Table Of ContentPajares Carlos, Shabelski Yuli Mechislavovich
Relativistic Nuclear Interactions
This book is devoted to the processes of elastic and inelastic hadron-nucleus inter
actions at moderate (- 1 Ge V) and high energies. After short di scussion of the main
properties of atomic nuclei we present the standard Glauber theory for elastic and quasi::
elastic hadron-nucleus and nucleus-nucleus scattering. Then the inelastic shadow correc
tions which appear at high energies are reviewed with several methods for their calcula
tion.
We consider the different absorptive parts of the elastic scattering amplitude by us
ing the unitarity condition to study the inelastic hadron-nucleus collisions. Several phe
nomenological models, including Intranuclear Cascade Model, Additive Quark Model,
and Quark-Gluon String Model are discussed in connection with the description of a
large range of experimental data on multiparticle production from nuclear targets. Thc
space-time picture of high energy interaction is given in detail.
Finally, the main features of high density parton matter, which is probably pro
duced in high energy heavy ion collisions, are considered. Topics as thermalization,
collective flow, parton saturation, colour glass condensate, energy loss, jet quenching,
quarkonium suppression, and percolation of colour sources arc introduced.
I13.1\BTCJlhCTBO "KOMKJrnTa». 117312, r. MOCKBB, rrp-T IlIcCTII.l\ccJITHJICTIUI OIcrJI5pJI, .1\. 9.
cJ)0pMaT 60x90116. TIeq. JI. 17,5. 3al<. N. 1242.
OrrreqaTIlHO B 000 «JlEHAH.ll».
117312. r. MoeKBa, rrp-T IlIecTHl\ccJITHJICTIDl OKTlI6p",.1\. IIA, C'rp. II.
ISBN 978-5-484-01015-8 © KomKniga, 2007
SCIENTIFIC LITERATURE
AND TEXTBOOKS 1,1
'~.""~_~'.N 53731061934
Our catalogua on tha Internet:
http://URSS.ru
Phone/fax: 7(495)135-42-16 111111111111111111111111111
Phone/fax: 7(495)135-42--46 URSS 9 785484 0101581
AU rights reserved. No part of this book may be used ur reproduced in any manner whatsoever
without written permission of the publisher.
Contents
Introduction ... 7
Part I
Elastic hA Scattering 10
Chapter 1. General Features of Interaction with Nuclei . . . . . . . . . . .. 11
1.1. Nuclear stTIlcture, protons, neutrons, meson currents ...... 11
1.2. Elastic eA scattering and nuclear form factors . . . . . . . . . . . . . . 12
1. 3. Hofstadter experiments and the geometry of the nucleus. . . . . . .. 13
1.4. Unitarity condition, optical model and hA cross sections. . . . . .. 15
1.5. Relativistic kinematics and Lobachevski geometry ... . ... . ... , 17
Chapter 2. Elastic and Quasi-elastic hA Scattering
at Intermediate Energies. . . . . . . . . . . . . . . . . . . . . . .. 19
2.1. Eikonal approximation and Glauber theolY . . . . . . . . . . . . . . . . 19
2.2. Elastic scattering on a deuteron target . . . . . . . . . . . . . . . . . .. 20
2.3. Elastic scatterings on heavy nuclei. . . . . . . . . . . . . . . . . . . . .. 22
2.4. Quasielastic scattering ..... . . . . . . . . . . . . . . . . . . . . . . .. 24
2.5. Experimental proton and neutron distributions in the nuclei . . . .. 25
2.6. Parameters of N N amplitude for bound target nucleon . . . . . . .. 28
Chapter 3. Hadron-Nucleon Collisions at High Energies. . . . . . . . . . . 29
3.1. Elastic hN scattering via Regge-poJe exchange .. .. . .. . ..... 29
3.2. The change of the space-time picture of interactions
at high energies ......................... 34
3.3. Diffractive dissociation in hN collisions. . . . . . . . . . 37
3.4. Inclusive reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
3.5. The AGK cutting TIlles for Reggeon diagrams ........ . . . . .. 44
Chapter 4. Elastic Hadron-Nucleus Scattering at High Energies. . . . . . 47
4.1. High energy elastic hadron-deuteron scattering . .... ...... .. 47
4.2. Deuteron disintegration in AGK cutting rules ..... . . . . . . . .. 51
4.3. Inelastic screening for heavy nuclei. The two-channel approach. .. 54
4.4. Different hA cross sections in the two-channel approach . . . . . .. 57
Contents
.\. S. Total cross sections of unstable particles . . . . . . . . . . . . . . . . .. 59
'1.6. Multi-channel approach for high energy hA collisions. . . . . . . . . 61
4.7. Color transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62
('hallter 5. Elastic and QU3.sielastic Scattering
in the Additive Quark Model .... . ..... " 64
5.1. Total hN cross sections . . . .. ... .. . 64
5.2. Elastic hN scattering .. . .. . ...... . 66
5.3. Vector meson photoproduction ...... . 67
SA. Diffraction dissociation of hadrons . . . . . . . . . . . . . 72
5.5. Shadowing corrections for hadron-nucleus collisions. . . . . . . . .. 75
5.6. Coherent production from nuclei . . . . .. 77
Part II
Multiple Hadron Production on Nuclear Targets 79
Chapter 6. Production of Particles in hA Collisions at Low Energy
and in the Kinematics l<brbidden for hN Interactions .. 80
6.1. Secondary production on nucleon target near tiU'eshoJd . ... . 80
6.2. Pion production on nuclear targets at low energies . .. ..... . 81
6.3. Subthreshold production. . . . . . . . . . . . . . . . . . . . . . . . . . .. 82
6A. Cumulative effect ... ... .. ..... .. . . .... . 86
Chapter 7. Production of Secondaries in the Intranuclear
Cascade Model and the Arising Problems . . . . . . . . . . . .. 90
7.1. General features of the model . . . . . . . . . . . . . . . . . . 90
7.2. Multiplicities of secondaries produced on nuclear targets . . . . . .. 92
7.3. The comparison of v A and v N inelastic collisions . . . . . . . . . .. 94
7 A. Formation time ....... . . . ......... .. .... . . . 95
7.5. Secondary production in QCD ......... ... .... . . 97
7.6. Hadron resonance production ... . .. . ........ . . . . . . . 98
7.7. Weak absorption of low-energy antiprotons produced
on nuclear target. . . . . . . . . . . . . . . . . . . . . . . . . 100
7.8. The possibility to account some interference contributions
in the cascade model ...... ...... .. .... . .. .. . 102
Chapter 8. Parton Model and Fan Reggeon Diagram Approach . . . . . . . 106
II. I. Space-time picture of secondary production . ...... .. ...... 106
11.2. Inelastic hadron-nucleus interactions in the Palton Model ...... 107
!U. Reggeon Fan Diagrams Approach for description
the inclusive spectra of secondaries . . . . . . . . . . . . . . . . . . . . . 111
Contents 5
Chapter 9. High Energy Hadron-Nucleus Collisions
in Additive Quark Model. . . . . . . . . . . 113
9.1. Spectator mechanism and its consequences for secondary
production on nuclear targets .. .. . . . ....... .. . . . 113
9.2. The A -dependence of the inclusive spectra
in the fragmentation region ...... . .. .. ... .. ... . Jl3
9.3. Comparison with the data . ..... . . .. .. .. ...... . 117
9.4. Hadron content of secondaries in the fragmentation region ... .. . 121
Chapter 10. Multiple Scattering Theory for Secondary
Production in hA Collisions . . . . . . . . . . . . . 124
10.1. Non-coherent elastic scattering on nuclear target . .. . 124
10.2. The multiparticle absorptive parts .... . .. ..... .. . . 127
10.3. Self-shadowing effects . . . . . . . . . . . . . . 129
10.4. Energy division in multiple interaction .... 130
10.5. The problem of planar/non-planar diagrams 132
Chapter 1l. Phenomenological Predictions of the Multiple
Scattering Theory . . . . . . . . . . . . . . . . . . . . 137
11.1. Multiplicities and spectra of secondaries ..... ... . . 137
11.2. Cross sections of double scattering in inelastic
hadron-deuteron collisions .... . .. . ...... . . . .. . ..... . 142
11.3. Multinucleon interactions of incident hadron on nuclear target . .. 146
Chapter 12. Particle Production in the String Model . . . . . . . . . . . . . . 149
12.1. The motion of a relativistic string ... . .. . .. . ............ 149
12.2. The decay kinematics of the string .. .. ... ... ... . . . . . ... 151
12.3. Stochastic iterative fragmentation of the string .. . ....... .... 153
12.4. The Schwinger vacuum decay mechanism. . . . . . . . . . . 156
12.5. Dipole formalism . ..... . ......... . .. .... . .. . ... . . 157
Chapter 13. Quark-Gluon String Model for Multiparticle Production . .. . 161
13.1. General formalism . . .. . . ...... . ... . ...... . ... . ... 161
13.2. Quark and diquark distributions in hadrons in QGSM ........ . 164
13.3. Quark and diquark fragmentation runctions ................ 168
13.4. Comparison of QGSM predici ions with experimenlal data
for meson produclion from nucleon targd ....... . 172
13.5. Secondary baryon/antibaryon production .. . ....... . 176
13.6. Baryon diffusion 10 large distances in rapidily space 179
Contents
('hailler 14. Spectra of Secondaries on Nuclear Targets ... .... . ... . 184
111.1. Inell! ive spectra on nucleartarget in QGSM . .... ... . 184
1".2. omparison wit.h experimental data .. ............ . 185
14.3. Heavy flavour hadron production ..................... . 192
14.4. Parton distributions in the bound nucleons . .. . .. ... . 192
14.5. A-dependence of hard processes on nuclear targets
with accounting of nuclear shadowing .. ... ...... . . .. .... 196
14.6. Predictions for superhigh energies and violation
of Feynman scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
14.7. Passage of cosmic rays through the atmosphere in the QGSM .. .. 204
14.8. Increase of transverse momenta of secondaries
and its contribution to Feynman scaling violation . . . . . . . . . . . . 208
Chapter 15. Cross Sections and Probabilities in Heavy Ion Collisions . ... 211
15.1. Amplitude of elastic nucleus-nucleus scattering . . . . . . . 211
15.2. Monte Carlo simulation of nucleus-nucleus scattering . . . 215
15.3. Integrated cross sections. . . . . . . . . . . . . . . . . . . . 217
15.4. Disuibutions on the number of interacting nucleons . . . . 218
15.5. Multiplicity dist.libut.ions and dispersions . . ....... . .. . . . 220
Chapter 16. High Density Matter .............. .. .......... 225
16.1. Possibility of new physics in heavy ion collisions . . . .... . ... .. 225
16.2. Thermalization and Collective Flow ..... . . ... .. . ... . . 229
16.3. Parton saturation and Color Glass Condensate .............. 231
16.4. Jet quenching. . . . . . . . . . . . . . . . .. . ..... . . .. . . ... 242
J 6.5. Percolation of color sources. . . . . . . . . . . . . . . . . . . . . . . . 247
16.6. Charmonium supression . . . . . . . .. . . ..... .. . . . 254
Bibliography . . . . . . . . . . . . . . . . . . 256
Introduction
The subject of relativistic nuclear physics lies between classical nuclear
physics and high energy physics of elementary particles. There exists a natural
boundary between the classical and relativistic nuclear physics. The first one
considers mainly the nuclear structure and the nuclear excitations. In the case of
relativistic nuclear physics we often deal with quasi-free nucleons, and sometimes
it is possible to neglect their Femli-motion. Moreover, in many cases the internal
(quark-gluon) structure of the nucleons should be taken into account. Of course,
there are problems where the classical and relativistic nuclear physics overlap.
In the present book we will consider mainly the problems of relativistic nuclear
physics.
We wiU consider the energy boundary between classical and relativistic
nuclear physics as the energy where the production of at least one pion in the
considered hA reaction is possible.
However, the energy region of relativistic nuclear physics can also be divided
into two parts. The region of "intermediate energies" is that of energies where the
average multiplicity of the produced pings is rather small. The most important fact
is that in secondary production processes the momenta transferred to the nucleon
target are large compared to the Fermi motion momenta of nucleons. The region
of "high energies" is related to "Pomeron physics", where the multiplicities of
secondaries are large, and the diffractive production of secondaries with very
small momentum transferred to the target nucleon is possible. In the high
energy region a number of new effects appear; they will be considered later.
Relativistic nuclear physics is important in many respects in both energy
regions. First of all, we have an additional variable - the atomic weight of
the target that gives new possibilities for testing the models of multi particle
production. The size of a nuclear target is rather large in comparison with
QeD length scale AQ~D' Thus, we have here an interaction with an extended
target, and we can change its effective length by changing the atomic weight of
the target. There are also ideas about a possible new state or the matter, the
quark-gluon plasma, colour glass condensate, or another type of high density
parton matter, which may be produced in hcavy ion collisions. High energy
physics experimental data are obtained velY often on nllcle(lr targets; in these
cases we have to understand the possible nuclea I' correct ions. M ore examples,
both fundamental and applied, can be provided.
There exist many original papers and review on relat ivistic nuclear physics
which will be quoted in the course or our discussions. However. most of these
papers and reviews discuss only some specific problems. anc! sornl~ of the con
sidered approaches are in seriOlls disagreement with other ones. One of the most
Introduction
illl()llIlalivc book which discusses many phenomenological problems together
witil 'xpcrimental data is [1]. This book is, however, thirty years old, so many
l't' , 'nt qucstions are just not considered there. Another problem is that the main
part or II J is devoted to the cascade model and its application to the description
or the daLa. But there exist nowadays a lot of data which are in serious contra
diction with this model. These problems will be considered in detail mainly in
Pali 2 of our book.
The very useful book [2] considers mainly heavy ion collisions. Among
Ollier books and reviews for introducing and developing the discussed problems
we can suggest [3]-[7]. Many experimental data (mainly conceming multiple
secondary production on nuclear targets) can be found in [81.
The main idea of the present book is to give a more or less complete
analysis of the situation in relativistic nuclear physics without the discussion of
very special or purely mathematical details, So this book should be considered
as an introduction to the problem. However we cannot suggest it as a textbook
for students. We would like to recommend it, say, for post-graduate students and
young scientists who want to have some theoretical overview on the problem. So,
in many cases the results are presented without the formal detailed derivation.
More details can be found in the recommended References.
Most of the discussions are based on the so-called dispersion approach
which, foUowing from the unitarity condition, allows one to consider both the
processes of elastic scattering and of multiparticle production from the same
point of view.
We present rather simple picture which can explain a lot of experimental data
concerning mainly the high energy strong interactions with nuclear targets. This
pict.ure is based on the Multiple Scattering Theory with including the inelastic
screening effects, when it is necessary. Of course, the numerical predictions
needs, as a rule, some additional models and assumptions, however very often
all these assumptions and model parameters can be found from the interactions
with nucleons.
It is necessary to note that the understanding of secondary production
from nuclear targets has not only pure scientific importance. There exist several
practical projects which need the calculations of such processes. Among them
we can say about the neutrino exploration of the Earth for purposes of geological
research [9], muon catalysis for energy production by nuclear fusion [10] and
creation of high energy (14 MeV) and high intensity neutron beams (mainly for
material sciences) [11],
We assume that the reader is provided with the knowledge on quantum
mechanics and the basic topics of relativistic quantum theory and elementalY
particle physics. So we give just a very Sh0l1 introduction to these problems.
Let us outline the structure of this book. After discussing the general features
or
I he interactions with nuclei at high energies, we consider in more detail the
I110s1 important informations concerning elastic hadron-nucleus (itA) ~l,;ull\:J iIlg~
at intermediate and high energies; in the latter case some new cm::t'ib:::ic:-1:' {c. g.
1I1\:lilstic screening, coherent production, etc.) appear.
Introduction 9
One of the central topics of our discussions is multi particle production
in such processes. We consider different approaches, and show that some of
them are in contradiction with each other and with the description of the
experimental data. We demonstrate that the most logical way is to constrain the
multiple production reactions, in close connection with elastic processes, with
the help of unitarity conditions by using the technique of eigenstates and/or of
dispersi.on integrals. Again, we consider here both the regions of intermediate
and high energies.
In the last two Section of the book we discuss very shortly some well
established results for high energy heavy ion collisions coming from the Multiple
Scattering Theory and the effects whkh appear in the high density matter.
High energy heavy ion physics is developed now very fast, so, for our opinion,
a manuscript on this subject will be not complete at the moment of its printing.
Hadron-nucleus collisions were a fashionable subject mainly during the
seventies and eighties. So the reader should not be surprised that many references
and experimental data are rather old. Another reason for this is that we wanted
to recall, at least partly, the history of relativistic nuclear physics. The main
results in hadron-nucleus collisions are now more or less understandable, so it
is time to give a general view on the problem. Recently the most interesting
results in relativistic nuclear physics come from the heavy ion collisions, due to
the experimental programs at the CERN-SPS, at the BNL-RHIC and at the
forthcoming CERN-LHC.
Unfortunately, mainly due to the limited volume of the book we cannot
discuss many fields and ideas of the relativistic nuclear physics, for example such
as interplay between soft and hard particle production, hydrodynamical, kinetic
and transport equation approaches, etc.
Acknowledgments
We are indebted to N. Annesto and J. Nyiri for reading the manuscript and
many comments.
We are grateful to V. V. Anisovich, Ya. 1. Azimov, K. G. Boreskov, M. A. Braun,
V. M. Braun, A. Capella, L. G. Dakhno, J. Dias de Deus, V. Llsakov, A. B. Kaidalov,
O. V Kancheli, M. N. Kobrynsky, B. Z. KopeJiovich, E. M. Levin, L. N. Lipatov,
N. N. Nikolaev, M. G. Ryskin and AA Vorobyov.
We would like to clime that the background of our book i. based on the
ideas of V N. Gribov, V M. Shekhter and K. A. Ter-Martirosyan.
PART I
ELASTIC hA SCATTERING
his part is, in fact, an introduction to the second one. First of all, we present
~()lllC general informations about hadron-nucleus interactions. Many things here
are rather old, but practically aU new experimental data confirm them.
We recall the classical Glauber theory [12]-[15] for elastic and quasi-elastic
hadron-nucleus scattering. This theory can be also used for the case of very high
'l1crgies. The space-time picture of the interaction changes drastically, howeveT
Ihe Glauber low-energy theory can be rewritten by using the dispersion diagrams
in the framework of Gribov's Reggeon Field Theory [16]. Only some new (and
nUl11crically smaU) corrections appear this way.
We formulate our main approach based on the unitarity condition and
dispersion diagrams which allow us to consider clastic and quasi-elastic hadron
nucleus scattering as weU as multiparticle production processes, see Part II, from
t he same point of view. A rather unusual tlick is that in many places we consider
scries of diagrams the sum of which gives zero contribution to the interaction
cross section. However, these considerations provide a possibility to,estimate
Lhe contributions of some very important diagrams which can not be calculated
directly.
We present results for the high energy region both in the Gribov-Glauber
Multiple Scattering Theory and in the framework of the interaction eigenstate
melhod, and we demonstrate that they are equivalent. We show that virtually all
I hc important color transparency problems can be reduced to the multichannel
eigenstate problem.
The Additive Quark Model (AQM) is used for the illustration of its predictive
power for interactions with nuclear targets; one can see that its'results are in
qualitat.ive agreement with the data and quantitatively they are/as a-rule, within
Ihe standard 15..;-20 % accuracy. Probably, it is the accuracy of assumption
rv
thal a hadron consists from two (meson) or three (baryon) spatially separated
constituent quarks. More details about AQM and its application to high energy
interactions can be found in [17].
