# Kinematical higher-twist corrections in $\gamma^{*}\to M\bar{M}\gamma$
I. Neutral meson production

Bernard Pire<EMAIL_ADDRESS>CPHT, CNRS, École polytechnique,
Institut Polytechnique de Paris, 91128 Palaiseau, France  Qin-Tao Song
<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou
University, Zhengzhou, Henan 450001, China

###### Abstract

We carry out the calculation of kinematical higher-twist corrections to the
cross section of $\gamma^{*}\to M\bar{M}\gamma$ up to twist 4, where $M$ is a
scalar or pseudoscalar neutral meson. The three independant helicity
amplitudes are presented in terms of the twist-2 generalized distribution
amplitudes (GDAs), which are important non-perturbative quantities for
understanding the 3D structure of hadrons. Since this process can be measured
by BESIII in $e^{+}e^{-}$ collisions, we also perform the numerical estimate
of the kinematical higher-twist corrections by using the kinematics of BESIII.
We adopt the $\pi\pi$ GDA extracted from Belle measurements and the asymptotic
$\pi\pi$ GDA to study the size of the kinematical corrections in the case of
pion meson pair, and a model $\eta\eta$ GDA is used to see the impact of
target mass corrections $\mathcal{O}(m^{2}/s)$ for
$\gamma^{*}\to\eta\eta\gamma$. Our results show that the kinematical higher-
twist corrections account for $\sim 20\%$ of the cross sections at BESIII on
the average, and it is necessary to include them if one tries to extract GDAs
from experimental measurements precisely. Furthermore, the energy-momentum
tensor (EMT) form factors can be obtained for mesons with the help of their
GDAs, from which one can investigate interesting quantities such as the meson
mass radius and mass distribution.

##  I Introduction

Generalized distribution amplitudes (GDAs) Muller:1994ses ; Diehl:1998dk ;
Diehl:2000uv ; Polyakov:1998ze ; Pire:2002ut  are important non-perturbative
functions which reveal the 3D partonic structure of hadrons, and they
correspond to the amplitudes of the soft processes $q\bar{q}\to h\bar{h}$ or
$gg\to h\bar{h}$. GDAs were firstly investigated in $\gamma^{\ast}\gamma\to
h\bar{h}$ with large photon virtuality and small invariant mass of hadron pair
to satisfy the QCD factorization. This process is known as the $s$-$t$ crossed
channel of deeply virtual Compton scattering (DVCS), from which generalized
parton distributions (GPDs) Diehl:2003ny ; Belitsky:2005qn ; Boffi:2007yc ;
Goeke:2001tz  are probed. Recently, measurements of $\gamma^{\ast}\gamma\to
h\bar{h}$ were released for neutral pion pair Belle:2015oin  and neutral kaon
pair Belle:2017xsz  production by the Belle collaboration at KEK, and the
$\pi\pi$ quark GDA was extracted from the cross section of
$\gamma^{\ast}\gamma\to\pi^{0}\pi^{0}$ Kumano:2017lhr . In addition to
$\gamma^{\ast}\gamma\to h\bar{h}$, GDAs can also be accessed in the crossed
reaction Lu:2006ut :

$\gamma^{\ast}(q_{1})\to h(p_{1})\bar{h}(p_{2})\gamma(q_{2})\,,$ (1)

which may be studied in the electron-positron annihilation process

$e^{-}(k_{1})e^{+}(k_{2})\to h(p_{1})\bar{h}(p_{2})\gamma(q_{2})\,,$ (2)

where the large scale $Q^{2}=q_{1}^{2}$ is now timelike; a first access to
this process was released by BaBar BaBar:2015onb  in the charged meson channel
case, and future results should be obtained at BESIII and Belle (Belle II).

There exists a basic difference between the neutral (say $\pi^{0}\pi^{0}$)
production channel and the charged one (say $\pi^{+}\pi^{-}$), due to the
charge conjugation property of the $\pi\pi$ state. Since the $\pi^{+}\pi^{-}$
pair can be produced both with $C=+$ and $C=-$ charge conjugation, the QCD
amplitude (1) can interfere with the QED process, known as Initial State
Radiation (ISR) :

$e^{-}(k_{1})+e^{+}(k_{2})\to\gamma^{*}(q^{\prime}_{1})+\gamma(q_{2})~{}~{};~{}~{}\gamma^{*}(q^{\prime}_{1})\to
h(p_{1})\bar{h}(p_{2})\,,$ (3)

where the $h(p_{1})\bar{h}(p_{2})$ pair is produced in a $C=-$ state. The
amplitude of the ISR process (3) does not depend on GDAs, and is readily
calculated with the help of the measured $\pi$ meson timelike electromagnetic
form factor. The ISR process turns out to dominate the QCD process in most
kinematics, which renders inefficient the extraction of GDAs from integrated
$\pi^{+}\pi^{-}$ cross-sections, but allows us to extract the QCD contribution
- and hence the GDAs - at the amplitude level through cleverly defined
asymmetries, taking advantage of the different $C$ parities of the meson pair
selected by the two processes. This is quite reminiscent of the usual
procedure in DVCS or TCS measurements where the interference of the QCD
process with the Bethe-Heitler contribution populates interesting asymmetries.
We shall address these questions in a forthcoming study and restrict here to
the neutral pseudoscalar meson pair case, namely $\pi^{0}\pi^{0}$, $\eta\eta$
and $\pi^{0}\eta$ channels, where the process (3) does not contribute.

On the one hand, GDAs are important inputs for the three-body decays of $B$
mesons, which are used to study the Cabibbo-Kobayashi-Maskawa (CKM) matrix
Chen:2002th ; Wang:2015uea ; Li:2016tpn ; Jia:2021uhi . On the other hand,
GDAs and GPDs are key objects to investigate the matrix elements of the
energy-momentum tensor (EMT) for hadrons, which are expressed in terms of the
EMT form factors. In principle, one can not measure the EMT form factors of
hadrons directly by experiment since the gravitational interactions between
hadrons and gravitons are too tiny to probe. However, GDAs and GPDs can be
measured via electromagnetic interactions, as a consequence, the studies of
GDAs and GPDs are quite valuable. Many important physical quantities of
hadrons can be obtained through the study of the EMT form factors, e.g., mass,
pressure and shear force distributions of hadronsPolyakov:2002yz ;
Goeke:2007fp ; Mai:2012cx ; Polyakov:2018zvc ; Lorce:2018egm ; Burkert:2018bqq
; Kumericki:2019ddg ; Dutrieux:2021nlz ; Burkert:2023wzr . Let us also note
that the production of two different mesons, for example
$\gamma^{\ast}\to\gamma\pi\eta$ where the $\pi\eta$ GDA is measured, can be
also used to search for the hybrid meson ($J^{PC}=1^{-+}$)Anikin:2006du  and
to investigate the shear viscosity of quarks in hadronic matter
Teryaev:2022pke .

GPDs and GDAs can currently be measured at many experimental facilities, but
in a quite limited range of the hard scale $Q^{2}$ which is for instance the
virtuality of the incoming photon. Compared with the leading-twist cross
sections, the higher-twist corrections are thus not negligible considering the
energy scales of present and near future experimental measurements. In order
to extract GPDs and GDAs precisely, one needs to include the higher-twist
contributions to the cross sections. However, higher-twist GPDs and GDAs are
required to describe the higher-twist corrections, and this will make the
analysis difficult when one tries to extract GPDs and GDAs from the
experimental measurements. Recently, a separation of kinematical and dynamical
contributions in the operator product of two electromagnetic currents
$T\\{j_{\mu}^{\text{em}}(z_{1}x)j_{\nu}^{\text{em}}(z_{2}x)\\}$ was proven in
Refs. Braun:2011zr ; Braun:2011dg ; Braun:2011th ; Braun:2022qly , and these
operator results can be applied to the off-forward hard reactions such as
$\gamma^{\ast}h\to\gamma h$, $\gamma h\to\gamma^{\ast}h$,
$\gamma^{\ast}\gamma\to h\bar{h}$ and $\gamma^{\ast}\to h\bar{h}\gamma$, where
GPDs and GDAs can be measured. If one includes the kinematical higher-twist
contributions to the leading-twist cross sections, then only the leading-twist
GPDs and GDAs are involved. One can thus improve the theoretical description
of those processes without any knowledge of the higher-twist non-perturbative
functions. The kinematical higher-twist corrections can be considered as a
generalization of the target mass corrections Nachtmann:1973mr , which are
applied to the reaction of Deep Inelastic Scattering (DIS), and such
corrections were already included in Ref. Sato:2016tuz  where the parton
distribution functions were extracted. However, the kinematical higher-twist
corrections are more complicated in the off-forward hard reactions due to the
higher-twist operators which are reduced to the total derivatives of the
leading-twist ones, and these operators do not contribute in DIS since their
forward matrix elements vanish.

The kinematical higher-twist corrections were given up to the twist-4 accuracy
for the DVCS amplitude with a (pseudo)scalar target Braun:2012bg  and a
spin-1/2 target Braun:2012hq ; Braun:2014sta . The theoretical results were
applied to the recent DVCS measurements by the JLAB Hall A
collaborationJeffersonLabHallA:2022pnx . The authors of Ref. Lorce:2022tiq
also estimated the kinematical higher-twist corrections for
$\gamma^{*}\gamma\to M\bar{M}$ with a (pseudo)scalar meson pair. All thesse
theoretical studies suggest that the kinematical higher-twist corrections are
not negligible, besides, experimental measurements of DVCS also indicate that
there exists a gap between the measurements and the leading-twist cross
section, and this gap was diminished if the kinematical higher-twist
corrections were included JeffersonLabHallA:2015dwe . In this work, we intend
to calculate the kinematical higher-twist corrections in $\gamma^{*}\to
M\bar{M}\gamma$, and this process can be measured at BESIII in future. The
kinematics of BESIII measurements on this process will be similar to the Belle
(Belle II) measurements on $\gamma^{*}\gamma\to M\bar{M}$, whose cross
sections were released recently Belle:2015oin ; Belle:2017xsz . In this case,
the GDAs can be extracted by combining $\gamma^{*}\to M\bar{M}\gamma$ and
$\gamma^{*}\gamma\to M\bar{M}$. Moreover, one can study the university of GDAs
by the two processes, since the virtual photon is timelike in the former and
it is spacelike in the latter.

In Sec. II, we discuss the kinematics of $\gamma^{*}\to M\bar{M}\gamma$, and
the cross section is presented in terms of helicity amplitudes. We carry out a
complete calculation of kinematical higher-twist corrections to the helicity
amplitudes up to twist 4 in Sec. III, and the numerical estimate of the
kinematical higher-twist corrections are also presented. Our results are
summarized in Sec. IV.

##  II Kinematics and helicity amplitudes of $\gamma^{*}\to M\bar{M}\gamma$

Figure 1: Kinematics of $e^{-}(k_{1})e^{+}(k_{2})\to\gamma^{*}(q_{1})\to
M(p_{1})\bar{M}(p_{2})\gamma(q_{2})$ in the center-of-mass frame of the meson
pair, and the direction of the photons is chosen to the z axis.

If the center-of-mass energy $\sqrt{s}$ is large enough to satisfy the QCD
factorization in the process $e^{-}e^{+}\to\gamma^{*}\to M\bar{M}\gamma$, then
the amplitude can be factorized into a perturbative subprocess $\gamma^{*}\to
q\bar{q}\gamma$ and a two-meson GDA which describes the amplitude of
$q\bar{q}\to M\bar{M}$ Lu:2006ut . In Fig. 1, we show the polar angle $\theta$
and the azimuthal angle $\varphi$ in the center-of-mass frame of the meson
pair, for convenience we choose a coordinate system with the z axis along the
direction of photons, so that the momenta of the mesons lie in the x-z plane.
The polar angle $\theta$ can be expressed as

$\displaystyle\cos{\theta}=\frac{q_{1}\cdot(p_{2}-p_{1})}{\beta_{0}\,(q_{1}\cdot
q_{2})},\qquad\beta_{0}=\sqrt{1-\frac{4m^{2}}{\hat{s}}},$ (4)

where $m$ is the meson mass and
$\hat{s}=W^{2}=(q_{1}-q_{2})^{2}=(p_{1}+p_{2})^{2}$. Similarly, the azimuthal
angle $\varphi$ is also given in terms of Lorentz invariants,

$\displaystyle\sin{\varphi}=\frac{4\epsilon_{\alpha\beta\gamma\delta}q_{1}^{\alpha}q_{2}^{\beta}p_{1}^{\gamma}k_{1}^{\delta}}{\beta_{0}\sin{\theta}\sqrt{us\hat{s}(\hat{s}-u-s)}}$
(5)

with $\epsilon_{0123}=1$, $s=(k_{1}+k_{2})^{2}$ and $u=(k_{1}-q_{2})^{2}$. Two
lightlike vectors $n$ and $\tilde{n}$ are chosen with the help of the momenta
of the timelike virtual photon $q_{1}$ and the real photon $q_{2}$,

$\displaystyle\tilde{n}=q_{1}-(1+\tau)q_{2},\qquad n=q_{2},$ (6)

where $\tau=\hat{s}/(s-\hat{s})$. The momentum $\Delta=p_{2}-p_{1}$ can be
written as $\Delta=\zeta_{0}(\tilde{n}-\tau n)+\Delta_{T}$, and $\Delta_{T}$
is the transverse component. $\zeta_{0}$ is a parameter which is defined as

$\displaystyle\zeta_{0}=\frac{(p_{2}-p_{1})\cdot n}{(p_{2}+p_{1})\cdot n},$
(7)

and it is related to the polar angle $\theta$ as
$\zeta_{0}=-\beta_{0}\cos{\theta}$. We can obtain
$\Delta_{T}^{2}=4m^{2}-(1-\zeta_{0}^{2})\hat{s}$ by the on-shell condition.

To describe the process $\gamma^{*}\to M\bar{M}\gamma$, one needs to define
the amplitude

$\displaystyle A_{\mu\nu}=i\int d^{4}x\,e^{-ir\cdot
x}\langle\bar{M}(p_{2})M(p_{1})|\,T\\{j_{\mu}^{\text{em}}(z_{1}x)j_{\nu}^{\text{em}}(z_{2}x)\\}\,|0\rangle,\\!\\!$
(8)

where $z_{1}$ and $z_{2}$ are real constants with the constraint
$z_{1}-z_{2}=1$, and $r=z_{1}q_{1}-z_{2}q_{2}$ is used. This amplitude can be
further written in terms of helicity amplitudes by considering the
electromagnetic gauge invariance Braun:2012bg

$\displaystyle
A^{\mu\nu}=-A^{(0)}\,g_{\perp}^{\mu\nu}+A^{(1)}\,\left(\tilde{n}^{\mu}-(1+\tau)n^{\mu}\right)\frac{\Delta_{\alpha}g_{\perp}^{\alpha\nu}}{\sqrt{s}}+\frac{1}{2}\,A^{(2)}\,\Delta_{\alpha}\Delta_{\beta}(g_{\perp}^{\alpha\mu}g_{\perp}^{\beta\nu}-\epsilon_{\perp}^{\alpha\mu}\epsilon_{\perp}^{\beta\nu})+A^{(3)\mu}\,n^{\nu},$
(9)

and the transverse tensors $g_{\perp}^{\mu\nu}$ and
$\epsilon_{\perp}^{\mu\nu}$ are defined as

$\displaystyle
g_{\perp}^{\mu\nu}=g^{\mu\nu}-\frac{n^{\mu}\tilde{n}^{\nu}+n^{\nu}\tilde{n}^{\mu}}{n\cdot\tilde{n}},\qquad\epsilon_{\perp}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}\,\frac{\tilde{n}_{\alpha}n_{\beta}}{n\cdot\tilde{n}}.$
(10)

The longitudinal polarization vector $\epsilon_{0}$ exists in addition to the
transverse ones $\epsilon_{\pm}$ for the virtual photon. In the center-of-mass
frame of the meson pair as shown in Fig. 1, the polarization vectors read

$\displaystyle\epsilon_{0}^{\mu}=\frac{1}{\sqrt{s}}(|q_{1}^{3}|,0,0,q_{1}^{0}),\quad\epsilon_{\pm}^{\mu}=\frac{1}{\sqrt{2}}(0,\mp
1,-i,0),$ (11)

and the transverse polarization vectors $\tilde{\epsilon}_{\pm}$ of the real
photon are the same as the ones of the virtual photon. Then, the three
independent helicity amplitudes are given by

$\displaystyle A_{++}=A_{--}=A^{(0)},\quad\quad
A_{0\pm}=-A^{(1)}(\Delta\cdot\epsilon_{\mp}),$ $\displaystyle
A_{\pm\mp}=-A^{(2)}(\Delta\cdot\epsilon_{\pm})^{2}.$ (12)

where the notation
$A_{ij}=\epsilon_{i}^{\mu}\tilde{\epsilon}_{j}^{\ast\nu}A_{\mu\nu}$ is used,
and only $A_{++}$ can receive twist-2 contribution at leading order of
$\alpha_{s}$. In Ref. Lu:2006ut , $A_{++}$ was given in terms of the $\pi\pi$
GDA at twist-2 for the process of $e^{-}e^{+}\to\gamma^{*}\to\pi\pi\gamma$,
and the twist-2 $\pi\pi$ GDA is defined as Diehl:1998dk ; Diehl:2000uv

$\displaystyle\langle\bar{M}(p_{2})M(p_{1})|\,\bar{q}(z_{1}n)\not{n}q(z_{2}n)\,|0\rangle=2P\cdot
n\int dz\,e^{2i\left[zz_{1}+(1-z)z_{2}\right]P\cdot
n}\,\Phi_{q}(z,\zeta_{0},\hat{s}),$ (13)

where $z$ is the momentum fraction of the quark, $P$ is the average momentum
of the meson pair $P=(p_{1}+p_{2})/2$, and the real constants $z_{1}$ and
$z_{2}$ do not have to satisfy $z_{1}-z_{2}=1$. Note that the GDAs depend on a
renormalization scale $\mu^{2}$ which one usually takes as $\mu^{2}=s$.

The differential cross section can be expressed in terms of the helicity
amplitudes for $e^{-}e^{+}\to\gamma^{*}\to M\bar{M}\gamma$,

$\displaystyle\frac{d\sigma}{d\hat{s}\,du\,d(\cos\theta)\,d\varphi}=$
$\displaystyle\frac{\alpha_{\text{em}}^{3}\beta_{0}}{16\pi
s^{3}}\,\frac{1}{1+\epsilon}\,\Big{[}|A_{++}|^{2}+|A_{-+}|^{2}+2\epsilon\,|A_{0+}|^{2}-2\text{sgn}(\tau)\sqrt{\epsilon(1-\epsilon)}$
$\displaystyle\times\text{Re}(A_{++}^{\ast}A_{0+}-A_{-+}^{\ast}A_{0+})\cos\varphi+2\epsilon\,\text{Re}(A_{++}^{\ast}A_{-+})\cos(2\varphi)\Big{]},$
(14)

where $M\bar{M}$ is the pseudoscalar meson pair with even charge conjugation
and the parameter $\epsilon$ is defined as

$\displaystyle\epsilon=\frac{y-1}{1-y+\frac{y^{2}}{2}},\qquad
y=\frac{q_{1}\cdot q_{2}}{k_{1}\cdot q_{2}}.$ (15)

$\text{sgn}(\tau)=|\tau|/\tau$ is the sign function with $\tau=\hat{s}-s-2u$.

##  III Results

###  III.1 Theoretical amplitudes in terms of GDAs

Recently, Braun et al. have proved that the kinematical contributions can be
separated from dynamical ones in the time-ordered product of two
electromagnetic currents
$i\,T\\{j_{\mu}^{\text{em}}(z_{1}x)j_{\nu}^{\text{em}}(z_{2}x)\\}$
Braun:2011zr ; Braun:2011dg ; Braun:2011th . If we introduce the kinematical
higher-twist contributions to the twist-2 cross sections, one can improve the
description of reactions where two photons are involved without any knowledge
of the genuine higher-twist distributions. This is very helpful when one
intends to extract the twist-2 distributions from the measurements of the
cross sections, since taking into account of genuine higher-twist
distributions would imply more parameters to be used in the analysis. At
twist-4 accuracy, the kinematical contributions in
$i\,T\\{j_{\mu}^{\text{em}}(z_{1}x)j_{\nu}^{\text{em}}(z_{2}x)\\}$ can be
written as Braun:2011zr ; Braun:2011dg ; Braun:2011th

$\displaystyle
T_{\mu\nu}=\frac{-1}{\pi^{2}x^{4}z_{12}^{3}}\left\\{x^{\alpha}\left[S_{\mu\alpha\nu\beta}\mathbb{V}^{\beta}-i\epsilon_{\mu\alpha\nu\beta}\mathbb{W}^{\beta}\right]+x^{2}\left[(x_{\mu}\partial_{\nu}+x_{\nu}\partial_{\mu})\mathbb{X}+(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})\mathbb{Y}\right]\right\\},$
(16)

where the notation $z_{12}=z_{1}-z_{2}$ is used, and the tensor
$S^{\mu\alpha\nu\beta}$ is defined by

$\displaystyle
S^{\mu\alpha\nu\beta}=g^{\mu\alpha}g^{\nu\beta}-g^{\mu\nu}g^{\alpha\beta}+g^{\mu\beta}g^{\nu\alpha}.$
(17)

$\mathbb{V}_{\mu}$ and $\mathbb{W}_{\mu}$ contain all twists starting from
twist 2 to twist 4, whereas $\mathbb{X}$ and $\mathbb{Y}$ are purely twist 4
operators, the detailed expressions of them can be found in our previous work
Lorce:2022tiq .

In this work we shall use Eq. (16) to calculate the kinematical higher-twist
contributions in the reaction $\gamma^{*}\to M\bar{M}\gamma$, where $M\bar{M}$
is the scalar meson pair with even charge conjugation. The spinor formalism
Braun:2008ia ; Braun:2009vc  is used for $T_{\mu\nu}$ in the calculation, and
it will help us to figure out the twist of the matrix elements for the
operators much easier. Even though the final amplitudes are presented in terms
of GDAs, the double distributions Teryaev:2001qm  are used to calculate the
helicity amplitudes. Those techniques are well explained in Ref. Braun:2012bg
; Lorce:2022tiq , so we skip them in this work. The helicity amplitudes are
written as

$\displaystyle A^{(0)}$
$\displaystyle=\chi\left\\{\left(1+\frac{\hat{s}}{2s}\right)\int_{0}^{1}dz\,\frac{\Phi(z,\eta,\hat{s})}{1-z}+\frac{\hat{s}}{s}\int_{0}^{1}dz\,\frac{\Phi(z,\eta,\hat{s})}{z}\,\ln(1-z)\right.$
$\displaystyle\qquad+\left.\left(\frac{2\hat{s}}{s}\,\eta+\frac{\Delta_{T}^{2}}{\beta_{0}^{2}s}\frac{\partial}{\partial\eta}\right)\frac{\partial}{\partial\eta}\int_{0}^{1}dz\,\frac{\Phi(z,\eta,\hat{s})}{z}\left[\frac{\ln(1-z)}{2}+\text{Li}_{2}(1-z)-\text{Li}_{2}(1)\right]\right\\},$
$\displaystyle A^{(1)}$
$\displaystyle=-\frac{2\chi}{\beta_{0}\sqrt{s}}\frac{\partial}{\partial\eta}\int_{0}^{1}dz\,\Phi(z,\eta,\hat{s})\,\frac{\ln(1-z)}{z},$
$\displaystyle A^{(2)}$
$\displaystyle=\frac{2\chi}{\beta_{0}^{2}s}\frac{\partial^{2}}{\partial\eta^{2}}\int_{0}^{1}dz\,\Phi(z,\eta,\hat{s})\,\frac{2z-1}{z}\,\ln(1-z),$
(18)

where $\eta=\cos\theta$, $\chi=5e^{2}/18$, and $\Phi=\Phi_{u}+\Phi_{d}$ for
the isosinglet meson pair such as $\pi\pi$ and $\eta\eta$. In case of a charge
conjugation-even $K$ meson pair, we just need to replace $\chi\Phi$ with
$e_{u}^{2}\Phi_{u}+e_{d}^{2}\Phi_{d}+e_{s}^{2}\Phi_{s}$ in the above
amplitudes since the GDA for $s$ quark is also required. We note that although
Eq. (16) is dependent on $z_{1}$ and $z_{2}$, this dependence is eliminated by
the constraint $z_{1}-z_{2}=1$ in physical amplitudes, which indicates that
translation invariance is recovered. The existence of correction
$\mathcal{O}(\hat{s}/s)$ is obvious in the amplitudes, while the correction
$\mathcal{O}(m^{2}/s)$ exists in the term $\Delta_{T}^{2}/s$ due to
$\Delta_{T}^{2}=4m^{2}-(1-\zeta_{0}^{2})\hat{s}$.

The general expressions of GDAs were firstly derived in Diehl:2000uv , in this
work we only need the charge conjugation-even ones, and they are given by

$\displaystyle\Phi(z,\cos\theta,\hat{s})=6\,z(1-z)\sum_{\begin{subarray}{c}n=1\\\
n\,\text{odd}\end{subarray}}^{\infty}\sum_{\begin{subarray}{c}l=0\\\
l\,\text{even}\end{subarray}}^{n+1}\tilde{B}_{nl}(\hat{s})\,C_{n}^{(3/2)}(2z-1)\,P_{l}(\cos\theta),$
(19)

where $C_{n}^{(3/2)}(x)$ are Gegenbauer polynomials, and $P_{l}(x)$ are
Legendre polynomials with $l$ indicates the angular momentum of the scalar
meson pair. The nonvanishing helicity-flip amplitudes $A_{-+}(A^{(2)})$ and
$A_{0+}(A^{(1)})$ indicate the existence of the D-wave or higher-wave GDA. In
Eq. (18), it seems that the amplitudes are divergent if $z\rightarrow 0$ or
$z\rightarrow 1$. However, the singularities will be compensated by the
general expression of GDAs, since one can see
$\Phi(z,\cos\theta,\hat{s})\rightarrow 0$ if $z\rightarrow 0$ or $z\rightarrow
1$. The dependence on energy scale $s$ is abbreviated in Eq. (19), in the
limit of $s\rightarrow\infty$, only the terms with $n=1$ survive in the GDAs
Diehl:2000uv ,

$\displaystyle\Phi(z,\cos\theta,\hat{s})=18\,z(1-z)(2z-1)\left[\tilde{B}_{10}(\hat{s})+\tilde{B}_{12}(\hat{s})P_{2}(\cos\theta)\right],$
(20)

which is the so-called asymptotic form of GDAs. The first and second terms
correspond to the S-wave and D-wave production of a meson pair, respectively.

###  III.2 Numerical estimates of the kinematical higher-twist contributions

Figure 2:  Differential cross section of $e^{-}e^{+}\to\pi^{0}\pi^{0}\gamma$
is dependent on the invariant mass of pion pair $W=\sqrt{\hat{s}}$, using the
$\pi\pi$ GDA extracted from Belle measurements Kumano:2017lhr . The dashed
lines are the twist-2 cross sections, and the solid lines include the
kinematical higher-twist contributions.

In principle, the process $\gamma^{*}\to M\bar{M}\gamma$ can be measured by
BESIII and Belle (Belle II) in $e^{+}e^{-}$ collisions. The center-of-mass
energy is $\sqrt{s}=3-5$ GeV at BESIII, while it is $\sqrt{s}=8-10$ GeV at
Belle (Belle II). It should be much easier to measure this process at BESIII
due to the larger cross section which can be seen from Eq. (14). As a
consequence, we shall use the kinematics of BESIII in the numerical estimate
of the kinematical higher-twist contributions, and the differential cross
section of Eq. (14) is used by integrating over $\varphi$,

$\displaystyle\frac{d\sigma}{du\,dW^{2}\,d(\cos\theta)}=$
$\displaystyle\frac{\alpha_{\text{em}}^{3}\beta_{0}}{8s^{3}}\,\frac{1}{1+\epsilon}\,\Big{[}|A_{++}|^{2}+|A_{-+}|^{2}+2\epsilon\,|A_{0+}|^{2}\Big{]},$
(21)

where the helicity amplitudes are given by Eq. (18) including the kinematical
higher-twist contributions up to twist 4.

We firstly calculate the cross section of
$e^{-}e^{+}\to\gamma^{*}\to\pi^{0}\pi^{0}\gamma$ with the $\pi\pi$ GDA
extracted from Belle measurements Kumano:2017lhr . In Fig. 2, the twist-2
cross sections are depicted as dashed lines, and the solid lines include the
kinematical higher-twist contributions. The kinematics is set according to the
BESIII experiment as $s=12$ GeV2 and $W\in(0.5,2)$ GeV. The colors of the
lines (black, orange, red, blue) represent different values of $\cos\theta$
(0,2, 0.4, 0.6, 0.8), and $u$ is chosen as $u=-3$ GeV2 and $u=-6$ GeV2. We can
clearly see that the kinematical higher-twist corrections are always positive
in the cross section, and this is different from the case of $e^{-}\gamma\to
e^{-}\pi^{0}\pi^{0}$ where the corrections can be positive or negative
Lorce:2022tiq . In the region $W>1$ GeV, the kinematical higher-twist
corrections turn out to be important and it is thus crucial to include them to
extract in a valuable way GDAs from the cross section of
$e^{-}e^{+}\to\gamma^{*}\to\pi^{0}\pi^{0}\gamma$, and then access both the
timelike pion EMT form factors, and the spacelike ones, obtained from the
timelike ones by using dispersion relations requiring reliable information at
$W>1$ GeV so as to make the integrals convergent.

Figure 3:  Ratio $d\sigma(2+3+4)/d\sigma(2)$ with the $\pi\pi$ GDA extracted
from Belle measurements, same conventions as in Fig. 2.

The ratio $d\sigma(2+3+4)/d\sigma(2)$ is also shown in Fig. 3, where
$d\sigma(i)$ ($i=2,3,4$) denotes the twist-$i$ contribution to the cross
section. The colors of the lines indicate different values of $\cos\theta$ as
in Fig. 2. We can see that the kinematical higher-twist contributions have a
significant impact on the cross section when $W>1$ GeV. The ratios just
slightly change from $u=-3$ GeV2 to $u=-6$ GeV2, and this is because the
ratios are dependent on $u$ through the parameter $\epsilon$, namely, only the
contribution from the amplitude $A_{0+}$ is affected in the ratios as one
changes $u$. The peaks around $W\sim 1.1$ GeV and $W\sim 1.5$ GeV with
$\cos\theta=0.8$ in Fig. 3 are due to the fact that the twist-2 cross sections
are quite tiny in this region as indicated by Fig. 2; however, the extracted
GDA used in this estimate may not be accurate enough due to the large
uncertainties of Belle measurements, and these peaks in the ratio may thus not
reflect real physics.

Figure 4:  Differential cross section for
$e^{-}e^{+}\to\gamma\to\pi^{0}\pi^{0}\gamma$ with the asymptotic $\pi\pi$ GDA
described in the text, same conventions as in Fig. 2.

For comparison, we also present our results in Fig. 4 when we employ the
asymptotic pion GDA in the analysis of
$e^{-}e^{+}\to\gamma^{\ast}\to\pi^{0}\pi^{0}\gamma$. In Ref. Diehl:1998dk ,
the asymptotic GDA was given when the energy scale $s\rightarrow\infty$,

$\displaystyle\Phi(z,\cos\theta,\hat{s})=20\,z(1-z)(2z-1)R_{\pi}\left[\frac{-3+\beta_{0}^{2}}{2}\,e^{i\delta_{0}}+\beta_{0}^{2}e^{i\delta_{2}}P_{2}(\cos\theta)\right],$
(22)

where $R_{\pi}=0.5$ is the momentum fraction carried by quarks in the pion
meson. $\delta_{0}$ is $\pi\pi$ the elastic scattering phase shift for S wave,
and $\delta_{2}$ is the one for D wave Bydzovsky:2016vdx ; Bydzovsky:2014cda ;
Surovtsev:2010cjf . In Fig. 4, $u$ is chosen as $u=$-3 GeV2 and $u=$-6 GeV2
together with 0.5 GeV $\leq W\leq$ 2.1 GeV and $s=12$ GeV2. The dashed lines
represent the twist-2 cross sections, and the solid ones include kinematical
higher-twist contributions. Black lines denote $\cos\theta=0.2$ and orange
lines correspond to $\cos\theta=0.4$, while $\cos\theta=0.6$ and
$\cos\theta=0.8$ are depicted as red and blue, respectively. We can clearly
see that the kinematical higher-twist corrections become more and more
important as one increases $W=\sqrt{\hat{s}}$ in this figure, which is evident
since the corrections are expected to be proportional to $\hat{s}/s$. As for
the case of the extracted GDA from Belle measurements, it is thus necessary to
include the kinematical higher-twist corrections to describe the cross section
in the region of $W>1$ GeV. Both GDAs indicate a similar magnitude of the
cross section for $e^{-}e^{+}\to\gamma^{\ast}\to\pi^{0}\pi^{0}\gamma$.

Figure 5: Ratio $d\sigma(2+3+4)/d\sigma(2)$ with the asymptotic $\pi\pi$ GDA
described in the text, same conventions as in Fig. 2.

In Fig. 5, the ratio of $d\sigma(2+3+4)/d\sigma(2)$ is also presented so as to
see the proportion of the kinematical higher-twist corrections in the cross
section clearly. $u$ is set as $u=$-3 GeV2 and $u=$-6 GeV2 for the left panel
and right panel, respectively, and the colors of lines indicate the values of
$\cos\theta$ as in Fig. 4. The ratios increase rapidly as $W$ goes up, and the
kinematical higher-twist corrections account for more than $40\%$ of the cross
section around $W\sim 2$ GeV, which proves that they need to be included in
any reliable GDA extraction from the cross section.

Figure 6:  Differential cross section for
$e^{-}e^{+}\to\gamma^{\ast}\to\eta\eta\gamma$ with the model $\eta\eta$ GDA
described in the text, same conventions as in Fig. 2.

There are two types of corrections $m^{2}/s$ and $\hat{s}/s$ in the
kinematical corrections, and only the latter contributes to the cross section
of $e^{-}e^{+}\to\gamma^{\ast}\to\pi^{0}\pi^{0}\gamma$ due to the small mass
of pion meson in comparison with the value of $s$. In order to see the impact
of the target mass correction $m^{2}/s$, we now consider the production of a
pair of slightly heavier mesons, namely $\eta\eta$. However, very little
information is known about their GDAs at the current stage, and we thus
estimate the kinematical higher-twist corrections for
$e^{-}e^{+}\to\gamma^{\ast}\to\eta\eta\gamma$ by using a simple model GDA
identical to the asymptotic $\pi\pi$ GDA except that the $\eta$ mass is used.
The center-of-mass energy of $e^{-}e^{+}$ is again chosen as $s=12$ GeV2. In
Fig. 6, the cross sections are shown with the range of 1.2 GeV $\leq W\leq$
2.1 GeV. The colors of the lines denote different values of $\cos\theta$ as
indicated on the different panels of the figure. The dashed lines represent
the twist-2 cross sections, and the solid ones include kinematical higher-
twist contributions. The gaps between the dashed lines and the solid ones
increase along with $W$ as expected, and one thus cannot neglect the
kinematical higher-twist contributions in the cross section. The kinematical
higher-twist contributions are always positive in the cross section. We
present the ratios of $d\sigma(2+3+4)/d\sigma(2)$ in Fig. 7. The kinematical
higher-twist contributions account for less than $40\%$ of the cross section
around $W\sim 2$ GeV. Compared with the results in Fig. 5, the ratios decrease
if one replace the mass of pion meson with the one of the $\eta$ meson in the
asymptotic $\pi\pi$ GDA; indeed, the target mass corrections are negative and
thus diminish the positive corrections of order $\hat{s}/s$ in the cross
section.

Figure 7: Ratio $d\sigma(2+3+4)/d\sigma(2)$ with the model $\eta\eta$ GDA
described in the text, same conventions as in Fig. 2.

##  IV Summary

GDAs can be studied in both $\gamma^{*}\gamma\to M\bar{M}$ and $\gamma^{*}\to
M\bar{M}\gamma$. The former process has been measured by Belle for $\pi\pi$
Belle:2015oin  and $KK$ Belle:2017xsz  with large uncertainties; in the near
future we can expect more precise measurements from Belle II due to the much
higher luminosity. It will be more advantageous to measure the latter process
at BESIII, since its cross section will be suppressed by the larger center-of-
mass energy of electron-positron pair at Belle (Belle II). In this case, the
measurements of $\gamma^{*}\gamma\to M\bar{M}$ at Belle and Belle II can be
cross checked by the ones of $\gamma^{*}\to M\bar{M}\gamma$ at BESIII due to
the similar kinematics, and the GDAs can be extracted by combining the
measurements of the two pocesses. Besides, since GDAs are probed by a
spacelike photon and a timelike photon in $\gamma^{*}\gamma\to M\bar{M}$ and
$\gamma^{*}\to M\bar{M}\gamma$, respectively, we can also check the university
of GDAs.

In this work we calculate the kinematical higher-twist corrections for
$\gamma^{*}\to M\bar{M}\gamma$ up to twist 4, and three helicity amplitudes
are expressed in term of the twist-2 GDA. A precise description of the cross
section is obtained without the knowledge of the higher-twist GDAs. We adopt
two types of GDAs to estimate the kinematical higher-twist contributions for
$\gamma^{*}\to\pi^{0}\pi^{0}\gamma$ numerically. In the calculation, the
center-of-mass energy of the electron-positron pair is chosen as $s=12$ GeV2,
which is typical for BESIII. All the numerical results indicate that the
kinematical higher-twist corrections have a significant impact on the cross
section of $\gamma^{*}\to M\bar{M}\gamma$ as in the case of
$\gamma^{*}\gamma\to M\bar{M}$ Lorce:2022tiq ; Lorce:2022cze . However, the
corrections are always positive in the cross section of the former, and this
is different from the latter process where kinematical higher-twist
corrections can go both ways. A model $\eta\eta$ GDA is used to see the impact
of the target mass corrections of $\mathcal{O}(m^{2}/s)$, and the kinematical
higher-twist corrections account for about $20\%$ of the cross section in the
region of 1.2 GeV $\leq W\leq$2.1 GeV on the average, which are not
negligible. As a consequence, it is important to use the accurate description
of the cross section with the inclusion of kinematical contributions when one
tries to extract GDAs from experimental measurements. The extracted GDAs can
be used to study the EMT form factors of hadrons, which are important physical
quantities to investigate mass, pressure and shear force distributions of
hadrons.

The present study was performed at lowest order in the strong coupling, but it
would be interesting to include higher order corrections which are known - at
leading twist - to be very sensitive to the timelike vs spacelike nature of
the probe Mueller:2012sma .

## Acknowledgments

We acknowledge useful discussions with Cédric Lorcé, Wen-Cheng Yan and Ya-Teng
Zhang. Qin-Tao Song was supported by the National Natural Science Foundation
of China under Grant Number 12005191.

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# Evaluating Generalizability of Fine-Tuned Models for Fake News Detection

Abhijit Suprem<EMAIL_ADDRESS>Georgia Institute of TechnologyAtlantaGAUSA
and  Calton Pu<EMAIL_ADDRESS>Georgia Institute of
TechnologyAtlantaGAUSA

###### Abstract.

The Covid-19 pandemic has caused a dramatic and parallel rise in dangerous
misinformation, denoted an ‘infodemic’ by the CDC and WHO. Misinformation tied
to the Covid-19 infodemic changes continuously; this can lead to performance
degradation of fine-tuned models due to concept drift. Degredation can be
mitigated if models generalize well-enough to capture some cyclical aspects of
drifted data. In this paper, we explore generalizability of pre-trained and
fine-tuned fake news detectors across 9 fake news datasets. We show that
existing models often overfit on their training dataset and have poor
performance on unseen data. However, on some subsets of unseen data that
overlap with training data, models have higher accuracy. Based on this
observation, we also present KMeans-Proxy, a fast and effective method based
on K-Means clustering for quickly identifying these overlapping subsets of
unseen data. KMeans-Proxy improves generalizability on unseen fake news
datasets by 0.1-0.2 f1-points across datasets. We present both our
generalizability experiments as well as KMeans-Proxy to further research in
tackling the fake news problem.

##  1\. Introduction

The rapid spread of the Covid-19 virus has led to a parallel surge in
misinformation and disinformation (Enders et al., 2020a) This surge of false
information, coined an ‘infodemic’ by the CDC (Enders et al., 2020b) can be
life-threatening, destabilizing, and potentially dangerous (Quinn et al.,
2021). The infodemic is multimodal, meaning associated fake news can take
forms of social media posts, tweets, articles, blogs, commentary,
misrepresented titles and headlines, videos, and audio content.

Current Research..Current Research.Current Research.. There is significant
progress on developing domain-specific automated misinformation detection and
classification tools (Cui and Lee, 2020; Weinzierl et al., 2021; Hossain et
al., 2020; Serrano et al., 2020; Kou et al., 2021; Wahle et al., 2022). Such
tools analyze labeled datasets in aforementioned modalities to classify fake
news. Recent approaches focus on transformer-based classifiers and language
modelers (Wahle et al., 2022; Serrano et al., 2020).

Such fake news detectors are specific to the datasets they are trained with
(Hossain et al., 2020; Wahle et al., 2022). More recently, there is a focus on
addressing generalizability concerns in these models (Wahle et al., 2022; Bang
et al., 2021; Kaliyar et al., 2021; Walambe et al., 2022; Das et al., 2021;
Chen et al., 2021). For example, (Wahle et al., 2022) explores the impact of
generalization of 15 transformer models on 5 fake news datasets. The results
show there is a generalizability gap in fake news detection: a fine-tuned
model trained on one fake news dataset performs poorly on other unseen, but
related, fake news datasets.

Generalization Study..Generalization Study.Generalization Study.. In this
paper, we study the generalizability and fine-tuning tradeoff and present our
findings for furthering the research interest. We study several fake news
detection architectures across 9 fake news text datasets of different
modalities. We find that fine-tuned models often have reduced accuracy on any
unseen dataset. However, when paired with a reject option to abstain from low-
confidence predictions, fine-tuned models perform significantly better. These
abstention results can then be labeled with active learning, crowdsourcing,
weak label integration, or a variety of other methods present in literature.

KMeans-Proxy.KMeans-ProxyKMeans-Proxy. Through observations on our
generalizability results, we present a simple ‘reject option’ (Hendrickx et
al., 2021; Brinkrolf and Hammer, 2018; Geifman and El-Yaniv, 2019) for fake
news detectors, called KMeans-Proxy. KMeans-Proxy is based on KMeans
clustering, and is inspired by research into proxy losses (Movshovitz-Attias
et al., 2017; Wern Teh et al., 2020) and foundation models (Bommasani et al.,
2021; Chen et al., 2022; Orr et al., 2021). It is written as a PyTorch layer
and requires only a few lines of code to implement for most feature
extractors. We show in our results that KMeans-Proxy improves generalization
on fake news datasets by 0.1 to 0.2 f1 points across several experiments.

Contributions.ContributionsContributions. In summary, our contributions are:

  * 1\.

Extensive set of experiments across 9 fake news datasets on the
generalizability/fine-tuning trade off.

  * 2\.

KMeans-Proxy, a simple reject-option for feature extractors. KMeans-Proxy uses
cluster proxies from ProxyNCA to estimate embedding cluster centers of the
training data. During prediction, KMeans-Proxy provides a reject option based
on label difference between sample prediction and nearest training data
cluster center.

Our code for running experiments and for KMeans-Proxy are
provided111https://github.com/asuprem/GLAMOR/blob/colabel-
multicb/src/ednaml/utils/blocks/KMeansProxy.py.

##  2\. Related Work

###  2.1. Generalizability and Fine-Tuning

Since 2002, there are increasing numbers of Covid-19 fake news datasets and
associated models for these datasets (Cui and Lee, 2020; Kou et al., 2021).
Recently, there is increasing interest in gauging the effectiveness of each of
these fine-tuned models on related, but unseen datasets (Wahle et al., 2022).
The authors of (Wahle et al., 2022) conduct a generalization study over 15
model architectures over 5 datasets, and find that fine-tuning offers little
advantage in classification accuracy. There is also an abundance of research
in unsupervised domain adaptation to recover accuracy under changing domains
or concept drift (Suprem et al., 2020; Gama et al., 2014; Kouw and Loog, 2018;
Žliobaitė, 2010).

###  2.2. Concept Drift

Concept drift occurs when testing or prediction data exhibits distribution
shift (Gama et al., 2014), either in the data domain, or in the label domain
(Žliobaitė, 2010). Data domain shift can include introduction of new
vocabularies, disappearance of existing words, and word polysemy (Suprem et
al., 2019). Label domain shift occurs when the label space itself changes for
the same type of data (Kouw and Loog, 2018; Pu et al., 2020). For example,
when new types of misinformation are detected, then the boundary between
misinformation and true information must be adjusted (Pu et al., 2020). We
show label shift in Figure 1, where subsets of true and fake news occupy the
same embedding space across datasets due to fine-grained differences.

###  2.3. Reject Options

One drawback of classification models is that they provide a prediction for
every data point (Hendrickx et al., 2021), regardless of confidence. Reject
options perform external or internal diagnosing. This can help detect either
low confidence due to low coverage or divergence from training data
distributions due to concept drift. Several approaches are covered in a recent
survey (Hendrickx et al., 2021). We present a reject option that uses recent
findings in (Chen et al., 2022) and (Urner and Ben-David, 2013) on the
topology of the embedding space: (i) find that local smoothness of the label
space is indicative of local accuracy and coverage (Urner and Ben-David,
2013), and (ii) local label shift, where nearby samples have different labels,
is a good predictor of local smoothness (Chen et al., 2022). Our reject
option, described in Section 3.3 is a clustering approach that calculates
cluster centers in the feature extractor embedding space for the training
data. Then, during prediction, a model can provide a prediction as well the
label for the nearest training data cluster center. Flipped, or different,
labels can indicate reduced local smoothness, confidence, and coverage,
leading to a reject decision.

###  2.4. Motivation

It is well known that fine-tuned models suffer performance degradation over
time due to data domain shift (Pu et al., 2020; Yang et al., 2015; Lazer et
al., 2014). Usually, this performance degradation is detected, and a new model
is trained on new labeled data. Recently, the velocity and size of new data
makes obtaining labeled data quickly and at scale, very expensive (Li et al.,
2021). Updating models during data domain shift requires relying on weak
labels, authoritative sources, and hierarchical models (Li et al., 2021;
Ratner et al., 2017; Rühling Cachay et al., 2021). In such cases, a team of
prediction models is pruned and updated with new training data (Pu et al.,
2020). However, we still need predictions in the period when data domain shift
is occurring, and new models have not been trained. Our work, as well as
recent research in generalizability (Wahle et al., 2022), weak labeling
(Ratner et al., 2017), foundation models (Chen et al., 2022), and rapid fake
news detection (Li et al., 2021) falls in this period. Our generalizability
experiments in Section 3 show that fine-tuned models, while having lower
performance on unseen data, do have better accuracy on some subsets. Our
KMeans-Proxy solution finds these subsets where fine-tuned models have higher
accuracy.

##  3\. Generalization Experiments

We transformer-based text feature extractors for fake news classification
generalizability with several experiments. We cover the datasets,
architectures, and experiments below.

Datasets..Datasets.Datasets.. We used 9 fake news datasets, consisting of blog
articles, news headlines, news content, tweets, social media posts, and
article headlines. We have described our dataset below in Table 1.

Table 1. Datasets used for our experiments. If splits were not available, we used a random class-balanced split. For Tweet datasets, sample counts are after rehydration, which removed some samples due to missing tweets. Dataset | Training | Testing | Type
---|---|---|---
k_title (Patel, 2021) | 31k | 9K | Article Titles
coaid (Cui and Lee, 2020) | 5K | 1K | News summary
c19_text (Agarwal, 2020) | 2.5K | 0.5K | Articles
cq (Mutlu et al., 2020) | 12.5K | 2K | Tweets
miscov (Memon and Carley, 2020) | 4K | 0.6K | Headlines
k_text (Patel, 2021) | 31k | 9K | Articles
rumor (Cheng et al., 2021) | 4.5K | 1K | Social Posts
cov_fn (Das et al., 2021) | 4K | 2K | Tweets
c19_title (Agarwal, 2020) | 2.5K | 0.5K | Article Titles

Where possible, we have used the provided training and validation sets;
otherwise, we performed a random, class-balanced 70-30 split for training and
testing. For (Mutlu et al., 2020) and (Das et al., 2021) datasets, we
performed tweet rehydration, which removed some samples due to missing tweets.
We show example of label shift due to label overlap in Figure 1. Here, samples
from each dataset are passed through a pre-trained BERT classifier. The BERT
embeddings are then reduced to 50 components with PCA then to 2 components
with tSNE. There are several regions with label overlaps, where samples with
positive and negative labels occupy similar spaces.

Figure 1. Overlap of label embeddings: We show label embeddings for all 9
datasets here. There are several regions where true news (blue) overlap with
fake news (red) across different datasets. This can make models for one
dataset perform worse on unseen, but related text-based fake news datasets.

Architectures..Architectures.Architectures.. We use BERT and AlBERT
architectures for our experiments (Devlin et al., 2018; Lan et al., 2019).
Each transformer architecture converts input tokens to a classification
feature vector. We used pretrained architectures as starting points; our
selections include the BERT (Devlin et al., 2018), AlBERT (Lan et al., 2019),
and COVID-Twitter-BERT (Müller et al., 2020).

Experiments.ExperimentsExperiments. We performed 3 experiments to evaluate
generalizability of covid fake news detectors. In each case, our starting
point is a pre-trained foundation model, described in previous section. We
then conduct the following experiments:

  * 1\.

Static-Backbone. We freeze the pre-trained feature extractor backbone, and
train only the classifier head. This is analogous to using a static foundation
model.

  * 2\.

Static-Embedding. We fine-tune the transformer part of the pre-trained feature
extractor along with the classifier head together with a single optimizer, and
freeze the embedding module

  * 3\.

Fine-Tuned Backbone. We fine-tune the entire feature extractor backbone along
with the classifier head.

Evaluation..Evaluation.Evaluation.. We average results across multiple runs of
each transformer architecture. To show results in limited space, we have
provided complete evaluation results for backbones using Covid-Twitter-BERT.
To test generalizability, we train each model on a single dataset, and
evaluate on the test-sets of the remaining, unseen datasets as well as its own
testing dataset. Our results are presented as a confusion matrix. All
approaches are trained for 5 epochs with an AdamW optimizer, with a learning
rate of 1e-4, with a batch size of 64.

###  3.1. Generalizability Results

We show generalizability results using the COVID-Twitter backbone in for
static-backbone training in Figure 2, static-embedding training in Figure 3,
and fine-tuned backbone training in Figure 4.

Figure 2. Confusion Matrix for Static Backbone Figure 3. Confusion Matrix for
Static-Embedding Backbone Figure 4. Confusion Matrix for FIne-Tuned Backbone

Static backbone vs Fine-Tuning..Static backbone vs Fine-Tuning.Static backbone
vs Fine-Tuning.. The confusion matrices show that upon fine-tuning, each model
increases accuracy on its corresponding test dataset. For example, accuracy on
‘k_short’ increases from 0.90 to 0.97 between the static backbone and the
fine-tuned backbone. However, this fine-tuning comes at the cost of
generalization in some cases: across several datasets, model accuracy on
unseen data suffers in the fine-tuned backbone experiments. For example, a
model trained on ‘cov_fn’ achieves f1 of 0.72 when tested on ‘c19_title’ in
the static-backbone experiment in Figure 2. On the fine-tuned experiment, the
same trained model achieves f1 of 0.57, approximately a 20% drop. SImilarly, a
model trained on ‘miscov’ and tested on ‘c19_text’ achieves f1 of 0.71 with
static backbone, versus f1 of 0.43 with fine-tuned backbone. This indicates
once a model is fine-tuned on a specific covid dataset, it loses some
generalization information compared to the static-backbone version. However,
this is not consistent. In some cases, generalization accuracy increases:
‘rumor’ performs better on ‘cov_fn’ after fine-tuning, with f1 of 0.52 on the
static backbone, versus an f1 of 0.75 after fine-tuning. Furthermore, ‘rumor’
achieves f1 of 0.67 when tested on ‘c19_title‘ on the static backbone, and f1
of 0.81 on the fine-tuned backbone (conversely, it performs worse on ‘coaid‘,
with f1 dropping from 0.83 to 0.40).

(a) Static-Backbone

(b) Static-Embedding

(c) Fine-tuned

(d) F1-Score Variance

Figure 5. Overlap vs Accuracy for each of our experiment approaches. In Figure
5(c), we have also provided the trendlines from the other experiments for
comparison.

Static Embedding..Static Embedding.Static Embedding.. A middle ground between
complete fine-tuning and a fully static foundation feature extractor is to
freeze the embedding layer and fine-tune the transformer layer of the
backbone. Recent work finds freezing the embedding layer during training can
reduce computation costs while achieving 90% of the accuracy of a fully fine-
tuned model (Lee et al., 2019; Hulburd, 2020; Moon et al., 2019; Jiao et al.,
2021; Merchant et al., 2020). We find similar results, where freezing the
embedding layer achieves accuracy similar to the corresponding fine-tuned
model on the testing dataset, shown in Figure 3 However, on unseen data,
accuracy drop has higher variance. It is not immediately clear what impact
embedding freezing has on unseen data accuracy. For this, we must explore the
actual overlap between datasets.

###  3.2. Data Overlap and Accuracy

We have seen that there is label overlap between datasets in Figure 1. We have
also seen accuracy variances on unseen data: rather than a linear drop across
unseen data, some models perform better and some perform worse after fine-
tuning. These can be explained by directly measuring dataset overlap.

O-Metric..O-Metric.O-Metric.. To compute overlap, we use the O-metric to
calculate point-proximity overlap from (Cardillo and L. Warren, 2016). The
O-metric computes overlap between 2 sets of points in n-dimensional space
using a distance-metric. We find overlap as follows: given two datasets A and
B, we compute the fraction of points in each dataset where the nearest
neighbor is not from the same dataset. So, for each point $x\in A$, first we
obtain:

$O_{A}(x)=w_{A}(x)/b_{A}(x)$

where $w_{A}()$ is the distance to the nearest point to $x$ in A, and
$b_{A}()$ is the distance to the nearest point to $x$ in B. Then we can
compute the ratio $p_{A}=|O_{A}>1|/|A|$ to find overlap of B in A. $p_{A}$ is
bounded in $[0,1]$; as $p_{A}$ approaches 1, this indicates most points in $A$
are closer to a point in $B$ than in $A$. The O-metric is bidirectional in
computing overlap and includes both $p_{A}$ and $p_{B}$:

$O=\frac{p_{A}+p_{B}}{2}$

Since we are interested in evaluating generalization, where we want to see
only the overlap of unseen data on training data, we use a directional
O-metric. That is, we let $O=p_{A}$ for the final overlap value in a context
where A is the training dataset and B is the unseen dataset. We compute the
overlap value between each dataset pair using cosine similarity on the
embeddings of each data point. So, for each model, we compute embeddings of
every sample across all 9 datasets, then compute the directional O-metric
overlap of each dataset on the model’s training dataset [ALGO???].

Accuracy and Overlap.Accuracy and OverlapAccuracy and Overlap. We show in
Figure 5 the comparison of overlap to accuracy across all 9 datasets on
static-backbone, static-embedding, and fine-tuned backbone experiments. First,
we see that as overlap increases, accuracy also increases for the fine-tuned
backbone in Figure 5(c). Second, and perhaps more striking, there is higher
variance in accuracy on the static-embedding versus fine-tuned backbone
models. This fits the observations from (Merchant et al., 2020) that the
embedding layers capture more domain-specific information than the deeper
layers. In this cases, because the embedding layers were frozen, they never
learned the domain-specific fake news information. Consequently, the static-
embedding models’ accuracy on unseen data sees significant variance, compared
to the fine-tuned backbone and static-backbone models, shown in Figure 5(d).

Generalization and Overlap..Generalization and Overlap.Generalization and
Overlap.. Clearly, higher overlap between evaluation data and training data is
indicative of accuracy. During testing, however, it may be difficult to
calculate this overlap on data for each sample. Further, evaluation data
changes continuously, so the overlap may itself change due to concept drift.
Recent research has shown the importance of exploring a model’s feature space
to identify embedding clusters (Chen et al., 2022; Urner and Ben-David, 2013).
These embedding clusters signify regions of the data space a model has
captured. Metrics such as probabilistic Lipschitzness show that accuracy on
embedding clusters can be bounded using the smoothness, or gradient, in the
embedding space (Urner and Ben-David, 2013). Further, LIGER (Chen et al.,
2022) shows that nondeterministic label regions, i.e. where labels overlap,
indicate non-smoothness. We extend these findings to present KMeans-Proxy - a
plug-and-play pytorch layer.

###  3.3. KMeans-Proxy

Intuitively, if we can store the coverage of a model’s embedding space, then
for any sample point, we can check if it falls inside the coverage. Further,
we can also check if a model’s prediction on the sample matches the prediction
for the coverage. We can pair this with a coverage radius, e.g. by computing
$r$ that constitutes coverage of all points in a cluster that are 1 standard
deviation away from the cluster center with respect to a distance metric, such
as the l2 norm. Then, if the predictions do not match or a point falls outside
the single standard deviation coverage radius, this is a strong
abstention/reject signal.

Implementation..Implementation.Implementation.. We can capture the coverage of
the embedding space by using embedding proxies. Proxies are common in cluster
and proxy NCA losses (Movshovitz-Attias et al., 2017; Wern Teh et al., 2020).
We adapt them as the KMeans cluster centroids by acting as proxies for the
cluster centers. This allows our approach to extend to online or continuous
learning domains as well. The findings in (Chen et al., 2022) suggest
increased partitioning of the embedding space can yield better local region
coverage. So, KMeans-Proxy is initialized with 2 parameters: the number of
classes $c$, and a proxy factor $k$. Then, we then obtain $k\cdot c$ centers,
with $k$ proxies for each class, so that each cluster is a smaller, more
representative local region.

⬇

1self.proxy = KMeansProxy(proxy_factor = 4, classes = 3, dimensions = 768)

During model training, KMeans-Proxy performs minibatch online KMeans
clustering to obtain the embedding space proxies for cluster centers. Online
clustering converges asymptotically, per (So et al., 2022).

⬇

1def forward(x): # Proxy forward function

2 if self.training:

3 self.update_proxies(x)

4 return x, None, None

5 return x, self.nearest_proxy(x), self.nearest_proxy_label(x)

During prediction, a model using KMeans-Proxy can predection, as well as
nearest proxy label and nearest proxy. A meta abstention policy can review for
label flipping, or coverage radius.

⬇

1def forward(x): # Model forward function

2 x = self.feature_extractor(x)

3 x, proxy, proxy_label = self.proxy(x)

4 label = self.classifier(x)

5 return label, proxy, proxy_label

Table 2. Generalization improvement with KMP for model trained on ‘coaid’.
Trained on ‘coaid’
---
Testing Dataset | Approach
SB | SE | FT | FT+KMP
cov_fn | 0.57 | 0.44 | 0.51 |  0.53
k_short | 0.49 |  0.57 | 0.56 | 0.57
coaid | 0.85 | 0.95 |  0.97 | 0.98
cq | 0.42 | 0.59 | 0.55 |  0.57
k_long | 0.47 | 0.51 | 0.58 |  0.55
rumor | 0.34 | 0.31 |  0.55 | 0.68
c19_text | 0.43 | 0.64 |  0.79 | 0.94
miscov |  0.54 | 0.45 | 0.47 | 0.57
c19_title | 0.49 | 0.75 |  0.77 | 0.90
Table 3. Generalization improvement with KMP for model trained on ‘rumor’
Trained on ‘rumor’
---
Testing Dataset | Approach
SB | SE | FT | FT+KMP
cov_fn |  0.76 | 0.48 | 0.75 | 0.77
k_short | 0.49 |  0.52 | 0.52 | 0.54
coaid | 0.17 | 0.17 |  0.40 | 0.76
cq | 0.47 | 0.59 | 0.42 |  0.58
k_long | 0.59 | 0.52 | 0.43 |  0.53
rumor | 0.70 | 0.66 |  0.83 | 0.86
c19_text | 0.17 | 0.57 |  0.58 | 0.73
miscov | 0.52 | 0.45 |  0.54 | 0.54
c19_title | 0.74 | 0.57 | 0.81 |  0.76

###  3.4. KMeans-Proxy Results

We now show results from using KMeans-Proxy as a reject option in Table 2 and
Table 3. With rejection, we can increase generalization accuracy on unseen
data. This is a faster approach than domain adaptation, since proxies are
updated during training. Further, it is a plug-and-play solution, allowing for
faster iteration on overall model design.

With KMeans-Proxy, we are able to improve generalization performance across
the board. Here, we compare models trained on ‘coaid’ and ‘rumor’ in Table 2
and Table 3, respectively. Models are compared across static backbone (SB),
static embedding (SE), fine-tuned (FT), and fine-tuned with KMeans-Proxy
(FT+KMP).

For models trained on ‘coaid’ (in Table 2) and tested on all datasets,
incorporating KMeans-Proxy improves generalization performance. In each case,
FT+KMP is either the best or the runner-up model by at most 0.05 f1 f1 points.
We see similar result for models trained on ‘rumor’ in Table 3, where KMeans-
Proxy is either the best model or runner up for every testing dataset.

Choice of Proxy Factor.Choice of Proxy FactorChoice of Proxy Factor.
Increasing the proxy factor leads to better generalization performance. Table
4 shows performance of a model trained on ‘c19_text‘ that has poor
generalization without KMeans-Proxy (see Figure 4). As we increase the proxy
factor, we gain better generalization across testing datasets. We test with
different proxy factors and compare generalization performance of each model.
We find that increasing the proxy factor leads to small, but measurable
increase in accuracy.

Table 4. Impact of changing proxy factor: increasing the proxy factor increases accuracy, since more proxies allow tighter bounds on local coverage estimates. Testing Dataset | Trained on c19_text
---|---
FT | k=1 | k=2 | k=3 | k=5 | k=10
cov_fn | 0.44 | 0.50 | 0.54 | 0.58 | 0.58 | 0.58
k_short | 0.59 | 0.70 | 0.73 | 0.73 | 0.73 | 0.75
coaid | 0.76 | 0.88 | 0.84 | 0.89 | 0.89 | 0.90
cq | 0.51 | 0.56 | 0.58 | 0.58 | 0.60 | 0.63
k_long | 0.60 | 0.70 | 0.72 | 0.73 | 0.73 | 0.73
rumor | 0.47 | 0.45 | 0.51 | 0.58 | 0.58 | 0.67
c19_text | 0.97 | 0.98 | 0.99 | 0.99 | 0.99 | 0.99
miscov | 0.48 | 0.45 | 0.44 | 0.53 | 0.58 | 0.58
c19_title | 0.58 | 0.61 | 0.64 | 0.69 | 0.68 | 0.73

###  3.5. Discussion

There are several observations we can make from our generalization studies and
KMeans-Proxy experiments. Generalization.GeneralizationGeneralization. For
fake news detection, fine-tuned models must be used carefully to take advntage
of learned parameters. As we showed in the confusion matrices, fine-tuning
improves performance only on subsets of unseen data. These subsets are regions
of the data space where the unseen data overlaps with training data. On
completely new regions of the data space, fine-tuned models make mistakes.
These mistakes are because of label overlap.

We must make a distinction between label and data overlap. Data overlap means
a model has coverage on the unseen data, and can make predictions with higher
confidence. Label coverage, as we showed in Fig, indicates where different
labels occur close to each other in the embedding space. Both can coincide:
unseen data points can have both data and label overlap. For these points,
fine-tuned models that have better captured a local region with training data
are better poised to provide high-confidence labels.

KMeans-Proxy.KMeans-ProxyKMeans-Proxy. KMeans-Proxy allows us to identify
these regions. With KMeans-Proxy, we partition the data space into clusters
representing model coverage and labels. Our inclusion of the proxy factor $k$,
where we create $k$ clusters for each class label, allows fine-grained
partitioning of the embedding space. This means we can better capture local
characteristics of the embedding space (Chen et al., 2022). In our
experiments, we focus on 2 such characteristics: (i) whether the label for an
unseen point matches the label for nearest proxy, and (ii) whether this unseen
point is within one standard-deviation radius of the proxy. In our
experiments, we show that using these provides improvements in generalizing to
unseen data points.

Clearly, there is significant progress to be made in capturing local
characteristics. For example, when using an ensemble of fine-tuned models,
local smoothness (Urner and Ben-David, 2013; Chen et al., 2022) can be
computed for each non-abstaining model to rank them on coverage. There may
also be advantages in using dynamic proxy allocation. If prior class balance
is known, then we could use a class-specific proxy factor.

##  4\. Conclusion

In this paper, we have presented generalizability experiments and KMeans-
Proxy. We perform generalization studies across 9 fake news datasets using
several transformer-based fake news detector models. Our generalizability
experiments show that fine-tuned models generalize well to unseen data when
there is overlap between unseen and training data. On unseen data that does
not overlap, fine-tuned models make mistakes due to poor coverage, label
flipping, and concept drift.

Using our observations and recent research into local embedding regions, we
develop and present KMeans-Proxy, a simple online KMeans clusterer paired with
a proxy factor. With KMeans-Proxy, we partition the embedding space into local
regions and use local characteristics to create a reject option for models. We
show that KMeans-Proxy improves generalization accuracy for fine-tuned models
across all 9 fake news datasets. We welcome future research in this area to
better explore the generalizability and fine-tuning tradeoff.

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