Diffraction and the Pomeron 1 Introduction

Home , H1 (particle detector), Pomeron

Di raction and the Pomeron

Halina Abramowicz

School of Physics and Astronomy

Raymond and Beverly Sackler Faculty of Exact Sciences

Tel Aviv University, Israel

1 Intro duction

In hadron-hadron scattering, interactions are classi ed by the characteristics of the

nal states. In elastic scattering, b oth hadrons emerge unscathed and no other par-

ticles are pro duced. In di ractive scattering, the energy transfer between the two

interacting hadrons remains small, but one single disso ciation or b oth double dis-

so ciation hadrons disso ciate into multi-particle nal states, preserving the quantum

numb ers of the asso ciated initial hadron. The remaining con gurations corresp ond

to inelastic interactions.

The rst interpretation of di raction, develop ed byFeinb erg and Pomeranchuk [1]

and Good and Walker [2], was that di erent comp onents of the pro jectile were

di erently absorb ed by the target, leading to the creation of new physical states.

This was one of the early indications of the comp osite nature of hadrons.

In the Regge theory of strong interactions [3], di raction is the result of ex-

changing a universal tra jectory with the quantum numb ers of the vacuum, the soft

Pomeron, IP ,intro duced by Grib ov[4].

In the language of Quantum Chromo dynamics, the candidate for vacuum ex-

change with prop erties similar to the soft Pomeron is two gluon exchange [5, 6].

As a result of interactions between the two gluons, a ladder structure develops. In

p erturbative QCD, the prop erties of this ladder dep end on the energy and scales

involved in the interaction, implying its non-universal character. In the high-energy

limit, the prop erties of the ladder have b een derived for multi-Regge kinematics and

the resulting exchange is called the hard BFKL p omeron [7, 8, 9].

A renewed interest in di ractive scattering followed the observation of a copious

pro duction of di ractive-like events in deep inelastic scattering DIS at the HERA

ep collider [10, 11] as well as the earlier observation of jet pro duction asso ciated

with a leading proton in p p at CERN [12]. The presence of a large scale op ens the

p ossibility of studying the partonic structure of the di ractive exchange as suggested

by Ingelman and Schlein [13] and testing QCD dynamics. Moreover, the study of

di ractive scattering o ers a unique opp ortunity to understand the relation b etween 495

Halina Abramowicz Di raction and the Pomeron

the fundamental degrees of freedom prevailing in soft interactions { hadrons and

Regge tra jectories, and those of QCD { quarks and gluons. One of the challenges is to

establish theoretically and exp erimentally the reactions in which the soft comp onent

is dominant and those in which the p erturbative QCD formalism is applicable. In

this rep ort, the discussion will fo cus on single di ractive pro cesses in the presence of

a large scale, mainly studied at HERA and at FNAL.

2 Kinematics of di ractive scattering

The variables used to analyze di ractive scattering will be intro duced for ep DIS.

Because DIS is p erceived as a two-step pro cess in which the incoming lepton emits

a photon which then interacts with the proton target, the relevant variables can be

readily generalized to p p interactions. , k ke e q quark β MX

xIP,t LRG , pp

Figure 1: Schematic diagram for di ractive DIS in ep interactions.

A diagram for di ractive scattering in DIS, where the di racted state is separated

from the scattered proton by a large rapidity gap LRG, is presented in Fig. 1. All

the relevant four vectors are de ned therein. In addition to the usual DIS variables,

2 2 0 2 2 2 2

Q = q = k k , W = q + p , x = Q =2p q , and y = p q=p k, the

variables used to describ ed the di ractive nal state are,

0 2

t = p p ; 1

2 2 0

M + Q q p p

' ; 2 x =

2 2

q p W + Q

2 2

Q x Q

= = ' : 3

0 2

2q P P x Q + M

x is the fractional loss of the proton longitudinal momentum. It is sometimes de-

noted by . is the equivalent of Bjorken x, but relative to the exchanged ob ject. 496

Halina Abramowicz Di raction and the Pomeron

M is the invariant mass of the hadronic nal state recoiling against the leading

2 0 2

proton, M =q+pp . The approximate relations hold for small values of the

four-momentum transfer squared t and large W ,typical of high energy di raction.

The need to preserve the identity of the target in di ractive scattering limits

the square of the momentum transfer, jtj < 1=R , where R is the radius of the

target. The t distribution typically has an exp onential b ehavior, f t exp bjtj

with b ' R =6. The allowed M is also limited by the coherence requirement. The

minimum value of t required to pro duce a given M from a target with mass m is

X T

2 2 2 2 4 2 2

jt j ' m M + Q =W . For a typical hadronic radius of 1 fm, M < 0:2W

min

T X X

and the hadronic nal state exhibits a large rapidity gap between the fragments of

the di racted state and the unscathed target see Fig. 2. Therefore, in collider

dN dy

proton MX

large rapidity gap

p X

Figure 2: Schematic representation of rapidity y distribution for single di raction

events.

exp eriments, di ractive events are identi ed either by the presence of a fast proton

along the b eam direction or by the presence of a large rapidity gap in the central

detectors.

3 Formalism of di ractive scattering

To describ e di ractive DIS, it is customary to cho ose the variables x and t in addition

to the usual x and Q in the cross section formula. The di ractive contribution to F

is denoted by F and the corresp onding di erential contributions are

D D

dF dF

D 3 D 4

2 2

F = ; F = : 4

2 2

dx dx dt

IP IP

Throughout this pap er, the contribution from the longitudinal structure function

is omitted for simplicity. The four-fold di erential cross section for di ractive ep 497

Halina Abramowicz Di raction and the Pomeron

scattering can b e written as

4 D

h i

2 2

1+1y F x; Q ;x ;t: 5 =

2 4

dx dtdxdQ xQ

The structure function F is related to the absorption cross section of a virtual

photon by the proton, . For di ractive scattering, in the limit of high W low x,

2 D

D 4

F x; Q ;x ;t= : 6

4 dx dt

This relation allows predictions for di ractive scattering in DIS based on Regge phe-

nomenology applied to p scattering. In fact, many of the questions that are ad-

dressed in analyzing di ractive scattering are inspired by Regge phenomenology as

established in soft hadron-hadron interactions.

3.1 Regge phenomenology

For scattering of two hadrons a and b at squared center of mass energy s m ; t,

a;b

Regge phenomenology implies that,

ab s ; 7

tot

2 el 2

d ab

el 0

tot

b +2 ln st

ab ! ab = e ; 8

dt 16

2 01

2 D

D 0

b +2 ln t

1 d s

ab ! Xb e ; 9

2 2 2

dtdM M M

X X X

where t = 0 + t is the parameterization of the IP tra jectory. The uni-

IP IP

versality of this parameterization has b een p ointed out by Donnachie and Landsho .

0 2

The value of 0 = 1:081 [14] and =0:25 GeV [15] were derived based on to-

tal hadron-proton interaction cross sections and elastic proton-proton data. Recently,

the IP intercept has b een reevaluated [16] leading to a value of 0 = 1:096 0:03.

The IP intercept is sometimes presented as 0 = 1 + .

Three implications are worth noting.

1. The slop e of the t distribution is increasing with ln s. This fact, b orne out by

the data, is known as shrinkage of the t distribution. It is due to the fact that

6= 0 and has b een explained by Grib ov [4] as di usion of particles in the

0 2

exchange towards low transverse momenta, k , with 1=k see also [17].

IP T

2. A steep and universal x dep endence of the di ractive cross section is exp ected,

1+2

d =dx x .

el D

3. The ratios = and = rise like s . Because >0, this is b ound to lead

tot tot

to unitarity violation. 498

Halina Abramowicz Di raction and the Pomeron

3.2 Perturbative QCD

QCD factorization for the di ractive structure function of the proton, F , is exp ected

D D

to hold [18, 19, 20]. F is decomp osed into di ractive parton distributions, f , in a

2 i

way similar to the inclusive F ,

D 2 D

dF x; Q ;x ;t x df z ; ; x ;t

IP IP

2 i

= F ;Q ;; 10 dz

2;i

dx dt dx dt z

IP IP

where F is the universal structure function for DIS on parton i, is the factorization

2;i

scale at which f are prob ed, and z is the fraction of momentum of the proton carried

by the di ractive parton i. Di ractive partons are to be understo o d as those which

lead to a di ractive nal state. The DGLAP evolution equation applies in the same

way as for the inclusive case. For a xed value of x , the evolution in x and Q is

equivalent to the evolution in and Q .

Note that QCD factorization for the di ractive parton distributions is exp ected

to fail for hard di raction in hadron collisions [18].

If, following Ingelman and Schlein [13], one further assumes the validity of Regge

factorization, F may be decomp osed into a universal IP ux and the structure

function of IP ,

D 2

dF x; Q ;x ;t

IP 2

= f x ;tF ; Q ; 11

IP=p

dx dt

where the normalization of either of the two comp onents is arbitrary. It implies that

the x and t dep endence of the di ractive cross section is universal, indep endent of

Q and , and given by see Eq. 9

2 01

D 0

b 2 ln x t

f x ;t e : 12

IP=p

In this approach, the di ractive parton distributions would be obtained as a convo-

lution of the IP ux with parton distributions in the IP .

None of the approaches detailed ab ove address the issue of the dynamical origin of

the IP exchange in p erturbative QCD. The mechanism for pro ducing LRG is assumed

to b e present at some scale and the evolution formalism allows to prob e the underlying

partonic structure. A more dynamical approach will be discussed in the context of

the measurements p erformed up to date.

at HERA 4 Measurements of F

p runs with the proton b eam momentum of The data analyzed to date come from e

820 GeV and p ositron b eam of 27.5 GeV. The coverage of phase space in and Q

as well as and x is shown in Fig. 3.

IP 499

Halina Abramowicz Di raction and the Pomeron

2 D

Figure 3: Phase space coverage in , Q , and x for the measurements of F in ep

interactions. Also shown is the x and coverage of the recent CDF data with the

recoil proton measured in the Roman p ots see Section 6.2.

ZEUS 1996 Preliminary

(0) ZEUS BPC diffractive (THIS ANALYSIS)

IP ZEUS BPC total α ZEUS DIS diffractive H1 DIS diffractive ZEUS PHP diffractive H1 PHP diffractive

(⇐ Q2= 0GeV2)

ALLM97 parametrization

Q2 (GeV2)

Figure 4: Q dep endence of 0 derived from measurements of di ractive and total

p cross sections. The curve ALLM97 is a representation of the results obtained

in inclusive DIS measurements. 500

Halina Abramowicz Di raction and the Pomeron

4.1 Energy dep endence and Regge factorization

D ?

Following the lead of Regge phenomenology, the energy dep endence of p or

the x dep endence of F in the data is expressed in terms of 0. For x < 0:01,

IP IP IP

and within a given range of Q , the data are compatible with a universal x dep en-

dence [21, 22, 23]. The data used for these studies are integrated over t. To derive the

value of 0, it is assumed that the slop e of the t distribution is Q indep endent

D 2 0 2

and is given by b = 7 GeV [24]. For lack of other information =0:25 GeV is

used. The Q dep endence of 0 derived from di ractive measurements is shown

in Fig. 4. For comparison, also shown is the Q dep endence of 0 derived from

the inclusive DIS measurements and conveniently represented by the ALLM97 pa-

rameterization [25]. As has b een known for a while, the inclusive DIS data at high W

are not compatible with a universal IP tra jectory. They are, however, in very good

agreement with exp ectations of QCD evolution. The di ractive measurements seem

to p oint to some Q dep endence, but less pronounced than that in the inclusive case.

2 2

For Q > 10 GeV , the value of 0 is signi cantly higher than measured for soft

interactions. The imp ortant observation is that the W dep endence of di ractive DIS

is the same as for the inclusive cross section [22] and slower than would b e exp ected

from Regge phenomenology.

4.2 Q and dep endence

D 3

The Q dep endence of x F as measured by the H1 exp eriment [21, 26] for two

values of x and for a range of values is shown in Fig. 5. Also shown in the gure is

the distribution measured with the leading proton sp ectrometer LPS of the ZEUS

2 2

detector, at x =0:01 for two values of Q [27]. Both the Q and the dep endence

D 2

of F are very di erent from the one known for the proton F . There is almost no Q

dep endence for > 0:4 and the distribution itself is relatively at up to large ' 1.

This suggests a very di erent parton content of di ractive scattering compared to the

inclusive scattering. As we will see b elow, for the DGLAP evolution to hold, a large

gluon comp onent is required at large to comp ensate for the loss of quark momenta

in the evolution.

4.3 Di ractive parton distributions

The di ractive parton density functions DPDF are derived from F by tting the

DGLAP evolution equations with p ostulated input distributions at some low Q

scale. The t consists of adjusting the parameters of the input distributions to get

the b est description of all available data. No momentum sum-rule is available for

di ractive partons, however one assumes avor symmetry for light quarks. The results

of such a tting pro cedure [21] are shown in Fig. 6. Di erent solutions are found, 501

Halina Abramowicz Di raction and the Pomeron

xIP=0.005

D(3) H1 1994 2 D(3) 2 β H1 1994 = 0.01

F 0.05 F H1 Preliminary 1995-7 IP IP 0.05 H1 Preliminary 1995 D F2 fit 3 (IP + IR) x x D F2 fit 3 (IP) 0 β = 0.04 0.04 0.05 0.03 0 β = 0.1 0.05 0.02

0 β = 0.2 0.01 0.05 β = 0.4 xIP = 0.02 0 0 2 3 β = 0.4 10 10 10 0.05 Q2 (GeV2)

0 β = 0.65 0.05

0 β = 0.9 0.05

0 2 1 10 10

Q2 (GeV2)

D 3

Figure 5: Left and upp er right plots: Q dep endence of x F at xed values of

D 4

and x as denoted in the gure. Lower right plot: dep endence of x F for

IP IP

2 2

0:073 < jtj < 0:4 GeV at xed values of Q and x as denoted in the gure.

H1 1994

(a) Q2=4.5 GeV2 Gluon, fit 3 Gluon, fit 2 1.5 Light Quarks, fit 3 Light Quarks, fit 2 1

f 0.5

0 (b) Q2=12 GeV2 1.5

1 0.5

0 (c) Q2=75 GeV2 1.5

1 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6: Left: di ractive parton densities f as a function of obtained by tting

the DGLAP evolution equations. Right: di erence in Q evolution of the singlet

structure function for di ractive and inclusive parton distributions for =0:2 and

x =0:2 resp ectively. 502

Halina Abramowicz Di raction and the Pomeron

however for all cases a large gluon comp onent is required. The net result is that

while the density of inclusive quark distributions represented by the avor singlet

structure function decreases with Q at x =0:2, the density of di ractive partons

is still rising with Q at the equivalent =0:2 [28].

4.4 Jet and charm pro duction

The partonic structure of di ractive scattering can be further explored by studying

the hadronic nal states. A large gluon comp onent is exp ected to lead to a copious

pro duction of high p jets as well as of charm, through the b oson-gluon fusion diagram

depicted in Fig. 7. The H1 exp eriment has measured the cross section for dijet

γ q

q g (z)

IP (1-z)

p p

Figure 7: Left: b oson-gluon fusion diagram. Right: The dep endence of di ractive di-

jets

jet cross section on z , the exp erimental estimate of z as de ned in the left diagram.

The curves corresp ond to exp ectations for di erent DPDFs.

pro duction in di ractive scattering [29]. The observed rates are in agreement with

exp ectations based on DPDFs obtained from ts to F .

Di ractivecharm pro duction has also b een measured [30,31]. While ZEUS mea-

surements are in agreement with exp ectations, the H1 result seems to b e smaller than

exp ected. Note that the statistical and systematic errors of these measurements are

quite large and a de nite answer awaits more data.

5 Generic mo del for di ractive DIS

A tremendous theoretical e ort has b een, and still is, devoted to the understanding

of the dynamics b ehind di ractive DIS for recent reviews see [32, 33] and references 503

Halina Abramowicz Di raction and the Pomeron

therein . A simple picture of di raction emerges if the pro cess is viewed in the

rest frame of the proton. The virtual photon develops a partonic uctuation, whose

lifetime is =1=2m x [42]. At the small x typical of HERA, where 10 100 fm,

it is the partonic state rather than the photon that scatters o the proton. If the

scattering is elastic, the nal state will have the features of di raction.

The equivalence between the approach in the proton rest frame and in the Breit

frame is schematically depicted in Fig. 8 [43]. An interaction initiated by uctuation

of into a q q q qg state will contribute to the di ractive quark gluon distribution.

The uctuations of the are describ ed by the wave functions of the transversely and

longitudinally p olarized which are known from p erturbative QCD. Small and large

partonic con gurations of the photon uctuation are present. For large con gura-

tions, non-p erturbative e ects dominate in the interaction and the treatment of this

contribution is sub ject to mo deling. For a small con guration of partons large rel-

ative k the total interaction cross section of the created color dip ole on a proton

target is given by [44, 45]

= r xg x; ; 13

q qp S

' = ; 14

q qgp ggp qqp

where r is the transverse size of the color dip ole and 1=r is the scale at which

the gluon distribution g of the proton is prob ed. The corresp onding elastic cross

section is obtained from the optical theorem. In this picture, the gluon dominance in

di raction results from the dynamics of p erturbative QCD see Eq. 14.

The interesting observation is that all the mo dels, whether starting from a purely

p erturbative approach [28] or a semi-classical approach [36], give a good description

2 D

of the Q and dep endence of F . The leading-twist b ehavior of di ractive DIS is

determined by the hadronic nature of the p interactions. The dep endence seems

to b e mostly determined by the wave function of .

A p ossible explanation of the success of the p erturbative approach to di ractive

DIS has b een prop osed by Mueller [46]. If the large size uctuations were to be

absorb ed, di raction would be dominated by small size con gurations which lend

themselves to p erturbative calculations. The absorption would b e a manifestation of

unitarity e ects and would explain the slower than exp ected W dep endence of the p

di ractive cross section see Section 4.1. Absorption e ects have b een successfully

incorp orated in some of the mo dels [37, 38].

The p ossibility that unitarity e ects at low x are not negligible has b een p ointed

out by Frankfurt and Strikman [45]. The probability that an interaction on parton i

Here is a list of representative pap ers from which a thorough exploration can b egin: [34, 35,36,

37,38,39,40,41]. 504

Halina Abramowicz Di raction and the Pomeron

q q q q q P Rapidity gap a) quark distribution P

q q q q g g g P Rapidity gap

b) gluon distribution P

Figure 8: Schematic representation of the equivalence between di ractive DIS in the

proton rest frame left and in the Breit frame right.

is asso ciated with a di ractive nal state, P , is de ned as

D 2

f x=x ;Q ;x ;tdx dt

IP IP IP

D 2

P x; Q = ; 15

f x; Q

where f stands for the inclusive density of parton i. P may be interpreted as the

ratio of elastic to total cross sections, r i = i= i, induced either by a q q

el el tot

i = q orq qg i = g con gurations of . In the black disc limit, r =0:5. In DIS, for

2 2 3

Q =4:5 GeV and x<10 , the calculation using the ACTW [47] parameterization

D D D

leads to P ' 0:4 for gluons and P ' 0:15 for quarks. The value of P , close to

g q g

the black disc limit, may indicate the presence of unitarity e ects in the gluon sector

of the evolution.

6 Di raction in p p scattering

Di ractive-likeevents have b een observed in hard pp scattering by b oth the CDF and

the D0 exp eriments. The signature of these events, depicted in Fig. 9, is the presence

of a large rapidity gap in one of the forward regions asso ciated with the presence

of either large transverse momentum jets [48, 49], b eauty [50], J= [51], or the W

b oson [52].

6.1 Rates of di ractive pro cesses

A compilation of the published and preliminary results on the ratio of di ractive

to inclusive events with a given hadronic nal state is presented in Fig. 10 as a

function of the typical average scale of the pro cess. The measurements are compared 505

Halina Abramowicz Di raction and the Pomeron

p;p

p; p

φ Gap Jet+Jet

Figure 9: Schematic representation of single di ractiveevents in pp collisions and the

corresp onding density of the phase space of the azimuthal angle and of pseudora-

pidity .

Figure 10: Ratio of single di raction to inclusive pro duction in pp, SD/INCL, as a

function of the approximate hard scale in the pro cess. JJ stands for two jet pro duc-

tion while C and F stand for forward and central pro duction. Horizontal bars are

exp ectations obtained using the ACTW parameterization, t D [47], scaled down by

factor 20. 506

Halina Abramowicz Di raction and the Pomeron

to exp ectations obtained using the ACTW- t D parameterization [47] of DPDFs. In

order to repro duce vaguely the measured numb ers, the exp ectations have to b e scaled

down by a factor 20. This may not b e surprising as QCD factorization is not applicable

to hard di ractive scattering in hadron collisions [18]. What might be surprising is

that the overall scale dep endence is quite well repro duced. Unfortunately, there are

no calculations available for p p collisions at s = 630 GeV. The di erence in the

contribution of the di ractive quarks and gluons to di erent pro cesses has b een used

by CDF to constrain, to leading order, the momentum fraction f of gluons relative

+0:16

to all di ractive partons. Including W , jet, and b eauty pro duction, f = 0:54 ,

0:14

marginally less than in di ractive DIS, where this ratio is close to 90, for next-to-

leading order partons.

6.2 and dep endence

The fractional momentum loss of the proton in hard di ractive jet pro duction has

b een measured by D0 [49]. For s = 1800 630 GeV , the distribution p eaks at

' 0:03 0:07 and extends to = 0:1 0:2. This is a region where contributions

from sub-leading tra jectories may b ecome imp ortant.

CDF has measured and by detecting the leading p in a forward sp ectrometer

consisting of Roman p ots [53]. The results are expressed in terms of an e ective

structure function F derived from the rate of leading p and the two-jet inclusive cross

section. The shap e of the distribution was found to b e indep endentof, suggesting

Regge factorization. For the measured range, 0:04 < < 0:09, the dep endence of

D 0:87 D

F at =0:2 is describ ed by . The distribution of F is shown in Fig. 11

JJ JJ

and compared to exp ectations using the H1 - t 2 parameterization [21]. To get

the right order of magnitude for > 0:4, the exp ectations have to be scaled down

by a factor 20. In addition, CDF observes a relatively higher yield of low events.

Without the power of the QCD factorization theorem and given the very di erent

range, it is very dicult to drawany conclusions from the di erence in the p p and ep

data.

7 Vector meson pro duction

To test ideas b ehind the generic mo del of di ractive DIS see Section 5 and its hard

comp onent, one needs to devise a trigger to isolate when the photon uctuates into

q con guration. A small-size q q dip ole is most likely to be pro duced if a small-size q

the virtual photon is longitudinally p olarized or if the dip ole consists of heavy quarks

such as c c or bb . The exclusive J= [54] and photopro duction as well as vector

meson pro duction in DIS [55] are likely candidates for hard di ractive pro cesses fully

calculable in p erturbative QCD. The corresp onding diagram is shown in Fig. 12. The 507

Halina Abramowicz Di raction and the Pomeron

CDF Preliminary 6 D JJ F Jet1,2 ≥ H1 IP+IR / 20, ET 7 GeV 2 2 5 ( Q = 49 GeV ) 0.035 ≤ ξ ≤ 0.095 x > 3 × 10-4 ( IP : fit-2 ) | t | ≤ 1.0 GeV2 stat error only 4 0.4

0.3 3 x > 10-3 0.2

2 0.1

0 0.4 0.5 0.6 0.7 0.8 0.9 1 1 β

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 11: distribution of two jet pro duction with a leading p in pp collisions com-

pared to exp ectations based on di ractive parton distributions obtained at HERA [21]

scaled down by factor 20.

predictions of p erturbative QCD are very distinct:

1. The avor dep endence of = p ! Vp is determined by the photon quark

current and the SU4 relation

: : : =9:1:2:8; 16

which is badly broken in soft interactions see e.g. [56], is restored.

2. Exclusive V pro duction in DIS is dominated by the longitudinal comp onent

and [55]

d A

V;L

2 2 2

jxg x; Q j ; 17 Q =

dt Q

q V

Figure 12: Schematic diagram for exclusive vector meson, V , pro duction in hard

di ractive scattering. 508

Halina Abramowicz Di raction and the Pomeron

ρ 2

φ ρ ω ρ σ from ZEUS data

ZEUS96-97 (prelim.) / ZEUS96-97 (prelim.) / / φ ρ ω ρ

ZEUS96 (prelim.) / ZEUS93 / Ψ φ ρ ZEUS 95 ZEUS94 / J/ 0.3 φ ρ H1 / σ 1.5 H1 94 from H1 data 0.2 1

0.5 cross sections ratio 0.1

0 0 0 5 10 15 20 0 10203040

Q2 (GeV2) Q2 (GeV2)

Figure 13: Cross sections ratio = for V = , ! left and J= right as a

V 1

function of Q .

where A dep ends on the V wave function. At low x, the gluon distribution

increases rapidly when x decreases W increases and therefore a rapid increase

of with W is exp ected.

2 2

3. Because of the fast increase with Q of gluons at low x, the exp ected Q dep en-

dence is slower than Q .

4. The t distribution should b e universal, determined by the two-gluon form-factor,

0 2

with ' 1=Q , b ecause of the p erturbative nature of the interaction.

Below some of the recent exp erimental ndings are reviewed. A detailed discus-

sion of the HERA results can b e found in [32].

7.1 Flavor dep endence

A compilation of the ratio of , ! , and J= pro duction cross sections to that of the

0 2 0 0

as a function of Q is shown in Fig. 13. The : and J= : ratios increase

2 2

with Q and at high Q reach, within errors, the value exp ected from SU4. The

ratio remains constant. ! :

7.2 W dep endence

The W dep endence of exclusive V photopro duction is shown in Fig. 14. The striking

fact is the fast rise of the J= cross section, which has now b een measured by the H1

exp erimentuptoW = 300 GeV [57]. This rise is in go o d agreement with p erturbative

QCD calculations [59, 58]. For pro duction [61, 60], the rise of the cross section with

2 2 2

W dep ends on Q ,asshown in Fig. 14. As can b e seen, only at ab out Q = 20 GeV

the steepness of the rise is comparable to that of the J=. This may b e attributed to

2 2

the e ective scale in V pro duction, Q , which dep ends not only on Q , but also on

ef f

wave function e ects [59]. The rise with W of W can b e quanti ed by , which

V 509

Halina Abramowicz Di raction and the Pomeron

is shown in Fig. 15 as a function of Q . The results for di erent pro cesses seem to

ef f

2 2

line up along the same curve, which, ab ove Q ' 4 GeV , follows the exp ectations

ef f

from the W dep endence of p total cross section.

Contrary to exp ectations, the pro duction cross section was found to be larger

than predicted by p erturbative calculations [62, 63]. This disagreement led to two

ndings [64, 65]. The gluons exchanged in V pro duction do not have the same value

? 2

of x. For a V with mass M pro duced from a with virtuality Q , the di erence

x is given by

2 2

M + Q

x = : 18

2 2

W + Q

For pro duction, x can b e large and the skewed gluon distribution has to b e used.

The skewed also called o -forward parton distributions are hybrid ob jects, which

combine prop erties of form factors and ordinary parton distributions [66, 67]. The

use of the skewed distribution leads to an enhancement of by a factor ' 2. For

a pro cess with a fast rise with W , the relation between the elastic cross section and

the total cross section has the form

d ReA

tot

= 1+ ; 19

dt ImA 16

t=0

where the ratio of the real to imaginary part of the scattering amplitude, A, cannot

be neglected. This pro duces another enhancement factor of 1:5 1:7. The present

agreement of theoretical calculations with data is shown in Fig. 16.

7.3 Q dep endence

The Q dep endence of the cross section at xed W ranging from 75 to 90 GeV is

2 2 n

parameterized as Q + M [61, 60, 68, 69]. The power n as a function of

M is shown in Fig. 17 and is found to b e substantially smaller than 3 see Eq. 17.

7.4 t dep endence

A compilation of the exp onential slop es b of the t distributions at Q ' 0 as a function

0 2

of M is shown in Fig. 18. For pro duction, b is also shown as a function of Q . The

value of the slop e decreases b oth with the mass of the V and with Q , towards the

value measured for the J=, p ointing to a decreasing size of the interaction region.

The p erturbativecharacter of J= pro duction can b e b est established by deriving the

value of from the W dep endence of the cross section at xed t see Eq. 8

4 t1

W : 20

dt 510

Halina Abramowicz Di raction and the Pomeron

Figure 14: Left: W dep endence of the cross section p ! Vp for various vector

mesons, from xed target and HERA measurements. Right: W dep endence of

cross section for di erent Q values compared to J= photopro duction.

Figure 15: The power of the W dep endence of p ! Vp as a function of the

e ective scale of the scattering, Q . The curve represents the exp ectations from the

ef f

W dep endence of the total p cross sections. 511

Halina Abramowicz Di raction and the Pomeron

Figure 16: The photopro duction cross section as a function of W compared to

calculations of p erturbative QCD, FSM [64], MRT [65].

n ZEUS 95 H1 96-97 2.5

1.5

1 01234

MV (GeV)

2 2 n

Figure 17: The p ower n in the Q + M dep endence of V pro duction as a function

of mass, M .

Earlier measurements at HERA, combined with low energy data, indicated that

t ' const [70] see Fig. 19 left. H1 determined using only their own da-

0 2

ta [57]. The result, shown in Fig. 19 with =0:05 0:15 GeV , supp orts the hard

nature of exclusive J= pro duction at HERA.

8 Deeply virtual Compton scattering

Another example of hard di ractive scattering is the exclusive pro duction of a real

photon in DIS [71] see Fig. 20. In this deeply virtual Compton scattering DVCS

for recent discussions see [66, 67], the cross section is less suppressed by Q than in

exclusive vector meson pro duction [71], and the interpretation of the data is easier

due to the lackofV wave function e ects. DVCS interferes with the QED Compton

pro cess, which leads to an identical nal state, as depicted in Fig. 20. The interference

may allow a direct measurement of the real part of the QCD scattering amplitude. By 512

Halina Abramowicz Di raction and the Pomeron

] ) 2 -2 - 14 ZEUS (γp → ρp) 14 CHIO NMC E665 γ → ω ZEUS ( p p) H1 ZEUS [ (GeV ZEUS (γp → φp) γ → ψ b GeV 12 Vp) 12 ZEUS ( p J/ p) FKS97 → ρo H1 (γp → ρp) p γ ( H1 (γp → J/ψp) b 10 ω 10

8 8 φ 6 6 J/ψ 4 4 J/ ψ(Η1) 2 2 1 10 0 5 10 15 20 25 30 2 2 2 2

MV (GeV ) Q [GeV ]

Figure 18: Left: the exp onential slop e b of the t distribution in p ! Vp at HERA as

2 2 0

a function of M . Right: the exp onential slop e b as a function of Q for pro duction. V

1.6

1.4

1.2

0.8

0.6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Figure 19: The Pomeron tra jectory t as determined from low energy and HERA

data left and from H1 data alone right. 513

Halina Abramowicz Di raction and the Pomeron

e e γ q e e e γ γ e q

pp pp pp

Figure 20: Diagrams for real pro duction in DIS.

extracting the DVCS, skewed parton distributions can be measured and additional

information on the F of the proton at low x can be obtained [71].

2 2

The ZEUS exp eriment has searched for DVCS in DIS events with Q > 6 GeV

4 2

and for 5 10 < x < 10 [72]. Events with two electromagnetic, e= , candidates

were selected with no additional hadronic activity in the central detector. The second

candidate was de ned as the one pro duced closest to the original proton direction,

with angle , and was required to be in the region of the detector where tracking is

available. The data were compared to the exp ectations of a QED Compton MC [73]

and a MC which included DVCS GenDVCS [74]. The results are shown in Fig. 21.

A clear excess of candidates at low is observed over exp ectations from the QED

Compton pro cess, while their rate is well repro duced by GenDVCS. The signal for

DVCS is now established at HERA.

9 Outlo ok

This rep ort is far from a complete account of di ractive physics. Its main purp ose

was to review the progress achieved in understanding di ractive phenomena with

the language of p erturbative QCD. Much has b een learned from studying inclusive

di raction, where the gluons are found to be the main players. Di ractive DIS pro-

vides a complementary area to inclusive DIS in studying the dynamics of QCD at

high energy. The new class of hard di ractive pro cesses, such as exclusivevector me-

son pro duction and deeply virtual Compton scattering, ful ll the prop erties exp ected

from p erturbative QCD. A comprehensive study of these pro cesses may provide, in

the future, imp ortant information on the gluon content in the proton and on the wave

functions of vector mesons. The accumulated data cannot b e describ ed coherently by

one universal Pomeron tra jectory. The issue of what the Pomeron is requires further

study.

Two sub jects of p otential interest, which were not discussed due to time con-

straints, are the large t di ractive scattering studied at FNAL [75, 76] and HERA [77]

and the double-IP exchange studied at FNAL [78]. Here more theoretical work is nec- 514

Halina Abramowicz Di raction and the Pomeron

ZEUS 1996/97 Preliminary 1

TOTAL 0.9 / N

γ 0.8 N 0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 θ

2 (radians)

TOT AL

Figure 21: Fraction of photon candidates, N =N , for the second electromagnetic

candidate in the calorimeter with energy E > 5 GeV , pro duced at angle , closest to

the proton direction. The data dots are compared to MC exp ectations for GenDVCS

circles and for elastic Compton scattering only triangles.

essary. It is quite clear that the emerging picture of di raction will not b e satisfactory

without a complete account of all hard di ractive pro cesses.

I would like to thank all my colleagues who have help ed me in preparing this

rep ort. My sp ecial gratitude go es to M. Albrow, J. Bartels, K. Borras, A. Brandt,

J. Dainton, K. Goulianos, L. Frankfurt, H. Kowalski, A. Levy, M. McDermott, N.

Makins, K. Mauritz and M. Strikman who patiently discussed the data and their

interpretation with me. I would also like to acknowledge the partial supp ort of the

US-Israel Binational Foundation BSF, the German-Israel Foundation GIF and the

Israel Science Foundation ISF, which have made this contribution p ossible.

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Discussion

Harry Lipkin Weizmann Institute: I would like to p oint out that the Don-

nachie and Landsho t to the total cross section, whichwas accepted by the Particle

Data Group in 1966, was thrown out by them in 1998 with a di erent value of the

Pomeron intercept. I have t the same data some time ago with a still di erent val-

ue. The p oint is that there are still not enough data on total cross sections to tell

us whether there is a universal Pomeron with a universal intercept that is coupled to

meson-baryon and baryon-baryon total cross sections. What are needed are new data

at higher energies. The recent data on hyp eron-nucleus cross sections from SELEX

actually supp ort my mo del, but I don't b elieve that either. I think that what is

happ ening is that there is a non-leading contribution which confuses the issue and is

not understo o d. To resolve this, we just have to get more data at higher energies.

Abramowicz: There is no denying that a well \measured" soft Pomeron would help

to establish when the departure from universality b ecomes e ective. However, it is

dicult to imagine that it would alter in anyway the fact that the W dep endence of

? 2

the total p cross section changes with Q and follows the laws of QCD evolution. I

think that even the di ractive data, imprecise as they are, cannot b e accommo dated

by a universal tra jectory. It maywell b e that the idealized Pomeron of Grib ov is just

a mathematical construction, while the p omeron we try to investigate is just another

name for the nature of strong interactions. The latter, by virtue of QCD and the

interplay of soft and hard physics, remain versatile. 520