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Journal ArticleDOI

Target mass and finite momentum transfer corrections to unpolarized and polarized diffractive scattering

30 Oct 2006-Nuclear Physics (North-Holland)-Vol. 755, Iss: 1, pp 112-136

Abstract: A quantum field theoretic treatment of inclusive deep-inelastic diffractive scattering is given. The process can be described in the general framework of non-forward scattering processes using the light-cone expansion in the generalized Bjorken region. Target mass and finite t corrections of the diffractive hadronic tensor are derived at the level of the twist-2 contributions both for the unpolarized and the polarized case. They modify the expressions contributing in the limit t , M 2 → 0 for larger values of β or/and t in the region of low Q 2 . The different diffractive structure functions are expressed through integrals over the relative momentum of non-perturbative t -dependent 2-particle distribution functions. In the limit t , M 2 → 0 these distribution functions are the diffractive parton distribution. Relations between the different diffractive structure functions are derived.
Topics: Momentum transfer (54%), Parton (52%), Momentum (51%), Scattering (51%)

Content maybe subject to copyright    Report

arXiv:hep-ph/0605310v1 28 May 2006
DESY 05–008 hep-ph/0605310
SFB-CPP-06/25
May 2006
Target mass and finite momentum transfer
corrections to unpolarized and polarized
diffractive scattering
Johannes Bl¨umlein
a
, Bodo Geyer
b
and Dieter Roba schik
a,c
a
Deutsches Elektronen–Synchrotron, DESY,
Platanenallee 6, D–15738 Zeuthen, Germany
b
Center for Theoretical Studies and Institute of Theoretical Physics,
Leipzig University, Augustusplatz 10, D-04109 Leipzig, Germany
c
Brandenburgische Technische Universit¨at Cottbus, Fakult¨at 1,
PF 101344, D–03013 Cottbus, Germany
Abstract
A quantum field theoretic treatment of inclusive deep–inelastic diffr active scattering is
given. The process can be described in the general framework of non–forward scattering
processes using the light–cone expansion in the generalized Bjorken region. Target mass
and finite t corrections of the diffractive hadronic tensor are derived at the level of the
twist–2 contributions both for the unpolarized and the polarized case. They modify the
expressions contribu ting in the limit t, M
2
0 for larger values of β or/and t in the region
of low Q
2
. The different diffractive structure functions are expressed through integrals over
the relative momentum of non–perturbative t–dependent 2–particle distribution fun ctions.
In the limit t, M
2
0 these distribution functions are the diffractive parton distribution.
Relations between the different diffractive structure functions are derived.
PACS: 24.85.+p, 13.88.+e, 11.30.Cp
Keywords: Diffractive Scattering, Target Mass Effects, Finite m omentum transfer correc-
tions, Twist decomposition, Nonlocal light-cone operators, Multivalued distribution ampli-
tude, Generalized Bjorken limit.

1 Introduction
Deep inelastic diffractive lepton–nucleon scattering was observed at the electron–proton collider
HERA some years ago [1]. This process is measured in detail by now [2] and the structure
function F
D
2
(x, Q
2
) was extracted.
1
The experimental measurement s clearly showed that the
scaling violations of the deep-inelastic and the diffractive structure functions in the deep-inelastic
regime, aft er an appropriate change of kinematic variables, are the same. Furthermore the ratio
of the two quantities, did not vary strongly, cf. [4]. While the f ormer property is clearly of
perturbative nat ure, the latter is of non–perturbative origin.
The process of deep–inelastic diffractive scattering was first described phenomenologically
[5]. Diffractive events ar e characterized by a rapidity g ap between the diffractive nucleon a nd
the remaining part of the produced hadrons, which is sufficiently large. Actually it is this
experimental signature along with factorization for the twist–2 contributions [6] for this process,
which allows to give a consistent field theoretic description. Due to this phenomenological
considerations containing reference to sp ecific pomeron models can be thoroughly avoided. In
the limit of vanishing target masses the scattering cross sections and relations between the
diffractive structure functions were derived in Refs. [7–9] for unpolarized and polarized diffractive
scattering. In [7] we showed, that the scaling violations in the deep-inelastic and deeply inelastic
diffractive case have to be the same due to the fact that the scaling violations are actually those of
the operators which remain taking the respective matrix elements confirming the experimental
observation. The set of structure functions which emerge in both scattering cross sections is
actually larger t han measured in current experiments. At low scales of Q
2
target mass effects
become relevant similar to the case of deep–inelastic scattering [10–14], see also [15].
In the present paper we extend the picture developed in the massless case [7, 8] to the case
of finite target masses and finite values of t on the level o f twist–2 operators in the light cone
expansion [1 6]. While in absence of mass effects the two–particle problem could effectively be
reduced to a single particle description for the case t 0, this is no longer the case for finite values
of t and/or target masses. Here two particle effects become relevant, which do not allow for a
direct partonic description. The variables t = (p
i
p
f
)
2
and M
2
= p
2
i
= p
2
f
are closely connected
and the simplification emerges if these scales vanish. Yet one may still follow the field theoretic
picture developed in [7,8] in the general case M
2
, t 6= 0 and derive expressions for the diffractive
structure functions including relations between them. At low scales Q
2
and large values of β
target mass corrections have to be considered in the experimental analysis. This g enerally applies
also to finite values of t, unless the scale Q
2
is large enough. The diffractive structure functions are
found as integrals over two–particle correlation functions f (z
+
, z
; t) between the incoming and
outgoing nucleon. Here, z
±
denote the corresponding collinear light–cone momentum fractions
and t is the relative momentum transfer squared between the incoming and outg oing proton
momentum. We refer to the formalism of non–forward Compton scattering, cf. [17], and apply
the general group theoretical algorithm of decomposing off–cone tensor operators into operators
of definite geometric twist [18–20] to determine the contributions at twist 2. The analysis can
be generalized to operators o f higher twist. On the level of the various correlation functions
relations can be established. These relations correspo nd to relations between structure functions,
cf. [12,21–24]. In the case of deeply virtual Compton scattering relations of this type were found
in [25–27] before.
The present field–theoretic f ormalism for deep–inelastic diffractive scattering was also devel-
1
The measurement of the longitudinal diffractive structure function F
D
L
(x, Q
2
) has not yet been possible, but
would be important. For the DIS structure function c f. [3]. Likewise the polarized diffractive structure functions
g
D
1,2
(x, Q
2
) should be measured in the future to reveal the effects of nucleon polarization in this proces s.
2

oped in view of possible future measurements of the resp ective operator-matrixelements using
lattice techniques, as successfully applied in case of the moments of the deep–inelastic structure
functions [28]. We regard it as a challenge in future investigations to verify the experimentally
observed ratio of moments in the diffractive and deep-inelastic case with these technologies. This
ratio still awaits a rigorous non–perturbative explanation.
The paper is o rganized as follows. In section 2 we describe the basic for malism. In section 3
the symmetric part of the Compton a mplitude is dealt with, through which the diffractive
structure functions for unpolarized nucleons are derived. The polarized structure functions are
determined in section 4. In section 5 we derive relations between different structure functions
and section 6 contains the conclusions.
2 Basic Formalism
In order to compute the twist–2 target mass and finite momentum transfer corrections in po-
larized and unpolarized deep inelastic diffractive scattering we briefly recall the notations and
conventions used in previous papers [7, 8] by two of the present authors. The process of deep–
inelastic diffractive scattering belongs to the class o f semi–inclusive processes and is described
by the diagram in Figure 1. The differential scattering cross section for single–photon exchange
is given by
d
5
σ
diffr
=
1
2(s M
2
)
1
4
dP S
(3)
X
spins
e
4
Q
2
L
µν
W
µν
. (2.1)
Here s = (p
1
+ l)
2
is the cms energy squared of the process and M denotes the nucleon mass.
The phase space dP S
(3)
depends on five variables since one final state mass varies. We choose
as basic variables
x =
Q
2
Q
2
+ W
2
M
2
=
q
2
2 qp
1
, (2.2)
the photo n virtuality Q
2
= q
2
, t = (p
2
p
1
)
2
the 4–momentum difference squared between
incoming and outgoing nucleon, a variable describing the non-forwardness w.r.t. the incoming
proton direction,
x
P
=
Q
2
+ M
2
X
t
Q
2
+ W
2
M
2
=
qp
qp
1
x (2.3)
for M
2
X
> t, and the angle Φ between the lepton plane p
1
× l and the hadron plane p
1
× p
2
,
cos Φ =
(p
1
× l).(p
1
× p
2
)
|p
1
× l|| p
1
× p
2
|
, (2.4)
where p
±
= p
2
±p
1
, W
2
= (p
1
+ q)
2
denotes the hadronic mass squared and the diffractive mass
squared is given by M
2
X
= (q p
)
2
.
The momenta p
±
obey
(p
+
p
) = 0,
p
2
+
p
2
=
4M
2
t
1 . (2.5)
For later use we refer to the variables η and β defined by
η =
qp
qp
+
=
x
P
2 x
P
1 ,
x
2 x
, β =
q
2
2 qp
=
x
x
P
1, (2.6)
3

as well as to the transverse momentum variable
π
= p
+
p
η
, (qπ
) = 0 . (2.7)
The variables x, x
P
, β and η obey the inequalities
0 x x
P
1, 0 x β 1, (2.8)
−∞ 1
2
x
1
2β
x
=
1
η
1 η
x
2 x
0 . (2.9)
l
l
q
p
1
p
2
M
X
Figure 1: The virtual photon-hadron amplitude for diffractive ep scattering
For the spin averaged cross section, the leptonic tensor is symmetric. Taking into account
conservation of the electromagnetic current o ne obtains [7]
W
s
µν
= g
T
µν
W
s
1
+ p
T
1µ
p
T
1ν
W
s
2
M
2
+ p
T
2µ
p
T
2ν
W
s
3
M
2
+
p
T
1µ
p
T
2ν
+ p
T
2µ
p
T
1ν
W
s
4
M
2
. (2.10)
Here a nd in the following we do not assume implicitly, that azimuthal integrals are perfo rmed as
sometimes done in experiment. In the latter case the number of contributing structure function
reduces.
In the case of polarized nucleons we consider the initial state spin–vector S
1
S, S
2
= M
2
,
only and sum over the spin of the outgoing hadrons. One usually considers the longitudinal (||)
and transverse () spin projections choosing
S
||
= (0; 0, 0, 0, M) (2.11)
S
= (0; cos γ, sin γ, 0)M . (2.12)
Here γ denotes the azimuthal angle. The antisymmetric part of the hadronic t ensor was derived
in [8] and is given by
W
a
µν
= i
p
T
1µ
p
T
2ν
p
T
1ν
p
T
2µ
ε
p
1
p
2
qS
W
a
1
M
6
+ i
p
T
1µ
ε
νSp
1
q
p
T
1ν
ε
µSp
1
q
W
a
2
M
4
+ i
p
T
2µ
ε
νSp
1
q
p
T
2ν
ε
µSp
1
q
W
a
3
M
4
+ i
p
T
1µ
ε
νSp
2
q
p
T
1ν
ε
µSp
2
q
W
a
4
M
4
+ i
p
T
2µ
ε
νSp
2
q
p
T
2ν
ε
µSp
2
q
W
a
5
M
4
+ i
p
T
1µ
ε
T
νp
1
p
2
S
p
T
1ν
ε
T
µp
1
p
2
S
W
a
6
M
4
+ i
p
T
2µ
ε
T
νp
1
p
2
S
p
T
2ν
ε
T
µp
1
p
2
S
W
a
7
M
4
+ i ε
µνqS
W
a
8
M
2
. (2.13)
4

We will specify below which terms of this general structures contribute in case of deep–inelastic
diffractive scattering. The kinematic f actors above a re constructed out of the four–vectors q, p
1
, p
2
and S as well as g
µν
and ε
v
0
v
1
v
2
v
3
using
p
T
µ
= p
µ
q
µ
q.p
q
2
, g
T
µν
= g
µν
q
µ
q
ν
q
2
, (2.14)
ε
T
µv
1
v
2
v
3
= ε
µv
1
v
2
v
3
ε
qv
1
v
2
v
3
q
µ
q
2
, (2.15)
ε
T T
µνv
1
v
2
= ε
µνv
1
v
2
ε
qνv
1
v
2
q
µ
q
2
ε
µqv
1
v
2
q
ν
q
2
. (2.16)
At the level of the twist–2 contributions f actorization holds fo r diffractive scattering [6].
A. Mueller’s generalized optical theorem [29] allows to move the final state proton into an initial
state anti-proton, where both particle momenta are separated by t and form a ‘quasi two–
particle’ state. The correctness of this procedure within the light–cone expansion relies, first, on
the rapidity gap between the outgoing proto n and the remainder hadronic part with invaria nt
mass M
X
and, second, on the special property of matrix elements of the light-cone op erators
which contain no absorptive part. The structure functions for the diffractive process can thus
be obtained by analyzing the absorptive part of the expectation value
T
µν
(x) =
p
1
, p
2
, S; t
b
T
µν
(x)
p
1
, p
2
, S; t
, (2.17)
with
b
T
µν
defined as
b
T
µν
(x) iR T
h
J
µ
x
2
J
ν
x
2
S
i
. (2.18)
As shown in [17,30] the operator
b
T
µν
is represented in lowest or der of the non–local light–cone
expansion by
b
T
µν
(x) e
2
˜x
λ
2iπ
2
(x
2
)
2
S
µν|
αλ
O
α
˜x
2
,
˜x
2
ǫ
µν
αλ
O
5 α
˜x
2
,
˜x
2

, (2.19)
where
S
µν|αλ
= g
µα
g
νλ
+ g
µλ
g
να
g
µν
g
αλ
. (2.20)
˜x denotes a light–like vector related to x,
˜x = x n[(nx)
p
(nx)
2
n
2
x
2
] , (2.21)
with n a normalized time–like vector, n
2
= 1, and the bi–local light–ray operato r s O
α
and O
5
α
read
O
α
(κ
1
˜x, κ
2
˜x) = i
α
(κ
1
˜x, κ
2
˜x)
α
(κ
2
˜x, κ
1
˜x)
(2.22)
O
5
α
(κ
1
˜x, κ
2
˜x) =
5
α
(κ
1
˜x, κ
2
˜x) +
5
α
(κ
2
˜x, κ
1
˜x) , (2.23)
with
α
(κ
1
˜x, κ
2
˜x) = R T
:
¯
ψ (κ
1
˜x) γ
α
U (κ
1
˜x, κ
2
˜x) ψ (κ
2
˜x) : S
(2.24)
5
α
(κ
1
˜x, κ
2
˜x) = R T
:
¯
ψ (κ
1
˜x) γ
5
γ
α
U (κ
1
˜x, κ
2
˜x) ψ (κ
2
˜x) : S
, (2.25)
where κ
1
= κ
2
= 1/2, cf. [17]. As is well–known, these operators cont ain contributions of up
to twist four [18]. The scattering amplitude is obtained by the Fourier transform of the operator
5

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