EVIDENCE FOR GLUON RECOMBINATION IN DEEP INELASTIC SCATTERING

Kjell Prytz
University of Gavle
801 76 Gavle
Sweden
e-mail: kjell.prytz@hig.se
PACS: 12.38.Bx

Abstract
The pomeron structure function is extracted from the latest H1 data and are subject to a QCD
analysis. The result shows evidence for gluon recombination.

Introduction
We discuss the interpretation of the diffractive structure function as measured by the H1
collaboration at DESY [1]. Using a model proposed in ref. 2 we extract the pomeron structure
function, a notion that occur in Regge phenomenlogy [3]. Together with the Collins
factorization theorem for diffractive scattering [4] a framework for applying QCD evolution
equations to the pomeron parton dynamics can be constructed.
The procedure involves many assumptions but since it was first proposed some 10-15 years
ago [2] evidence and support for its correctness have been presented (see ref . 5 and
references therein).
In 1997, H1 published a paper [6] where the DGLAP QCD dynamics [7] was applied to their
own first data. Their result showed that the DGLAP evolution equations are able to fit these
data satisfactorily. However, in 1998 new data with an extended kinematic range were
published [8] and it was found that the DGLAP equations could not provide a decent
description of the data [1].
In ref. 5 it was explained why DGLAP doesn't work for the latest H1 data and it was
qualitatively shown that the inclusion of gluon recombination (or generally, a low-x higher
twist term) in the analysis is sufficient to explain the data. These equations are denoted GLR-
MQ [9] and corrects the DGLAP equations by adding terms accounting for gluon
recombination.
In this paper we apply the GLR-MQ equations quantitatively to the data by performing a full
QCD analysis of the H1 data in leading order. The sole aim is to check whether the DGLAP
equations can be ruled out in favour of the GLR-MQ equations.

Pomeron Phenomenology
In our model, the electromagnetic diffractive process is assumed to occur as depicted in fig. 1.





Figure 1: The diffractive process with pomeron exchange in electromagnetic electron-proton high energy
scattering.


There are three vertices which are subject to the factorization theorems of Collins (upper) and
of Regge (lower). The electromagnetic cross section can approximately be written [2]

ds(ep epX ) 4 2 2
pa y
= 1
[ - y + ]F D (x , t, , 2
x Q ) (1)
2 2
dx dtdxdQ xQ 2 2 Pom
Pom


where F D D
2 is called the diffractive structure function and the corresponding F1 structure
function has been neglected.
In the laboratory frame, y = /E where is the energy transfered by the photon in the process
and E is the energy carried by the incoming electron. Q2 = -(k-k')2 where k and k' are the
momenta of the incoming and outgoing electron respectively and x = Q2/2M where M is the
proton mass. is the electromagnetic coupling, t is the squared four-momentum lost by the
proton and xPom is the fraction of the incoming proton momentum lost by the proton which in
the model is the fractional momentum carried by the pomeron.
The Regge factorization theorem then leads to [2]

D Pom x
F ( x (2)
Pom , t, x, 2
2 Q ) = f Pom/ p (xPom, t)F ( , 2
2 Q )
xPom

were fPom/p is usually called "the pomeron flux" and describes the coupling between the
pomeron and proton. In a "quark-parton model jargong" one would say that fPom/p measures a
(unnormalized) probability to find a pomeron "inside" the proton [2].
We introduce = x/xPom which is interpreted (in the infinite momentum frame) as the fraction
of the pomeron momentum carried by the struck parton.
H1 presents their structure function data in terms of x D
PomF2 . In order to obtain the pomeron
structure function (F Pom
2 ), which will be subject to the QCD analyses, we need to divide by
xPom fPom/p.
For the pomeron flux we use the information obtained by H1 [6]:

bt
e
f (3)
Pom/ p ( x Pom , t) = 2a -1
Pom
xPom

where b = 4.6 GeV2 and a (t '
) = a (0 ) + a t = 1.2 + 0.26 t is the Regge trajectory
Pom Pom Pom
for the pomeron. An integration over t is performed from 1.0 GeV2 to 0 and xPom = 0.005. At
this low value of xPom, H1 showed that there is negligible contribution from other exchange
particles beyond the pomeron [6].
We use the following evolution equations

2 1 1
2
dG( ,
b Q ) a (Q )
s b b
= [ dy 2 dy 2
b G( y, Q ) Pgg ( ) + yq( y, Q ) Pgq ( ) ]
p
2 2 2
d log Q 2 y y y y
b b (4)
1
2 2 2 2 b
81 s
a (Q ) dy 2 81 s
a (Q ) dy 2
- [G( y, Q)] + [G( y, Q)]
2 2 2 2
16R Q y 16R Q y
b b / 2


d q
b ( ,
b Q2 ) a (Q2 1 1
)
s b b
= [ dy 2 dy
b G( y, Q )P 2
qg ( ) + yq( y, Q )Pqq ( ) ]
2
log 2p 2
d Q y y y 2 y
b b (5)
2 2
27 s
a Q
( )
- [G( ,
b Q2 2
)] + HT
160 R 2Q 2
where on the right hand side, the terms linear in gluon distribution G and quark distribution
yq are given by DGLAP [7] and the terms quadratic in G are given by GLR-MQ [9]. The
latter terms account for gluon recombination into gluons (eq. 1) and into quarks (eq. 2).
The fourth term in the gluon evolution equation was introduced in order to conserve gluon
momentum [10]. It was achieved by simply noting that a fusion of a couple of gluon results in
a new one with higher momentum.
The amount of gluon recombination is controlled by the pomeron size parameter R. The term
denoted `HT' was introduced by Mueller and Qiu [9] but not fully given. It was neglected in
our calculation.
For the initial gluon and quark distributions, given at the lowest used Q2 = 1.2 GeV2, we use
the following forms:

q = C1 + C2 + C3 2 + C4 3 + C5 4 + C6 5
G = g = C7 C8 (1-)C9

The rather complicated form of the quark distribution is dictated by data.
The pomeron is assumed to be a flavour singlet particle so that the evolution is purely singlet.
We use four flavours and the QCD parameter = 200 MeV.
We adopt a so called saturation procedure [11] meaning that if, in the gluon evolution, the
sum of the recombination terms get larger than the DGLAP term we set the gluon derivative
to zero. This actually happens in the low-, low-Q2 region of the data (see fig. 3) so a more
thorough theoretical investigation about this effect is asked for.


Results
The result of the fit using the GLR-MQ equations is shown in fig. 2 with 2/dof = 58/62.





Figure 2: Optimized fit of the H1 data based on the GLR-MQ equations.

There are three free quantities in the fit: the quark and gluon distributions and the pomeron
size. They mainly control the level, the Q2 dependence and the flattening of the data at low-Q2
respectively. There is no size parameter in the DGLAP equations and accordingly they cannot
account for the "flattening effect" seen at low-Q2.
The result on the quark and gluon distribution is shown in fig. 3.





Figure 3: The resulting gluon and quark distribution from the fit based on the GLR-MQ equations. The lines
corresponds to Q2=1.2, 1.5, 2.5, 3.5, 4.5, 5, 7.5 and 9 GeV2.

As has been noted several times earlier the gluons dominate in the pomeron. As can be seen in
fig. 3, the gluon distribution exhibits an unusual evolution at low-. This is due to the


saturation procedure adopted (see above). We expect that terms of twist-6 and higher, still
missing in the theory, would make the distribution smoother.
Furthermore, we note that the momentum sum rule is not fulfilled and momentum is not even
conserved. The latter is mainly due to the saturation effect and the former has been discussed
several times previously in the literature. In our view this is a theoretical issue which needs
more considerations.
As for the pomeron size we obtain R = 0.03 GeV-1 which is considerably smaller than
previously obtained [12]. But since we don't aim for any error and correlation analysis of the
fitting parameters in this paper we wish to refrain from drawing any conclusions from this. It
should at this moment only be considered as a fit parameter. Clearly, this parameter is highly
correlated with the gluon density and probably we need an extended Q2 lever arm in order to
disentangle the correlation satisfactorily.
We also perform a fit based on the DGLAP scheme shown in fig. 4 and resulting in 2/dof =
72/62. In view of the rather large errors of the data, we consider this as a poor fit.





Figure 4: Optimized fit of the H1 data based on the DGLAP equations.

The resulting parton distributions are shown in fig. 5.


Figure 5: The resulting gluon and quark distribution from the fit based on the DGLAP equations. The lines
corresponds to Q2=1.2, 1.5, 2.5, 3.5, 4.5, 5, 7.5 and 9 GeV2.

Since we look for gluon recombination in this analysis the region of low is most interesting.
We therefore in addition investigate the fit quality for the first four -bins only, where also the
data quality is best. For the GLR-MQ dynamics (figure 2) we then obtain 2/dof = 20/34
whereas for the DGLAP dynamics (figure 4) we get 2/dof = 33/34. This indicate a favour for
the GLR-MQ equations. Note that the errors on the data include systematics so that 2/dof < 1
is expected for a perfect fit.

Discussion
We have seen that a large contribution of gluon recombination is needed in order to describe
the data. Being used to discuss this effect in terms of `low-x effects' it might be surprising to
see that the whole range of in the gluon evolution (compare fig. 3 and 5) is affected.
However, the reason for associating gluon recombination with `low-x physics' is because it
was first discussed in connection with nucleon structure. What really governs this dynamic is
the gluon spatial density which is high at low-x in the nucleon. In the pomeron, however, due
to its small size and gluon domination its gluon density is high throughout the kinematical
region and gluon recombination is a natural part of its whole QCD evolution.
One can of course question the whole approach of applying the GLR-MQ equations under
such circumstances since we would expect even higher twist terms to become influential.
These terms are not yet available but at the moment they serve as an explanation for the fact
that the gluon distribution obtained in fig. 3 is rather odd, due to the kink at =0.15, and also
that the pomeron size obtained is much smaller than expected. Nevertheless, although we
could expect a more acceptable result when including terms of this kind, we certainly claim
that the main interpretation of the data is given in this analysis; the pomeron is gluon
dominated, exhibiting gluon recombination at such a level that saturation occurs in the
kinematical region of the data and that in this model the pomeron is much smaller than the
nucleon.
After all, we cannot deny the data where the huge effect of gluon recombination is visible
already `by eye'. Also, the effect of saturation is indicated through the flattening of data at
low-Q2.
We can also discuss and question the numerical result on the pomeron size which is, as
mentioned above, strongly correlated with the gluon distribution. Due to this correlation we
cannot increase the pomeron size since that just results in an increasing gluon density in order
to keep the amount of gluon recombination approximately unchanged. As a consequence, the
DGLAP terms will then increase contributions to the positive scaling violations and ruin the
fit. Alternatively, one could try to account for the flattening effect with the DGLAP dynamics
only and expect a very low gluon density at low-. This was done by H1 [6] with their old


data but it doesn't work with the present data set since the optimum gluon distribution is
shown in figure 5. Indeed, the density is low but not low enough to get a good fit.
We should mention though, that we haven't made any attempt to allow for negative values of
the gluon distribution. Such an approach could very well change the conclusions since it may
account for the flattening effect at low-Q2. However, our opinion is that this is unphysical
since our basis has been the physical quark-parton model. Since this model can account for
the data when including gluon recombination we see no reason for such an exotic adventure.

Conclusions and outlooks
We have for the first time found a model which fit the complete H1 diffractive data set for
xPom = 0.005. We have shown that the inclusion of gluon recombination solves the problem if
we at the same time introduce an effect of gluon saturation.
The parton densities found accordingly confirm the gluonic domination in the pomeron but
the shape of the gluon distribution differs from that found by H1 [6] (fit 3).
Our investigation also points to the needs of measurements extending the kinematical range of
the data. We could then learn more about gluon recombination and in particular get safer
information on the pomeron size.
The recent preliminary data from H1 [13] might add further information concerning gluon
recombination mainly due to improved precision. The general trend, however, is identical
with the data used here, i.e. for F2 vs Q2 we see a flattening effect at low Q2 for the lowest
bins at xPom < 0.005. Data at higher xPom is not considered by us due to contribution of other
exchange particles than the pomeron.
In the H1 paper [13] they claim that the DGLAP equations can describe the data. I disagree
with this conclusion. No 2 is given but one can see bye eye that the flattening effect is not
taken care of (see the bins =0.1 and =0.2 in figure 9 of this paper). No attempt to fit with
GLR-MQ is made. In this analysis minimum Q2 = 6 GeV2 whereas the most pronounced
effects of gluon recombination occur at lower Q2 explaining why they obtain an apparent
reasonable DGLAP fit.
As soon as these data are presented in tables one should use them to perform the same fits as
in this paper. Since these data seem to agree with those that have been used in this paper,
differing just in precision, and since no kinematical extension is presented, we cannot expect
any other kind of conclusion than what has been drawn here.

Acknowledgements
Many thanks to Gunnar Ingelman for helpful suggestions. The work has been supported
economically by the University of Gavle, Sweden.

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