The Gribov Theory of Quark Confinement

The

of Quark Confinement

itor

World Scientific

The

Gribov Theory of Quark Confinement

The

Gribov Theory of Quark Confinement Editor

J. Nyiri MTA KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary

V | f e World Scientific « •

New Jersev London* •Singapore Sine NewJerseyLondon *HongKong

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FOREWORD A. VAINSHTEIN Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455

Vladimir Gribov developed his ideas on quark confinement in Quantum Chromodynamics over a span of more than twenty years. When he died in 1997, he was in the process of writing the two papers that conclude this Volume. These two papers, the result of Gribov's long-term effort, present a theory that links confinement of colour to light quarks. Gribov started his challenging journey in 1976 with the discovery of the famous Gribov copies in gauge theories (papers 2 and 3 of this Volume), which was reported at the Leningrad Winter School in the beginning of 1977. The quark confinement was a crucial unfinished piece in Gribov's holistic picture of physics. The special role played by Goldstone particles in the Gribov theory led him to ideas about the origin of quark masses and non-elementary Higgs boson in the electroweak interactions (Paper 15 of this Volume). The starting point in Gribov's approach is that confinement is due to light quarks with underlying physics similar to supercritical phenomena in Quantum Electrodynamics in the presence of a strong electric field. The strong electric field is produced by a heavy nucleus with the electric charge Z larger than some critical value Z cr ~ C / a = 137 • C. This field leads to instability with respect to production of an electron-positron pair. The electron component of the pair goes away, and the positron and the nucleus form a bound state of small size. So, for distances larger than the size of the bound state it results in an effective screening of the electric charge: it becomes impossible to observe the charge larger that the critical one. In Quantum Chromodynamics, the strong chromoelectric field is due to the growth of the effective colour charge with distance. It leads to a pair

production of light quarks, i.e., to the reshaping of the fermion structure of the vacuum, which was the main object of Gribov's study. Related to this is an appearance of light mesons — Goldstone bosons associated with breaking of chiral symmetry. What is particular to Gribov's scenario is the presence of pointlike structure in these mesons. The Gribov theory of confinement does not belong to the mainstream of theoretical high energy physics. In a way, the community has given up efforts to construct a physical picture for confinement, taking a route of more formal studies instead. Gribov was virtually abandoned in his one-man struggle. Some of his conclusions are at variance with common beliefs, such that a theory containing only gluons is confining: Gribov claimed that pure gluodynamics presented scaling behaviour without glueballs. While it is not clear whether this is the case, I would like to emphasize that the depth of Gribov^ idea about the crucial role of light matter was. somewhat unexpectedly, confirmed in the recent progress of finding exact solutions in supersymmetric gauge theories. It has been demonstrated that supersymmetric gauge theories lead to very different phases depending on light matter content. Gribov's papers on confinement are not easy reading. Even having the privilege of multiple discussions with him. I know how difficult they sometimes are to follow. But to see the beautiful picture of the world Gribov created is also a joy and an inspiration to continue painting. Minneapolis, May 2, 2001

CONTENTS

Foreword (A. Vainshtein)

v

V. N. Gribov, Quantization of Non-Abelian Gauge Theories Preprint LENINGRAD-77-367; Proc. of the 12th LNPI Winter School on Nuclear and Elementary Particle Physics, 64 (1977); Nucl. Phys. B139, 1 (1978). Also in Quark Confinement and Gauge Theories, Phasis, Moscow (2001).

1

V. N. Gribov, Instability of Non-Abelian Gauge Theories and Impossibility of Choice of Coulomb Gauge Proc. of the 12th LNPI Winter School on Nuclear and Elementary Particle Physics, 147 (1977); SLAC-TRANS-0176 (1977). Also in Quark Confinement and Gauge Theories, Phasis, Moscow (2001).

24

V. N. Gribov, Local Confinement of Charge in Massless QED Proc. of the 15th LNPI Winter School on Nuclear and Elementary Particle Physics, 90 (1981); Preprint KFKI-1981-46 (1981); Nucl. Phys. B206, 103 (1982); Proc. of Neutrino '82 399 (1982).

39

V. N. Gribov, Outlook Proc. of Neutrino '82, 407 (1982).

68

V. N. Gribov, Anomalies as a Manifestation of the High Momentum Collective Motion in the Vacuum Preprint KFKI-1981-66 (1981); Proc. of the Workshop and Conference on Nonperturbative Methods in Quantum Theory (1986). Also in Quark Confinement and Gauge Theories, Phasis, Moscow (2001). V. N. Gribov, Anomalies and a Possible Solution of Problems of Zero-Charge and Infra-red Instability Preprint CERN-TH-4721 (1987); Phys. Lett. B194, 119 (1987). V. N. Gribov, A New Hypothesis on the Nature of Quark and Gluon Confinement Preprint KFKI-1986-29/A (1986); Phys. Scripta T15, 164 (1987); Nobel Sympos., 164 (1986).

74

92

98

V. N. Gribov, Possible Solution of the Problem of Quark Confinement Lund preprint LU-TP-91-7 (1991); presented at Perturbative QCD Workshop Meeting, Lund (1991).

103

V. N. Gribov, J. Nyiri, Supercritical Charge in Bosonic Vacuum Lund preprint LU-TP-91-15 (1991); presented at Perturbative QCD Workshop Meeting, Lund (1991).

155

Vlll

V. N. Gribov, Orsay Lectures on Confinement (I) Preprint LPTHE-ORSAY-92-60 (1993); e-Print Archive hep-ph/9403218. Also in Quantum Field Theory, Hindustan Book Agency, New Delhi (2000).

162

V. N. Gribov, Orsay Lectures on Confinement (II) Preprint LPTHE-ORSAY-94-20 (1994); e-Print Archive hep-ph/9404332. Also in Quantum Field Theory, Hindustan Book Agency, New Delhi (2000).

185

V. N. Gribov, Orsay Lectures on Confinement (III) Preprint LPT-ORSAY-99-37 (1999); e-Print Archive hep-ph/9905285. Also in Quantum Field Theory, Hindustan Book Agency, New Delhi (2000).

192

F. E. Close, Yu. L. Dokshitzer, V. N. Gribov, V. A. Khoze, M. G. Ryskin, /o(975), ao(980) as Eye-Witnesses of Confinement Preprint RAL-93-049 (1993); Phys. Lett. B319, 291 (1993). V. N. Gribov, Higgs and Top Quark Masses in the Standard Model Without Elementary Higgs Boson Preprint BONN-TK-94-11 (1994); Phys. Lett. B336, 243 (1994); e-Print Archive hep-ph/9407269.

204

213

V. N. Gribov, Bound States of Massless Fermions as a Source for New Physics Bonn preprint TK-95-35 (1995); e-Print Archive hep-ph/9512352; Lecture given at International School of Subnuclear Physics, 33rd Course: Vacuum and Vacua: The Physics of Nothing. Erice, Italy, 1 (1995).

218

V. N. Gribov, The Theory of Quark Confinement Lectures given at International School of Subnuclear Physics, 34th Course: Effective Theories and Fundamental Interactions. Erice, Italy, 30 (1996).

227

V. N. Gribov, QCD at Large and Short Distances Preprint BONN-TK-97-08 (1997); Eur. Phys. J. CIO, 70 (1999); e-Print Archive hep-ph/9807224.

239

V. N. Gribov, The Theory of Quark Confinement Preprint BONN-TK-98-09 (1999); Eur. Phys. J. CIO, 91 (1999); e-Print Archive hep-ph/9902279.

259

L. Montanet, ao(980) and /o(980) Revisited

275

V. N. Gribov

Quantization of non-Abelian Gauge Theories It is shown that the fixing of the divergence of the potential in nonAbelian theories does not fix its gauge. The ambiguity in the definition of the potential leads to the fact that, when integrating over the fields in the functional integral, it is apparently enough for us to restrict ourselves to the potentials for which the Faddeev-Popov determinants are positive. This limitation on the integration range over the potentials cancels the infrared singularity of perturbation theory and results in a linear increase of the charge interaction at large distances.

1. Introduction The quantization problem for non-Abelian gauge theories within the framework of perturbation theory was solved by Feynman [1], DeWitt [2] and Faddeev and Popov [3]. A subsequent analysis of perturbation theory in such theories (Politzer [4], Gross and Wilczek [5], Khriplovich [6]) has shown that they possess a remarkable property called asymptotic freedom. This property consists in the fact that zero-point field oscillations increase the effective charge not in the high-momentum region as in QED [7], but in the low-momentum region, i.e. at large distances between the charges. This gave hope that such theories may incorporate the phenomenon of colour confinement which is fundamental to present day ideas concerning the structure of hadrons. Answering the question as to whether colour confinement occurs in nonAbelian theories proved to be a very difficult problem since the non-Abelian fields possessing charges (colour) strongly interact in the large-wavelength region. Strong interaction between vacuum fluctuations in the region of large wavelengths means that at these wavelengths a significant role is played

Reprinted from Nuclear Physics, Vol. B139, No. 1 (1978), Gribov et al, with permission from Elsevier Science.

by field oscillations with large amplitudes, for which the substantially nonlinear character of non-Abelian theories is decisive. Thus, the problem of colour confinement is closely connected with that of the quantization of large non-linear oscillations. In this paper we show that in the region of large field amplitudes the prescription for quantization by Faddeev and Popov is to be made more precise. As will be demonstrated, it is very likely that this improvement reduces simply to an additional limitation on the integration range in the functional space of non-Abelian fields, which consists in integrating only over the fields for which the Faddeev-Popov determinant is positive. This additional limitation is not relevant for high-frequency oscillations, but substantially reduces the effective oscillation amplitudes in the low-frequency region. This in turn results in the fact that the «effective» charge interaction does not tend to infinity at finite distances as occurs in perturbation theory, but increases with the distance and goes to infinity at infinitely large distances between charges, if at all.

2. Non-uniqueness of gauge conditions The difficulties in the quantization of gauge fields are caused by the fact that the gauge field Lagrangian £ = _

4^

S P

^

F^u = d»Av - d ^

F

^'

+ [Ap, Au],

(1)

(2)

where A^ are antihermitian matrices, Sp>lM = 0, being invariant with respect to the transformation AM = S+A'^S + S+d^S,

S+ = S~\

(3)

contains non-physical variables which must be eliminated before quantization. The usual method of relativistic invariant quantization [3] is as follows. Let us consider the functional integral

W= [e-f^UdA'^

(4)

in Euclidean space-time and imagine the functional space A^ in the form shown in fig. 1 where the transverse and longitudinal components of the field

A"

UfiQzOvAv

d,Ah

A^ — Afj,

0^-^duAv

Fig. 1 An are plotted along the horizontal and vertical axes, respectively. Then for fixed A^ eq. (3) defines the line L (as a function of S) on which C is constant. The Faddeev and Popov idea is that, instead of integrating over A'p, one should integrate over matrices S and fields A^ which have a certain divergence / = d^A^. Then W is written in the form

W

x JdS-S+6[f

- S+{d^

+ [V^A^Sd^S^S],

(5)

where

A(A) = JdS- S+8[f - S+id.A'^ + [V/i(A')5^5+]}5],

(6)

V(i(A') = d» + A'tl. Since the variation with respect to S of the expression under the sign of the 6 function is d^V^A), S+dS],

^

= II°(A)H,

(7)

where the operator &(A) is defined by the equation

(8) Replacing in (4) the variables Aj1 = S , A /1 S + + S0 / 1 S + ,

(9)

we obtain W=

e-fCd4x6{f-dnAfj,)dA\\a(A)\\dS-S+.

(10)

Since (10) does not depend on / , we may integrate over / with any weight function, exp{(l/2ag2)S-p f f2dx} being commonly used for this purpose. In so doing, with the integration over S omitted, W is obtained in the form W = J exp (- J £dAx + ^ S p

[(d^d^x)

||D(A)||dA

(11)

This conclusion is correct under the essential condition that, given a field A'^, one can always find a unique field A^ with a given divergence / , i.e. there are neither situations where curve (3) crosses the line d^A^ = f several times (curve L') nor where it does not cross it at all (curve L"). We do not know any examples of situations of the type L", where one cannot find a field Ap, with an arbitrary divergence, which is gauge-equivalent to a given field A'^. However, a situation of the type V where many gauge-equivalent fields An with a given divergence correspond to a given field A' is typical in non-Abelian theories. Indeed, in order for two gauge-equivalent fields Ai^ and A2fj. with the same divergence to exist, there should be a unitary matrix S connecting Ai^ and ^2^, A2ll = S+AiltS

+ S+dltS,

(12)

and satisfying the equation d„S+[Vti(A1),S}

= 0,

(13)

or an equation obtained from it by substituting ^ ^ for Ai^ and S+ for S. In an Abelian theory, where S = etv is a unit matrix, eq. (13) reduces to the Laplace equation d \ = 0,

(14)

and to eliminate non-uniqueness it is sufficient for us to confine ourselves to the fields which decrease at infinity. In a non-Abelian case, the non-linear equation (13) cannot have growing solutions and hence even for Ai^ = 0 it has solutions for S leading to a decreasing A<m • In the appendix we

consider examples of the solution to eq. (13) for Ai^ = 0, from which it will be evident that a set of these solutions, i. e. of the transverse potentials equivalent to the vacuum, are in order of magnitude similar to a set of solutions to the Laplace equation, which grow at infinity, but that all of these, though corresponding to such S that do not tend to unity at infinity, result in the potentials A<m decreasing as 1/r. In the appendix we shall also show that, with values of Ax^ large enough, (13) has solutions for S which tend to unity at r —> oo and result in rapidly decreasing A-^i- Under these circumstances, to calculate correctly the functional integral in a non-Abelian theory, we must either replace eq. (11) by the expression

w

ru?i

&A

=l°- Wk '

(15)

where N is the number of fields gauge-equivalent to a given field A and having the same divergence, or restrict the integration range in the functional space so as to have no repetitions. An intermediate case, when both things are required, is possible. For instance, when integrating only over A^ vanishing at infinity faster than 1/r, we eliminate the fields gauge-equivalent to «small» fields, but for large enough A^ the gauge-equivalent fields will remain and hence N(A) in (15) will be needed. In this case, the problem of calculating N(A) reduces to the analysis of solutions of eq. (13) tending to unity under r —> oo which depend on the character and the magnitude of the field A^. This problem seems to be almost hopeless, but we shall demonstrate below that there exists a possibility of a sufficiently universal solution leading to interesting physical results. 3. A l i m i t a t i o n o n t h e integration range in t h e functional space In order to gain some insight into the nature of non-uniqueness in the functional space Afj,, let us see for what fields A^ there exist gauge-equivalent fields close to the former and having the same divergence, i. e. what are the conditions for solving eq. (13) with S close to unity. Substituting into (13) S = l + a,

a+ = — a,

(16)

we get n(A)a = dv\Vll(A),a]=0.

(17)

Since O(A) is the operator whose determinant enters into the functional integral, and eq. (17) is simply an equation for the eigenfunction of this operator with a zero eigenvalue, we draw the conclusion that the field A^ can only have a close equivalent field when the Faddeev-Popov determinant for this field turns into zero, or (which is the same) if the field is such that the Faddeev-Popov ghost has a zero-mass bound state. Clearly, if the field A^ is sufficiently small in the sense that the product of the width of the region where A^ differs from zero with its amplitude over the same region is small, then there are no bound states in such a field, i. e. the equation -n(A)ip

= eip

(18)

is solvable for positive e only. For a sufficiently large value and a definite sign of the field there appears a solution with e = 0, which becomes one with a negative e as the field increases further. For a particular still greater value of the field, the level with a zero e reappears, etc. Thus, one can imagine the fields for which eq. (17) is solvable as dividing the functional space into regions over each of which eq. (18) has a given number of eigenvalues, i. e. there exist a given number of bound states for the Faddeev-Popov ghosts,fig.2 shows this division of the field space into the regions Co, C i , . . . , Cn, over which the ghosts have 0 , 1 , 2 , . . . , n bound states, by the lines £i,£2,---,£n on which the ghosts have zero-mass levels. Hence, if we imagine the space of the fields Ap, in this way, it may be asked whether two near equivalent fields that can exist close to the line, say, £\, are located on different sides of this line, i. e. one field within the region Co, another in C\, or may be arbitrarily situated. We shall demonstrate below that, indeed, if there are only two near equivalent fields, they will always lie on different sides of the corresponding curve £\. Moreover, we shall show that for any field located within the region C\ close to the curve £\ there is an equivalent field within the Fig. 2 region Co close to the same curve.

If we could prove that not only for small neighbourhoods close to the curves £n, but also for any field in the region Cn there is an equivalent field in the region C„_i, we would prove that instead of integrating over the entire space of the fields A^, it would be sufficient for us to confine ourselves to the region Co, i. e. to integrating only with respect to the fields An which create no bound states for the ghosts (up to the first zero of the Faddeev-Popov determinant). However, even at the level of the things we can prove, there is a significant statement that for the functional integral, integration can be cut off on the boundary of the region Co, and if there exist fields A^ not equivalent to those over the region Co, they are separated by a finite region from the boundary i\ and are in no way connected with the region of small fields A^ lying within Co for which perturbation theory holds. We shall assume below (until the contrary is established) that these additions are either non-existent or insignificant and that the integral (15) is determined over the region Co- Generally speaking, we must retain N(A) in (15) because we have not proved that there are no equivalent fields over the region inside Co- We shall return to this subject below.

4. Proof of the field equivalence over the regions Co and C\ close to their boundary We shall first of all show that if the field A^, is close to £i, then there is always a similar field equivalent to the former, i. e. for such a field eq. (13) always has a solution with S little different from unity and tending to unity as r —>• oo. The condition for S -> 1 as r —> oo is required because a solution with S •/* 1 yields equivalent fields greatly different from the initial field. As shown in the appendix, these fields are located within the region Cx,. We write the field A^, in the form Afx = Cll + aM,

(19)

where C^ lies on £\, i. e. there exists ipo decreasing at infinity and satisfying the equation dlt[Vll(C), 0 can be easily avoided, and we shall not dwell on that. The derivation can just as readily be repeated for the fields close to any £n, imposing the orthogonality condition of the eigenfunction +00 only positive frequencies, respectively. Clearly, these conditions play the same part as those at infinity in the Euclidean case, and the linear equations will have no solution at A^ = 0 because of frequency conservation. Such solutions will exist for non-linear equations or for sufficiently large fields A^. For instance, eq. (18) at e = 0 is one for the ghost wave function in the external field A^. If the field A^ is situated on the line l\, such an equation has a solution under the boundary condition specified above, and defines the ghost transition from the state with negative energy to that with positive energy. Since the ghosts are quantized in the same way as fermions, the process is, doviously, interpreted as a classical formation of ghost pairs in the external field. In a similar manner it can be said that solutions of eq. (13) result in the fields A'^ which differ from A^, in pairs of the gauge quanta produced. The restriction of the integration in the functional integral to the region Co implies the restriction to the fields in which no classical ghost formation occurs because the formation of ghosts merely redefines the fields A^.

6. The effect of the field magnitude restriction on the zero-point oscillations and interaction in the low-momenta region In this section we shall try to analyze how a limitation on the integration range over the field in the functional integral affects the physical properties of non-Abelian theories. We shall proceed from eq. (15) for the action, disregarding the possibility for the equivalent fields to exist in CoW

Je-!Ed4x\\n(A)\\V(0)dA,

(31)

where V(O) means that the integration is performed only over the region Co- First of all, let us see whether the restriction V(D) is significant from the standpoint of what we know from the perturbation theory analysis. For this purpose, consider the Green function of the Faddeev-Popov ghost

G(fc) =

"^/e_/£~d4X(A:|n^)|fc)l|6||V(a)dA

(32)

It is well known that, if we calculate G(k) in perturbation theory, i.e., perform the integration over A in (32), omitting V(D) and expanding over the coupling constant, we get

G{k)

&(.

n ^ .

A2 x(3/22)(3/2-a/2)

(33)

where A is the ultraviolet cutoff, a is the gauge parameter in L. From this it is obvious that G(k) becomes large at a < 3 and physical k2 (in the Euclidean space) such that l l g 2 C 2 , A2

2

where (33) still holds. From the standpoint of (32), G(k) can be large only due to the integration range for the fields where Q is small, i. e. close to the lines £n.

It is interesting that transverse fields (small a) act on the ghosts as attractive fields and longitudinal fields as repulsive ones. Since the influence of longitudinal fields cancels in the calculations of gauge-invariant quantities, we may say that we study the contribution to the functional integral close to the curves tn when calculating G(k) near the «infrared pole», and hence V(D) is definitely significant at momenta below or of the order of the «infrared pole» position, whereas at large k we are within Co (low A), where V(D) is insignificant and perturbation theory works. Furthermore, V(D) makes it impossible for a singularity of G(k) to exist at finite k2 because, with k2 below the singularity position, G(k) would either reverse its sign or become complex. Both things would indicate that • has ceased to be a positive definite quantity, i. e. we have left the region Co when integrating over A^. The only possibility that now remains is that k2G(k) has a singularity at k2 = 0. Such a possibility would indicate that at k2 — 0 we feel the fields on the line l.\. Up to now, all attempts at finding the mechanism for removal of the «infrared pole» have not been successful. Higher corrections [8, 9, 10] and instantons [11, 12] only bring it nearer. If no other causes are found, V(d) will be the cause. The fact that there are no other causes for the interaction cutoff is equivalent to the statement that without V(n) zero fluctuations of the fields tend to leave the region Co- Hence it appears quite natural that the fields closest to the boundary of the region Co, i.e. connected with the singularity of k2G(k) at k2 = 0, will correspond to the real vacuum if V(D) is taken into account. For checking the above by a concrete calculation, one must write V(D) in a constructive way. Unfortunately, we have not succeeded in doing this. All we were able to do was to write this criterion to second order in perturbation theory and then calculate the functional integral taking no account of the interaction except for V(D). In this case it turns out that there appears a characteristic scale K2 defined by the condition g 2 l n A 2 / « 2 ~ 1, so that at k2 > K2 the gluon and ghost Green functions remain free. The gluon Green function D(k) has complex singularities and is non-singular at k2 —> 0. The ghost Green function under k2 -> 0 is G(k) ~ C/k4. If it were not for the roughness of the calculations and difficulties with complex singularities of D(k), this would be the right thing for the colour confinement theory. Let us show the way this is obtained. We write Gaa(k,A) =

-(k,a\l/n\k,a)

in the form of an expansion in perturbation theory (where a is the isotopic index) Gaa(k,A)=

+

I

+ I

I +...

(34)

The first-order term gives no contribution to the diagonal element. The second-order term is —

=:V

¥j

(2^

(F^p

= ¥°(k,A).

(35)

A^(q) is the Fourier component of the potential A^, V the volume of the system, a(k,A) defines positions of the poles G(k,A), if any, to a second Born approximation since

G(M)a,

(36)

Fidbr

In this case we assume, of course, that k is conserved in a typical field of zero-point fluctuations ((k\l/D\k')\k>=k is proportional to the volume of the system which is replaced by S(k-k') after averaging). The no-pole condition at a given A; is a(k, A) < 1. For simplicity, we choose a transverse gauge {a = 0). On averaging over the gluo» polarization directions A, we have

a(t A)-if

d q

'

|Aa A(g)|2

'

(l

M^

m)

If |^4a'A(g)|2 over the main range of integration with respect to q decreases monotonically with q2, as will prove to be the case in what follows, then a(k, A) decreases as k2 increases and hence as a no-level condition use can be made of

Taking (38) as a condition for V(D), replacing £ by £ 0 in (31) and omitting ||D||, instead of (31) we obtain a functional integral which is easy to calculate, if V(l - a(k, A)) is written in the form

V(l-a(0,A)) = J^Le^-^A»,.

(39)

w -/SWII^-*

-p{-^E^- A wi 2 -fE^}.

(40)

where V is the volume of the system. Calculating the integral over A, we get 1

W

7 27rt/9

2

Ai (q + Pg2/Vq2)3n/2'

(41)

n being the number of isotopic states. The integral over /3 can be obtained by the steepest-descent method, with the saddle-point value /3Q determined by 2

V2-.q4

+ /3og2/V

+ pQ

•

(42)

Setting fog2 /V = «4> with V -*• oo we get 2

"

5

/• d«g

1

_,

(43)

4

y (2TT)V + K

or 3n 2 A2 327* ^ = 1.

(44)

If the saddle-point value /?o is known, we can return to the functional integral (40), substituting /3 — /?o in it and omitting the integration over /3, so as to obtain an effective functional integral for calculating the correlation functions of the fields A. In this case, W is

W = [dAexp \ - 1 £ (k2 + u) I^WI'> .

(45)

Consequently, the gluon and ghost Green functions are

D£{k) = Aau(k)Abv(k) = 0 (T —>• — oo; otherwise the field Ai is singular as r —• 0) and tend to zero as r -» oo (r —>• +oo), then for a solution to exist at finite r, as r -> — oo, the pendulum should be in the position of unstable equilibrium, a = 0. In such an event, if its initial velocity at r —> —oo is not specifically selected, upon executing a number of oscillations in the field, the pendulum starts damping and once only the vertical force remains, it comes to stable equilibrium. Such a solution corresponds to S —> n, as r -*• oo, and the equivalent field Ai = -ndn/dxi

~ 1/r

(A.8)

decreases slowly at infinity. However, exceptional cases are possible. If sufficiently large, these forces can under specifically selected initial conditions restore the pendulum to its unstable equilibrium position. In this case, we obtain the equivalent field Ai which decreases fairly rapidly at infinity. We consider several versions of such a possibility. Let the forces and initial conditions be such that throughout the whole «time» — oo < r < +00, \a(r)\ IT and lying within the region Coo (it is easy to see that with / 2 = 1 as r —> oo, (A.9) has an infinite number of solutions) and the fields situated in Cn with a finite n. Finally, let us discuss the question as to whether for a particular field Ai an equivalent field A\ with a specified difference hAf in their divergence

can always be found. The equation for a corresponding a(r) will differ from (A.7) in the external force 2 A / e 2 r on the right-hand side perpendicular to the pendulum. In this case, it is likely that there exists, almost without exception, a solution with a tending to 2im as r —> oo because, as we have seen, if f\ and J2 are large, the solution comes into play through choosing the initial conditions; should f\,fc and A / be small, we have an inhomogeneous linear equation for which the choice is made in a trivial way. We now turn to the four-dimensional space. In this case, it is convenient to deal with the group 0(4) from which Sf7(2) is trivially separated. Instead of ir; as antihermitian matrices for infinitesimal transformations in the group 0(4), one may choose a^ = 5(7^71, — Ivl/i)- For constructing a scalar, we need an antisymmetric tensor, i.e. at least two vectors are required. This indicates that the field cannot be spherically symmetric. It can be axially symmetric if we choose as antisymmetric tensor b

v*> ~

A, yjl -

,

(A-10)

, x|» (nalay

where nM = x^/vx2 and l^ is a constant unit vector. The gauge transformation matrix between such axially symmetric fields can be written as

S = exp i 2^(r> nl)*l> >= cos-/3 + ip sin -(3, 4> = ^tiuFnu,

nt = l^rifjt, r = Jxf,.

(A.11)

The field A^ which preserves its shape under this transformation has the form Ap = fid^

+ MQ^

+ Wa.

(A. 12)

The transformation formulae between fi and fi coincide with (A.3), if a is replaced by /?. The equivalence condition is

d2P-

r 2 ( l — n{)2

(/ 2 + i ) s M - / i ( l - c o s ) 9 )

0.

(A.13)

With /1 = /2 = 0, there is a solution similar to (A.6) which is dependent on one variable p2 = r 2 ( l — nf) and has the same asymptotics. Despite

two variables, which make this equation more cumbersome, its structure is much the same as that of (A.6), and we do not see any reasons why the structure of its solutions should differ markedly from (A.6). References [1] R.P.Feynman, Acta Phys. Pol 24, 262 (1963). [2] B.DeWitt, Phys. Rev. 160, 113 (1967); 162, 1195, 1293 (1967). [3] L.D.Faddeev and V.N.Popov, Phys. Lett. 25B, 30 (1967). [4] H. D. Politzer. Phys. Rev. Lett. 30, 1346 (1973). [5] D.J. Gross and P.Wilczek, Phys. Rev. Lett 30, 1343 (1973). [6] I. B. Khriplovich, ZhETF 10, 409 (1969). [7] L. D. Landau, A. A. Abrikosov and I. M. Khalatnikov, DAN 95, 497 773, 1117 (1954). [8] A. A. Belavin and A. A. Migdal, ZhETF Pisma 19, 317 (1974). [9] D.R.T.Jones. Nucl. Phys. B75, 530 (1974). [10] W.Caswell, Phys. Rev. Lett. 33, 244 (1974). [11] A.A.Belavin, A.M.Polyakov, Yu.Tyupkin and A.S.Schwartz, Phys. Lett. 59B, 85 (1975). [12] C. G. Callan, R. Dashen and D. J. Gross, Princeton Univ. preprint C002220-115 (August 1977). [13] V.N. Gribov, Materials for the 12th LNPI Winter School. 1977, Vol. 1, p. 147.

Instability of non-Abelian gauge theories and impossibility of choice of Coulomb gauge V. N. Gribov In this lecture it is demonstrated that due to the impossibility of introducing Coulomb gauge for large fields and :o the growth of the invariant charge at large distances, a non-Abelian gauge theory can not be formulated as a theory of interacting massless particles. This assertion is a strong argument in favor of the idea that the spectrum of states in non-Abelian theories is substantially different from the spectrum of states in perturbation theory.

1. Introduction In the formulation of a free gauge theory corresponding to the group 5U(N). by analogy with quantum electrodynamics it is supposed that such a theory is describing -V2 — 1 sorts of interacting massless vector particles. Massless vector particles are described by three-dimensional transverse fields 3i{x):

IT-*OXi

In order to check that this is really the case, we have to demonstrate that all variables except Bi(x) which formally enter the gauge Lagrangian can be excluded, that is, the theory may be reduced to the interaction of only massless fields. At first glance it seems that such a proof exists and is trivial due to the possibility of formulating the theory in Coulomb gauge. In the present paper we demonstrate that this is not correct, and the usual Lagrangian in Coulomb gauge is not equivalent to the initial gauge invariant Lagrangian. The reason for this inequivalence is the fact that, contrary to electrodynamics, in non-Abelian theories it is impossible to introduce threedimensional transverse fields unambiguously (in particular, transverse fields

can be pure gauge fields). We will show also that the infrared instability of non-Abelian theories (the asymptotic freedom), demonstrated in perturbation theory, leads to the fact that this ambiguity is essential for scales where the invariant charge is of order of unity. These assertions make it most probable that the spectrum of states of non-Abelian theories does not contain massless particles. Because of charge conservation, the latter may lead to the confinement of colour. 2. C o u l o m b g a u g e Since the Lagrangian of the Yang-Mills field _1 ^ ~ TU^W^/"" L*fit/ — O^Ay

(i)

OiiA^ + lyijj) Av\

is invariant with respect to gauge transformations Alt = U-lA'U

+ U-1^-U,

(2)

where U is a unitary matrix, it is always possible to choose U in such a way that one of the components of the field goes to zero, for example, AQ. In this case C takes the form1 £ =

Hij

(3)

- 2? ^ 1\AiAi ^ - ~^ 2J ^ f ' =

dx~Aj~dx~Ai

+

[Al:Ajh

describing a mechanical system with potential energy HijHij/4. since HijH{j/A is unchanged by the transformation At = S-1BiS

However,

+ S-1-£-S,

(4)

OXi

the potential energy depends not on all components of Ai, i.e., the mechanical problem contains a cyclical variable. If we fix the potentials Bi in (4) by some condition. (4) may be understood as a formula determining, 1

The minus sign in front of C is caused by Ai being anti-Hamiltonian matrices

instead of the variables Ai, new variables: a cyclical coordinate S and noncyclical variables Bi. Passing to massless fields means that Bi is fixed by the condition

£:* =

(5)

a

If Bi is determined by the condition (5). then the cyclic coordinate is determined by the equation

where Vi(A)i> = | ^ + U , . rb] • OXi

If (6) determines S unambiguously, then (4) does the same for B{. In these variables the kinetic energy takes the form - - (Bi + Vi(B)f)

fa f =

+ V,-(B)/) = --EiEu

(7)

SS~l.

and the momentum corresponding to the cyclical coordinate 7T = V 2 ( £ ) / + ViBi = Vi(B)Ei

(8)

is conserved. Assuming TT = 0, we find the condition for the exclusion of the longitudinal component of E{. Dividing Ei into longitudinal and tranverse parts. 2* = * + ! ^ OXi

^ = 0 ,

(9)

OX{

and comparing (9) and (7), we get V V = diVi(B)f

= -•(£)/.

(10)

From the condition TT = 0 we obtain an equation for