Eur. Phys. J. C (2009) 61: 279–298
DOI 10.1140/epjc/s10052-009-0982-6
Regular Article - Theoretical Physics
Target-mass and finite t corrections to diffractive deep-inelastic
scattering
Johannes Blümlein
1,a
, Dieter Robaschik
1,2
, Bodo Geyer
3
1
Deutsches Elektronen–Synchrotron, DESY, Platanenallee 6, 15738 Zeuthen, Germany
2
Fakultät 1, Brandenburgische Technische Universität Cottbus, PF 101344, 03013 Cottbus, Germany
3
Center for Theoretical Studies and Institute of Theoretical Physics, Leipzig University, Augustusplatz 10, 04109 Leipzig, Germany
Received: 10 December 2008 / Revised: 12 February 2009 / Published online: 28 March 2009
© Springer-Verlag / Società Italiana di Fisica 2009
Abstract The quantum field theoretic treatment of inclu-
sive deep-inelastic diffractive scattering given in a previ-
ous paper (Blümlein et al. in Nucl. Phys. B 755:112–136,
2006) is discussed in detail using an equivalent formula-
tion with the aim to derive a representation suitable for data
analysis. We consider the off-cone twist-2 light-cone opera-
tors to derive the target-mass and finite t corrections to dif-
fractive deep-inelastic scattering and deep-inelastic scatter-
ing. The corrections turn out to be at most proportional to
x|t|/Q
2
, xM
2
/Q
2
, x = x
BJ
or x
P
, which suggests an ex-
pansion in these parameters. Their contribution varies in
size considering diffractive scattering or meson-exchange
processes. Relations between different kinematic amplitudes
which are determined by one and the same diffractive GPD
or its moments are derived. In the limit t,M
2
→ 0 one ob-
tains the results of (Blümlein and Robaschik in Phys. Lett. B
517:222, 2001) and (Blümlein and Robaschik in Phys. Rev.
D 65:096002, 2002).
PACS 24.85.+p · 13.88.+e · 11.30.Cp
1 Introduction
The process of deep-inelastic diffractive lepton–nucleon
scattering can be measured at high energy colliders and con-
stitutes a large fraction of the inclusive statistics, although
being a semi-inclusive process. It was first observed at the
electron–proton collider HERA some years ago [4, 5] and
is now measured in detail [6–8]. The structure function
F
D
2
(x, Q
2
) was extracted. In the same manner it is desir-
able to compare the longitudinal diffractive structure func-
tion F
D
L
(x, Q
2
) with the longitudinal structure function in
a
e-mail: johannes.bluemlein@desy.de
the inclusive case [9, 10]. The measurement of the polar-
ized diffractive structure functions g
D
1,2
(x, Q
2
) will be pos-
sible at future facilities like EIC [11], which are currently
planned. The experimental measurements clearly showed
that the scaling violations of the deep-inelastic and the dif-
fractive structure functions in the deep-inelastic regime, af-
ter an appropriate change of kinematic variables, are the
same. Furthermore, the ratio of the two quantities, did not
vary strongly, cf. [12]. While the former property is clearly
of perturbative nature, the latter is of non-perturbative ori-
gin. For diffractive scattering, however, another mass scale
is of importance, which is given by the invariant mass t =
(p
2
−p
1
)
2
.Herep
1(2)
denote the 4-momenta of the incom-
ing and outgoing proton, where for the latter a sufficiently
large rapidity gap between this particle and the remainder
final state hadrons is demanded as process signature. A sim-
ilar class of processes are the so-called meson-exchange
processes, cf. e.g. [13], where the finite rapidity gap is not
required, but the rôle of the formerly diffractive final state
proton is taken by a leading hadron, which distinguishes it-
self due to its high momentum from the remaining hadrons.
Also in this case one may try a leading twist description, al-
though the signature for this process is less clear than in the
diffractive case.
The process of deep-inelastic diffractive scattering was
first described phenomenologically [14–23]. A consistent
field-theoretic description of the process requires factoriza-
tion for the twist-2 contributions [24–26]. It is due to this
description that reference to specific pomeron models are
thoroughly avoided. In Refs. [2, 3, 27] two of the present au-
thors gave a corresponding field-theoretic description of the
process in the limit t,M
2
→0. In [2] we proved that under
these conditions the scaling violations for diffractive scatter-
ing and inclusive deep-inelastic scattering are the same, up
to a change in the momentum-fraction variable in the former
case.
280 Eur. Phys. J. C (2009) 61: 279–298
At low 4-momentum transfer Q
2
both target-mass (M
2
)
and finite momentum transfer (t) corrections have to be con-
sidered for the diffractive and leading hadron processes with
meson exchange. In the deep-inelastic case the target-mass
corrections were studied in Refs. [28–32], see also [33–
38]. The kinematics of the diffractive and leading hadron
processes is similar to that in deeply virtual non-forward
scattering. Considering this general class of processes, one
finds that the treatment of target-mass effects and finite t-
effects can only be performed by combining both, see [39,
40]. If compared with the deep-inelastic case the number of
hadronic structure functions enlarges in the diffractive case
from two to four for unpolarized scattering and to eight for
polarized scattering, as shown in [2, 3], if the general kine-
matics is considered. In Ref. [1] we worked out these correc-
tions for the hadronic tensor in general, yet without quan-
tifying the result. If one departs form the limit t,M
2
→ 0
the corresponding representations require to carry out a one-
dimensional
definite integral which kinematically relates the
two proton momenta p
1
and p
2
. As the integration is to be
performed over unknown non-perturbative functions there is
no a priori experimental way to unfold the non-perturbative
distributions, which also would invalidate the partonic de-
scription in case of diffractive scattering. Moreover, the M
2
and t effects dealt with in this case are not yet complete,
since there emerge other contributions more in the scat-
tering cross section. One may expand the complete solu-
tion in two variables t/Q
2
,M
2
/Q
2
. It is found that these
terms multiply at least with a factor x = x
BJ(P)
, which is
bounded in the diffractive case to values below 0.01 and in
the meson-exchange case 0.3. Thus the leading terms be-
yond t,M
2
=0 give a good first estimate for the corrections.
The further corrections turn out to be widely suppressed in
the diffractive case, while they are larger for leading particle
cross sections in the meson-exchange case.
In the present paper we will discuss both the unpolar-
ized and polarized case. The paper is organized as follows.
In Sect. 2 we derive the differential scattering cross section
for inclusive diffractive scattering at the Lorentz level. Main
aspects of the relation of this process to the Compton ampli-
tude within the light-cone expansion including finite M
2
and
t effects are summarized in Sect. 3. The hadronic tensors for
the unpolarized and polarized case are expanded in terms of
the variables t/Q
2
, M
2
/Q
2
in Sect. 4 to show the size of
the correction terms. Section 5 contains the conclusions. In
Appendix A we summarize some kinematic relations. The
present formalism is specified to the case of deep-inelastic
forward scattering (DIS) in Appendix B, where we obtain
the target-mass corrections given in [29–32] before.
2 The Lorentz structure
The process of deep-inelastic diffractive scattering belongs
to the class of semi-inclusive processes. It is described by an
effective 2 →3 diagram, cf. Fig. 1 of Ref. [2], with incom-
ing and outgoing charged lepton and nucleon lines and an
effective 4-vector for all the other hadron lines in the final
state, which are well separated in rapidity from the outgoing
diffractive nucleon line.
The differential scattering cross section for single-photon
exchange is given by
d
5
σ
diffr
=
1
2(s −M
2
)
1
4
dPS
(3)
spins
e
4
Q
2
L
μν
W
μν
. (2.1)
Here s =(p
1
+l
1
)
2
is the cms energy squared of the process
and M denotes the nucleon mass. The phase space dPS
(3)
depends on five variables since the mass M
X
of the diffrac-
tively produced inclusive set of hadrons varies. We choose
as basic variables
x
BJ
=
Q
2
Q
2
+W
2
−M
2
=−
q
2
2qp
1
, (2.2)
y =
Q
2
x
BJ
(s −M
2
)
, (2.3)
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
BJ
, (2.4)
and the angle φ
b
between the lepton plane p
1
×l
1
and the
hadron plane p
1
×p
2
,
cos(φ
b
) =
(p
1
×l
1
).(p
1
×p
2
)
|p
1
×l
1
||p
1
×p
2
|
(2.5)
in the frame p
1
+q =0. Here Q
2
=−q
2
denotes the photon
virtuality and W is the hadronic mass with W
2
=(p
1
+q)
2
.
We also refer to x = Q
2
/qp
+
. It is useful to introduce the
4-vectors
p
±
=p
2
±p
1
. (2.6)
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. (2.7)
Eur. Phys. J. C (2009) 61: 279–298 281
For later use we refer to the non-forwardness η and the vari-
able β defined by
η =
qp
−
qp
+
=
−x
P
2 −x
P
∈
−1,
−x
2 −x
,
β =
q
2
2 qp
−
=
x
BJ
x
P
≤1.
(2.8)
The variable x
P
is directly related to η but is more com-
monly used in experimental analyzes,
x
P
=
2η
η −1
. (2.9)
More kinematic invariants are given in Appendix A.
The transverse momentum variable, introduced as ˆπ
−
,
[1], or π
−
=−η ˆπ
−
is of special importance,
π
−
=p
−
−p
+
η, (qπ
−
) =0. (2.10)
Later on it will play the role of an expansion parameter. The
variables x
BJ
,x
P
,β and η obey the inequalities
0 ≤x
BJ
≤x
P
≤1, 0 ≤x
BJ
≤β ≤1, (2.11)
−∞≤1 −
2
x
BJ
≤1 −
2β
x
BJ
=
1
η
≤−1 ≤η ≤
−x
BJ
2 −x
BJ
≤0. (2.12)
For the spin averaged cross section, the leptonic tensor is
symmetric. Taking into account conservation of the electro-
magnetic current one obtains [2]
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
4
M
2
+
p
T
1μ
p
T
2ν
+p
T
2μ
p
T
1ν
W
s
5
M
2
. (2.13)
Here and in the following we do not assume that azimuthal
integrals are performed as sometimes is done in experiment.
In the latter case the number of contributing structure func-
tion 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 refers to the lon-
gitudinal () and transverse (⊥) spin projections choosing
S
=
E
2
−M
2
;0, 0, 0,E
, (2.14)
S
⊥
=(0;cos γ,sin γ,0)M, (2.15)
in the laboratory frame with p
1
= (E;0, 0,
√
E
2
−M
2
),
with S.p
1
= 0. Here γ denotes the azimuthal angle. In the
case of longitudinal polarization the contraction of S
with
l
1
and p
2
being nearly collinear to p
1
are of O(μ
2
/Q
2
),
μ
2
=|t|, M
2
, see Appendix A.
The antisymmetric part of the hadronic tensor was de-
rived in [3] 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
+iε
μνqS
W
a
8
M
2
, (2.16)
where ε
μναβ
denotes the Levi-Civita symbol. The kinematic
factors above are constructed out of the 4-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
, (2.17)
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
, (2.18)
ε
TT
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
. (2.19)
One may rewrite (2.16) into an equivalent form using the
Schouten identities [41, 42].
Target-mass and finite t corrections to the differential
scattering cross section (2.1) in the leading twist approxima-
tion emerge from three sources: (i) from kinematic terms at
the Lorentz level after contracting the leptonic and hadronic
tensor; (ii) from the expectation value of the Compton oper-
ator; (iii) the t-behavior of the non-perturbative distribution
functions.
We will first consider the contributions (i) and discuss the
terms (ii) in Sect. 4. The non-perturbative effects cannot be
calculated by rigorous methods within Quantum Chromody-
namics at present, but are left to phenomenological models
or are determined through fits to data, cf. [14–23].
For pure photon exchange the leptonic tensor is given by
L
μν
=2
l
1μ
l
2ν
+l
2μ
l
1ν
−g
μν
l
1
.l
2
−iε
μναβ
l
α
1
q
β
, (2.20)
cf. [43], in case of longitudinal lepton polarization.
We consider the Bjorken limit,
282 Eur. Phys. J. C (2009) 61: 279–298
2p
1
.q =2Mν →∞,p
2
.q →∞,Q
2
→∞,
with x
BJ
and x
P
=fixed. (2.21)
Here,
MW
s
1
→F
1
, (2.22)
νW
s
k
→F
k
,k=2, 4, 5, (2.23)
with ν =y(s −M
2
)/(2M).
In the unpolarized case we obtain in the limit M
2
,t →0
w.r.t. the kinematics of the momenta p
1
and p
2
, keeping the
target-mass dependence
d
s
σ
unpol
dx
BJ
dQ
2
=
2πα
2
Q
4
x
BJ
2xF
1
·y
2
+
F
2
+(1 −x
P
)F
4
+(1 −x
P
)
2
F
5
·2
1 −y −
x
2
BJ
y
2
M
2
Q
4
, (2.24)
where F
k
= F
k
(x
BJ
,x
P
,Q
2
;t) are the diffractive structure
functions, cf. [2]. The correction terms are of O(M
2
/Q
2
,
t/Q
2
). In the limit M
2
,t →0 the azimuthal dependence on
φ
b
vanishes.
Likewise we obtain in the polarized case for longitudinal
nucleon polarization,
d
3
σ
pol
(λ, ±S
)
dx
BJ
dQ
2
dx
P
=∓4πsλ
α
2
Q
4
y
2 −y −
2x
BJ
yM
2
s
xg
1
−4x
BJ
y
M
2
s
g
2
, (2.25)
d
4
σ
pol
(λ, ±S
⊥
)
dx
BJ
dQ
2
dx
P
dΦ
=∓4πsλ
M
2
s
α
2
Q
2
x
BJ
y
1 −y −
x
BJ
yM
2
s
×cos(γ −Φ)[yx
BJ
g
1
+2x
BJ
g
2
]. (2.26)
Here Φ denotes the angle between the
l
1
−
S and the
l
1
−
l
2
plane and γ is the angle between
l
1
and
S.The
structure functions g
1,2
(x
BJ
,x
P
,Q
2
;t) are obtained from
W
a
2
,W
a
3
,W
a
4
,W
a
5
and W
a
8
by
g
1
=
p.q
1
M
2
W
a
8
, (2.27)
g
2
=
(p.q
1
)
3
q
2
M
4
W
a
2
+(1 −x
P
)
W
a
3
+W
a
4
+(1 −x
P
)
2
W
a
5
(2.28)
and the different structure functions F
i
and g
i
depend on the
variables x
BJ
,x
P
,Q
2
and t.
3 The Compton amplitude
The hadronic tensor for deep-inelastic diffractive scattering
can be obtained from a Compton amplitude as has been out-
lined in [1–3] before. We limit the description to the level
of the twist-2 contributions, where factorization holds for
the semi-inclusive diffractive process [24–26]. Furthermore,
Mueller’s generalized optical theorem [44–48] allows one to
move the final state proton into an initial state anti-proton,
where both particle momenta are separated by t and form
a formal ‘quasi two-particle’ state |p
1
, −p
2
,S;t. These
states are used to form the operator matrix elements. The
correctness of this procedure within the light-cone expan-
sion relies, first, on the rapidity gap between the outgo-
ing proton and the remaining hadronic part with invariant
mass M
X
and, second, on the special property of matrix el-
ements of the contributing light-cone operators to contain
no absorptive part. Independently, one could argue that the
corresponding matrix element is a pure phenomenological
quantity satisfying restrictions imposed by quantum field
theory. The general structure of the scattering amplitude
is completely determined by the off-cone structure of the
twist-2 Compton operator (3.4), cf. [49]. We mention that in
the present process the electromagnetic current is conserved
unlike the case in [39, 40] since the operator expectation
value (3.2) is taken for forward scattering using a quasi two-
particle state.
The structure functions for the diffractive process can
thus be obtained by analyzing the absorptive part
W
μν
=Im T
μν
(3.1)
of the expectation value
T
μν
(x) =p
1
, −p
2
,S;t|
ˆ
T
μν
(x)|p
1
, −p
2
,S;t, (3.2)
with the well-known operator
ˆ
T
μν
of (virtual) Compton scat-
tering defined as
ˆ
T
μν
(x) ≡iRT
J
μ
x
2
J
ν
−
x
2
S
. (3.3)
In [1], based on a general quantum field theoretic consider-
ation of virtual Compton scattering at twist 2 [40, 50–52],
we specified the various terms which contribute to the gen-
eral structure of the hadronic tensor W
μν
= Im T
μν
in case
of deep-inelastic diffractive scattering. As shown in [53, 54]
the operator
ˆ
T
μν
in lowest order of the non-local light-cone
expansion [55, 56] contains the vector or axial vector opera-
tors only. The scattering amplitude is obtained by the Fourier
transform of the operator
ˆ
T
{μν}
(x) and forming the matrix
element (3.2). Here, we want to study its twist-2 contribu-
tions including target-mass and finite momentum transfer
corrections. This is obtained by harmonic extension [51, 52,
Eur. Phys. J. C (2009) 61: 279–298 283
57, 58] of the twist-2 light-cone operators to twist-2 off-cone
operators [59], leading to
ˆ
T
tw2
μν
(q) =−e
2
d
4
x
2iπ
2
e
iqx
x
λ
(x
2
−i)
2
S
μν|
αλ
O
tw2
α
(κx, −κx)
+
μν
αλ
O
tw2
5α
(κx, −κx)
, (3.4)
with
O
tw2
α
(κx, −κx) = i
ψ(κx)γ
α
ψ(−κx)
−
ψ(−κx)γ
α
ψ(κx)
tw2
,
O
tw2
5α
(κx, −κx) =
ψ(κx)γ
5
γ
α
ψ(−κx)
+
ψ(−κx)γ
5
γ
α
ψ(κx)
tw2
,
and κ = 1/2. The matrix elements can be written in terms
of vectors K
a
μ,(5)
and 2-dimensional Fourier integrals over
partonic twist-2 distributions f
a(5)
(z
+
,z
−
,t)summing over
a,
p
1
, −p
2
;t|e
2
O
tw2
μ
(κx, −κx)|p
1
, −p
2
;t
=K
a
μ
(p
±
)
DZ
(2π)
4
e
iκx(p
−
z
−
+p
+
z
+
)
f
a
(z
+
,z
−
,t), (3.5)
p
1
, −p
2
,S;t|e
2
O
tw2
5μ
(κx, −κx)|p
1
, −p
2
,S;t
=K
a
5μ
(p
±
,S)
DZ
(2π)
4
e
iκx(p
−
z
−
+p
+
z
+
)
f
5a
(z
+
,z
−
,t),
(3.6)
which is defined as asymptotic expression on the light-cone
at x
2
=0.
We choose as kinematic factors for the representation of
the matrix element of the non-local operator for the symmet-
ric part (3.5)
K
1μ
=p
μ
+
, K
2μ
=π
μ
−
≡p
μ
−
−ηp
μ
+
, (3.7)
and for its antisymmetric part (3.6)
K
1μ
5
=S
μ
, K
2μ
5
=p
μ
+
(p
2
S)/M
2
,
K
3μ
5
=π
μ
−
(p
2
S)/M
2
.
(3.8)
The normalization to M
2
in (3.8) is arbitrary and has to
be arranged with the definition of the corresponding dis-
tribution functions f
a,(5a)
(z
+
,z
−
), respectively. The cor-
responding Lorentz-invariant has to be formed out of the
hadronic momenta, except the spin vector, since the polar-
ization symmetries are assumed to be linear in the spin.
The momentum fractions z
±
in (3.5), (3.6) corresponding
to the momenta p
±
are
P =(p
+
,p
−
) =(p
2
+p
1
,p
2
−p
1
),
Z =(z
+
,z
−
) =
(z
2
+z
1
)/2,(z
2
−z
1
)/2
,
(3.9)
with the measure DZ
DZ = 2 dz
+
dz
−
θ(1 −z
+
+z
−
)θ(1 +z
+
−z
−
)
×θ(1 −z
+
−z
−
)θ(1 +z
+
+z
−
). (3.10)
We refer to f
a(5)
(z
+
,z
−
,t) as diffractive generalized
parton distribution functions (dGPD), in distinction to the
GPDs emerging in deeply virtual Compton scattering [60–
65]. These amplitudes are directly connected to the total
cross sections and polarization asymmetries, respectively.
Both kinds of GPDs are expectation values of the same light-
cone operator, however, between different states. Interest-
ing limiting cases can be derived from them. For the dGPDs
these are the quasi collinear limit: π
−
→0,M
2
→0, [2, 3],
and the limit of deep-inelastic scattering, see Appendix B.
Furthermore, for both types of GPDs the evolution equations
are derived from the renormalization group equation for the
same light-cone operators. It is remarkable, that the evo-
lution equations for the dGPDs are two-variable equations
which reduce to the simple evolution equation for forward
scattering in the quasi collinear limit, cf. [2].
The (dimensionless) amplitudes f
(5)a
(z
+
,z
−
,t) depend
on t and η explicitly. In addition, there appears a t- and
M
2
-dependence of the amplitude (3.2) in momentum space,
which finally, on the one hand, results from the Fourier
transform in (3.4) where the operator O
tw2
(5)α
(κx, −κx) is off
the light cone, i.e. with all trace subtractions. On the other
hand, the dependence results from the kinematic pre-factors
K
a
(5)μ
(p
±
,S).
1
Concerning the independent kinematic factors one has
two possibilities, which are mathematically equivalent, de-
pending on whether one chooses p
−
or p
+
as essential vari-
able as we did in our previous papers [1] and [40], respec-
tively. The corresponding choices lead to different dGPDs.
(1) In the first case, which we considered in [1], cf. also
[2] and [3], p
−
was chosen as essential variable, by start-
ing from the physical picture using the generalized optical
theorem, and the parameterization
2
p
−
z
−
+p
+
z
+
=
ˆ
λ [p
−
+
ˆ
ζ(p
+
−p
−
/η)]
=
ˆ
λ
p
−
+
ˆ
ζ ˆπ
−
≡
ˆ
λ
ˆ
P, (3.11)
with
ˆ
λ =z
−
+z
+
/η,
z
+
=
ˆ
λ
ˆ
ζ, (3.12)
z
−
=
ˆ
λ(1 −
ˆ
ζ/η).
1
In the following the explicit t -dependence of the distribution functions
is always understood and we drop this variable to lighten the notation.
2
For later convenience the notation (ϑ, ζ ) of Ref. [1] has been changed
into (
ˆ
λ,
ˆ
ζ).