aainstitutetext: KEK Theory Center, Institute of Particle and Nuclear Studies,
High Energy Accelerator Research Organization (KEK),
Oho 1-1, Tsukuba, Ibaraki, 305-0801, Japanbbinstitutetext: J-PARC Branch, KEK
Theory Center, Institute of Particle and Nuclear Studies, KEK,
and Theory Group, Particle and Nuclear Physics Division, J-PARC Center,
Shirakata 203-1, Tokai, Ibaraki, 319-1106, Japanccinstitutetext: School of
Physics and Microelectronics, Zhengzhou University,
Zhengzhou, Henan 450001, China

# Twist-2 relation and sum rule for tensor-polarized parton distribution
functions of spin-1 hadrons

S. Kumano c and Qin-Tao Song<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Sum rules for structure functions and their twist-2 relations have important
roles in constraining their magnitudes and $x$ dependencies and in studying
higher-twist effects. The Wandzura-Wilczek (WW) relation and the Burkhardt-
Cottingham (BC) sum rule are such examples for the polarized structure
functions $g_{1}$ and $g_{2}$. Recently, new twist-3 and twist-4 parton
distribution functions were proposed for spin-1 hadrons, so that it became
possible to investigate spin-1 structure functions including higher-twist
ones. We show in this work that an analogous twist-2 relation and a sum rule
exist for the tensor-polarized parton distribution functions $f_{1LL}$ and
$f_{LT}$, where $f_{1LL}$ is a twist-2 function and $f_{LT}$ is a twist-3 one.
Namely, the twist-2 part of $f_{LT}$ is expressed by an integral of $f_{1LL}$
(or $b_{1}$) and the integral of the function $f_{2LT}=(2/3)f_{LT}-f_{1LL}$
over $x$ vanishes. If the parton-model sum rule for $f_{1LL}$ ($b_{1}$) is
applied by assuming vanishing tensor-polarized antiquark distributions,
another sum rule also exists for $f_{LT}$ itself. These relations should be
valuable for studying tensor-polarized distribution functions of spin-1
hadrons and for separating twist-2 components from higher-twist terms, as the
WW relation and BC sum rule have been used for investigating $x$ dependence
and higher-twist effects in $g_{2}$. In deriving these relations, we indicate
that four twist-3 multiparton distribution functions $F_{LT}$, $G_{LT}$,
$H_{LL}^{\perp}$, and $H_{TT}$ exist for tensor-polarized spin-1 hadrons.
These multiparton distribution functions are also interesting to probe
multiparton correlations in spin-1 hadrons. In the near future, we expect that
physics of spin-1 hadrons will become a popular topic, since there are
experimental projects to investigate spin structure of the spin-1 deuteron at
the Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider
facility, the electron-ion colliders in US and China in 2020’s and 2030’s.

##  1 Introduction

High-energy spin physics has been one of exciting fields in hadron physics
since the late 1980’s for clarifying the origin of nucleon spin. In addition
to longitudinally-polarized collinear structure functions, we investigate
transverse-spin and three-dimensional (3D) structure functions nowadays.
Furthermore, structure functions of spin-1 hadrons will be investigated
extensively in the near future due to the existence of new tensor-polarized
observables.

We have been investigating structure functions of spin-1 hadrons, especially
on the spin-1 deuteron. There are four tensor-polarized structure functions
$b_{1-4}$ for a spin-1 hadron in charged-lepton deep inelastic scattering fs83
; Hoodbhoy:1988am . The leading-twist functions $b_{1}$ and $b_{2}$ are
related with each other by the Callan-Gross type relation $2xb_{1}=b_{2}$, and
other functions $b_{3}$ and $b_{4}$ are higher-twist ones. For finding the
overall $x$-dependent functional form, there is a useful parton-model sum rule
for $b_{1}$ b1-sum . The $b_{1}$ measurement by the HERMES collaboration
obtained a finite sum $\int dxb_{1}\neq 0$ Airapetian:2005cb , which indicated
a finite tensor-polarized antiquark distribution. Possible tensor-polarized
parton distribution functions (PDFs) were proposed for explaining the HERMES
data tensor-pdfs . In future, an accurate determination of $b_{1}$ will be
made by the experiment at the Thomas Jefferson National Accelerator Facility
(JLab) jlab-b1 . This $b_{1}$ project is interesting in the sense that the
HERMES data are different from a conventional theoretical estimate based on a
convolution description b1-convolution . In addition, the gluon transversity
$\Delta_{T}g$ exists for the spin-1 deuteron, although it does not exist for
the spin-1/2 nucleons. This is also an interesting observable to find new
hadron physics in the deuteron beyond the simple bound system of nucleons. It
is expected to be measured by an JLab experiment jlab-gluon-trans ; ma-
spin-1-2013 . These $b_{1}$ and $\Delta_{T}g$ measurements could be continued
at the electron-ion collider (EIC) eic  and the Electron-ion collider in China
(EicC) eicc .

Such tensor-polarized structure functions can be also investigated at hadron
accelerator facilities. In fact, the proton-deuteron Drell-Yan process could
probe the tensor-polarized PDFs, especially the tensor-polarized antiquark
distributions pd-drell-yan ; Kumano:2016ude  and the gluon transversity ks-
trans-g-2020 . It could be realized as a Fermilab-SpinQuest experiment
Fermilab-dy  when a polarized-deuteron target becomes ready Keller:2020wan .
In addition, since the Nuclotron-based Ion Collider fAcility (NICA) will have
a polarized-deuteron beam, the tensor-polarized structure functions and the
gluon transversity will be investigated, for example, by observing J/$\psi$
production nica . These structure functions are interesting for investigating
especially exotic aspects of hadron physics, such as hidden color miller-b1
and non-nucleonic component transversity-model  in the deuteron.

On the other hand, transverse-momentum-dependent parton distribution functions
(TMDs) became one of hot topics in hadron physics. It is intended to
understand not only basics 3D structure of hadrons but also to find explicit
color degrees of freedom in terms of color flow. Because the color is confined
in hadrons, it is not easy to find its explicit signature in observables. The
TMDs are such quantities to probe the color directly ks-tmd-2021 . For
example, a color Aharonov-Bohm effect and color entanglement phenomena could
be investigated by the TMDs. In addition, gluon condensates are now
investigated with the understanding of gluon TMDs in the nucleons and nuclei.

Recently, we investigated the tensor-polarized TMDs for spin-1 hadrons and
found 30 new TMDs at the twist 3 and twist 4 ks-tmd-2021  in addition to the
twist-2 functions bm-2000 . Integrating the TMDs over the partonic transverse
momentum, we also found that there are three new collinear PDFs at the twist 3
and twist 4. Therefore, including the twist-2 PDF, we have the collinear PDFs
$f_{1LL}$, $e_{LL}$, $f_{LT}$, and $f_{3LL}$ for spin-1 hadrons. By
considering this situation, the purposes of this work are the following.

  * (1)

We derive a useful twist-2 relation and a sum rule for $f_{1LL}$ and $f_{LT}$
in analogy to the Wandzura-Wilczek relation ww-1977  and the Burkhardt-
Cottingham sum rule bc-1970  for the structure functions $g_{1}$ and $g_{2}$.
Here, $f_{1LL}$ is a twist-2 function and $f_{LT}$ is a twist-3 one, and
$f_{1LL}$ is often used as the tensor-polarized structure function $b_{1}$.

  * (2)

We show that four twist-3 multiparton distribution functions exist in a
tensor-polarized spin-1 hadron.

  * (3)

We show that the leading deviation of the twist-2 relation, namely the higher-
twist term, is expressed by these twist-3 multiparton distribution functions,
so that we try to obtain the full-decomposition expression of $f_{LT}$ into
twist-2 and twist-3 terms.

For simply deriving the twist-2 relation and sum rule, the studies of the
twist-3 multiparton distributions functions are not necessary. However, we
investigate one more step to express the higher-twist term of $f_{LT}$ by the
twist-3 multiparton distributions functions in this work.

In this paper, the Wandzura-Wilczek relation and Burkhardt-Cottingham sum rule
are introduced by the operator-product-expansion formalism in Sec. 2. Then,
the details are explained on our derivations of analogous relations for
tensor-polarized PDFs in Sec. 3. First, correlation functions and collinear
PDFs are introduced, and the matrix element of a nonlocal vector operator is
expressed by the tensor-polarized collinear PDFs in Sec. 3.1. Next, the
nonlocal operator is written in term of the gluon field tensor and it is
related to the multiparton distribution functions in Sec. 3.2. A useful
twist-2 relation and a sum rule are derived for $f_{1LL}$ and $f_{LT}$ in Sec.
3.3. Our studies are summarized in Sec. 4.

##  2 Wandzura-Wilczek relation and Burkhardt-Cottingham sum rule

In investigating structure functions, there are useful sum rules and relations
among them for finding their functional behavior. For example, there is a
useful relation for the polarized structure function $g_{2}$, which exists in
the spin-1/2 nucleons. Since we intend to derive a relation which is analogous
to the Wandzura-Wilczek (WW) relation and also a sum rule like the Burkhardt-
Cottingham (BC) sum rule, we introduce their outline within the formalism of
operator product expansion.

The polarized distribution functions $g_{1L}$, $g_{T}$, and $g_{3L}$ are
defined by the matrix element of a nonlocal operator as

$\displaystyle\int\frac{d(P^{+}\xi^{-})}{2\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,S\left|\,\bar{\psi}(0)\,\gamma^{\mu}\gamma_{5}\psi(\xi)\,\right|P,\,S\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}$
$\displaystyle\ \ \ \ \ =2M_{N}\left[\,g_{1L}(x)\bar{n}^{\mu}S\cdot
n+g_{T}(x)S_{T}^{\mu}+g_{3L}(x)\frac{M^{2}_{N}}{(P^{+})^{2}}n^{\mu}S\cdot
n\,\right].$ (1)

Here, $M_{N}$ is the nucleon mass, $\psi$ is the quark field, the lightcone
vectors $n$ and $\bar{n}$ are defined by

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

$P$ and $S$ are the nucleon momentum and spin, $S_{T}^{\mu}$ is the
transverse-spin vector kt-1999 ; ks-tmd-2021 , $\xi$ is the space-time
coordinate, the variable $x$ is the momentum fraction carried by a parton and
it is defined by $k^{+}=xP^{+}$, and the lightcone variables $a^{\pm}$
indicate $a^{\pm}=(a^{0}\pm a^{3})/\sqrt{2}$. In Eq. (1), the gauge link is
abbreviated. In this paper, the momentum and renormalization scale dependence
($Q^{2}$, $\mu^{2}$) is not explicitly written in the PDFs and structure
functions.

The moments of Eq. (1) become

$\displaystyle\frac{1}{2M_{N}(P^{+})^{n-1}}\,n_{\mu_{1}}\cdots
n_{\mu_{n-1}}\langle\,P,S\left|\,R^{\sigma\\{\mu_{1}\cdots\mu_{n-1}\\}}\,\right|P,\,S\,\rangle$
$\displaystyle=\bar{n}^{\sigma}(S\cdot
n)\int_{-1}^{1}dxx^{n-1}g_{1L}(x)+S_{T}^{\sigma}\int_{-1}^{1}dxx^{n-1}g_{T}(x),$
(3)

by keeping the terms up to twist 3. Hereafter, the twist-4 distribution
function $g_{3L}$ and twist-4 terms are neglected in this section. The
structure functions are classified by the twist, which is defined by the mass
dimension minus spin, in the operator product expansion Anselmino-1995 ;
Blumlein-1997-1999 ; Braun-2001 ; kt-1999 ; lr-2000 ; leader-book . The local
operators $R^{\sigma\\{\mu_{1}\cdots\mu_{n-1}\\}}$ for describing the
polarized structure functions are defined by

$\displaystyle
R^{\sigma\\{\mu_{1}\cdots\mu_{n-1}\\}}=i^{n-1}\bar{\psi}\gamma^{\sigma}\gamma_{5}D^{\\{\mu_{1}}\cdots
D^{\mu_{n-1}\\}}\psi,$ (4)

where $D^{\mu}$ is the QCD covariant derivative given by
$D^{\mu}=\partial^{\mu}-igA^{\mu}$ with the QCD coupling constant $g$ and the
gluon field $A^{\mu}$, and the curly bracket $\\{\ \\}$ indicates the
symmetrization of all the Lorentz indices as defined in Eq. (5). The higher-
twist ($\geq 4$) trace terms
$g_{\mu_{i}\mu_{j}}R^{\sigma\\{\mu_{1}\cdots\mu_{n-1}\\}}$ and
$g_{\sigma\mu_{i}}R^{\sigma\\{\mu_{1}\cdots\mu_{n-1}\\}}$ should be subtracted
to make the operator traeless; however, they are not explicitly written in Eq.
(4). The gluon field $A^{\mu}$ contains the SU(3) generator $t^{a}$, defined
by the Gell-Mann matrix $\lambda^{a}$ as $t^{a}=\lambda^{a}/2$ with the color
index $a$, as $A^{\mu}=A^{\mu}_{a}t^{a}$. Since these operators contain both
twist-2 and twist-3 components, they should be separated so as to have
definite twists as jaffe-ji-1991 ; jaffe-1996 ; Blumlein-1997-1999 ; kt-1999 ;
Braun-2001

$\displaystyle R^{\sigma\\{\mu_{1}\cdots\mu_{n-1}\\}}$
$\displaystyle=R^{\\{\sigma\mu_{1}\cdots\mu_{n-1}\\}}+R^{[\sigma\\{\mu_{1}]\cdots\mu_{n-1}\\}},$
$\displaystyle R^{\\{\sigma\mu_{1}\cdots\mu_{n-1}\\}}$
$\displaystyle=\frac{1}{n}\left[R^{\sigma\\{\mu_{1}\mu_{2}\cdots\mu_{n-1}\\}}+R^{\mu_{1}\\{\sigma\mu_{2}\cdots\mu_{n-1}\\}}+R^{\mu_{2}\\{\mu_{1}\sigma\cdots\mu_{n-1}\\}}+\cdots\right],$
$\displaystyle R^{[\sigma\\{\mu_{1}]\cdots\mu_{n-1}\\}}$
$\displaystyle=\frac{1}{n}\left[(n-1)R^{\sigma\\{\mu_{1}\mu_{2}\cdots\mu_{n-1}\\}}-R^{\mu_{1}\\{\sigma\mu_{2}\cdots\mu_{n-1}\\}}-R^{\mu_{2}\\{\mu_{1}\sigma\cdots\mu_{n-1}\\}}-\cdots\right],$
(5)

where the first term $R^{\\{\sigma\mu_{1}\cdots\mu_{n-1}\\}}$ has spin $n$ and
the second one $R^{[\sigma\\{\mu_{1}]\cdots\mu_{n-1}\\}}$ has $n-1$ so that
they are definite twist-2 and twist-3 operators. The twist-3 operator
$R^{[\sigma\\{\mu_{1}]\cdots\mu_{n-1}\\}}$ is obtained by expanding the
nonlocal operator in the Taylor series as

$\displaystyle\xi_{\mu}\,\bar{\psi}(0)\left(\partial^{\mu}\gamma^{\sigma}-\partial^{\sigma}\gamma^{\mu}\right)\gamma_{5}\,\psi(\xi)\Longrightarrow\text{twist-3:}\
R^{[\sigma\\{\mu_{1}]\cdots\mu_{n-1}\\}},$ (6)

where $\partial^{\mu}=\partial/\partial\xi_{\mu}$. Then, it is written by the
following operators with the gluon field tensor $G^{\mu\nu}$, the quark mass
($m_{q}$), and equation of motion as kt-1999

$\displaystyle\\!\\!\\!\xi_{\mu}\,\bar{\psi}(0)\left(\partial^{\mu}\gamma^{\sigma}-\partial^{\sigma}\gamma^{\mu}\right)\gamma_{5}\,\psi(\xi)=g\int^{1}_{0}dt\,\bar{\psi}(0)\left\\{i\gamma_{5}\left(t-\frac{1}{2}\right)G^{\sigma\rho}(t\xi)-\frac{1}{2}\tilde{G}^{\sigma\rho}(t\xi)\right\\}\xi_{\rho}\not{\xi}\psi(\xi)$
$\displaystyle\ \hskip
85.35826pt+\,2m_{q}\bar{\psi}(0)\gamma_{5}\sigma^{\sigma\rho}\xi_{\rho}\psi(\xi)$
$\displaystyle\ \hskip
85.35826pt+\,\bar{\psi}(0)\gamma_{5}\sigma^{\sigma\rho}\xi_{\rho}(i\not{D}-m_{q})\psi(\xi)-\bar{\psi}(0)(i\overleftarrow{\not{D}}+m_{q})\gamma_{5}\sigma^{\sigma\rho}\xi_{\rho}\psi(\xi),$
(7)

where the dual field tensor $\tilde{G}^{\mu\nu}$ is defined by
$\tilde{G}^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}G_{\rho\sigma}/2$ Itzykson-
Zuber-book  with the convention $\epsilon^{0123}=+1$. The first line of the
right-hand side is from the quark-gluon-quark correlation, the second one is
the quark-mass term, and the third one is the equation-of-motion term. Scale
evolution was studied by using these three types of operators; however, we do
not step into such details. Interested readers may read, for example, the
summary article of Ref. kt-1999 .

In this way, the matrix elements of these operators are generally expressed
for the nucleons as

$\displaystyle\langle\,P,S\,\big{|}\,R^{\\{\sigma\mu_{1}\cdots\mu_{n-1}\\}}\,\big{|}\,P,S\,\rangle$
$\displaystyle=\frac{2}{n}a_{n}M_{N}\left[S^{\sigma}P^{\mu_{1}}\cdots
P^{\mu_{n-1}}+S^{\mu_{1}}P^{\sigma}\cdots P^{\mu_{n-1}}\cdots\right],$
$\displaystyle\langle\,P,S\,\big{|}\,R^{[\sigma\\{\mu_{1}]\cdots\mu_{n-1}\\}}\,\big{|}\,P,S\,\rangle$
$\displaystyle=\frac{2}{n}d_{n}M_{N}\big{[}(S^{\sigma}P^{\mu_{1}}-S^{\mu_{1}}P^{\sigma})\,P^{\mu_{2}}\cdots
P^{\mu_{n-1}}$ $\displaystyle\ \hskip
42.67912pt+(S^{\sigma}P^{\mu_{2}}-S^{\mu_{2}}P^{\sigma})\,P^{\mu_{1}}\cdots
P^{\mu_{n-1}}+\cdots\big{]},$ (8)

where $a_{n}$ and $d_{n}$ are constants to indicate the magnitudes of the
twist-2 and twist-3 matrix elements. The polarized structure functions $g_{1}$
and $g_{2}$ are given by $g_{1L}$ and $g_{T}$ as
$g_{1}(x)=[g_{1L}(x)+g_{1L}(-x)]/2$ and
$g_{1}(x)+g_{2}(x)=[g_{T}(x)+g_{T}(-x)]/2$. From Eqs. (3), (4), (5), and (8),
we obtain the moments as

$\displaystyle\int_{0}^{1}dxx^{n-1}g_{1}(x)$ $\displaystyle=\frac{1}{2}a_{n},$
$\displaystyle\int_{0}^{1}dxx^{n-1}\left[\,g_{1}(x)+g_{2}(x)\,\right]$
$\displaystyle=\frac{1}{2}\left(\frac{1}{n}a_{n}+\frac{n-1}{n}d_{n}\right).$
(9)

In these equations, we use $g_{1}$ and $g_{2}$ for a single flavor. The second
equation is written by using the first one as

$\displaystyle\int_{0}^{1}dxx^{n-1}g_{2}(x)=\int_{0}^{1}dxx^{n-1}\left[-g_{1}(x)+\int_{x}^{1}\frac{dy}{y}g_{1}(y)\right]+\frac{n-1}{2n}d_{n}.$
(10)

From this equation, the structure function $g_{2}$ is written in terms of
twist-2 and twist-3 parts separately as

$\displaystyle g_{2}(x)=g_{2}^{WW}(x)$ $\displaystyle+\bar{g}_{2}(x),$ (11)
$\displaystyle g_{2}^{WW}(x)$
$\displaystyle=-g_{1}(x)+\int_{x}^{1}\frac{dy}{y}g_{1}(y),$ (12)
$\displaystyle\int_{0}^{1}dx\,x^{n-1}\bar{g}_{2}(x)$
$\displaystyle=\frac{n-1}{2n}d_{n}.$ (13)

Equation (12) is the WW relation which is valid in the twist-2 level by
neglecting higher-twist effects. There is also a similar relation for the
chiral-odd twist-3 structure function $h_{L}$ and the twist-2 one $h_{1}$
jj-1991 ; chiral-odd-twist-3 . In addition, higher-twist terms are explicitly
written by multiparton distribution functions chiral-odd-twist-3 . If the WW
relation of Eq.(12) is integrated over $x$, it becomes

$\displaystyle\int_{0}^{1}dx\,g_{2}^{WW}(x)=0.$ (14)

This relation is the BC sum rule, which was originally derived by using the
dispersion relation for the virtual Compton amplitude. Here, the convergence
of this sum could depend on the $x$-dependent functional form of $g_{2}$ at
small $x$. Furthermore, no operator is defined for $n=1$ in Eq. (4), so that
the WW relation and BC sum rule could not be rigorously proven in the
operator-product-expansion formalism Anselmino-1995 ; Blumlein-1997-1999 ;
Braun-2001 ; br-book ; accardi-2009 ; Deur-2019 . On the other hand, these
relations are satisfied even if perturbative QCD corrections are included in
coefficient functions Braun-2001 .

##  3 Parton distribution functions of spin-1 hadrons and their twist-2
relation and sum rule

For studying structure functions, sum rules provide useful information on
their $x$-dependent functional forms. There are sum rules for the structure
functions and TMDs of spin-1 hadrons b1-sum ; ks-tmd-2021 . If tensor-
polarized antiquark distributions vanish, there is a sum rule for the twist-2
collinear structure function as $\int dxb_{1}(x)=0$. In addition, due to the
time-reversal invariance of collinear parton distributions, there exist the
sum rules
$\int\\!d^{2}k_{T}h_{1LT}(x,k_{T}^{\,2})=\\!\int\\!d^{2}k_{T}g_{LT}(x,k_{T}^{\,2})=\\!\int\\!d^{2}k_{T}h_{LL}(x,k_{T}^{\,2})=\\!\int\\!d^{2}k_{T}h_{3LL}(x,k_{T}^{\,2})=0$
for the TMDs of tensor-polarized spin-1 hadrons. In this subsection, we show
the existence of a new sum rule and a twist-2 relation, which are analogous to
the BC sum rule and the WW relation, respectively.

###  3.1 Matrix elements of nonlocal operators and parton distribution
functions

The PDFs of hadrons are often discussed by correlation functions. The PDFs of
spin-1/2 nucleons are now theoretically investigated including higher-twist
ones. On the other hand, the PDFs of spin-1 hadrons are not well studied
especially for the tensor-polarized part. The TMDs and PDFs of spin-1 hadrons
were investigated for the twist-2 in Ref. bm-2000 , and twist-3 and twist-4
functions were recently proposed in Ref. ks-tmd-2021 . The PDFs of hadrons are
generally defined from the correlation function

$\displaystyle\Phi_{ij}^{[c]}(k,P,T\,|\,n)=\int\frac{d^{4}\xi}{(2\pi)^{4}}\,e^{ik\cdot\xi}\langle\,P,T\left|\,\bar{\psi}_{j}(0)\,W^{[c]}(0,\xi)\psi_{i}(\xi)\,\right|P,T\,\rangle,$
(15)

which is related to the amplitude to extract a parton from a hadron and then
to insert it into the hadron at a different spacetime point $\xi$. Here, $k$
is the quark momentum, the hadron momentum and tensor polarization are denoted
by $P$ and $T$, respectively, $W^{[c]}$ is the gauge link for satisfying the
color gauge invariance, and $c$ indicates the integral path. We do not write
the spin vector polarization $S$ explicitly in Eq. (15) because only the
tensor polarization $T$, which is specific to hadrons with spin$\geq 1$, is
investigated in this paper. The vector polarization part is essentially the
same as the one for the spin-1/2 nucleons. From the general correlation
function in Eq. (15), we obtain the collinear correlation function by
integrating it over the lightcone momentum $k^{-}$ and the transverse momentum
$k_{T}$, and fixing the $k^{+}$ component as

$\displaystyle\Phi_{ij}(x,P,T)$ $\displaystyle=\int
d^{2}k_{T}\,dk^{+}dk^{-}\Phi^{[c]}_{ij}(k,P,T\,|n)\,\delta(k^{+}-xP^{+}),$
$\displaystyle=\int\frac{d\xi^{-}}{2\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}_{j}(0)\,W(0,\xi\,|n)\psi_{i}(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}.$
(16)

Here, $W(a,b\,|n)$ indicates the gauge line connecting
$a=(a^{+},a^{-},\vec{a}_{T})$ to $b=(b^{+},b^{-},\vec{b}_{T})$ along the
straight lightcone direction of $\xi^{-}$. Since the link is along the
straight line, there is no path-$c$ dependence.

The TMDs and collinear PDFs for spin-1 hadrons are defined in various traces
of the TMD and collinear correlation function as $\text{Tr}(\Phi\Gamma)$,
where $\Gamma$ is expressed by $\gamma$ matrices, in Ref. ks-tmd-2021 . In
this subsection, the tensor-polarized collinear PDFs are studied and the gauge
link $W(0,\xi\,|n)$ is not explicitly written. Later, the Fock-Schwinger gauge
$\xi_{\mu}A^{\mu}(\xi)=0$ bb-1988  is used from Sec. 3.2 and the gauge link is
unity $W(0,\xi\,|n)=1$ in any case for the collinear PDFs. However, this link
appears in Sec. 3.2, for example in Eq. (36), for obtaining our WW- and BC-
like relations because its total derivative does not vanish. Here, we are
interested in collinear PDFs which are defined in the trace as

$\displaystyle\Phi^{[\Gamma]}(x,P,T)$
$\displaystyle\equiv\frac{1}{2}\,\text{Tr}\left[\,\Phi(x,P,T)\,\Gamma\,\right]=\frac{1}{2}\Phi_{ij}(x,P,T)\,(\Gamma)_{ji}$
$\displaystyle=\frac{1}{2}\int\frac{d\xi^{-}}{2\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}(0)\,\Gamma\,\psi(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}.$
(17)

The tensor polarization $T^{\mu\nu}$ is generally expressed by the
polarizations $S_{LL}$, $S_{LT}^{\mu}$, and $S_{TT}^{\mu\nu}$ as ks-
trans-g-2020 ; ks-tmd-2021

$\displaystyle T^{\mu\nu}=\frac{1}{2}$
$\displaystyle\left[\frac{4}{3}S_{LL}\frac{(P^{+})^{2}}{M^{2}}\bar{n}^{\mu}\bar{n}^{\nu}-\frac{2}{3}S_{LL}(\bar{n}^{\\{\mu}n^{\nu\\}}-g_{T}^{\mu\nu})+\frac{1}{3}S_{LL}\frac{M^{2}}{(P^{+})^{2}}n^{\mu}n^{\nu}\right.$
$\displaystyle\
\left.+\frac{P^{+}}{M}\bar{n}^{\\{\mu}S_{LT}^{\nu\\}}-\frac{M}{2P^{+}}n^{\\{\mu}S_{LT}^{\nu\\}}+S_{TT}^{\mu\nu}\right],$
(18)

where $a^{\\{\mu}b^{\nu\\}}$ is the symmetrized combination
$a^{\\{\mu}b^{\nu\\}}=a^{\mu}b^{\nu}+a^{\nu}b^{\mu}$. The tensor-polarization
parameters $S_{LL}$, $S_{LT}^{\mu}$, and $S_{TT}^{\mu\nu}$ are explained in
Appendix of Ref. bm-2000 . The $S_{LL}$ is associated with the tensor
polarization along the $z$ direction, $S_{TT}^{\mu\nu}$ is the linear
polarization in the transverse plane, and $S_{LT}^{\mu}$ is the polarization
in the plane in-between.

We investigate possible relations of tensor-polarized PDFs $f_{1LL}$ and
$f_{LT}$, where the function $f_{1LL}$ is twist 2 and $f_{LT}$ is twist 3, in
analogy to the WW and BC relations for $g_{1}$ and $g_{2}$. The function
$f_{1LL}$ is defined in $\Phi^{[\gamma^{+}]}(x,P,T)$, and $f_{LT}$ is in
$\Phi^{[\gamma^{i}]}(x,P,T)$, and the actual expressions are given in the TMD
form in Ref. ks-tmd-2021 . The twist-4 function $f_{3LL}$ exists in another
vector type correlation function $\Phi^{[\gamma^{-}]}(x,P,T)$, so that it is
also included in the following discussion of this subsection. There is another
twist-3 function $e_{LL}$ defined in $\Phi^{[\mathbf{1}]}(x,P,T)$, where
$\mathbf{1}$ is the $4\times 4$ identity matrix, and it is also listed in this
subsection. There are related works on the TMDs and PDFs of spin-1 hadrons
Hoodbhoy:1988am ; bm-2000 ; ma-spin-1-2013 . Integrating the TMD expressions
of Eqs. (33), (43), and (52) in Ref. ks-tmd-2021  for $\Gamma=\gamma^{+}$,
$\gamma^{i}$, $\gamma^{-}$, and $\mathbf{1}$ over the transverse momentum
$\vec{k}_{T}$, we obtain PDF expressions in terms of operator matrix elements
as

$\displaystyle\\!\\!\Phi^{[\gamma^{+}]}(x,P,T)$
$\displaystyle=\\!\int\\!\frac{d\xi^{-}}{4\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}(0)\,\gamma^{+}\psi(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}=S_{LL}\,f_{1LL}(x),$
$\displaystyle\\!\\!\Phi^{[\gamma^{\alpha}]}(x,P,T)$
$\displaystyle=\\!\int\\!\frac{d\xi^{-}}{4\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}(0)\,\gamma^{\alpha}\,\psi(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}=\frac{M}{P^{+}}S_{LT}^{\,\alpha}\,f_{LT}(x),$
$\displaystyle\\!\\!\Phi^{[\gamma^{-}]}(x,P,T)$
$\displaystyle=\\!\int\\!\frac{d\xi^{-}}{4\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}(0)\,\gamma^{-}\psi(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}=\frac{M^{2}}{(P^{+})^{2}}\,S_{LL}\,f_{3LL}(x),$
$\displaystyle\\!\\!\Phi^{[\mathbf{1}]}(x,P,T)$
$\displaystyle=\\!\int\\!\frac{d\xi^{-}}{4\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}(0)\,\psi(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}=\frac{M}{P^{+}}\,S_{LL}\,e_{LL}(x),$
(19)

where $\alpha$ is the transverse index $\alpha=1$ or 2 and this transverse
$\alpha$ notation is used throughout this paper. The tensor-polarized PDFs
$f_{1LL}(x)$, $f_{LT}(x)$, $f_{3LL}(x)$, and $e_{LL}(x)$ are expressed by the
matrix elements of the nonlocal operators with different $\gamma$ matrices in
Eq. (19). The collinear PDFs are often written from the TMDs as

$\displaystyle f(x)=\int d^{2}k_{T}f(x,k_{T}^{\,2}).$ (20)

However, we should note that this integral equation could not be a general
relation due to the ultraviolet divergence in the transverse-momentum integral
rogers-2020 . Futhermore, the TMDs $f^{\,\prime}_{LT}(x,k_{T}^{\,2})$ and
$f^{\perp}_{LT}(x,k^{\,2}_{T})$ are combined to express them as ks-tmd-2021
$f_{LT}(x,k_{T}^{\,2})\equiv
f^{\,\prime}_{LT}(x,k_{T}^{\,2})-\frac{k_{T}^{\,2}}{2M^{2}}\,f^{\perp}_{LT}(x,k^{\,2}_{T})$.
Therefore, two functions among $f_{LT}$, $f^{\,\prime}_{LT}$, and
$f^{\perp}_{LT}$ are independent, and $f_{LT}$ and $f^{\perp}_{LT}$ are
selected in the table IV of Ref. ks-tmd-2021 . In Eq. (19), $M$ is the spin-1
hadron mass, $f_{1LL}$, $f_{LT}$, and $f_{4LL}$ are twist-2, twist-3, and
twist-4 distribution functions, respectively. In this way, the collinear
correlation function is written up to twist-4 as

$\displaystyle\Phi(x,P,T)=\frac{1}{2}\bigg{[}S_{LL}\,\not{\bar{n}}\,f_{1LL}(x)+\frac{M}{P^{+}}\,S_{LL}\,e_{LL}(x)+\frac{M}{P^{+}}\,\not{S}_{LT}\,f_{LT}(x)+\frac{M^{2}}{(P^{+})^{2}}\,S_{LL}\,\not{n}\,f_{3LL}(x)\bigg{]}.$
(21)

Then, the matrix element of Eq. (17) is expressed by the Fourier transform of
$\Phi^{[\gamma^{\mu}]}(x,P,T)$ expressed by the PDFs as

$\displaystyle\\!\\!\\!\\!\\!\\!\langle\,P,T\left|\,\bar{\psi}(0)\,\gamma^{\mu}\,\psi(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!=\\!\int_{-1}^{1}dxe^{-ixP^{+}\xi^{-}}2P^{+}\left[S_{LL}\,\bar{n}^{\mu}\,f_{1LL}(x)+\frac{M}{P^{+}}\,S_{LT}^{\,\mu}\,f_{LT}(x)+\frac{M^{2}}{(P^{+})^{2}}\,S_{LL}\,n^{\mu}\,f_{3LL}(x)\right].$
(22)

Therefore, the matrix element of the vector operator is given by the three
collinear PDFs $f_{1LL}(x)$, $f_{LT}(x)$, and $f_{3LL}(x)$ for the tensor-
polarized hadron. In particular, we derive a useful relation and a sum rule
for the twist-3 functions $f_{LT}(x)$ in this work.

Since antiquark distribution are discussed later in deriving twist-2
relations, we briefly explain them. We define the collinear antiquark
correlation function in the same way as mulders-2014-lect ; ks-trans-g-2020

$\displaystyle\bar{\Phi}_{ij}(x,P,T)$
$\displaystyle=-\int\frac{d\xi^{-}}{2\pi}\,e^{-ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}_{j}(0)\,W(0,\xi\,|n)\,\psi_{i}(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}$
$\displaystyle=-\Phi_{ij}(-x,P,T).$ (23)

Therefore, the antiquark correlation function is related to the quark
correlation function at negative $x$, so that the antiquark distributions are
described by the quark distributions at negative $x$, $q(x<0)$. On the other
hand, the charge-conjugate correlation function, in which the antiquark
distributions $\bar{q}(x)$ are defined, is given by the conjugate spinor
$\psi^{C}\equiv C\bar{\psi}^{T}$ with $C=i\gamma^{2}\gamma^{0}$ and
$A_{\mu}^{C}=-A_{\mu}$ as

$\displaystyle\Phi_{ij}^{C}(x,P,T)=\int\frac{d\xi^{-}}{2\pi}\,e^{ixP^{+}\xi^{-}}\langle\,P,T\left|\,\bar{\psi}_{j}^{C}(0)\,W^{C}(0,\xi\,|n)\,\psi_{i}^{C}(\xi)\,\right|\\!P,\,T\,\rangle_{\xi^{+}=0,\,\vec{\xi}_{T}=0}.$
(24)

These equations indicate the relation between them as

$\displaystyle\Phi^{C}(x,P,T)=-C\,[\bar{\Phi}(x,P,T)]^{\cal T}C^{\dagger},$
(25)

where ${\cal T}$ indicates the transposed matrix. The antiquark distributions
$\bar{q}(x)$ are then defined by using this conjugate correlation function.
The relations between the various antiquark distributions $\bar{q}(x)$ and the
corresponding “quark” distributions $q(x<0)$ are explicitly written in the end
of Sec. 3.1. Next, we show relations between the antiquark distributions and
the quark ones at negative $x$. The relation between the correlation function
$\bar{\Phi}$ and the conjugate one $\Phi^{C}$ is

$\displaystyle\Phi^{C[\Gamma]}(x,P,T)=\left\\{\begin{array}[]{l}+\bar{\Phi}^{[\Gamma]}(x,P,T)=-\Phi^{[\Gamma]}(-x,P,T)\
\ \text{for }\Gamma=\gamma^{\mu},\ \sigma^{\mu\nu},\
i\gamma_{5}\sigma^{\mu\nu}\\\
-\bar{\Phi}^{[\Gamma]}(x,P,T)=+\Phi^{[\Gamma]}(-x,P,T)\ \ \text{for
}\Gamma=\mathbf{1},\ i\gamma_{5},\ \gamma_{5}\gamma^{\mu}\end{array}\right.,$
(28)

from Eqs. (23), (24), (25), and (17). By taking $\Gamma=\gamma^{\mu}$ or
$\mathbf{1}$ in Eqs. (21) and (28) the antiquark distributions are related to
the quark distributions as

$\displaystyle\bar{f}_{1LL}(x)$ $\displaystyle=-f_{1LL}(-x),\ \
\bar{f}_{LT}(x)=-f_{LT}(-x),\ \ \bar{f}_{3LL}(x)=-f_{3LL}(-x),$
$\displaystyle\bar{e}_{LL}(x)$ $\displaystyle=e_{LL}(-x),$ (29)

where the sign is opposite for the chiral-odd distribution function
$e_{LL}(x)$.

Let us consider the matrix element of non-local vector operator
$\bar{\psi}(0)\gamma^{\mu}\psi(\xi)$ in the region which is not necessarily on
the lightcone, where Eq. (22) was obtained, for calculating the matrix element
with its derivative. The Fourier transform of the vector-operator matrix
element is the correlation function, which is expanded by the linear terms of
the tensor polarization $T^{\mu\nu}$ as given in Eq. (20) of Ref. ks-tmd-2021
. Therefore, the matrix element should be generally expressed in terms of
three terms $(\xi\cdot T\cdot\xi)P^{\mu}$, $(\xi\cdot T\cdot\xi)\xi^{\mu}$,
and $T^{\mu\nu}\xi_{\nu}$ which are linear in the tensor polarization with the
available Lorentz vectors $P^{\mu}$, $\xi^{\mu}$, and $T^{\mu\nu}\xi_{\nu}$ as

$\displaystyle\langle\,P,T\left|\,\bar{\psi}(0)\gamma^{\mu}\psi(\xi)\,\right|P\,,T\,\rangle$
$\displaystyle\ \hskip
28.45274pt=\int_{-1}^{1}dx\,e^{-ixP\cdot\xi}\left[\,\xi\cdot
T\cdot\xi\left\\{A(x)\,P^{\mu}+B(x)\,\xi^{\mu}\right\\}+C(x)\,T^{\mu\nu}\xi_{\nu}\,\right],$
(30)

where $\xi\cdot T\cdot\xi$ is defined by $\xi\cdot
T\cdot\xi=\xi_{\mu}T^{\mu\nu}\xi_{\nu}$ and $\xi$ may not be on the lightcone.
The term $T^{\mu\nu}P_{\nu}$ does not exist in Eq. (30) because it vanishes
identically $T^{\mu\nu}P_{\nu}=0$. Here, $A(x)$, $B(x)$, and $C(x)$ are
coefficients to be determined. In this expansion, twist-4 effects are
neglected, so that the twist-4 function $f_{3LL}(x)$ does not appear in the
following discussions. The tensor polarization $T^{\mu\nu}$ contains the three
types of polarizations $S_{LL}$, $S_{LT}^{\mu}$, and $S_{TT}^{\mu\nu}$, and
the right-hand side of Eq. (30) should be expressed by them. We find the
factors $A(x)$, $B(x)$, and $C(x)$ so that Eq. (30) becomes Eq. (22) in the
lightcone limit $\xi^{2}\to 0$ ($\xi^{\mu}=\xi^{-}n^{\mu}$,
$\xi^{+}=\vec{\xi}_{T}=0$). In this limit, the factors $\xi\cdot T\cdot\xi$
and $T^{\mu\nu}\xi_{\nu}$ are expressed by the tensor polarization factors
$S_{LL}$ and $S_{LT}^{\mu}$ by using Eq. (18) as

$\displaystyle\xi\cdot T\cdot\xi$
$\displaystyle=\frac{2}{3\,M^{2}}(P^{+}\xi^{-})^{2}S_{LL},$ $\displaystyle
T^{\mu\nu}\xi_{\nu}$
$\displaystyle=\frac{2}{3\,M^{2}}\left[(P^{+})^{2}\xi^{-}\bar{n}^{\mu}-\frac{1}{2}M^{2}\xi^{-}n^{\mu}\right]S_{LL}+\frac{1}{2\,M}P^{+}\xi^{-}S_{LT}^{\mu}.$
(31)

Then, $A(x)$, $B(x)$, and $C(x)$ are obtained as

$\displaystyle A(x)$
$\displaystyle=\frac{3\,M^{2}}{(P\cdot\xi)^{2}}\left[f_{1LL}(x)-\frac{4}{3}f_{LT}(x)\right],\
B(x)=\frac{3\,M^{4}}{2(P\cdot\xi)^{3}}\left[-f_{1LL}(x)+\frac{8}{3}f_{LT}(x)\right],$
$\displaystyle C(x)$ $\displaystyle=\frac{4\,M^{2}}{P\cdot\xi}f_{LT}(x),$ (32)

in the lightcone limit with $P\cdot\xi=P^{+}\xi^{-}$. In order to derive a
twist-2 relation and a sum rule, we need to investigate the twist-3 matrix
element
$\xi_{\mu}\langle\,P,T\big{|}\,\bar{\psi}(0)(\partial^{\mu}\gamma^{\alpha}-\partial^{\alpha}\gamma^{\mu})\psi(\xi)\,\big{|}P\,,T\,\rangle$
as the derivative operator is given in Eqs. (6) and (7) for $g_{2}$. Taking
the transverse index $\alpha=1$ or 2 and considering the lightcone limit, we
obtain the relation

$\displaystyle\xi_{\mu}\langle\,P,T\left|\,\bar{\psi}(0)(\partial^{\mu}\gamma^{\alpha}-\partial^{\alpha}\gamma^{\mu})\psi(\xi)\,\right|P,T\,\rangle$
$\displaystyle=2MS_{LT}^{\,\alpha}\int_{-1}^{1}dx\,e^{-ixP^{+}\xi^{-}}\left[-\frac{3}{2}\,f_{1LL}(x)+f_{LT}(x)-\frac{d}{dx}\left\\{xf_{LT}(x)\right\\}\right],$
(33)

from Eq. (30). In this way, it becomes possible to identify the twist-3 part
of the function $f_{LT}(x)$ in connection with the twist-2 function
$f_{1LL}(x)$ .

Since the nonlocal operator of the left-hand side in Eq. (33) gives rise to
the twist-3 operators as explained in Sec. 2, the right-hand side should
vanish if higher-twist effects are neglected. It leads to the WW- and BC-like
twist-2 relations for $f_{LT}$ and $f_{1LL}$. However, we investigate further
in this work by defining possible multiparton distribution functions for the
tensor-polarized spin-1 hadron, and then we explicitly show that the left-hand
side is expressed by these multiparton distribution functions. Namely, we try
to obtain the full decomposition of $f_{LT}$ into the twist-2 and twist-3
terms.

###  3.2 Twist-3 matrix element and multiparton distribution functions

For specifying twist-3 effects, we derived the expression of twist-3 terms in
Eq. (33) in terms of the tensor-polarized distribution functions. In general,
the twist-3 terms are described by multiparton (three-parton in this work)
distribution functions kt-1999 ; bb-1988 ; Ball:1998ff . In order to derive a
twist-2 relation and a sum rule for the tensor-polarized PDFs, we try to
connect the derivative terms in the left-hand side of Eq. (33) to the
multiparton distribution functions. For this purpose, we try to express the
derivative terms by the nonlocal quark-gluon operators in this subsection.

The Fock-Schwinger gauge $x_{\mu}A^{\mu}(x)=0$ is used in our formalism, so
that the gluon field is expressed by the field strength tensor by introducing
the variable $t$ as

$\displaystyle A^{\nu}(\xi)=\int^{1}_{0}dt\,t\,\xi_{\mu}G^{\mu\nu}(t\xi),$
(34)

where
$G^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}-ig\left[A^{\mu},A^{\nu}\right]$.
The gauge link is generally expressed as

$\displaystyle
W(0,\xi)=\pazocal{P}\exp\left[-ig\int^{1}_{0}\\!dt\,\xi_{\mu}A^{\mu}(t\xi)\right],$
(35)

where the integral path is the direct one from $0$ to $\xi$. If the Fock-
Schwinger gauge is taken, the gauge link becomes unity. However, the total
derivative $\bar{\partial}^{\alpha}W$ does not vanish and it is used for
relating the derivative relation of Eq. (33) to the field tensor and
subsequently to the multiparton distribution functions. We consider that the
total derivative
$\left.\frac{\partial}{\partial(\Delta\xi^{\rho})}W(\Delta\xi,\xi+\Delta\xi)\right|_{\Delta\xi\rightarrow
0}$, which is given by the field tensor $G^{\rho\mu}$ and the gluon field
$A^{\rho}$ as

$\displaystyle\bar{\partial}_{\rho}W(0,\xi)\equiv\left.\frac{\partial}{\partial(\Delta\xi^{\rho})}W(\Delta\xi,\xi+\Delta\xi)\right|_{\Delta\xi\rightarrow
0}=-ig\int^{1}_{0}dt\,\xi^{\nu}G_{\rho\nu}(t\xi)-ig\left[A_{\rho}(\xi)-A_{\rho}(0)\right].$
(36)

From Eq. (36), the derivative of the local operator
$\bar{\psi}(0)W\Gamma\psi(\xi)$ becomes

$\displaystyle\\!\bar{\partial}_{\rho}\bar{\psi}(0)$ $\displaystyle
W(0,\xi)\Gamma\psi(\xi)=\bar{\psi}(0)\left(\overleftarrow{D}_{\rho}+\overrightarrow{D}_{\rho}\right)\Gamma\psi(\xi)-ig\int^{1}_{0}dt\,\xi^{\nu}G_{\rho\nu}(t\xi)\bar{\psi}(0)\Gamma\psi(\xi),$
(37)

where the covariant derivatives are given by
$\overleftarrow{D}_{\rho}=\overleftarrow{\partial}_{\rho}+igA_{\rho}$ and
$\overrightarrow{D}_{\rho}=\overrightarrow{\partial}_{\rho}-igA_{\rho}$.

Next, we express the left-hand side of Eq. (33) in terms of the covariant
derivatives and the field tensor as

$\displaystyle\bar{\psi}(0)(\partial^{\mu}\gamma^{\alpha}-\partial^{\alpha}\gamma^{\mu})\psi(\xi)$
$\displaystyle=\bar{\psi}(0)(\overrightarrow{D}^{\mu}\gamma^{\alpha}-\overrightarrow{D}^{\alpha}\gamma^{\mu})\psi(\xi)$
$\displaystyle-\bar{\psi}(0)\gamma^{\mu}\psi(\xi)\,ig\int^{1}_{0}dt\,t\,\xi_{\rho}G^{\rho\alpha}(t\xi).$
(38)

The index $\alpha$ is the transverse one ($\alpha=1,\,2$); however, all the
following equations within this subsection are valid as a general
4-dimensional Lorentz index. By the identity

$\displaystyle\gamma^{\rho}\sigma^{\alpha\mu}=i(g^{\alpha\rho}\gamma^{\mu}-g^{\mu\rho}\gamma^{\alpha})-\epsilon^{\alpha\mu\rho\sigma}\gamma_{\sigma}\gamma_{5},$
(39)

the covariant derivative term becomes
$\vec{D}^{\mu}\gamma^{\alpha}-\vec{D}^{\alpha}\gamma^{\mu}=i(\overrightarrow{\not{D}}\sigma^{\alpha\mu}-\sigma^{\alpha\mu}\overrightarrow{\not{D}})/2$.
Then, using Eq. (37) with the $\Gamma$ factor
$\Gamma=\gamma^{\rho}\sigma^{\alpha\mu}$, we obtain the first derivative terms
in the right-hand side of Eq. (38) as

$\displaystyle\bar{\psi}(0)\big{(}\overrightarrow{D}^{\mu}\gamma^{\alpha}$
$\displaystyle-\overrightarrow{D}^{\alpha}\gamma^{\mu}\big{)}\psi(\xi)=-\frac{i}{2}\bar{\psi}(0)\big{(}\sigma^{\alpha\mu}\overrightarrow{\not{D}}+\overleftarrow{\not{D}}\sigma^{\alpha\mu}\big{)}\psi(\xi)$
$\displaystyle\ \ \
-\frac{g}{2}\int^{1}_{0}dt\xi_{\nu}G^{\rho\nu}(t\xi)\bar{\psi}(0)\gamma_{\rho}\sigma^{\alpha\mu}\psi(\xi)+\frac{i}{2}\bar{\partial}_{\rho}\\{\bar{\psi}(0)\gamma^{\rho}\sigma^{\alpha\mu}\psi(\xi)\\}.$
(40)

The third term of the right-hand side ($\gamma_{\rho}\sigma^{\alpha\mu}$) is
written by the antisymmetric tensor $\epsilon^{\alpha\mu\rho\sigma}$ of Eq.
(39), and then it is given by the dual field tensor $\widetilde{G}^{\mu\nu}$
as

$\displaystyle\xi_{\mu}\xi_{\nu}G_{\rho\tau}(t\xi)g^{\nu\tau}\epsilon^{\alpha\mu\rho\sigma}=2\xi_{\mu}\xi^{\alpha}\widetilde{G}^{\mu\sigma}-2\xi^{2}\widetilde{G}^{\alpha\sigma}+\xi_{\mu}\xi^{\tau}G_{\tau\rho}\epsilon^{\alpha\mu\rho\sigma}+2\xi_{\mu}\xi^{\sigma}\widetilde{G}^{\alpha\mu},$
(41)

where the relation
$g^{\nu\tau}\epsilon^{\alpha\mu\rho\sigma}=g^{\nu\alpha}\epsilon^{\tau\mu\rho\sigma}+g^{\nu\mu}\epsilon^{\alpha\tau\rho\sigma}+g^{\nu\rho}\epsilon^{\alpha\mu\tau\sigma}+g^{\nu\sigma}\epsilon^{\alpha\mu\rho\tau}$
was used. We may note that the third term in the right-hand side of Eq. (41)
is identical to the left-hand side with the minus sign, so that the factor of
2 in front of the dual field tensor is dropped for calculating the left-hand-
side term. Therefore, the third term of Eq. (40) contracted with $\xi_{\mu}$
becomes

$\displaystyle\\!\\!\\!\\!\\!-\frac{g}{2}\xi_{\mu}\int^{1}_{0}\\!dt\,\xi^{\nu}G_{\rho\nu}(t\xi)\bar{\psi}(0)\gamma^{\rho}\sigma^{\alpha\mu}\psi(\xi)$
$\displaystyle=\frac{g}{2}\int^{1}_{0}\\!dt\,\bar{\psi}(0)\bigg{[}-i\xi_{\mu}G^{\alpha\mu}(t\xi)\not{\xi}+\xi_{\mu}\widetilde{G}^{\alpha\mu}(t\xi)\not{\xi}\gamma_{5}$
$\displaystyle\ \ \
-\left\\{\xi^{2}\widetilde{G}^{\alpha\sigma}(t\xi)-\xi_{\mu}\xi^{\alpha}\widetilde{G}^{\mu\sigma}(t\xi)\right\\}\gamma_{\sigma}\gamma_{5}\,\bigg{]}\psi(\xi).$
(42)

Substituting Eqs. (40) and (42) into Eq. (38), we obtain

$\displaystyle\\!\\!\\!\\!\\!\xi_{\mu}\,\bar{\psi}(0)\,(\overrightarrow{\partial}^{\mu}\gamma^{\alpha}-\overrightarrow{\partial}^{\alpha}\gamma^{\mu})\,\psi(\xi)=\\!g\\!\int^{1}_{0}\\!dt\,\bar{\psi}(0)\\!\left[i(t-\frac{1}{2})G^{\alpha\mu}(t\xi)-\frac{1}{2}\gamma_{5}\widetilde{G}^{\alpha\mu}(t\xi)\right]\\!\xi_{\mu}\not{\xi}\psi(\xi)$
$\displaystyle\
+\frac{g}{2}\int^{1}_{0}dt\,\bar{\psi}(0)\left[\xi_{\mu}\xi^{\alpha}\widetilde{G}^{\mu\sigma}(t\xi)-\xi^{2}\widetilde{G}^{\alpha\sigma}(t\xi)\right]\\!\gamma_{\sigma}\gamma_{5}\psi(\xi)$
$\displaystyle\
-\frac{i}{2}\xi_{\mu}\bar{\psi}(0)\sigma^{\alpha\mu}(\overrightarrow{\not{D}}-m_{q})\psi(\xi)-\frac{i}{2}\xi_{\mu}\bar{\psi}(0)(\overleftarrow{\not{D}}+m_{q})\sigma^{\alpha\mu}\psi(\xi)+\frac{i}{2}\xi_{\mu}\bar{\partial}_{\rho}\\{\bar{\psi}(0)\gamma^{\rho}\sigma^{\alpha\mu}\psi(\xi)\\},$
(43)

where the quark-mass term like the second line of Eq. (7) does not exist due
to the operator difference. In this equation, the first term in the second
line of RHS is a twist-4 contribution and the next term vanishes in the
lightcone limit $\xi^{2}=0$. The first two terms in the third line are the
equation-of-motion terms, and the last term could be neglected since the total
derivative term vanishes in the forward matrix elements bb-1988 ; kt-1999 .
Therefore, at the twist-3 level, the relation is given by

$\displaystyle\xi_{\mu}\,\bar{\psi}(0)\big{(}\partial^{\mu}\gamma^{\alpha}-\partial^{\alpha}\gamma^{\mu}\big{)}\psi(\xi)=g\int^{1}_{0}dt\,\bar{\psi}(0)\\!\bigg{[}\,i\left(t-\frac{1}{2}\right)G^{\alpha\mu}(t\xi)-\frac{1}{2}\gamma_{5}\widetilde{G}^{\alpha\mu}(t\xi)\,\bigg{]}\xi_{\mu}\not{\xi}\,\psi(\xi),$
(44)

In this way, the derivative terms in the left-hand side of Eq. (33) are given
by the field tensor $G^{\mu\nu}$ and the dual one $\tilde{G}^{\mu\nu}$.

The next step is to relate the field-tensor terms of Eq. (44) to twist-3
multiparton (three-parton) distribution functions. For this purpose, we define
a quark-gluon-quark correlation function $\Phi_{G}^{\mu}(x_{1},x_{2})$ for a
tensor-polarized spin-1 hadron in terms of the field tensor $G^{\mu\nu}$. For
defining the correlation function, only the transverse component of the gluon
field is considered in the lightcone formalism as the leading term jaffe-1996
; brodsky-1998 ; mulders-2014-lect . We have been using the Fock-Schwinger
gauge ($\xi\cdot A=0$) in this paper; however, it is identical to the
lightcone gauge ($n\cdot A=0$) in the lightcone limit because of the relation
$\xi\cdot A=\xi^{-}n\cdot A=\xi^{-}A^{+}=0$. The multiparton (quark-gluon-
quark) correlation function is defined by using the transverse gluon field
$A^{\alpha}(=A_{T}^{\alpha})$, where $\alpha$ is taken as the transverse index
$\alpha=1$ or 2. In the lightcone gauge $A^{+}=0$, the field tensor is
expressed by the gluon field as $G^{+\alpha}=\partial^{+}A^{\alpha}$, so that
the correlation function is defined with $G^{+\alpha}$ as

$\displaystyle(\Phi_{G}^{\alpha})_{ij}(x_{1},x_{2})=\int\\!\frac{d\xi_{1}^{-}}{2\pi}\frac{d\xi_{2}^{-}}{2\pi}\,e^{ix_{1}P^{+}\xi_{1}^{-}}e^{i(x_{2}-x_{1})P^{+}\xi_{2}^{-}}\langle\,P,T\left|\,\bar{\psi}_{j}(0)\,g\,G^{+\alpha}(\xi_{2}^{-})\,\psi_{i}(\xi_{1}^{-})\,\right|P,T\,\rangle.$
(45)

Due to the relation $G^{+\alpha}=\partial^{+}A^{\alpha}$, the same
correlation-function expression, which is derived in the following, should be
valid for the correlation function defined with the gluon field $A^{\alpha}$
instead of $G^{+\alpha}$, except for minor changes discussed after Eq. (49).
Here, the gauge link is not written because of the lightcone gauge. The
conditions of Hermiticity, parity invariance, and time-reversal invariance are
given in Refs.mulders-2014-lect ; ks-tmd-2021 .

Next, we try to express the multiparton correlation function in terms of
possible Lorentz vectors with the index $\alpha$. There was some study on the
multiparton correlation function for the deuteron in Ref. ma-spin-1-2013 . In
this work, we try to provide the full expression for the multiparton
correlation function for spin-1 hadrons, and then we relate them to the
twist-2 relation and sum rule for $f_{LT}$. Here, only the tensor-polarization
is considered since the unpolarized and vector-polarization correlation
functions have already investigated for the spin-1/2 nucleons. Then, the
correlation function of Eq. (45) should be proportional to the tensor
polarization $T^{\mu\nu}$ as shown in Eq. (20) of Ref. ks-tmd-2021 . The
correlation function in Eq. (45) has Dirac spinor indices $i$ and $j$, so that
it is expressed by the Dirac $\gamma$ matrices such as $\gamma^{\mu}$ and
$\gamma^{\mu}\gamma^{\nu}$. Three and more $\gamma$ types are not independent
from these one- and two-$\gamma$ types because of the relation
$\gamma^{\mu}\gamma^{\rho}\gamma^{\nu}=S^{\mu\rho\nu\beta}\gamma_{\beta}+i\epsilon^{\mu\rho\nu\beta}\gamma_{\beta}\gamma_{5}$
with
$S^{\mu\rho\nu\beta}=g^{\mu\rho}g^{\nu\beta}+g^{\mu\beta}g^{\rho\nu}-g^{\mu\nu}g^{\rho\beta}$.

In terms with $T^{\mu\alpha}$, the Lorentz index $\mu$ should be contracted
with other vectors. We note that the hadron momentum $P^{\mu}$ is expressed by
the two lightcone vectors $n^{\mu}$ and $\bar{n}^{\mu}$
($P^{\mu}=P^{+}\bar{n}^{\mu}+M^{2}n^{\mu}/(2P^{+})$), where the $n^{\mu}$ term
is suppressed by the factor $O(M^{2}/(P^{+})^{2})$. First, let us take the
contraction of $T^{\mu\alpha}$ in Eq. (18) with the lightcone vector $n_{\mu}$
to obtain
$n_{\mu}T^{\mu\alpha}\not{\bar{n}}=\frac{P^{+}}{2M}S_{LT}^{\,\alpha}\not{\bar{n}}$,
where $\not{\bar{n}}$ is multiplied for including the $\gamma$ matrix, and the
higher-twist $\not{n}$ terms are not included. Namely, the term
$n_{\mu}T^{\mu\alpha}$ is given by the tensor polarization vector
$S_{LT}^{\,\alpha}$. The term $\bar{n}_{\mu}T^{\mu\alpha}$ is expressed as
$\bar{n}_{\mu}T^{\mu\alpha}=-\frac{M^{2}}{2(P^{+})^{2}}n_{\mu}T^{\mu\alpha}$,
so that it is not an independent term. In the same way, the term
$\gamma_{\mu}T^{\mu\alpha}$ is not independent from $n_{\mu}T^{\mu\alpha}$ as
they are related with each other by
$n_{\mu}T^{\mu\alpha}\not{\bar{n}}=\gamma_{\mu}T^{\mu\alpha}$ in the lightcone
limit. On the other hand, $\gamma_{T\mu}T^{\mu\alpha}\not{\bar{n}}$ is an
independent term. Therefore, we have two independent terms associated with
$T^{\mu\alpha}$:

$\displaystyle
n_{\mu}T^{\mu\alpha}\not{\bar{n}}=\frac{P^{+}}{2M}S_{LT}^{\,\alpha}\not{\bar{n}},\
\
\gamma_{T\mu}T^{\mu\alpha}\not{\bar{n}}=\left(\frac{1}{3}S_{LL}\gamma_{T}^{\alpha}+\frac{1}{2}\gamma_{T\mu}S_{TT}^{\mu\alpha}\right)\not{\bar{n}}.$
(46)

As for independent $T^{\mu\nu}$ terms where the index $\alpha$ comes from
other vectors, we have two possibilities by noting the transverse components
$\epsilon_{T}^{\alpha\nu}$ ($\epsilon_{T}^{11}=-\epsilon_{T}^{22}=1$) and
$\gamma_{T}^{\alpha}$ as

$\displaystyle
n_{\mu}T^{\mu\nu}\epsilon_{T\nu}^{\,\alpha}i\gamma_{5}\not{\bar{n}}=\frac{P^{+}}{2M}\epsilon_{T}^{\alpha\mu}S_{LT\mu}i\gamma_{5}\not{\bar{n}},\
\
n_{\mu}T^{\mu\nu}n_{\nu}\gamma_{T}^{\alpha}\not{\bar{n}}=\frac{2}{3}S_{LL}\frac{(P^{+})^{2}}{M^{2}}\gamma_{T}^{\alpha}\not{\bar{n}}.$
(47)

From these considerations, the correlation function
$\Phi_{G}^{\alpha}(x_{1},x_{2})$ of Eq. (45) is generally expressed by four
terms with the tensor polarizations as

$\displaystyle\Phi_{G}^{\alpha}(x_{1},x_{2})=\frac{M}{2}\bigg{[}\,$
$\displaystyle
iS_{LT}^{\alpha}\,F_{G,LT}(x_{1},x_{2})-\epsilon_{T}^{\alpha\mu}S_{LT\mu}\gamma_{5}G_{G,LT}(x_{1},x_{2})$
$\displaystyle+iS_{LL}\gamma^{\alpha}H_{G,LL}^{\perp}(x_{1},x_{2})+iS_{TT}^{\alpha\mu}\gamma_{\mu}H_{G,TT}(x_{1},x_{2})\bigg{]}\not{\bar{n}}.$
(48)

All the terms in the right-hand side of Eq. (48) satisfy the parity and time-
reversal invariances. The functions $F_{G,LT}(x_{1},x_{2})$,
$G_{G,LT}(x_{1},x_{2})$, $H_{G,LL}^{\perp}(x_{1},x_{2})$,
$H_{G,TT}(x_{1},x_{2})$ are twist-3 multiparton distribution functions. The
correlation function $\Phi_{A}^{\alpha}$ is defined by using the gluon field
$A^{\alpha}$ instead of the field thensor $G^{+\alpha}$ in Eq. (45), and it is
expressed by real multiparton correlation functions. Therefore, the right-hand
side of Eq. (48) contains the $i$ factor due to the derivative
$\partial^{+}(\to iP^{+})$ in $G^{+\alpha}$.

We try to find properties of the multiparton distribution functions
$F_{G,LT}(x_{1},x_{2})$, $G_{G,LT}(x_{1},x_{2})$,
$H_{G,LL}^{\perp}(x_{1},x_{2})$, $H_{G,TT}(x_{1},x_{2})$ under the exchange of
variables $x_{1}$ and $x_{2}$ by using the Hermiticity condition. For example,
this Hermiticity relation is given for the first term of Eq. (48) as
$-iS_{LT}^{\nu}\not{\bar{n}}^{\dagger}F_{G,LT}(x_{1},x_{2})^{*}=iS_{LT}^{\nu}\gamma^{0}\not{\bar{n}}\gamma^{0}F_{G,LT}(x_{2},x_{1})$.
For the real function of $F_{G,LT}(x_{1},x_{2})$, it becomes
$F_{G,LT}(x_{1},x_{2})=-F_{G,LT}(x_{2},x_{1})$. In this way, the Hermiticity
condition is satisfied if the functions $F_{G,LT}$, $G_{G,LT}$,
$H_{G,LL}^{\perp}$, and $H_{G,TT}$ are real and they have the properties

$\displaystyle F_{G,LT}(x_{1},x_{2})$ $\displaystyle=-F_{G,LT}(x_{2},x_{1}),\
\ $ $\displaystyle G_{G,LT}(x_{1},x_{2})$
$\displaystyle=G_{G,LT}(x_{2},x_{1}),$ $\displaystyle
H_{G,LL}^{\perp}(x_{1},x_{2})$ $\displaystyle=H_{G,LL}^{\perp}(x_{2},x_{1}),\
\ $ $\displaystyle H_{G,TT}(x_{1},x_{2})$
$\displaystyle=H_{G,TT}(x_{2},x_{1}),$ (49)

under the exchange of the variables $x_{1}$ and $x_{2}$. If the multiparton
distribution functions are defined by taking the gluon field $A^{\alpha}$
instead of $G^{+\alpha}$ in Eq. (45) and the functions ($F_{A,LT}$,
$G_{A,LT}$, $H_{A,LL}^{\perp}$, and $H_{A,TT}$) are defined without the $i$
factors, these symmetric properties have opposite signs, namely
$F_{A,LT}(x_{1},x_{2})=F_{A,LT}(x_{2},x_{1})$,
$G_{A,LT}(x_{1},x_{2})=-G_{A,LT}(x_{2},x_{1})$,
$H_{A,LL}^{\perp}(x_{1},x_{2})=-H_{A,LL}^{\perp}(x_{2},x_{1})$, and
$H_{A,TT}(x_{1},x_{2})=-H_{A,TT}(x_{2},x_{1})$.

From Eqs. (45) and (48), the functions $F_{G,LT}$ and $G_{G,LT}$ are expressed
as the correlation matrix-element forms as

$\displaystyle
S_{LT}^{\nu}F_{G,LT}(x_{1},x_{2})=-\frac{i}{2M}\,g\\!\int\frac{d\xi_{1}^{-}}{2\pi}\frac{d\xi_{2}^{-}}{2\pi}\,e^{ix_{1}P^{+}\xi_{1}^{-}}\,e^{i(x_{2}-x_{1})P^{+}\xi_{2}^{-}}$
$\displaystyle\ \hskip
113.81102pt\times\langle\,P,T\left|\,\bar{\psi}(0)\,\not{n}n_{\mu}G^{\mu\nu}(\xi_{2}^{-})\psi(\xi_{1}^{-})\,\right|P,T\,\rangle,$
(50)

and

$\displaystyle
S_{LT}^{\nu}G_{G,LT}(x_{1},x_{2})=\frac{i}{2M}\,g\\!\int\frac{d\xi_{1}^{-}}{2\pi}\frac{d\xi_{2}^{-}}{2\pi}\,e^{ix_{1}P^{+}\xi_{1}^{-}}\,e^{i(x_{2}-x_{1})P^{+}\xi_{2}^{-}}$
$\displaystyle\ \hskip
113.81102pt\times\langle\,P,T\left|\,\bar{\psi}(0)\,i\gamma_{5}\not{n}n_{\mu}\tilde{G}^{\mu\nu}(\xi_{2}^{-})\psi(\xi_{1}^{-})\,\right|P,T\,\rangle.$
(51)

Equations (50) and (51) were obtained by the traces of
$(\Phi_{G}^{\alpha})_{ij}(x_{1},x_{2})(\not{n})_{ji}$ and
$(\Phi_{G}^{\alpha})_{ij}\\\ (x_{1},x_{2})(i\gamma_{5}\not{n})_{ji}$,
respectively. Then, the relations $\epsilon_{T}^{\alpha\mu}S_{LT\mu}=g^{\alpha
1}S_{LT}^{2}-g^{\alpha 2}S_{LT}^{1}$ and
$n_{\mu}\tilde{G}^{\beta\gamma}\epsilon_{\beta\gamma}^{\ \ \mu\alpha}\\\
=g^{\alpha 1}2\tilde{G}^{+2}-g^{\alpha 2}2\tilde{G}^{+1}$ were used to reach
to Eq. (51). Defining the variable $t$ by $\xi_{2}^{-}=t\xi_{1}^{-}$ and
calculating derivatives of $F_{G,LT}(x_{1},x_{2})$ and $G_{G,LT}(x_{1},x_{2})$
with respect to $x_{1}$ and $x_{2}$, we obtain the relation with the field
tensors $G^{\alpha\mu}$ and $\tilde{G}^{\alpha\mu}$ in Eq. (44). In this
calculation, the principal integral expressed by the sign function

$\displaystyle\frac{i}{\pi}{\cal
P}\int_{-\infty}^{\infty}d\omega\,\frac{1}{\omega}\,e^{-i\omega
z}=\epsilon(z)=\left\\{\begin{array}[]{l}+1\ \ \text{for }z>0\\\ -1\ \
\text{for }z<0\end{array}\right.,$ (54)

is used. The integral region of $x_{2}$ is from $-1$ to $1$; however, the
integrand with the distribution functions vanish in the region $|x_{2}|\geq
1$, so that the integral region is extended to the one from $-\infty$ to
$\infty$. Here, ${\cal P}$ indicates the principal integral. Using Eqs. (50)
and (51), we obtain the matrix element of the field tensors in the right-hand
side of Eq. (44) as

$\displaystyle\\!\\!\\!\\!\int\frac{d(P\cdot\xi)}{2\pi}\,e^{ix_{1}P\cdot\xi}\,\bigg{\langle}\,P,T\,\bigg{|}\,g\int^{1}_{0}dt\,\bar{\psi}(0)\bigg{[}i\left(t-\frac{1}{2}\right)G^{\alpha\mu}(t\xi)-\frac{1}{2}\gamma_{5}\,\tilde{G}^{\alpha\mu}(t\xi)\bigg{]}\xi_{\mu}\not{\xi}\psi(\xi)\,\bigg{|}\,P,T\,\bigg{\rangle}_{\xi^{+}=\vec{\xi}_{T}=0}$
$\displaystyle\\!\\!\\!=-2MS_{LT}^{\nu}{\cal
P}\int_{-1}^{1}dx_{2}\frac{1}{x_{1}-x_{2}}\bigg{[}\frac{\partial}{\partial
x_{1}}\left\\{F_{G,LT}(x_{1},x_{2})+G_{G,LT}(x_{1},x_{2})\right\\}$
$\displaystyle\ \hskip 128.0374pt+\frac{\partial}{\partial
x_{2}}\left\\{F_{G,LT}(x_{2},x_{1})+G_{G,LT}(x_{2},x_{1})\right\\}\bigg{]}.$
(55)

In this way, it becomes possible to separate the twist-3 effects expressed by
the multiparton distribution functions from the twist-2 ones, which is
essential in deriving the WW-like relation for the distribution function
$f_{LT}$.

The multiparton correction functions $\bar{\Phi}_{G}^{\alpha}$ and
$\Phi_{G}^{\alpha,C}$ are defined for antiquarks in the same way with Eqs.
(23), (24), and (25) by considering the definition for quarks in Eq. (45). The
only extra factor is the gluon tensor $G^{+\alpha}$. Noting the charge-
conjugation property for the gluon field $A_{\mu}^{C}=-A_{\mu}$, we find the
relation which has the opposite sign of Eq. (25) as

$\displaystyle\Phi_{G}^{\alpha,C}(x_{1},x_{2},P,T)=C\,[\bar{\Phi}_{G}^{\alpha}(x_{1},x_{2},P,T)]^{\cal
T}C^{\dagger}.$ (56)

Therefore, the relations between the multiparton correlation functions for
antiquarks have different signs from Eq. (28) as

$\displaystyle\Phi_{G}^{\alpha,C[\Gamma]}(x_{1},x_{2})=\left\\{\begin{array}[]{l}-\bar{\Phi}_{G}^{\alpha,[\Gamma]}(x_{1},x_{2})=+\Phi_{G}^{\alpha,[\Gamma]}(-x_{2},-x_{1})\
\ \text{for }\Gamma=\gamma^{\mu},\ \sigma^{\mu\nu},\
i\gamma_{5}\sigma^{\mu\nu}\\\
+\bar{\Phi}_{G}^{\alpha,[\Gamma]}(x_{1},x_{2})=-\Phi_{G}^{\alpha,[\Gamma]}(-x_{2},-x_{1})\
\ \text{for }\Gamma=1,\ i\gamma_{5},\
\gamma_{5}\gamma^{\mu}\end{array}\right..$ (59)

Then, the multiparton distribution functions for antiquark are obtained by
noting these correlation-function relations as

$\displaystyle\bar{F}_{G,LT}(x_{1},x_{2})$
$\displaystyle=F_{G,LT}(-x_{2},-x_{1}),\ \ $
$\displaystyle\bar{G}_{G,LT}(x_{1},x_{2})$
$\displaystyle=-G_{G,LT}(-x_{2},-x_{1}),$
$\displaystyle\bar{H}_{G,LL}^{\perp}(x_{1},x_{2})$
$\displaystyle=H_{G,LL}^{\perp}(-x_{2},-x_{1}),\ \ $
$\displaystyle\bar{H}_{G,TT}(x_{1},x_{2})$
$\displaystyle=H_{G,TT}(-x_{2},-x_{1}).$ (60)

Therefore, the multiparton distribution functions for antiquarks are described
by the functions for quarks at negative $x_{1}$ and $x_{2}$; however, they are
also expressed by the functions $\bar{F}_{G,LT}$, $\bar{G}_{G,LT}$,
$\bar{H}_{G,LL}^{\perp}$, and $\bar{H}_{G,TT}$. Similar relations were given
in the multiparton distribution functions for the spin-1/2 nucleons in Ref.
Kanazawa-2016 .

###  3.3 Twist-2 relation and sum rule

After all these preparations on the tensor-polarized PDFs of spin-1 hadrons,
we are now ready to drive a twist-2 relation and a sum rule for the twist-3
distribution function $f_{LT}$. First, the matrix element of the nonlocal
operator
$\bar{\psi}(0)(\partial^{\mu}\gamma^{\alpha}-\partial^{\alpha}\gamma^{\mu})\psi(\xi)$
with the transverse index $\alpha$ was expressed in terms of the tensor-
polarized PDFs $f_{1LL}(x)$ and $f_{LT}(x)$ in Eq. (33). Second, this nonlocal
operator was expressed by the field tensor $G^{\alpha\mu}$ and its dual one
$\tilde{G}^{\alpha\mu}$ in Eq. (44). Third, the matrix element of the field
tensors was given by the twist-3 multiparton distribution functions $G$ and
$F$ in Eq. (55). Combining Eqs. (33), (44), and (55), we obtain

$\displaystyle
x\,\frac{df_{LT}(x)}{dx}=-\frac{3}{2}f_{1LL}(x)-f_{LT}^{(HT)}(x),$ (61)

where the twist-3 multiparton-distribution part is defined as

$\displaystyle f_{LT}^{(HT)}(x)=-{\cal
P}\int_{-1}^{1}dy\frac{1}{x-y}\bigg{[}\,$
$\displaystyle\frac{\partial}{\partial
x}\left\\{F_{G,LT}(x,y)+G_{G,LT}(x,y)\right\\}$
$\displaystyle+\frac{\partial}{\partial
y}\left\\{F_{G,LT}(y,x)+G_{G,LT}(y,x)\right\\}\bigg{]}.$ (62)

Here, $HT$ indicates the higher twist. Integrating Eq. (61) over $x$, we
obtain

$\displaystyle
f_{LT}(x)=\frac{3}{2}\int^{\epsilon(x)}_{x}\frac{dy}{y}f_{1LL}(y)+\int^{\epsilon(x)}_{x}\frac{dy}{y}f_{LT}^{(HT)}(y),$
(63)

by using the sign function of Eq. (54). Namely, the integral is from $x$ to
$1$ if $x$ is positive for the quark distribution, and it is from $x$ to $-1$
if $x$ is negative for the antiquark distribution. In Eq. (63), we obtained
the full decomposition $f_{LT}$ into the twist-2 term and the twist-3
multiparton distribution functions, as investigated for $g_{2}$ and $h_{L}$ in
Refs. jj-1991 ; chiral-odd-twist-3 ; kt-1999 .

In Sec. 2, the structure function $g_{1}$ is given by the function $g_{1L}$ as
$g_{1}(x)=[g_{1L}(x)+g_{1L}(-x)]/2$ for describing both the quark and
antiquark distributions in the $x$ range of $0\leq x\leq 1$. In the same way,
we define the distribution functions $f_{1LL}^{+}$, $f_{LT}^{+}$, and
$f_{LT}^{(HT)+}$ as

$\displaystyle f^{+}(x)\equiv f(x)+\bar{f}(x)=f(x)-f(-x),\ \ f=f_{1LL},\
f_{LT},\ f_{LT}^{(HT)},\ \ x>0.$ (64)

Here, these functions are given for a single flavor. From Eqs. (63) and (64),
we obtain a relation for $f_{LT}^{+}(x)$ and $f_{1LL}^{+}(x)$ as

$\displaystyle
f_{LT}^{+}(x)=\frac{3}{2}\int^{1}_{x}\frac{dy}{y}\,f_{1LL}^{+}(y)+\int^{1}_{x}\frac{dy}{y}\,f_{LT}^{(HT)+}(y).$
(65)

Since the function $f_{LT}^{(HT)+}(y)$ indicates the twist-3 effects as given
by the multiparton correlation functions, this equation indicates that the
twist-2 part of the function $f_{LT}^{+}(x)$ is expressed by the integral of
$f_{1LL}^{+}(x)$. In this way, the twist-3 function $f_{LT}^{+}(x)$ is
expressed by the twist-2 function and the remaining twist-3 one. If the
twist-3 part is neglected, the relation becomes

$\displaystyle
f_{LT}^{+}(x)=\frac{3}{2}\int^{1}_{x}\frac{dy}{y}\,f_{1LL}^{+}(y).$ (66)

This equation corresponds to the WW relation in Eq. (12). It should be noted
that the structure function $b_{1}$ or the tensor-polarized PDF $\delta_{T}q$
is given by the $f_{1LL}$ as
$-(3/2)f_{1LL}^{+}=b_{1}^{q}+b_{1}^{\bar{q}}=\delta_{T}q+\delta_{T}\bar{q}$
ks-tmd-2021 . This new relation suggests that the tensor-polarized
distribution function $f_{LT}(x)$ is expressed by the integral of $f_{1LL}(x)$
or $b_{1}(x)$ if higher-twist effects are ignored. If the function
$f_{2LT}(x)$ is defined by

$\displaystyle f_{2LT}(x)\equiv\frac{2}{3}f_{LT}(x)-f_{1LL}(x),$ (67)

Eq. (65) becomes

$\displaystyle
f_{2LT}^{+}(x)=-f_{1LL}^{+}(x)+\int^{1}_{x}\frac{dy}{y}\,f_{1LL}^{+}(y)+\frac{2}{3}\int^{1}_{x}\frac{dy}{y}\,f_{LT}^{(HT)}(y).$
(68)

The distribution function $f_{2LL}^{+}(x)$ is expressed by the twist-2 and
twist-3 terms. If the twist-3 term is neglected, we obtain a relation

$\displaystyle
f_{2LT}^{+}(x)=-f_{1LL}^{+}(x)+\int^{1}_{x}\frac{dy}{y}\,f_{1LL}^{+}(y),$ (69)

which is analogous to the WW relation for $g_{1}$ and $g_{2}$ in Eq. (12).
Furthermore, integrating this equation, we obtain

$\displaystyle\int_{0}^{1}dx\,f_{2LT}^{+}(x)=0,$ (70)

which is analogous to the BC sum rule. These relation and sum rule are useful
in studying the tensor-polarized distribution function $f_{2LT}^{+}(x)$ (or
original $f_{LT}(x)$) as the WW relation and the BC sum rule provide strong
constraints on determining the structure function $g_{2}(x)$ for the nucleons.
Furthermore, considering the sum rule based on the parton model $\int
dxb_{1}(x)=0$ (or $\int dxf_{1LL}^{+}(x)=0$ by the notation in this paper)
b1-sum ; Airapetian:2005cb , which is valid if tensor-polarized antiquark
distributions vanish, we have the sum rule for the twist-3 function $f_{LT}$
itself as

$\displaystyle\int_{0}^{1}dx\,f_{LT}^{+}(x)=0.$ (71)

Equations (70) and (71) could be affected by the small-$x$ behavior of the
distribution functions in the same way with the BC sum rule for $g_{2}$. We
proved these twist-2 relations in the tree level and have not discussed
perturbative QCD corrections. At this stage, it is not obvious whether these
relations are satisfied in the structure-function level by including
coefficient functions as investigated in $g_{2}$ Braun-2001 . We leave this
issue for a future project.

The tensor-polarized structure functions of spin-1 hadrons and nuclei have
been investigated since the end of 1980’s. Due to lack of experimental
measurements except for the HERMES experiment for $b_{1}$, theoretical
developments of this field were rather slow in comparison with the spin
physics of spin-1/2 nucleons. However, we believe that bright future is ahead
of us in the sense that the tensor-polarized structure function $b_{1}$ and
the gluon transversity, which are specific observables in the spin-1 hadrons,
will be measured at JLab in the middle of 2020’s jlab-b1 ; jlab-gluon-trans
and such experiments will be proposed at Fermilab Fermilab-dy  as the proton-
deuteron Drell-Yan process. The NICA facility will have the polarized-deuteron
beam in the near future nica , so that they could focus their studies on the
structure functions of the spin-1 deuteron. In addition, there are EIC
projects in US and China eic ; eicc  to investigate the structure functions of
the spin-1 hadrons and nuclei in 2030’s. In the JLab measurements, the scale
$Q^{2}$ is not large in general, which enables to probe the twist-3 structure
functions such as $g_{2}$. In the same way, higher-twist tensor-polarized
structure functions could become accessible at JLab or future high-intensity
accelerator facilities. In this sense, our previous studies on general twist-3
and twist-4 distribution functions ks-tmd-2021  as well as this work should
become useful in future. In particular, the new twist-2 relation of Eq. (69)
[or (66)] and the sum rule of Eq. (70) [also Eq. (71) in addition] could
become important for constraining the twist-3 function $f_{2LT}(x)$ or
$f_{LT}(x)$, although the integral relations are always difficult to be tested
due to the experimental inaccessibility at small $x$. The tensor-polarized
PDFs and structure functions will be measured at JLab, Fermilab, and NICA at
relatively large $x$, and the small-$x$ part will be investigated at EIC and
other high-energy lepton facilities, for example, by a possible fixed target
project of a linear or circular lepton collider. According to the theoretical
estimate on higher-twist tensor-polarized structure functions in a few-GeV
$Q^{2}$ region b1-convolution , which is the typical kinematical region of
JLab, they are not be much smaller than the leading-twist ones, in the similar
way to the 15–40% breaking of the WW relation for $g_{2}$ accardi-2009 .
Therefore, the understanding of the higher-twist structure functions are
valuable also in determining the leading-twist functions from actual
measurements in future. At this stage, the available information is very
limited for the tensor-polarized PDFs and structure functions. However, we
hope to make progress on numerical studies of them by considering the sum
rules derived in this work.

##  4 Summary

There are tensor-polarized PDFs and structure functions for spin-1 hadrons. In
this work, we derived a new useful twist-2 relation

$\displaystyle
f_{LT}(x)=\frac{3}{2}\int^{\epsilon(x)}_{x}dy\frac{f_{1LL}(y)}{y}+\int^{\epsilon(x)}_{x}dy\frac{f_{LT}^{(HT)}(y)}{y},$

where $\epsilon(x)=1$ ($-1$) at $x>0$ ($x<0$), for the twist-3 distribution
function $f_{LT}(x)$ and twist-2 one $f_{1LL}(x)$. This equation indicates the
quark and antiquark distributions at $x>0$ and $x<0$, respectively. Defining
the plus function by the quark and antiquark distributions as
$f^{+}(x)=f(x)+\bar{f}(x)$ and neglecting the higher-twist term, this equation
is written as

$\displaystyle
f_{LT}^{+}(x)=\frac{3}{2}\int^{1}_{x}\frac{dy}{y}\,f_{1LL}^{+}(y).$ (72)

Namely, the twist-2 part of $f_{LT}(x)$ is expressed by the integral of
$f_{1LL}(x)$. Since the integrand is given by the structure function
$b_{1}^{q+\bar{q}}=-(3/2)f_{1LL}^{+}$, the twist-2 of $f_{LT}(x)$ is expressed
by the function $b_{1}$. Using the function $f_{2LL}$ defined by
$f_{2LL}=\frac{2}{3}f_{LT}-f_{1LL}$, we obtained

$\displaystyle
f_{2LT}^{+}(x)=-f_{1LL}^{+}(x)+\int^{1}_{x}\frac{dy}{y}f_{1LL}^{+}(y).$

This relation is similar to the Wandzura-Wilczek relation for the polarized
structure functions $g_{1}$ and $g_{2}$ for the spin-1/2 nucleons. It is
useful in the sense that the twist-2 part is constrained and the separation of
higher-twist effects become clear. In addition, we showed that the sum rule

$\displaystyle\int_{0}^{1}dx\,f_{2LT}^{+}(x)=0,$

exists for $f_{2LL}$, and it constrains the overall $x$-dependent functional
form of $f_{2LL}$. It is similar to the Burkhardt-Cottingham sum rule for
$g_{2}$. Furthermore, if the parton-model sum rule $\int dxf_{1LL}^{+}(x)=0$
($\int dxb_{1}^{q+\bar{q}}(x)=0$) is applied in the case where the tensor-
polarized antiquark distributions vanish, it led to another sum rule

$\displaystyle\int_{0}^{1}dx\,f_{LT}^{+}(x)=0.$

All these relations are valuable in investigating the tensor-polarized PDFs
and structure functions in future. For specifying twist-3 terms in deriving
these relations, we explained that four twist-3 multiparton distribution
functions

$\displaystyle F_{LT}(x_{1},x_{2}),\ \ G_{LT}(x_{1},x_{2}),\ \
H_{LL}^{\perp}(x_{1},x_{2}),\ \ H_{TT}(x_{1},x_{2}),$

exist for tensor-polarized spin-1 hadrons. These multiparton distribution
functions are also interesting for probing multiparton correlations in spin-1
hadrons.

###### Acknowledgements.

The authors thank N. Sato for suggestions. S. Kumano was partially supported
by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for
Scientific Research (KAKENHI) Grant Number 19K03830. Qin-Tao Song was
supported by the National Natural Science Foundation of China under Grant
Number 12005191 and the Academic Improvement Project of Zhengzhou University.

## References

  * (1) L. L. Frankfurt and M. I. Strikman, _High-momentum-transfer processes with polarized deuterons_ , _Nucl. Phys._ A 405 (1983) 557.
  * (2) P. Hoodbhoy, R. L. Jaffe, and A. Manohar, _Novel effects in deep inelastic scattering from spin-one hadrons_ , _Nucl. Phys._ B 312 (1989) 571.
  * (3) F. E. Close and S. Kumano, _Sum rule for the spin-dependent structure function $b_{1}(x)$ for spin-one hadrons_, _Phys. Rev._ D 42 (1990) 2377;
S. Kumano, _Tensor-polarized structure functions: Tensor structure of deuteron
in 2020’s_ , _J. Phys.: Conf. Series_ 543 (2014) 012001.

  * (4) A. Airapetian et al. (HERMES Collaboration), _Measurement of the Tensor Structure Function $b_{1}$ of the Deuteron_, _Phys. Rev. Lett._ 95 (2005) 242001.
  * (5) S. Kumano, _Tensor-polarized quark and antiquark distribution functions in a spin-one hadron_ , _Phys. Rev._ D 82 (2010) 017501.
  * (6) J.-P. Chen et al., _The Deuteron Tensor Structure Function $b_{1}$_, _Proposal to Jefferson Lab PAC-38_ , PR12-11-110 (2011).
  * (7) W. Cosyn, Yu-Bing Dong, S. Kumano, and M. Sargsian, _Tensor-polarized structure function $b_{1}$ in the standard convolution description of the deuteron_, _Phys. Rev._ D 95 (2017) 074036.
  * (8) R. L. Jaffe and A. Manohar, _Nuclear gluonometry_ , _Phys. Lett._ B 223 (1989) 218;
M. Jones et al., _Search for Exotic Gluonic States in the Nucleus_ , _A Letter
of Intent to Jefferson Lab PAC 44_ , LOI12-16-006 (2016).

  * (9) J. P. Ma, C. Wang, and G. P. Zhang, _Azimuthal Dependence of DIS with Spin-1 Target_ , arXiv:1306.6693, unpublished (2013).
  * (10) R. Abdul Khalek et al., _Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report_ , arXiv:2103.05419 (2021).
  * (11) D. P. Anderle et al., _Electron-Ion Collider in China_ , _Front. Phys._ 16 (2021) 64701.
  * (12) S. Hino and S. Kumano, _Structure functions in the polarized Drell-Yan processes with spin-1/2 and spin-1 hadrons. I. General formalism_ , _Phys. Rev._ D 59 (1999) 094026;
S. Hino and S. Kumano, _Structure functions in the polarized Drell-Yan
processes with spin-1/2 and spin-1 hadrons. II. Parton model_ , _Phys. Rev._ D
60 (1999) 054018.

  * (13) S. Kumano and Qin-Tao Song, _Theoretical estimate on tensor-polarization asymmetry in proton-deuteron Drell-Yan process_ , _Phys. Rev._ D 94 (2016) 054022.
  * (14) S. Kumano, Qin-Tao Song, _Gluon transversity in polarized proton-deuteron Drell-Yan process_ , _Phys. Rev._ D 101 (2020) 054011;
S. Kumano, Qin-Tao Song, _Deuteron polarizations in the proton-deuteron Drell-
Yan process for finding the gluon transversity_ , _Phys. Rev._ D 101 (2020)
094013.

  * (15) D. Geesaman et al., _Letter of Intent for a Drell-Yan experiment with a polarized proton target_ , _Proposal to Fermilab PAC_ , P-1039 (2013).
  * (16) D. Keller, D. Crabb, and D. Day, _Enhanced tensor polarization in solid-state targets_ , _Nucl. Instrum. Meth._ A 981 (2020) 164504.
  * (17) For the Spin Physics Detector (SPD) project at NICA, see http://spd.jinr.ru/;
A. Arbuzov et al., _On the physics potential to study the gluon content of
proton and deuteron at NICA SPD_ , _Prog. Nucl. Part. Phys._ 119 (2021)
103858.

  * (18) G. A. Miller, _Pionic and hidden-color, six-quark contributions to the deuteron $b_{1}$ structure function_, _Phys. Rev._ C 89 (2014) 045203.
  * (19) M. Nzar and P. Hoodbhoy, _Estimation of the double-helicity-flip deuteron structure function_ , _Phys. Rev._ D 45 (1992) 2264.
  * (20) S. Kumano and Qin-Tao Song, _Transverse-momentum-dependent parton distribution functions up to twist 4 for spin-1 hadrons_ , _Phys. Rev._ D 103 (2021) 014025.
  * (21) A. Bacchetta and P. Mulders _Deep inelastic leptoproduction of spin-one hadrons_ , _Phys. Rev._ D 62 (2000) 114004; A. Bacchetta, _Probing the transverse spin of quarks in deep inelastic scattering_ , Ph. D. thesis, Free University Amsterdam (2002), arXiv:hep-ph/0212025 (2002); T. van Daal, _Mapping the internal structure of hadrons through color and spin effects_ , Ph. D. thesis, University of Groningen (2018), arXiv:1812.07336 (2018); S. Cotogno, _Polarized partons in hadrons at high energy_ , Ph. D. thesis, Free University Amsterdam (2018).
  * (22) S. Wandzura and F. Wilczek, _Sum rules for spin-dependent electroproduction- test of relativistic constituent quarks_ , _Phys. Lett._ B 72 (1977) 195.
  * (23) H. Burkhardt and W. N. Cottingham, _Sum rules for forward virtual compton scattering_ , _Annals Phys._ 56 (1970) 453.
  * (24) J. Kodaira and K. Tanaka, _Polarized Structure Functions in QCD_ , _Prog. Theor. Phys._ 101 (1999) 191.
  * (25) J. Blümlein and N. Kochelev, _On the twist-2 and twist-3 contributions to the spin-dependent electroweak structure functions_ , _Nucl. Phys._ B 498 (1997) 285; J. Blümlein and A. Tkabladze, _Target mass corrections for polarized structure functions and new sum rules_ , _Nucl. Phys._ B 553 (1999) 427.
  * (26) V. M. Braun, G. P. Korchemsky, and A. N. Manashov, _Evolution equation for the structure function $g_{2}(x,Q^{2})$_, _Nucl. Phys._ B 603 (2001) 69.
  * (27) M. Anselmino, A. Efremov, and E. Leader, _The theory and phenomenology of polarized deep inelastic scattering_ , _Phys. Rept._ 261 (1995) 1.
  * (28) B. Lampe and E. Reya, _Spin physics and polarized structure functions_ , _Phys. Rept._ 332 (2000) 1.
  * (29) E. Leader, _Spin in Particle Physics_ (Cambridge University Press, 2001).
  * (30) R. L. Jaffe and X. Ji, _Studies of the transverse spin-dependent structure function $g_{2}(x,Q^{2})$_, _Phys. Rev._ D 43 (1991) 724.
  * (31) R. L. Jaffe, _Spin, Twist and Hadron Structure in Deep Inelastic Processes_ , arXiv:hep-ph/9602236 (1996), in _The Spin Structure of the Nucleon: International School of Nucleon Structure_ , edited by B. Frois et al. (World Scientific, 1997).
  * (32) C. Itzkyson and J.-B. Zuber, _Quantum Field Theory_ (McGraw-Hill Inc., 1980).
  * (33) R. L. Jaffe and X. Ji, _Chiral-odd parton distributions and polarized Drell-Yan process_ , _Phys. Rev. Lett._ 67 (1991) 552;
R. L. Jaffe and X. Ji, _Chiral-odd parton distributions and Drell-Yan
processes_ , _Nucl. Phys._ B 375 (1992) 527.

  * (34) A. V. Belitsky and D. Müller, _Scale dependence of the chiral-odd twist-3 distributions $h_{L}(x)$ and $e(x)$_, _Nucl. Phys._ B 503 (1997) 279; A. V. Belitsky, _Leading-order analysis of the twist-3 space- and time-like cut vertices in QCD_ , arXiv:hep-ph/9703432 (1997).
  * (35) V. Barone and R. G. Ratcliffe, _Transverse Spin Physics_ (World Scientific, Singapore, 2003).
  * (36) A. Accardi, A. Bacchetta, W. Melnitchouk, and M. Schlegel, _What can break the Wandzura-Wilczek relation?_ , _JHEP_ 11 (2009) 093;
N. Sato, W. Melnitchouk, S. E. Kuhn, J. J. Ethier, and A. Accardi, _Iterative
Monte Carlo analysis of spin-dependent parton distributions_ , _Phys. Rev._ D
93 (2016) 074005.

  * (37) A. Deur, S. J. Brodsky, and G. F de Teramond, _The spin structure of the nucleon_ , _Rep. Prog. Phys._ 82 (2019) 076201.
  * (38) I. I. Balitsky and V. M. Braun, _Evolution equations for QCD string operators_ , _Nucl. Phys._ B 311 (1988/89) 541.
  * (39) T. Rogers, _Transverse moments of TMD parton densities and ultraviolet divergences_ , _Mod. Phys. Lett._ A 35 (2020) 37.
  * (40) P. J. Mulders, Lectures at the 17th Taiwan Nuclear Physics School, Institute of Physics, Academia Sinica, Taipei, August 25-28, 2014, see the background material in http://www.nat.vu.nl/$\sim$mulders/; R. D. Tangerman, _Higher-twist correlations in polarized hadrons_ , Ph.D. thesis, Free University Amsterdam (1996), https://inis.iaea.org/collection
/NCLCollectionStore_Public/28/015/28015381.pdf.

  * (41) P. Ball and V. M. Braun, _Higher twist distribution amplitudes of vector mesons in QCD: twist-4 distributions and meson mass corrections_ , _Nucl. Phys._ B 543 (1999) 201.
  * (42) S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, _Quantum chromodynamics and other field theories on the light cone_ , _Phys. Rept._ 301 (1998) 299.
  * (43) K. Kanazawa, Y. Koike, A. Metz, D. Pitonyak, and M. Schlegel, _Operator constraints for twist-3 functions and Lorentz invariance properties of twist-3 observables_ , _Phys. Rev._ D 93 (2016) 054024.


# Narrow quadrupolar surface lattice resonances and band reversal in vertical
metal-insulator-metal gratings

Xinyu Fang1,2,4,6, Lei Xiong1,3,4,6, Jianping Shi2, Hongwei Ding3, and
Guangyuan Li1,4,5,∗

###### Abstract

We report narrow quadrupolar surface lattice resonances (SLRs) under normal
incidence, and the observation, for the first time, of the band reversal
effect of SLRs supported by a vertical metal-insulator-metal nanograting,
which is embedded in a homogeneous dielectric environment. Simulation results
show that under normal incidence, quadrupolar SLR with linewidth of 1 nm and
high quality factor of 979 can be excited in the near-infrared regime, and
that under oblique incidence, out-of-plane dipolar SLRs of relatively large
quality factors ($\geq 150$) can be launched. By varying the incidence angle,
the SLR wavelength can be continuously tuned over an extremely broadband range
of 750 nm, covering most of the near-infrared regime, and the quality factor
decreases exponentially. Remarkably, the resonance lineshape can also be
dynamically tuned from an asymmetric Fano-shaped dip to a peak, a dip/peak
pair, and a perfect symmetric Lorentzian peak, suggesting the appearance of
the band reversal effect. We expect the high-$Q$ SLRs with broadband
tunability and tunable lineshapes will find potential applications in enhanced
nanoscale light-matter interactions in nanolasers, nonlinear optics and
sensing.

1CAS Key Laboratory of Human-Machine Intelligence-Synergy Systems, Shenzhen
Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen
518055, China

2College of Physics and Electronic Technology, Anhui Normal University, Wuhu
241000, China

3School of Information Science and Engineering, Yunnan University, Kunming
650500, China

4Guangdong-Hong Kong-Macao Joint Laboratory of Human-Machine Intelligence-
Synergy Systems, Chinese Academy of Sciences, Shenzhen Institute of Advanced
Technology, Shenzhen 518055, China

5Shenzhen College of Advanced Technology, University of Chinese Academy of
Sciences, Shenzhen 518055, China

6These authors contributed equally.

*Corresponding author<EMAIL_ADDRESS>
##  1 Introduction

Plasmonic surface lattice resonances (SLRs) supported by periodic arrays of
metallic nanostructures have many appealing merits, such as spectrally narrow
optical responses, large quality factors, strong field enhancements extended
over large volumes, and large wavelength tunability, and thus have emerged as
a new platform for enhancing light-matter interactions on the nanoscale [1, 2,
3, 4]. Over the years, SLRs have found attractive applications in nanolasers
[5, 6, 7], nonlinear optics [8, 9, 10], and optical sensing [11, 12, 13].

In order to achieve high quality factors, quadrupolar SLRs have received
increasing attention recently. Burrows and Barnes [14] made use of large
nanoparticles and symmetry-broken conditions (oblique incidences or asymmetric
structures) to achieve quadrupolar SLRs. Humphrey et al. [15] employed arrays
of asymmetric disc dimers and controlled the relative strength of the dipolar
and the quadrupolar SLRs through the degree of asymmetry. Yang et al. [16]
obtained both dipolar and quadrupolar SLRs in aluminum nanoparticle arrays
with diameter larger than 120 nm. However, these quadrupolar SLRs are weakly
excited on a dipolar SLR background and have relatively small quality factors.

In order to tackle this problem, quite recently, the authors [17] proposed a
horizontal metal-insulator-metal (MIM) nanograting for achieving high-$Q$ out-
of-plane quadrupolar SLR without the dipolar SLR background. We showed that
the quality factor obtained under oblique incidence can reach up to 1036 in
the near-infrared, which is three times of that of the out-of-plane dipolar
SLR excited in the vertical MIM nanograting under the same condition. Although
these results are encouraging, a question arises: can we excite high-$Q$
quadrupolar SLR without the dipolar SLR background under normal incidence? We
are interested in normal incidence since it is the most convenient in
experiments.

In this work, we will show that high-$Q$ quadrupolar SLR without dipolar SLR
background can be excited in the vertical MIM nanograting under normal
incidence. Simulation results will show that the quality factor can reach up
to 979 in the near-infrared regime, which is comparable to that of the out-of-
plane quadrupolar SLR excited in the horizontal MIM grating under oblique
incidence. We will show that, as the incidence angle increases, high-$Q$ out-
of-plane dipolar SLRs can be excited, and the SLR wavelength can be tuned over
an extremely broadband range, covering 979 nm to 1735 nm. Interestingly,
results will show that the SLR quality factor decreases exponentially with the
incidence angle but remains above 150. Surprisingly, we will observe, for the
first time, the band reversal effect in SLRs, which manifests as the
transition of resonance lineshapes from an asymmetric Fano-shape dip to a
dip/peak pair, a peak, and a symmetric Lorentzian peak.

##  2 Simulation setup

Figure 1 illustrates the 1D vertical MIM nanograting under study, which has
period $\Lambda$ in the $x$ direction and is embedded in a homogeneous
dielectric environment with refractive index $n_{0}$. The nanograting unit
cell is composed of three gold, silica, and gold nanoridges from the top to
the bottom with width of $w$, and heights of $h_{mt}$, $h_{d}$ and $h_{mb}$,
respectively. The structure is illuminated by plane wave with unitary electric
field amplitude ($|E_{0}|=1$) at the incidence angle of $\theta$. We restrict
ourselves to transverse magnetic (TM) polarization only.

Figure 1: Schematics of the vertical MIM nanograting with period $\Lambda$
supports (a) quadrupolar SLR under normal incidence and (b) out-of-plane
dipolar SLR under oblique incidence of incidence angle $\theta$ and TM
polarization. The nanograting unit cell is composed of vertical gold-silica-
gold nanoridges of width $w$ and heights $h_{\rm mt}$, $h_{\rm d}$ and $h_{\rm
mb}$ from the top to the bottom. Red arrows in gold nanoridges indicate the
directions of the induced electric polarizations.

The zeroth-order reflectance and transmittance spectra, as well as the near-
field distributions of the MIM nanograting were simulated with a home-
developed package for fully vectorial rigorous coupled-wave analysis (RCWA),
which was developed following [18, 19]. In the simulations, we adopted 301
retained orders, which are tested to be large enough to reach the convergence
regime [17]. All the calculations were performed with $n_{0}=n_{\rm d}=1.45$
($n_{0}$ is for the silica substrate and the index-matching oil, and $n_{\rm
d}$ is for the silica nanoridge), $\Lambda=675$ nm, $w=165$ nm, and $h_{\rm
mb}=h_{\rm d}=h_{\rm mt}=100$ nm. Wavelength-dependent permittivities of gold
were taken from Johnson and Christy [20].

The MIM nanograting can be fabricated using the state-of-the-art
nanofabrication processes. The thin layers of gold, silica and gold films are
first deposited on the silica substrate in sequence. A photoresist is spin-
coated on the top and patterned using the electron beam lithography. The
pattern is then transferred to the gold-silica-gold films using the ion beam
etching. Finally, the remaining photoresist is removed, resulting in the
vertical MIM nanograting under study.

##  3 Results and discussion

###  3.1 Spectra and near-fields for quadrupolar SLRs

We first investigate the spectral and near-field responses of the MIM
nanograting under normal incidence ($\theta=0^{\circ}$). Figure 2(a) shows
that there exists an extremely narrow Fano-shaped dip (or peak) locating at
$\lambda_{0}=979.4$ nm with a small linewidth of $\Delta\lambda=1.0$ nm in the
reflectance (or transmittance) spectrum. Correspondingly, the quality factor
is estimated to be $Q=\lambda_{0}/\Delta\lambda=979.4$. This value is
comparable to that ($Q=1036$) of the out-of-plane quadrupolar SLR supported by
the horizontal MIM nanograting under oblique incidence [17].

As a reference, we also study the scenario of the oblique incidence of
$\theta=45^{\circ}$, which was done in our previous work [17]. Figure 2(b)
shows that the reflectance (or transmittance) spectrum exhibits a relatively
broad Fano-shaped peak (or dip) locating at 1678.8 nm. The corresponding
linewidth is estimated to be 10.6 nm and the quality factor is calculated to
be $Q=158.4$. Compared with our previous work [17], the reflectance and
transmittance spectra are similar, and the quality factors are also
comparable.

Figure 2: Simulated zeroth-order reflectance (red) and transmittance (blue)
spectra of the vertical MIM nanograting under (a) normal incidence and (b)
oblique incidence of $\theta=45^{\circ}$.

According to our previous study [17], the asymmetric spectral features in
Figure 2 should originate from the Fano-type interference between the
localized surface plasmon resonance (LSPR) and the RA, resulting in SLRs. In
order to understand the large difference on the spectral linewidth and the
quality factor between the two resonances under normal incidence and under the
oblique incidence of $\theta=45^{\circ}$, we plot the simulated near-field
electric field $|E|^{2}$ and Poynting vector $|S|$ maps at the resonance
wavelengths of 979.4 nm and 1678.8 nm. Figures 3(a)(c) show that the electric
fields are greatly enhanced over extended volumes, which are typical
characteristics of SLRs, that two quadrupoles are excited in the top and
bottom gold nanoridges under normal incidence, and that two out-of-plane
dipoles in phase are excited under the oblique incidence. The local electric
filed enhancement reaches 210 and 60 for normal incidence and the oblique
incidence of $\theta=45^{\circ}$, respectively. Under normal incidence, the
Poynting vector map in Figure 3(b) shows circulating energy flows around the
gold nanoridges, suggesting standing waves formed by the interference of two
counterpropagating surface waves. Under the oblique incidence of
$\theta=45^{\circ}$, however, the energy flux propagates along the $-x$
direction. Therefore, the quadrupolar SLR is excited under normal incidence
and the out-of-plane dipolar SLR is launched under the oblique incidence of
$\theta=45^{\circ}$.

Figure 3: (a)(c) Simulated electric field $|E|^{2}$ and (b)(d) Poynting vector
$|S|$ distributions (color for intensity and arrows for directions) at
resonance wavelengths of (a)(b) 979.4 nm under normal incidence and (c)(d)
1678.8 nm under the oblique incidence of $\theta=45^{\circ}$. Symbols “$+$”
and “$-$” in (a)(c) indicate charge distributions. Gold nanoridges are
outlined by white rectangles.

Note that under normal incidence, the near-field electric field maps in Figure
3(a) are distinct from those of MIM nanopillars in our previous work [21],
where a dipole and a quadrupole are excited in the top and the bottom gold
nanoridges, respectively. This difference is induced because here we adopt
symmetric dielectric environment, whereas in ref. [21] we considered
asymmetric environment. Under oblique incidence, the near-field electric field
and Poynting vector distributions in Figures 3(c)(d) are similar to our
previous work[17]. The only difference is that, here the SLR wavelength of
1678.8 nm is close to the $-1$ diffraction order of Rayleigh anomaly (RA),
because in this work we adopt a much smaller period.

###  3.2 Spectral tunability and band reversal effect

In Figure 2, we find that, as the incidence angle changes from
$\theta=0^{\circ}$ to $45^{\circ}$, the SLR wavelength is tuned from 979.4 nm
to 1678.8 nm, suggesting extremely broadband tunability. Remarkably, we
observe an impressive transition from a Fano-shaped reflectance dip (or a
transmittance peak) under normal incidence to a peak (or a dip) under the
oblique incidence of $\theta=45^{\circ}$. This suggests the appearance of the
band reversal effect.

We emphasize that for both the quadrupolar SLRs excited in the vertical MIM
nanograting under normal incidence and in the horizontal MIM nanograting under
oblique incidence [17], there is no dipolar SLR background. The difference in
their spectral features is that, the former has a reflectance dip (or
transmittance peak), as shown in Figure 2(a), whereas the latter has a
transmittance dip (or reflectance peak) [17], which is similar to Figure 2(b).

Figure 4: (a) Simulated zeroth-order reflectance spectra of the vertical MIM
nanograting under incidence of different angles. (b) The corresponding
resonance wavelengths and (c) the quality factors as function of the incidence
angle. $\lambda_{{\rm RA},m}$ in (b) is given by Equation (1) and fitted $Q$
is defined by Equation (2).

To further study the spectral tunability and the band reversal effect, we
calculated the reflectance (as shown in Figure 4) and the transmittance (not
shown for clarity) spectra for more incidence angles. Figures 4(a)(b) show
that, as the incidence angle increases from $\theta=0^{\circ}$ to $50^{\circ}$
in a step of 5∘, the SLR wavelength increases continuously from
$\lambda_{0}=979.4$ nm to 1735.7 nm. This suggests extremely broadband
tunability over 756.3 nm, covering most of the near-infrared regime. These SLR
wavelengths generally follow the RA wavelengths of the $m=-1$ order, which are
determined by

$\lambda_{{\rm RA},m}=n_{0}\Lambda(\pm 1-\sin\theta)/m\,.$ (1)

Figure 4(a) also shows that the SLR has an increasing linewidth as the
incidence angle increases. We estimate the quality factors for these SLRs that
are excited under different incidence angles and plot the results in Figure
4(c). Results show that as the incidence angle increases, the quality factor
first decreases dramatically from 979 for $\theta=0^{\circ}$ to 551 for
$\theta=5^{\circ}$, and then gradually converges to 150 for
$\theta=50^{\circ}$. Interestingly, we further find that the quality factor
decreases exponentially following a fitted expressing,

$Q=173+796.2\times\exp{(-0.135\theta)}\,.$ (2)

Besides the broadband wavelength tunability, which is intrinsic for the SLR,
and the exponentially decreasing quality factor as the incidence angle
increases, which is interesting, in Figure 4(a) we observe, for the first
time, an impressive phenomenon of the band reversal effect in SLRs, which
manifest itself as the transition from a Fano-shaped dip to a peak in the
reflectance spectra. Although similar effect was reported by Rybin et al. in
photonic crystals with Fano resonance between continuum Mie scattering and a
narrow Bragg band [22], it has not been reported in the field of SLRs yet. A
typical example is the SLR supported by a 1D metal nanoridge grating, for
which the transmittance spectra always exhibit a Fano-shaped dip as the
incidence angle increases [23].

Figure 5: Evolution of the resonance lineshape as the incidence angle
increases. Solid curves are simulation results taken from Figure 4, and dashed
curves are fitted results with equation 3 using Fano parameters $q$ labelled
on the top right.

In order to focus on the evolution of the resonance lineshape, we replot the
reflectance spectra around the SLR wavelengths in Figure 5. It is clear that
the resonance lineshape evolves from a Fano-shaped dip for $\theta=0^{\circ}$
and $5^{\circ}$, to a dip/peak pair with a distinct asymmetric Fano lineshape
for $\theta=10^{\circ}$ and $15^{\circ}$, then a reflectance peak with
symmetric Lorentzian lineshape for $\theta=25^{\circ}$, and finally a peak
with quasi-Lorentzian lineshape for larger incidence angles.

The evolution of the resonance lineshape can be quantified using the Fano
parameter $q$, which can be extracted by fitting the reflectance spectra
with[24, 25]

$R_{0}=B+A\frac{(q\gamma+\lambda-\lambda_{0})^{2}}{(\lambda-\lambda_{0})^{2}+\gamma^{2}}\,.$
(3)

Here the fitting parameters include the background spectrum $B$, the
coefficient $A$, the Fano parameter $q$, and the nonradiative linewidth
$\gamma$.

Figure 6: Fano parameter $q$ as a function of the incidence angle.

The extracted Fano parameters for the different incidence angles are labelled
in Figure 5, and the evolution as a function of the incidence angle is
depicted in Figure 6. We find that $q=0.019$ and 0.047 for $\theta=0^{\circ}$
and $5^{\circ}$, respectively, corresponding Fano-shaped dips in the
reflectance spectra. $q=1.132$ and 2.644 for $\theta=10^{\circ}$,
$15^{\circ}$, respectively. These correspond to a distinct asymmetric Fano
dip/peak lineshape. For $\theta=20^{\circ}$, $q=9.434$, since the reflectance
peak is less asymmetric. Particularly, for $\theta=25^{\circ}$,
$q\rightarrow\infty$, since the reflectance peak has a perfect Lorentzian
lineshape. As the incidence angle further increases, the Fano parameters $q$
have negative values with large magnitudes, which correspond to less
asymmetric Fano lineshapes. Therefore, we show that by varying the incidence
angle, the SLR lineshape can be dynamically tuned ranging from a Fano-shaped
dip, a Fano-shaped dip/peak pair, a Fano-shaped peak, to a perfect Lorentzian
peak. This suggests that the vertical MIM nanograting can act as a platform
for realizing Lorentzian and Fano resonance lineshapes, which is similar to
the microring resonator/bus-waveguide structure [26].

##  4 Conclusions

In summary, we have shown that narrow quadrupolar SLR with high quality factor
of 979 in the near-infrared regime can be excited under normal incidence in
the vertical MIM nanograting. Under oblique incidence, out-of-plane dipolar
SLRs can be excited. We have found that these SLRs have extremely broadband
tunability (over 756 nm) while maintaining relatively high quality factors
(above 150). Interestingly, we have found that the quality factor decreases
exponentially as the incidence angle increases. Remarkably, we have observed,
for the first time, the band reversal effect in the field of SLRs, which
manifests as the lineshape evolution from a Fano-shaped dip, to a dip/peak
pair of distinct Fano lineshape, a Fano-shaped peak, and a peak with perfect
Lorentzian lineshape. This makes the vertical MIM nanograting an attractive
platform for realizing dynamically tunable Fano and Lorentzian resonance
lineshapes. We expect the SLRs with high quality factors, extremely broadband
tunability, and tunable lineshapes will find promising applications in light-
matter interactions on the nanoscale.

## Acknowledgments

This work was funded by the State Key Laboratory of Advanced Optical
Communication Systems and Networks, China (2019GZKF2).

## References

  * [1] M. B. Ross, C. A. Mirkin, and G. C. Schatz.  Optical properties of one-, two-, and three-dimensional arrays of plasmonic nanostructures.  J. Phys. Chem. C, 120:816–830, 2016.
  * [2] W. Wang, M. Ramezani, A. I. Väkeväinen, P. Törmä, J. G. Rivas, and T. W. Odom.  The rich photonic world of plasmonic nanoparticle arrays.  Mater. Today, 21:303–314, 2018.
  * [3] V. G. Kravets, A. V. Kabashin, W. L. Barnes, and A. N. Grigorenko.  Plasmonic surface lattice resonances: A review of properties and applications.  Chem. Rev., 118:5912–5951, 2018.
  * [4] D. Wang, J. Guan, J. Hu, M. R. Bourgeois, and T. W. Odom.  Manipulating light-matter interactions in plasmonic nanoparticle lattices.  Acc. Chem Res., 52:2997–3007, 2019.
  * [5] W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom.  Lasing action in strongly coupled plasmonic nanocavity arrays.  Nat. Nanotechnol., 8:506–511, 2013.
  * [6] A. Fernandez-Bravo et al.  Ultralow-threshold, continuous-wave upconverting lasing from subwavelength plasmons.  Nat. Mat., 18:1172–1176, 2019.
  * [7] Jan M. Winkler, Max J. Ruckriegel, Henar Rojo, Robert C. Keitel, Eva De Leo, Freddy T. Rabouw, and David J. Norris.  Dual-wavelength lasing in quantum-dot plasmonic lattice lasers.  ACS Nano, 14:5223–5232, 2020.
  * [8] Mohammad Ramezani, Quynh Le-Van, and Alexei Halpin.  Nonlinear emission of molecular ensembles strongly coupled to plasmonic lattices with structural imperfections.  Phys. Rev. Lett., 121:243904, 2018.
  * [9] R. Czaplicki, A. Kiviniemi, M. J. Huttunen, X. Zang, T. Stolt, I. Vartiainen, J. Butet, M. Kuittinen, O. J. F. Martin, and M. Kauranen.  Less is more: Enhancement of second-harmonic generation from metasurfaces by reduced nanoparticle density.  Nano Lett., 18:7709–7714, 2018.
  * [10] D. C. Hooper, C. Kuppe, D. Wang, W. Wang, J. Guan, T. W. Odom, and V. K. Valev.  Second harmonic spectroscopy of surface lattice resonances.  Nano Lett., 19:165–172, 2019.
  * [11] Zhengqi Liu, Guiqiang Liu, Guolan Fu, Xiaoshan Liu, Zhenping Huang, and Gang Gu.  All-metal meta-surfaces for narrowband light absorption and high performance sensing.  J. Phys. D: Appl. Phys., 49:445104, 2016.
  * [12] A. Danilov, G. Tselikov, F. Wu, V. G. Kravets, I. Ozerov, F. Bedu, A. N. Grigorenko, and A. V. Kabashin.  Ultra-narrow surface lattice resonances in plasmonic metamaterial arrays for biosensing applications.  Biosens. Bioelectron., 104:102–112, 2018.
  * [13] Mohammad Reza Rakhshani.  Tunable and sensitive refractive index sensors by plasmonic absorbers with circular arrays of nanorods and nanotubes for detecting cancerous cells.  Plasmonics, 15:2071–2080, 2020.
  * [14] Christopher P. Burrows and William L. Barnes.  Large spectral extinction due to overlap of dipolar and quadrupolar plasmonic modes of metallic nanoparticles in arrays.  Opt. Express, 18:3187–3198, 2010.
  * [15] Alastair D. Humphrey, Nina Meinzer, Timothy A. Starkey, and William L. Barnes.  Surface lattice resonances in plasmonic arrays of asymmetric disc dimers.  ACS Photon., 3:634–639, 2016.
  * [16] Ankun Yang, Alexander J Hryn, Marc R Bourgeois, Won-Kyu Lee, Jingtian Hu, George C Schatz, and Teri W Odom.  Programmable and reversible plasmon mode engineering.  Proc. Natl. Acad. Sci. U. S. A., 113(50):14201–14206, 2016.
  * [17] Xinyu Fang, Lei Xiong, Jianping Shi, and Guangyuan Li.  High-$q$ quadrupolar plasmonic lattice resonances in horizontal metal-insulator-metal gratings.  Opt. Lett., 46(7):1546–1549, 2021.
  * [18] M. G. Moharam, Drew A. Pommet, Eric B. Grann, and T. K. Gaylord.  Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach.  J. Opt. Soc. Am. A, 12(5):1077–1086, 1995.
  * [19] Philippe Lalanne and G. Michael Morris.  Highly improved convergence of the coupled-wave method for tm polarization.  J. Opt. Soc. Am. A, 13:779–784, 1996.
  * [20] P. B. Johnson and R. W. Christy.  Optical constants of the noble metals.  Phys. Rev. B, 6:4370–4379, 1972.
  * [21] Xiuhua Yang, Gongli Xiao, Yuanfu Lu, and Guangyuan Li.  Narrow plasmonic surface lattice resonances with preference to asymmetric dielectric environment.  Opt. Express, 27:25384–25394, 2019.
  * [22] M.V. Rybin, A. B. Khanikaev, M. Inoue, K. B. Samusev, M. J. Steel, G. Yushin, and M. F. Limonov.  Fano resonance between mie and bragg scattering in photonic crystals.  Phys. Rev. Lett., 103:023901, 2009.
  * [23] Y. Hua, A. K. Fumani, and T. W. Odom.  Tunable lattice plasmon resonances in 1D nanogratings.  ACS Photon., 6(2):322–326, 2019.
  * [24] Andrey E. Miroshnichenko, Sergej Flach, and Yuri S. Kivshar.  Fano resonances in nanoscale structures.  Rev. Mod. Phys., 82:2257–2298, 2010.
  * [25] Mikhail F. Limonov, Mikhail V. Rybin, Alexander N. Poddubny, and Yuri S. Kivshar.  Fano resonances in photonics.  Nat. Photon., 11:543–554, 2017.
  * [26] Linpeng Gu, Hanlin Fang, Juntao Li, Liang Fang, Soo Jin Chua, Jianlin Zhao, and Xuetao Gan.  A compact structure for realizing lorentzian, fano, and electromagnetically induced transparency resonance lineshapes in a microring resonator.  Nanophotonics, 8:841–848, 2019.