Diffractive parton distributions from the analysis
with higher twist
Krzysztof Golec-Biernat(a,b)∗ and Agnieszka  Luszczak(b)†
aInstitute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland
bInstitute of Physics, University of Rzeszów, Rzeszów, Poland
(October 29, 2018)
Abstract
Diffractive parton distributions of the proton are determined from fits to diffractive data
from HERA. In addition to the twist–2 contribution, the twist–4 contribution from longi-
tudinally polarised virtual photons is considered, which is important in the region of small
diffractive masses. A new prediction for the longitudinal diffractive structure function is
presented which differs significantly from that obtained in the pure twist–2 analyses.
e-mail: golec@ifj.edu.pl
e-mail: agnieszka.luszczak@ifj.edu.pl
http://arxiv.org/abs/0704.1608v3
1 Introduction
The diffractive deep inelastic scattering (DDIS) at HERA provide a very interesting example of
the interplay between hard and soft aspects of QCD interactions. On one side, the virtuality of
the photon probe is large (Q2 ≫ Λ2QCD), while on the other side, the scattered proton remains
almost intact, loosing only a small fraction of its initial momentum. Its transverse momentum
with respect to the photon-proton collision axis is also small. In addition to the scattered incident
particles, a diffractive system forms which is well separated in rapidity from the scattered proton.
The most important observation made at HERA is that diffractive processes in DIS are not rare,
quite the contrary, they constitute up to 15% of deep inelastic events. What’s more, the ratio of
the diffractive and inclusive cross sections is constant as a function of energy of the γ∗p system
or as a function of the photon virtuality. The latter fact reflects the logarithmic dependence on
Q2 of diffractive structure functions in the Bjorken limit.
In the t-channel picture, the diffractive interactions can be viewed as a vacuum quantum
number exchange between the diffractive system and the proton. In old days of Regge phe-
nomenology such a mechanism of interactions was termed a pomeron. With the advent of
quantum chromodynamic we gain a new way of understanding the pomeron by modelling it
with the help of gluon exchanges projected onto the color singlet state. In the lowest approxi-
mation, the pomeron is a two gluon exchange which is independent of energy. By considering
radiative corrections to this process in the high energy limit, the famous BFKL pomeron [1–4]
was constructed with a strong, power-like dependence on energy. This dependence ultimately
violates unitarity which means that exchanges with more gluons have to be considered. A sys-
tematic program to sum exchanges with gluon number changing vertices was formulated in [5,6]
and developed in [7–10]. Other, somewhat more intuitive formulation, called Color Glass Con-
densate [11–14], is based on the idea of parton saturation [15] in which deep inelastic scattering
occurs on a dense gluonic system in the proton. In these approaches unitarization is supposed
to change the asymptotic energy behaviour of the cross sections involving the pomeron from
power-like to logarithmic.
DDIS is particularly sensitive to the pomeron energy behaviour since diffractive scattering
amplitudes are squared in diffractive cross sections. Thus, unitarization effects play more im-
portant role than for the total cross section which is proportional to the imaginary part of the
scattering amplitude. This observation is a basis of a successful description of the first diffractive
data from HERA in which the diffractive system was formed by the quark-antiquark (qq) and
quark-antiquark-gluon (qqg) systems. They can be viewed in the space of Fourier transformed
transverse momenta as color dipoles [16,17]. In the approach we follow in the forthcoming pre-
sentation, the pomeron interaction is modelled by a two-gluon exchange which is subsequently
substituted by the effective dipole–proton cross section fitted to inclusive DIS data [18, 19]. In
this way unitary is achieved.
In an alternative approach to DDIS, the diffractive structure functions are defined in terms of
diffractive parton distributions (DPD). They are evolved in Q2 with the help of the Dokshitzer-
Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [20–22]. Thus in the Bjorken limit, the
diffractive structure functions depend logarithmically on Q2, i.e. they provide the twist–2 de-
scription of DDIS. The theoretical justification of this approach is provided by the collinear fac-
torisation theorem which is valid for hard diffractive scattering in ep collisions [23–27]. However,
collinear factorisation fails in hadron–hadron scattering due to nonfactorizable soft interactions
between incident hadrons [28,29]. Thus, unlike inclusive parton distributions, the DPD are not
Figure 1: Kinematic variables relevant for diffractive DIS.
universal objects and in general can only be used for diffractive processes in the ep deep inelas-
tic scattering. Nevertheless, the scale of nonuniversality can be estimated by applying them to
hadronic reactions.
The relation between the color dipole approach with the qq and qqg diffractive components
and the DGLAP based description was studied in detail in [30]. In short, after extracting the
twist–2 part, the dipole approach provides Q2-independent quark and gluon DPD. In addition,
the qqg component, which was computed assuming strong ordering between transverse momenta
of the gluon and the qq pair, gives the first step in the Q2-evolution of the gluon distribution.
The twist–2 approach, which is based on the DGLAP evolution equations, extends the two
component dipole picture by taking into account more complicated diffractive final state. In
the performed up till now twist-2 analyses, the diffractive parton distributions are determined
through fits to diffractive data from HERA [31] We will follow this approach with an important
modification.
The dipole approach teaches us one important lesson concerning the seemingly subleading
twist–4 contribution, given by the qq pair from longitudinally polarised virtual photons (Lqq).
Formally, it is is suppressed by a power of 1/Q2 with respect to the leading twist–2 transverse
contribution. However, the perturbative QCD calculation shows that for small diffractive masses,
M2 ≪ Q2, the longitudinal contribution dominates over the twist–2 one which tends to zero
in this limit. The effect of the Lqq component is particularly important for the longitudinal
diffractive structure function FDL which is supposed to be determined from the high luminosity
run data at HERA. Thus, we claim that it is absolutely necessary to consider the twist–4
contribution in the determination of the diffractive parton distributions. The analysis which we
present confirms its relevance for the prediction for FDL , which differs significantly from that
based on the pure DGLAP analysis. This is the main result of our paper.
The paper is organised as follows. In Section 2 we provide basic formulae for the kinematical
variables and quantities measured in diffractive deep inelastic scattering. We also describe the
three contributions which we include in the description of the diffractive structure functions, i.e.
the twist–2, twist–4 and Regge contributions. In Section 3 we describe performed fits while in
Section 4 we present their impact on the determination of the diffractive parton distributions
and diffractive structure functions. We finish with conclusions and outlook.
2 Basic formulae
We consider diffractive deep inelastic scattering: ep → e′p′X, shown schematically in Fig. 1.
After averaging over the azimuthal angle of the scattered proton, the four-fold differential cross
section is given in terms of the diffractive structure functions FD2 and F
dβ dQ2 dxIP dt
2πα2em
1 + (1− y)2
FD2 −
1 + (1− y)2
where y = Q2/(xBs) and s is the ep centre-of-mass energy squared. The expression in the curly
bracket is called reduced cross section:
σDr = F
1 + (1− y)2
FDL . (2)
Both structure functions depend on four kinematic variables (β,Q2, xIP , t), defined as follows
xIP =
Q2 +M2 − t
Q2 +W 2
, β =
Q2 +M2 − t
, (3)
where −Q2 is virtuality of the photon, t = (p − p′)2 < 0 is the square of four-momentum
transferred into the diffractive system, M is invariant diffractive mass and W is invariant energy
of the γ∗p system. The Bjorken variable xB = xIP β. For most of the diffractive events |t| is
much smaller then other scales, thus it can be neglected in eqs. (3). The diffractive structure
functions are measured in a limited range of t, thus the integrated structure functions are defined
2,L (β,Q
2, xIP ) =
∫ tmax
dt FD2,L(β,Q
2, xIP , t) , (4)
The integrated reduced cross section σ
r is defined in a similar way.
2.1 Twist–2 contribution
In the QCD approach based on collinear factorisation, the diffractive structure functions are
decomposed into the leading and higher twist contributions
FD2,L = F
D(tw2)
2,L + F
D(tw4)
2,L + . . . . (5)
The twist–2 part is given in terms of the diffractive parton distributions through the standard
collinear factorisation formulae [23,32–34]. In the next-to-leading logarithmic approximation we
D(tw2)
2 (x,Q
2, xIP , t) = SD +
CS2 ⊗ SD +C
2 ⊗GD
D(tw2)
L (x,Q
2, xIP , t) =
CSL ⊗ SD + C
L ⊗GD
where αs is the strong coupling constant and C
2,L are coefficients functions known from inclusive
DIS [35,36]. The integral convolution is performed for the longitudinal momentum fraction
(C ⊗ F )(β) =
dz C (β/z)F (z) . (8)
Notice that in the leading order, when terms proportional to αs are neglected, the longitudinal
structure function F
D(tw2)
L = 0. The functions SD and GD are given by diffractive quark and
gluon distributions, q
D and gD:
e2f β
D(β,Q
2, xIP , t) + q
D(β,Q
2, xIP , t)
GD = βgD(β,Q
2, xIP , t) (10)
Note that β = x/xIP plays the role of the Bjorken variable in DDIS. In the infinite momentum
frame, the DPD are interpreted as conditional probabilities to find a parton with the momentum
fraction x = βxIP in a proton under the condition that the incoming proton stays intact losing
a small fraction xIP of its momentum. A formal definition of the diffractive parton distributions
based on the quark and gluon twist-2 operators is given in [23,25].
The DPD are evolved in Q2 by the DGLAP evolution equations [37] for which the variables
(xIP , t) play the role of external parameters. In this analysis we assume Regge factorisation for
these variables:
D(β,Q
2, xIP , t) = fIP (xIP , t) q
IP (β,Q
2) (11)
gD(β,Q
2, xIP , t) = fIP (xIP , t) gIP (β,Q
2) . (12)
For convenience, the functions q
IP (β,Q
2) and gIP (β,Q
2) are called pomeron parton distributions.
The motivation for such a factorisation is a model of diffractive interactions with a pomeron
exchange [38]. In this model fIP is the pomeron flux
fIP (xIP , t) =
F 2IP (t)
1−2αIP (t)
IP , (13)
where αIP (t) = αIP (0) + α
IP t is the pomeron Regge trajectory and the formfactor
F 2IP (t) = F
IP (0) e
−BD |t| (14)
describes the pomeron coupling to the proton. We set F 2IP (0) = 54.4 GeV
−2 [34], BD =
5.5 GeV−2 and α′IP = 0.06 GeV
−2 [31], while the pomeron intercept αIP (0) is fitted to data.
The pomeron quark distributions are flavour independent, thus they are given by one function,
a singlet quark distribution ΣIP :
IP (β,Q
2) = q
IP (β,Q
ΣIP (β,Q
2) (15)
where Nf is a number of active flavours. The question about Regge factorisation is an issue
which should be tested experimentally. In our approach, the pomeron is a model of diffractive
interactions which only provides energy dependence through the xIP -dependent pomeron flux.
Its normalisation is only a useful convention since the normalisations of the pomeron parton
distributions in eqs. (11) are fitted to data.
2.2 Twist-2 charm contribution
We describe the charm quark diffractive production using twist-2 formulae for the cc pair gener-
ation from a gluon. These are formulae analogous to the inclusive case in which the diffractive
Figure 2: The qq̄ and qq̄g components of the diffractive system in the dipole approach.
gluon distribution gD is substituted for the inclusive one [39]:
D(cc)
2,L (β,Q
2, xIP , t) = 2β e
gD(z, µ2c , xIP , t) , (16)
where a = 1 + 4m2c/Q
2 and the factorisation scale µ2c = 4m
c with the charm quark mass
mc = 1.4 GeV. The coefficient functions read
C2(z, r) =
z2 + (1− z)2 + 4z(1− 3z)r − 8z2r2
1 + α
α {−1 + 8z(1 − z)− 4z(1 − z)r} (17)
CL(z, r) = −4z
2r ln
1 + α
+ 2αz(1 − z) (18)
with α =
1− 4rz/(1 − z). The cc pair can only be produced if invariant mass of the diffractive
system M2 fulfils the following condition
M2 = Q2
> 4m2c . (19)
2.3 Twist–4 contribution
The computation of the twist–4 contribution, proportional to 1/Q2, is a nontrivial task and
one could be tempted to assume that this contribution is suppressed at large Q2 as in inclusive
DIS. However, by analysing diffractive final states in the dipole approach it was found that
for diffractive mass M2 ≪ Q2 (β → 1), the twist–4 contribution dominates over the vanishing
twist–2 one [19,40,41].
This observation is made on the basis of the perturbative QCD calculations in which the
diffractive state is formed by the qq and qqg systems interacting with a proton through a colorless
gluonic exchange which is a model of the pomeron interactions in QCD. In the simplest case, two
gluons projected onto the color singlet state are exchanged, see Fig. 2. The computed amplitudes
do not depend on energy in such a case which problem can be cured in a more sophisticated
approach by modelling the dipole-proton cross section which fulfils unitarity conditions [19].
Independ of the details of the pomeron description, the diffractive mass (or β) dependence
is a genuine prediction of pQCD calculations. It appears that the leading in Q2 behaviour
components, qq and qqg from transverse virtual photons, vanish for β → 1. This is not the case
for the qq production from longitudinal photons (Lqq) which is formally suppressed by 1/Q2
with respect to the leading components. Thus, this particular β-dependence makes the Lqq
contribution dominant for β → 1, see Fig. 3.
The presence of the Lqq component has important consequence for the longitudinal diffractive
structure function which is supposed to be determined from the HERA data. The formula given
below is an important element in the description of FDL in the region of large β:
FDLqq̄ =
16π4xIP
e−BD |t|
(1− β)4
2(1−β)
k2/Q2
φ20(k, xIP ) (20)
where the function φ0(k, xIP ) is given in terms of the dipole cross section σ̂(xIP , r) and the Bessel
functions K0 and J0:
φ0(k, xIP ) = k
dr r K0
J0(kr) σ̂(xIP , r) . (21)
Strictly speaking, eq. (20) contains all inverse powers of Q2 but the part proportional to 1/Q2
(called twist–4) dominates. The dipole-proton cross section describes the interaction of a color
dipole, formed by the qq or qqg systems, with a proton. Following [18] we choose
σ̂(xIP , r) = σ0 {1− exp (−r
2Q2s/4)} (22)
where Q2s = (xIP /x0)
−λ GeV2 is a saturation scale which provides the energy dependence of
the twist–4 contribution. The parameters σ0 = 29 mb, x0 = 4 · 10
−5 and λ = 0.28 are taken
from [18] (Fit 2 with charm). This form of the dipole cross section provides successful description
of the first HERA data on both inclusive and diffractive structure functions [18,19]. A different
parametrisation of σ̂, without the saturation scale, is also given in [42–44]. We checked that a
very similar description of FDLqq̄ was found in a recent analysis [45] based on the Color Glass
Condensate parameterisation of the dipole scattering amplitude [46].
The relation between the dipole approach with three diffractive components and the DGLAP
approach with diffractive parton distributions was analysed at length in [30]. Summarising this
relation, the twist–2 part of the qq component gives a diffractive quark distribution. The twist-2
part of the qqg component forms a first step of the DGLAP evolution which starts from a given
gluon distribution. Both diffractive parton distributions do not depend on Q2, thus they may
serve as initial conditions for the DGLAP equations at the scale which is not determined. From
this perspective, the DGLAP approach offers a description of more complicated diffractive state
with any number of partons ordered in transverse momenta. However, the pQCD calculations
tell us that the twist–2 analysis of diffractive data should include the twist–4 contribution since
it cannot be neglected at large β. This is the strategy which we follow in our analysis.
We also borrow from the dipole approach a general form in β of the initial quark distribution
which vanishes at the endpoints β = 0, 1 (see eq. (31) in which Aq and Cq are positive). A very
important aspect of Regge factorisation (11) can also be motivated by the dipole approach. It
is a consequence of geometric scaling of the dipole cross section (22) [30,47].
2.4 Reggeon contribution
The diffractive data from the H1 collaboration for higher values of xIP hints towards a contribu-
tion which decreases with energy. This effect can be described by reggeon exchanges in addition
to the rising with energy pomeron exchange. Following [48, 49], we consider the dominant
isoscalar (f2, ω) reggeon exchanges which lead to the following contribution to F
fR(xIP , t)FR(β,Q
2) . (23)
This contribution breaks Regge factorisation of the diffractive structure function, however, its
presence is necessary for xIP > 0.01 [50]. The reggeon flux fR is given by the formula analogous
to eq. (13)
fR(xIP , t) =
F 2R(0)
e−|t|/Λ
R |ηR(t)|
1−2αR(t)
IP , (24)
where αR(t) = 0.5475 + 1 · t is the reggeon trajectory. From the Regge phenomenology of
hadronic reactions ΛR = 0.65GeV and the reggeon–proton couplings are given by [49]: F
(0) =
194GeV−2 and F 2ω(0) = 52GeV
−2. The functions
|ηR(t)|
2 = 4cos2[παR(t)/2] , |ηR(t)|
2 = 4 sin2[παR(t)/2] (25)
are signature factors for even (f2) and odd (ω) reggeons, respectively. We could also consider
isovector reggeons (a2, ρ) but their couplings to the proton are much smaller and we neglect
them. Finally, the reggeon structure function FR is given by [49]
FR(β) = AR β
−0.08 (1− β)2 , (26)
where the normalisation AR is a fitted parameter. Thus, in the first approximation, we neglect
the Q2-dependence of the reggeon contribution.
3 Fit details
Collab. No. points Data |t|-range Q2-range β-range
H1 [50] 72 LP [0.08, 0.5] [2 , 50] [0.02 , 0.7]
ZEUS [51] 80 LP [0.075 , 0.35] [2 , 100] [0.007 , 0.48]
H1 [31] 461 MY < 1.6 [|tmin| , 1] [3.5 , 1600] [0.01 , 0.9]
ZEUS [52] 198 MY < 2.3 [|tmin| ,∞] [2.2 , 80] [0.003 , 0.975]
Table 1: Kinematic regions of diffractive data from HERA. LP means leading proton data and
MY is invariant mass of a dissociated proton. Dimensionfull quantities are in units of 1 GeV.
In our analysis we use diffractive data from the H1 [31,50] and ZEUS [51,52] collaborations.
In Table 1 we show their kinematic limits in which they have been measured. The minimal value
value of |t| is given by
|tmin| ≃
1− xIP
m2p , (27)
where mp is the proton mass. The leading proton data from H1, measured in the range given
in Table 1, were corrected by the H1 collaboration to the range |tmin| < |t| < 1 GeV
The ZEUS data are given for the diffractive structure function FD2 , thus we use in our
analysis the following formulae
FD2 = F
D(tw2)
2 + F
2 + F
Lqq̄ (28)
FDL = F
D(tw2)
L + F
Lqq̄ . (29)
No Data Fit αIP (0) AR Aq Bq Cq Ag Bg Cg χ
1 H1 (LP) tw-2 1.098 0.29 1.75 1.49 0.5∗ 2.09 0.67 0.80 0.48
2 ZEUS (LP) tw-2 1.145 1.05 2.13 1.51 0.5∗ 10.0* 1.03 2.26 0.40
3 H1 tw-2 1.117 0.49 1.33 1.63 0.34 0.17 -0.16 -1.10 1.04
4 tw-(2+4) 1.119 0.48 1.62 1.98 0.59 0.04 -0.56 -1.68 1.17
5 ZEUS tw-2 1.093 0.0∗ 1.68 1.01 0.5∗ 0.49 -0.03 -0.40 1.35
6 tw-(2+4) 1.092 0.0∗ 1.20 0.85 0.57 0.07 -0.52 -1.48 1.82
Table 2: The fit parameters to H1 nd ZEUS data. The presence of twist–4 in the fits is marked
by tw-(2+4). The parameters with an asterisk are fixed in the fits.
The longitudinal twist-4 contribution is present on the r.h.s. of eq. (28) since FD2 is the sum of
the contributions from the transverse and longitudinal polarised virtual photon. The H1 data,
however, are presented for the reduced cross section (2). Thus we substitute relations (28) and
(29) in there and use
σDr =
D(tw2)
2 + F
1 + (1− y)2
D(tw2)
2(1− y)
1 + (1− y)2
FDLqq . (30)
The expression in the curly brackets is the twist–2 contribution while the last term is the twist–4
one. Notice that the difference between FD2 and σ
r is most important for y → 1.
We fit the diffractive parton distributions at the initial scale Q20 = 1.5 GeV
2, assuming the
Regge factorised form (11) with the following pomeron parton distributions [31]:
βΣIP (β) = Aq β
Bq (1− β)Cq (31)
βgIP (β) = Ag β
Bg (1− β)Cg . (32)
The six indicated parameters are fitted to data. We additionally multiplied both distributions
by a factor exp{−a/(1 − β)} with a = 0.01 to secure their vanishing for β = 1. This factor
is only important when Cq or Cg becomes negative in the fits. For the evolution, we use the
next-to-leading order DGLAP equations with ΛQCD = 407 MeV for Nf = 3 flavours [53].
The pomeron flux in eq. (11) is integrated over t in the limits given in Table 1 which leads
to the form
fIP (xIP ) =
F 2IP (0)
e−B|tmin| − e−B|tmax|
1−2αIP (0)
IP . (33)
The shrinkage parameter B equals
B = BD + 2α
IP ln(1/xIP ) (34)
with BD = 5.5GeV
−2 and α′IP = 0.06GeV
−2 [50].
In summary, we have eight fit parameters altogether: the pomeron intercept αIP (0), reggeon
normalisation AR in eq. (26) and six parameters in eqs. (31,32)
4 Fit results
The data sets from Table 1 were obtained in different kinematical regions, using different methods
of their analysis. Thus, we decided to perform fits to each data set separately. The values of
the fit parameter are shown in Table 2. The difference between them can be attributed to the
scale of uncertainty of our analysis. In each case we preformed two fits: with and without the
twist–4 formula added to the twist–2 contribution.
4.1 Leading proton data
We started from fits to the leading proton data. The fit parameters in this case are displayed
in the first two rows of Table 2. We only show the twist–2 fit results since they are not changed
in fits with the twist–4 term. This happenes because the leading proton data comes from the
region of β values where the twist–4 contribution is small (β ≤ 0.7 for H1 and β < 0.5 for
ZEUS), see Fig. 3.
The data with a dissociated proton (DP) which are measured in the region of large β influence
most the value of the parameter Cg which controls the behaviour of the gluon distribution at
β → 1. For the LP data Cg is positive and the gluon distribution is suppressed near β ≈ 1, while
for the DP data Cg is negative and the gluon distribution is strongly enhanced. This shows that
the data with β > 0.7 are crucial for the proper analysis. Without this kinematic region we lose
important information about diffractive interactions. Thus, from now on we concentrate on the
analysis of the DP data.
4.2 H1 data
The fit parameters to the H1 data with a dissociate proton are given in the third and fourth rows
of Table 2. We see that the fit quality is practically the same for both fits, with and without the
twist–4 contribution. The presence of the reggeon term improves fit quality by 30 units of χ2
for 461 experimental points. A good quality of the fits is illustrated in Fig. 4 which also shows
that the reduced cross sections (30) from the twist–2 (solid lines) and twist–(2+4) fits (dashed
lines) are very close to each other.
In Fig. 5 we show our results on the reduced cross section for the largest measured value of
β = 0.9. In this region, the twist–4 contribution, shown by the dotted lines, cannot be neglected.
We see that the curves from both the twist–2 (solid) and twist–(2+4) (dashed) fits describe data
reasonable well. However, the curves with twist–4 have a steeper dependence on xIP (energy)
than in the pure twist–2 analysis. This observation is by far more pronounced in the analysis of
the ZEUS data performed for the structure function FD2 .
The diffractive parton distributions from our fits are shown in Fig. 6 in terms of the pomeron
parton distributions, βΣIP (β,Q
2) and βgIP (β,Q
2). Being independent of the pomeron flux, such
a presentation allows for a direct comparison of the results from fits to different data sets. We see
that the singlet quark distributions are quite similar while the gluon distributions are different.
In the fit with twist–4, the gluon distribution is stronger peaked near β ≈ 1. This somewhat
surprising result can be understood by looking at the logarithmic slope of FD2 for fixed values
of β. From the LO DGLAP equations we have schematically:
∂ lnQ2
∂ lnQ2
= Pqq ⊗ ΣIP + PqG ⊗GIP − ΣIP
Pqq (35)
where the negative term describes virtual corrections. For large β, the measured slope is negative
which means that the virtual emission term must dominate over the real emission ones. The
addition of the twist–4 contribution to FD2 , proportional to 1/Q
2, contributes a negative value
to the slope which has to be compensated by a larger gluon distribution in order to describe the
same data.
In Fig. 7 we present our most important results. On the left panel, the FD2 structure function
is shown from both fits, with and without the twist–4 contribution (shown by the dotted lines).
We see no significant difference between these two results. However, the longitudinal structure
function FDL differs significantly for the two fits (right panel) due to the twist–4 contribution.
Let us emphasise that both sets of curves were found in the fits which well describe the existing
data on σDr , including the large β region. Thus, an independent measurement of F
L in this
region would be an important test of the QCD mechanism of diffraction.
4.3 ZEUS data
The results of same fits performed for the ZEUS data are shown in the last two rows of Table 2.
This time the Regge contribution (26) is not necessary since fits give the reggeon normalisation
AR ≈ 0. In general, the fit quality is worse than for the H1 data.
As shown in Fig. 8, the biggest difference between the twist–2 and twist–(2+4) results occurs
at large β values. This is analysed in detail in Fig. 9. We see that the presence of the twist–4
term in the fit (dashed lines) improves the agreement with the data in this region. In particular,
a steep dependence of FD2 on xIP is better reproduced by the twist–(2+4) fit then by the twist–2
one (solid lines). This dependence is to large extend driven by the twist–4 contribution (dotted
lines).
The behaviour of the diffractive parton distributions and structure functions, shown in
Figs. 10 and 11, respectively, is very similar to that found for the H1 data. The gluon dis-
tribution from the fit with twist–4 is stronger peaked near β ≈ 1 and the longitudinal structure
functions in the large β region is dominated by the twist–4 contribution.
We summarise the effect of the twist–4 contribution in Fig. 12 showing the predictions for
the longitudinal diffractive structure function FDL . Ignoring this contribution, we find the two
solid curves coming from the pure twist–2 analysis of the H1 (upper) and ZEUS (lower) data.
With twist–4, the dashed curves are found, the upper one from the H1 data and the lower one
from the ZEUS data. There is a significant difference between these two predictions in the
region of large β. We believe that the effect of the twist–4 contribution will be confirmed by the
forthcoming analysis of the HERA data.
5 Conclusions
We performed fits of the diffractive parton distributions to new diffractive data from the H1 and
ZEUS collaborations at HERA. In addition to the standard twist–2 formulae, we also considered
the twist–4 contribution which dominates in the region of large β. This contribution comes from
the diffractive production of the qq pair by the longitudinally polarised virtual photons. The
effect of the twist–4 contribution on the diffractive parton distributions and structure functions
was carefully examined. The twist–4 contribution leads to the gluon distribution which is peaked
stronger at β ≈ 1 than in the case without twist–4.
The main result of our analysis is a new prediction for the longitudinal diffractive struc-
ture function FDL . The twist–4 term significantly enhances F
L in the region of large β. A
measurement of this function at HERA in the region of large β should confirm the presented
expectations which are based on the perturbative QCD calculations. The obtained diffractive
parton distributions can also be used in the analysis of diffractive processes at the LHC, in
particular, to the estimation of the background to the diffractive Higgs production, see [54] for
a recent discussion.
Acknowledgements
This research has been partly supported by MEiN research grant 1 P03B 028 28 (2005-08)
and by EU grant HEPTOOLS, MRTN-CT-2006-035505. The hospitality of the Galileo Galilei
Institute for Theoretical Physics in Florence and partial support of INFN during the completion
of this work is gratefully acknowledged.
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Diffractive structure function
Q2=8 GeV2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 3: Three contributions to FD2 from: qq and qqg from transverse (T) and longitudinal (L)
photons [19] for xIP = 0.003. The twist–4 contribution Lqq̄ is indicated by the yellow band. Old
ZEUS data points are shown.
H1 DATA
β=0.01
β=0.01 β=0.04β=0.04 β=0.1β=0.1 β=0.2β=0.2 β=0.4β=0.4 β=0.65β=0.65 β=0.9 Q
β=0.9 Q
6.56.5
8.58.5
200200
Figure 4: Reduced cross section σ
r for H1 data as a function of xIP . Solid lines: twist–2 fit,
dashed lines: twist–(2+4) fit.
H1 DATA (β=0.9)
Q2= 3.5 GeV2
dashed: tw-(2+4) fit
solid:  tw-2 fit
Q2= 5 GeV2
dotted:  tw-4 contribution
Q2= 6.5 GeV2
Q2= 8.5 GeV2
Figure 5: Reduced cross section σ
r for H1 data at β = 0.9 for four values of Q
2 against fit
curves.
DPD (H1)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 6: Pomeron parton distributions: singlet βΣIP (β,Q
2) (left) and gluon βgIP (β,Q
2) (right)
from H1 data.
DSF (H1)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
dotted: tw-4 contribution
dotted: tw-4 contribution
dotted: tw-4 contribution
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 7: Diffractive structure functions F
2 (left) and F
L (right) from fits to H1 data for
xIP = 10
−3. The band shows the effect of twist–4 on the predictions for F
ZEUS DATA
β=0.652
β=0.652 β=0.231β=0.231 β=0.07β=0.07 β=0.022β=0.022 β=0.007β=0.007
β=0.003
β=0.003
β=0.735β=0.735 β=0.308β=0.308 β=0.1β=0.1 β=0.032β=0.032 β=0.010β=0.010
β=0.004
β=0.004
β=0.807β=0.807 β=0.4β=0.4 β=0.143β=0.143 β=0.047β=0.047 β=0.015β=0.015
β=0.007
β=0.007
β=0.848β=0.848 β=0.471β=0.471 β=0.182β=0.182 β=0.062β=0.062 β=0.020β=0.020
β=0.009
β=0.009
β=0.907β=0.907 β=0.61β=0.61 β=0.28β=0.28 β=0.104β=0.104 β=0.034β=0.034
β=0.015
β=0.015
β=0.949β=0.949 β=0.75β=0.75 β=0.43β=0.43 β=0.182β=0.182 β=0.063β=0.063
β=0.029
β=0.029
β=0.975β=0.975 β=0.86β=0.86 β=0.604β=0.604 β=0.313β=0.313 β=0.121β=0.121
Figure 8: Diffractive structure function F
2 as a function xIP for ZEUS data. Solid lines:
twist–2 fit, dashed lines: twist–(2+4) fit.
ZEUS DATA
dotted: tw-4 contribution
β = 0.86 Q2= 55 GeV2
solid:     tw-2 fit
dashed: tw-(2+4) fit
β = 0.91 Q2= 14 GeV2
β = 0.95 Q2= 27 GeV2
β = 0.975 Q2= 55 GeV2
Figure 9: Diffractive structure function F
2 for ZEUS data at large values of β against fit
curves.
DPD (ZEUS)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 10: Pomeron parton distributions βΣIP (β,Q
2) (left) and βgIP (β,Q
2) (right) from fits to
ZEUS data.
DSF (ZEUS)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
dotted: tw-4 contribution
dotted: tw-4 contribution
dotted: tw-4 contribution
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 11: Diffractive structure functions F
2 (left) and F
L (right) from fits to ZEUS data
for xIP = 10
−3. The band shows the effect of twist–4 on the predictions for F
Diffractive FL
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
0.005
0.015
0.025
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 12: Predictions for F
L for xIP = 10
−3 and Q2 = 10 GeV2 from the twist–(2+4) fits to
the H1 (upper dashed line) and ZEUS (lower dashed line) data. The solid lines show predictions
from pure twist–2 fits to the H1 (upper) and ZEUS (lower) data.
	Introduction
	Basic formulae
	Twist–2 contribution
	Twist-2 charm contribution
	Twist–4 contribution
	Reggeon contribution
	Fit details
	Fit results
	Leading proton data
	H1 data
	ZEUS data
	Conclusions

  Diffractive parton distributions of the proton are determined from fits to
diffractive data from HERA. In addition to the twist--2 contribution, the
twist--4 contribution from longitudinally polarised virtual photons is
considered, which is important in the region of small diffractive masses. A new
prediction for the longitudinal diffractive structure function is presented
which differs significantly from that obtained in the pure twist--2 analyses.

Introduction
The diffractive deep inelastic scattering (DDIS) at HERA provide a very interesting example of
the interplay between hard and soft aspects of QCD interactions. On one side, the virtuality of
the photon probe is large (Q2 ≫ Λ2QCD), while on the other side, the scattered proton remains
almost intact, loosing only a small fraction of its initial momentum. Its transverse momentum
with respect to the photon-proton collision axis is also small. In addition to the scattered incident
particles, a diffractive system forms which is well separated in rapidity from the scattered proton.
The most important observation made at HERA is that diffractive processes in DIS are not rare,
quite the contrary, they constitute up to 15% of deep inelastic events. What’s more, the ratio of
the diffractive and inclusive cross sections is constant as a function of energy of the γ∗p system
or as a function of the photon virtuality. The latter fact reflects the logarithmic dependence on
Q2 of diffractive structure functions in the Bjorken limit.
In the t-channel picture, the diffractive interactions can be viewed as a vacuum quantum
number exchange between the diffractive system and the proton. In old days of Regge phe-
nomenology such a mechanism of interactions was termed a pomeron. With the advent of
quantum chromodynamic we gain a new way of understanding the pomeron by modelling it
with the help of gluon exchanges projected onto the color singlet state. In the lowest approxi-
mation, the pomeron is a two gluon exchange which is independent of energy. By considering
radiative corrections to this process in the high energy limit, the famous BFKL pomeron [1–4]
was constructed with a strong, power-like dependence on energy. This dependence ultimately
violates unitarity which means that exchanges with more gluons have to be considered. A sys-
tematic program to sum exchanges with gluon number changing vertices was formulated in [5,6]
and developed in [7–10]. Other, somewhat more intuitive formulation, called Color Glass Con-
densate [11–14], is based on the idea of parton saturation [15] in which deep inelastic scattering
occurs on a dense gluonic system in the proton. In these approaches unitarization is supposed
to change the asymptotic energy behaviour of the cross sections involving the pomeron from
power-like to logarithmic.
DDIS is particularly sensitive to the pomeron energy behaviour since diffractive scattering
amplitudes are squared in diffractive cross sections. Thus, unitarization effects play more im-
portant role than for the total cross section which is proportional to the imaginary part of the
scattering amplitude. This observation is a basis of a successful description of the first diffractive
data from HERA in which the diffractive system was formed by the quark-antiquark (qq) and
quark-antiquark-gluon (qqg) systems. They can be viewed in the space of Fourier transformed
transverse momenta as color dipoles [16,17]. In the approach we follow in the forthcoming pre-
sentation, the pomeron interaction is modelled by a two-gluon exchange which is subsequently
substituted by the effective dipole–proton cross section fitted to inclusive DIS data [18, 19]. In
this way unitary is achieved.
In an alternative approach to DDIS, the diffractive structure functions are defined in terms of
diffractive parton distributions (DPD). They are evolved in Q2 with the help of the Dokshitzer-
Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [20–22]. Thus in the Bjorken limit, the
diffractive structure functions depend logarithmically on Q2, i.e. they provide the twist–2 de-
scription of DDIS. The theoretical justification of this approach is provided by the collinear fac-
torisation theorem which is valid for hard diffractive scattering in ep collisions [23–27]. However,
collinear factorisation fails in hadron–hadron scattering due to nonfactorizable soft interactions
between incident hadrons [28,29]. Thus, unlike inclusive parton distributions, the DPD are not
Figure 1: Kinematic variables relevant for diffractive DIS.
universal objects and in general can only be used for diffractive processes in the ep deep inelas-
tic scattering. Nevertheless, the scale of nonuniversality can be estimated by applying them to
hadronic reactions.
The relation between the color dipole approach with the qq and qqg diffractive components
and the DGLAP based description was studied in detail in [30]. In short, after extracting the
twist–2 part, the dipole approach provides Q2-independent quark and gluon DPD. In addition,
the qqg component, which was computed assuming strong ordering between transverse momenta
of the gluon and the qq pair, gives the first step in the Q2-evolution of the gluon distribution.
The twist–2 approach, which is based on the DGLAP evolution equations, extends the two
component dipole picture by taking into account more complicated diffractive final state. In
the performed up till now twist-2 analyses, the diffractive parton distributions are determined
through fits to diffractive data from HERA [31] We will follow this approach with an important
modification.
The dipole approach teaches us one important lesson concerning the seemingly subleading
twist–4 contribution, given by the qq pair from longitudinally polarised virtual photons (Lqq).
Formally, it is is suppressed by a power of 1/Q2 with respect to the leading twist–2 transverse
contribution. However, the perturbative QCD calculation shows that for small diffractive masses,
M2 ≪ Q2, the longitudinal contribution dominates over the twist–2 one which tends to zero
in this limit. The effect of the Lqq component is particularly important for the longitudinal
diffractive structure function FDL which is supposed to be determined from the high luminosity
run data at HERA. Thus, we claim that it is absolutely necessary to consider the twist–4
contribution in the determination of the diffractive parton distributions. The analysis which we
present confirms its relevance for the prediction for FDL , which differs significantly from that
based on the pure DGLAP analysis. This is the main result of our paper.
The paper is organised as follows. In Section 2 we provide basic formulae for the kinematical
variables and quantities measured in diffractive deep inelastic scattering. We also describe the
three contributions which we include in the description of the diffractive structure functions, i.e.
the twist–2, twist–4 and Regge contributions. In Section 3 we describe performed fits while in
Section 4 we present their impact on the determination of the diffractive parton distributions
and diffractive structure functions. We finish with conclusions and outlook.
2 Basic formulae
We consider diffractive deep inelastic scattering: ep → e′p′X, shown schematically in Fig. 1.
After averaging over the azimuthal angle of the scattered proton, the four-fold differential cross
section is given in terms of the diffractive structure functions FD2 and F
dβ dQ2 dxIP dt
2πα2em
1 + (1− y)2
FD2 −
1 + (1− y)2
where y = Q2/(xBs) and s is the ep centre-of-mass energy squared. The expression in the curly
bracket is called reduced cross section:
σDr = F
1 + (1− y)2
FDL . (2)
Both structure functions depend on four kinematic variables (β,Q2, xIP , t), defined as follows
xIP =
Q2 +M2 − t
Q2 +W 2
, β =
Q2 +M2 − t
, (3)
where −Q2 is virtuality of the photon, t = (p − p′)2 < 0 is the square of four-momentum
transferred into the diffractive system, M is invariant diffractive mass and W is invariant energy
of the γ∗p system. The Bjorken variable xB = xIP β. For most of the diffractive events |t| is
much smaller then other scales, thus it can be neglected in eqs. (3). The diffractive structure
functions are measured in a limited range of t, thus the integrated structure functions are defined
2,L (β,Q
2, xIP ) =
∫ tmax
dt FD2,L(β,Q
2, xIP , t) , (4)
The integrated reduced cross section σ
r is defined in a similar way.
2.1 Twist–2 contribution
In the QCD approach based on collinear factorisation, the diffractive structure functions are
decomposed into the leading and higher twist contributions
FD2,L = F
D(tw2)
2,L + F
D(tw4)
2,L + . . . . (5)
The twist–2 part is given in terms of the diffractive parton distributions through the standard
collinear factorisation formulae [23,32–34]. In the next-to-leading logarithmic approximation we
D(tw2)
2 (x,Q
2, xIP , t) = SD +
CS2 ⊗ SD +C
2 ⊗GD
D(tw2)
L (x,Q
2, xIP , t) =
CSL ⊗ SD + C
L ⊗GD
where αs is the strong coupling constant and C
2,L are coefficients functions known from inclusive
DIS [35,36]. The integral convolution is performed for the longitudinal momentum fraction
(C ⊗ F )(β) =
dz C (β/z)F (z) . (8)
Notice that in the leading order, when terms proportional to αs are neglected, the longitudinal
structure function F
D(tw2)
L = 0. The functions SD and GD are given by diffractive quark and
gluon distributions, q
D and gD:
e2f β
D(β,Q
2, xIP , t) + q
D(β,Q
2, xIP , t)
GD = βgD(β,Q
2, xIP , t) (10)
Note that β = x/xIP plays the role of the Bjorken variable in DDIS. In the infinite momentum
frame, the DPD are interpreted as conditional probabilities to find a parton with the momentum
fraction x = βxIP in a proton under the condition that the incoming proton stays intact losing
a small fraction xIP of its momentum. A formal definition of the diffractive parton distributions
based on the quark and gluon twist-2 operators is given in [23,25].
The DPD are evolved in Q2 by the DGLAP evolution equations [37] for which the variables
(xIP , t) play the role of external parameters. In this analysis we assume Regge factorisation for
these variables:
D(β,Q
2, xIP , t) = fIP (xIP , t) q
IP (β,Q
2) (11)
gD(β,Q
2, xIP , t) = fIP (xIP , t) gIP (β,Q
2) . (12)
For convenience, the functions q
IP (β,Q
2) and gIP (β,Q
2) are called pomeron parton distributions.
The motivation for such a factorisation is a model of diffractive interactions with a pomeron
exchange [38]. In this model fIP is the pomeron flux
fIP (xIP , t) =
F 2IP (t)
1−2αIP (t)
IP , (13)
where αIP (t) = αIP (0) + α
IP t is the pomeron Regge trajectory and the formfactor
F 2IP (t) = F
IP (0) e
−BD |t| (14)
describes the pomeron coupling to the proton. We set F 2IP (0) = 54.4 GeV
−2 [34], BD =
5.5 GeV−2 and α′IP = 0.06 GeV
−2 [31], while the pomeron intercept αIP (0) is fitted to data.
The pomeron quark distributions are flavour independent, thus they are given by one function,
a singlet quark distribution ΣIP :
IP (β,Q
2) = q
IP (β,Q
ΣIP (β,Q
2) (15)
where Nf is a number of active flavours. The question about Regge factorisation is an issue
which should be tested experimentally. In our approach, the pomeron is a model of diffractive
interactions which only provides energy dependence through the xIP -dependent pomeron flux.
Its normalisation is only a useful convention since the normalisations of the pomeron parton
distributions in eqs. (11) are fitted to data.
2.2 Twist-2 charm contribution
We describe the charm quark diffractive production using twist-2 formulae for the cc pair gener-
ation from a gluon. These are formulae analogous to the inclusive case in which the diffractive
Figure 2: The qq̄ and qq̄g components of the diffractive system in the dipole approach.
gluon distribution gD is substituted for the inclusive one [39]:
D(cc)
2,L (β,Q
2, xIP , t) = 2β e
gD(z, µ2c , xIP , t) , (16)
where a = 1 + 4m2c/Q
2 and the factorisation scale µ2c = 4m
c with the charm quark mass
mc = 1.4 GeV. The coefficient functions read
C2(z, r) =
z2 + (1− z)2 + 4z(1− 3z)r − 8z2r2
1 + α
α {−1 + 8z(1 − z)− 4z(1 − z)r} (17)
CL(z, r) = −4z
2r ln
1 + α
+ 2αz(1 − z) (18)
with α =
1− 4rz/(1 − z). The cc pair can only be produced if invariant mass of the diffractive
system M2 fulfils the following condition
M2 = Q2
> 4m2c . (19)
2.3 Twist–4 contribution
The computation of the twist–4 contribution, proportional to 1/Q2, is a nontrivial task and
one could be tempted to assume that this contribution is suppressed at large Q2 as in inclusive
DIS. However, by analysing diffractive final states in the dipole approach it was found that
for diffractive mass M2 ≪ Q2 (β → 1), the twist–4 contribution dominates over the vanishing
twist–2 one [19,40,41].
This observation is made on the basis of the perturbative QCD calculations in which the
diffractive state is formed by the qq and qqg systems interacting with a proton through a colorless
gluonic exchange which is a model of the pomeron interactions in QCD. In the simplest case, two
gluons projected onto the color singlet state are exchanged, see Fig. 2. The computed amplitudes
do not depend on energy in such a case which problem can be cured in a more sophisticated
approach by modelling the dipole-proton cross section which fulfils unitarity conditions [19].
Independ of the details of the pomeron description, the diffractive mass (or β) dependence
is a genuine prediction of pQCD calculations. It appears that the leading in Q2 behaviour
components, qq and qqg from transverse virtual photons, vanish for β → 1. This is not the case
for the qq production from longitudinal photons (Lqq) which is formally suppressed by 1/Q2
with respect to the leading components. Thus, this particular β-dependence makes the Lqq
contribution dominant for β → 1, see Fig. 3.
The presence of the Lqq component has important consequence for the longitudinal diffractive
structure function which is supposed to be determined from the HERA data. The formula given
below is an important element in the description of FDL in the region of large β:
FDLqq̄ =
16π4xIP
e−BD |t|
(1− β)4
2(1−β)
k2/Q2
φ20(k, xIP ) (20)
where the function φ0(k, xIP ) is given in terms of the dipole cross section σ̂(xIP , r) and the Bessel
functions K0 and J0:
φ0(k, xIP ) = k
dr r K0
J0(kr) σ̂(xIP , r) . (21)
Strictly speaking, eq. (20) contains all inverse powers of Q2 but the part proportional to 1/Q2
(called twist–4) dominates. The dipole-proton cross section describes the interaction of a color
dipole, formed by the qq or qqg systems, with a proton. Following [18] we choose
σ̂(xIP , r) = σ0 {1− exp (−r
2Q2s/4)} (22)
where Q2s = (xIP /x0)
−λ GeV2 is a saturation scale which provides the energy dependence of
the twist–4 contribution. The parameters σ0 = 29 mb, x0 = 4 · 10
−5 and λ = 0.28 are taken
from [18] (Fit 2 with charm). This form of the dipole cross section provides successful description
of the first HERA data on both inclusive and diffractive structure functions [18,19]. A different
parametrisation of σ̂, without the saturation scale, is also given in [42–44]. We checked that a
very similar description of FDLqq̄ was found in a recent analysis [45] based on the Color Glass
Condensate parameterisation of the dipole scattering amplitude [46].
The relation between the dipole approach with three diffractive components and the DGLAP
approach with diffractive parton distributions was analysed at length in [30]. Summarising this
relation, the twist–2 part of the qq component gives a diffractive quark distribution. The twist-2
part of the qqg component forms a first step of the DGLAP evolution which starts from a given
gluon distribution. Both diffractive parton distributions do not depend on Q2, thus they may
serve as initial conditions for the DGLAP equations at the scale which is not determined. From
this perspective, the DGLAP approach offers a description of more complicated diffractive state
with any number of partons ordered in transverse momenta. However, the pQCD calculations
tell us that the twist–2 analysis of diffractive data should include the twist–4 contribution since
it cannot be neglected at large β. This is the strategy which we follow in our analysis.
We also borrow from the dipole approach a general form in β of the initial quark distribution
which vanishes at the endpoints β = 0, 1 (see eq. (31) in which Aq and Cq are positive). A very
important aspect of Regge factorisation (11) can also be motivated by the dipole approach. It
is a consequence of geometric scaling of the dipole cross section (22) [30,47].
2.4 Reggeon contribution
The diffractive data from the H1 collaboration for higher values of xIP hints towards a contribu-
tion which decreases with energy. This effect can be described by reggeon exchanges in addition
to the rising with energy pomeron exchange. Following [48, 49], we consider the dominant
isoscalar (f2, ω) reggeon exchanges which lead to the following contribution to F
fR(xIP , t)FR(β,Q
2) . (23)
This contribution breaks Regge factorisation of the diffractive structure function, however, its
presence is necessary for xIP > 0.01 [50]. The reggeon flux fR is given by the formula analogous
to eq. (13)
fR(xIP , t) =
F 2R(0)
e−|t|/Λ
R |ηR(t)|
1−2αR(t)
IP , (24)
where αR(t) = 0.5475 + 1 · t is the reggeon trajectory. From the Regge phenomenology of
hadronic reactions ΛR = 0.65GeV and the reggeon–proton couplings are given by [49]: F
(0) =
194GeV−2 and F 2ω(0) = 52GeV
−2. The functions
|ηR(t)|
2 = 4cos2[παR(t)/2] , |ηR(t)|
2 = 4 sin2[παR(t)/2] (25)
are signature factors for even (f2) and odd (ω) reggeons, respectively. We could also consider
isovector reggeons (a2, ρ) but their couplings to the proton are much smaller and we neglect
them. Finally, the reggeon structure function FR is given by [49]
FR(β) = AR β
−0.08 (1− β)2 , (26)
where the normalisation AR is a fitted parameter. Thus, in the first approximation, we neglect
the Q2-dependence of the reggeon contribution.
3 Fit details
Collab. No. points Data |t|-range Q2-range β-range
H1 [50] 72 LP [0.08, 0.5] [2 , 50] [0.02 , 0.7]
ZEUS [51] 80 LP [0.075 , 0.35] [2 , 100] [0.007 , 0.48]
H1 [31] 461 MY < 1.6 [|tmin| , 1] [3.5 , 1600] [0.01 , 0.9]
ZEUS [52] 198 MY < 2.3 [|tmin| ,∞] [2.2 , 80] [0.003 , 0.975]
Table 1: Kinematic regions of diffractive data from HERA. LP means leading proton data and
MY is invariant mass of a dissociated proton. Dimensionfull quantities are in units of 1 GeV.
In our analysis we use diffractive data from the H1 [31,50] and ZEUS [51,52] collaborations.
In Table 1 we show their kinematic limits in which they have been measured. The minimal value
value of |t| is given by
|tmin| ≃
1− xIP
m2p , (27)
where mp is the proton mass. The leading proton data from H1, measured in the range given
in Table 1, were corrected by the H1 collaboration to the range |tmin| < |t| < 1 GeV
The ZEUS data are given for the diffractive structure function FD2 , thus we use in our
analysis the following formulae
FD2 = F
D(tw2)
2 + F
2 + F
Lqq̄ (28)
FDL = F
D(tw2)
L + F
Lqq̄ . (29)
No Data Fit αIP (0) AR Aq Bq Cq Ag Bg Cg χ
1 H1 (LP) tw-2 1.098 0.29 1.75 1.49 0.5∗ 2.09 0.67 0.80 0.48
2 ZEUS (LP) tw-2 1.145 1.05 2.13 1.51 0.5∗ 10.0* 1.03 2.26 0.40
3 H1 tw-2 1.117 0.49 1.33 1.63 0.34 0.17 -0.16 -1.10 1.04
4 tw-(2+4) 1.119 0.48 1.62 1.98 0.59 0.04 -0.56 -1.68 1.17
5 ZEUS tw-2 1.093 0.0∗ 1.68 1.01 0.5∗ 0.49 -0.03 -0.40 1.35
6 tw-(2+4) 1.092 0.0∗ 1.20 0.85 0.57 0.07 -0.52 -1.48 1.82
Table 2: The fit parameters to H1 nd ZEUS data. The presence of twist–4 in the fits is marked
by tw-(2+4). The parameters with an asterisk are fixed in the fits.
The longitudinal twist-4 contribution is present on the r.h.s. of eq. (28) since FD2 is the sum of
the contributions from the transverse and longitudinal polarised virtual photon. The H1 data,
however, are presented for the reduced cross section (2). Thus we substitute relations (28) and
(29) in there and use
σDr =
D(tw2)
2 + F
1 + (1− y)2
D(tw2)
2(1− y)
1 + (1− y)2
FDLqq . (30)
The expression in the curly brackets is the twist–2 contribution while the last term is the twist–4
one. Notice that the difference between FD2 and σ
r is most important for y → 1.
We fit the diffractive parton distributions at the initial scale Q20 = 1.5 GeV
2, assuming the
Regge factorised form (11) with the following pomeron parton distributions [31]:
βΣIP (β) = Aq β
Bq (1− β)Cq (31)
βgIP (β) = Ag β
Bg (1− β)Cg . (32)
The six indicated parameters are fitted to data. We additionally multiplied both distributions
by a factor exp{−a/(1 − β)} with a = 0.01 to secure their vanishing for β = 1. This factor
is only important when Cq or Cg becomes negative in the fits. For the evolution, we use the
next-to-leading order DGLAP equations with ΛQCD = 407 MeV for Nf = 3 flavours [53].
The pomeron flux in eq. (11) is integrated over t in the limits given in Table 1 which leads
to the form
fIP (xIP ) =
F 2IP (0)
e−B|tmin| − e−B|tmax|
1−2αIP (0)
IP . (33)
The shrinkage parameter B equals
B = BD + 2α
IP ln(1/xIP ) (34)
with BD = 5.5GeV
−2 and α′IP = 0.06GeV
−2 [50].
In summary, we have eight fit parameters altogether: the pomeron intercept αIP (0), reggeon
normalisation AR in eq. (26) and six parameters in eqs. (31,32)
4 Fit results
The data sets from Table 1 were obtained in different kinematical regions, using different methods
of their analysis. Thus, we decided to perform fits to each data set separately. The values of
the fit parameter are shown in Table 2. The difference between them can be attributed to the
scale of uncertainty of our analysis. In each case we preformed two fits: with and without the
twist–4 formula added to the twist–2 contribution.
4.1 Leading proton data
We started from fits to the leading proton data. The fit parameters in this case are displayed
in the first two rows of Table 2. We only show the twist–2 fit results since they are not changed
in fits with the twist–4 term. This happenes because the leading proton data comes from the
region of β values where the twist–4 contribution is small (β ≤ 0.7 for H1 and β < 0.5 for
ZEUS), see Fig. 3.
The data with a dissociated proton (DP) which are measured in the region of large β influence
most the value of the parameter Cg which controls the behaviour of the gluon distribution at
β → 1. For the LP data Cg is positive and the gluon distribution is suppressed near β ≈ 1, while
for the DP data Cg is negative and the gluon distribution is strongly enhanced. This shows that
the data with β > 0.7 are crucial for the proper analysis. Without this kinematic region we lose
important information about diffractive interactions. Thus, from now on we concentrate on the
analysis of the DP data.
4.2 H1 data
The fit parameters to the H1 data with a dissociate proton are given in the third and fourth rows
of Table 2. We see that the fit quality is practically the same for both fits, with and without the
twist–4 contribution. The presence of the reggeon term improves fit quality by 30 units of χ2
for 461 experimental points. A good quality of the fits is illustrated in Fig. 4 which also shows
that the reduced cross sections (30) from the twist–2 (solid lines) and twist–(2+4) fits (dashed
lines) are very close to each other.
In Fig. 5 we show our results on the reduced cross section for the largest measured value of
β = 0.9. In this region, the twist–4 contribution, shown by the dotted lines, cannot be neglected.
We see that the curves from both the twist–2 (solid) and twist–(2+4) (dashed) fits describe data
reasonable well. However, the curves with twist–4 have a steeper dependence on xIP (energy)
than in the pure twist–2 analysis. This observation is by far more pronounced in the analysis of
the ZEUS data performed for the structure function FD2 .
The diffractive parton distributions from our fits are shown in Fig. 6 in terms of the pomeron
parton distributions, βΣIP (β,Q
2) and βgIP (β,Q
2). Being independent of the pomeron flux, such
a presentation allows for a direct comparison of the results from fits to different data sets. We see
that the singlet quark distributions are quite similar while the gluon distributions are different.
In the fit with twist–4, the gluon distribution is stronger peaked near β ≈ 1. This somewhat
surprising result can be understood by looking at the logarithmic slope of FD2 for fixed values
of β. From the LO DGLAP equations we have schematically:
∂ lnQ2
∂ lnQ2
= Pqq ⊗ ΣIP + PqG ⊗GIP − ΣIP
Pqq (35)
where the negative term describes virtual corrections. For large β, the measured slope is negative
which means that the virtual emission term must dominate over the real emission ones. The
addition of the twist–4 contribution to FD2 , proportional to 1/Q
2, contributes a negative value
to the slope which has to be compensated by a larger gluon distribution in order to describe the
same data.
In Fig. 7 we present our most important results. On the left panel, the FD2 structure function
is shown from both fits, with and without the twist–4 contribution (shown by the dotted lines).
We see no significant difference between these two results. However, the longitudinal structure
function FDL differs significantly for the two fits (right panel) due to the twist–4 contribution.
Let us emphasise that both sets of curves were found in the fits which well describe the existing
data on σDr , including the large β region. Thus, an independent measurement of F
L in this
region would be an important test of the QCD mechanism of diffraction.
4.3 ZEUS data
The results of same fits performed for the ZEUS data are shown in the last two rows of Table 2.
This time the Regge contribution (26) is not necessary since fits give the reggeon normalisation
AR ≈ 0. In general, the fit quality is worse than for the H1 data.
As shown in Fig. 8, the biggest difference between the twist–2 and twist–(2+4) results occurs
at large β values. This is analysed in detail in Fig. 9. We see that the presence of the twist–4
term in the fit (dashed lines) improves the agreement with the data in this region. In particular,
a steep dependence of FD2 on xIP is better reproduced by the twist–(2+4) fit then by the twist–2
one (solid lines). This dependence is to large extend driven by the twist–4 contribution (dotted
lines).
The behaviour of the diffractive parton distributions and structure functions, shown in
Figs. 10 and 11, respectively, is very similar to that found for the H1 data. The gluon dis-
tribution from the fit with twist–4 is stronger peaked near β ≈ 1 and the longitudinal structure
functions in the large β region is dominated by the twist–4 contribution.
We summarise the effect of the twist–4 contribution in Fig. 12 showing the predictions for
the longitudinal diffractive structure function FDL . Ignoring this contribution, we find the two
solid curves coming from the pure twist–2 analysis of the H1 (upper) and ZEUS (lower) data.
With twist–4, the dashed curves are found, the upper one from the H1 data and the lower one
from the ZEUS data. There is a significant difference between these two predictions in the
region of large β. We believe that the effect of the twist–4 contribution will be confirmed by the
forthcoming analysis of the HERA data.
5 Conclusions
We performed fits of the diffractive parton distributions to new diffractive data from the H1 and
ZEUS collaborations at HERA. In addition to the standard twist–2 formulae, we also considered
the twist–4 contribution which dominates in the region of large β. This contribution comes from
the diffractive production of the qq pair by the longitudinally polarised virtual photons. The
effect of the twist–4 contribution on the diffractive parton distributions and structure functions
was carefully examined. The twist–4 contribution leads to the gluon distribution which is peaked
stronger at β ≈ 1 than in the case without twist–4.
The main result of our analysis is a new prediction for the longitudinal diffractive struc-
ture function FDL . The twist–4 term significantly enhances F
L in the region of large β. A
measurement of this function at HERA in the region of large β should confirm the presented
expectations which are based on the perturbative QCD calculations. The obtained diffractive
parton distributions can also be used in the analysis of diffractive processes at the LHC, in
particular, to the estimation of the background to the diffractive Higgs production, see [54] for
a recent discussion.
Acknowledgements
This research has been partly supported by MEiN research grant 1 P03B 028 28 (2005-08)
and by EU grant HEPTOOLS, MRTN-CT-2006-035505. The hospitality of the Galileo Galilei
Institute for Theoretical Physics in Florence and partial support of INFN during the completion
of this work is gratefully acknowledged.
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Diffractive structure function
Q2=8 GeV2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 3: Three contributions to FD2 from: qq and qqg from transverse (T) and longitudinal (L)
photons [19] for xIP = 0.003. The twist–4 contribution Lqq̄ is indicated by the yellow band. Old
ZEUS data points are shown.
H1 DATA
β=0.01
β=0.01 β=0.04β=0.04 β=0.1β=0.1 β=0.2β=0.2 β=0.4β=0.4 β=0.65β=0.65 β=0.9 Q
β=0.9 Q
6.56.5
8.58.5
200200
Figure 4: Reduced cross section σ
r for H1 data as a function of xIP . Solid lines: twist–2 fit,
dashed lines: twist–(2+4) fit.
H1 DATA (β=0.9)
Q2= 3.5 GeV2
dashed: tw-(2+4) fit
solid:  tw-2 fit
Q2= 5 GeV2
dotted:  tw-4 contribution
Q2= 6.5 GeV2
Q2= 8.5 GeV2
Figure 5: Reduced cross section σ
r for H1 data at β = 0.9 for four values of Q
2 against fit
curves.
DPD (H1)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 6: Pomeron parton distributions: singlet βΣIP (β,Q
2) (left) and gluon βgIP (β,Q
2) (right)
from H1 data.
DSF (H1)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
dotted: tw-4 contribution
dotted: tw-4 contribution
dotted: tw-4 contribution
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 7: Diffractive structure functions F
2 (left) and F
L (right) from fits to H1 data for
xIP = 10
−3. The band shows the effect of twist–4 on the predictions for F
ZEUS DATA
β=0.652
β=0.652 β=0.231β=0.231 β=0.07β=0.07 β=0.022β=0.022 β=0.007β=0.007
β=0.003
β=0.003
β=0.735β=0.735 β=0.308β=0.308 β=0.1β=0.1 β=0.032β=0.032 β=0.010β=0.010
β=0.004
β=0.004
β=0.807β=0.807 β=0.4β=0.4 β=0.143β=0.143 β=0.047β=0.047 β=0.015β=0.015
β=0.007
β=0.007
β=0.848β=0.848 β=0.471β=0.471 β=0.182β=0.182 β=0.062β=0.062 β=0.020β=0.020
β=0.009
β=0.009
β=0.907β=0.907 β=0.61β=0.61 β=0.28β=0.28 β=0.104β=0.104 β=0.034β=0.034
β=0.015
β=0.015
β=0.949β=0.949 β=0.75β=0.75 β=0.43β=0.43 β=0.182β=0.182 β=0.063β=0.063
β=0.029
β=0.029
β=0.975β=0.975 β=0.86β=0.86 β=0.604β=0.604 β=0.313β=0.313 β=0.121β=0.121
Figure 8: Diffractive structure function F
2 as a function xIP for ZEUS data. Solid lines:
twist–2 fit, dashed lines: twist–(2+4) fit.
ZEUS DATA
dotted: tw-4 contribution
β = 0.86 Q2= 55 GeV2
solid:     tw-2 fit
dashed: tw-(2+4) fit
β = 0.91 Q2= 14 GeV2
β = 0.95 Q2= 27 GeV2
β = 0.975 Q2= 55 GeV2
Figure 9: Diffractive structure function F
2 for ZEUS data at large values of β against fit
curves.
DPD (ZEUS)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 10: Pomeron parton distributions βΣIP (β,Q
2) (left) and βgIP (β,Q
2) (right) from fits to
ZEUS data.
DSF (ZEUS)
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
solid:     tw-2 fit
dashed: tw-(2+4) fit
dotted: tw-4 contribution
dotted: tw-4 contribution
dotted: tw-4 contribution
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1
Figure 11: Diffractive structure functions F
2 (left) and F
L (right) from fits to ZEUS data
for xIP = 10
−3. The band shows the effect of twist–4 on the predictions for F
Diffractive FL
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
twist-(2+4)
twist-2
Q2=10 GeV2
xP=10
0.005
0.015
0.025
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 12: Predictions for F
L for xIP = 10
−3 and Q2 = 10 GeV2 from the twist–(2+4) fits to
the H1 (upper dashed line) and ZEUS (lower dashed line) data. The solid lines show predictions
from pure twist–2 fits to the H1 (upper) and ZEUS (lower) data.
	Introduction
	Basic formulae
	Twist–2 contribution
	Twist-2 charm contribution
	Twist–4 contribution
	Reggeon contribution
	Fit details
	Fit results
	Leading proton data
	H1 data
	ZEUS data
	Conclusions
