It is demonstrated that the global properties of the rapidity gap events at
HERA can be understood based on electron-gluon scattering and a
non-perturbative mechanism of colour neutralization. Using the measured
inclusive structure function $F_2$ to determine the parameters of the parton
model, the diffractive structure function $F_2^D$ is predicted. The ratio of
diffractive and inclusive cross sections, $R_D = \sigma_D/\sigma_{incl}\simeq
1/9$, is determined by the probability of the produced quark-antiquark pair to
evolve into a colour singlet state.Comment: talk at Workshop on DIS and QCD, Paris, April 1995, 3 pages LaTeX,
  uses qcdparis.sty, 2 figures (uuencoded

Buchmuller, W.

Hebecker, A.

English

arXiv

It is demonstrated that the global properties of the rapidity gap events at HERA can be understood based on electron-gluon scattering and a non-perturbative mechanism of colour neutralization. Using the measured inclusive structure function F_2 to determine the parameters of the parton model, the diffractive structure function F_2^D is predicted. The ratio of diffractive and inclusive cross sections, R_D = \sigma_D/\sigma_{incl}\simeq 1/9, is determined by the probability of the produced quark-antiquark pair to evolve into a colour singlet state

Buchmüller, Wilfried

Hebecker, A

CERN Document Server

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Calculating the diractive from the inclusive structure function
W. Buchmuller and A. Hebecker
Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany
Abstract
It is demonstrated that the global properties of the rapidity gap events at HERA can be understood
based on electron-gluon scattering and a non-perturbative mechanism of colour neutralization.
Using the measured inclusive structure function F
2
to determine the parameters of the parton
model, the diractive structure function F
D
2
is predicted. The ratio of diractive and inclusive
cross sections, R
D
= 
D
=
incl
' 1=9, is determined by the probability of the produced quark-
antiquark pair to evolve into a colour singlet state.
Resume
Les evenements a gap de rapidite observes a HERA peuvent e^tre attribues a la diusion d'un
gluon et d'un electron combine avec un mechanisme non-perturbative pour la neutralisation de la
couleur. Nous utilisons la fonction de structure inclusive F
2
pour determiner les parametres du
modele de partons et pour prevoir la fonction de structure diractive F
D
2
. Le quotient des sections
ecaces diractives et inclusives, R
D
= 
D
=
incl
' 1=9 correspond a la probabilite que le pair
quark-antiquark produit dans ce proces evolue dans un etat neutre par rapport a la couleur.
Recently, detailed analyses of rapidity gap events in deep
inelastic scattering at HERA have appeared conrming
the previously observed leading twist behaviour of
the diractive cross section [1]-[4]. The absence of
a hadronic energy ow between proton remnant and
current fragment suggests that in the scattering process
a colour neutral part of the proton is stripped o. A
leading twist behaviour is usually regarded as evidence
for scattering on point-like constituents. This, however,
appears to be in conict with the fact that quarks and
gluons carry colour.
In the following we shall demonstrate that this
puzzle can be resolved considering the production of
a quark-antiquark pair in electron-gluon scattering as
basic partonic process and taking non-perturbative
fragmentation eects into account [5]. Our approach
is related to previous work on \aligned jet models" [6]

Talk given in the Session on Diraction at the Workshop on
Deep Inelastic Scattering and QCD, Paris, April 1995
and \wee parton lumps" in deep inelastic scattering
[7, 8]. Photon-gluon fusion has been independently
considered as basic process underlying rapidity gap
events by Kowalski [9] and Edin et al. [10].
The relevant kinematic variables are dened in
gure 1. With ~p
g
= 
~
P ,  p
2
g
= m
2
g
 Q
2
and the
invariant mass square M
2
= (q + p
g
)
2
one has,
 
Q
2
Q
2
+M
2
'
x

: (1)
The dierential cross section for the inclusive
production of quark-antiquark pairs is given by [11]
d(ep! e(qq)X)
dxdQ
2
d
= (2)
=

xQ
2
g()

1  y +
y
2
2

(
T
+ 
L
) 
y
2
2

L

:
Here g() is the gluon density, and the cross section

T (L)
is obtained by integrating the dierential parton
k
k´
q
l´
lpg
P
Figure 1. Quark-antiquark pair production in electron-gluon
scattering.
cross section over the kinematically allowed range for
the momentum transfer t = (q   l
0
)
2
. The dierential
partonic cross sections for the absorption of a virtual
photon 
T
and 
L
, calculated in the massive gluon
scheme, can be found in ref. [11].
At small values of  the quark and antiquark
densities are much smaller than the gluon density.
Therefore we shall neglect the quark contribution to F
2
in the following. Note, that in our model this essentially
amounts to calculating the sea quark densities at scale
Q in terms of a gluon density g() at a scale m
g
= O(1
GeV).
From eq. (2) together with the virtual photon
cross sections one then obtains the inclusive structure
function F
2
(x;Q
2
),
F
2
(x;Q
2
) = x

s
2
X
q
e
2
q
Z
1
x
d

g() (3)


(
2
+ (1  )
2
) ln
Q
2
m
2
g

2
  2 + 6(1   )

:
The virtuality m
2
g
of the gluon regularizes the collinear
divergence at t = 0.
For the gluon density at small values of  we use the
usual parameterization,
g() = A
g

 1 
; (4)
where A
g
is a constant. Inserting eq. (4) into eq. (3) we
can now evaluate the inclusive structure function F
2
.
In the small{x region the obtained expression can be
further simplied for small values of the exponent .
This yields
F
2
(x;Q
2
) '

s
3
X
q
e
2
q
xg(x)

2
3
+ ln
Q
2
m
2
g

: (5)
For the parameters  = 0:23, m
g
= 1:0 GeV and
A
g

s
P
q
e
2
q
= 0:61 the above expression provides a
good description of the H1 measurement of the structure
function (compare the phenomenological t of F
2
in
[12]).
We are now ready to calculate the diractive
structure function. The main idea is that the
quark-antiquark pair, originally produced in a colour
octet state, changes its colour through further soft
interactions with the colour eld of the proton remnant.
Hence, the quark-antiquark pair evolves into a parton
cluster which separates from the proton remnant with
some probability P
8
in a colour octet state, and with
probability P
1
= 1 P
8
in a colour singlet state. In
the rst case a colour ow between proton remnant
and current fragment leads to the typical hadronic nal
state. In the latter case, however, the colour singlet nal
state fragments independently of the proton remnant,
yielding a gap in rapidity. For a suciently fast
rotation of the colour spin of quark and antiquark, the
probabilities should simply be given by the statistical
weight factor accounting for the possible states of the
quark-antiquark pair, i.e., P
1
' 1=9 ; P
8
' 8=9.
Similar ideas concerning the rotation of quarks in
colour space have been discussed by Nachtmann and
Reiter [13] in connection with QCD-vacuum eects on
hadron-hadron scattering. Another approach describing
the colour rotation by a soft gluonic eld is based on
the eikonal approximation to the quark propagator [14].
Here a non-abelian phase is introduced through a path
ordered integral along the fermion line.
The diractive structure functions are dened as
d
D
dxdQ
2
d
=
4
2
xQ
4

1  y +
y
2
2

F
D
2
 
y
2
2
F
D
L

: (6)
F
D
2
(x;Q
2
; ) is easily obtained from eq. (3). With
x = , and including the statistical weight factor P
1
,
one obtains
F
D
2
(x;Q
2
; ) '
1
9

s
2
X
q
e
2
q
g()

F
D
2
(;Q
2
) ; (7)
where

F
D
2
(;Q
2
)=

(
2
+(1 )
2
) ln
Q
2
m
2
g

2
 2+ 6(1 )

:
Since the gluon density g() and the mass scale m
g
have been determined by the t to the inclusive
structure function F
2
, the diractive structure function
is unambiguously predicted, including its normalization.
As an immediate consequence the ratio of diractive
and inclusive cross sections follows from eqs. (3) and (7),
R
D
=
R
1
x
dF
D
2
(x;Q
2
; )
F
2
(x;Q
2
)
'
1
9
: (8)
This ratio directly measures the probability of forming
a colour singlet parton cluster in the scattering process.
2
Figure 2. Dependence of the diractive structure function on 
and Q
2
.
The form of the diractive structure function (7)
is identical with expressions obtained based on the
idea of a \pomeron structure function" [15, 16]. The
interpretation of the ingredients, however, is rather
dierent. The \pomeron ux factor" is replaced by the
density of gluons inside the proton, which factorizes.
The \pomeron structure function" for partons with
momentum fraction  inside the \pomeron" is identied
as the dierential distribution for the production of a
quark-antiquark pair with invariant mass M
2
.
As it can be seen from gure 2, the function

F
D
2
(;Q
2
) is rather at for intermediate values of 
between 0:2 and 0:6. Approximating

F
D
2
(;Q
2
) in this
interval by

F
D
2
(0:4; Q
2
), a comparison of eqs. (3) and
(7) yields the scaling relation [8]
F
D
2
(x;Q
2
; ) '
0:04

F
2
(;Q
2
) : (9)
This scaling relation provides a rather accurate
description [5] of recent measurements of the diractive
structure function by the H1 collaboration [2].
Let us nally verify the appearance of rapidity gaps
within the above model. The rapidity of the antiquark
with momentum l in the 

p-rest frame is related to
other kinematical variables by
 =
1
2
ln
"
(1   )
u +m
2
g

t +m
2
g

#
: (10)
Using this relation the dierential cross section as
a function of the rapidity of the antiquark can be
calculated from eqs. (2) and (10). The total diractive
cross section for a maximum rapidity 
max
can now
be obtained by integrating over the kinematic domain
where the rapidity of the antiquark is larger than the
rapidity of the quark. The reverse congurations yield
the same contribution. Using the kinematic boundaries
of ref. [1] the 
max
-distribution has been calculated
in the described manner [5]. Above 
max
 2 the
diractive cross section is found to be negligible.
In summary, we have demonstrated that electron-
gluon scattering can account for the global properties
of the rapidity gap events observed at HERA provided
the following two hypotheses are correct: First, the
initially produced quark-antiquark pair evolves with a
probabilityP
1
' 1=9 into a colour singlet parton cluster;
second, the rapidity range of the diractive hadronic
nal state is essentially given by the rapidity interval
spanned by the produced quark-antiquark pair.
This simple picture appears to provide a rather
accurate description of the observed diractive events,
including the total rate, the -dependence and the Q
2
-
dependence. However, a number of theoretical issues
still remain to be settled. Our results indicate, that
a semiclassical approach to the small-x region might
be appropriate, where \wee partons" are treated as a
classical colour eld.
References
[1] H1 collaboration, T. Ahmed at al., Phys. Lett. B348 (1995)
681
[2] J. Dainton, these proceedings
[3] B. Foster, these proceedings
[4] ZEUS collaboration, M. Derrick et al., DESY 95-093 (1995)
[5] W. Buchmuller and A. Hebecker, DESY 95-077, to appear
in Phys. Lett. B
[6] M. Strikman, these proceedings
[7] W. Buchmuller, Phys. Lett. B335 (1994) 479
[8] W. Buchmuller, Phys. Lett. B353 (1995) 335
[9] H. Kowalski, talk at the Int. Workshop on Deep Inelastic
Scattering and Related Subjects (Eilat, 1994), unpublished
[10] A. Edin, G. Ingelman and J. Rathsman, these proceedings
[11] R.D. Field, Applications of Perturbative QCD, Addison
Wesley, New York, 1989
[12] H1 collaboration, T. Ahmed et al., Nucl. Phys. B439 (1995)
471
[13] O. Nachtmann and A. Reiter, Z. Phys. C24 (1984) 283
[14] O. Nachtmann, Ann. Phys. 209 (1991) 436
[15] G. Ingelman and P. Schlein, Phys. Lett. B152 (1985) 256
[16] K. Golec-Biernat; A. Kaidalov; H.-G. Kohrs;
P.V. Landsho; these proceedings
3


Calculating the diffractive from the inclusive structure function

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