DESY 11-191

BFKL equation for the adjoint representation of the gauge group

in the next-to-leading approximation at $N=4$ SUSY †

V.S. Fadin ∗, L.N. Lipatov ∗∗

Universität Hamburg,

II. Institut für Theoretische Physik,

Luruper Chaussee, 149, D-22761 Hamburg,

* Budker Nuclear Physics Institute

and Novosibirsk State University,

630090, Novosibirsk, Russia

** Petersburg Nuclear Physics Institute

and St.Petersburg State University,

Gatchina, 188300, St.Petersburg, Russia

Abstract

We calculate the eigenvalues of the next-to-leading kernel for the BFKL
equation in the adjoint representation of the gauge group $SU(N_{c})$ in the
N=4 supersymmetric Yang-Mills model. These eigenvalues are used to obtain the
high energy behavior of the remainder function for the 6-point scattering
amplitude with the maximal helicity violation in the kinematical regions
containing the Mandelstam cut contribution. The leading and next-to-leading
singularities of the corresponding collinear anomalous dimension are
calculated in all orders of perturbation theory. We compare our result with
the known collinear limit and with the recently suggested ansatz for the
remainder function in three loops and obtain the full agreement providing that
the numerical parameters in this anzatz are chosen in an appropriate way.

†The work was supported in part by grant 14.740.11.0082 of Federal Program
“Personnel of Innovational Russia,” in part by RFBR grants 10-02-01238,
10-02-01338-a.

##  1 Introduction

In the Regge pole model scattering amplitudes at large energies $\sqrt{s}$ and
fixed momentum transfers $\sqrt{-t}$ have the form [1]

$A_{Regge}^{p}(s,t)=\xi_{p}(t)\,s^{1+\omega_{p}(t)}\,\gamma^{2}(t)\,,\,\,\xi_{p}(t)=e^{-i\pi\omega_{p}(t)}-p\,,$
(1)

where $p=\pm 1$ is the signature of the reggeon with the trajectory
$\omega_{p}(t)$ and $\gamma^{2}(t)$ represents the product of reggeon
vertices. The Pomeron is the Regge pole of the $t$-channel partial wave
$f_{\omega}(t)$ with vacuum quantum numbers and the positive signature
describing an approximately constant behaviour of total cross-sections for the
hadron-hadron scattering. S. Mandelstam demonstrated, that the Regge poles
generate cut singularities in the $\omega$-plane [2].

In the leading logarithmic approximation (LLA) the scattering amplitude at
high energies in QCD has the Regge form [3]

$M_{AB}^{A^{\prime}B^{\prime}}(s,t)=M_{AB}^{A^{\prime}B^{\prime}}(s,t)|_{Born}\,s^{\omega(t)}\,,$
(2)

where $M_{Born}$ is the Born amplitude and the gluon Regge trajectory is given
below

$\omega(-|q|^{2})=-\frac{\alpha_{s}N_{c}}{4\pi^{2}}\,\int
d^{2}k\,\frac{|q|^{2}}{|k|^{2}|q-k|^{2}}\approx-\frac{\alpha_{s}N_{c}}{2\pi}\,\ln\frac{|q^{2}|}{\lambda^{2}}\,.$
(3)

Here $\lambda$ is the infrared cut-off. In the multi-Regge kinematics, where
the pair energies $\sqrt{s_{k}}$ of the produced gluons are large in
comparison with momentum transfers $|q_{i}|$, the production amplitudes in LLA
are constructed from products of the Regge factors $s_{k}^{\omega(t_{k})}$ and
effective reggeon-reggeon-gluon vertices $C_{\mu}(q_{r},q_{r+1})$ [3]. The
amplitudes satisfy the Steinmann relations and the $s$-channel unitarity
incorporated in bootstrap equations [4].

The knowledge of $M_{2\rightarrow 2+n}$ allows one to construct the BFKL
equation for the Pomeron wave function using analyticity, unitarity,
renormalizability and crossing symmetry [3]. The integral kernel of this
equation has the property of the holomorphic separability [5] and is invariant
under the Möbius transformations [6]. The generalization of this equation to a
composite state of several gluons [7] in the multi-color QCD leads to an
integrable XXX model [8] having a duality symmetry [9].

The next-to-leading correction to the color singlet kernel in QCD is also
calculated [10]. Its eigenvalue contains non-analytic terms proportional to
$\delta_{n,0}$ and $\delta_{n,2}$, where $n$ is the conformal spin of the
Möbius group. But in the case of the $N=4$ extended supersymmetric gauge model
these Kronecker symbols are canceled leading to an expression having the
properties of the hermitian separability [11] and maximal transcendentality
[12]. The last property allowed to calculate the anomalous dimensions of
twist-two operators up to three loops [13, 14]. It turns out, that evolution
equations for the so-called quasi-partonic operators are integrable in $N=4$
SUSY at the multi-color limit [15]. The $N=4$ four-dimensional conformal field
theory according to the Maldacena guess is equivalent to the superstring model
living on the anti-de-Sitter 10-dimensional space [16, 17, 18]. Therefore the
Pomeron in N=4 SUSY is equivalent to the reggeized graviton in this space. The
equivalence gives a possibility to calculate the intercept of the BFKL Pomeron
at large coupling constants [14, 19]. The Möbius invariance of the BFKL kernel
was demonstrated also in two loops [20]. For next-to-leading calculations one
can use the effective field theory for reggeized gluons [21]. The generalized
bootstrap equation gives a possibility to prove the multi-Regge form of
production amplitudes in the next-to-leading approximation [22].

Another application of the BFKL approach is a verification of the BDS ansatz
[23] for the inelastic amplitudes in $N=4$ SUSY. It was demonstrated [24, 25],
that the BDS amplitude $M^{BDS}_{2\rightarrow 4}$ should be multiplied by the
factor containing the contribution of the Mandelstam cuts [2] in LLA. In the
two-loop approximation this factor can be found also from properties of
analyticity and factorization [26] or directly from recently obtained exact
result [27] for $M_{2\rightarrow 4}$ (see [28]). In a general case the wave
function in LLA for the composite $n$-gluon state in the adjoint
representation satisfies the Schrödinger equation for an open integrable
Heisenberg spin chain [29].

In this paper we shall calculate the eigenvalues $\omega(t)$ of the kernel $K$
for the BFKL equation in the adjoint representation of the gauge group at
$N=4$ SUSY in the next-to-leading approximation. The Green function of this
equation allows one to find the asymptotic behavior of the inelastic amplitude
in the Regge kinematics. There is a hypothesis [30, 31], that the inelastic
amplitude with the maximal helicity violation in a planar approximation is
factorized in the product of the BDS amplitude $M^{BDS}$, containing in
crossing channels the Regge factors with corresponding infrared divergencies,
and the remainder function $R$ depending on the anharmonic ratios

$M=R\,M^{BDS}\,.$ (4)

In an accordance with this hypothesis the $q^{2}$-dependence of the
eigenvalues of the octet BFKL equation is given by the expression (cf. [25] in
LLA)

$\omega(-q^{2})=\omega_{g}(-q^{2})+\omega_{0}\,,$ (5)

where $\omega_{g}(t)$ is the gluon Regge trajectory, which can be expressed in
all orders of the perturbation theory of $N=4$ SUSY in terms of two functions
entering in the expression for the BDS amplitude [24]. The ”intercept”
$\omega_{0}$ does not depend on $q^{2}$ due to the conformal invariance of
$N=4$ SUSY and can be written in terms of the ”energy” $E=-\omega_{0}$ being
the eigenvalue of the BFKL kernel discussed in the next section.

##  2 Integral kernel in the adjoint representation

The homogeneous BFKL equation can be written in the form

$\omega_{0}\phi=\hat{K}\phi\,,$ (6)

where $\hat{K}$ is the integral operator from which the gluon Regge trajectory
is subtracted. In the momentum representation it has the form

$\hat{K}\phi(\vec{q}_{1},\vec{q}_{2})=\int\frac{d^{2}q_{1}^{\;\prime}}{|q_{1}^{\;\prime}|^{2}|q_{2}^{\;\prime}|^{2}}\,K(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})\,\phi(\vec{q}_{1}^{\;\prime},\vec{q}_{2}^{\;\prime})\,,\,\,\vec{q}=\vec{q}_{1}+\vec{q}_{2}=\vec{q}^{\;\prime}_{1}+\vec{q}^{\;\prime}_{2}\,.$
(7)

The integral kernel for $N=4$ SUSY can be presented as follows (cf. [3, 22]
for the QCD case)

$K(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})=\delta^{2}(\vec{q}_{1}-\vec{q}_{1}^{\;\prime})\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}\left(\omega_{g}(-\vec{q}_{1}^{\;2})+\omega_{g}(-\vec{q}_{2}^{\;2})-\omega_{g}(-\vec{q}^{\;2})\right)+K_{r}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})\,,$
(8)

where the first term corresponds to virtual corrections with the gluon regge
trajectory subtraction (see (5)) and the second term appears from the real
intermediate states in the $s$-channel. The total contribution does not
contain infrared divergencies. Using results of Refs. [32] it can be written
in the form

$K(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})=\frac{1}{2}\delta^{2}(\vec{q}_{1}-\vec{q}_{1}^{\;\prime})\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}\left(\omega_{g}(-\vec{q}_{1}^{\;2})+\omega_{g}(-\vec{q}_{2}^{\;2})-2\omega_{g}(-\vec{q}^{\;2})\right)+K^{ns}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q}),$
(9)

where

$\omega_{g}(-\vec{q}_{1}^{\;2})+\omega_{g}(-\vec{q}_{2}^{\;2})-2\omega_{g}(-\vec{q}^{\;2})=-\frac{\alpha\,N_{c}}{2\pi}\left(1-\zeta(2)\,\frac{\alpha\,N_{c}}{2\pi}\right)\,\ln\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}{\vec{q}^{\;4}}\right)$
(10)

and

$K^{ns}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})=-\delta^{2}(\vec{q}_{1}-\vec{q}^{\;\prime}_{1})\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}\frac{\alpha\,N_{c}}{8\pi^{2}}\Biggl{(}\left(1-\zeta(2)\,\frac{\alpha\,N_{c}}{2\pi}\right)\int
d^{2}k\;\left(\frac{2}{\vec{k}^{\;2}}+2\frac{\vec{k}(\vec{q}_{1}-\vec{k})}{\vec{k}^{\;2}(\vec{q}_{1}-\vec{k})^{2}}\right)$
$-3\alpha\,N_{c}\zeta(3)\Biggr{)}+\frac{\alpha\,N_{c}}{8\pi^{2}}\left\\{\Biggl{(}1-\zeta(2)\,\frac{\alpha\,N_{c}}{2\pi}\Biggr{)}\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}+\vec{q}_{1}^{\;\prime\;2}\vec{q}_{2}^{\;2}}{\vec{k}^{\;2}}-\vec{q}^{\;2}\right)+\right.$
$\frac{\alpha\,N_{c}}{4\pi}\Biggl{[}\frac{\vec{q}^{\,2}}{2}\left(\ln\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_{2}^{\;2}}{\vec{q}^{\;2}}\right)+\ln\left(\frac{\vec{q}_{1}^{\;\prime\;2}}{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_{2}^{\;\prime
2}}{\vec{q}^{\;2}}\right)+\ln^{2}\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}_{1}^{\;\prime\;2}}\right)\right)-\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}+\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;2}}\ln^{2}\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}_{1}^{\;\prime\;2}}\right)$
$-\frac{1}{2}\,\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}-\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;2}}\ln\left(\frac{\vec{q}_{1}^{\;2}}{\vec{q}_{1}^{\;\prime\;2}}\right)\,\ln\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;4}}\right)+\biggl{[}\vec{q}^{\;2}(\vec{k}^{\;2}-\vec{q}_{1}^{\;2}-\vec{q}_{1}^{\;\prime\;2})$
$\left.+2\vec{q}_{1}^{\;2}\vec{q}_{1}^{\;\prime\;2}-\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}-\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}+\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\;2}-\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\;2}}{\vec{k}^{\;2}}(\vec{q}_{1}^{\;2}-\vec{q}_{1}^{\;\prime\;2})\biggr{]}I(\vec{q}_{1}^{\;2},\vec{q}_{1}^{\;\prime\;2},\vec{k}^{\;2})\Biggr{]}\\!\\!\right\\}$
$+\left(\vec{q}_{1}\leftrightarrow\vec{q}_{2},\;\;\vec{q}_{1}^{\;\prime}\leftrightarrow\vec{q}_{2}^{\;\prime}\right)~{},$
(11)

where $\vec{k}=\vec{q}_{1}-\vec{q}_{1}^{\;\prime}$ and the function $I$ is
given below

$I(\vec{q}_{1}^{\;2},\vec{q}_{1}^{\;\prime\;2},\vec{k}^{\;2})=\int_{0}^{1}\frac{dx}{\vec{q}_{1}^{\;2}(1-x)+\vec{q}_{1}^{\;\prime\;2}x-\vec{k}^{\;2}x(1-x)}\ln\left(\frac{\vec{q}_{1}^{\;2}(1-x)+\vec{q}_{1}^{\;\prime\;2}x}{\vec{k}^{\;2}x(1-x)}\right)~{}.$
(12)

Note that $I(a,b,c)$ is a totally symmetric function of the variables $a,\;b$
and $c$.

One could expect, that the BFKL kernel in $N=4$ SUSY is Möbius invariant in
the momentum representation, which would lead to the following simple form of
its eigenfunctions (cf. [25])

$\phi_{\nu
n}(\vec{q}_{1},\vec{q}_{2})=\left|\frac{q_{1}}{q_{2}}\right|^{2i\nu}\,e^{i\,n\,\phi}\,,$
(13)

where $\phi$ is the azimuthal angle of the complex number constructed from
transverse components of the vectors $\vec{q}_{1}$ and $\vec{q}_{2}$

$\frac{q_{1}}{q_{2}}=\left|\frac{q_{1}}{q_{2}}\right|\,e^{i\phi}\,.$ (14)

However, in the existing form the kernel is not Möbius invariant and in future
one should construct the similarity transformation to the invariant form (cf.
[20]). Such transformation exists because the remainder function $R$,
corresponding to the correction factor for the BDS expression, should be
invariant under four-dimensional dual conformal transformations and the Green
function obtained from the BFKL equation in the adjoint representation allows
to find the asymptotic behavior of the remainder function in the Mandelstam
kinematical regions [25].

##  3 Eigenvalues of the kernel

It is important, that the eigenvalues of the BFKL kernel do not depend on its
representation and can be found from our expression (8). To calculate these
eigenvalues we consider the BFKL equation in the limit (cf. [25])

$|q_{1}|\sim|q^{\prime}_{1}|\ll|q|\approx|q_{2}|\approx|q^{\prime}_{2}|\,.$
(15)

Denoting the two dimensional vectors $\vec{q}_{1}$ and
$\vec{q}^{\;\prime}_{1}$ by $\vec{p}$ and $\vec{p}^{\;\prime}$, respectively,
we write the BFKL equation in the form

$\int\frac{d^{2}p^{\;\prime}}{|p^{\;\prime}|^{2}}\,K(\vec{p},\vec{p}^{\;\prime})\,\Phi(\vec{p}^{\;\prime})=\omega_{0}\,\Phi(\vec{p})\,.$
(16)

Its kernel is given below

$K(\vec{p},\vec{p}^{\;\prime})=-\delta^{2}(\vec{p}-\vec{p}^{\;\prime})\,|p|^{2}\,\frac{\alpha
N_{c}}{4\pi^{2}}\,\left(\left(1-\frac{\alpha N_{c}}{2\pi}\zeta(2)\right)\,\int
d^{2}p^{\;\prime}\,\left(\frac{2}{|p^{\;\prime}|^{2}}+\frac{2(p^{\;\prime},p-p^{\;\prime})}{|p^{\;\prime}|^{2}|p-p^{\;\prime}|^{2}}\right)-3\alpha\,\zeta(3)\right)$
$+\frac{\alpha N_{c}}{4\pi^{2}}\,\left(1-\frac{\alpha
N_{c}}{2\pi}\zeta(2)\right)\,\left(\frac{|p|^{2}+|p^{\;\prime}|^{2}}{|p-p^{\;\prime}|^{2}}-1\right)+\frac{\alpha^{2}N_{c}^{2}}{32\,\pi^{3}}\,R(\vec{p},\vec{p}^{\;\prime})\,.$
(17)

Here $\vec{p}$ and $\vec{p}^{\;\prime}$ are momenta of the same reggeized
gluon before and after its scattering in the $t_{2}$-channel (momenta of
another gluon tend to infinity together with $q$). The reduced kernel
$R(\vec{p},\vec{p}^{\;\prime})$ is given below

$R(\vec{p},\vec{p}^{\;\prime})=\left(\frac{1}{2}-\frac{|p|^{2}+|p^{\;\prime}|^{2}}{|p-p^{\;\prime}|^{2}}\right)\,\ln^{2}\frac{|p|^{2}}{|p^{\;\prime}|^{2}}-\frac{|p|^{2}-|p^{\;\prime}|^{2}}{2|p-p^{\;\prime}|^{2}}\,\ln\frac{|p|^{2}}{|p^{\;\prime}|^{2}}\,\ln\frac{|p|^{2}|p^{\;\prime}|^{2}}{|p-p^{\;\prime}|^{4}}$
(18)
$+\left(-|p+p^{\;\prime}|^{2}+\frac{(|p|^{2}-|p^{\;\prime}|^{2})^{2}}{|p-p^{\;\prime}|^{2}}\right)\,\int_{0}^{1}dx\,\frac{1}{|(1-x)p+xp^{\;\prime}|^{2}}\,\ln\frac{|(1-x)p+xp^{\;\prime}|^{2}}{x(1-x)|p-p^{\;\prime}|^{2}}\,.$
(19)

From the rotational and dilatational invariance of the kernel we obtain its
eigenfunctions in the simple form

$\Phi_{\nu n}(\vec{p})=|p|^{2i\nu}e^{i\phi n}\,,$ (20)

where $\phi$ is the angle of the transverse vector $\overrightarrow{p}$ with
respect to the axis $x$. Note, that $\nu$ is real and $n$ is integer.

The orthonormality condition for this set of functions is obvious

$\frac{1}{2\pi^{2}}\int\frac{d^{2}p}{|p|^{2}}\,\Phi^{*}_{\mu
m}(\vec{p})\,\Phi_{\nu
n}(\vec{p}^{\;\prime})=\delta(\mu-\nu)\,\delta_{m,n}\,.$ (21)

The corresponding eigenvalues can be calculated with the action of the BFKL
kernel on the eigenfunctions and are given below

$\omega(\nu,n)=-a\left(E_{\nu n}+a\,\epsilon_{\nu
n}\right)\,,\,\,a=\frac{\alpha N_{c}}{2\pi}\,,$ (22)

where $E_{\nu n}$ is the ”energy” in the leading approximation [25]

$E_{\nu
n}=-\frac{1}{2}\,\frac{|n|}{\nu^{2}+\frac{n^{2}}{4}}+\psi(1+i\nu+\frac{|n|}{2})+\psi(1-i\nu+\frac{|n|}{2})-2\psi(1)\,,\,\,\psi(x)=(\ln\Gamma(x))^{\prime}$
(23)

and the next-to-leading correction $\epsilon_{\nu n}$ can be written as
follows

$\epsilon_{\nu
n}=-\frac{1}{4}\left(\psi^{\prime\prime}(1+i\nu+\frac{|n|}{2})+\psi^{\prime\prime}(1-i\nu+\frac{|n|}{2})+\frac{2i\nu\left(\psi^{\prime}(1-i\nu+\frac{|n|}{2})-\psi^{\prime}(1+i\nu+\frac{|n|}{2})\right)}{\nu^{2}+\frac{n^{2}}{4}}\right)$
$-\zeta(2)\,E_{\nu
n}-3\zeta(3)-\frac{1}{4}\,\frac{|n|\,\left(\nu^{2}-\frac{n^{2}}{4}\right)}{\left(\nu^{2}+\frac{n^{2}}{4}\right)^{3}}\,.$
(24)

Here the $\zeta$-functions are expressed in terms of polylogarithms

$Li_{n}(x)=\sum_{k=1}^{\infty}\frac{x^{k}}{k^{n}}\,,\,\,\zeta(n)=Li_{n}(1)\,.$
(25)

Note, that $\omega(\nu,n)$ has the important property

$\omega(0,0)=0\,.$ (26)

It is in an agreement with the existence of the eigenfunction $\Phi=1$ with a
vanishing eigenvalue, which is a consequence of the bootstrap relation [3,
22].

##  4 Corrections to the remainder function

One can easily construct the Green function for the conformally invariant BFKL
kernel in terms of its eigenvalues. This Green function allows us to calculate
the remainder functions $R_{n}$ for an arbitrary number of external legs in
the regions, where there are Mandelstam’s cuts corresponding to the composite
states of two reggeized gluons. For simplicity we consider the remainder
function $R_{6}$ for the gluon transition $2\rightarrow 4$ depending on three
anharmonic ratios (cf. [28])

$u_{1}=\frac{ss_{2}}{s_{012}s_{123}}\,,\,\,u_{2}=\frac{s_{1}t_{3}}{s_{012}t_{2}}\,,\,\,u_{3}=\frac{s_{3}t_{1}}{s_{123}t_{2}}\,.$
(27)

In the multi-regge kinematics one obtains

$s\gg s_{012},s_{123}\gg s_{1},s_{2},s_{3}\gg t_{1},t_{2},t_{3}\,,$ (28)

which corresponds to the following restrictions on the variables $u_{k}$

$1-u_{1}\rightarrow 0\,,\,\,\tilde{u}_{2}=\frac{u_{2}}{1-u_{1}}\sim
1\,,\,\,\tilde{u}_{3}=\frac{u_{3}}{1-u_{1}}\sim 1\,.$ (29)

It is convenient also to introduce the complex variable $w$ [28]

$w=|w|e^{i\phi_{23}}\,,\,\,|w|^{2}=\frac{u_{2}}{u_{3}}\,,\,\,\cos\phi_{23}=\frac{1-u_{1}-u_{2}-u_{3}}{2\sqrt{u_{2}u_{3}}}$
(30)

expressed in terms of transverse momenta of produced particles $k_{1}$,
$k_{2}$ and momentum transfers $q_{1},q_{2},q_{3}$

$w=\frac{q_{3}k_{1}}{k_{2}q_{1}}\,.$ (31)

In this case the remainder function $R$ in the Mandelstam region, where

$s,s_{2}\rightarrow+\infty\,,\,\,s_{1},s_{3}\rightarrow-\infty\,,$ (32)

can be presented in the form of a dispersion-like relation [26]

$R\,e^{i\pi\delta}=cos\,\pi\omega_{ab}+i\,\frac{a}{2}\sum_{n=-\infty}^{\infty}(-1)^{n}\left(\frac{w}{w^{*}}\right)^{\frac{n}{2}}\int_{-\infty}^{\infty}\frac{|w|^{2i\nu}d\nu}{\nu^{2}+\frac{n^{2}}{4}}\,\Phi_{Reg}(\nu,n)\left(-\frac{1}{\sqrt{u_{2}u_{3}}}\right)^{\omega(\nu,n)},$
(33)

where

$\delta=\frac{\gamma_{K}}{8}\,\ln(\tilde{u}_{2}\tilde{u}_{3})=\frac{\gamma_{K}}{8}\,\ln\frac{|w|^{2}}{|1+w|^{4}}\,,\,\,\omega_{ab}=\frac{\gamma_{K}}{8}\,\ln\frac{\tilde{u}_{2}}{\tilde{u}_{3}}=\frac{\gamma_{K}}{8}\,\ln|w|^{2}$
(34)

and the cusp anomalous dimensions

$\gamma_{K}=4a-4\,a^{2}\,\zeta(2)+22\,\zeta(4)\,a^{3}+...$ (35)

is known in all orders of perturbation theory [33].

Further, instead of the traditional variable $1/(1-u_{1})$ (see [24, 25]) we
used in eq. (33) the following energy invariant

$\frac{1}{\sqrt{u_{2}u_{3}}}=s_{2}\,\frac{|q_{2}|^{2}}{\sqrt{|k_{1}|^{2}|q_{1}|^{2}}\,|k_{2}|^{2}|q_{3}|^{2}}=\frac{1}{1-u_{1}}\,\frac{|1+w|^{2}}{|w|}\,,$
(36)

because according to the Regge theory the amplitude should be factorized in
the $t_{2}$-channel. As a result, by expanding this expression for $R$ in the
perturbation theory

$R=1+i\,a^{2}\left(b_{1}\ln\frac{1}{1-u_{1}}+b_{2}\right)+a^{3}\left(ic_{1}\ln^{2}\frac{1}{1-u_{1}}+(d_{1}+ic_{2})\ln\frac{1}{1-u_{1}}+d_{2}+ic_{3}\right)+...=$
$1+i\,a^{2}\left(\widetilde{b}_{1}\ln\frac{1}{\sqrt{u_{2}u_{3}}}+\widetilde{b}_{2}\right)+a^{3}\left(i\widetilde{c}_{1}\ln^{2}\frac{1}{\sqrt{u_{2}u_{3}}}+(\widetilde{d}_{1}+i\widetilde{c}_{2})\ln\frac{1}{\sqrt{u_{2}u_{3}}}+\widetilde{d}_{2}+i\widetilde{c_{3}}\right)+...\,,$
(37)

we obtain [25, 28]

$\widetilde{b}_{1}=b_{1}=-\frac{\pi}{2}\,\ln|1+w|^{2}\ln\frac{|1+w|^{2}}{|w|^{2}}\,,$
(38)
$\widetilde{b_{2}}=b_{2}-b_{1}\ln\frac{|1+w|^{2}}{|w|}\,,\,\,\frac{1}{\pi}\,b_{2}=\frac{1}{2}\ln|w|^{2}\ln^{2}|1+w|^{2}$
$-\frac{1}{3}\ln^{3}|1+w|^{2}+\ln|w|^{2}\left(Li_{2}(-w)+Li_{2}(-w^{*})\right)-2\left(Li_{3}(-w)+L_{3}(-w^{*})\right)\,,$
(39)

and (see ref. [28])

$\frac{4}{\pi}\,\widetilde{c}_{1}=\frac{4}{\pi}\,c_{1}=\ln|w|^{2}\,\ln^{2}|1+w|^{2}-\frac{2}{3}\ln^{3}|1+w|^{2}-\frac{1}{4}\ln^{2}|w|^{2}\ln|1+w|^{2}$
$+\frac{1}{2}\,\ln|w|^{2}\left(Li_{2}(-w)+Li_{2}(-w^{*})\right)-Li_{3}(-w)-Li_{3}(-w^{*})\,,$
(40)
$\frac{4}{\pi^{2}}\,\widetilde{d}_{1}=\frac{4}{\pi^{2}}\,d_{1}=-\ln|w|^{2}\,\ln^{2}|1+w|^{2}+\frac{2}{3}\ln^{3}|1+w|^{2}+\frac{1}{2}\ln^{2}|w|^{2}\ln|1+w|^{2}$
$+\ln|w|^{2}\left(Li_{2}(-w)+Li_{2}(-w^{*})\right)-2Li_{3}(-w)-2Li_{3}(-w^{*})\,.$
(41)

Note, that in the second order the real contribution to $R$ is absent [26].

The product of two impact factors $\Phi_{Reg}(\nu,n)$ can be obtained with the
use of the Fourier transformation of the function $\widetilde{b}_{2}$

$\Phi_{Reg}(\nu,n)=1+\Phi_{Reg}^{(1)}(\nu,n)\,a+\Phi_{Reg}^{(2)}(\nu,n)\,a^{2}+...\,,$
(42)
$\Phi_{Reg}^{(1)}(\nu,n)=\Phi^{(1)}(\nu,n)+\Delta\Phi(\nu,n)=-\frac{1}{2}E_{\nu
n}^{2}-\frac{3}{8}\,\frac{n^{2}}{\left(\nu^{2}+\frac{n^{2}}{4}\right)^{2}}-\zeta(2)\,,$
(43)

where $\Delta\Phi(\nu,n)$ is the contribution of the term
$-b_{1}\ln\frac{|1+w|^{2}}{|w|}$ in $\widetilde{b}_{2}$ (39) and the
contribution $\Phi^{(1)}(\nu,n)$ appearing from the term $b_{2}$ was
calculated in ref. [28] 111In the reference [28] the quantity
$\Phi^{(1)}(\nu,n)$ was found for the remainder function, but here we need it
for the full amplitude. According to (33) they differ by the term appearing
from the expansion of $\exp(i\pi\delta)$ and proportional to the second order
contribution to the anomalous dimension $\gamma_{K}$ (35).

$\Phi^{(1)}(\nu,n)=E_{\nu
n}^{2}-\frac{1}{4}\,\frac{n^{2}}{\left(\nu^{2}+\frac{n^{2}}{4}\right)^{2}}-\zeta(2)\,.$
(44)

The knowledge of eigenvalues (24) in the next-to-leading approximation gives a
possibility to calculate the coefficients $\widetilde{c}_{2}$ and
$\widetilde{d}_{2}$ from expression (33)

$\frac{1}{\pi}\,\widetilde{c}_{2}=-\frac{1}{4}\,\ln|w|^{2}\left(S_{1,2}(-w)+S_{1,2}(-w^{*})+\ln(1+w)\,Li_{2}(-w)+\ln(1+w^{*})\,Li_{2}(-w*)\right)$
$+\frac{\zeta(3)}{2}\,\ln|1+w|^{2}-\ln\frac{|1+w|^{2}}{|w|}\left(Li_{3}(-w)+Li_{3}(-w^{*})-\frac{1}{2}\ln|w|^{2}(Li_{2}(-w)+Li_{2}(-w^{*}))\right)$
$+\frac{1}{4}\,\ln|1+w|^{2}(Li_{3}(-w)+Li_{3}(-w^{*}))+\frac{1}{16}\,\ln^{2}|w|^{2}\ln|1+w|^{2}\ln\frac{|1+w|^{2}}{|w|^{2}}$
$+\frac{1}{8}\,\ln^{2}|1+w|^{2}\ln^{2}\frac{|1+w|^{2}}{|w|^{2}}+\frac{1}{8}\ln^{2}|w|^{2}\ln(1+w)\,\ln(1+w^{*})+\zeta(2)\,\ln|1+w|^{2}\ln\frac{|1+w|^{2}}{|w|^{2}}\,,$
(45)
$\widetilde{d}_{2}=\pi\left(\widetilde{c}_{2}-\ln\frac{|1+w|^{2}}{|w|}\,\widetilde{b}_{2}+2\,\zeta(2)\,\widetilde{b}_{1}\right)\,.$
(46)

In the above expression the function $S_{1,2}(-x)$ has the following
representation

$S_{1,2}(-x)=\int_{0}^{x}\frac{dx^{\prime}}{2x^{\prime}}\,\ln^{2}(1+x^{\prime})=Li_{3}(\frac{x}{1+x})+Li_{3}(-x)-\ln(1+x)Li_{2}(-x)-\frac{1}{6}\ln^{3}(1+x)\,.$
(47)

One can verify with the use of the known relations among polylogarithms
$Li_{n}(x)$, that the coefficients $\widetilde{c}_{2}$ and $\widetilde{d}_{2}$
are single-valued functions on the two-dimensional plane $\overrightarrow{w}$
and are symmetric to the inversion $w\rightarrow 1/w$. We can calculate also
the coefficients $c_{2}$ and $d_{2}$ in (37) using the relations

$c_{2}=\widetilde{c}_{2}+2\widetilde{c}_{1}\ln\frac{|1+w|^{2}}{|w|}\,,\,\,d_{2}=\widetilde{d}_{2}+\widetilde{d}_{1}\,\ln\frac{|1+w|^{2}}{|w|}\,.$
(48)

Note, that recently the authors of ref. [35] suggested an anzatz for the
remainder function $R_{6}$ in three loops based on the theory of symbols. They
calculated its high energy behavior in our Mandelstam region in the form of
the polynomial expansion in $\log(1-u_{1})$. It turns out, that up to three
loops their results are completely coincides with our perturbative expansion
(37). In particular, one can derive the expressions (58) and (66) from the
paper [35] using the fact, that the corresponding functions
$g_{1}^{(2)}(w,w^{*})$ and $h_{0}^{(3)}(w,w^{*})$ are related with our
coefficients $c_{2}$ and $d_{2}$ in (37) as follows

$g_{1}^{(2)}(w,w^{*})=-\frac{c_{2}}{2\pi}\,,\,\,h_{0}^{(3)}(w,w^{*})=-\frac{d_{2}}{(2\pi)^{2}}\,.$
(49)

It gives a possibility to fix the parameters $\gamma^{\prime}$ and
$\gamma^{\prime\prime\prime}$ appearing in ref. [35] in the form

$\gamma^{\prime}=-\frac{9}{2}\,,\,\,\gamma^{\prime\prime\prime}=0\,.$ (50)

In expression (63) of the paper [35] also the additional function
$g_{0}^{(3)}(w,w^{*})$ was calculated. This function contains three unknown
parameters appearing in the last line of (63). Our coefficients $c_{3}$ and
$\widetilde{c}_{3}$ in (37) can be expressed in terms of it

$c_{3}=2\pi\,g_{0}^{(3)}(w,w^{*})\,,\,\,\widetilde{c}_{3}=c_{3}-\ln\frac{|1+w|^{2}}{|w|}\,c_{2}+\ln^{2}\frac{|1+w|^{2}}{|w|}\,c_{1}\,.$
(51)

It gives a possibility to construct the following function

$\rho(w,w^{*})=\frac{\widetilde{c}_{3}}{\pi}+\pi\,\widetilde{c}_{1}+\ln\frac{|1+w|^{2}}{|w|}\,\left(\zeta(2)\,\ln^{2}\frac{|1+w|^{2}}{|w|}-\frac{11}{2}\,\zeta(4)\right),$
(52)

where the term proportional to $\zeta(4)$ appears from the third order
contribution to $\gamma_{K}$ (35) which was calculated firstly in ref. [13].
The important next-to-next-to-leading corrections to the product of impact-
factors $\Phi_{Reg}(\nu,n)$ (42) can be expressed through $\rho(w,w^{*})$

$\Phi^{(2)}_{Reg}(\nu,n)=(-1)^{n}\left(\nu^{2}+\frac{n^{2}}{4}\right)\int\frac{d^{2}w}{\pi}\,\rho(w,w^{*})\,|w|^{-2i\nu-2}\,\left(\frac{w^{*}}{w}\right)^{\frac{n}{2}}\,.$
(53)

We are going to calculate $\Phi^{(2)}_{Reg}(\nu,n)$ in future.

Similar results can be obtained for the remainder function describing the
$3\rightarrow 3$ transitions in the corresponding Mandelstam regions [34].

##  5 Collinear limit

It is well known, that the BFKL equation for the Pomeron wave function gives a
possibility to predict the leading singularities of the anomalous dimensions
$\gamma$ of the twist-2 operators at $\omega\rightarrow 0$ in all orders of
perturbation theory [3, 10]. In particular, for the case of $N=4$ SUSY the
predictions of Ref. [11] are in a full agreement with the direct calculations
of $\gamma$ up to 5 loops [13, 36, 37]. As it follows from the previous
section, the BFKL kernel for the adjoint representation of the gauge group
allows one to find the high energy corrections to the remainder functions. On
the other hand, in the collinear limit the remainder functions obey the
renormalization group-like equations [38, 39]. The analytic continuation of
the collinear expressions for $R$ to the Mandelstam regions was performed in
Ref. [40]. The leading asymptotics corresponds to the unit conformal spin
$|n|=1$. The anomalous dimensions $\gamma_{col}$ for the collinear limit in
the Euclidean region were constructed [39] and the relation between the Regge
and collinear limits was investigated [40].

To calculate $\gamma_{col}$ in the Mandelstam region we present expression
(33) in the following form with the use of the Fourier transformation

$R\,e^{i\pi\delta}=cos\,\pi\omega_{ab}+i\,\frac{a}{2}\sum_{n=-\infty}^{\infty}(-1)^{n}\left(\frac{w}{w^{*}}\right)^{\frac{n}{2}}\int_{-\infty}^{\infty}d\nu\,|w|^{2i\nu}\,L_{\nu
n}\left(-\frac{1}{1-u_{1}}\right),$ (54)

where

$L_{\nu
n}\left(-\frac{1}{1-u_{1}}\right)=\sum_{n^{\prime}=-\infty}^{\infty}(-1)^{n^{\prime}-n}\int_{-\infty}^{\infty}\frac{\Phi_{reg}(\nu^{\prime},n^{\prime})d\nu^{\prime}}{\nu^{\prime
2}+\frac{n^{\prime 2}}{4}}\,S_{\nu^{\prime}n^{\prime}}^{\nu
n}\,\left(-\frac{1}{1-u_{1}}\right)^{\omega(\nu^{\prime},n^{\prime})}$ (55)

and

$S_{\nu^{\prime}n^{\prime}}^{\nu
n}=\int\frac{d^{2}w}{2\pi^{2}}\,|w|^{2i(\nu^{\prime}-\nu)-2}\,\left(\frac{w}{w^{*}}\right)^{\frac{n^{\prime}-n}{2}}\,\left(\frac{|1+w|^{2}}{|w|}\right)^{\omega(\nu^{\prime}n^{\prime})}\,.$
(56)

The collinear limit $w\rightarrow 0$ or $w\rightarrow\infty$ of the remainder
function (54) should be performed at fixed $1-u_{1}$ [39, 40]. Generally
expressions (54) and (55) correspond to the collinear renormalization with an
infinite number of the multiplicatively renormalizable operators (cf. [40]).
But in the case, when we take into account only the asymptotic terms at
$|w|\rightarrow\infty$ with the conformal spin $|n|=1$, we can obtain for $R$
the simple expression

$R\,e^{i\pi\delta}\approx
cos\,\pi\omega_{ab}-ia\,\cos\phi_{23}\,|w|^{-1}\int_{-i\infty}^{i\infty}d\omega\,\frac{\Phi^{Reg}(\nu,1)}{\left(\nu^{2}+\frac{1}{4}\right)\,\frac{d\omega}{d\nu}}\,|w|^{2\gamma_{col}(\omega)}\,\left(-\frac{1}{1-u_{1}}\right)^{\omega}\,,$
(57)

where the contour of integration goes to the right of the BFKL singularity
$\nu\sim\sqrt{\omega-\omega(0,1)}$ present in the integrand in an accordance
with the fact, that the functions
$\gamma=\gamma_{col}(\omega),\,\nu=\nu(\omega)$ satisfy the set of equations
222Note, that our definition of the collinear anomalous dimension
$\gamma_{col}$ differs with the factor $-1/2$ from that used in ref. [39].

$\gamma=\frac{1}{2}+i\nu+\frac{\omega}{2}\,,\,\,\omega=\omega(\nu,1)\,.$ (58)

For finding $\gamma_{col}$ in perturbation theory the function $\omega(\nu,1)$
(22) should be expanded near the point $\nu=i/2$

$\lim_{\nu\rightarrow\frac{i}{2}}\omega(\nu,1)=\frac{a}{2}\,f_{1}\left(i\nu+\frac{1}{2}\right)-\frac{a^{2}}{8}\,f_{2}\left(i\nu+\frac{1}{2}\right)\,,$
(59)

where

$f_{1}(x)=\frac{1}{x}-1-x-x^{2}(1-4\zeta(3))-x^{3}+O(x^{4})\,,$ (60)
$f_{2}(x)=\frac{1}{x^{3}}+\frac{1}{x^{2}}+\frac{4\zeta(2)}{x}-8\zeta(3)-4\zeta(2)-2+O(x)\,,$
(61)

Thus, we obtain the following equation for $\gamma=\gamma_{col}(\omega)$

$\omega=\frac{a}{2}\,f_{1}(\gamma)-\frac{a^{2}}{8}\,\left(f^{\prime}_{1}(\gamma)f_{1}(\gamma)+f_{2}(\gamma)\right)\,.$
(62)

Its perturbative solution is given below

$\gamma_{col}(\omega)=\frac{a}{2}\,\left(\frac{1}{\omega}-1\right)-\frac{a^{2}}{4}\left(\frac{1}{\omega^{2}}+2\,\frac{\zeta(2)}{\omega}\right)+\frac{a^{3}}{4\,\omega^{2}}\left(1+2\zeta(2)+\zeta(3)\right)+O(a^{4})\,.$
(63)

The above approach is similar to that for the singlet BFKL kernel, but in that
case one obtained the main contribution to the Bjorken limit from $n=0$ [10].

The collinear anomalous dimension $\gamma_{col}(\omega)$ in the Mandelstam
region $s,s_{2}>0,s_{1},s_{3}<0$ can be found in one loop using the results of
the paper [40]. We start with the perturbative expansion of the remainder
function in the collinear limit $|w|\rightarrow\infty$ in LLA of the Operator
Product Expansion (OPE) [38]

$R_{OPE}\approx
a\cos\phi\,\frac{e^{-\sigma}}{2|w|}\,\sum_{k=0}^{\infty}\frac{(-a\ln|w|)^{k}}{k!}\,h_{k}(\sigma)\,,\,\,\sigma=\frac{1}{2}\,\ln\frac{u_{1}}{1-u_{1}}\,,$
(64)

where we expressed the world sheet coordinates $\tau$ and $\sigma$ in terms of
our variables $w$ and $u_{1}$ (see eqs (76)-(79) from ref. [40]) and included
one loop contribution contained in the BDS amplitude. The analytic
continuation of the two loop remainder function calculated in ref. [27] to the
Mandelstam region $s,s_{2}>0,\,s_{1},s_{3}<0$ gives the result (see eqs. (51),
(C.12)-(C.16) from ref. [40])

$\cos\phi\,\rightarrow\cos\phi_{23};\;\;h_{k}(\sigma)\rightarrow-
h_{k}(\sigma)+\Delta_{k}(\sigma)\,,\,\,\frac{\Delta_{0}(\sigma)}{2\pi
i}=-2\,e^{\sigma}\,,\,\,$ $\frac{\Delta_{1}(\sigma)}{2\pi
i}=4\left(\cosh\sigma\ln(1+e^{2\sigma})-e^{\sigma}\right).$ (65)

Here the functions $h_{k}(\sigma)$ for $k=0,1$ in the right hand side of the
first relation are known from ref. [39]. They are not essential for the
calculation of $\gamma_{col}$ because they are real and fall at large
$\sigma$. The contributions $\Delta_{k}(\sigma)$ appear from the analytic
continuation of the corresponding discontinuities of the functions
$h_{k}(\sigma)$ on the cut $-1<\widetilde{s}_{2}<0$, where
$\widetilde{s}_{2}=\exp(2\sigma)$ [40]. After the continuation we can write
this discontinuity using the collinear renormalization group in the form

$\Delta
R_{OPE}=-\,a\,\cos\phi_{23}\,\frac{1}{|w|}\,\int_{-i\infty}^{i\infty}\frac{d\omega}{\omega(\omega+1)}\,|w|^{2\gamma_{col}(\omega)}e^{2\omega\sigma}\,,$
(66)

where

$\gamma_{col}(\omega)=a\,\omega\,(1+\omega)\,\int_{0}^{\infty}d(2\sigma)\,e^{-2\sigma\omega}\left(e^{-\sigma}\cosh(\sigma)\ln\left(1+e^{2\sigma}\right)-1\right)=$
$\frac{a}{2}\left(\frac{1}{\omega(\omega+1)}-2\omega+(\omega+1)\left(\psi(\omega+1)-\psi(\frac{\omega+2}{2})\right)+\omega\,\psi(\omega+2)-\omega\,\psi(\frac{\omega+3}{2})\right).$
(67)

As one can see from expression (63), the BFKL approach reproduces correctly
the first two terms of $\gamma_{col}$ at $\omega\rightarrow 0$.

##  6 Conclusion

In this paper we solved the BFKL equation for the channel with color octet
quantum numbers in the next-to-leading approximation. The eigenvalues of its
integral kernel were used to calculate in the next-to-leading logarithmic
approximation the remainder function for the production amplitude
$2\rightarrow 4$ in the multi-Regge kinematics at the Mandelstam channels. The
obtained result in three loops is in an agreement with the recently suggested
anzatz [35] for the remainder function. This anzatz allowed us to construct
the product of corresponding impact-factors in the next-to-next-to-leading
approximation. The collinear anomalous dimension in the Mandelstam region was
calculated explicitly in one loop. Its leading and next-to-leading
singularities are found in all loops.

Acknowledgments. We thank J. Bartels, A. Prygarin and G. Vacca for helpful
discussions, the Hamburg University and DESY for the warm hospitality and
support. This work was done in the framework of the program LEXI ”Connecting
Particles with the Cosmos”.

## References

  * [1] V. N. Gribov, Sov. Phys. JETP 14 (1962) 478.
  * [2] S. Mandelstam, Nuovo Cim. 30 (1963) 1148.
  * [3] L. N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338;
V. S. Fadin, E. A. Kuraev and L. N. Lipatov, Phys. Lett. B 60 (1975) 50;
E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Sov. Phys. JETP 44 (1976) 443; 45
(1977) 199;
Ya. Ya. Balitskii and L. N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822.

  * [4] Ya. Ya. Balitskii, L. N. Lipatov and V. S. Fadin, in Materials of IV Winter School of LNPI (Leningrad, 1979) p.109.
  * [5] L. N. Lipatov, Phys. Lett. B 309 (1993) 394.
  * [6] L. N. Lipatov, Sov. Phys. JETP 63 (1986) 904.
  * [7] J. Bartels, Nucl. Phys. B 175 (1980) 365;
J. Kwiecinskii and M. Praszalowicz, Phys. Lett. B 94 (1980) 413.

  * [8] L. N. Lipatov, High energy asymptotics of multi-colour QCD and exactly solvable lattice models, arXiv:9311037 [hep-th].
  * [9] L. N. Lipatov, Nucl. Phys. B 548 (1999) 328.
  * [10] V. S. Fadin and L. N. Lipatov, Phys. Lett. B 429 (1998) 127;
M. Ciafaloni and G. Camici, Phys. Lett. B 430 (1998) 349.

  * [11] A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B 582 (2000) 19.
  * [12] A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B 661 (2003) 19.
  * [13] A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko and V. N. Velizhanin, Phys. Lett. B 595 (2004) 521;
  * [14] A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko and V. N. Velizhanin, Phys. Lett. B 632 (2006) 754.
  * [15] L. N. Lipatov, talk at ”Perspectives in Hadronic Physics”, Proc. of Conf. ICTP, Triest, Italy, May 1997.
  * [16] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231.
  * [17] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428 (1998) 105.
  * [18] E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253.
  * [19] R. C. Brower, J. Polchinski, M. J. Strassler, C. I. Tan, JHEP 0712  (2007) 005.
  * [20] I. Balitsky and G. A. Chirilli, Nucl. Phys. B 822 (2009) 45;
V. S. Fadin, R. Fiore and A. V. Grabovsky, Nucl. Phys. B 831 (2010) 248.

  * [21] L. N. Lipatov, Nucl. Phys. B 452 (1995) 369; Phys. Rept. 286 (1997) 131.
  * [22] V. S. Fadin, R. Fiore, M. G. Kozlov and A. V. Reznichenko, Phys. Lett.B 639 (2006) 74.
  * [23] Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D 72 (2005) 085001.
  * [24] J. Bartels, L. N. Lipatov and A. Sabio Vera, Phys. Rev. D 80 (2009) 045002.
  * [25] J. Bartels, L. N. Lipatov and A. Sabio Vera, Eur. Phys. J. C 65 (2009) 587.
  * [26] L. N. Lipatov, arXiv:1008.1015 [hep-th], Proceedings of the conference ”Quarks-2010”.
  * [27] A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, arXiv:1006.5703 [hep-th].
  * [28] L. N. Lipatov and A. Prygarin, Phys. Rev. D 83 (2011) 045020; D 83 (2011) 125001.
  * [29] L. N. Lipatov, J. Phys. A 42 (2009) 304020.
  * [30] L. F. Alday and J. M. Maldacena, IHEP 0706 (2007) 064.
  * [31] J. M. Drummond, J. Henn, G. P.Korchemsky and E. Sokatchev, Nucl. Phys. B 826 (2010) 337 ; B 828 (2010) 3179.
  * [32] V. S. Fadin, R. Fiore, A. V. Grabovsky and A. Papa, Nucl. Phys. B 784 (2007) 49;
R. E. Gerasimov and V. S. Fadin, Phys. Atom. Nucl. 73  (2010) 1214;
V.S. Fadin, R. Fiore and A. Papa, Phys. Rev. D 60 (1999) 074025.

  * [33] N. Beisert, B. Eden, M. Staudacher, J. Stat. Mech. 0701  (2007) P01021.
  * [34] J. Bartels, L. N. Lipatov and A. Prygarin, arXiv:1012.3178 [hep-th].
  * [35] L. J. Dixon, J. M. Drummond and J. M. Henn, arXiv:1108.4461 [hep-th].
  * [36] A. V. Kotikov, L. N. Lipatov, A. Rej, M. Staudacher and V. N. Velizhanin, J. Stat. Mech. 0710 (2007) P10003;
Z. Bajnok, R. A. Janik and T. Lukovski, Nucl. Phys. B 816 (2009) 376.

  * [37] T. Lukovski, A. Rej and V. N. Velizhanin, Nucl. Phys. B 831 (2010) 105.
  * [38] L. F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, IHEP 1104 (2011) 088.
  * [39] D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, arXiv:1102.0062 [hep-th].
  * [40] J. Bartels, L. N. Lipatov and A. Prygarin, arXiv:1104.4709 [hep-th].


# 1Cademy @ Causal News Corpus 2022: Leveraging Self-Training in Causality
Classification of Socio-Political Event Data

Adam Nik2 4 , Ge Zhang1 2 3††footnotemark:  , Xingran Chen3, Mingyu Li2 3, Jie
Fu 1
1 Beijing Academy of Artificial Intelligence, China
2 1Cademy Community, USA
3 University of Michigan Ann Arbor, USA
4 Carleton College, USA
fujie AT baai.ac.cn The two authors contributed equally to this work.
Corresponding Author

###### Abstract

This paper details our participation in the Challenges and Applications of
Automated Extraction of Socio-political Events from Text (CASE) workshop @
EMNLP 2022, where we take part in Subtask 1 of Shared Task 3 Tan et al.
(2022a). We approach the given task of event causality detection by proposing
a self-training pipeline that follows a teacher-student classifier method.
More specifically, we initially train a teacher model on the true, original
task data, and use that teacher model to self-label data to be used in the
training of a separate student model for the final task prediction. We test
how restricting the number of positive or negative self-labeled examples in
the self-training process affects classification performance. Our final
results show that using self-training produces a comprehensive performance
improvement across all models and self-labeled training sets tested within the
task of event causality sequence classification. On top of that, we find that
self-training performance did not diminish even when restricting either
positive/negative examples used in training. Our code is be publicly available
at https://github.com/Gzhang-umich/1CademyTeamOfCASE.

Figure 1: A) Self-training pipeline with Teacher Model. B) We use the self-
labeled examples as part of the training when training in Student Models for
the task of causality classification

##  1 Introduction

Task 1 of the CASE workshop @ EMNLP 2022 works to identify and classify event
causality in socio-political event (SPE) data, with subtask 1 being a binary
classification of causality. In other words, participants are tasked with
answering: Does an event sentence contain cause-effect meaning? The workshop
provides data from Causal News Corpus (CNC) Tan et al. (2022b) for training
and evaluation of the subtask. Causality itself aims to identify a semantic
relationship between two events where one event (the cause) is responsible for
the production of the other event (the effect). Utilizing the CNC dataset
serves as a benchmark for participants to evaluate the ability of a given
model or process to identify causality in event data.

We approach the problem of causality sequence classification by applying self-
training Ouali et al. (2020); Van Engelen and Hoos (2020); Triguero et al.
(2015) as a means to improve the performance of language models in this task.
The goal of self-training is to generate proxy labels for previously unlabeled
data to enhance the learning process. The self-training process works by
iteratively labeling previously unpredicted data, and then using the new
pseudo-labels as truthful labels in the next training stage. The intuition
behind self-training comes from the fact that it can pseudo-expand the
training space to basically an unlimited size in a very cheap manner, as no
hand-labeling is required in the process.

Additionally, we run supplementary experiments to test the effectiveness of
self-training against various transformer-based data augmentation techniques
Feng et al. (2021) and separate multi-task learning approaches Caruana (1997)
that we originally designed for the competition. The description and results
of these additional experiments can be found in the Appendix.

In summary, our main contributions are as follows.

1) We propose a self-training pipeline for the task of causality detection in
SPE data for the purposes of competing in Subtask 1 of Shared Task 3 of the
CASE workshop @ EMNLP 2022. Our best model achieved 0.8135 accuracy and a
0.8398 F1 score on the competition’s test set.

2) We evaluate our self-training pipeline with collected self-labeled datasets
of highly positive samples, highly negative samples, and even distributed
positive and negative samples. We show that using self-labeled datasets
improves performance across the board on all tested models, and that the
performance increase provided by self-training did not significantly change
based on the ratio of positive to negative self-labeled samples used in
training.

For all implementations of our code, we use the HuggingFace Transformers
library Wolf et al. (2020) (version 4.21.2) and all models are built using
PyTorch Paszke et al. (2019) (version 1.12.1).

Organization. As for how the rest of the paper is outlined, §2 describes the
data used in the training, evaluation, and final testing of our models, §3
recounts the procedures used in our self-training approach, §4 discusses our
findings, and §5 wraps up the paper with our final remarks and ideas for
future direction.

##  2 Data

###  2.1 Causal News Corpus

The CNC dataset Tan et al. (2022b) is a corpus of 3,559 event sentences from
protest event news labeled on whether a given sentence contains causal
relations or not. The data of the CNC comes from two workshops focused on
mining socio-political data: Automated Extraction of Socio-political Events
from News (AESPEN) Hürriyetoğlu et al. (2020) in 2020 and the CASE 2021
workshop @ ACL-IJCNLP Hürriyetoğlu et al. (2021). For the purposes of subtask
1, the data is split into a training set of 2925 examples, a development set
of 323 examples, and a final test set of 311 examples that is used as an
evaluation benchmark for the competition.

###  2.2 Self-labeled Training Data

Sample sentences used in the self-labeling phase of self-training are gathered
from 205,328 articles on Wikipedia. The Wikipedia dataset is built from the
Wikipedia dump 111https://dumps.wikimedia.org and is available as on
HuggingFace Dataset library Lhoest et al. (2021). We use the 20220301.simple
training split to generate our self-labeled examples.

##  3 Methodology

In this section, we review the methods we used in our approach to the sequence
causality classification subtask.

###  3.1 Self-Training

We follow a similar teacher-student pipeline as Yalniz et al., 2019 that
includes using a teacher model to generate a new labeled dataset
$\mathcal{D^{\prime}}$ from the original dataset $\mathcal{D}$ and then
training a new student model on both the new labeled dataset
$\mathcal{D^{\prime}}$ and the original dataset $\mathcal{D}$. We use the
training split provided of 2925 CNC samples Tan et al. (2022b) as the original
dataset $\mathcal{D}$, and fine-tune a BERT base-cased model Devlin et al.
(2019) for sequence classification, which serves as our teacher model. Figure
1a shows the full pipeline from Wikipedia data collection to saving self-
labeled samples. These self-labeled examples are used as training data for the
separate student models later in the experimentation process, as shown in
Figure 1b.

Baseline Training vs. Self-Training Results
---
Baseline Training (simple fine-tuning, no self-training) |  |  | Accuracy | F1 | Recall | Precision | MCC
BERT |  | 0.8204 | 0.8394 | 0.8516 | 0.8276 | 0.6363
RoBERTa |  | 0.8390 | 0.8543 | 0.8561 | 0.8525 | 0.6745
Google ELECTRA Discriminator |  | 0.8365 | 0.8535 | 0.8640 | 0.8432 | 0.6689
Self-Training |  |  | Ratio of Positive to Negative Self-Labeled
---
Examples used in training
Accuracy | F1 | Recall | Precision | MCC
BERT | 1:3 | 0.8380 | 0.8531 | 0.8539 | 0.8525 | 0.6726
| 1:1 | 0.8225 | 0.8377 | 0.8315 | 0.8468 | 0.6425
| 3:1 | 0.8380 | 0.8526 | 0.8502 | 0.8552 | 0.6728
RoBERTa | 1:3 | 0.8576 | 0.8715 | 0.8764 | 0.8671 | 0.7123
| 1:1 | 0.8586 | 0.8711 | 0.8670 | 0.8755 | 0.7149
| 3:1 | 0.8586 | 0.8719 | 0.8727 | 0.8711 | 0.7142
Google ELECTRA Discriminator | 1:3 | 0.8400 | 0.8579 | 0.8764 | 0.8415 | 0.6760
| 1:1 | 0.8524 | 0.8665 | 0.8689 | 0.8641 | 0.7016
| 3:1 | 0.8421 | 0.8580 | 0.8652 | 0.8510 | 0.6806

Table 1: Results of the evaluating the CNC development set on both simple
fine-tuning with only CNC training data (top) and fine-tuning classifiers on
training sets of self-labeled data in addition to CNC training data (bottom).
Bold indicates highest performance across all splits and model types,
underline indicates the highest performance of the specific model type.

####  3.1.1 Data Preprocessing

To preprocess Wikipedia data (§ 2.2), we first split the articles into
individual sentences and discarded all sentences of less than 50 characters
and more than 500 characters. To self-label the sentences, we feed the
sentences into the teacher model and keep all examples with a softmax
classifier over a predetermined threshold $\mathcal{T}$. For the purposes of
our experiments, we choose a $\mathcal{T}$ of 0.9. In total, we collect a pool
of 77,748 positive (causal) examples and 77,940 negative (non-causal)
examples. The large total number of examples collected for this data pool is
done to minimize the overlap of examples between the later created self-
labeled training splits.

####  3.1.2 Training Splits

From the pools of self-labeled Wikipedia examples, we collect 5 different
training sets, all with the size of 10,000 samples but with varying ratios of
positive to negative self-labeled examples. We collect sets with positive to
negative proportions of 1:3, 1:1, and 3:1 (that is, for a positive to negative
proportion of 1:3, we include 2,500 self-labeled positive examples in the
training set and 7,500 negative samples). We design this set-up to test how
the different polarity proportions of self-labeled data used in training
affect not only overall model accuracy, but also if there is a discrepancy
between model precision and recall with the varying polarity splits. We chose
a training split size of 10,000 examples as we notice that self-training
performance does not continue to improve with training with splits larger than
this 222Observed in our initial internal testing phase. When formulating each
set, we randomly reshuffle the positive and negative self-labeled sets and
chose the first s and t positive and negative samples for a training set that
require s positive examples and t negative examples. From there, we combine
the s positives and the t negatives and again shuffle the concatenated
training set.

####  3.1.3 Fine-tuning on Self-labeled data

For each self-labeled dataset, we fine-tune a classifier—which serves as our
student model—on one epoch of the self-labeled dataset and then five epochs of
the CNC provided training data. The predictions generated after the final
epoch of training are used for evaluation. We run our experiments with student
classifiers built on BERT base-cased Devlin et al. (2019), RoBERTa base Liu et
al. (2019), and Google ELECTRA-base-discriminator Clark et al. (2020) pre-
trained models.

###  3.2 Transformer-based Data Augmentation and Multi-task Learning

In our participation of the CASE workshop, we also explore both Transformer-
based data augmentation and multi-task learning as a means to improve
performance on causality classification. While our both of these approaches
are out-performed by our self-training approaches and thus are not the main
focus of this paper, we still find significant results with these methods and
implement both a Transformer-based data augmentation technique and a multi-
task architecture that comprehensively outperform the baseline classifier for
the given task. The full methodology and experimentation of our Transformer-
based data augmentation and multi-task learning approaches are available in
the Appendix.

##  4 Experiments and Results

###  4.1 Experiment Set up

In our experimentation setup, we test all three backbone models (BERT,
RoBERTa, and Google ELECTRA Discriminator) with both the self-training
pipeline and a simple fine-tuning process that only uses the provided CNC
training set that served as the baseline. In the baseline experiments, the
classifiers are trained solely on five epochs of the CNC training data. We
conduct five trials of each setup, each trial having a randomly initialized
seed. We use the CNC development set as our testing benchmark due to the
limited number of allowed workshop testing phase submissions.

###  4.2 Classifier Set up

In our experiments, we run all trials on a Tesla V100-SXM2-16GB GPU device. We
use an AdamW optimizer with $\beta_{1}$ = 0.9, $\beta_{2}$ = 0.999, a learning
rate of $5e-5$, and a linear decay rate. Finally, all experiments are run with
a batch size of 8.

###  4.3 Findings

Table 1 displays the results from our experiments, which include the averages
of 5 trials for each set-up. From the table, we can see that every self-
training setup outperforms the baseline classifier in terms of accuracy, with
an average accuracy improvement of 1.33% across all models and polarity
splits. Furthermore, for all but one self-training set-up, there is an
improvement of the F1 score from the baseline, with an average improvement of
0.011.

Other key takeaways from our results are that 1) there is very little overall
performance degradation across the polarity splits (1:3, 1:1, 3:1) in the
self-labeled datasets (only the BERT model shows a range of F1 scores above
0.01) and 2) there is low discrepancy between recall and precision among the
splits (only the 1:3 split with an ELECTRA backbone shows a recall-precision
discrepancy > 0.015.)

###  4.4 Competition Results

Our best-performing prediction set of the final competition testing comes from
a RoBERTa classifier trained on a self-labeled training set with a polarity
ratio of 1:1. The results of our all of our competition submissions
333Workshop competition limited participants to five submissions for the
testing phase are shown in Table 2. All of our competition submissions
comprehensively outperform the provided baseline, and our best overall
performing submission achieve competition rankings of 6th in accuracy, 10th in
F1, 7th in recall, 7th in precision, and 10th in MCC.

Competition Results (CNC Test Set)
---
|  | Ratio of Positive to Negative
---
Self-Labeled Examples
Accuracy | F1 | Recall | Precision | MCC
RoBERTa | 1:3 | 0.8071 | 0.8256 | 0.8068 | 0.8452 | 0.6108
| 1:1 | 0.8135 | 0.8398 | 0.8636 | 0.8172 | 0.6185
| 3:1 | 0.7974 | 0.8215 | 0.8239 | 0.8192 | 0.5873
ELECTRA | 1:1 | 0.8135 | 0.8324 | 0.8181 | 0.8471 | 0.6228
3:1 | 0.7942 | 0.8107 | 0.7784 | 0.8457 | 0.5886
| Provided Competition Baseline
---
(BERT baseline model)
0.7781 | 0.8120 | 0.8466 | 0.7801 | 0.5452

Table 2: Results of competition submissions on CNC test set. Bold indicates
highest performer.

##  5 Conclusion and Discussion

This paper explores how training a classifier on self-labeled data can improve
the performance of sequence classification tasks. In our case, we examine the
effect of self-training on the task of event causality in socio-political
event data as part of Subtask 1 of Shared Task 3 of the CASE workshop @ EMNLP
2022.

Our results show that training a classifier on self-labeled data using a
teacher-student approach comprehensively improves task performance.
Furthermore, we find that performance improvement from self-training did not
differ significantly between self-labeled training sets with varying levels of
example polarity. This indicates that the model is capable of reaping the full
benefits of self-training despite having limited access to positive or
negative samples. One thing that could help explain this is our relatively
high threshold $\mathcal{T}$ of 0.9 which determines whether or not to keep an
example during the initial self-labeled process. Further research should
explore whether a lower $\mathcal{T}$ could alter the benefits of self-
training, especially when self-labeled examples would have a higher chance of
being incorrectly labeled.

Next, given that our self-labeled examples are gathered from an assortment of
articles from Wikipedia, it should be well noted that the benefits of self-
training are apparent even when the self-labeled examples are not domain
specific to the original labeled data. We decide to use Wikipedia as the
source of our self-labeled examples as we view it as a more accessible source
with far greater amounts available unlabeled data. Thus, our findings indicate
that performance improvements from self-training work with non-domain specific
data, which alleviates us from the restriction of confining our self-labeled
data to the single domain of the original labeled data.

Finally, one more aspect of our experiments that should be further explored is
the classifier’s actual dependence on the self-labeled data versus the
originally provided training data. In our setup, we choose to train our models
on one epoch of self-labeled data and then on five epochs of the original
training data in order to prioritize the true labeled training data. We
believe that it would be worthwhile to explore training classifiers with a
higher training priority on the self-labeled data, or even to test the
performance of classifiers trained solely on the self-labeled data, without
the original true data.

## Acknowledgments

The work for this study is supported by the National Key R&D Program of China
(2020AAA0105200) and the Sam ’75 and Meg Woodside Fund for Career Exploration
provided through Carleton College.

## References

  * Caruana (1997) Rich Caruana. 1997.  Multitask learning.  _Machine learning_ , 28(1):41–75.
  * Clark et al. (2020) Kevin Clark, Minh-Thang Luong, Quoc V Le, and Christopher D Manning. 2020.  Electra: Pre-training text encoders as discriminators rather than generators.  _arXiv preprint arXiv:2003.10555_.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Feng et al. (2021) Steven Y Feng, Varun Gangal, Jason Wei, Sarath Chandar, Soroush Vosoughi, Teruko Mitamura, and Eduard Hovy. 2021.  A survey of data augmentation approaches for nlp.  In _Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021_ , pages 968–988.
  * Guo et al. (2021) Pengxin Guo, Feiyang Ye, and Yu Zhang. 2021.  Safe multi-task learning.  _arXiv preprint arXiv:2111.10601_.
  * Hürriyetoğlu et al. (2021) Ali Hürriyetoğlu, Hristo Tanev, Vanni Zavarella, Jakub Piskorski, Reyyan Yeniterzi, Osman Mutlu, Deniz Yuret, and Aline Villavicencio. 2021.  Challenges and applications of automated extraction of socio-political events from text (CASE 2021): Workshop and shared task report.  In _Proceedings of the 4th Workshop on Challenges and Applications of Automated Extraction of Socio-political Events from Text (CASE 2021)_ , pages 1–9, Online. Association for Computational Linguistics.
  * Hürriyetoğlu et al. (2020) Ali Hürriyetoğlu, Vanni Zavarella, Hristo Tanev, Erdem Yörük, Ali Safaya, and Osman Mutlu. 2020.  Automated extraction of socio-political events from news (AESPEN): Workshop and shared task report.  In _Proceedings of the Workshop on Automated Extraction of Socio-political Events from News 2020_ , pages 1–6, Marseille, France. European Language Resources Association (ELRA).
  * Lhoest et al. (2021) Quentin Lhoest, Albert Villanova del Moral, Yacine Jernite, Abhishek Thakur, Patrick von Platen, Suraj Patil, Julien Chaumond, Mariama Drame, Julien Plu, Lewis Tunstall, Joe Davison, Mario Šaško, Gunjan Chhablani, Bhavitvya Malik, Simon Brandeis, Teven Le Scao, Victor Sanh, Canwen Xu, Nicolas Patry, Angelina McMillan-Major, Philipp Schmid, Sylvain Gugger, Clément Delangue, Théo Matussière, Lysandre Debut, Stas Bekman, Pierric Cistac, Thibault Goehringer, Victor Mustar, François Lagunas, Alexander Rush, and Thomas Wolf. 2021.  Datasets: A community library for natural language processing.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 175–184, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.  _arXiv preprint arXiv:1907.11692_.
  * Mikheev et al. (1999) Andrei Mikheev, Marc Moens, and Claire Grover. 1999.  Named entity recognition without gazetteers.  In _Ninth Conference of the European Chapter of the Association for Computational Linguistics_ , pages 1–8.
  * Mohit (2014) Behrang Mohit. 2014.  Named entity recognition.  In _Natural language processing of semitic languages_ , pages 221–245. Springer.
  * Ng et al. (2019) Nathan Ng, Kyra Yee, Alexei Baevski, Myle Ott, Michael Auli, and Sergey Edunov. 2019\.  Facebook fair’s wmt19 news translation task submission.  _arXiv preprint arXiv:1907.06616_.
  * Ouali et al. (2020) Yassine Ouali, Céline Hudelot, and Myriam Tami. 2020.  An overview of deep semi-supervised learning.  _arXiv preprint arXiv:2006.05278_.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. 2019.  Pytorch: An imperative style, high-performance deep learning library.  In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, _Advances in Neural Information Processing Systems 32_ , pages 8024–8035. Curran Associates, Inc.
  * Raleigh et al. (2010) Clionadh Raleigh, Andrew Linke, Håvard Hegre, and Joakim Karlsen. 2010.  Introducing acled: an armed conflict location and event dataset: special data feature.  _Journal of peace research_ , 47(5):651–660.
  * Ruder (2017) Sebastian Ruder. 2017.  An overview of multi-task learning in deep neural networks.  _arXiv preprint arXiv:1706.05098_.
  * Tan et al. (2022a) Fiona Anting Tan, Ali Hürriyetoğlu, Tommaso Caselli, Nelleke Oostdijk, Hansi Hettiarachchi, Tadashi Nomoto, Onur Uca, and Farhana Ferdousi Liza. 2022a.  Event causality identification with causal news corpus - shared task 3, CASE 2022.  In _Proceedings of the 5th Workshop on Challenges and Applications of Automated Extraction of Socio-political Events from Text (CASE 2022)_ , Online. Association for Computational Linguistics.
  * Tan et al. (2022b) Fiona Anting Tan, Ali Hürriyetoğlu, Tommaso Caselli, Nelleke Oostdijk, Tadashi Nomoto, Hansi Hettiarachchi, Iqra Ameer, Onur Uca, Farhana Ferdousi Liza, and Tiancheng Hu. 2022b.  The causal news corpus: Annotating causal relations in event sentences from news.  In _Proceedings of the Language Resources and Evaluation Conference_ , pages 2298–2310, Marseille, France. European Language Resources Association.
  * Triguero et al. (2015) Isaac Triguero, Salvador García, and Francisco Herrera. 2015.  Self-labeled techniques for semi-supervised learning: Taxonomy, software and empirical study.  _Knowl. Inf. Syst._ , 42(2):245–284.
  * Van Engelen and Hoos (2020) Jesper E Van Engelen and Holger H Hoos. 2020.  A survey on semi-supervised learning.  _Machine Learning_ , 109(2):373–440.
  * Wang et al. (2018) Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel Bowman. 2018.  Glue: A multi-task benchmark and analysis platform for natural language understanding.  In _Proceedings of the 2018 EMNLP Workshop BlackboxNLP: Analyzing and Interpreting Neural Networks for NLP_ , pages 353–355.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45, Online. Association for Computational Linguistics.
  * Yalniz et al. (2019) I Zeki Yalniz, Hervé Jégou, Kan Chen, Manohar Paluri, and Dhruv Mahajan. 2019.  Billion-scale semi-supervised learning for image classification.  _arXiv preprint arXiv:1905.00546_.
  * Zhang and Yang (2021) Yu Zhang and Qiang Yang. 2021.  A survey on multi-task learning.  _IEEE Transactions on Knowledge and Data Engineering_.

##  6 Appendix

Figure 2: Multi-task learning architectures used in supplementary testing.

Here, we outline the supplementary experimentation we conducted to compare our
self-training results with other methods we explored in the CASE event
causality competition. These methods include a few popular transformer-based
textual data augmentation techniques and two multi-task learning-based
classifier architectures.

###  6.1 Transformer-based Data Augmentation

In general, data augmentation—within the context of textual data—works by
altering a given labeled example and attaching the label of the original
example to the augmented one. Each of the transformer-based data augmentation
techniques is considered with the same goal of increasing the training data
space to improve the model performance on the task of causality
classification. We use the CNC training split of 2925 as the original data to
be augmented in our experiments.

####  6.1.1 Sequence to Sequence Data Augmentation

Sequence-to-sequence text augmentation works by taking the sentence of the
original example (all of our data examples are English examples), translating
the sentence into a foreign language, and then finally translating the
rendered sentence back to the original language. This works by altering some
words or clusters of words in a sentence while preserving the original
structure and semantics. For the purposes of our experiments, we use two
foreign languages to augment the data, German and Russian, using HuggingFace’s
ported versions of the Facebook FAIR’s WMT19 News Translation Task Submission
Ng et al. (2019). The sequence-to-sequence augmented training set has 8,775
examples; 2,925 from the original training set and 5,850 augmented examples.

####  6.1.2 Random Fill-mask Data Augmentation

In random fill-mask augmentation, we first randomly select a word from the
original. From there, we replace the selected word with a masking token and
use the new sentence with masking as input to a pre-trained RoBERTa fill-mask
language model Liu et al. (2019) to select the three most likely fill-mask
options for the masked word. With the three selected substitutions for the
masked word, we create three new sentences by replacing each respective
substitution with the original masked word and keeping the original label of
the sentence with the new augmented examples. The final random fill-mask
augmented set has 11,700 total samples.

####  6.1.3 NER Fill-Mask Data Augmentation

The NER fill-mask data augmentation functions in a similar fashion to the
random fill-mask data augmentation, but instead of selecting a single random
word to replace, we make substitutions to any named entities identified by
Named Entity Recognition (NER) Mikheev et al. (1999); Mohit (2014).
Specifically, we use the EntityRecognizer module from spaCy
444https://spacy.io/ to identify which tokens in a sentence corresponded to
named entities. For each example sentence from the original training data that
contained named entities, we create three augmented sentences by substituting
the best unused fill-mask option for each named entity in the text. The final
NER augmented dataset has 10,443 example sentences in total.

###  6.2 Multi-Task Learning Approaches

Supplementary Experiments Results
---
|  | Accuracy | F1 | Recall | Precision | MCC
Baseline | 0.8390 | 0.8543 | 0.8561 | 0.8525 | 0.6745
Self-Training (1:1 polarity) | 0.8586 | 0.8711 | 0.8670 | 0.8755 | 0.7149
| Sequence to Sequence | 0.8235 | 0.8430 | 0.8596 | 0.8270 | 0.6424
| Random Fill-Mask | 0.8406 | 0.8562 | 0.8574 | 0.8556 | 0.6778
Transformer-based Data Augmentations | NER Fill-Mask | 0.8452 | 0.8571 | 0.8427 | 0.8721 | 0.6888
| Architecture 1 | 0.8498 | 0.8655 | 0.8764 | 0.8548 | 0.6960
Multi-Task Learning | Architecture 2 | 0.8313 | 0.8489 | 0.8596 | 0.8385 | 0.6583
Table 3: Results from supplementary testing done on CNC development set. All
runs use a RoBERTa backbone model. The baseline and self-training results are
taken from the main experiments of the paper. Bold indicates outperforming the
baseline.

Multi-task learning (MTL) Caruana (1997); Zhang and Yang (2021); Ruder (2017)
is a paradigm of machine learning that improves the performance of a model in
a given task by leveraging simultaneous learning of other distinct but related
tasks. Our MTL architectures learn the distinct tasks of entailment
classification (binary classification of whether the meaning of one sentence
can be inferred from another sentence) and event detection (whether a sentence
contains information about a socio-political event), then combine the prior
knowledge of those two tasks to help supplement the classifier’s prediction to
the task of causality classification.

####  6.2.1 MTL Datasets

We used two distinct datasets for the multi-task learning of entailment
detection and event detection.

Entailment Detection Dataset We evaluate using the Recognizing Textual
Entailment (RTE) task provided in the GLUE Benchmark Wang et al. (2018) for
the entailment detection task. In training, we used the given training set
that consisted of 2490 examples. Each example from the RTE dataset consisted
of two sentences and a binary label on whether or not one of the two sentences
holds logical entailment to the other. To better fit the structure of the
other data, we concatenated the two provided sentences into a single text to
be used as input into the models.

Event Detection Dataset In order to learn the task of event detection, we used
data provided in the second shared task of CASE @ ACL-IJCNLP 2021 Hürriyetoğlu
et al. (2021), which provided data to the object of sentence-level event
classification. The data provided from subtask 2 of CASE 2021 included 1023
examples sentences of socio-political events, labeled using the Armed Conflict
Location & Event Data Project (ACLED) Raleigh et al. (2010) event taxonomy,
which consists of 25 fine-grained event subtypes. These 1023 example sentences
are concatenated with 720 non-event-specific English sentences to create an
event detection dataset, with all sentences coming from the event
classification receiving a label of ’1’, denoting that the sentence contained
information about an event.

####  6.2.2 MTL Pre-training

Prior to fine-tuning our models for the task of causality classification, we
train a shared encoder Guo et al. (2021)—a RoBERTa pre-trained model—on the
separate tasks of event detection and entailment detection by fine-tuning the
shared encoder on the respective datasets for each task. We fine-tune three
epochs for both tasks.

####  6.2.3 MTL Architectures

We experiment with two similar but different architectures in MTL testing. In
both architectures, we first simultaneously fine-tune a classifier on the two
tasks of entailment detection and event detection. Because we have distinct
datasets for each respective task, we implement this by using the shared
encoder approach, where model parameters are hard-shared and each task has its
own task-specific classification head.

The distinction between our two MTL architectures comes from how we choose to
combine prior knowledge. The architectures we build are shown in Figure 2.
Both architectures include task-specific classification heads for the tasks of
entailment detection and event detection. The distinction between the two
architectures comes in where Architecture no. 2 also includes a causality-
specific classification head; the outputs of all three task heads are combined
and inputted into a final linear layer to output the final logits prediction.
Architecture no. 1 omits the causality-specific classification head and simply
combines the outputs of the entailment detection and event detection heads
before the linear layer.

###  6.3 Supplementary Experiments and Results

####  6.3.1 Set up

For the supplementary experiments, we follow the same setup as in the main
study to maintain consistency. Thus, models trained on a transformer-augmented
dataset are trained on five epochs of the respective dataset, and each MTL
architecture is trained on five epochs of the CNC training set. The
evaluations are calculated on the predictions made after the final epoch of
training. Likewise, we use the same hyperparameter setup as the main
experiments, meaning that we run all trials on a Tesla V100-SXM2-16GB GPU
device. Hyperparameters are listed in Table 4. For purposes of the
supplementary experiments, we run all trials using a RoBERTa backbone.

AdamW Optimizer w/ Linear Decay
---
$\beta_{1}$ | 0.9
$\beta_{2}$ | 0.999
Per device batch size | 8
Table 4: Classifier hyperparameter settings.

####  6.3.2 Results

Table 3 displays the results of our supplementary tests. Consistent with the
main study, the results are the averages over five trials for each of the
setups on the CNC development set. Between the transformer-augmented
experiments, the random fill-mask and NER fill-mask experiments outperformed
the baseline in terms of both accuracy and F1 score. Similarly, Architecture
no. 1 of the MTL approaches also outperformed the baseline in terms of
accuracy and F1.

####  6.3.3 Discussion

We include the supplementary experiments to 1) show how our self-training
results compared to popular state-of-the-art data augmentation techniques
using contemporary NLP, and 2) propose the multi-task learning architectures
we originally developed for the Subtask 1 of the competition. Although the
final results of the MTL approaches did not reach the same level of
performance as the self-training approaches and therefore did not belong in
the main paper, we believe the MTL experiments and results are still notable
and worth mentioning for further investigation.


the geometry and the topology of the background solutions. We have constructed
the necessary energy current by hand, which satisfies all the required
properties including positivity (for arbitrarily large data), coercivity, and
convexity (for small data) in addition to the cancellation of principal terms
in its divergence expression that is essential to close the energy argument.
Application of CMCSH gauge turns the Einstein’s equations into an elliptic-
hyperbolic system and together with the aforementioned energy current for the
fluid degrees of freedom, we obtain a coupled elliptic-hyperbolic system. This
is an advantage of using this particular choice of gauge, where the analysis
of Einstein’s equations simplifies substantially.

In addition, the re-scaled Einstein-Euler system considered in the current
context is an example of a non-autonomous (infinite-dimensional) dynamical
system since the explicitly time-dependent scale factor $a$ appears in the
equations. This makes the analysis non-trivial since there are terms in the
equations of motion which involve factors such as $\frac{\dot{a}}{a}$,
$\frac{1}{a}$, $\dot{a}a^{1-n\gamma_{ad}}$, and $a^{1-n\gamma_{ad}}$. The
decay estimates that we obtained in the last section heavily depended on the
fact that these factors exhibit a particular pattern in their appearances such
that $\frac{\dot{a}}{a}$ should only multiply the term that generates decay or
higher-order terms. Remarkably, the structure of Einstein’s equations is such
that all these factors appear in the field equations in an innocuous way. The
stability is then found to be dependent on an integrability criteria of this
scale factor namely
$\int_{t_{0}}^{\infty}a^{1-n\gamma_{ad}}dt<\infty,~{}\int_{t_{0}}^{\infty}a^{-1}dt<\infty$.
Notice that the crucial integrability criterion
$\int_{t_{0}}^{\infty}a^{-1}dt<\infty$ does not appear at the linear level (in
the study of [23], this integrability criterion arises due to the
consideration of the motion tangential to the center manifold, and as such is
absent for the $n=3$ case). It is therefore tempting to conjecture these
integrability criteria to be the conditions to avoid shock and naked
singularity formation. However, there is much left to be done. It is worth
mentioning that there are studies in the literature [74, 75] that deal with
such issue of shock formation on a background expanding spacetimes.

Our results only provide a notion of the Lyapunov stability of the FLRW
solutions considered here. In other words, the perturbed solutions to the
Einstein-Euler field equations around this class of spatially compact FLRW
metrics (spatial slices are compact negative Einstein spaces in general and
hyperbolic for the physically relevant $n=3$ case) arising from regular Cauchy
data remain uniformly bounded and decay to a family of metrics with constant
negative spatial scalar curvature. Since the metrics with constant scalar
curvature do not imply constant sectional curvature for dimensions greater
than 2 and the final mass-energy density is not spatially constant, the
asymptotic state is essentially described by an inhomogeneous and anisotropic
physical universe filled with dispersed fluid. This may be slightly
unsatisfactory since one would expect the perturbed solutions to return to the
FLRW family. However, since the physical universe does consist of large-scale
structures, it is tempting to conjecture that our result indicates a
possibility of structure formation through Einstein flow. In addition, it
provides yet another reason to believe that FLRW models are physical contrary
to their isometric cousin Milne model (which is devoid of background energy
density) which does not exhibit a possibility of an inhomogeneous and
anisotropic final state in the absence of a positive cosmological constant.
This notion of structure formation through perturbing the FLRW solution was
noted in several heuristic studies (e.g., [7, 8, 45, 76]). It would be worth
investigating this issue further in future studies due to its tremendous
physical implications.

Lastly, it is worth pointing out that our stability result heavily relies on
the assumption on the allowed range of the adiabatic index $\gamma_{ad}$ in
the chosen barotropic equation of state ($P=(\gamma_{ad}-1)\rho$). The non-
linear stability result holds if $\gamma_{ad}$ lies in the range
$(1,\frac{n+1}{n})$ ($(1,4/3)$ for the physically relevant $n=3$ case). We
recall that the same condition was obtained in the linear stability analysis
by [23]. Prior to our study, Rodnianski and Speck [35] and Speck [36] obtained
the same condition for nonlinear stability of small data on
$\mathbb{R}\times\mathbb{T}^{3}$. In addition to [35] and [36], avoidance of
shock formation by exponential expansion was also studied by [77] for a
radiation fluid (on the $\mathbb{S}^{3}$ spatial topology), by [78] for
$\gamma_{ad}\in(4/3,3/2)$ (with $\mathbb{T}^{3}$ spatial topology), and [79]
for dust (with $\mathbb{T}^{3}$ spatial topology). It would be interesting to
extend our study to these special cases and include different types of
equations of state as well.

##  6 Acknowledgement

P.M would like to thank Prof. Vincent Moncrief for numerous useful discussions
related to this project and for his help improving the manuscript. This work
was supported by CMSA at Harvard University.

##  7 Bibliography

## References

  * [1] D. Christodoulou, The formation of shocks in 3-dimensional fluids, European Mathematical Society, vol. 2, 2007.
  * [2] J.E. Peebles, B. Ratra, The cosmological constant and dark energy, Reviews of modern physics, vol. 75, 559, 2003.
  * [3] S.M. Carrol, The cosmological constant, Living reviews in relativity, vol. 4, 1-56, 2001.
  * [4] C. LeBrun, Einstein metrics and Mostow rigidity, Mathematical Research Letters, vol. 2, 1-8, 1995
  * [5] N. Aghanim, Y. Akrami, M. Ashdown, J. Aumot and others, Planck 2018 results. VI. Cosmological parameters, Astronomy and Astrophysics, vol. 641, A6, 2020.
  * [6] J. Ryan, S. Doshi, B. Ratra, Constraints on dark energy dynamics and spatial curvature from Hubble parameter and baryon acoustic oscillation data, Monthly Notices of the Royal Astronomical Society, vol. 480, pages 759-767, 2018.
  * [7] J.W. Percival, Cosmological structure formation in a homogeneous dark energy background, Astronomy & Astrophysics, vol. 443, 819-830, 2005.
  * [8] J.H. Macpherson, D.J. Price, P.D., Lasky, Einstein’s Universe: Cosmological structure formation in numerical relativity, Physical Review D, vol. 99, 063522, 2019,
  * [9] E. W. East, R. Wojtak, T. Abel, Comparing fully general relativistic and Newtonian calculations of structure formation, Physical Review D, vol. 97, 043509, 2018.
  * [10] E. Bertschinger, J.M. Gelb, Cosmological N-Body Simulations: Modern Simulations Use Millions of Particles to Follow Thousands of Galaxies in a Large Volume of Space, Computers in Physics, vol. 5, 164-179, 1991
  * [11] V. Springel, S. White, A. Jenkins, C.S. Frenk, N. Yoshida, L. Gao, J. Navarro, R. Thacker, D. Croton, J. Helly, Simulations of the formation, evolution and clustering of galaxies and quasars, nature, vol. 435, 629-636, 2005.
  * [12] L. L-Smith, H.V. Peiris, A. Pontzen, M. Lochner, Machine learning cosmological structure formation, Monthly Notices of the Royal Astronomical Society, vol. 479, 3405-3414, 2018.
  * [13] F. Kitaura, S. Hess, Cosmological structure formation with augmented Lagrangian perturbation theory, Monthly Notices of the Royal Astronomical Society: Letters, vol. 435, 78-82, 2013.
  * [14] T. Buchert, S. Räsänen, Backreaction in late-time cosmology, Annual Review of Nuclear and Particle Science, 62, 57-79, 2012.
  * [15] T. Buchert, A.A. Coley, H. Kleinert, F.B. Roukema, D.L. Wiltshire, Observational challenges for the standard FLRW model, The Fourteenth Marcel Grossmann Meeting On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories: Proceedings of the MG14 Meeting on General Relativity, University of Rome “La Sapienza”, Italy, 622-638, 2018
  * [16] A. Heinesen, T. Buchert, Solving the curvature and Hubble parameter inconsistencies through structure formation-induced curvature, Classical and Quantum Gravity, vol. 37, 164001, 2020.
  * [17] S.W. Hawking, Perturbations of an expanding universe, The Astrophysical Journal, vol. 145, page 544, 1966.
  * [18] S. Weinberg, Gravitation and cosmology: principles and applications of the general theory of relativity, 1972, Wiley New York.
  * [19] J.M. Bardeen,Gauge-invariant cosmological perturbations, Physical Review D, vol. 22, page 1882, 1980.
  * [20] H. Kodama, M. Sasaki, Cosmological perturbation theory, Progress of Theoretical Physics Supplement, vol. 78, pages 1-166, 1984.
  * [21] C. Ma, E. Bertschinger, Cosmological perturbation theory in the synchronous and conformal Newtonian gauges, arXiv preprint astro-ph/9506072, 1995
  * [22] E.Lifshitz, On the gravitational stability of the expanding universe, Perspectives in Theoretical Physics, pages 191-205, 1992.
  * [23] P. Mondal, The linear stability of the n+ 1 dimensional FLRW spacetimes, Classical and Quantum Gravity, vol. 38, 225009, 2021.
  * [24] H. Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Communications in Mathematical Physics, vol. 107, 587-609, 1986.
  * [25] D. Christodoulou, S. Klainerman, The global nonlinear stability of the Minkowski space, Séminaire Equations aux dérivées partielles (Polytechnique), pages 1-29, 1993.
  * [26] H. Lindblad, I. Rodnianski, The global stability of Minkowski space-time in harmonic gauge, Annals of mathematics, pages 1401-1477, 2010.
  * [27] L. Andersson, V. Moncrief, Future complete vacuum spacetimes, The Einstein equations and the large scale behavior of gravitational fields, 299-330, 2004, Springer.
  * [28] L. Andersson, V. Moncrief, Einstein spaces as attractors for the Einstein flow, Journal of differential geometry, vol. 89, 1-47, 2011.
  * [29] T.A. Oliynyk, Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant, Communications in Mathematical Physics, vol. 346, 293-312, 2016
  * [30] H. Ringström, Future stability of the Einstein-nonlinear scalar field system, Inventiones mathematicae, vol. 173, pages 123-208, 2008.
  * [31] D. Fajman, Z. Wyatt, Attractors of the Einstein-Klein-Gordon system, Communications in Partial Differential Equations, vol. 46, pages 1-30, 2021.
  * [32] D. Fajman, The nonvacuum Einstein flow on surfaces of negative curvature and nonlinear stability, Communications in Mathematical Physics, vol. 353, pages 905-961, 2017.
  * [33] P.G. LeFloch, C. Wei, Nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FLRW geometry, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 38, 787-814, 2021.
  * [34] I. Rodnianski, J. Speck, A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation, Annals of Mathematics, pages 65-156, 2018.
  * [35] I. Rodnianski, J. Speck, The stability of the irrotational Euler-Einstein system with a positive cosmological constant, Journal of the EMS, 15, 2013.
  * [36] J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Mathematica, vol. 18, pages 633-715, 2012.
  * [37] D. Christodoulou, The formation of shocks in 3-dimensional fluids, European Mathematical Society, vol. 2, 2007.
  * [38] D. Fajman, M. Ofner, Z. Wyatt, Slowly expanding stable dust spacetimes, arXiv preprint arXiv:2107.00457, 2021.
  * [39] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Courier Corporation, 2013.
  * [40] V. Moncrief, Convergence and Stability Issues in Mathematical Cosmology, in: General Relativity and Gravitation-A Centennial Perspective, A. Asthekar, B. Berger, J. Isenberg and M. MacCallum (editors), Cambridge University Press, Cambridge, UK, 480-498, 2015.
  * [41] V. Moncrief, P. Mondal, Could the Universe have an Exotic Topology?, Pure and Applied Mathematics Quarterly, vol. 15, 921-965, 2019.
  * [42] V. Moncrief, P. Mondal, Einstein flow with matter sources: stability and convergence, Philosophical Transactions of Royal Society A, 2021, accepted.
  * [43] M.T. Anderson, Geometrization of 3-manifolds via the Ricci flow, Notices AMS, vol. 51, 184-193, 2004.
  * [44] P.J.E Peebles, The large-scale structure of the universe, Princeton university press, vol. 98, 2020.
  * [45] X. Roy, T. Buchert, S. Carloni, N. Obadia, Global gravitational instability of FLRW backgrounds—interpreting the dark sectors, Classical and Quantum Gravity, vol. 28, pages 165004, 2011.
  * [46] P. Mondal, Attractors of the ‘n+ 1’dimensional Einstein-$\Lambda$ flow, Classical and Quantum Gravity, vol. 37, pages 235002, 2020.
  * [47] V. Moncrief, Hamiltonian formalism for relativistic perfect fluids, Physical Review D, vol. 16, 1702, 1977.
  * [48] J. Demaret, V. Moncrief, Hamiltonian formalism for perfect fluids in general relativity, Physical Review D, vol. 21, 2785, 1980.
  * [49] D.Bao, J. Marsden, R. Walton, The Hamiltonian structure of general relativistic perfect fluids, Communications in mathematical physics, vol. 99, pages 319-345, 1985.
  * [50] J.D, Brown, Action functionals for relativistic perfect fluids, Action functionals for relativistic perfect fluids, Classical and Quantum Gravity, vol. 10, 1579, 1993.
  * [51] E. Babichev, V. Dokuchaev, Y. Eroshenko, Dark energy cosmology with generalized linear equation of state, Classical and Quantum Gravity, vol. 22, pages 143, 2004.
  * [52] P. Chavanis, A simple model of universe with a polytropic equation of state, Journal of Physics: Conference Series, vol. 1030, pages 012009, 2018.
  * [53] T. Nakamura, T. Chiba, Determining the equation of state of the expanding universe: Inverse problem in cosmology, Monthly Notices of the Royal Astronomical Society, vol. 306, pages 696-700, 1999.
  * [54] P. Chavani, Models of universe with a polytropic equation of state: II. The late universe, The European physical journal plus, vol. 129, 1-39, 2014.
  * [55] R. Bartnik, Remarks on cosmological spacetimes and constant mean curvature surfaces, Communications in mathematical physics, vol. 117, pages 615-624, 1988.
  * [56] G. Galloway, E. Ling, Existence of CMC Cauchy surfaces from a spacetime curvature condition, General Relativity and Gravitation, vol. 50, pages 1-7, 2018.
  * [57] A.D Rendall, Constant mean curvature foliations in cosmological spacetimes, Helvetica Physica Acta, vol. 69, pages 490-500, 1996.
  * [58] L. Andersson, M. Iriondo, Existence of constant mean curvature hypersurfaces in asymptotically flat spacetimes, Annals of Global Analysis and Geometry, vol. 17, pages 503-538, 1999.
  * [59] D. Fajman, Local Well-Posedness for the Einstein–Vlasov System, SIAM Journal on Mathematical Analysis, vol. 48, 3270-3321, 2016.
  * [60] I. Rodnianski, J. Speck, Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems, Selecta Mathematica, vol. 24, pages 4293-4459, 2018.
  * [61] L. Andersson, V. Moncrief, Elliptic-hyperbolic systems and the Einstein equations, Annales Henri Poincaré, vol. 4, 1-34, 2003.
  * [62] R.M. Schoen, S.T. Yau, Lectures on harmonic maps, AMS, vol. 2, 1997
  * [63] J. Eells, J.H Sampson, Harmonic mappings of Riemannian manifolds, American journal of mathematics, vol. 86, pages 109-160, 1964
  * [64] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer Science & Business Media, vol. 53, 2012.
  * [65] J.E Mardsen, M. McCracken, The Hopf bifurcation and its applications, vol. 19, 2012, Springer Science.
  * [66] O. E. Lanford, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens, Nonlinear problems in the physical sciences and biology, pages 159-192, 1973, Springer.
  * [67] Y. Choquet-Bruhat, General relativity and the Einstein equations, OUP Oxford, 2008.
  * [68] D. Christodoulou, The Action Principle and Partial Differential Equations, Princeton University Press, vol. 146, 2016.
  * [69] R.S. Palais, Foundations of global non-linear analysis, W.A. Benjamin, Inc., New York-Amsterdam, 1968.
  * [70] M. Taylor, Pseudodifferential operators and nonlinear PDE, Springer, vol. 100, 2012.
  * [71] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Communications on Pure and Applied Mathematics, vol. 41, 891-907, 1988.
  * [72] V. Moncrief, Gribov degeneracies: Coulomb gauge conditions and initial value constraints, Journal of Mathematical Physics, vol. 20, 579-585, 1979
  * [73] A. Fischer, V. Moncrief, Quantum conformal superspace, General Relativity and Gravitation, vol. 28, 221-237, 1996.
  * [74] D. Fajman, T.A. Oliynyk, Z. Wyatt, Stabilizing relativistic fluids on spacetimes with non-accelerated expansion, Communications in Mathematical Physics, vol. 383, 401-426, 2021.
  * [75] J. Speck, The stabilizing effect of spacetime expansion on relativistic fluids with sharp results for the radiation equation of state, Archive for Rational Mechanics and Analysis, vol. 210, 535-579, 2013.
  * [76] A.K. Raychaudhuri, S. Banerji, A. Banerjee, General relativity, astrophysics, and cosmology, Springer Science, 2003.
  * [77] C. Lübbe, J.A.V. Kroon, A conformal approach for the analysis of the non-linear stability of radiation cosmologies, Annals of Physics, vol. 328, 2013.
  * [78] T. A. Oliynyk, Future Global Stability for Relativistic Perfect Fluids with Linear Equations of State $p=K\rho$ where $1/3<K<1/2$, SIAM Journal on Mathematical Analysis, vol. 53, 4118-4141.
  * [79] M. Hadžić, J. Speck, The global future stability of the FLRW solutions to the Dust-Einstein system with a positive cosmological constant, Journal of Hyperbolic Differential Equations, vol. 12, 87-188, 2015.