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Lev Nikolaevich Lipatov was born on May 2nd, 1940. His early background did not seem
likely to suggest a brilliant academic career. The path to Big Science was determined by a
resolute military veteran in the backyard who played checkers with a curious boy. That turned
Lev away from the lightminded milieu of comrade youngsters and lead him to the
municipal club where he learned to play checkers and later chess astonishingly well. His
choice of higher education at the Physical Faculty of Leningrad (now St. Petersburg)
State University was, then, not a matter of chance but an outcome of his continuous
curiosity.
Lev graduated from the University in 1962 and after some time became a graduate student
at the Ioffe PhysicalTechnical Institute in Leningrad, one of the best in the country. He joined
the group of Vladimir Naumovich Gribov, a rising star in elementary particles theory. In
late 1950’s and early 1960’s there was a general disappointment in Quantum Field
Theory (QFT) as applied to strong interactions. New methods were developing to
circumvent traditional QFT by means on studying the unitarity and analyticity
constraints on the scattering amplitudes, and on extending Regge’s ideas on complex
angular momenta to the relativistic theory. In the 1960’s the Leningrad group lead by
Gribov probably became a world leader in these matters, and Lev witnessed the
development firsthand. Later on he contributed more and more to the fame of the
group.
However, Quantum Field Theory was not given up even by such a proponent of the new
approach as Gribov. Quantum Electrodynamics (QED) was used to check the new ideas of
particle ‘Reggeization’. In the famous paper by Gorshkov, Gribov, Lipatov and Frolov [1] the
socalled Leading Logarithm Approximation to processes at high energies was developed and
“double logs” discovered, different from the already known Sudakov formfactor. Subsequently,
using QED as an example, it was demonstrated in Refs. [2] and [3] that Quantum Field
Theory leads to a total cross section that was not decreasing with energy  actually it
was the first example of a pomeron exchange. It corresponded to a fixed cut in the
complex angular momentum plane for the scattering amplitude, and its location
was calculated. Moreover, in the particular case of QED the main features of the
Reggeon diagram technique (also called the Reggeon Field Theory) were checked and
confirmed [3, 4].
Looking ahead, methods mastered by Lev in those early days of his career paved the way to
the great accomplishments in future. Before describing that work, let me mention a couple of
direct applications of his profound knowledge of QED.
When the J/ meson was first discovered in 1974 simultaneously in e^{+}e^{} annihilation at
SLAC and in DrellYan muon pair production at Fermilab, in both experiments the peak
substantially deviated from the BreitWigner shape: there was a considerable enhancement of
events to the right from the peak position in the first case, and to the left in the second case.
Together with Yakov Azimov, Valery Khoze and Arkady Vainshtein, Lev quickly realized that
it was QED at work: the radiation of unregistered bremsstrahlung photons lead to typical
doublelogarithmic ‘tails’ in the observed events [5]. Owing to a very small width of J/
resonance and its large mass, the value of the “large logarithm” there was indeed so large
that the QED radiative corrections became an important effect clearly visible in the
experiments.
A generalization of a QED calculation of the e^{+}e^{} ^{+}^{} annihilation was
used later by Lev with Roland Kirschner to study the asymptotic behaviour of the
amplitudes in Quantum Chromodynamics (QCD), with the exchange of a quark and an
antiquark in the tchannel. That was an example of a nonvacuum Reggeon exchange in
QCD [6].
At the end of the 1960’s Bjorken scaling was discovered in deep inelastic leptonhadron
scattering at SLAC. Feynman and Bjorken introduced nucleon constituents  ‘partons’ which
later on turned to be nothing but quarks, antiquarks and gluons. Gribov became
interested if Bjorken scaling could be reproduced in Quantum Field Theory. He studied
a fermion theory with a pseudoscalar coupling and also QED as examples, in the
kinematical conditions when there was a large momentum transfer Q^{2} to the fermion.
The task was to select and sum up all leading Feynman diagrams giving rise to the
logarithmically enhanced ( log Q^{2})^{n} contributions to the cross section, at fixed value of the
Bjorken variable x = Q^{2}/(s + Q^{2}) (0,1) where s is the invariant energy of the
reaction.
At some point Lev joined Gribov in that project; they also studied the inclusive e^{+}e^{} h
annihilation to a particle h in two fieldtheoretic models, one of them being QED [7]. They
showed that in a renormalizable QFT one has to expect the violation of Bjorken scaling in
structure functions. They obtained relations between the structure functions describing
deep inelastic scattering and those describing jet fragmentation in e^{+}e^{} annihilation
(GribovLipatov reciprocity relations). It is instructive to note that this work appeared at the
time when experiments did not yet detect any violations of the Bjorken scaling, nor did the
characteristic transverse momenta in “hard” hadronic reactions rise with the momentum
transfer, as it would follow from a renormalizable field theory. This paradox lead to
continuous, sometimes hot discussions at the newly born Theory Division of the Leningrad
Nuclear Physics Institute in Gatchina (now PNPI); in 1971 it was separated from the
Ioffe Institute and became an independent institute within the Russian Academy of
Sciences.
Somewhat later Lev reformulated GribovLipatov results for QED in the form
of the evolution equations for parton densities [8]. It differed from the real thing,
QCD, only by colour factors and by the absence of the gluontogluon splitting kernel
provided later independently by Dokshitzer at PNPI, and Altarelli and Parisi. Today
the GribovLipatovDokshitzerAltarelliParisi (DGLAP) evolution equations are
the basis for all phenomenological approaches used to describe hadron interactions
at short distances. A more general evolution equation for quasipartonic operators
obtained by Lev and coauthors [9] allow to consider more complicated reactions
including high twist operators and the polarization phenomena in hard hadronic
processes.
In the middle of the seventies Lev made a memorable detour from his favourite ‘large
logarithms’ and started studying properties of perturbation theory in QFT in a more
general context. He suggested, through arguments based on analyticity, that the
behaviour of the high orders in a perturbation theory is governed by the Euclidean
classical solutions in the theory, like instantons [10]. This work was immediately highly
appreciated by Brezin and ZinnJustin (and later by ’t Hooft), who wrote: “[Lipatov]
made the beautiful observation that the large orders of perturbation theory may be
described by a classical structure with small quantum fluctuations, but around a
pseudoparticle solution of the classical field equations. Such results are not only of
conceptual but also of practical importance.” The corresponding classical solution in
the scalar field theory was named the ‘lipaton’. This work by Lev created a whole
industry of evaluating high orders in various field theories from classical solutions
therein.
Returning to his principle subject, high energy scattering in field theory, Lev started from
showing that the gauge vector boson in the YangMills theory is ‘reggeized’ [11]: with
radiative corrections included the vector boson becomes a moving pole in the complex angular
momentum plane near j = 1. In QCD, however, this pole is not directly observable by itself as
it corresponds to colour exchange. A meaningful thing is an exchange of two or more reggeized
gluons, leading to ‘colourless’ exchange in the t channel, either with vacuum quantum numbers
(then it is called a pomeron) [12] or nonvacuum ones (then it can be a socalled ‘odderon’
or a secondary nonleading reggeon). Lev and his collaborators showed that the
pomeron corresponds not to a pole but to a cut in the complex angular momentum
plane.
In contrast to the DGLAP approach where one sums up higher order _{s} contributions
enhanced by the logarithm of virtuality, lnQ^{2}, in the case of the high energy scattering one
looks for contributions enhanced by the logarithm of energy, lns, or by the logarithm of a small
momentum fraction carried by gluons, x. The leading log contributions of the type
(_{s} ln(1/x))^{n} are summed up by the famous BFKL equation [13, 14]. In comparison with
DGLAP this is a much more complicated problem since the BFKL equation actually includes
contributions of operators of higher twists.
In its general form the BFKL equation describes not only the high energy behaviour of the
cross sections but also the amplitudes at nonzero momentum transfer. Lev discovered
beautiful symmetries of that equation, which enabled him to find solutions in terms of the
conformalsymmetric eigenfunctions. This completed the construction of the “bare pomeron in
QCD”, a fundamental entity of high energy physics [15]. An interesting new property of this
‘bare pomeron’ (not known in the old Reggeon Field Theory) is the diffusion of the emitted
particles in the lnk_{t} space.
Later on, in the 1990’s the nexttoleading corrections to the BFKL equation were
calculated by Lev together with Victor Fadin, and the “BFKL pomeron in the nexttoleading
approximation” was obtained [16]. Technically, it was probably one of the most involved works
by Lev, that took many years. Besides, Lev studied the higher order amplitudes
with an arbitrary number of gluons exchanged in the tchannel. In particular, the
odderon exchange was described in perturbative QCD [17]. The significance of that
study was, however, much greater. It lead Lev to the discovery of the connection
between high energy scattering and the exactly solvable twodimensional fieldtheoretic
models [18, 19].
In the last decade Lev works mainly on topics related to the new hot field in
theoretical physics: the socalled ADS/CFT correspondence  a hypothesis put forward
by Juan Maldacena in 1997. It states that there is a correspondence, a duality in
the description of the maximally supersymmetric N = 4 modification of QCD from
the standard fieldtheoretic side, and from the “gravity” side, when one studies a
seemingly unrelated problem of the spectrum of a string moving in a peculiar curved
antide Sitter background. Lev’s unique experience in and deep understanding of
resummed perturbation theory enabled him to move fast into the new territory where he
developed and tested new ideas, first of all by considering the BFKL and DGLAP
equations in the N = 4 theory, and computing the anomalous dimensions of various
operators [20, 21, 22, 23]. The high symmetry of that theory, as compared to the
standard QCD, allows one to make calculations at unprecedented high orders, and to
compare the results with the ‘dual’ predictions of string theory. It also facilitates to
find the integrable structures in the theory, in which Lev is also a longstanding
expert [24].
In that work, Lev collaborated with many people: Vitaly Velizhanin, Alexander Kotikov,
Alexander Onishchenko, Jochen Bartels, Matthias Staudacher, Victor Kim, and others. It is to
a great extent owing to the work of Lev and his collaborators that the duality hypothesis is
today practically beyond doubts. That opens a new horizon in studying Quantum Field Theory
at strong couplings  something one would never dream of at the time when Lev started his
academic career.
A few formal details: Apart from numerous visiting positions abroad, Lev always remained
a staff member of the Petersburg Nuclear Physics Institute (for the first few years, before the
separation of the institutes, it was part of the Ioffe PhysicalTechnical Institute in Leningrad).
In 1980 he became Head of the Quantum Field Theory Department and in 1997 he was elected
Head of the Theory Division at PNPI. He is a member of the Coordination and Scientific
Councils at PNPI and at several other institutions. In 1997 he was elected Corresponding
Member of the Russian Academy of Sciences.
Lev’s research was awarded the Alexander von Humboldt prize (Germany) in 1995 and
the prestigious Marie Curie Fellowship (European Union) in 2006. In 2001 he got
the International Pomeranchuk prize, simultaneously with Tullio Regge  quite a
symbol.
SPIRES database lists more than 200 papers by Lev (as of April 2010),
enjoying more than 16000 citations. Among them are 10 papers
listed as ``renowned'' having more than 500 citations each, out of which
three papers
[7], [13] and [14]
have about 2000
citations. A recently published
Festschrift
entitled
``Subtleties in Quantum Field Theory'',
a collection of papers by Lev's friends, colleagues
and collaborators, pays tribute to his pioneering
contributions
to various topics in modern theory.Probably, the word ``subtlety'' is
characteristic of Lev's work in general. He typically becomes interested
in certain fine features of a theory (some would say they are ``technical''),
and elaborates on them. He builds a new formalism based on subtleties,
and ultimately it becomes overwhelming and commands our understanding.
It's like a beautiful chess party. Methods are of primary importance, the
results will follow.
Dmitri Diakonov
[1] V.G. Gorshkov, V.N. Gribov, L.N. Lipatov and G.V. Frolov, “Double
logarithmic asymptotics of quantum electrodynamics”, Phys. Lett. 22, 671 (1966);
“Doubly logarithmic asymptotic behavior in quantum electrodynamics”, Sov. J.
Nucl. Phys. 6, 95 (1968) [Yad. Fiz. 6, 129 (1967)].
[2] G.V. Frolov, V.N. Gribov, L.N. Lipatov, “On the vacuum pole in Quantum Electrodynamics”, Phys. Lett. B31, 34 (1970).
[3] V.N. Gribov, L.N. Lipatov, G.V. Frolov, “The leading singularity in the j plane in quantum electrodynamics”, Sov. J. Nucl. Phys. 12, 543 (1971) [Yad. Fiz. 12, 994 (1970)].
[4] L.N. Lipatov, G.V. Frolov, “Some processes in quantum electrodynamics at high energies”, Yad. Fiz. 13, 588 (1971).
[5] Ya.I. Azimov, A.I. Vainshtein, L.N. Lipatov and V.A. Khoze, “Electromagnetic corrections to the production of narrow resonances in colliding e^{+}e^{} beams”, JETP Lett. 21, 172 (1975) [Pisma Zh. Eksp. Teor. Fiz. 21, 378 (1975)].
[6] R. Kirschner and L.N. Lipatov, “Double logarithmic asymptotics and Regge singularities of quark amplitudes with flavor exchange”, Nucl. Phys. B213, 122 (1983).
[7] V.N. Gribov, L.N. Lipatov, “Deep inelastic electron scattering in perturbation
theory”, Phys. Lett. B37, 78 (1971);
“Deep inelastic ep scattering in perturbation theory”, Sov. J. Nucl. Phys. 15, 438
(1972) [Yad. Fiz. 15, 781 (1972)];
“e^{+}e^{} pair annihilation and deep inelastic ep scattering in perturbation theory”,
Sov. J. Nucl. Phys. 15, 675 (1972) [Yad. Fiz. 15, 1218 (1972)].
[8] L.N. Lipatov, “The parton model and perturbation theory”, Sov. J. Nucl. Phys.
20, 94 (1975) [Yad. Fiz. 20, 181 (1974)];
A.P. Bukhvostov, L.N. Lipatov and N.P. Popov, “Parton distribution functions in
perturbation theory”, Yad. Fiz. 20, 532 (1974).
[9] A.P. Bukhvostov, G.V. Frolov, E.A. Kuraev and L.N. Lipatov, “Evolution
equations for quasipartonic operators”, Nucl. Phys. B258, 601 (1985);
A.P. Bukhvostov, E.A. Kuraev, L.N. Lipatov, “Deep inelastic electron scattering
by a polarized target in quantum chromodynamics” JETP Lett. 37, 482 (1983)
[Pisma Zh. Eksp. Teor. Fiz. 37, 406 (1983); Sov. Phys. JETP 60, 22 (1984).
[10] L.N. Lipatov, “Divergence of the perturbation theory series and the
quasiclassical theory”, Sov. Phys. JETP 45, 216 (1977) [Zh. Eksp. Teor. Fiz. 72,
411 (1977)];
L.N. Lipatov, A.P. Bukhvostov and E.I. Malkov, “Large order estimates for
perturbation theory of a YangMills field coupled to a scalar field”, Phys. Rev.
D19, 2974 (1979).
[11] L.N. Lipatov, “Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories”, Sov. J. Nucl. Phys. 23, 338 (1976) [Yad. Fiz. 23, 642 (1976)].
[12] V.S. Fadin, E.A. Kuraev and L.N. Lipatov, “On the Pomeranchuk singularity in asymptotically free theories”, Phys. Lett. B60, 50 (1975); “MultiReggeon processes in the YangMills theory”, Sov. Phys. JETP 44, 443 (1976) [Zh. Eksp. Teor. Fiz. 71, 840 (1976)].
[13] E.A. Kuraev, L.N. Lipatov and V.S. Fadin, “The Pomeranchuk singularity in nonabelian gauge theories”, Sov. Phys. JETP 45, 199 (1977) [Zh. Eksp. Teor. Fiz. 72, 377 (1977)].
[14] I.I. Balitsky and L.N. Lipatov, “The Pomeranchuk singularity in Quantum Chromodynamics”, Sov. J. Nucl. Phys. 28, 822 (1978) [Yad. Fiz. 28, 1597 (1978)].
[15] L.N. Lipatov, “The bare pomeron in Quantum Chromodynamics”, Sov. Phys. JETP 63, 904 (1986) [Zh. Eksp. Teor. Fiz. 90, 1536 (1986)].
[16] V.S. Fadin and L.N. Lipatov, “BFKL pomeron in the nexttoleading approximation”, Phys. Lett. B429, 127 (1998).
[17] J. Bartels, L.N. Lipatov and G.P. Vacca, “New odderon solution in perturbative QCD” Phys. Lett. B477, 178 (2000).
[18] L.N. Lipatov, “Asymptotic behavior of multicolor QCD at high energies in connection with exactly solvable spin models”, JETP Lett. 59, 596 (1994) [Pisma Zh. Eksp. Teor. Fiz. 59, 571 (1994)].
[19] H.J. de Vega and L.N. Lipatov, “Exact resolution of the Baxter equation for reggeized gluon interactions”, Phys. Rev. D66, 074013 (2002).
[20] A.V. Kotikov and L.N. Lipatov, “NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories”, Nucl. Phys. B582, 19 (2000); “DGLAP and BFKL equations in the N=4 supersymmetric gauge theory”, Nucl. Phys. B661, 19 (2003), Erratum: ibid. B685, 405 (2004).
[21] A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, “Three loop universal anomalous dimension of the Wilson operators in N=4 SUSY YangMills model”, Phys. Lett. B595, 521 (2004), Erratum: ibid. B632, 754 (2006).
[22] A.V. Kotikov, L.N. Lipatov, A. Rej, M. Staudacher and V.N. Velizhanin, “Dressing and wrapping”, J. Stat. Mech. 0710, P10003 (2007).
[23] J. Bartels, L.N. Lipatov and A.S. Vera, “BFKL pomeron, reggeized gluons and BernDixonSmirnov amplitudes”, Phys. Rev. D80, 045002 (2009).
[24] L.N. Lipatov, “Integrability of scattering amplitudes in N=4 SUSY”, J. Phys. A42, 304020 (2009).