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 light-minded 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 Physical-Technical 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 so-called 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 [34].

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/y meson was first discovered in 1974 simultaneously in e+e- annihilation at SLAC and in Drell-Yan muon pair production at Fermilab, in both experiments the peak substantially deviated from the Breit-Wigner 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 double-logarithmic ‘tails’ in the observed events [5]. Owing to a very small width of J/y 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- --> m+m- 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 t-channel. That was an example of a non-vacuum Reggeon exchange in QCD [6].

At the end of the 1960’s Bjorken scaling was discovered in deep inelastic lepton-hadron 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 Q2 to the fermion. The task was to select and sum up all leading Feynman diagrams giving rise to the logarithmically enhanced (a log Q2)n contributions to the cross section, at fixed value of the Bjorken variable x = Q2/(s + Q2)  (- (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 field-theoretic 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 (Gribov-Lipatov 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 Gribov-Lipatov 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 gluon-to-gluon splitting kernel provided later independently by Dokshitzer at PNPI, and Altarelli and Parisi. Today the Gribov-Lipatov-Dokshitzer-Altarelli-Parisi (DGLAP) evolution equations are the basis for all phenomenological approaches used to describe hadron interactions at short distances. A more general evolution equation for quasi-partonic operators obtained by Lev and co-authors [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 Zinn-Justin (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 Yang-Mills 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 non-vacuum ones (then it can be a so-called ‘odderon’ or a secondary non-leading 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 as contributions enhanced by the logarithm of virtuality, lnQ2, 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 (as ln(1/x))n are summed up by the famous BFKL equation [1314]. 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 non-zero momentum transfer. Lev discovered beautiful symmetries of that equation, which enabled him to find solutions in terms of the conformal-symmetric 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 lnkt space.

Later on, in the 1990’s the next-to-leading corrections to the BFKL equation were calculated by Lev together with Victor Fadin, and the “BFKL pomeron in the next-to-leading 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 t-channel. 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 two-dimensional field-theoretic models [1819].

In the last decade Lev works mainly on topics related to the new hot field in theoretical physics: the so-called 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 field-theoretic side, and from the “gravity” side, when one studies a seemingly unrelated problem of the spectrum of a string moving in a peculiar curved anti-de 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 [20212223]. 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 long-standing 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 Physical-Technical 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 quasi-partonic 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 Yang-Mills 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); “Multi-Reggeon processes in the Yang-Mills 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 next-to-leading 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 Yang-Mills 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 Bern-Dixon-Smirnov amplitudes”, Phys. Rev. D80, 045002 (2009).

[24]   L.N. Lipatov, “Integrability of scattering amplitudes in N=4 SUSY”, J. Phys. A42, 304020 (2009).