# Novel relations for twist-3 tensor-polarized fragmentation functions in
spin-1 hadrons

Qin-Tao Song<EMAIL_ADDRESS>School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou, Henan 450001, China

###### Abstract

There are three types of fragmentation functions (FFs) which are used to
describe the twist-3 cross sections of the hard semi-inclusive processes under
QCD collinear factorization, and they are called intrinsic, kinematical, and
dynamical FFs. In this work, we investigate the theoretical relations among
these FFs for a tensor-polarized spin-1 hadron. Three Lorentz-invariance
relations (LIRs) are derived by using the identities between the nonlocal
quark-quark and quark-gluon-quark operators, which guarantee the frame
independence of the twist-3 spin observables. The QCD equation of motion
(e.o.m.) relations are also presented for the tensor-polarized FFs. In
addition, we also show that the intrinsic and kinematical twist-3 FFs can be
decomposed into the contributions of twist-2 FFs and twist-3 three-parton FFs,
and the latter are also called dynamical FFs. If one neglects the dynamical
FFs, we can obtain relations which are analogous to the Wandzura-Wilczek (WW)
relation. Then, the intrinsic and kinematical twist-3 FFs are expressed in
terms of the leading-twist ones. Since the FFs of a spin-1 hadron can be
measured at various experimental facilities in the near future, these
theoretical relations will play an important role in the analysis of the
collinear tensor-polarized FFs.

##  I Introduction

Parton distribution functions (PDFs) are key physical quantities in hadron
spin physics, since they are used to solve the proton spin puzzle and to
understand the inner structure of hadrons. For a spin-1/2 hadron, the
theoretical relations of PDFs and fragmentation functions (FFs) have been well
studied. Starting with the Wandzura-Wilczek (WW) relation, it is known that if
one neglects the three-parton PDFs, the twist-3 PDF $g_{2}$ can be expressed
in terms of the leading-twist one $g_{1}$ which has been well measured
Wandzura:1977qf . The violation of the WW relation comes from the three-parton
PDFs, and it was shown that such violation can be as large as 15-40% of the
size of $g_{2}$ Accardi:2009au . There also exist the so-called Lorentz-
invariance relations (LIRs) for the PDFs in a spin-1/2 hadron, which were
investigated in Refs. Mulders:1995dh ; Belitsky:1997ay ; Tangerman:1994bb ;
Kundu:2001pk ; Goeke:2003az ; Metz:2008ib ; Accardi:2009au ; Kanazawa:2015ajw
. In addition to PDFs, LIRs were also derived for the quark FFs
Kanazawa:2015ajw . Recently, the authors of Ref. Koike:2019zxc  performed a
systematic study on the gluon PDFs and FFs, where the intrinsic and
kinematical twist-3 gluon distributions are written in terms of the twist-2
distributions and the twist-3 dynamical distributions, and the latter are
actually three-parton distributions; moreover, the complete LIRs are also
listed for the gluon part. On the one hand, those interesting relations can be
used as constraints for the analysis of twist-3 distributions. On the other
hand, they are also crucial to describe the spin observables, for example, the
LIRs can be used to guarantee the frame independence of the twist-3 cross
sections, such as the single-spin asymmetries (SSAs) in the semi-inclusive
deep-inelastic scattering (SIDIS) and the hadron production of hadronic
collisions ($pp\rightarrow\Lambda^{\uparrow}X$) Kanazawa:2015ajw ;
Koike:2017fxr ; Koike:2021awj .

For a spin-1 hadron, there are unpolarized, vector-polarized and tensor-
polarized distributions. The former two also exist for a spin-1/2 hadron,
while the tensor-polarized distributions are the new ones. Among the tensor-
polarized PDFs, $b_{1}(x)$ (or $f_{1LL}(x)$) Hoodbhoy:1988am ;
Frankfurt:1983qs  and the gluon transversity $\Delta_{T}g(x)$ Jaffe:1989xy ;
Nzar:1992ax  are the most interesting ones. The sum rule of $\int
dxb_{1}(x)=0$ was derived for an isoscalar object such as the deuteron, and
the breaking of this sum rule is related to the contribution of a tensor-
polarized component of the sea quarks and antiquarks Close:1990zw . In 2005,
the HERMES collaboration did the first measurement of $b_{1}(x)$ for deuteron
with slightly large uncertainties HERMES:2005pon , and it indicates that
$b_{1}(x)$ is much larger than the theoretical prediction Cosyn:2017fbo .
Since theoretical estimate of $b_{1}(x)$ was given by considering deuteron as
a weakly bound state of proton and neutron, the large $b_{1}(x)$ could
indicate exotic components of deuteron such as six-quark state and hidden
color state Miller:2013hla . As for the gluon transversity $\Delta_{T}g(x)$,
it is related to the helicity flipped amplitude, so it only exists in a hadron
with spin more than or equal to 1 due to the angular momentum conservation. In
this case, one can infer that there are non-nucleonic components in the
deuteron by the nonzero $\Delta_{T}g(x)$, which means that it is interesting
to investigate the gluon transversity by experiment, for example, it can be
extracted from the cross sections of deep-inelastic scattering (DIS)
Ma:2013yba and Drell-Yan process Kumano:2019igu ; Kumano:2020gfk  with a
tensor-polarized deuteron target. In the near future, $b_{1}(x)$ and
$\Delta_{T}g(x)$ will be measured at the Thomas Jefferson National Accelerator
Facility (JLab) jlab-b1 ; jlab-b2 , Fermilab (Fermi National Accelerator
Laboratory) Keller:2020wan ; Keller:2022abm ; Clement:2023eun  and Nuclotron-
based Ion Collider fAcility (NICA)Arbuzov:2020cqg . There are also interesting
theoretical relations for the tensor-polarized PDFs, in Ref. Kumano:2021fem
the twist-3 PDF $f_{LT}(x)$ was decomposed into the contributions of twist-2
PDF $b_{1}(x)$ ($f_{1LL}(x)$) and the three-parton PDFs, moreover, the WW-type
relation was obtained by dropping the latter. The QCD equation of motion
(e.o.m.) relations and LIR were derived in Ref. Kumano:2021xau  for tensor-
polarized PDFs. Recently, the gluon transversity generalized parton
distribution (GPD) was also investigated for a spin-1 hadron Cosyn:2018rdm ,
which becomes the gluon transversity $\Delta_{T}g(x)$ in the forward limit. In
addition to the collinear PDFs, one can find the tensor-polarized transverse-
momentum dependent (TMD) PDFs up to twist 4 for a spin-1 hadron in Refs.
Bacchetta:2000jk ; kumano:2021 ; Ninomiya:2017ggn ; Boer:2016xqr .

The spin-1 hadrons are produced in the hard semi-inclusive processes, such as
$\rho$, $\phi$, $K^{\ast}$ and deuteron. In order to describe those processes,
the tensor-polarized FFs are needed. The quark collinear FFs are defined in
Ref. Ji:1993vw  up to twist 4 for a spin-1 hadron, and the tensor-polarized
TMD FFs can be also found in Refs. Bacchetta:2000jk ; Chen:2016moq . In
future, the tensor-polarized FFs can be measured at BESIII and Belle II.
Actually, such measurement is now in progress, for example the FFs of $\phi$
in the process $e^{+}e^{-}\rightarrow\phi X$ by the BESIII collaboration bes-
phi . However, the theoretical relations of tensor-polarized FFs have not been
completely investigated. In this work, we intend to derive the LIRs, QCD
e.o.m. and WW-type relations for the tensor-polarized FFs in a spin-1 hadron,
which can provide constraints for the future experimental and theoretical
studies of these FFs.

This paper is organized as follows. In Sec. II, we define the intrinsic,
kinematical, and dynamical twist-3 FFs, and general properties of them are
discussed. We derive the theoretical relations among tensor-polarized FFs
using QCD e.o.m. for quarks in Sec. III. The operator identities are obtained
for the nonlocal quark-quark and quark-gluon-quark operators, then LIRs and
WW-type relations are also given based on the matrix elements of the operator
identities in Sec. IV. A brief summary of this work is presented in Sec. V.

##  II Tensor-polarized fragmentation functions

The tensor polarization is often indicated by the matrix $T$ for a spin-1
hadron, and the covariant form of $T^{\mu\nu}$ is expressed as
Bacchetta:2000jk ; Boer:2016xqr

$\displaystyle T^{\mu\nu}$
$\displaystyle=\frac{1}{2}\left[\frac{4}{3}S_{LL}\frac{(P_{h}^{-})^{2}}{M^{2}}n^{\mu}n^{\nu}-\frac{2}{3}S_{LL}(n^{\\{\mu}\bar{n}^{\nu\\}}-g_{T}^{\mu\nu})+\frac{1}{3}S_{LL}\frac{M^{2}}{(P_{h}^{-})^{2}}\bar{n}^{\mu}\bar{n}^{\nu}+\frac{P_{h}^{-}}{M}n^{\\{\mu}S_{LT}^{\nu\\}}-\frac{M}{2P_{h}^{-}}\bar{n}^{\\{\mu}S_{LT}^{\nu\\}}+S_{TT}^{\mu\nu}\right],$
(1)

where $P_{h}$ and $M$ are momentum and mass for the produced hadron,
respectively.
$a^{\left\\{\mu\right.}b^{\left.\nu\right\\}}=a^{\mu}b^{\nu}+a^{\nu}b^{\mu}$
denotes symmetrization of the indices. The lightcone vectors $n$ and $\bar{n}$
are given by

$\displaystyle n^{\mu}=\frac{1}{\sqrt{2}}(\,1,\,0,\,0,\,-1\,),\ \
\bar{n}^{\mu}=\frac{1}{\sqrt{2}}(\,1,\,0,\,0,\,1\,),$ (2)

and $P_{h}$ can be written as
$P_{h}=P_{h}^{-}n+\frac{M^{2}}{2P_{h}^{-}}\bar{n}$. For a Lorentz vector
$a^{\mu}$, the lightcone components $a^{\pm}$ and transverse component $a_{T}$
are defined by

$\displaystyle a^{+}=a\cdot
n,\,\,a^{-}=a\cdot\bar{n},\,\,a_{T}^{\mu}=g_{T}^{\mu\nu}a_{\nu}$ (3)

with

$\displaystyle
g_{T}^{\mu\nu}=g^{\mu\nu}-n^{\mu}\bar{n}^{\nu}-n^{\nu}\bar{n}^{\mu}.$ (4)

In Eq. (1), $S_{LL}$, $S_{LT}^{\mu}$ and $S_{TT}^{\mu\nu}$ are the parameters
which indicate different types of tensor polarization.

For a spin-1 hadron, the fragmentation correlator is defined as kumano:2021 ;
Chen:2016moq ; Bacchetta:2000jk ; Ji:1993vw

$\displaystyle\Delta_{ij}(z)=$
$\displaystyle\frac{1}{N_{c}}\int\\!\frac{d\xi^{+}}{2\pi}\,e^{i\frac{P_{h}^{-}\xi^{+}}{z}}\langle
0\left|\mathcal{W}\left[\infty^{+};\xi^{+}\right]q_{i}(\xi^{+})\right|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\bar{q}_{j}(0)\,\mathcal{W}\left[0^{+};\infty^{+}\right]\right|0\rangle,$
$\displaystyle=$
$\displaystyle\frac{1}{z}\Big{\\{}S_{LL}\not{n}F_{1LL}(z)+\frac{M}{P_{h}^{-}}\left[\not{S_{LT}}F_{LT}(z)+S_{LL}E_{LL}(z)\right]+\sigma^{i+}S_{LT,i}H_{1LT}$
$\displaystyle+\frac{M}{P_{h}^{-}}\left[S_{LL}\sigma^{-+}H_{LL}(z)+\gamma_{5}\gamma_{i}\epsilon_{T}^{ij}S_{LT,j}G_{LT}\right]\Big{\\}}$
(5)

where $z$ is the longitudinal momentum fraction carried by the produced
hadron, $N_{c}$ is the number of color, and $\mathcal{W}$ is a Wilson line
which ensures color gauge invariance. The transverse tensor
$\epsilon_{T}^{\alpha\beta}$ is given by

$\displaystyle\epsilon_{T}^{\alpha\beta}=\epsilon^{\alpha\beta\mu\nu}n_{\mu}\bar{n}_{\nu}$
(6)

with the convention $\epsilon^{0123}=1$. In Eq. (5), the correlator is
expressed in terms of 6 tensor-polarized FFs up to twist 3, and the FFs are
real functions with the support region of $0<z<1$. $F_{1LL}(z)$ and
$H_{1LT}(z)$ are leading-twist FFs, and the rest ones are also called
intrinsic twist-3 FFs Kanazawa:2015ajw . Since time-reversal invariance is not
a necessary constraint for the fragmentation correlator, the last three FFs
are actually time-reversal odd FFs. Note that there are also unpolarized and
vector-polarized FFs in the correlator, which are neglected here since we are
interested in the tensor-polarized ones.

The kinematical twist-3 FFs are related to the TMD FFs. In case of a tensor-
polarized hadron, the TMD fragmentation correlator reads Collins:1981uw ;
Collins:1992kk ; Boer:2003bw ; Bacchetta:2006tn ; Metz:2016swz

$\displaystyle\Delta_{ij}(z,k_{T})=$
$\displaystyle\frac{1}{N_{c}}\int\\!\frac{d\xi^{+}d^{2}\xi_{T}}{(2\pi)^{3}}\,e^{i(k^{-}\xi^{+}+k_{T}\cdot\xi_{T})}\langle
0\left|\mathcal{W}_{1}\left[\infty;\xi\right]q_{i}(\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\bar{q}_{j}(0)\,\mathcal{W}_{2}\left[0;\infty\right]\right|0\rangle_{\xi^{-}=0}$
(7)

with

$\displaystyle\mathcal{W}_{1}\left[\infty;\xi\right]$
$\displaystyle=\mathcal{W}\left[\infty^{+},\infty_{T};\infty^{+},\xi_{T}\right]\mathcal{W}\left[\infty^{+},\xi_{T};\xi^{+},\xi_{T}\right],$
$\displaystyle\mathcal{W}_{2}\left[0;\infty\right]$
$\displaystyle=\mathcal{W}\left[0^{+},0_{T};\infty^{+},0_{T}\right]\mathcal{W}\left[\infty^{+},0_{T};\infty^{+},\infty_{T}\right],$
(8)

and the correlator can be written in terms of TMD FFs Bacchetta:2000jk ;
Chen:2016moq ; Chen:2023kqw . The $k_{T}$-weighted FFs are defined with the
help of the TMD fragmentation correlator,

$\displaystyle\Delta_{\partial,ij}^{\nu}(z)=\int
d^{2}k_{T}k_{T}^{\nu}\Delta_{ij}(z,k_{T}),$ (9)

which is parameterized by four $k_{T}$-weighted FFs at twist 3
Bacchetta:2000jk ,

$\displaystyle\Delta_{\partial}^{\nu}(z)=\frac{M}{z}\left[-S_{LT}^{\nu}\not{n}F_{1LT}^{(1)}(z)-\epsilon_{\bot}^{\nu\rho}S_{LT\rho}\gamma_{5}\not{n}G_{1LT}^{(1)}(z)+S_{LL}\sigma^{\nu\alpha}n_{\alpha}H_{1LL}^{(1)}(z)-S_{TT}^{\nu\alpha}\sigma_{\alpha\beta}n^{\beta}H_{1TT}^{(1)}(z)\right],$
(10)

and these FFs are also called kinematical twist-3 FFs in Ref. Kanazawa:2015ajw
. Due to Eq. (9), the kinematical twist-3 FFs are given by TMD FFs,

$\displaystyle F^{(1)}(z)=-z^{2}\int
d^{2}k_{T}\frac{k_{T}^{2}}{2M^{2}}F(z,z^{2}k_{T}^{2}),$ (11)

where $F(z,z^{2}k_{T}^{2})$ is a TMD FF.

Similarly, we define the collinear there-parton fragmentation correlator
Kumano:2021fem ,

$\displaystyle\Delta^{\nu}_{F,ij}(z,z_{1})=$
$\displaystyle\frac{1}{N_{c}}\int\\!\frac{d\xi^{+}}{2\pi}\frac{d\xi_{1}^{+}}{2\pi}\,e^{iP_{h}^{-}\xi^{+}\frac{1}{z_{1}}+iP_{h}^{-}\xi_{1}^{+}(\frac{1}{z}-\frac{1}{z_{1}})}\langle
0\left|\mathcal{W}\left[\infty^{+};\xi_{1}^{+}\right]igF^{-\nu}(\xi_{1}^{+})\mathcal{W}\left[\xi_{1}^{+};\xi^{+}\right]q_{i}(\xi^{+})\right|P_{h},T;X\rangle$
$\displaystyle\times\langle\,P_{h},T;X\left|\,\bar{q}_{j}(0)\mathcal{W}\left[0^{+};\infty^{+}\right]\right|0\rangle.$
(12)

By inserting a complete set of intermediate states, one can prove that

$\displaystyle\Delta^{\nu}_{F}(z,z)=0,\quad\Delta^{\nu}_{F}(z,0)=0,$ (13)

and this corresponds to the vanishing partonic pole matrix elements which are
important to understand the SSAs in the hard semi-inclusive processes
Meissner:2008yf . Then, the support region of $\Delta^{\nu}_{F}(z,z_{1})$ is

$\displaystyle 0\leq z\leq 1,\quad 0<\frac{z}{z_{1}}<1.$ (14)

Taking the derivative of this correlator with respect to $1/z_{1}$ and then
setting $z_{1}=z$, one can also obtain Kanazawa:2015ajw

$\displaystyle\frac{\partial\Delta^{\nu}_{F}(z,z_{1})}{\partial(1/z_{1})}|_{z_{1}=z}=0.$
(15)

The parameterization of $\Delta^{\nu}_{F}(z,z_{1})$ is just a copy of the
corresponding three-parton distribution correlator Kumano:2021fem , and it can
be expressed in terms of four dynamical FFs at twist 3,

$\displaystyle\Delta^{\nu}_{F,ij}(z,z_{1})=\frac{M}{z}\left[-S_{LT}^{\nu}\not{n}\hat{F}_{LT}(z,z_{1})-i\epsilon_{T}^{\nu\rho}S_{LT\rho}\gamma_{5}\not{n}\hat{G}_{LT}(z,z_{1})-S_{LL}\gamma^{\nu}\not{n}\hat{H}_{LL}^{\bot}(z,z_{1})-S_{TT}^{\nu\rho}\gamma_{\rho}\not{n}\hat{H}_{TT}(z,z_{1})\right].$
(16)

Note that the dynamical FFs are complex functions which are different from the
intrinsic and kinematical ones.

##  III Equation of motion relations for FFs

The intrinsic, kinematical and dynamical FFs are not independent functions,
since they can be related to each other by the e.o.m. relations. For a
spin-1/2 hadron, the e.o.m. relations for FFs were derived in Refs.
Kanazawa:2015ajw ; Metz:2012ct  based on the QCD e.o.m. for quarks, namely,
$(i\not{D}-m_{q})q(x)=0$. In the following, we will investigate the e.o.m.
relations for tensor-polarized FFs. After some algebra, the QCD e.o.m. for
quarks becomes

$\displaystyle(iD^{\mu}+\sigma^{\mu\nu}D_{\nu}+m_{q}\gamma^{\mu})q(x)=0,$ (17)

where $m_{q}$ is the mass of the quark. If we set $\mu=-$ and take the
corresponding matrix element for Eq. (17), an e.o.m. relation can be obtained
for the intrinsic, kinematical and dynamical FFs,

$\displaystyle\frac{E_{LL}(z)}{z}+\frac{iH_{LL}(z)}{z}-\frac{m_{q}}{M}F_{1LL}(z)=2\left[-iH^{(1)}_{1LL}(z)+\mathcal{P}\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\hat{H}_{LL}^{\bot}(z,z_{1})}{\frac{1}{z}-\frac{1}{z_{1}}}\right],$
(18)

where $\mathcal{P}$ stands for the principal integral, and it can be neglected
due to Eq. (13). All the FFs appear in Eq. (18) are related to the
$S_{LL}$-type tensor polarization. Furthermore, this relation can be
reexpressed in terms of the real and complex parts of the dynamical FF,

$\displaystyle\frac{E_{LL}(z)}{z}=$ $\displaystyle
2\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Re}\left[\hat{H}_{LL}^{\bot}(z,z_{1})\right]}{\frac{1}{z}-\frac{1}{z_{1}}}+\frac{m_{q}}{M}F_{1LL}(z),$
(19) $\displaystyle\frac{H_{LL}(z)}{z}=$ $\displaystyle
2\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Im}\left[\hat{H}_{LL}^{\bot}(z,z_{1})\right]}{\frac{1}{z}-\frac{1}{z_{1}}}-2H_{1LL}^{(1)}(z).$
(20)

We can see that the time-reversal even and odd intrinsic FFs are related to
the real and imaginary parts of the dynamical FFs, respectively. If we neglect
the quark mass, the intrinsic twist-3 FFs $E_{LL}(z)$ and $H_{LL}(z)$ are
given by the kinematical and dynamical twist-3 FFs.

Multiplying $\gamma^{\nu}$ on the l.h.s. of Eq. (17), then antisymmetrizing
$\mu$ and $\nu$, we can obtain the identity as

$\displaystyle\left[i(\gamma^{\mu}D^{\nu}-\gamma^{\nu}D^{\mu})-\epsilon^{\mu\nu\rho\sigma}\gamma_{\sigma}\gamma_{5}D_{\rho}+im_{q}\sigma^{\mu\nu}\right]q(x)=0.$
(21)

Analogously, we set $\mu=-$ and consider $\nu$ as a transverse component, then
the matrix element of Eq. (21) leads to

$\displaystyle\frac{F_{LT}(z)}{z}+\frac{iG_{LT}(z)}{z}+\frac{im_{q}}{M}H_{1LT}(z)$
$\displaystyle=$ $\displaystyle-
iG_{1LT}^{(1)}(z)+\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\hat{G}_{LT}(z,z_{1})}{\frac{1}{z}-\frac{1}{z_{1}}}-\left[F_{1LT}^{(1)}(z)+\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\hat{F}_{LT}(z,z_{1})}{\frac{1}{z}-\frac{1}{z_{1}}}\right],$
(22)

and it indicates the relation among the intrinsic, kinematical and dynamical
FFs for the $S_{LT}$-type tensor polarization. Furthermore, Eq. (22) can be
divided into two identities,

$\displaystyle\frac{F_{LT}(z)}{z}=$
$\displaystyle-\int_{z}^{\infty}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Re}\left[\hat{F}_{LT}(z,z_{1})-\hat{G}_{LT}(z,z_{1})\right]}{\frac{1}{z}-\frac{1}{z_{1}}}-F^{(1)}_{1LT}(z),$
(23) $\displaystyle\frac{G_{LT}(z)}{z}=$
$\displaystyle-\int_{z}^{\infty}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Im}\left[\hat{F}_{LT}(z,z_{1})-\hat{G}_{LT}(z,z_{1})\right]}{\frac{1}{z}-\frac{1}{z_{1}}}-G^{(1)}_{1LT}(z)-\frac{m_{q}}{M}H_{1LT}(z).$
(24)

As indicated by Eq. (5), there are no intrinsic FFs for the $S_{TT}$-type
tensor polarization. However, we can also derive the following identity using
Eq. (21),

$\displaystyle
iH^{(1)}_{1TT}(z)+\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\hat{H}_{TT}(z,z_{1})}{\frac{1}{z}-\frac{1}{z_{1}}}=0,$
(25)

and it implies

$\displaystyle\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Re}\left[\hat{H}_{TT}(z,z_{1})\right]}{\frac{1}{z}-\frac{1}{z_{1}}}=$
$\displaystyle 0,$ (26) $\displaystyle
H^{(1)}_{1TT}(z)+\int^{\infty}_{z}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Im}\left[\hat{H}_{TT}(z,z_{1})\right]}{\frac{1}{z}-\frac{1}{z_{1}}}=$
$\displaystyle 0,$ (27)

which complete the derivation of the QCD e.o.m. relations for tensor-polarized
FFs.

##  IV Lorentz invariance and Wandzura-Wilczek-type relations

Taking derivative of nonlocal quark-quark operators, one can obtain the
identities where the quark-quark operators are expressed in terms of the
quark-gluon-quark ones. The theoretical relations have been investigated for
PDFs, FFs and distribution amplitudes (DAs) by using these identities of
nonlocal operators, and this method was well explained in Refs.
Kanazawa:2015ajw ; Balitsky:1987bk ; Balitsky:1989ry ; Balitsky:1990ck ;
Ball:2001uk ; Ball:1998ff ; Ball:1996tb ; Braun:1989iv ; Kodaira:1998jn ;
Eguchi:2006qz . In this section, we adopt the same method to derive the
theoretical relations for twist-3 tensor-polarized FFs such as LIRs and WW-
type relations. We firstly consider the derivative of the nonlocal quark-quark
operator Kanazawa:2015ajw ,

$\displaystyle\frac{\partial}{\partial\xi_{\alpha}}\langle
0|\mathcal{W}\left[\infty\xi;-\xi\right]q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\Gamma\mathcal{W}\left[\xi;\infty\xi\right]|0\rangle$
$\displaystyle=$ $\displaystyle-\langle
0|\mathcal{W}\left[\infty\xi;-\xi\right]\overset{\rightarrow}{D}\\!\\!\\!\\!\\!\phantom{D}^{\alpha}(-\xi)q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\Gamma\mathcal{W}\left[\xi;\infty\xi\right]|0\rangle$
$\displaystyle+\langle
0\left|\mathcal{W}\left[\infty\xi;-\xi\right]q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\overset{\leftarrow}{D}\\!\\!\\!\\!\\!\phantom{D}^{\alpha}(\xi)\Gamma\mathcal{W}\left[\xi;\infty\xi\right]|0\rangle$
$\displaystyle+i\int^{\infty}_{-1}dtt\langle
0\left|\mathcal{W}\left[\infty\xi;t\xi\right]gF^{\alpha\xi}(t\xi)\mathcal{W}\left[t\xi;-\xi\right]q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\Gamma\mathcal{W}\left[\xi;\infty\xi\right]|0\rangle$
$\displaystyle+i\int_{\infty}^{1}dtt\langle
0|\mathcal{W}\left[\infty\xi;-\xi\right]q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\overline{q}(\xi)\Gamma\mathcal{W}\left[\xi;t\xi\right]gF^{\alpha\xi}(t\xi)\mathcal{W}\left[t\xi;\infty\xi\right]\right|0\rangle$
(28)

where $\xi$ is not necessarily a lightcone vector and $\Gamma$ is a gamma
matrix. In Eq. (28), the terms with the covariant derivative $D^{\alpha}$ can
be replaced by the total derivative of the nonlocal quark-quark operator,
which is related to the translation of the operator, and its matrix element
can be expressed as Kanazawa:2015ajw

$\displaystyle\bar{\partial}^{\rho}\langle
0\left|\mathcal{W}\left[\infty\xi;-\xi\right]q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\overline{q}(\xi)\Gamma_{1}\mathcal{W}\left[\xi;\infty\xi\right]\right|0\rangle$
$\displaystyle=$ $\displaystyle\lim_{x_{\rho}\rightarrow
0}\frac{d}{dx_{\rho}}\langle
0\left|\mathcal{W}\left[\infty\xi+x;-\xi+x\right]q(-\xi+x)\right|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\overline{q}(\xi+x)\Gamma_{1}\mathcal{W}\left[\xi+x;\infty\xi+x\right]\right|0\rangle$
$\displaystyle=$ $\displaystyle\langle
0|\mathcal{W}\left[\infty\xi;-\xi\right]\overset{\rightarrow}{D}\\!\\!\\!\\!\\!\phantom{D}^{\rho}(-\xi)q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\bar{q}(\xi)\Gamma_{1}\mathcal{W}\left[\xi;\infty\xi\right]|0\rangle$
$\displaystyle+\langle
0\left|\mathcal{W}\left[\infty\xi;-\xi\right]q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\bar{q}(\xi)\overset{\leftarrow}{D}\\!\\!\\!\\!\\!\phantom{D}^{\rho}(\xi)\Gamma_{1}\mathcal{W}\left[\xi;\infty\xi\right]|0\rangle$
$\displaystyle+\int^{\infty}_{-1}dt\langle
0\left|\mathcal{W}\left[\infty\xi;t\xi\right]igF^{\rho\xi}(t\xi)\mathcal{W}\left[t\xi;-\xi\right]q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\bar{q}(\xi)\Gamma_{1}\mathcal{W}\left[\xi;\infty\xi\right]\right|0\rangle$
$\displaystyle+\int_{\infty}^{1}dt\langle
0\left|\mathcal{W}\left[\infty\xi;-\xi\right]q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\bar{q}(\xi)\Gamma_{1}\mathcal{W}\left[\xi;t\xi\right]igF^{\rho\xi}(t\xi)\mathcal{W}\left[t\xi;\infty\xi\right]\right|0\rangle$
(29)

where $\Gamma_{1}$ stands for a gamma matrix such as $\gamma^{\mu}$ and
$\sigma^{\mu\nu}$. Due to the translation invariance, the matrix element in
Eq. (29) should vanish.

In the following, the Wilson lines are neglected in the operator identities,
since this will not cause confusion. We derive a relation between quark-quark
and quark-gluon-quark operators by choosing
$\Gamma=(g^{\rho\alpha}g^{\lambda}_{\ \,\sigma}-g^{\alpha}_{\
\,\sigma}g^{\rho\lambda})\gamma_{\lambda}$ in Eq. (28) and
$\Gamma_{1}=(\sigma^{\sigma\beta}\gamma^{\rho}+\gamma^{\rho}\sigma^{\sigma\beta})$
in Eq. (29),

$\displaystyle\xi_{\alpha}\left[\frac{\partial}{\partial\xi_{\alpha}}\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\gamma^{\sigma}|0\rangle-\frac{\partial}{\partial\xi_{\sigma}}\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\gamma^{\alpha}|0\rangle\right]$
$\displaystyle=$ $\displaystyle\left[\int^{\infty}_{-1}dt\langle
0\left|gF_{\rho\xi}(t\xi)q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\gamma_{\tau}\gamma_{5}|0\rangle+\int_{\infty}^{1}dt\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\overline{q}(\xi)\gamma_{\tau}\gamma_{5}gF_{\rho\xi}(t\xi)\right|0\rangle\right]$
$\displaystyle\times\epsilon^{\sigma\xi\rho\tau}-i\int^{\infty}_{-1}dtt\langle
0\left|gF^{\sigma\xi}(t\xi)q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\not{\xi}|0\rangle-i\int_{\infty}^{1}dtt\langle
0|q(-\xi)|P_{h},T;X\rangle$
$\displaystyle\times\langle\,P_{h},T;X\left|\,\overline{q}(\xi)\not{\xi}gF^{\sigma\xi}(t\xi)\right|0\rangle,$
(30)

where the total derivative operator is neglected, and the quark mass terms
vanish. The quark-quark operator appears in the l.h.s. of Eq. (30), which can
be written in terms of the intrinsic tensor-polarized FFs as shown in Eq. (5).
If the vector $\xi$ is not necessarily on the lightcone, the matrix element of
the nonlocal quark-quark operator can be expressed as

$\displaystyle\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\gamma^{\sigma}|0\rangle$
$\displaystyle=$ $\displaystyle 8N_{c}M^{2}\int
d\left(\frac{1}{z}\right)e^{\frac{2iP\xi}{z}}\left[\frac{3}{4}A^{\sigma}\frac{F_{1LL}(z)}{z}+B^{\sigma}\frac{F_{LT}(z)}{z}\right]$
(31)

with

$\displaystyle A^{\sigma}=$ $\displaystyle\frac{\xi\cdot
T\cdot\xi}{(P_{h}\cdot\xi)^{2}}P_{h}^{\sigma}-M^{2}\frac{\xi\cdot
T\cdot\xi}{(P_{h}\cdot\xi)^{3}}\xi^{\sigma},$ (32) $\displaystyle B^{\sigma}=$
$\displaystyle\frac{T^{\sigma\mu}\xi_{\mu}}{P_{h}\cdot\xi}-\frac{\xi\cdot
T\cdot\xi}{(P_{h}\cdot\xi)^{2}}P_{h}^{\sigma}+M^{2}\frac{\xi\cdot
T\cdot\xi}{(P_{h}\cdot\xi)^{3}}\xi^{\sigma}.$ (33)

Eq. (31) is exact at twist 3 since the twist-4 FFs are not included. We
substitute Eq. (31) into Eq. (30) to estimate the derivative, and take the
lightcone limt of $\xi^{2}\rightarrow 0$, then, the l.h.s. of Eq. (30) is
given by $F_{1LL}(z)$ and $F_{LT}(z)$. The r.h.s. of Eq. (30) can be directly
calculated with the help of Eq. (16), and we obtain the following relation by
combining the l.h.s. and r.h.s.,

$\displaystyle\frac{3}{2}\tilde{F}_{1LL}(z)+\frac{1}{z}\frac{d\tilde{F}_{LT}(z)}{d(1/z)}$
$\displaystyle=$ $\displaystyle\int
d\left(\frac{1}{z_{1}}\right)\mathcal{P}\left(\frac{1}{\frac{1}{z}-\frac{1}{z_{1}}}\right)\left\\{\Big{(}\frac{\partial}{\partial(1/z)}+\frac{\partial}{\partial(1/z_{1})}\Big{)}\mathrm{Re}\left[\tilde{G}_{LT}(z,z_{1})\right]-\Big{(}\frac{\partial}{\partial(1/z)}-\frac{\partial}{\partial(1/z_{1})}\Big{)}\mathrm{Re}\left[\tilde{F}_{LT}(z,z_{1})\right]\right\\},$
(34)

where the convention of $\tilde{F}(z)=F(z)/z$ is used for a intrinsic or
kinematical FF $F(z)$, and $\tilde{F}(z,z_{1})=\hat{F}(z,z_{1})/z$ for a
dynamical one. Combining Eq. (34) with the e.o.m relation of Eq. (23), one can
obtain

$\displaystyle\frac{3}{2}\tilde{F}_{1LL}(z)-\tilde{F}_{LT}(z)-(1-z\frac{d}{dz})F_{1LT}^{(1)}(z)=-2\int_{z}^{\infty}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Re}\left[\tilde{F}_{LT}(z,z_{1})\right]}{(\frac{1}{z}-\frac{1}{z_{1}})^{2}},$
(35)

and this is a new LIR for tensor-polarized FFs. If we integrate Eq. (34) over
the momentum fraction $z$, one can have

$\displaystyle F_{LT}(z)=$
$\displaystyle-\frac{3z}{2}\int_{z}^{1}dz_{1}\frac{F_{1LL}\left(z_{1}\right)}{(z_{1})^{2}}+z\int_{z}^{1}\frac{dz_{1}}{z_{1}}\int_{z_{1}}^{\infty}\frac{dz_{2}}{(z_{2})^{2}}\Bigg{\\{}\frac{\left[1+\frac{1}{z_{1}}\delta(\frac{1}{z_{1}}-\frac{1}{z})\right]\mathrm{Re}\left[\hat{G}_{LT}(z_{1},z_{2})\right]}{\frac{1}{z_{1}}-\frac{1}{z_{2}}}$
$\displaystyle-\frac{\left[\frac{3}{z_{1}}-\frac{1}{z_{2}}+\frac{1}{z_{1}}(\frac{1}{z_{1}}-\frac{1}{z_{2}})\delta(\frac{1}{z_{1}}-\frac{1}{z})\right]\mathrm{Re}\left[\hat{F}_{LT}\left(z_{1},z_{2}\right)\right]}{(\frac{1}{z_{1}}-\frac{1}{z_{2}})^{2}}\Bigg{\\}},$
(36)

where it should be understood that $z_{1}$ falls within the range of
integration $(z,1)$, namely,
$\int_{z}^{1}dz_{1}F(z_{1})\delta(1/z_{1}-1/z)=z^{2}F(z)$. The intrinsic
twist-3 FF $F_{LT}(z)$ is decomposed into the contributions of a twist-2 FF
$F_{1LL}$ and the dynamical FFs. We can obtain a similar expression for the
kinematical twist-3 FF $F_{1LT}^{(1)}(z)$ by inserting Eq. (36) into the
e.o.m. relation of Eq. (23),

$\displaystyle
F_{1LT}^{(1)}(z)=\frac{3}{2}\int_{z}^{1}dz_{1}\frac{F_{1LL}\left(z_{1}\right)}{(z_{1})^{2}}+\int_{z}^{1}\frac{dz_{1}}{z_{1}}\int_{z_{1}}^{\infty}\frac{dz_{2}}{(z_{2})^{2}}\Bigg{\\{}\frac{\left(\frac{3}{z_{1}}-\frac{1}{z_{2}}\right)\mathrm{Re}\left[\hat{F}_{LT}\left(z_{1},z_{2}\right)\right]}{\left(\frac{1}{z_{1}}-\frac{1}{z_{2}}\right)^{2}}-\frac{\mathrm{Re}\left[\hat{G}_{LT}\left(z_{1},z_{2}\right)\right]}{\frac{1}{z_{1}}-\frac{1}{z_{2}}}\Bigg{\\}}.$
(37)

By dropping the contributions of dynamical FFs in Eqs. (36) and (37), they
become the WW-type relations. Then, the twist-3 intrinsic and kinematical FFs
can be estimated by using the twist-2 FF $F_{1LL}(z)$, and the latter should
be much easier to be extracted from experimental measurements compared with
the former.

If we choose $\Gamma=\sigma^{\mu\alpha}$ in Eq. (28) and $\Gamma_{1}=1$ in Eq.
(29), the following identity can be derived Kanazawa:2015ajw ,

$\displaystyle\frac{\partial}{\partial\xi_{\alpha}}\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\sigma^{\xi\alpha}|0\rangle$
$\displaystyle=$ $\displaystyle\int^{\infty}_{-1}dtt\langle
0\left|igF_{\alpha\xi}(t\xi)q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\sigma^{\xi\alpha}|0\rangle$
$\displaystyle+\int_{\infty}^{1}dtt\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\overline{q}(\xi)\sigma^{\xi\alpha}igF_{\alpha\xi}(t\xi)\right|0\rangle.$
(38)

Similarly, the matrix element of the nonlocal operator
$\overline{q}(\xi)\sigma^{\xi\sigma}q(-\xi)$ is expressed in terms of the FFs
$\tilde{H}_{1LT}(z)$ and $\tilde{H}_{LT}(z)$ at twist 3,

$\displaystyle\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\sigma^{\xi\sigma}|0\rangle$
$\displaystyle=$ $\displaystyle-4N_{c}M\int
d\left(\frac{1}{z}\right)e^{\frac{2iP\xi}{z}}\left[2(W^{\sigma}+V^{\sigma})\tilde{H}_{1LT}(z)+\frac{3}{2}V^{\sigma}\tilde{H}_{LL}(z)\right],$
(39)

where $W^{\sigma}$ and $V^{\sigma}$ are defined as

$\displaystyle W^{\sigma}=$ $\displaystyle
T^{\sigma\mu}\xi_{\mu}-\frac{\xi\cdot
T\cdot\xi}{P_{h}\cdot\xi}P_{h}^{\sigma},$ (40) $\displaystyle V^{\sigma}=$
$\displaystyle M^{2}\frac{\xi\cdot
T\cdot\xi}{(P_{h}\cdot\xi)^{2}}\left[\xi^{\sigma}-\frac{\xi^{2}}{P_{h}\cdot\xi}P_{h}^{\sigma}\right],$
(41)

and they satisfy the relations of $W\cdot\xi=0$ and $V\cdot\xi=0$. In the
lightcone limit $\xi^{2}\rightarrow 0$, Eq. (39) goes back to Eq. (5). We
obtain the following identity by calculating the matrix element in Eq. (38),

$\displaystyle
4\tilde{H}_{1LT}(z)-\frac{1}{z^{2}}\frac{dH_{LL}(z)}{d(1/z)}=-2\int
d\left(\frac{1}{z_{1}}\right)\mathcal{P}\left(\frac{1}{\frac{1}{z}-\frac{1}{z_{1}}}\right)\Big{(}\frac{\partial}{\partial(1/z)}-\frac{\partial}{\partial(1/z_{1})}\Big{)}\mathrm{Im}\left[\tilde{H}_{LL}^{\bot}(z,z_{1})\right].$
(42)

Moreover, we obtain $d(\tilde{H}_{LL}(z)/z)/d(1/z)$ by using the expression in
Eq. (20), and the sum of $d(\tilde{H}_{LL}(z)/z)/d(1/z)$ and Eq. (42) leads to

$\displaystyle\tilde{H}_{LL}(z)+2\tilde{H}_{1LT}(z)+(1-z\frac{d}{dz})H_{1LL}^{(1)}(z)=-2\int_{z}^{\infty}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Im}\left[\tilde{H}_{LL}^{\bot}(z,z_{1})\right]}{(\frac{1}{z}-\frac{1}{z_{1}})^{2}},$
(43)

which is also a LIR for tensor-polarized FFs. The integration of Eq. (42)
gives

$\displaystyle H_{LL}(z)=$ $\displaystyle
4\int_{z}^{1}dz_{1}\frac{H_{1LT}\left(z_{1}\right)}{z_{1}}+4\int_{z}^{1}dz_{1}\int_{z_{1}}^{\infty}\frac{dz_{2}}{(z_{2})^{2}}\frac{\frac{2}{z_{1}}-\frac{1}{z_{2}}+\frac{1}{2z_{1}}(\frac{1}{z_{1}}-\frac{1}{z_{2}})\delta(\frac{1}{z_{1}}-\frac{1}{z})}{(\frac{1}{z_{1}}-\frac{1}{z_{2}})^{2}}\mathrm{Im}\left[\hat{H}_{LL}^{\bot}\left(z_{1},z_{2}\right)\right],$
(44)

and the intrinsic twist-3 FF $H_{LL}(z)$ is expressed in terms of the twist-2
FF $H_{1LT}(z)$ and the dynamical FF
$\hat{H}_{LL}^{\bot}\left(z_{1},z_{2}\right)$. If we combine Eq. (44) with the
e.o.m. relation of Eq. (18),

$\displaystyle H_{1LL}^{(1)}(z)=$
$\displaystyle-\frac{2}{z}\int_{z}^{1}dz_{1}\frac{H_{1LT}\left(z_{1}\right)}{z_{1}}-\frac{2}{z}\int_{z}^{1}dz_{1}\int_{z_{1}}^{\infty}\frac{dz_{2}}{(z_{2})^{2}}\frac{\frac{2}{z_{1}}-\frac{1}{z_{2}}}{(\frac{1}{z_{1}}-\frac{1}{z_{2}})^{2}}\mathrm{Im}\left[\hat{H}_{LL}^{\bot}\left(z_{1},z_{2}\right)\right],$
(45)

which also decompose the kinematical twist-3 FF $H_{1LL}^{(1)}(z)$ into the
contributions of $H_{1LT}(z)$ and
$\hat{H}_{LL}^{\bot}\left(z_{1},z_{2}\right)$. We can obtain the WW-type
relations for $H_{LL}(z)$ and $H_{1LL}^{(1)}(z)$ by dropping the terms of the
dynamical FF in Eqs. (44) and (45).

If we consider the matrix elements of Eqs. (28) and (29) with
$\Gamma=\epsilon^{\alpha\mu\rho S_{LT}}\gamma_{\mu}\gamma_{5}$ and
$\Gamma_{1}=\frac{i}{2}(\gamma^{\rho}\sigma^{S_{LT}\xi}-\sigma^{S_{LT}\xi}\gamma^{\rho})$,
respectively, one can derive

$\displaystyle\epsilon^{\alpha\mu\rho
S_{LT}}\xi_{\rho}\frac{\partial}{\partial\xi^{\alpha}}\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\gamma_{\mu}\gamma_{5}|0\rangle$
$\displaystyle=$ $\displaystyle\int^{\infty}_{-1}dt\langle 0\left|gF_{\xi
S_{LT}}(t\xi)q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\not{\xi}|0\rangle+\int_{\infty}^{1}dt\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X\left|\,\overline{q}(\xi)\not{\xi}gF_{\xi
S_{LT}}(t\xi)\right|0\rangle$ $\displaystyle+i\epsilon^{\alpha\mu\xi
S_{LT}}\Big{[}\int^{\infty}_{-1}dtt\langle
0\left|gF_{\alpha\xi}(t\xi)q(-\xi)\right|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\gamma_{\mu}\gamma_{5}|0\rangle+\int_{\infty}^{1}dtt\langle
0|q(-\xi)|P_{h},T;X\rangle$
$\displaystyle\times\langle\,P_{h},T;X\left|\,\overline{q}(\xi)\gamma_{\mu}\gamma_{5}gF_{\alpha\xi}(t\xi)\right|0\rangle\Big{]}-2im_{q}\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\sigma^{\xi
S_{LT}}|0\rangle,$ (46)

and the l.h.s. is related to the matric element of
$\overline{q}(\xi)\gamma^{\mu}\gamma_{5}q(-\xi)$, which is given by

$\displaystyle\langle
0|q(-\xi)|P_{h},T;X\rangle\langle\,P_{h},T;X|\,\overline{q}(\xi)\gamma^{\mu}\gamma_{5}|0\rangle=4N_{c}M\frac{\epsilon^{\mu\xi\alpha
P_{h}}}{P_{h}\cdot\xi}Y_{\alpha}\int
d\left(\frac{1}{z}\right)e^{\frac{2iP\xi}{z}}\tilde{G}_{LT}(z),$ (47)

and the vector $Y$ is defined as

$\displaystyle
Y^{\alpha}=\frac{2M}{P_{h}\cdot\xi}\left[T^{\alpha\mu}\xi_{\mu}-\frac{\xi\cdot
T\cdot\xi}{P_{h}\cdot\xi}P_{h}^{\alpha}+M^{2}\frac{\xi\cdot
T\cdot\xi}{(P_{h}\cdot\xi)^{2}}(\xi^{\alpha}-\frac{\xi^{2}}{P_{h}\cdot\xi}P_{h}^{\alpha})\right].$
(48)

If we take the lightcone limit $\xi^{2}\rightarrow 0$, one can obtain
$Y^{\alpha}\rightarrow S_{LT}^{\alpha}$. Thus, Eq. (46) leads to the following
identity,

$\displaystyle\frac{1}{z}\frac{d\tilde{G}_{LT}(z)}{d(1/z)}+\frac{m_{q}}{M}\frac{d\tilde{H}_{1LT}(z)}{d(1/z)}$
$\displaystyle=$ $\displaystyle\int
d\left(\frac{1}{z_{1}}\right)\mathcal{P}\left(\frac{1}{\frac{1}{z}-\frac{1}{z_{1}}}\right)\left\\{\Big{(}\frac{\partial}{\partial(1/z)}-\frac{\partial}{\partial(1/z_{1})}\Big{)}\mathrm{Im}\left[\tilde{G}_{LT}(z,z_{1})\right]-\left(\frac{\partial}{\partial(1/z)}+\frac{\partial}{\partial(1/z_{1})}\right)\mathrm{Im}\left[\tilde{F}_{LT}(z,z_{1})\right]\right\\}.$
(49)

Combining Eq. (49) with the e.o.m. relation of Eq. (24), another LIR can be
derived for tensor-polarized FFs,

$\displaystyle\tilde{G}_{LT}(z)+(1-z\frac{d}{dz})G_{1LT}^{(1)}(z)=-2\int_{z}^{\infty}\frac{dz_{1}}{(z_{1})^{2}}\frac{\mathrm{Im}\left[\tilde{G}_{LT}(z,z_{1})\right]}{(\frac{1}{z}-\frac{1}{z_{1}})^{2}}.$
(50)

and the quark mass term in Eq. (49) is cancelled in this LIR. From Eqs. (49)
and (24), one can also express the twist-3 FFs $G_{LT}(z)$ and
$G_{1LT}^{(1)}(z)$ in terms of $H_{1LT}(z)$,
$\hat{F}_{LT}\left(z_{1},z_{2}\right)$ and
$\hat{G}_{LT}\left(z_{1},z_{2}\right)$,

$\displaystyle G_{LT}(z)=$
$\displaystyle-\frac{m_{q}}{M}\left[zH_{1LT}(z)+z\int_{z}^{1}dz_{1}\frac{H_{1LT}(z_{1})}{z_{1}}\right]-z\int_{z}^{1}\frac{dz_{1}}{z_{1}}\int_{z_{1}}^{\infty}\frac{dz_{2}}{(z_{2})^{2}}\Bigg{\\{}\frac{\left[1+\frac{1}{z_{1}}\delta(\frac{1}{z_{1}}-\frac{1}{z})\right]\mathrm{Im}\left[\hat{F}_{LT}\left(z_{1},z_{2}\right)\right]}{\frac{1}{z_{1}}-\frac{1}{z_{2}}}$
$\displaystyle-\frac{\left[\frac{3}{z_{1}}-\frac{1}{z_{2}}+\frac{1}{z_{1}}(\frac{1}{z_{1}}-\frac{1}{z_{2}})\delta(\frac{1}{z_{1}}-\frac{1}{z})\right]\mathrm{Im}\left[\hat{G}_{LT}\left(z_{1},z_{2}\right)\right]}{(\frac{1}{z_{1}}-\frac{1}{z_{2}})^{2}}\Bigg{\\}},$
(51)

$\displaystyle G_{1LT}^{(1)}(z)=$
$\displaystyle\frac{m_{q}}{M}\int_{z}^{1}dz_{1}\frac{H_{1LT}(z_{1})}{z_{1}}+\int_{z}^{1}\frac{dz_{1}}{z_{1}}\int_{z_{1}}^{\infty}\frac{dz_{2}}{(z_{2})^{2}}\Bigg{\\{}\frac{\mathrm{Im}\left[\hat{F}_{LT}(z_{1},z_{2})\right]}{\frac{1}{z_{1}}-\frac{1}{z_{2}}}-\frac{(\frac{3}{z_{1}}-\frac{1}{z_{2}})\mathrm{Im}\left[\hat{G}_{LT}\left(z_{1},z_{2}\right)\right]}{(\frac{1}{z_{1}}-\frac{1}{z_{2}})^{2}}\Bigg{\\}}.$
(52)

##  V Summary

The tensor-polarized FFs of a spin-1 hadron ($h$) can be measured in the
various hard semi-inclusive processes such as $e^{+}e^{-}\rightarrow hX$ and
$ep\rightarrow ehX$ (SIDIS), and the former process is accessible at BESIII
and Belle II, while the latter is possible at JLab and the Electron-Ion
Colliders in the US and China. Inspired by the ongoing measurement of the
tensor-polarized FFs for $\phi$ at BESIII, we investigate the theoretical
relations among the tensor-polarized intrinsic, kinematical and dynamical FFs
for a spin-1 hadron in this work. Firstly, the QCD e.o.m. relations are
obtained for the tensor-polarized FFs. Secondly, we derive the operator
identities where the nonlocal quark-quark operators are expressed in terms of
quark-gluon-quark operators. Three new Lorentz invariance relations (LIRs) are
presented for the tensor-polarized FFs, and they can be used to remove the
frame dependence of the twist-3 spin observables in the hard semi-inclusive
reactions so that Lorentz invariance properties are satisfied. Finally, we
also show that the intrinsic and kinematical twist-3 FFs are expressed in
terms of the twist-2 FFs and the dynamical twist-3 FFs, and the Wandzura-
Wilczek-type relations are obtained by neglecting the dynamical FFs. Since the
twist-2 FFs are much more easier to be accessed in experiment than the twist-3
ones, one can give a rough estimate for the twist-3 FFs by such relations. Our
results will be valuable for the future experimental measurements and
theoretical studies of tensor-polarized FFs.

## Acknowledgments

We acknowledge useful discussions with Shunzo Kumano, Bernard Pire, Ji Xu 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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#  Graph Agent: Explicit Reasoning Agent for Graphs

Qinyong Wang  Center for Artificial Intelligence in Drug Discovery, Case
Western Reserve University  Department of Computer and Data Sciences, Case
Western Reserve University  Zhenxiang Gao  Center for Artificial Intelligence
in Drug Discovery, Case Western Reserve University  Rong Xu  Center for
Artificial Intelligence in Drug Discovery, Case Western Reserve University

###### Abstract

Graph embedding methods such as Graph Neural Networks (GNNs) and Graph
Transformers have contributed to the development of graph reasoning algorithms
for various tasks on knowledge graphs. However, the lack of interpretability
and explainability of graph embedding methods has limited their applicability
in scenarios requiring explicit reasoning. In this paper, we introduce the
Graph Agent (GA), an intelligent agent methodology of leveraging large
language models (LLMs), inductive-deductive reasoning modules, and long-term
memory for knowledge graph reasoning tasks. GA integrates aspects of symbolic
reasoning and existing graph embedding methods to provide an innovative
approach for complex graph reasoning tasks. By converting graph structures
into textual data, GA enables LLMs to process, reason, and provide predictions
alongside human-interpretable explanations. The effectiveness of the GA was
evaluated on node classification and link prediction tasks. Results showed
that GA reached state-of-the-art performance, demonstrating accuracy of
90.65%, 95.48%, and 89.32% on Cora, PubMed, and PrimeKG datasets,
respectively. Compared to existing GNN and transformer models, GA offered
advantages of explicit reasoning ability, free-of-training, easy adaption to
various graph reasoning tasks.

Figure 1: Overview of Graph Agent methodology

##  1 Introduction

Knowledge graph Hogan et al. (2021) has emerged as a pivotal structure for
organizing and representing vast amounts of human knowledge in explicit form
Pan et al. (2023). Graph embedding methods, or predominantly Graph Neural
Networks (GNNs) Zhou et al. (2020) in recent years, have been used in
capturing intricate graph structures and node features based on their local
neighborhoods, making them adept at various graph reasoning tasks.

GNNs have demonstrated proficiency across various tasks, yet their
explainability of predictions remains a significant challenge Dai et al.
(2022). Such techniques, we termed "implicit reasoning methods," depend on
entangled representations Wang et al. (2019a). GNNs utilize message passing to
integrate information from adjacent nodes. However, during this process, both
node features and the GNN kernels are numeric vectors, which are hard to
interpret and understand by humans. This disadvantage poses a significant
concern, especially in scientific discovery using graph dataZeng et al.
(2022); Zhao et al. (2021), where mere prediction outcomes are insufficient; a
comprehensive understanding of the underlying rationale is imperative.
Moreover, given that information in KGs possesses an explicit form Pan et al.
(2023), a rationale exists for pursuing explicit graph reasoning.

On the other hand, there are symbolic reasoning frameworks available. These
might involve rule-based systems Jamian et al. (2019); Yao et al. (2019) or
using first-order logic Belle (2020); Jane and Ganesh (2019) to reason with
intricate data. Such reasoning processes are more transparent and human-
interpretable. Hence, we term them "explicit reasoning methods". Yet, they
come with their own set of problems. Rules within these systems are static,
making adaptability to varied tasks challenging Belle (2020). The current
rule-based and symbolic reasoning methods could only work on a limited number
of data sets. The intricate nature of heterogeneous graph data often proves
too complex to be encapsulated solely by a simple logical system. These
explicit logical reasoning methods often under-perform when benchmarked on
various data sets Belle (2020).

Large language models(LLM) demonstrate commendable reasoning capabilities
Bubeck et al. (2023), with their reasoning processes in natural language or
symbolic language Gao et al. (2023). Recent investigationsGuo et al. (2023)
have illuminated that LLMs can understand graph structures and analyze the
encompassed information. Such ability paved the way for formulating graph
reasoning algorithms with high efficacy and applicability. Researchers have
employed LLMs as controllers or agents across a spectrum of tasks Park et al.
(2023), such as software development Hong et al. (2023) and robotic
planningFan et al. (2022); Shah et al. (2023). However, the potential of LLM
agent systems on complex graph reasoning is yet to be discovered.

We introduce the Graph Agent methodology that leverages the reasoning
capabilities of LLMs and long-term memoryZhong et al. (2023) for KG tasks.
Given a graph data set, nodes or edges are embedded and stored in long-term
memory. For each input link prediction or node classification sample, similar
nodes or edges from the training set are fetched from this long-term memory.
The LLM then undergoes a two-phase inductive-deductive reasoning process.
During induction, the LLM is provided with similar nodes or edges, their
neighbors, and associated labels from the prompt and concludes the rationale
behind their labeling. In the deduction phase, the LLM incorporates these
concluded reasons to reason and predict the presented sample. The underlying
principle is similar to human cognitive processes: recalling similar past
instances to inform decisions on unfamiliar problems. Such analogical
reasoning, which widely exists in human cognition Bartha (2013), has
previously been validated as effective in knowledge graph contexts Yuan et al.
(2023). GA employs explicit reasoning, producing human-interpretable natural
language outputs. Contrasting GA with GNNs, while GNNs retain learned patterns
within graph convolution kernels, GA preserves it in textual format after
inductive reasoning. Instead of the entangled representation utilized for
message passing in GNNs, GA conveys neighbor information through prompts in
natural language. Essentially, GA shares some mechanics with GNNs but in an
explicit manner. The overview of GA is shown in Figure 1.

We evaluate GA’s performance on node classification and link prediction
datasets, notably Cora, PubMed, and PrimeKG data-sets, which are challenging
to GNN-based approaches. Our findings indicate that GA excels in these tasks
and offers enhanced prediction explainability compared to prior methods.

##  2 Related Work

###  2.1 Graph neural networks and graph transformers

A standard GNN Zhou et al. (2020) is structured with layers that aggregate
information from neighboring nodes. Message Passing NetworksHamilton et al.
(2017) are a foundational framework for many GNNs. The core idea is to
iteratively update node representations by "passing messages" between nodes.
At each iteration, each node aggregates information (messages) from its
neighbors. The node then updates its representation based on its previous
state and the aggregated messages. The final node representations can be used
for various tasks, such as node classification, graph classification, or link
prediction. Graph Convolutional Networks (GCN) Wu et al. (2019) are one of the
most popular and effective GNN architectures. Graph Attention Network (GAT)
Velickovic et al. (2017)introduces attention mechanisms Vaswani et al. (2017)
to the world of GNNs. Standard GNNs, designed for homogeneous graphs, may not
be optimal for Heterogeneous Graph. Heterogeneous GNNs are designed to handle
multiple node and edge types. They often involve multiple relation-specific
aggregation functions. An example is the Relational Graph Convolutional
Network (R-GCN) Schlichtkrull et al. (2018), which uses different weight
matrices for different relation types. There are also other GNNs such as
Heterogeneous Graph Transformer Hu et al. (2020), Hypergraph Convolution and
Hypergraph Attention Bai et al. (2021). Transformer models were adapted to
graph-structured data by covert both nodes and edges as tokens Min et al.
(2022). As previously discussed, these methods exhibit a deficiency in the
explainability of predictions.

###  2.2 Generative language model agents

Recently, there has been a growing interest in enhancing LLMs with additional
tools, memory, and sophisticated reasoning frameworks. Techniques such as the
"Chain of Thoughts," Wei et al. (2022a) "Self-Consistency," Wang et al.
(2022)and "Tree of Thoughts" Yao et al. (2023) have been introduced to boost
the reasoning capabilities of LLMs. A common practice among LLMs is the
utilization of vector databases to maintain their long-term memory Wang et al.
(2023), which is crucial for retaining comprehensive graph entities.
Furthermore, LLMs are now being trained to use tools, with notable
developments like Toolformer Schick et al. (2023), Visual Programming Gupta
and Kembhavi (2023), and GeneGPT Jin et al. (2023) leading the way. These
advancements have significantly amplified the competencies of LLMs, leading to
a surge in research exploring LLMs as intelligent agentsLiu et al. (2023a). In
experimental setups, these agents are immersed in virtual environments,
allowing researchers to observe and analyze their behaviors Park et al.
(2023). Remarkably, LLMs have even been employed to simulate a software
development Hong et al. (2023). It would be interesting to see how LLM agents
behave in complex graph reasoning tasks.

###  2.3 Graph reasoning with LLMs

In recent advancements in graph-based reasoning, the synergistic combination
of LLMs with graphs has showcased enhanced performance compared to the
exclusive GNNs. Two primary methodologies have been identified in the realm of
LLM-integrated graph techniques Chen et al. (2023a):

LLM as an Augmentor: LLMs are pivotal in augmenting graph data. They are adept
at generating text contextually related to the graph nodes. This results in
the enrichment of each node with additional textual features, thereby
amplifying the depth and quality of information associated with each node.
TAPE leveraged the capabilities of ChatGPT to enhance text-attributed graphs,
demonstrating state-of-the-art performance in node classification tasks He et
al. (2023). LLMs exhibit potential as annotators, enhancing performance in
graph tasks where labels are absent Chen et al. (2023b).

LLM as a Predictor: Deploying LLMs as predictors has also been a significant
stride forward. By feeding the LLM with information about a node and its
neighboring nodes, it is feasible to anticipate the class of the given node or
infer the likelihood of a link existing between two nodes Ye et al. (2023).
GraphText-ICL used a graph-to-text encoder and leveraged the In-context
Learning(ICL) ability of LLM for node classification Zhao et al. (2023).
Additionally, integrating LLMs with graph structures for fine-tuning has shown
to have refined outcomes in tasks such as Substructure Counting and shortest
path identification Chai et al. (2023).

Previous studies have investigated fine-tuning LLMs for graph reasoning tasks
Ye et al. (2023); Zhao et al. (2023), and have explored the utilization of
advanced reasoning techniques to enhance the graph reasoning capabilities of
LLMs. However, fine-tuning LLMs for specific tasks can be computationally
intensive and time-consuming, posing challenges when adapting them to diverse
datasets. Furthermore, approaches centered on prompt engineering have yielded
suboptimal results compared to training-based methods Zhao et al. (2023).

Figure 2: Workflow of the proposed Graph Agent methodology in link prediction
task with an information-centered view. Graph Agent first memorized the train
set of an input graph, then inferenced on the test set without model training.
Cosine similarity was used to similar edges for inductive reasoning, then
predictions and explanation was generated after deductive reasoning. Text
boxes with gray backgrounds indicate input text for LLM; blue background
indicate output text from LLM.

###  2.4 Graph to text encoder and node sampling

To feed structured graph data to LLMs, we would need covert sub-graphs to text
sequences. The efficacy of LLMs in graph reasoning tasks was influenced by the
encoding method Zhao et al. (2023). Our study leveraged an encoding strategy
optimized for computational efficiency.

For a given node $v$ with attributes $A$ and n-hop neighbors $N_{h}$, the
encoding process integrated the node’s attributes $\\{a_{n}\in A\\}$ and
sampled information from its n-hop neighbors $\\{n\in N\\}$ using a sampling
function denoted as $f_{\text{sample}}()$. The formulation of the node’s
encoder function is as follows:

$\text{encoder}(v,A,N)=[\text{{"node:"}},v,\text{{"attributes:"}}A,\text{{"n-hop-
neighbours: "}}[f_{\text{sample}}(N_{h})]]$

For graph edges, represented by vertices $x$ and $y$, the encoding encompassed
the attributes of the two edge nodes ($A_{x},A_{y}$) and information
pertaining to the n-hop neighbors for each vertex ($N_{x},N_{y}$). The edge
encoder function is as follows:

$\displaystyle\text{encoder}((x,y),(A_{x},A_{y}),N_{x},N_{y})=[$
$\displaystyle\text{{``edge: ''}},(x,y);$ $\displaystyle\text{{``attributes:
''}},(A_{x},A_{y});$ $\displaystyle y\text{{-``n-hop-neighbours:
''}}[f_{\text{sample}}(N_{y_{h}})],$ $\displaystyle x\text{{-``n-hop-
neighbours: ''}}[f_{\text{sample}}(N_{x_{h}})]]$

This encoder intentionally omitted the interconnections among the neighboring
nodes, which might affect the performance due to information loss. Including
these connections would substantially increase the text length. It would
prolong inference time and risk the LLM being overwhelmed by over-complicated
graph structures.

Our experiments underscored the pivotal role of information sampling in
optimizing GA, especially within graphs with dense connectivities. For
instance, nodes in biomedical knowledge graphs or individuals in social
networks often have connections exceeding hundreds of edges, posing a risk of
overloading information beyond the working memory Bubeck et al. (2023) of
LLMs, or exceeding the maximum context length of LLMs.

To mitigate this, we leveraged a sampling technique based on node degrees. For
heterogeneous graphs, we computed the average degree $D_{\text{avg}}$ for each
category of nodes. Subsequently, a node’s importance was quantified as the
ratio of its degree to $D_{\text{avg}}$ pertinent to its category. This
relative importance metric guided the selective encoding of neighbor
information, specifically incorporating only the top $k$ most significant
nodes, where $k$ was tailored to the task, dataset, and LLM in use. The
sampling function is formalized as:

$f_{\text{sample}}(N)=\text{select\\_top\\_k}\left(\left\\{\frac{\text{degree}(n)}{D_{\text{avg,
type}(n)}}:n\in N\right\\},k\right)$

where $\text{degree}(n)$ denotes the degree of node $n$, The function
select_top_k is an operation selecting the $k$ nodes with the highest node
importance.

While more advanced sampling and encoding techniques could be used, our
research prioritized the initial development of Graph Agent. Consequently, we
adopted a straightforward method to facilitate implementation.

###  2.5 Long-term memory

Given a graph task, the first step of GA was memorization of the graph.
Compared with other methods with a training phase, the train set was not used
for back-propagation training; instead, Graph Agent embedded all training
samples and stored them in a vector database. During inference on test
samples, we retrieved similar samples from the long-term memory.

We leveraged two methods to embed samples. The first was language model
embedding, and the second was GNN embedding. In language model embedding, for
each sample, we used the graph to text encoder previously discussed and passed
the output text of the encoder to an embedding-optimized language model. In
GNN embedding, we trained a GNN on the training data set and stored the node
embedding in a vector database. For edge embedding, we simply contacted the
embedding of two nodes. We used the Cosine similarity of embedding to retrieve
similar node or edge examples.

###  2.6 Inductive reasoning

During the inference phase, GA initiated a retrieval of analogous examples
from its long-term memory for a target sample; each example was denoted as
example_n. The aggregation of these examples formed a structured prompt
augmented with task instructions. The prompt was formulated as follows: "Given
the provided examples and your existing knowledge, identify reasons why
example nodes are categorized as labeled or why a connection exists in example
edges. List the reasons concisely."

Selecting these analogous examples was critical, as the LMM sought patterns or
commonalities within the examples. This mechanism, termed _explicit learning_
, where the learned patterns were in the output text.

###  2.7 Deductive reasoning

In the deductive reasoning phase, all selected examples, outcomes from
inductive reasoning, and the designated sample were integrated into a
comprehensive prompt, and the prompt was followed with instructions and a
question. For node classification, the LLM was queried with a crafted
question: "Given the reasons and examples, determine the type of node_a from
the following options: [options…], think step by step then choose one of the
options"

Similarly, for link prediction, the LLM was instructed: "Considering the
reasons and examples, does a connection exist between node_a and node_b? think
step by step, and choose either TRUE or FALSE."

The LLM would respond with explicit reasoning in natural language and a
definitive prediction. This output represented transparent, human-
interpretable logic, demonstrating how GA leveraged patterns discerned through
inductive reasoning to inform their predictions. The Deductive reasoning phase
could be considered a form of the chain of thought method Wei et al. (2022a).
The difference was the examples were dynamically generated and tailored to the
designated sample.

##  3 Experiments

###  3.1 Node Classification

Our experimentation with GA encompassed node classification tasks juxtaposing
its efficacy against previous methodologies. A direct comparison between GA
and GNNs is inequitable, primarily because text-attributed graphHe et al.
(2023) node classification could be transformed into document classification
Minaee et al. (2021), an LLM innately excels. Consequently, this section aims
at establishing a robust benchmark against preceding methods.

Datasets Conscious of the high inference costs and long inference time of
LLMs, our study utilized comparatively smaller datasets—specifically, the
widely recognized Cora and PubMed graph datasets Yang et al. (2016). These
selections facilitated comparisons with state-of-the-art techniques but also
adhered to manageable sizes, with Cora comprising 2,706 nodes and PubMed
encompassing 19,711. Consistent with previous works, we implemented a
60%/20%/20% partitioning for train/validation/test sets. Our experiments
engaged versions of Cora and PubMed retaining their original textual data.

Baselines Our performance assessment of GA involved juxtapositions with
preceding GNN models, transformer models, and those LLM-related methods. The
GNN models include GCNWu et al. (2019), GATVeličković et al. (2017), RevGATLi
et al. (2021), etc. We also compared with Transformer-based graph learns,
including CoarFormer Kuang et al. (2021), GraphormerYing et al. (2021), and GT
Dwivedi and Bresson (2020).

More importantly, we compared with graph algorithms that leveraged LLMs in
recently published papersYe et al. (2023), including IntructGLM, TAPEHe et al.
(2023), and GraphText-ICLZhao et al. (2023).

Implementation Details For our experiments, we employed the gpt-4-0613 model
as the LLM backend, and used embedding-ada-002 model for graph text embedding.
Our methodology included sampling the top 8 neighboring nodes. The prompts for
the target nodes comprised the title, abstract, authors, and keywords, while
for neighboring nodes, we confined the information to the title, authors, and
node type label. We masked the labels of the target node within the prompts
since similar node examples were often the neighbors of the target node and
revealed labels of the target node. After memorizing the training dataset, GA
directly inferred the test dataset without training.

Results GA outperformed GNN and transformer models, achieving state-of-the-art
results on the Cora and PubMed datasets. It demonstrated superior accuracy,
attaining 95.48% on PubMed— the highest in this category—and 90.65% on Cora,
ranking second only to InstructGLM. Notably, both TAPE and InstructGLM require
a training phase, highlighting GA’s efficiency as it yields competitive
results without the necessity of model training. Furthermore, when compared
against the free-of-training method GraphText-ICL, which also utilized GPT-4,
GA observed an increase of approximately 20 points in accuracy.

|  | Cora (Acc) | PubMed (Acc)
---|---|---|---
GAT | Veličković et al. (2017) | 76.70 | 83.28
GraphSAGE | Hamilton et al. (2017) | 86.58 | 86.85
GCN | Wu et al. (2019) | 87.78 | 88.90
RevGAT | Li et al. (2021) | 89.11 | 88.50
ACM-Snowball-3 | Luan et al. (2022) | 89.59 | 90.96
ACM-GCN+ | Luan et al. (2022) | 89.75 | 91.44
Graphormer | Ying et al. (2021) | 80.41 | 88.75
GT | Dwivedi and Bresson (2020) | 86.42 | 88.24
CoarFormer | Kuang et al. (2021) | 88.69 | 89.75
InstructGLM | Ye et al. (2023) | 90.77 | 94.62
TAPE | He et al. (2023) | 89.30 | 95.30
GraphText-ICL | Zhao et al. (2023) | 68.3 | -
Graph Agent |  | 90.65 | 95.48
Table 1: Results on Cora and PubMed node classification

###  3.2 Link Prediction

In pursuit of real-world applicability, we explored drug-gene link prediction
within a heterogeneous biomedical knowledge graph. The interrelations of
genes, drugs, diseases, and biological processes were complex, which made it
an ideal task for us to test the comprehensive graph reasoning ability of GA.
Secondly, as stated previously, the inference cost of GA was high. It would
make more sense to test GA for high-value graph reasoning tasks.

Dataset We adopted the Precision Medicine Oriented Knowledge Graph(PrimeKG)
dataset Chandak et al. (2023), recognizing it as one of the latest and most
complex biomedical graphs available. Our primary focus centered on the
prediction of drug-gene edges. As delineated earlier, LLMs were bound by input
length. Consequently, we confined our attention to certain node and edge
types; we used a subset of PrimeKG node types, including drugs, genes,
biological processes, pathways, and diseases. Only edge types that
interconnected these node types were taken into consideration. The filtered
version of PrimeKG utilized for our analysis comprised 2,085 drug nodes,
19,001 gene nodes, 7,161 biological process nodes, 1,625 pathway nodes, and
2,658 disease nodes. This configuration encapsulated a complex network with a
total of 954,438 edges of various types. Within PrimeKG, 20,417 drug-gene
edges were identified, indicating existing associations. An equivalent number
of non-associated drug-gene pairs were randomly generated and reintegrated
into the dataset. These newly created links were labeled as negative,
contrasting with the original positive associations. The data set was
partitioned into 80%, 10%, and 10% segments for training, validation, and
testing.

Baselines We benchmarked GA’s performance against established GNNs and prompt-
engineering methods. Comparisons were drawn with GCNWu et al. (2019) and
Heterogeneous graph attention network(HGAT)Wang et al. (2019b), both in
isolation and in conjunction with text augmentation. We also compared GA with
simple asking and chain-of-thoughWei et al. (2022a) prompt methods.

Evaluation Metrics A notable limitation of LLMs in link prediction is their
binary response format, wherein they can only output labels such as "True" or
"False". Consequently, metrics like mean reciprocal rank (MRR) and the hit
rate for the top k candidates (hits@K) are incompatible with our specific
scenario Pan et al. (2023). Given this constraint, our evaluation adopted
precision, recall, the F1 score, and the accuracy of the positive edges.

Implementation Details For comparative analysis, we trained 2-layer GCN and
HGAT models on the training dataset. We also used the graph embedding from
GNNs for retrieving similar edges in addition to language model embedding.
Capitalizing on PrimeKG’s text-associated nodes, we also integrated text
embedding—generated from text-embedding-ada-002 model —as initial node
embedding in GNN training, which was a text-augmented training. We used both
Gpt-4 and LLaMa2-70B-chat-hf as our LLM backend. Our sampling strategy
prioritized the top 15 neighbor nodes for GPT-4, and the top 5 neighbor nodes
for LLaMa2-70B, since LLaMa had much fewer parameters and could only process a
smaller local graph. The prompt included only node names and types, excluding
other node attributes. Similar to node classification, edge examples could
contain the targeted edge label. We only leveraged edge examples that had
different nodes with targeted edges. In our approach, we extracted three
analogous positive edge instances from long-term memory and arbitrarily
selected two negative edges for inclusion in the prompt. We also explored
alternative methodologies for comparison: the Simple Ask approach and the
5-shot Chain-of-Thought (COT) Wei et al. (2022a) technique. The Simple Ask
method involved presenting the LLM with text after graph-to-text encoding and
straightforwardly inquiring about the existence of a connection. The 5-shot
COT, similar to the previous GraphText-ICL strategy, employed a consistent set
of three positive and two negative edge examples.

Results Table 2 illustrates that GA outperformed competing methods in link
prediction, achieving an F1 score of 0.889 and an accuracy of 0.893. Compared
to the 5-shot COT’s accuracy of 0.803, GA enhanced GPT-4’s graph reasoning
capabilities in the nearly 10-point increase in both F1 and accuracy metrics.
Notably, the ’Simple Ask’ method yielded a mere 0.196 recall, suggesting
GPT-4’s initial inclination to deny the existence of most drug-gene
associations. Minor improvements were witnessed with text augmentation for
GNNs, which aligned with findings from prior studiesChen et al. (2023a).
Furthermore, the efficacy of COT methods was confirmed, with a 20-point surge
in accuracy over the ’Simple Ask’ approach.

Table 2: Results on PrimeKG drug-gene link prediction |  PrimeKG | Precision | Recall | F1 | Accuracy
---|---|---|---|---|---
GNN methods |  HGAT | 0.831 | 0.826 | 0.829 | 0.837
|  HGAT with text augmentation | 0.836 | 0.842 | 0.839 | 0.846
|  GCN | 0.826 | 0.838 | 0.832 | 0.832
|  GCN with text augmentation | 0.830 | 0.842 | 0.836 | 0.837
LLM predictors |  Simple ask + Graph2text encoder | 0.964 | 0.196 | 0.325 | 0.599
|  5-shot COT + Graph2text encoder | 0.962 | 0.673 | 0.792 | 0.803
|  Graph Agent | 0.926 | 0.854 | 0.889 | 0.893

Ablation Results Our ablation study, summarized in Table 3, evaluated the
influence of n-hop information and the specific embeddings and LLM utilized.
Contrary to expectations, incorporating more n-hop information into GA
diminished performance: 1-hop, 2-hop, and 3-hop configurations yielded
accuracies of 0.8902, 0.8202, and 0.774, respectively. We found that the LLM
was significantly affected by the shared neighbor pattern, with more neighbors
for each drug and gene node. LLMs found more shared neighbors, leading the LLM
to think there were associations. However, this is not true for many
biomedical knowledge graphs. For example, a common gene could be associated
with many genes and biological processes, and within 2-hop of this common
gene, many nodes do not have associations. Furthermore, leveraging GNN
embeddings for similar edge example retrieval proved effective, with an
accuracy of 0.869, surpassing the 0.846 achieved by HGAT. It showed we could
use GA to enhance existing GNN methods. However, GNN embedding under-performed
compared with language model embedding, which could be due to our GNN model
was not refined for information retrieval. Testing with LLama2-70B had a
precision of only 0.5464. Hallucination of LLama2 was frequently observed, and
it stated most drugs and genes had associations. Our explanation was that
LLama2, with only 70B parameters, had not yet developed the emergent graph
reasoning abilityWei et al. (2022b) required for a complex biomedical graph.

| Precision | Recall | F1 | Accuracy
---|---|---|---|---
3-hop with LM embedding | 0.698 | 0.955 | 0.807 | 0.774
2-hop with LM embedding | 0.730 | 0.932 | 0.821 | 0.820
1-hop with LM embedding (proposed) | 0.926 | 0.854 | 0.889 | 0.893
1-hop with GNN embedding | 0.958 | 0.767 | 0.852 | 0.869
1-hop with LM embedding and LLama2 70B | 0.546 | 0.956 | 0.696 | 0.565
Table 3: Ablation test results

Qualitative analyses In the link prediction experiments, Graph Agent,
empowered by GPT-4, demonstrated commendable reasoning capabilities. A manual
review of reasoning instances revealed GPT-4 had hallucinations with uncommon
genes and diseases. We found GPT-4 was confused with genes within the same
family, which often shared similar names and biological processes. Since those
genes share similar functions, they usually have links with the same drugs, so
even if the thinking processing was wrong, the prediction was correct. This
process correctness was hard to evaluate Bubeck et al. (2023), so we could not
provide a quantitative score for the reasoning quality. Despite the factual
error, GA was good at identifying common reasons for drug-gene associations,
such as shared neighbors, key biological processes, and pathways. This
behavior could give researchers insights during drug discovery.

##  4 Discussion and limitations

Deep learning has frequently been criticized as a "black box". Now we have
seen this very "black box" offer compelling reasoning and elucidations Bubeck
et al. (2023). This poses a contemplative question: can we trust explanations
derived from black box models? Addressing this quandary on philosophical
grounds remains elusive. Our analysis of many graph reasoning outcomes
produced by the LLM, indeed affirms a commendable quality of reasoning. Future
studies could investigate the reliability of LLM reasoning.

The computational intensity of LLMs translates to formidable latencies and
costs, rendering the current GA impractical for large-scale graph reasoning. A
potential solution could lie in a hybrid system, wherein the LLM is harnessed
exclusively for hard samples or high-value cases. Explorations could also go
towards determining if smaller fine-tuned LLM can emulate the efficacy of its
larger version of LLM. This, however, surfaces additional complications,
including curating the dataset and methodology for fine-tuning and
subsequently ascertaining the generalizability of the fine-tuned graph LLMs.

Our current Graph Agent methodology encounters limitations regarding
information coverage and flow. Unlike multi-layered GNNs, which facilitate
good node coverage and employ neural networks to regulate inter-node
information flow, GA is constrained by its reliance on sampled local graphs
and naive sampling methods for information flow control. Future research
directions could be the exploration of advanced information retrieval and
control techniques to enhance GA’s efficacy.

An additional limitation pertains to the issue of redundant reasoning. GA
often uncovers rationales that are applicable across multiple samples,
rendering the repetitive inductive reasoning for each instance. Subsequent
research might explore the development of reasoning modules with higher
efficiency.

Our approach did not address the hallucination problem of LLMs Rawte et al.
(2023). While mitigating this phenomenon typically involves empowering agents
with factual information retrieval tools, incorporating such tools within GA
was avoided to prevent data leakage. However, for practical implementations of
GA in open-world applications, the integration of tools is an indispensable
consideration, underscoring an avenue for subsequent studies.

While considerable scholarship has been dedicated to generative agents in
robotics, software development, and human-like conversation Wu et al. (2023),
a void exists in discussions centered on agents with graph reasoning
capabilities. Knowledge Graphs, representing structures of human knowledge,
remain important. Efficacious reasoning atop KGs has significance. Our current
Graph Agent only exhibits capacities for shallow reasoning on knowledge
graphs, thereby accessing the surface layers of graph information. There
exists an exigency for concerted efforts to formulate graph reasoning datasets
and train graph foundation model Liu et al. (2023b) for graph agents. The
development of evaluative metrics that truly reflect the logical and factual
correctness of graph reasoning remains a challenge.

##  5 Conclusion

Existing implicit graph reasoning methods lack explainability. To address this
issue, we proposed the Graph Agent methodology with long-term memory and an
inductive-deductive reasoning module. Graph Agent has demonstrated high
efficacy and transparency of predictions within Knowledge graphs. Beyond the
promising results, the uniqueness of our approach lies in addressing the
longstanding explainability challenge in graph reasoning. As a first-of-its-
kind framework, the current Graph Agent implementation is naive and primitive.
Future studies could investigate advanced Graph Agents that can truly voyage
and learn in human knowledge networks.

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