The Gribov Theory of Quark Confinement

(51)

(52)

A{A)(p = \yi{A),di
p = [Ai,iri].

(53)

In this case, the integration should be performed over gauge-inequivalent transverse fields, as for invariant gauges in (51). The number of fields equivalent to a given field Ai is determined by the number of solutions of the equation similar to eq. (13) [Vi(A),SdiS+]

= 0.

(54)

The condition for the existence of two equivalent fields is the existence of a solution of the equation [Vi(A),di
(55)

i.e. that there is a zero eigenvalue of the operator A (A) defining the Coulomb potential ip according to eq. (53). Repeating the arguments given above for the four-dimensional case (see also the appendix), we come to

the conclusion that the integration range in (51) should be restricted to the region Co where the operator A(A) has no eigenvalues. This region coincides with the one where the Coulomb energy density VCoul = \pjd2jP

( 56 )

does not go to infinity anywhere except for its boundary. In the Coulomb gauge, instead of the functional integral we can use the Schrodinger equation

2

9J

-.# • d di 63xH [ -i9-Qj;,Ai

) 1>{Ai) = EiP(Ai),

(57)

bearing in mind that % is determined only for the region CoLet us discuss now how a limitation by the fields within CQ affects the spectrum and zero oscillations defined by eq. (57). As before, for the no-level condition in the field A; we set

}

(2TT)3

*2

< *'

<58)

obtained in a similar manner from the Green function. Omitting in H(n, A) all terms except for Ho = — ^{nf — A{d2Ai), we obtain, instead of (57), an equation for a set of oscillators E K^)

+ k2alq{k)}^{a) = E1>(a)t

(59)

k,\,q

provided that

w 2- " H F ~ <

L

(60)

k,\,q

Taking no account of (60), for free oscillations \a\tq(k)\2 ~ 1/fc and hence the left-hand side of (60) is infinite. This means that free oscillations correspond to the fields far outside the region Co- eqs. (59), (60) can be solved approximately by the variational method assuming that

v>=n fMwk,X,q

(6i)

Calculating the energy minimum for the system with the fixed value of the left-hand side in (60), we get an oscillator equation for / with k2 + Ki/k4 instead of A;4, where K 4 is the variational parameter. In this case, the groundstate energy and average squares of oscillation amplitudes will be

* 4 Ek,\,qV * 2 + ^ K 9 (*0I 2

k

(62)

Vk* + K4

The energy is at a minimum with K = 0, but as this takes place, the left-hand side in (60) equals infinity. Therefore, K is determined from the condition d3A;

1 V 3" J (2n (SiO'tVFTF

2 2

* ln£2 = l.

3TT2

k

(63)

The ghost Green function is G{k) = (k

k

I f d3fc/

k24g2n [J

(27T)3

(k2-2kk')(l-(kk')2/k2k,z)

(64)

k'{k' - A ; ) V f c ' 4 + K4

as k2 -> 0

G(k) =

552nA;4 ln(K 2 /^ 2 )'

(65)

Thus, this rough calculation shows that again there is a characteristic scale beyond which zero-point oscillations become small and the Coulomb potential increases linearly with distance. In conclusion, it should be noted that since zero-point oscillations of the fields in vacuum turn out to be on the boundary of the region Co, we have no right to ignore the Coulomb energy which may go to infinity on this boundary. However, considering that the number of zero oscillation modes pushing the system outside the region Co is infinitely large and that the boundary equation comprises one condition (60), we have good cause to think that the system will remain close to the boundary despite the

Coulomb interaction. In this case, the condition (60) will be satisfied all the same, which is indicative of a decrease in the zero oscillation amplitudes for momenta below a particular value and of a linear increase in the Coulomb potential with distance. Finally, I wish to express my sincere gratitude to A. A. Belavin, A. M. Polyakov, L. N. Lipatov and Yu. L. Dokshitzer for numerous most helpful discussions. Appendix In this appendix we consider the properties of gauge-equivalent fields with equal divergence, giving the simplest examples. Let us begin with the case of the three-dimensional space (Coulomb gauge) and the group SU(2). We consider «spherically symmetric)) fields A{, i.e., the fields dependent on one unit vector tii = Xi/r (r = ^/x). The general expression for such a field has the form Ai(x) = hR—

+ / 2 ( r ) n ^ + f3(r)nni,

(A.l)

n = inaoa, aa are Pauli matrices, n 2 = —1. Under the spherically symmetric gauge transformation of the form 5 = exp < -a(r)n

> = cos I -a J + nsin ( -a ] .

(A.2)

Ai{x) goes into Ai{x) = S+AiS + S+diS so that / i = / i cosa + f f2 + - j sina, h + 2 = ~h sina + f h + ^ )

cosa

h = f'z + l^,

'

(A.3)

(9/4

^ The condition

= h [ti + ( 2 / r ) / 8 - 2/ x /r 2 ] .

(A.4)

is equivalent to the equation a" + ( 2 / r V - (4/r 2 ) { ( / 2 + h sin a + / i (cos a - 1 ) 1 = 0 .

(A.6)

If we introduce the variable T = lnr, eq. (A.6) reduces to the equation for a pendulum (fig. 4) with damping in the field of the vertical force 4 / + 2 , horizontal force 4/i, and the force perpendicular ^ •'1 to the pendulum, —4/i \

/ >f-

4^

a + a - (2 + 4/ 2 ) sin a + 4/i (1 - cos a) = 0.

(A.7)

+•

j ' 2 , 4f

Prom this analogy the general properties of solutions to the equations of the equivalence conditions (A.5), (A.6) are readily seen. Fig. 4 If the forces f\ and / 2 are equal to zero as r —> 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)\
(A.9)

should be satisfied. This equation is simply eq. (20) for the threedimensional case. In order for eq. (A.9) to hold, - / / 2 d r should have a

particular and sufficiently large value. The field with a corresponding / 2 lies on the curve l\ independently of the values for f\ and / 3 . Taking into account the quadratic term in (A.7) enables us to get a solution with a somewhat larger or smaller / 2 (within or outside the region Co) and to determine the amplitude of a as demonstrated in the text. If / 1 = 0, there are no second-order terms and a solution exists only for / 2 larger than the value required for a zero energy level. The amplitude of a is determined by taking into account third-order terms. It is obvious that in this case there are two solutions that differ from one another by the sign of a. It can easily be shown that they are in the region Co- This situation corresponds to the phenomenon of intersection of the equivalent field lines discussed in the text and illustrated in fig. 3. The field for which f\ = 0, lies on the intersection of such lines. The occurrence of two equivalent fields in Co in this example does not point to the necessity of introducing N(A) in the region Co because it is manifestation of the symmetry of the problem with respect to reflection. It is easily shown that the fields / i , f%, / 2 and —/i, —fa,—/2 are gauge-equivalent and have equal divergence. Hence the functional integral should be divided by two, independently of the values of the fields / i , / 3 , / 2 , which do not matter. Accordingly, eq. (A.6) always has two solutions ct\ = a(fi,f2) and a 2 = — a ( — / i , / 2 ) . The second solution in the field —/i, — fs, / 2 , a 2 (—/i,/ 2 ) leads to an equivalent, reflected field obtained from / i , / 2 , /3 using the solution ai(/i,/2).With / i = 0, both reflected fields are obtained from one initial field. Two other solutions «i(—/ij/2) and cx2{fi,fz) are outside Co anyway, for small f\ and / 2 close to£i. Another possibility for the pendulum to regain its unstable equilibrium position is to execute one complete revolution (or more). This possibility may exist even with / 2 = 0, i.e., deep inside the region Co- Such solutions exist at large enough / i , but the equivalent fields corresponding to them differ greatly from the initial field and are likely to lie outside Co. For example, with / 2 = 0, / 2 is of the order of unity according to (A.3) because a in a complete revolution changes from 0 to 2TT. Hence, the fields inside Co have their equivalents of two types, i. e. the fields which possess asymptotics at oo of the type (A.8) a —> 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

(11)

The Hamiltonian of the system may be written m this case as H=^

[*i*i + f&f + \HijHij - 2pf\ , A = D^D.

(12) (13)

or. using (11),

H=

~{*i*i-P±P^\HijHijy

(14)

This completes the usual verification of the fact that the initial Lagrangian describes interacting massless gluons. However, for proving (14) we made an important assumption, namely, that the procedure of introducing transverse fields, determined by Eqs. (4) and (6), is unambiguous. If this procedure is not unambiguous, i.e. one and the same Ai corresponds to several fields Bi. then using Bi as an independent variable, we will, by summing the fields Bi. repeat the same Ai several times. If for different fields Ai the number of repetitions will be different, then the obtained result will not make sense. 3. Impossibility of unique introduction of transverse fields The question about the unique introduction of transverse fields leads to the following. Imagine that we found such a Bi, which appears to obey Eq. (4), (6). Does there exist another, also transverse field Bi: which is connected with the field Bi by the gauge transformation B'i = U+BiU + U - ^ l axi Since B{ and Bi are transverse, then U must satisfy the equation Vi(B)^U+ = 0.

(15)

(16)

This equation may be obtained as the condition for the extremum of the action 2

2

The author is indebted to A. M. Polyakov for bringing attention to the possibility of writing the action in this form.

with the additional condition U+U = 1. From this expression for the action, it is almost obvious that for sufficiently large fields B; there exist solutions of Eq. (16) with U -> 1 at r -> oc. In order to prove this, let us note that for a sufficiently small Bi W > 0 and it has an absolute minimum at U = 1. Consider the value of W on «trajectories»> near 1: ( 7 = 1 + iv,

v~ = v.

(18)

On these trajectories W = [ d3xTr(vO(B)v),

(19)

0(*),~g-

(20)

Bi,— " dxi

Since the operator 0(B) is analogous to the Schrodinger operator for a particle in the field [B,d/dx], dependent on the velocity, it is evident that for fields Bi with a sufficiently large product of the field amplitude and the width of the region where the field Bi is different from zero, the operator 0(Bi) will have negative eigenvalues. The action W will be negative for the eigenfunctions corresponding to these negative eigenvalues. Since the minimum value of the action (17) Wm is less than or equal to Wj^ (W^ is the minimum value on the «trajectories» near 1). we come to the conclusion that for the fields Bi for which O(B) has negative eigenvalues, the action (17) achieves a non-trivial extremum, and consequently. (16) has a solution with U -+ 1 at x —> oo. Hence, we can conclude that for such fields the Coulomb gauge is not determined unambiguosly. Moreover, as is well known, the linear action (19) achieves n extrema, if O(B) has n eigenvalues. If this situation persisted for the action on all «trajectories», we would have the following picture of the ambiguity of transverse potentials. Let us divide the whole space of fields Bi into regions depending on the spectrum of the operator 0(B) (Fig. 1) so that to each region Cn belongs a field for which O(B) achieves «n» eigenvalues. The boundaries between regions, Li, correspond to fields for which O(Bi) has a zero eigenvalue. Thus, in the region CQ the transverse potentials are determined uniquely. In all other regions Cn we find from each potential Bi at least n potentials in the same or any other region, except Co, connected with the potential Bi via a gauge transformation. We shall

demonstrate below these phenomena in an example of particular solutions of Eq. (16). Now we note that similar problems of uniqueness arise also in a relativistically invariant formulation of the theory. For example, in the gauge dAn/dXfj, = 0 the condition of uniqueness is determined by the same Eq. (16), but for four variables. If, as it is often the case, the theory is formulated in four-dimensional Euclidean space, the situation coincides exactly with that considered above. Consequently, in the Euclidean case the gauge OA^/dXfj, = 0 can also not be introduced for large fields. In the pseudo-Euclidean case the situation is not so clear and an additional analysis is required. 4. I n c r e a s e of t h e invariant charge i n n o n - A b e l i a n t h e o r i e s The calculation of the invariant charge in Coulomb gauge was carried out by I. B. Khriplovich [1] even before the discovery of asymptotic freedom in non-Abelian theories [2]. We present this calculation in order to explain the reason of the growth of the invariant charge and to demonstrate that the large values of the invariant charge originate from averaging over fields for which the Coulomb gauge cannot be introduced. For the calculation of the invariant charge we use, following I. B. Khriplovich, one of its possible definitions, namely that of a quantity which determines the interaction energy of two heavy sources. For this end we substitute p by p + ph in the expression for the energy of the system, and calculate the second order correction in ph to the vacuum energy. This correction takes the form ^cou. = ~

f d3xph(x)(0\±\0)pk(x)

+£

^ 3 ^ -

(21)

where VOn = \Jd:ixTrph(x)(0\±p(x)\0),

A ^ = • ^ • V = d2i> + 2[BU diV) + [Budi^[Bi,

(22)

dM.

The first term describes the usual Coulomb interaction, connected with the propagation of the Coulomb field from one charge to another with the only

30

difference that the Coulomb field does not propagate freely (the propagator is not equal to d~2) but in the field of zero-point oscillations of the gluons B{. The second term, which is negative and therefore decreases the interaction, is the usual vacuum polarization in an external field. Hence, the increase of the interaction can originate only from the first term, i.e. from the change in the conditions of propagation of the Coulomb field. The fact that this change of the conditions of propagation leads to an increase of the interaction at large distances is clear from the following. Consider the operator -1/A before averaging over the field B{\

where e(k), ipk are the eigenvalues and eigenfunctions of the operator A with momentum k at infinity. It is clear that if one calculates G(r. r ) for a field Bi lying near Ln (the boundary of the region Cn in Fig. 1), where the operator • has a zero eigenvalue, then the eigenvalue of A will also be zero, and G(r, r) will go to oo. If we calculated G for a potential within the region Co, where the potential is either small or repulsive, then the change in the density of states would not be essential. This is so even if the average value of Bi equals zero, since the attractive action is more essential than the repulsive one, and the presence of zero point oscillations leads to an increase of the interaction. It is also clear that the intensifying effect of the medium grows with growing distance between the charges, because in this case the probability increases that in the region between the charges there is a zero point oscillation field lying near Ln, guaranteeing a level at zero energy of the operator A. The discussed phenomena are explicitly seen in the calculation of the invariant charge in perturbation theory. For the calculation of the first term in (21) we must expand the operator 1/A and, consequently, 1/D in a series of powers of B:

-k = -h + h[Bi'di]h~h[5"di] h[B-di] h-

(24)

After averaging over the fields Bi, we obtain the following expression for the contribution of this term to the Coulomb energy in momentum representation (with ph = g25(x)): d3k'C2 (2TT)3{k-k')22k'

U) _ 5 l 2 Kcoul k2 l + 3 5

1-

~2l

{kk')< k2k" n2r2

A2

(25) 4TT 2 A:2 ~ k2 corresponding to the diagrams in Fig. 2, where the solid line corresponds to the Coulomb propagator, and the dashed line to the average value of the products of fields B{, i.e., to the transverse Green function of gluons at equal times. For the calculation of the second term in (21), one can substitute A by d2. If so, V^Jj reduces to the usual expression for the vacuum polarization, corresponding to the diagram in Fig. 3: (2) _

c\2

1+

,nA2

(26)

H^4)

(27)

v:coul

^

k 487T"

Combining (25) and (26), we obtain 9 V'coul = ^2

which coincides with the usual expression for the invariant charge in first order of g2. In the calculation of higher orders of g2, in logarithmic approximation the corrections to the vertex parts and the Green functions behave in the same way as in QED, and Koul = 9

k2

i-cl

2,l n A2 Tk2? 487T29 11

(28)

Cojnparing (21) and (28) and remembering that the second term in (21) is negative, we see that in the region 2 1 _ C 7 2

11 2, A2 7 ln-rr k2 48^

+ Fig. 2

Fig. 3

where the logarithmic approximation is still applicable, begins a huge (for g2 < 1) increase in the strength of the interaction, so that (0|1/A|0) ~ ~ l/k2g2. Such an increase in (0|1/A|0) can take place only because fields for which 0(B) has eigenvalues close to zero, play an essential role in the averaging, i.e. near to LQ or even to other Li in Fig. 1. Hence, the invariant charge of the order of unity emerges from averaging over fields for which, generally speaking, the usual theory is already incorrect due to the ambiguity in the introduction of transverse fields. The same may be said in another way. On the space scale at which the invariant charge is of the order of unity, the fluctuations of the transverse field are so large that they cannot be described as interactions of massless transverse fields of gluons. As was already noted in the introduction, this may mean that at such scales massless gluons do not exist. The latter, in its turn, may mean that these scales determine the radius of colour confinement. 5. P a r t i c u l a r s o l u t i o n s of t h e chiral e q u a t i o n Henceforth, we shall restrict ourselves to the SU{2) group and shall consider first the case 5* = 0, i.e., we shall ascertain whether there exist purely transverse gauge fields. We will seek in this case a very simple spherically symmetric solution in the form U _= e ia(r)ftT*

(29)

where 7v are Pauli matrices, and ,

_

^

i_

2

_

r Substituting (29) into (17), we get W

-I

r dr

fda\2

2 sin 2 a

\dr J

(30)

Hence, the equation for a(r) is d2 r--^(ar) or1

— sin 2a = 0.

(31)

v(a) a • >

Fig. 4

Fig. 5

If we introduce the variable t = lnr, we come to the equation for a pendulum in a gravitational field with damping (Fig. 4): a + a — sin 2a = 0.

(32)

The potential energy of the pendulum is v(a) = —2 sin a (Fig. 5). It is obvious that the solution, nonsingular for r -»• 0 (t —>• — x), corresponds to a pendulum which at t -*• —oo is in the position of unstable equilibrium a = 0 (or = rnr). When t —> — oo, we have a(t) -+ 7r —»• jel.

(33)

When t -> +oo, we have a(t) -> ± | , and, owing to the presence of the damping, for any t we get \a(t)\ < it. The field B{ corresponding to such an a(r) is not singular for finite values of r, and for r —y oc it has the form: Q n

f

-2

(34)

i.e., it decreases as 1/r. The solution found for ?7(r), and consequently for B (r). is characterized by four parameters: three parameters T^o* which define the reference point, and a parameter 7, which has the meaning of an inverse radius of the region beyond which B\ decreases. We have shown that there is an ambiguity characterized by at least four parameters, i.e., corresponding to the zeroth field Ai we have at least a four-parameter family of fields B{. To understand how significant this ambiguity is, it is useful to find out where the fields Bi are situated from the point of view of describing the space of the fields Bi with the help of Fig. 1. The first thing which is

obvious is that the equation 0(B)tp = 0 in the field Bi has a solution with t/}, decreasing as r -> oo. This is clear from the fact that Bi depends on four arbitrary parameters. With an infinitely small variation of the parameters, we switch to a field that satisfies the same conditions, i.e., we perform a gauge transformation with U close to unity, without altering the behavior of the field as r —> oo, which does not depend on the parameters. The latter means that ^ ( r ) —>• 0 as r -*• oo. We are then left with two possibilities: either i/j(r) is square integrable, in which case Bi lies on one of the lines £ n , or ijj{r) is not square integrable. In this latter case, in the field Bi there should be a bunching of the levels toward zero, i. e., an infinite number of levels, and Bi(r) lies in the region Coo. A direct substitution of Bj into the equation Dtp — 0 leads to an equation which reduces to the Schrodinger equation with a potential that decreases as 1/r and has an infinite number of levels. Hence, the fields Bt corresponding to the zeroth Ai from the point of view of Fig. 1, are situated infinitely far away» from Bi = 0. One can adduce arguments in support of the contention that all solutions of Eq. (16) for Bi = 0 possess this property. Let us now discuss Eq. (16) in the presence of the field Bi, and for the sake of simplicity consider the field Bi as having a simple form convenient for solving the equation. We let Bi(x) have the form:

Bf{x) = \eapi^f(r),

(35)

where / ( r ) is an arbitrary function. This field is transverse. Selecting U in the same form (29) and substituting (35) and (29) into (17), we get an action in the form

l7 = r2

/ *{(^) 2+2! ^ [1 - /(r)1 }

(36)

and an equation a + a - sin2a[l - / ( r ) ] = 0,

(37)

i.e., the equation for a pendulum in a time-dependent field. This equation has solutions of the same type as does the equation for / = 0, i. e., solutions in which a -> f as r —• oo. However, if Bi is sufficiently large, Eq. (37) has

other solutions too: a —> 0 as r —>• oo or a —y n as T —> oc, which will lead to fields Bt decreasing rapidly as r —> oc. It is clear that one can select a field such that the pendulum, after starting to move from a position of unstable equilibrium with an energy equal to zero, will at a certain instant change its direction of velocity under the influence of the field and, after acquiring a positive energy, will start to damp, approaching asymptotically its starting position [a(oo) = 0]. One can also select a field / such that the pendulum, after increasing its velocity at a moment r, will start to approach the starting position from the opposite direction ot(oc) = TV. If f(r) has a more complex dependence on r, then the pendulum can pass several times through a position of unstable equilibrium and only then start to approach it asymptotically. Hence, we come to the conclusion that, if the transverse field B{ is sufficiently large, corresponding to it there can be a whole set of transverse fields B{ related to the field B{ through a gauge transformation and decreasing fairly rapidly as r —• oo. Let us now discuss what determines the magnitude of the field f(r) in which the pendulum returns to the equilibrium position. Numerical analysis shows that a minimal restriction on / ( T ) occurs if / ( r ) acts at moments r at which the deviation a(r) is still much smaller than 1. In this case, instead of (37), we have d + d - 2 a ( l - / ) = 0.

(38)

Equation (38) is nothing but the equation D ^ = 0 when ip = a(r)h, with the particular form of B{ in expression (35). Consequently, the «minimal» field is the field on the curve LQ. However, for «minimal» / = /o, B{ = Bi, and in order to have an ambiguity, / has to be larger than /o. There will then exist a negative — e eigenvalue of the operator • . To determine a(r) in this case, we set a(r) =ae + a,

(39)

d £ + d e - 2a e (l - / ) = +ea(,

(40)

where

in which ae decreases at large r. Substituting (39) into (37) and considering d and ae to be small, and that d
e

-^(l-/).

(41)

For (41) to have a non-singular solution at e —> 0, it is necessary that the right-hand side be orthogonal to ae: dr

e Q

2_4

a

4

( w )

= 0.

(42)

This relation determines the normalization of ae: a 2 ~ e. i.e.. ae ~ >/e. Then a ~ e2 also decreases with large r. As / further increases, e (the binding energy) increases, and the expansion in powers of e deteriorates. However, the solution still exists. When / increases to rhe point where a second level with a small binding energy e appears, then, by repeating the above operations, we shall show that a second solution appears. We therefore get a degree of ambiguity equal to the number of eigenvalues of the operator • . Actually, owing to the fact that a(oc) can equal not only zero but mr as well (the pendulum can sail through the equilibrium position several times from different directions), the degree of ambiguity of the fields B{ is even larger. 6. T r a n s v e r s e g a u g e fields In conclusion, let us discuss whether a significant role in the interaction of gluons may not be played by the purely transverse gauge fields which slowly decrease as r —• oo (34), found in the preceding section. The existence of such transverse fields means that even in transverse gauge we have, along with the vacuum Bi = 0, vacua described by transverse fields with the asymptotic form (34). From the topological point of view, these fields have a half-integer topologic charge. If between such vacua there existed tunneling transitions (analogous to the tunneling transitions between the vacua with integer topologic charges, achieved by the instanton solutions of Ref. 4), then, regardless of the magnitude of the coupling constant and of the instability of perturbation theory, large transverse fields would always be present in a physical vacuum. Then the entire theory would change completely. But if the probability amplitude for such transitions equals zero, we can confine ourselves to the transverse fields which decrease more slowly than 1/r, and we shall have one vacuum in the transverse gauge. Attempting to calculate the probability of such transitions, one gets the impression that this probability equals zero in a purely local theory. To understand the reason for this, we shall take as an example a field of the

form B?(T>,t) = ±ea(ri%(l-U(r,t)),

(43)

where U — 1 corresponds to the zeroth field, and U = — 1 to the transverse gauge field with a zero radius. The transition from the field with U — 1 to the field / with U = — 1 is the one that interests us. The action for a Yang-Mills field of the form (43) has the form W = ±fdrdt{u*-U'2-±(1-Ur}.

(44)

This action describes a string with the two equilibrium positions U = 1, U = — 1. Between these equilibrium positions, and beyond them, there is a potential barrier whose height at the point r = 0 is infinite and tends toward zero as r —> oo . The energy of the system is finite only under the condition {7(0, t) = ± 1 . If 17(0, t) is fixed by one of these values, e. g., U = 1. then for any excitations of the string involving jumps across the barrier for arbitrary values of r except r = 0, the string will slip into the region of large r and, after a finite length of time, will arrive at the equilibrium state U = 1 for any finite r. However, even a tunneling variation of U(0, t) is impossible, for the barrier has a finite width and an infinite height. Hence, any state of the string is characterized by the values {7(0, t) = ± 1 , between which there are no transitions. Transitions would occur only if we introduced a cutoff of the interaction at r = 0. The situation changes it there are monopoles in the theory. Since the monopole field is defined by (43) for U = 0 and decreases at infinity only as 1/r. it is already impossible for us to restrict ourselves to fields which decrease more slowly than 1/r. Then any solution of Eq. (16) is possible, and, consequently, there is no way tofixthe transverse gauge in the presence of a monopole. It is interesting to note, though, that at large distances the monopole field is invariant with respect to a gauge transformation with a matrix having the form (29). In conclusion, I wish to express my profound thanks to A. A. Belavin, Yu. L. Dokshitzer, A. M. Polyakov, V. Korepin, and I. B. Khriplovich for their numerous and extremely helpful comments. Bibliography [1] L B . Khriplovich. YaF 10 409 (1969).

[2] D.I. Gross, P. Wilczek. Phys. Rev. Lett.30 1343 (1973). H. D. Politzer. Phys. Rev. Lett. 30 1346 (1973). [3] A. M. Altukhov. I. B. Khriplovich. YaF 11 902 (1970). [4] A.A.Belavin. A. M. Polyakov, A.S.Schwartz, Yu.S.Tyupkin. Phys. Lett. 59B 85 (1975).

Nuclear Physics B206 (1982) 103-131 © North-Holland Publishing Company

LOCAL CONFINEMENT OF CHARGE IN MASSLESS QED V.N. GRIBOV L.D. Landau Institute for Theoretical Physics, USSR Academy of Sciences, Moscow 117334, Vorobjevskoe shosse 2, USSR Received 27 July 1981 It is shown that massless electric charges cannot exist in nature as they are completely locally screened in the process of formation.

1. Introduction The possibility that massless charged particles exist has always been doubted, yet no detailed analysis of the problem has ever been performed. It was not clear what would happen if such particles were introduced in the theory. The introduction of massless charged particles in the theory occurs permanently and it is a very attractive approach since the introduction of particles with bare masses different from zero leads to the problem of eigenmasses of particles for the solution of which we have so far no ideas. If bare particles have no masses, the mechanism of spontaneous symmetry breaking (condensate formation) can possibly provide a reasonable explanation of the appearance of mass of the observed particles. To understand the essence of the problem associated with the masslessness of particles it is sufficient to recall what problems were encountered in the thirties in the description of massive particles and how they were solved. In quantum electrodynamics there exist bare massive particles before the interaction is switched on. The amplitudes of creation and scattering of such particles are easily calculated in the Born approximation and have a simple and beautiful structure. However, the calculation of photon emission amplitudes in such processes leads to infrared divergences which show that the number of photons created in these processes is large. This has been interpreted as the creation of the particle's classical field. Besides, it has been found that this classical field polarizes the vacuum. As a result now every student knows that an electron is created with a definite charge distribution and a classical field around it. The difficulty of the problem associated with the masslessness of particles becomes evident if one remembers that the probability of photon emission for time M n a process with sufficiently high momentum transfer q at m2t/q > 1 is determined by 2

a In —j

m

m

Q1

103

Reprinted from Nuclear Physics, Vol. B206, 103 - 131 (1982), Gribov et al, with permission from Elsevier Science.

104

V.N. Gribov / Local confinement of charge 2

which at m -» 0 tends to oo. That means that the number of photons appearing in the creation process of massless particles is essentially larger than that for massive ones. This leads to the fact that the classical field of massless particles is rather singular, which would not have been terrible, had it not resulted in a strong vacuum polarization. In the present paper we show that massless charged particles created (in a pair with an antiparticle) in a region of the size of the order of 1/q will be screened due to vacuum polarization for the time t determined by the condition ^\nqt~\,

(1)

where a =a(q ) is the charge of the particle. As a result a massless neutral object (a pair of neutral objects) of size of the order of l/q will be formed. This result seems natural from the point of view of the analysis of zero charge in quantum electrodynamics. The phenomenon is the following. Let us study the creation process of two classical particles with charges e0 and —e0 moving in opposite directions. The current corresponding to this process is of the form f* = — [v^S(vx)-vlS(v'x)]S(x2)^(x0),

(2)

where v^, v'^ are light-like four-vectors of the velocities of the charges. The electromagnetic field, created by this current is the following*:

Al = a0(^—V-A#(x2)-&{x0), \VX

Flxv=2a0[

; VX

\

(3)

V X)

)S(x).

V X

(4)

J

The strength of the field is different from zero only on the light cone or on the sphere of the radius t, where t is the time from the moment the charge was created (fig. 1). Thanks to this simple form of the field it is possible to calculate exactly the current appearing in the vacuum of free massless fermions. It proves to be equal to 3ir \vx

v x)

IT

x^jvn) , (6) nx where n is a light-like vector normal to the cone in the point x. The current is oriented along the meridians of the sphere, the poles of which are determined by ±

vIJ.=vl,

* In this paper AK denotes egA^; ]„. is the density of the number of particles.

V.N. Gribov / Local confinement of charge

FIG

105

1.

the instantaneous position of the charges. The charge density on the sphere equals zero. At the points where the charges are positioned there is a 5-like diverging contribution to the current (r-»0), il.ll(x) = -~\n(^)jT(x),

(7)

necessary from the point of view of charge conservation. The divergence is due to the fact that the value of the charge transferred from one pole to the other for time t is proportional not to t but to In? and is infinite at any finite moment of time. It implies that physically it is impossible to create two charges instantaneously and it is necessary to introduce at least an infinitely small duration of the formation process T. This is done when the final expression (7) is derived. So far we have described the result for vacuum polarization in a pure bremsstrahlung field of two massless charges created for an arbitrarily small time T. In fact, this is not the solution of the problem of what happens in the creation process of two massless charges since vacuum polarization changes the field of the created charges. To get a correct answer it is necessary to solve a self-consistent problem, inserting into the Maxwell equations the current in the form of a sum of external and polarization currents. This problem can also be solved and the result for the total current on the sphere /M = ; ' " ' + / ^ is the following. The current on the sphere outside the points where the particles are positioned, is equal to •o ;

=

^ (l + yoinV)) 2 '

(8)

where ; ° , ± is the current in a pure bremsstrahlung field (5), y 0 = a 0 /37r. The local contribution at the points where the particles are positioned is •ext

' • ^ - ( l + yoMr/T))The field on the sphere is of the form

(9)

106

V.N. Gribov / Local confinement of charge

FIG

2.

This means that massless charges created for time r will be completely locally screened in a sufficiently large time when y In (t/r) » 1 . In the self-consistent field apart from the currents and field on the sphere (8)-(10), there are also currents and fields inside the sphere which we shall discuss. Let us consider how this classical approach is connected with the quantum description of the creation process of massless charges. The field A M (x) of the charged particles created by the external field (for example, e + e~ annihilation) and observed in the points xi, x2 is determined by the two diagrams of fig. 2. It is not difficult to show that the contribution of these two diagrams leads to eq. (3) multiplied by the amplitude of fig. 3 everywhere inside the sphere of fig. 1 with the exception of the shaded region. The width of this shaded region is of the order of \/q if the external field is a packet of size \/q. The bremsstrahlung field corresponds to the limit q -* oo. Calculating the vacuum polarization in the creation process of massless charges in the first approximation (fig. 4) and doing it in the formal limit q -> oo, one gets a current different from zero in the shaded region. This current has the same structure as (5) differing only by the replacement of u£ by —uM. This replacement implies that the density of the charge on the sphere is non-zero; its distribution is determined by the denominators vx ~2(1 - c o s + cos •&) and is singular near the charges. However, as is shown in the text, the charge distribution in the shaded region is defined by the details of the behaviour of the external current during the early stage of particle creation (t<\/q). This means, that the formal calculation in the limit q -* 00, i.e. the replacement of calculations of the real diagrams by the calculations of the current in the singular bremsstrahlung field, is not correct. The structure (5) of the vacuum current is not due to higher order corrections but is a result of more thorough calculations of the integral features of the singular current. In the region near the external charges (black region) one can use the bremsstrahlung approximation. The result (7) literally coincides with the perturbation theory.

FIG

3.

FIG

4.

V.N. Gribov I Local confinement of charge

107

a) FIG

5.

Higher order corrections (fig. 5 a) are very essential if one wants to calculate the detailed behaviour inside both regions, but they do not influence the integral features of the singular current. Both results (5) and (7) are analogous to the sum rules for inclusive processes, but in coordinate space. The self-consistent solution (8)-(10) comes from summing the diagrams of fig. 5b. The described result is interesting from the point of view of the zero-charge problem [1] and the possible features of quantum electrodynamics at small distances. According to [1] the renormalized charge as a function of the cut-off A and mass m of the particle can be written in the form a(m ) =

«(A) l + (a(A)/37r)ln(A 2 /m 2 )'

a(m ) goes to zero when either A goes to infinity or m goes to zero. The zero value of a(m2) in the limit A -»oo means total screening of the particle's charge by the vacuum polarization charge distributed in the region from 1/A up to 1/m. This means, that the charged particles we are talking about and the masses of which enter the formula cannot exist. The above result was considered as a contradiction inside the theory due to its local structure. It was not clear how to handle the limit ra2-»0 because of difficulties connected with the infrared divergences of the theory. These difficulties can be avoided by considering the process for a final time t and introducing the classical field instead of the photons. Final time considerations of the process are preferable also because in this case one does not need any hypotheses about the character of the stationary states existing in the theory. The statement in the present paper on the total screening of the charge in the limit m -» 0 is in accordance with the idea about the equivalence of both the limits m -» 0 and A -* oo. The essential difference is due to the fact that, in our case, the screening charge is distributed not over the region 1/A
108

V.N. Gribov / Local confinement of charge

is not in contradiction with the equivalence of the limits m -» 0 and A -» oo, since the charge distribution from 1/A up to 1/m, which appears in the usual approach is, in fact, the charge distribution inside the stationary state (charged particles) not existing in the limit A -» oo. In our approach it is quite natural that point-like neutral particles are created in the limit A -> oo too. Indeed, applying the calculation described above not to the creation of massless charges but to that of massive ones, the only differences would be that (instead of being valid for an arbitrary t) the formulae would be correct only for a time t less than q/m2, where q is the momentum of the charge. At time less than this q/m2 the masses of the particles do not figure at all in the calculations. If the momentum q of the particle is of the order of A, then at the moment t = A/m 2 the changing charge of the created object becomes, according to (9), «(A) (l + (a(A)/37r)ln(A 2 /m 2 )) 2 ' which is going to zero as A -» oo. From all that we have said, it looks natural to suppose that the solution of the zero-charge problem is the following in both massless and massive QED in the limit A -»oo. In the theory there appear massless neutral point-like particles (size of the order of 1/A) which are "bound states" of bare charged particles. If one proposes such a hypothesis, it is interesting to discuss what happens, with these particles because of the existence of vacuum fluctuations which we have neglected up to now. It is quite natural to have among these particles scalar ones. However, scalar massless particles must condense and create an average field in the vacuum. Since the time these particles require for their formation is of the order of ~l/x e~ 1 / a ( x ) , we do not see any reasonable equations for the density of the condensate except for equations of the type «(A), A2 ——In— ~ 1 . J7T

jX

The interactions of massless neutral particles with this condensate would give them a mass probably of the same order as /x. If the bare charged particles of the theory are massless, they also can obtain a mass m, but of the order of a V because of their coupling constant a. The value of n depends on the details of the mechanism h produces the mass. The result would be the usual QED with massive charged particles plus heavy neutral particles with mass of order (1 / a ") m. This theory would be very similar to the standard theory of electroweak interactions. In conclusion we should like to stress that, independently of the hypothesis about the existence of neutral massless particles in the theory, formulae (8)-(10) describe in the framework of the usual QED the early stages (t
V.N. Gribov / Local confinement of charge

109

from the usual electrons and positrons (the charge of these states is much less than the electron charge). Therefore it is natural to expect that decays of these states would produce jets analogous to the jets in hadronic processes. 2. Solution of the Dirac equation for massless particles in a field, singular on the light cone To calculate the vacuum current in the bremsstrahlung field (3), (4) let us find the solution of the Cauchy problem for the Dirac equation in such a field. Outside the light cone A^ = 0 the wave function of a fermion satisfies the free equation: d

(11)

Inside the cone V<J, = (d-iA)4, = 0.

(12)

The field (4) inside the cone is a pure gauge field: /> df A»=T~,

VX / = a0ln — ,

OX^

(13)

V X

and the wave function ip may be written as tp = eif
-4(l,J).

*-*(!.!).

a <(3, (3>0 inside the cone; z2 = n2 = 0, zn = 1:

110

V.N. Gribov / Local confinement of charge

3 =

i

adl> (16)

a —p V = y_Va + r + V / 3 +V- L , y+ = y ^ , Writing down i// as

y- = yMrtM,

4i = 4>+ + 4>-,

y+=y!l=0,
y+y_ + y_y + = 2 . (Jz-^hy+y-ip,

(17)

and multiplying (12) by y + and y^, respectively, we shall obtain

Va
(17')

V ^ + ly-Vi/^O. Taking into account that __,_ 1 _2 y+" = — - y + y „ v „ = a— (3

1 _z -y„v„y+ a —p

1 dz„ ^y-ypT—a—p dzp

7— = ovp — zvnn — znnv = o» aZp

(18) dzp

dzp '

y ± V V = y ± V-V ± ±

(/rT , a —p

then the Dirac equation is of the form V a ( a - / 3 ) ^ - + 5y + VV + = 0 , (19) V (3 (a-/3)(A + + | y _ V > _ = 0 . Since eq. (19) does not involve the derivative of - may be discontinuous on the cone while ip+ should be continuous on the cone because in the opposite case the second equation would have contained <5(/3). Thus, t//+ = + and the potential

'P- = ^T±TAadaciF^')V^+,

(20)

2 ( a - j S ) Jp

F(x,x')=

( da"A M 0c")z M . •*a'

(21)

V.N. Gribov I Local confinement of charge

111

On the cone (at /? = 0) we have, therefore,

J

i a

da e

V
o

(22) near the cone (at small /?), perturbation theory is always applicable, since ip+ is equal to

<Ma,|8) = *>+(«, 0=0)- —^—- f e ^ ' M / S ^ a - p ' ) * - , 2 ( a - / 3 ) Jo FQ3,/3') = f

(23)

d/TA^,

and at small /3 the correction to 1^+ is small independently of the form of the field. Despite the fact that we are interested in the current near the cone, to employ formula (23) is not convenient. Iterations of this formula are in essence the expansion of i/f in a power series of @. If

(24)

where GR{x, x') is the retarded Green function satisfying the equation VxGR(x,x')

= 8(x-x').

(25)

Due to the multiplier x', eq. (24) involves only
eHfM-f{*'))GR(x,x'), (26)

f(x) is determined by the integration on the straight line C in fig. 6. G R satisfies the equation (d-iB(x))GR(x,x')

= 8(x-x'),

(27)

(28) * * = ** + ( « ' - a ) * * ,

112

V.N. Gribov / Local confinement of charge

GK(x, x') = G0(x, x') + J" G0(x, x")iB(x")G*(x", x') d'x".

(29)

Since B is determined by the strength of the field inside the cone, which is assumed to be finite at /3 -*• 0, and the integration region in (29) tends to zero at /3 -» 0, the second term in (29) is of the order @B: G0 =

^-dS((x-xf). LIT

Formula (24) for
f d4o

7^3^(-Po)S(pX"(x2)y^p(xi),

J (27TT (2ir)

L=Spy„G-(xi,x2)

2i (JCI — x2)

(30)

= —-, rztr \x\-xi) jp equals zero after the averaging with respect to eM = (*i - x2)^. At t > 0

(31)

j^S(p2)^(-p0)^p(x2)ylxiJ,Kp(x1)expi{ij

(32)

L=2zZj

dx^J ,

where i/fp are wave functions in the external field coinciding with oo. In the region outside the light cone the current is equal to zero. Inside the cone, if one employs (24), the current may be written in the form /„ = 4 f d V d4x"S(x,2)S(x"2) xexp[/[

Sp y»GR(x, x')x'G-(x',

df^4J.

x")x"GR(x2, x") (33)

This expression corresponds to the diagram of fig. 7, which shows that to make a contribution to the current, the particle should enter the region inside the cone and go out of it in the Feynman sense. The expression x'G(x', x")x" involved in (32) has a wonderful property: x'G(x',x")x"~.

, 1 ,.4[x'2x"-x"2x']. (x'-x")

(34)

It is equal to zero with the exception of the situation where x' and x" are on the same cone ray (the straight line coming from the vertex). It means that if the

V.N. Gribov / Local confinement of charge

FIG

7.

113

8.

FIG

principle term of (26), (29) is inserted as G R -j(f-n

(35)

G0(x,x'),

we arrive at fig. 8, x'->x", since a free propagation of the wave coming into the point x = Xi = x2 occurs along the cone ray with its vertex at the point x. Then the potential in GK{x, x') and G^(x, x") is cancelled and we obtain an expression for the current in the absence of the field, which yields zero after averaging over e. Corrections to G R being taken into account leads to diagrams of fig. 9 type which are different from zero inside the cone. Thus, we come to the conclusion that in the case of the bremsstrahlung field when the gauge-invariant potential 2?M = 0 inside the cone, the current inside the cone also equals zero. In the general case the current inside the cone is determined by the strength of the field inside the cone. To calculate the current according to (33) it is necessary to calculate (34) for x'G — (x1, x")x" in the limit x' ,x" -» 0 which results in »,„ , ,

»x-»

'

x'G-(x', x")x" = - — j 47T

z'S(z'z") a

—r—r.

(36)

—a

and, accordingly, the expression for the current inside the cone acquires the form / f A xd x 'M = 2 —; r 8(x'2)8(x"2)8(z'z") IT J a —a xSp y„GK(x, x')z'GR(x2, x").

C exp [i

2

Av dx„] J,, (37)

Let us now discuss the singular part of the current on the cone. In this case (37) corresponds to the diagram of fig. 10. In the pure bremsstrahlung field J,

77 J

a

- a

xSp y^Goix, x')z'G0(x2,

FIG

9.

x").

(38)

114

V.N. Gribov / Local confinement of charge

Yet, in the pure bremsstrahlung field ti

\

v x

i

i

v z

fix) = a0 In —f- = a0 In -— vx v z does not depend on a and consequently fix')-fix") = 0 in (38). Hence the current is again independent of the external field and consequently is equal to zero. Nevertheless this does not imply that the current on the cone is actually equal to zero. This is explained by the fact that we have not sufficiently accurately taken into account the separation of the points xi,x2. Let us assume that X\ — x+\e, Xi = x— 2e, where e is a space-like vector, over the direction of which averaging should be performed. If the vector e is in the plane parallel to the plane tangent to the cone (perpendicular to the plane z, n) then the points x\ and x2 are either both inside the cone or outside it. Then in the first case we get (38) and consequently zero in the bremsstrahlung field. In the second case the current is trivially equal to zero. However, if the vector e is in the plane (z, n) perpendicular to the plane tangent to the cone, then one of the points x\, x2 may be outside the cone and the other inside it. Then both (33) and (38) will be invalid for the current since one of the functions ip or ijj in (32) should be replaced by

x2) exp M * Av dx„)

+ hermitian conjugation .

(39)

Eq. (39) corresponds to fig. 11. If for G R we insert (26), then (39) becomes of the gauge-invariant form U = 2 | dVsOt' 2 ) Qmx-Xl Sp y^GAx, x')x'G-ix', + hermitian conjugation ,

x2) (40)

where F= [ Avdx„, Jc

(41)

where the contour C is given in fig. 11. The quantity F involves the 5-like part of the strength of the field.

FIG

11.

V.N. Gribov I Local confinement of charge

115

4. Vacuum current in the pure bremsstrahlung field The vacuum current in this case is determined completely by formula (40) with G R = (l/2v)dS(x2) and G_ = (l/4ir2i)d(l/x2). It can be written in the form Jv, = —3 d x S(x ) sin F(x, x ) Sp y^x 2n J

d2 —

jrr~, (xi-x) (42)

vx vx' F = <*o In — ~ «o In -y-,, V X

V X

where x is the coordinate of the point, which is the intersection of the light cone with the straight line connecting x\ and x2: 2

. __

_

*M~***

2

_*

E

e^

4

e x v-ot

\3 •

(43)

2ex 8(ex) In the light-cone variables x'= a'z'+ (3'n', $'x'= \(a'-(3'f out the integration on /3' and a', we have

^

=

1 ^7

f d/2' . T~ Sln F

Sp

. ,* ^ 3 l f *2 7

32-n- J 47r

da' d/3'd/2'. Carrying

x? ^^^

^

(jtiz ) LUxZ )x2 -\x2z

;r^j,

(44)

Mi]

or Q„

d/(y,*)

2 2

//*=z—2*1*2—: 27T dy„

,.e.

(45)

'

f d/2' 1 / - j — sinF(f,z')-^2, y 2 = e 2 x?xl,

y,u = * i M * 2 - * 2 ( i * i ,

CM„ = J:IMX2„ +*i^*2M — S,j.v{xix2).

(46) (47) (48)

2

The integral / can be easily calculated at small y and is equal to 2 / = — sin F(x,y).

(49)

It is also easy to check that y* ~ (*2 - * i ) \ x

L

v

+ -

——j

2T2 >

(50)

2e;c(jt2 — * i ) J

and therefore dJ 2y„ . A„(y) 1 — --^sinF +^ ^ c o s F — - 5 - 5 2-A^f), dy„ y y y (*2-*i) A^(x)=A„(x)-^(eA(x)). ex

(51) (52)

116

V.N. Gribov / Local confinement of charge

If we put (51) into (45), taking into account that x^At = 0, e,At = 0, we obtain . /M=

(*ix2)

A±(~

2 1, 2 2T>M*)=7r e (JCI — x2)

\2(x2 + ke2)

1 1 1 .^.,I

....

27 C

- ^ ^ W i . exJ TT

(53)

L e (ex)

The current (53) is different from zero only if Xi=x

x\ = X2

+4£ +ex>0,

+ \E2-EX

<0;

i.e. - e x < x 2 + | e 2 < e x .

This means, that the first term in (53) has the character of S'(x2), because 1 / 2 . 1 2 \ . 2 r> 1 f 2/ 2 . 1 2\ j 2 2 (eX ) . —2~.—r (x +5e )dx = 0 , —J;—r x (x +4S )ax =3—2~-* const. E (XE) J

E (XE) J

E 2

(54)

2

The second term in (53) is5 proportional to S(x ). In the limit e -»0 the cu: current can be written in the form -.2

L=^[i^p-8'(x2)

+ S(x2)JAi(x).

(55)

Taking into account, that 2(-^-A^(x)S(x2) E

= ^E,,dv[S(x2)(Ex)Ai(x)]~S(x2)A^—j E

(ex)ev-^dpAt, E

dx E„-R = Q,

{Ex)Ai(x)8{x2)

= hPFP„{x),

dXv

(56)

0XU

we have 7M = 5—2 I g"2 3„e„epFpM +|5(x 2 )A^ I .

(57)

The first term in (57) depends on the direction of eM. Carrying out a relativistically invariant averaging over the direction of eM when EIJLEV = \8^VE , this term is proportional to 3^F„M, i.e. is equal to zero because we are calculating the current outside the position of the external charges. Skipping this term, we obtain L=T^Ai(x)8(x2),

(58)

37T A

±

A

ZAnA)

(59)

nx where n is the null-vector normal to the light cone at the point x. In order to get (59) from (52), one can write s in the form eM =n f i (ez) + z M (en) + e i .

V.N. Gribov / Local confinement of charge

117

The second term in this expression gives zero because of the condition A^x^ = 0, while the third term turns to zero after averaging over the directions of g£. Hence formula (52) leads to (59) and we obtain for the current the expression which was discussed in the introduction. Let us now consider the calculation of the 5-like terms in the current, which are different from zero at the points where the external charges are placed (vx = 0, v'x = 0). The expression for the integral / is not valid in the neighbourhood of these points because of the following reasons. Carrying out the integration over the azimuthal angle

vy vy

vx vx

The parameter y is of the order of e 2 at vx and v'x of the order of unity, and it becomes a quantity of the order of e (or 1) if either vx or v'x is of the order of e (or e 2 ). The above-mentioned expression (49) for / is obtained in the zeroth-order approximation in y. It is easy to calculate / for an arbitrary y. Doing this, one finds that the longitudinal part of the current is strongly localized around the positions of the external charges. The charge distribution is essentially different from zero inside the region r < e. The integral value of this current contains even divergences. Because of this we will calculate this current in a different manner to make clear the source of these divergences. For the sake of simplicity we assume that *i,2n=ai,2*M+0i,2«n.

(60)

Let us now return to the integration which was performed when we derived (46). Multiplying (46) by x\[

J a

^r8ix\-2a'{xyz')-\

and interchanging the integrations over a' and Q\ we get 1 x\ f a'do' fd/2' , . n . » ^Ti~7rra ^ ~ 1 TTT^ -T—C^ZvSmFSip^-zz), 2-rr Xi(pi — p2) a J (ai —a +5) J 4-n-

] =

£=

ffi-g'

«i«

n

r/3i.

S = a/32-

"2-Qi

Pi-Pi

At the choice of e according to (60) we have sin F(x, z') = sin F(z,

—.

z').

. , _

(61)

118

V.N. Gribov / Local confinement of charge

If P!-»0,

z|,->z„,

C„.vz'v=*2a piz^,

(ai==a2~a),

• - lit "02 f a'da'p! 77 /3i(/8i-/8 2 ) J ( a i - a +5)

_ vz D2

(62)

Let us calculate J{puz) in the region of small u = vz/v'z. The region of large u(v'z -»0) is calculated in the same fashion. Calculating the integral over

(63)

where we have neglected u and H' in comparison with unity. Assuming that u' = zu, u = p\x we get TI \ it \ f dzsin(a0lnz) J(p1,z) = J(x) = -x \ . (64) J V4zx2-[>(l+*)-l] It is easy to make sure that at x-»oo, / ( * ) is of the order 1/x2 = p\/u2, i.e. is actually equal to zero at finite u and /8i-»0. As a result, ;_ is non-zero only at vz ~ e / a , |/3| < e and in this region it is of the order 1/e 2 , i.e. has the character of 8(vz)8(p) and equals U = z,jC8{vz)8[fi),

(65)

where C = \ \ 7 - ^ 2 f (a P \ ,3 dyg f J{pu z) d(vz). (66) 7T J ( a - a ) J (/3i-0 2 ) J The integral over d(uz) = dw involved in (66) can easily be calculated by means of (63) and equals ra

r

J

J(piz)du=pi\

dz sin (a0 In z){\ + z)

\l-z?

This expression formally is equal to zero due to the antisymmetry at z -»1/z. However, the expression is singular at z close to unity. To separate the singularity it is sufficient to calculate instead of (63) (u/u') Ti \ f du'a0ln ln(u/u') ri(Pi, * ) = ,

,_ 1 =aoln-fl(l-s).

J •JAuu''-(£I-U-U') *J Auu' — (?•,—u — u'\

x

(67)

Hence /(p 1 z)dM = a 0 /3i,

(68)

V.N. Gribov / Local confinement of charge

119

A&K-JZT

dfl8iff2 (01-02) 3

(69)

and accordingly /»*(*) =

_

«o

(70)

J a (a — a )

The integral in (70) diverges in the lower and upper limits. The divergence at the upper limit is accounted for by the fact that calculating it we have omitted S which is of higher order with respect to e. The meaning of the divergence at the lower limit has been discussed in the introduction and is associated with the instantaneous character of the acceleration. The fact that the divergence must appear in both the limits is clear from the following. If the acceleration occurs during a finite time T, the particle trajectory in the plane (r, t) would have the form of the curve C given in fig. 12. The irradiation field would have been between the cone r = t and the c u r v e C and would have reached its total value only in the region v 2 r < a ' < a — v2r. Hence instead of (70) we can write (71)

;M(x)=-^ln(^)/r(x)

Thus, we have completed the derivation of the expressions for the vacuum current in the bremsstrahlung field discussed in the introduction. 5. Vacuum current in perturbation theory In this section we shall compare the obtained results with the calculation in perturbation theory. As is well known, the current in a given time-dependent field is not determined by the Feynman technique since the integrals from —oo up to t are not reduced to integrals from t up to +oo and consequently we cannot replace the retarded Green functions by the Feynman functions. In first order of perturbation theory the current is determined by the two diagrams given in fig. 13. The expressions for the current corresponding to these diagrams are of the form /„ = - / j d V Sp (yJ1A(x')d2)DR(x1-x')D-(x2-x')

+ (x1^X2) •

(72)

120

V.N. Gribov I Local confinement of charge

x x

x x

FIG

13.

Calculating Sp and substituting £>R and £>- we can write /M as

/»(*) = a »'(2fl-ru-" „ ^ {J d V 8{(x2-x') (x2-x')2}

Avix ] + (xi

'

"* * z ) } '

(73)

dIJ.v = dlidv + dfldI, — Stiv{d d ) = 4{dMd„ — 5M„5 — 2dl,dl+2S^de} }, ,

(74) Xi=x+e,

x2 = x — e,

3^ =

dl=-

dxu

de„

Eq. (73) is explicitly singular with respect to e, (x2-x')2 = 4[e3 - (ex -x')]. To get rid of the singularity it is convenient to transform (73). Let us rewrite (73) in the form

U = ^^s

1

f

, c , / ,2N \A„(x+x' + e) 2 : e +ex'

\ d4x'S(x'2)[ j4

Expanding Av(x+x'±e) and (l±e2/ex') expression in the brace as 2

t

t

+ 3epea€TdpdtTdT

t •**-v\X > X )

ex

;

1

_

Av(x+x'-e)}

^}

(75) 2 e —ex in powers of e, we can write the

f

l-2£pdp

•""

**-v\X

ex

>X )

;

. 3

+U{e

),

(76)

Integrating (76) by parts, we get i^^d^^\d\'8\x'2)\AAx+x')

+ \epe^y
\ • (77)

Integrating the last term of (77) once more by parts, taking into account that 4S"(x2) = -d28'(x2) and performing differentiation over e, we obtain L = - A

f d*x'S'((x

-x')2)d^Ax'),

(78)

or, otherwise, /M

=

2a0 °2^d4x'S'((x-xf)]T(x').

(79)

V.N. Gribov / Local confinement of charge

121

In terms of the cone variables (79) can be written in the form l^(x)=-—— 7—-8[(a-a){p-p)-aazz]-]» 5tr dp J a —a AIT

(x).

(80)

Inserting (2) for / " ' , we get Ux)=-^r$d0\-^18[(a-a')p--aa'vz]-(v^v'J. 07T

(81)

Ja—a

The integral which enters (81) in equal to ft(/3)/a2vz, which leads to

/.(*) =-A(^-^W). 3TT

\VX

(82)

v xJ

It is essential in this integral, however, that, at, (i = 0, vz # 0, a' = a(3/vx is equal to zero. In other words, outside the points where the charges are placed, the vacuum current calculated due to (79) is determined by the detailed behaviour of/"' during the time of acceleration. This is not surprising, since in the derivation of expression (79) for the vacuum current we in fact supposed that the potential was of nonsingular character and expanded it in powers of e. After that, we applied (79) to a singular potential. If the real field is different from zero in the region of the order \/q around the cone, such a procedure of calculation with eq« 1 corresponds to the inclusion of the current contributions from all particles in the vacuum sea with momenta from zero to 1/e. However, no particles with momenta from q to 1/e can be produced in the real process corresponding to the diagrams offig.2. Therefore, if we want to imitate the real process we have to use e larger than l/q in order to calculate the contribution to the vacuum current only from the particles with momenta less than q. In this approach we can take 5-functions for the field (q -* oo) if e -*• 0. This is exactly the approach we used in the previous paragraph where we got the result which does not depend on the details of the acceleration process.

6. Structure of the self-consistent field So far we have dealt with the vacuum current in a given external field. Let us now define the field and the current from the self-consistent equation 3 >iV = elj'T + elil,

(83)

dJ7^ = 0 .

(84)

We will suppose, that the vacuum current does not contain stronger singularities than S(x2). Let us write FM„ in the form F„.v - {n^zv - rivZ^F + z^F' - zj7^

+ n^.F+v - nj7^ +FX^V,

(85)

122

V.N. Gribov / Local confinement of charge

where zj^t = 0, nj^t = 0, F„v is transverse in both indices, F£„ = 0 due to the axial symmetry of the problem. The first Maxwell equation in the light-cone variables is of the form danj7^ + dfizj7^ + dtF^ = elj^.

(86)

It follows from this equation that zj7^ = -z^F-F^ can not contain 5(/8) because the current does not contain 8{fi). It means that only F ~ contains S(x2). Hence, we can try to find self-consistent fields and currents in the form L=^r^+Jl, a F^^izJ-v-zj-J^

(87)

+ FlMP).

(88)

a If we insert these expressions into the Maxwell equations, we obtain z^dlf^-adaf^+a

F'IJ.vzv = e0JIJ.,

adJl-a2Fl'

=0

on the cone and the Maxwell equations for F^„ inside the cone are d ^ U = eo/li. dvF'ia,=0.

(89) (90) (91)

If one denotes ft(a,z)

= a2F+ (a, z, 0 = 0 ) ,


(92)

Eqs. (89), (90) acquire the form - 2 a d o / " = e%J^L , dj:-
= e20J-,

ft=adJl,

(93) (94) (95)

/+ = 0. (96) Eq. (93) determines the dependence of the singular field on the cone / " upon the time (a). Eqs. (94), (95) determine the quantities/^ and
(97)

The quantity ;'+ on the cone is expressed by J^ thanks to current conservation -a j'+=adaJ-

+ dUllLt±.

(98)

V.N. Gribov / Local confinement of charge

123

Therefore, current conservation is a self-consistency condition of the entire construction. The pure bremsstrahlung field is obtained when j'^ =0,J^= 0, /_ = /I"', hence, it follows that

dlf-=elJey.

(99)

Let us now discuss properties of the quantities (26), (28), introduced above and determining the vacuum current: f(x)=\ Blx{x)=-\

da'AiOc')**,

-^F„(x')dx'v,

(26)

x'li=xll + (a'-a)zli.

(28)

The strength of the field (88) in terms of the discontinuous potential AM may be written in the form F^

= (Z»AV-Z,AJ8(I3)+FIM(3)

•

(100)

For FILVzv not containing 5(/3) one has to take zliAli = 0

at 0 = 0 .

(101)

Hence, at first glance it follows that f(x) = 0 at (3 -» 0. Yet the integral may diverge at a'-* 0 and yield a finite f(x) independent of a, as occurs for the bremsstrahlung field. Moreover, it is clear that at a' -» 0 the field coincides with the bremsstrahlung field and therefore f(x) is exactly of the previous form at /3 -» 0: / = a0ln-^. v z The potential B^ at (3 -> 0 is expressed by /£ and

(102)

^ = ^ f

^ < p ( a ' , z ) + - f ^ft(z,a'). (103) Jo a a Jo a The fact that the expression for the phase f(x) is the same as for the bremsstrahlung field (102) proves that in the self-consistent field of the form (88) at zero order with respect to B^ the current will contain no terms of the type 5'(/3) while first order contains additional smallness and therefore does not contain such terms either.

7. Vacuum current in the self-consistent field The difference between the calculation of the vacuum current in the self-consistent field and the calculation in the bremsstrahlung field lies in the fact that the Green functions G R involved in (33) and (40) should be calculated not in the zeroth but in the first order approximation with respect to B^. As a result the current is

124

V.N. Gribov / Local confinement of charge

FIG

14.

FIG

15.

represented in the form of a sum of the current in the bremsstrahlung field ;'° and current / ^ denned by the diagrams of fig. 14. Due to the fact that, as is seen from the above, the phase in the integrals corresponding to these diagrams does not depend on a, the differences f(x")-f(x') and f(x")—f(xi) in these diagrams are exactly cancelled in the case of fig. 14a, since they enter at the same zM and are cancelled at any a, with the exception of the smallest a' ~/3 in the case of fig. 14b because at large a they also enter at the same z^. As it is shown by explicit calculation, a'~(5 are not important and therefore the phase factors may be replaced by unity. However, then the diagrams of fig. 14 become diagrams of the conventional perturbation theory corresponding to fig. 15, since 2 | d4x"S(x"2)GR(x',

x")x"G-(x", x2) = G-(x',

x2),

4 [ dV'SOc"2) dV"S(jc'" 2 )G R (x', x")x"G-(x", x'")x'"GA(x2, x'") = G-(x',

(104) x2).

The fact that the integration over x' in the diagram of fig. 15 is limited by the condition x' > 0 can be taken into account if instead of B^{x) the potential A^ is introduced: A,W=5,W#(i2).

(105)

Since we are interested in first order with respect to B^{x) we can employ formula (78) and write j ^ in the form JV = - A a

h = -Vl

[ d4x'8'((x

£• -(•'

JW>

-x')2)dj.„(x'), i\

+zIL\dJv (a, z)-Bvfv

(106)

2

(107)

(0, z)-
(108)

in the notation of the preceding sections. Unlike eq. (89) for the perturbative currents in the bremsstrahlung field, the singular part of (106) does not depend on the behaviour of currents during the time of acceleration because the dangerous part J^ [the second term in (108)] is equal to zero at a = 0. To calculate the singular •d) vacuum current on the cone we have to calculate the singular part of j ^ ' coming from the second term of (107).

V.N. Gribov / Local confinement of charge

125

Substituting it into (106), we obtain / t V s i n g = T T ^ { -^-,^8((a-a')t3-aa'zz')Ua',z').

(109)

YlTT dp J a —a A-rr At 0 = 0 ,(1) ;

-sinB"

d eld{P)

f

da'

• , ,

5/3 12TT 2 J a a ' ( a - a ' )

,_

"(a*Z)~

8(p)el

f

a da'

12TTV J a'(a-a')

.

f

,

"la '^ '

(HO) Summing this expression with the current in the bremsstrahlung field (56), (57), (80), we get /.a =A ( / ^ ( 0 , z ) + [ ,"da',./;(a',z)), 07T \ J a (a —a ) I /- =/ ! * - T A IZTV

,"da',,[a/:(a',z)-y(a',z)],

f

(111) (112)

J a (a — a )

where we have taken into account that d ^ = /~(0, z). Comparing (111) with (93), (95) and (112) with (99), we find that the current on the cone satisfies the simple equations /~(0, z)

a0 f

a da'

,

• ^ - e ^ " 3 ^ J «Xa^)7-(a'Z)'

(113)

/_(a,z)=/--f f-f^-/_(«', z).

(114)

37T J a (a —a ) These equations are easily solved in the logarithmic approximation. Designating £ = y0\n(t/T)(t = <Jla), y0 = a/3-ir,

V = -2/-J^pa f J(f).

(H5)

In the logarithmic approximation the second term on the right-hand side of (115) equals 3f/(f). Consequently, (l + £ ) V = - 2 7 ;

(116)

hence, /W

= (T^>

^

j-ext

/

• ^

- = (TTi^

_/-(0,z)^_ 6 ^ 2 (1 + f) 2 "

(118

>

(119)

126

V.N. Gribov / Local confinement of charge

Accordingly, the singular field on the cone is

f-Jjj°±±l.

(120)

These results have been discussed in the introduction. The fields non-singular on the cone are /•+

/M

(,"> Z)

.

.

(122)

8. Currents and fields inside the cone So far we have been interested only in the part of the vacuum current which is singular on the cone. Let us now pass over to the discussion of currents inside the cone. The possibility of calculating currents inside the cone depends on the behaviour of the potentials AM inside the cone. If the vacuum current j'^ (87) inside the cone is small, it is easy to calculate the potentials A^ due to the current on the cone according to the formula

HP')

V^ = -el\d\'D^{x-x'f)^-Ux'), J a

(123)

and then calculating currents in this field it is easy to find, by means of iterations, j'u. and corrections to A^. Yet, without doing this, it is also possible to obtain information on the current inside the cone employing the current conservation law. From the current conservation law (98) or, which is the same, from eq. (97), we know xJl = j + at jS = 0, j + = -[8(vz)-8(v'z)]-3^-^.

(124) a

7T ( l + £)

This expression is of first order in y 0 at £ ~ 1 and should be valid not only at /? = 0 but also at /3/a « 1 since otherwise the current must have been singular close to the cone and the self-consistent solution would have become senseless. As a result we arrive at the interesting conclusion that the vacuum current is singular along the line connecting the created charges. Eq. (124) may be written in the form ^=-7^[fi(w)-5(«'*)]-77^5<125) (nx) IT (1 + | ) However, this current is not conserved. This can be improved, if corrections of the

V.N. Gribov / Local confinement of charge

127

order of (3/a are included. An explicitly conserved expression coinciding with (125) at /J -* 0 can be written in the form .,=_2y2niig(p2) U 77 nx (l + £)

;

;

(126)

2 , ,-, (UJC)(t>'x)

2

p =-x

+2———. (l>U )

In the reference system where v and v' are antiparallel, (126) differs from (125) by the factor a2/(a -ft)2; p2 is the distance from the observation point to the line connecting the charges. The singular character of the current j'^ makes it impossible to use perturbation theory for calculating it by means of the potential A(^\ since the corresponding integrals for j'^(x) are determined by small times. To obtain the current we should perform a calculation similar to the one for the current in the bremsstrahlung field. This has not been done so far. If one accepts (126) on the basis of current conservation then with (118) and (119) one has the following expression for the vacuum current in first order in y 0 and /3/a«l: z 2\

, ,N1 SS(x (X ) ) (v-*v)\

f

c s K = v„S(vx) -y0 l

VX

J

77

*/• 2 \ „ «„ M S(p") (1 +>.x2~2yo tf 'nx v

11!

(1 + f)

2£_ 3

(1 + iY (127)

This expression can be written in the form of a sum of two conserved currents: / =;(1)+/<2) J fJ-

//A

•d)

r

st

.(2) /M

\

' st '

f/«i Toil i\vx

M

g

(128)

* J fJ,

1 (*2) 77 ( l + £ )

n» Sif')) S(p

v'J-\ 80c) —J v xJ

*V g ( p 2 ) 1 «* 77 ( 1 + | )

+1 77

nx

rr

(ITfl-


The force lines of these currents are given in fig. 16. Thus j(2) is not involved in the charge transfer between the outgoing particles and, essentially, the problem is reduced to the fact that particles send their charges along the line connecting them. The particles then remain neutral. Let us now calculate the fields inside the cone. Calculating the potentials A K according to a formula similar to (123) with the current (127) it is easy to come

FIG

16.

128

V.N. Gribov / Local confinement of charge

to the following expression for the potentials inside the cone: ,2/2vx

A

2

"~ a*M(i + 4 ) V u i + £ ) dfir

1

1 -i

2

df„

r

1 + &J

/„ = a 0 In vx,

') J

°-l J0

(v -> U ) ,

axM i + f„ / p = a 0 In p 2

a £ = ro In - ,

P & = yo In - .

2UXT '

f 3

x u ' - a ' (1+1') X

ft =ffoIn vz ,

& = To In ;

T

(130)

T

Accordingy, the strength of the field F^ equals 2 dfv , 2 ^ 3/„s(1 3x^1(1 +&)3 dxv +-( 'a+&)3dx,„ \

3& I" F =—

""

,

l

H

ajc^Li + f

S a

x\

p2/

3/p

1

1

a*„(i + &)2J

(131)

, \ Bj l 36 l 1 a# 2 2 .a*M (i + £) a*. (i + &) Jax„

,

.

The expression is of the order of a 0 but singular both in x and p . The part singular in x can easily be written since it originates from (Ix^

vA

3MU = yo

z„ a M §s= — y 0

1,

and is equal to f n» - yo

r

2 a/„+_ 24 a/„. a/p l a/£ a/, 1l 1 [(1 + &Y dxv (l + ^ r a j c ax„(l + & r a*„ a*„ (1 (! + &,) &)•2/

- (n-+i>,

(132)

v-+v').

At small /3 and not small vz, v'z we have £v = 4' = £e As a result F* „ can be written in the form C2

\XfjVy

j.v = a o y o |

•xvv^ , , J 1 1—_3& -(»-*«') } vx J 0(1 + 6)

(133)

This expression at T -» 0 has the character of S(/3), since at r-> 0 and /3-finite, £p-><x> F*t d/3' = - a 0

V;„-x^_ ( ^ u , ) J|(^) ;

(134)

ux

and, consequently, in the limit T -* 0 ,-.x2

-i

\xiiPv

F*„ = -2a0

XVV^

— L

,.\

. 2

- - (u -»t/) 5(* tut

J

(135)

V.N. Gribov I Local confinement of charge

FIG

129

17.

coincides with the original bremsstrahlung field. Yet, at fixed r the distribution F* „ over /3 is essentially wider than the distribution of the original bremsstrahlung field F~„, the width of which is of the order of r. The distribution of both fields F~„ and F*„ is given in fig. 17. This distribution satisfies a remarkable sum rule stating that the integral value of the bremsstrahlung field at sufficiently large £ does not depend on the fact that the charge is really screened and is equal to the original bremsstrahlung field. This fact completely contradicts the conventional understanding of the zero charge due to which the vanishing of the charge is regarded as the absence of interaction, i.e. the impossibility of photon formation. Let us now discuss the part of F^„ singular in p2. The transverse parts FM„ have weak singularities in p. The most singular quantity is F = njr^z^ According to (131) the singular part F at small vz equals F = yet at /3/a«l

2a0y0(vv') (vx)(v'x)

£, - 1 (l+^r

the small value of y 0 is compensated and _ F

2 jaoSivz)

f*df(g'-l) : n,,,,3

— a

a08(vz)———2(1 + 1)

(136)

In conclusion, let us point out the following interesting phenomenon. Despite the fact that the strengths of the field at £ -»<x> and /3 ~T tend to zero, the potential under these conditions does not tend to zero. It can be written in the form -An =

-a0zll.[8(vz)-8{v'z)]

(137) 1+f

This is a pure gauge potential. The fact that the gauge potential singular in the region where the screened initial charges are placed must necessarily be retained at t-*co is clear from the following. At K O w e have a vacuum, i.e. all the levels with negative energies are occupied. At t -* +oo we have a state where the negative charge goes in one direction and the positive in the other direction and they are localized at the points where our initial particles are positioned which themselves are not included in the system but represent the external field. This means that the

130

V.N. Gribov / Local confinement of charge

external field (thefieldof the initial charges) affects the density of levels at the points where the initial charges are positioned and therefore the appearance of screening charges is not surprising. The paradox lies in the fact that there exists no field affecting the density of levels at t -* oo, since the self-consistent field, as a result of screening, tends to zero and, consequently, there is nothing that could change the density of levels. The key to the paradox is that in the process of charge screening there should always be a gauge potential which formally is responsible for the density variation of levels. In other words, in the process of complete charge screening the vacuum must transit into a new state with a gauge field and a different number of particles (at the same charge). This is analogous to what is observed at instanton transitions. 9. Analogy with the one-dimensional case In this section we shall discuss the process of formation of two massless charges in one-dimensional Schwinger electrodynamics. This may be interesting since we know the fate of massless charges in the model - they become neutral mesons. The problem of formation of two massless charges in the one-dimensional case has been studied in a very interesting paper by Kogut, Casher and Susskind [2], Yet, while interpreting what takes place at the formation of two massless charges, they believe that the formulae derived by them correspond to the process of creation of the neutral mesons. However, in fact their formulae describe charge screening very similar to that which we have investigated above. As has been shown by Schwinger, the vacuum current in an arbitrary external field in the transverse gauge is of a very simple form: /;(*)=--AM(x),

(138)

and the equation of the self-consistentfieldis -d2A^eljT--A.{x).

(139)

77"

The current of massless charge formation is i7 = v„8{x-t)-v'„8(.x + t).

(140)

The solution of eq. (139) obtained in [3] at f->oo and at the correct initial conditions is A„{x) = ir[v„8(x-t)-vl8(x

+ t)],

(141)

i.e. corresponds to the gauge potential d^A^ = 0, F^ = 0 and the local current. The difference from the three-dimensional case is that this asymptotics is reached not logarithmically in time but as a function of e\t since the charge is dimensional.

V.N. Gribov I Local confinement of charge

131

Thus, at the formation of two massless charges there appear two massless neutral classical packets and the fact that the physical state has a mass is accounted for by the fact that the interaction of this state with vacuum oscillations makes it massive. In conclusion I would like to express by deep gratitude to G.S. Danilov, I.G. Dyatlov, D.I. Diakonov, V.Yu. Petrov, L.V. Gribov, Yu.L. Dokshitzer, Ya.I. Azimov, L.B. Okun, A. Vainshtein, Ya.A. Smorodinsky, J. Nyiri and F. Niedermayer for interesting and useful discussions. References [1] L.D. Landau, I.Ya. Pomeranchuk [2] A. Casher, J. Kogut and L. Susskind, Phys. Rev. D10 (1974) 732

- 407 OUTLOOK V.N. Gribov* Central Research Institute for Physics, Budapest

I was asked to give some outlook of the theoretical situation in particle physics and it was very difficult for me to choose the topic and especially for this conference, because the main direction of this conference is to go — 13 — 16 far beyond our usual space scale, from 10 -10 cm. To go far to the neutrino mass, to proton decay and so on. I certainly agree with Lobashev that now it is very important to search for new particles, new phenomena connected with the smallest distances. Of course this is true for the experimentators, who have done so much for exploring the world of the distances from !0~ -10~ cm. But for theoreticians the situation is not so clear, because the world of smaller distances was not explored by the experimentalists, and dealing with it we must predict the structure of this world. Unfortunately, the possibility of such a prediction looks doubtful from the point of view of our previous experience. It is obvious that nobody would be able to predict the existence of quarks, gluons etc. without experiment. In this sense the theoreticians now are in a new situation: they can wonder about the new world, or chey can try to understand better the world which is already explored for us. Of course, work has to be done in both directions. In the new range of distances we can rely mainly on the symmetry approach asking ourselves how the symmetry of the observed world can be derived from more fundamental symmetries at small distances. In the old range of the distances the situation is the opposite: this symmetry approach was very successful, much more successful than the dynamics. But it gave its results and now, in this old range there is a problem of dynamics: how do things happen? In this situation I decided to talk about an old problem and only after that to connect this problem with the problem of the smallest distances. In order to be understood, I would like to remind you what kind of problem we had in the beginning of the exploration of the world of the elementary particles. The first discovery was that there exist strong and weak interactions, leptons and hadrons. What is the main difference between them? Leptons *0n leave from the L. D. Landau Institute for

Theoretical Physics, Moscow

- 408 arc point like within our accuracy. 10

Madrons have a size, of the order of

cm. How did we start to deal with these objects? For leptons, the situa-

tion was in a sense clear from the beginning. The wan who first recognized this was Fermi. He undcrtood that if particles are point.like, the only thing which can happen is that they come freely to a point, at this point they scatter or they produce something, and we can write this interaction phenomcnologically in a simple form. After that we will discuss what is the structure o( this interaction, what is its true form, what is the amount of the created particles and so on. This is what people call a field theory, but in fact it is a simple thing: poiutlike particles can interact only in a point. The situation with the hadrons was very difficult from the beginning. First, in the sense how to write the interaction, second, how to imagine that something has a size? From the relativistic point of view, if something has a size, one can divide it. From this point of view one had the first puzzle: if two hadrons collide, we would expect from classical intuition that they can be divided,that we can find the things from which they consist of. However, in any hadronic collision we are producing particles, perhaps new ones, each of which has the same size.

This means that it is impossible to divide them.

As a reflection of this, it was absolutely impossible to write the true interaction of hadrons. There were attempts in the fifties, like the bootstrap, to try to find a theory without considering the constituents of this big objects. Hut these attempts practically failed. They failed not only theoretically but also experimentally because people started to investigate the Bpectrum of these particles, their quantum numbers, and came to the idea of the quarks. After that, in deep inelastic the experimentators could see precisely these quarks. So, the situation is that some constituents in some region really do exist. After the successes in different fields, we pretend to know what these constituents are. These are quarks and gluons. In many respects they are very similar to leptons. They enter the weak interaction in a very symmetrical way, and so on. They are almost like leptons, but the strange thing about them is that they are always living together, they can never go free: they were never observed to be free up to now, and this is a puzzle. Not only in the sense that they always arrange themselves in some volume, but also that the objects they form have quantum numbers which are different from the quantum number of the constituents. Thu constituents have color, non-integer baryou and electric charges, while the directly observed particles are colorless and have integer quantum numbers. How can this be? How can it be that quarks and

- 4<)') -

gluons nri' so similar to (ho leptotis, aiul ;it the same time they are living in very different condil inns? The loinwr answer to i ho quest i DII why they nre living together was that I hey ;iri- interacting st m u g I y. Hut imw we know that, for ;| short period of t. imc ;i I t <• r we create I hem I hoy will

i 111 < • I -1 < t very weakly, JUHL like leptotis.

This is just a emit nuli il ion. People said I ha( of course the interaction depends on the distance. II the particles are close enough they interact weakly il I hey an* far enough, lhey inleracl mcaiii nglcss concept,

literally

strongly. Hut (his seems to he a

speaking.

II

they

interact weakly when

they are close to each other, how can they know that they will have to interact strongly when they will he far from each other? This seems to contradict the causality. What we know is that

the system is very simple for a short

period of time, and something happens with the system if we leave il alone lor some time. This idea existed even before, quarks and gluons were introduced. This is the old story which was the essence of Keggo pole theory, partou model and so on. Suppose we have some constituents. There, is a probability ol their decay. A quark can emit a gluon. If we wait a little bit, it can emit again and again. This means that

in the relativislic situation if we have some constitu-

ents and if they can inleracl at a point, after some time we have a hunch ol the same constituents and we have the situation which we expected. The problem is, why they do not blow up, and not, how to produce a sufficiently

large amount

o| the constituents.

In the old theory we supposed, that the probability

for the decay is of

the order of unity and supposed that after some time the constituents will fill some volume and their configuration will become stable because ol the interaction. This led to successes of the Kcggu pole theory in multiple particle production, in scaling, and so on, but all these successes were very qualitative. If you come to quantitative calculations and try to adjust the parameters to agree with experiment, you find, that the'pure asymptotic behaviour of cross sections occurs at very-very high energies. What

is the reason? We did not know then, but now the situation seems to

be the following. Putting the decay probability of the order of 1, you immediately produce this stable situation, and should expect that the asymptotic behaviour is reached very soon. After a few GcV everything will be very simple. Taking a very small probability, the obtained object will be very what

will

have

nothing

diluted,

to do with experiment where the cross sections

-


-

are still large, so this is also bed. What is the concept which comes now? It is that the probability for decay depends on time. For small times the probability

for decay in RIII.'I I I , Tor larger I. i inert I lie probability

in increased.

How to understand this? Here the new concept comes. The essence of the situation is that this process of decaying and creatine the parton is not taking place in the empty space, but

in the vacuum. In the vacuum a very striking

phenomenon was discovered. Creating two charged objects which are receding from each other, at the same time the Bremsstrahlung field is also created. H

we created a pair of quarks, we create also gluons; in the electromagnetic

case, we create photons. If the particles are massless then the field created together with the particles is smeared on the same sphere where the particles are located, and the field creates in the vacuum a vacuum polarisation current. Due to this current It turns out

the charges of the original particles will change.

that

in QF.U the vacuum current

decreases the charge of the

particles. Hut in ()'!!) I he gluon cut rent ha.'; the opposite si en, and it

increases

the charge. The value of this current, in the vacuum depends only on the m,i|;nelic moment of the particles in the vacuum and their statistics. For the given type of particles it is defined by a simple

formula

(i

.,. 4 £„(,,- I) where
is the coupling constant, it - the number of spin slates of the par-

ticles, it - their magnetic moment: si en ' stands for bosons and - for fcrmions. u For the spin 1/2 fcrmions u - 2, |l = 1/2, i = ~ r.—. For the gluons n « 2, , . II ' p » 2, j » — - (K . ' J f»n o The value of this increasing local charge defines the probability of the omission of new gluons and quarks and plays the role of the coupling constant, growing with the time. If the particles have a mass, the picture is a little bit different, but. the result is the same. This phenomenon explains the peculiar feature of gluons and quarks which makes them different from leptons and which can be important

for the understanding of hadron interactions, but does

not explain the confinement

in the sense, that only colorless states (confine-

ment of color) with integer baryon numbers (quark confinement) really exist. The progress in the solution of the confinement problem is at the moment rather modest. What

T describe upto now looks even opposite to the color con-

finement, because the color charge is increasing due to the vacuum polarization, tn order to have a confinement we need first the mechanism for screening

- 411 -

the color and second the mechanism for screening the baryon charge of the quarks. The first mechanism exists in nny gauge theory; it is the following. Tho Hri'infiHi i ah lung glimn field, emitted during the time of creation of color rlinrj'.i'M ;md pi mine i ng the vacuum polarization current

has his own color charge.

This charge caii screen the color charge of the created colored particles, and, n« it can he shown, it really docs it. The situation with the screening of the baryon charge of the quarks is more difficult. In order to screen the baryon charge of the created particles the gluon field must produce a colorless baryon current in the vacuum. Such a current can not exist in e.g. SU(2) gauge theory, because here the fields of quarks and nntiquarks are the same. In theories in which the number of colors in more than two (e.g. SU(3)) such a current

exists. However, in order to screen the baryon charge of the quarks

completely or to make it integer this baryon current must be large enough. It can lie shown, that

lor quarks in the vacuum with bare mass much less than

the characteristic mass of the strong interactions (about 150 MeV) such a current: can be sufficiently

large, but if only heavy quarks exist it is small.

This suggests, that, maybe, the confinement

in the real world

is not a feature

of any color interact ions but the particular feature of the gauge theory with light quarks and with a number of colors more than two. The problems of confinement and of the structure of the hadronic states look very difficult at the moment, but I believe they can be solved. The main reason for this belief in the fact, that, as it is known from experiment, this problem has a simple answer. It is very important, that, it is impossible to avoid the solution of this problem even going to the physics of short distances, because in this physics we are facing the same problem. Indeed, we have a very good theory of the weak interaction and it is in fact a phenomenological theory for distances much larger then the characteristic size of leptons. For example, if we look at the electron it looks simple: a small charge, emitting a small radiation and so on.

However, from the point of view of bare particles it is simple be-

cause its charge is screened very well. If we created two bare electrons with very high momenta they will have very big charges and they can produce anything. This means, that if we go to the smallest distances we shall find the same problem. A manifestation of this problem phenomenologically

in a bad

feature of the Weinberg Salam theory in the sense of the mass of the particles. The problem of the mass in the Weinberg Salam theory is unsolved. The masses

- 412 ore introduced ns free parameters nnd one does not understand how they appear. The reason to this is just what I said. The masses come from the shortest cli Hi iinccn due In noun* dyiiami rn

mid ! u order to underst and this wo have to

solve a problem similar to that of confinement. In order to make contact with the topic of the conference, 1 uouLd like to say a few words about this mass problem. Purine the Ins I. ten years our uderstanding of the notion of the mass changed very strongly. Some time ago, one thought that the mass is some intrinsic property of the particles. Like in the old theory of the electron, where some additional

forces hold the

charge together inside the electron. This idea may be not so bad but it gave us nothing up to now. Another idea about the mass is that the mass is just the scattering amplitude of the particle on the vacuum fluctuation: on some lliggs field or on something like that. There

is no intrinsic property which

we call mass, but of course it has the intrinsic properly

like a coupling

constant; the particle interacts with the average field in the vacuum, it scatters, its velocity becomes smaller, and it: gets a mass. This concept is more fruitful. When people start

to discuss the possibility oT the dill'erent

masses, it is clear that without

neutrino

solving (he dynamical problem we shall not

be able to say much. For example, if on small distances we have some unperturbative phenomena, and we say from the beginning chat, we have right and left neulrions, due to this unpert urhnl. i vo phenomena left neutrions can become right. What would be the amplitude lor this transition? ft can be quite large 2 in the sense that it can be q /A where A is a momentum where the charge is large and if we put this into the calculation we shall find that left and right will oscillate into each other, and we shall get two states: one will have .1 mass of order A, the other one will be massless. Then it will be impossible to have a massive neutrino. Only two components survive. This is just to show that our expectations could be quite different little bit more about

relativistic dynamics.

if we knew a

V. N. Gribov

Anomalies, as a manifestation of the high momentum collective motion in the vacuum It is shown that the source of any anomalies in the theory is the collective motion of the vacuum particles with arbitrarily large moments.

1. General description of the phenomenon This paper has a mainly pedagogical character. Still, I feel that it is not absolutely meaningless to rediscuss in another language one of the most beautiful and non-trivial phenomena in the modern field theory. What is the mystery of the phenomenon we call anomalies? We have a theory in which the energy-momentum tensor and for example the axial charge are conserved on the classical level. We also have the quantum perturbation expansion in the theory which preserves conservations of energy-momentum of the particles and their helicities in any order. But, when we start to calculate the energy-momentum tensor and the axial current explicitly, for example in a given external field, we find out, that these quantities are not conserved even in second order in the external field. This discovery [1] becomes very important, when we are outside the region where perturbation theory is valid [2]. In order to understand what is going on, let us consider the simplest case — free vacuum of massless fermions in external electromagnetic field. Even more, let us imagine this vacuum, as a gas of classical massless particles, moving independently in external field. Such a movement can be described as a movement in the phase space ~p^, ~# (see fig. 1) where each particle has a trajectory p(t), x(t). In the relativists notation we can introduce four-component vectors Pn(r), X^T) as functions of some invariant parameter T. These vectors satisfy the equations

of motion: xll=Pp(T) PVL =

(1) (2)

FIU,PV{T).

The current and energy-momentum of each particle are 3»{x) = / Puir^Xy, Tp,(x) = J

- Xn(T)]dT

PH{T)PV{T)5{X^

(3)

- x^T)]dT

(4)

These quantities are conserved in a sense:

dtf? = 0 dvT^ = F^{x)ju(x).

¥ x

If we are not interested in looking for each particle, we can introduce the densities of jfj, and T^v in phase space j»(p,x) = Q{p,x)pli T»v(p,x) = g(p,x)pltpv

Fig. 1

(5) (6)

(7) (8)

where

e(p, x )

dftpdftx

is the number of particles in a small volume near p, x. Due to the Liouville theorem g(p, x) satisfies the equation dg{p, x) . dgjp, x) -x H ~ Pn = U u dx. dp*

(9)

where x^ and pM are defined by (1) and (2). If we want to calculate the densities of j ^ and T^ in the normal space-time x^, we must integrate (7) and (8) over the momenta p^ and of course we get infinity. Does this infinity have physical significance? The essence of Dirac's idea to describe the vacuum, as a sea of the particles with negative energies is that the particles with infinite momenta could not influence any physics, because they

produce the homogeneous distribution of the current, which could not be observable. This means, that if we have an external field, the characteristic frequencies of which are much less than K, we could consider j ^ (p, x) and T^vip, x) integrated over some volume V in the momentum space, which is large enough to contain any momenta of the order of x and no physics would depend on this volume. Let us consider such quantities

=

^

d4p

j Wfe{p,x)p>l v

T

(10)

2TT)4

d4p

X

]L( ) = J j2^QiP^)PnPu i

(11)

v and ask: are these quantities conserved or not? According to (9) f d4 d PP

^ -v

^

dd

,. _u. x,

=-} v Wdp-^ i

=

A ff dd3°a

,

s

(12)

"J ( 2 ^ ^

v

WT^dp-^

.

s =

-J (2^^^

+F

^-

(13)

These equations show that d ^ and dvT^v are defined by integrals over a large surface in momentum space of the flow of j^ip, x) and TX,(p,x) from the region of large momenta to the small one. It is obvious, that Diran's idea can be literally true only, if there is no flow of any quantities like charge, energy etc. from the world of infinitely large momenta to our world of finite momenta. As we will show below this flow is not always absent. In QED it is absent for the vector current and not absent for the axial current and for the energy momentum tensor. The question arises, whether this situation is hopeless and we have no theory for our poor world of finite momenta. Is there any example of similar situation in physics? The answer is «yes». We know that in thermodynamics, when we are considering a system which is the part of a large system and is in thermodynamical equilibrium with it, the energy of our system is of course not conserved. However, since the amount of energy change is uniquely defined in equilibrium, one is able to introduce free energy which is conserved in a quasi-equilibrium process. We can use the same trick in our case. If we will calculate surface terms (12) (13) in an unique way and if they will have the structure of divergencies in the usual

space-time, we will be able to introduce the conserved free axial current and the free energy-momentum tensor. Fortunately this is the case. As we will show, this description of the vacuum of fermions could be extended in order to take into account the quantum nature of particle motion in the vacuum. Doing this one gets that the surface terms lead to well-known results.

= -0/4,

-—e

JfJ,

""

Ov-Lfiv —

6TT

Ov-Lfiu ~l~ *'\iv3v

A F

Tt-rinr-rvo- ~

<J7Oiiu^.

i

(14) (15) (16)

where a is the fine-structure constant. These results give us the possibility to introduce the conserved free axial current jl and the conserved energy-momentum T/fj, 3l=%

+ il

TIL=T^/ + Tli.

(17) (18)

Conservation of these quantities leads in many cases to the same results as conservation of the original j ^ and T^u but in some cases there might be a difference. The main difference can come from the fact, that T/ 0 is not positively defined. This could lead to a real instability of the theory, when the charges of the particles would be increased infinitely in spite of the conservation of T/f„. The description of the anomalies we have presented shows that the source of anomalies is a collective motion of particles with arbitrarily large momenta in the vacuum. This collective motion transfers the axial charge and the energy-momentum from the world of particles with infinitely large momenta to our world of finite momenta. This description makes the main features of the phenomenon very obvious. First of all it is clear, that if we represent our field F^ by a finite number of photons we will never see the phenomenon, because one photon can interact only with one particle and therefore a finite number of photons is not able to change the momenta of an infinite number of particles. It does not mean, however, that Feyman's diagrams contain no such effects. Indeed, infrared divergencies of the theory are in fact the manifestation of the existence of the classical fields with infinite number of photons in the processes of particle interaction. Calculating probabilities of different infrared divergent processes, we are adding to the initial or to the final

states an infinite amount of photons in order to take into account the contribution of the classical field to the structure of these states. The classical field is important not only for the diagrams which are really infrared divergent, but also for those which contain no infrared divergencies. The reason for this is the fact that the absence of infrared divergencies of the diagrams is very often due to calculations of different photon corrections to the process, what means the existence of some sum rule, reflecting in a correct way the presence of classical fields during the intermediate stages of the process. For example, we could calculate the axial charge of a photon simply calculating the diagram of . _ fig. 2. The general picture of any process of particle scatpjg 2 tering or creation looks as follows. A classical electromagnetic field created during the time of interaction produces a collective motion, and as a result of it particles get some axial charge and additional energy. After a sufficiently large time particles will be far apart from each other and states of the system become the product of independent stationary states. Any of these states will have a fixed value of axial charge and energy, as a function of other quantum numbers of the states. Any collective motions must be stopped at that time because these states are stationary. If we would look for the features of this stationary states and compare for example their axial charges before and after the interaction, we would find that charge is not conserved. It is easy to calculate photon and fermion charges by means of the diagrams on fig. 2. The result is a Qph = -^Zph, (19) where £p/i is the helicity of the photon. If we calculate free axial charges for photon and fermion according to (17) we naturally find qfph = 0 and qfferm = £ferm. Up to now we discussed only the role of anomalies in the manifestation of the importance of classical fields and collective motions in the vacuum during the process of particle interactions. Let us discuss now some properties of jjj, and T/i/ which are of significance. One interesting feature, which is difficult to understand without our interpretation of anomalies, is the non-zero value of the trace of 7}{„.

This non-zero value of T^ is not so surprising, as it would be if T/L were the energy-momentum tensor of any collective of massless particles, but not some useful effective quantity as it is. It is interesting to notice also that from this point of view T/{„ is a quite improbable candidate to determine the interaction of a system with gravitational field. May be this interpretation gives some insight on the problem of the cosmological term in the theory of gravitation. One of the problems with the cosmological term is the following. Having a lot of different condensates in the vacuum (Higgs field condensate, strong interactions condensate), why have we no cosmological term in the gravitation? The answer, may be, lies in the fact, that in all these condensation processes the energy only transfers from the region of large momenta to that of small ones, not changing the gravitation field of the vacuum. The most important property of jfi which was widely discussed in the literature is that it is not gauge invariant. What could mean that the amount of the axial current transferred from the region of large momenta is not gauge invariant? The answer is, that the amount of current transferred during some processes from the region of the large momenta depends not only on parameters of the initial and final states, but also on the details of these processes. In particular, if the system went from the initial state to the final one and back through different stages, the amount of transferred current would be different. It is like hysteresis in macrophysics. In QED, although jl is not gauge invariant, the free axial charge qf = f j[d?x is gauge invariant and therefore our previous discussion makes sense. In QCD even qf is not gauge invariant for non-perturbative fields and these fields could lead to observable nonconservation of the helicity iii a number of processes.

2. Quasi-classical description of the solutions of the Dirac equation for massless particles in electromagnetic field In this section we will show how to formulate the picture we have described for massless fermions, obeying not the classical equation of motion, but the Dirac equation. We are used to the fact that in quantum mechanics particles cannot be characterized by coordinates and momenta simultaneously. This is, however, not in contradiction with the possibility of introducing the local momentum Pn(x) in such a way that the current density V,P0A7A»V,POAix)

would be proportional to p^yx). Jtiipox) = VVoA^h^VVoA = Q{x,Po)p^(x,po)

(21)

If Q(X,PO) is chosen in an appropriate way, Pn{x,po) is obviously a local momentum of the classical particles in the quasiclassical approximation for if). Outside the quasiclassical approximation we have a freedom in the definition of Pn(x,po) up to a factor a(x,po), which could be included in g(x,po)If we fix this factor in some way, we will know Pn(x,po) solving the Dirac equation for ippo\(x) . The interesting feature of the massless fermion is that we can proceed also in the opposite way. We can find out ^Po\{x) up to the phase factor, if we know p^ (x). It means that we must have some equation for p^x), which is an equivalent of the Dirac equation for ip. In the quasiclassical approximation this equation will be, of course, the classical equation (2). Anyhow, if we know p^(x,po), we can write down the vacuum current for right or left particles separately in the form (10). jfo)

= 2 J 0~36(p2o)ti(-poo)QX(Pox)p*(x,po)

If we introduce in integral (23) a new variable pM = p^(x,po) we get the expression for j^{x):

ifa) = *f ?x(p,x) =

(22) instead of p°

d4p X Q (p,x)Pfl (2*)«

2n6(pl)d(-p0o)QX(po,x)

(23) dpo dp

(24)

In (24) p0il must be expressed through p^ by means of the equations PIM=PH(X,PO)-

(25)

The expression (23) coincides with (10) for V -> oo. In the same way we can write down the energy-momentum tensor T^,(x,po) = \ {4>p0\l»V»i>poX + Herm. conj.} = gx(p0,x)Tx„(p,x)

T% = J

(26)

^S{plM-Poo)T^Po)

= f £^eX(P,x)T^(p,x). (21T)

(27)

Here g (p, x) is defined by (24). The expression for TMj,(p, x) we will calculate below. The conservation law for j^{po-,x) and for T*u(po,x) immediately leads to the equations (12) (13), independently of the explicit form of P[i(x,po) or Tpvip,x). From the equation d^(x,P0)

=0

(28)

and from properties of the Jacobians it is easy to find the equality p^d^ip,

x) + -Q—Q^ip, x)pildtlpv

=0

(29)

which is equivalent to (9) and leads to (12) with p^ = pudup^. Similarly, from the equation 3„T*,(po, x) = F^rfbo,

x)

(30)

- —(T^dvPe)

(31)

we have dvf[p,x)Tr,[p,x)

= F»u0x(p,x)p„(x)

which leads to (13) with PuPe -»

T

v.vdvpg.

Let us discuss now the concrete structure of the equations for p^(x,po) and i>(x) as a function of p^x). If we have for example a right-handed spinor j5tp{x) = tp(x), then for any if>e we can find Pn(x) satisfying the condition p(x)il>(x) = 0.

(32)

From (32) p2{x)=0,

Ihrf

~ p»{x).

(33)

If ip(x) satisfies the Dirac equation Vip(x) = 0 + *i)V» = 0

(34)

with Afj,(x) vanishing at space infinity, it can be written in the form rl>{x) = e-^+K-x

S(xMP0)

(35)

where x a n d X a r e r e a ^ functions of x and f(po) is a free spinor with momentum p0fl. Here Po
S =

j0S+io.

From (32) and (35) we can define p^ and S(x) in such a way, that P = SpoS.

(36)

One can propose the following idea to find ip(x) and Pn(x) in a quasi-classical manner. Let us introduce Ffiv(x) = dvPp, - d^pv and suppose for a moment that we know ^^(x). condition p2 (x) = 0 we have

(37) Then from (37), using the

PvdvPnix) = Fvv(x)p„(x).

(38)

We can understand this equation as the Newton equation if we will determine the classical particle's trajectory in the field T^ by the equations i/i = Pv, Pn = ?v.vPv

(39) (40)

and will remember that p^ — the derivative Pfi(x) along the trajectory — is equal to the derivative pM(x) along pv(x) (pvduPn)- If we know T^x) we can find Pp,(x), solving the classical equations (39) (40) with the condition Pn(x) 7^ P° at a; -^ oo. We also could find S(x), because from (40) and (36) we have S = FS, 1 T = 7i/4

(41)

From (41) o

- /

S = Te - »

dT?[x{T)\

,

(42)

where the integration in (42) is performed along the trajectory going from the point x^ to infinity and having at infinity the momentum p°. Now we can try to define the effective force T^ from the Dirac equation. Prom (35)

V ^ = H(pJ - d^x ~ AJ - d^x + d„SS}i>;

(43)

substituting (43) into (34) and using the condition 75V' = ip, we have (pi + ifaW = 0,

(44)

Pin = Pj ~ dnX ~A^ + fin,

(45)

where

(46) ^

=

^SP(5757M^),

(47) (48)

From (44) Pl=P2,

PiP2 = 0;

(49)

PP2 = 0.

(50)

from (32), (44) PPi = 0,

In order to determine a spinor completely, we have to fix the system of coordinates. The three vectors Pi,P2,P are just the self-vectors which provide a natural coordinate system for spinor ip. Really, if we multiply (44) by pi, we get the equation [p? + *
(51)

In a coordinate system where two space-like (42) vectors pi^, p2^ have only two components piM = ( 0 , ^ , 0 , 0 ) , p 2/J = (0,0,p y ,0), (1 - ot)rl> = 0.

(52)

This means that in this coordinate system a two component spinor describes the particle with the momentum along the z axis. From (52) p^ is just this

momentum vector. If the initial spinor ip(po) is determined by vector p° and two space-like vectors £j , £2^; £?2 = ^22 = — 1 m s u c n a w a y * n a *

-(1+75)^ = VPoo L J in the coordinate system of vectors PUJ^IMJ^UJ then the matrix S which transforms p° ->• pM (p = Spotf) will transform £ ^ , ^ into £1^, £2fJ. l i = 5e?5,

(53)

Pin = aipp + Ai£i/i + A26M,

(54)

P2/i = 02P/i + #2l£lp + /3226i/,

(55)

A i = /?22,

(56)

Now, we can say that

A2 = -/3i2,

because the self-vectors pi^ and P2n can differ from the self-vectors £iM, ^2/i by a scale factor, by rotations in the plane £1^, £2^ and by addition of the vector proportional to p^. These conditions are enough to determine •7>i/(£) if we fix the scale of p^, what was not done up to now. We will do this putting ot\ = 1. As a result we have Pn = Pin + Aifi/i + &2&M

(57)

and according to (45) •7>f = Ffju/ + h^v + £ ^ , h^v = dvh^ - d^K,

(58) (59)

Zv, = &(Ai£i„ + fah») - (M -> ")•

(60)

The equation (55) is the equation for ^^{x) because, as we will see, h^ and £M„ is determined by Tfj,v(x) as integrals over classical trajectories. The hfj,v is expressed explicitely by (47) and (42). The calculation of ^ is a little bit more complicated. Before doing this, let us calculate tf) i. e. x a n d X . Prom (50) and (45), (46) we have X = -PnAii + KPV. + PnP%

(61)

X =P/i9n-

( 62 )

If we will write X

P° = *V ~ / Fnvdxv

(63)

—oo

and notice that from (48)

we can write X = - / ^Ma/jcte^ + / h„(x)dxv-

I

dXydx^T^x

),

(65)

X >I

0

X = 2 / drduPuix) = - ^ P v .

(66)

—00

By <£ we denote the integration of any quantity ip along the trajectory going from point x to —00 . Using these expressions for x and x we can write pifi and p^fj, in the form Pin = P° + / dx'u-^-[Feu{x)

+ hev{x)} + d^

/

dx^dx'^T^ (x"),

(67)

X >X

1

/ dxp

/

dxv—gQV{x ). (68) These formulae give us a possibility to find f3\\, f3\2 and consequently ^v. dx^ dx0

/

,

/

dxvi2^{x)^-geu{x),

dxD

1

(69)

1

/

dxvixii{x)-^-gev{x), , dXa

(70) ,

/

This expression for ^ £2 in their plane.

dxa-^-gap(x ) - (n -»• 1/). (71) shows that £M„ is invariant under rotation of £1 and

In order to make it explicit it is natural to introduce the fourth vector £M of the coordinate system defining the spinor.

C2 = o,

piC = o, p2C = o, K = I,

(72) (73)

C = S(0S;

Zofi = (£,0,0, | J \ in the coordinate system, where p0ll - (-p°,0,0,p). ZlvilQ — ZlQ&v = £veySPfC(T,

(74) (75)

/<£CTT = /

dx'agpa(x)

dxa

dx

dx.

nrP"& + sr-PrC* + 5x ^TP«C. 5a; ftr. T

(76)

3. Current and e n e r g y - m o m e n t u m t e n s o r at large m o m e n t a Now we can calculate th^e current in terms of integrals on the classical trajectories. According to (35) and (66) jpfapo)

=

e-°eEiPli(x).

(77)

In order to calculate the vacuum current (24) we have to know the Jacobian (25). This Jacobian is the determinant of the matrix C^ — g^jf. Due to the equation of motion (40), C^u satisfies the equation Pa C,QV

Cfiv —

(78)

In (78) we consider T^v not as a function ^^(x^po), but as a function T^u{x,p) = ^^{X^PQIP^X)). Denoting the expression in the brackets as a matrix (p^g, we have (79)

C = Te* and ||C'|| = e ^ = e " ^

d^lioPo £ +

-^_.

(80)

Therefore the vacuum current (24) is equal U*)

=2/ ^%2W-Poo)p

M

e~^;

(81)

We = (FQO — Fec)pa is the non-classical part of the accelerations along the trajectory. It is interesting that the classical motion produces no current in the vacuum in the way classical electrons produce no diamagnetism in the metals. According to (58) and (59) "p

= £ IWOTPVSOPOLQOLT "•" h/j.vPu ~ £nvcTT\yePv)JQOT•

\°ty

The first term in (82) is of the order of K and could lead to linear divergency of the current, but this does not occur by symmetry reasons. Other terms give the usual logarithmically divergent expression for the current. Let us consider now the divergence of the current

V'--*f£fr^£[w-*»)<~

(83)

If we introduce j ^ which is defined as the integral on a finite volume in the momentum space, then d ^ will reduce to the integral over a large surface in this space r

^3

9W

e

s Unfortunately this integral depends on the form of the surface. This is natural, because the integral (81) is defined not well enough to differentiate it before the integration. In order to define j ^ correctly we could proceed in the usual way introducing a displacement of the point in the current. We could write, instead of (21) rx+e/2

^ ( x , p 0 ) = ? P o ( x - | ) 7 ^ p o ( ^ + | ) e ^-£/2

"

"

(85)

Introducing (35) into (85), we have Ju(Po,x) = - S p [5 (x - | ) 7 „ S (x + | ) p(l + 75) x e-x'(I+f)-x'(^-f)-ip%-^)

( 86 )

88

—

/

-ti-i/UiXi/

—

dx'^F^e^ + Oie5).

i + a(rf)'

The conture C is defined by fig. 3. In order to calculate dvjX and d^T^ we will need the explicit expression for the singular part of j ^ linear in the field. If we multiplied (86) by ezpe, we could expand the obtained expression in the power of e. Doing this up to the third order in e and linearly in the field, we have

eip£tf(P(hx) = e -d-^

\pu

Fi

S-

(87)

3

ieW--(ed)2eRFliaPaj

(l +

K 1 J a T ^ +. -{edyiTuaPg /^\2r + -ydg-7W>
- Pud^gPff]

\ ,

(88)

J~va — n^V'Va.fiJ'a.li-

(89)

•J~voPa — FvoPa 4- £vaPcn

(90)

Linearly in the field

because, as it is easy to show from (47), h^ Prom (48) and (75), (76) iuoPcr = •^•[TyaPa

-pvdjfjgPo]

}V — Of(P'yJ''UTPT

The quantity {ed^e^F^p,, form

is quadratic in the field.

=

(92)

PU-'ITPT)-

which entered in (82) could be written in the

PvjedfsuF^pg

+ if,,,

(93)

- e 7 F„ T p T ].

(94)

= —TpiedfU

fl = •7p(£d)2d1\p„Fieee

In (91), (92) and in the last term in (88) we can replace T^ ing (88)-(94) we can write the vacuum current in the form

K = 2 J ^e-^8{p2)e~^(-p00) ^d2fu 4

(91)

^f^

m

+

by F^.

{p„(l - ieW) + %^dpFvapa \{edffv

+

-^Mdf

HL ' 2432

Us-

In order to calculate the singular part of the current coming from the region of large momenta we can replace e — ^ - by 1 — -g—£. Using the relations (96) = \d2F^p^

W„ - p„a g W g = \&Tvaya

+

(97)

±d%

which are consequences of the definition

jX = 2 1 ^ e -^(-p 0 0 )«5(p 2 ) jp„ + i [ 5 2 ^ ^ + i ^ / U p J

+-

d% + i(ed)d2^

+ \(ed)%

+ -^{edffu

+ \f'v

}•

(98)

Using relations similar to (96), we can write the singular current of the right-hand particles in a compact form

This expression can be calculated explicitly. d ;, de-£-Fvapa5tf)

= 2p0deFvoPaS

(p2) = 2FvaP(T8 (p 2 ),

(100) (101)

Finally we have jX = g ^ y™e"^2

+ \diFi»ln£2

+ ^ 2 d 4 ^ F ^ e r - evF1Ter] \ • (102)

If we will average this expression over the directions of e^, the linearly divergent part of the current will be equal to zero, and the logarithmically

divergent part will have the usual form jl = •^I\n.ed1Flu. The current (102) is conserved for any eM. Using (102) we will find d^j^ and dJT^ very easily. According to (85) dvjvv = iFveeefv.

(103)

On the right-hand side we can use (102) and only the first term of it will contribute. Therefore d

A = -^F^J-^-

(104)

This expression does not depend on e^ because the matrix FvqFva portional to the unit matrix

is pro-

(105)

rf = ^hFp-

d

In order to calculate <9„TM„ and to find T^v we have to define T^v. simplest definition is the following: TL = i-z-3l{e,x),

The

(106)

-p

0„T;„ = -Fve—eett

= F^jv

+ eeFeu^-.

(107)

In the last term of (107) we can again use (102). The first term of (94) will not contribute because it is odd in e^ and we have d(TVu+T*„) = Fflwj„,

T

^

~ 48^

\ ^

F

+

\6^

~ ~^~)

(108)

?(£F)

-2^Fn„F^ 2 x ea* v°

£

J^FmFTU]

. (109)

As we see, T^u is finite, but it depends on the direction of e^. T/„ is conserved independently of the form of the cut-off and probably could be made independent of it. If we average T*v over directions of eM we find Td ""

— - — —2 ^liarua 48TT

24 "V

(no)

This expression was written in the first section. The important feature of the expression (109) for T^v is that the trace T^v does not depend on e^ and therefore Tf

=Td

=

F2

Cllll

independently of any averaging. In conclusion, I would like to express my deep gratitude to A. Valnshtein, S. Voioshin, V. Zakharov, L. B. Okun, D. G. Dyakonov, L. V. Gribov, C. Callan for interesting and useful discussions. References [1] S.L.Adler, Phys. Rev. 177, 2426 (1969). J. S.Bell, R. Jackiw, Nuovo Cimento 60A, 47 (1969). C.C. Callan, Phys. Rev. D2, 1541 (1970). K.Symanzik, Comm. Math. Phys. 18, 227 (1970). S.Coleman, R. Jackiw Ann. Phys. (N.Y.) 67, 552 (1972). [2] G. 't Hooft. Phys. Rev. D14, 3432 (1976).

92 Volume 194, number I

PHYSICS LETTERS B

30 July 1987

ANOMALIES AND A POSSIBLE SOLUTION OF PROBLEMS OF ZERO-CHARGE AND INFRA-RED INSTABILITY V.N. GRIBOV Landau Institute, 117334 Moscow, CSSR andCERN, CH-1211 Geneva 23, Switzerland Received 30 April 1987

We discuss a possible solution of the problems of zero-charge in QED and of infra-red instability in QCD. The solution is based on the introduction of extra currents which compensate the Pauli-Villars subtraction in the amplitudes of photon-photon or gluon-gluon scattering.

/. Introduction. One of the main hypotheses of modern field theory is the hypothesis that the theory can be defined by a lagrangian with an ultra-violet cut-off A. Practically, this means that when calculating any loops we can neglect all contributions of particles of momenta larger than v '/„ The only manifestation of momenta beyond V'A is that the coupling constant must depend on A. This hypothesis is commonly accepted as a very natural one. Theories which do not satisfy this condition, such as those which contain an axial anomaly, are considered as not self-consistent. But even in a usual theory like QED or QCD, the existence of anomalies in the axial current and the energy-momentum tensor shows that fluctuations with frequencies higher than sfk can be essential. In this paper we shall show that in usual gauge theory (QED, QCD) the vector current generally contains an "anomalous" part at third order in an external field, which comes from particles with momenta larger than ^fk. Unlike the case of an axial current, contributions to the vector current from momenta above yfk and below ^/J. are separately conserved, so that the vector current coming from above ^/X could have been neglected. .Another reason why this "anomalous" part of the current has hitherto been neglected is that it is impossible to write it down in a gauge invariant form in the framework ot the conventional lagrangian or Feynman formulations of the field theory. This is due to the fact that this current could not be ascribed, in perturbation

theory, to some degrees of freedom known a priori. It is not a current of real particles of the theory, because all the particles have a momentum less than Jx. It could be a current of some condensate, but condensates do not show up in perturbation theory. The existence of this anomalous part of the current means that the bare vacuum is unstable and that inclusion of interactions will lead to the creation of some condensates and, maybe, of some new physical states. If we knew these condensates and states, we could in principle formulate the theory in terms of new fields in the usual lagrangian way and write the current down as a gauge invariant form. In perturbation theory we have no choice at the moment, but to look at the development of the vacuum state as a function of time in the process of switching on the interaction or some external fields. In this time-dependent formulation of the theory the anomalous part of the current can be written down in a gauge invariant form. In this paper we shall present the arguments for the existence of an anomalous current in QED and QCD and propose a well-defined expression for calculating it. Qualitatively, the result is very simple. Calculations of the vacuum polarization to third order in the external field require a Pauli-Villars subtraction in order to maintain gauge invariance. Adding in the anomalous current cancels exactly the regulator contribution and the total current is really defined by a Feynman diagram without regulariza-

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Reprinted from Physics Letters B194, 1 1 9 - 124 (1987), Gribov, with permission from Elsevier Science.

119

93 Volume 194, number 1

PHYSICS LETTERS B

30 July 198^

q

i

,0

P

qi///q2

— ' —

Fig. 1.

— I

a

tion. The anomalous current corresponds to the diagrams of fig. 1 with Pauli-Villars loops, but is of opposite sign. Using this result we have been able to calculate the change in charge renormalization which is due to the anomalous current. In QED we find, instead of the usual formula discovered by Landau, Abrikosov and Khalatnikov, aL(«?2) = 37r/fn(L2/
(1)

as new expression for the '"running charge" a(q2) = aL(q2)/[Lai(q2)+\yn2

(2)

at q IL < 1, and a(q2)=^2,

q2>L2,

(3)

where aL(<7') is parametrized in terms of the position of the Landau pole L2, it is assumed that 9 : » m 2 , m2 is the electron mass. This new expression has no pole as a function of q2 and a(q2) is of the order of unity in the vicinity of the Landau pole. q2~L2. Of course, a(q2) does not depend on the initial ultra-violet cut-off /., because contributions from momenta both below and above N//. have been taken into account. Here L2 is playing the role of a physical cut-off. At this cut-off, contrary to the usual lagrangian formulation, the interaction does not vanish, but becomes strong. If one tried to write down the effective lagrangian in the region of momenta of the order of L, it would be a complicated and non-local one. This is the region where the condensates and new degrees of freedom can show up. If, by virtue of strong interactions in the region of momenta q2-L2, effective confinement takes place, in the sense that at q2ye>L2 there will be no charged states, then there will be essentially no current associated with particles with momenta above L. Then, we will have an effective lagrangian theory with physical cut-off L, which would differ from the usual one only at q2-L2. In QCD, the answer for the "running charge" a(q2) has a similar structure, but quite a different 120

•

'

b Fig. 2.

meaning. If we introduce a„(q2) in the form found by Politzer, Gross and Wilczek which can be written as, at q2>/.;, ap(q2)=4n/b\n(q2/;.2),

(4)

where in QCD without quarks

*=¥««

(5)

nc the number of colours, and then the new "running charge" would be a(q2) = ap(q2)/l-ha2(q2)

2

'

a(<j 2 )= N /lT,

q2<)2.

+ \]u2,

q2>>2,

(6) (7)

This charge is smalj_at large q2 and approaches a constant value ( y ' l l ) of the order of unity at This means that the usual infra-red instability is stabilized, if we add the "anomalous" current to the usual one. Again, as in QED, we come to the region q2~)rq where the coupling is strong and where newstates (presumably hadrons and condensates such as , (y/yy) can be created). If it is true, we can associate "anomalous" currents with the hadronic states and these condensates. 2. Vacuum polarization in external Coulomb field. In order to understand the reason for introducing the anomalous current, let us consider the vacuum polarization by an external field in the usual Feynman way. The vacuum polarization current is defined by the set of diagrams shown in fig. 2, with one-electron loops. In the standard formulation of the theory, only the first-order diagram leads to charge renormalization. The third- and higher-order diagrams are supposed to give no contribution to the charge renormalization. This means that the contribution of this diagram to the current in the momentum representation j„(n) must vanish at 0=0. However, if we calculate

94 PHYSICS LETTERS B

Volume 194, number 1

30Julv 1987

£1 3k"0

b

3f 3k^

ko

^7

Fig. 3.

explicitly the third-order diagram we will find, as is well known, that it does not vanish at q=0. It can be seen clearly, if one writes down the contribution of the third-order diagram to the zeroth component at the current j0 at q->0 in the form

,6

>/-0

-\*' -k P =J

/(oo)

qiqKq-qx-qiY

(9)

JeuM=ij/(x-t)yu

k

»-o

d 3 ?,d 3 ? ;

Ma) = I (2n)9 17-0 J

In order to get an expression fox jQ(q) which vanishes at q=0 we can make a subtraction of the Pauli-Villars type or, more generally, introduce a cutoff which would force f(k0, q^2) to vanish at k0->cc. Instead of fig. 3a we would then have fig. 3b, and the counterpart of fig. 4a will be fig. 4b. This cut-off can be introduced, for example, if we consider the current as the limit at €->0 of the expression

k

Ml)

Fig. 4.

n./

/

xexp(-i

! I

dko—f(k0,q,
(8)

and notices thatyt^o. ?i> 1z) at k0'»qiq2 does not depend on ko by dimensional considerations. Therefore f(k(j, q, q2) as a function offco,has the structure shown in fig. 3a, with a derivative as depicted in fig. 4a. As a result, from (8) one gets

Av dx'„ J \j/(x+t).

(10)

Of course, we can proceed this way, but it is clear that this procedure is a very unnatural one. The quantityytoo) is defined by the scattering amplitude of vacuum fluctuations on an external field and by the density of panicle states in the vacuum. In order foxfiko, q,,q2) to start vanish as &b-»oo, some physical interaction must change the number of states, or the scattering amplitude. To understand this phenomenon it is reasonable to ask what are the bad features of the vacuum polarization charge density, if we calculate it without sub121

95 Volume 194, number 1

PHYSICS LETTERS B

traction. For a point-like external charge, the calculation of the third-order diagram is very simple, if we are only interested in the charge density at distances from the source which are much larger than the Compton wavelength of the particles in the loops. At such distances, unlike the contribution of the firstorder diagram, which decreases exponentially, the third-order vacuum polarization density is equal to 2

p(r)= alfin

r

3

(11)

and exhibits instead a power-like decrease. Here a0=l2l/4n, l0 is the charge of the external source, taken equal to the electron charge. Such a slow decrease of the charge density means that the bare vacuum has an infra-red instability, because an external field produces infinite charge. Of course, we would not like to live with such a theory. But, it may be that by eliminating this term we go to another extreme and ask for a good theory too early, at a price of encountering the zero-charge problem. As was discussed above, an increase of the charge at large momenta can diminish the number of charged states at momenta of the order of the Landau scale L. That would mean a real physical cut-off. Contributions of various momenta to the current would have the form of fig. 4b with V '/. = L and the infra-red instability would be killed. Now it is clear what we mean when speaking of an anomalous current. If the contributions of various momenta to the current have a structure corresponding to figs. 3a and 4a, but the lagrangian part of the current has the structure of figs. 3b and 4b, then the difference between these two currents is the anomalous current. One reason for choosing this wording is that an anomaly, i.e., the non-conservation of some charge (axial,...) is due to the same states with infinite momenta we are concerned with. 3. "Running" charge of an external source. The usual way of calculating the running coupling constant consists of calculating the renormalization of the vertex parts and Green functions in a logarithmic approximation. It can be done in a simpler and more transparent way. Because we are interested in the ultra-violet behaviour of QED, we can neglect the electron mass. In this case, the calculation is especially simple. As is well known since the old paper of Kroll and 122

30 July 1987

Wiechman, all the diagrams, except for the first-order one, give a polarization density p( r) which is a deltafunction, S(r), for massless electrons and a point-like source, if one makes a Pauli-Villars type subtraction. The polarization density, corresponding to the first-order diagram, can be written in the form

*->--£/"£$•

(12)

where p'"(r) =<5(r). In order to find the charge density distribution in the logarithmic approximation it is sufficient to rewrite (12) in a self-consistent equation, putting in (12), instead of p'"(r), the total charge density: p(r)=p"'+pP(r).

(13)

One then obtains the equation P(r)=p<

6TT

dV

P(r') \r-r'\2

(14)

This equation contains the logarithmic divergence coming from the region \r—r' | -»0. If we introduce a small radius of the source r0, corresponding to the ultra-violet cut-off /. : ~ Mr5, and exclude the region \r— r' I r0 can be rewritten as r' < r

'<'> = - §

ln

£ - f i £ b j ^r-p(r').

(15)

in rg on r J This equation can be solved very easily. If we introduce r' < r

a(r)=a0

J d V p(r'),

p(r) = An^a

rdra(r), (16)

[3n/a0+\n(r2/rl)]r2dr2a(r)

= -a(r),

a(r) = 3n/[3n/a0 + \n(r2/rl)].

(17) (18)

It is the same expression as (1) with r2->l/q2,

L2=>(\/r20)exp(3n/a0).

Up to this moment we have been calculating the current in the usual way, with the Pauli-Villars type regularization which permitted us to throw out all the higher-order diagrams.

96 Volume 194, number 1

PHYSICS LETTERS B

30 July 1987

As we discussed above, if we do not make any subtractions, the third order would give us the non-zero contribution (11). It can be shown that only this contribution survives at r>r0. This means that we must add (11) to (14), changing again the external charge a0 to the total charge a(r), as we did when deriving (14). We get, instead of (14), P(r)=pc

6n ^

cPr'

p(r')

*\r) 3KV

(19)

Fig. 5.

and, instead of (17), we obtain {(3n/a0)+\n(r2/r2)]r2dr,a(r) = - a ( r ) + }a3(/-)-

AU(X, Z):

(20)

This equation can be solved easily. If a L (r) satisfies (17), then a(r)=a L (r)/[iaE(r) + l]'«

(21)

satisfies (20).

This is just expression (2) whose properties were discussed in section 1. 4. Gauge invariance and a general expression for the anomalous current. While considering the vacuum current in a static field we did not encounter the problem of gauge invariance of the anomalous current (11) explicitly. But, of course, in the general case it exists. Moreover, we do not know how in general to find a definition of this current. What is the underlying principle of this definition? We have not solved this problem. We just guess a general gaugeinvariant expression for this anomalous current using in the analysis the vacuum current in an explicitly gauge-invariant form and the limiting case of static field. The current (11) can be written in the form p(r) = (l/3n2)
(22)

where (r) is the Coulomb potential. But the Coulomb potential can always be written as an integral of the field strength along some line r) = -

\ dEidXj,

(23)

and does not depend on the direction of the line. For a general field, the natural generalization of (22) would be the expression

dxlF^(x')

(24)

where the integration is performed along some straight line where z„ is a vector defining the direction of the line. But now, A^(x, z) would depend on zu. If we want to write the current of the particles of infinitely large momenta, we can use the fact that the particles propagate along the light cone. It is natural to expect that the A,, which defines the current would be an integral of F^x) along a straight line on the light-cone. Therefore, the natural generalization of (22) would be Jl =

1 3K2

f di2 Afl(x,z)A2(x,z), J 4;r

(25)

where the angular integration is performed over the directions of null-vectors zm corresponding to different straight trajectories from — oo to the top of the light cone (see fig. 5). However, the expression (25) does not have the right Lorentz transformation properties; because if we parametrize z„ in the form (1, z), |z| 2 = 1 and d£2 is then the integration over the orientations of the three-dimensional unit vector z, after Lorentz transformation (1, z) goes to y( 1, z') and dX2 goes to d£2'y2. If we would write, instead of A2{x, z) in (25), [M„(*, z)] 2 where n= ±(1, - z ) is the space or time reflection of the vector (1, z). This «„ is a vector normal to the light-cone which transforms as follows under the Lorentz transformation: H„-»l/y(l, — z'), and would compensate the change in <M2. Therefore we would like to rewrite (25) as 1 CdQ A^x,z)[nA(x,z)]2 in2 J Ait

(26)

123

97 Volume 194. number 1

PHYSICS

Unfortunately, this is covariant but not conserved. A conserved current can be written down in the form

y

'" = 3^!^f(^ U 2 ) t ^ U ' r ) ] 2 .v

-

J dx,dlA,(x,,z)(nA)2),

(27)

— oc

where the integration over x' is performed along the same line defined by z. This expression is our final hypothesis on the form and magnitude of the anomalous current in QED. It is easy to show that in the case of a time-independent field, F^ reduces to (23) because b^A^n^A^) =0 and A0 and n,A„ do not depend on z. The expression (27) can be written down in an explicitly covariant form if we introduce, instead of (24), A"

A„{x,x') = j dx''Fm(x"),

(28)

where the integration is performed along a straight line, in which case (27) is equivalent to 67T3 J

(«, X-X )-

- ^dx:d:.AAx"x')(nA)A,

(29)

with nu a space or time reflection of (x—x')u. Because (29) contains a total derivative, one can reduce the integration to the surface JC0-+— oc and get (27). If the expressions (27) and (29) are a correct guess, then the non-lagrangian theory can be formulated by writing down the Maxwell equation for second-quantized operators in the form bvF„ll(x)=ell(vyllV)>+tt}.

(30)

TTERSB

30 July 1987

because all operators are on the light cone. The expression (29) for the current appears comparatively natural if one is considering the current (.Hin\Jii(x)\£2i„y as a function of time in an external field. But this we would like to postpone for a more lengthy paper. We also will postpone to this more lengthy paper the derivation of eq. (6) for a(q2) and the expressions analogous to (27) and (29) in QCD. In connection with QCD we want only to draw the reader's attention to the following interesting problem. The calculation of the gluonic contribution to the anomalous current is straightforward, if we know the formulae of type (27) and (29) in QCD, because the gluons are asymptotically free. But quarks are not asymptotically free, and have electromagnetic interactions which are strong for large momenta. As we have discussed, the charged one-particle states must be suppressed at large momenta in order to produce an effective Pauli-Villars cut-off for low-momentum physics. But, if one-particle states were suppressed, the vacuum colour current of quarks would also be suppressed at large momenta and we would have no anomalous vacuum colour current of quarks. This means that the formula (6) for the QCD coupling constant could be correct even with the inclusion of quarks, if we change the b=ij-nc to b=1±ns. — f«f, where n{ is the number of flavours. Another interesting thing to mention is that in scalar electrodynamics, instead of \ in eq. (2) for QED, one has — 1, and the theory is unstable even after the inclusion of the anomalous current. In conclusion, we would like to point out that the approach we have described here gives us the possibility of considering a theory with axial anomaly. If we add to the usual axial current with non-zero divergence a current of the form Jl = --£p\-^FxAx)Mx,z),

(31)

where «„ is an arbitrary vector, similarly to (28), the total current would be conserved, and the theory could make sense.

where (
Physica Scnpta. Vol. T15, 164-168, 1987.

A N e w Hypothesis on t h e Nature of Quark and Gluon Confinement V. N. Gribov L. D. Landau Institute for Theoretical Physics. Academy of Sciences of the USSR. Moscow 117334, ul. Kossygina 2; Central Research Institute for Physics, H-1525 Budapest 114. P.O.B. 49, Hungary Received June 1, 1986

Abstract

2. The vacuum of light quarks in the field of a heavy

A new hypothesis on the nature of quark confinement and the structure of infrared behaviour in gluodynamics is proposed.

1. Introduction

The essence of the phenomenon is the following. Let us see, what happens with light quark states of negative energies occupying the Dirac sea in the field of a heavy antiquark, which has the form (1) because of the polarization of gluon and fermion vacua. Neglecting the interaction of quarks with each other one can divide the quarks into two groups: quarks which are in singlet colour states with respect to the antiquark-quarks qs — and those forming an adjoint representation with the antiquark-quarks q„. The fields which act *>n quarks qs and q, are

The generally accepted view on the nature of confinement of quarks and gluons in quantum chromodynamics is based on the hypothesis according to which, in any non-abelian gauge theory, the asymptotic freedom leads to an infinite charge increase, and, consequently, only colourless states can survive. The analysis of vacuum currents (which will be presented in a more detailed paper) shows, however, that there is no reason for such a strong hypothesis. From the point of view of this analysis it seems to be most natural, that in a nonand albelian theory without light quarks (i.e., quarks of mass m0 much smaller than fi where I In is the strong interaction scale) at large space-time scales the conformal invariance, destroyed by the introducion of ultraviolet cut-off, will be reestablished. In such a conformal invariant theory asymptotic gluon states respectively. Since the Hamiltonian of the quark in the external are absent (the Green functions are of scaling character field A looks like similarly to those in the theory of phase transitions). How- H = //ki0 + A, (5) ever, the Coulomb field of external charges does exist, since the conserved charge operator has no anomalous dimension. the energies of quarks qs decrease [Fig. 1(a)] while those of qa increase [Fig. 1(b)] inside the region where the field is acting. The value of the charge ag of an arbitrary source turns out to As a result, free places for the quarks qs appear which can be be universal and has to be determined by the equations of the occupied by quarks with positive kinetic energies. On the theory. The field of a source at rest is in this case other hand, for the quarks q, having negative kinetic energies it becomes possible to go to infinity with positive kinetic A = *!;•. (1) energies. For a weak field neither of these processes can take r place because of the uncertainty principle. If, however, the At fir <§ 1 the value <x(r) is defined by the usual expression of field is sufficiently large, a particle with positive kinetic energy p in the field /f, can be in a stationary state with a negative asymptotic freedom energy provided that |/l s | > p. Since p Z 1/r, it is natural to expect the relation aW ft (2)

*--?«

" ¥mW)-

= ""<->"'

while at fir P 1, ct(l) -» ag. In the present paper we show, that for confinement in the real world with light quarks, it is sufficient if ag is larger than a finite critical value ac C * 2( „ , - ! ) " .

l(„c-i)>.

(6)

to be the condition for a real appearance of a free place for qs. In the same way, quark q, with a negative kinetic energy — p can go to infinity with a positive kinetic energy if A, — p is larger than zero, i.e., again A, > \p\ > l/r and, conse(3) quently,

In this case there exists a phenomenon which can be called quantum screening; it is caused by the formation of superbound states in the light quark vacuum at ag > ac analogical to electron states in the field of nuclei with Z ~ 180 [1], Physica Scripta TI5

V

Let us discuss, what will happen, if a, fulfils the weaker condition (6) and does not satisfy the condition (7). Obviously,

A New Hypothesis on the Nature of Quark and Gluon Confinement

165

SU(2). The mistake is, that - according to traditions taking their origin from Dirac - we suppose that the vacuum is occupied by particles with negative energies. In fact this vacuum consists of particles and antiparticles with negative energies; only such a situation corresponds to the calculation of loops with two directions of arrows in the usual approach. Adding to the previous considerations in the interaction of the heavy antiquark with the light antiquarks which might be in (0, 2) and (1,0) SU(3) states with respect to the heavy one, and taking into account, that Fig. la

Fig. lb

:j(,.I

^ld-21

if there is a free place in region I in Fig. 1 (a), one of the quarks q, with a negative kinetic energy in region II will transfer to region I, obtaining a positive kinetic energy. Instead of this quark, in region II remains a hole — an antiquark — which passes with a positive energy to infinity. This process resembles the emission of a positron and the subsequent formation of a supercritical atom in the case of a supercharged nucleus. In our case the situation is essentially different. The heavy antiquark is a particle with the same charge a0 as the quarks in the vacuum, and its large field is due only to the vacuum polarization which is a consequence of the fact, that this antiquark was the only charged object against the background of the neutral vacuum. Because of that, as soon as the antiquark of the sea goes to infinity, and the quark with the negative energy being a colour singlet state with respect to the heavy antiquark, begins to rotate around it, the vacuum polarization of this colourless system will change drastically. The field inside the orbit of the light quark remains, of course, almost the same, but outside it the screening of the heavy antiquark by the light quark will provide a field equal to zero. At the same time, obviously, a large number of gluons will be emitted besides the emission of the light antiquark which took place before. Direct calculations confirm the presented qualitative explanation of the phenomenon. The only difference is, that if eq. (6) is satisfied, there appear a whole series of quasistationary Coulomb states in the field (4) among which only the lowest one is stable. Consequently, the light antiquark may be emitted even with a small momentum. This would correspond to the capture of the quark on a far-lying orbit, with its subsequent transition to the lowest orbit by gluon emission. Thus we see, that no stable coloured state of the heavy antiquark can exist if the polarization field is sufficiently large. The appearance of a coloured state is metastable (with a complex energy) and it does not differ in anything from a highly excited colourless state — a meson — with a light quark on a very far orbit. At the same time we come to an interesting statement on the structure of the Z)-meson state, being a heavy antiquark and a light quark with a negative total energy which does not go to infinity only because there is no place for it: all the levels with negative energies at infinity are occupied. (Obviously, the same can be told about the Z>-meson state, substituting the word quark by antiquark.) We will construct this state explicitly in another paper, taking into account the finiteness of the heavy quark mass. Still, such an approach is clearly not satisfactory. Applying it, e.g., to the colour group SU(2), we obtain meson states qq but not baryon states qq although the features of the latter ones should not differ from those of the meson states in

(8)

<xl_

74

n,

it becomes clear, that the system (q, q) will be bound at larger a values. If, however, a was large enough and such a state could be formed, its field would coincide at large distances with the quark field and the system would be unstable with respect to the capture of an additional antiquark. As a result, a colourless antibaryon state would be produced. 3. Attenuation of quarks in the vacuum The instability of the coloured quark state can be established in fact without assuming that the quark is a heavy one. If a quark with a mass m and a supercritical colour change existed in the vacuum, its motion could always be described in a quasi-classical way by means of a wave packet, the longitudinal size of which is of the order of \\\p\, where p is the quark momentum. The quark field outside the packet is of the form A\ =

sJiPX?

= «•;.*

(9)

where pt, is the four-momentum of the packet; a = a(r, p), f2 = (Po/ m2 )( 2 — vl)2 + Q2\ z> Q a r e t n e longitudinal and transverse distances from the center of the packet. The behaviour of the vacuum states in such a field will differ from that in the case of a heavy quark only in the following respect. Since in comparison to the field (1) the field (9) is shrunk in the longitudinal direction while the value of A is proportionally increased, for pr ~ 1 the new state formed in the vacuum with negative energy but with positive kinetic energy will have a longitudinal size of the order of (m/p)(\/p), and, consequently, a momentum p'. ~ p(p/m); still, it will exist, as before, independently of the value of the quark momentum (Fig. 2). Difficulties arise only for quark masses m < p, when the longitudinal size of the bound state turns out to be less than the size of the packet, while the field outside the packet is of the form (9). However, as it was mentioned above, there exists a whole series of levels in the vacuum at a(oo) > a t . The frequencies of these levels cu„ can be determined roughly by the condition - 1 • In — = 27t(n + 1)

(10)

(for states with momenta./ = J) supposing
166

V. N. Gribov

W o - -*n0-

-

Fig. 3

Fig. 2

contains a loop describing the vacuum polarization in the quark field. If the quark charge is larger than the critical one, this loop turns out to be complex in the momentum representation because of the described instability, and, consequently, the Gree functions of q and q become exponentially decreasing functions of .r2 — x, (i2 — l,). Accordingly, the production amplitude of qq at l2, /, -• oo will be zero. The same is true for an arbitrary number of quarks and antiquarks (and, as it will be seen later, for gluons too). Hence, a t ; -> oo only the amplitudes of meson production turn out to be different from zero; naturally, the sum of their squares has to be unity. As far as the presence of the coloured and the baryon currents at finite t needed for the transformation of the original qq pair into a meson is concerned, it is essentially secured by the quantum mechanical interference due to which only stationary states survive at large times.

m > m„ outside it; the latter lead then to the instability of the quark state. As soon as the renormalized mass is always larger than the bare one, even the lightest quarks turn out to be instable, i.e., non-propagating in the vacuum. This leads to a paradoxon: a quark decays into a quark, gluons and a meson, which contradicts the law of energy conservation. The solution of this puzzle is the following. We have supposed, that the real free quark exists in a vacuum consisting obviously of bare quarks and gluons. If such a quark existed, its energy would have been determined by its renormalized mass which contained a large energy of its Coulomb field. As was mentioned above, when a superbound state is formed the 4. Analogies from the field of solid state physics system becomes neutral and the Coulomb field of the initial quark vanishes outside this neutral state. It is just the energy Because of the importance of the investigated question and of this field which transfers into the energy of bare gluons and because of its paradoxical properties it would be useful to quarks. We can, however, ask ourselves what will happen know whether there exist analogous phenomena in other with the conservation of energy when these gluons and fields of physics. From a discussion with A. A. Abrikosov quarks acquire polarization charges and the energy of each and L. A. Falkovski I learned about the phenomenon of one becomes equal to the energy of the initial quark. There Andreev's reflection. Andreev has investigated the behaviour are two equivalent answers to this question. Since — as we of electrons near the junction of conductors in normal and intend to prove - there are no free quarks and gluons, this superconducting states [3]. In a superconductor there is a process of dressing up will not take place, the system of forbidden zone due to the pairing of electrons with momenta gluons and quarks changes into a system of mesons, while larger than the Fermi momentum p? and smaller than pr + A their baryon and colour changes go there where the initial (Fig. 5). The question is what happens to an electron of the hypthetic "real" quark, the instability of which we investigate, normal conductor when it approaches the junction surface with a momentum corresponding to the forbidden zone. comes from. According to Ref. [3], the superconductor will be entered by The second answer is based on the observation, that the a ee pair while a hole will proceed back to the normal question of energy conservation for the non-stationary states conductor from the junction. This phenomenon is analogous in the modern field theory is not simple at all [2]. Because of to that which was considered before, namely that a single the existence of the anomaly in the energy-momentum tensor charge can not distribute into the forbidden zone contrary to the polarization energy of the charge comes from the region a double one. Consequently, the single charge transforms of momenta of vacuum particles larger than the ultra-violet into a double one, while the charge — 1 becomes reflected. cut-off. Only the stationary states are not exchanging energy In the case of the production of a qq pair in, e.g., e + e~ with this reservoir and only for them the conservation of free annihilation the region between the quarks will play the role energy [2] leads to the same results as the usual conservation of the normal metal. The external region — the vacuum — law. will correspond to the superconductor. Since the quarks are The instability of the quark with a colour charge larger produced with charges a0, they can freely distribute at the than xc manifests itself formally in the following way. Suppose beginning, until their increasing charges do not reach the that we want to calculate the amplitude of production of a q, q pair in e + e" annihilation. In the Born approximation to this amplitude corresponds the graph in Fig. 3, where q and q are described by free Green functions. Taking into account the self-energy corrections to each Green function, among them there will be corrections of the form Fig. 4 which Fig. 4 Physica Scripla 775

A New Hypothesis on the Nature of Quark and Gluon Confinement

167

wonderful property: the Coulomb field of any external charge exists due to the fact, that the anomal dimension of the conserved current operator is zero. Besides, the charge of an arbitrary external source is at. If there were no light fermions, our world would look, p p +A presumably, just like this. If, however, light quarks exist and r K the considered crtical charge <xc with respect to the fermion r normal vacuum is less than ag, the situation changes entirely. state superconductor From the point of view of the action on the fermion vacuum a gluon with life-time / ~ pin2 behaves like a particle with a mass of the order of p.. The field of such a particle will differ from eq. (9) only by a substitution of the matrices Fig. 5 belonging to the fundamental representation by the matrices of the adjoint representation. Being charge symmetrical, this coloured field acts in the same way on quarks and antiquarks critical value. At the moment when this happens the quark in the vacuum. One can divide, as before, the vacuum quarks transforms into a meson moving in the same direction and and antiquarks corresponding to the representations (1,0) into a reflected quark. and (0, 1) into quarks and antiquarks which together with the There is, of course, an essential difference: the Andreev gluons (1,1) form the representations (1,0), (0, 2), (2, 1) and reflection takes place only in a small momentum interval near (0,1), (2,0), (1, 2), respectively. If so, the fields acting on these the Fermi momentum, while the reflection of a quark in the quarks and antiquarks will have the following charges: vacuum goes on at arbitrary momentum values. The phenomenon of the quantum screening which means n, that the colour charge of the quark increases with the time, = "a2; and, reaching a critical value, abruptly becomes zero because 2 3 of the appearance of a bound state in the vacuum, seems to (11) have also an analogy in solid state physics: the Kondo effect. «2.0 = «(0.2) = - J ( 1 - + Here the impurity spin is screened by an electron being in a bound state with an energy equal to the Fermi energy. Hence at a value ac = 2/nc even somewhat smaller than in the quark case, the gluon becomes unstable with respect to the 5. Gluon confinement emission of a quark-antiquark pair and gluons with the If the gluon existed as a real massless particle, the described creation of a gqq state, which, being colourless as a whole, mechanism — the instability of the coloured state and the provides another parton component of the same meson state formation of the colourless one — would not work. Indeed, which is produced together with the quark. as it was shown above, the necessary condition for this mechanism is that the mass of the particle has to be larger than the 6. Gluonic zero point fluctuations in the Coulomb field bare mass of the lightest quark. However, a real massless gluon does not exist even in a theory without light quarks. The basic assumption of the present paper is the hypothesis As was discussed in the introduction, the most natural according to which in a theory without light quarks the structure for a theory without light quarks is the following charge of the particles reaches a value <xg larger, than (details will be given in another paper). In the region of large virtual momenta k2 $> p? the Green functions and the vertex nc\ n\j parts are of the usual form of free gluodynamics. For virtual momenta k2 < p}, however, the initial conformal invariance, Upto now we did not bring up any arguments supporting which was destroyed by the introduction of a cut-off, becomes such a relation between at and <xc. A serious reason for such reestabilished and all the Green functions obtain conformal a relation arises when one investigates the behaviour of gluon invariant, scaling forms. The gluonic Greens function, e.g., is zero point fluctuations in the external Coulomb field. in this case of the form (k2/p2)A'\ A being its anomal dimenAs it was realized by A. B. Migdal [4], the vacuum of sion. In this expression there is no gluon pole; this means, that bosonic zero point fluctuations behaves in a way significantly in such a theory the gluon as an asymptotic state is absent. different from that of the Dirac vacuum with regard to the In the Green functions remain only cuts, and because of external critical field. Let us suppose zero fermion and boson that, the theory contains no asymptotic states. The physical masses, then the difference will be the following. In the spectra meaning of this phenomenon is, that a gluon produced with of vacuum particles in a supercritical field states with complex momentum p and charge a0 decays into gluons with smaller frequencies w, = w[ + ico" having negative real parts w\ momenta and smaller charges after a time t of the order of arise in both cases. The imaginary parts m" are, however, of piI*2, when its charge reaches some critical value ag. The opposite signs for fermions and bosons. The sign of m" for newly produced gluons will behave in the same way, and as fermions corresponds to a usual unstable state which is a result the whole energy transformations into the energy of damping with the time because of transitions into the ground oscillation with wave lengths which are of the order of the size state having a different charge. For bosons the sign of cu" is of the system. This reminds the waves of the parameter of in accordance with the increase of the wave function in time. This increase corresponds to the formation of a boson conorder at critical temperatures. The described conformal invariant theory has, however, a densate in the region where the field is acting. The density of

%mmwm¥///

Physica Scripta TIS

168

V. N. Gribov

the condensate will be growing [4] until the sum of the two fields (the external one and that of the condensate) decreases to the critical value at which w" — 0. In other words, the boson vacuum screens the external field in such a way that it cannot be larger than a definite critical value. This fact provides by itself, without detailed investigations of charge renormalizing currents, a strong argument in favour of the hypothesis, that the charge in gluodynamics can not approach infinity; it is limited by a critical value ag. The critical value of the external Coulomb field of the form (1) can be easily calculated if one neglects the interaction of gluons in the vacuum. States with different angular momenta y turn out to be unstable at different a values. Consequently,

«,(./) - - a + i)-

d2)

If gluons withy = 0 existed, as would be equal to l/wc, i.e., it would be smaller than ac. Luckily enough gluons with y ' = 0 are pure gauge degrees of freedom and they give no contribution to the critical field. Thus our estimation gives a

3 = - > ac.

(13)

7. The structure of meson states In the framework of the considered picture the structure of the ground states of heavy mesons (D, F, . . .) does not differ significantly from the usual Coulomb states similarly to the structure of supercritical atoms which is not very different from the usual atom structure. The states of light mesons might have, however, very unusual features, especially in fastly moving reference frames. The reason is, that in the presented picture there is an asymmetry between the antiquark (quark) with positive energy, creating a large field, and its partner quark (antiquark) with negative energy, which, in contrast with the former one, can be easily exchanged by vacuum quarks (antiquarks). This difference in the exchange interactions of the partners has to lead to a difference between their effective masses. Because of that the heavier antiquark (quark) is placed in the center, while the lighter quark (antiquark) is at the periphery. It seems to be natural, that these two configurations - an antiquark in the center and a quark at the periphery, and a quark in the center, an antiquark at the periphery — have to be divided by a barrier. In a fastly moving reference frame one transforms into the other rarely, since for the exchange of these configurations the sign of the self-consistent field has to be changed. From this point of view the baryon state is less unusual, because the exchange

Physica Scripta 775

between its constituent quarks does not require a change in the self-consistent field. As we have seen before when we have considered the instability of the gluon, a third (may be most probable) component of the meson wave function is also possible. Namely: in the center a gluon field is localized, while both the quark and the antiquark have negative energies and they are placed at the periphery, forming an adjoint representation. The features of this component is similar to those of a meson state in the bag model. In the presented picture unusual configurations are created even in the case of heavy mesons, if highly excited states are considered. Indeed, if a quark with a negative energy is in a highly excited orbit with a large radius, then inside the orbit the field will be a supercritical one. This leads either to the emission of mesons and a transition into the ground state, or to the possible creation of a state of molecular or quasi-string type in which the field does not exceed anywhere the critical value. Finally, let us note, that in the previous considerations we have ignored all along the interaction of vacuum quarks and the existence of finite densities of meson states and condensates of the type ijiip which is connected with the interaction. The appearance of condensates in the vacuum is quite natural from our point of view. Indeed, at m0
2. 3. 4.

Pomeranchuk, I. and Smorodinsky, Ya., Journ. Fiz. USSR 9. 97 (1945); Pieper, W. and Greiner, W., Zcitschr. fur Physik, 218. 327 (1969); Zeldovich, Ya. B. and Popov, V. S.. Uspektii Fiz. Nauk. 105, 4(1971). Gribov, V. N., KFK1-1981-66 (1981). Andreev, A. F., ZhETF, 46. 1823 (1964). Migdal. A. B., ZhETF. 61. 2209 (1971).

March 1991 LU TP 91-7

pert

QCD WORKSHOP - MEETING Meeting May 21-24,1991 Department of Theoretical Physics, University of Lund, Sweden A Joint Arrangement Lund University and NORDITA

P o s s i b l e S o l u t i o n of t h e P r o b l e m of Q u a r k Confinement V.N. Gribov1 Landau Institute for Theoretical Physics, Moscow Abstract An approach to the problem of how a confined particle can provide long range forces in QCD is suggested, which is based on both the existence of a strong Coulomb-like interaction beyond the scale 1/A and the existence of light quarks with bare masses less then A. In contrast with the well-known phenomenon related to supercharged nuclei in QSD the colourless bound states with negative energy occur in QCD. This results in the phase transition and leads to the crucial change of Dirac's picture: positive energy states become occupied instead of negative energy states. In the new state colourless states have positive energy, but quarks get complex masses. The quark Green functions acquire analytic properties similar to the analytic properties of the Green functions at linite temperatures.

'This work was completed during visits to the Institute for Advanced Study, Princeton, Stanford Linear Accelerator Center, Central Physical Research Institute, Budapest and the Department of Theoretical Physics, University of Lund.

104

1

Introduction

The problem of quark confinement has been for many years the most challenging problem for the theory of strong interactions. We know now almost certainly that quarks exist as point-like particles with definite quantum numbers. At the same time, they have never been observed as real stable particles. The first problem for the theory was to provide forces strong enough to prevent quarks from separating. This problem is especially serious, since the bound states we observe — hadrons — are very loosely bound and can easily be destroyed in scattering processes. That means that the confinement forces have to be long range ones. The problem of t h e existence of sufficiently strong long range forces for the interaction of quarks was essentially solved by the discovery of QCD [1] and of asymptotic freedom in QCD [2]. QCD provides a Coulomb interaction between quarks via gluon exchange. As asymptotic freedom shows, this interaction can be large enough at large distances because of the instability of the gluonic vacuum fluctuations. The discovery of QCD was a very important development, but it created a new question: that of colour confinement. For a long time the suggested solution was the hypothesis that the instability of gluonic vacuum fluctuations produced forces growing with distance which confine both quarks and gluons. Numerical calculations in Euclidean space led to some success in observing the forces increasing with distance. However, the most obvious problems with this hypothesis, namely, how to see t h a t such forces are not in contradiction with relativistic invariance, and how confined gluons can produce these forces, remained unsolved. Especially the second question — how a confined particle can provide long range forces — was left almost without attention. But this problem is very severe. In QCD we have only one parameter A (XQCD), which is supposed to define the size of hadrons and the radius of confinement. T h a t means that at distances larger t h a n 1/A quarks and gluons (even if they are not free particles but bound states) are not interacting at all. If they are light (and we know they are light inside hadrons), how can these bound states — the hadrons — be stable. If they are heavy, what is the scale at which they acquired large masses? This paper is an a t t e m p t to solve these problems using an idea proposed several years ago [3]. The essence of the idea is that the strong interactions in our world are defined by two, equally important, ingredients: • the existence of gluons which, due to asymptotic freedom (or infrared instability) vide a strong Coulomb interaction beyond the QCD scale 1/A, and

pro-

• the existence of light quarks with masses mo much smaller than A. If these two ingredients are present, the main phenomenon which occurs is t h a t the strong colour Coulomb field (a/q2 , a ~ 1) of any coloured particles causes "falling into the center" in the light fermionic vacuum, similar to what happens in QED around a heavy nucleus with charge Z exceeding 137 ^ '. T h e latter leads to the existence of superbound atoms with

1

105 atomic electrons inside the Dirac sea and to the instability of nuclei with Z > 137. In this case the "falling into the center" happens in the region of space between the radius ro of the nucleus and the Compton wave length 1/ra of the electron. In QCD the "falling into the center" occurs in the perturbative vacuum in the region between 1/A and l / m 0 and produces colourless bound states with negative energy and instability of any coloured states. There is one important difference compared to the QED case, namely, the existence of negative energy bound states must change the features of fermionic vacuum seriously. T h e most obvious possibility for changing the fermionic vacuum is the creation of a bosonic condensate from these negative energy bound states. This would lead to the increase of effective fermion masses up to values big enough to prevent the "falling into the center". But, as we show in this paper, if m 0 / A is sufficiently small and provided that a > ac (with ac a critical value), another possibility emerges, which corresponds to a really dramatic change in the structure of the fermionic vacuum. Due to the strong interaction, the average value of the number of quarks (n+) and antiquarks (TI_) is not zero in the appearing new vacuum state. The easiest way to understand this state is to use the language of the old Dirac theory in which fermions can have both positive and negative energies, but the vacuum corresponds to a state with all negative energy levels occupied. Because of this structure of the vacuum, one can forget about negative energy states and talk only about positive energy excitations. Essentially, Dirac created a model in which the sign of the kinetic energy of fermions was conserved, and the source of this conservation was the Pauli principle. However, if mesontype qq states with negative total energy exist in the bare vacuum everything can change. The existence of a state of two particles (e.^. , qq ) having positive kinetic energies which have acquired negative total energy due to the strong Coulomb interaction implies that, in addition to t h e free negative energy states, new negative energy states appear in the vacuum which have to be occupied. If we now fill these new states, our vacuum will contain not only negative, b u t also positive kinetic energy fermions. The second important phenomenon is the following. The same interaction t h a t produces a negative energy state of two positive kinetic energy particles, will also produce a state with positive total energy of two negative kinetic energy particles. This positive energy state has to be empty. This means, t h a t besides the occupied positive kinetic energy fermionic states we shall have many empty negative kinetic energy fermionic states. As a result, Dirac's model for the conservation of the sign of energy will be destroyed, and it becomes necessary to consider all four solutions of t h e Dirac equation. Fortunately this does not mean that one can observe negative energy fermions; neither positive nor negative energy fermions are observable because they are unstable (having complex masses). As we shall see, a positive energy quark will decay into a positive energy meson and a negative energy quark, while a quark with negative energy will decay into several negative energy quarks and antiquarks and a positive energy meson. There is a possibility that the Dirac model for conservation of sign of energy will be recovered in a very peculiar way. Positive energy quark does not exist (all positive energy

2

106 levels being occupied), but negative energy quarks are unstable, decaying in described way. As a result neither colour quantum numbers nor quark quantum numbers are propagating. Gluons, like any other coloured states, will also be unstable, each decaying into a meson and a quark and antiquark with negative energy. The only real excitations in such a vacuum will be hadron-type colour singlet states. T h e structure of hadron states and vacuum fluctuations is defined in the theory by two scales. One scale is A. The other, smaller scale mq generated by m0, can be written in the form / A V mq = m0

7<1.

(1.1)

At 7 ^ | we have mq « ^/m^X and mq ft! m T , with m* the 7r-meson mass. In the space-time region between 1/A and l/mq a strong Coulomb field is present. In the region outside l/mq the colour field is small and produces only residual hadronic interactions. All these unusual features of the fermionic vacuum can be summarized in a simple formal statement: sufficiently strong Coulomb interactions can produce transitions in the fermionic vacuum which lead to Green functions of quarks and gluons having complex poles on unphysical sheets. T h e cuts, which define the imaginary parts of the poles, have properties different from those of the usual unitarity cuts generated by transitions to other physical states. These unusual analytical properties are in contradiction with the analytical properties of the Green functions in perturbation theory. T h e reason for this difference is t h a t perturbation theory is an expansion which takes into account small fluctuations around a stable ground state. In the usual perturbative approach we start with a bare vacuum, introducing t h e interaction adiabatically, and our bare vacuum gradually becomes the real vacuum (curve (a) in Fig.l). However, if the perturbative vacuum is unstable and the interaction induces a phase transition, we shall never reach a real vacuum adiabatically starting from t h e bare vacuum. In order to do so, we have to start with an excited state. At a certain value of the coupling constant levels will cross, and at a large value of a we shall reach the ground state (curve (6) in Fig.l). The structure of the initial state has to be found in a self-consistent way from the condition t h a t we reach the ground state at a large value of t h e coupling. T h e Green function which will appear in this new adiabatic perturbation will b e the expectation value of the T-product or of the commutator of Heisenberg operators in t h e excited state 6: ~ (fl&l Ti/)(xi)if>(x2) jJTib). The analytic features of these Green functions will be different from those in a field theory, but they will be similar to those of the Green functions at finite temperature, where t h e excited state we are averaging over is defined by t h e temperature. For example, if the state "b" contains n+(k) particles with positive energy and n_(A;) particles with negative energy, the free Feynman Green function will contain four poles: two poles in the lower half-plane: ha =

yk2+m2

— ie

with residue

3

1 — n+ ,

107

excited state free vacuum

- a

Figure 1: Intersection of levels corresponding to a free perturbative vacuum (a) and to an excited bound state (b) at a certain value of coupling constant a.

to

=

2 2 - V ik + m — ie

with residue

n_

and two in the upper half-plane (Fig.2): k0

=

yk2 + m2 -f ie,

k0

=

-\/fc

with residue

+ m2 + ie

n+ ,

with residue

1—n._ .

If, as usual, the Green function for interacting particles can be obtained by integrating over masses (generally speaking, from zero to infinity) with a weight, the poles induce cuts and the Green function will have analytic properties as described in Fig.3. Having in mind the analogy with a system at finite temperature, it is clear that these analytic properties can not contradict causality, since finite t e m p e r a t u r e physics does not. Indeed, causality is ensured by the good analytic properties of t h e retarded Green function, which is denned as a limit on all cuts in Fig.3 approached from the upper half-plane (Section 2).

® .l-n_

• n+

. 7l_

. l — n+

Figure 2: Situation of the poles of G°F.

4

108

-|&|+te

W

1 -* 1

\k\+ie

\k\-ie

-\k\-ie

Figure 3: Analyticity cuts of Gp, induced by the qq interaction.

One can come to all these unusual conclusions in the following framework. We assume that the charge (the coupling constant) in gluodynamics without light quarks, which increases with the decrease of the momentum transfer q, becomes a constant at a q value less than a certain value of A that we call XQCD (see Fig.4). Making this assumption we have neglected the gluon-gluon interactions, and moreover, the existence of gluons as real particles at all. Our aim was to understand how the Coulomb-like interaction of the form a(q)/q2 influences the features of the vacuum of light quarks and the spectrum of its excitations. In order to solve this problem, we succeeded in obtaining approximate equations for the fermion Green function and for the bound states (Section 3). The solution of the equation for the fermion Green function has shown, that if a(0)/ic > 1 —
i/a.

i/q

Figure 4: The supposed dependence of the coupling constant on a distance.

5

109

m3

-m3 m2

~mi Figure 5: Regularized fermion mass versus the bare mass mo-

any fermionic excitations.

2

The Green function of confined fermions

Before giving a constructive analysis of fermionic interactions leading to their confinement, let us discuss what general properties are required for the fermion Green functions assuming that no asymptotically free states with quantum numbers of the considered fermions exist. The Feynman Green function is defined as the expectation value in the vacuum state of the T-product of the operators ip and ip: GF(x, x') = 4(x - x') (0|tf( * $ ( « ' ) |0) - 0(*' - *) (0| ftx'Mx) |0) In the momentum representation the expression (2.1) defines GF(PO,P)

6

(2.1)

as the sum of two

110

Figure 6:

Analytic structure of the usual GF{P)-

analytic functions: dp

GF(P)

'°

= - / -7

f
l

d

7

7 - / + I P 0 . P J - - / -;

— ww

Po x ,„/ ^ „ , ..,/-(PoiP)

(2.2)

—OO

where /+ and /_ are the Fourier transforms of (0| tpip |0) and (0| tpift |0), respectively: /+(?)

=

y/e-^^IVW^OjlO)^

(2.3)

/-(?)

=

^/e*-(0|^(*)^(0)|0)i4a!.

(2.4)

In the usual theory, where states with fermion number different from zero are observable physical states,

(0| M')M°)

1°) =

E e~ipnX (0| MO) In) (n| ^(0) |0)

(2.5)

n

(o|^(o)v-.(*)|o)

=

p* > m2

and

£e«>«-(o|^(o)|n)Mv>a(o)|o)

(2.6) (2.7)

p„i0 > 0

This means that f±(po,p)

= ^(poMp2)

f±(P)

and, consequently, -IPI

GF(p) = - [dp'°f+(p'<»P) + I / •K J p'n — po - ie -K J IPI

d Pof-(Po,P) p'Q — po + ie

(2.8)

(2.9)

This corresponds to the usual analytic properties of the Green function (Fig.6) and to the Kallen-Lehmann representation, taking into account that due to (2.3) and (2.4) f-(p) = Cfl(-p)C

7

(2.10)

Ill

\p\+ie

|p*|+ie —•

GR

\p\ — ie

\p\-ie

GA

Figure 7: Anal tic structure of GF(P) in case of supercritically charged states.

where C is t h e matrix of charge conjugation, and U = 7± = 7 ° / ± V •

(2.11)

However, if states which have fermionic quantum numbers and at t h e same time are coloured states happen to be confined, i.e. do not correspond asymptotically to states of free particles, the conditions (2.5)-(2.7) will generally not be satisfied. Assuming that the reason for the confinement of coloured states (and of those with fermion number, in particular) is that these states are supercritically charged with respect to the fermionic vacuum, it is natural to expect that the effective Hamiltonian will not b e Hermitian in the coloured sector, as happens in QED for states with charges larger than 137. In this case t h e momenta pn in the expressions (2.5) and (2.6) will be, generally speaking, complex, and because of this the Fourier components f+(p) and f-(p) will be different from zero throughout —oo < po < °°- Consequently, the Feynman Green function has two cuts from —oo t o + o o with a gap of 2ie between them (Fig.7) and appears to be, essentially, a non-analytic function. There is, however, no contradiction with causality, since t h e retarded Green function GR(x - x') = i?(s - x') (0| {l>{z)rf(x')} |0) (2.12) which determines the causality properties of the theory, has t h e usual features:

f

GR(P)=-.r v°v+M»n- -u.f)].

(2.13)

2m J-oo Po ~ Po — *e The value f+(po,p) — f-(po,p) differ from zero only for time-like p2 > 0. This is a consequence of the relation fR = f+-f= -iJei"(0\{i,{x)rf(0)}\0) d4x (2.14) where (0| { ^ ( x ) , ^ ( 0 ) } |0) is a matrix different from zero at x2 > 0 and is an antisymmetric function of x, in the sense that {x = 'fixxli) (0|{^(x),^(0)}|0)

= a(x)ix

8

+ b(x),

(2.15)

112

b(-z)

= -b(x)

,

a(-x)

Hence, /+ — / _ can be written as ( a(x) = a(x2)e(x0), fR

f+-f_

=

-i Jd4xeipxe(x0)J{x2)

(2.16)

b(x) = 6(a;2)e(a;o))

= -ijeipx^{x2)[a{x2)ix

=

= -a(x).

b{x2)]e{x0)d4x

+

b{x2) + lv-^-J

dAx eipxe{x0)#(x2)

a{x2) .

(2.17)

It is easy to prove that for p2 = — v2 < 0 -ijd*xeiF'e(xoMx2)f{x2)

dx2 f{x2)

= -^-^-J

j

oo

= 2TT-^'

dX c o s h * c o s ( i V ^ s i n h x )

°? 2

2

J dX-Q-sm(vVx~2smhX)

fdx Vx^f{x )

' 0

Q

= 0- (2.18)

-oo

Substituting /R in the form /fi = e(poWp 2 ) [a(p2)p + b(p2)}

(2.19)

we obtain the usual Kallen-Lehmann representation for GR in the massless theory: = aRp + bR

GR

1 7 R= - I ~i

a

dn2 a(n2) 7 T^Ti

Fi2

'

, 1 7 R= -5

b

dK2 b(K2) 7 T7^ Fii •

(^UJ

z z ir J K2 — (po + te)2 — \p \2 -K J K2 — (po + ie) — |p | On the other hand, the Feynman Green function is equal to

GF = GR + 2if_(p).

(2.21)

The analytical features described correspond t o the idea of confinement of fermionic states, in the sense t h a t the cuts make it impossible for the Green functions GR and GF to have poles on the real axis which would describe particles with finite masses. The Feynman Green function contains b o t h positive and negative energy contributions independently of the sign of time, i.e. t h e Dirac picture is ruined, if the particles are confined. As was discussed in the Introduction, the analytical properties corresponding to Fig.7 are the same as those of the Green functions at finite temperatures, and they are natural for T-products of operators which are averaged not over the vacuum, but over an excited state. The usual Green functions, which do not contain additional cuts, correspond to normal perturbation theory, which is correct only if the exact vacuum states can be obtained from vacuum states of non-interacting particles by switching on t h e interaction adiabatically.

9

113 The Green function of the free vacuum has two poles at p0 = ±>/p2 + m 2 — * e a s shown in Fig.6. As a result of the adiabatic introduction of the interaction, cuts appear in addition to the poles but only in the same quadrants where those poles are situated. This corresponds to the fact that only states with positive energies can propagate forwards in time. It is obvious, however, t h a t the introduction of the interaction will transform the free vacuum into t h e exact one only if the change of the adiabatic parameter will not lead to the crossing of levels. On the other hand, if at a certain value of the parameter a phase transition takes place, a crossing of levels becomes obligatory (see above F i g . l ) . Nevertheless, the presence of crossed levels does not necessarily mean that the adiabatic perturbation theory is wrong. One can try to apply it starting not from the ground state (curve a in Fig.l), but from an excited state of the free vacuum (curve b), which has to be chosen in a self-consistent way from the condition of being adiabatically transformable into the exact ground state. It is worth trying to use the usual Feynman perturbation theory with a free Green function which will now have four poles: _ /

1 — n+

\E+ - p0 — ie

7i+

\

/

E+ — po + ie)

\E_

1 — 7i_

— p0 + ie

n_

E_ — p0 — ie

(2.22) where E± = ± V p 2 + m 2 , and u± are spinors with positive and negative energies; n± are t h e average numbers of particles with charges ± 1 in the chosen state. In this case GF will acquire additional cuts even in perturbation theory. As before, however, it will satisfy the Dyson-Schwinger equation and the Ward identity. If the exact solution leaves one-particle states with fermion q u a n t u m numbers in the theory, the exact Green function will not contain additional cuts. This situation might correspond to n+ = n_ = 0. If, however, there is confinement in the theory, the additional cuts will be forced to remain in t h e exact Green function. This could happen even at arbitrarily small n± values. If so, a non-perturbative solution of the Dyson-Schwinger equation should exist with t h e analytical features described above. In the next Section we shall obtain an approximate equation from the Dyson-Schwinger equation. We shall show that this equation has the solutions we are looking for, if the charge of the fermion is sufficiently large while its bare mass is small. As a conclusion of this Section, let us note, that t h e non-Hermiticity of the Hamiltonian for a sufficiently large colour charge in the coloured sector seems to be absolutely natural. Indeed, due to the existence of divergences in the theory which lead to anomalies in t h e energy-momentum tensor, the Hamiltonian is always an effective one (see, for example, Ref. [5]). On the other hand, in an effective Hamiltonian the charge renormalization is acting, essentially, as an external field. Consequently, for a large charge it is natural to expect phenomena like those which appear in the electron vacuum in the field of supercharged nuclei '*'.

10

114

3

Equations for the fermion Green function and for the wave functions of bound states

As was explained in the Introduction, gluons as real objects are ignored in this paper, and we consider fermion interactions due to t h e Green function of effective "photons" D(S) = *&

(3.1)

where a(q2) — a(q2/X2) is an effective charge of t h e form shown in Fig.4. At q2/X2 > 1 we have ^X2)

bln(q/X)

where A is t h e QCD scale. In order to obtain an approximate equation for t h e Green function of fermions, we make an assumption which will be justified later. Namely, we suppose that non-trivial solutions for t h e fermionic Green function appear at t h e value g = a(0)/7r A2, which leads to corrections of t h e order of g]n(q2/X2). As a result, t h e Dyson2 2 Schwinger equation becomes simplified in t h e region q /X >• 1 and it is transformed into the renormalization group equation which defines t h e renormalization of t h e fermion mass. If, however, we want to learn how t h e fermion mass is changing in reality, we have t o take into account t h e fact that, due t o effective "photons" with small momenta, there are infrared logarithms of t h e type gin * ^ j " at q ^ j "
dq^dq,

G-l(q) = d2G-\q).

11

(3.2)

115

The Dyson-Schwinger equation has the form

G-\q)

= G?{q)-^

!_

\T„.

(3.3)

Let us see how the operator d2 will act on the simplest diagram (in Feynman gauge):

G(q') It is easy to prove that 92

1 7 7vl^- = - 4 T 2 ^ 4 ( ? - ?') • W - 9') + *e

(3-5)

Consequently,

fi2Grr1=^7MG(g)7M. Consider now more complicated diagrams, containing for example two photons: q-q[

q-q'2

/

/ _i

(3-6)

/

N

/ 1

(3.7)

N

\

\ 1

I

q[ 9i+?2 _ 9 ?2 Applying the operator d2 to this diagram, we obtain several contributions. The double differentiation of any of the photon lines will lead to ^-functions. In other words, the momentum of one of the photons has to be set equal to zero, while all the other integrations remain unchanged. Retaining only contributions of this sort in all diagrams, we can conclude that d2G~l is equal to the amplitude of Compton scattering of a zero momentum photon by a virtual fermion.

92G~l =

C//yyM/%/y)\

12

•

(3-8)

116

This amplitude contains a quasi-pole contribution

(3.9) r

n

which is expressed in terms of the exact fermion Green function and vertices at zero photon momentum; it is obviously the most singular one in the region of q near the fermion mass m. Considering the remaining contributions to d2G~~1, such as those from single differentiations of each photon line in (3.7), they will be determined by integration over the region in which the momenta of both photons are of the same order. This means that if we initially had a logarithmic integration over photon momenta in which one of the momenta was much smaller than the others, then after differentiating twice with respect to the smallest momentum we lose one logarithm, and after differentiating with respect to two different momenta, we lose two logarithms. Hence, at least for small g, these contributions will be less singular near q ~ m. We can conclude that the most singular part of d2G~l is defined by the diagram (3.9), and therefore, ignoring the less singular terms, we get &G-1 = g U ? , 0) G(q) T^q, 0 ) .

(3.10)

Due to the Ward identity, the vertex part T^q, 0) can be expressed in terms of the Green function: T^q^^d.G-'iq) (3.11) and, consequently, 8>G-'(q)

= gd„G-\tiG{q)d»G-\q)

(3.12)

, _ 2fl2 7T

In the present paper the equation (3.12) is taken to describe the behaviour of the Green function in the region q2 -C A2. (Important corrections to (3.12) arising from the influence of real hadron states will be discussed later.) This equation has several nice features. It is scale invariant, but not explicitly gauge invariant, since for extracting the contributions of small momentum "photons" a photon Green function of the form D^ = S^/k2 was used. If we added to this D^ an additional term of the form (k^ku/k4 and calculated d2(kltku/k4), we would get

d2^^

2 = 42 \d ± + 20,Ai -"f"" W 2L2^ vKd U2 -"•-#*' —**~"1.2

(6.16)

Because d2k~2 ~ S(k) (see Eq.(3.5)), the first term in (3.13) equals zero, and the last one has the same structure as the contribution of DIAU = 6^u/k2. The contribution of the second term

13

117 is, however, not universal, and it depends entirely on the behaviour of the Green function at small k. For example, supposing that in

J q'-m

" U{q-q')2

H

J

\k*

V Jq - k- m

small k are relevant, and extracting l / ( g — m ) from the integral, we get zero. In reality this integral is of the order of 1/(5 — m ) at small q—m, i.e. of the same order as the terms we were taking into account, and values k ~ q — m turn out to be essential. Hence, the Feynman gauge is exceptional in the sense that it enables us to extract the contribution of small "photon" momenta in a universal way, independently of the concrete behaviour of the fermion Green function and of the vertex parts. As was discussed in the Introduction, in order to understand the structure of the vacuum state we need to know not only the features of t h e Green function but also the character of fermion bound states, because these bound states determine the real excitations and the stability of the vacuum. To be able to see the connection between different solutions for the Green function, corresponding to different vacuum states and structure of bound states, it is necessary to have an equation for the bound states formulated under the same assumptions as the equation for the Green function. Formally the equation for the energies and wave functions of the bound states can be obtained by considering t h e interaction of fermions with an external field having the same quantum numbers as the s t a t e in which we are interested . In this case one has to sum up the perturbation expansion for the vertex part of the fermion interaction with the external field, in the limit that the external field tends to zero. This vertex part

(3.14)

P) = -Z + q can be described in perturbation theory by the sum of diagrams

+

+

(3.15)

In order to obtain an approximate equation, we shall apply the same procedure as before in obtaining the equation for the Green function. This means that we shall take into account

14

118 the contribution of soft "photons", calculating d2(q,p) again. For the simplest ladder-type diagrams of Eq.(3.15) we have d2(q,p) = ~Tr(qlt

0) G(gi) (q,p) G(q2) T^faO)

P

,

(3.16)

P

The differentiation of non-ladder type diagrams extracts the contribution of "photons" with zero momentum which are emitted by one of the external lines and absorbed at an arbitrary place. As it is well known, the summation of such "photons" from, e.g. , the upper line will lead to the contribution of two diagrams:

+

(3.17)

The sum of these two diagrams can be written as d22 a= •K

d

-T(1(q1,0)G(q1)-z-(q,p)

(3.18)

8q„

Adding to this expression the contribution of "photons" emitted by the lower line, we obtain an equation for the vertex part (q,p) which takes into account the contribution of soft "photons", of the form

0V(?,P) = - {M(q,p) + d^(q,p)All(q2) - MtoWlrfM*)} All{q) = dl>G-\q)G(q)

A^q) = G(q) ^ G " X ( ? ) .

,

(3-19) (3.20)

2

The eigensolutions of Eqs.(3.19) and (3.20) which decrease at q —• oo and have proper singularities will define the energies and two-particle wave functions of the bound states. Unlike the case of the equation for the Green function, the accuracy of this equation depends not only on the parameter q2/X2 but also on pq/X2 and p 2 /A 2 , i.e. it can be correct only for relatively small p. Essentially, the virtualities of both quarks, q\ and q2,, must be small. The equation (3.19) has the following nice feature: it follows from Eq.(3.12) that at p = 0 it turns out to be an equation for the variation of G _ 1 . Consequently, if due to the symmetry of the equation there exists a class of solutions for G~x depending on a parameter £,

G? = G £ + (6G-*) ,

15

(3.21)

119 and we choose G^ 1 as the solution, thus breaking the symmetry, then = ( 5 G - 1 ) ^ will satisfy Eq.(3.19) at p — 0. If all G~£l fulfill the same condition at infinity, = (6G~1)^ will decrease at q —• oo and because of this, (q,p) will be the wave function of a bound state with p — 0. In other words, due to relativistic invariance of the equation, a Goldstone boson with p2 = 0 must exist. For example, if the bare fermions were massless, the equation for G _ 1 as well as its boundary conditions would be 75-invariant. Nonetheless there has to be a solution of the form

G-'=atf)q

+ b{q>)

and in this case = C"[sb(q2) decreases at q2 —> 00 and is the wave function of the Goldstone boson. In the following, Eqs.(3.12) and (3.19) will be used to provide a constructive approximate description of the properties of the fermion spectrum and of the mesonic excitations.

4

Solution of the equation for the fermion Green function in the Euclidean region

The fermion Green function is a 4 X 4 matrix. Due to relativistic invariance, it can be written in the form G-'iq)

= q

=

-b+aq (4.1)

7„gM

and therefore, essentially, turns out to be a 2 x 2 matrix. From the point of view of solving Eq.(3.13) it is useful to express G _ 1 as

(4.2) and to look for a solution of the form G-1(q) = v* =p+vi(q)

+

(4.3)

p-v-(q)

where /3 is an unknown constant. Substituting (4.3) in (3.13) we make use of d.G-'G

= =

duV-

- P+^

w

.

r

+ P-- l

+

+w

r

1

1

V

3

(4.4)

B/i

16

120 where d'^v contains differentiations only over the scale q,
=

(4.5)

d v Siva-" q2 Oqv = ~5 q2 ( '

£=lng; whle

5, = - ^ -

1

—

«+

P+ +

(4.6) 1>_

is an operator acting on the spin variables, which depends only on the ratio v+/v. In these terms we have

d2G~l = i ( i - 1)5> ff^v • vh~2 + ±d'2v • v?-1 + d^ g d»G~l G d^G'1 = g—ff^v • vT>~2 + gB^v?

v* +

B^v*

(4.7)

.

If we choose 0 =

(4.8)

1-9,

then the equation

82G~1 = g dfi-1

G d^G-1

(4.9)

is transformed into a system of linear equations for v+ and t;_ with a potential depending on their ratio v + /u_. The resulting form of the equation for v is: d'2v + Bd^Bp v + 32BI1B(1 v = 0

(4.10)

It can easily be proven that B^

B B

»»

=

=

3 q Aq2q 4^

(4.11)

fe) +

1 ^ - 2

d'2u = - 1 ( ^ + 2 ) ^ .

(4.12) (4.13)

Consequently, we obtain a set of equations for v+ and i;_ as follows:

(di + 2)div+ + ^-f^jv+v(d( + 2)dtu. + -£-f (?±\

17

v+v_

=0

(4.14a)

=0

(4.14b)

121 where

f = -f

V

-±

G;1

V-

+

G:

v+

G:1

1 +

G;

'

1

(4.15)

We can denote v — e ^u

(4.16)

and rewrite the equations in the form (<9f-l)U± + —

/u+u_

=0

(4.17)

for which there is an obvious constant of the motion E = d(u+ d(ii- + ( / — l ) u + t i _ = const

(4.18)

the existence of which will considerably simplify the analysis of the solution. The simplest way to write / ( v + / v _ ) , which as in (4.15) defines the non-linearity of t h e equation, is to express it in terms of the components G j 1 . Hence, it will be convenient to return to the original G - 1 in the form

ti

G-1 = - ( e - ^ ) p e x p


„ — (e *p) cosh — — - sinh V *V I 2 q 2

"2?

(4.19)

which one gets after the substitution u±=

>, p exp ( T ^ l

(4.20)

•

We can then obtain the following equations for p and : (4.21)

<ji + 2 - < £ - 3 s i n h < £ = 0 P

P-P+P* 9

3sinh^ + ^ 2 4

dz '

p

0,

(4.22)

di

In these terms the constant of the motion becomes E =

2 p<-p*+02 3 sinh — 2

18

— 4

(4.23)

122 First of all, let us determine the value of E for which we are looking. If the solution for G~l corresponds to a finite fermion mass, it should have no singularity at g - * 0 (£—>— oo). Consequently, at q—»0 - • 27n'ra

p ~ e(

,

;

n = 0,1,... .

(4.24)

Under these conditions Eq.(4.22) leads to p= P

,

Ce±(

Pi,2 =

and we have to choose the first solution. Because p/p —• 1, equation (4.21) for 4> gives the non-singular solution — 2ivin + const • e^ . Hence, all the terms in the expression for E tend to zero as £ —> — oo. This means that p/p can be expressed in terms of (f> and <J) as P P

3 sinh 2

^

2

(4.25)

• 4

and consequently only one non-linear equation for remains to be solved. In order to have a complete understanding of the structure of the solutions of this equation, it would be useful to know their properties at arbitrary complex q. We begin with the investigation of the solution at purely imaginary values of g,that is, in the Euclidean region. In the transition to Euclidean q and 7M the ratio q/q remains unchanged, but cf> changes into i. Hence, in the Euclidean region we have G-1 = ( e - « / , ) ? e i H ,

<j> — 3 s i n = — 2 1 + / 3 2

N

3 sin 2 2

(4.26)

P 4

4>.

(4.27)

It is useful to introduce the energy e connected with the ^-degree of freedom: e =

_ _

6 s i n

_,

(4.28)

which enables us to write Eq.(4.27) in the form e = - 2

A

/ l - ^ e . ^ .

(4.29)

This expression shows that the problem we are considering is in fact that of the motion of a particle in a periodic potential with damping (Fig.8).

19

123

0

-7C

n


Figure 8: The oscillating potential corresponding to the equation in Euclidean region. At q—>0 (£—• - c o ) the particle is situated at one of the maxima of the potential, having e = 0 (e.g. , (f> = 0). As £ increases its energy will decrease. Consequently, the particle will remain within the limits of one well independently of the value of q, and as £ —*• oo it will approach the minimum of the potential (e.g. the point cf> — ir). Trajectories corresponding to different initial momenta and to different values of (32 may differ only in their behaviour in approaching <j> = ic: they will reach this point either monotonically (curve 1) or in an oscillating way (curve 2). The character of this asymptotic behaviour can easily be found from an analysis of the solution near the minimum, where the equation becomes linear. Indeed, taking <j) = ir—Tp,il) + 2 , / l + 3/32ip + 3V> = 0 ,

(4.30)

which results in

i> = dent with

7i,2 =

+ Cte**

- \ / l + 3/32±v/3/32-2,

(4.31)

When 3/32 — 2 > 0 we get a solution that approaches t h e point = % monotonically. On the other hand, when 3/32 — 2 < 0, i.e. if

l + ^/f>*>l-/f = *.

(4.32)

the solution will oscillate before reaching this point. It is easy to show, that the solution of the equation for leads, as usual, to the existence of a critical = c which separates the regions of monotonic behaviour and oscillation:

cos

&

1

(^)^T

+ ^/(l + 3/? 2 )(l-/3 2 )

20

(4.33)

124

I n k Y K

.Z.

-v 0_ Figure 9: An oscillating solution in case of /32 < 2/3.

At @ near the critical value /3C = J|,

^ c approaches 7r and

V>c = *r - & « ±2 J l - | / ? 2 .

(4.34)

If 01 > §, all the solutions are monotonic. If (32 < | , the solutions are of the form shown in Fig.9. For i\>c

=

qe

=

i>e cos

V6

qe\ WJ

(4.35)

mtjjt '75

where m is defined by the behaviour of the solution at small q values ( « q/m.) and can be expressed in terms of the parameters of the Green function (4.1) 6

m = a

(4.36) ,-o

while At is a number to be determined by the solution of the equation in the non-linear region. In the next chapter we shall show that the renormalized fermion mass is expressed unambiguously in terms of m. Hence, in what follows we shall simply call m the renormalized fermion mass.

21

125

As we discussed earlier, the parameter m and thus the fermion Green function can be obtained by matching the solution of Eq.(4.9), which is valid at q < A, with the solution of the renormalization group equation, which coincides with Eq.(4.9) at q2 ^> A if g = g(q) is assumed. In order to be convinced that this coincidence really takes place, it is necessary to compare the initial Eq.(4.9) with a running g at large q (i.e. as <j> —• 7r) with the renormalization group equation. But in doing this one must not use Eqs.(4.21) and (4.22), because they were derived using the assumption t h a t g was a fixed constant. It is necessary to return from these equations to Eq.(4.9), which does not contain the assumption of a constant g. This can be done just by writing

(4.37)

p — &*• (P

which corresponds to the representation x

G

(4.38)

— —/>exp I —(f)}

2q

In this case the equations for <j) and p in the Euclidean region will be of the form

+ 2 i+/?4 — 3 s i n = 0 ,

(JL + 2)±p-fi

. 2

n

2

1.9.., p di

(4.39)

(4.40)

A s —> 7T, <j) = 7T — lj),

^ + 2 1 + /9- i> + 3V» = o , ( | If p = e^Z expression

+

i.d 2)(|-1)^ + 3 ^ - ^ ( ^ = 0

(4.41) (4.42)

x

(£), where Z(£) is a slowly changing function, we obtain from Eq.(4.42) the ^Z~l

+ \gZ~x

= 0,

(4.43)

which is just the equation for the renormalization of the Green function. Its solution Z = {g)s- , Z

dig~

2'

(4.44) (4.45)

coincides with the wave function renormalization in Feynman gauge. Similarly, i> = 2e"< m{i)

22

= 2m(0/? ,

(4.46)

126

^m(0 = -Igmtf)

(4.47)

is the gauge invariant equation for the fermion mass renormalization ~ (g)S™ d 1 3 5 ^ 2-

m(0 m

(4.48) (4.49)

Denning the quark mass m = m0 at a scale ( = ^ 0l we get m(C) = mQ (±\

(4.50)

which proves that the solution of the equation with a running g leads to the renormalization group equation at q2/X2 ~S> 1. Consequently, Eq.(4.9) with a slowly changing g can be used to find G~l at any q. As was shown above, the solution for G~l (i.e. and p) that is nonsingular at q —• 0 is defined unambiguously, up to an irrelevant normalization factor, by the parameter m (4.36). According to Eqs.(4.41) and (4.42) the general structure of if> and p takes the form (as q—>oo)

P =

q

q3

qZ-i(C)

+ -sr

l

(4-52)

and consequently the regularity conditions at q —• oo do not impose any restrictions upon the solutions. This means that the parameters m(£) and f 3 (£), which determine ip as q —> oo, are defined unambiguously by the renormalized mass m (4.36) and, of course, by A. Hence, by solving the equations with slowly-varying g we obtain m0 as a function of m and A in a unique way: m 0 = m0(m, A) = A f(m/\) . (4.53) The character of the solution for (see Fig.9) reveals, however, t h a t the "renormalized mass" m can be a single-valued function only if g < gc. Indeed, let us remember that in this case the solution for <j) has the behaviour shown in Fig.(10). Let g decrease at q > A. In this case the slope of <j>(q), which defines m0 at large q, is not so steep, but it still remains a monotonic function of m. Consequently, not only is / ( m / A ) unique, but so is / - 1 ( m / A ) . This means, in particular, that if m0 = 0 then m = 0 also, i.e. when g < gc there is no spontaneous symmetry breaking. When g(0) > gc, however, the situation changes entirely. Suppose that g decreases so that at q = A it becomes ^(A) = gc. In this case ipc denned by (4.33, 4.34) goes to zero, and the region of non-monotonicity takes the form shown in Fig.11.

23

127

Figure 10: A single-valued solution (q) in case of g < gc

Outside this region, when q > A, the solution is determined by the renormalization group equation. We shall assume that the bare fermion mass is small enough that m(£) at £ = In A, which we shall call the perturbation theory mass mp m

P = ™(OI*=lnA .

is much less than A. It is obvious from Fig. 11 that one can always find a curve of the type of fa, which matches the perturbation theory and does not cross = ir. In the limiting case that the value mp/X, which roughly equals the distance between the renormalization group curve and the axis (j> = x, tends to zero, fa will be close to the solution of the equation with a constant g, which has its first node at q = A. Consequently, there exists for ^(0) > gc a solution fa of the equation for the Green function with a finite renormalized mass of the order of A, if the bare mass of the fermion is zero: mp = 0. It is also obvious that in this limiting case there must be many solutions with nodes at q = A and an arbitrary number of nodes for q < A. This, however, is not true, if mp/X is small but finite. Indeed, due to equation (4.35), in the region of non-monotonicity the solutions oscillate with decreasing amplitude. Because of this, these solutions, having a given amplitude when entering the region, may get nearer to the axis than mp/X and thus matching will then turn out to be impossible. The situation in which the second solution appears for the first time at a given mp/X is shown in Fig.ll as a curve <j>* starting at = 2n, having a node at q = qK and a minimum near q = A. For a smaller value of mp/X, two solutions fa and fa appear simultaneously. They correspond to the cases that the minimum does not reach the point q = A and that it passes q = A, respectively. If g approaches the critical value, the equation which defines the parameters m for the solutions can be obtained by setting mp/X equal to the expression (4.35) at q = A. As a result, the equation for the masses corresponding to the other solutions

24

128

Figure 11: The solutions fa and fa for the case of ^(0) > gc.

that appear at smaller mp/X values can be written as

T = l—J

cos

V3

run

J

(4.54)

The structure of solutions of this equation is shown in Fig. 12. For the sake of completeness, we have added here the region of negative mp values and the solution m\(mp) which is defined by the behaviour of the solutions in the region of monotonicity. We see that for a charge larger than gc the decrease of the "bare" fermion mass can lead to a phase transition at a critical value m£. As a result, the appearance of two different vacuum states with negative fermion masses becomes possible. As we continue to decrease mp, in one of these states (^3) the fermion obtains a large negative mass of the order of A, and at nip = 0 its Green function will be just the 75-reflection of the fermion Green function corresponding to the initial vacuum. In the second vacuum state (fa) the fermion has a small negative mass. The subsequent decrease of the bare mass leads to a series of new phase transitions (fa, fa, etc. ). Hence, the limit mo —» 0 of the theory makes sense only if one fixes a definite curve m = m(mp). In the last section the stability of different vacuum states will be discussed. It will be shown that the solutions corresponding to light fermions have a remarkable property which will eventually lead to the confinement of fermionic states. Prior to this analysis we present the quantitative characteristics of different solutions.

25

129

Figure 12: The structure of the multi-valued solution.

The critical value mc is defined by the equation coth

Vo

\rn c /c

3 '

/

(4.55)

Consequently,

- ^ 2 ^

A

?-C AJ

+ 1) +

V3

(4.56) (4.57)

3

Re-writing the last expression as 1 /3mf 771,c = — — I

;

K V Vc

26

1A/3

•A1-^

(4.58)

130

$£q, 0) Figure 13: Different behaviour of <j>2(q) at different values of m

we see that mp is close to the geometric mean of A and m£. As mp changes from the critical value m.p to zero while moving along the curve that corresponds to small fermionic masses, the value of m remains of the order of m c (m(0) w 0.44mc). At the same time the structure of the solution for the Green function changes significantly (Fig.13).

5

The solution of t h e equation for t h e Green function at time-like q2

The behaviour of the Green function for any complex value of q can, of course, be determined uniquely from the solution obtained for purely imaginary q values in the previous Section. However, no explicit analytic expression for G~x has been derived and so analytic continuation is not possible. One can nevertheless find C?"1 at complex q values by making use of differential equations written along an arbitrary ray in the complex q—plane and considering the explicitly known behaviour for q close to the origin as the initial condition. Let-us begin with the solution of Eqs.(4.21) and (4.22) on the real axis. For real q both and p are real. Making use of Eq. (4.21) we obtain the following equation for <j> : — 3sinh = —21 - P2 3 sinh

i

27

4>

?) 4>-

(5.1)

131 Introducing the energy 1 •„ e= ^2-6sinh2^

(5.2)

we again get a system with damping:

d

i=-*H^

(5.3)

There are, however, differences: the potential energy is no longer periodic (see Fig. 14) and, furthermore, at a value of e = —2//32 the damping vanishes and the system starts to accelerate.

Figure 14: The non-oscillating potential corresponding to the equation in Minkowski region and the trajectory corresponding to the fa- solution (dashed line).

We shall show that the trajectory of the solution which begins at £ —• oo, <j> —* 0 (solution fa in the previous Section) has the shape of the dashed line in Fig. 14, going to infinity at a finite value q = m*. This means that G - 1 has a singularity on the real axis at q = m", and that m* is the fermion mass. In the case that q < 0, the singularity appears at q = —m*. For the solutions fa, fa of Section 4 we can write <j> =

<j) + 27TZ

(j> =

q —— m

at

(5.4) q —* 0 ,

(5.5)

and the trajectory turns out to be of the same shape as before, except that the values of q are reversed in sign. The standard approach to solving Eq.(5.1) is to reduce it to a first order equation by introducing Hamilton variables. It is convenient to write the momentum in the form 2 p P=

0-p

=

i-12sinh2^ + ^

28

(5.6)

132

P(

\

/

\ *

Figure 15: Phase-plane corresponding to Minkowski region and the trajectory (dashed line).

with f3 = 1 — g and to consider p as a function of . From Eq.(5.1) we then obtain (5.7)

where

<£ = \/u~+;

(5.8)

4 , d> u = - - + 12sinh2|.

(5.9)

If Eq.(5.7) is solved then Eq.(5.8) determines the dependence of on £. The phase plane of Eq.(5.7) is relatively simple (see Fig.15). The trajectories p() in which, we are interested go from infinity to infinity, passing on their way through the point = 0, p = 2//3. Near ^ = 0 w e have V

fi

2

(5.10)

v

For large it will be convenient to write p = —y/uy s=s —\/3e*y

(5.11)

which leads to (5.12) There are two possible solutions of this equation: i) if 0 > 1/2 (g < 1/2), then y —• co as -*oo v y ~ eJfi-l*

-+ oo

(5.13)

and consequently p = -Ce?+,

29

C>0,

(5.14)

133 ii) if f3 < 1/2 (g > 1/2) then j/-> const, 2/3 2/ = v/1 - 4/32 +

Ce'

.(1-/3^1-4/3')*

(5.15)

and, hence, 3 ± 1 - 4 / 9 2 e*

p « -2/3

(5.16)

In both cases Eq.(5.8) enables us to find easily how depends on £. Re-writing Eq.(5.8) in the form __ (5.17)

Vp2 + u '

« " < • - /

it becomes clear that since p2 and u are increasing rapidly at <j> —> co, the integral on the right-hand side will converge at <j>' —> oo and therefore ^>(£) —• oo at £ —• £* (g —• m*). For # < 1/2, substituting Eq.(5.14) in Eq.(5.17), we obtain as gr —• m"

i-J-

= ±e-e+

m*

(5.18)

C/3

From Eq.(5.6)

C/3

P _ _121 e ^ p i

=

1

1

(5.19)

2 g/m* - 1 '

Consequently, (5.20a)

P = Po = 53

(5.20b)

and

f#+i)(

m"\

»V

2V

1

9 )

2 C2a2 0

•\-W-br (5.21)

Since /3 = 1 — g, ^ = 1 + g for J < 1 , and choosing Zo = 1 at q = m" we get

G=if1 + 2\

*)_JL___Lf 1 _2)ff 1 _^ ( 7 ' qjm'-q

2m* \

qj

[\

(5.22)

q)

thus G has a pole for the state u such that qu = qu and a weak singularity for qu = — qu.

30

134

In the case that g > 1/2 2 -L^/lT^e-

1 -

(5.23a)

73

m*

2/32

p

G

-2/33

lit*,

= *(}-£)

p0[l

=

(5.23b)

)

1

1

11g7 4(1 - 4 / 3 2 ) V~ T W J ^ ' ' 2 V g/ ?n* — q 2m A similar expression can be easily written for q —> —m". G proves to be singular for the state such that qu = —qu. Since we assume that (j> ~ q/m when q/m < 1, we are dealing in fact with the solution 4>i(q) of the previous Section. The solutions faiq), ^3(9) are of the form ^2,3(9) =

-q/m

at small values of q. For example, instead of (5.21) one obtains •JC+B)

f

+

1 /

2

1+

g

f

1--

771

C22/3/ j 2

771

l2a2

•&-h) (5.25)

as 5 —> 77i*, and

m* I 2 V

1+

g

771

-*(* + *)

+ 2'1

1+-

cp

•W-bY (5.26)

as q —* —m . In both cases it is easy to find the relation between m* and 771 from Eq.(5.17). It may be written for example in the form 771

In-

771

/ # '

1 y/p>W) + u{(f>')

(5.27) sinh(2<^')

In a conclusion, we can state that G~l has reasonable analytical properties as a function of q for both solutions shown in Fig. 16. However, we were not able to prove in general the absence of singularities of G off the real axis. Such singularities would mean t h a t a solution of Eq.(5.7) with complex initial conditions had gone to infinity at a finite value of £. At the same time we could not detect any solutions of this kind. We shall assume that they do not exist and t h a t consequently the analytical properties of G~l are those illustrated by Fig. 16.

31

135

d) m*

—TO

Figure 16: Plausible structure of singularities of G x.

6

Phase transitions and structure of qq bound states

The two solutions obtained for the Green function differ from each other by the value of the quark mass compared to A, and by the sign of the mass at a given m0 value. Let us see how these differences influence the character of meson-type bound states. For this purpose we shall use the equation for the bound states which was considered in Section 3 (3.19,3.20): 92(q,p) = 9{A^qi)d^{q,p)

A„(g) = d.G-\q) &q) ,

+d ^ p ) ! ^ )

A^q) = G^d.G'^q)

- Al±{ql)<j>{q,p)A^q2)} ,

= -{C

(6.1)

A^-q^C^f

assuming, as in the case of the Green function, that g is a slowly changing function of q/X. This equation can be re-written in an explicitly Hermitian form. Introducing the operator AJ^q) such that {A„, } = A^q^cj) + 4>A^{q2) [A» 4>) = A^qi ) - M.(<7 2 ),

(6.2)

the right-hand side of Eq.(6.1) takes the form g{A„ d^} - f { A ^ A , , *}} + f K [ A M , ]). Now using the covariant derivative VM<£ = c U - | { A ^ } ,

(6.3)

we get W

= d2
}} - | { ^ , } .

According to the equation for the Green function, we have VW
+ 8llG-1(q1)dflG{ql)

32

= (g - l)A M ( gi )A M ( gi )

(6.4)

136

and a similar expression for d^A^fa). {d^,

Consequently,

<j>} = \{g - 1) {{AM„ # > + [AM*, ) ]) •

(6-5)

Thus Eq.(6.1) takes the form

VV = ! ( ! - £ ) [4.14.,*]]

(6-6)

H(f> = 0.

(6.7)

which can be written as The kinetic energy part —V2 of the operator H is a Hermitian one if the scalar product in the Euclidean space is defined as

(2fc)= Jd\ ^{q)G'{qx)4>{q)G«{q2) .

(6.8)

Indeed, if one writes in (6.6), (6.7) in the form ^G-ol\qi)i,{q)G-ol\q2),

(6.9)

then H<j> transforms into H'i> = - V ' 2 V- + | ( 1 - § ) Ma^]]

=0

(6.10)

where V ^ =^

+ {6 M ,^}

6M = Gal2d»G-a'2

- |op

aM = G fl/2 (a M G _ 1 ) • G • G-* /2 .

(6.11) (6.12) (6.13)

The scalar product (tj)x ^2) is chosen to fulfil the condition <7A = 0

(6.14)

which is, obviously, satisfied in (6.12), (6.13) since q^d^G'9^2 commutes with G, and q^dfiG'8^2 can be calculated in an Abelian way. Due to (6.14) 6M can be written in the form K(q) = %-f(iq) + d^f(iq). 9

9

(6.15)

9

At the same time (6.13) and (6.14) lead to CblC~l

33

= bl

(6.16)

137 while according to (6.15) , - il _ M « 1 Cb*C=[ldJ.f{iq)\

-Id^ftiq)

(6.17)

This means that / = 0, and therefore

*i=-K

(6.18)

In the notations of Section 4 (see (4.38), (j) —> i<^) the expHcit calculation gives (6.19) The relation (6.18) corresponds to the fact that the kinetic energy part of the Hamiltonian H' is Hermitian, with the scalar product (6.20)

(^aVi) = T r | d 4 q ^ V i and

(V'2|^'|V'i) = T r / A ( ( V ^ ) t V ^ + [^,^][V',^]f(l-f)^

(6.21)

The operator a^ which defines the potential energy can be presented in the form

°M = K

+ ai + C*

(6.22)

where

2 CM = —i sin — dnh exp

W 1q

9

(6.23)

The resulting potential energy is 1

(6.24) (6.25) 1

tf_ = - ( i - / ? ' ) [ < [ < v]]

(6.26)

Here U+ is Hermitian, £/_ - anti-Hermitian, and since a£ is a C-number, # ' is Hermitian for p = 0.

34

138

/

s

/

'

;

21

\

S

[

\

'2 i / y

/

Figure 17: Illustration of a potential having the structure of a static external field with the two centres ±p/2.

In the four-dimensional Euclidean space of q, Eq.(6.1) describes the motion of a particle with definite spin properties and zero energy in the static external field of two centres at the points ± p / 2 (Fig.17). The equation for a non-zero "energy" value would be H'4>€ = e€.

(6.27)

Our task of finding the bound qq states means that we have to choose a distance p between the sources of the field which leads to an eigenvalue e — 0 of the energy. Since the potentials AM(<ji) and Ali(q2) are not singular as q-y —+ 0, q2 —• 0, the possibility for bound states to exist is determined, as usual, by the behaviour of the solution as q —• oo. If one can choose p so that the solution is decreasing at infinity, bound states will exist. The picture of two centres enables us to imagine at once the properties of bound states for two different Green functions, i.e. for two different potentials A^. For q > m, the potentials Afj. contained in Eq.(6.27) have at the simple form , 1 Ay. « 5 7 M 1 and create a field of the order of g/q2 at large "distances" g > p . In other words, at large values of q the two sources produce a Coulomb field with a charge g which leads to a collapse at g > gc similar to that which takes place in the field of a heavy nucleus with a charge larger than 137.

35

139

For q> A , however, g begins to decrease and the field becomes less than critical. Consequently, a large Coulomb-like field can exist only in the region m < q < A. If X/m S> 1, as for the free Green function, then there must be many levels of Eq.(6.27) which can be gradually shifted to the point e = 0, by changing the distance p between the sources. This will lead to a large number of eigenvalues of p. If they are real in the Euclidean space it would mean instability of the vacuum. If the Green function which defines the potentials A^ corresponds to the solution with heavy fermions, then, as we have seen, m ~ A and the region of large Coulomb field in (6.27) vanishes. Consequently, the vacuum corresponding to the solution with heavy fermions can be stable. If, on the other hand, the Green function corresponds to the second solution with light fermions, m/X may be less than unity, and consequently the region of large field may not disappear at p < A. We shall see that in this case there will be some remaining levels, but corresponding to a real meson type excitation. It turns out to be possible to analyse the structure of bound states in these non-trivial vacua with almost no calculations. In order to do so, we begin with mp values near ro£, where the solutions 3 just appear. Obviously, there exists a zero mass Goldstone state at mp = m£, at the point where the phase transition takes place. This follows from the fact that at mp values close to, but still less than, m i there are two solutions G2, G3 of the equation for the Green function, which coincide as mp —* m£. Since, as we have seen in Section 3, at p = 0 Eq.(6.1) for the bound states is the same as that for the variation of G - 1 , the difference 4> = (%l-G? (6.28) will in this case also satisfy Eq.(6.1). Both GJ 1 and G^1 fulfill the same conditions as q —• 00. Hence (f> will decrease as q —* 00, and consequently there will exist a Goldstone-type bound state (p = 0) with wave function 9o =

dG~x

da

Ob +

= 7r-* 7r--

am

am

6 29)

-

am

The presence of Goldstone states near phase transition points is quite natural. Indeed, the existence of fluctuations of an arbitrary large scale is essentially the reason for phase transitions. Let us show now that if there exists a solution <j>0 for Eq.(6.1) at p = 0, this leads to the existence of a Goldstone state with p2 = 0 for m = mc and of a bound state with finite mass for mp<mcp. Let us consider first the solution of the Eq.(6.10) for m = mc and p ^ 0 and assume, for the beginning, that p^/m,. «C 1. In this case we shall look for a solution in the form ^ = •^0 + ^ .

(6.30)

H'(me,0)i> = H'{mc,p)4>0.

(6.31)

The first order equation for ip is

36

140 The inhomogeneous Eq.(6.31) has a solution only if the r.h.s. is orthogonal to the eigenfunction tpo with a zero eigenvalue, i.e. if the condition

TTJd4q^oH'{mc,p)rPo = 0

(6.32)

is satisfied. Here i/> can be written as an expansion J

=

y ' (A\H'{mc,p)

\ip0)

(6.33)

where ipe are eigenfunctions of the Eq.(6.27) at p = 0 with e ^ 0. Eq.(6.32) is, obviously, always fulfilled at p2 = 0, since the integral in the l.h.s. is invariant and therefore proportional to p2. If Pft/m is not small, b u t p2 — 0, t h e solution can be derived by means of the Lorentz transformation of the solutions (6.30) and (6.33) with small p^ values. Consequently, the existence of the solution of H'tp = 0 at p = 0 indeed means that a massless bound state exists. Let us consider now mp < m£ for small value of mp — m£. In this case we obtain H'{mc,0)i>

=

(6.34)

H'{m,p)iPo

instead of (6.31). The condition of the existence of a solution for the inhomogeneous equation now means that Tr / d4q ipl H'(m, p) rf)0 = 0 . (6.35) Expanding H'(m, p) into a power series of m — mc and pM we get H'(m, p) = H\mc,

0) + (m - m c )

8H' dm

dH' m—Tnc p=0

VPn

m=Tn c p=0

+

(6.36)

The first order terms give no contribution t o (6.35). For the linear term in the momentum this is obvious because of the invariance of (6.35). The t e r m which is linear in m — mc gives no contribution due to a less trivial reason, namely, I/J0 satisfies the equation (6.37)

H'(m, 0) V>o(m, 0) = 0

not only for m = m c , but for any m > mc. In case of m ^ mc ip0 is not an eigenfunction since it does not decrease when q —* oo. Differentiating the Eq.(6.37) with respect to m we obtain (m — mc)

dH'(m, dm

0)

i,o(mc,0)+

H'(mc,0)

37

dip0{m, 0) dm

= 0

(6.38)

141 Making use of (6.35) and (6.36) we can conclude t h a t the first order contribution in m — mc in Eq.(6.35) is -Tr/A^(mC)0)F'(mc,0) ^ J om and equals zero, because H'(mc,Q) is Hermitian and H'(m, 0)^0 = 0, i.e. indeed, the first order terms give no contribution. The second order contribution to the Eq.(6.35) can be written in the form 2

(m-m.rj*q

}d

H T r ^ ^ o

d2H' + \ JA ^ ^ g ^ *

= 0

(6.39)

where p and q are Euclidean variables. Thus the equation for the mass of the bound state, H2 = -p2, is H2 = K2{m-mc)2 (6.40) where K2 is the ratio of the integrals in (6.39). In order to obtain the sign of K2 let us note that the anti-Hermitian part of the operator H' does not contribute to (6.39) since

Jd4qTI4i<[ai^o}} = -\\
(a£(gi) - aftq,)) Tr ^ o ^ - ^ ( g i ) -

^0-d^(q2) = 0

(6.41)

due to Ti tpip^q/q = 0. Therefore we substitute H' by H'+ in (6.39) and consider the solution of the equation H'+(m,p)1>+(m,p) = 0 (6.42) and the matrix element

Jd4q0 Ti4,lH'+(m,p)i,+ (m,p) = 0.

(6.43)

Expanding (6.43) up to the second order terms in m — mc and pv, it is easy t o prove that (6.39) is equivalent to the equation

( m

*& „„

_roc)./ATl^(m„0)^

+

^ / ^ T

9

r

9 |^+± XJK ^ , 0n\ )^+ ^ = 0.

(6.44)

The integrals which enter (6.44) are the expectation values of t h e operator H'(mc, 0) calculated over the functions -^ and -^ which are not its eigenfunctions. H'(mc,0) has a zero eigenvalue. If this eigenvalue is the minimal one, its expectation values over any other states will be positive. The minimality of the zero eigenvalue follows from the fact t h a t , as it is easy to see, ^o has no nodes. Hence, we come to t h e conclusion, that the integrals in (6.44).

38

142

1 1 —2mp

1 1

! i

1 1

(i_

Figure 18: Spectrum of bound states at mp > mp.

are positive and, consequently, K2 in (6.40) is positive, i.e. for mp < mp there are two bound states with masses /x± = ± « ( m — mc) . (6.45) The consequences of this equation will be discussed in detail in the next Section. At present we want to note only the following. Expressing /x± in (6.45) in terms of mp — mp, we have fj,± « ±K<Jmp — mp .

(6.46)

It can now be seen t h a t for mp > mp the masses are purely imaginary. For mp = mp they coincide, becoming zero, and after t h a t they move apart along the real axis in two possible ways, depending on which of the two solutions <j>2 and z is realized. The structure of the Goldstone spectrum near the other critical points <£4, 5, etc. has exactly the same features. There is a weak point in the analysis of t h e structure of the bound states near the phase transition point. Namely, all the transformations of Eq.(6.1) were carried out for a constant <7, i.e. it was supposed, that the region where g is changing would not alter the obtained results and therefore it can be substituted by suitable boundary conditions. Finally, it will be useful to investigate t h e connection between t h e real spectrum and the spectrum of bound states which we would obtain by calculating their energies using the perturbative Green functions. In order to do this, let us note, that unlike (f>\, the solutions (f>2 and 3 do not differ essentially from the Green functions of perturbation theory in the broad region mc < 5 < I Hence, the equation for the bound states with these Green functions is similar to t h a t obtained from t h e perturbative Green functions. Consequently, the spectrum of the bound states (fi+,[i-) which we have obtained resembles the spectrum in perturbation theory. Assuming that at mp > mp the perturbative bound state spectrum can be described as shown in Fig. 18, our calculations show that at mp —• mp the values of fi± move closer together, at mp = m_ they coincide at zero, and for mp < mp they move apart again. One of the solutions for the Green function (as will become clear, fa) corresponds to the situation that fi+ and /z_ are reflected; the second solution (<^2) to t h e situation that they

39

143

cross and change places. If at an early stage, for mp much larger t h a n mp, condensation occurs, which leads to an increase in the fermion mass, preventing the levels fi± from going to zero, the corresponding solution is fa. The phase transition means t h a t condensation is called off and there need not be a large fermion mass any more. If for mp < mp the vacuum state changes into the vacuum described by fa, then with the further decrease of mp the absolute value of the fermion mass will increase together with the masses of the bound states which have been reflected. As a result at mp = 0 we arrive at a state which differs from fa only by the sign of t h e fermion mass. If, on the contrary, we reach the state fa, the absolute value of the fermion mass will continue to decrease. It would be also interesting to see what happens near the next critical point mp = 7Tip(<^4, fa). Here the Green functions G$ and G$ are also close to the perturbative ones and hence are also closely connected with the structure of the bound states in the perturbation theory. The behaviour of ^4,^5 differs from that of fa, fa by the fact t h a t <j>i,fa have two intersections with t h e line = ir while fa, fa have only one. This, as one could see, leads to a situation in which the Goldstone wave functions which correspond to fa,3 have the structure of a wave function for the ground state (having no nodes), while the Goldstone wave functions corresponding to ±,fahave the structure of the first excited state (with one node). This gives us a hint that the second phase transition as well as t h e subsequent ones are connected with t h e fact that in perturbation theory, at sufficiently small mp, not only one Coulomb level b u t several exist, and the spectrum can be described as shown in Fig. 19. In the phases fa or fa, in which fi± are repelled from the point fi = 0, the excited states have even less opportunity to cross each other. In fa, however, [i+ and fi- have changed places and the states (i± are able to cross t h e states p,±, and thus can b e followed by a second phase transition corresponding to crossing of /2± at mp = rhp.

I 1

t

1 1

1 1

—2mp fTl fi^

•

1 1

•

fi+ fa. 2mp

Figure 19: Spectrum of bound states at large m p .

40

144

7

The stability of the vacuum and the instability of fermion states

Due to their relativistic invariance, Eq.(6.1) have solutions which correspond to both positive and negative energy of bound states. Nevertheless, physical states always have positive energies. This is usually achieved by supposing that the masses of all particles in the theory have negative imaginary parts, and therefore only particles with positive energies propagate forwards in time. States with negative energies describe the absorption of the same particles or antiparticles (propagating backwards in time). For fermions this means, in the language of the old Dirac theory, that all states with negative energies are occupied in the vacuum. In contrast to phenomenological field theory, where particles are not considered to be compound and their masses have arbitrary imaginary parts, here t h e masses of bound states have to be calculated in terms of the masses of their constituents, and therefore the imaginary parts are necessarily connected. If the masses of the bound states are increasing monotonically with the masses of their constituents, the addition of negative imaginary parts to the constituent masses lead to negative imaginary parts for the masses of the bound states. In our case, however, near the phase transition point the masses /x± of the bound states do not depend on m 0 monotonically, and this fact leads to t h e following remarkable phenomena. Let us consider the expression (6.45) for the masses of the bound states corresponding to light fermions (curve 2)- In this case the mass of the bound state with positive energy will be fi = ( m 2 — m c ) « . (7.1) Due to Eq.(7.1) fi and m 2 have imaginary parts of the same sign, b u t their real parts are of opposite sign: fi > 0, while m 2 < 0. If t h e vacuum corresponding t o t h e curve 2 is stable, i.e. its excitations have positive energy fi t h e n fj, should have a negative imaginary part. T h e Green function in such a vacuum would describe the propagation of fermions with negative energies forward in time (having cuts shifted with respect to the real axis, see Fig.20).

I©

Figure 20: Situation of singularities of G{q) if the vacuum state corresponds to the solution (f>2-

41

145

The same would happen in the case of the curve it which corresponds to light fermions after the second phase transition. Here the positive /2 is ji = ( m c — m 4 ) K .

(7.2)

with n and m 4 of the same sign, b u t the signs of their imaginary parts are opposite. For t h e curves ^3 and s, the relation between the signs of t h e masses and their imaginary parts is the normal one, and the features of the states described by these solutions do not differ essentially from those of z and 5 correspond to the case t h a t (i+ and fi- repel each other at the phase transition point, while 2 and $4 describe the case of their crossing. We have made no attempt to calculate the vacuum energies for the different solutions and to compare them, deciding t h a t the unusual properties of the solutions which lead to the confinement of light fermions gave us reason enough to study them. Let us now present a more detailed discussion of the physics of the states described by the Green functions with the analytic properties shown in Fig.20. From the point of view of our considerations in the Introduction, the Green function obtained is a limiting case in which the normal cuts disappear entirely. The unusual cuts correspond to a situation in which the average number of quarks ( n + ) and antiquaries (n_) differ from zero, being in the limit (n+) =
Figure 21: Mass-shell hyperbolae for a fermion in the Dirac picture.

42

146

Switching on the interaction between quarks and antiquaries, and choosing this interaction in such a way t h a t each quark-antiquark pair with the same m o m e n t u m appears to be a bound state with mass /z_, the energy of the vacuum, being - c o before t h e crossing, becomes + 0 0 as soon as t h e levels cross and /x_ becomes larger than zero. In order to obtain the ground state after the crossing, it becomes necessary to move all particles to the upper hyperbola and to put a quark and an antiquark with t h e same momentum into the state fi+ having negative mass. As a result all quark states with positive kinetic energies will now be occupied, and those with negative kinetic energies will be empty. The simplest excited state of such a system would be the b o u n d state (i_ formed by two interacting quarks of negative kinetic energy. Tbis bound state would have positive energy, because the interaction of the quarks corresponds to repulsion, a n d it would be stable, since if it decayed two particles (q, q) of positive kinetic energy would b e produced; this, however, is impossible, because all the positive energy states are occupied. The situation described means, however, not only that all vacuum quarks and antiquarks are inside t h e mesonic states fi+ with different momenta, but also t h a t all possible bosonic states with negative energies are occupied. The restriction on the n u m b e r of possible bosonic states is connected with their non-local fermionic structure. At t h e same time, the number of occupied bosonic states has to fluctuate. If we take into account these bosonic fluctuations, then free fermion states with positive energy and occupied states with negative energy can appear, and this will mean t h a t the normal cuts for the Green functions of fermions also have to appear. This fact can easily be seen formally, without t h e use of the Dirac picture. Indeed, our Green function is the solution of Eq.(3.12), which takes into account diagrams of the type

/

*

/

/

1

1

x \

/ 1

1

s \

\ 1

1

.

Calculating those corrections to this equation connected with pair creation of the type

L

>

*

we obtain a diagram corresponding to the interactions of fermions

43

147

m*—fi+ie

© ie

—ie

' fi—m —

Figure 22: Analytic structure of m*. The latter, however, does not change the analytic properties of the Green function, since all the fermionic propagators in it are of anti-Feynman character (i.e. have the wrong sign of ie) and, consequently, the whole diagram has anti-Feynman analytic properties. If, however, we introduce the qq interaction

/

\

X

\

in the diagram for pair creation, and substitute the contribution of the meson-type bound state qq for ladder-type diagram, we obtain

(7.3)

This diagram contains one anti-Feynman Green function G 2 and a meson Green function with normal Feynman-type structure. The analytic properties of this diagram are rather unusual and we shall consider them in detail. The main result of this consideration is the following: if fi is the meson mass and m* the positive quark mass, then for fi > m* the contribution of this diagram has two Feynman-type cuts going from q = 0 to q = ±^ _ m * ) ( w hich are shown in Fig.22. As a result, if we write the fermion Green function as G'1 = G^1 -
44

(7.4)

148

m' — fi

1® m

fi — m*

—771

Figure 23: Analytic structure of G(q) for /z > m*. where Gj"1 is the solution of our equation with non-Feynman poles and cuts, while 2m, the pole of Gi turns out to be on the cut, and G _ 1 will have no nodes at real q values. As we shall see later, the sign of a is such that the pole of G will occur on the nonphysical sheet corresponding to a new Feynman-type cut, i.e. the quark becomes a resonance with a life time determined by the imaginary part of a, and it decays into a meson and a quark of negative energy. If \i < 2m, the pole will be outside the cut; if \i < m, the singularities of G - 1 are of the form described in Fig.24, and no Feynman-type cuts occur at all. In the two last cases, however, if instead of (7.3) one considers diagrams corresponding to the creation of two or more mesons

il) 771*—fl

fi—m~ 1

Figure 24: Analytic structure of cr(q) for fi < m.

45

149

1® „• le

m

—ie -m*

Figure 25: Analytic structure of

G2(q).

the singularities of G~l will again be those shown in Fig.23, with fi —• oo. Consequently, the life time of the quark will be determined by decays into two or more mesons. This means that in both cases the quark is a resonance. If correct, this statement is the solution of the problem of quark confinement. We have described the analytic properties of G2l with the assumption that in the diagram (7.3) the quark line corresponds to the contribution of the fermion pole. As we know, in reality the quark Green function contains cuts in q from q = ± m * to q = ± o o . Introducing the contribution of these cuts in the diagram (7.3) for cr and denoting this contribution as <x2, we see t h a t cr2 will have the form of an integral over m* from m* to oo of the expression which takes into account the pole contribution to the quark Green function (<7i). As a result, oo, i.e. the cuts from 0 + ie to oo and 0 — ie to —oo. Here the poles again turn out to be on the cuts and, consequently, they move to t h e unphysical sheets corresponding to these cuts. In this case the quark of negative energy, propagating forwards in time, decays into a positive energy meson and quarks and gluons of negative total energy. Hence, taking into account the contribution of both the pole and the cut to <x, the analytical properties of G2 may be described by Fig.25. Which particular unphysical sheet will be chosen depends on t h e relative value of <7i and 1 in Eqs.(5.25) and (5.26),

G?=m" + q.

(7.5)

The meson Green function is of the usual form. As a result,

= ij

d4k {2TV)H (m*

— k

TO* 2

- k

2

+ ie) [f

46

- (g - k)2 - ie] '

(7.6)

150

where <7M is the quark-meson coupling constant. The signs of ie in (7.6) correspond to the properties of the quark and meson which were considered above. In the reference frame in which q — 0, the denominator in (7.6) takes the form (mk

- k0 + ie) (mk + A;0 + ie) (fik - k0 + q — ie) (fj,k + k0 - q - ie)

(7.7)

where mk = ym*2 + k2, fik = VM2 + k2. The positions of the corresponding poles with respect to the real axis are crucially different from the usual ones. Closing the contour of integration around the poles ko = —mk — ie, ko = fik + q — ie in the lower half-plane, we get 2 f

d3k 2mk{iik

mk~fo +m* — mk-q-

2ie)

(fik + q)-f0 - rn* 1 2fik(mk — \ik - q + 2ze)J jik + mk + q

(7-8)

The integrand in Eq.(7.8) has no singularities at \ik + mk + q = 0. Since for \L > m

fik - mk = \/fi2 + k2 - \Jm*2 + k2 > 0 ,

(7.9)

in this case the first term in Eq.(7.8) has an imaginary part for 0 < q < fi — m , a n d a is determined as the limit from above on this cut, while t h e second term has an imaginary part for m — fi < q < 0 and <x is determined as the limit from below. This corresponds to the analytic properties shown in Fig.23. In the case t h a t 0 < q < fi — m one gets

w =

lih^™'+^ ~q2] [{l1" m*)2 ~q2] {m*" /x2 ~™r~
The expression for Retr can be obtained, using dispersion relations. If one writes a as
1

*-toJc

t dg'2 Imo-O>i, 2 2

q' -q

(7-11)

(? 12)

-

where Im
47

151

Figure 26: A contour of integration in the dispersion integral for a. strong singularity might be a fictitious one, and it can be compensated by adding terms like Cb/q2, Ca/q4 to expression (7.12). Let us now consider the position of the pole of G for q > 0. As soon as G^1 is written in the form G j 1 = m* -f q, the pole will be in the state such that qu = — qu if q is positive. In this state G~l =m"-q-tr. (7.13) At q = — q Imc->0

for

I m g > 0,

and

Im
for

lmq<0.

Hence, Im(g + a) cannot be zero on the physical sheet, and the poles of G are on unphysical sheets in the r.h.s. lower quadrant and the l.h.s. upper quadrant. The features of fJ-k, ».c. by the second term in (7.8). As it is seen from (7.8), this term determines the limit of the function o-2 from below and has Im 0. Hence, <x2 possesses the same features as the considered contribution of t h e lower cut, i.e. it leads to a pole on the unphysical sheet. In Section 2 the analytic properties of the Green functions of confined particles, as functions of qo, have been investigated. Let us discuss in conclusion how these properties, for example the existence of cuts at — \q\ < q0 < \q\, manifest themselves in 0), will be situated to the right from the point q0 — \q\, going up to the point q0 = ^Jq2 + (fi — m ) 2 . Moving along the path Cqo corresponds to moving along the contour Cq in the g-plane (see Fig.28)

48

152

if 1o

a

90

4 »

m

•I«l

—(-

y/q* + (fi-my

Figure 27: A contour of the integration in the complex q0 -plane. which can be drawn only by separating the tips of the right that if we want to reproduce the left-cut-induced singularity of kinematic cut in addition, stretching from — \q\ + ie to |g*| + properties of
and left cuts. This means a(q) we must introduce one ie. As a result, the analytic where the signs ± stand for

the signs of the imaginary parts of Jql — q2 on the corresponding sides of the kinematic cuts. The function
V>'k1

L J

-*l?l

Figure 28: A contour of the integration in the complex iq0 -plane

49

153

9o)

-Jp + bi-my

— \q\+ie — — \q\—ie

|g|—ie

1 y/p+ifi-my

v

Figure 29: The analytic structure of cr(q). Green function must correspond. From the point of view of Eq.(7.7) the imaginary part appears for — \q\ < q0 < \q\ in a trivial way because q is purely imaginary. If, however, we would calculate the integral (7.6) not in the reference frame where 9 = 0, which exists only for q2 > 0 of course, but for example in the frame where qo = 0, existing only at negative q2, then we should get the same imaginary p a r t as a consequence of the fact that for the space-like q2 decay into two particles with opposite signs of energy always is possible. It can be shown, that since the poles of the Green function move to unphysical sheets, the large-time asymptotic behaviour of the Green function is defined by the cut — \q\ < q0 < |
8

Conclusion

In the present work we have discussed only the general theoretical properties of a mechanism which leads to the existence of confined quarks and stable mesons. We, however, did not analyse the meson spectrum which would result from the introduction of flavours, of real quark masses, and of gluonic decays of singlet flavour states. We have not discuss effect of the complexity of the quark mass on the spectrum of meson states either. All these questions will be considered in the next paper. Here we note only that the confinement of heavy quarks (may be including the strange quark too) proceeds according to a mechanism described in [3]. T h e confinement of gluons is due to their decay into a meson and a qq pair with negative energies. As we have seen, there is no threshold for such decays.

50

154

Acknowledgements In conclusion I want to express my special gratitude to J. Nyiri for constant help and support. I am very grateful to all my colleagues from different institutions with whom I had opportunity to discuss the problem of quark confinement for many years. I am especially thankful to Yu. Dokshitzer, S. Troyan and B. Webber for their great help at the last stage of the work.

References [1] H. Fritzsch and M. Gell-Mann: in 16th Intern. Conf. on High Energy Physics, Batavia (1972), vol 2, p. 135 H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys.Lett. 47B (1973) 365 [2] G.'t Hooft, (1973) unpublished H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346 D.I. Gross and F.W. Wilczek, Phys. Rev. Lett. 30 (1973) 1343 LB. Khriplovich, Yad. Fiz. 10 (1969) 409 [3] V.N. Gribov, Physica Scripta T15 (1987) 164 [4] I. Pomeranchuk and Ya. Smorodinsky, Journ. Fiz. USSR 9 (1945) 97 W. Pieper and W. Greiner, Z. Phys. 218 (1969) 327 Ya.B. Zeldovich and V.S. Popov, Uspekhi Fiz. Nauk 105 (1971) 4 [5] V.N. Gribov, Budapest preprint KFKI-1981-66 (1981)

51

155

pert

QCD

May 1991 LU TP 91-15

WORKSHOP - MEETING Meeting May 21-24,1991 Department of Theoretical Physics, University of Lund, Sweden A Joint Arrangement Lund University ind NORDITA

Supercritical Charge in Bosonic Vacuum V.N. Gribov, Landau Institute for Theoretical Physics, Moscow and J. Nyiri, Central Research Institute for Physics, Budapest

Abstract The behaviour of the boson vacuum in the presence of a heavy supercritical charge is investigated. It is shown that the supercritical charge is totally screened in the limit when its radius is much smaller than the Compton wave length of the bosons in the vacuum.

156

1

Introduction

In the previous works of one of the authors [1,2] the quark confinement in quantum chromodynamics has been connected with phenomena occurring in the fermionic vacuum in a field of a supercritical charge (Z > 137). As soon as the gluons in QCD are also "charged" particles and since the anti-screening of the charge in QCD leads, generally speaking, to the appearance of supercritical charges, the changes of the boson vacuum in the field of a supercritical charge might turn out to be significant for the understanding of the structure of QCD. In the present paper we consider the simplest case, namely: the structure of the vacuum state of non-interacting scalar particles with masses m in the field of a supercritical charge of the radius r0 provided that mr0
1

157

2

T h e instability of the vacuum of scalar particles in the field of a supercritical charge

Consider the interaction between charged scalar particles described by the field y>(x) and the e.m. field A„(x) having a source j ' * 1 ^ ) . The corresponding Lagrangian can be written in the form

C(x) = -±Fta/(x)F^(x)

+ {V&YV&

- m W -

M*)JT(9)

W

where VMy> = drf - iAptp.

(2)

The equations for A^ and

- e » £ " ( « ) - «e' (f'V^

- (V&)V)

= 0

(3)

and

VV + mV = ° »

(4)

respectively. Assuming, that e ' < 1, but j**1 is proportional to a large number Z so that Ze2 ~ 1, the vacuum polarization (the last term in Eq.(3)) can be neglected in the case of a stable vacuum. If the source is considered to be a static one, and we choose A^ = (A, 0,0,0), we obtain the usual equation - d'A = e2p"t

(5)

for A(r) and, respectively,

Mr) - £ ,

(6)

if r > r 0 , where r 0 is the radius of the external charge, a = |^. The stationary zero point fluctuations r 0 this equation is the usual one for the motion of a particle in the Coulomb field. The finiteness of the radius of the charge leads to the decrease of rA(r), if r < r 0 , and thus the region r < r0 does not create any ambiguities. The eigenstates of this equation have a continuous spectrum LJ < — m and u> > m, and, for small Za values, discrete Coulomb-type bound states which at u; values close to m condense towards the point u» = m ( Fig. 1 ). These discrete states describe physical states of atomic type and give no contribution to the zero-point fluctuations, which are determined by states belonging to the continuous spectrum.

2

158 U)

m

-m

Figure 1

As Za is increasing, the eigenvalues situated at u> close to m are moving to the left, and the deepest u> state reaches the point w = 0 at a critical Za. However, before reaching this value of Za a bound state arises with a charge of the opposite sign corresponding to a state with u> near —m, which moves to the right. At Za = Z„a the two poles come to the point u> = 0, and at Z > ZCT the poles move to the complex plane. Such a behaviour of the singularities which differs significantly from that in the Dirac equation where no second pole appears while the first pole moves to the unphysical sheet at u> = —m, is formally due to the fact, that for u> —» 0 the Klein-Gordon equation (7) does not depend on the sign of the field A. From the described picture of the motion of the singularities it is clear, that at Z —» Z„ there exists an infinite number of states with the same energy, since the states w+ and u/_ of zero energy can be filled by an arbitrary number of particles. The move of the poles to the complex plane at Z > Z„ expresses the instability of the perturbation theory vacuum, since a solution of the equation (7) appears which increases in time. This solution which corresponds to « = wi + iwj at u>j > 0 has rather interesting features. As soon as it is of the form
=

J
dwm{T%t)dw{r,t)

+

m'v'ir,

t)
+

Qi
(9)

and its total charge Q = -Jd3r

[(r, t))

(10)

we obtain H = e^'J

d*r {(| u f +m2 - A3) | ^(r)

\3 + d^r)

Q = -2e 3 < "' j d \ (ui, -I- A{T)) I ^ ( r ) | J

3

d^Jr)}

(11) (12)

159

increasing in time, what contradicts the conservation of the energy and of the charge, except for the case when H and Q are zero. Hence, the instability occurring at Z > Z„ corresponds to the growth in time of the field

l a - m a M r ) = 0 ,

(13)

and the field A will be determined by the equation - d3A = e%p"\r) - A{r)
(14)

where we substituted ip by y/2e
3

The solution of the equation for the stationary state

The equations (13), (14) contain three parameters: Za, r0 and m. We will suppose, that mr0 <. 1 and consider first Za near Z„a . As it will be seen, ZCT equals 1/4 . Under these conditions (p will be a slowly changing function of r/r0 and therefore it is reasonable to look for A(r) in the form Mr) - ^

^ (15) r with a slowly changing d. If r/r0 ~ 1, we have a = Za. Introducing (15) in (13) and neglecting m2 in the region r0 < r < 1/m, we obtain rs0V + « V = O V> = Ttp .

A

(16)

160

The quasi-classical solution of this equation in the region a(r) > 1/4 is

jClR* + 6

(17)

where 6 is to be defined by matching with the region r < r0. If a(r) did not decrease with the increase of r, ip{r) would have nodes in those points r„ for which

Cf^v

+ * = nir.

(18)

This would mean, that V> i" not a ground state and there exist solutions with complex w values. Consequently, a*{r) has to become less than 1/4 before the left hand side of Eq. (18) becomes ir. If a1 = 1/4 at r = ri, then for r > ri the function ^ can be written in a quasi-classical way as a sum of two terms V>~ae x + +&e x (19) where

^-i^jCvi-**

(20)

In order to obtain a solution tp = * which decreases at r —• oo, only one of the exponents in (19), e x ~, has to be different from zero. The quasi-classical condition for the decrease of the solution requires, that

If at r —• oo

d a —»const < 1/4, (22)

(7)

i.e. it decreases more slowly than -. This, however, is in contradiction with the equation (14) for the field A, which in the case of d?a = ab2

5

(24)

161

with the decreasing solution

«-*(?)'

(25)

If (p(r) is given, C and r^ can be defined in terms of Za and r0 from the equation for a. The quantity b7 has to be determined from the equation (21) which provides us with the decreasing solution for
References [1] V. N. Gribov, Physica Scripta T15 (1987) 164. [2] V. N. Gribov, Lund preprint LU TP 91-7 (1991). [3] A. B. Migdal, ZhETF 61 (1971) 2209 .

O R S A Y L E C T U R E S ON C O N F I N E M E N T (I) by Vladimir N . Gribov* L. D. Landau Institute for Theoretical Physics Acad, of Sciences of the USSR. Leninsky pr. 53. 117 9*24 Moscow. Russia and KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences H - 1525 Budapest 114, P.O.B. 49. Hungary and Laboratoire de Physique Theorique et Hautes Energies** Universite de Paris XI. batiment 211. 91405 Orsav Cedex. France

LPTHE Orsay 92/60 June 1993

* Supported in part by Landau Institute-ENS Departement de Physique exchange program ** Laboratoire associe au Centre National de la Recherche Scientifique

Lectures On Confinement (I)

159

1. The theory of supercharged nucleus This talk is just introductory. Literally it is not about confinement, but it is important since it is closely connected to the theory of quark confinement. In fact, this theory deals with two different problems. The first one is, how to confine particles, i.e. the problem of producing forces strong enough to prevent quarks from separating, which will be discussed later. But there is also another, very severe problem. We know, that quarks are light, almost massless particles. The question is, how to bind massless particles in a volume which is much smaller than the Compton wave length. Usually this is not easy to do, because the wave i

function is decreasing exponentially like e~Vm

5

y

~u r , which in the best case

can be e~ m r . But if the mass is very small, the state is very broad, and it is in complete contradiction with what we know about hadrons. The mechanism of supercharged nuclei, which we will discuss in the present talk, seems to be a unique possibility to bind a particle in a small region in space. The theory of the supercharged nucleus is very old. It was initiated in the forties by the work of Pomeranchuk and Smorodinsky [1], and it became very well developed, with no unsolved questions left. Since we, however, want to apply this theory to quarks, we will talk about it in a way slightly different from what people are used to, combining the picture of the Dirac sea in external field and the language of Feynman's Green function. The problem of the supercharged nucleus is the following. If there is a nucleus Nz with a charge Z, and if Z is larger than a critical value Zc

Z>ZC

160

Vladimir N Gribov

(theoretically Zc = 137, practically it is around Zc ~ 180), then this nucleus will decay to an a t o m with a charge Z — 1 and a positron:

Nz -» Az-i

+ e+ .

This a t o m can be stable or unstable. If it is unstable,it can decay again u p to a situation, when the total charge of the atomic state Az-n

becomes

sufficiently small. This is a peculiar thing : it means, t h a t the nucleus behaves like a resonance. It does not exist in the n a t u r e freely, b u t it exists inside the atom. In this sense it is analogous to what we know about the quarks. This, of course, is not confinement, b u t in some respects it is not so different. Indeed, the nucleus has a baryon number B and a lepton number zero. But this state with lepton number 0 does not exist, there are only states for which the lepton number equals unity. In this sense, it is a confinement of states with zero lepton number. Theoretically the described problem is very well defined. If we want to understand the new type of atomic states, we have to consider the interaction of the electron with the nucleus, taking into account all possible interactions.

""-O-

Lectures On Confinement (I)

161

In this case there are some simplifications. First,

a « l

,

(1)

aZ ~ 1 .

(2)

but

T h e third important simplification is t h a t the ratio of the electron mass m e and the nucleon mass m^

is much smaller t h a n unity :

"' mN

«

1 •

(3)

Because of these conditions, the problem can be solved exactly. First, due to (1), we can neglect all the corrections to electron propagation, since they would be of the order of a, and leave those diagrams which contain lines connecting electron and nucleus lines and radiative correction to the nucleus line. But also, because of (3), the recoil of the nucleus due to photon emission will be very small, and instead of writing this interaction and taking into account all the sequences, including recoil, we can write t h a t this is equal to the electronic Green function in the external field of the nucleus, multiplied by the nuclear Green function with radiative corrections

—*

*

*

->

x-

Ge(A) GN

o- x --" (4)

162

Here Ge{A)

Vladimir N Gribov

is the Green function of the electron in the external field ; the

Green function of the nucleus Gjv, which corresponds to the diagram a

x^.--'

b

contains a lot of radiative corrections. We have to include loops like b since they are of the order of 01Z, and diagrams c being aZ2.

Fortunately, all this

t u r n s out to be very simple. The reason is, t h a t all the photons in these diagrams cause no recoil for t h e nucleus. It means, t h a t we can write the following :

+

< • . . '

G

0.e

N= V

where everything can be factorized.

(5) \

,

will be t h e symbolic expres-

sion for the electrostatic energy of the nucleus, and

P

\\

gives us the

mass renormalization for the nucleus due to electron vacuum polarization. And thus we have an exact description for GN. Further, the first step is, of course, to calculate the electron Green function in an external field, which also defines the nucleus mass renormalization entering the exponent in (5).

Lectures On Confinement (I)

163

This means, t h a t with the knowledge of the Green function of the electron in the Coulomb field, we can calculate everything. T h e equation for the electron Green function is V G e = -i6 ^(d„-iAll)Ge

(6) = -iS

(7)

Since the potential does not depend on the time, we can always write

Ge(x2,Xl) = J ^ " ' ^ - ^ ^ ( r ^ i ^ )

(8)

and, because of t h a t , we have 7M(3M-*^)^ = 0 ,

(9)

or, in a more convenient way, [ujpj + m 7 o -(u-A)]il)u

= 0 .

(10)

Before discussing the solution of the equation, let us consider the well-known features of the spectrum of this system.

(0

-m

m

164

Vladimir N Gribov

There are cuts going from m to oo a n d from —m to —oo, corresponding to the continuous spectrum, and sequences of poles corresponding to Coulomb levels. Now - how does the critical phenomenon come in ? If we increase Z , the pole which corresponds to the a t o m in the ground state moves to the left, passing zero without any troubles (for a finite-size nucleus) and reaches the point — m at the value Z — Zc. W i t h the further growth of Z > Zc the position of the level UQ is going to the complex plane and the state becomes unstable. However, at this point we come to a paradox : this LOQ is supposed to be the energy of an atomic state, i.e.

the energy of the atomic state

becomes complex, contradicting the physics we have expected. Indeed, we thought t h a t the nucleus was unstable and would decay on a stable atom and a positron. This means, t h a t the described simple solution has to be essentially changed, since, apparently, the Green function Ge(A)

of the electron does

not reflect the whole physics, and the features of GN in (4) t u r n out to be i m p o r t a n t . We will show, in fact, t h a t there is a cancellation between and

Ge(A)

GN-

Let us consider now, what is happening from the point of view of the Dirac sea. From m to oo and from — m to — oo there is a continuous spectrum of electron states in the Coulomb field. Also, as we said before, there is a resonance state with complex energy wo + «T/2, where UQ < —m. W h a t is the physics of this resonance ?

165

Lectures On Confinement (I)

CO

. ...

i

1

^

—

The Dirac equation can be written in the form (T + A)V> = 0

(11)

where T is the kinetic energy and A the potential energy. This kinetic energy can have two signs T — ±^p2

+ m2, and therefore the total energy can be

quasi-classically either V V +m2 + A(r) = E

(12)

or -y/p2+m2

+ A(r) = E .

(13)

Obviously, (12) can be fulfilled only close to the origin. In this case the potential is negative, the square root is positive, and the total energy is negative. Classically, the electron will be stopped at a return point ri ,where the energy E is the sum of the potential and the mass m : m + A(r1) = E.

(14)

Vladimir N Gribov

166

This means, that in this region there is a normal wave function which oscillates at r < n .

After that it has to decrease exponentially because of

the absence of classical trajectories. For the equation (13) the situation is the opposite, it can exist only at large distances, since the potential at large distances is small, and the energy is negative ; consequently, there will be another return point r2, where

-m + A(r2) — E .

(15)

Again, there will be a plane wave, i.e. normal oscillation at r > r2 with a subsequent exponential decrease due to the absence of classical motion between the points r\ and r2

CO

m r. l

r

2

y^—

-m A=E+m ..•..../ s E A=E-m • - - ! • /

I

The situation is somewhat similar to that with a barrier in the non-relativistic problem :

Lectures On Confinement (I)

167

This barrier-type behaviour can be easily seen, if one writes a second order equation instead of t h e first order one.

Indeed, let us re-write the Dirac

equation in the form

V " + -F^a^

(16)

)V = 0

Here V 2 - (d2 - iA)2

= --d2rr

- J1 + 2UJA -

A2

(17)

In this equation the effective potential energy is 2uA — A2. If A is negative, the first t e r m 2uiA corresponds to attraction at large distances, and the second t e r m —A2 to repulsion at short distances. It is natural to expect, t h a t this negative energy state is just a resonance. But : in the previous discussion we forgot about the Pauli principle. We have been talking only about one state and not about the Dirac sea. However, Dirac sea means, that all these normal negative levels have to be occupied.

168

Vladimir N Gribov

In this case, if our particle will try to go out and pass through this barrier, there will be no place for it. When we find, that the Green function Ge has a complex singularity, this reflects the fact, that we did not say up to now, what type of Green function we are using. Imagine, that the Dirac sea is not filled up. In this case it would be natural for our state to have complex energy. On the other hand, we are used to the fact, that the Feynman Green function reflects the Pauli principle in the correct way, and therefore, if we are looking for the Feynman Green function, we expect to find the proper answer. Suppose now, that we will calculate the Feynman Green function. Still, we said that there is a general solution for the Green function with a complex pole. The question is, how this happens. The answer is essentially very simple. Indeed, we are looking for a Green function which is a sum of diagrams of the type

(18)

If the Feynman Green function contained only positive energy propagating in positive time, all the energy denominators in (18) 1 £ — y/m2

— p2

would be real at negative total energy. But the Feynman Green function contains also negative energy, propagating in negative time. It corresponds to the so-called Z-diagram

Lectures On Confinement (I)

169

Perturbatively this diagram contains also only positive energy intermediate states. However, if the incoming and outgoing lines will correspond to negative energy bound states, then the three-particle intermediate state containing two negative energy particles and one positive energy particle can have negative total energy :

e — \ / m 2 + p2 — 2e

e + \Jm2 + p2

This means, that the Feynman Green function in this case obtains an imaginary position of the pole in contradiction with the Pauli principle. (This contradiction is obvious from the second of the Z-diagrams, since we have there two particles in the same state at the same time). It is also clear, that this diagram reflects not the decay of a state, but that of the vacuum. Let us see, how the Green function of the nucleus will recover the Pauli principle. Consider the diagram

170

Vladimir N Giibov

<0>' N

The Green function of the electron is in this case

Ge(t,r,r') = J ^ e - i w , i W i ( ' ' ' ) •

(19)

In order to calculate it, we have to write the following contour of integration (for Z


With Z growing, the pole will move (as indicated by the dotted line) to the complex plane,

175 Lectures On Confinement (I)

171

CO

-co+ir X

-m

...X

X X I

m

a n d we will have to change the contour of integration:

CO

...x x x i m

172

Vladimir N Gribov

As we see, the Green function will contain a growing exponent which will be defined by the imaginary part of the pole. At large t we will be left with an unstable pole contribution

G,

J

Jni

Obviously, the total of the diagrams can not contradict the Pauli principle, and due to this fact there has to be a cancellation between the electron Green function and the Green function of the nucleus. T h e latter contains contributions of the type

a n d has singularities connected with atomic states

This means, t h a t the Green function of the nucleus will contain this loop, which, definitely, has an imaginary part, because the nucleus becomes heavier t h a n the a t o m and the positron. It gives us a contribution which describes

Lectures On Confinement (I)

173

the instability and results in an exponent e~Tt corresponding to the decay of the nucleus

Hence, e

will cancel and


e

G

N

will be proportional to e-*(uo+M)t

^ j ^ corresponds to the propagation of

an atomic state with real energy u>o + M

<\AAAAAAAAAA*

t,r 1

0,r' l

So, indeed, we have shown, that in this case the bound state is stable not only due to the existence of binding forces, b u t also because of the Pauli principle, which reflects the antisymmetry between our electron a n d the electron in the vacuum. In the electron-nucleus scattering amplitude (4) asymptotically (i.e. at large t) only the part

174

Vladimir N Gribov

survives in the sense

t'

because the Green function of the nucleus decays always exponentially, and it has to be compensated by the increase of the electron Green function. This is possible only, if

r(*2 - * i ) < i ,r(
179 Lectures On Confinement (I)

175

X

X

•*JL_>

.1.

How can we derive Feynman rules from this ? It will be, indeed, very simple. In this picture our Green function

G X 1

X 2

has, of course, to be retarded :

G X 1

X 2

The real propagation is second order in external field :

G

\

*

A

S-*-

A

x2

176

Vladimir N Gribov

We start with the usual calculation, but we have to include the Dirac sea. T h e external field A is acting not only on the particles, b u t also on the Dirac sea. This means, t h a t we have to consider our particle and a particle from the Dirac see ;

X

x

X

x

1

2

will be a possible diagram. However, we can write also

X X

a n d consider the diagrams

••—X x

•x

1

2

-X(21)

and

181 Lectures On Confinement (I)

177

-

*

•

X

X •*r

(22)

Averaging over all particles in the Dirac sea, the diagram (21) leads to X

which corresponds to the first order contribution to vacuum polarization and is in fact zero. The diagram (22) gives

c and we get

Vladimir N Gribov

178

X

* -

- *

x

G

(23)

summing all negative energy levels. Adding the diagram with GR (20) to that with G- (23), we obtain the Feynman diagram. By these two diagrams, however, symmetry is not reflected. In order to be symmetrical, we have to add

X •

*

*

•

which means that the external field is interacting twice with the vacuum. But averaging this, we get

X

X

i.e. we have to add a diagram which corresponds to vacuum polarization in external field. In other words, the Pauli principle is taken into account by adding vacuum polarization to the normal Feynman propagator.

Lectures On Confinement (I)

179

It is, of course, always necessary t o consider vacuum polarization, but sometimes it is relevant, sometimes not. In the case of weak coupling the vacuum polarization was real and, when added to the Feynman propagator, did not change the picture essentially. But in our case, in the case of strong coupling, the vacuum polarization becomes complex and because of this, it defines everything. In conclusion, let us make it clear, which region of distances between the nucleus and the electron is important in our calculation. As it is shown in [2], the solution tpu(r) of the Dirac equation (10) is

il>u(r) ~ —j=cos

y/(Za)2 - Un— + 6

(24)

in the region

r0 < < r < <

where ro is the radius of the nucleus. If |a>|ro < < 1, the function ^ w ( r ) will oscillate and at ro —• 0 this corresponds to "falling into the center". If ro is finite, the "falling into the center" does not occur. However, the existence of oscillations is a sign which indicates t h a t the levels passed through the point u) = —m. T h e number of oscillations n in the region TQ < r < — which determines the number of levels passing through is defined by the condition

Vladimir N Gribov

180

At n = 1 formula (25) provides the condition for the charge to be supercritical (with ro and m finite) and shows, that the region where the supercritical phenomenon occurs is

r0 < < r < — . m Acknowledgements The text of these lectures was written by Julia Nyiri. She substantially expanded and edited the original notes she took during the lectures. I am very grateful to Ph. Boucaud, A. Dudas and J. Mourad who formated the text and the figures. Also, I would like to thank the Lab. de Physique Theorique et Hautes Energies, and especially A. Capella, M. Fontannaz, A. Krzywicki, D. Schiff and Tran Than Van for their hospitality, and the warm and inspiring atmosphere during my stay at Orsay. References [1] I. Pomeranchuk, Ya. Smorodinsky, Journ. Fiz. USSR 9 (1945) 97. [2] Ya. Zeldovich, V. Popov, Uspekhi Fiz. Naukl05 (1971) 4.

Orsay Lectures On Confinement (II) Vladimir N. Gribov* L. D. Landau Institute for Theoretical Physics Acad, of Sciences of the USSR , Leninsky pr. 53, 117 924 Moscow, Russia and KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy of Sciences, H-1525 Budapest 114, P.O.B. 49, Hungary and Laboratoire de Physique Theorique et Hautes Energies* Universite de Paris XI, batiment 211, 91405 Orsay Cedex, France

LPTHE Orsay 94-20 February 1994

'Supported in part by Landau Institute-ENS Departement de Physique exchange program t Laboratoire associe1 au Centre National de la Recherche Scientifique

182

1

Vladimir N Gribov

The confinement of t h e heavy quark

In the previous talk [1] we have considered some aspects of the theory of supercharged nuclei, and came to the conclusion, that the superbound atoms are stable mainly due to the Pauli principle. Before going to the heavy quark, let us discuss briefly, how the problem of the supercharged nucleus can be handled practically. One possibility to formalize it is the following. We have to calculate the Green function of the Dirac equation in external field : V G(x, x') = -S(x - x')

,

(1)

i

where V = 7 „ ( ^ - ^ ) • (2) The initial external field is that of the considered nucleus. However, in the presence of a charge Z the stationary states correspond to atomic states with a charge Z — N where N is large enough to fulfil Z — N < Zcr, i.e. there have to be N electrons rotating around the nucleus. But if so, we will have to take into account, that the field which acts on each electron is also changing. The ground state in our case is an atomic type state. This means, that although we don't know what our field is, we can find it. The gradient squared of the zeroth component A0 = A of the potential has to be equal V2A0 = el (pnuci + pei)

.

(3)

where pet = < ipjoip >• The charge density will depend on the potential, and the equation becomes a Thomas-Fermi type equation for the potential existing in this system. If we find a wave function satisfying the equation we will know A, the Green function and p. This means, that the formal way of solution is just to solve a self-consistent, Thomas-Fermi type equation for the effective atomic potential. Let us consider the total charge Q

Q = Jpd2r

;

(4)

the density is the average value p = < ip{r)ip{r) >= ] T Tpn{r)ipn(r)

(5)

which in the Dirac picture is the sum of all negative energy levels. The total charge Q will be the sum of the energy levels over all ujn. .The sum is divergent, and we have to make a cut-off and subtract the bare particles. This, however, will not be enough. Indeed, if we just subtract the value of the charge of the free vacuum, then at Z / 0 the charge of the vacuum (i.e. of the nucleus) will not be equal Z and will continue to diverge logarithmically. We have to subtract the value at small Z in a way which gives zero for the total charge of the electron vacuum (this corresponds to the correct renormalization of the charge of the nucleus). The above procedure does not reflect literally the subtraction of the vacuum charge without external field. This becomes especially obvious, if one makes use of the Levinson theorem which connects the number of states of a particle in external field in a given energy interval with the phase of scattering of this particle in the same field. The number N of additional states in the energy interval between E\ and E2 is defined by the difference of phase shift N=-[S(E2)-S(E1)]

(6)

7T

Because of this, the number of new states in the Dirac vacuum equals N=-

[5(-m) - <S(-oo)]

(7)

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For Z < Zc this number should be zero. However, for the Dirac equation in external field this condition is not fulfilled, because <5(—oo) ^ 0, and in order to obtain the proper definition of the vacuum charge we have to subtract a quantity which, generally speaking, depends on Z (<5(—oo)). This subtraction means, in fact, that we have to change the interaction with the external field when E —> —oo so that <5(—oo) = 0. (<5(—m) can always be considered to be zero if the field is small). If, however, Z will be increased and becomes more than critical, then, as we saw, the levels will pass the point E = —m and move to the complex plane. It is easy to check, that every time this happens the phase 5(—m) is changing by n and the move of n levels into the complex plane changes the number of states by n so that the charge of the vacuum (i.e. of the atom) becomes Z — n. The value of pe which enters the equation for the self-consistent field is to be defined by the contribution of levels which passed through E = — m. We discussed in detail this concrete structure in order to refer to it in the following, talking about the heavy quark in the vacuum of the light quark. We shall suppose, that due to gluonic vacuum polarization the effective coupling a(r) which at small r has the usual perturbative behaviour, reaches a constant value at r > > r0 (r0 ~ 1/A), and this constant value will be more than unity. (Without this ansatz, allowing that a continues to grow, things are more complicated, but nothing essentially changes).

Figure 1 The supposed behaviour of the effective coupling.

It is natural to expect such a behaviour of the charge in QCD. But even an Abelian theory can reveal such a behaviour of charge, if it originates from a non-Abelian theory via spontaneous colour symmetry breaking. In this case the charge will increase, and as a result, 6-7 gluons acquire masses. After that there remain one or two massless objects - "photons" - and the behaviour of charge will be exactly as discussed, because after the symmetry breaking we have ro = -^ (heavy gluon). We can ask now : what happens in the vacuum of light quarks under these circumstances ? Outside the region where a is growing, we will have an Abelian theory and we can consider the quark states in the normal way.

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Vladimir N Gribov

Vm (symmetry breaking) If a becomes more than critical, the corresponding level goes to the Dirac sea, which, consequently, will have to be filled up and we will find an atomic type state. If this is so, the charge density will be exactly the same as in the case of a supercharged nucleus. The XQCD is analogous to the radius of the nucleus. There is, however, an important difference between QED and this case. Indeed, in QED we consider a nucleus with a charge Z in the centre, and we put one or two electrons around it to organize an atom. In QCD we have a heavy quark with a very small intrinsic charge.

_qr

\s^ Due to vacuum polarization the charge becomes large inside 1/A. The system creates an empty level, and we have to fill it. This means, that we have to add an intrinsic charge, equal to that of the heavy quark, but with the opposite sign (outside the region of 1/A). So we have here two intrinsic charges with the total charge zero, and of course the vacuum polarization can never change the total charge. In other words, in QCD the supercritical atom would be a meson-type state with zero colour charge. Our task is to show, that these are not only words - we have to include the mechanism of vacuum polarization formally. In order to do so, let us discuss the problem in a language similar to QED, neglecting non-Abelian fluctuations of the colour field A - the average field of heavy and light quarks inside the meson. In this case Ao is defined by the equation V2A = e20 (ph - pi)

(8)

where ph is the density of the heavy quark, and pe that of the light quark. Again, we can write the equation for the Green function

(y-rnjG(x,x')

= \s(x)

(9)

But now the problem is, how to calculate this. What means, e. g., pheavy ? We can not use the expression ph = e\5(r), since it does not include the bosonic vacuum polarization, which has to be taken into account. It is well known, that in principle vacuum polarization means summing a diagram like

Lectures On Confinement [II]

185

0

*

a

with gluonic loops inside. The problem is, however, how to write this in normal space-time language. There is a good way to sum this diagram by writing an equation for currents. Knowing the external current and wishing to calculate the total current, we have to solve the equation j»(x)=Jl*t(x)

+ ^Js'{(x-x')2)j»(x')dix'

,

(10)

which corresponds to the summation of the diagram. We will not derive this equation, because it is almost obvious for the static charge which we are interested in. Let us consider a static charge, not depending on time. In this case P(r)=Pr(r)

+^ l ^ ^ p ( r ' )

,

\r-r'\>e

,

(11)

where 1/e is the ultraviolet cut-off, 6 = ^ n c — | n / . The solution of this equation is very simple and leads to the usual expression for charge renormalization in QCD. In order to see this, let us introduce the quantity Q(r) : Q(r) = [ p(r')4nr'2dr' Jo The logarithmic derivative of Q(r) is

.

diQ(r)=r^l=4nr3p(r)

(12)

.

(13)

For Q{r) we can write the equation

*Q(0 = d,Qext(r) + g / _ | > £ ^

£W )

•

(14)

The integration in the right hand-side of this expression contains two logarithmic regions : \r — r'\ « r and r » r'. In the first region, d^Q(r') can be substituted by d^Q(r) ; integrating over the second region, r' in the denominator \r — r'\3 can be neglected. As a result, we obtain dtQir) = d, Qext(r) + ^

2

[

^

Q(r) + ^Q(r)

,

which is equivalent to d( (l - f ^ )

Q(r) = di Qext{r)

(15)

or Q(r)=

Qext r)

[

.

For a point-like charge Qext{r) = 1

,

(16)

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Vladimir N Gribov

and we have A

(r) = ~

= a(r)Aext

.

(17)

The concrete expression 1-

^£nL

obtained from perturbation theory has, of course, an unphysical singularity. For a point-like charge (18) has to be substituted by an expression corresponding to the behaviour of a(r) as shown in Fig. 1. However, for the distributed charge the relation between the external field and the field which takes into account the polarization is non-local in coordinate space. The correct expression for the relation between the external field and the observable field is local in the momentum space : A(q) = a(q2)Aext(q)

(19)

which leads in the coordinate space to an expression of the following type : A(r)= JK(r-r')Aext(r')d3r'

,

(20)

where

e

^ = / ^(0 • Similarly, 3 3 p(r) = If K(r - r')pr')p r' r' ext(r')d ext(r')d

.

(21)

If we now suppose, that a(q) as a function of 1/q behaves according to Fig. 1, and at large q values it/is defined by perturbation theory, then the equation (8) has to be understood as an equation for Aext '.

V2Aext{r)

= S(r) - ^ ( r ) 7 o ^ ( 0

(22)

where ipi{r) is the solution of the Dirac equation (H + A)iP = Eip for the light antiquark in a superbound state in the field A(r) defined by equation (20). The solution of this problem gives the energy and the features of the meson qnqi with zero total charge. Due to (21), Q = j K(r)d3r Qext = a(q = 0)Qext

(23)

with Q = 0 if Qext = 0. We have just proved, that because of the big charge which appears through vacuum polarization, in the case of QCD the "atomic" bound state will, indeed, be a meson. The heavy quark will decay into a Qh^-meson and a light quark : qh -»• M {qhqi) + qt

.

(24)

In the next lecture we shall consider light quark states. So far there is one important thing to stress, namely if we don't include essential interactions between light quarks in the vacuum, we come to a reasonable conclusion for the case of the heavy quark but, as we will just see, to an unreasonable one about the light quark. Indeed, let us try to extend the considered procedure to

Lectures On Confinement [II]

187

the latter case. Suppose, that there is a light quark moving, and a potential acts on its vacuum. What will we see classically ? Since the Coulomb field is a vector field, it is shrinking, but the total integral remains the same. Because of this, we will find immediately, that there is a bound state in this potential even for fastly moving particles. This, however, means, that we have here an unstable state, which has to be filled, and as a consequence the light quark will decay into a meson and a light quark again : qi -> M + qe which, of course, contradicts the energy conservation, unless the appearing meson is of negative energy. In order to have a self-consistent picture, we have to suppose that the light quark in the vacuum will interact so strongly that there have to be negative energy levels and the whole vacuum has to be rearranged. So from the picture we described we come quite naturally to light quark interactions. We will see, that these interactions are, indeed, very strong and lead to the confinement of light quarks which will take place at relatively small — values ; this means, that the overall corrections for vacuum polarization will not be large. This is for the future. What we have to add now, is, that even in the language which was accepted so far, with no strong interactions between particles in the vacuum, the problem in real QCD which is non-Abelian is more complicated. In QED we have one charge, and all the electrons in the vacuum interact with this charge, independently from each other. In QCD this can take place only, if the field of the heavy quark is an Abelian one. It is highly probable, that this is, indeed, the case, when the field of the heavy quark becomes large as a result of gluonic vacuum polarization.

References [1] V.N. Gribov, Orsay Lectures on confinement (I): The theory of supercharged nucleus, LPTHE ORSAY 92/60 (1993).

Orsay Lectures On Confinement (III) V. N . Gribov L.D. Landau Institute for Theoretical Physics, Moscow

and KFKI Research Institute for Particle and Nuclear Physics, Budapest and Laboratoire de Physique Theorique et Haute Energies Universite de Paris XI, batiment 211, 91405 Orsay Cedex, France

LPT-ORSAY-99-37

hep-ph/9905285

Lectures On Confinement (III)

189

Light quark confinement 1

2

We have described the confinement of heavy quarks in an analogy with the theory of the supercharged nucleus [1,2]. Let us now suppose again that a is behaving like

Fig. 1. Making this assumption, we are neglecting gluon-gluon interactions and the existence of gluons as real particles. Our aim is to see, what can arise from the discussion of light quarks only. We introduce A corresponding to ac and consider quark masses m0 <SC A. The interactions of light quarks (for which m0
—s— \ \

i /

s

where the gluon propagator (considered as an effective photon), corresponding to the dotted line is Du (1) -20'"'' Further, we look for a model which enables us to see, what happens to the fermions if there is an interaction between them, as indicated above. The question is, how the bound states or the Green function behave in such a case. Let us consider the energy of two quarks, u and d, for example. Without interaction there will be positive energy states with E > 2m and negative energy states with E < —2m:

E(ud) 2m

a -2m ' T h i s is the third lecture on quark confinement given by V.N.Gribov in 1992 in Orsay. An extensive discussion of the consequences of all this for the structure of the Green function can be found in [5,6] - in the two last papers concluding his 20 years long study of the problem of quark confinement in QCD. 2 T h e text was prepared for publication by Yu. Dokshitzer, B. Metsch and J. Nyiri on the basis of a tape recording and notes taken during the lecture

190

Vladimir N Gribov

Introducing the interaction, for small a we will find that there are some bound states near 2m and -2m.

E(ud) 2m boundstates-

a

-2m So far, we consider the usual Dirac vacuum: the negative energy states are occupied, and the positive ones are empty. Increasing the coupling, i.e. increasing a, we could expect that the magnitude of the energy is decreasing and the levels corresponding to the bound states will come closer and closer to zero.

E [ud)

0

a

-2m With a further increase of the coupling up to a critical value, one possibility for the levels will be just to approach the zero line and never cross. There is, however, also a possibility of crossing. We will see that the first case corresponds to normal spontaneous symmetry breaking. But, if the levels cross, and especially in the most clear case, when they pass the lines 2m and —2m, respectively, everything will change and we arrive at very different phenomena:

E{ud) 2m 0 -2m Indeed, now the original vacuum which corresponds to the case when level 2 is empty and level 1 is occupied, is absolutely unstable. We have to fill the new negative energy state and leave the positive energy level empty. But by filling this new state, we get an excitation, a meson-type state with a mass fi. For free quarks this would mean that the quark with negative energy decays into a negative energy meson (filling the negative energy levels) and creates a positive energy quark. As a result, the Dirac picture in which all negative energy levels are filled up and all positive energy levels are empty, is destroyed. But if so, a positive energy quark also decays into a positive energy meson and a quark with negative energy. This means that both decays q~ q+

-> -+

n- + q+ n++ q~

Lectures On Confinement (III)

191

are possible, and both q and q+ are unstable. The question is now, how to deal with the bound state problem. Of course, we could just start to calculate the bound states, considering the interactions without corrections to the Green function. However, one has to take into account that the fermion-fermion interaction changes the effective mass of the quarks and this in its turn will change the bound states considerably,

which makes the problem more complicated. We thus will have to consider bound states and the Green functions on equal footing. Up to now, the only approach which deals with this problem and is self-consistent is the Nambu - Jona-Lasinio model [3]. It considers the fermion Green function corrections due to a four-fermion interaction:

In spite of the strong dependence on the cut-off, the model preserves all symmetries in the Green function and in two-particle interactions. Let us present the result of Nambu and Jona-Lasinio in a way somewhat different from what is given in [3]. We express it as the dependence of the renormalized mass m on the bare mass m0. They found that if the effective coupling (it depends 2

on the definition in their case) jj^- is less than unity, the curve will be just the usual one:

< 1.

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Vladimir N Giibov

If, however, the coupling | - is larger than unity, the dependence will be like 3 :

m

spontaneous

Fig. 2.

According to the interpretation of Nambu and Jona-Lasinio, the upper part of the curve, which at mo = 0 reaches a finite point, corresponds to the spontaneous symmetry breaking. But there are three solutions at mo = 0 and at sufficiently small m 0 values. What Nambu and Jona-Lasinio claim is that the lower part of the curve is unstable, and there is a real vacuum. I agree with this, if mo > Tncrit. We can ask: what is the source of instability of this curve? The general argumentation is the following. Talking about spontaneous symmetry breaking, m is like a magnetic field in a ferromagnetic; we just choose a definite direction. But mo is like an external field, and the system is like a compass. If the external field and the induced field are pointing at the same direction, the situation is stable. If they point to opposite directions, the compass will change. I am, however, not sure that the instability of the almost perturbative solution which contains no condensate at all can be explained in such a way. The explanation may be right for the part b of the curve in Fig.2 which corresponds to a big spontaneous magnetic moment and the opposite direction of m 0 . It does not work for the part a close to perturbation theory which has no spontaneous magnetic moment. And, looking more carefully at the curve, we see that the part b corresponds to pseudoscalar states inside the Dirac see, while on the piece a both the pseudoscalar and scalar states are inside, both levels passed. Recognizing this, one can conclude that indeed, the mentioned state is unstable, but for a trivial reason: the corresponding level is inside the Dirac see and it is not filled up. The problem is, what happens if we fill up this level. It remains an open question, what can be considered as a ground state under these conditions. And it is a problem how to get these results in a more self-consistent way, not depending on the cut-off so strongly. The Nambu - Jona-Lasinio model can be reproduced in our picture. For this purpose, just as

3

T h i s result is not always quoted, but it is present in their paper.

Lectures On Confinement (III)

193

a theoretical exercise, let us use a not going to unity, i.e. draw a

Fig.3. instead of the curve in Fig.l. In this case there will be a second scale A2, and outside this scale we will have just point-like interaction. This reminds of the Nambu - Jona-Lasinio picture, which, apparently, can be reached somehow in our approach. The problem is, how to write constructively the corresponding equation, and whether this can be done at all. Of course, this constructive part can be only approximate. But if we recognize that it can be written, then we will be able to develop a theory in which we put the main ingredient of our discussion as an input into our solution and try to find the real construction. The main difference will appear in the analytic properties of the Green function. The Green function of a fermion for such a case would be quite different in its analytic structure compared to the usual one. I am afraid I will not have the time to come to this point today, but I would like to explain just the physics. How to write the equation? What happens in the real case and how to deal with it? Let us start with with the Green function. What do we know, what are we supposed to know about this Green function as a function of q2 ? renorm group equation m

Beyond a certain A in the region where the coupling is small, it is asymptotically free; here the Green function has to satisfy the renorm group equation. But if as a result of the interaction a mass is acquired, this mass would be somewhere at smaller q2\ here the equation becomes essentially very complicated and we are not able to extract a reasonable structure. The idea to write an equation which is correct in both regions, near the threshold and at large q2, and to match these two solutions, comes from the following consideration. Suppose that acrit/^ is small: Otcrit

0.2.

7T

Now, however, we may ask: how could new masses, new solutions etc. appear at all at such a small a. Obviously, this a has to be multiplied by something large. What happens, for example, at large q? We know, that there is always a logarithm of q2/\2 and the real parameter becomes 1 r a0ln^-

194

Vladimir N Gribov

which is, in spite of the smallness of a, big enough to change the Green function essentially. But near the threshold there is also a logarithm: g2 — m 2 In —= , with some scale A = A or m A2 which is always present. In other words, in this region there could be also a quantity which changes seriously in spite of the relative smallness of a. Hence, we want to write the equation which is correct near the threshold, taking into account correctly the singularity of a supposed mass, and after that compare this with the renorm group equation; we shall see whether it is possible to write an equation which is correct in both regions, and if yes, we will try to solve it. In order to get the singularity correctly, we take the second derivative of G~l with respect to the momenta. (fl-4':

/ \ G

G

~ °

+

i

G(q>)

(2)


v'

The contribution of the first term is trivial, the second derivative of q + m gives zero. Taking the second derivative of the first graph, it can be easily seen that

This gives for the first diagram 7/iG(g)7M J - just by direct calculation. In other words, we make it local. From this diagram we take the contribution where k = (q — q') - the momentum of the photon - essentially equals zero. Let us now look at the second diagram. We have here the choice of taking the second derivative at one of the photon lines, or to differentiate once at one line and once at the second line. Having in mind that all the integrations would have a structure which need some logarithmic enhancement, it would mean that the most important regions of integration in this integral would be those where k\ <S &2 and i j < i i . We take the derivative at ki and then set it to be zero, but for fe the integration will give the same as before. If this integration gives us two logarithms, we kill one and recover it after the integration of our differential equation; but we still have the first one. But if we differentiate once one line and once the other, we will always sit on the region k\ ~ k2, because they have to be of the same order. And, in this case, there is no logarithm at all; after the integration, we will recover one, but one order will be lost. Clearly, a possible approach is to try not to choose different diagrams, but to use the small k region of integration. Ordering the integration inside the diagram in such a way that one momentum is much smaller than the others, and differentiating this diagram, we will find a relatively simple answer. Indeed, suppose that we have any diagrams with any loops. If we differentiate some lines twice (it can be any line) and neglect all first derivatives, we get an amplitude of the following structure:

k=0

k=0

This is just the Compton scattering of a zero momentum photon k — 0, and for this quantity the most singular contribution is obviously

TJ0,q)G(q)T^0,q)

199

195

Lectures On Confinement (III)

which corresponds to the diagram

G(q)

But the vertex T is at zero momentum and hence r^(0,q) = d^G we can write a very simple equation: d2G-l(l)

a{q)

dliG-\q)G{q)dtlG-1{q)

1

(q). In this approximation

(4)

which differs essentially from any Bethe-Salpeter t y p e equation. Indeed, using a Bethe-Salpeter t y p e equation, we do not change t h e vertex p a r t and end u p with rather b a d properties. Equation (4) is scale invariant, it is 75-invariant, it has many nice symmetry properties and, what is most important, it has a correct behaviour near t h e threshold. T h e gauge is fixed, because we used

Du

<5U

It is an i m p o r t a n t question, what we would get in different gauges. In Feynman gauge we are very lucky: we find an expression which does not depend explicitly on t h e expression for the Green function. Using a different gauge, we would find t h e infrared behaviour of this diagram to be more complicated, and we would not be able t o extract universally the region of small k. We would have integrals over q which are also possible t o use, b u t with t h e necessity t o think a b o u t t h e behaviour near the threshold. We, however, have chosen this gauge; we did not destroy t h e general features and used t h e current conservation which just corresponds t o T^ = d^G"1. Accepting this, we can now ask, what is t h e relation t o t h e renorm group equation. Suppose t h a t we would like t o write t h e renorm group equation in t h e same spirit. Let us take again t h e second derivatives. In this case we would be definitely correct, because we know t h a t it is a logarithmic approximation. In our logarithmic approximation we will do exactly t h e same with t h e only difference t h a t a would be a(q2) . In t h e renorm group equation a is a function of q2. B u t , of course, a in general depends on two momenta: k2 and q2. And in the renorm group equation a t large momenta, in the ultraviolet region, a depends on t h e variable which is t h e largest. Since we are close t o k = 0, this means t h a t here we will have a(q2), and we will recover t h e renorm group equation at large q2. If we solve this equation with a slowly varying a, we will be correct in b o t h t h e threshold region and t h e ultraviolet region. We also have t o formulate an equation for t h e bound states under t h e same assumption. Looking for bound states, we consider scalar and pseudoscalar vertices. This vertex


has to be equal to

+

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Vladimir N Gribov

Here
,

(5)

where A^ = dliG~1G, AM = G<9MG_1. It means that we have two equations in this approximation. We used this approximation just to be constructive and to study what will result if we make this approximation. In principle, solving both equations we will get everything what is necessary: we know G and A^, we have a linear equation for bound states, we can see what is the type of the energy etc. The equation for the bound states has very nice features. It is beautiful from the point of view of the Goldstone theorem in the following sense. Suppose I have some symmetry in my equation, e.g. 75-invariance. Since there is no 75 in equation (5), it is 75-invariant. But of course the boundary condition for G _ 1 at q —• 00 is just G^ 1 = (q — mo), and thus destroys the symmetry. But suppose that mo = 0. In this case there would be symmetry here, which means that the Green function will not be unique, since it can be G~l

+SG'1,

where G _ 1 is some solution and <5G_1 oc 7 5 G - 1 . This means that the variation <5G_1 also is important. What would be the equation for the variation? If we calculate the variation on both sides of equation (4) we obtain 5„ (5G- 1 ) = - (dptfG-^GdpG-1

+6llG-1Gdll{6G-1)

- d^GifiG-^Gd^G-1)

,

so we find that ip = <5G_1 fulfils equation (5) at p = 0. It means that if some symmetry is broken, i.e. if there are multiple solutions of the equation for the Green function, we always will have some solution of the equation for the vertex at p = 0, which is the Goldstone. It is clear, that in the present model we can discuss many questions, use a running coupling a as in Fig. 1 and reproduce the NJL-features without any essentials depending on a cutoff. Before discussing this point further, we will first look for the solution of (4) and discuss the result. Above, we introduced AM = (d^G~1)G, which is a very useful quantity. Since G _ 1 = a^ + b is essentially a 2 x 2 matrix, AM is a U(2) gauge potential: 3 2 G- X =dlt{{d„G-1)GG-1)

= {dpAJG-1

+ A^dpG-1)

= ^A^G'1

(6)

where in the last step we used Eq. (4). Multiplying from the right by G we thus find TX

This means that VliAp =

pApA.^',

and Ap is a pure t/(2)-gauge potential with a condition /3 = 1— ^. Of course, this is just a useful trick. Important is to write down the real equation for the Green function. The most natural thing is to express G _ 1 in the form G-1 =pe$h, where n is a 2 x 2-matrix

n= l Q

It is just easier to use this form for our purpose: we can find an equation for p and an equation for ip. Both are functions of q2: p(q2),
Lectures On Confinement (III)

197

we will find an equation in which £ can be considered as "time", and which is an oscillator equation. In fact there are two oscillators, one for p, the other for ip, and they will satisfy non-linear equations. For (p we find ip + 2 ( 1 + / 3 - )
(8)

This is just an oscillator with damping; a similar equation can be written for p. Important is that that there has to be "energy" conservation in this equation. Indeed, we said that £ plays the role of time; it, however, did not enter the equation explicitly. Thus there has to be a conservation law which, as it is easy to show, leads to 1 +

" ^ =

1+


3 sinh2

V

(9)

We thus can eliminate p altogether, and find the equation for ip Cp + 2\l\-(32

f 3 s i n h 2 | - ^ J v J - 3 s i n h V 3 = 0,

which is an oscillator with damping. Having this in mind is sufficient to understand the structure of the solution. Indeed, what is this

(10)

The perturbative solution is ip close to iir. In this case the first term is zero, the other is proportional to q/q - this corresponds to the massless solution. Since mo is small, we have to have solutions like this at q -> oo. Now we have to find the solution everywhere. Let us first investigate the equation without damping; we get (p — 3 sinh ip = 0. If we go to the Euclidean space, ip = iip, the potential becomes a periodical potential: V> — 3sinV> = 0.

Fig. 4. We have to look for a possible solution for this structure with damping. What does this mean? For the oscillator with damping any solution at £ -4 oo has to be in a minimum, because the energy is decreasing. But if £ is going to —oo, the energy is growing. What could be in this case a normal, reasonable solution? It is almost clear that the only possibility is to put at f - • —oo the "particles" in this oscillator at the maximum, and start to move them slowly; eventually, they will appear inside the well.

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Vladimir N Gribov

There is a most important question, namely: what is the critical coupling in this case? What do we know about an oscillator with damping? If the damping is large enough, all the trajectories will go monotonically to the minimum. If the damping is sufficiently small, the solution will start to oscillate. In order to see when this transition happens, we have to look for the equation just near the minimum ip = IT. With 0 = ix — %p we have for small <j>

4> + 2^/1+

3/32<j> + 3 0 = 0 ,

with fundamental solutions i,h2 = e " 1 ^ where j / l i 2 = - y/l + 3/32 ± s/lp1

- 2.

So we have monotonic behaviour for 3/32 — 2 > 0. On the other hand if /32 < f, i.e.

1-

2 a 3 < ^

, < 1 +

V 3

we will have oscillations before reaching the minimum. The critical angle ipc, which separates the regions where the solution is monotonic and where it oscillates can be shown (see e.g. (4)) to be given by . 2 V> sin — = 2

-P

1 + 3/32 1 - / 3 2 l + v/(l + 3/3 2 )(l-/3 2 )

*

a > critical 7T +

*

C

IT

0

a < critical Fig. 5.

Up to now we have considered a constant coupling a. We know that for q > A the Green function is determined by perturbation theory, which has to match the solutions in the region of for q —> 0 matches the solution smaller q. If p2 > 4 for all
Lectures On Confinement (III)

199

oscillate and m0(mCi) = 0 for some mCi as indicated in Fig.6.

Fig.6 This then is a solution corresponding to broken chiral symmetry.

References 1 V. N. Gribov, Orsay lectures on confinement (I), preprint LPTHE Orsay 92-60 (1993); hepph/9403218 2 V. N. Gribov, Orsay lectures on confinement (II), preprint LPTHE Orsay 94-20 (1994); hepph/9407269 3 Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122 (1965), 345 4 V. N. Gribov, Lund preprint LU 91-7 (1991) 5 V. N. Gribov, QCD at large and short distances, Bonn preprint TK 97-08 (1997), hep-ph/9807224. 6 V. N. Gribov, The theory of quark confinement, Bonn preprint TK 98-09 (1998), hep-ph/9902279

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fo (97 5), a o (9 80) as eye-witnesses of confinement F.E. Close", Yu.L. Dokshitzer^ 1 , V.N. Gribov c ' 2 , V.A. K h o z e < u and M.G. Ryskin 1 a b c d

Rutherford Appleton Lab, Chilton, Didcot, 0X11 OQX, UK Department of Theoretical Physics, University of Lund, Sohegatan 14A, S-22362 Lund, Sweden KFKI Research Institute for Particle and Nuclear Physics, Konkoly Thege ut 29, Budapest XII, Hungary Center for Particle Theory, University of Durham, Durham DH1 3LE, UK

Received 30 September 1993; revised manuscript received 23 October 1993 Editor: P.V. Landshoff We investigate some phenomenological consequences of an idea [see V.N. Gribov, Lund preprint LU-TP 91-7 (March 1991), and in preparation] that the f0(975) and a0(980) play a special role in the dynamics of quark confinement. 1. Super-critical confinement and "novel" hadrons Recently one of us [ 1 ] has proposed a theory of confinement in QCD in which light quarks interact strongly enough that the total energy of quark antiquark pairs becomes less than zero. This results in the appearance of a new type of "condensate" consisting of strongly interacting pairs of light flavours q, ~q with positive kinetic energy but negative total energy. In the present paper we suggest some phenomenological tests of the theory (section 2 et seq.). First we briefly summarise the ideas in order to motivate and define terms for the subsequent phenomenology. For details of the theory we refer to refs. [1,2]. It is the existence of quarks with very small (current) mass m < X ~ 1 GeV that are essential to the theory. These lead to a radical change of the perturbative vacuum in the region between \jk and 1/w, the Compton wavelength of the light quarks, analogous to the phenomenon of "super-charged" ions in QED [3]. In QED, when the electric charge of a nucleus exceeds some critical value Z > 1 8 0 ( Z > 1 3 7 f o r a point-like charge), light fermions in the vacuum start to "fall on the centre" creating stationary states with negative electron energy, e < -m. This causes insta1 2

Permanent address: Nuclear Physics Institute, Gatchina, St. Petersburg 188350, Russia. Permanent address: Landau Institute, Moscow, Russia.

bility of the perturbative vacuum. One consequence is that nuclei with Z > Zcrit cannot survive free, and they decay: Z —> (Z - 1) + e+. In QCD the Coulomb-like attraction between fermions leads to a similar falling on the centre. It is important to notice that this happens not only for a light quark in an external field of a heavy quark (as in the above QED example) but for interaction between light quarks as well. Any coloured particle in QCD acquires a spatial colour charge distribution due to gluonic vacuum polarisation. The "super-critical" phenomena develop when the size of the volume ro = A -1 in which the total charge as(X) exceeds some critical value a crit(~ 0.6), is much smaller than the light quark Compton wavelength w _ 1 ~ (5-10 MeV) - 1 ; i.e. the parameter mjk < 1. Contrary to the QED case where the nuclear charge would decrease by one unit, in the QCD context this results in producing a colourless bound state with negative total energy which causes instability of any coloured state. The existence in the theory of the small parameter m/1 « 1 makes it possible to construct a nonlinear equation for the quark Green function [ 1 ] that is a relativistic analogue of the gap equations in the theory of super-conductivity and the Nambu-JonaLasino model. Starting from as/n > aCTn/n ~ 0.2 this equation has two types of solutions. The first solution corre-

0370-2693/93/$ 06.00 © 1993-Elsevier Science Publishers B.V. All rights reserved SSDI0370-2693(93)E1376-9 Reprinted from Physics Letters B319, 291 - 2 9 9 (1993), Close et al, with permission from Elsevier Science.

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sponds to a fast increase of fermion masses and resembles the properties of the well-known spontaneous symmetry breaking solutions (for m = 0). The second solution corresponds to a new type of phase transition and has the following features. In Dirac theory fermions can have both positive or negative energies q0 > e = (m\ + q2)l/2 and qQ ^ - e ; in Dirac's language, the former zone is empty and the latter is occupied. The energy range - e < qo < £ may be looked upon, by analogy with the solid state physics, as the forbidden zone: Normal excitations correspond to the creation of qq~ pairs with energy > 2m\. As far as positive energy quarks are concerned, the fact that the lower zone 0 ^ e empty, implies that the mean number of quarks (and antiquarks) in the vacuum state is zero. In the solution corresponding to the new phase transition, the latter statement is no longer true. In the vacuum, quarks with positive kinetic energy are present but are clustered into "super-bound" qq pairs with negative total energy. Thus a picture emerges of the vacuum as a conductor (instead of that of "Dirac's insulator"); there is also a new mass scale HF ~ A which reflects the position of the "Fermi surface" separating the occupied and empty zones. The former parameter m, now describes the effective mass of quark excitations in the upper zone and becomes complex (in a conductor single electron excitations decay with time). The specific relation between Rem! > /i f and the corresponding complex effective quark mass in the lower zone, Re m2 5 HF, depends crucially on the dynamics near the Fermi surface and on current quark masses (if any): this must be studied quantitatively but lies beyond the present work. This form of the quark spectrum suggests the possibility of new types of vacuum excitations which correspond to pair creation in the new zones below the Fermi surface (see fig. 1) and manifested as / = 0,1 nonstrange mesons. These excitations correspond to quarks and antiquarks with negative kinetic energy that are interacting repulsively leading to positive total energy. (We shall denote these quark states by ?(_> in contrast to the conventional positive energy states, q(+>.) It is the existence of the ?(_) modes that causes instability of 292

23 December 1993

Fig. 1. Structure of the quark vacuum in the light quark confinement theory of refs. [1,2]. Upper new zone is filled up with positive kinetic energy quarks bound into colourless pairs with negative total energy. Excitation of such a pair corresponds to creation of a VS meson state. Dashed lines show the "Fermi surface" that separates the zones with standard and inverse population. isolated quarks and results in colour "neutralisation" through "decay" into hadrons; g<±) —• Af(+) + #(_>. Normal excitations corresponding to creation of qq~ pairs with mass-scale m\ also have to exist, of course. Thus in the theory there are two types of hadrons ("normal" and "novel") and since the new condensate consists only of u,d quark flavours, SU(3) flavour symmetry will be badly broken for these novel states. In reality the normal hadrons will have some admixture of the "novel" (negative kinetic energy) quark states and vice versa. For baryons one expects the superposition of the two types of state corresponding to three quarks with large mass, m\ which could be described by the nonrelativistic quark model and another state which corresponds to three negative energy quarks with repulsion. The latter is essentially an excitation of the condensate that has a chance to be described in terms of self-consistent pion field as in the Skyrmion model. A notable feature of the new states is that as their constituents have negative kinetic energy and the interaction that elevates the meson mass to a positive value must be repulsive, one expects that the system's mass is large for small systems and small for spatially extended systems. Here "large" and "small" are with respect to 0(X ~ 1 GeV) where a (A) = acrit =

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O (1). The lowest mass strongly interacting states of the new type are expected to be scalars and pseudoscalars [2]. The purpose of this paper is to abstract some phenomenological consequences of the above "axioms" [1,2]. 2. /0(975) and a0(980) as "vacuum scalars" Given the above qualitative features, it is tempting to identify the/o (975) anda0(980) as compact nearly degenerate systems of (negative kinetic energy) u and d flavours: Jo = (uu + dd) j\fl,

a0 = (tiu-dd)

/%/2.

As we shall discuss below, it is possible that the corresponding pseudoscalar pair consists of 7r( 1300) and one of the nearby n states. The light pseudoscalars, it and n, are expected to be mainly normal states; any admixture of negative energy u and d pairs in the light pseudoscalars will presumably be at long range. We expect that the "vacuum scalars" (VS) Jo(975) and ao(980) are compact states with a size r

vs' ~ ^ ~ 1 GeV > nti, ms > mi,

where m\,ms are respectively the masses of the constituent positive energy u.^ands flavours. In such circumstances, decays of Jo which require the creation of pairs of light flavours (mi) or strange quarks (ms) have a good chance to be free from any "strangeness suppression". Empirically the coupling to KK. and nn for the Jo appear to be comparable [4,5]. The hadronic couplings of the ao are currently unclear, see e.g. ref. [6]. 2.1. Coupling to "normal" hadrons 2.1.1. J"o (975) decay width The suppressed widths of the Jo and ao have been a long-standing problem for the ^o 09 interpretation of these states in the quark model. For S-wave decays of ¥0 qq into nn one would expect [7] a width of 500-1000 MeVin contrast to that of the J"0(975), for example, whose width is « 50 MeV. A recent analysis of nn and KK processes [4] finds that the Jb( 1400) of the PDG [8] is better described

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23 December 1993

as "a very broad f0 (1000) of width around 700 MeV", which is thereby a good candidate for the conventional ¥0 state; in addition this state is strongly decoupled from the narrow Jo (975) even though they have superficially the same quantum numbers # 1 . The yy width of the /(IOOO/1400) state is consistent with it being the ¥0 [9,5]. Both the suppressed width of the Jo (975) and the negligible coupling to the /(1000/1400) arise rather naturally in our picture of these states. The decay Jo(975) -» nn occurs either by the "direct" transition from the compact Jo [#<-> ] system to the long range 0(_) component in the n wavefunction, or by "tunnelling" to the dominant #(+> component of the n. Each of these dynamically causes a suppressed width and nugatory mixing with the / (1000/1400) 'Po state. In the "direct" transitions to the ?<_) component, the small width is due to the small overlap between the Jo (975) compact C(l GeV-1) wavefunction and that of the more extended pseudoscalar n. The alternative dynamics is that the transition is from the
r(/ 0 (975)->*B)~(2d«£j

2

r(ay^PK)

K±r{ai^pn),

(1)

where we have made a measure of the ratio of widths of the "novel" and "normal" states by a comparison of Jo (975) —* nn and the similar S wave decay of the "normal" P-state meson a\ —* pn; the power of as has been taken from the leading two-gluon channel, p. is some hadronic reference mass scale on dimensional grounds, and we have used the empirical values for the widths in the final equation. If the /(IOOO/1400) were established as the "normal" 'Po state, then the Jb(1000/i4oo) —• nn would be a more direct compar#1

Hereafter we denote this state as /(IOOO/1400) . 293

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ison; note that this would give essentially the same result as eq. (1).

no reported sighting of n + / 0 (975). To test this we suggest systematic study of

2.1.2. / 0 (975) production in decays of "normal" hadrons At leading order the f0 (975), / (1000/MOO) are and #<+) excitations respectively, and hence are orthogonal states. Wavefunction mixing contains the helicity suppression factor mijm\ and an extra power of small as (X) in tunnelling and thus should be smaller than would be expected were the /o(975) a glueball, for example. Hence, extreme decoupling may be more natural in the present picture. However, VS do decay (e.g. f0(915) —> mi) with width of order tens of MeV. Taking this number as a quantitative estimate for the phenomenological magnitude of the "normal"/"novel" (ad)mixture, we anticipate that such a width should be typical for the VS "appearance" in the decays of "normal" hadrons. If A,B are such hadrons, then by analogy with eq. (1) we expect that (apart from phase space effects)

/o(975)//o(1000/i4oo) + meson(s)

r U ^ 5 + / 0 (975)) « ^ r U ^ B

+ / 0 (1000/i4oo)).

Thus we would expect qualitatively that /o(975) and flo(980) will be suppressed relative to /(IOOO/1400) say in processes where the decaying hadron has a "normal" hadronic width (of order 150 MeV or more). At the same time, in processes where selection rules and/or phase space suppress the total width to be of order tens of MeV or less, the a 0 (980) and / 0 (975) may show up (the "normal competition" has been removed). A survey of the Particle Data tables shows consistency with this picture (though this may be due to lack of search as much as lack of signal for the fo (975)). As a first example we note the conventional quark model state, 7C2(1670), whose width is 250 MeV: the n + /0(IOOO/1400) channel* 2 is seen with a branching ratio at the level of ten percent whereas there is *2 Recall that the PDG call it 1400 but we use the new Morgan-Pennington numbers. 294

in the decay products of "conventional" T(/ 0 (975) -+ nn) (even after allowing for the reduced phase space). First, the smaller momentum transferred to the final state in the ao case causes less dramatic overlap suppression between the nn be large enough for the yy width to be small (as required by eq. (9), see below). 2.2. The fo-ao mass splitting The fo-ao mass splitting due to (helicity violating) 04 —"• gg annihilation can be estimated as us, \ **if\ M(ao)-M(fo)

[aHM)m2]2 JJ^

.

... (2)

The suppression of a ( _ ( —•#(+) transitions expressed in eq. (1) implies that

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r 2,,,,

,2

2

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r ( / 0 ( 9 7 5 ) ->nn) i (/o(1000/1400) —• 7I7t)

If the parameter /i is naturally associated with the constituent mass scale mi ~ \ GeV then we would have M(a0)-M(fQ)&O(5M€V)

(4)

in accord with the data [8]. However, one should exercise some caution here as numerical factors in a detailed modelling could change these conclusions by an order of magnitude in either direction. The actual mass difference for ao-/o is rather poorly determined empirically. At this stage we note only that the picture appears to have some consistency. 2.3. Coupling to photons The yy couplings to fo,flopromise to be rather sharp probes of the mesons' substructure. Their yy widths are up to an order of magnitude smaller than expected for conventional ¥0(39) states [10,9]. The yy branching ratio is similar to that expected for the ¥0 states, suggesting that a common suppression is at work for the yy and total widths of the fo (975). This is natural in the present picture. In the quark model these decays violate helicity conservation in the zero mass limit. If we extract the "nonstrange" component of the n' width into yy (approximately 8/9 of the total) then the suppression of the helicity violating r(0+ —> yy) decay can be used to estimate the (smaller) "novel" mass m^. ryy (n'„) « 4 keV;

_

_r„if0)

mj

ryy{n')

/W m22\

r„ (fo) <, 670 eV;

< 1 ~ 6'

(5a) (5b)

Note that the essential dynamics here is exactly as for the case of the Jo (975) —• nn suppression discussed above. A detailed comparison is clearly rather naive as there are S and P-wave factors that will influence the yy example in contrast to the S-wave dominated hadronic decays eq. (1). Note that this implies the welcome result that m^ < m\, in particular, if we identify m\ ss 350 MeV as the scale of constituent quark masses then m2 S 150 MeV. However, at this stage we emphasize that more detailed modelling is required before too strong conclusions are drawn.

23 December 1993

The most immediate test of the entire hypothesis and one that is free from detailed dynamical assumptions (such as mass scales) comes from the ratio of the fo and ao amplitudes to yy as this probes primarily the squared charges of the mesons' constituents as well as their relative phases. In the present theory where the fo,ao are compact, the yy decays are short distance dominated and the relevant couplings are directly to the u, d flavoured quarks; thus one expects

r(fo^yy) / ' ( a o - * yy)

25 9 '

(6)

This is to be contrasted with the hypothesis [11] that the fo, ao are diffuse KK molecular states in which case the yy coupling will be dominated by the KK loop [12] and the widths be equal. The values quoted by the Particle Data Group [8] have large uncertainties and do not discriminate among models, r(f0^yy)

= 0.30 ±0.10 keV,

r(aQ^yy)

= 0A9±l\l

keV.

(7)

However, a recent analysis by Morgan and Pennington [4] has modified these numbers. The current world average is [8] r(a0^yy)B(a0-^ri7t) r(fo^yy)

= 0.24±0.08 keV;

= 0.56 ±0.11 keV,

(8)

and the absolute magnitude of the ao width and the ratio of the two depends on the poorly determined B(ao —• nn). This has often been approximated by unity (as in the numbers cited above in eq. (7)) whereas the value could be below 50 percent. Given that B(a0 —> nn) < 1 we can only limit the ratio (6) r(fo^yy) < 21 ± 4 r(ao^yy) " 9±3

(9)

Information on the ao —• nn couplings from LEAR [13], Fermilab [14] and from the impending (fr -y yao -> ynn at DAONE [15,12] will be important in helping to clarify this important datum. We anticipate [12] that the branching ratio for production of (j> —» yao in the present picture will be of order 10~6 due to the quark line disconnected production process. It is possible that the -» yfo could be somewhat larger due to its spatially compact nature 295

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a manner analogous to the >/(1440), namely by the which may enable enhanced production from the vacchain 7r(1300) —• 7t/o(975) and with a comparable uum (see also the subsection on "Onium decays"). In width (some 50 MeV). Isolating a nf> contribution any event, the branching ratios and the ratio of the from the nnn continuum or in KKn is a challenge. branching ratios for <j> —> yfo/4> —* V
r(r,1 (960)

^yy)ir(f0^yy)

(see eq. (5)) suggests that the ?/'(960) is not dominantly a compact /(1440)" on page VII.42 of ref. [8].) As for the photon width, we expect that (cf. eq. (5)) r(n(\440)^yy)

< 1 keV«r(/0(975) -»yy),

which is consistent with the present experimental limit. If the ft (1300) is the isovector partner there is the problem of explaining its broad width. By analogy with the /o (1000/Moo)-/0(975) and possibly also the ?/(1440) system we suggest therefore that the broad illdefined structure called 7r (1300) may in fact be a combination of a conventional # ( + ) state and a "novel" ?(_) state which are strongly decoupled from one another. We expect that this latter state decays in 296

This picture of the fo, do as compact "vacuum scalars" suggests conditions where their production and isolation may be favourable. The spatial separation of coloured objects produced in a hard interaction causes a collective negativeenergy quark current built up from consecutive decays « (+ ) ->Ml+) +?<_),

(10a)

«(_) ->Mi+) +?(_)«<_,?(_).

(10b)

It is the existence of the quark modes in the theory of [ 1 ] that causes instability of isolated quarks and results in colour neutralisation (or "confinement"). Within the sequence of the (10b) decays one could expect "vacuum scalars" to show up. To illustrate the idea here, consider the qq~ produced in e+e~ annihilation. As the coloured qq move apart from one another, a colour "neutralisation" current flows between them. Initially this is due to (perturbative) gluons, then at larger distances as the force increases a negative-energy quark current develops. For inclusive production at high energy the final state is dominated by soft production of many particles for which traditional ideas apply; it is in the (near-) exclusive processes, which are relatively improbable at high energies and where abnormal energy goes into the production of relatively few particles, that the negative energy quark current flows and consequent production of the "vacuum scalars" fo, cio is expected to become more important. The essential sequence is that the
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- "Next-to-the-leader" topology. (i)e+e~ annihilation: (a) Measure the relative yield of (2n) resonances (e.g., p,fo,fi) next to the leading particle in jets as a function of relative rapidity An with respect to, say, thrust axis; (b) ibid, with respect to leading D, B triggered in heavy quark jets in the Z° peak. With An increasing one could expect an increase of the ratios

fo/P,

h/fi,

etc.

(ii) large-pi hadron-hadron interactions: Measure the relative amount of /o production as a function of Pi of the leading hadron. - "Isolated meson" topology. In e+e~ annihilation, yy —»jets, pp annihilation and Onium decays measure the relative production of vacuum-scalar mesons that are (i) isolated in rapidity from other hadrons, (e.g. central production) as a function of the size of "rapidity window". We note the recent report of f0, fa, p production in Z° decays at LEP [16] and the first h seen at DESY [17] and urge that these events be analysed as a function of the rapidity window. (ii) as a function of total event multiplicity (less particles —> larger "isolation" —> larger yield of VS). (iii) as a function of total energy (e.g., 5 of pp annihilation at LEAR [13] or Fermilab [14]). (iv) in e+e~ annihilation immediately above flavour threshold at a r-Charm-Factory or a Bfactory. As 5 increases we expect that DDfo, DDCIQ and BBfo, BB~a0 final states will be significant among low multiplicity events. Measure the relative importance of heavy hadrons accompanied by OQ, h, fi, p in such experiments. - Proton-antiproton annihilation at rest. It is reasonable to expect that the proton will have a component containing three quarks with negative kinetic energy [2]. If this is true, then pp annihilation at rest can contain three quarks and three antiquarks with negative energy. But these correspond to the excitation of the new condensate and in such circumstances it will be natural that the products are nn and also the scalar states h, OQ. We note that these

23 December 1993

states are rather clearly seen in the LEAR pp final states [13]. It will be interesting to study their production rate as a function of energy and, at higher energies, rapidity. - Short distance annihilation: Onium decays. As the vacuum scalars are expected to be spatially compact relative to "normal" hadrons, their production may be relatively enhanced in processes that are short distance dominated, such as y/ and T decays. The h (975) was seen in i// decay as a resonance peak recoiling against the co and cj> in W ~* cofo(975) and V -> /o(975). ARGUS reported recently [17] the first measurement of h (975) production in the Tenergy range: a significant inclusive production rate has been found, namely, /o(975) _ f (7.2 ± 1.8)% continuum, />°(770) ~ \ (11.7 ±3.0)% r . At the Z° the corresponding ratio measured by DELPHI in the x p > 0.05 momentum range [16], is /o(975) _ 0.10±0.04 />°(770) ~ 0.83 ± 0 . 1 4 ' We urge study of this ratio as a function of rapidity and the separation from other hadrons in phase space. It is noticeable that the ^(1440) system is prominent in if/ —• y + X (which is why we were attracted to this as a possible novel candidate). If these ideas are right, then we anticipate that the S-wave decay of the t}c into a pair of novel states >Z e -/o(975) + »7(1440)

(11a)

may be significant. The rate for this should dominate, for example, that for >/ c ^/o(1000/i40o)+ >/(1440).

(lib)

Interesting combinations of novel states might be anticipated in the decays of the x states and should be searched for at BEBC or at future r-Charm Factories. S-wave decays of x(0 + ) should be examined for the presence of/ 0 (975) + / 0 (975) and for ^(1440) + >/(1440). In general, * ( 7 + ) -» /o(975) + Xj is a promising pathway to studying the nature of light spectroscopy with spiny and can be used to test the 297

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flavour content of the fo (975). For example, this may be probed in the relative strengths of Z-+/o/2(1270):aofl2(1320):/a/2(1525).

(12)

We anticipate that the first pair are similar in rate and the latter is relatively supressed * 3 .

4. Other dynamical implications - In the usual approach to hadronization, gluons from QCD cascades are highly virtual - of 0 ( l GeV2) due to the sequence g —> qq —» hadrons. This is especially true for the production of heavy particles such as baryons. This causes a conceptual problem for understanding, from "first principles", the yield of massive hadrons and the structure of their inclusive momentum spectra. In the present picture, by contrast, there is no need for extreme gluon virtuality due to the existence of the #(_) modes: g —» M( + >g , (_)^ ( _ ) . As a result it is rather natural to have a direct correspondence between the distribution of partons and hadrons and thereby give a theoretical basis for the phenomenologically successful "local parton hadron duality" [19]. - The present picture also may offer the promise of explaining the observed similarity of K and A inclusive log xp distributions [18]. The negative-energy "strange" vacuum current is absent. Because of that we would expect the doublestrange ct> and E production to be strongly suppressed (stronger than the "single strangeness suppression squared") and their momentum spectra to be stiffened as compared to K,A. - Nuclear targets may help identify the dynamical structure of the scalars. A loosely bound KK "molecular" a0, f0 will propagate very differently in nuclei compared to a compact vacuum scalar. A compact state has a good probability to propagate and depart the nucleus intact with the result that equal numbers of K+, K~ will be observed from the decay fo —• K+K~. For a loosely bound KK molecule, by contrast, the K~ has a high chance of disappearance through K~p —• nA whereas the K+ continues to propagate in essentially the same direction as the * 3 Note that the fa (1525) is a "normal" (si) tensor meson. 298

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initial scalar. The A dependence and other kinematic dependences of scalar production on nuclei should be able to distinguish between dynamical structures rather clearly [20,21 ]. For example, it would be interesting to study energy and atomic number dependence of the effective width of ao, fo mesons produced on nuclear targets. As is well known in atomic spectroscopy and has been discussed by Bugg in a particle physics context [ 22 ], the lifetime of a particle resonance propagating through a nucleus is reduced by collisions, and, consequently, its width increases. Indeed, interactions between the resonance and the nucleons inside nuclear matter leads to absorption of the resonance and diminishes its lifetime. If the absorptive cross section is denoted by a and the nucleon density p, then the inverse lifetime inside a nucleus (neglecting the natural width of a resonance, i.e., in the r —* 0 approximation) would be \/XN = ATlv = ap • c. In the vacuum the inverse lifetime of a fast hadron is 1/r = Fjy, with y = E/mKS the Lorentz factor. So the effective width of a resonance produced on a heavy nuclear target should be reff = r + AT* • y = r + - f -

ape.

If ao and fo mesons are KK "molecules", the absorptive cross section would be > 30 mb and at initial energy is as 5 GeV one would expect a huge effective width ~ 500 MeV ; at higher energies the ref! will be even bigger. In practice, the effect will be not so strongly pronounced because a part of the resonance production occurs on the periphery of the nucleus where the nucleon density p is much smaller. Only when the resonance is produced in this way will it actually be observed as a peak in the mass distribution. The Adependence of the cross section in this peak will be rather weak: Al/* instead of the conventional A11* for asymptotically large nuclei (A —» oo). On the other hand, if ao and fo as VS are small size objects, their cross sections should be of the order of 1-4 mb (a = lit (1 GeV) ~ 2 ). In such a case one would observe normal nuclear dependence and only a small increase of VS-width with energy. - The present picture could provide a clue for understanding the phenomenological success of the so called "frozen couplant" approach, based on the idea that

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the effective QCD interaction strength (strong coupling) "freezes" at low energies (for a recent review see [23]). It is interesting to note that the optimized low energy value as/n = 0.26 (below 300 MeV) found within the framework of this approach is in accord with acrjt/7T.

5. Outlook We have limited ourselves to a first survey of phenomenological consequences of the idea [1,2] that the ./o(975) andao(980) play a special role in the dynamics of confinement. At this stage we have restricted attention to the more well defined tests (e.g. yy widths and ratios) and currently active experiments (such as Z° decays and pp at LEAR). The advent of r-Charm or B factories would open up new possibilities for developing and testing these primitive ideas both above the heavy flavour threshold and in the i// or T decays.

Acknowledgement This work was supported in part by European Economic Community contract number CHRX-CT920026. References [1]V.N. Gribov, Lund preprint LU-TP 91-7 (March 1991). [2] V.N. Gribov, in preparation. [3] I. Pomeranchuk and Ya. Smorodinsky, Journ. Fiz. USSR 9 (1945) 97;

23 December 1993

W. Pieper and W. Greiner, Z. Phys. 218 (1969) 327; Ya.B. Zeldovich and V.S. Popov, Uspekhi Fiz. Nauk 105 (1971) 4. [4] D. Morgan and M.R. Pennington, Rutherford Lab report RAL-92-070, Phys Rev D, in press. [ 5 ] D. Morgan and M.R. Pennington, Z. Phys. C 48 (1990) 623. [6] R.N. Cahn and P.V. Landshoff, Nucl. Phys. B 266 (1986) 451. [7] S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985) 189. [8] Particle Data Group, Phys. Rev. D 45 (1992) SI. [9] Z.P. Li, F.E. Close and T. Barnes, Phys. Rev. D 43 (1991) 2161; F.E. Close and Z.P. Li, Z. Phys. C 54 (1992) 147. [10] T. Barnes, Phys. Lett. B 165 (1985) 434. [11] J. Weinstein and N. Isgur, Phys. Rev. D 27 (1983) 588; D 41 (1990) 2236. [12] F.E. Close, N. Isgur and S. Kumano, Nucl. Phys. B 389 (1993) 513. [13] Crystal Barrel Collab. (at LEAR), Phys. Lett. B 291 (1992) 347; B 294 (1992) 451. [14] G. Smith, Proc. Dallas HEP Conf. (1992). [15] F.E. Close, in: Proc. Workshop on Physics and Detectors for DAFNE (Frascati, April 1991) ed. G. Pancheri, p. 310 [16] DELPHI Collab., CERN-PPE/92-183 (October 1992). [17] ARGUS Collab., DESY 92 174 (December 1992). [18] OPAL Collab., CERN-PPE/92-118 (July 1992). [19]Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller and S.I. Troyan, Basics of Perturbative QCD (Editions Frontieres, Gif-sur-Yvette, 1991) p. 175; Yu.L. Dokshitzer, S.I. Troyan and V.A. Khoze, Z. Phys. C 55 (1992) 107. [20] W.R. Gibbs and W.B. Kaufmann, in: Proc. Physics with Light Mesons, Los Alamos report LA-11184-C (1987) 28; C. Alexander and T. Sato, Phys. Rev. C 36 (1987) 1732. [21] H. Lipkin, Phys. Lett. B 124 (1983) 509. [22] D.V. Bugg, Nucl. Phys. B 88 (1975) 381. [23] A.C. Mattingly and P.M. Stevenson, preprint DEFG05-92ER40717-7.

299

213 22 September 1994

= PHYSICS LETTERS B

Physics Letters B 336 (1994) 243-247

—————______

Higgs and top quark masses in the standard model without elementary Higgs boson V.N. Gribov lnstitut fur Theorelische Kernphysik, Universitdt Bonn, Nufiallee 14-16, 53115 Bonn, Germany and Research Institute for Particle and Nuclear Physics, Budapest, Hungary and L.D. Landau Institute for Theoretical Physics, Moscow, Russia Received 5 July 1994 Editor: P.V. Landshoff

Abstract In this short note I present a simple calculation of the top quark and Higgs masses, based on the idea that in the standard model without elementary Higgs the fact that the U( 1 )y coupling becomes of the order of unity at the Landau scale A leads to spontaneous symmetry breaking and generation of masses.

1. Introduction The basis of this paper is the idea that in the standard model without elementary Higgs particle the U(\)y coupling becomes of the order of unity at the Landau scale A, thus resulting in a spontaneous symmetry breaking and generation of masses. In this case the longitudinal components of the W, Z° and the Higgs have to be bound states similar to the deuteron in the zero-radius limit of nuclear forces. The usual SU(2) and SU(3) interactions would act like the electromagnetic interaction between nucleons in the deuteron, and the situation could be treated perturbatively. The existence of the longitudinal components of the W, Z° is connected to the fact that the generation of quark and lepton masses leads to non-conservation of the two components of the SU(2) left-handed current and the left-handed component of the t / ( l ) current. The existence of the Higgs is due to the degeneracy of states with positive and negative parity at short distances. In a more detailed paper we will show that

there is a good chance for such a theory to exist.

2. The W and Z° masses We start with the calculation of the W mass. The fermionic part of the polarization operator of the SU(2) bosons is given by

^

4 ZfJ

(2TT)4;

xiT(ray^Gj(q)^TbGj(q-p))

(1)

•vOwhere y^ = y^ • \ (1 — 75) and the sum extends over all fermionic generations. In the absence of quark and lepton masses n# = n£ = ( w 2

- P*PV) n (p 2 )

*«»

Elsevier Science B.V. SSDI0370-2693(94)00942-2

Reprinted from Physics Letters B336, 243 - 247 (1994), Gribov, with permission from Elsevier Science.

(2)

214

244

V.N. Gribov / Physics Letters B 336 (1994) 243-247

due to current conservation. For massive fermions 1

Gj =

(3)

can therefore neglect it when integrating up to the Landau scale. Taking into account only the contribution of the heaviest quark - the top quark - and using the as dependence on q2,

Here mj is a function of T3 and can be written as mj = my

i ( l + r 3 ) + mu • ±(1 - r 3 ) .

(4)

Because of (3), the form (2) is no longer valid. Instead, in second order in m, Yl^ can be written for g 3> m and p2 —> 0 as nab

fXV

1

xtr(r^my

+ 477-

2 r

mj«;
(11)

<*,(A 2 )

Kv=Kv+sig^8ab^mi{\

as(m2)

2

'2g

(12)

(lirfi

^ ) - ?ab

= n"* + Anafc

(5)

where g2 = 4Tras(m2). (Sab - <5a3<S«) we find

(6)

replacing Sab

~~4

by

(13) as(X2) as(mf)

v

n^„ + gfivSab- *&

After

••gi-

Applying (4),

n?

ln

and (8) we obtain

d4q

s\

nab

=

{XV

as(q2)=a;l(m2)

2g2s

(14)

1677-2

5/* 1>

„2

K + m j, + I^2x W'

(7)

™Z

The lower limit in (7) is defined by the physical masses of quarks and leptons. At first sight this integral looks divergent. But if we take into account the dependence of m on q2 the integral turns out to be convergent for the main contribution, given by the top quark mass, if the upper limit is the Landau scale A. The dependence of the masses on q2 is well-known in the standard model (see for example [ 1]). It is defined by the SU(3) and U{\) interactions only and reads (for three generations)

miaiq2)

=m]q

as(q2) as(m^q)

(8)

In order to calculate the Z° mass, we have to consider not only the transition between the zero-components of W, given by AII /li j,(2,2). We also have to consider the transitions between Wo and the U( 1) boson - A n ^ ( l , 2 ) and AII M „(2,1) - and between U(l) bosons, AIl^,, ( 1 , 1 ) . We can do this very easily when noticing that the U( 1) current has the structure •y

R YR

Qy„

„L

T

3

(15)

where Q is the operator of electric charge and Qy^ is the conserved electric current. Therefore, because the non-conserved part A I I ^ ( 2 , 2 ) is equal (16)

An ^ ( 2 , 2 ) = ^ then

2

miq(q

)

=miq

mi(q ) = m ,

as(q ) as(tn2iq)

a'(m2) a'(q2)

(9)

(10)

The dependence on the [/(1) coupling a' is very slow and becomes essential only near the Landau scale. We

A n „ „ ( l , 2 ) = A n ™ ( 2 , 1 ) = -gig1 AIV(l,l)=s/z-

-

(17)

(18)

and this will reproduce the usual features of Z° and y with v/2= 123.11 GeV.

215

245

V.JV. Gribov / Physics Letters B 336 (1994) 243-247

Up to now we did not discuss Goldstone particles. As a result our An M „ is not transverse. But if mass generation is due to spontaneous symmetry breaking, currents have to be conserved via the existence of Goldstone particles and so A I l ^ has to be transverse. In order to find the correct expression for A I l ^ , we have to add to the left-handed SU(2) current the Goldstone contribution:

(28)

£3 = 175 T3 — •

Now it is easy to write down the correct expression for A I l ^ . We have to replace both vertices in the diagram of formula (1) by the two terms in (19) or (20). As a result AIl^,, would be represented by four diagrams: An.

I

X 91

••yu •

/

<

abt

(19)

+ -+ •

92

Pn

(20)

+

(29)

-•

Here gb denotes the Goldstone-fermion coupling and fab is the current-Goldstone transition amplitude. These couplings are defined by the Ward identity for the new current

'wnQ

/ r ; = i(i+r5)yG-

AIV = f

G-\q2)

id

'(?i)

: 7s) .-,

(21)

For G having the form (3) the Ward identity for p = 0 gives us a

i[r

,m]

+ \y5{Ta,m}

= -~fab

(22)

: / {2ir)Ai

Kiy^-GjiGj)

facfd, = fiSab

(26)

2

with f = v /4, as defined in (14) from (7) and (12). Further we can simplify ga by choosing an appropriate basis in the a£>-12-plane, {ra,m})

1 2/'

Py.Pv

(30)

The last two diagrams have to cancel each other. The cancellation condition requires that the new loop in the fourth term of (29) fulfills 'ah

?a(

)gb=P

Sab,

(31)

3. The Higgs Mass

(25)

+ y5

g^

(24)

ifab = /<5fl3<5(,3 + fab

ga = i([ra,m]

terms give the correct expression for AII^,,

(23)

Combining (22) and (24) and calculating the integral (24) in the same way as we did in the calculation of AIl^,,, one can derive

~ <5a3<5«)

j — which actually defines it. The first two

which is a natural condition for the self-energy 2 G of a Goldstone type bound state. It is this cancellation condition that will enable us to calculate the Higgs mass.

For fab being a transition amplitude and because we suppose the Goldstone to be a fermionic bound state

d4q

where in turn we have replaced -VA/V^— — by

(27)

The existence of a Higgs particle is suggested by the fact that at short distances, where all particle masses are zero, bound states have to be degenerate in parity. Therefore, if a pseudoscalar Goldstone particle exists, according to (28), there also has to be a scalar bound state which we will call Higgs particle. Due to large distance effects the mass of this state would not be zero. Because of the degeneracy at short distances, the self-energies of these two states would have the same structure: Xs = j

(32)

216

246

V.N. Gribov / Physics Utters B 336 (1994) 243-247

~Zps =

iy5gT3(

J iysgT3

(33)

or A2

with

2

m

The mass of the scalar state is defined by the condition ls(p2=m2H)=0.

m\ =

1S(P2)-ZPS(P2)+P2=0

(36)

ls(p2) = -Y/[7~TMsGjgGj)

~?/

(37)

(2TT) I (Z7T)

d4q -—T-MiysgGjiysgGj) (38)

:tr(g{y5,Gj}y5gGj) 4

(39)

(2TT) (

2-3 f2 • 1677-2

(m\q +m\q+\m\)

. (40)

2

Using the expression for f (see (14), (7) and (12)) we can write (40) in the form m2H=A

(41)

Neglecting the contributions of all quarks and leptons except the top quark we have

2

Jjrt^l

(44)

Formulas (14) and (44) are the predictions we mentioned in the beginning of the paper. The quantity as(A2) can be expressed through the electromagnetic coupling ae(m2). From (a'(A2))-1 = ( a ' ( m 2 ) ) - 1 - A l n ^ L :

In A22 m

(45)

(42)

(46)

3TT 1

5

a'(m2)

Using (11) and the relation ae(m2) (9w - Weinberg angle), we have

The integral on the right-hand side is convergent in the same sense as we discussed before and we find m2H =

« S (A 2 ) .ocs(mf)

8m? 38y

in order to have a'(A 2 ) ~ 1

(Z7T) I

p2=XPS(p2)-Xs(p2)

"?/

(43)

37r mf

so that

dAq

4 2

Using (8) and (11) and neglecting the a' dependence we obtain

(35)

This condition can be written in another form, if we know that the pseudoscalar Goldstone state satisfies (31):

lps(p2)

f dq2

(34)

#Higgs = 8 = ^ 7 •

•~ J

3

= a'(m2)

cos 2 6\y

<*s(m2) _ 1 21 rnQ2 a as(m2) 2 TT^T 1 + TO COS 0W ae(m—2)= - . « S (A )

(47)

The concrete values for m, and m# depend essentially on the value of the strong coupling at the top quark scale, which is not very well known. For example, for as(m2) =0.11 (48)

m,=215GeV

(49)

mH = 255 GeV 2

in contradiction to Fermilab data, and for as{m ) = 0.09 m, = 201 GeV

(50)

mH = 252 GeV.

(51)

Accepting the Fermilab result m, = 174 GeV [ 2 ] , we find mH = 167 GeV from (44) for as(m2) = 0 . 1 1 . However, the fact that the masses calculated for the Higgs and top quark very roughly, but essentially from first principles, turn out to be in the correct region seems to me very encouraging. At the same time,

217

V.N. Gribov / Physics Utters B 336 (1994) 243-247

there may be an indication that some finite contribution to v2/4 from the region where q2 is of order of A2 could survive. In this respect the prediction of the Higgs mass becomes very important, because it is less dependent on the large momentum region of the theory than the top quark mass.

4. Conclusion The idea of a non-elementary nature of the Higgs particle was widely discussed in connection with the so-called top quark condensate [ 3 ] . The essential difference to these approaches is that in my discussion the top quark has no special interaction. It becomes important only because it is the heaviest fermion. My Goldstone and Higgs particles consist of all existing quarks and leptons on equal footing. The ratio of the contributions of the different fermions is defined by their masses and the U( 1) interaction at short distances. It is interesting to notice that the Golstone and the Higgs interact weakly with fermions at the scale m,. The largest of the Goldstone couplings - the one to the top - is, according to (14) and (34),

<*s(m2) =I 47T-167r/2-61_[aj(A2)M(m2)]r

iL =

w

?

If we accept this formula for an arbitrary scale m, —* yu., we would find that g2 decreases with the momentum scale, similar to the QCD coupling as. After passing the "grand unification" point as ~ a', the behavior of g2 changes. It starts to increase with momentum and asymptotically becomes proportional to the U(l) coupling

^

=f«V).

247

(53)

The Higgs self-interaction has the usual properties of the Glashow-Weinberg-Salam model, but here there is no question of fine-tuning the Higgs mass, because it is fixed by the symmetry condition (36). These and other details of the present theory, such as the influence of the Goldstone couplings on the quark mass dependence, will be discussed in a forthcoming paper [4].

Acknowledgments I would like to express my deep gratitude to Yu. Dokshitzer for numerous discussions of various aspects of the theory without an elementary Higgs particle. I am also grateful to C. Ewerz for discussions and his help in preparing the manuscript. I wish to thank all the members of the Institute for Theoretical Nuclear Physics for the kind and inspiring atmosphere I experienced during my stay in Bonn.

References [ 1 ] T.-P. Cheng, L.-F. Li, Gauge theory of elementary particle physics, Clarendon Press, Oxford, 1984; Ch. 14.5. [2] CDF Collaboration, FERM1LAB-PUB-94/097-E. [3] Y. Nambu, in: New Theories in Physics, Proc. XI Int. Symp. on Elementary Particle Physics, World Scientific, Singapore, 1989; A. Miransky, M. Tanabashi, K. Yamawaki, Mod. Phys. Lett. A 4 (1989) 1043; Phys. Lett. B 221 (1989) 177; For recent developments see for example M. Lindner, Int. J. Mod. Phys. A 8,3 (1993) 2167, and references therein. [4] V. Gribov, Yu. Dokshitzer, in preparation.

Bound states of massless fermions as a source for new physics Opening lecture given at the 33rd course of the International School of Subnuclear Physics, Erice, Sicily (1995)

V. N. Gribov*-1 a

Landau Institute for Theoretical Physics, ul. Kosygina Moscow, Russia and KFKI Research Institute for Particle and Nuclear Physics, H-1525 Budapest 114, P.O.B. 49, Hungary

Abstract The contribution of interactions at short and large distances to particle masses is discussed in the framework of the standard model.

1

Introduction

In this opening lecture I would like to discuss the standard model from a point of view different from the usual discussion of this theory. What is the Weinberg-Salam-Glashow standard model ? We have three interactions: 5C/(3), SU(2) and U(l) and three generations of quarks and leptons. All particles in this theory are supposed to be intrinsically massless. In order to connect this theory with the real world, the so-called Higgs sector is introduced, which is responsible for fermionic masses and the masses of W+, W~

and Z°. While the first part of the theory feels beautiful, the second part of it looks ugly to almost everybody. What is the reason for that ? The first part - massless gauge bosons and massless fermions - has several remarkable properties. At the first sight it contains no dimensional parameters, only dimensionless couplings gx, g2, #3. But, as it was discovered in 1957 by Landau, Abrikosov and Khalatnikov [1], the Abelia.n coupling is increasing with momentum, while

1

E-ma.il: [email protected]

Preprint submitted to Elsevier Science

20 December 1995

the non-Abelian couplings g2, 93, as found by Gross, Politzer and Wilczek [2], 2

are decreasing with it. Experimentally we know, that a$ = |^ is of the order 2

of unity in the momentum region A3 ~ 1 GeV and a i = |£ is of the order of unity at the scale XL = 10 38 GeV; g2 is small everywhere. As a result, instead of dimensionless couplings we have here two scales, short and large distance ones different by 103S orders of magnitude. Between these two scales all interactions are weak. Introducing the Higgs sector, we are introducing a new scale due to the running of the Higgs self-coupling A - a scale, where this A is of the order of unity. If the Higgs boson was light enough, this new scale would be relatively large. However, no light Higgs boson was found so far and in this sense the theory of the trivial Higgs sector is close to being in trouble. There were many proposals - supersymmetry, grand unification, supergravity etc. - which corresponded to the introduction of new interactions between the two scales I mentioned. Some of them may be correct - we don't know. But to my knowledge nobody was able yet to construct a theory which is asymptotically completely free in the sense that it does not contain strong interactions at short distances like U(l) interactions. The aim of my lecture is to raise the question, whether it is really necessary to introduce new interactions between those two scales in order to explain the nature, and to present some observations which indicate that this might not be necessary. We believe, that strong SU(3) interactions are able to prevent quarks from being observed as real particles and to organize bound states which we call hadrons. Why then the strong U(l) interaction at the scale q2 ~ \ \ is unable to give masses to quarks and leptons and their bound states which would be the Higgs boson and the longitudinal components of W+, W~ and Z° ? It is important to understand that the two phenomena. - the existence of bound states and the generation of masses - are very strongly connected. If the interaction is strong enough to create massless bound states, these bound states will condensate and the scattering of massless fermions on this condensate will result in the propagation of fermions as massive objects. In this sense the source for the mass generation is the existence of massless bound states of massless fermions. What kind of bound states can we expect ? If we have a doublet of left-handed quarks interacting with right-handed antiquarks through the U(l) exchange, they could create a massless spinless doublet. Together with an antidoublet it will give us four states which, eventually, become massive bosons the Higgs boson and three longitudinal components of W+, W~ and Z°. The last step occurs at large distances: between the Compton wave length of the quark and A^ 1 . In order to understand how this can happen, let us discuss the space-time structure of these "intrinsically" massless bosons. Due to the fact,

0

that these bound states are created by the strong interactions on the distances 1 0 - 5 2 cm, one might expect that they will be point-like. However, although there is very little knowledge about the structure of relativistic bound states, these expectations do not seem to be correct. Even in the non-relativistic quantum mechanics this expectation is not true in general, since the size of a bound state is determined by the binding energy and not by the radius of the forces. A famous example for this is the deuteron in the limit when the radius of nuclear forces is going to zero. In this limit the proton and the neutron inside the deuteron are always outside the region where the interaction is different from zero. In reality, in addition to nuclear forces there exist always long-range electromagnetic interactions between the proton and the neutron. The binding energy depends on both interactions and would not be zero even if the binding energy due to solely nuclear interactions were exactly zero. This is just an analogue of the mechanism by which a massless boson acquires a mass. In the latter case the role of e.m. interactions between the proton and the neutron is played by the interaction with the condensate and with the colour field. The size of the new massive state would be determined by the masses of the constituent fermions. These considerations suggest, that the boson masses could be calculated as functions of fermionic masses, with little sensitivity to the lack of knowledge about the interaction structure at the distances of the order of 1/AL. In the A2 '

next section I will present these calculations. They contain a factor In -$• and give reasonable predictions for masses of the top quark and the Higgs meson. In the third section of this lecture we shall discuss another topics - the n meson mass. The TT meson is a bound state of a light q and a light q interacting via the exchange of coloured gluons, which becomes a massless Goldstone in the limit when the masses of light quarks tend to zero. But the light quark participates in both U(l) and SU(3) interactions and thus plays a role in the creation not only of pions but also of massless Goldstones which are the source of the longitudinal components of W and Z. This means, that these two interactions, the long range SU(S) and the short range U(l) produce two types of zero mass bound states. It is natural to expect, that these states will strongly influence each other. It is possible to calculate the interaction between these two states. As we will show, this gives an expression for the ix meson mass as an integral A2

over light quark masses and this integral contains the same In -^ as the Higgs mass. Knowing the TT meson mass and using the same value for the Landau scale, we find a reasonable value for the constituent light quark mass. The conclusion of sections 2 and 3 is, that measuring masses of some bound states of quarks and leptons we obtain information about the scale where the fermionic masses are created, and this information is roughly in agreement with the idea that masses are generated at the Landau scale. If proved to be true, this would suggest, that the other interactions, even if they exist between

3

two scales, are not important in the process of creation of fermion masses. Before going to these rather simple calculations, let us discuss, what kind of problems will arise if one assumes that U(l) interactions are responsible for the fermionic masses. The source of the problem is obvious: there are many quarks and leptons, so why do we expect to have only four intrinsically massless bosons ? Indeed, these bosons are proved to be complicated superpositions of different quarks and leptons. Where are the other massless states which we have to expect on the basis of the U{1) symmetry ? A possible answer is the following. The interaction I call U{1) is well defined only in the region of momenta, much smaller than A/,; in the region q2 > X\ where the coupling is large we are not able to write any Lagrangians, since in this region the U{\) interaction could be considered as a very complicated fermionic interaction or even an interaction which contains an additional horizontal gauge field connecting different generations. The only condition imposed on this interaction is, that it has to have no logarithmic tails into the q2
2

Masses of vector bosons and Higgs bosons

Let us consider the polarization operator for SU(2) vector bosons. It can be written as

IW*) = _ _ -T-"£<\ - - - with vertices in n„„ of the form fr„±(l - 75)7^. For massless fermions IIM1/ has the form ]!„„ = Il£, = (S^k2 - k^k„) II(fc2), and it corresponds to the massless W. Fermionic masses lead to an additional term II' in !!„„: 4

(i)

222

IT

rrii

rrii

=

= ^ 3 ^ E / d4 K-(
(2)

g2>m2

where £ t m (2) stands for the summation over all generations of up (t) and down (4-) quarks and leptons (/). Correspondingly, the square of the W mass is equal

2

7 ^

2

32TT 2

2/ 2x

(3)

y -^-^(o

Here mt is the top-quark mass, if we keep only the contribution of the heaviest quarks. The behaviour of m2(q2) is well-known in the region where all couplings are small: 8/7

9z{q2) gsimt)

2^2\ m {q t

9\{™2t)

1/10

.9i(l2)

(4)

According to (3), m2t{q2) changes with q2 very slowly, and therefore, roughly, 1 « #2 ^7T^ 2

m

W

m

32TT

<

ln

\2 2

(5)

mi

In spite of the fantastically large value of the Landau scale, we have \2 16rr2

m2

1

(6)

and thus mw ~ rnt. Taking into account the explicit dependence of m2(q2) on q2 according to (4), we find [3]

2

_ 3 ff|(m2) J

m w — - ^ 7 2— j 7 nv
W -

^(AI

I1/7!

(7)

glim2)

The accuracy of this result depends on the unknown contribution of the region g2 > ^ i and on the strong interaction in the region q2 < m2 as well as on the effective Yukawa couplings which appear in the theory due to non-zero quark masses.

5

Because of the smallness of ^ ? , which was compensated by the large value of In -\, the corrections coming from the region q2 < m2 are small. Corrections from the region q2 > X2L will also be small provided m2(q2) starts to decrease faster (not logarithmically) at q2 > \\. In the work which was done together with Yu. Dokshitzer [4] we have estimated the possible value of mt from the known value of raw by taking into account contributions of the Yukawa couplings, reasonable cut-off values in the region q2 > X2 and different values for gz(m,t). We obtained the interval

ISO GeV <-mt < 200 GeV

(8)

for mt. If fermionic masses were created at a scale smaller than the Landau scale, the resulting top quark mass would, respectively, be larger. Written in the form (2), the polarization operator IIM1/ for W is not transverse. In order to make it transverse, we have to add to the expression (1) the Goldstone contribution:

n„„ =

<•->')

<\zw-

(9)

Strictly speaking, it is necessary to have three Goldstone states in (9), and in this sense these three massless states are responsible for the masses of W and Z. At short distances, where the masses of fermions are small, the existence of three massless bosons implies, that there has to be a fourth massless state corresponding to a scalar boson, which, however, could become massive due to long distance interactions; this state is called the Higgs boson. In terms of the mass of the heaviest quark (the top quark) the mass squared of the Higgs boson is equal [3]

m%=4—J^. J rnt -^

(10)

In the approximation which leads to the restriction (8), expression (10) gives the Higgs mass in the same range

180 GeV < mH < 200 GeV .

(11)

6

3

T h e IT m e s o n mass

In order to see how the mass of the n meson could be influenced by the interaction which is responsible for the generation of fermionic masses, we can start with an SU(2) current. Due to the Goldstone contribution

-

(12)

G m7 5 -

to this current (where m is the quark mass), it has to be conserved for any fermionic states, and for the light quark states in particular. For simplicity, we will neglect the mass difference of the u and d quarks. If we include the SU(3) colour interaction and suppose, that this interaction produces a bound state called TT, equation (12) will take the form

(13)

The current conservation k^T^ = 0 implies that on the pion mass shell we have *V7M751

V*

2777-/5 f

¥>*

(14)

In the first order in light quark masses ipv can be calculated in the limit of zero quark masses; in this case tp* is connected with the quark Green's function in a simple way. It is easy to show, that, if the latter is written in the form

G-1(q) =

Z(m-q),

then at large q2 we have, in general:

(15)

m ( 9 ) = m(?) + ^

,

(16)

where m and i/3 are slowly changing functions of q2; the zero quark mass limit means m = 0. In this case n is the Goldstone and its coupling to fermions (i.e. its wave function) is

V* = y ^ 7 5 .

(17)

Combined with the expressions (15)-(17) the equation (14) leads to

**-£/£•»>.

(«)

where the integration goes over the region q2 > i/2 or q2 > XQCD f° r which (16) is correct; k2 is the square of the pion mass and f„ ~ 93 MeV. The right-hand-side of (18) has the same structure as the expression (3) for the W A2

mass: for slowly varying m and v values it is proportional to In

xi

'• and this

*QCD

large In factor is compensated by 16TT2. The form (18) shows explicitly, how the pion mass feels the scale where the quark masses are created. One can show, that as a function of the strong coupling g$ the product mi/3 behaves in the following way: m{q)u\q) = mo!/3 (^) V 530 J

' ;

(19)

here m 0 , VQ and #30 are the values of the quantities m, v, g$ at the minimal q2 where (16) is correct, (q2 w UQ). Using (18) and (19), we will find m 2 in the form 12m0*0>

A|

/93(Xl)\S'"

Since we do not know 1/, the value of ra, cannot be checked yet. However, being the part of the quark Green's function, it has to enter another physical quantity, like the p meson mass and therefore it can be measured. Calculating I/Q by making use of (20) and a reasonable value for p | 0 , we find 230 MeV < 1/ < 260 MeV

(21)

and 1/2 < a3(i/) < 1. This value of 1/ is in excellent qualitative agreement with experiment, because, according to (15), (16), 1/ is the mass of the constituent

8

quark (G x (g = v) = 0 if mo
(22)

ml = ^ < W > .

Indeed, (22) can be obtained from (18), if we forget about Z in (15) and about the running of m.

References

[1] Landau L.D., Abrikosov A.A., Khalatnikov I.M., Dokl. AN SSSR 95 (1954) 773, Dokl. AN SSSR 95 (1954) 1177 [2] Gross D.J., Wilczek F., Phys. Rev. D8 (1973) 3633, Phys. Rev. D9 (1974) 980; Politzer H.D., Phys. Rev. Lett. 30 (1973) 1346 [3] Gribov V.N., Phys. Lett. B336 (1994) 243 [4] Dokshitzer Yu.L., Gribov V.N., unpublished

9

30

The theory of quark confinement* Vladimir Gribov Landau Institute for Theoretical Physics, Moscow and Research Institute for Particle and Nuclear Physics, Budapest January 13, 1997 Abstract The structure of QCD and the theory of quark confinement are discussed.

1

Introduction

Almost ten years ago [1] I have proposed a hypothesis according to which quarks and gluons are confined due to the existence of light quarks (u and d) with Compton wave lenghts much larger than the radius of strong interaction defined by X QCD. The essence of this hypothesis is the supercritical phenomenon which is well-known in QED where it is the following. If there exists a heavy nucleus of radius R with charge Z exceeding a critical value Z„ (which is of the order of 137), the vacuum of the light charged fermions (electrons) becomes unstable due to the process of (e~, e + ) pair creation. The negative component of the pair, the electron, falls into the heavy charge and the positive component goes to infinity. The condition for this phenomenon to occur is that the Compton wave length 1/m of the electron has to be much larger than the radius of the heavy charge: a / Z 2 - Z*. ln(mfl) > it.

(1)

The electron which falls into the center forms together with the source Z an ion of charge Zi = Z — 1. If Z — 1 > Zcr, this process will continue until the vacuum becomes stable i.e. when Zn = Z — n becomes less than the critical charge. From the point of view of the Dirac equation the falling electron has negative energy. On the other hand, the supercritical ion with Zn < Z„ is stable because "Lecture given at the 34th course of the International School of Subnuclear Physics, Erice, Sicily (1996)

31 of the Pauli principle. Indeed, the electron cannot leave the ion, since all negative energy states outside it are occupied. In QCD this phenomenon can occur since, due to asymptotic freedom, the colour charge can reach the critical value for any object. Let us suppose that two heavy quarks are created with opposite colours.

Fig. 1 The gluonic vacuum polarization increases the colour charges of the quarks. When these colour charges become large enough, light antiquarks start to fall onto the heavy quark. However, the bound state formed by a light antiquark and a heavy quark will be very different from the bound states which appeared in QED. In QED we had an ion with a charge Z — 1. In QCD the corresponding state can be colourless. The difference is due to the fact, that while the large colour charge of the heavy quark results from gluonic vacuum polarization, the bound state is formed by two particles the total charge of which equals zero, and this charge cannot be changed by vacuum polarization. This phenomenon can be called quantum screening. Formally it means, that the heavy quark is unstable and has to decay into a meson and a light quark qh -+ M + qi.

(2)

This case - the confinement of the heavy quark - is relatively simple and almost independent of the properties of the light quark [2]. The problem of the light quark is more difficult. At the first sight it is not clear whether the "falling into the center" and the formation of a supercritical state can occur in the system of a light quark and a light antiquark. It was shown [3], however, that indeed the critical phenomenon exists in the light qq system. Moreover, the critical coupling constant is proved to be sufficiently small:

^-(i-i/i)-!•"•«•

<»>

There is an even more complicated question, namely: what kind of states can be formed as a result of this phenomenon. If the appearing new state is a normal meson, the "falling into the center" leads to qi -» M + qi

(4)

32

instead of (2). Equation (4) can be satisfied if there are not only positive energy quarks q+ but also negative energy quarks g_. This result, although it looks very strange, is not unexpected. Indeed, we already learned in QED that particles which fall into the center correspond to the negative energy solution of the Dirac equation. To resolve the problem of the negative energy states in his equation, Dirac supposed that they are all occupied and therefore not observable. The necessity to consider now both the positive and negative solutions of the Dirac equation means only, that Dirac's hypothesis is not always true. If the interaction is strong enough, there is another possibility: negative energy states might be only partly occupied and positive energy states only partly empty. There is, however, no contradiction between the existence of a stable supercritical bound state (a meson) and the absence of negative energy particles in the real world, since in this case both the positive and negative energy quarks will be unstable q+ -¥ M + 9_ q. -> M + q-q-q-

(5)

and, consequently, unobservable. From the point of view of the fermionic spectrum in the vacuum (5) means the following. In the case of weak interactions the fermionic spectrum has a structure corresponding to Fig.2.

•+ m

All levels above <jo = V<12 + m 2 a^e empty, and the levels below q0 = — \/q2 + m 2 are occupied. If the interaction is more than critical, quark-antiquark pairs are present in the vacuum and the spectrum has a structure which roughly corresponds to Fig.3.: all levels are occupied in the shaded regions and empty in the unshaded ones.

*o

Fig. 3 In terms of condensed matter physics, the curves qo = ±Jq2 + rn2, correspond to the Fermi surface. The change in the number of constituents in the theory is also natural from the point of view of the physics of condensed matter. In a theory with weak coupling we had two constituents q and q; a theory with supercritical coupling contains four states q+, ?_, q+ and g_. In non-relativistic physics there are only particles (electrons) and no antiparticles. But in a conductor we have particles and holes, and holes are negative energy states (counting the energy of the holes from zero and not from the Fermi energy as it is usually done). In a relativistic theory it is impossible to add a constant to the energy of particles, and therefore we are forced to talk about negative energy states which exist only inside our matter (vacuum). This unusual quark spectrum has been discussed for several years. Still, I was not able to formulate a constructive theory because I didn't understand what meson has to be introduced as a supercritical bound state. From the structure of the spectrum shown in Fig.3. it is clear that there have to be different types of excitations corresponding to different types of mesons. It is natural to identify hole - particle excitations of the type 1 near the Fermi surface with mesons like p, ui etc. Hole particle excitations near the light cone can be identified with the ao and /o mesons [4] and also with rj''. But all these mesons do not look as natural candidates for the lowest supercritical bound states. Only recently I recognized that there are strong reasons to believe that the pseudoscalar octet, and the 7r-meson in particular, are in fact the lowest supercritical bound states. Most of these reasons are connected with the discussion of the nature of the 7r-meson state which I presented at this school last year [5]. As I have told before, the supercritical atom in QED is not a usual Bohr-type atom, containing a definite number of electrons. Rather, it is a collective state in which only the electron density is localized around the heavy charge, and only at large distances (much larger than the atomic radius) it looks like a state with a definite electron number. A quasi-Goldstone state like a if- meson with a finite mass has the same properties. At short distances (distances less than \QCD) ' ' c a n D e considered as a collective state of the type qiqi —q r q r (divergence of the pseudovector current; qi and qr stand for left-handed and right-handed quarks), whereas at large distances it is the two-particle state ql,qr — q^ft. Accepting this identification of the ff-meson as the lowest supercritical state by introducing it explicitly in the equation for the quark Green function (which will be discussed in the next part of the lecture) it is proved to be possible to find a solution for this Green function which has properties corresponding to the spectrum in Fig.3. This solution has two complex poles as functions of energy in the complex energy plane in accordance with the two types of'quarks (constituent and current) we have discussed. It has no pole on the real

34 axis which guarantees that quarks as propagating states do not exist. It has a soft singularity when q7 —> 0 reflecting the fact that quark currents exist in the region where the hadrons are created. It is important to stress that in all these considerations I assume the coupling constant a being saturated at a value not much larger than OCT. Quark masses corresponding to the positions of the two complex poles m ± are of the order of m± ~ e vo-aer \QCD. The solution for the Green's function is self-consistent if the IT- meson mass is close to T7i_. This means that at least in the case when a is not very large, the 7r-meson mass is defined by strong interaction dynamics. This result contradicts the usual point of view according to which the 7r-meson mass squared is the product of the bare quark mass mo defined by weak interactions and the condensate density {$*!>)/f% defined by strong interactions. In the next part of the lecture I will explain what type of equations have to be written and solved in order to come to the conclusion I have stated, and to calculate quantities which have not been analyzed until now. For example, quark-gluon vertices and the gluon Green's function have not been calculated yet. It is clear, however, that the gluon Green's function will also have a complex singularity due to the gluon decay into a qq-p&h. Consequently, the gluon will also be confined. Equations for hadronic amplitudes can also be written constructively if a is not very large. It can be shown that if the quark Green's function has properties as described before, the hadronic amplitude will have no singularities connected with intermediate qq states, but will have singularities related to the 7>meson thresholds. The fact that the 7r-meson is included intrinsically in the equation for the quark Green's function and has a mass of the order of m_ makes this statement much less mysterious than it would look without it.

2

Divergence-free equations for Green's functions in QED and QCD

The question I would like to discuss now is how to formulate QCD as an asymptotically free theory in a well-defined way. The usual formulation based on the QCD Lagrangian is not defining the theory even perturbatively because of uncertainties connected with divergences and their regularization. But even believing that the perturbative theory can be well defined we would have to answer the question where non-perturbative phenomena like confinement and spontaneous symmetry breaking come from. Obviously, it is necessary to find a formulation of the theory in which perturbative and non-perturbative phenomena exist simultaneously. This can be done in the following way. We know that in any renormalized theory the divergences can

35

be included only in the Green's functions of the particles and in the vertices which correspond to their interactions. Hence, if it is possible to write equations for Green's functions and vertices not containing any divergences even in the form of series in effective interactions, all the other amplitudes can be found in the usual diagramatical way. It is proved to be not very difficult to formulate such equations. In order to avoid technical complications let us first show how to derive the equations and what kind of structure they will have in Abelian theories. Consider the polarization operator Ullu(p) defining the photon Green's function D^ D^ = D% + D^U^D^.

(6)

Due to current conservation, the expression for IIM1/(p) can be written in the form

n^(p) = ( < W - j w v W ) .

(7)

In Feynman gauge

The polarization operator IIMI/ can be represented symbolically as the sum of quadratically divergent Feynman diagrams: P 3 -

n,,(p)=

p -^Q

y-y--+

-f^

>-

+

•••

(9)

Taking the second derivative of II(iI/(p) as a function of momentum p we will find

* n-(ri. . . '-jC> , t£K + <...3 > + < S > + < 3. > ,101

dpadpa:

c

v The diagrams in the right-hand side of (10) diverge only logarithmically. The notation corresponds to differentiations of the fermionic line

-*— =44 Taking now the derivative of (10) and multiplying by p„ we get

^-*S (12) and thus the divergences in the right-hand side disappear. We can rearrange the diagrams by introducing exact Green's functions and vertices. The right-hand side will then have the structure _ „ (13)

36 where to each line corresponds an exact Green's function and to each point of interaction corresponds an exact vertex. To any point which appeared from the differentiation a vertex with zero momentum has to be related - it is equal to the derivative of the inverse fermionic Green's function, i.e. dfG~l{k). The left-hand side of the equation will be

s*. - ^

(d( + 6)0
(i4)

where

Including the coupling constant el in the definition of the photon Green's function D^y and writing

^ " ?i^u = — •

(15)

we obtain

This is the equation we have been looking for. It is an integro- differential equation for the photon Green's function, containing in the right-hand side exact Green's functions and vertices, and completely free of divergences. The equation for the fermionic Green's function has to be of second order since the fermionic self-energy is linearly divergent. It can be written in the form d2G~l =

—^-*-—^-r • • (17) At 6 a v The equation for the corresponding vertex function which is divergent only logarithmically will be of first order: •

d

^

>

This system does not contain the coupling constant explicitly because the effective interaction is defined by the photon Green's function; a(pi) is a well-defined running coupling. The equation (16) coincides with the renorm group equation in two leading orders in a. Analogous equations can be written in QCD too. It is, however, technically more complicated to derive them. There are two reasons responsible for the difficulties. First, the interaction vertices for gluons depend on the momenta of the particles. The second problem is connected with the fact that the effective running coupling is defined

37 not only by gluonic Green's functions, but by combinations of Green's functions and vertices too. Both complications can be avoided if we replace the usual description of the gluons and their interactions by a Duffin-Kemmer type description in which the potential A^ and the field strength FM„ are considered as independent quantities. In the framework of this description the interaction vertices are momentum independent. The gluonic Green's function D is a matrix, different components of which correspond to the average values (F ( i l / (i),/lp(x'))) (•^•I>(X)I^I>'(X')) and (F^x), Fli-U>(x')). In momentum space the gauge invariant quantity ( F ^ z ) , Fti
(19)

the function a(p 2 ) is the running coupling in QCD. The equation for a(p2) is of the same structure as the equation for ct(p') in QED, having, of course, contributions from gluons and quarks with different signs. The structure of the equations for the two other components of D coincides with that of the equation for the fermionic Green's function. We will not write these equations explicitly because, on the one hand, only the general structure is relevant for the discussion of the perturbative and uon-perturbative contributions to the solutions, and on the other hand, we in fact did not use these equations upto now. In QED the running coupling is small in the region of small momenta and therefore we can consider the perturbative solutions of these equations. The equation for 1/e2 defines e2 upto a constant; this enables us to fix e2 at<72 = 0 a s a = ^ = Y5^. The equation for the vertex allows tofixT^ = 7M when k = 0, p2 = m 2 where m is the mass of the electron. The equation for the fermionic Green's function provide us with more freedom. G~l is defined by the equation upto the solution of the free equation d2G0(q) = 0.

(20) 2

The general solution of this equation depending only on q and q = "/t,q^ and has the Go1(q) = Z:1(m1-q)+Q-^t)

(21)

where m ^ Zi, v\ and vi are constants. If we believe that there are no unphysical singularities in the region of small momenta, we have to put ux = 0, v? = 0,fixmy as the electron mass and take Z\ = 1 to be consistent with the Ward identity. This means, that the zeroth approximation is fixed and we can iterate the equation in a unique way. In QCD the situation is completely different. The coupling constant is small when momenta are large and thus we can iterate the equations in the large momentum region. In this case we have a lot of uncertainties. In the limit q2 -t oo

38 m, = (e 2 r»mo

,

Z, = ( e 2 ) ^ - 1 ,

(22)

where A, m 0 , Zo and Z3 are arbitrary values; 71, 72 and 73 are the so-called anomalous dimensions. These uncertainties are trivial in the sense that we have the same number of parameters as in QED, only is not easy to give them direct experimental definitions. There are, however, additional parameters J/I and v>i. Differently from the case of QED, here we can't put them equal to zero because in the region of large q1 their contributions are small and do not contradict anything. These parameters are connected with densities of different condensates ({tytf'), {T^)). The same type of uncertainties exist also in the equations for gluonic Green's functions; they correspond to different gluonic condensates. All these considerations show that contrary to an infrared free theory, in an asymptotically free theory the "perturbative" solution is not defined uniquely. It contains arbitrary parameters which can be determined only by self-consistence conditions in the region of small momenta. These conditions are a priori not known. In concrete cases they are defined by the stability of particle states. There is another serious problem in asymptotically free theories, where the spectrum of the particles is a priori unknown and it has to contain bound states. Namely: how to treat bound states in general, and those in particular, which are stable if there are no weak interactions (the pseudoscalar octet and some baryons). I don't know how to approach the baryons, but for the pseudoscalar octet 1 believe the solution is the following. These states have to be considered as elementary objects and must be introduced in the equations explicitly. The reason for this is the quasi- Goldstone nature of these states. However, by this same reason the couplings of these states to the quarks can not be arbitrary: they are defined by the Goldberger-Treiman relation. The masses of these states are not supposed to be calculated in the theory of strong interactions a priori. But at least in some cases, like that of the
3

The theory of the self-consistent field

The equation (16) for the running coupling and the equations for quark Green's functions and vertices have very different properties. The expansion in the right-hand side of (16) is in fact an expansion in powers of ^ ; hence, for small - the corrections remain small. Contrary to this case, higher order corrections for equations (17) and (IS) can be large even if a is small. For example, the first diagram in (17)

39

-+-£•—.

.—:>

Fig. 4 is finite, but singular. It contains a fermion propagator in the third power. The contribution of this diagram to d2G~l at a value of q close to the fermion mass will be of the order of (q — m)~l. It gives a correction of the type a In ? ~™ to G~l which 2

2

is large when 3-~r~ <3C 1 even if a is small. The reason for this singularity is the contribution of small momentum gluons to the diagram. There is another singularity connected with large gluon momenta. We have to find a way to include both the main infrared effect and the large q2 behaviour in the first term of the expansion. This can be done by using the fact that there are different possibilities to differentiate diagrams. It is the following. Let us consider the first diagram Fig.4 and relate momenta to the lines as it is shown in Fig.5 1 MZ N

1

Fig. 5 It is easv to see that ; = ^i8\q

- q')

(23)

(
and therefore &G;1

= —

7MG(?)7M-

(24)

IT

This result is kind of surprising, because it shows that zero momentum gluons give a hundred percent contribution to the second derivative inspite of the fact that Gf 1 itself is infrared finite. Let us consider diagrams of higher order, for example

/


40 only the second derivatives of the gluon lines and including everything else in the correction terms we will have >\q)

= ?^-d»G-1GdliG-1+

<'

where the two quantities d„,G~x are emission and momentum gluons. The corrections can be redefined of a(0); in this case the first term of d2G~l will a In 2 -jjj?- ^ d asymptotic corrections of the type of In an Abelian theory the equation for T^ can be d2Tll(ql,q2) +

=

<' ^ — —

+

•-.

(25)

absorption amplitudes for zero so that a(q2) will appear instead contain all infrared corrections In 2^~written in the form

-{AMdpTM+

dpritAp(q2)-A,(q1)rilAp(q2)}

+ ---

(26)

where A^ = dliG~1(q)G, AM = Gd^G. In QCD the same equation (25) is correct for the quark Green's function if we replace a by | a ; the equation (26) (with the same change of a) is correct for any colourless vertex (e.g. a nqq-vertex). In first order (without corrections) the equation (25) can be considered as an equations for quark propagation in a self- consistent vacuum field, because it takes into account the interactions with gluonic vacuum fluctuations the momenta of which are much smaller than those of the quarks. According to this equation, the value of the self-consistent field is defined by the quark Green's function itself. The argument to accept equation (25) as the first approximation for the description of quark propagation in the vacuum would be very convincing if the gluonic fluctuations were the only small momentum fluctuations in the vacuum. In reality, due to the existence of almost massless (compared to the QCD scale) n- mesons there are long range pion fluctuations in the vacuum and these fluctuations have to be included in the equation. Fortunately this can be done in a very natural way, at least for massless pions. Deriving (25), we have considered diagrams of the type Figs. 5,6 which contain massless gluonic propagators. If we now include in the diagram for G~l not only gluons but also massless pions interacting with quarks through the Yukawa coupling, and perform the same differentiation, we shall get an additional term coming from the differentiation of the pion propagator. As a result, instead of (25) we obtain d2G-1(q) = where i"ysI2(/(q'i)G(q)

=

^G-\q)G{q)d.G-'-{i1J^9)G(q){i1J^cj)±-2 is the quark-pion Yukawa coupling.

(27)

41

According to the Goldberger-Treiman relation for virtual quarks, i^^g{q2) be written as

,,/p^

=

fcrfpHs.

can

(28)

Here fn is defined in the usual way as the transition amplitude of n to the axial current; it normalizes the pion wave function to unity. Instead of (25) we finally get the equation for quark propagation in gluon and pion fields in the form d2G-l(q) = = ^dllG-\q)G{q)dtiG-i

- T±^[il5G-l(q)}G(q){i1sG-1(q)}

(29)

All the results I discussed in the Introduction come from the analysis of this equation and the investigation of the analytic properties of G(q) based on unitarity with the inclusion of finite mass corrections for the pion.

4

References

1 V.N.Gribov, Physica Scripta T15 (1987) 164 2 Vladimir N. Gribov, Orsay lectures on confinement (II). LPTHE Orsay 94-20 (1994). 3 V.N. Gribov, Lund preprint LU-TP 91-7 (1991). 4 F.E. Close, Yu.L.Dokshitzer, V.N. Gribov, V.A. Khoze, M.G. Ryskin, Phys. Lett. B 319 (1993) 291. 5 V.N.Gribov, Bonn preprint TK-95-35 (1995).

239 Eur. Phys. J. C 10, 71-90 (1999) D O I 10.1007/sl00529900051

T H F n c

'

C,

| R n p F A tun rcH

U

M ™

PHYSICAL JOURNAL C © Springer-Verlag 1999

Editorial Note The following two papers had been nearly completed by V.N. Gribov before he passed away in summer 1997. Since these papers are the only documents on his ideas about quark confinement in QCD, the editors welcomed the proposal to publish the two manuscripts post mortem after they were edited by his wife and some of his closest friends. Even though Gribov's work on quark confinement has not led yet to a generally accepted solution of this problem - in fact some fundamental aspects are rather controversial - the publication of these unconventional and novel ideas may initiate new theoretical developments in this area. The editors consider this point a compelling reason for the publication of the two manuscripts in Bur. Phys. J. C. Hamburg, February 26, 1999

P.M. Zerwas J. Bartels

QCD at large and short distances* (annotated version)* Vladimir Gribov 1 , 2 ' 3 1

3

Landau Institute for Theoretical Physics, Moscow, Russia Research Institute for Particle and Nuclear Physics, Budapest, Hungary Institut fur Theoretische Kernphysik der Universitat Bonn, Germany Received: 15 February 1999 / Published online: 15 July 1999 Abstract. A formulation of QCD which contains no divergences and no renormalization procedure is presented. It contains both perturbative and nonperturbative phenomena. It is shown that, due to its asymptotically free nature, the theory is not denned uniquely. The chiral symmetry breaking and the nature of the octet of pseudoscalar particles as quasi-Goldstone states are analyzed in the theory with massless and massive quarks. The U(l) problem is discussed.

1 Introduction In this paper I show how to formulate an asymptotically free theory in such a way that it includes perturbative and nonperturbative phenomena simultaneously. This work was completed in Germany under the Humboldt T h e i d e a i s t h e f 0 u o w i r i g. Contrary to an infrared free Research Award Program t h e o r y ) i n a n a s y m p t o t i c a l l y free t h e o r t h e divergences ' lne original version of this paper, completed by the au4. t •>-• 4. i , 4.• • ii, • . , , » n , . -4.4. j 4 , , , . T, prevent us from writing even perturbation expansions in thor in April 1997, was submitted to the hep-ph archive (hep- a . 11 j c j iir i_ 1 i./n-7no/inA\ c J n T . 4 7 - . r x T .-, •. , unique, well-defined way. We can, however, make use of ph/9708424) a few days after Prof. V.N. Gribov passed away ., , ', , , . ., ,, ' , . , on August 13 1997 divergences in the theory occur only in the Green S functions t h e vertices 0 n This is the first of'two papers concluding his 20-year study , ' ^ the other hand, of the problem of quark confinement in QCD. This annotated k l l o w I I 1 S t h e G r e e n s f a c t i o n s and the vertices, we can version is the result of an attempt by a group of his colleagues e x P r e s s a 1 1 t h e o t h e r amplitudes through them perturbato understand the paper, starting in November 1997 in Or- t i v e l y m a unique way. Thus, we have to formulate equasay. A number of misprints were eliminated, most of the equa- t l o n s f o r Green's functions and vertices in a form that tions were checked, and some corrected. Comments have been does not contain any divergences. If this is done, the soadded in order to make the text easier to read. These comments lutions of these equations will contain both perturbative are displayed in square brackets. Many theorists participated and nonperturbative phenomena. in the process; the comments were assembled, and the final To avoid technical complications, in Sect. 2 of the paversion prepared, by Yu. Dokshitzer, C. Ewerz, A. Kaidalov, per we will derive these equations in an Abelian theory, A. Mueller, J. Nyiri and A. Vainshtein. in which usual perturbation theory is also well defined. In

Reprinted from Eur. Phys. J CIO. 71 - 9 0 (1999), Gribov, with permission from Springer-Verlag.

240

72

V. Gribov: QCD at large and short distances

Sect. 3, we generalize the equations for a non-Abelian theory and discuss the connection between perturbative and nonperturbative effects. The equations have an integrodifferential structure in which the asymptotic behaviour of the Green's functions is defined by the boundary conditions. The main conclusion in this section is that in the region of large momenta the Green's functions of quarks and gluons contain additional parameters compared to normal perturbation theory. These parameters can be associated with different types of "condensates". These nonperturbative parameters can be defined by solving the equations and finding nonsingular solutions in the infrared region. A priori, it is not clear what type of additional conditions have to be imposed on this system of equations in order to fix a nonsingular solution. For example, it can be the conservation of the axial current or of other currents that is formally satisfied from the point of view of the Lagrangian but is not ensured because of the divergences. In Sect. 4, we will show that these equations make it possible, for the first time, to analyze the problem of spontaneous symmetry breaking in an asymptotically free theory and to find the approximate value of the critical coupling at which symmetry breaking occurs. Also, these equations allow us to answer the fundamental question for asymptotically free theories, namely, how the bound states - the hadrons - have to be treated in these theories and how these bound states influence the equations for the Green's functions. The analysis leads to the following conclusion. The conditions for axial current conservation of flavour nonsinglet currents (in the limit of zero bare-quark masses) require that eight Goldstone bosons (the pseudoscalar octet) have to be regarded as elementary objects with couplings defined by Ward identities. This is so in spite of the fact that the couplings of these states to fermions decrease at large fermion virtualities. The same analysis provides a new possibility for the solution of the U(l) problem. In this solution, the flavour singlet pseudoscalar rj is a normal bound state of qq without a point-like structure. The mass of this bound state is different from zero and can be calculated in the limit of massless quarks. For massive quarks, the pseudoscalar octet becomes massive. The masses of the pseudoscalar mesons, however, are not calculable in terms of bare-quark masses, because of logarithmic divergences, and have to be regarded as unknown parameters which, in the real case of confined quarks, are defined by the self-consistence of the solution of the infrared problem. These states have an essential influence on the equations for the Green's functions, which, as it will be shown in the next paper, can be used constructively in solving the confinement problem if t h e effective coupling in the infrared region is not too large. In this case, the integro-differential equations can be reduced to a system of nonlinear differential equations, and the theory looks like a theory of particle propagation in self-consistent fields defined by the Green's functions themselves (as is the case in Landau's Fermi liquid theory) . The self-consistent fields are fields of gluons and 7r mesons.

2 Equations for Green's functions in QED In QED, we have two Green's functions - those of the photon D^(k) and electron G(q) - and a vertex function r)j(fc,g). The photon Green's function is defined by the vacuum polarization operator iIM„(fc), which can be expressed symbolically by a sum of Feynman diagrams n^(k)

= el ( 7 M d Z ^ > + 7 M < C X ^ > 1" + •••}•

(1) In order to obtain series not containing divergences, we will try to consider the derivatives of II^(k) as functions of momenta. Due to current conservation, we have

n^{k) = (s^k2 -

k^nik2)

(2)

where II{k2) contains only logarithmic divergences. Differentiating (2) twice, we will have

d2n^{k) = m{k2)5liU+[

(% + 6)cV7(fc2);

v

(3) we here use

The second term in (3) does not contain divergences; the first one does. In order to obtain a finite expression, consider

d^aanav{k) = ~3S^n(k2) - 3^d %(n(k ^ i 27).( * ^

(4)

From (3) and (4) it follows that the quantity

d2n^(k) + 2d»d„nav{k) 2

-e^d(n(k )

(5)

+ [ su

does not contain divergences. In Feynman gauge, Dllu(k2)

D^{k)

+ 6)din(k2"

^){di

is as follows:

<W

2

k„k

_ i

2

e2

(fc2)

~ k e {k2)1

k (l-n{k ))

2 2

Any diagram contains only the product e2)Dll,i/(k), therefore e2D^(k)

=

<9e/7(/fc2)

1

el

7—^7
3

k2

+

and

^e2(k2)S^,

In first order, this means

1

(6)

7X^A,

(7)

241

V. Gribov: QCD at large and short distances 7u^-^

h

73

*la

7X^y>

(8)

-SO7MGI7M>

the integrals on the right-hand side are convergent. This has to hold in any order. Hence, we can include in (8) the exact electron Green's functions and the exact vertex functions and add all the corresponding diagrams which were not included. As a result, we can write

T-*

--,<--

r~

-So

,

o

1

1 k^kv

V~~*

d„G-x±?

:

•o'

V G

7M

7M

7M

f

7M

d4fc

Restricting ourselves to d2l/k2,

7M

r r«M^si(fc 2 )

7M-

we can write

d„G~ 2

d (J\

G

= -go7M o7M>

(9)

92u2 = - 9 O 7 M G O ^ < T 1 where J^ denotes the sum over the permutations of the indices similarly to (8); a quantity e2/k2 corresponds to each photon line. Equation (9) is an equation for e 2 (in the form of series in e 2 ) provided that G(k) and F^(k,q) are known. In order to obtain equations for electron Green's functions and vertex functions, we have to remember that these functions can change rapidly, even if e 2 is small, because they can have infrared singularities of the type e 2 ln((g 2 — m2)/m2) or ultraviolet singularities of the type ( a / a o ) 7 . It has been proven that it is possible to arrange the differentiation in such a way that there will be an expansion only in e 2 . To write an equation for the fermion Green's function not containing any divergences, we have to differentiate twice the self-energy of the fermion or, equivalently, its inverse Green's function. Let us consider dflflG~1(q); the diagrammatic expression for G^ 1 will be the following. The simplest diagram is

G

god^Goff, G

- 0 O 7 M I 7 M - SI7M O7M

and, consequently, d2G~1 = g d^G'1

G d^G-1,

G " 1 = GQ 1 + G ^ 1 4- G ^ 1 .

The term 62 describes the contribution to , " ,"" \ \ , which does not contain overlapping divergences, and is of the form 2*y ~W

2k' "fc73"

&2 = la

la

2k„

2kL

let (12) -<-&-

For the calculation of higher-order diagrams, it is convenient to adopt the following principle. Beginning from the first point of interaction, we shall relate the external momentum to the photon line:

Diagrams of the next order are

-~-k

^K

'q — k

la

q-k'-k

(13) q-q

It can be easily shown that 1 fc2 + ie

-47T2ia4(fc)

(10)

The next photon interaction can be expressed as

Using this equality, we have (14) " J ^ 7 M G O 7 M = -9O7M G O7M>

q-q

242 V. Gribov: QCD at large and short distances

74

Now we relate the external momentum to the positron line. If the positron is emitting a photon, we keep sending the external momentum along the positron line up to its annihilation. As a result, we get a tree structure

which has the simple diagrammatical meaning

fc = 0

fc' = 0

= 2

(17) in which the initial photon line can end only at the final electron. It can be shown that every line in the diagram will be passed only once if the photons included in the fermion self-energy are not taken into account. This means that in this approach, exact electron Green's functions have to be used with bare vertices, since the idea is basically the exclusion of overlapping divergences. Taking the second derivative of one of the photon propagators and restricting ourselves to the contribution 4w2S4,(k)g(0), we obtain both on the left-hand side and the right-hand side photon-emission amplitudes with zero momentum. Due to gauge invariance, however, a zero-momentum photon cannot change the state of the system (it is emitted from the external lines). The photon- emission amplitude equals F^lq, 0) = dllG~l{q) [within this convention, the bare vertex is — e7M]. The final contribution of the differentiation has the structure (11). The above statement, which essentially means that the emission of a zero-momentum photon is determined by the total charge, can be proven by the Ward identity. Thus, we have

The second term on the right-hand side of (17) represents the contribution of photons which, if we carry out the aforementioned differentiation, corresponds to photons inside the Green's function, which are not present in Mvv. Hence, M„„(g,0) = 2dllG-1 G d„G-1. All the diagrams for d2G~l

(18) The sum of the diagrams (18) is built up of the exact photon Green's functions, each of which equals (6), which equal An2g(k)(l/k2). By differentiating the photon lines, we have calculated the contribution of 2 2 d l/k . Thus, there remains 1 =

d2G~1 =

g^d^G^Gd^G-

—-^SSN—

(16)

The remaining diagrams contain first derivatives of photon and positron lines and second derivatives of positron lines. All these diagrams can be expressed in terms of the exact r^, G and D functions. They do not contain divergences, except for those graphs which correspond to the photon self-energy. The first term in (16) has the structure g{Q)Mvv(q,k

=

G-1d2GG-

= 0).

•• 2 9 „ G

^Gd^G'1

I

AAkMvv{q,k) 47T 2 i

d"

a2g(k)-g(0) 2

k

(19)

where we have introduced g(0) in order to avoid a contribution from 54(fc). This expression contains logarithmic corrections coming from the ultraviolet region (k > q). Replacing g(k) - p(0) by g(q) - p(0) 4- g(k) - g(q) and performing an integration by parts, we get (9(9)-9(P))

"/

Mvv{q,Q) 2

d4fe g(k) 4.31 k2

~

,(q,k) 2 •

<2°)

The first term in (20) can be explicitly calculated, while the second one does not contain any logarithms because of the presence of the difference g(k) — g(q)- We can rewrite (20) in the form

where Muv{q,k = 0) is a quantity close to the Compton scattering amplitude, in the sense that they would be equal if we differentiated all fermion propagators including those inside the electron Green's function. (The factor 1/2 is present because the Compton amplitude contains the sum of diagrams with momenta k and — k). The Compton scattering amplitude Mvv{q, k = 0) satisfies the Ward identity Mvv{q)

are of the form

d2G~

Z = (g(q) consequently, d2G~1 2

l

d G- (q)

S(0))a>JG-

1

G a / i G - 1 + 6U

can be expressed as

= g{q)dliG-1Gdy.G-1

+5X+ S2,

(21)

where <5i is defined by [the second line of] (20), and 52 contains first-order derivatives of the photon and positron lines and second-order derivatives of the positron lines, as has been explained above. The first term contains all singularities of the types a\n((q2 —m2)/m2) and ( a / a o ) 7 .

243

V. Gribov: QCD at large and short distances

3 Equations for the vertex function and for the amplitudes of interaction with the external field

Inserting (24) into (23) and replacing #(0) by g(q) we get 9 2 r " ( p , q) = g(q) {A„(Q2)

To obtain the equation for the vertex function, let us express / ^ ( p , ? ) as a set of diagrams containing exact Green's functions:

-Av(q2)

Ali{q) =

r"b,q)

(22)

We shall relate the external momenta to the lines in the diagrams in the same way as we did when we derived the equation for the Green's function. Calculating the secondorder derivative in q, we obtain d qr» (p, q) = - g(0)M& (p, q,k)\ where M^l/(p,q,k)\/c=o tude of the process

+

v

q) + 0 „ r " ( p , q)

r»(p, q) A„{qi))

+ a 2 f " ( p , q) .

k2

which corresponds to the emission of photons with momenta k\, k2 = 0 from the external legs:

All(q) = ai,G-1{q)G(q), G(q)dliG-\q).

(26)

The correction terms d2r^ are defined as a set of diagrams with exact Green's functions and vertices. They contain first-order derivatives of both the photon lines and the positron lines, second-order derivatives of the positron lines and, as in the case of the Green's function, corrections due to the replacement of g(0) by g(q). The same equation is valid for the interaction amplitude of fermions with the external field, provided that this interaction does not depend on the relative momentum q of the fermions. The equations for the interactions with external fields are essential. Indeed, if these equations have solutions that decrease at large virtualities of the fermions - i.e., solutions which do not require driving terms - this means the existence of bound states. For the sake of convenience, we rewrite the equation (25) in the form 924>{p,q) = g(q) {A„{q2)dv(p,q) + ~Al/(q2)<j>(p,q)Al,(q1)}

d„4>(p,q)Av{qi) ,

r»

(27)

which will be understood as the equation for the bound state, with a spin that is defined by the invariant structure of the matrix <j>. It is important to note that the accuracy of the equations for the vertex r ^ ( p , q) and for the bound states differs from the accuracy of the equation for the Green's function. The functions r^{p, q) and {p, q) contain, in general, the so-called Sudakov logarithms, which are not included in the equations (25) and (27). Hence, the equations are valid only if

9T

dvG-

(25)

and

2 2 a iIn P—=i InP —^^
WMq) 1

Au{qi)

(23)

is the contribution to the ampli-

fei/

d^(p,

Here we have introduced the notations 9i,2 = g ± | ,

2

75

(28)

92

duG~

4 Equations for Green's functions in QCD d„G~

d„r»

dvr"

d„G~

(24)

QCD is the theory of interacting quarks and gluons. The description of quarks is more or less the same as in QED. Gluons, however, are very different. Even the fact that a gluon has to have a multi-component wave function because it is a spin-1 particle is not seen explicitly. In

244 V. Gribov: QCD at large and short distances

76

the usual approach, it is described by a Green's function D^v = (Ap(x), Av(y)), which, in momentum space (in Feynman gauge), can always be written in the form (29)

This equation contains only one unknown function. All spin properties of the gluon are included in the moment u m dependence of its interaction. In order to introduce multi-component Green's functions, we have to formulate the theory such that the interaction is momentumindependent: We have to replace the usual description of the gluons and their interactions by a Duffin-Kemmer type formulation. In the framework of this description, the gluon Lagrangian is C(x)

-{dllA„-d„Ahl+g[Av,All)}Fllv

+ -Flll,Fllv,

(30)

[where the trace over colour indices is implied]. The potential A^ and the field strength F^v are independent quantities. The interaction is momentum-independent and equals [A^, AV]F^„. In this formulation, we have three independent Green's functions (A^AJ,

{F^,AP),

(37)

(SM)~ = K<W<5,

6, ^C(k2).

Du

[Their nonzero components, the 4 x 6 matrices S^] have a simple representation

(F^,FP„).

(31)

The quantities 7± are projection operators, projected onto the upper and lower components of