (2023) 222 [2301.00233].
  * [14] C.R. Mafra and O. Schlotterer, _Tree-level amplitudes from the pure spinor superstring_ , _Phys. Rept._ 1020 (2023) 1 [2210.14241].
  * [15] C.-H. Fu, P. Vanhove and Y. Wang, _A Vertex Operator Algebra Construction of the Colour-Kinematics Dual numerator_ , _JHEP_ 09 (2018) 141 [1806.09584].
  * [16] M. Ben-Shahar and M. Guillen, _10D super-Yang-Mills scattering amplitudes from its pure spinor action_ , _JHEP_ 12 (2021) 014 [2108.11708].
  * [17] L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Saemann and M. Wolf, _Becchi-Rouet-Stora-Tyutin-Lagrangian Double Copy of Yang-Mills Theory_ , _Phys. Rev. Lett._ 126 (2021) 191601 [2007.13803].
  * [18] L. Borsten and S. Nagy, _The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order_ , _JHEP_ 07 (2020) 093 [2004.14945].
  * [19] R. Bonezzi, F. Diaz-Jaramillo and S. Nagy, _Gauge independent kinematic algebra of self-dual Yang-Mills theory_ , _Phys. Rev. D_ 108 (2023) 065007 [2306.08558].
  * [20] G. Chen, H. Johansson, F. Teng and T. Wang, _On the kinematic algebra for BCJ numerators beyond the MHV sector_ , _JHEP_ 11 (2019) 055 [1906.10683].
  * [21] G. Chen, H. Johansson, F. Teng and T. Wang, _Next-to-MHV Yang-Mills kinematic algebra_ , _JHEP_ 10 (2021) 042 [2104.12726].
  * [22] A. Brandhuber, G. Chen, G. Travaglini and C. Wen, _A new gauge-invariant double copy for heavy-mass effective theory_ , _JHEP_ 07 (2021) 047 [2104.11206].
  * [23] H. Georgi, _An Effective Field Theory for Heavy Quarks at Low-energies_ , _Phys. Lett. B_ 240 (1990) 447.
  * [24] M.E. Luke and A.V. Manohar, _Reparametrization invariance constraints on heavy particle effective field theories_ , _Phys. Lett. B_ 286 (1992) 348 [hep-ph/9205228].
  * [25] M. Neubert, _Heavy quark symmetry_ , _Phys. Rept._ 245 (1994) 259 [hep-ph/9306320].
  * [26] A.V. Manohar and M.B. Wise, _Heavy quark physics_ , vol. 10, Cambridge University Press (2000).
  * [27] P.H. Damgaard, K. Haddad and A. Helset, _Heavy Black Hole Effective Theory_ , _JHEP_ 11 (2019) 070 [1908.10308].
  * [28] A. Brandhuber, G. Chen, G. Travaglini and C. Wen, _Classical gravitational scattering from a gauge-invariant double copy_ , _JHEP_ 10 (2021) 118 [2108.04216].
  * [29] A. Brandhuber, G.R. Brown, G. Chen, S. De Angelis, J. Gowdy and G. Travaglini, _One-loop gravitational bremsstrahlung and waveforms from a heavy-mass effective field theory_ , _JHEP_ 06 (2023) 048 [2303.06111].
  * [30] N.E.J. Bjerrum-Bohr, G. Chen and M. Skowronek, _Covariant Compton Amplitudes in Gravity with Classical Spin_ , 2309.11249.
  * [31] N.E.J. Bjerrum-Bohr, G. Chen and M. Skowronek, _Classical spin gravitational Compton scattering_ , _JHEP_ 06 (2023) 170 [2302.00498].
  * [32] A. Brandhuber, G.R. Brown, G. Chen, J. Gowdy and G. Travaglini, _Resummed spinning waveforms from five-point amplitudes_ , 2310.04405.
  * [33] A. Herderschee, R. Roiban and F. Teng, _The sub-leading scattering waveform from amplitudes_ , _JHEP_ 06 (2023) 004 [2303.06112].
  * [34] K. Haddad, _Recursion in the classical limit and the neutron-star Compton amplitude_ , 2303.02624.
  * [35] R. Aoude, K. Haddad and A. Helset, _On-shell heavy particle effective theories_ , _JHEP_ 05 (2020) 051 [2001.09164].
  * [36] R. Aoude, K. Haddad and A. Helset, _Classical gravitational scattering at $\mathcal{O}(G^{2}S_{1}^{\infty}S_{2}^{\infty})$_, 2304.13740.
  * [37] T. Adamo and R. Gonzo, _Bethe-Salpeter equation for classical gravitational bound states_ , _JHEP_ 05 (2023) 088 [2212.13269].
  * [38] A. Brandhuber, G. Chen, H. Johansson, G. Travaglini and C. Wen, _Kinematic Hopf Algebra for Bern-Carrasco-Johansson Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory_ , _Phys. Rev. Lett._ 128 (2022) 121601 [2111.15649].
  * [39] A. Brandhuber, G.R. Brown, G. Chen, J. Gowdy, G. Travaglini and C. Wen, _Amplitudes, Hopf algebras and the colour-kinematics duality_ , _JHEP_ 12 (2022) 101 [2208.05886].
  * [40] M.E. Hoffman, _Quasi-shuffle products_ , _Journal of Algebraic Combinatorics_ 11 (2000) 49.
  * [41] J. Blumlein, _Algebraic relations between harmonic sums and associated quantities_ , _Comput. Phys. Commun._ 159 (2004) 19 [hep-ph/0311046].
  * [42] M. Aguiar and S.A. Mahajan, _Monoidal functors, species and Hopf algebras_ , vol. 29, American Mathematical Society Providence, RI (2010).
  * [43] M.E. Hoffman and K. Ihara, _Quasi-shuffle products revisited_ , _Journal of Algebra_ 481 (2017) 293–326.
  * [44] F. Fauvet, L. Foissy and D. Manchon, _The Hopf algebra of finite topologies and mould composition_, in _Annales de l’Institut Fourier_ , vol. 67, pp. 911–945, 2017.
  * [45] G. Chen, G. Lin and C. Wen, _Kinematic Hopf algebra for amplitudes and form factors_ , _Phys. Rev. D_ 107 (2023) L081701 [2208.05519].
  * [46] Q. Cao, J. Dong, S. He and Y.-Q. Zhang, _Covariant color-kinematics duality, Hopf algebras, and permutohedra_ , _Phys. Rev. D_ 107 (2023) 026022 [2211.05404].
  * [47] J. Broedel and L.J. Dixon, _Color-kinematics duality and double-copy construction for amplitudes from higher-dimension operators_ , _JHEP_ 10 (2012) 091 [1208.0876].
  * [48] G. Chen, _https://github.com/amplitudegravity/kinematichopfalgebra_ , 2022\.
  * [49] J.J.M. Carrasco, M. Lewandowski and N.H. Pavao, _Color-Dual Fates of F3, R3, and N=4 Supergravity_ , _Phys. Rev. Lett._ 131 (2023) 051601 [2203.03592].
  * [50] F.A. Berends and W.T. Giele, _Recursive Calculations for Processes with n Gluons_ , _Nucl. Phys._ B306 (1988) 759.
  * [51] L.M. Garozzo, L. Queimada and O. Schlotterer, _Berends-Giele currents in Bern-Carrasco-Johansson gauge for $F^{3}$\- and $F^{4}$-deformed Yang-Mills amplitudes_, _JHEP_ 02 (2019) 078 [1809.08103].
  * [52] C. Cheung, C.-H. Shen and C. Wen, _Unifying Relations for Scattering Amplitudes_ , _JHEP_ 02 (2018) 095 [1705.03025].
  * [53] M. Chiodaroli, M. Gunaydin, H. Johansson and R. Roiban, _Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies_ , _JHEP_ 06 (2017) 064 [1511.01740].
  * [54] H. Kawai, D.C. Lewellen and S.H.H. Tye, _A Relation Between Tree Amplitudes of Closed and Open Strings_ , _Nucl. Phys._ B269 (1986) 1.
  * [55] N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, _The Momentum Kernel of Gauge and Gravity Theories_ , _JHEP_ 01 (2011) 001 [1010.3933].
  * [56] S. He and Y. Zhang, _New Formulas for Amplitudes from Higher-Dimensional Operators_ , _JHEP_ 02 (2017) 019 [1608.08448].
  * [57] A. Brandhuber and G. Travaglini, _On higher-derivative effects on the gravitational potential and particle bending_ , _JHEP_ 01 (2020) 010 [1905.05657].
  * [58] X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, _Causality Constraints on Corrections to the Graviton Three-Point Coupling_ , _JHEP_ 02 (2016) 020 [1407.5597].
  * [59] Y.-t. Huang, J.-Y. Liu, L. Rodina and Y. Wang, _Carving out the Space of Open-String S-matrix_ , _JHEP_ 04 (2021) 195 [2008.02293].
  * [60] Z. Bern, D. Kosmopoulos and A. Zhiboedov, _Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude_ , _J. Phys. A_ 54 (2021) 344002 [2103.12728].
  * [61] S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, _Causality constraints on corrections to Einstein gravity_ , _JHEP_ 05 (2023) 122 [2201.06602].
  * [62] L.-Y. Chiang, Y.-t. Huang, W. Li, L. Rodina and H.-C. Weng, _(Non)-projective bounds on gravitational EFT_ , 2201.07177.
  * [63] L.-Y. Chiang, Y.-t. Huang and H.-C. Weng, _Bootstrapping string theory EFT_ , 2310.10710.
  * [64] J. Berman, H. Elvang and A. Herderschee, _Flattening of the EFT-Hedron: Supersymmetric Positivity Bounds and the Search for String Theory_ , 2310.10729.
  * [65] J.J.M. Carrasco and N.H. Pavao, _UV Massive Resonance from IR Double Copy Consistency_ , 2310.06316.
  * [66] A.S.-K. Chen, H. Elvang and A. Herderschee, _Emergence of String Monodromy in Effective Field Theory_ , 2212.13998.
  * [67] A.S.-K. Chen, H. Elvang and A. Herderschee, _Bootstrapping the String Kawai-Lewellen-Tye Kernel_ , _Phys. Rev. Lett._ 131 (2023) 031602 [2302.04895].
  * [68] T.V. Brown, K. Kampf, U. Oktem, S. Paranjape and J. Trnka, _Scalar BCJ Bootstrap_ , 2305.05688.
  * [69] J.J.M. Carrasco, L. Rodina, Z. Yin and S. Zekioglu, _Simple encoding of higher derivative gauge and gravity counterterms_ , _Phys. Rev. Lett._ 125 (2020) 251602 [1910.12850].
  * [70] J.J.M. Carrasco, L. Rodina and S. Zekioglu, _Composing effective prediction at five points_ , _JHEP_ 06 (2021) 169 [2104.08370].
  * [71] J.J.M. Carrasco and N.H. Pavao, _Virtues of a symmetric-structure double copy_ , _Phys. Rev. D_ 107 (2023) 065005 [2211.04431].
  * [72] M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, _Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity_ , _Phys. Rev. D_ 102 (2020) 046014 [2006.02375].


11institutetext: Y. Zou 22institutetext: Department of Physics, East China
Normal University, Shanghai 200062, China 33institutetext: Y. Zou
44institutetext: J. Kurths 55institutetext: Potsdam Institute for Climate
Impact Research, P. O. Box 60 12 03, 14412 Potsdam, Germany 66institutetext:
E. E. N. Macau 77institutetext: A. Mário 88institutetext: Instituto Nacional
de Pesquisas Espaciais, São José dos Campos, São Paulo, Brazil
99institutetext: G. Sampaio 1010institutetext: Instituto Nacional de Pesquisas
Espaciais, Cachoeira Paulista, São Paulo, Brazil 1111institutetext: J. Kurths
1212institutetext: Department of Physics, Humboldt University Berlin,
Newtonstraße 15, 12489 Berlin, Germany 1313institutetext: Institute for
Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen
AB243UE, United Kingdom 1414institutetext: Department of Control Theory,
Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod,
Russia

# Do the recent severe droughts in the Amazonia have the same period of
length?

Yong Zou  Elbert E. N. Macau  Gilvan Sampaio  Antônio Mário  Jürgen Kurths

###### Abstract

We propose a new measure based on drought period length to assess the temporal
difference between the recent two severe droughts of 2005 and 2010 in the
Amazonia. The sensitivity of the measure is demonstrated by disclosing the
distinct spatial responding mechanisms of the Northeastern and Southwestern
Amazon (NA, SA) to the surrounding sea surface temperature (SST)
variabilities. The Pacific and Atlantic oceans have different roles on the
precipitation patterns in Amazonia. More specifically, the very dry periods in
the NA are influenced by El Niño events, while the very dry periods in the SA
are affected by the anomalously warming of the SST in the North Atlantic. We
show convincingly that the drought 2005 hit SA, which is caused by the North
Atlantic only. There are two phases in the drought 2010: (i) it was started in
the NA in August 2009 affected by the El Niño event, and (ii) later shifted
the center of action to SA resulted from anomalously high SST in North
Atlantic, which further intensifies the impacts on the spatial coverage.

###### Keywords:

Amazonia droughts; Drougth period length; SST; Tropical Atlantic; ENSO

††journal: Climate Dynamics

##  1 Introduction

In Amazonia there remains the largest contiguous tropical forest of the
planet. The vegetation, including its deep root system, is efficient in
recycling water vapor, which is an important mechanism for the forest s
maintenance and helps to maintain evergreen canopies during dry seasons
(Nepstad et al, 1994; Cox et al, 2008), demonstrating the adaptation of Amazon
forest species to seasonal drought (Davidson et al, 2012). However, multi-year
or extreme droughts can affect this adaptation to seasonal drought and
eventually leading to mortality when roots are unable to extract enough soil
water during multi-year droughts (Fisher et al, 2006; Davidson et al, 2012).
Understanding drought events in Amazonia has fundamental importance especially
because the droughts may result in increased length of the dry season and
become more frequent during this century as a result of anthropogenic climate
change (Salazar et al, 2007; Saatchi et al, 2013; Phillips et al, 2009)).

The sea surface temperature (SST) variabilities in both the tropical Pacific
and Atlantic Oceans play important roles in forming the climate conditions in
the inter-annual rainfall variations over the Amazon (Aceituno, 1988; Marengo,
1992; Ronchail et al, 2002). Rainfall reductions and droughts in Amazonia have
been associated with El Niño Southern Oscillation (ENSO) events, or with
anomalous warming of SSTs in the tropical North Atlantic during the austral
winter-spring, or both (Richey et al, 1989; Uvo et al, 1998; de Souza et al,
2005; Marengo et al, 2011; Coelho et al, 2012). In 2005, the drought affected
the southern two-thirds of Amazonia and especially the southwest through
reduced precipitation as well as higher-than-average temperatures, and this
event was driven by elevated tropical North Atlantic SSTs (Marengo et al,
2008; Phillips et al, 2009). On the other hand, the drought event observed in
2010 was mainly caused by large-scale atmospheric circulation pattern forced
by tropical SST anomalies in the equatorial Pacific and the drought affected
more than half of the basin, resulting in the lowest discharge ever recorded
at Manaus (Marengo et al, 2011; Lewis et al, 2011; Xu et al, 2011).

In this study we characterize the main statistical differences between the
recent two severe droughts of 2005 and 2010 in the Amazonia by deriving
drought period lengths, which are equivalent to analyzing waiting times in
nonlinear time series (Zou et al, 2014). Therefore, our analysis essentially
provides some nonlinear perspectives on the underlying process.

##  2 Data and methods

Precipitation is evaluated using the Princeton Global Forcings dataset at
$0.5^{\circ}$ resolution (Sheffield et al, 2006). This dataset blends surface
and satellite observations with reanalysis and is available for 1948-2010. Sea
Surface Temperature (SST) is evaluated using NOAA High Resolution SST data
products (Reynolds et al, 2007). We subdivide the Amazon domain into
Northeastern Amazonia (NA: $65^{\circ}W-48^{\circ}W$,
$5^{\circ}N-10^{\circ}S$) and Southwestern Amazonia (SA:
$75^{\circ}W-50^{\circ}W$, $15^{\circ}S-4^{\circ}S$) regions. Time series for
precipitation of daily resolution were built for these two regions, while time
series for SST, are respectively, obtained by area averaging over El Niño3.4,
North Atlantic (n-Atlan) and South Atlantic (s-Atlan) domains, according to
the definitions of (Yoon and Zeng, 2009). Note that the following results do
not change if the El Niño3 area is used instead.

Daily precipitation anomalies for the NA and SA are calculated relative to a
base period of 1961-2000. This 40-year base period is chosen as it is
representative of the record of the 20th century. From the daily rainfall
anomaly series $A(t)$, we propose to calculate a drought period length
$DPL(t)$ that characterizes the waiting time of the present day to have the
next first non-negative rainfall anomaly. This captures the waiting time when
anomaly series goes from negative to positive, characterizing the expectation
to have a positive rain anomaly. More specifically,
$DPL(t)=\tau,\tau=\min\\{\tau:A(t+\tau)\geq 0,\tau\in[0,\infty)\\}$.

##  3 Annual variabilities of precipitation

We calculate $DPL(t)$ for the precipitation anomalies from NA and SA regions.
As shown in Figs. 1, $DPL(t)$ captures the starting dates of the recent two
severe droughts convincingly, in particular, 22 June 2005 and, respectively, 1
August 2009, as highlighted in Fig. 1.

Figure 1: Annual $DPL$. The drought periods longer than $20$ days are
highlighted, especially the extreme event starting from the 173rd day of 2005
(22 June 2005), and the event from the 213th day of 2009 (1 August 2009). (A)
Northeastern Amazon, (B) Southwestern Amazon.

Based on $DPL(t)$, in the following, we identify a date as mild dry, dry, and
very dry condition if $DPL\leq 1/3\max{DPL(t)}$, $1/3\max{DPL(t)}<DPL\leq
2/3\max{DPL(t)}$, and $DPL>2/3\max{DPL(t)}$, where $t\in[1,365(6)]$. With
respect to the climatological average period of 40 years, we find that
$\max{DPL(t)}=42$ days (shown in Fig. 1), which yields that the mild dry days
correspond to drought periods less than 2 weeks (14 days), while the drought
periods for very dry days are typically longer than one month (29 days) of
negative anomalies continuously. Furthermore, it is reasonable to regard mild
dry conditions as for normal years, and very dry days as extreme events.

For both NA and SA, the dry periods vary considerably over annual scales.
There are two ways to characterize the difference in the annual variabilities
of the drought period lengths: (i) temporal variations of $DPL(t)$, and (ii)
frequency plots for $DPL$ (shown in Figs. 1, 2, respectively).

Figure 2: Histograms of $DPL$ in each year.

The results are summarized as follow:

  1. 1.

Years 2004, 2006, 2007, and 2008 show no signs for very dry periods in either
NA nor SA since all dry periods are less than 20 days. Only mild dry and dry
days are observed in these years. Therefore, we term them as normal years of
dry conditions.

  2. 2.

In 2005: In NA, no signature for very dry period longer than 20 days is found.
However, in SA, there is an event of very dry period starting from the 173rd
day (22 June) (Fig. 1), reaching over 40 days of negative precipitation
anomalies. About 60 days of rainfall less than 1mm/day in SA has been observed
(similar statistics based on precipitation anomalies, not shown), which leads
to the very dry period over 40 days of continuous negative rainfall anomalies
as shown in Fig. 2.

  3. 3.

In 2009: In NA, there is an event of dry period starting from the 213th day (1
August), reaching over 40 days of negative precipitation anomaly (Fig. 1).
From the beginning of the year till this big event, NA has experienced much
floods of rainfall (very small $DPL$ values). In contrast, there is plenty of
rain in SA throughout the year since the dry period is much less than 10 days
(Fig. 2).

  4. 4.

In 2010: In NA, no signature for very dry period longer than one month is
found. Instead, we observe several dry periods longer than 2 weeks (but less
than 20 days) in the first half year. In SA, however, starting from the 213th
day (1 August), there are at least two big events of very dry periods reaching
over 35 days of no rains (Fig. 2).

From the daily variation of $DPL$, we infer the crucial difference between the
droughts 2005 and 2010 as follows. The drought 2005 affected mostly the SA
area over 40 days of continuous negative rain anomalies, leading to very dry
periods once-in-a-century event (Marengo et al, 2008). It has less serious
effects on NA, resulting in dry periods less than 20 days.

The drought 2010 is rather special in the following sense. It may started as
early as in the beginning of August 2009, hitting NA of very dry period over
40 days of negative rain anomalies, which forms the first phase of this
drought event and affects NA only. This initial dry condition in 2009
essentially extends to the first half (about 160th day) of 2010. Therefore, we
conclude that the drought 2010 in Amazonia started as early as in the
beginning of August 2009\. This detected starting time for the drought was
earlier as was previously reported by the austral summer. Another important
point is that, when this drought started in NA, there is plenty of rainfall in
SA since $DPL$ is much less than 10 days. The extension of the dry condition
to August 2010 forms the second phase of the drought, affecting mainly in SA.
Between these two phases, both NA and SA have experienced dry periods less
than 20 days. In consequence, the whole Amazonian basin is severely affected
by this basin-wide drought, which leads to a much larger spatial and temporal
coverage of rainfall shortage than the year 2005.

##  4 Correlations to surrounding oceanic SST

Now, we investigate the relationship between the surrounding oceanic SST
anomalies and the Amazonian rainfall. There are some hypothesis regarding to a
possible causality of the respective droughts in 2005 and 2010. The drought
2005 is not linked to ENSO, but to the Atlantic Ocean, while the drought of
2010 started in early austral summer during El Niño and then was intensified
as the warming of the tropical North Atlantic. We show these intricate
correlations by the observed time series and disclose the distinct roles of
ENSO and Atlantic over the spatial patterns of precipitation in the Amazonia.

The hypothetical relationship between the oceanic SSTs and precipitation
anomalies has been confirmed over the entire time span from 2004 till 2010. In
Fig. 3, we highlight those time periods when we have El Niño events and
warming in the Atlantic (Fig.3A,B). One traditional way to define these
anomalously warming periods from the SSTs is to choose a threshold value for
the temperature anomalies, i.e., $0.4^{\circ}C$ (Trenberth, 1997). Typically,
an El Niño event is identified if the 5-month running mean of SST in the
Niño3.4 area is above this threshold for at least six consecutive months. We
adopt the same threshold for defining an anomalously warming period in the
Atlantic. We find that there are 2-3 months of delay for the precipitation in
the Amazonian region in responding to the warming in the SSTs of ENSO,
respectively, 1 month of delay in response to the warming in the North
Atlantic. The temporal correlation patterns are validated by the
differentiated SSTs between the North and South Atlantic Ocean (Fig.3C).

Figure 3: Sensitivity of $DPL$ to the SSTs of the Pacific and Atlantic. (A)
$DPL$ for the NA and SA. (B) SSTs of Niño3.4 and the tropical North Atlantic.
The horizontal dashed line corresponds to a threshold of $0.4^{\circ}C$, which
is often used to define an El Niño event. Both a weak in 2006 and a strong El
Niño events in 2010 are highlighted. Two time periods when the SST of the
Atlantic Ocean is significantly above $0.4^{\circ}C$ are highlighted, together
with a weak event in 2008. In 2010, a significantly overlapped window is
highlighted between the El Niño event and the warming of SSTs in the North
Atlantic. (C) Differential SST between the North and South Atlantic (a
threshold of $0.4^{\circ}C$ is plotted for comparison).

As discussed previously, mild dry days could be regarded as normal conditions,
since the dry period is less than 2 weeks. Next, we focus on investigating
details how the SST anomalies in the surrounding oceans vary when the
Amazonian basin experiences dry and very dry days as observed in 2005 and
2010, respectively. Scatter plots are shown in Fig. 4 and the main results are
summarized as follows (the results do not change if the Pacific Niño3 region
is used).

  1. 1.

In 2005, ENSO plays no role in linking the very dry days in the Amazon. The
very dry days happen only in the SA (Fig. 4(A,B)). However, the time period
for the SSTs in the Niño3.4 to exceed $0.4^{\circ}C$ is negligible (with small
fluctuations around the threshold), which therefore suggests that the Pacific
region can not explain the drought event in 2005.

  2. 2.

In 2010, ENSO has a fundamental role in linking both dry and very dry days in
the NA region as the SSTs in the Niño3.4 area are well above the threshold.
This explains the phase (i) of drought 2010. However, ENSO has no link to the
dry and very dry days in the SA, since there is a cooling period of SSTs in
the Pacific.

  3. 3.

In 2005, the anomalously warming of the SSTs of the tropical North Atlantic
has a dominant link to the dry and very dry days of the drought in the SA.
Since no significant changes of rainfall related to the very dry days as
observed in the NA (in Fig. 4(C,D)), it suggests a rather weak role of the
North Atlantic on the drought conditions in the NA.

  4. 4.

In 2010, there is anomalously warming in the SSTs in the tropical North
Atlantic. This abnormal rise of SSTs explains phase (ii) of the drought 2010,
intensifying the drought initiated by the ENSO. Importantly, the influence of
the North Atlantic on the very dry days in the NA is negligible, although
having some mild effects only on dry days.

  5. 5.

The tropical South Atlantic essentially plays no role in changing the very dry
day patterns in either NA or SA. Although there is no direct influence on the
rainfalls, we notice that it has a significant role in forming the temperature
gradient between the North and South Atlantic, since the SSTs in the South
Atlantic experience an anomalously cooling phase when the very dry days
happened in both 2005 and 2010 (in Fig. 4(E,F)).

Figure 4: Correlations between dry, very dry periods and SSTs of the Niño3.4
and Atlantic Ocean, respectively. Black dots are for the year 2005, while
triangles are for 2010. The horizontal dashed lines are thresholds for
temperature anomaly of $0.4^{\circ}C$. Note, normal mild dry days (drought
periods less than 2 weeks) are suppressed. (A, B) Niño3.4, (C, D) tropical
North Atlantic, (E, F) South Atlantic. (A, C, E) NA, (B, D, F) SA.

##  5 Discussions and Conclusions

We identify the annual variability of precipitation anomalies over
Northeastern and Southwestern Amazonia, in particular disclosing a substantial
difference between the droughts 2005 and 2010. By extracting the drought
period length, we found that the drought condition in 2010 started as early as
in August 2009 in the NA region forming phase (i) of this event. The phase
(ii) of this event corresponds to the intensified drought in the SA due to the
anomalous warming of the SSTs in the tropical North Atlantic. The two phases
of the drought 2010 explain the severity on the Amazon basin from both
temporal time span and spatially extensive coverage perspectives.

The influences by the Pacific and Atlantic have a different center of action
within the Amazon basin. The anomalous Walker circulation is a key mechanism
linking the Pacific to the dry conditions over the Amazon basin, but it is
limited to the lower latitude of northeast. On the other hand, the Hadley
circulation is a key mechanism linking the tropical North Atlantic and the
precipitation patterns over the Amazonia, and the warming of SST in the
Atlantic Ocean plays a dominant role in the dry conditions in the SA. Our
results are consistent with the results reported by other studies (Marengo et
al, 2008; Yoon and Zeng, 2009; Coelho et al, 2012; Andreoli et al, 2012). As
discussed in (Coelho et al, 2012), “particularly in 2010 SST in the tropical
North Atlantic reached the highest values in history, contributing to the
establishment of a local meridional Hadley circulation with upward vertical
motion over the North Atlantic and downward vertical motion (subsidence) over
the Amazon, and the combination of El Niño conditions in the Pacific with warm
SST in the North Atlantic reinforce subsidence conditions in the Amazon, which
are unfavorable to the occurrence of precipitation”. This might be the unique
properties of phase transition from El Niño to La Nina in 2010. La Nina event
corresponds to large northward temperature gradient in the Atlantic, which
favors the high SST in the tropical North Atlantic. The El Niño condition in
2010 not only broke the record of the highest SST in the central Pacific, but
also it went through the fastest phase transition to La Niña (Kim et al,
2011).

The tropical South Atlantic essentially has no clear direct causal
relationship with the very dry days over the Amazonian region. It shows a mild
influence on dry days. However, the tropical South Atlantic is important to
form the temperature gradient in the Atlantic Ocean. More specifically,
shortage of rainfall over the SA region is associated with anomalously warm of
SSTs in the tropical North Atlantic, coupled with anomalously cooling of SSTs
in the tropical South Atlantic. Such a situation resulted in the anomalous
northward displacement of the Inter-Tropical Convergence Zone (ITCZ)
associated with the north-south differential gradient circulation, which is a
mechanism linking the Atlantic SSTs to SA rainfall (Nobre and Srukla, 1996).

A key question is whether there is a general trend towards drought conditions
and, if so, whether this is associated with anthropogenic climate change,
namely deforestation (Fu et al, 2013). (Li et al, 2008) suggests a more
pervasive drying trend over the southern Amazon between 1970 – 1999.
Previously, tendencies studied by (Marengo, 2009) during 1929 – 1998 suggest
that no unidirectional rainfall trend has been identified in the entire Amazon
region, but a slight negative/positive trend has been identified in
northern/southern Amazonia. Perhaps, the most important aspect is the presence
of inter-annual and interdecadal variability in rainfall, more noticeable than
any trend. This decadal variability is linked to interdecadal variations in
the SST in the tropical Atlantic (Wagner, 1996).

Projections of IPCC AR4 and AR5 and regional climate models (Chou et al, 2012;
Joetzjer et al, 2013; Marengo et al, 2009) suggest that the eastern Amazon may
become drier in the future, and that this drying could be exacerbated by
positive feedbacks with the vegetation. At the broadest temporal and spatial
scale, most global circulation models predict that greenhouse gas accumulation
and associated increases in the radiative forcing of the atmosphere will cause
a substantial (more than 20%) decline in rainfall in the eastern Amazonia by
the end of the century, with the strongest decline occurring at the end of the
rainy season and in the dry season (Malhi et al, 2009; Marengo, 2009; Hilker
et al, 2014). If severe droughts like those of 2005 and 2010 do become more
frequent in the future, this demands adaptation measures to avoid impacts on
the population, particularly those living on the river’s bank. The impacts
felt during the droughts in 2005 and 2010 show how local population are
vulnerable to climate extremes (Marengo et al, 2011): local farmers are
affected by drought due to high temperatures and drought conditions; and river
levels are extremely low, making it impossible to transport along the main
channels, which in many cases is the only way for populations to move around
and becoming isolated. Two extreme record droughts in less than five years
period is something that have brought the negative impacts of extremes of
climate variability and climate change in the region. Therefore, a proper
assessment of the intensity, spatial coverage, and climatic impacts of the
future droughts is of fundamental importance to the society (Phillips et al,
2009; Malhi et al, 2008).

###### Acknowledgements.

This work was partially supported by the NNSFC (Grant No. 11305062, 11135001),
and the DFG/FAPESP (Grant No. IRTG 1740/TRP 2011/50151-0). All data for this
paper is properly cited and referred to in the reference list.

## References

  * Aceituno (1988) Aceituno P (1988) On the functioning of the southern oscillation in the south american sector. part i: Surface climate. Mon Wea Rev 116:505 – 524
  * Andreoli et al (2012) Andreoli RV, Ferreira de Souza RA, Kayano MT, Candido LA (2012) Seasonal anomalous rainfall in the central and eastern Amazon and associated anomalous oceanic and atmospheric patterns. Int J Climatology 32(8):1193–1205
  * Chou et al (2012) Chou S, Marengo J, Lyra A, Sueiro G, Pesquero J, Alves L, Kay G, Betts R, Chagas D, Gomes J, Bustamante J, Tavares P (2012) Downscaling of south america present climate driven by 4-member hadcm3 runs. Climate Dynamics 38(3-4):635–653
  * Coelho et al (2012) Coelho CAS, Cavalcanti IAF, Costa SMS, Freitas SR, Ito ER, Luz G, Santos AF, Nobre CA, Marengo JA, Pezza AB (2012) Climate diagnostics of three major drought events in the amazon and illustrations of their seasonal precipitation predictions. Meteorol Appl 19(2):237–255
  * Cox et al (2008) Cox PM, Harris PP, Huntingford C, Betts RA, Collins M, Jones CD, Jupp TE, Marengo JA, Nobre CA (2008) Increasing risk of amazonian drought due to decreasing aerosol pollution. Nature 453:212 – 215
  * Davidson et al (2012) Davidson EA, de Araujo AC, Artaxo P, Balch JK, Brown IF, C Bustamante MM, Coe MT, DeFries RS, Keller M, Longo M, Munger JW, Schroeder W, Soares-Filho BS, Souza CM, Wofsy SC (2012) The amazon basin in transition. Nature 481:321 – 328
  * Fisher et al (2006) Fisher RA, Williams M, do Vale RL, da Costa AL, Meir P (2006) Evidence from amazonian forests is consistent with isohydric control of leaf water potential. Plant, Cell & Environment 29(2):151–165
  * Fu et al (2013) Fu R, Yin L, Li W, Arias PA, Dickinson RE, Huang L, Chakraborty S, Fernandes K, Liebmann B, Fisher R, Myneni RB (2013) Increased dry-season length over southern amazonia in recent decades and its implication for future climate projection. Proc Natl Acad Sci USA 110(45):18,110–18,115
  * Hilker et al (2014) Hilker T, Lyapustin AI, Tucker CJ, Hall FG, Myneni RB, Wang Y, Bi J, Mendes de Moura Y, Sellers PJ (2014) Vegetation dynamics and rainfall sensitivity of the amazon. Proc Natl Acad Sci USA 111(45):16,041–16,046
  * Joetzjer et al (2013) Joetzjer E, Douville H, Delire C, Ciais P (2013) Present-day and future amazonian precipitation in global climate models: Cmip5 versus cmip3. Climate Dynamics 41(11-12):2921–2936
  * Kim et al (2011) Kim W, Yeh SW, Kim JH, Kug JS, Kwon M (2011) The unique 2009-2010 El Niño event: A fast phase transition of warm pool El Niño to La Niña. Geophys Res Lett 38(15):L048,521
  * Lewis et al (2011) Lewis SL, Brando PM, Phillips OL, van der Heijden GMF, Nepstad D (2011) The 2010 Amazon drought. Science 331(6017):554
  * Li et al (2008) Li W, Fu R, Juárez RIN, Fernandes K (2008) Observed change of the standardized precipitation index, its potential cause and implications to future climate change in the amazon region. Philos Trans R Soc Lond B Biol Sci 363:1767 – 1772
  * Malhi et al (2008) Malhi Y, Roberts JT, Betts RA, Killeen TJ, Li W, Nobre CA (2008) Climate change, deforestation, and the fate of the amazon. Science 319(5860):169–172
  * Malhi et al (2009) Malhi Y, Arag o LEOC, Galbraith D, Huntingford C, Fisher R, Zelazowski P, Sitch S, McSweeney C, Meir P (2009) Exploring the likelihood and mechanism of a climate-change-induced dieback of the amazon rainforest. Proc Natl Acad Sci USA 106(49):20,610–20,615
  * Marengo (1992) Marengo JA (1992) Interannual variability of surface climate in the amazon basin. Int J Climatology 12(8):853–863
  * Marengo (2009) Marengo JA (2009) Long-term trends and cycles in the hydrometeorology of the amazon basin since the late 1920s. Hydrol Processes 23(22):3236–3244
  * Marengo et al (2008) Marengo JA, Nobre Ca, Tomasella J, Oyama MD, Sampaio de Oliveira G, de Oliveira R, Camargo H, Alves LM, Brown IF (2008) The Drought of Amazonia in 2005. J Clim 21(3):495–516
  * Marengo et al (2009) Marengo JA, Jones R, Alves LM, Valverde MC (2009) Future change of temperature and precipitation extremes in south america as derived from the precis regional climate modeling system. Int J Climatology 29(15):2241–2255
  * Marengo et al (2011) Marengo JA, Tomasella J, Alves LM, Soares WR, Rodriguez Da (2011) The drought of 2010 in the context of historical droughts in the Amazon region. Geophys Res Lett 38(12):L12,703
  * Nepstad et al (1994) Nepstad DC, de Carvalho CR, Davidson EA, Jipp PH, Lefebvre PA, Negreiros GH, da Silva ED, Stone TA, Trumbore SE, Vieira S (1994) The role of deep roots in the hydrological and carbon cycles of amazonian forests and pastures. Nature 372:666 – 669
  * Nobre and Srukla (1996) Nobre P, Srukla J (1996) Variations of sea surface temperature, wind stress, and rainfall over the tropical atlantic and south america. J Clim 9:2464 – 2479
  * Phillips et al (2009) Phillips OL, Aragão LEOC, Lewis SL, Fisher JB, Lloyd J, López-gonzález G, Malhi Y, Monteagudo A, Peacock J, Quesada CA, Heijden GVD, Almeida S, Amaral I, Arroyo L, Aymard G, Baker TR, Bánki O, Blanc L, Bonal D, Brando P, Chave J, Cristina A, Oliveira AD, Cardozo ND, Czimczik CI, Feldpausch TR, Freitas MA, Gloor E, Higuchi N, Jiménez E, Lloyd G, Meir P, Mendoza C, Morel A, Neill DA, Nepstad D, Patiño S, Peñuela MC, Prieto A, Ramírez F, Schwarz M, Silva J, Silveira M, Thomas AS, Steege H, Stropp J, Vásquez R, Zelazowski P, Dávila EA, Andelman S, Andrade A, Chao Kj, Erwin T, Fiore AD, C EH, Keeling H, Killeen TJ, Laurance WF, Cruz APn, Pitman NCA, Vargas PNn, Ramírez-angulo H, Rudas A, Salamão R (2009) Drougth Sensitivity of the Amazon Rainforest. Science 323:1344–1347
  * Reynolds et al (2007) Reynolds RW, Smith TM, Liu C, Chelton DB, Casey KS, Schlax MG (2007) Daily high-resolution-blended analyses for sea surface temperature. J Clim 20:5473–5496
  * Richey et al (1989) Richey JE, Nobre C, Deser C (1989) Amazon river discharge and climate variability: 1903 to 1985. Science 246(4926):101–103
  * Ronchail et al (2002) Ronchail J, Cochonneau G, Molinier M, Guyot JL, De Miranda Chaves AG, Guimarães V, de Oliveira E (2002) Interannual rainfall variability in the amazon basin and sea-surface temperatures in the equatorial pacific and the tropical atlantic oceans. Int J Climatology 22(13):1663–1686
  * Saatchi et al (2013) Saatchi S, Asefi-Najafabady S, Malhi Y, Aragão LEOC, Anderson LO, Myneni RB, Nemani R (2013) Persistent effects of a severe drought on amazonian forest canopy. Proc Natl Acad Sci USA 110(2):565–570
  * Salazar et al (2007) Salazar LF, Nobre CA, Oyama MD (2007) Climate change consequences on the biome distribution in tropical south america. Geophys Res Lett 34(9):L09,708
  * Sheffield et al (2006) Sheffield J, Goteti G, Wood EF (2006) Development of a 50-year high-resolution global dataset of meteorological forcings for land surface modeling. J Clim 19:3088 – 3111
  * de Souza et al (2005) de Souza EB, Kayano MT, Ambrizzi T (2005) Intraseasonal and submonthly variability over the Eastern Amazon and Northeast Brazil during the autumn rainy season. Theor Appl Climatol 81:177–191
  * Trenberth (1997) Trenberth KE (1997) The Definition of El Niño. Bull Amer Meteor Soc 78:2771–2777
  * Uvo et al (1998) Uvo CB, Repelli CA, Zebiak SE, Kushnir Y (1998) The relationships between tropical pacific and atlantic sst and northeast brazil monthly precipitation. J Clim 11:551– 562
  * Wagner (1996) Wagner RG (1996) Decadal-scale trends in mechanisms controlling meridional sea surface temperature gradients in the tropical atlantic. J Geophys Res: Oceans 101:16,683–16,694
  * Xu et al (2011) Xu L, Samanta A, Costa MH, Ganguly S, Nemani RR, Myneni RB (2011) Widespread decline in greenness of amazonian vegetation due to the 2010 drought. Geophys Res Lett 38(7):L07,402
  * Yoon and Zeng (2009) Yoon JH, Zeng N (2009) An Atlantic influence on Amazon rainfall. Clim Dyn 34(2-3):249–264
  * Zou et al (2014) Zou Y, Small M, Liu Z, Kurths J (2014) Complex network approach to characterize the statistical features of the sunspot series. New Journal of Physics 16(1):013,051