# Equable Triangles on the Eisenstein Lattice

Christian Aebi and Grant Cairns  Collège Calvin, Geneva, Switzerland 1211
<EMAIL_ADDRESS>Department of Mathematical and Physical Sciences, La
Trobe University, Melbourne, Australia 3086<EMAIL_ADDRESS>
##  1\. Introduction

This paper concerns triangles whose vertices lie on the _Eisenstein lattice_ ,
which is the lattice in the complex plane generated by the elements $1$ and
$\omega=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$. We investigate triangles that are
_equable_ , that is, they have equal perimeter and area. Our study is inspired
by the classification of equable Heron triangles, which have integer sides and
area and are known to be realisable on the integer lattice [1]. There are
precisely five triangles of the preceding type, up to Euclidean motions; see
[2] and [3]. By comparison, on the Eisenstein lattice, we find the following
result.

######  Theorem 1.

There are only two equable triangles having vertices on the Eisenstein
lattice, up to Euclidean motions. They are realized by the following vertices.

  1. (a)

$A=8+4\omega,\ B=4+8\omega,\ C=0$,

  2. (b)

$A=6+3\omega,\ B=8+16\omega,\ C=0$.

Figure 1 shows the two triangles, with the first (equilateral) triangle
translated by $5$ to the right.

Figure 1. Two equable triangles on the Eisenstein lattice

##  2\. Proof of the Theorem

Consider an equable triangle $T$ with vertices
$A=a_{1}+a_{2}\omega,B=b_{1}+b_{2}\omega$ and $C=0$, where
$a_{1},a_{2},b_{1},b_{2}\in\mathbb{N}\cup\\{0\\}$. Let $a,b,c$ denote the
lengths of the sides $AC,BC,AB$, respectively. Notice that the squares of the
side lengths are integers; for example,
$a^{2}=a_{1}^{2}-a_{1}a_{2}+a_{2}^{2}$. Moreover, the signed area of $T$ is
$\frac{\sqrt{3}}{4}(a_{1}b_{2}-a_{2}b_{1})$.

######  Lemma 1.

The side lengths $a,b,c$ are each of the form $\sqrt{3}n$, for some
$n\in\mathbb{N}$.

###### Proof.

From above, as $T$ is equable,
$a+b+c=\frac{\sqrt{3}}{4}(a_{1}b_{2}-a_{2}b_{1})$, so

$\sqrt{3a^{2}}+\sqrt{3b^{2}}+\sqrt{3c^{2}}=\sqrt{3}(a+b+c)=\frac{3}{4}(a_{1}b_{2}-a_{2}b_{1}),$

which is rational. But it is well known that if $\sum_{i=1}^{n}\sqrt{m_{i}}$
is rational for integers $m_{1},\dots,m_{n}$, then $\sqrt{m_{i}}$ is rational
for each $i$; see for example [4] or [5]. So
$\sqrt{3a^{2}},\sqrt{3b^{2}},\sqrt{3c^{2}}$ are each rational. Hence, as
$3a^{2},3b^{2},3c^{2}$ are integers, it follows that
$\sqrt{3a^{2}},\sqrt{3b^{2}},\sqrt{3c^{2}}$ are also integers. So the side
lengths $a,b,c$ are each of the form $\frac{n}{\sqrt{3}}$, for some
$n\in\mathbb{N}$. Thus, since the squares of the side lengths are integers,
the required result follows. ∎

We follow the reasoning used in the proof of the equable Heron triangle
theorem given in the Appendix in [6]. For equable triangles with sides
$a,b,c$, Heron’s formula gives

(1) $(a+b+c)(-a+b+c)(a-b+c)(a+b-c)=16(a+b+c)^{2}.$

Let
$u=\frac{-a+b+c}{\sqrt{3}},v=\frac{a-b+c}{\sqrt{3}},w=\frac{a+b-c}{\sqrt{3}}$,
so that $u,v,w\in\mathbb{N}$ by the above lemma, and
$a=\frac{\sqrt{3}(v+w)}{2},b=\frac{\sqrt{3}(u+w)}{2},c=\frac{\sqrt{3}(u+v)}{2}$.
Then Equation (1) gives

(2) $3uvw=16(u+v+w),$

and we may assume without loss of generality that $u\leq v\leq w$. Note that
by construction, $u,v,w$ have the same parity, and from (2), $u,v,w$ are
necessarily even. Let $u=2x,v=2y,w=2z$, so
$a=\sqrt{3}(y+z),b=\sqrt{3}(x+z),c=\sqrt{3}(x+y)$. Then $3xyz=4(x+y+z)$. Thus

$y\leq z=\frac{4(x+y)}{3xy-4},$

so $3xy^{2}-8y-4x\leq 0$. Hence

$x\leq y\leq\frac{4+\sqrt{16+12x^{2}}}{3x}$

so $3x^{2}\leq 4+\sqrt{16+12x^{2}}$. Hence $(3x^{2}-4)^{2}\leq(16+12x^{2})$.
Thus $9x^{4}-36x^{2}\leq 0$, which gives $x\leq 2$. Then

$y\leq\frac{4+\sqrt{16+12x^{2}}}{3x}\leq\frac{4+\sqrt{16+12}}{3},$

since the function $\frac{4+\sqrt{16+4x^{2}}}{3x}$ is decreasing for positive
$x$. So, as $y$ is an integer, $y\leq 3$. Then, considering the values $x\leq
2$, $y\leq 3$ and $z=\frac{4(x+y)}{3xy-4}$, we find there are only two
solutions, which have the following integer values for $x,y,z$:

$1,2,6\quad\text{and}\quad 2,2,2,$

which imply finally the values for $a,b,c$:

$8\sqrt{3},7\sqrt{3},3\sqrt{3}\quad\text{and}\quad
4\sqrt{3},4\sqrt{3},4\sqrt{3}.$

## References

  * [1] Paul Yiu, Heronian triangles are lattice triangles, _Amer. Math. Monthly_ 108 (2001), no. 3, 261–263.
  * [2] Arthur H. Foss, Integer-sided triangles, _Math. Teacher_ 73 (1980), no. 5, 390–392.
  * [3] Christopher J. Bradley, _Challenges in geometry_ , Oxford University Press, Oxford, 2005\.
  * [4] Victor Wang, _The Art of problem Solving_ , https://artofproblemsolving.com/community/c1461h1035155.
  * [5] Qiaochu Yuan, _Annoying Precision_ , https://qchu.wordpress.com/2009/07/02/square-roots-have-no-unexpected-linear-relationships/.
  * [6] Christian Aebi and Grant Cairns, _Lattice equable quadrilaterals I: parallelograms_ , L’Enseign. Math. 67 (2021), no. 3/4, 369–401.


# Six-dimensional light-front Wigner distribution of hadrons

Yingda Han  Tianbo Liu  Bo-Qiang Ma  School of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China  Key Laboratory of Particle Physics and Particle Irradiation
(MOE), Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao, Shandong 266237, China  Center for High Energy Physics,
Peking University, Beijing 100871, China

###### Abstract

We propose a six-dimensional light-front Wigner distribution for the complete
description of partonic structures of a hadron such as pion and proton, taking
advantage of the recently proposed light-front variable $\tilde{z}$ by Miller
and Brodsky. Quantities derived from the Wigner distribution contain the most
general information of partonic structures, including also new quantities
correlating longitudinal coordinate with transverse momenta or transverse
coordinates, together with spins. The new Wigner distribution can be viewed as
a relativistic version of the original Wigner distribution in hadron physics
and an extension of widely utilized five-dimensional light-front Wigner
distribution.

###### keywords:

Parton distribution , Wigner distribution , Light front coordinate

##  1 Introduction

Unraveling the structure of hadrons in terms of quarks and gluons, the
fundamental degrees of freedom in quantum chromodynamics (QCD), is one of the
key issues in modern particle physics. The parton model, together with QCD
factorization, is proven a powerful tool in explaining high-energy hadron
scatterings with parton distribution function (PDF) describing the probability
density of finding a parton (quark or gluon) carrying light-front (LF)
longitudinal momentum fraction $x$ of the parent hadron. For the description
of observables that are also sensitive to the transverse kinematics of a
parton, the concept of PDF has been generalized to transverse momentum
dependent parton distributions (TMDs) and generalized parton distributions
(GPDs) to include the information of transverse momentum and transverse
coordinate distributions respectively, towards a multi-dimensional tomography
of the nucleon, as one of the main goals of the upcoming Electron-Ion Collider
[1, 2, 3].

As has been known for a long time, the most complete information of a system
is encoded in quantum phase space distributions, among which the Wigner
distribution is the most widely used. It is originally introduced in non-
relativistic quantum mechanics [4],

$\displaystyle\rho({\bm{b}},{\bm{k}})=\int{\rm
d}^{3}{\bm{y}}e^{i{\bm{k}}\cdot{\bm{y}}}\psi^{*}\Big{(}{\bm{b}}-\frac{\bm{y}}{2}\Big{)}\psi\Big{(}{\bm{b}}+\frac{\bm{y}}{2}\Big{)},$
(1)

and the expectation value of any physical operator can be obtained from the
average over the Wigner distribution [5]. Due to the Heisenberg’s uncertainty
principle in quantum theories, the Wigner distribution is a quasi-
distribution, describing non-positive definite density of finding the particle
at average position $\bm{b}$ with average momentum $\bm{k}$. This concept has
been applied in various areas, such as the quantum information, quantum
molecular dynamics, optics, nonlinear dynamics, quantum transport, and heavy
ion collisions [6], and was introduced to hadron physics in Refs. [7, 8] as
six-dimensional functions by neglecting relativistic effects. For a proper
description of the nucleon, which is a relativistic bound state of the strong
interaction, the LF Wigner distribution is defined [9] as five-dimensional
function of the longitudinal momentum fraction $x$, transverse position
${\bm{b}}_{\perp}$, and transverse momentum ${\bm{k}}_{\perp}$. Taking
advantage of the LF dynamics [10], or sometimes interpreted as the physics in
the infinite momentum frame [11], the LF Wigner distribution is boost-
invariant, connecting to TMDs by integrating over transverse position and to
GPDs at zero skewness, $\xi=0$, by integrating over transverse momentum.
Although the measurement of Wigner distributions is much more challenging than
the measurements of TMDs and GPDs, several processes such as the diffractive
di-jet production in deep inelastic scattering [12], the virtual photon-
nucleus quasi-elastic scattering [13], and the exclusive double Drell-Yan
process [14], have been proposed in recent years, which increases our
confidence in extracting the rich information encoded in Wigner distributions
from future experiments. The measurements of generalized transverse momentum
dependent parton distributions (GTMDs) at nonzero skewness through these
processes may also shed light on the extraction of the proposed six-
dimentional Wigner distribution via Fourier transform.

Comparing with the originally Wigner distribution in non-relativistic quantum
mechanics, one easily notices that the LF Wigner distribution does not include
all position and momentum variables in pair, missing the longitudinal
position. This seems a natural consequence of the Lorentz contraction in
special relativity. However, the LF coordinate $b^{-}$, which contracts to
zero in the infinite momentum frame, is not the right variable to describe the
longitudinal position of a parton. One needs to introduce a boost-invariant
variable related to $b^{-}$, similar to the case that we use $x$, instead of
$k^{+}$, to describe the longitudinal momentum of the parton. Such variable is
recently proposed by Miller and Brodsky [15], $\tilde{z}=b^{-}P^{+}$, where
$P^{+}$ is the LF momentum of the hadron. We note that this variable was first
considered in the study of deeply virtual Compton scattering (DVCS) process in
Refs. [16, 17], which provide an analogy to the diffraction pattern in optics.
The boost-invariant LF longitudinal position variable $\tilde{z}$ is merely a
parameter, which is not associated with an operator. This inherits the general
statement that one cannot construct a position operator in relativistic
quantum field theory, because any attempt to localize a particle will involve
high frequency modes whose energy is enough to create additional particles.
The closest concept one may consider as a position observable is the center of
inertia [18]. In this letter, we demonstrate that the Miller-Brodsky variable
is exactly the conjugate variable to the skewness $\xi$ in GPDs or GTMDs.
Hence, the LF Wigner distribution can be extended by pairing the longitudinal
momentum with the Miller-Brodsky variable to provide a complete information of
parton distributions inside the hadron.

##  2 Extended light-front Wigner distributions.

We start from the Wigner operator, similar to the one in Ref. [9] but
including the dependence on LF longitudinal coordinate,

$\displaystyle\widehat{W}^{[\Gamma]}$
$\displaystyle(b^{-},{\bm{b}}_{\perp},k^{+},{\bm{k}}_{\perp})=\int\frac{{\rm
d}y^{-}{\rm
d}^{2}{\bm{y}}_{\perp}}{2(2\pi)^{3}}e^{ik^{+}y^{-}-i{\bm{k}}_{\perp}\cdot{\bm{y}}_{\perp}}$
$\displaystyle\times\bar{\psi}\left(b-\frac{y}{2}\right)\Gamma{\cal
L}_{[b-y/2,b+y/2]}\psi\left(b+\frac{y}{2}\right)\Bigg{|}_{y^{+}=0},$ (2)

where $b=(0,b^{-},{\bm{b}}_{\perp})$ is the average LF position at some LF
time which we set as $0$, and ${\cal L}_{[b-y/2,b+y/2]}$ is the Wilson line
connecting the quark operators to ensure the color gauge invariance. $\Gamma$
is a Dirac matrix projecting the quark spin state, and at the leading twist,
$\Gamma=\gamma^{+}$, $\gamma^{+}\gamma_{5}$, and $i\sigma^{i+}\gamma_{5}$,
corresponds to unpolarized, longitudinally polarized, and transversely
polarized quarks respectively. The condition $y^{+}=0$ reflects the probe at a
fixed LF time. The extended LF Wigner distribution is then defined by
interpolating the Wigner operator (2) between initial and final nucleon states
with a momentum transfer
$\Delta=(\Delta^{+},\Delta^{-},{\bm{\Delta}}_{\perp})$,

$\displaystyle\rho^{[\Gamma]}$
$\displaystyle(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S)=\int\frac{{\rm
d}\xi{\rm d}^{2}{\bm{\Delta}}_{\perp}}{4\pi^{3}}$
$\displaystyle\times\Big{\langle}P+\frac{\Delta}{2},S\Big{|}\widehat{W}^{[\Gamma]}(b^{-},{\bm{b}}_{\perp},k^{+},{\bm{k}}_{\perp})\Big{|}P-\frac{\Delta}{2},S\Big{\rangle},$
(3)

where $\xi=-\Delta^{+}/(2P^{+})$ is the skewness variable representing the
longitudinal momentum transfer in a physical process, $P^{+}$ is the average
LF momentum of the nucleon, $x=k^{+}/P^{+}$ is the average LF momentum
fraction carried by the quark, $\tilde{z}=b^{-}P^{+}$ is the Miller-Brodsky
variable reflecting the longitudinal position, and $S$ is the spin state of
the nucleon. Keeping quark momentum fraction non-negative, the integral over
$\xi$ runs from $-x$ to $x$, which corresponds to the so-called DGLAP region
[19], i.e., the transferred momentum $\Delta^{+}$ is not enough to produce a
quark pair each carrying average momentum fraction $x$. Then the function
defined in (3) represents the quantum phase space distribution of a single
quark inside the nucleon.

To see the relation between
$\rho^{[\Gamma]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S)$ and the
widely used LF Wigner distribution, one can insert translation operations
between the Wigner operator and the nucleon state and integrate over
$\tilde{z}$,

$\displaystyle\int{\rm
d}\tilde{z}\,\rho^{[\Gamma]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S)=-\int{\rm
d}b^{-}\int\frac{{\rm d}{\Delta^{+}}{\rm
d}^{2}{\bm{\Delta}}_{\perp}}{(2\pi)^{3}}$ $\displaystyle\times
e^{ib^{-}\Delta^{+}}\Big{\langle}P+\frac{\Delta}{2},S\Big{|}\widehat{W}^{[\Gamma]}(0,{\bm{b}}_{\perp},k^{+},{\bm{k}}_{\perp})\Big{|}P-\frac{\Delta}{2},S\Big{\rangle}$
$\displaystyle=\int\frac{{\rm
d}^{2}{\bm{\Delta}}_{\perp}}{(2\pi)^{2}}\Big{\langle}P+\frac{{\bm{\Delta}}_{\perp}}{2},S\Big{|}\widehat{W}^{[\Gamma]}(0,{\bm{b}}_{\perp},k^{+},{\bm{k}}_{\perp})\Big{|}P-\frac{{\bm{\Delta}}_{\perp}}{2},S\Big{\rangle},$
(4)

which is just the five-dimensional LF Wigner distribution as introduced in
Ref. [9]. Therefore, the extend Wigner distribution in Eq. (3) is a
generalization of the well-known five-dimensional Wigner distribution without
losing any information, but including also the longitudinal position
distribution and its correlations with $x$, ${\bm{b}}_{\perp}$,
${\bm{k}}_{\perp}$, and spins.

Similar to the five-dimensional version, the extended LF Wigner distributions
have a direct connection with the GTMDs,

$\displaystyle\rho^{[\Gamma]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S)$
$\displaystyle=\int\frac{{\rm d}\xi{\rm
d}^{2}{\bm{\Delta}}_{\perp}}{4\pi^{3}}e^{-2i\xi\tilde{z}-i{\bm{b}}_{\perp}\cdot{\bm{\Delta}}_{\perp}}W^{[\Gamma]}\big{(}\xi,x,{\bm{\Delta}}_{\perp},{\bm{k}}_{\perp},S\big{)},$
(5)

where $W^{[\Gamma]}(\xi,x,{\bm{\Delta}}_{\perp},{\bm{k}}_{\perp},S)$ are the
GTMDs introduced in Refs. [20, 21],

$\displaystyle
W^{[\Gamma]}\big{(}\xi,x,{\bm{\Delta}}_{\perp},{\bm{k}}_{\perp},S\big{)}=\int\frac{{\rm
d}y^{-}{\rm
d}^{2}{\bm{y}}_{\perp}}{2(2\pi)^{3}}e^{ixP^{+}y^{-}-i{\bm{k}}_{\perp}\cdot{\bm{y}}_{\perp}}$
$\displaystyle\times\Big{\langle}P+\frac{\Delta}{2},S\Big{|}\bar{\psi}\left(-\frac{y}{2}\right)\Gamma{\cal
L}_{[-y/2,+y/2]}\psi\left(+\frac{y}{2}\right)\Big{|}P-\frac{\Delta}{2},S\Big{\rangle}\Big{|}_{y^{+}=0}.$
(6)

While GTMDs are complex functions, one can easily find that the extended
Wigner distributions are real functions. From (5), one can observe that the
Miller-Brodsky variable $\tilde{z}$ is conjugate to the skewness $\xi$.
Integrating out the transverse momentum ${\bm{k}}_{\perp}$ and the transverse
position ${\bm{b}}_{\perp}$ of (3), one can define the longitudinal LF Wigner
distribution,

$\displaystyle\rho^{[\Gamma]}(\tilde{z},x)=\int{\rm d}^{2}{\bm{k}}_{\perp}{\rm
d}^{2}{\bm{b}}_{\perp}\rho^{[\Gamma]}$
$\displaystyle(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S),$ (7)

which is a two-dimensional quasi-distribution function encoding partonic
distribution information along the longitudinal direction. As we will show
later, this function is not positive definite. The probability interpretation
clearly fails for negative values. It is due to the non-classicality in
quantum theory, or in other words, it reflects how well a classical
description works for a parton at certain kinematics. The non-negativity is
equivalent to non-contextuality, originating from the description of quantum
phenomena by so-called hidden variable models in which each observable has a
pre-determined value merely revealed by the act of measurement. As explained
in Refs. [22, 23, 24], it is impossible in a quantum theory. Alternatively,
one can view the Wigner distribution as the kernel of the density matrix [25].

For a multi-dimensional tomography of the hadron, one needs distribution
functions including transverse kinematics. Apart from TMDs and GPDs (at
$\xi=0$) that can be obtained via the integrals of the five-dimensional LF
Wigner distributions, we can also define three-dimensional parton distribution
functions by integrating the extended LF Wigner distributions over
${\bm{k}}_{\perp}$ and $x$,

$\displaystyle\rho^{[\Gamma]}(\tilde{z},{\bm{b}}_{\perp},S)=\int{\rm d}x{\rm
d}^{2}{\bm{k}}_{\perp}\rho^{[\Gamma]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S),$
(8)

as position-space counterparts to TMDs. One can also define longitudinal
position-transverse momentum joint three-dimensional parton distribution
functions by integrating over ${\bm{b}}_{\perp}$ and $x$,

$\displaystyle\rho^{[\Gamma]}(\tilde{z},{\bm{k}}_{\perp},S)=\int{\rm d}x{\rm
d}^{2}{\bm{b}}_{\perp}\rho^{[\Gamma]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S),$
(9)

as counterparts to GPDs which describe the parton distribution in a joint
three-dimensional space of longitudinal momentum and transverse position. In
addition, one can learn the correlation of the longitudinal position with
transverse momentum, transverse position, and spins directly from the extended
Wigner distributions. Unlike TMDs and GPDs at zero skewness, the three-
dimensional distributions in Eqs. (8) and (9) do not have probability
interpretations. The on-shell condition of the initial and final state
hadrons, $(P-\Delta/2)^{2}=(P+\Delta/2)^{2}=M^{2}$, requires
$P\cdot\Delta=P^{+}\Delta^{-}+P^{-}\Delta^{+}=0$. To obtain the distribution
in $\tilde{z}$, one needs the amplitude at nonzero skewness
$\xi=-\Delta^{+}/(2P^{+})\neq 0$, which also leads to nonzero $\Delta^{-}$ and
thus $x^{+}$ dependence.

##  3 Modeling the extended LF Wigner distribution.

To further our understanding, we perform calculation of the extended LF Wigner
distribution of the pion and the proton in the spectator model, which is
proven successful in qualitatively describing many physical quantities, such
as form factors, unpolarized and polarized quark PDFs, and TMDs. Applying the
LF Fock-state expansion to the hadron state,

$\displaystyle\left|P,S\right\rangle=$ $\displaystyle\sum_{n}\int[{\rm d}x{\rm
d}^{2}{\bm{k}}_{\perp}]\Psi\left(x_{i},{\bm{k}}_{i\perp},\lambda_{i};n\right)$
$\displaystyle\times\left|n:x_{i}P^{+},x_{i}{\bm{P}}_{\perp}+{\bm{k}}_{i\perp},\lambda_{i}\right\rangle,$
(10)

where $n$ runs over all possible Fock states, $x_{i}$, ${\bm{k}}_{i\perp}$ and
$\lambda_{i}$ represent the LF momentum fraction, intrinsic transverse
momentum and helicity of the constituent within the Fock state,
$\Psi\left(x_{i},{\bm{k}}_{i\perp},\lambda_{i};n\right)$ is the LF wave
function (LFWF), and the integral measure is

$\displaystyle[{\rm d}x{\rm d}^{2}{\bm{k}}_{\perp}]$
$\displaystyle=\left(16\pi^{3}\right)\left(\prod_{i\in n}\frac{{\rm
d}x_{i}{\rm d}^{2}{\bm{k}}_{i\perp}}{2\sqrt{x_{i}}(2\pi)^{3}}\right)$
$\displaystyle\times\delta\left(1-\sum_{i\in
n}x_{i}\right)\delta^{(2)}\left(\sum_{i\in n}{\bm{k}}_{i\perp}\right),$ (11)

one can in principle calculate the extend LF Wigner distributions following
the definition in (3) if LFWFs are provided.

Since solving LFWFs is not the scope of this work, we adopt the spectator
model LFWFs, which have been utilized in many phenomenological studies. For
the pion, we take the valence state LFWFs [26, 27],

$\displaystyle\Psi\left(x,{\bm{k}}_{\perp},\uparrow,\downarrow\right)$
$\displaystyle=-\Psi\left(x,{\bm{k}}_{\perp},\downarrow,\uparrow\right)=\frac{m}{\sqrt{2{\cal
M}_{T}^{2}}}\varphi_{\pi},$ (12a)
$\displaystyle\Psi\left(x,{\bm{k}}_{\perp},\uparrow,\uparrow\right)$
$\displaystyle=\Psi^{*}\left(x,{\bm{k}}_{\perp},\downarrow,\downarrow\right)=-\frac{k_{1}-ik_{2}}{\sqrt{2{\cal
M}_{T}^{2}}}\varphi_{\pi},$ (12b)

where ${\cal M}_{T}^{2}=m^{2}+{\bm{k}}_{\perp}^{2}$, the arrows represent the
helicities of quark and the spectator antiquark, and $\varphi_{\pi}$ is the
spin-independent wave function which we choose the Brodsky-Huang-Lepage
prescription [28], $\varphi_{\pi}=A_{\pi}\exp\left\\{-{\cal
M}_{T}^{2}/[8\beta^{2}x(1-x)]\right\\}$ with $A_{\pi}$ as the normalization
factor. The parameters are $m=0.2\,\rm GeV$ and $\beta=0.41\,\rm GeV$ are
chosen according to Ref. [27].

For the proton, we consider the valence component as a quark-diquark
configuration, including both scalar and axial-vector diquarks which serve as
a spectator and absorb part of high Fock-state contributions into the
effective masses. The explicit expressions of LFWFs are chosen the same as
those in Ref. [29], where the momentum space wave functions are taken as real
functions sharing the same form of the Brodsky-Huang-Lepage prescription.

For simplicity, we consider the unpolarized extended Wigner distribution,
which for the pion case is defined as $\rho_{\rm
UU}^{\pi}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp})=\rho^{[\gamma^{+}]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp})$,
and for the proton case is defined as the average over spin states,

$\displaystyle\rho_{\rm UU}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp})$
$\displaystyle=\frac{1}{2}\Big{[}\rho^{[\gamma^{+}]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},S)$
$\displaystyle+\rho^{[\gamma^{+}]}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp},-S)\Big{]}.$
(13)

Substituting the Fock expansion (10) into (3), one can derive the overlap
representation of the extended LF Wigner distribution of the pion as

$\displaystyle\rho_{\rm
UU}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp})=\sum_{\lambda_{q},\lambda_{\bar{q}}}\int\frac{{\rm
d}\xi{\rm
d}^{2}{\bm{\Delta}}_{\perp}}{4\pi^{3}}e^{-2i\xi\tilde{z}-i{\bm{b}}_{\perp}\cdot{\bm{\Delta}}_{\perp}}$
$\displaystyle~{}\times\frac{1}{16\pi^{3}}\Psi^{*}\left(x^{\rm
out},{\bm{k}}_{\perp}^{\rm
out},\lambda_{q},\lambda_{\bar{q}}\right)\Psi\left(x^{\rm
in},{\bm{k}}_{\perp}^{\rm in},\lambda_{q},\lambda_{\bar{q}}\right),$ (14)

where $x^{\rm out}=(x-\xi)/(1-\xi)$ and $x^{\rm in}=(x+\xi)/(1+\xi)$ are
longitudinal momentum fractions carried by the struck quark in the final and
initial states. The transverse momenta ${\bm{k}}_{\perp}^{\rm
out}={\bm{k}}_{\perp}+(1-x){\bm{\Delta}}_{\perp}/[2(1-\xi)]$ and
${\bm{k}}_{\perp}^{\rm
in}={\bm{k}}_{\perp}-(1-x){\bm{\Delta}}_{\perp}/[2(1+\xi)]$ in the LFWFs are
intrinsic transverse momenta of the struck quark with respect to the final and
initial state hadrons. Their relations to ${\bm{k}}_{\perp}$ are derived by
substituting the Fock state expansion (10) into Eq. (3). In the frame where
the final state hadron has transverse momentum ${\bm{\Delta}}_{\perp}/2$ and
correspondingly the initial state hadron has transverse momentum
$-{\bm{\Delta}}_{\perp}/2$, the transverse momenta of the struck quark can be
evaluated from the intrinsic transverse momenta via

$\displaystyle{\bm{p}}_{\perp}^{\rm out}$ $\displaystyle={\bm{k}}_{\perp}^{\rm
out}+x^{\rm
out}\frac{{\bm{\Delta}}_{\perp}}{2}={\bm{k}}_{\perp}+\frac{{\bm{\Delta}}_{\perp}}{2},$
(15) $\displaystyle{\bm{p}}_{\perp}^{\rm in}$
$\displaystyle={\bm{k}}_{\perp}^{\rm in}-x^{\rm
in}\frac{{\bm{\Delta}}_{\perp}}{2}={\bm{k}}_{\perp}-\frac{{\bm{\Delta}}_{\perp}}{2},$
(16)

with the difference equaling to the transferred transverse momentum
${\bm{p}}_{\perp}^{\rm out}-{\bm{p}}_{\perp}^{\rm in}={\bm{\Delta}}_{\perp}$.
The $\lambda_{q,\bar{q}}$ are helicities of the quark and the spectator
antiquark. Similar expressions can be obtained for the proton case, where one
needs to sum over the spectator types and spin states. Corresponding
unpolarized longitudinal LF Wigner distribution $\rho_{\rm
UU}(\tilde{z},{x})$, three-dimensional position distribution function
$\rho_{\rm UU}(\tilde{z},{\bm{b}}_{\perp})$, and longitudinal position-
transverse momentum joint three-dimensional distribution function $\rho_{\rm
UU}(\tilde{z},{\bm{k}}_{\perp})$ are defined as the integrals of (13)
following (7), (8), and (9) respectively.

Figure 1: Six-dimensional unpolarized LF Wigner distribution $\rho_{\rm
UU}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp})$ of the pion (left) and the
proton (right). Upper : the distribution in $\tilde{z}$-$b_{x}$ plane with
fixed $\bm{k}_{\perp}=0.6\,{\rm GeV}\,{\bm{e}}_{x}$ (${\bm{e}}_{x}$ is the
unit vector in $x$ direction) and $b_{y}=0.4\,\rm GeV^{-1}$. Lower : the
distribution in $\tilde{z}$-$k_{x}$ plane with fixed $\bm{b}_{\perp}=0.4\,{\rm
GeV^{-1}}\,{\bm{e}}_{x}$ and $k_{y}=0.6\,\rm GeV$.

The numerical results of unpolarized extend LF Wigner distribution $\rho_{\rm
UU}(\tilde{z},x,{\bm{b}}_{\perp},{\bm{k}}_{\perp})$ of the pion and the proton
are shown in Fig. 1. To see the correlation of longitudinal position with
transverse momentum and transverse position, the function is plotted at some
fixed values of ${\bm{k}}_{\perp}$ or ${\bm{b}}_{\perp}$ as specified in the
caption. The unpolarized distribution is centro-symmetry in the
$\tilde{z}$-$\bm{b}_{\perp}$ and $\tilde{z}$-$\bm{k}_{\perp}$ subspaces, which
can be derived from (14). One can also observe in Fig. 1 that the distribution
around the central region increases at large $x$, which can be roughly
interpreted with the picture that a parton carrying larger momentum is less
likely to appear far from the center. Since the range of $\xi$ becomes wider
with increasing $x$, the distribution in its Fourier conjugate $\tilde{z}$ is
more centralized as a typical feature of the Fourier transform. We should note
that the value of the distribution around the $\tilde{z}$ center is a balance
between the decrease of $\tilde{z}$-integrated distribution (i.e. the PDF) at
large $x$ and the centralization of the distribution in $\tilde{z}$. This
feature is more clearly shown in Fig. 2, the unpolarized longitudinal LF
Wigner distribution $\rho_{\rm UU}(\tilde{z},x)$. With increasing $x$ values,
the distribution becomes more and more centralized in $\tilde{z}$. At
longitudinal positions away from the center (the classical limit), the Wigner
distribution is negative in some regions, which is interpreted as nonclassical
behavior of the quantum system. In addition, the oscillating behavior of the
distribution in $\tilde{z}$ can be viewed as an analog to the diffraction
pattern in optics. Such analogy was first found in the study of the DVCS
amplitude in the longitudinal distance [16, 17]. The finite range of the
skewness $\xi$, from $-x$ to $x$, is like the width of the slit in the single
slit experiment, which leads to the oscillating behavior in the Fourier
transform. As can be observed from the curves at different $x$ values in Fig.
2, the distribution oscillates in $\tilde{z}$ and achieves its first local
minimal point at smaller $\tilde{z}$ value with increasing $x$ value.

Figure 2: Unpolarized longitudinal LF Wigner distributions $\rho_{\rm
UU}(\tilde{z},x)$ of the pion (left) and the proton (right).

Figure 3: Three-dimensional parton distribution function $\rho_{\rm
UU}(\tilde{z},{\bm{b}}_{\perp})$ of the pion (left) and the proton (right).

Figure 4: Three-dimensional parton distribution function $\rho_{\rm
UU}(\tilde{z},{\bm{k}}_{\perp})$ of the pion (left) and the proton (right).

The unpolarized three-dimensional position distribution functions $\rho_{\rm
UU}(\tilde{z},{\bm{b}}_{\perp})$ of the pion and the proton are shown in Fig.
3, and the unpolarized longitudinal position-transverse momentum joint three-
dimensional distribution functions $\rho_{\rm UU}(\tilde{z},{\bm{k}}_{\perp})$
of the pion and the proton are shown in Fig. 4. These functions are symmetric
in the longitudinal position, transverse position and transverse momentum, and
have a positive peak at the center. They represent the correlations of the
longitudinal position with transverse momentum and transverse position
respectively. As mentioned in section 2, $\rho_{\rm
UU}(\tilde{z},{\bm{b}}_{\perp})$ and $\rho_{\rm
UU}(\tilde{z},{\bm{k}}_{\perp})$ do not have probability interpretations. In
Figs. 3 and 4, these distributions are only plotted in the region
$|\tilde{z}|\leq 8$, where the values are positive. If going to large
$\tilde{z}$, one will find nonpositive values and oscillating behavior. Such
long and oscillating tail crossing zero of the distribution in $\tilde{z}$ was
also found in the form factor calculation in the $(1+1)$-dimensional
$\phi^{3}$ model [30].

##  4 Summary and outlook.

We have proposed extended LF Wigner distributions for the description of
hadron structures. With all position and momentum variables appear in pair,
they are six-dimensional functions, which contain the most complete
information of parton distributions in the hadron. The boost-invariant LF
variable $\tilde{z}$, referred to as the Miller-Brodsky variable, is conjugate
to the skewness $\xi$ in GPDs and GTMDs, allowing us to define relativistic
Wigner distributions beyond fixing $\xi=0$. As an extension of the widely used
five-dimensional LF Wigner distribution, one can learn richer information of
partonic structures, particularly the longitudinal position distribution and
its correlation with transverse momentum, transverse position, and spins. From
these extended LF Wigner distributions, we are able to define new three-
dimensional parton functions that encodes the correlation of longitudinal
position with transverse momentum or transverse position, in addition to the
TMDs and GPDs that are being actively studied.

For an illustration of these new quantities, we take the unpolarized case as
an example with spectator model calculation. As expected, non-positivity is
observed in longitudinal LF Wigner distribution, but from the quasi-
distribution point of view a general feature that quarks at larger $x$ have
more centralized distribution in the longitudinal position $\tilde{z}$, is
obtained in consistent with intuitions.

When the spin degree of freedom is taken into account, one is able to learn
much richer partonic structures of the nucleon, and to have a more profound
understanding of the strong interaction in confining phase. Considering a
transversely polarized nucleon, the longitudinal position-transverse momentum
joint three-dimensional function will shed light on the transverse orbital
angular momentum, which is an indispensable part of the nucleon spin
structure. The measurements of GTMDs at nonzero skewness via the diffractive
di-jet production in deep inelastic scattering [12], the virtual photon-
nucleus quasi-elastic scattering [13], and the exclusive double Drell-Yan
process [14] will provide valuable information, and it is necessary to measure
GTMDs at nonzero skewness for the extended LF Wigner distributions. We leave
these topics to more careful future studies.

## Acknowledgements

This work is supported in part by National Natural Science Foundation of China
under Grant No. 12075003 and No. 12175117.

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