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The Gribov Conception of Quantum Chromodynamics

A major contribution to the quest of constructing quantum dynamics of non-Abelian fieds is due to V.N. Gribov. Perturbative approach to the colour confinement, both in gluodynamics and the real world, was long considered heretic but is gaining ground. We discuss Gribov's approach to the confinement problem, centered around the role played by light quarks - the supercritical light quark confinement scenario. We also review some recent developments that are motivated, directly or indirectly, by his ideas.


INTRODUCTION
One of the most challenging problems for modern theoretical physics is understanding the confinement of colour -the selection rule suggested by quantum chromodynamics (QCD) to explain the fact that quarks (and gluons) appear in the physical spectrum only imprisoned inside composite states -colourless mesons and baryons.
An amazing success of the relativistic theory of electron and photon fieldsquantum electrodynamics (QED) -has produced a long-lasting negative impact: it taught the generations of physicists that came into the business in/after the 70's to "not to worry". Indeed, today one takes a lot of things for granted.
One rarely questions whether the alternative roads -secondary quantization, functional integral and the Feynman diagram approach -really lead to the same quantum theory of interacting fields.
One feels ashamed to doubt an elegant and powerful, but potentially deceiving, technology of translating the dynamics of quantum fields into that of statistical systems (imaginary time = Euclidean rotation trick, resulting in the coupling ↔ temperature analogy).
One takes the original concept of the "Dirac sea" -the picture of the fermionic content of the vacuum -as an anachronistic model, a sort of Maxwell's mechanical ether, I'd-rather-do-without.
One was taught to consider the problems with field-theoretical description of point-like objects and their interactions at very small distances (ultraviolet divergences) as purely technical: renormalize it and forget it.
So far so good for QED, where the physical objects -electrons and photons -are direct images of the fundamental fields that one put into the local La-grangian of the theory. In other words, interacting fields closely resemble their bare counterparts. In these circumstances, the ground state of the theory -the physical vacuum -turns out to be, in a certain sense, trivial. The rôle played by the vacuum fluctuations of electromagnetic fields and by the "Dirac sea" of negative-energy electron states, is as important as it is straightforward. Interaction with the vacuum makes the effective interaction strength α e.m. (as well as the effective electron mass) run with the distance. At the same time, it does not affect the nature of the interacting fields.
Not so clear in QCD. Here the physical hadronic states are not in one-to-one correspondence with the fundamental quarks and gluons: interaction with the QCD vacuum changes the bare fields beyond recognition. In such an environment, our preconceived ideas about QFT dynamics may turn out to be a handicap, a serious hidden obstacle on the way to attacking the confinement problem.
Vladimir Naumovich Gribov (1930Gribov ( -1997 belonged to a generation of physicists (now almost extinct) that thought about the Quantum Field Theory (QFT), that was used to questioning its foundations, concepts and means. An invitation to the "Gribov conception of QCD" is an invitation to unlearn. Learning to unlearn isn't easy. This possibly explains why the programme that Gribov formulated and was pursuing in the 90's of explaining the confinement of colour as "supercritical binding" of light quarks has yet to receive the attention it deserves from the physics community at large.
It goes without saying that Gribov could be wrong in his vision about the nature of the QCD confinement. However, it does not seem to us a good policy to be simply indifferent to what one of the creators of the modern theoretical physics had to say on the subject, to the ideas and tools he has developed during the last half of his professional life.
This review does not attempt to cover Gribov's impact on theoretical physics.
Suffices it to mention that his name is attached to many a key notion of the theory arsenal: Gribov-Froissart projection  His last works remain to be discovered, understood and developed.
For V.N. Gribov, the problem of confinement as the problem of understanding the dynamics of vacuum fluctuations, and of the structure of hadrons as physical states of the theory, was always inseparable from the problem of understanding the physics of high energy hadron scattering (the Pomeron picture). Therefore, we start by an overview of the Gribov's development of high-energy physics. The two subsequent Sections are devoted to his contribution to the quantum dynamics of non-Abelian Yang-Mills gauge fields in general, and to the basics of the Gribov light-quark confinement picture. The last Section briefly describes the byproducts of the QCD programme: an unpublished study of QED at short distances (a potential resolution of the notorious "Landau ghost" problem) and his picture of a composite (super-critically bound) Higgs boson.

HADRON INTERACTIONS AT HIGH ENERGIES
In the late fifties, when Vladimir Gribov, then a young researcher at the Ioffe Physico-Technical Institute (Leningrad, USSR) became interested in the physics of strong hadron interactions, not much was understood about processes at high energies. The only theoretical result, derived from the first principles, was the Pomeranchuk theorem -an asymptotic equality of particle and antiparticle total scattering cross sections.

Asymptotic Behaviour s α(t)
Gribov's 1960 paper Asymptotic behaviour of the scattering amplitude at high energies (1) was a breakthrough. Using the so-called double-dispersion representation for the scattering amplitude, suggested by S. Mandelstam back in 1958 (2), Gribov proved an inconsistency of the then popular black disk model of hadronhadron scattering. This analysis may be considered as the first building block put into the edifice of the modern theory of hadron interactions. It has demonstrated the combined power of the general principles of relativistic quantum theoryunitarity (conservation of probability), analyticity (causality) and the relativistic nature (crossing symmetry) -as applied to high-energy scattering phenomena.
By studying the analytic properties in the cross channels, he showed that the imaginary part of the scattering amplitude in the form that constituted the black disk model of diffraction in the physical region of s-channel scattering, contradicts the unitarity relation for partial waves in the crossing t-channel. To solve the puzzle, Gribov suggested the behaviour of the amplitude (for large s and finite t) in the general form where B t is a slow (non-exponential) function of ln s (decreasing fast with t) and q(0) = 1 ensures the approximate constancy of the total cross section with energy.
In this first paper Gribov only analysed the constant exponent, q(t) = 1, having remarked on the possibility q(t) = const as "extremely unlikely". Indeed, considering the t-dependence of the scattering amplitude, this would correspond to a strange picture of the radius of a hadron infinitely increasing with energy. 2 (In this particular case he proved that the cross section has to decrease at high energies, B t (ln s) < 1/ ln s, to be consistent with the t-channel unitarity.) Soon after that Gribov became aware of the finding by T. Regge (3) that in non-relativistic quantum mechanics, in the unphysical region |t| ≫ s (momentum transfer much larger than the energy, corresponding to large imaginary scattering angles | cos Θ| → ∞), the scattering amplitude has a form where ℓ(s) is the position of the pole of the partial wave f ℓ in the complex plane of the orbital momentum ℓ. 3 T. Regge  with q(t) = const. "You are too young to judge" the blessing went, according to legend. 3 Gribov apparently learned about the Regge result from a paper by G. Chew (6), Thus, having accepted ℓ(0) = 1 for the rightmost pole in the ℓ-plane -the vacuum pole -as the condition "that the strongest possible interaction is realized", Gribov formulated and discussed "the main properties of such an asymptotic scattering behaviour . . . which in spite of having a few unusual features is theoretically feasible and does not contradict the experimental data": • the total interaction cross section is constant at high energies, • the elastic cross section tends to zero as 1/ ln s, • the scattering amplitude is essentially imaginary, • the significant region of momentum transfer in elastic scattering shrinks with energy increasing, He also analysed the s-channel partial waves to show that for small impact parameters ρ < R their amplitudes fall as 1/ ln s, while the interaction radius R increases with energy as ρ ∝ √ ln s. He concluded: "this behaviour means that the particles become grey with respect to high-energy interaction, but increase in size, so that the total cross section remains constant." Thus, shrinkage of the diffractive peak was predicted, which was then experimentally verified at particle accelerator experiments in Russia (IHEP, Serpukhov), Switzerland (CERN) and the US (FNAL, Chicago), as were the general relations between the cross sections of different processes, that followed from the Gribov factorization theorem (7).
These were the key features of what has become known, quite ironically, as the "Regge theory" of strong interactions at high energies.
On the opposite side of the curtain, the basic properties of the Regge pole picture of forward/backward scattering were formulated half a year later by G. Chew and S. Frautschi (8). In particular, they suggested "the possibility that the recently discovered ρ meson is associated with a Regge pole whose internal quantum numbers are those of an I = 1 two-pion configuration," and conjectured the universal high-energy behaviour of backward π + π 0 , K + K 0 and pn scattering due to ρ-reggeon exchange. Chew and Frautschi also stressed that the hypothetical Regge pole with α(0) = 1 responsible for forward scattering possesses quantum numbers of the vacuum.
Dominance of the Gribov vacuum pole automatically validates the Pomeranchuk theorem. The name "Pomeron" for the vacuum pole was coined by Murray Gell-Mann, who referred to Geoffrey Chew as an inventor.

Complex Angular Momenta in Relativistic Theory
In non-relativistic quantum mechanics the interaction Hamiltonian allows for scattering partial waves to be considered as analytic functions of complex angular momentum ℓ. Gribov's paper Partial waves with complex orbital angular momenta and the asymptotic behaviour of the scattering amplitude showed that the partial waves with complex angular momenta can be introduced in a relativistic theory as well. Here it is the unitarity in the crossing channel that replaces the Hamiltonian and leads to analyticity of the partial waves in ℓ. The corresponding construction is known as the "Gribov-Froissart projection" (9).
Few months later Gribov demonstrated that the simplest (two-particle) tchannel unitarity condition indeed generates moving poles in the complex ℓ-plane.
This was the proof of the Regge hypothesis in relativistic theory (10).
The "Regge trajectories" α(t) combine hadrons into families: where s h and m h are the spin and the mass of a hadron (hadronic resonance) with given quantum numbers (baryon number, isotopic spin, strangeness, etc.) (8).
Moreover, at negative values of t, that is in the physical region of the s-channel, the very same function α(t) determines the scattering amplitude, according to Eq. (2). It looks as if high-energy scattering was due to t-channel exchange of a "particle" with spin α(t) that varies with momentum transfer t -the "reggeon".
Thus, the high-energy behaviour of the scattering process a + b → c + d is

Interacting Pomerons
A lot of theoretical effort was invested into understanding of the approximately constant behaviour of total cross sections at high energies.
To construct a full theory that would include the Pomeron trajectory with the maximal "intercept" α P (0) = 1 that respects the Froissart bound, and would be consistent with unitarity and analyticity proved to be very difficult. This is because multi-Pomeron exchanges become essential, which generate branch points in the complex ℓ-plane, which singularities were first discovered by Mandelstam in his seminal paper (11). Moreover, the study of particle production processes  Gribov's earlier work Interaction of γ-quanta and electrons with nuclei at high energies (14) had been a precursor to the famous Feynman paper. Gribov described the photon interaction in the rest frame of the target nucleus. An incident real photon (or a virtual photon in the deep inelastic scattering case) fluctuates into hadrons before the target, at the longitudinal distance L increasing with energy. 4 Therefore, at sufficiently large energy, when the fluctuation distance exceeds the size of the target, the photon no longer behaves as a point-like weekly interacting particle. Its interaction resembles that of a hadron and becomes "black", corresponding to complete absorption on a large nucleus.

Gribov Partons and Feynman Partons
Being formally equivalent to Feynman's treatment, Gribov's approach is better suited for the analysis of deep inelastic phenomena at very small Bjorken x, where the interaction becomes actually strong, and the perturbative QCD treatment is bound to fail. Gribov diffusion in the impact parameter space giving rise to energy increase of the interaction radius and to the reggeon exchange amplitude, coexisting fluctuations as a source of branch cuts, duality between hadrons and partons, a common basis for hard and soft elastic, diffractive and inelastic process -these are some of the key features of high energy phenomena in QFTs, which are still too hard a nut for QCD to crack.

Gribov Reggeon Field Theory
The two best known applications of the Gribov RFT are • general quantitative relation between the shadowing phenomenon in hadronhadron scattering, the cross section of diffractive processes and inelastic multi-particle production, known as "Abramovsky-Gribov-Kanchelli cutting rules" (AGK) (16), and

Reggeization and Pomeron Singularity in Gauge Theories
In the mid-sixties Gribov initiated the study of double logarithmic asymptotics of various processes in QED, making use of the powerful technique he had developed for the analysis of the asymptotic behaviour of Feynman diagrams (22).
In particular, in 1975 Gribov, Lipatov and Frolov studied the high energy behaviour of QED processes from the point of view of "Regge theory". High energy scattering amplitudes with exchange of an electron in the t-channel acquire, in higher orders in QED coupling, a characteristic behaviour A ∝ s j(t) with j(m 2 e ) = 1/2. This means that electron becomes a part of the Regge trajectory: reggeizes. 5 For the vacuum channel, however, they found (23) that the rightmost singularity in the complex j-plane is not a moving pole (as it is for electron) but, instead, a fixed branch point singularity positioned to the right from unity, This was a precursor of a similar result found later by Fadin, Lipatov and Kuraev (21) in non-Abelian theories, and QCD in particular. The problem of apparent anti-Froissart behaviour of the perturbative "hard Pomeron" in QCD still awaits resolution.
With the advent of QCD as a microscopic theory of hadrons and their interactions, the focus of theoretical studies has temporarily shifted away from Gribov-Regge problematics to "hard" small-distance phenomena.

Anatomy of Asymptotic Freedom
In The anti-screening phenomenon had been first observed back in 1969 (27) for the non-Abelian SU (2) theory in the ghost-free Coulomb gauge within the Hamiltonian approach. In the Hamiltonian language, there are (or rather seem to be) N 2 −1 massless B quanta (transverse gluons, ⊥) and, as in QED, an additional Coulomb field (0). It is important to stress that the latter is not a physical quantum degree of freedom but a means for describing classical instantaneous interaction between colour charges. Unlike QED, the non-Abelian Coulomb field has a colour charge of its own. Therefore, traversing the space between two external charge, it may virtually decay into two transverse fields, or into a qq pair in the same manner as the QED Coulomb field fluctuates in the vacuum into an e + e − pair. Both these effects lead to screening of the colour charge of the external sources, in a perfect accord with one's physical intuition.
This (anti-asymptotically-free) behaviour of the running coupling was first found by Landau, Abrikosov and Khalatnikov for QED (28). It turned out to be common for all then-known renormalizable field theories: with scalar (λφ 4 ), Yukawa, four-fermion interactions (29) as well as for pedagogically valuable Lee model (30).
This observation had dramatic consequences: the physical interaction (renormalized coupling) was predicted to vanish in the limit of a point-like bare interaction, Λ UV → ∞. In the late 1950s the problem was known as "Moscow Zero". The depth of the crisis can be measured by the Dyson prophecy (31) that the correct "meson" theory -the theory of strong interactions -"will not be found in the next hundred years" and by the Landau conclusion (32) that "the Hamiltonian method for strong interactions is dead and must be buried, although of course with deserved honour." Universality of the screening phenomenon was readily understood as a consequence of unitarity in the cross-channel. Indeed, the discontinuity of the loop describes, according to Cutkosky rules, production cross section of two physical (sic!) particles in the t-channel. Positivity of the imaginary part at t > 0 then dictates the screening sign of the virtual loop in the s-channel (s > 0, t < 0).
Thus, the fact that there are "physical" fields in the intermediate state in Eq. (5) -quark-antiquark or two transverse gluons -fixes the sign of the virtual correction to correspond to screening, via the unitarity relation in the cross-channel. Technically speaking, these virtual decay processes contribute to the QCD β-function as that is, make the effective coupling decrease at large distances R between the external charges, as in QED.
Where then the anti-screening comes from? It originates from another, specifically non-Abelian, effect namely, interaction of the Coulomb field with the field of "zero-fluctuations" of transverse gluons in the vacuum, In a course of such multiple rescattering, the Coulomb "quantum" preserves its identity as an instantaneous interaction mediator, and therefore is not affected by the unitarity constraints. For n = 1 the contribution vanishes upon averaging. Statistical average over the transverse vacuum fields in the second order of perturbation theory (n = 2) results in an additional contribution to the Coulomb interaction energy which, translated into the running coupling language, gives According to Gribov, an anti-intuitive minus sign in Eq. (6b) has its own simple explanation. It is of the same origin as the minus sign in the shift of the energy of the ground state of a quantum-mechanical system under the second order in perturbation: The rôle of perturbation δV is played by the vacuum field of transverse gluons.
Taken together, the two contributions Eq. (6) combine into the standard QCD β-function:

Infrared Instability
The three-dimensional transversality condition is usually imposed on the field potential to describe massless vector particles (the Coulomb gauge). Being antisymmetric, the field strength tensor F a µν does not contain time derivative of the zero-component of the potential A a 0 . In the Hamiltonian language, this puts A a 0 in a position of a cyclic variable which does not constitute a physical degree of freedom. It can be eliminated from field dynamics (24,33), contributing to the Hamiltonian that describes transverse gluons, an additional term responsible for Coulomb interaction between "charges", where ρ a is given by the sum of the colour charge density of external sources (e.g., static quarks) and that induced by the transverse gluons themselves, Introducing the covariant derivative D, We see that the propagation of a Coulomb field A 0 in the "external" transverse gluon field A ⊥ is governed by the operator that resembles the propagator of the Faddeev-Popov ghost, The factor J in Eq. (8) is the determinant of this operator, Taking the expectation value of K(x, y; A) over the vacuum fields A ⊥ produces an instantaneous interaction term, This expression describes an interaction generated by exchange of a Coulomb gluon dressed by the fluctuations of transverse gluon fields in the vacuum.
By setting A ⊥ = 0 we would return to the Laplace operator G = −1/∇ 2 ∝ 1/k 2 that corresponds, in the coordinate space, to the canonical Coulomb potential 1/ |x − y|. For small vacuum fields, g s A ⊥ /∇ ≪ 1, expanding perturbatively the Coulomb propagator G in Eq. (11) to the second order in g s produces the one-loop anti-screening effect as stated in Eq. (6b).
If we take, however, gluon fields in the QCD vacuum as large as (with L a spatial extent of the field), the perturbative expansion in g s A ⊥ of the denominators in Eq. (11) is no longer justified. Under such circumstances a qualitatively new phenomenon takes place namely, the Coulomb (ghost) propagator may become singular: Appearance of a "zero-mode" solution C 0 in the external field is a sign of an infrared instability of the theory.
An illuminating way to see the instability of the perturbative vacuum, and to re-derive the running of the coupling in QCD, is to look at the quantum correction to the vacuum energy V 0 (H) = 1 2 H 2 in a constant chromo-magnetic field H.
The one-loop corrected energy density reads (34) where µ is the renormalization scale of the bare coupling, g s = g s (µ 2 ), and b = 11N c /3 (gluodynamics). For relatively small fields, the potential Eq. (13) formally turns negative and develops a minimum -"true vacuum" (?) -corresponding to non-zero expectation value of the chromomagnetic field. It was soon realized, however, that this new "vacuum" is unstable (35).
In pure gluodynamics the vacuum correction can be computed by summing over the Landau levels of gluons moving in the external field: The real part of (14) yields (13). Thus, the asymptotic freedom in this approach can be seen as arising from the paramagnetic response of the vacuum.
At the same time, we readily see that one specific state in Eq. (14) namely, n = 0 with gluon spin parallel to the direction of the field, s z = 1, gives an imaginary contribution: (15) Non-zero imaginary part of the effective potential signals that the vacuum is unstable (decays with time) and does not correspond to the true ground state of the theory. Note also that n = 0 corresponds to the Landau level of the largest transverse radius, i.e. to the infrared region, as had to be expected.
Once again, we come to the conclusion that naïve perturbative treatment of non-Abelian gauge fields is flawed due to problems in the infrared.

Gribov Copies
In (24) Gribov localized the problem. He showed that the three-dimensional transversality condition Eq. (7) actually does not solve the problem of gauge fixing.
Consider a group Ω, and let ω(x) be a function with values in this group. The basic principle that defines the corresponding gauge theory is that the vector potentials A µ (x) and describe physically identical fields. Therefore, to avoid multiple counting, one has to impose a gauge-fixing condition of the general form To do the job, Eq. (17) should have a unique solution for arbitrary A. This can be achieved for a variety of gauge fixings within perturbation theory, when the fields are weak. In general, however, solutions may appears copious since the gauge fixing condition Eq. (17) constitutes a system of nonlinear equations for the function ω(x).
Indeed, Gribov found that due to essential non-linearity of the gauge transformation Eq. (16), a "transverse" field potential satisfying Eq. (7) may actually happen to be a pure gauge field which should not be separately counted as an additional physical degree of freedom. He explicitly constructed such "transverse gauge fields" for the SU (2) gauge group and showed that the uncertainty in gauge fixing arises when the effective magnitude of the field becomes large, or, in other words, when the effective interaction strength (QCD coupling) becomes of the order of unity, that is, in the non-perturbative region. More precisely, he found that it happens exactly when the Faddeev-Popov operator acquires a zero mode solution Eq. (12) that is, as we discussed above, in the infrared region where the vacuum enhancement of the dressed Coulomb gluon propagator (11) becomes catastrophically large. A promising attempt to implement the Gribov fundamental domain in the 5dimensional formulation of gluodynamics can be found in (39).

Coulomb Confinement
Gribov himself addressed the quest of possible modification of the QCD quantization procedure in the second lecture "Quantization of non-Abelian gauge theories" (25). The paper (40) The gluon Green function coincides with the perturbative one at large momenta (small distances), D(k) ∝ k −2 but vanishes off at k = 0, instead of having a pole corresponding to massless gluons: The new non-perturbative parameter σ 2 in Eq. (18) has dimension (and the meaning) of the familiar vacuum gluon condensate, σ 2 ∼ α s (F a µν ) 2 , that emerged in the context of QCD sum rules (41).
Literally speaking, the ansatz Eq. (18b) cannot be correct since such a Green function would violate causality. 6 In reality, the gluon (as well as the quark) propagator should have a more sophisticated analytic structure with singularities on unphysical sheets, which would correspond, in the standard field-theoretical language, to unstable particles.
At the same time, the modification of the Coulomb (ghost) propagator due to restricting the functional integral to the fundamental domain (25) resulted in the singular small momentum behaviour corresponding to a linear increase of the interaction energy at large distances The idea of confinement emerging from dressed Coulomb exchange was further explored by Zwanziger (42) who has recently shown (43) that for the static interaction potential V (R) the following inequality holds: This means that if confinement exists in pure gluodynamics, it should arise already at the dressed one-gluon exchange level -"no confinement without Coulomb confinement" (43). The inequality Eq. (19) appears inevitable: an inclusion of "quantum" (gluons, quarks) degrees of freedom can only screen, that is, suppress, the classical (Coulomb) interaction, as we have discussed in the beginning of this Section.
Recent lattice studies of the correlator of timelike link variables in Coulomb gauge (44) show that the Coulomb interaction energy of static quarks indeed 6 It is unfortunate, therefore, that the form Eq. (18b) which Gribov suggested and discussed for illustrative purposes only, is often referred to in the literature as "the Gribov propagator". grows linearly at large quark separations. This simulation was designed to measure the pure Coulomb energy, without admixture of additional "constituent" gluons, The corresponding string tension σ turned out to be several times greater than the generally accepted value that originated from lattice measurements of the full static interaction energy.
This fact is remarkable. It shows that the Coulomb exchange excites transverse vacuum gluons inclusion of which results in the energy gain. These gluons "shake" the initial string Eq. (20) and soften it.
In this picture the typical transverse size of the string grows logarithmically with quark separation (the so-called "roughening"). It is worthwhile to notice that this phenomenon is similar to the increase of the size of a hadron at high energies as described by "Gribov diffusion", multiplication of the number of gluons being the physical reason for both.

Perturbative (Gluon) Confinement
The spatial distribution of the gluons is then determined by minimizing the total Monte Carlo simulations showed. If E gluon is the kinetic plus nearest-neighbor interaction energy of each gluon, then the total energy of the system is where the the rôle of the string tension is played by gluon energy per unit length, on the hard side -the famous Feynman plateau, xD(x) ≃const., characteristic for parton distributions that explain spacelike deep inelastic scattering (and particle content of timelike jets) at small Bjorken (Feynman) x.
This important link is not a monopoly of the gluon chain model. Indeed, we seem to be retelling the good old Kogut-Susskind confinement story: gluon field lines forming a string, digging up quark-antiquark pairs from the vacuum, a'la Schwinger tunneling mechanism in a constant field, assembling colourless hadrons in the final state. There is a crucial difference, however. We were used to look at the Kogut-Susskind scenario as calling for essentially non-perturbative dynamics: strong fields, "superconductivity" vortex picture to explain one-dimensional nature of the "string", etc. Now we are lead to think that the same physical picture can be achieved by purely perturbative means, that is, by employing the language of the fundamental degrees of freedom of the theory -gluons -which gluons interact in a Coulomb-like (again, perturbative) fashion.
't Hooft (48) has advanced the Greensite-Thorn arguments, and pointed out that the "gluon chains" cannot provide a complete set of states to confine the sources. The problem becomes apparent if we turn to the real world with light quarks present. Indeed, by pulling apart two heavy quarks we expect to find ourselves holding two colourless D-mesons, in the end of the day. Obviously, this cannot be achieved without including light vacuum quarks in the game.
(Formally speaking, the adjoint representation gluons cannot fully blanch colour charges in the fundamental representation -the external quarks.) Nevertheless, since gluon chain states appear to be energetically profitable, as recent lattice results have indicated (45), this model can serve as a good starting point, as a "first approximation" to the problem of quantitatively approaching the physics of hadrons. In particular, it may help to understand puzzling softness of the transformation of partons into hadrons. These phenomena that one observed studying energy and angular distributions of soft particles produced in hard interactions known as "local parton-hadron duality" (49), for recent reviews see (50,51).

't Hooft's (unpublished) remark (48) bears an elegant strikingly simple title
Perturbative Confinement. Looking back into historical perspective, we find this rather ironic, since this very endeavour -perturbative approach to the confinement problem -motivated the NPB referee to initially reject 7 the pioneering Gribov paper (40).
Instability of the perturbative vacuum tells us that in order to have a chance to approach the true groung state of the theory by adiabatically switching on the interaction, one has to start from an excited state, in terms of non-interacting perturbative vacuum. Implementation of this idea is being developed by P. Hoyer and S. Peigne, who are trying to model non-trivial structure of the vacuum by 7 The second reason being, that "the confinement problem had been already solved and isn't worth talking about." explicitly adding condensate terms to perturbative quark and gluon propagators (52).

OPE and Linear Potential
Another way of approaching the problem of confinement is to start at short distances, use the operator product expansion (OPE) and to study the structure of power corrections. For the case of potential acting between massive quarks inside a colourless bound state, the expansion takes the form (53), see (54) for a recent survey, where the second term in the brackets describes the higher order perturbative corrections, and the third one is the leading power correction. The latter is motivated Within perturbation theory, its origin can be traced to "infrared renormalons", for a review see (55).
What is most remarkable about the expression Eq. (23) in the context of our discussion is that it does not contain a term linear in R and thus contradicts the expected form borne out both by phenomenology of heavy quarkonia and by the lattice studies, as we have discussed above. We therefore come to the troubling conclusion that the OPE is inconsistent with the linear confining potential. Alternatively, one may look at the QCD coupling itself; in other words, to search for 1/Q 2 corrections emerging from the dressed gluon propagator.

Infrared-Finite QCD Coupling
It is interesting that such terms naturally appear when one tries to implement the idea of analyticity in the Q 2 variable by assuming the existence of spectral representation for the running coupling (58,59). For example, a 1/Q 2 -suppressed correction emerges as a result of simple removal of the Landau pole from the one-loop coupling (60): The ansatz Eq. (25) may turn out to be too simplistic. 8 Nevertheless, the idea of enforcing analyticity (causality) on α s turned out to be very efficient, see in particular (61). The corresponding technology that improves perturbative expansions is known under the name of "Analytic Perturbation Theory" (62). Impressive recent results of refurbishing of perturbative series can be found in (63).
A QCD couling that stays finite all over the complex Q 2 -plane (as an analytic α s (Q 2 ) obviously does) allows one to peek into the strong interaction domain.
One can aim at quantifying genuine confinement effects in QCD observables by linking power-suppressed non-perturbative contributions with momentum integrals (moments) of the coupling in the infrared region (59).
Markedly, the average value of the coupling that emerges from the study of the leading 1/Q power corrections to various jet shape observables in e + e − annihilation and DIS turns out to be (For a recent review see (51) and references therein.) Let us mention, in passing, that this number turns out to be suspiciously close to the corresponding integral over Eq. (25). Even more interesting a "coincidence" is that the "measured" value Eq. (26) is comfortably above the so-called critical value of the strong coupling which is necessary, as we shall discuss in the next Section, to trigger the light quark confinement, according to Gribov, see Eq. (28) below.
The quest for defining α s at large distances has a long and turbulent history, for reviews see (64,65,66). On one hand, it goes without saying that the "Landau

GRIBOV CONFINEMENT
Returning to the task of constructing consistent QFT dynamics of non-Abelian gauge fields, we must conclude that, in spite of many attempts, the problem of Gribov copies ("Gribov horizon", "Gribov uncertainties") remains essentially open today.
By the mid-80's Gribov decided, however, to change direction and not pursue pure gluodynamics. He did it not because of severe difficulties in describing the fundamental domain in the functional space: he always had his ways around technical obstacles. Gribov convinced himself (though not yet the physics community at large) that the solution to the confinement problem lies not in the understand-9 for a recent emotional review see (66) ing of the interaction of "large gluon fields" but instead in the understanding of how the QCD dynamics can be arranged as to prevent the non-Abelian fields from growing real big. There was a deep reason for this turn, which he formulated in the following words: "I found I don't know how to bind massless bosons" (read: how to dynamically construct glueballs).
As for fermions, there is a corresponding mechanism provided by the Fermi-Dirac statistics and the concept of the "Dirac sea". Spin-1 2 particles, even massless which are difficult to localize, can be held together simply by the fact that, if pulled apart, they would correspond to the free-fermion states that are occupied as belonging to the Dirac sea.

Perturbative Light Quark Confinement
As a result of the search for a possible solution of the confinement puzzle Gribov formulated for himself the key ingredients of the problem and, correspondingly, the lines to approach it: • The question of interest is not of "a" confinement, but that of "the" confinement in the real world, namely, in the world with two very light quarks -u and d -whose Compton wave lengths are much larger than the characteristic confinement scale (m q ∼ 5 − 10 MeV ≪ 1 GeV).
• No mechanism for binding massless bosons (gluons) seems to exist in Quantum Field Theory (QFT), while the Pauli exclusion principle may provide means for binding together massless fermions (light quarks).
• The problem of ultraviolet regularization may be more than a technical trick in a QFT with apparently infrared-unstable dynamics: the ultraviolet and infrared regimes of the theory may be closely linked. Example: the pion field as a Goldsone boson emerging due to spontaneous chiral symmetry breaking (short distances) and as a quark bound state (large distances). The question becomes quantitative: how strong is strong? How much of a tension does one need to break the vacuum and organize such a screening?

Supercritical Binding
In QCD one has to account for the colour Casimir operator. Then the value of the coupling above which restructuring of the perturbative vacuum leads to chiral symmetry breaking and, likely, to confinement (see (72) and references therein), translates into This number, apart from being easy to memorize, has another important quality: it is numerically small. Gribov's ideas, being understood and pursued, offer an intriguing possibly to address all the diversity and complexity of the hadron world from within the field theory with a reasonably small effective interaction strength. Take the first order self-energy diagram Σ 1 (q): a fermion (quark/electron) with momentum q virtually decays into a quark (electron) with momentum q ′ and a massless vector boson (gluon/photon) with momentum k = q − q ′ :

Gribov Equation
with G and D the fermion and boson propagators, respectively. The corresponding Feynman integral diverges linearly at q ′ → ∞. To kill the ultraviolet divergences (both linear and logarithmic), it suffices to differentiate it twice over the external momentum q.
The first Gribov's observation was that 1/k 2 of the boson propagator happens to be the Green function of the four-dimensional Laplace operator, where ∂ µ denotes the momentum differentiation. Therefore, the operation ∂ 2 µ applied to Eq. (29) takes away the internal momentum integration and leads to an algebraic expression which is local in the momentum space, k = q − q ′ = 0, This is the "Born approximation". With account of higher order radiative corrections, the first thing that happens is that the bare fermion propagator G 0 dresses up, G 0 (q) → G(q), and so do the Born vertices γ µ → Γ µ (q, q, 0). The second crucial observation was that the exact vertex function Γ µ (q, q − k, k) describing the emission of a zero momentum vector boson, k µ ≡ 0, is not an independent entity but is related with the fermion propagator by the Ward identity, This statement is literally true in Abelian theory (QED), and, after some reflection, can be made true in the non-Abelian case (QCD) as well.
Thus, we arrived to the Gribov equation for the quark Green function (71,72) where . . . stand for less singular O g 2 integral terms.
In principle, the whole PT series expansion may be constructed for the right hand side in terms of exact Green functions (and their momentum derivatives).
In particular, with account of the first subleading terms Eq. (32) becomes an integro-differential equation and its r.h.s. may be represented in the following graphic form: with black dots standing for the momentum gradient of the inverse propagatorthe exact zero-momentum boson emission vertex (69). Yet another set of higher order corrections makes the coupling run, g → g(q 2 ). In the q 2 → ∞ limit the QCD coupling vanishes due to the asymptotic freedom, and Eq. (32) turns into the free equation, ∂ 2 µ G −1 = 0, whose general solution has the form This general perturbative solution has two new arbitrary parameters ν 1 and ν 2 in addition to the familiar two (bare mass m 0 and the wave function renormalization constant Z 0 ), since the master equation is now the second order differential equation, unlike in the standard renormalization group (RG) approach.
The new terms are singular at q 2 → 0. Therefore in QED, for example, we simply drop them as unwanted, thus returning to the RG structure. Such a prescription, however, exploits the knowledge that nothing dramatic happens in the infrared domain, so that the real electron in the physical spectrum of the theory, whose propagation we seek to describe, is inherently that very same object that we put into the Lagrangian as a fundamental bare field.
Not so clear in an infrared unstable theory (QCD). Here we better keep all four terms in Eq. (33), wait and see.
At large virtualities, q 2 ≫ m 2 , the two additional terms can be looked upon as power suppressed corrections that emerge due to non-trivial structure of the QCD vacuum. The corresponding dimensional parameters can be directly linked with the non-perturbative vacuum condensates, which constituted the core of the "ITEP sum rules" famous for successful phenomenology of low-lying hadron resonances, see (73).

Critical Coupling and Chiral Symmetry Breaking
In this section we suggest a simple algebraic exercise that should help a reader to see the origin of the critical coupling and to appreciate the Gribov equation as a tool for grasping dynamical breaking of the chiral symmetry.

Let us introduce
Observing that the master equation Eq. (32) can be cast in the following compact form, Using the standard representation for the fermion propagator in terms of the wave function renormalization factor Z and the running mass m, and introducing anomalous dimensions it is straightforward to derive an explicit expression for the "vector field" A ν : Here n ν = q ν q , n 2 = 1; σ νµ = 1 2 (γ ν γ µ − γ µ γ ν ), σ νµ σ µρ = −3g νρ .
In the region of relatively large momenta, q ≫ m, we have Applying this approximation to Eq. (35) gives Equating the scalar andn terms in Eq. (35) produces then a system of coupled differential equations for anomalous dimensions: It is its stable pointΓ =Γ m = 0 which determines the ultraviolet anomalous dimensions Γ * and Γ * m . The first equationΓ = 0 is self-contained and results in where the sign of the square root was chosen such as to select among the two fixed points in Eq. (39) the stable one. 11 The anomalous dimension of the mass in Eq. (40) is driven by that of the wave function, since fromΓ m = 0 we have Substituting Γ from Eq. (41), for the stable point we finally obtain

Quarks, Pions and Confinement
As far as confinement is concerned, the approximation Eq.
It is important to notice that since pions have emerged dynamically in the theory, their coupling to quarks is not arbitrary but is tightly linked with the quark propagator itself (search for an anti-commutator of γ 5 with G −1 in Eq. (44)).
Moreover, the pion-axial current transition constant f π is not arbitrary either, but has to satisfy a definite relation which, once again, is driven by the behaviour of the exact quark Green function: The second of the two papers (

Gluon Sector
Another important open problem is to construct and to analyse an equation for the gluon similar to that for the quark Green function, from which a consistent picture of the coupling g(q) rising above the critical value in the infrared momentum region should emerge.
The difficulty with the gluon sector of the theory lies in separating the running coupling effects from an unphysical gauge dependent phase that are both present in the gluon Green function. To analyse renormalization of the gluon Gribov has used in (72) the Duffin-Kemmer formalism (also known as "linear formalism") which treats the gluon potential A a µ and the field strength The solution may lie in using the "background gauge" or in exploiting the unitarity motivated "pinch" technique developed by J. Papavassiliou, N.J. Watson and others (see (76) and references therein), as in both approaches the QCD coupling is directly related with the renormalization of the gluon Green function. Gribov has carried out this programme for QED. Here the coupling becomes supercritical at extremely short distances. Gribov's analysis aimed at resolving the long-standing problem of the "Landau pole" in the running QED coupling.

QED AT SHORT DISTANCES
The "Landau ghost" in the photon propagator at academically large momenta seems to be a formal problem so long as QED is actually a part of a broader field theoretical scheme. Nevertheless, its resolution is extremely important since it demonstrates what type of new, non-perturbative, phenomena in QFT one might expect when the strength of the coupling becomes large (dynamical symmetry breaking and appearance of Goldstone states, condensation, confinement, etc.).
Gribov did not finish the paper entitled "Quantum Electrodynamics at Short Distances". Handwritten notes he left behind, deciphered and translated into English by J. Nyiri, appeared in (77). It is important to mention that no attempt has been made by the editorial team to "debug" these notes. 12 12 This was done on purpose. By correcting omnipresent mistakes/misprints in obvious places the editors could have misled a potential reader into putting too much trust into not-so-obvious derivations and conclusions. As the matter stands now, it should be not read but rather worked through with a pencil and an A4-pad at hand. We strongly encourage to do so everyone willing Here we will reproduce essential steps of the part of Gribov's analysis that directly addressed the "Moscow zero" problem. By so doing we hope to be able to ignite enough curiosity so that our readers will be tempted to look themselves into more delicate issues (such as appearance of light super-bound states, condensation, the Goldstone phenomenon) that were also considered in (77).

Equation for Photon Polarization Operator
Electron loop Feynman diagrams for the polarization operator Π µν (k) for a photon with momentum k diverges quadratically. Therefore in order to obtain a convergent integral (and thus a finite answer) we need to differentiate it thrice over k. At the same time, conservation of current dictates where the factor Π diverges logarithmically in the ultraviolet; in the Born approximation, Let us apply successively two differential momentum operations to Eq. (46): (recall that the dot marks ξ-derivative). To get rid of the remaining logarithmic divergence it suffices to differentiate Eq. (47) over ξ. This is equivalent to applying the operator k σ ∂ σ which produceṡ to play a 1 The l.h.s. can be expressed directly through the running QED coupling. Indeed, observing that, by definition, we may write As for the r.h.s. of Eq. (48), we treat Π µν as a sum of Feynman diagrams and apply the first two differentiations diagrammatically. This results in where crosses mark momentum derivative of the bare electron propagator, (Integral over the loop momentum q diverges only logarithmically which justifies integration by parts used to derive Eq. (49).) After differentiation, the overlapping divergencies disappear that plagued the original Schwinger-Dyson equation for the polarization operator. Therefore, higher order corrections combine to renormalize four fermion propagators, G 0 (q) → G(q), both external vertices, γ µ → Γ(q, q + k, k), as well as the crosses: We thus arrive at the renormalized graph For the correction term ∆ = O (g) a perturbative series expansion can be constructed in terms of two-particle irreducible multi-loop graphs built of exact propagators and interaction vertices, Feynman integral corresponding to the leading contribution in Eq. (52) reads where we have combined the adjacent dot-operator = ∂ µ G −1 (q) and the Green function G(q) into A µ of Eq. (34).
We are interested solely in the logarithmically divergent contribution to Eq. (54) since finite pieces, after applying the last differentiation (k σ ∂ σ ) in Eq. (48), will produce but negligible corrections that are power suppressed for large external photon momenta, k ≫ m. Bearing this in mind, in the relevant integration region k ≪ q ≪ Λ UV we can omit k in the loop propagators, G(q + k) ≃ G(q), and in the exact photon vertices, Γ µ (q, q+k, k) ≃ Γ µ (q, q, 0) = −∂ µ G −1 (q), In the large momentum region we invoke Eq. (38b) to obtain Transforming the integration phase space, Tr[ ] 4 .
Substituting into Eq. (48) and taking the ξ-derivative we finally arrive at This second order differential equation generalises the standard RG equation, with the β-function expressed via anomalous dimension of the (Feynman gauge) electron wave function, Γ = Γ * (g).

Small and Large Coupling Regimes
In the small coupling regime we have Γ = O (g) ≪ 1 and Eq. (55) gives which is the standard one-loop expression for the running QED coupling that formally develops "Landau pole" in the deep ultraviolet, k ∼ m e exp( 3π 2 · 137).
However, for g = O (1) there is no reason to neglect Γ. The latter is given by From this representation, which is equivalent to the original Eq. (41), we observe that Γ is monotonically increasing with g, crosses unity at g = 1 and tends to Γ(g) We observe that in the large coupling limit the r.h.s. of Eq. (55) becomes small, resulting in logarithmic increase of the coupling at large photon virtualities: g(ξ) ≃ 2 · ξ = ln k 2 ≫ 1. Numerically large coupling g ≫ 1 does not mean that the interaction becomes really strong. If this were the case, the approximation based on neglecting higher order terms ∆ in Eq. (52) that was adopted in Eq. (55) would have been undermined. Gribov argued that since adding a photon is accompanied by additional factors A µ in each vertex, the relative magnitude of radiative corrections, in spite of g ≫ 1, may turn out to be finite, g · (A µ ) 2 = O (1).
In fact, there is no need to analyse the structure of ∆ for g ≫ 1, which regime is actually of little interest. Indeed, as have we discussed above, already at g = O (1) a new phenomenon occurs, which is supercritical binding of fermion pairs. It leads to appearance of scalar/pseudoscalar "mesons" and changes drastically the behaviour of the Abelian theory in the ultraviolet.

Composite Higgs as a Super-critically Boundtt Pair
As an interesting byproduct of his supercritical quark confinement in the infrared, at the top quark mass scale (78).
The mass of such composite Higgs particle (whose binding is similar to that of a deuteron in the zero-radius limit of nuclear forces) would have been m H = (170±7) GeV, given the experimental top quark mass m t ≃ 175 GeV (and varying α s between 2m t and 1 2 m t ).

CONCLUSIONS
Hadron phenomenology has accumulated a very impressive dossier of puzzles and hints, ranging from unexplained regularities in hadron spectroscopy to soft "forceless" hadroproduction in hard processes.
The reason why one keeps talking, 30 years later, about puzzles & hints, about constructing QCD rather than routinely applying it, lies in the conceptually new problem one faces when dealing with a non-Abelian theory with unbroken symmetry. We have to learn to master a Quantum Field Theory whose dynamics is intrinsically unstable in the infrared domain so that the objects belonging to the physical spectrum of the theory (supposedly, colourless hadrons, in the QCD con-text) have no direct one-to-one correspondence with the fundamental fields the microscopic Lagrangian of the theory is made of (coloured quarks and gluons).
In these circumstances we don't even know how to formulate at the level of the microscopic fields the fundamental properties of the theory, such as conservation of probability (unitarity) and analyticity (causality). Indeed, • What does Unitarity imply for confined objects? Gribov works on gauge theories and, in particular, all his papers, talks and lectures devoted to anomalies and the QCD confinement (including the lectures 13 The issue of quark masses is particularly damaging since a mismatch between quark and hadron thresholds (whatever the former means) significantly affects predicting the yield of heavyflavoured hadrons in hadron collisions, markedly at LHC. Needless to say, the Gribov approach and recent developments reviewed here did not yet provide a consistent recipe for dealing with QCD at large distances.
Nevertheless, an impressive progress has been made, and Gribov's ideas continue to inspire many of those who have contributed to it.
Literature Cited