Proceedings of the 12th Lomonosov Conference on Elementary Particle Physics
PARTICLE PHYSICS at the Year of 250th Anniversary of Moscow University Alexander I Studenikin
PARTICLE PHYSICS at the Year of 250th Anniversary of Moscow University
Faculty of Physics of Moscow State University
JPINTERREGIONAL CENTRE J I L F Q R ADVANCED STUDIES
Proceedings of the 12th Lomonosov Conference on Elementary Particle Physics
PARTICLE PHYSICS at the Year of 250th Anniversary of Moscow University Moscow, Russia
25 - 31 August 2005
Editor
Alexander I Studenikin Department of Theoretical Physics Moscow State University, Russia
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PARTICLE PHYSICS AT THE YEAR OF 250TH ANNIVERSARY OF MOSCOW UNIVERSITY Proceedings of the 12th Lomonosov Conference on Elementary Particle Physics Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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Moscow State University Faculty of Physics Interregional Centre for Advanced Studies Dedicated to the ?.50th Anniversary of Moscow State University
TWELFTH LOMONOSOV CONFERENCE ON ELEM ENTARY PART CLE PHYS CS
Moscow, August 25-31, 2005
Mikhail Lomonosov 1711-1765
Sponsors Russian Foundation for Basic Research Russian Agency for Science and Innovation Russian Academy of Sciences Russian Agency for Atomic Energy Supporting Institutions Faculty of Physics of Moscow State University Skobeltsyn Institute of Nuclear Physics, Moscow State University Interregional Centre for Advanced Studies Joint Institute for Nuclear Research (Dubna) Institute of Theoretical and Experimental Physics (Moscow) Institute for Nuclear Research (Moscow) Budker Institute of Nuclear Physics (Novosibirsk) v
VI
International Advisory Committee E.Akhmedov (ICTP, Trieste & Kurchatov Inst.,Moscow), S.Belayev (Kurchatov Inst.,Moscow), V.Berezinsky (LNGS, Gran Sasso), S.Bilenky (JINR, Dubna), J.BIeimaier (Princeton), M.Danilov (ITEP, Moscow), GDiambrini-Palazzi (Univ. of Rome), A.Dolgov (INFN, Ferrara & ITEP, Moscow), V.Kadyshevsky (JINR, Dubna), S.Kapitza (EAPS, Moscow) A.Logunov (IHEP, Protvino), V.Matveev (INR, Moscow), P.Nowosad (Univ. of Sao Paulo), L.Okun (ITEP, Moscow), V.Rubakov (INR, Moscow), D.Shirkov (JINR, Dubna), J.Silk (Univ. of Oxford), A.Skrinsky (INP, Novosibirsk), A.SIavnov (MSU & Steklov Math.lnst, Moscow) A.Smirnov (ICTP, Trieste & INR, Moscow), PSpillantini (INFN, Florence), Organizing Committee V.Bagrov (Tomsk State Univ.), V.Belokurov (MSU), V.Braginsky (MSU), A.Egorov (ICAS, Moscow), D.Galtsov (MSU), A.Grigoriev (MSU & ICAS, Moscow), P.Kashkarov (MSU), A.Kataev (INR, Moscow), O.Khrustalev (MSU), V.Mikhailin (MSU & ICAS, Moscow) A.Mourao (IST/CENTRA, Lisbon), N.Narozhny (MEPHI, Moscow), A.Nikishov (Lebedev Physical Inst., Moscow), N.Nikiforova (MSU), V.Ritus (Lebedev Physical Inst., Moscow), Yu.Popov (MSU) , VSavrin (MSU), D.Shirkov (JINR, Dubna), Yu.Simonov (ITEP, Moscow), A.Sissakian (JINR.Dubna), A.Studenikin (MSU & ICAS, Moscow), VTrukhin (MSU)
Moscow State University Interregional Centre for Advanced Studies Centre of International Educational Projects Ministry of Education and Science of Russia
SIXTH INTERNA IONAL MEET NG ON PROBLEMS OF INTELLIGENTSIA "INTELLIGENTSIA and VIOLENCE: Responses to Repression and Terrorism" Moscow, August 31, 2005
Presidium of the Meeting V.A.Sadovnichy (MSU) - Chairman V.V.Belokurov (MSU) J.BIeimaier (Princeton) GDiambrini-Palazzi (Universiry of Rome) V.GKadyshevsky (JINR) S.P.Kapitza (Russian Academy of Sciensies) N.S.Khrustaleva (Ministry of Education and Science, Russia) A.I.Studenikin (MSU & ICAS) - Vice Chairman V.l.Trukhin (MSU) VII
FOREWORD The 12th Lomonosov Conference on Elementary Particle Physics was held at the Moscow State University (Moscow, Russia) on August 25-31, 2005. The conference was dedicated to the 250th Anniversary of the Moscow State University. The conference was organized by the Faculty of Physics of the Moscow State University and the Interregional Centre for Advanced Studies and supported by the Joint Institute for Nuclear Research (Dubna), the Institute of Theoretical and Experimental Physics (Moscow), the Institute for Nuclear Research (Moscow), the Budker Institute of Nuclear Physics (Novosibirsk) and the Skobeltsyn Institute of Nuclear Physics (Moscow State University). The Russian Foundation for Basic Research, the Russian Agency for Science and Innovation, the Russian Academy of Sciences and the Russian Agency for Atomic Energy sponsored the conference. It was more than twenty years ago when the first of the series of conferences (from 1993 called the "Lomonosov Conferences"), was held at the Department of Theoretical Physics of the Moscow State University (June 1983, Moscow). The second conference was held in Kishinev, Republic of Moldavia, USSR (May 1985). After the four years break this series was resumed on a new conceptual basis for the conference programme focus. During the preparation of the third conference (that was held in Maykop, Russia, 1989) a desire to broaden the programme to include more general issues in particle physics became apparent. During the conference of the year 1992 held in Yaroslavl it was proposed by myself and approved by numerous participants that these irregularly held meetings should be transformed into regular events under the title "Lomonosov Conferences on Elementary Particle Physics". Since then at subsequent meetings of this series a wide variety of interesting things both in theory and experiment of particle physics, field theory, astrophysics, gravitation and cosmology were included into the programmes. It was also decided to enlarge the number of institutions that would take part in preparation of future conferences. Mikhail Lomonosov (1711-1765), a brilliant Russian encyclopaedias of the era of the Russian Empress Catherine the 2nd, was world renowned for his distinguished contributions in the fields of science and art. He also helped establish the high school educational system in Russia. The Moscow State University was founded in 1755 based on his plan and initiative, and the University now bears the name of Lomonosov. The 6th Lomonosov Conference on Elementary Particle Physics (1993) and all of the subsequent conferences of this series were held at the Moscow State University on each of the odd years. Publication of the volume "Particle Physics, Gauge Fields and Astrophysics" containing articles written on the basis of presentations at the 5th and 6th Lomonosov Conferences was supported by the Accademia Nazionale dei Lincei (Rome, 1994). Proceedings of the 7th and 8th Lomonosov Conference (entitled "Problems of Fundamental Physics" and "Elementary Particle Physics") were published by the Interregional Centre for Advanced Studies (Moscow, 1997 and 1999). Proceedings of the 9th, 10th and 11th Lomonosov Conferences (entitled "Particle Physics at the Start of the New IX
X
Millennium", "Frontiers of Particle Physics" and "Particle Physics in Laboratory, Space and Universe") were published by World Scientific Publishing Co. (Singapore) in 2001, 2003 and 2005, correspondently. The physics programme of the 12 Lomonosov Conference on Elementary Particle Physics (August, 2005) included review and original talks on wide range of items such as neutrino and astroparticle physics, electroweak theory, fundamental symmetries, tests of standard model and beyond, heavy quark physics, nonperturbative QCD, quantum gravity effects, physics at the future accelerators. Totally there were more than 300 participants with 107 talks including 34 plenary (30 min) talks, 38 session (25-20 min) talks and 35 brief (15 min) reports. One of the goals of the conference was to bring together scientists, both theoreticians and experimentalists, working in different fields, so that no parallel sessions were organized at the conference. Following the tradition that has started in 1995, each of the Lomonosov Conferences on particle physics has been accompanied by a conference on problems of intellectuals. The 6* International Meeting on Problems of Intelligentsia held during the 12th Lomonosov Conference (August 31, 2003) was dedicated to discussions on the issue "Intelligentsia and Violence: Responses to Repression and Terrorism ". Three papers on this subject are included into this volume. The Round table discussion on "Neutrino Physics and Astrophysics" was held during the last day of the 12th Lomonosov Conference. The main results of the of the Round Table discussion were summarized in the Memorandum on "Neutrino and Astroparticle Physics" approved by the participants of the conference: "The progress in particle physics comes from both the high energy frontier and precision experiments. This applies to accelerator and non-accelerator physics. In the last years, field like neutrino physics, astroparticle physics and cosmology have had an spectacular development. One may anticipate that these clues to the knowledge of nature will develop more along the XXI century, particular when taking into account the longer time periods involved in the construction of higher energy facilities. There has been important progress in neutrino and astroparticle physics achieved during the last several years: • The non-vanishing neutrino mass and flavour violation has been observed in neutrino oscillation experiments. • The validity of the Standard Solar Model has also been proven. • The non-zero neutrino mass can have an important impact on cosmology, in particular, for our understanding of the baryon asymmetry of the universe. On the other hand, the upper boundary of the sum of three neutrino masses can now be constrained on the level of the order of 1 eV from cosmology. • Observations of tritium beta-decay have lowered the neutrino upper mass limit to the level of 2.1 eV. • Double beta decay experiments have reached a sensitivity ~ (0.5 - 1) eV for effective Majorana mass of the neutrino.
XI
World-wide recognition of the obtained results has been evidenced by two Nobel Prizes which have been recently awarded for research in neutrino and astroparticle physics. Further progress in the study of the fundamental properties of neutrinos will open the window to a new physics. Application of these studies could also play a very important role in our understanding of the inner structure of stellar cores as well as of the early stages of evolution of the universe. Studies of geo neutrinos also open promising possibilities for the future. More accurate measurements of neutrino characteristics will make further progress in the field possible. Our conference also focused on the need to train specialized manpower in this field for the future". The success of the 12* Lomonosov Conference was due in a large part to contributions of the International Advisory Committee and Organizing Committee. On behalf of these Committees I would like to warmly thank the session chairpersons, the speakers and all of the participants of the 12th Lomonosov Conference and the 6th International Meeting on Problems of Intelligentsia. We are grateful to the Rector of the Moscow State University, Victor Sadovnichy, the Vice Rector of the Moscow State University, Vladimir Belokurov, the Dean of the Faculty of Physics, Vladimir Trukhin, the Director of the Skobeltsyn Institute of Nuclear Physics, Mikhail Panasyuk, the Directors of the Joint Institute for Nuclear Research, Vladimir Kadyshevsky and Alexey Sissakian, the Director of the Institute for Nuclear Research, Victor Matveev, the Director of the Budker Institute of Nuclear Physics, Alexander Skrinsky, the Vice Director of the Institute of Theoretical and Experimental Physics, Mikhael Danilov, and the Vice Deans of the Faculty of Physics of the Moscow State University, Anatoly Kozar and Pavel Kashkarov for the support in organizing these two conferences. Special thanks are due to Inna Bilenkina and Alexander Suvorinov (the Russian Agency for Science and Innovations), Nelli Khrustaleva (the Russian Agency of Education), Boris F. Myasoedov (the Russian Academy of Sciences) and Oleg Patarakin (the Russian Agency for Atomic Energy) for their valuable help. I would like to thank Concezio Bozzi, Alexander Dolgov, Andrey Kataev, Catherine Leluc, Lev Okun, Alexander Olshevsky and Alexey Smirnov for their help in planning of the scientific programme of the conference and inviting speakers for the topical sessions of the meeting. Furthermore, I am very pleased to mention Alexander Grigoriev, the Scientific Secretary of the conference, Andrey Egorov, Artyem Ivanov, Sergey Shinkevich, and Marina Mescheraykova, Maria Moiseeva, Olya Moiseeva, Nastay Sutormina for their very efficient work in preparing and running the meeting. These Proceedings were prepared for publication at the Interregional Centre for Advanced Studies with support by the Russian Foundation for Basic Research, the Russian Agencies for Science and Innovations and Education, the Russian Academy of Sciences and the Russian Agency for Atomic Energy. Alexander Studenikin
CONTENTS
Twelfth Lomonosov Conference on Elementary Particle Physics Sponsors and Committees
v
Sixth International Meeting on Problems of Intelligentsia - Presidium
vii
Foreword
ix
World Year of Physics 2005 The Concept of Mass in the Einstein Year
1
L.B. Okun Neutrino Physics Accelerator Neutrino Experiments T. Kobayashi Searching for Neutrino Oscillations with OPERA N. Sawinov Reactor Neutrinos and KamLAND J. Shirai
16 24 29
Double Beta Decay Experiments A. Barabash
37
Spontaneous and Induced Two-Beta Processes S. Semenov, Yu. Gaponov, F. Simkovic, V. Khruschov
45
Neutrinoless Double Beta Decay in Theories Beyond the Standard Model: Electron Angular Distributions A. Ali, A. Borisov, D. Zhuridov
50
Search for Phenomena Outside the Standard Model with Prototype of the Borexino Detector A. Derbin, O. Smirnov
54
Cross-Section Measurements in the NOMAD Experiment R. Petti
59
Solar Neutrinos: Spin Flavour Precession and LMA J. Pulido, R. Raghavan, B. Chauhan
64
XIII
XIV
Neutrino Spin-Flavor Oscillations in Rapidly Varying Magnetic Fields M. Dvornikov
69
Spin Light of Electron in Matter A. Grigoriev, S. Shinkevich, A. Studenikin, A. Ternov, I. Trofimov
73
Neutrino Physics, BBN, LSS and CMBR
78
A. Dolgov Astroparticle Physics and Cosmology Dip in UHECR Spectrum as Signature of Proton Interactions with CMB V. Berezinsky The Science of PAMELA Space Mission P. Picozza, A. Morselli
87 96
Current Status and Prospects of the AMS Experiment D. Rapin
104
Astroparticle Physics with AMS-02: the Quest of Antimatter C. Sbarra
112
Dark Matter Investigations R. Bernabei, P. Belli, F. Montecchia, F. Nozzoli, F. Cappella, A. Incicchitti, D. Prosperi, R. Cerulli, C.J. Dai, H.L. He, H.H. Kuang, J.M. Ma, Z.P. Ye
120
Mirror Dark Matter R. Volkas
130
Indirect Dark Matter Search V. Zhukov
138
Dark Energy and Black Holes E. Babichev, V. Dokuchaev, Yu. Eroshenko
143
Microlensing with the Radioastron Space Telescope A. Zakharov
147
Search for Gravitational Waves by LIGO Scientific Collaboration V. Mitrofanov
152
Quantum Gravity as Twistorial Unification of Quantum and Gravity A. Burinskii
159
Quantum Cosmology and the Global Rotation Problem M. Fil'chenkov
163
XV
Self-Interaction of Charged Particles Outside Brane Topological Defects Yu. Grats, V. Dmitriev
167
Physical Degrees of Freedom in a Stabilized Randall-Sundrum Model E. Boos, Yu. Mikhailov, M.N. Smolyakov, I. Volobuev
171
Gravitational Energy-Momentum Tensors According to Belinfante and Rosenfeld A. Nikishov
175
Stable Matter of 4th Generation: Hidden in the Universe and Close to Detection? K. Belotsky, M. Khlopov, K. Shibaev
180
Small-Scale Fluctuations of Extensive Air Showers as the Origin of Energy Estimation Systematics G. Rubtsov
185
CP Violation and Rare Decays o Status of E39la Experiment for the Rare Decay K T. lnagaki
o —> n vv
189
On CP Effects Generated by Electroweak Penguin Diagrams in Non-Leptonic A: Decays E. Shabalin
197
CP Violation in Kr->n0K°KL G. Faldt
201
Study of K~ - » 7T°e~vey and K~ -> KQ/J.~v^y Decay with ISTRA+ Setup
206
V. Bolotov, E. Guschin, V. Duk, S. Laptev, V. Lebedev, A. Mazurov, A. Polyarush V. Postoev, S. Akimenko, G. Britvich, K. Datsko, A. Filin, A. Inyakin, V. Konstantinov, A. Konstantinov, I. Korolkov, V. Khmelnikov, V. Leontiev, V. Novikov, V. Obraztsov, V. Polyakov, V. Romanovsky, V. Shelikhov, O. Tchikilev, V. Uvarov, O. Yushchenko Rare Semileptonic Meson Decays in R-Parity Violating MSSM
215
A. Ali, A. Borisov, M. Sidorova Hadron Physics Next to Leading Order in Semi-Inclusive Deep Inelastic Scattering Processes A. Sissakian, O. Shevchenko, O. Ivanov
219
Bd(Bd)-±p±7t+,p+p~,n+n~~ M. Vysotsky
224
: Hunting for Alpha
XVI
Deep Inelastic Spin Structure Functions at Small x B. Ermolaev, M. Greco, S. Troyan
232
B° - B° Mixing A. Pivovarov
236
Mass Spectra of Radially and Orbitally Excited States of Mesons V. Khruschov, V. Savrin, S. Semenov
240
Inclusive ©+ and A(1520) Production in Hadron Collisions at High Energy /. Narodetskii, M. Trusov
244
Testing the ^j-Factorisation Approach at the LHC in Quarkonium Production Processes S. Baranov
249
On the Relation Between x-Dependence of the Higher Twist Contribution to the Structure Functions F3 and gf - g" A. Sidorov
253
Production of Heavy Baryons S. Baranov, V. Slad
257
Self-Energy of Kaons in Pion Matter M. Krivoruchenko
261
Structure Functions Result in CHORUS Experiment M. Serin
266
New Relations Between the QCD Sum Rules for Meson - Baryon Couplings T. Aliev, A. Ozpineci, S. Yakovlev, V. Zamiralov
270
Analytical Approach to Constructing Effective Hadron-Hadron Interaction Operators and its Application to Nucleon-Nucleon Scattering at Low and Intermediate Energies A. Safronov
274
Physics at Accelerators and Studies in SM and Beyond Top Quark Physics E. Boos
278
Precision Measurement of the Top Quark Mass From MM Distribution in t-^bbi Decays M. Nekrasov
286
Review of CKM Results from BaBar E. Rosenberg
290
XVII
Lepton Flavor Violation in T Decays at BaBar S. Banerjee
298
ISR Physics at BaBar V. Druzhinin
303
Semileptonic B Decays at BaBar V. Azzolini
308
Dalitz Plot Analysis of D° - » K°K+K~ M. Pappagallo
and D^ States at BaBar
313
Rare Muonic B-Decays at Atlas K. Toms, N. Nikitine, S. Sivoklokov, L. Smirnova, D. Tlisov
318
Z-Scaling and Strange High-pr Particle Production in p-p Collisions at RHIC M. Tokarev
326
Grid for Exploration of High Energy Physics: RDIG and the EGEE/LCG Projects
330
V. Ilyin, A. Kryukov, A. Demichev New Developments in Quantum Field Theory Predictions for the Muon g-2 M. Passera The Comments on QED Contributions to (G-2)^ A. Kataev
338
Localization of Scalar and Fermionic Eigenmodes in SU(2) Lattice Gauge Theory M. Polikarpov, F. Gubarev, S. Morozov, S. Syritsyn, V. Zakharov
350
Resummation of Large Logarithms within the Method of Effective Charges C. Maxwell
358
Generalized Dual Symmetry of Nonabelian Theories, Monopoles and Dyons C. Das, L. Laperashvili, H. Nielsen
363
Summation of Feynman Diagrams in N=l Supersymmetric Electrodynamics K. Stepanyantz
367
Spectrum of Higgs Particles in the Exceptional Supersymmetric Standard Model S. King, S. Moretti, R. Nevzorov
371
Quantization of Nonlinear Fields on Classical Background M. Chichikina
376
345
XVIII
Collapse of Positronium and Vacuum Instability A. Shabad On Influence of Intensive Stationary Electromagnetic Field on the Behavior of Fermionic Systems V. Rodionov, A. Mandel, E. Arbuzova Radiative Effects in the Standard Model Extension V. Zhukovsky, A. Lobanov, E. Murchikova Two-Frequency Undulator and Harmonic Generation of an Ultrarelativistic Electron V. Mikhailin, K. Zhukovsky
380
385
389
393
Influence of the Electron Spin on the Angular Distribution of the Photoelectrons /. Andrievskiy, V. Tlyachev
398
From Radiation Theory to the Dynamics of Relativistic Charged Particles V. Bordovitsyn, T. Pozdeeva
402
On the Wave Zone of Uniformly Accelerated Charge V. Bordovitsyn, B. Bulenok, T. Pozdeeva
406
Dynamical Methods of Investigation in Application to Quantum Systems Yu. Rylov
410
Some Aspects of Multitrace Matrix Models
414
A. Shishanin Problems of Intelligentsia Resisters and Pacifists, the Intelligentsia Fights Back J. Bleimaier Clinical Approach to Investigation and Solution of International Conflicts or Paranoia in the Age of Mass-Media M. Reshetnikov I was a Terrorist
418
426
430
M. Mockers Conference Programme
435
List of Participants
441
T H E C O N C E P T OF MASS IN T H E EINSTEIN Y E A R L.B. Okuif State Research Center, Institute for Theoretical and Experimental Physics, 117218, Moscow, Russia Abstract. Various facets of the concept of mass are discussed. The masses of elementary particles and the search for higgs. The masses of hadrons. The pedagogical virus of relativistic mass. 1
From "Principia" t o Large Hadron Collider (LHC)
The term "mass' was introduced into mechanics by Newton in 1687 in his aPrincipia" [1]. He defined it as the amount of matter. The generally accepted definition of matter does not exist even today. Some authors of physics text-books do not consider photons - particles of light - as particles of matter, because they are massless. For the same reason they do not consider as matter the electromagnetic field. It is not quite clear whether they consider as matter almost massless neutrinos, which usually move with velocity close to that of light. Of course it is impossible to collect a handful of neutrinos similarly to a handful of coins. But in many other respects both photons and neutrinos behave like classical particles, while the electromagnetic field is the basis of our understanding of the structure of atoms. On the other hand, the socalled weak bosons W+, W~, Z° are often not considered as particles of matter because they are too heavy and too short-lived. Even more unusual are such particles as gluons and quarks. Unlike atoms, nucleons, and leptons, they do not exist in a free state: they are permanently confined inside nucleons and other hadrons. There is no doubt that the problem of mass is one of the key problems of modern physics. Though there is no common opinion even among the experts what is the essence of this problem. For most of particle theorists, as well as members of LHC community, the solution of the problem is connected with the quest and discovery of the higgs - scalar boson which in the Standard Model is responsible for the masses of leptons and quarks and their electroweak messengers: W and Z. The discovery of higgs and the study of higgs sector might elucidate the problem of the pattern of hierarchy of masses of leptons and quarks: from milli electron Volts for neutrinos to about 180 GeV for t-quark. For many physicists it is a QCD problem: how light quarks and massless gluons form massive nucleons and atomic nuclei. Still for majority of confused students and science journalists there is no difference between mass of a body m and its energy E divided by c : they believe in the "most famous formula E = mc2". If higgs exists, its discovery will depend on the funding of the particle physics. In 1993 the termination of the SSC project sent the quest for the higgs into a painful knockdown. The decision not to order in 1995 a few dozen of extra superconducting cavities prevented, a few years later, LEP II from crossing the 115 GeV threshold for the mass of the higgs. If we are lucky and higgs is discovered around year 2010 at LHC, then the next instrument needed to understand what keeps the masses of the higgs below 1 TeV scale, is ILC (International linear collider). This machine would provide a clean environment for the study of higgs production and decays. It could also be used for discovery and study of light supersymmetric particles (SUSY). A prototype of ILC was suggested a few years ago by DESY as the project TESLA. There was no doubt that if funded, TESLA would work, but the funding was not provided by the German government. The new variant of ILC envisions increasing the maximal center of mass a
e-mail:
[email protected] 1
2 energy of colliding electron and positron from ) 0.5 TeV to 1 TeV. If everything goes well, ILC can start before 2020. Further increase of energy, to say, 5 TeV, would call for a machine of the type of CLIC (Compact linear collider) the project of which is under discussion at CERN for more than a decade. In this machine the role of clystrons is supposed to play a low energy but very high current "decelerator" the energy of which would be pumped into the high energy accelerator part of CLIC. Unlike situation with ILC, even the mere feasibility of CLIC is not clear now. Special experimental research to ascertain the feasibility is going on at CERN. The discussion of higgses, neutrinos and QCD in connection with the fundamental problems of mass is often accompanied and even overshadowed by a "pseudoproblem" of the so-called "relativistic mass" (see section 5). 2
M a s s in N e w t o n i a n Mechanics
The more basic is a physical notion, the more difficult to define it in words. A good example give the 1960s editions of "Encyclopedia Britannica" where energy is defined in terms of work, while the entry "work" refers to labour and professional unions. Most people have intuitive notions of space and time. Every physicist has intuitive notions of energy, mass, and momentum. But practically everybody has difficulties in casting these notions into words without using mathematics. Though the definition of mass ("Definition I: The quantity of matter is the measure of the same, arising from its density and bulk conjointly") given by Newton in his "Principia" [1] was so unclear that scholars are discussing its logical consistency even today, the equations of Newtonian mechanics are absolutely self-consistent. Mass m enters in the relations of velocity v = dr/dt and momentum p: P = mv ,
(1)
as well as acceleration a = dv/dt and force F: F = dp/di = m a .
(2)
It also enters in the equation defining the force of gravity with which a body with mass mi at point ri attracts another body with mass m,2 at point w. F s = -Gmim2r/r3
.
(3)
Here r = r2 — r i , r = |r|, while G is the famous Newton constant: G = 6.67-10"nm3kg-1s~2 .
(4)
The kinetic energy of a body is defined as Ek = p 2 / 2 m = m v 2 / 2 .
(5)
The potential gravitational energy: Ug = —Gm\m2/r
,
(6)
while the total energy in this case is E = Ek + Ug .
(7)
The total energy is conserved. When a stone falls on the earth, its potential energy decreases (becomes more negative), kinetic energy increases: so that the total energy
3 does not change. When the stone hits the ground, its kinetic energy is shared by the ambient molecules raising the local temperature. One of the greatest achievements of the XIX century was the formulation of the laws of conservation of energy and momentum in all kinds of processes. At the beginning of the XX century it was realized that conservation of energy is predetermined by uniformity of time, while conservation of momentum - by uniformity of space. But let us return to the notion of force. People strongly felt the force of gravity throughout the history of mankind, but only in XVII century the equations (3) and (6) were formulated. An important notion in this formulation is the notion of gravitational potential ipg. The gravitational potential of a body with mass mi is
