

 29 Jun 1995

HARD DIFFRACTION AT HERA AND THE

GLUONIC CONTENT OF THE POMERON

A. Capella, A. Kaidalov*, C. Merino, D. Pertermann** and J. Tran Thanh Van

Laboratoire de Physique Th'eorique et Hautes Energies*** B^atiment 211, Universit'e de Paris-Sud, 91405 Orsay cedex, France

Abstract

We show that the previously introduced CKMT model, based on conventional Regge theory, gives a good description of the HERA data on the structure function F D2 for large rapidity gap (diffractive) events. These data allow, not only to determine the valence and sea quark content of the Pomeron, but also, through their Q2 dependence, give information on its gluonic content. Using DGLAP evolution, we find that the gluon distribution in the Pomeron is very hard and the gluons carry more momentum than the quarks. This indicates that the Pomeron, unlike ordinary hadrons, is a mostly gluonic object. With our definition of the Pomeron flux factor the total momentum carried by quarks and gluons turns out to be 0.3 \Xi 0.4 - strongly violating the momentum sum rule.

LPTHE Orsay 95-33 May 1995

* Permanent address : ITEP, B. Cheremushkinskaya ulitsa 25, 117259 Moscow, Russia ** Permanent address : University of Siegen, 5900 Siegen, Germany *** Laboratoire associ'e au Centre National de la Recherche Scientifique. URA D0063

2 1. Introduction

Diffraction dissociation of virtual photons, observed by the H1 and ZEUS collaborations at HERA [1-4] provides information on the nature of the Pomeron and on its partonic structure. In a previous paper [5] a model (CKMT model) based on Regge theory has been proposed for the description of diffractive dissociation of both real and virtual photons. In this model the structure function of the Pomeron FP (fi; Q2) is related to that of the deuteron [6] via Regge factorization. Recently, experimental information on the Pomeron structure function at different values of Q2 has been obtained [2, 4]. These data are in reasonable agreement with the predictions of the model.

In this paper, we extend our analysis of the Pomeron structure function and determine the gluonic content of the Pomeron. We use our model as initial condition in DGLAP evolution. We show that the Q2-dependence of the diffractive structure function, F D2 , observed experimentally, can only be understood if the gluon distribution in the Pomeron is very hard.It is also shown that the gluons carry more momentum than quarks. This observation confirms theoretical expectations that the Pomeron is mainly connected with gluonic degrees of freedom.

The plan of the paper is as follows. In Section 2 we recall the ingredients of the CKMT model. In Section 3 we perform the QCD evolution and in Section 4 we present our numerical results. In Section 5 we discuss other recent QCD analysis and give our conclusions.

2. - The CKMT model

The contribution of diffractive processes to the deep inelastic scattering (DIS) crosssection, corresponding to the Pomeron exchange diagram of Figs. 1 and 2, can be written in the form [4]

d4oeD dx dQ2 dxP dt =

4ssff2em

x Q4 ae1 \Gamma y +

y2 2 [1 + RD (x; Q2; xP ; t)] oe F

D2 (x; Q2; xP ; t) (1)

where x and y are standard DIS variables, t is the invariant momentum transfer t =

3 \Gamma (p \Gamma p0)2 and xP is the fraction of the proton momentum carried by the Pomeron :

xP = pP \Delta pflp

p \Delta pfl

' M

2 + Q2

W 2 + Q2 j

x fi : (2)

We have argued in Ref. [5] that the Pomeron in the diagrams of Figs. 1 and 2 can be considered as a Regge pole with a trajectory ffP (t) = ffP (0) + ff0t determined from an analysis of soft processes, in which absorptive correction (Regge cuts) are taken into account. We have [5], [7]

ffP (0) = 1:13 ; ff

0

P (0) = 0:25 GeV

\Gamma 2 : (3)

In this case the diffractive contribution to DIS can be written in a factorized form

F D2 (x; Q2; xP ; t) = \Gamma g

Ppp(t)\Delta 2

16ss x

1\Gamma 2ffP (t) P FP (fi; Q

2; t) (4)

where gPpp(t) is the Pomeron-proton coupling gPpp(t) = gPpp(0) exp(Ct) with (gPpp(0))2 = 23 mb and C = 2.2 GeV\Gamma 2. Eq. (4) is the definition* of the Pomeron structure function FP (fi; Q2; t) with the variable fi = x=xP playing the same role as the Bjorken variable x in F p2 . As emphasized in Ref. [5], the definition of FP depends on the particular choice of the Pomeron flux factor. For instance in Ref. [8] the flux factor differs from ours by a factor 2=ss. (See also Ref. [9]).

At large Q2 we have

FP (fi; Q2; t) = X

i

e2i fi \Theta qPi (fi; Q2; t) + _qPi (fi; Q2; t)\Lambda : (5)

Due to the arbitrariness in the normalization of FP discussed above, the partonic distributions qPi and _qPi do not satisfy, in general, the energy-momentum sum rule.

Many models of the Pomeron structure function are available in the literature [8,10- 14]. In the CKMT model [5] FP is determined using Regge factorization together with the

* Note that the definition of FP in Eq. (4) differs from the one in Ref. [5] by a factor (1 \Gamma fi). The definition in Eq. (4), which coincides with the one used in experimental papers [2, 4], is more appropriate for a partonic interpretation. We thank X. Artru for an enlightening correspondence on this point.

4 values of the triple Regge couplings determined from soft diffraction data. More precisely, the proton and Pomeron structure functions are given in terms of the diagrams of Figs. 3 and 4, respectively. (Note that Fig. 4 is the upper part of the diagram obtained by squaring, in the sense of unitarity, the flP amplitude in Fig. 1). We see that, due to factorization, F P2 can be obtained from F p2 (or more precisely from the combination F d2 = 12 (F p2 + F n2 )), by replacing the Reggeon-proton couplings gPpp and gfpp appearing in the lower part of Fig. 3, by the corresponding triple Reggeon couplings rP P P and rP P f which appear in the lower part of Fig. 4. The latter can be determined from soft diffraction data (see below). Another difference between the two structure functions is the fi ! 1 behaviour. Indeed, as far as the dimensional counting rules are concerned, the Pomeron is like a pion and, therefore, there is one spectator less than in the proton case. Thus the power of 1 \Gamma fi is smaller by two units in the Pomeron case. (The same result is obtained at Q2 = 0 in terms of Regge intercepts). The CKMT model can be summarized with the following formulae [5, 6], valid in the region 1 ^ Q2 ^ 5 GeV2 :

FP (fi; Q2) = F d2 \Gamma fi; Q2; A ! eA; B ! f B; n ! n \Gamma 2\Delta ; (6)

F d2 (x; Q2) = A(Q2) x

\Gamma \Delta (Q2) (1 \Gamma x)n(Q2)+4 + B x1\Gamma ffR(0) (1 \Gamma x)n(Q2) (7)

where e = rPP P (0)=gPpp(0) and f = rfP P (0)=gfpp(0). The values of all other parameters are given in Refs. [5] [6]. A main feature of the CKMT model is the Q2 dependence of the effective Pomeron intercept, \Delta ,= ffeffP \Gamma 1, which appears in the upper part of the diagrams in Figs. 3 and 4. It was argued in Ref. [6] that this is due to the fact that the size of the absorptive corrections decreases when Q2 increases. As a consequence, when using a parametrization with a simple power x\Gamma \Delta , \Delta must depend on Q2.

The parameters e and f can be determined from soft diffraction data. Here also absorptive corrections, which are very important in diffractive processes [15], have to be taken into account when extracting the values of e and f from the data. In an analysis of soft diffraction without absorption [16] one obtains e ,= f ,= 0.025. In the first paper of Ref. [15] it was shown that these values have to be multiplied by a factor of about three in order to take into account absorptive effects. The values of e and f have, of course, some

5 uncertainties-the largest being in the value of f [5]. Experimentally, the t-dependence of the triple Reggeon couplings is very small. We have taken it to be the same for rPP P and rP P f and included it in the function C. In this way the t-dependence of FP , which is expected to be very small, is factored out.

The comparison of the prediction [5] of the CKMT model with recent data from the H1 and ZEUS collaborations is quite satisfactory. The data confirm the factorized dependence of the hard diffractive cross-section on xP (Eq. (1)) with the parameter of the Pomeron trajectory in Eq. (2). Agreement in absolute values is also good for most values of fi and Q2, favoring our hypothesis that soft and hard diffraction are governed by the same triple Regge couplings.

3. QCD evolution

A closer look at the model predictions shows however a systematic decrease with Q2 at fi * 0:2. Such a trend is not present in the data. The reason for such a decrease is the following. As explained in Ref. [5] the model predictions at large Q2 have been obtained from DGLAP evolution [17], using Eqs. (6) and (7) at Q20 as initial condition. The result depends, of course, on the input gluon distribution function. In Ref. [5] it was assumed that the relation between gluon and sea quark distributions in the Pomeron was the same as in the proton. This corresponds to a rather soft gluon distribution in (1 \Gamma fi)3 at Q2 ss 5 GeV2. It was already pointed out in Ref. [5] that this assumption was not justified and should be changed. Indeed, such a behaviour is valid for ordinary mesons with a large valence quark content. However, the Pomeron can have a much harder gluon distribution ("valence gluons"). This is also in agreement with UA8 data [18].

In the following we will perform DGLAP Q2-evolution using the valence and sea quark initial distribution given by Eqs. (6) and (7) at Q20 = 5 GeV2, with a harder gluon distribution. More precisely, we use the same fi ! 0 behaviour as for sea quark [5], but we leave the power ng of 1 \Gamma fi as a free parameter, i.e. we put

fi ggP (fi) = e Ag fi

\Gamma \Delta (Q20)(1 \Gamma fi)ng(Q20) (8)

where Ag = 1:71 is the gluon normalization constant in a nucleon [6] and e and \Delta are

6 defined in Eq. (7). Note that the gluon normalization is obtained from Regge factorization and the energy-momentum sum rule is not used. For ng small enough, the Q2 behaviour of FP will be very different from the one of F p2 . The latter exhibits a decrease with Q2 for x * 0:2 due to a softening of the quark distribution resulting from gluon emission. Even though the same effect is present for the Pomeron, a harder distribution of gluons will lead in this case, through gluon decay, to an increase with Q2 in the number of quarks up to comparatively large values of fi.

Numerical calculations have been performed using the QCD evolution program of Refs. [19, 20, 14]. We present the results in a one loop approximaton but we have checked that practically the same results are obtained in two loops. The input and evolved partonic distributions are given in Figs. 5-8.

4. Numerical results

We have found [21] that a good description of the Q2 dependence of the HERA data is achieved with either ng = 0 or \Gamma 0.5 at Q20 = 5 GeV2. We have also allowed the parameters e and f in Eqs. (6) and (8) to vary within limits consistent with the results from the analysis of soft diffraction. A good description of the data is obtained with e = f = 0:07. All other parameters in Eq. (7) have the values given in Ref. [6].

The 3-dimensional structure function F D2 (x; Q2; xP ), integrated over t, is plotted versus xP in Fig. 9 and compared with H1 data. The results for the Q2 and fi dependence of F D2 (integrated over t and xP ) are given in Fig. 10 and compared with H1 data. In both cases, unpublished data from the ZEUS collaboration are also available. These data are in agreement, within errors, with the ones of H1. We see that the Q2-dependence at large fi is flat, in agreement with experiment - whereas in Ref. [5] there was a substantial decrease with increasing Q2.

In Fig. 11 we give the Pomeron structure function defined in Eq. 4 at different values of Q2. We see that the cross-over, which is at fi , 0:2 when the gluon distribution in the Pomeron is taken to be proportional to that of the proton [5], is shifted to larger values of fi. Comparison of Figs. 11a (ng = 0) and 11b (ng = \Gamma 0:5) shows that this shift is larger

7 when the gluon distribution is harder.

In all figures, except Fig. 11, we have presented the results only for ng = \Gamma 0:5. However, for ng = 0 there is very little change in our results - at least in the range of variables where data are available.

Finally, we give in Fig. 12 the fractions of the Pomeron momentum carried by quarks and gluons. With ng = \Gamma 0.5 (ng = 0). For Q20 = 5 GeV2, we have

Ng = Z

1

0 fi g

P (fi)dfi = 0:27 (0:15) (9)

Nq = Z

1

0 fi Xi \Gamma q

P i (fi) + _q

P i (fi) dfi = 0:13 (0:13) : (10)

We see that for both values of ng the momentum fraction carried by gluons is larger than the one carried by quarks, indicating that the Pomeron has a substantial gluonic component with a very hard gluon distribution. The energy-momentum sum rule is strongly violated.

5. Comparison with other QCD analysis and conclusions

Very recently, QCD-based analysis of the Pomeron structure function and its Q2 evolution have been carried out [13, 22, 23, 24]. In these papers the initial condition in DGLAP evolution equation and, in particular, the gluon momentum distribution is different from ours. However, in all cases conclusions similar to ours are reached, namely, the Q2 dependence of the data can be reproduced only if the gluon momentum distribution in the Pomeron is hard - the gluons carrying more momentum than the quarks. Note that in Refs. [22, 23] the gluon normalization relative to that of the quarks is determined from the momentum sum rule. This leads to a large normalization constant for gluons. It is known from UA8 results [18] that the normalization of partons satisfying the momentum sum rule leads to absolute predictions which exceed experiment by a factor close to five (see also Ref. [9]).

In conclusion, recent HERA data confirm the predictions of the CKMT model of hard diffraction based on Regge theory, factorization, and the assumption that not only the Pomeron intercept but also the triple Reggeon couplings are the same in soft and hard

8 diffraction. Although a QCD analysis of the Q2-dependence of the present data does not allow to extract the shape of the gluon distribution in the Pomeron, it requires a hard gluon distribution - the gluons carrying substantially more momentum than the quarks. This conclusion can be reached without reference to the momentum sum rule, using a normalization constant of the gluon distribution determined from Regge factorization.

Acknowledgements

It is a pleasure to thank P. Aurenche and M. Fontannaz for providing an updated versions of the code for DGLAP evolution used in Refs. [19, 20], and R. Engel for providing the version of the same code used in Ref. [14], adapted to our initial condition.

The present work has been realized with the help of an INTAS contract 93-0079. One of the authors (C. M.) has benefitted of a EEC postdoctoral project ERBCHBICT 930547.

9 References

[1] M. Derrick et al (Zeus Collaboration), Phys. Lett. B315, 481 (1993) ; B332, 228

(1994) ; B338, 477 (1994). [2] B. Foster (Zeus Collaboration), Workshop on DIS and QCD, Paris, April 1995. [3] T. Ahmed et al (H1 Collaboration), Nucl. Phys. B429, 477 (1994). [4] T. Ahmed et al (H1 Collaboration), Phys. Lett. B348, 681 (1995) . [5] A. Capella, A. Kaidalov, C. Merino and J. Tran Thanh Van, Phys. Lett. B343, 403

(1995). [6] A. Capella, A. Kaidalov, C. Merino and J. Tran Thanh Van, Phys. Lett. B337, 358

(1994). [7] A. Capella, J. Kaplan and J. Tran Thanh Van, Nucl. Phys. B97, 493 (1975).

K. A. Ter-Martirosyan, Sov. J. Nucl. Phys. 44, 817 (1986). A. B. Kaidalov, K. A. Ter-Martirosyan and Yu. M. Shabelski, Sov. J. Nucl. Phys. 44, 822 (1986). [8] A. Donnachie and P. V. Landshoff, Nucl. Phys. B303, 634 (1988). [9] K. Goulianos, preprint RU 95/E-06 (HEP-PH 950 2356). [10] G. Ingelman and P. E. Schlein, Phys. Lett. B152, 256 (1985). [11] G. Ingelman and K. Prytz, Z. Phys. C58, 289 (1993). [12] E. L. Berger et al, Nucl. Phys. B286, 704 (1987). [13] N. N. Nikolaev and B. G. Zakharov, Z. Phys. C53, 331 (1992).

M. Genovese, N. N. Nikolaev and B. G. Zakharov, preprint KFA-IKP (Th) 1994-37. [14] R. Engel, J. Ranft and S. Roesler, to appear in Phys. Rev. D. [15] A. Capella, J. Kaplan and J. Tran Thanh Van, Nucl. Phys. B105, 333 (1976).

A. B. Kaidalov, L.A. Ponomarev and K. A. Ter-Martirosyan, Sov. Journal of Nucl. Phys. 44, 468 (1986). [16] A. B. Kaidalov, Phys. Rep. 50 (1979) 157. [17] For a review see E. Reya, Phys. Rep. B69, 195 (1981). [18] A. Brandt et al (UA8 Collaboration), Phys. Lett. B297, 417 (1992) and to be

published.

10 [19] A. Devoto, D. W. Duke and J. F. Owens, Phys. Rev. D27, 508 (1983). [20] P. Aurenche, R. Baier, M. Fontannaz, M. N. Kienzle-Focacci and M. Werlen, Phys.

Lett. B233, 517 (1989), in preparation. [21] A. Kaidalov, Workshop on DIS and QCD, Paris, April 1995. [22] T. Gehrmann and W. J. Stirling, DTP/95/26. [23] K. Golec-Biernat and J. Kwiecinski, Krakow INP Report No 1670/PH. [24] J. Dainton (H1 Collaboration), Workshop on DIS and QCD, Paris, April 1995.

11 Figure Captions

Fig. 1 Pomeron exchange diagram for single diffraction dissociation of a virtual photon. Fig. 2 Pomeron exchange diagram for a double diffraction dissociation in which both the

hard current and the target proton are excited.

Fig. 3 Regge exchange diagram for the proton structure function.

Fig. 4 Regge exchange diagram for the Pomeron structure function. Fig. 5 Initial gluon distribution (Q2O = 5 GeV2) and its perturbative QCD evolution for

ng = \Gamma 0:5.

Fig. 6 Same as Fig. 5 for a u quark.

Fig. 7 Same as Fig. 5 for an s-quark. Fig. 8 Same as Fig. 5 for a c-quark. The charm distribution is taken to be zero at Q20 and

is generated, at larger Q2, via DGLAP evolution.

Fig. 9 The 3-dimensional proton structure function for diffractive events F D2 (x; xP ; Q2) is

plotted versus xP at various values of Q2 and fi = x=xP and compared with H1 data [4]. The model results are for ng = \Gamma 0:5. Very similar results are obtained with ng = 0.

Fig. 10 The 2-dimensional proton structure function for diffractive events F D2 (fi; Q2) is plotted versus Q2 at fixed fi and versus fi at fixed Q2 and compared with H1 data [4]. The model results are for ng = \Gamma 0:5. Very similar results are obtained with ng = 0. In order to take into account the presence of double diffraction in the data, the theoretical result has been multiplied by 1.3.

Fig. 11 The Pomeron structure function F P2 (fi; Q2), defined in Eq. (4), is plotted versus fi at

several values of Q2 for ng = \Gamma 0:5 (Fig. 11a) and ng = 0 (Fig. 11b).

Fig. 12 The functions of the Pomeron momentum carried by quarks (Nq ) and gluons (Ng )

defined by eqs. (9) and (10) are plotted versus Q2 for two different values of ng.

12 g* e

e'

M2 =s1

t

P

p p'

Figure 1

g* e

e'

M2 =s1

t

P

p

Figure 2

g* g*

P,R(f) p p

Figure 3

g* g*

P,R(f) P P

Figure 4

13 0:15

0:2 0:25

0:3 0:35

0:4 0:45

0:5 0:55

10 100 figp(fi; Q2)

Q2

figp(fi; Q2) figp(fi; Q20) = 0:12fi\Delta (1 \Gamma fi)ng

ng = \Gamma 0:5; Q20 = 5:0GeV 2

fi = 0:065

fi = 0:175 fi = 0:375 fi = 0:650

0:15

0:2 0:25

0:3 0:35

0:4 0:45

0:5 0:55

0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 figp(fi; Q2)

fi

Q20 = 5:0 GeV 2 Q2 = 8:5 GeV 2 Q2 = 12:0 GeV 2 Q2 = 25:0 GeV 2 Q2 = 50:0 GeV 2

Fig.5

14 0:02 0:025

0:03 0:035

0:04

10 100 fiup(fi; Q2)

Q2

fiup(fi; Q2) ng = \Gamma 0:5; Q20 = 5:0GeV 2

fi = 0:065 GeV 2 fi = 0:175 GeV 2

fi = 0:375 GeV 2 fi = 0:650 GeV 2

0:02 0:025

0:03 0:035

0:04

0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 fiup(fi; Q2)

fi Q20 = 5:0 GeV 2 Q2 = 8:5 GeV 2 Q2 = 12:0 GeV 2 Q2 = 25:0 GeV 2 Q2 = 50:0 GeV 2

Fig.6

15 1e \Gamma 05

0:0001

0:001

0:01

0:1

10 100 fisp(fi; Q2)

Q2

fisp(fi; Q2) ng = \Gamma 0:5; Q20 = 5:0GeV 2

fi = 0:065 GeV 2 fi = 0:175 GeV 2 fi = 0:375 GeV 2 fi = 0:650 GeV 2

1e \Gamma 05

0:0001

0:001

0:01

0:1

0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 fisp(fi; Q2)

fi Q20 = 5:0 GeV 2 Q2 = 8:5 GeV 2 Q2 = 12:0 GeV 2 Q2 = 25:0 GeV 2 Q2 = 50:0 GeV 2

Fig.7

16 0:0001

0:001

0:01

10 100 ficp(fi; Q2)

Q2

ficp(fi; Q2) ng = \Gamma 0:5; Q20 = 5:0GeV 2

fi = 0:065 GeV 2 fi = 0:175 GeV 2 fi = 0:375 GeV 2 fi = 0:650 GeV 2

0:0001

0:001

0:01

0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 ficp(fi; Q2)

fi

Q20 = 5:0 GeV 2 : ficp(fi; Q20) j 0 Q2 = 8:5 GeV 2 Q2 = 12:0 GeV 2 Q2 = 25:0 GeV 2 Q2 = 50:0 GeV 2

Fig.8

17 0:1

1 10 100 1000

F

D(3)

2

Q2 = 8:5GeV 2 fi = 0:065

r r

0:1

1 10 100 1000

F

D(3)

2

Q2 = 8:5GeV 2 fi = 0:175

r r

0:1

1 10 100 1000

F

D(3)

2

Q2 = 8:5GeV 2 fi = 0:375

r

r

0:1

1 10 100 1000

0:001 0:01 0:1 F

D(3)

2

xP

Q2 = 8:5GeV 2 fi = 0:650

r

r

Q2 = 12:0GeV 2 fi = 0:065

r

r r

r

r

Q2 = 12:0GeV 2 fi = 0:175

r

r

r

r

r

r

Q2 = 12:0GeV 2 fi = 0:375

r

r r

r

r

r

0:001 0:01 0:1x

P

Q2 = 12:0GeV 2 fi = 0:650

r

r

r r

r r

Fig.9

Q2 = 25:0GeV 2 fi = 0:065

r

r r

Q2 = 25:0GeV 2 fi = 0:175

r

r r

r

r

Q2 = 25:0GeV 2 fi = 0:375

r

r

r

r r

0:001 0:01 0:1x

P

Q2 = 25:0GeV 2 fi = 0:650

r

r

r

r

r

Q2 = 50:0GeV 2 fi = 0:175

r

r r

r

Q2 = 50:0GeV 2 fi = 0:375

r

r

r

r

r

0:001 0:01 0:1x

P

Q2 = 50:0GeV 2 fi = 0:650

r r

r

r

r

F D(3)2 (x; xP ; Q2) ng = \Gamma 0:5 ffl H1-Data

18

0 0:05

0:1 0:15

0:2 0:25

0:3 0:35

0:4 0:45

0:5

F D2

F D2 (fi; Q2 = const)

Q20 = 5:0GeV 2

Q2 = 8:5GeV 2 ng = \Gamma 0:5; e = 0:07; f = 0:07

e

e e

e

0 0:05

0:1 0:15

0:2 0:25

0:3 0:35

0:4 0:45

0:5

F D2

Q2 = 12:0GeV 2

e e e e

0 0:05

0:1 0:15

0:2 0:25

0:3 0:35

0:4 0:45

0:5

F D2

Q2 = 25:0GeV 2

e

e e e

0 0:05

0:1 0:15

0:2 0:25

0:3 0:35

0:4 0:45

0:5

0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 F D2

fi

Q2 = 50:0GeV 2 e

e

e

F D2 (fi = const; Q2)

Q20 = 5:0GeV 2

fi = 0:065 ng = \Gamma 0:5; e = 0:07; f = 0:07

e

e e

fi = 0:175 e e

e

e

fi = 0:375 e e

e

e

10 100

Q2(GeV 2)

fi = 0:650 e

e e

e

Fig.10

19 0:01 0:015

0:02 0:025

0:03 0:035

0:04 0:045

0:05 0:055

0:06

0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 FP

fi

FP (fi; Q2) a) ng = \Gamma 0:5

e = 0:07 f = 0:07

Q20 = 5:0 GeV 2 Q2 = 8:5 GeV 2 Q2 = 12:0 GeV 2 Q2 = 25:0 GeV 2 Q2 = 50:0 GeV 2

0:01 0:015

0:02 0:025

0:03 0:035

0:04 0:045

0:05 0:055

0:06

0 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 1 FP

fi

b) ng = 0:0

e = 0:07 f = 0:07

Q20 = 5:0 GeV 2 Q2 = 8:5 GeV 2 Q2 = 12:0 GeV 2 Q2 = 25:0 GeV 2 Q2 = 50:0 GeV 2

Fig.11

20 0 0:05

0:1 0:15

0:2 0:25

0:3

10 100 Ni

Q2(GeV 2)

Parton Momentum Fraction in the Pomeron

a) ng = \Gamma 0:5

Ng

Nq ng = \Gamma 0:5 e(0) = 0:07 f(0) = 0:07

Q20 = 5:0GeV 2

0 0:05

0:1 0:15

0:2 0:25

0:3

10 100 Ni

Q2(GeV 2)

b) ng = 0:0

Ng Nq ng = 0:0 e(0) = 0:07 f(0) = 0:07

Q20 = 5:0GeV 2

Fig.12

