Quark Wigner Distributions and Orbital Angular Momentum

Abstract

We study the Wigner functions of the nucleon which provide multidimensional
images of the quark distributions in phase space. These functions can be
obtained through a Fourier transform in the transverse space of the generalized
transverse-momentum dependent parton distributions. They depend on both the
transverse position and the three-momentum of the quark relative to the
nucleon, and therefore combine in a single picture all the information
contained in the generalized parton distributions and the transverse-momentum
dependent parton distributions. We focus the discussion on the distributions of
unpolarized/longitudinally polarized quark in an unpolarized/longitudinally
polarized nucleon. In this way, we can study the role of the orbital angular
momentum of the quark in shaping the nucleon and its correlations with the
quark and nucleon polarizations. The quark orbital angular momentum is also
calculated from its phase-space average weighted with the Wigner distribution
of unpolarized quarks in a longitudinally polarized nucleon. The corresponding
results obtained within different light-cone quark models are compared with
alternative definitions of the quark orbital angular momentum, as given in
terms of generalized parton distributions and transverse-momentum dependent
parton distributions.

Full text

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arXiv:1106.0139v1 [hep-ph] 1 Jun 2011
Quark Wigner Distributions and Orbital Angular Momentum
C. Lorc´ e∗1and B. Pasquini†2
1Institut f¨ ur Kernphysik, Johannes Gutenberg-Universit¨ at,
D-55099 Mainz, Germany
2Dipartimento di Fisica Nucleare e Teorica, Universit` a degli Studi di Pavia,
and INFN, Sezione di Pavia, I-27100 Pavia, Italy
Abstract
We study the Wigner functions of the nucleon which provide multidimensional images of the
quark distributions in phase space. These functions can be obtained through a Fourier transform
in the transverse space of the generalized transverse-momentum dependent parton distributions.
They depend on both the transverse position and the three-momentum of the quark relative to
the nucleon, and therefore combine in a single picture all the information contained in the gen-
eralized parton distributions and the transverse-momentum dependent parton distributions. We
focus the discussion on the distributions of unpolarized/longitudinally polarized quark in an unpo-
larized/longitudinally polarized nucleon. In this way, we can study the role of the orbital angular
momentum of the quark in shaping the nucleon and its correlations with the quark and nucleon
polarizations. The quark orbital angular momentum is also calculated from its phase-space average
weighted with the Wigner distribution of unpolarized quarks in a longitudinally polarized nucleon.
The corresponding results obtained within different light-cone quark models are compared with
alternative definitions of the quark orbital angular momentum, as given in terms of generalized
parton distributions and transverse-momentum dependent parton distributions.
PACS numbers: 12.38.-t,12.39.-x,14.20.Dh
∗E-mail: lorce@kph.uni-mainz.de
†E-mail: pasquini@pv.infn.it
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I. INTRODUCTION
One of the most challenging tasks for unravelling the partonic structure of hadrons is
mapping the distribution of momentum and spin of the proton onto its constituents. To this
aim, generalized parton distributions (GPDs) [1–6] and transverse-momentum dependent
parton distributions (TMDs) [7–12] have proven to be among the most useful tools. GPDs
provide a new method of spatial imaging of the nucleon [13–17], through the definition
of impact-parameter dependent densities (IPDs) which reveal the correlations between the
quark distributions in transverse-coordinate (or impact-parameter) space and longitudinal
momentum for different quark and target polarizations. On the other hand, TMDs contain
novel and direct three-dimensional information about the strength of different spin-spin and
spin-orbit correlations in the momentum space [18–21]. The ultimate understanding of the
partonic structure of the nucleon can be gained by means of joint position-and-momentum
(or phase-space) distributions such as the Wigner distributions. These distributions contain
the most general one-body information of partons, corresponding to the full one-body den-
sity matrix in both momentum and position space, and reduce in certain limits to TMDs
and GPDs. Because of the uncertainty principle which prevents to know simultaneously the
position and momentum of a quantum-mechanical system, the phase-space distributions do
not have a density interpretation. Only in the classical limit they become positive definite.
Nonetheless, the physics of a phase-space distribution is very rich and one can try to select
certain situations where a semi-classical interpretation is still possible. Wigner distributions
have already been applied in many physics areas like heavy ion collisions, quantum molec-
ular dynamics, signal analysis, quantum information, optics, image processing, nonlinear
dynamics, ... [22–24], and can even be measured directly in some experiments [25–28].
The concept of Wigner distributions in QCD for quarks and gluons was first explored
in Refs. [29, 30], introducing the definition of a Wigner operator whose matrix elements
in the nucleon states were interpreted as distributions of the partons in a six-dimensional
space (three position and three momentum coordinates). The link with GPDs was exploited
to obtain three-dimensional spatial images of the proton which were interpreted as charge
distributions of the quarks for fixed values of the Feynman variable x. This interpretation
relies however on a nonrelativistic approximation.
Wigner distributions have a direct connection with the generalized parton correlation
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functions (GPCFs) which were recently introduced in Ref. [31]. The GPCFs are the dis-
tributions that parametrize the fully unintegrated, off-diagonal quark-quark correlator for a
hadron. In the case of the nucleon and after integration over the light-cone energy of the
quark, one finds the so-called generalized transverse-momentum dependent parton distri-
butions (GTMDs). At leading-twist there are 16 GTMDs which depend on the light-cone
three-momentum of the quark and, in addition, on the momentum transfer to the nucleon
∆µ. After two-dimensional Fourier transform from?∆⊥to the impact-parameter space co-
ordinates?b⊥, in a frame without momentum transfer along the light-cone direction, one
obtains the Wigner distributions which are completely consistent with special relativity.
The purpose of this paper is to investigate the phenomenology of the quark Wigner distri-
butions. As a matter of fact, since it is not known how to access these distributions directly
from experiments, phenomenological models are very powerful in this context. Collecting
the information that one can learn from quark models which were built up on the basis
of available experimental information on GPDs and TMDs, one can hope to reconstruct
a faithful description of the physics of the Wigner distributions. To this aim we will rely
on models for the light-cone wave functions (LCWFs) which have already been used for
the description of the generalized parton distributions (GPDs) [6, 32–34], the transverse-
momentum dependent parton distributions (TMDs) [21, 34–37] and electroweak properties
of the nucleon [34, 38–41].
The plan of the manuscript is as follows. In sec. IIA, we review the definition of the
Wigner distributions obtained by Fourier transform of the GTMDs to the impact-parameter
space. Although the Wigner distributions cannot have a strict probabilistic interpretation,
they reduce to genuine probability distributions after integration over position and/or mo-
mentum variables. As discussed in sec. IIB, one can obtain four types of three dimensional
densities: in addition to momentum distributions given by the TMDs and to IPDs related
to the GPDs, there are two new distributions mapping the nucleon as functions of one
transverse-space coordinate and one transverse-momentum component which are not conju-
gated and therefore not constrained by the Heisenberg uncertainty principle. Whereas the
GTMDs are in general complex-valued functions, the two-dimensional Fourier transforms
of the GTMDs are always real-valued functions, in accordance with their interpretation as
phase-space distributions. These 16 functions can be disentangled by selecting different
configurations, along three orthogonal directions, of the nucleon and quark polarizations.
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In order to simplify the discussion, in sec. IIC we focus on the cases without transverse
polarizations. In sec. IID we discuss and compare different definitions of the quark orbital
angular momentum, as obtained from GPDs, TMDs and Wigner distributions. In particular,
by treating the Wigner functions as if they were classical distributions, we can obtain the
expectation value of the orbital angular momentum operator from its phase-space average
weighted with the Wigner distribution of unpolarized quarks in a longitudinally polarized
nucleon. In sec. III we explicitly calculate the Wigner distributions in two light-cone quark
models, showing the results for the first x moments in the four-dimensional phase space (two
transverse position and two transverse momentum coordinates). In particular, we discuss
specific situations where the density matrices have a quasi-probabilistic interpretation, giv-
ing a semi-classical picture for multidimensional images of the nucleon. In section IV we
draw our conclusions.
II. WIGNER DISTRIBUTIONS
A. Wigner Operators and Wigner Distributions
Wigner distributions in QCD were first explored in Refs. [29, 30]. Neglecting relativistic
effects, the authors used the standard three-dimensional Fourier transform in the Breit frame
and introduced six-dimensional Wigner distributions (three position and three momentum
coordinates). We propose to study instead five-dimensional Wigner distributions (two posi-
tion and three momentum coordinates) as seen from the infinite momentum frame (IMF).
The advantages of working in the IMF have already been emphasized in the derivation of
transverse charge densities [42–44] and IPDs [13–16] from form factors (FFs) and GPDs,
respectively. Analogously, they will be exploited here to arrive at a definition of Wigner
distributions which is not spoiled by relativistic corrections.
Introducing two lightlike four-vectors n± satisfying n+· n− = 1, we write the light-
cone components of a generic four-vector a as [a+,a−,? a⊥] with a±= a · n∓. Similarly to
Refs. [29, 30], we define the Wigner operators for quarks at a fixed light-cone time y+= 0
as follows
?
W[Γ](?b⊥,?k⊥,x) ≡1
2
?
dz−d2z⊥
(2π)3
ei(xp+z−−?k⊥·? z⊥)ψ(y −z
2)ΓW ψ(y +z
2)??
z+=0
(1)
with yµ= [0,0,?b⊥], p+the average nucleon longitudinal momentum and x = k+/p+the
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average fraction of nucleon longitudinal momentum carried by the active quark. The su-
perscript Γ stands for any twist-two Dirac operator Γ = γ+,γ+γ5,iσj+γ5with j = 1,2. A
Wilson line W ≡ W(y −z
2,y +z
2|n) ensures the color gauge invariance of the Wigner oper-
ator, connecting the points (y −z
2) and (y +z
2) via the intermediary points (y −z
2) + ∞ · n
and (y +z
2) + ∞ · n by straight lines [31].
We define the Wigner distributions in terms of the matrix elements of the Wigner oper-
ators sandwiched between nucleon states with polarization?S as follows
?
ρ[Γ](?b⊥,?k⊥,x,?S) ≡
d2∆⊥
(2π)2?p+,
?∆⊥
2,?S|?
W[Γ](?b⊥,?k⊥,x)|p+,−
?∆⊥
2,?S?. (2)
Thanks to the properties of the Galilean subgroup of transverse boosts in the IMF [15, 45], we
can form a localized nucleon state in the transverse direction, in the sense that its transverse
center of momentum is at the position ? r⊥:
|p+,? r⊥? =
?
d2p⊥
(2π)2e−i? p⊥·? r⊥|p+,? p⊥?. (3)
The Wigner distributions defined according to Eq. (2) can then be written in terms of these
localized nucleon states as
ρ[Γ](?b⊥,?k⊥,x,?S) =
?
d2D⊥?p+,−
?D⊥
2,?S|?
W[Γ](?b⊥,?k⊥,x)|p+,
?D⊥
2,?S?, (4)
where?D⊥ is the transverse distance between the initial and final centers of momentum.
Note that the nucleon in our definition of the Wigner distributions has vanishing average
transverse position and average transverse momentum, see Eqs. (2) and (4). This allows
us to interpret the variables?b⊥and?k⊥as the relative average transverse position and the
relative average transverse momentum of the quark, respectively.
Using a transverse translation in Eq. (2), we find
ρ[Γ](?b⊥,?k⊥,x,?S) =
?
d2∆⊥
(2π)2e−i?∆⊥·?b⊥W[Γ](?∆⊥,?k⊥,x,?S),(5)
where W[Γ]are the quark-quark correlators defining the GTMDs [31] for ∆+= 0
W[Γ](?∆⊥,?k⊥,x,?S) = ?p+,
?
?∆⊥
2,?S|?
ei(xp+z−−?k⊥·? z⊥)?p+,
W[Γ](?0⊥,?k⊥,x)|p+,−
?∆⊥
2,?S?
=1
2
dz−d2z⊥
(2π)3
?∆⊥
2,?S|ψ(−z
2)ΓW ψ(z
2)|p+,−
?∆⊥
2,?S???
z+=0.
(6)
This means that the Wigner distributions defined as in Eq. (2) are the two-dimensional
Fourier transforms of GTMDs, just like transverse densities and IPDs are two-dimensional
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Fourier transforms of FFs and GPDs, respectively. Contrarily to all the other distribu-
tion functions, the GTMDs are in general complex-valued functions. However the two-
dimensional Fourier transforms of the GTMDs are always real-valued functions, in accor-
dance with their interpretation as phase-space distributions.
B. Three-Dimensional Probability Densities
Wigner distributions cannot have a strict probabilistic interpretation, since Heisenberg
uncertainty relations prevent to determine at the same time position and momentum of a
particle. Accordingly, Wigner distributions are not positive definite. Nevertheless, inte-
grating out position and/or momentum variables, Wigner distributions reduce to genuine
probability distributions. There are in particular four types of three-dimensional probability
densities:
• Integrating over?b⊥amounts to set?∆⊥=?0⊥, and so the Wigner distributions reduce
to the standard TMD correlators Φ[Γ][31, 34]
?
d2b⊥ρ[Γ](?b⊥,?k⊥,x,?S) = W[Γ](?0⊥,?k⊥,x,?S)
≡ Φ[Γ](?k⊥,x,?S),
(7)
which can be interpreted as quark densities in three-dimensional momentum space;
• Integrating over?k⊥amounts to set ? z⊥=?0⊥, and so the Wigner distributions reduce
to two-dimensional Fourier transforms of the standard GPD correlators [31, 34]
?
d2k⊥ρ[Γ](?b⊥,?k⊥,x,?S) =
?
d2∆⊥
(2π)2e−i?∆⊥·?b⊥F[Γ](?∆⊥,x,?S) (8)
with
F[Γ](?∆⊥,x,?S) ≡1
2
?
dz−
2πeixp+z−?p+,
?∆⊥
2,?S|ψ(−z
2)ΓW ψ(z
2)|p+,−
?∆⊥
2,?S???
z+=z⊥=0,
(9)
where the modulus of a general transverse vector ? a⊥is indicated as a⊥. In other words,
one recovers the IPDs which can be interpreted as quark densities in the transverse
position space and longitudinal momentum space;
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• Integrating over byand kxamounts to set ∆y= zx= 0, and so the Wigner distributions
reduce to new three-dimensional quark densities
?
dbydkxρ[Γ](?b⊥,?k⊥,x,?S) ≡ ˜ ρ[Γ](bx,ky,x,?S). (10)
The variables bxand kyrefer to two orthogonal directions in the transverse plane and
so are not subjected to Heisenberg uncertainty relations;
• Integrating over bxand kyamounts to set ∆x= zy= 0, and so the Wigner distributions
reduce to other new three-dimensional quark densities
?
dbxdkyρ[Γ](?b⊥,?k⊥,x,?S) ≡ ¯ ρ[Γ](by,kx,x,?S).(11)
There are a priori no simple relations between the quark densities in Eqs. (10) and
(11), except when the quark and nucleon polarizations have no transverse components.
In this case, the only privileged directions in the transverse plane are?b⊥and?k⊥, and
we have ˜ ρ[Γ](b⊥,k⊥,x,? ez) = ¯ ρ[Γ](−b⊥,k⊥,x,? ez) for Γ = γ+,γ+γ5.
C. Wigner Distribution with Longitudinal Polarizations
On the one hand, there are in total 16 GTMDs at twist-two level [31]. On the other
hand, the quark and nucleon can be either unpolarized or polarized along three orthogonal
directions, which means 16 configurations. All the 16 configurations can be written in terms
of 16 independent linear combinations of the GTMDs. We will not present all of them in this
study. To keep the discussion relatively simple, we focus on cases without any transverse
polarization.
The Wigner distribution of quarks with longitudinal polarization λ in a nucleon with
longitudinal polarization Λ is obtained for Γ = γ+1+λγ5
2
and?S = Λ? ez
ρΛλ(?b⊥,?k⊥,x) ≡1
2
?
ρ[γ+](?b⊥,?k⊥,x,Λ? ez) + λρ[γ+γ5](?b⊥,?k⊥,x,Λ? ez)
?
. (12)
We decompose it as follows
ρΛλ(?b⊥,?k⊥,x)
=1
2
?
ρUU(?b⊥,?k⊥,x) + ΛρLU(?b⊥,?k⊥,x) + λρUL(?b⊥,?k⊥,x) + ΛλρLL(?b⊥,?k⊥,x)
?
, (13)
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where
ρUU(?b⊥,?k⊥,x) =1
2
?
ρ[γ+](?b⊥,?k⊥,x,+? ez) + ρ[γ+](?b⊥,?k⊥,x,−? ez)
?
(14)
is the Wigner distribution of unpolarized quarks in an unpolarized nucleon;
?
represents the distortion due to the longitudinal polarization of the nucleon;
?
represents the distortion due to the longitudinal polarization of the quarks, and
?
represents the distortion due to the correlation between quark and nucleon longitudinal
ρLU(?b⊥,?k⊥,x) =1
2
ρ[γ+](?b⊥,?k⊥,x,+? ez) − ρ[γ+](?b⊥,?k⊥,x,−? ez)
?
(15)
ρUL(?b⊥,?k⊥,x) =1
2
ρ[γ+γ5](?b⊥,?k⊥,x,+? ez) + ρ[γ+γ5](?b⊥,?k⊥,x,−? ez)
?
(16)
ρLL(?b⊥,?k⊥,x) =1
2
ρ[γ+γ5](?b⊥,?k⊥,x,+? ez) − ρ[γ+γ5](?b⊥,?k⊥,x,−? ez)
?
(17)
polarizations. These four contributions can be written as
ρUU(?b⊥,?k⊥,x) = F1,1(x,0,?k2
ρLU(?b⊥,?k⊥,x) = −1
⊥,?k⊥·?b⊥,?b2
∂
∂bj
⊥
∂
∂bj
⊥
⊥),(18a)
M2ǫij
1
M2ǫij
⊥ki
⊥
F1,4(x,0,?k2
⊥,?k⊥·?b⊥,?b2
⊥), (18b)
ρUL(?b⊥,?k⊥,x) =
⊥ki
⊥
G1,1(x,0,?k2
⊥,?k⊥·?b⊥,?b2
⊥), (18c)
ρLL(?b⊥,?k⊥,x) = G1,4(x,0,?k2
⊥,?k⊥·?b⊥,?b2
⊥),(18d)
where the distributions X = F1,1,F1,4,G1,1,G1,4are the Fourier transforms of the correspond-
ing GTMDs X = F1,1,F1,4,G1,1,G1,4introduced in Ref. [31]
X(x,ξ,?k2
⊥,?k⊥·?b⊥,?b2
⊥) =
?
d2∆⊥
(2π)2e−i?∆⊥·?b⊥X(x,ξ,?k2
⊥,?k⊥·?∆⊥,?∆2
⊥). (19)
In Eq. (18) the two-dimensional antisymmetric tensor ǫij
M is the nucleon mass and roman indices are to be summed over. Integrating out?b⊥
or?k⊥ kills the contributions ρLU and ρUL, showing that there exists no TMD or GPD
⊥has been used with ǫ12= −ǫ21= 1,
corresponding to F1,4and G1,1. These GTMDs carry therefore completely new information
about the nucleon structure. On the other hand, the contributions ρUU and ρLL survive
both integrations. It follows that the GTMD F1,1can be seen as the mother distribution of
the TMD f1and the GPD H
f1(x,?k2
⊥) =
?
?
d2b⊥F1,1(x,0,?k2
⊥,?k⊥·?b⊥,?b2
⊥) = F1,1(x,0,?k2
⊥,0,0),(20a)
H(x,0,?∆2
⊥) =d2k⊥F1,1(x,0,?k2
⊥,?k⊥·?∆⊥,?∆2
⊥),(20b)
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and the GTMD G1,4as the mother distribution of the TMD g1Land the GPD˜H
?
?
g1L(x,?k2
⊥) =d2b⊥G1,4(x,0,?k2
⊥,?k⊥·?b⊥,?b2
⊥) = G1,4(x,0,?k2
⊥,0,0), (20c)
˜H(x,0,?∆2
⊥) =d2k⊥G1,4(x,0,?k2
⊥,?k⊥·?∆⊥,?∆2
⊥). (20d)
Integrating out all the variables, one naturally gets
?
?
?
?
dxd2k⊥d2b⊥ρq
UU(?b⊥,?k⊥,x) = Nq, (21a)
dxd2k⊥d2b⊥ρq
LU(?b⊥,?k⊥,x) = 0, (21b)
dxd2k⊥d2b⊥ρq
UL(?b⊥,?k⊥,x) = 0, (21c)
dxd2k⊥d2b⊥ρq
LL(?b⊥,?k⊥,x) = ∆q,(21d)
where the index q indicates the contribution of the quark of flavor q, Nqis the valence-quark
number (Nu= 2 and Nd= 1 in the proton) and ∆q is the axial charge. Note that Eq. (21b)
tells us that the valence-quark number does not depend on the nucleon polarization and
Eq. (21c) means that in an unpolarized nucleon there is no net quark polarization.
D. Quark Orbital Angular Momentum
Quantifying quark orbital angular momentum (OAM) inside the nucleon is essential in
order to solve the so-called “spin crisis”, see e.g. [46, 47]. Almost 15 years ago, Ji derived a
sum rule that allows one to extract the total quark contribution to the nucleon spin from a
combination of GPDs [48]
Jq
z=1
2
?
dxx[Hq(x,0,0) + Eq(x,0,0)]. (22)
By subtracting half of the axial charge ∆q =
?dx˜Hq(x,0,0) which is interpreted as the
spin contribution of quarks with flavor q to the nucleon spin, one gets the quark OAM
contribution
Lq
z=1
2
?
dx
?
x[Hq(x,0,0) + Eq(x,0,0)] −˜Hq(x,0,0)
?
.(23)
From a density point of view, this result is surprising in the sense that the extraction of
the quark OAM along the z-axis involves the GPD E which appears only in a transversely
polarized nucleon. Note however that E describes the amplitude where the nucleon spin
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flips while the quark light-cone helicities remain unaffected, implying therefore a change by
one unit of OAM between the initial and final nucleon states.
More recently it has been suggested, based on some quark models, that the TMD h⊥
1T
may also be related to the quark OAM [49–52]
Lq
z= −
?
dxd2k⊥
?k2
2M2h⊥q
⊥
1T(x,?k2
⊥). (24)
Note that one expects in general Lq
z?= Lq
zin a gauge theory, see e.g. [53]. Once again,
from a density point of view, this expression is surprising in the sense that it involves the
TMD h⊥
1Twhich describes the distribution of transversely polarized quarks in a transversely
polarized nucleon. Note however that h⊥
1Tcorresponds to the amplitude where the nucleon
and active quark longitudinal polarizations flip in opposite directions, involving therefore a
change by two units of OAM between the initial and final nucleon states.
Clearly, Wigner distributions provide much more information than GPDs and TMDs
as they contain also the full correlations between quark transverse position and three-
momentum. Furthermore, once the Wigner distributions are known, it is rather straightfor-
ward to compute physical observables. One has just to take the phase-space average as if
the Wigner distributions were classical distributions
?? A?[Γ](?S) =
?
dxd2k⊥d2b⊥A(?b⊥,?k⊥,x)ρ[Γ](?b⊥,?k⊥,x,?S). (25)
In particular, we can write the average quark OAM in a nucleon polarized in the z-direction
as
ℓq
z≡ ??Lq
z?[γ+](? ez) =
?
?
dxd2k⊥d2b⊥
??b⊥×?k⊥
??b⊥×?k⊥
?
?
zρ[γ+]q(?b⊥,?k⊥,x,? ez)
=dxd2k⊥d2b⊥
z[ρq
UU(?b⊥,?k⊥,x) + ρq
LU(?b⊥,?k⊥,x)]. (26)
From Eq. (18a), it is clear that
?
dxd2k⊥d2b⊥
??b⊥×?k⊥
?
zρq
UU(?b⊥,?k⊥,x) = 0,(27)
which means that in an unpolarized nucleon there is no net quark OAM1. Using now
1An unpolarized nucleon has no spin, which means that the total quark and gluon angular momentum
contributions have to sum up to zero. By rotational invariance, one expects all the four contributions
(spin and OAM of quarks and gluons) to vanish identically. The angular momentum sum rule for an
unpolarized nucleon is therefore trivially satisfied.
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Eq. (18b) and integrating by parts, we find that the quark OAM ℓq
zreads
ℓq
z= −
?
dxd2k⊥
?k2
M2Fq
⊥
1,4(x,0,?k2
⊥,0,0). (28)
An interesting issue which deserves further investigation is the relation between Lq
zin
Eq. (23) and ℓq
zin Eq. (28). As discussed in the following sections, in models without
gauge-field degrees of freedom one finds that the two definitions give the same results for
the total quark contribution to the OAM, but not for the separate quark-flavor contributions.
However, this remains to be confirmed in more complex systems, when the contribution of
the Wilson line is explicitly taken into account.
Wigner distributions allow us also to study the correlation between quark spin and OAM,
which we define as
Cq
z≡
?
?
dxd2k⊥d2b⊥
??b⊥×?k⊥
?
zρq
UL(?b⊥,?k⊥,x)
=dxd2k⊥
?k2
M2Gq
⊥
1,1(x,0,?k2
⊥,0,0),
(29)
where we have used Eq. (18c). For Cq
z> 0 the quark spin and OAM tend to be aligned,
while for Cq
z< 0 they tend to be antialigned. Finally, note that from Eq. (18d) one has
?
dxd2k⊥d2b⊥
??b⊥×?k⊥
?
zρLL(?b⊥,?k⊥,x) = 0, (30)
reflecting like Eqs. (21b), (21c) and (27) the isotropy of space.
III. RESULTS AND DISCUSSIONS
Since it is not known how to extract Wigner distributions or GTMDs from experiments,
one has to rely on phenomenological models. We studied the Wigner distributions in the
light-cone constituent quark model (LCCQM) [32, 33, 35] and the light-cone version of the
chiral quark-soliton model (χQSM) restricted to the three-quark sector [38, 40, 41, 54, 55],
using the general formalism developed in Ref. [34] for the overlap representation of the quark-
quark correlator in terms of light-cone wave functions. We neglect the contribution from
gauge degrees of freedom, and in particular from the Wilson line in the Wigner operator (1).
As the resulting distributions are very similar in both models, we will present only those
from the LCCQM. However, when discussing more quantitative aspects, we will also report
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the numerical values from the χQSM. Furthermore, we will discuss only the first x moment
of the Wigner distributions
ρ(?b⊥,?k⊥) ≡
?
dxρ(?b⊥,?k⊥,x), (31)
i.e. purely transverse four-dimensional phase-space distributions (two transverse position
and two transverse momentum coordinates), referred to as transverse Wigner distributions.
A.Unpolarized Quarks in an Unpolarized Nucleon
We start the discussions with ρUU(?b⊥,?k⊥), the transverse Wigner distribution of unpo-
larized quarks in an unpolarized proton. In Fig. 1 we present the distributions in impact-
parameter space with fixed transverse momentum?k⊥ = k⊥ˆ ey and k⊥ = 0.3 GeV (upper
panels), and compare them with the distribution in transverse-momentum space with fixed
impact parameter?b⊥= b⊥ˆ ey and b⊥= 0.4 fm (lower panels). The left (right) panels re-
fer to the u (d) quarks. We observe a distortion in all these distributions which indicates
that the configuration?b⊥⊥?k⊥is favored with respect to the configuration?b⊥??k⊥. This
can be understood with naive semi-classical arguments. The radial momentum (?k⊥·ˆb⊥)ˆb⊥
(ˆb⊥≡?b⊥/b⊥) of a quark is expected to decrease rapidly in the periphery because of confine-
ment. The polar momentum?k⊥−(?k⊥·ˆb⊥)ˆb⊥receives a contribution from the orbital motion
of the quark which can still be significant in the periphery (in an orbital motion, one does
not need to reduce the momentum to avoid a quark escape). This naive picture also suggests
that this phenomenon should be stronger as we go to peripheral regions (b⊥≫) and to high
quark momenta (k⊥≫). Such a behavior is indeed observed in our model calculations and
can be quantified in terms of the average quadrupole distortions Qij
b(?k⊥) and Qij
k(?b⊥) defined
as
Qij
b(?k⊥) = Qb(k⊥)
?
?
2ˆkiˆkj− δij?
2ˆbiˆbj− δij?
=
?d2b⊥
?d2k⊥
?2bi
?d2b⊥b2
?2ki
?d2k⊥k2
⊥bj
⊥− δijb2
⊥ρUU(?b⊥,?k⊥)
⊥kj
⊥ρUU(?b⊥,?k⊥)
⊥
?ρUU(?b⊥,?k⊥)
?ρUU(?b⊥,?k⊥)
,(32a)
Qij
k(?b⊥) = Qk(b⊥)=
⊥− δijk2
⊥
, (32b)
where i,j = x,y. The distortions calculated in the LCCQM are tabulated in Table I. We
note that the quadrupole distortions of u and d quarks are very similar. For increasing
values of k⊥and b⊥the distortions get more and more pronounced, and at the same time
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FIG. 1: The transverse Wigner distributions of unpolarized quarks in an unpolarized proton. Upper panels:
distributions in impact-parameter space with fixed transverse momentum?k⊥= k⊥ˆ ey and k⊥= 0.3 GeV.
Lower panels: distributions in transverse-momentum space with fixed impact parameter?b⊥ = b⊥ˆ ey and
b⊥= 0.4 fm. The left (right) panels show the results for u (d) quarks.
the spread of the distributions shrinks towards the center. From Fig. 1 we also note that
the spread of the distributions is smaller for u quarks than for d quarks, especially in the
transverse-coordinate space. This reflects the fact that u quarks are more concentrated at
the center of the proton, while the d-quark distribution has a tail which extends further at
the periphery of the proton.
From Eq. (18a) we see that ρUU(?b⊥,?k⊥) = ρUU(b⊥,k⊥,?k⊥·?b⊥). This explains the left-right
symmetry in Fig. 1 and implies that the quark is as likely to rotate clockwise as to rotate
anticlockwise. In Fig. 1 we also observe a top-bottom symmetry. Such a symmetry is not
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TABLE I: The average quadrupole distortions of the transverse Wigner distribution of unpolarized quarks
in an unpolarized proton from the LCCQM. See Eq. (32) for the definition of Qb(k⊥) and Qk(b⊥).
k⊥[GeV]
Qb(k⊥)
b⊥[fm]
Qk(b⊥)
udud
000000
0.1−0.04−0.030.2−0.15−0.10
0.2
−0.14−0.130.4−0.24−0.19
0.3−0.30−0.270.6−0.29−0.27
a general property of the Wigner distribution ρUU, but follows from the fact that in our
calculations there are no explicit gluons. Indeed, time-reversal invariance implies that the
real part of the GTMDs is T-even (e) while the imaginary part is T-odd (o). Hermiticity
tells us that the four GTMDs X = F1,1,F1,4,G1,1,G1,4satisfy the relations
Xe(x,ξ,?k2
⊥,?k⊥·?∆⊥,?∆2
⊥) = Xe(x,−ξ,?k2
⊥,−?k⊥·?∆⊥,?∆2
⊥),
Xo(x,ξ,?k2
⊥,?k⊥·?∆⊥,?∆2
⊥) = −Xo(x,−ξ,?k2
⊥,−?k⊥·?∆⊥,?∆2
⊥).
This means that for ξ = 0, Xeis an even function of?k⊥·?∆⊥while Xois an odd function of
?k⊥·?∆⊥. It follows that the Fourier transforms X of these GTMDs with respect to?∆⊥are
real-valued functions. The contribution Xeis an even function of?k⊥·?b⊥while Xois an odd
function of?k⊥·?b⊥. Since we have no explicit gluons and therefore no final-state interactions,
our GTMDs are real. It follows that X = Xeexplaining the top-bottom symmetry of Fig. 1.
The dominant effect of final-state interactions would be to shift up or down the distributions.
As mentioned earlier, Wigner distributions have only a quasi-probabilistic interpretation
due to Heisenberg uncertainty relations. A genuine probabilistic interpretation can be re-
covered only when integrating out certain variables. If we integrate out?b⊥or?k⊥, we reduce
to the unpolarized TMD and GPD, respectively. In these cases, the distortion we observed
in the Wigner distribution ρUUis completely absent and we are left with axially symmetric
distributions, see Eq. (20). By integrating over one momentum and one coordinate variables
which are not conjugated, we obtain the probability densities ˜ ρ of Eqs. (10) and (11). In
Fig. 2 we show the probability density ˜ ρUU(bx,ky) which gives the correlation between bx
and ky. We observe that ˜ ρUU(bx,ky) is maximum at the center, bx= ky= 0, and decreases
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FIG. 2: The mixed transverse densities ˜ ρ(bx,ky) of unpolarized u quarks (left panel) and unpolarized d
quarks (right panel) in an unpolarized proton.
in the outer regions of the phase space, with the equidensity lines in each quadrant of Fig. 2
having approximately a linear dependence in bxand ky. Furthermore, we clearly see that
the width of the densities in kyis similar for u and d quarks, while it is more extended in bx
for d quarks than for u quarks.
B. Unpolarized Quarks in a Longitudinally Polarized Nucleon
We now consider ρLU(?b⊥,?k⊥), the distortion of the transverse Wigner distribution of un-
polarized quarks due to the longitudinal polarization of the proton. In Fig. 3, the upper
panels show the distortions in impact-parameter space for u (left panels) and d (right panels)
quarks with fixed transverse momentum?k⊥= k⊥ˆ eyand k⊥= 0.3 GeV, while the lower pan-
els give the corresponding distortions in the transverse-momentum space with fixed impact
parameter?b⊥= b⊥ˆ eyand b⊥= 0.4 fm. We observe a clear dipole structure in both these
distributions, with opposite sign for u and d quarks. The corresponding distortions of the
mixed transverse densities ˜ ρ(bx,ky) are shown in Fig. 4 for the u (left panel) and d (right
panel) quarks. In this case, we observe a quadrupole structure. These multipole structures
are due to the explicit factor ǫij
⊥ki
⊥
∂
∂bj
⊥
in Eq. (18b) which breaks the left-right symmetry
in Fig. 3 allowing therefore non-vanishing net OAM. We learn from these figures that the
OAM of u quarks tends be aligned with the nucleon spin, while the OAM of d quarks tends
15