PHYSICAL REVIEW D 79, 034006 (2009)
Chiral-odd generalized parton distributions in transverse
and longitudinal impact parameter spaces
D. Chakrabarti,1 R. Manohar,2 and A. Mukherjee2
1
Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
2
Department of Physics, Indian Institute of Technology, Powai, Mumbai 400076, India
(Received 5 November 2008; published 6 February 2009)
We investigate the chiral-odd generalized parton distributions for nonzero skewness in transverse and
longitudinal position spaces by taking Fourier transform with respect to the transverse and longitudinal
momentum transfer, respectively. We present overlap formulas for the chiral-odd generalized parton
distributions in terms of light-front wave functions (LFWFs) of the proton both in the EfremovRadyushkin-Brodsky-Lepage and Dokshitzer-Gribov-Lipatov-Altarelli-Parisi regions. We calculate
them in a field theory inspired model of a relativistic spin-1=2 composite state with the correct correlation
between the different LFWFs in Fock space, namely, that of the quantum fluctuations of an electron in a
generalized form of QED. We show the spin-orbit correlation effect of the two-particle LFWF as well as
the correlation between the constituent spin and the transverse spin of the target.
DOI: 10.1103/PhysRevD.79.034006
PACS numbers: 12.38.Bx, 12.38.Aw, 13.88.+e
I. INTRODUCTION
Generalized parton distributions (GPDs) give a unified
picture of the nucleon in the sense that x moments of them
give the form factors accessible in exclusive processes
whereas in the forward limit they reduce to parton distributions accessible in inclusive processes (see [1] for example). At zero skewness , if one performs a Fourier
transform (FT) of the GPDs with respect to the momentum
transfer in the transverse direction ? , one gets the socalled impact parameter dependent parton distributions,
which tell us how the partons of a given longitudinal
momentum are distributed in transverse position (or impact
parameter b? ) space. These obey certain positivity constraints and unlike the GPDs themselves, have probabilistic
interpretation [2]. As transverse boost on the light front is a
Galilean boost, there are no relativistic corrections. Impact
parameter dependent parton distribution functions (PDFs)
are defined for nucleon states localized in the transverse
position space at R? . In order to avoid a singular normalization constant, one can take a wave packet state. A wave
packet state which is transversely polarized is shifted sideways in the impact parameter space [3]. An interesting
interpretation of Ji’s angular momentum sum rule [4] is
obtained in terms of the impact parameter dependent PDFs
[3]. On the other hand, in [5], real and imaginary parts of
the deeply virtual Compton scattering (DVCS) amplitudes
are expressed in longitudinal position space by introducing
a longitudinal impact parameter conjugate to the skewness , and it was shown that the DVCS amplitude show a
certain diffraction pattern in the longitudinal position
space. Since Lorentz boosts are kinematical in the front
form, the correlation determined in the three-dimensional
b? , space is frame independent. As GPDs depend on a
sharp x, the Heisenberg uncertainty relation restricts the
longitudinal position space interpretation of GPDs themselves. It has, however, been shown in [6] that one can
1550-7998= 2009=79(3)=034006(11)
define a quantum mechanical Wigner distribution for the
relativistic quarks and gluons inside the proton. Integrating
over k and k? , one obtains a four-dimensional quantum
distribution which is a function of r~ and kþ where r~ is the
quark position vector defined in the rest frame of the
proton. These distributions are related to the FT of GPDs
in the same frame. This gives a 3D position space picture of
the GPDs and of the proton, within the limitations mentioned above.
At leading twist, there are three forward parton distributions (PDFs), namely the unpolarized, helicity, and
transversity distributions. Similarly, three leading twist
generalized quark distributions can be defined which in
the forward limit reduce to these three forward PDFs. The
third one is chiral-odd and is called the generalized transversity distribution FT . This is defined as the off-forward
matrix element of the bilocal tensor charge operator. It is
~ T , ET ,
parametrized in terms of four GPDs, namely HT , H
~
and ET in the most general way [3,7,8]. Unlike E, which
gives a sideways shift in the unpolarized quark density in a
transversely polarized nucleon, the chiral-odd GPDs affect
the transversely polarized quark distribution both in unpolarized and in the transversely polarized nucleon in various
ways. A relation for the transverse total angular momentum of the quarks has been proposed in [3], in analogy with
Ji’s relation, which involves a combination of second mo~ T in the forward limit. E~T does not
ments of HT , ET , and H
contribute when skewness ¼ 0, as it is an odd function of
. HT reduces to the transversity distribution in the forward
limit when the momentum transfer is zero. Unlike the
chiral-even GPDs, information about which can be and
has been obtained from deeply virtual Compton scattering
and hard exclusive meson production, it is very difficult to
measure the chiral-odd GPDs. That is because, being chiral
odd, they have to combine with another chiral-odd object
in the amplitude. In [9], a proposal to measure HT has been
034006-1
Ó 2009 The American Physical Society
D. CHAKRABARTI, R. MANOHAR, AND A. MUKHERJEE
PHYSICAL REVIEW D 79, 034006 (2009)
given in photo or electroproduction of a longitudinally
polarized vector meson 0 via two gluon fusion; this
meson is separated by a large rapidity gap from other
transversely polarized þ and the scattered neutron. The
scattering amplitude factorizes and involves HT ðx; ; 0Þ at
zero momentum transfer as well as the chiral-odd lightcone distribution amplitude for the transversely polarized
meson. In [10], the exclusive process P ! 0 P has been
suggested to measure the tensor charge. However, one has
to look at the helicity flip part which is a higher twist
contribution. There is also a prospect of gaining information about the Mellin moments of chiral-odd GPDs from
lattice QCD [11]. They have been investigated in several
models, the first being the bag model [12], where only HT
has been found to be nonzero. In [13], they have been
calculated in a constituent quark model, a modelindependent overlap in the Dokshitzer-Gribov-LipatovAltarelli-Parisi (DGLAP) region is also given. HT ðx; ; 0Þ
is also modeled in the Efremov-Radyushkin-BrodskyLepage (ERBL) region x < in [9]. The possibility of
getting model-independent relations between transverse
momentum-dependent parton distributions and impact parameter representation of GPDs at ¼ 0 has been investigated in [14]; however it was concluded that such
relations are model dependent. In a previous work we
have investigated the chiral-odd GPDs for a simple spin-12
composite particle for ¼ 0 in impact parameter space
[15].
In this work, we present overlap formulas for the chiralodd GPDs in the terms of the light-front wave functions
(LFWFs) both in the DGLAP (n ! n) and ERBL (n þ
1 ! n 1) regions. We investigate them in a simple
model, namely, for the quantum fluctuations of a lepton
in QED at one-loop order [16], the same system which
gives the Schwinger anomalous moment =2. We generalize this analysis by assigning a mass M to the external
electrons and a different mass m to the internal electron
lines and a mass to the internal photon lines with M <
m þ for stability. In effect, we shall represent a spin- 12
system as a composite of a spin- 12 fermion and a spin-1
vector boson [5,17–20]. This field theory inspired model
has the correct correlation between the Fock components
of the state as governed by the light-front eigenvalue
equation, something that is extremely difficult to achieve
in phenomenological models. Also, it gives an intuitive
understanding of the spin and orbital angular momentum
of a composite relativistic system [21]. GPDs in this model
satisfy general properties like polynomiality and positivity.
So it is interesting to investigate the properties of GPDs in
this model. By taking the Fourier transform (FT) with
respect to ? , we express the GPDs in transverse position
space and by taking a FT with respect to we express them
in longitudinal position space.
0
FjT ;
II. OVERLAP REPRESENTATION
The chiral-odd GPDs are expressed as the off-forward
matrix element of the bilocal tensor charge operator on the
light cone. These involve a helicity flip of the quark.
We use the parametrization of [3] for the chiral-odd
GPDs:
Z dz þ z
z
iP
z
=2
0
0
þj
e
5 c
j P; i
¼
hP ; j c
2
2
2
P
~ T ðx; ; tÞþj uðP
0Þ
0Þ
0 Þþj 5 uðPÞ þ H
uðPÞ
uðPÞ þ ET ðx; ; tÞþj uðP
¼ HT ðx; ; tÞuðP
2
2M
M
P 0Þ
uðPÞ:
þ E~T ðx; ; tÞþj uðP
M
Pþ
We choose the frame where the initial and final momenta
of the proton with mass M are
M2
þ
P ¼ P ; 0? ; þ ;
(2)
P
M2 þ 2?
:
P0 ¼ ð1 ÞPþ ; ? ;
ð1 ÞPþ
So, the momentum transferred from the target is
t þ 2?
¼ P P0 ¼ Pþ ; ? ;
;
Pþ
(3)
(4)
where t ¼ 2 . Following [22] we expand the proton state
of momentum P and helicity in terms of multiparticle
light-front wave functions:
(1)
X
n
n
n
XY
X
dxi d2 k?i
3
2
j P; i ¼
16 1 xi k?i
pffiffiffiffi
3
n i¼1 xi 16
i¼1
i¼1
c n ðxi ; k?i ; i Þ j n; xi Pþ ; xi P? þ k?i ; i i; (5)
þ
here xi ¼ kþ
i =P is the light cone momentum fraction and
k?i represent the relative transverse momentum of the ith
constituent. The physical transverse momenta are p?i ¼
xi P? þ k?i . i are the light cone helicities. The light-front
wave functions c n are independent of Pþ and P? and are
boost invariant.
Like the chiral-even GPDs, here too there are diagonal
n ! n overlaps in the kinematical region < x < 1 and
1 < x < 0. In the region 0 < x < there are offdiagonal n þ 1 ! n 1 overlaps. The overlap representation of the chiral-odd GPDs in terms of light-front wave
034006-2
CHIRAL-ODD GENERALIZED PARTON DISTRIBUTIONS . . .
PHYSICAL REVIEW D 79, 034006 (2009)
functions is given by
0
T ;
¼ ð1 Þ1ðn=2Þ
F1;n!n
XZ
ni¼1
n;i
n
X X
dxi d2 ki?
j
3
2
0 0 0i
0
i
16
1
x
k
j
? ðx x1 Þ c n ðxi ; k? ; i Þ c n ðxi ; k? ; i Þ
3
16
j
j¼1
01 ;1 ½0i ;i ði ¼ 2; . . . ; nÞ;
(6)
xi
xi
1x1
1
1 where x0i ¼ 1
, k0i? ¼ ki? þ 1
? for i ¼ 2; . . . ; n and x01 ¼ x1
, k01
? ¼ k? 1 ? .
0
T ;
¼ ð1 Þ3=2n=2
F1;nþ1!n1
XZ
n;i
nþ1
i¼1
nþ1
X nþ1
X dxi d2 ki?
3 Þ2 1 2
ð16
x
ki?
j
163
j¼1
j¼1
0
ðx0 ; k0i ; 0 Þ c ðx ; ki ; Þ
ðxnþ1 þ x1 Þ2 ðk?nþ1 þ k1? ? Þðx x1 Þ c n1
i
i ?
i
nþ1 i ?
01 ;nþ1 ½0i ;i ði ¼ 2; . . . ; nÞ;
(7)
xi
xi
where x0i ¼ 1
, k0i? ¼ ki? þ 1
? , for i ¼ 2; . . . ; n label the n 1 spectators. The overlaps are different from the chiraleven GPDs as there is a helicity flip of the quark. The above overlap formulas can be used in any model calculation of the
chiral-odd GPDs using LFWFs.
III. CHIRAL-ODD GENERALIZED PARTON DISTRIBUTIONS IN QED AT ONE LOOP
Following [5,16], we take a simple composite spin- 12 state, namely, an electron in QED at one loop to investigate the
GPDs. The light-front Fock state wave functions corresponding to the quantum fluctuations of a physical electron can be
systematically evaluated in QED perturbation theory. The state is expanded in Fock space and there are contributions from
je i and je e eþ i in addition to renormalizing the one-electron state. The two-particle state is expanded as
j"two particle ðPþ ; P~ ?
¼ 0~ ? Þi ¼
dxd2 k~?
1
"
þ
~
~
c þð1=2Þþ1 ðx; k? Þ
þ þ 1; xP ; k?
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
xð1 xÞ163
1
1
"
þ ~
þ ~
~
;
k
c
ðx;
k
Þ
;
k
1;
xP
þ
1;
xP
þ c "þð1=2Þ1 ðx; k~? Þ
þ
þ
?
? ?
þð1=2Þþ1
2
2
1
þ 1 1; xPþ ; k~? þ c "
þ 1; xPþ ; k~?
~? Þ
þ c "þð1=2Þ1 ðx; k~? Þ
ðx;
k
ð1=2Þþ1
2
2
1 1; xPþ ; k~? ;
þ c "ð1=2Þ1 ðx; k~? Þ
2
Z
(8)
where the two-particle states jszf ; szb ; x; k~? i are normalized as in [22]. szf and szb denote the z component of the spins of the
constituent fermion and boson, respectively,
and the variables x and k~? refer to the momentum of the fermion. The light
P
kþ
i
cone momentum fraction xi ¼ Pþ satisfy 0 < xi 1 and i xi ¼ 1. We employ the light cone gauge Aþ ¼ 0, so that the
gauge boson polarizations are physical. The three-particle state has a similar expansion. Both the two- and three-particle
Fock state components are given in [22]. Here we give the two-particle wave function for the spin-up electron [16,21,22]
pffiffiffi k1 þ ik2
’;
c "þð1=2Þþ1 ðx; k~? Þ ¼ 2
xð1 xÞ
pffiffiffi
m
"
~
c ð1=2Þþ1 ðx; k? Þ ¼ 2 M ’;
x
pffiffiffi k1 þ ik2
’;
c "þð1=2Þ1 ðx; k~? Þ ¼ 2
1x
c "ð1=2Þ1 ðx; k~? Þ
¼ 0;
e
1
’ðx; k~? Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi
:
2
2
~
þm
k
1 x M2 ? 2 k~? þ2
x
1x
Similarly, the wave function for an electron with negative helicity can also be obtained.
034006-3
(9)
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D. CHAKRABARTI, R. MANOHAR, AND A. MUKHERJEE
PHYSICAL REVIEW D 79, 034006 (2009)
Following the same references, we work in a generalized
form of QED by assigning a mass M to the external
electrons, a different mass m to the internal electron lines,
and a mass to the internal photon lines. The idea behind
this is to model the structure of a composite fermion state
with mass M by a fermion and a vector ‘‘diquark’’ constituent with respective masses m and . The electron in
QED also has a one-particle component
sector of the Fock space, which is strictly at x ¼ 1 (wave
function renormalization) [19]. We exclude x ¼ 1 by imposing a cutoff and we do not consider this contribution in
this work. It contributes also in the 3 ! 1 overlap in the
ERBL region. However, the single particle contribution is
important as it cancels the singularity as x ! 1. This has
been shown explicitly in the forward limit in [23], namely,
for the transversity distribution h1 ðxÞ. The coefficient of
the logarithmic term in the expression of h1 ðxÞ gives the
correct splitting function for leading order evolution of
h1 ðxÞ; the delta function from the single particle sector
providing the necessary ‘‘plus’’ prescription. In the offforward case, the cancellation occurs similarly, as shown
for Fðx; ; tÞ in [20]. The behavior at x ¼ 0; 1 can be
improved by differentiating the LFWFs with respect to
M2 [24]. The x ! 0 limit can be improved as well by a
different method [25].
The GPDs are zero in the domain 1 < x < 0, which
corresponds to emission and reabsorption of an eþ from a
physical electron. Contributions to the GPDs in that domain only appear beyond the one-loop level.
In our model F1T"# ¼ F1T#" and thus
j";#
one
¼
þ ~
particle ðP ; P?
¼ 0~ ? Þi
Z dxd2 k~?
1
3
2 ~
þ ~
pffiffiffi
;
xP
16
ð1
xÞ
ð
k
Þ
c
;
k
?
ð1Þ ? ;
2
x163
(11)
where the one-constituent wave function is given by
pffiffiffiffi
c ð1Þ ¼ Z:
(12)
pffiffiffiffi
Here Z is the wave function renormalization of the oneparticle state and ensures overall probability conservation.
Also, in order to regulate the ultraviolet divergences we use
a cutoff on the transverse momentum k? . If instead of
imposing a cutoff on transverse momentum we imposed
a cutoff on the invariant mass [21], then the divergences at
x ¼ 1 would have been regulated by the nonzero photon
mass.
In the domain < x < 1, there are diagonal 2 ! 2 overlaps. These correspond to [setting j ¼ 1 in Eq. (1)] the
helicity nonflip F1T"" , F1T## , and helicity flip contributions,
F1T"# and F1T#" , respectively, which can be written as
F1T"" ¼
F1T## ¼
(13)
ð1 ÞA2 ¼
From the above equations we have
M
ET ðx; ; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ðA1 A2 Þ;
1
Z d2 k?
¼
½ c 1=2 ðx0 ; k0? Þ c 1=2
1=21 ðx; k? Þ
163 1=21
1=2
0 0
þ c 1=2
1=2þ1 ðx ; k? Þ c 1=2þ1 ðx; k? Þ;
M
E~ T ðx; ; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi fð2 ÞA2 A1 g;
2 1
(16)
where
x
;
x0 ¼
1
k0? ¼ k? ð1 x0 Þ? :
1
HT ðx; ; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ðA3 þ MA2 Þ:
2 1
(17)
F1T"# and F1T#" receive contribution from the single particle
(20)
F1T"# þ F1T#"
2
pffiffiffiffiffiffiffiffiffiffiffiffi
2
¼ i 2HT 1 pffiffiffiffiffiffiffiffiffiffiffiffi ET pffiffiffiffiffiffiffiffiffiffiffiffi E~T :
2 1
1
(21)
(14)
(15)
F1T"" F1T##
1
pffiffiffiffiffiffiffi ½ET þ 2E~T ;
¼
2
2M (19)
iA3 ¼
Z d2 k?
½ c 1=2 ðx0 ; k0? Þ c 1=2
1=21 ðx; k? Þ
163 1=21
1=2
0 0
þ c 1=2
1=21 ðx ; k? Þ c 1=21 ðx; k? Þ;
F1T#"
F1T"" þ F1T##
2
i2
pffiffiffiffiffiffiffiffiffiffiffiffi ½ð2 ÞET þ 2E~T ;
¼
2M 1 ði2 ÞA1 ¼
Z d2 k?
1=2
0 0
½ c 1=2
1=2þ1 ðx ; k? Þ c 1=2þ1 ðx; k? Þ
3
16
1=2
0 0
þ c 1=2
1=21 ðx ; k? Þ c 1=21 ðx; k? Þ;
(18)
Since the above equation is true for arbitrary 1 and 2 , it
~ T ¼ 0 in our model. We define the combinations:
implies H
Z d2 k?
½ c 1=2 ðx0 ; k0? Þ c 1=2
1=2þ1 ðx; k? Þ
163 1=2þ1
1=2
0 0
þ c 1=2
1=2þ1 ðx ; k? Þ c 1=2þ1 ðx; k? Þ;
F1T"# ¼
1
1 2
F1T"# F1T#" ¼ H
~ T pffiffiffiffiffiffiffiffiffiffiffiffi
¼ 0:
1 M2
2
(22)
(23)
(24)
Explicit matrix element calculation using Eqs. (13)–(16) in
our model gives
034006-4
CHIRAL-ODD GENERALIZED PARTON DISTRIBUTIONS . . .
e2
m
m
A1 ¼ 3 pffiffiffiffiffiffiffiffiffiffiffiffi
M xð1 Þ M 0 x0
x
x
8 1 m
(25)
ð1 x0 ÞI1 M xð1 xÞI2 ;
x
PHYSICAL REVIEW D 79, 034006 (2009)
QðyÞ ¼ yð1 yÞð1 x0 Þ2 2? yðM2 xð1 xÞ
m2 ð1 xÞ 2 xÞ ð1 yÞðM2 x0 ð1 x0 Þ
m2 ð1 x0 Þ 2 x0 Þ
and
A2 ¼
e2
m
p
ffiffiffiffiffiffiffiffiffiffiffiffi
ð1 xÞxI2
M
x
83 1 m
m
M ð1 Þx þ M 0 x0 ð1 x0 ÞI1 ;
x
x
(26)
A3 ¼
I1 ¼
I2 ¼
0
Z1
0
(28)
1
;
QðyÞ
(29)
dy
Z
d2 k?
0
d2 k?
So, we have the expressions for the GPDs
here
Bðx; Þ ¼ M2 x0 ð1 x0 Þ m2 ð1 x0 Þ 2 x0 þ
2
M xð1 xÞ m2 ð1 xÞ 2 x ð1 x0 Þ2 2? .
1y
dy
;
QðyÞ
0
1
2 0
0
k02
þ
M
x
ð1
x
Þ
m2 ð1 x0 Þ 2 x0
?
2
¼ ln
: (31)
2
0
2
0
2
0
0
m ð1 x Þ þ x M x ð1 x Þ I4 ¼
e2
ðx þ x0 Þ
½I3 I4 þ Bðx; ÞI2 ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
16 ð1 xÞð1 x0 Þ
(27)
Z1
Z
1
k2? þ M2 xð1 xÞ m2 ð1 xÞ 2 x
2
¼ ln
(30)
;
2
2
2
m ð1 xÞ þ x M xð1 xÞ I3 ¼
ET ðx; ; tÞ ¼ e2 2M
m
M
xð1 xÞI5 ;
x
83 1 e2 M
m
ð1 xÞ M x
E~T ðx; ; tÞ ¼ 3
x
8 1 m
m
þ M 0 x0 I1 þ M xð1 xÞI2 ;
x
x
(33)
where
3
4
(a)
(b)
-t=1.0
3
2.5
E T (x, ζ, t)
E T (x, ζ, t)
(32)
2
x=0.1
x=0.9
x=0.95
2
1
1.5
0
-t=0.1
-t=1.0
-t=5.0
ζ=0.2
1
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
ζ
x
FIG. 1 (color online). Plot of ET ðx; ; tÞ (a) vs x for a fixed value of ¼ 0:2 and different values of t in MeV2 ; (b) vs for different
values of x and for fixed t ¼ 1 MeV2 .
034006-5
D. CHAKRABARTI, R. MANOHAR, AND A. MUKHERJEE
PHYSICAL REVIEW D 79, 034006 (2009)
2
0.3
-t=0.1
-t=1.0
-t=5.0
ζ=0.2
(a)
1.5
x=0.1
x=0.5
x=0.9
-t=1.0
0.2
0.4
(b)
0.2
1
0.1
0
0.2
0.5
0.4
0.8
0.6
x
0
1
0
0.8
0.6
ζ
FIG. 2 (color online). Plot of E~T ðx; ; tÞ (a) vs x for a fixed value of ¼ 0:2 and different values of t in MeV2 ; (b) vs for different
values of x and for fixed t ¼ 1 MeV2 .
HT ðx; ; tÞ ¼
4
e2 x þ x0
ln
3 2 2ð1 xÞ
DD0
8
x þ x0
M
m
M
Bðx; Þ þ
þ
1
x
2ð1 xÞ
M
m
M xð1 Þ
xð1 xÞ I2 1
x
m
þ M 0 x0 ð1 x0 ÞI1 ;
(34)
x
I5 ¼
100
-t=0.1
-t=1.0
-t=5.0
(b)
(a)
ζ=0.2
Λ=10
40
HT(x,ζ,t)
HT(x,ζ,t)
y
;
QðyÞ
50
60
40
30
20
-t=1.0, Λ=10.0
x=0.2
x=0.5
x=0.7
10
20
0
0.2
0
dy
D ¼ M2 xð1 xÞ m2 ð1 xÞ 2 x,
and
D0 ¼
M2 x0 ð1 x0 Þ m2 ð1 x0 Þ 2 x0 .
All the numerical plots are performed in units of
e2 =ð83 Þ. We took M ¼ 0:51 MeV, m ¼ 0:5 MeV,
and ¼ 0:02 MeV. In Figs. 1(a), 2(a), and 3(a), we have
plotted the GPDs ET , E~T , and HT , as functions of x for
fixed values of ¼ 0:2 and a fixed value of t. HT has
logarithmic divergence similar to the transversity distribution [23]. For numerical analysis, we have used a cutoff
with
80
Z1
0.4
0.6
0.8
x
0
0
0.1
0.2
0.3
0.4
0.5
0.6
ζ
FIG. 3 (color online). Plot of HT ðx; ; tÞ (a) vs x for a fixed value of ¼ 0:2 and different values of t in MeV2 ; (b) vs for different
values of x and for fixed t ¼ 1 MeV2 . is in MeV.
034006-6
CHIRAL-ODD GENERALIZED PARTON DISTRIBUTIONS . . .
120
HT(x,ζ,t)
100
Λ=10
Λ=20
Λ=50
-t=1.0
ζ=0.2
80
60
40
20
0
0.2
0.4
0.6
0.8
x
FIG. 4 (color online).
MeV).
HT ðx; ; tÞ for different UV cutoff (in
¼ 10 MeV ( M) on transverse momentum. In
Fig. 4, we show the cutoff dependence of HT . However,
one has to incorporate the single particle contribution to get
the correct þ distribution and also to get a finite answer at
x ¼ 1, as stated above. Both ET and E~T are independent of
x at small and medium x and at x ! 1, they are independent of t [2] and approach zero. Note that our analytic
expressions for HT and ET agree with the quark model
calculation of [14] without the color factors, in the limit
¼ 0, M ¼ m, and ¼ 0. Figures 1(b), 2(b), and 3(b)
present the behavior of the above GPDs for fixed t ¼
1:0 MeV2 and different values of x. Note that as we are
plotting in the DGLAP region, x > . Both ET and E~T
increases in magnitude with the increase of , but HT
decreases. E~T is zero at ¼ 0.
IV. GENERALIZED PARTON DISTRIBUTIONS IN
POSITION SPACE
Introducing the Fourier conjugate b? (impact parameter) of the transverse momentum transfer ? , the GPDs
can be expressed in impact parameter space. Like the
chiral-even counterparts, chiral-odd GPDs, as well, have
interesting interpretation in impact parameter space. The
~ T is related to the
second moment of HT , ET , and H
transverse component of the total angular momentum carried by transversely polarized quarks in an unpolarized
proton:
si Z
~ T ðx; 0; 0Þ þ ET ðx; 0; 0Þ:
hJ i i ¼
dxx½HT ðx; 0; 0Þ þ 2H
4
(35)
In the impact parameter space, at ¼ 0, the second two
terms denote a deformation in the transversity asymmetry
of quarks in an unpolarized target. This deformation is due
PHYSICAL REVIEW D 79, 034006 (2009)
to the spin-orbit correlation of the constituents [3,8]. This
is similar to the role played by the GPD Eðx; 0; 0Þ in Ji’s
sum rule for the longitudinal angular momentum. On the
~ T in impact paother hand, the combination HT 2Mt 2 H
rameter space gives the correlation between the transverse
quark spin and the spin of the transversely polarized
nucleon.
The above picture was proposed in the limit of zero
skewness in [3,8]. In most experiments is nonzero,
and it is of interest to investigate the chiral-odd GPDs in b?
space with nonzero . The probability interpretation is no
longer possible as now the transverse position of the initial
and final protons are different as there is a finite momentum
transfer in the longitudinal direction. The GPDs in impact
parameter space probe partons at transverse position j b? j
with the initial and final proton shifted by an amount of
order j b? j . Note that this is independent of x and even
when GPDs are integrated over x in an amplitude, this
information is still there [26]. Thus the chiral-odd GPDs in
impact parameter space gives the spin-orbit correlations of
partons in protons with their centers shifted with respect to
each other.
Taking the Fourier transform with respect to the transverse momentum transfer ? we get the GPDs in the
transverse impact parameter space.
1 Z 2
E T ðx; ; b? Þ ¼
d ? ei? b? ET ðx; ; tÞ
ð2Þ2
1 Z
dJ0 ðbÞET ðx; ; tÞ;
¼
(36)
2
where ¼ j? j and b ¼ jb? j. The other impact parameter dependent GPDs E~ T ðx; ; b? Þ and H T ðx; ; b? Þ can
also be defined in the same way.
Figure 5 shows the chiral-odd GPDs for nonzero in
impact parameter space for different and fixed x ¼ 0:5 as
a function of j b? j . As increases the peak at j b? j¼ 0
increases for E~ T ðx; ; b? Þ but decreases for H T ðx; ; b? Þ
and E T ðx; ; b? Þ. It is to be noted that H T ðx; ; b? Þ for a
free Dirac particle is expected to be a delta function; the
smearing in j b? j space is due to the spin correlation in the
two-particle LFWFs. Figure 6 shows the plots of the above
three functions for fixed and different values of x. For
given , the peak of H T ðx; ; b? Þ as well as E~ T ðx; ; b? Þ
increases with an increase of x; however, for E T ðx; ; b? Þ it
decreases.
So far, we discussed the chiral-odd GPDs in transverse
position space. In [6], a phase space distribution of quarks
and gluons in the proton is given in terms of the quantum
~ in the rest frame of
mechanical Wigner distribution Wðr~; pÞ
the proton, which are functions of three-position and threemomentum coordinates. Wigner distributions are not accessible in the experiment. However, if one integrates two
momentum components one gets a reduced Wigner distribution W ð~r; xÞ which is related to the GPDs by a Fourier
transform. For given x, this gives a 3D position space
034006-7
D. CHAKRABARTI, R. MANOHAR, AND A. MUKHERJEE
PHYSICAL REVIEW D 79, 034006 (2009)
0.08
(a)
x=0.5
0.06
0.04
ζ=0.1
ζ=0.2
ζ=0.45
0.02
0
0
0.2
0.4
0.6
0.8
1
b
0.015
(b)
(c)
ζ=0.1
ζ=0.2
ζ=0.45
x=0.5
x=0.5
1.5
ζ=0.1
ζ=0.2
ζ=0.45
0.01
1
0.005
0
0.5
0
0
0.2
0.4
0.6
0.8
0
1
FIG. 5 (color online).
0.2
0.4
0.6
0.8
1
b
b
Fourier spectrum of the chiral-odd GPDs vs j b? j for fixed x ¼ 0:5 and different values of .
picture of the partons inside the proton. In the infinite
momentum frame, rotational symmetry is not there.
Nevertheless, one can still define such a 3D position space
distribution by taking a Fourier transform of the GPDs.
Another point is the dynamical effect of Lorentz boosts.
If the probing wavelength is comparable to or smaller than
the Compton wavelength M1 , where M is the mass of the
proton, electron-positron pairs will be created; as a result,
the static size of the system cannot be probed to a precision
better than M1 in relativistic quantum theory. However, in
light-front theory, transverse boosts are Galilean boosts,
which do not involve dynamics. So one can still express the
GPDs in transverse position or impact parameter space and
this picture is not spoilt by relativistic corrections.
However, rotation involves dynamics here and rotational
symmetry is lost. In [5], a longitudinal boost invariant
impact parameter has been introduced which is conjugate to the longitudinal momentum transfer . It was
shown that the DVCS amplitude expressed in terms of
the variables and b? show a diffraction pattern analogous to diffractive scattering of a wave in optics where the
distribution in measures the physical size of the scattering center in a 1D system. In analogy with optics, it was
concluded that the finite size of the integration of the FT
acts as a slit of finite width and produces the diffraction
pattern.
We define a boost invariant impact parameter conjugate
to the longitudinal momentum transfer as ¼ 12 b Pþ [5].
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CHIRAL-ODD GENERALIZED PARTON DISTRIBUTIONS . . .
PHYSICAL REVIEW D 79, 034006 (2009)
0.08
(a)
0.06
ζ=0.1
0.04
x=0.15
x=0.7
x=0.9
0.02
0
0
0.2
0.4
0.6
0.8
1
b
0.003
(b)
(c)
x=0.15
x=0.7
x=0.9
0.0025
x=0.3
x=0.7
x=0.8
10
ζ=0.1
0.002
ζ=0.1
0.0015
5
0.001
0.0005
0
0
0.2
0.4
0.6
0.8
0
1
0
b
0.2
0.4
0.6
0.8
1
b
FIG. 6 (color online). Fourier spectrum of the chiral-odd GPDs vs j b? j for fixed and different values of x.
The chiral-odd GPD ET in longitudinal position space is
given by
1 Z f
þ
deið1=2ÞP b ET ðx; ; tÞ
2 0
1 Z f
¼
dei ET ðx; ; tÞ:
2 0
t:
max
E T ðx; ; tÞ ¼
(37)
Since we are concentrating only in the region < x < 1,
the upper limit of integration f is given by max if x is
larger then max , otherwise by x if x is smaller than max ,
where max is the maximum value of allowed for a fixed
ðtÞ
¼
2M2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi
4M
1 :
1þ
ðtÞ
(38)
Similarly, one can obtain H T ðx; ; tÞ and E~ T ðx; ; tÞ as
well. Figure 7 shows the plots of the Fourier spectrum of
chiral-odd GPDs in longitudinal position space as a function of for fixed x ¼ 0:5 and different values of t. Both
E T ðx; ; tÞ and H T ðx; ; tÞ show a diffraction pattern as
observed for the DVCS amplitude in [5]; the minima occur
at the same values of in both cases. However E~ T ðx; ; tÞ
034006-9
D. CHAKRABARTI, R. MANOHAR, AND A. MUKHERJEE
PHYSICAL REVIEW D 79, 034006 (2009)
(a)
-t=5.0
-t=1.0
-t=0.1
0.5
0.4
x=0.5
0.3
0.2
0.1
0
-20
-10
0
σ
20
3
0.03
(b)
(c)
-t=5.0
-t=1.0
-t=0.1
0.025
-t=5.0
-t=1.0
-t=0.1
2.5
x=0.5
0.02
x=0.5
2
0.015
1.5
0.01
1
0.005
0
10
0.5
-20
-10
0
σ
10
20
0
-20
-10
0
σ
10
20
FIG. 7 (color online). Fourier spectrum of the chiral-odd GPDs vs for fixed x and different values of t in MeV2 .
does not show a diffraction pattern. This is due to the
distinctively different behavior of E~T ðx; ; tÞ with compared to that of ET ðx; ; tÞ and HT ðx; ; tÞ. E~T ðx; ; tÞ rises
smoothly from zero and has no flat plateau in and thus
does not exhibit any diffraction pattern when Fourier transformed with respect to . The position of the first minima
in Fig. 7 is determined by f . For t ¼ 5:0 and 1.0, f x ¼ 0:5 and thus the first minimum appears at the same
position while for t ¼ 0:1, f ¼ max 0:45 and the
minimum appears slightly shifted. This is analogous to
the single slit optical diffraction pattern. f here plays the
role of the slit width. Since the positions of the minima
(measured from the center of the diffraction pattern) are
inversely proportional to the slit width, the minima move
away from the center as the slit width (i.e., f ) decreases.
The optical analogy of the diffraction pattern in space
has been discussed in detail in [5] in the context of DVCS
amplitudes.
V. CONCLUSION
In this work, we have studied the chiral-odd GPDs in
transverse and longitudinal position space. Working in
light-front gauge, we presented overlap formulas for the
chiral-odd GPDs in terms of proton light-front wave functions both in the DGLAP and ERBL regions. In the first
case there is parton number conserving n ! n overlap,
whereas in the latter case, the parton number changes by
two in n þ 1 ! n 1 overlap. We investigated them in the
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CHIRAL-ODD GENERALIZED PARTON DISTRIBUTIONS . . .
PHYSICAL REVIEW D 79, 034006 (2009)
DGLAP region when the skewness is less than x. We
used a self-consistent relativistic two-body model, namely,
the quantum fluctuation of an electron at one loop in QED.
We used its most general form [16], where we have a
different mass for the external electron and different
masses for the internal electron and photon. The impact
parameter space representations are obtained by taking a
Fourier transform of the GPDs with respect to the transverse momentum transfer. It is known that [3,8] the chiralodd GPDs provide important information on the spin-orbit
correlations of the transversely polarized partons in an
unpolarized nucleon as well as the correlations between
the transverse quark spin and the nucleon spin in the
transverse polarized nucleon. When is nonzero, the
initial and final proton are displaced in the impact parame-
ter space relative to each other by an amount proportional
to . As this is the region probed by most experiments, it is
of interest to investigate this. By taking a Fourier transform
with respect to we presented the GPDs in the boost
invariant longitudinal position space variable . HT and
ET show a diffraction pattern in space. Further work is
needed to investigate this behavior and to study its model
dependence.
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ACKNOWLEDGMENTS
The work of D. C. is supported by Marie-Curie (IIF)
Fellowship. A. M. thanks DST Fasttrack scheme,
Government of India for financial support for completing
this work.
034006-11