On the vacuum rearrangement in massless chromodynamics

Volume 76B, number 5

PHYSICS LETTERS

3 July 1978

ON THE VACUUM REARRANGEMENT IN MASSLESS CHROMODYNAMICS V.P. GUSYNIN and V.A. MIRANSKY

Institute for TheoreticalPhysics,Academy of Sciencesof the UkrainianSSR, Kiev-130, USSR Received 31 January 1978

It is shown that in the weak coupling limit the Bethe-Salpeter equation of massless chromodynamics admits a colourless tachyon solution when the number of quark multiplets n < nc, where for the colour group SU(N) the critical value n c 0.14Na/(N2 - 1). When n ~ n c all particles remain massless.

The normal phase instability in superconductivity theory is displayed in the appearance of a tachyon (already in the ladder approximation) in the spectrum of two-particle bound states [1 ]. In this paper we investigate the possibility of a similar situation in massless chromodynamics [2], in the normal phase of which the masses of all particles equal zero. The set o f the Bethe-Salpeter (BS) equations for wave functions ~i(q, P) of the bound b o s o n - b o s o n (i = V), ghost-ghost (i = G) and fermion-antifermion (i = F) states has the form

xI~-¢/~v (q, P) = f d 4x eiqx(OITA~ (x/2)A bu(-x/2)113

= ~abI(g~v --'quqv~ q~qv A2 +P P A 3 q2 ] A1 +----Tq

(2)

+ (quPv +Pqv)A4 + (qu P - Pqv)A5J,

~Gb(q, P) =fd4x

eiqx,(01Tc"(x/2)c*b(-x/2)lp)

= 6abB,

(3)

~mn .F.~tq,p~j=f d4xe~ (01T~m (x/2) ~(-x/2)lP) qti(q'P)= f

(d4k ~)

4 Gi(q + ~P)Ti](P;q, k)

X q~(k,P)Gi( q - ~e),

= 8mn[x S + 7uPUx1 + "yuqUx2 (1)

where Gi is the propagator, q and P are the relative and total momentum, respectively, A is a cut-off parameter, which can be removed in the final expressions by passing to the renormalized parameters. All the Lorentz and group indices are omitted here. The kernels vii = 5Mi/fG] [3], where the self-energy Mi is defined by the relation GT1 = G~1 - M i (Goi is a free propagator). A general structure of the wave functions ~ i for the colourless bound state jPc = 0++ is as follows:

+ °uv(PUqu - PVqu) XT ] ,

(4)

where the indices a, b, m, n refer to the colour group; A l - A 5 , B, XS, X1, X2, XT are the scalar functions of the invariants q2, p2, (pq). Charge C-invariance as well as Bose and Fermi-statistics require that the functions XT, A1, A2, A3, AS, XS, X2 are even, and the fimctions A 4, B, X1 odd under Pq. Hence and from eqs. ( 2 ) - ( 4 ) it follows that in the limit P = 0 only the functions A 1, A2, X2, Xs, XT remain. Moreover, in the Landau gauge the transversality condition for q'Vuv has as a result that the function A 2 equals zero too when P = 0. The set of eqs. (1) will be further studied in the ladder approximation in the Landau gauge providing

p2/4A2 -~a ~ 1. 585

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It is easy to see that in this approximation the BS equations split into two groups of coupled equations corresponding to the functions ×s, XT and Ai, Xi, B, respectively. The set of equations for the latter in the lowest order of the parameter a includes only two functions A1, X2 and in the euclidean domain (q0 ~ iq0, P0 "+ iP0) can be transformed to the form G 1 =A 1 ,

(s) y = k2/A 2,

I 6 ~ N1

(6) 3 y

T12(x,y)=_T21(y,x)=

7x2]} 1 , 3y2' @ - + a l / ~ - 1) 1 " 3N x/%-Ta (7)

586

and 1

0 r[i(x, y) - r;i(x, y),

,61 3Nln(1/a-----) + O

~>X1

(8)

(9)

(10)

Hence *' p 2 ~ 4A 2 exp [-16/3NX1[ .

where n is the number of quark multiplets. Let us first discuss a pure Yang-Mills theory (n = 0). The BS equations for this theory have already been treated in refs. [4]. Our equations, however, are different from those given in refs. [4[, since there charge invariance was not taken into account, and it was assumed that the wave function for ghosts B :/= 0 when P = 0. This difference is essential, since for n = 0 eq. (5) turns out to be the Hilbert-Schmidt integral equation [5] with symmetric positive kernel (Tll(x, y) = T 11(y, x)), while the kernel of equations including wave functions for ghosts (or fermions) are not symmetric due to Fermi statistics (for more details see our further discussion). As is well known, for the Hilbert-Schmidt equation with positive kernel all eigenvalues X are positive, and for the minimum possible eigenvalue ~1 the estimate [51

x/rff2/S4 ~>X1 >/ 1 / ~ 4

0

16 1 0 ( _ _ 1___) 3N In (1/a) + ln2(1/a)

x,I r22(x, y) = 0,

=~i fdxTii2,,, (x,x) '

are the iterated kernels. Performing the necessary calculations we get

{O(x-y) [ 3 + 570yyX-qx21 -~-13

+O(y_x)[3+50x

S2m =--Tr T2m

X = g2/47r2.

The kernels T//for the colour group SU(N) are:

Tll(x,y) -

1

0 2 = X 2,

o

x = q2/A2,

holds, whe re

rgm(X'Y)=f rigCXm", t)T~'(t, y)dt,

1

Gi(x)=Xf dyri:(x,y)G/y),

3 July 1978

(11)

Since the vector P is euclidean, the tachyon is present in the spectrum of the system, i.e. the vacuum of the normal phase is unstable [1,4]. The tachyon quantum numbers coincide with the quantum numbers of the mass operator AaA a • For this reason the value #--/~ (p2)-1/2 determines a vacuum rearrangement time, and (p2)1/2/2 the mass of the vector bosom When the fermions are incorporated, the kernel of eq. (5) is no longer symmetric (T12(x, y) = -T21(y, x)), the minus sign appears here due to Fermi statistics. The eigenvalues X of this system are complex. However, when the number of quark multiplets is sufficiently small, n ~
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eigenvalues ~'1,2 -_ ~i c _+ 4½c 2 - b 2 are real when b 2 ~< c2/4). In order to determine the value n c we should note that eq. (5) is equivalent to the operator equation *~ (7q l + Xf'12~F21)G1 = (1/X)G 1 ,

(12)

hence for the function G1, which is normalized to unity, we obtain 1/X = ½(G117q IIG1)

-+ ½[((G 1 Ii "11 IG1))2 + 4(Gllf'121"211G1)]

112 .

(13)

Since the operators ~11, _(7~12 ]b21) are positive, then if ((GII/qlIG1)) 2 < - 4 ( G I I / q 2 : p 2 1 I G 1 ) ,

(14)

the imaginary part Im (1/X) is different from zero, and Re ( 1/X) = ½(G 1 if,11[G1)"

(15)

The explicit form of the kernels ( 6 ) - ( 8 ) implies that, for the eigenvalue of interest, Xl, with Re Xl 1/ln(1/a), the normalized function G 1 has the form [x/xTa lnl/2(l/a)] -1 in the vicinity o f x ~ 0. Hence and from eq. (14) we find that the imaginary part is different from zero when

n >n e ~9-~N3/(N2

-

1).

(16)

When n > n c, the only acceptable solution to eq. (5) will thus be Gi = 0, and a tachyon is absent in the spectrum of the system. We emphasize that when n ~
3 July 1978

merit g2(N2 - 1)/87rN~ a is performed, the set of equations for XS, XT coincides with the corresponding equations of massless electrodynamics considered by us in ref. [8]. It is shown there that in the weak coupling limit a tachyon solution to these equations is impossible. The bound-state problem for massless electrodynamics and massless chromodynamics thus appears to be entirely different: when the number of quark multiplets is small enough, in the ladder approximation for an arbitrary coupling g there is a colourless tachyon in the bound-state spectrum of chromodynamics. What we can say about the tachyon solution when going beyond the ladder approximation is as follows. At first sight the fact that the constant g can be small seems to be a justification of this approximation. The present pape} (see also [4]), however, shows that the infrared region of small momenta is primarily responsible for the tachyon solution. As is known [6], the effective coupling constant in this region grows when higher orders of perturbation theory are taken into account. This point complicates the bound-state problem in chromodynamics to a very great extent. Nevertheless, if we take a heuristic point of view [9] and assume that in chromodynamics the interaction dynamics at large distances is basically determined by replacing the bare constant g with the effective coupling g(k 2) (which grows in the infrared region, k 2 ~ 0), it is reasonable to believe that an effective increase in the coupling results only in an increased tachyon mass m t. Here m t can even take infinitely large values. The vacuum rearrangement could then lead to an infinitely large mass for colour gluons (a colour confinement?). What is the role of fermions in this problem? As follows from the well-known renormalization group arguments, only asymptotically free theories can produce masses dynamicaUy [10]. If the number of quark multiplets is large enough (n > ~1_N), the theory cannot be asymptotically free [6]. In the ladder approximation the sufficient condition (see eq. (16)) for existence of a dynamical mass generation mechanism is stronger than the requirement of asymptotic freedom. When going beyond the ladder approximation this condition may change. But it still must remain stronger than the requirement of asymptotic freedom. M1 this enables us to hope that the results obtained in the present paper reflect some features of the vacuum rearrangement in massless chromodynamics in a qualitatively correct manner. 587

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We thank P.I. Fomin for fruitful discussions.

References [ 1] J.R. Schrieffer, Theory of superconductivity (Benjamin, New York, 1964) § 7. [2] H. Fritzsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 47B (1973) 365. [3] See, for example: M. Baker, J. Johnson and B.W. Lee, Phys. Rev. 133B (1964) 209. [4] R. Fukuda, preprint RIFP-301, Kyoto Univ. (1977); Phys. Lett. 73B (1978) 33.

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3 July 1978

[5] F.G. Tdcomi, Integral equations (Interscience, New York, London, 1957). [6] D.J. Gross and F. Wilczek, Phys. Rev. D8 (1973) 3633. [7] See, for example: I.C. Gohberg and M.G. Krein, Introduction to the theory of linear non-self-adjoint operators (in Russian) (Nauka, Moscow, 1965) Ch. V, § 12. [8] V.P. Gusynin, V.A. Miransky and P.I. Fomin, preprint ITP-139R-77 (1977). [9] See, for example: J.M. Cornwall, Nuel. Phys. B128 (1977) 75; H. Pagels, Phys. Rev. D14 (1976) 2747. [10] D.J. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235; K. Lane, Phys. Rev. D10 (1974) 1353.