# A Relativistic Quantum Approach to Neutrino and Antineutrino Emission via
the Direct Urca Process in Strongly Magnetized Neutron-Star Matter

Tomoyuki Maruyama  A. Baha Balantekin  Myung-Ki Cheoun  Toshitaka Kajino
Motohiko Kusakabe  Grant J. Mathews  College of Bioresource Sciences, Nihon
University, Fujisawa 252-0880, Japan  National Astronomical Observatory of
Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Advanced Science Research
Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
Department of Physics and OMEG Institute, Soongsil University, Seoul, 156-743,
Korea  School of Physics, Int. Research Center for Big-Bang Cosmology and
Element Genesis, Beihang University, Beijing 100083, China  Graduate School of
Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Center of Astrophysics, Department of Physics, University of Notre Dame, Notre
Dame, IN 46556, USA

###### Abstract

We study neutrino and antineutrino emission from the direct Urca process in
neutron-star matter in the presence of strong magnetic fields. We calculate
the neutrino emissivity of the direct Urca process, whereby a neutron converts
to a proton, an electron and an antineutrino, or a proton-electron pair
converts to a neutron-neutrino pair. We solve exact wave functions for protons
and electrons in the states described with Landau levels. We find that the
direct Urca process can satisfy the kinematic constraints even in density
regions where this process could not normally occur in the absence of a
magnetic field.

###### keywords:

Neutron-Star, Direct Urca, Neutrino and Antineutrino Emissions, Strong
Magnetic Field, Relativistic Quantum Approach

Neutron stars (NSs) are cooled by neutrino and/or anti-electron neutrino
emission. At very low temperatures the neutrino mean-free-path is very long.
In this case neutrinos are easily emitted from the core regions of NSs. Since
neutrino emission rates depend on circumstances inside NSs, the study of NS
cooling through neutrino emission gives important information for constraining
internal NS structure [1].

There are several kinds of the cooling processes [2]. Previous works have
studied decay processes for the neutrino or the anti-neutrino emission such as
the direct Urca (DU) process ($n\rightarrow p+e^{-}+{\bar{\nu}}_{e}$,
$p+e^{-}\rightarrow n+\nu_{e}$), the modified Urca process (MU)
($n+N\rightarrow p+e^{-}+N^{\prime}+{\bar{\nu}}_{e}$, $p+e^{-}+N\rightarrow
n+N^{\prime}+{\nu}_{e}$ ) [3], the neutrino-pair emission process
($N_{1}+N_{2}\rightarrow N_{1}^{\prime}+N_{2}^{\prime}+\nu+{\bar{\nu}}$) [4,
5] and so on. Furthermore, there are many approaches which combine the above
processes with other mechanisms. For example, one of them is to consider the
superfluidity [6] produced by neutron and proton pairing correlations. This is
known to lead to a strong reduction of the neutrino emissivity and affect the
neutrino-antineutrino emission by the Cooper pair breaking and formation
mechanism [7].

The DU process is one of the most feasible candidates to explain the rapid
cooling of NSs [8, 9]. One usually examines the rapid cooling associated with
the DU process to compare temperatures and ages of isolated NSs. Recently
another method has been realized by observing NSs in binary systems. Indeed,
by analyzing the x-ray emission in MXB 1659-29, evidence of the direct Urca
process has been found [10]

Because of energy-momentum conservation and Fermi statistics for NS matter
composed of protons, neutrons and electrons, the DU process occurs in a
density region, where the proton density $\rho_{p}$ is larger than 1/8 of the
neutron density $\rho_{n}$ ($\rho_{p}\geq\rho_{n}/8$); this condition cannot
be achieved in low density regions. However, if the symmetry energy is
proportional to the density, and its slope is sufficiently large, the
condition for the DU process can be satisfied even at rather low density. Many
other approaches have been proposed to satisfy this condition by considering
other phases such as pion [11] or kaon condensation [12, 13] and matter
including hyperon degrees of freedom [6, 14]. However, the latter case turns
out to affect the masses of NSs, which is closely related to the hyperon
puzzle in NSs [15].

On the other hand, magnetars, which are associated with a very strong magnetic
field [16, 17], have properties different from normal NSs. The strength of
their magnetic field is about $10^{14}-10^{15}$ G in the surface region, and
can reach $10^{17}$ G inside the star. Soft gamma repeaters (SGR) and
anomalous X-ray pulsars (AXPs) correspond to magnetars [18]. Magnetars emit
energetic photons. Furthermore, the surface temperature of magnetars is
$T\approx 0.28-0.72~{}$keV, which is larger than those of normal NSs $T\approx
0.01-0.15~{}$keV at a similar age [19].

There are a number of theoretical works that show the influence of the
magnetic field on the equation of state (EOS) of NS matter [20, 21, 22, 23,
24, 25, 26, 27] and how the structure of NSs is altered by including magnetic
fields [28, 29] . We note however, that since our purpose here is to explore
effects of the DU in NS matter and not the structure of NSs, we do not re-
examine NS structure in this study. For the reader interested in magnetic
effects on NS structure a list of works can be found in Ref. [28] (see also
the ”Lorene” website: https://lorene.obspm.fr/data/magnetNS.html).

A strong magnetic field can supply energy and momentum into the cooling
process and changes the restriction caused by energy-momentum conservation by
introducing transitions between Landau levels [30, 31, 32]. Indeed, the
magnetic field influences the beta-decay process in NSs [24] and axion
emission from magnetars [33]. Leinson [34] studied the effect of magnetic
fields perturbatively and showed that a magnetic field increases the
antineutrino emissivity. Thus, the associated strong magnetic fields may play
a significant role in the cooling of magnetars.

In a previous paper [30] we have studied $\nu{\bar{\nu}}$-pair emissions from
the transition between two Landau levels in strongly magnetized NS matter. In
that work, when the strength of the magnetic field is about $B\sim 10^{15}$ G,
emission energies of this process are larger than those of the MU process with
zero magnetic field, $B=0$. All of this suggests that the magnetic field may
increase the emission of particles because the strong magnetic field can
supply momentum into the emission processes.

Leinson and Pérez [35] calculated the neutrino emissivity from the DU process
in strong magnetic fields, and showed that the emissivity becomes larger as
the magnetic field increases. In that calculation the magnetic field was so
strong that all charged particles populated the lowest Landau level. This
means that one has not yet estimated realistic magnetic effects in NSs with
realistic magnetic field strengths.

In the present paper, therefore, we apply our quantum theoretical approach to
the DU process in strong magnetic fields and calculate it through the
transition between Landau levels for electrons and protons. As noted above,
only this quantum approach can exactly describe the momentum transfer from the
magnetic field.

The low energy Lagrangian of the weak interaction between baryons and leptons
is written as

$\displaystyle{\cal L}_{W}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\sum_{l_{1},l_{2}}\mbox{$\overline{\psi}$}_{l_{1}}\gamma_{\mu}(1-\gamma_{5})\psi_{l_{2}}\sum_{\alpha_{1}\alpha_{2}}\mbox{$\overline{\psi}$}_{\alpha_{1}}\gamma_{\mu}(c_{V}-c_{A}\gamma_{5})\psi_{\alpha_{2}},$
(1)

where $\psi_{l_{1.2}}$ is the field of the lepton $l_{1,2}$ which denotes the
electron or muon or neutrino, and $\psi_{\alpha}$ is the field of the baryon
$\alpha$ which denotes the proton or neutron, $G_{F}$ is the Fermi coupling
constant, while $c_{V}$ and $c_{A}$ are the vector and axial-vector coupling
constants which include the contribution from the Cabbibo angle [36].

In the single particle model, the $l$-neutrino emissivity of the DU process is
written as

$\displaystyle\epsilon_{DU}$ $\displaystyle=$ $\displaystyle
2\sum_{\alpha_{n},\alpha_{p},\alpha_{l},\alpha_{\nu}}{\cal
F}_{n}(E_{n},E_{p},E_{l})e_{\nu}(2\pi)\delta(E_{n}-E_{p}-E_{l}-e_{\nu})$ (2)
$\displaystyle\qquad\times\left|G_{F}\int
d^{3}\mbox{$\bm{r}$}\left[\mbox{$\overline{\psi}$}^{(l)}_{\alpha,_{l}}(\mbox{$\bm{r}$})\gamma_{\mu}(1-\gamma_{5})\psi^{(\nu)}_{\alpha_{\nu}}(\mbox{$\bm{r}$})\right]\left[\mbox{$\overline{\psi}$}^{(p)}_{n_{p}}(\mbox{$\bm{r}$})\gamma_{\mu}(c_{V}-c_{A}\gamma_{5})\psi^{(n)}_{\alpha_{n}}(\mbox{$\bm{r}$})\right]\right|^{2},$

with

${\cal
F}_{n}(E_{n},E_{p},E_{l})=n_{n}(E_{n})[1-n_{p}(E_{p})][1-n_{e}(E_{l})],$ (3)

where the subscript $l$ indicates the electron or muon, while $\nu$ indicates
the $\nu_{e}$ or $\nu_{\mu}$, $\psi_{a}$ is the single particle wave-function,
$E_{a}$ is the energy of the particle $a(=n,p,e,\mu,\nu_{e},\nu_{\mu})$, and
$\alpha_{a}$ denotes the quantum numbers such as momentum, spin, iso-spin and
Landau-number. The quantity $n_{a}(E)$ is the Fermi-distribution function for
the particle $a$,

$n_{a}(E)=\frac{1}{1+e^{(E-\mu_{a})/T}},$ (4)

where $T$ is the temperature, and $\mu_{a}$ is the chemical potential of
particle $a$.

We assume a uniform magnetic field along the $z$-direction,
$\mbox{$\bm{B}$}=(0,0,B)$, and take the electro-magnetic vector potential
$A^{\mu}$ to be $A=(0,0,xB,0)$ at the position $\mbox{$\bm{r}$}\equiv(x,y,z)$.
The relativistic wave function $\psi$ is obtained from the following Dirac
equation:

$\left[\gamma_{\mu}\cdot(i\partial^{\mu}-\zeta
eA^{\mu}-U_{0}\delta_{0}^{\mu})-M+U_{s}-\frac{e\kappa}{4M}\sigma_{\mu\nu}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})\right]\psi_{a}(x)=0,$
(5)

where $\kappa$ is the anomalous magnetic moment (AMM), $e$ is the elementary
charge and $\zeta=\pm 1$ or $0$ is the sign of the particle charge. $U_{s}$
and $U_{0}$ are the scalar field and time component of the vector field,
respectively. These are determined from relativistic mean-field (RMF) theory
[37], although the AMM and the mean-fields are taken to be zero for charged
leptons.

The single particle energy for a charged particle ($\zeta=\pm 1$) is then
written as

$E(n,p_{z},s)=E^{*}+U_{0}=\sqrt{p_{z}^{2}+\left(\sqrt{2eBn+M^{*2}}+se\kappa
B/2M\right)^{2}}+U_{0},$ (6)

with $M^{*}=M-U_{s}$, where $n$ is the Landau number, $p_{z}$ is the
$z$-component of momentum, and $s=\pm 1$ is the spin. Furthermore, the wave-
function overlap at points $\mbox{$\bm{r}$}_{1}$ and $\mbox{$\bm{r}$}_{2}$ for
charged particles such as the proton and electron are written as

$\psi^{(a)}_{n,s,p_{z}}(\mbox{$\bm{r}$}_{1})\mbox{$\overline{\psi}$}^{(a)}_{n,s,p_{z}}(\mbox{$\bm{r}$}_{2})=\frac{e^{i(p_{y}y+p_{z}z)}}{\sqrt{R_{y}R_{z}}}{\hat{F}}\left(x_{1}-p_{y}/eB\right)\frac{\rho^{(a)}_{M}}{4E}{\hat{F}}\left(x_{2}-p_{y}/eB\right)$
(7)

with

$\displaystyle\rho_{M}(n,s,P_{z})$ $\displaystyle=$
$\displaystyle\left[E^{*}\gamma^{0}-\zeta\sqrt{2eBn}\gamma^{2}-p_{z}\gamma^{3}+M^{*}+\frac{e\kappa
B}{2M}\Sigma_{z}\right]$
$\displaystyle\qquad\times\left[1+\frac{s}{\sqrt{2eBn+{M^{*}}^{2}}}\left(\frac{e\kappa
B}{2M}+p_{z}\gamma_{5}\gamma^{0}-E^{*}\gamma_{5}\gamma^{3}\right)\right],$
$\displaystyle{\hat{F}}$ $\displaystyle=$ $\displaystyle{\rm
diag}\left(f_{n},f_{n-1},f_{n},f_{n-1}\right)=f_{n}\frac{1+\Sigma_{z}}{2}+f_{n-1}\frac{1-\Sigma_{z}}{2}\quad({\rm
proton}),$ (8) $\displaystyle{\hat{F}}$ $\displaystyle=$ $\displaystyle{\rm
diag}\left(f_{n-1},f_{n},f_{n-1},f_{n}\right)=f_{n-1}\frac{1+\Sigma_{z}}{2}+f_{n}\frac{1-\Sigma_{z}}{2}\quad({\rm
electron}),$ (9)

where $\Sigma_{z}={\rm diag}(1,-1,1,-1)$, and $R_{a}$ ($a=x,y,x$) is the size
of the system along the $a$-direction.

For neutral particles ($\zeta=0$) such as the neutron, the wave function is
written as an eigenstate of momentum
$\mbox{$\bm{p}$}\equiv(\mbox{$\bm{p}$}_{T},p_{z})$. So, the single particle
energy is given by

$E(\mbox{$\bm{p}$},s)=E^{*}+U_{0}=\sqrt{p_{z}^{2}+\left(\sqrt{\mbox{$\bm{p}$}_{T}^{2}+M^{*2}}+se\kappa
B/2M\right)^{2}}+U_{0},$ (10)

and the wave-function overlap becomes

$\displaystyle\psi_{\bf p,s}(\mbox{$\bm{r}$}_{1})\mbox{$\overline{\psi}$}_{\bf
p,s}(\mbox{$\bm{r}$}_{2})$ $\displaystyle=$ $\displaystyle\frac{e^{i\bf
p\cdot(\bf r_{1}-\bf
r_{2})}}{\sqrt{R_{x}R_{y}R_{z}}}\left[E^{*}\gamma_{0}-\mbox{$\bm{p}$}\cdot\mbox{$\bm{\gamma}$}+M^{*}+\frac{e\kappa
B}{2M}\Sigma_{z}\right]$ (11)
$\displaystyle\qquad\times\left\\{1+\frac{s}{\sqrt{{\bf
p}_{T}^{2}+{M^{*}}^{2}}}\left[\frac{e\kappa
B}{2M}+\gamma_{5}\left(p_{z}\gamma^{0}-E^{*}\gamma^{3}\right)\right]\right\\}.$

Substituting the above wave-functions into Eq. (2), we obtain

$\displaystyle\epsilon_{DU}$ $\displaystyle=$
$\displaystyle\frac{2(2\pi)^{3}G_{F}^{2}(eB)^{2}}{R_{x}^{2}R_{y}^{4}R_{z}^{4}}\sum_{n_{p},n_{e}}\sum_{s_{n},s_{p}s_{e}}\left\\{\prod_{a=n,\nu}\left[R_{x}R_{y}R_{z}\int\frac{d^{3}p_{a}}{(2\pi)^{3}}\right]\right\\}\left\\{\prod_{b=p,e}\left[R_{z}\int\frac{dp_{bz}}{2\pi}R_{y}\int^{eBR_{x}/2}_{-eBR_{x}/2}\frac{dp_{by}}{2\pi}\right]\right\\}$
(12) $\displaystyle\quad\times e_{\nu}{\cal
F}_{n}(E_{n},E_{p},E_{l})\delta(E_{n}-E_{p}-E_{l}-e_{\nu})\delta(p_{ny}-p_{py}-p_{ly}-p_{\nu
y})\delta(p_{nz}-p_{pz}-p_{lz}-p_{\nu z})$
$\displaystyle\quad\times\frac{1}{16^{3}E_{n}E_{p}E_{l}e_{\nu}}\sum_{i_{1},j_{1},i_{2},j_{2}(=\pm
1)}{\cal M}(j_{1},i_{1}){\cal M}^{*}(j_{2},i_{2})L_{\mu\nu}N^{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{G_{F}^{2}(eB)^{3}}{4(2\pi)^{5}}\sum_{n_{p},n_{l}}\sum_{s_{n},s_{p},s_{l}}\int\frac{dp_{nT}p_{nT}dp_{pz}dp_{lz}de_{\nu}e_{\nu}^{3}}{E^{*}_{n}E^{*}_{p}E_{l}e_{\nu}}\frac{d\Omega_{\nu}}{4\pi}{\cal
F}_{n}(E_{n},E_{p},E_{l})$
$\displaystyle\qquad\times\delta(E_{n}-E_{p}-E_{l}-e_{\nu})\sum_{i_{1},j_{1},i_{2},j_{2}}{\cal
M}(j_{1},i_{1}){\cal
M}^{*}(j_{2},i_{2})L_{\mu\nu}(i_{1},i_{2})N^{\mu\nu}(j_{1},j_{2}),$

with

$\displaystyle{\cal M}(j_{p},j_{l})$ $\displaystyle=$ $\displaystyle\int
dxf_{n_{e}+(j_{l}-1)/2}\left(x+\frac{p_{nT}}{\sqrt{2eB}}\right)f_{n_{p}+(j_{p}-1)/2}\left(x-\frac{p_{nT}}{\sqrt{2eB}}\right),$
(13) $\displaystyle L_{\mu\nu}(j_{1},j_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{16}{\rm
Tr}\left\\{\rho_{M}^{(e)}(n_{f},s_{f},k_{e})(1+j_{1}\Sigma_{z})\gamma_{\mu}(1-\gamma_{5})\ooalign{\hfil/\hfil\crcr{k}}_{\nu}\gamma_{\nu}(1-\gamma_{5})(1+j_{2}\Sigma_{z})\right\\},$
(16) $\displaystyle N_{\mu\nu}(j_{1},j_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{16}{\rm
Tr}\left\\{(1+j_{2}\Sigma_{z})\rho_{M}^{(p)}(n_{i},s_{i},P_{iz})(1+j_{1}\Sigma_{z})\gamma_{\mu}(c_{V}-c_{A}\gamma_{5})\right.$
(17)
$\displaystyle\qquad\quad\times\left.\rho_{M}^{(n)}\gamma_{\nu}(c_{V}-c_{A}\gamma_{5})\right\\},$

where $R_{x}=R_{y}=R_{z}$ is assumed, and $f_{n}$ is the $n$-th harmonic
oscillator wave function. In addition, the neutron momentum is taken to be
$p_{n}=(E_{n},0,p_{nT},p_{nz})$, where without loss of generality the neutron
transverse momentum is assumed to be directed along the $y$-axis.

In cool NSs the temperature is less than 1 keV, and the emitted neutrino
energy is of the order of the temperature $e_{\nu}\sim T$, so that we can take
the lowest order term of the temperature by using the following approximation

$e_{\nu}^{3}{\cal F}_{n}(E_{n},E_{p},E_{l})\approx{\cal
I}_{DU}\delta(E_{n}-\mu_{n})\delta(E_{p}-\mu_{p})\delta(e_{e}-\mu_{e}),$ (18)

with

${\cal I}_{DU}=\int dE_{1}dE_{2}dE_{3}(E_{1}-E_{2}-E_{3})^{3}{\cal
F}_{n}(E_{1},E_{2},E_{3})\approx\frac{457}{5040}\pi^{6}T^{6}.$ (19)

Therefore, the neutrino emissivity in the low temperature limit can be written
as

$\displaystyle\epsilon_{DU}$ $\displaystyle=$
$\displaystyle\frac{457\pi}{10080}G_{F}^{2}T^{6}\sum_{n_{l},n_{p}}\frac{p_{nT}}{p_{pz}p_{lz}\sqrt{p_{nT}^{2}+{M_{n}^{*}}^{2}}}$
(20)
$\displaystyle\qquad\qquad\qquad\times\frac{1}{2^{6}}\sum_{s}\sum_{i,j}\int\frac{d\Omega_{\nu}}{4\pi}{\cal
M}(j_{1},i_{1}){\cal M}^{*}(j_{2},i_{2})\frac{L_{\mu\nu}N^{\mu\nu}}{e_{\nu}}.$

In addition, we assume chemical equilibrium for the system. Hence, the
chemical potentials satisfy the following relations:

$\mu_{n}=\mu_{n}^{*}+U_{0}(n)=\mu_{p}+\mu_{l}=\mu_{p}^{*}+\mu_{l}+U_{0}(p).$
(21)

At the low temperature limit, the chemical potentials are taken to be the
Fermi energies at zero temperature, and the effective chemical potentials for
baryons $\mu^{*}$ in the presence of a magnetic field are written as

$\displaystyle\mu_{p}^{*}(n_{p},k_{pz},s)$ $\displaystyle=$
$\displaystyle\sqrt{\left(\sqrt{2eBn_{p}+{M_{p}^{*}}^{2}}+se\kappa_{p}/2M_{p}\right)^{2}+p_{pz}^{2}},$
(22) $\displaystyle\mu_{n}^{*}(k_{nT},k_{nz},s)$ $\displaystyle=$
$\displaystyle\sqrt{\left(\sqrt{k_{nT}^{2}+{M_{n}^{*}}^{2}}+se\kappa_{n}/2M_{n}\right)^{2}+k_{nz}^{2}},$
(23)

where $s=\pm 1$, $n_{p}$ is the Landau number of the proton, and $k_{pz}$ is
the $z$-component of the proton Fermi momentum which depends on $n_{p}$. In
addition, $k_{nT}$ and $k_{nz}$ are the transverse component and $z$-component
of the neutron Fermi momentum, whose values are not the same. That is the
momentum-distribution breaks the spherical symmetry through the AMM.

For comparison we also show the neutrino emissivity from the DU process when
$B=0$ as

$\displaystyle\epsilon_{DU}$ $\displaystyle=$
$\displaystyle\frac{457\pi}{10080}G_{F}^{2}T^{6}\left\\{\frac{(c_{V}-c_{A})^{2}}{2}\mu^{*}_{p}\left({M_{n}^{*}}^{2}-{M_{p}^{*}}^{2}+m_{l}^{2}-2\mu^{*}_{p}\Delta
U_{0}-\Delta U_{0}^{2}\right)\right.$ (24)
$\displaystyle\left.\quad+\frac{(c_{V}+c_{A})^{2}}{2}\mu^{*}_{n}\left({M_{n}^{*}}^{2}-{M_{p}^{*}}^{2}-m_{l}^{2}-2\mu^{*}_{n}\Delta
U_{0}+\Delta
U_{0}^{2}\right)-(c_{V}^{2}-c_{A}^{2})\mu_{l}M^{*}_{p}M^{*}_{n}\right\\},$

where $\Delta U_{0}=U_{0}(p)-U_{0}(n)$; this expression is equivalent to that
of Ref. [38]. This emissivity is non-zero $\epsilon_{DU}>0$ when $p_{F}(n)\leq
p_{F}(p)+p_{F}(l)$ with $p_{F}(a)$, the Fermi momentum of the particle $a$.
This momentum matching condition determines the density region for the DU
process when $B=0$. In contrast, for $B\neq 0$, the magnetic field carries
momentum and breaks the momentum matching condition, and the DU process can
occur in any density region.

In this work we discuss the relation between the DU emissivity and the density
dependence of the symmetry energy as well as the effects of the magnetic
field. For the calculation of nuclear matter, we use the following Lagrangian
density in the relativistic mean-field approach:

$\displaystyle{\cal L}_{RMF}$ $\displaystyle=$
$\displaystyle\mbox{$\overline{\psi}$}\left(i\gamma\cdot\partial-M+g_{s}\sigma\right)\psi+\tilde{U}[\sigma]+\frac{g_{V}^{2}}{2m_{V}^{2}}\left(\mbox{$\overline{\psi}$}\gamma\psi\right)^{2}-\frac{C_{s}^{IV}}{2M^{2}}\left(\mbox{$\overline{\psi}$}\tau\psi\right)^{2}-\frac{C_{v}^{IV}}{2M^{2}}\left(\mbox{$\overline{\psi}$}\gamma\tau\psi\right)^{2},$
(25)

where $\tilde{U}[\sigma]$ is the self-energy term of the sigma meson which
includes the non-linear effect for the scalar mean-field, and the other terms
are written as the zero-range interaction between two nucleons. In addition,
$C_{s}^{IV}$ and $C_{v}^{IV}$ are the coupling constants for the iso-vector
interaction in the Lorentz scalar and vector channels, respectively [39].
Then, the scalar-field $U_{s}$ and the time component of the vector-field are
given by

$\displaystyle U_{s}$ $\displaystyle=$ $\displaystyle
g_{s}\sigma+\frac{C_{s}^{IV}}{2M^{2}}(\rho_{s}(p)-\rho_{s}(n))\tau_{z}~{}~{},$
(26) $\displaystyle U_{0}$ $\displaystyle=$
$\displaystyle\frac{g_{V}^{2}}{m_{V}^{2}}(\rho_{p}+\rho_{n})+\frac{C_{v}^{IV}}{2M^{2}}(\rho_{p}-\rho_{n})\tau_{z}.$
(27)

In the above formalism we omit the magnetic field energy density
$B^{2}/(8\pi)$ in the EOS. This term contributes to the pressure and affects
the composition of NS matter significantly when $B\gtrsim 10^{18}$ G [27]. In
the present calculation we adopt a smaller strength of the magnetic field, and
this term does not contribute to the neutrino emissivity.

Parameter-Set |  $C_{s}^{IV}$ | $C_{v}^{IV}$ |  $L$ (MeV)
---|---|---|---
SF1 | 0 | 20.42 | 92.7
SF2 | 23.61 | 0 | 84.1
SF3 | 33.02 | -8.154 | 81.0
Table 1: The parameter-sets for the iso-vector parts of the mean-fields. All
parameter-sets give a symmetry energy of 32 MeV at normal nuclear density. The
fourth column shows the slope parameter of the symmetry energy.  Figure 1:
The density dependence of the symmetry energy. The solid, dot-dashed and
dashed lines represent the results with the parameter sets SF1, SF2 and SF3,
respectively.

We use the parameter-set PM1 for symmetric matter as given in Ref. [40]. In
addition, we give the three parameter-sets, SF1, SF2 and SF3, for the other
parameters of the iso-vector parts, $C_{s}^{IV}$ and $C_{v}^{IV}$, whose
detailed values are shown in Table 1 . The symmetry energy is fixed to be
$e_{sym}=32$ MeV at normal nuclear density though the parameter-sets give
different density dependences of the symmetry energy. The SF1 includes only
the Lorentz vector channel ($C_{s}^{IV}=0$), the SF2 includes only the iso-
vector Lorentz scalar channel ($C_{v}^{IV}=0$), and the SF3 includes the
negative value of the Lorentz vector channel ($C_{v}^{IV}<0<C_{s}^{IV}$). In
addition, we list the values of the slope parameter $L$ for these parameter-
sets.

In Fig. 1 we show the density-dependence of the symmetry energy, We see that
the three parameter-sets give quite different density dependences of the
symmetry energy though all of them satisfy 32 MeV for the symmetry energy at
normal nuclear density. This means that the parameter-sets, SF1, SF2 and SF3,
represent different EOSs.

In actual calculations we substitute the mean-fields and the chemical
potentials at $B=0$ with these parameter-sets into Eq. (20) and calculate the
neutrino emissivity. As mentioned above, surface temperatures of magnetars are
$T\approx 0.28-0.72~{}$keV [19]. Thus, we choose $T=0.50$ keV for the present
calculations.

Figure 2:  (a) The density dependence of the neutrino emissivity in the DU
process at $T=0.50$ keV and (b) that of the proton fraction of NS matter at
$B=0$. The vertical lines represent the results with the SF1, SF2 and SF3,
respectively. In the upper panel (a) the solid and dashed lines indicate the
results when $B=10^{17}$G and $B=0$, respectively. Note that the dashed line
for the SF3 is not shown because $\epsilon_{DU}=$0.

First, we restrict the matter to consist of only protons, neutrons and
electrons. In Fig. 2 we show the density dependence of the neutrino emissivity
in the DU process (a) and the density dependence of the proton fraction in NS
matter, $x_{p}$ (b). In Fig. 2(a), the dashed line indicates the results for
$B=0$, which appear in the density region of $x_{p}\geq 1/9$; $x_{p}=1/9$ is
denoted by the horizontal dashed line in the lower panel (b). For future
reference, we define $\rho_{DU}$ as the critical density at which the proton
fraction $x_{p}=1/9$.

The calculations result in large fluctuations, which also appears the in
$\nu{\bar{\nu}}$-pair [30] and axion emission [31]. These fluctuations reflect
the density of states in the $xy$-plane given by the Landau levels [32, 41].

Figure 3:  The density dependence of neutrino emissivity in the DU process at
$T=500$ eV and $B=10^{17}$ G (a) for PM1-SF1. The thin solid blue and dashed
lines indicate the same quantities as those in Fig. 2a while the scale is
linear. The thick solid red line represents averaging over the density
dependence of the emissivities (thin solid blue line) using a width of $0.01$
fm-3.

When $B=0$, the results of the neutrino emissivity for SF1 and SF2 suddenly
appear at the density $\rho_{DU}$. They increase very rapidly and become
almost flat as the density increases. When $B=10^{17}$G, the density
dependence of the neutrino emissivity has comb-like-shapes which are due to
the fact that the neutrinos are emitted through transitions between Landau
levels. The emissivity appears at a density of $\rho<\rho_{DU}$ and becomes
larger gradually with increasing density. In the density region
$\rho\geq\rho_{DU}$, they undergo large oscillations though their local minima
almost agree with the $B=0$ results.

For SF3, on the other hand, the proton fraction does not exceed $1/9$, and the
DU does not occur when $B=0$. When $B=10^{17}$G, however, neutrinos are
emitted, and thus, we can confirm that the magnetic-field increases the
neutrino emission. In addition, the peak position of the neutrino emissivity
agrees with that of the proton fraction, namely the effect of the magnetic
field is related to the proton fraction and can appear when the proton
fraction is close to $1/9$.

In Fig. 3 we plot the results for SF1 on a linear scale. Here one can see that
the spike of the neutrino emissivity is very narrow, and that its height is
much larger than that with zero-magnetic field. To estimate the strength of
the neutrino emission, we smooth the emissivity according to the following
equation:

${\bar{\epsilon}}_{DU}(\rho_{B})=\frac{1}{\Delta\rho}\int_{\rho_{B}-\Delta\rho/2}^{\rho_{B}+\Delta\rho/2}d\rho\epsilon_{DU}(\rho).$
(28)

We show this result for SF1 when $\Delta\rho=0.01$ fm-3 with a solid line. The
smoothed value for the emissivity is still larger than that when $B=0$, but
the difference is not very drastic.

Figure 4:  Proton fractions (a) and lepton fractions (b) in NS matter with
muons. In the upper panel the solid and dashed lines represent the proton
fractions in NS matter with and without muons, respectively. In the lower
panel the thin solid and dotted lines represent the electron and muon
fractions. Lines represent the results from the SF1, SF2 and SF3 EoS as
labeled.

Here, we add the following comment. In the study of pion [42, 43], axion [31]
and $\nu{\bar{\nu}}$-pair production [30] we showed a very large difference in
the results between cases including the AMM of the protons and those not
including it. In this calculation there is no significant difference between
the cases with and without the AMM. Without a magnetic field the above
particle production is forbidden by energy-momentum conservation. Thus, only
the argument of the overlap function of Eq. (13) in the tail region
contributes to the transition strength, and a small difference in input values
makes a very large difference in the calculation results. In the DU process
the transitions are not completely forbidden by the kinematics, and the value
of the argument $p_{nT}/\sqrt{2eB}$ is not located in the tail region.

As a next step we study the DU process in NS matter including muons. In Fig. 4
we show the proton fraction in such matter in the upper panel and the electron
and muon densities per baryon in the lower panel. The solid and open circles
show the critical densities of $\rho_{DU}$ for the electron and muon neutrino
emission, respectively. In matter containing muons the proton fraction becomes
larger, and the critical density $\rho_{DU}$ becomes a little lower than in
matter without muons. Here, we note that the proton fraction in the SF3 EOS is
also larger and exceeds $1/9$, but the DU does not appear.

In order to examine the dependence on the magnetic field strength, in Fig. 5
we show the results at $B=10^{17}$, $B=10^{16}$ and $B=10^{15}$ G for NS
matter without muons (a) and with muons (b). The results for SF3 when
$B=10^{15}$ G are too small to appear in this figure. For comparison, we also
show the neutrino luminosities from the MU process [44] with the long-dashed
lines. When $B\lesssim 10^{15}$ G, the emissivities do not appear for
$\rho_{B}<\rho_{DU}$, and the magnetic field does not play an important role
for the DU process.

The emissivities from the NS matter with muons are larger than those without
muons. Particularly, the emissivity for SF3 at $B=10^{16}$ G is much larger in
the matter with muons than that without muons though the contribution from the
muon process is negligibly small.

As the strength of the magnetic field decreases, the amplitudes of the
oscillations in the density dependence become smaller for both magnetic field
strengths in NS matter. In the density region, $\rho_{B}<\rho_{DU}$, the
neutrino emissivities are largely suppressed, and the DU process dominates in
a narrower density region, as shown in the results by SF3, where the
emissivities in the DU process are still much larger than that of the MU
process.

Here, we observe the following facts: As the magnetic field strength
decreases, the emission rate becomes larger for the $\nu{\bar{\nu}}$-pairs
[30] despite a smaller momentum transfer from the magnetic field. The
$\nu{\bar{\nu}}$-pairs are produced via the transition of the proton and
electron between different Landau levels; namely the initial and final
particles are the same. As the magnetic field strength decreases, the energy
interval between the initial and final states also becomes smaller. Hence, the
emission strength becomes larger. In the DU process, however, the initial
particle is a neutron which does not stay in the Landau level, and the energy
interval is continuous, so that this effect does not appear.

Figure 5:  Density dependence of the neutrino emissivity in the DU process at
$T=500$ eV for the PM1-SF1 EoS and for matter containing only protons,
neutrons and electrons (a) and matter also containing muons (b). The solid,
dotted and long-dashed lines represent the results for magnetic field
strengths, $B=10^{17}$, $10^{16}$ and $10^{15}$ G, respectively, and the
dashed lines indicate the results without a magnetic field. The long-dashed
line shows the neutrino emissivity from the MU process.

In summary, we have used a relativistic quantum approach to study neutrino and
anti-neutrino emission in the DU process from NS matter with strong magnetic
fields, $B=10^{15}-10^{17}$ G. We use three parameter-sets for the symmetric
nuclear force to illustrate the relation between the proton fraction and the
neutrino emissivity. If the proton fraction satisfies the DU condition
$x_{p}\geq 1/9$, the neutrino emissivities are not much different from the
case of $B=0$, and the magnetic field does not significantly amplify the
emission, though it causes very large fluctuations in the density dependence
of the neutrino emissivity. In the usual forbidden region $x_{p}<1/9$,
however, the magnetic field contributes to the emission and changes the
kinematical condition. The effect is larger when the magnetic field strength
is larger and the proton fraction is closer to $1/9$.

If the magnetic field is rather weak $B\lesssim 10^{15}$ G, the DU process
does not appear when $x_{p}<1/9$, so that it does not occur in the surface
region of NSs. However, neutrino emission from the Landau quantization may
occur in the inner core of magnetars which are thought to have such strong
magnetic fields. Those neutrinos are able to escape from the matter because
they have very low energy, and their mean-free-paths are very long.

Magneto-hydrodynamic proto-NS simulations [45, 46, 47] have demonstrated that
the magnetic field inside a NS can obtain a toroidal configuration. It has
also been demonstrated [46] that the field strength of toroidal magnetic
fields can be $\sim$100 times stronger than that of a poloidal magnetic field.
Indeed, by analyzing the hard X-ray data and precession of pulsars. Makishima
et al. [48] suggested evidence for the existence of a toroidal magnetic field
whose strength is about one hundred times that of poloidal magnetic fields for
cold NSs.

In the present calculation the neutrino emissivities vanish when $x_{p}\ll
1/9$ because we use the low temperature approximation whereby the Fermi
distribution functions are approximated by a step function. As mentioned
above, we exactly calculated the $\nu{\bar{\nu}}$-pair production in strong
magnetic fields [30] and showed that the neutrino emissivity is larger than
that of the MU process in the case of zero magnetic field, even though the
production rates described here are zero in the low temperature approximation.
This implies that there may be much larger neutrino emissivities in the DU
process when $x_{p}\ll 1/9$; it may be possible to explicitly show that with
an exact calculation. We defer exploring this point to a future work

Our final remark is about the EOS as there are many different versions from
many nuclear models. A salient feature is the density-dependence of the
symmetry energy, which is related to the proton fraction in NS matter. One
needs to explore using different EOSs [49, 50] which we also defer to a future
work.

This work was supported in part by the Grants-in-Aid for the Scientific
Research from the Ministry of Education, Science and Culture of Japan
(JP20K03958, JP19K03833, JP17K05459, JP16K05360, JP15H03665). ABB is supported
in part by the U.S. National Science Foundation Grants No. PHY-1806368,
PHY-2020275, PHY-2108339. MKC is supported by the National Research Foundation
of Korea (Grant Nos. NRF-2021R1A6A1A03043957 and 2020K1A3A7A09080134,
NRF-2020R1A2C3006177). Work of GJM supported in part by the U.S. Department of
Energy under Nuclear Theory Grant DE-FG02-95-ER40934.

## References

  * [1] D.G. Yakolev and C.J. Pethick, Ann. Rev. Astron. Astrophys, 42 (2004) 169.
  * [2] A. Dohi. K. Nakazato, M. Hashimoto, Y. Matsuo and T. Noda, Prog. Theor. Exp. Phys. 2019 (2019) 113E01.
  * [3] For a review, D.G. Yakovlev, A. D. Kaminker, O.Y. Gnedin, P. Haensel, Phys. Rep. 354 (2001) 1.
  * [4] A.D. Kaminker, C.J. Pethick, A.Y. Potekhin, V. Thorsson, and D.G. Yakovlev, Astron. Astrophys. 343 (1999) 1009.
  * [5] D.D. Ofengeim, A.D. Kaminker and D.G. Yakovlev, EPL 108 (2014) 31002
  * [6] Y. Lim, C. H. Hyun, and C.-H. Lee, Int. J. Mod. Phys. E26 (2017) 1750015.
  * [7] J.-B. Wei, F. Burgio and H.-J. Schulze, Universe 2020, 6 (2020) 115.
  * [8] J. Boguta, Phys. Lett. B106 (1981) 255.
  * [9] J.M. Lattimer, C.J. Pethick, M. Prakash and P. Haensel, Phys. Rev. Lett. 66 (1991) 2701.
  * [10] E.F. Brown, A. Cumming, F.J. Fattoyev, C.J. Horowitz, D. Page, and S. Reddy, Phys. Rev. Lett. 120, 182701 (2018)
  * [11] O. Maxwell, G.E. Brown, D.K. Campbell, R.F. Dashen and J.T. Manassah, Astrophys. J. 216 (1977) 77.
  * [12] G.E. Brown, K. Kubodera, D. Page, and P. Pizzochero, Phys. Rev. D37 (1988) 2042\.
  * [13] H. Fujii, T. Muto, T. Tatsumi, and R. Tamagaki Phys. Rev. C50 (1994) 3140.
  * [14] M. Prakash, M. Prakash, J. M. Lattimer and C. J. Pethick, Astrophys. J. Lett, 390 (1992) L77.
  * [15] S. Choi, M.-K. Cheoun, K.S. Kim, H. Kim, H. Sagawa, PTEP 2020 (2020) 013D04.
  * [16] B. Paczyński, Acta. Astron. 41 (1992) 145.
  * [17] For a review, G. Chanmugam, Annu. Rev. Astron. Astrophys. 30 (1992) 143.
  * [18] S. Mereghetti, Annu. Rev. Astron. Astrophys. 15 (2008)225.
  * [19] A.D. Kaminker et al. MNRAS 395 (2009) 2257.
  * [20] I. Fushiki, E.H. Gudmundsson and C.J. Pethick, Astrophys, J. 342 (1989) 958\.
  * [21] I. Fushiki, E.H. Gudmundsson, C.J. Pethick and J. Yngvason, Ann of Phys. 216 (1992) 29.
  * [22] A.M. Abrahams and S.L. Shapiro, Astrophys. J. 374 (1991) 652.
  * [23] Ö.E. Rögnvaldsson, I Fushiki, E.H. Gudmundsson, C.J. Pethick and J. Yngvason, Astrophys. J. 416 (1993) 276.
  * [24] D. Lai and S.L. Shapiro, Astrophys. J. 383 (1991) 745..
  * [25] S. Chakrabarty, Phys. Rev. D54 (1996) 1302.
  * [26] S. Chakrabarty, D. Bandyopadhyay and S. Pal, Phys. Rev. Lett. 78 (1997) 2898\.
  * [27] A. Broderick, M. Parakash and J. M. Lattimer, Astrophys. J. 537 (2000) 351\.
  * [28] C.Y. Cardall, M. Prakash and J.M. Lattimer, Astrophys. J. 554 (2001) 322.
  * [29] A. Thorolfsson, O.E. Rognvaldsson, J. Yngvason, E.H. Gudmundsson, Astrophys. J. 502 (1988) 847.
  * [30] T. Maruyama, A B. Balantekin, M.-K. Cheoun, T. Kajino and G.J. Mathews, Phys. Lett. B80 (2020) 135413.
  * [31] T. Maruyama, A.B. Balantekin, M.-K. Cheoun, T. Kajino, G.J. Mathews, Phys. Lett. B779 (2018) 160.
  * [32] V. Canuto and J. Ventura, Fundam. Cosm. Phys. 2 (1977) 203\.
  * [33] J.-F. Fortin, H.-K. Guoy, K. Sinha, S.P. Harr, D. Kim and C. Sun, arXiv:2102.12503 [hep-ph].
  * [34] L.B. Leinson, Phys. Lett. B532 (2002) 267.
  * [35] L.B. Leinson and A. Pérez, JHEP 09 (1998) 020
  * [36] S. Reddy, M. Prakash and J.M. Lattimer, Phys. Rev. D58 (1998) 013009.
  * [37] B.D. Serot and J.D. Walecka, Int. J. Mod. Phys. E6 (1997) 515.
  * [38] L.B. Leinson and A. Pérez, Phys. Lett. B518 (2001) 15.
  * [39] T.Maruyama and S.Chiba, J. Phy. G25 (1999) 2361.
  * [40] T. Maruyama, H. Fujii, T. Muto and T. Tatsumi, Phys. Lett. B337, 19 (1994).
  * [41] For example, D.K. Ferry, ”Transport in Semiconductor Mesoscopic Devices”, IOP Publishing Ltd (2015).
  * [42] T. Maruyama, M.-K. Cheoun, T. Kajino, Y. Kwon, G.J. Mathews, C.Y. Ryu, Phys. Rev. D91 (2015) 123007.
  * [43] T. Maruyama, M.-K. Cheoun, T. Kajino, G.J. Mathews, Phys. Lett. B75 (2016) 125.
  * [44] O.V. Maxwell, ApJ 319, 691 (1987); D.G. Yakolev and K.P. Levenfish Astron. Astrophys, 297 (1995) 717.
  * [45] J. Braithwaite and H.C. Spruit, Nature 431 (2004) 89.
  * [46] T. Takiwaki, K. Katake and K. Sato, Astrophys. J. 691 (2009) 1360.
  * [47] T. Kuroda and H. Umeda, Astrophys. J. Suppl. 191 (2010) 439.
  * [48] K. Makishima, T. Enoto, J.S. Hiraga, T. Nakano, K. Nakazawa, S. Sakurai, M. Sasano, and H. Murakami, Phys. Rev. Lett. 112 (2014) 171102.
  * [49] M.-K. Cheoun, C. Deliduman, C. Güngör, V. Keleş, C.Y. Ryu, T. Kajino and G.J. Mathews., JCAP 10 (2013) 021.
  * [50] X. Roca-Maza, X. Viñas, M. Centelles, P. Ring and P. Schuck, Phys. Rev. C84 (2011) 054309.


# Overview of AI-Debater 2023: The Challenges of Argument Generation Tasks

Jiayu Lin1, Guanrong Chen2∗, Bojun Jin2∗, Chenyang Li3∗, Shutong Jia4∗,
Wancong Lin5∗, Yang Sun2,
Yuhang He2, Caihua Yang2, Jianzhu Bao2, Jipeng Wu2, Wen Su4, Jinglu Chen4,
Xinyi Li4,
Tianyu Chen4, Mingjie Han5, Shuaiwen Du5, Zijian Wang5, Jiyin Li5, Fuzhong
Suo5,
Hao Wang5, Nuanchen Lin6, Xuanjing Huang1, Changjian Jiang1, Ruifeng Xu2, Long
Zhang3†,
Jiuxin Cao4†, Ting Jin5†, Zhongyu Wei1†
1Fudan University
2Harbin Insitute of Technology, Shenzhen
3Zhongyuan University of Technology, 4Southeast University
5Hainan University, 6South China Agricultural University
<EMAIL_ADDRESS>{23S051030<EMAIL_ADDRESS>
{lcy<EMAIL_ADDRESS>{shutong_jia<EMAIL_ADDRESS>
{wanconglin<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>Equal contribution. Corresponding author.

###### Abstract

In this paper we present the results of the AI-Debater 2023 Challenge held by
the Chinese Conference on Affect Computing (CCAC 2023), and introduce the
related datasets. We organize two tracks to handle the argumentative
generation tasks in different scenarios, namely, Counter-Argument Generation
(Track 1) and Claim-based Argument Generation (Track 2). Each track is
equipped with its distinct dataset and baseline model respectively. In total,
32 competing teams register for the challenge, from which we received 11
successful submissions. In this paper, we will present the results of the
challenge and a summary of the systems, highlighting commonalities and
innovations among participating systems. Datasets and baseline models of the
AI-Debater 2023 Challenge have been already released and can be accessed
through the official website111http://www.fudan-
disc.com/sharedtask/AIDebater23/index.html of the challenge.

_K_ eywords Computational Argumentation $\cdot$ AI-Debater $\cdot$ Natural
Language Processing

##  1 Introduction

Argument and debate are fundamental capabilities of human intelligence,
essential for a wide range of human activities, and common to all human
societies. Argumentation [1, 2, 3] takes the human logical argumentation
process as the research object, and is a research field involving logic,
philosophy, language, rhetoric, computer science and education. Striving to
enable models to automatically understand and generate argument texts,
computational argumentation, a newly emerging research field, is obtaining
increasing attention from the research community [4]. Depending on the task
objectives, computational argumentation tasks can be divided into two aspects,
argument mining and argument generation.

With the rapid development of modern technology, online forums like
ChangeMyView allow people to freely exchange opinions on specific topics,
making them suitable data sources for argument generation tasks, especially
for designing artificial debaters, as online forums closely resemble real-
world debates. Initial research in this field has focused on analyzing
ChangeMyView data [5, 6] to summarize the key factors of persuasive arguments.

For an extended period, the field of argument mining has been particularly
active. Li et al. [7] proposed a Structure-Aware Argument Encoder (SAE) in
their work, which enhances the ability to capture structural information in
the analysis of scientific literature discourse by distinguishing between
framing words and topic words in sentences and incorporating paragraph-level
positional information. Additionally, some researchers have integrated
knowledge graph structures into argument mining tasks. Yuan et al. [8]
constructed a knowledge graph for external knowledge, improving the model’s
ability to identify interactive argument pairs. Liang et al. [9] proposed a
hierarchical argumentation graph structure and introduced a text-graph multi-
modal pre-training framework.

Recently, large language models, such as OpenAI ChatGPT and GPT-4 [10], PaLM
[11], and LlaMAs [12, 13] have achieved great success and demonstrated
remarkable performance in text generation tasks. Therefore, to align the field
of computational argumentation with the development trend of large language
models, we have organized the AI-Debater 2023 Challenge222This event is an
CCAC 2023 task sponsored by Fudan University.. This challenge focuses on
generation tasks, including two tracks: counter-argument generation (Track 1)
and claim-based argument generation (Track 2). In Track 1, we introduce the
task of generating counter-argument based on given topic; while in Track 2, we
introduce the task of generating argument based on given claim. We provide two
datasets in this task, one for each track.

In total, 32 teams from over 10 colleges and corporates enter for the
challenge, 11 of which successfully submit their models and obtain their
model’s performance. We hope that we can prompt the computational
argumentation community to align itself with mainstream text generation
technologies through this challenge.

In this paper, we present a detailed description for each track and their
dataset, along with technical solutions of the winning team, and discuss the
possible future research directions of the task.

##  2 Related Works

###  2.1 Counter-Argument Generation

Datasets for counter-argument generation mainly establish the rebuttal
relationship in the conversation using automatic methods such as citation or
reply detection [14, 15]. Seaman et al. [5] proposed CMV dataset, including
the citation relationship between original posts and their corresponding
replies. Bolton et al. [16] introduced Kialo, a dataset for sentence-level
argument stance classification, which can also be applied to counter-argument
generation task. Lin et al. [17] introduced ArgTersely, a dataset for
sentence-level counter-argument generation, this dataset is obtained by manual
annotation.

Early work [15, 18] focus on how to introduce external knowledge into the
system; Alshomary et al. [19] developed a system to identify weak points in
arguments; Schiller et al. [20] developed a controlled argument generation
system, which is able to generate arguments based on given information;
Alshomary et al. [21] completed it through multi-task and multi-step
reasoning. Lin et al. [17] constructed argumentation instructions, and fine-
tuned a large language model for this task.

###  2.2 Claim-based Argument Generation

Claim-based argument generation is a burgeoning field within NLP that aims to
construct persuasive arguments automatically. This involves not only
comprehending the topic but also aligning the generated claims with the
audience’s beliefs for increased effectiveness.

Alshomary et al. [22] address the challenge of tailoring arguments to an
audience’s beliefs by generating claims that are both topic-relevant and
belief-aligned. Hu et al. [23] propose AMERICANO framework and innovate
argument generation through discourse-driven decomposition and agent
interaction, enhancing the coherence and persuasiveness of generated
arguments. Alkhawaldeh et al. [24] introduces a deep learning and
reinforcement learning-based approach for generating Toulmin arguments,
focusing on claim and warrant components to enhance stance detection and
factuality checking in NLP tasks.

##  3 Task Description and Result

In this section, we formally define the specific task, introduce the
construction of the corresponding dataset, scoring metrics as well as the
baseline model for each track respectively. The results of this competition
can be found in the Appendix A.

###  3.1 Track 1: Counter-Argument Generation

##### Task Formulation

We formulate our task according to Lin et al. [17]’s setting. For a given
topic $\tau$ and original argument$x$, the participating model automatically
generates one sentence $y$ that refutes the original argument (referred to as
a counter-argument).

$y=F_{1}(\tau,x)$ (1)

##### Data Construction

We created ArgTersely dataset for counter-argument generation task by
extracting data from the ChangeMyView (CMV) debate forum and manually
annotating them. The process began with data preprocessing to segment replies
into sentences and remove invalid content. Annotators then selected sentences
that countered the original arguments during trial annotation, which also
served as training and consistency testing with reference annotations. The
formal annotation phase used a cross-annotation strategy with two annotators
per triplet and a third to resolve disagreements, ensuring dataset quality.
During AI-Debater 2023 challenge, we used a subset of this dataset with 10,000
training and 4,000 test samples.

##### Scoring Metric

We use ROUGE-L score as the automatic evaluation metrics.

##### Baseline Model

We fine-tuned GPT-2 [25] as a baseline model. Specifically, we concatenated
the debate topic, original argument, and counter-argument into a continuous
text, applied mask processing to the debate topic and original argument, and
then conducted auto-regressive training targeting the counter-argument part
with a cross-entropy loss function.

###  3.2 Track 2: Claim-based Argument Generation

##### Task Formulation

In this task, for the given claim $c$, the participating model automatically
generates 5 independent arguments, $Z=[z_{1},z_{2},...,z_{5}]$ that fit the
claim.

$z_{i}=F_{2}(c),i=1,2,...,5$ (2)

##### Data Construction

The dataset is derived from nearly 700 renowned Chinese debate competitions
held between 2007 and 2021. Each debate match’s segment and monologue text
were obtained through speech-to-text transcription and subsequent manual
verification. The monologue texts were chunked based on punctuation marks such
as periods and question marks, and then annotators marked the argument
sentences. Each argument sentence corresponds to the claim of the current
debate round, resulting in pairs of claim-argument data. During AI-Debater
2023 challenge, the training set includes 33 claims with 3455 arguments, and
the test set comprises 41 claims with 930 arguments.

##### Scoring Metric

We use ROUGE-L score as the automatic evaluation metrics.

##### Baseline Model

We fine-tuned Mengzi-T5-base [26] as a baseline model. Specifically, we
concatenated the claim and the argument into a continuous text, applied mask
processing to the debate topic, and then conducted auto-regressive training
targeting the argument part with a cross-entropy loss function.

##  4 Technical Approaches

###  4.1 Track 1: Data Augmentation and Instruction Tuning in Counter-Argument
Generation

This subsection will introduce the details of the model submitted by HITSZ-HLT
team in Track 1.

####  4.1.1 Analysis of the Problem

The competition’s objective is to create a model capable of automatically
generating counter-arguments for a given topic and original argument. The
training data set presents challenges such as duplicate topics and sources,
and a skewed distribution of counter-arguments in length and frequency.

The original arguments typically range from 30 to 200 words, averaging
108.9877 words, while counter-arguments range from 30 to 250 words, averaging
118.8507 words. The counter-argument length distribution is notably uneven,
with a few excessively long sentences that can introduce noise into the
training process.

Additionally, very short sentences can impede the model’s ability to learn
complex logical expressions. The topic distribution is also uneven, with the
majority of topics having more than forty counter-arguments, and the least
having only a few.

####  4.1.2 Methodology

As is shown in figure 1, our methodology encompasses a two-part approach: a
data augmentation module and a generative language model based on instruction
tuning. The data augmentation module addresses the imbalance in the training
data through two-tiered expansion. Firstly, we utilized ChatGPT [27] to
generate novel counter-arguments for existing topics, adding 6171 new data
points after filtering. Secondly, we incorporated human debate data from the
Kialo forum, manually curating and labeling topics to add 9987 new data points
and 98 new topics. We also refined the data by removing extreme lengths and
low-quality text, such as profanity and non-argumentative sentences, to
enhance model performance.

For the generative language model, we employed instruction tuning on a pre-
trained model, selecting Tk-INSTRUCT [28] over Flan-T5 [29] for its superior
performance. Tk-INSTRUCT was fine-tuned using a dataset covering 1616 diverse
NLP tasks. As is shown in figure 2, we crafted instruction templates tailored
to the counter-argument task, consisting of a task definition, positive
example demonstrations, and reasoning cases. The instruction template was
designed with two positive cases, and we explored the use of connective
adverbs to promote syntactic diversity in the output.

Figure 1: The overall architecture of the proposed method. Figure 2:
Instruction fine-tuning template for generating counter-arguments.

####  4.1.3 Experiments

In our experiments, the validation set was structured to reflect the original
dataset’s length distribution. We experimented with theme-based division but
found it less effective due to the uneven distribution of themes and counter-
arguments. The model implementation utilized tk-instruct-large-def-pos, a
model with 770 million parameters, and applied a minimum generation length of
50 words and a maximum of 256 words to prevent the generation of overly short
sentences that could reduce model performance. We employed Beam Search with
three beams for decoding, balancing the decoding effect with training time
efficiency. To mitigate repetitive word generation, we set
no_repeat_ngram_size to 2. The result is shown in table 1.

Our model achieved a ROUGE-L score of 0.252 on the official test set, and
manual inspection of the validation set output demonstrated that the model
could understand and generate counter-arguments with good logical and thematic
relevance. Ablation studies confirmed the positive impact of our data
augmentation module, with the addition of real human debate text from Kialo
proving most effective.

Model | ROUGE-L
---|---
w/o D | 0.2301
w/o ChatGPT | 0.2351
w/o Kialo | 0.2389
Our model | 0.2400
Table 1: The impact of different data augmentation approaches.

###  4.2 Track 1: Pre- and Post-Processing in Counter-Argument Generation

This subsection will introduce the details of the model submitted by huashui
team in Track 1.

####  4.2.1 Framework

Pre-processing and post-processing are pivotal in NLP, particularly for text
generation where they encompass tokenization, template design, and decoding
strategies. Despite the prevalence of pre-trained models fine-tuned for
specific tasks, these methods fall short in low-resource or under-equipped
settings. Our approach circumvents this by optimizing performance through
strategic pre-processing and post-processing, without structural model
changes. Experimental results validate the efficacy of our strategies against
those reliant on extensive data or model modifications.

The overall framework of our study is as follows. Initially, the original text
is transformed into an input that is more easily understood by the model
through a predefined template. Subsequently, the tokenizer completes the basic
word embedding and inputs it into the GPT-2 model to extract text features and
predict the probability of generating words. Generally, greedy algorithms are
used as the default decoding strategy in current research. However, the demand
in this track is to allow the model to output multiple sentences
simultaneously. Therefore, this study introduces diverse beam search [30] and
contrastive search [31] into the model decoding process.

Diverse beam search [30] improves upon the limitations of the "single-point
departure" inherent in traditional beam search strategies. It draws on the
ideas of breadth-first search (BFS), exploring paths from multiple different
starting points, effectively enhancing the diversity of the model’s
generation. Contrastive search, a new concept proposed in 2022, involves
judging the text similarity matrix at each decoding moment to incorporate a
similarity penalty, resulting in non-repetitive yet coherent output.

Figure 3: Overall model framework.

####  4.2.2 Experiments

In our experimental setup, we used the GPT-2 base version as the foundational
model. For diverse beam search, the beam width was set to 5, the number of
groups to 5, and the diversity penalty to 1. For contrastive search, the
penalty factor was set to 0.6, and the top-k value to 5. The experimental
results are as follows:

Decoding Strategy | ROUGE-L
---|---
Greedy Search (Baseline) | 0.158
Contrastive Search | 0.162
Diverse Beam Search | 0.172
Table 2: Experimental results of different decoding strategies.

It can be observed that the two strategies adopted in this study have
outperformed the method used by the baseline model, thereby proving the
rationality of the starting point of this study.

###  4.3 Track 1: A Diffusion Framework for Counter-Argument Generation

This subsection will introduce the details of the model submitted by ZUT team
in Track 1.

####  4.3.1 Controlled Text Generation Task Formulation

The problem addressed in this document can be defined as follows: Given
control attributes (arguments, claims) $w^{x}$ and a target text (counter-
argument) $w^{y}$, train a language model to output high-quality $w^{y}$ that
aligns with the control attributes upon input $w^{x}$.

$p(w^{y^{\prime}}|w^{x})\propto p(w^{y^{\prime}})\cdot
p(w^{x}|w^{y^{\prime}})$ (3)

The controlled text generation task is formalized as sampling from a
conditional distribution $p(w^{y^{\prime}}|w^{x})$, where $w^{x}$ represents
control attributes, $p(w^{y^{\prime}})$ ensuring fluency to complete the
attribute control process $p(w^{x}|w^{y^{\prime}})$.

Figure 4: Sequence diffusion generation model integrating pre-trained models.

####  4.3.2 Sequence Diffusion Process

Inspired by D3PM, we use a method for diffusing disorganized text by treating
mask tokens as noise addition and decode tokens as noise removal during the
diffusion process. The forward diffusion process involves progressively
masking tokens, while the reverse diffusion process decodes the masked tokens
back into text.

Figure 5: Sequence diffusion process.

####  4.3.3 Model Integration

We introduces a combined sequence diffusion model that integrates a pretrained
model (BERT) with a diffusion model. The model uses BERT’s encoding and
decoding capabilities in conjunction with the diffusion model’s noise addition
and removal processes. This integration allows for the establishment of a
connection between control attributes and corresponding text within different
feature spaces.

The diffusion process is guided by a posterior distribution, with specific
steps outlined for optimization and regularization of fluency. The model aims
to generate high-quality text with controlled attributes without the need for
a separate attribute classifier, thus avoiding errors and reducing training
time.

Figure 6: Integration process of diffusion model and pre-training model.

####  4.3.4 Experiments

The document presents experimental results comparing the proposed model with
baseline models GPT-2. The performance is measured using the ROUGE-L metric,
which evaluates the quality of generated text.

Method | Pretrained | Step | ROUGE-L
---|---|---|---
Baseline | GPT-2 | 1 | 0.143
Our Model | BERT | 256 | 0.159
512 | 0.188
Table 3: Experimental results in Track 1.

###  4.4 Track 2: Enhancing Argument Diversity for Claim-based Argument
Generation

This subsection will introduce the details of the model submitted by HITSZ-HLT
team in Track 2.

####  4.4.1 Task Analysis

The competition’s objective was to create an automated system capable of
generating five relevant arguments for a given claim. The analysis of the
competition data revealed that while claims were brief, the corresponding
arguments were more extensive. The challenge lay in producing lengthy and
varied texts from short inputs. With an average of 104.70 arguments per claim,
the task was to efficiently utilize this wealth to generate five distinct
arguments. Additionally, the dataset included some arguments that were short
and lacked substance, necessitating a strategy to address these issues.

####  4.4.2 Methods

To overcome the identified challenges, a two-part framework (figure 7) was
devised.

The first component, Diverse Generation Strategy Based on Subset Division
(DiverGS), involved splitting the training data into five exclusive subsets to
train five individual BART models, each aimed at generating a single, diverse
argument per claim.

The second component, Generation Enhancement Strategy Based on Keyword
Guidance (KeyGuide), introduced keywords for each argument to guide the model
during generation. These keywords, extracted using the TF-IDF algorithm, were
concatenated to the argument’s beginning and served as prompts. This approach
resulted in a higher diversity and quality of generated arguments.

Figure 7: Illustration of our method.

####  4.4.3 Experiments

The experimental phase began with data preprocessing, where TF-IDF was used to
generate and append three keywords to each argument. The arguments for each
claim were evenly divided into five subsets, creating five sub-training sets.
The pre-trained models used for the generation task include Mengzi-T5 [26], T5
[32], BART [33], CPT [34], etc. And the bart-base-chinese model was selected
for its performance in preliminary experiments. Each subset was then used to
fine-tune a separate BART model, resulting in five distinct generation models.

Figure 8: Illustration of subset division.

After obtaining the five subsets, each subset is processed into the form of
“source sequence to target sequence” and then used separately to fine-tune
five Bart models with different parameters, resulting in five fine-tuned
generation models. During the inference phase, the given claims are inputted
into the aforementioned five generation models separately, and each model
generates one argument. The decoding strategy is beam search, with num_beams
set to 5, maximum sequence length set to 128. Additionally, during decoding,
repetition_penalty is set to 5.0 to alleviate repetition issues, and
length_penalty is set to 5.0.

Our framework achieved a performance of ROUGE-L=0.167 on the official unseen
test set provided by the competition committee.

Furthermore, to validate the necessity and effectiveness of each module in our
framework, we conducted ablation experiments on the validation set, and the
results are shown in Table 4. It can be seen that compared to not using
keywords as guidance, our proposed keyword-guided generation enhancement
method leads to a significant improvement in performance. This is because the
keywords generated by the model can guide the generation of subsequent
arguments. Moreover, our proposed strategy of generating diversity based on
subset partitioning shows some improvement in ROUGE-1 and ROUGE-2 scores. This
experiment validates the effectiveness of the two modules we proposed.

| BLEU | ROUGE-1 | ROUGE-2 | ROUGE-L
---|---|---|---|---
Bart | 0.069 | 0.206 | 0.010 | 0.166
Bart w/ KeyGuide | 0.075 | 0.212 | 0.012 | 0.176
Bart-DiverGS | 0.072 | 0.210 | 0.011 | 0.164
Bart-DiverGS w/ KeyGuide | 0.062 | 0.216 | 0.014 | 0.176
Table 4: Ablation study results.

###  4.5 Track 2: Longest Common Subsequence Search for Claim-based Argument
Generation

This subsection will introduce the details of the model submitted by
tingzhidui team in Track 2.

####  4.5.1 Solution Description

The T5 model is used as the baseline model, with the input being "Topic:
claim," where the claim is replaced with a specific text. If the length of the
argument is less than 128, other arguments are copied to increase the length.
The loss function is the cross-entropy function, with a learning rate of 1e-4
and the optimizer being Adamw. Parameters were adjusted for the T5 and GPT-2
models to achieve the best configuration. After analyzing the experimental
results, the GPT-2 model was chosen as the final model.

Our team proposed a method based on two algorithms, a best score (BS)
calculation algorithm based on longest common subsequence (LCS) and a best
standard argument (BSA) calculation algorithm. Using these algorithms, the
best score and the best standard argument for each claim can be obtained.

####  4.5.2 Experiments

Table 5 shows the Rouge-L scores of the T5 and GPT-2 models on the validation
set and their performance under different parameter configurations.

Table 6 shows some of the best arguments, the scores of the best arguments,
the best subsequences, and their scores for certain claims.

Model | ALC | MCL | MAL | Beam Size | RP | Rouge-L
---|---|---|---|---|---|---
T5 | T | 32 | 256 | 20 | 5.0 | 0.102
T5 | F | 32 | 256 | 20 | 5.0 | 0.1573
T5 | F | 25 | 256 | 20 | 5.0 | 0.1729
T5 | F | 32 | 128 | 20 | 5.0 | 0.1559
T5 | F | 32 | 256 | 3 | 5.0 | 0.1803
T5 | F | 32 | 256 | 4 | 5.0 | 0.1828
T5 | F | 32 | 256 | 5 | 5.0 | 0.1666
T5 | F | 32 | 256 | 10 | 5.0 | 0.1679
T5 | F | 25 | 256 | 4 | 5.0 | 0.1757
T5 | F | 32 | 256 | 20 | 2.0 | 0.1745
T5 | F | 32 | 256 | 20 | 3.0 | 0.1627
T5 | F | 32 | 256 | 4 | 2.0 | 0.1767
GPT-2 | F | 32 | 256 | 4 | / | 0.1884
Table 5: The T5 model and the GPT-2 model’s Rouge-L scores on the validation set. We report Argument Length Completion (ALC), Maximum Claim Length (MCL), Maximum Argument Length (MAL), Beam Size during beam search, Repetition Penalty (RP) and corresponding Rouge-L score. Claim |  BSA | BSA score |  BS | BS Score
---|---|---|---|---
“佛系”标签对青年人成长弊大于利 |  “首先是在心理学层面，佛系标签不利于青年人成长的人格全面发展。” | 0.2115 |  “佛系标签是的，在的不的。” | 0.2449
“佛系”标签对青年人成长利大于弊 |  “综上佛系标签对青年人的成长利大于弊，谢谢。” | 0.2085 |  “我们是佛系，是的，的。” | 0.2419
短视频的火爆是精神文化匮乏的表现 |  “而体现二字，一方面指短视频火爆的成因是精神文化的匮乏，另一方面是说短视频本身的精神文化也是匮乏的。” | 0.2294 |  “方精神文化的，是短视频的精神文化是的。” | 0.2734
对知识网红的崇拜让我们对真知更远 |  “如果它是提供给你知识可以，可是如果他提供给你的知识方式是告诉你这是你思维的终点，提供知识的同时剥夺你的思维，这不可以。” | 0.2123 |  “知识网红是，是的，我。” | 0.2539
Table 6: Best Subsequence and Best Standard Argument Results. The claim, best
standard argument (BSA), best subsequence (BS) and corresponding scores are
reported in the table.

It was found that the evaluation index did not reflect the differences between
predicted arguments and proposed a method based on high-frequency words and
keywords. The GPT-2 model was used for training and prediction, achieving a
Rouge-L score of 0.2035, and the score after submission was 0.125.

##  5 Conclusion

The AI-Debater 2023 Challenge moves towards argument generation tasks. We set
up counter-argument generation and claim-based argument generation tasks. In
this challenge, we build and release a new counter-argument generation
dataset, enriching argument generation tasks.

The winning approaches, which included data augmentation, instruction tuning,
and diffusion model integration, have demonstrated the potential of current AI
technologies to understand and construct arguments. These methods have not
only improved the performance of the models but also provided insights into
how AI can be further developed for complex language tasks.

Looking ahead, the challenge has identified key areas for future research,
including enhancing argument quality, addressing data imbalance, and improving
coherence in generated texts. As the field progresses, it is expected that AI
will increasingly contribute to nuanced debates, offering new possibilities
for AI applications in various domains.

## Acknowledgements

This work is supported by National Natural Science Foundation of China (No.
62176058) and National Key R & D Program of China (2023YFF1204800). The
project’s computational resources are supported by CFFF platform of Fudan
University.

## References

  * [1] Philippe Besnard and Anthony Hunter.  Elements of argumentation, volume 47.  MIT press Cambridge, 2008.
  * [2] James Milton O’Neill, Craven Laycock, and Robert Leighton Scales.  Argumentation and debate.  Macmillan, 1927.
  * [3] Frans H Van Eemeren.  Reasonableness and effectiveness in argumentative discourse.  Argumentation library, 27, 2015.
  * [4] Noam Slonim, Yonatan Bilu, Carlos Alzate, Roy Bar-Haim, Ben Bogin, Francesca Bonin, Leshem Choshen, Edo Cohen-Karlik, Lena Dankin, Lilach Edelstein, et al.  An autonomous debating system.  Nature, 591(7850):379–384, 2021.
  * [5] Julie A Seaman.  Winning arguments.  Law & Psychol. Rev., 41:1, 2016.
  * [6] Zhongyu Wei, Yang Liu, and Yi Li.  Is this post persuasive? ranking argumentative comments in online forum.  In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pages 195–200, 2016.
  * [7] Yinzi Li, Wei Chen, Zhongyu Wei, Yujun Huang, Chujun Wang, Siyuan Wang, Qi Zhang, Xuanjing Huang, and Libo Wu.  A structure-aware argument encoder for literature discourse analysis.  In Nicoletta Calzolari, Chu-Ren Huang, Hansaem Kim, James Pustejovsky, Leo Wanner, Key-Sun Choi, Pum-Mo Ryu, Hsin-Hsi Chen, Lucia Donatelli, Heng Ji, Sadao Kurohashi, Patrizia Paggio, Nianwen Xue, Seokhwan Kim, Younggyun Hahm, Zhong He, Tony Kyungil Lee, Enrico Santus, Francis Bond, and Seung-Hoon Na, editors, Proceedings of the 29th International Conference on Computational Linguistics, pages 7093–7098, Gyeongju, Republic of Korea, October 2022. International Committee on Computational Linguistics.
  * [8] Jian Yuan, Zhongyu Wei, Donghua Zhao, Qi Zhang, and Changjian Jiang.  Leveraging argumentation knowledge graph for interactive argument pair identification.  In Chengqing Zong, Fei Xia, Wenjie Li, and Roberto Navigli, editors, Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pages 2310–2319, Online, August 2021. Association for Computational Linguistics.
  * [9] Jingcong Liang, Rong Ye, Meng Han, Qi Zhang, Ruofei Lai, Xinyu Zhang, Zhao Cao, Xuanjing Huang, and Zhongyu Wei.  Hi-ArG: Exploring the integration of hierarchical argumentation graphs in language pretraining.  In Houda Bouamor, Juan Pino, and Kalika Bali, editors, Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 14606–14620, Singapore, December 2023. Association for Computational Linguistics.
  * [10] Sébastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott Lundberg, et al.  Sparks of artificial general intelligence: Early experiments with gpt-4.  arXiv preprint arXiv:2303.12712, 2023.
  * [11] Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles Sutton, Sebastian Gehrmann, et al.  Palm: Scaling language modeling with pathways.  Journal of Machine Learning Research, 24(240):1–113, 2023.
  * [12] Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée Lacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, et al.  Llama: Open and efficient foundation language models.  arXiv preprint arXiv:2302.13971, 2023.
  * [13] Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al.  Llama 2: Open foundation and fine-tuned chat models.  arXiv preprint arXiv:2307.09288, 2023.
  * [14] Lu Ji, Zhongyu Wei, Jing Li, Qi Zhang, and Xuanjing Huang.  Discrete argument representation learning for interactive argument pair identification.  In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 5467–5478, Online, June 2021. Association for Computational Linguistics.
  * [15] Xinyu Hua and Lu Wang.  Neural argument generation augmented with externally retrieved evidence.  In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 219–230, Melbourne, Australia, July 2018. Association for Computational Linguistics.
  * [16] Eric Bolton, Alex Calderwood, Niles Christensen, Jerome Kafrouni, and Iddo Drori.  High quality real-time structured debate generation.  arXiv preprint arXiv:2012.00209, 2020.
  * [17] Jiayu Lin, Rong Ye, Meng Han, Qi Zhang, Ruofei Lai, Xinyu Zhang, Zhao Cao, Xuan-Jing Huang, and Zhongyu Wei.  Argue with me tersely: Towards sentence-level counter-argument generation.  In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 16705–16720, 2023.
  * [18] Xinyu Hua, Zhe Hu, and Lu Wang.  Argument generation with retrieval, planning, and realization.  In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 2661–2672, Florence, Italy, July 2019. Association for Computational Linguistics.
  * [19] Milad Alshomary, Shahbaz Syed, Arkajit Dhar, Martin Potthast, and Henning Wachsmuth.  Counter-argument generation by attacking weak premises.  In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pages 1816–1827, Online, August 2021. Association for Computational Linguistics.
  * [20] Benjamin Schiller, Johannes Daxenberger, and Iryna Gurevych.  Aspect-controlled neural argument generation.  In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 380–396, Online, June 2021. Association for Computational Linguistics.
  * [21] Milad Alshomary and Henning Wachsmuth.  Conclusion-based counter-argument generation.  In Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics, pages 957–967, Dubrovnik, Croatia, May 2023. Association for Computational Linguistics.
  * [22] Milad Alshomary, Wei-Fan Chen, Timon Gurcke, and Henning Wachsmuth.  Belief-based generation of argumentative claims.  In Paola Merlo, Jorg Tiedemann, and Reut Tsarfaty, editors, Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pages 224–233, Online, April 2021. Association for Computational Linguistics.
  * [23] Zhe Hu, Hou Pong Chan, and Yu Yin.  Americano: Argument generation with discourse-driven decomposition and agent interaction, 2023.
  * [24] Fatima Alkhawaldeh, Tommy Yuan, and Dimitar Kazakov.  Rl-gan based toulmin argument generation.  7:15, 04 2020.
  * [25] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.  Language models are unsupervised multitask learners.  OpenAI blog, 1(8):9, 2019.
  * [26] Zhuosheng Zhang, Hanqing Zhang, Keming Chen, Yuhang Guo, Jingyun Hua, Yulong Wang, and Ming Zhou.  Mengzi: Towards lightweight yet ingenious pre-trained models for chinese.  arXiv preprint arXiv:2110.06696, 2021.
  * [27] TB OpenAI.  Chatgpt: Optimizing language models for dialogue. openai, 2022.
  * [28] Yizhong Wang, Swaroop Mishra, Pegah Alipoormolabashi, Yeganeh Kordi, Amirreza Mirzaei, Atharva Naik, Arjun Ashok, Arut Selvan Dhanasekaran, Anjana Arunkumar, David Stap, Eshaan Pathak, Giannis Karamanolakis, Haizhi Lai, Ishan Purohit, Ishani Mondal, Jacob Anderson, Kirby Kuznia, Krima Doshi, Kuntal Kumar Pal, Maitreya Patel, Mehrad Moradshahi, Mihir Parmar, Mirali Purohit, Neeraj Varshney, Phani Rohitha Kaza, Pulkit Verma, Ravsehaj Singh Puri, Rushang Karia, Savan Doshi, Shailaja Keyur Sampat, Siddhartha Mishra, Sujan Reddy A, Sumanta Patro, Tanay Dixit, and Xudong Shen.  Super-NaturalInstructions: Generalization via declarative instructions on 1600+ NLP tasks.  In Yoav Goldberg, Zornitsa Kozareva, and Yue Zhang, editors, Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, pages 5085–5109, Abu Dhabi, United Arab Emirates, December 2022. Association for Computational Linguistics.
  * [29] Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Yunxuan Li, Xuezhi Wang, Mostafa Dehghani, Siddhartha Brahma, et al.  Scaling instruction-finetuned language models.  Journal of Machine Learning Research, 25(70):1–53, 2024.
  * [30] Ashwin K Vijayakumar, Michael Cogswell, Ramprasath R Selvaraju, Qing Sun, Stefan Lee, David Crandall, and Dhruv Batra.  Diverse beam search: Decoding diverse solutions from neural sequence models.  arXiv preprint arXiv:1610.02424, 2016.
  * [31] Yixuan Su, Tian Lan, Yan Wang, Dani Yogatama, Lingpeng Kong, and Nigel Collier.  A contrastive framework for neural text generation.  In Advances in Neural Information Processing Systems, 2022.
  * [32] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu.  Exploring the limits of transfer learning with a unified text-to-text transformer.  Journal of machine learning research, 21(140):1–67, 2020.
  * [33] Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer.  BART: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension.  In Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel Tetreault, editors, Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 7871–7880, Online, July 2020. Association for Computational Linguistics.
  * [34] Yunfan Shao, Zhichao Geng, Yitao Liu, Junqi Dai, Hang Yan, Fei Yang, Zhe Li, Hujun Bao, and Xipeng Qiu.  Cpt: A pre-trained unbalanced transformer for both chinese language understanding and generation.  Science China Information Sciences, 67(5):1–13, 2024.

##  Appendix A Challenge Result

###  A.1 Track 1: Counter-Argument Generation

Team | Score
---|---
HITSZ-HLT | 25.2
ZUT | 18.8
huashui | 17.2
tingzhidui | 16.3
baseline | 14.3
Table 7: Performance of participants on Track 1.

###  A.2 Track 2: Claim-based Argument Generation

Team | Score
---|---
HITSZ-HLT | 16.7
ZUT | 15.4
tingzhidui | 12.5
baseline | 10.1
Table 8: Performance of participants on Track 2.


survives the randomizing influence of flux compactifications.

The lesson, then, is that one must exercise care when invoking random field
theories as a model for the space of low energy compactified string dynamics.
More particularly, our results, and generalizations thereof to higher
dimensional moduli spaces, provide a sharper ensemble for accurate statistical
modeling of the features of low energy string theory.

Going forward, these results suggest a number of research directions. For ease
of computation we have focused on a Calabi-Yau compactification with a single
complex structure modulus. One would like to acquire an understanding of the
distributions we have studied in more generic cases with higher dimensional
moduli spaces. Explicit analysis of the sort we’ve undertaken here would be
difficult. However, in the vicinity of a conifold locus – where vacua
generally accumulate – we’ve reduced the statistical dependence to the three
dominant control parameters introduced earlier. These each have natural higher
dimensional generalizations and so it would be of interest to see if we can
gain insight into more general Calabi-Yau compactificaitons guided by the
results we found here, and thus avoiding direct calculation. We hope to return
to this shortly. It would also be interesting to revisit the works tumbling
which have investigated the quantum stability of vacua in random high
dimensional scalar field theories as a surrogate for the stability of the
string landscape. Are those results modified by studying a collection of field
theories whose distribution more closely aligns with that of low energy
dynamics of string theory? We intend to return to this question as well.

###### Acknowledgements.

It is a pleasure to thank Thomas Bachlechner, David Kagan, Ruben Monten, and
especially Frederik Denef, for numerous insightful conversations and
observations, which greatly assisted the completion of the research reported
herein. We also gratefully acknowledge the support of the Department of Energy
through grant DE-FG02-92-ER40699.

## References

  * (1) M. R. Douglas, JHEP 0305, 046 (2003).
  * (2) F. Denef and M. R. Douglas, JHEP 0405, 072 (2004).
  * (3) F. Denef, and M. R. Douglas, JHEP 0503, 061 (2005).
  * (4) F. Denef, M. R. Douglas, and B. Florea, JHEP 0406, 034 (2004).
  * (5) N. Arkani-Hamed, S. Dimopoulos, and S. Kachru, SLAC-PUB 10928 (2005) arXiv:hep-th/0501082 [hep-th].
  * (6) J. Distler, and U. Varadarajan, (2005) arXiv:hep-th/0507090 [hep-th].
  * (7) B. Greene, D. Kagan, A. Masoumi, D. Mehta, E. J. Weinberg and X. Xiao, Phys. Rev. D 88, no. 2, 026005 (2013) arXiv:1303.4428 [hep-th].
  * (8) X. Chen, G. Shiu, Y. Sumitomo, and S. H. H. Tye, JHEP 1204, 026 (2012).
  * (9) D. Marsh, L. McAllister, and T. Wrase, JHEP 1203, 102 (2012) arXiv:1112.3034 [hep-th].
  * (10) D. Marsh, B. Vercnocke, and T. Wrase, JHEP 1505, 081 (2015) arXiv:1411.6625 [hep-th].
  * (11) U. H. Danielsson, N. Johansson, and M. Larfors, JHEP 0703, 080 (2007).
  * (12) D. I. Podolsky, J. Majumder, and N. Jokela, JCAP 0805, 024 (2008) arXiv:0804.2263 [hep-th].
  * (13) M. Dine, and S. Paban, (2015) arXiv:1506.06428 [hep-th].
  * (14) M. Dine, (2015) arXiv:1512.08125 [hep-th].
  * (15) S. B. Giddings, S. Kachru, and J. Polchinski, Phys. Rev. D 66, 106006 (2002).
  * (16) P. Ahlqvist, B. R. Greene, D. Kagan, E. A. Lim, S. Sarangi, and I-S. Yang, JHEP 1103, 119 (2011).
  * (17) P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes, Nucl. Phys. B 359 (1991), 21-74.

##  Appendix A Near Conifold Period Expansion Coefficients

Recall that our integral and symplectic basis for mirror quintic’s period
functions are denoted $\Pi_{i}$, with $i=0$ and $3$ an intersecting pair, and
$i=1$ and $2$ the other. $\Pi_{0}$ is the only non-analytic period at the
conifold point. It’s partner, $\Pi_{3}$, is the period obtained by integrating
the holomorphic $3$-form over the cycle that collapses. This period is
nevertheless well-behaved (it simply vanishes at the conifold). The two
periods associated with the other intersecting pair of cycles are also
analytic. These are nonvanishing. The following expansions about the conifold
point at $z=1$ hold,

$\displaystyle\Pi_{1}(z)$ $\displaystyle=\sum_{n=0}^{q}b_{n}(z-1)^{n}$ (163)
$\displaystyle\Pi_{2}(z)$ $\displaystyle=\sum_{n=0}^{q}c_{n}(z-1)^{n}$ (164)
$\displaystyle\Pi_{3}(z)$ $\displaystyle=\sum_{n=1}^{q}d_{n}(z-1)^{n}.$ (165)

The values for the coefficients were computed in Mathematica by evaluating
derivatives of the expressions for the $\Pi_{i}$ in terms of the Meijer-G
functions (the $U_{i}$) at the conifold. We used $40$ digit accuracy in these
computations. The values are listed in Table 3.

$n$ | $b_{n}$ | $c_{n}$ | $d_{n}$
---|---|---|---
$0$ | $+1.293574i$ | $6.19502-7.11466i$ | $0$
$1$ | $-0.150767i$ | $-1.016605+0.829217i$ | $-0.355881i$
$2$ | $+0.0777445i$ | $0.570733-0.427595i$ | $0.249117i$
$3$ | $-0.0522815i$ | $-0.401804+0.287548i$ | $-0.194548i$
$4$ | $+0.0393684i$ | $0.312044-0.216526i$ | $0.161285i$
$5$ | $-0.0315669i$ | $-0.256050+0.173618i$ | $-0.138686i$
$6$ | $+0.0263447i$ | $0.217649-0.144896i$ | $0.122217i$
$7$ | $-0.0226046i$ | $-0.189607+0.124325i$ | $-0.109620i$
$8$ | $0.0197941i$ | $0.168193-0.108868i$ | $0.0996353i$
Table 3: Expansion coefficients for period functions $\Pi_{1}$, $\Pi_{2}$ and
$\Pi_{3}$.

The remaining period, $\Pi_{0}$, is multiple-valued at the conifold point. The
cycle it is associated with picks up one copy of its vanishing partner for
each revolution around the conifold in moduli space. This fixes the form of
$\Pi_{0}$ to 78. To match the branch cuts of the logarithm in the expansion
with that of the relevant Mejer-G ($U_{0}$) in Mathematica we must negate the
argument of the logarithm. Ultimately the expression we write for $\Pi_{0}$
is,

$\Pi_{0}(z)=\Pi_{3}(z)\left(\frac{\log(-(z-1))}{2\pi
i}-\frac{1}{2}\right)+f(z)$ (166)

with $f(z)$ analytic.

Its expansion coefficients, $a_{n}$, were computed using a recursion relation
based on the fact $\Pi_{0}$ satisfies the Picard-Fuchs equation. This is
discussed at length in section 2.3. Here we simply tabulate the resulting
values. The first three $a_{n}$ are needed by the recursion to generate the
rest. They are obtained numerically, as discussed in section 2.3. All
coefficients were calculated using 30–40 digit accuracy computations and are
listed in Table 4.

$n$ | $a_{n}$
---|---
$0$ | $1.07073$
$1$ | $0.024708-0.177941i$
$2$ | $-0.0115108+0.1245584i$
$3$ | $0.0065650-0.0972742i$
$4$ | $-0.0042768+0.0806427i$
$5$ | $0.0030290-0.0693428i$
$6$ | $-0.0022701+0.0611087i$
$7$ | $0.0017719-0.0548102i$
$8$ | $-0.0014261+0.0498177i$
Table 4: Expansion coefficients for analytic contribution to $\Pi_{0}$