PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 25B (1992) 142-151 North-Holland

T H E O D D E R O N - P A S T , P R E S E N T A N D F U T U R E

Basarab NICOLESCU

Division de Physique Th~-ique*, IPN, 91406 Orsay Cedex and LPTPE, Universit~ Pierre et Marie Curie, 4 Place Jnssieu, 75252 Paris Cedex 05, France

After presenting some short historical considerations concerning the evolution in time of the Odderon approach (from asymptotic theorems to QCD), we discuss the experimental finite-energy effects induced by the presence of the Odderon. The general agreement with the present experimental data is also briefly discussed. The last part of the talk is devoted to the predictions of the Odderon approach at LHC energies. We expect new physics to occur at LHC energies. We will therefore present strong arguments in favour of the adoption of both pp a n d /3p options at LHC.

1. I N T R O D U C T I O N

The Odderon approach became popular after the

discovery, in 1987, of a large pPP value at the CERN

SppS collider I and it was widely discussed in the lit-

erature. I will therefore not give in this talk historical

details of the Odderon approach (a short description of

its long and complex history can be found in Ref. 2).

Also, I will not present here the formalism and the phe-

nomenological consequences of the maximal odderon

approach, because they were already discussed by E.

Leader ~, B. Nicolescu 4 and P. Gauroa s at the previous

Blois Workshops.

In this talk I prefere to concentrate upon some

basic qualitative aspects of the Odderon approach lead-

ing to the conclusion that one can safely expect new

physics to occur at LHC energies. In order to test this

new physics both pp and #p options at LHC have to be

adopted.

2. S H O R T H I S T O R I C A L C O N S I D E R A T I O N S

I present in "fable 1 a list of some relevant dates

and references from the point of view of the Odderon ap-

proach. Of course, a bibliographical selection is always

a dangerous exercise and involves a somewhat subjec-

tive choice. However, what I wanted to point out in

Table 1 is the more and more pronounced link in time

between the Odderon concept and QCD : in our clays

the Odderon (or, at last, the perturbative Odderon)

naturally arises in QCD (LLA) as a compound state

of 3 reggeized gluons. Recently, by studying the con-

formal properties in QCD, we gave a semi-quantitative

argument 19 indicating that the intercept of the Odd-

eron in LLA is probably bigger than 1. We therefore

expect at very high energies important experimental ef-

fects, induced by the presence of the Odderon. The fact

that there are already quite strong indications of Odd-

eron effects at ISR and CERN-collider energies is, of

course, very encouraging.

* Unit~ de Recherche des Universit~s Paris 11 et Paris 6 Associ~e au CNRS

0920-5632/92/$0.5.00 ~ 1992 - Els~'ier Science Publishers B.V. All rights roserved.

B. Nicolescu/Oddemn -past, present and future 143

Let me go backwards in time in order to make

some observations about the initial pre-QCD dynamical

basis of the maximal Odderon idea. I will mahe just two

remarks :

i) Stimulated by the extraordinary climate gener-

ated in 1972-1973 by the discovery at ISR of increasing

pp total cross-sections and also stimulated by exciting

discussions with Andr~ Martin and Jean-Louis Basde-

vant, Lulmszuk and I thought that the @ increase is

just a sign of a very general property of "maxlmallty"

of strong irteractions ~. By "maximality" I understand

here the fol owing simple, but powerful statement : ev-

erything that can happen, happens. The analytic self-

consistent saturation of axiomatic bounds becomes, in

such a way, a dynamical principle. There are many con-

sequences of this principle. One of them is the maximal

Odderon.

| | ) One may think that the maximality principle

involves early asymptoticity. This statement is simply

wrong. The axiomatic bounds are used only as bound-

ary conditions as s -~ co for the scattering amplitudes.

In fact, there is a very delicate balance between the non-

asymptotic (Regge) contributions and the asymptotic

contributions. The Regge contributions are dominating

at low energies and become smaller and smaller at high

energies, while the asymptotic contributions are small

at low energies and become bigger and bigger at high

energies. This delicate balance generates spectacular

finite-energy effects, which are generally very different

from the asymptotic etfects.

Let me give just one example. Let us suppose that

A~ ~ - c o as s ~ co, where

However, A¢ > 0 at the expedmmt~ pre-ISR ~ ,

while Ao. is already very small at ISR energies. One

condedes that A¢ must go through zero and dmage itB

sign just beyond ISR energies. In fact, tiffs featme of

Act was already present in our first 1973-1975 palmm 1'7

and became much later indirectly confirmed by the high

UA4 value I .

3. E X P E R I M E N T A L F I N I T E - E N E R G Y EF-

F E C T S I N D U C E D B Y T H E P R E S E N C E OF

T H E O D D E R O N

In this section, for the sake of generality, I will ¢ma-

sider all kinds of Odderons, not only the maximal one.

By "Oddemn" I mean a J-plane singularity near J -- 1

in the odd-under-erossing amplitude F - . The Oddenm

is the natural companion of the Pomero~ which cor-

responds to a J-plane singularity near J ---- 1 in the

even-under-crossing amplitude F+. Both are expressed

dyvamically in terms of multigluon exchanges.

Taking into account that

Fpp f F+ + F= and F~p = F+ - F_ (2)

we can define two important cases :

a) There is no Odderon, i.e. F_ -- 0 at high

energies, leading to Fpp = F~p. The physical me~n;r,~ is

that pp and/~p scattering are identical at high energies.

Let me car this case the "symmetrical" case.

b) The Odderon is present, i.e. F - ~ 0 at high

energies, leading to Fpp ~ F~p. The physical me~n;n~ is

that pp is difl'erent from pp scattering at high energies.

Let me call this case the "asymmetrical" case.

, x o = O T' - ( I )

144 B. Nicolescu / Odder~n -past, present and future

T A B L E I . Some relevant dates and references 1990 1T concerning the Odderon approach -1991

Year Re~ C o m m e n t s

1973 6

1975

1975

1 9 8 0

1984

1985

1987

1987

1987

1988

1988

1988 -1990

First paper on the Odderon as a con- sequence of the maximality principle

T The only model that predicted a high value of pP~ at CERN-colUder energies (seen 12 years later). Derivation of the derivative relation for the ease J -- i inF_ .

8 The case of a simple pole at J = 1 in F - was considered (later identified as 3-ghon Odderon in QCD). The name ~Odderon" was invented.

9 The Odderon naturally arises in QCD (LLA) as a compound state of 3 regge- ized ghons

10 Minimal Odderon-pole finked to 3 ghon exchange, leading to a quantita- tive prediction of A(d~/dt ) ~ 0 at ISR e n e r g i e s

11 ExperlmeEtal discovery of A(d¢ /d t ) 0 at vr~ = 52.8 GeV in the structure region (It[ _ 1.3 GeV 2)

I Experimental discovery of a high OPP value at v ~ = 546, in agreement with predictions made in 1975 (Ref. 7)

12 Discussion of the Odderon interpreta- tion of the UA4 pPP datum (Ref. 1)

13 A dynamical quark-model interpreta- tica of the Odderon

14 A rigorous extension of the maximal Odderon approach at t ~ 0

15 Multiparticle unitarity indicates that probably the slope b( s, t = -0.02 Ge V 2 ) crosses over the slope b~p(s,t = -0.02 GeV 2) above the ISR energies

16 Tevatron data - ET10 Collaboration. High values of aT, as expected by al- most M1 threshold models, are excluded

1989 18,19 -1991

en d 20,21 1990

T A B L E 2.

O'T

~el

6reliCT

Detailed comparison of the Odderon approach with all the existing experi- mental data

Conformaily invaxiant approach in QCD of the Odderon problem

Theoretical challenge for having both pp and pp options at LHC

Maximal Odderon (GLN) approach pre-

dictions 17 at v ~ = 17 TeV.

O~T P = 121 rob, ~T ~ = 127 mb

A n T = o~ p -- ~T p = --6 mb

o~' ---- 27.4 rob, ceP[ ' = 29.6 mb

vP -- -2 .2 mb /~O'el --~ ~ f -- 0"¢1

(~el/~rT) ~p = 0 .227 , (~eI/IYT) pp = 0 .233

A(~Fel/~TT) = (Oel/~7T) pp -- ( f fel /¢T)PP ----

-0.006

p~P ---- 0.25, pPP = 0 . 0 5 - ~

Ap = ~P - f P = 0.2

Menn-slope in the region 0.01 <. It] <

0.16 GeV 2,

bpp = 29 G c V - 2 , bpp = 30 G e V -2 ..-*

]Ab[ = 1 GeV -2

B. Nicolescu / Odderon - past, present and future 145

In the asymmetrical case almost all experimental

quantities will feel the dynamical difference between the

pp and #p. I give some examples :

i) For a large class of Odderons, Aa --. coast, or

[Aa[ .--, oo, as s --* vo, i.e.

A a # 0, at TeV energies. (3)

Of course the most spectacular case is A a < 0, which

has crucial dynamical implications. One can note that

an asymptotic symmetry is still generally kept :

o P P I ~PP T I V T ""~ 1, a s 8 " + 0 0 . ( 4 )

ii) Similar finite-energy effects concern da/dt :

A ( ~ ( d a V " ( d a V ' k dt / = \ ~ 1 - k ~ / ¢ 0. (5)

One expects spectacular differences A(da/dt) at TeV

energies. It is important to note that the ISR results tl

at v ~ = 52.8 GeV are the first experimental indica-

tion in favour of (5). Let us also note the asymptotic

symmetry

(da/dt)PP ~ 1, as s ~ co, (6) (d~/dt)"

at least inside the diffraction peak.

Re F(s, t = O) PC*) = z , , , F ( 8 , t = O) '

i.e.

iii) Very interesting finite-energy effects concern

(~)

ppv--_ ReF+ + ReF_ ~v = ReF+-ReF- ImF++ImF_ and ImF+-ImF-" (8)

In the symmetrical case p --~ 0 as s ~ oo and f P = p~P

at finite high-el2ergies, while in the asymmetrical ease

p --* const, or even p --* co as s --* oo.

The case p --* 0o is, of course, uninteresting in

the light of present experiraental indications. Let us

therefore consider the ease p --* coast. (with different

constants for pp and l~p scattering). One thus expects

Ap = fep _ p~p # 0 at TeV energies. (9)

For example, if Re F_ < 0 it is seen from eq. (8) that

the Odderon pushes f up and f down, i.e. Ap >

0. This is precisely the cas~ corresponding to the UA4

data I. Moreover the phase of the F - amplitude directly

forces A a < 0 at these energies.

A pedagogical way in looking for all these effects

in connection with general principles is to consider the

toy model of Cornille 22, valid at t -- 0, 8 --* c~ :

F+(s) --* iC+, [ln(se-i=/2)] ~+ , (10)

- c _ , , (11)

where an overall scale factor is assumed.

Analytieity, unitarlty and positivity imply

/~+<2, /~_</~+/2+1, /~_<~++1. (12)

The (/~+, /3_) domain allowed by general principles

is shown in Fig. 1, where the behaviour of A a and

p in different regions is also indicated. The maximal

Odderon corresponds to the point (/3+ = 2, /~- = 2).

Different classes of other possible Odderons fl- < 2

can be directly explored on Fig. 1. I think that the

most interesting region from the point of view of QCD

would be proved in future to be the upper-right triangle

0 + < 2 , / L > I , /3+ > ~_).

|46 B. Nicolescu / Odderoa - past, present and [uture

2 L

t"

Fig. 1 Cornille's plot

4. C O M P A R I S O N W I T H E X P E R I M E N T A L

D A T A

Detailed considerations concerning the compari-

son of the Odderon approach with the experimental

da ta (including the Tevatron data published before this

Conference) were made elsewhere 12'l~ and there is no

need to repeat them hare.

I will just quote our general conclusion : the max-

imal Odderon (GLN, ~ approach is the only existing the-

oretical picture which fits all the present elastic pp and

#p data, including ~(d¢ /d t ) at vr~ -- 52.8 GeV and

f P at ~ -- 546 GeV. This conclusion is not changed

by the interesting Tevatron data presented at this Con-

~'erence by the E71023 and CDF 24 Collaborations, as

e~:plained below.

A first remark concerns the o" T Tevatron data. I

hope that it is obvious to everybody that, because A~

is expected to be small at these energies, the Tevatrou

CT data do not check at all the Odderon's presence or

absence : they check just the rate of increase of I m F+.

So the real question is : are the Tevatron da ta still com-

patible with a ]n 2 s growth of uT ? A negative answer

to this ql,.estion was given at this Conference 2s, but it is

in fact based on hidden assumptions and on a particu-

lar way of defining the procedure of fitting the data. It

therefore has no general validity. (Note also that a In s

fit leads to Regge intercepts 2s which have to be rejected

on physical grounds). In fact a In 2 s growth predicts

~T ~ 7 6 - 78 rob, the maximal Odderon case being

just a particular example of this class of models. So,

the Tevatron da ta presented at this Conference, even

if they look rather low, do not exclude a In 2 s growth

of ~rT. What they really exclude is the entire class of

threshold models (with one exception2e), which predict

large and even huge O" T at these energies. On an ex-

perimental level, it would be very important to clarify

the p - b - aT correlations, even if they would lead

to larger experimental error bars. In particular, the

t-dependence of the slope b(t) is crucial in getting ~T.

The experimental quantity which is more rele~xLt

for the Odderon case is p~P. We predicted f = 0.22

at V~ = 1.8 TeV. In Fig. 2 we show our prediction

together with the four p-data sets of the E710 Collabo-

ration (data read from graph 23) :

pl = 0.215 ~ 0.108, p2 = 0.120 4- 0.173,

p3 = 0.027 + 0.104, p4 = 0.200:1: 0.177. (13)

It is seen from Fig. 2 that our prediction is compat i~e

with sets 1, 2 and 4. The set 3 corresponds to very low

values of p, which are in fact much lower than those

predicted by dispersion relations (with or without the

Odderon). Moreover, the overall p-value 23, computed

B. Nicolescu / Odderon - past, present and future 147

0.40

0.32

0.24

0.16

0.08

-0.98

~'s = 1 .8 T e V

I I , I

! 2 3 4 Data .set number

Fig. 2

GLN prediction of pPP at v ~ = 1.8 TeV (dashed line) as compa,~d with experimental data 23. The domain covered by the UA4 data 1 at VG = 546 GeV is also shown.

under the assumption that the four sets are uncorre-

lated,

p = 0.126~0.067 (14)

is almost entirely determined by the sets 1 and 3, cor-

responding to almost incompatible p-data (see Fig. 2).

It is clear that I can not draw any definite con-

clusion concerning p : the Tevatron data are neither

against nor for the 0dderon approach. We have to

wait for the precise UA4/2 experiment planned 27 at

-- 546 GeV. One can hope that similar experiments

would be done in tile future at Tevatron energies.

Let me also mention our prediction for

(o~I~T)~ = 0.223, at v/a -- 1.8 TeV (15)

compared with the experimental value reported by the

E710 Collaboration at this Conference 23 :

(ael/~T)~£~ ---- 0.224 ~ 0,012. (16)

5. C O M P A R I S O N W I T H O T H E R T H E O R -

E T I C A L A P P R O A C H E S

I will consider very briefly just two approaches.

The Cheng and Wu appro'~,ch 2s is typical for the

symmetrical case. Having F - = 0 at high energies, it

145 B. Nicolescu / Odderon -past, present and future

does not fit neither A(d~/dt) at V~ ----- 52.8 GeV nor

p~' at Vf~ --- 546 GeV. However, it beautifully fits all

the other data.

The Donnachie-Landchoff (DL) approach 1s'2~,

like the maximal Odderon (GLN) approach, belongs to

the asymmetrical class of models.

There are a lot of similarities in detail between the

DL and GLN approaches. I will mention here only the

main similarities :

| ) Both approaches take into account all the pre-

sent data from ~/s ~ I0 GcV up to ~ ~- 1.8 TeV,

including the cosmic ray data. Our common ph'dosophy

is that all data have to constrain all the contributions.

(However, an elementary rule hss to be respected in the

GLN case : the asymptotic contributions do not have

to be comp|etely determined by the low-energy very

precise data).

In fact, the only one real difference is the maximal

Odderon, which is present in the GLN approach and is

absent in the DL approach. There are therefore two

ways of deciding between the DL and GLN approaches

: 1) see if mnltighon exchanges with C -- :k in QCD

lead to poles or to more complicated singularities; 2)

test predictions at LHC with both l~p and pp options

: there are dramatic differences between the DL and

GLN at ultra-high energies.

In fact, the crucial question behind an that is the

problem of the coupling of the Odderon in the forward

direction. There is now a quite general agreement about

the existence of the Odderon in the non-forward scat-

tering. However, there is not yet a general and rigorous

proof about the coupling of the Odderon at t = 0. For

the moment, we have to allow us to be guided by the

experimental data.

ii) Another important common feature is that

both DL and GLN approaches take the Re~:~e approach

(poles + cuts) as a valid dynamical basis.

iii) The 3rd important common feature is the

presence of the Odderon : a simple pole in the DL case

(allowing tofit A(dcr/dt) ~ 0 but not the UA4 p~v data)

and a more comp~cate singularity in the GLN case.

Concerning the differences between the two ap-

proaches, one can think that the DL approach has a

small number of free parameters, while the GLN ap-

proach has a much bigger number of free parameters.

This is not quite true. The number of Regge parameters

in the GLN case can be reduced precisely by using the

DL procedure. Also, the value~ of most of the param-

eters ~sociated with the asymptotic contributions are

highly constrained by the present data. Paradoxically,

there is very little freedom also in the GLN case.

6. P R E D I C T I O N S OF T H E O D D E R O N AP-

P R O A C H A T L H C E N E R G I E S

In the future, interesting results concerning the

Odderon approach are expected from UNK, Tevatron

and SSC machines. Of special interest are two of the

nine already presented proposals at RHIC. The first

one 3° concerns the pp measurement (in particular, f 'P)

at V~ -- 0.5 TeV. This would give a wonderful com-

plementary information to those of UA4/1 and UA4/2

measurements ofp ~p at practically the same energy : we

could directly check if Ap ~ 0. The second proposal al

concerns the measurement of cry? at several energie~ up

to v ~ = 0.5 TeV. These data would fill the huge

empty range in pp scattering and would therefore give

an invaluable information concerning the precise rate of

increase of ~T with increasing energies.

B. Nicolescu / Odderon - past ~r~,nt and [uCure 149

I have to stress that the Odderon effects are not

confined to pp and ~p interactions. Very interesting

Odderon effects are also expected in other reactions.

Important suggestions in this direction were made by

several speakers at this Conference 3s.

However, it must be realized that the only way to

have a crucial test of the hadron-hadron - antihadron-

hadron symmetry or asymmetry is to perform the ap-

propriate experiments at LHC with both pp and /~p

options.

16

12

-8

- 12

be', mb

t

RmIEt ~o~:c

. . . . . . . . . . . . , , i , , ,

I show in Figs. 3-5 the predictions of the maximal

Odderon (GLN) approach z2'z4,z7 at ultra-high energies,

as an illustrative case of a large class of Odderon models

: O'T in Fig. 3 , / ~ o in Fig. 4 and p in Fig. 5.

@T • rnb

200

I0(

ot 1

. . . . . . pp - - ~p

/ l / / /

. . . . . . . . ~ H I C

. . . . . . . . . . . . . . . . . . . . . ! ..... t . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 3

Fig. 4

GLN prediction of A ¢ at ultra-high energies

03

O.2

0.1

0

-2 I

1

. . . . . p p

/ / /

/ #

~M,c c~ic

.................... T.., ~ .......... I ........

10 10 ~ ~3 ~ ..~6 ,u ~% ,~-\

Fig. 5

GLN prediction of p at ultra-high energies

GLN prediction of G T at ultra-hlgh energies

The maximal Odderon predictions at LHC ener-

gies (other than those for d~r/dt) are summarized in

Table 2. One can see from Table 2 that very inter-

esting and measurable Odderon effects occur at LHC

energies, e.g. A~ _-- --6 mb and ~p = 0.2. In some

sense the Odderon effects shown in Table 2 are mini-

mal experimental effects. If one relaxes the constraints

coming from low energies, one expects even more copi-

ous experimental effects. For example, the value of A¢

can be as large as - 2 0 mb at LHC energies 33.

I also show in Fig. 6 our predictions for d~/dt at

LHC energies. We expect a spectacular finite-euergy

Odderon effect to occur at these energies : A(d~/dt)

is predicted to be large and positive in the range Itl "-"

0.2 - 0.7 GeV 2.

150 O. Nicolescu / Odderon -past, present and future

~'~ = 17 TeV

. . . . . . pp

10

0 05 10 1.5

Fig. 6

GLN predictions of the pp and ~p differential cross sec-

tions at the LHC energy v/~ = 17 TeV

7. CONCLUSIONS

1) The LHC with both pp and/~p options will open

the gate towards new physics, presumably intimately

related with C = - multigluon exchanges in QCD.

2) The LHC with both pp and pp options will offer

the unique and ultimate crucial test of the fundamental

theoretical problem o f ~ p - p p symmetry or asymmetry.

This crucial test could lead to a radical change in our

understanding of the annihilation mechanism in strong

interactions.

3) The LHC with both pp and ~p options is surely

of interest for other fields of physics, e.g. new physics

connected with the search for Z*'s.

4) The/~p option at LHC is fortunately inexpen-

sive. Our community would find itself in an ideal situ-

ation : able to do fundamental physics for low sums of

money.

R E F E R E N C E S

1. UA4 Collaboration, D. Bernard et al., Phys. Lett. B198 (1987) 583

2. B. Nicolescu, The Odderon today, in : High En- ~gy Hadronic Interactions, Proc. of the XXVth Rencontre de Morlond, ed. J. Tran Thanh Van (Fronti~res, Gif sur Yvette, 1990) pp. 421-430

3. E. Leader, Comparison of proton-proton and proton-antiproton scattering at very high ener- gies, in : Elastic and Diffractive Scattering at the Collider and beyond, Proc. of the 1st Blois Work- shop, Blois, France, eds. B. Nicolescu and J. Tran Tbanh Van (Fronti~res, Gif sur Yvette, 198,5) pp. 183-194

4. B. Nicolescu, The new UA4 pPP datum and the Odderon, in : Elastic and Diffractive Scattering, Proe. of the 2nd Blois Workshop, New York, USA, ed. K. Goulianos (Fronti~res, Gif sur Yvette, 1987) pp.133-144

5. P. Gauron, Oddcron description of the pp and ~p data, in : Elastic and Diffractive Scattering o the Interface of Soft and Hard Processes in QCD, Proc. of the 3rd Blois Workshop, Evanston, USA, edso M.M. Block and A.R. White, Nucl. Phys. B (Proc. Suppl.) 12 (1990) 80-89

6. L. Lulmszuk and B. Nicolescu, Nuovo Cimento Lett. 8 (1973) 405

7. K. Kang and B. Nicolescu, Phys. Rev. D l l (1975) 2461

8. D. Joynson, E. Leader, C. Lopez and B. Nicolescu, Nuovo Cimento 30A (1975) 345

9. J. Bartels, Nucl. Phys. B175 (1980) 365; J. Kwiecinski and M. Praszalowicz, Phys. Lett. B94 (1980) 413

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11. A. Breakstone et al., Phys. Rev. Lett. 54 (1985) 2180; S. Ethan et al., Phys. Lett. B152 (1985) 131

12. D. Bernard, P. Gauron and B. Nicolescu, Phys. Lett. B199 (1987) 125

B. Nicolescu / O / #eron - past, present and future 151

13. M.M. Islam, Europhys. Lett. 4 (1987) 183; M.M. Islam, V. Innocente and T. Fearnley, Euro- phys. Lett. 4 (1987) 189

14. P. Ganron, E. Leader and B. Nicolescu, Nucl. Phys. B299 (1988)640; Phys. Rev. Lett. 54 (1985) 2656; 55 (1985) 639

15. M. Giffon, ILS. Nahabetian and E. Predazzi, Phys. Lett. B205 (1988) 363

16. E710 Collaboration, N.A. Amos et al., Phys. Rev. Lett. 61 (1988) 525; 63 (1989) 2784; Phys. Lett. B243 (1990) 158; B247 (1990) 127; E710 Collaboration, results presented by S. Shulda, in : New Results in Hadronie Interactions, Proc. of the XXIVth Rencontre de Moriond, ed. J. Tran Thanh Van (Fronti~res, Gif sur Yvette, 1989) pp. 87-90

17. P. G~itron, E. Leader and B. Nicoiescu, Phys. Lett. B238 (1990) 406; P. Gauron and B. Nicolescu, Phys. Lett. B258 (1991) 482

18. L. Lipatov, Phys. Lett. B251 (1990) 284; for a review see L. Lipatov, in : Perturbative QCD, ed. A.H. Mueller (World Scientific, Sin- gapore, 1989)

19. P. Gattron, L. Lipatov and B. Nicolescu, Phys. Lett. B260 (1991) 407; L. Lipatov, talk at this Conference.

20. A. Martin, Elastic scattering and total cross- sections : present and future, Talk given at CERN-Dubna-Serpukhov Triangle Seminar, Dubna, USSR, October 1990, CERN-TH 5630/91 preprint

21. Proceedings of the ECFA Large Hadron Collider Workshop, Aachen, Federal Republic of Germany, 4-9 October 1990, vols. I-III, edited by G. Jarl- skog and D. Rein, CERN 9O-10/ECFA 90-133, 3 December 1990. See, in particular, the talks given by D. Denegri (vol. I, pp. 56-117) and E. Leader (vol. II, pp. 22-36)

22. H. Cornille, Thdor~mes asymptotiques et sections efflcaces totales croissantes, Comptes-Rendus du Colloque d'Aussois (Institut de Physique Nucldaire, Orsay, 1973) pp. II. 1-24; Nuovo Ci- mento 70A (1970) 165

23. Preliminary results on pPP, E710 Collaboration, presented by S. Shukla at this Conference

24. Preliminary results on O~T P, CDF Collaboration, presented by S. White at this Conforence

25. M.M. Block, talk at this Conference; M.M. Block and A.R. White NUHEP 152-91/ ANL-HEP-PR 91-38 preprint

26. K. Kang, talk at this Conference; K. Kang and A.R. White, ANL-HEP-PR 91-32/ BROWN-HET-805 preprint

27. Genoa-Palaiseau-Praha-Roma-Valencia Collabor- tion, D. Bernard et al., Minute of the CERN P,~- search Board CERN/DG/RB 90-156

28. H. Cheng and T.T. Wu, Phys. Rev. Lett. 24 (1970) 1456; C. Bourrcly, J. Softer and T.T. Wu, Phys. Lctt. B252 (1990) 287

29. P.V. Landshoff, talk at this Conference; for a short review see A. Donnachie and P.V. Landshoff, Particle World 2 (1991) 1

30. W. Guryn, S.Y. Lee, S. Majewski, M. Sakitt and A. Sambamur.'i, BNL-45162 preprint (August 1990), in : Proc. 4th Workshop for a Relativistic Heavy Ion Collider (BNL, Upton, NY, July 1990), to be published

31. CERN Courrier 31 (1991) 12

32. I.F. Ginzburg, B.V. StnLminsky and I.R. Zhinlt- sky, talks at this Conference

33. E. Leader, in Ref. 21.