arXiv:hepph/0605310v1 28 May 2006
DESY 05–008 hepph/0605310
SFBCPP06/25
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, D04109 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 lightcone 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 deepinelastic and the diﬀractive structure functions in the deepinelastic
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 deepinelastic 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 operatormatrixelements 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 deepinelastic 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 nonforwardness 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 photonhadron 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 antiproton, 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 lightcone 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
− iǫ)
2
S
µν
αλ
O
α
˜x
2
, −
˜x
2
− ǫ
µν
αλ
O
5 α
˜x
2
, −
˜x
2
, (2.19)
where
S
µναλ
= 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
Ω
α
(κ
1
˜x, κ
2
˜x) − Ω
α
(κ
2
˜x, κ
1
˜x)
(2.22)
O
5
α
(κ
1
˜x, κ
2
˜x) = Ω
5
α
(κ
1
˜x, κ
2
˜x) + Ω
5
α
(κ
2
˜x, κ
1
˜x) , (2.23)
with
Ω
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)
Ω
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