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 Theoretical Physics, Academy of Sciences of the Ukrainian SSR, 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/(N 2 - 1). When n ~< n c, the vacuum rearrangement results in the gluons acquiring a mass; when n > n c all particles remain massless.

The normal phase instability in superconductivity theory is displayed in the appearance of a tachyon (al- ready in the ladder approximation) in the spectrum of two-particle bound states [1 ]. In this paper we investi- gate 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 boson -boson (i = V), ghost-ghost (i = G) and fermion-ant ifermion (i = F) states has the form

d4k + ~P)Ti](P;q, k) qti(q'P)= f ( ~ ) 4 Gi(q

X q~(k,P)Gi( q - ~e), (1)

where G i is the propagator, q and P are the relative and total momentum, respectively, A is a cut-off pa- rameter, which can be removed in the final expressions by passing to the renormalized parameters. All the Lorentz and group indices are omitted here. The ker- nels vii = 5Mi/fG ] [3], where the self-energy M i is de- fined 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 fol- lows:

xI~-¢/~ v (q, P) = f d 4x e iqx(OI TA~ (x/2)A bu(-x/2)113

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

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

~Gb(q, P) =fd4x eiqx,(01Tc"(x/2)c*b(-x/2)lp)

(2)

= 6abB, (3)

~mn .F.~tq, p~j = f d4xe ~ (01T~ m (x/2) ~(-x/2)lP)

= 8mn[x S + 7uPUx1 + "yuqUx2

+ °uv(PUq u - PVq u) X T ] , (4)

where the indices a, b, m, n refer to the colour group; A l -A5 , 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

Volume 76B, number 5 PHYSICS LETTERS 3 July 1978

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

1

Gi(x)=X f dyri:(x,y)G/y), G 1 = A 1 , 0 2 = X 2,

o (s)

x = q2/A2, y = k2/A 2, X = g2/47r2.

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

I 6 ~ N 1 { [ 3 5 0 y 7 y - ~ - 1 3 Tll(x,y) - O(x-y) + X-qx21 (6)

+O(y_x ) [3+50x 7x2]} 1 , 3 y 3y2' @-+a

T12(x,y)=_T21(y,x)= l / ~ - 1) 1 " 3N x/%-Ta (7)

x,I r22(x, y) = 0, (8)

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 differ- ent from those given in refs. [4[, since there charge in- variance was not taken into account, and it was as- sumed 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 sym- metric 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 esti- mate [51

x/rff2/S4 ~> X 1 >/ 1 / ~ 4 (9)

holds, whe re

1 =~i fdxTii (x,x) S2m =-- Tr T2m 2,,, '

0

and

1

rgm(X' Y)= f rigCXm" , t)T~'(t, y )d t , 0

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

are the iterated kernels. Performing the necessary calculations we get

, 6 1 3Nln (1/a-----) + O ~> X 1

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

Hence *'

p2~ 4A 2 exp [-16/3NX1[ .

(10)

(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. How- ever, when the number of quark multiplets is suffi- ciently small, n ~< nc, X may remain real (a simple ex- ample is the two-dimensional matrix ( ~ 8); their

,1 The cut-off parameter can be removed by passing in eq. (11) to the renormalized charge

hi (u) = k l / [ 1 - (3N/16) ?,lln(A 2/ t f l ) ] ,

which differs from the renormalization group charge [61 by a coefficient (3/16 instead of 1116). This is so because the ladder approximation is not invariant under the renormal- ization group.

586

Volume 76B, number 5 PHYSICS LETTERS 3 July 1978

_ i _+ 4½c 2 b 2 eigenvalues ~'1,2 - ~ c - 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 equa- tion *~

(7 q l + Xf'12~F21)G1 = (1/X)G 1 , (12)

hence for the function G1, which is normalized to uni- ty, we obtain

1/X = ½ (G117q IIG1)

-+ ½[((G 1 I i "11 IG1)) 2 + 4 ( G l l f ' 1 2 1 " 2 1 1 G 1 ) ] 112 . (13)

Since the operators ~11, _(7~12 ]b21) are positive, then if

((GII/qlIG1)) 2 <-4(GII /q2 :p21IG1 ) , (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 fx ~ 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 G i = 0, and a tachyon is absent in the spectrum of the system.

We emphasize that when n ~< n c the vacuum rear- rangement in the weak coupling limit results in the fact that only the bosons, but not the fermions, ac- quire a mass. Really, the mass vertex ~ = ~L~R + ~R~L (~JL, R -~- 1 (1 -+75)ff) connects fermions and antifermions of the same chirality. On the other hand, the wave function X = 7u×~ belonging to eq. (5) satis- fies the condition (1 -+ 75)X(1 -+ 3'5) = 0 and thus refers to the bound state of a fermion and an antifermion of opposite chirality. The bound state of a fermion and an antifermion of the same chirality is described by the function X = Xs + au,,X~ v. When the replace-

,2 Equations of this type belonging to the operator bundle theory were studied by Keldysh and Krein [7].

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 cou- pling limit a tachyon solution to these equations is im- possible.

The bound-state problem for massless electrodynam- ics and massless chromodynamics thus appears to be entirely different: when the number of quark multi- plets 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 be- yond 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 or- ders 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 re- sults only in an increased tachyon mass m t. Here m t can even take infinitely large values. The vacuum rear- rangement 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 dy- namicaUy [10]. If the number of quark multiplets is large enough (n > ~1_ N), the theory cannot be asymp- totically free [6]. In the ladder approximation the suf- ficient condition (see eq. (16)) for existence of a dy- namical 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 vacu- um rearrangement in massless chromodynamics in a qualitatively correct manner.

587

Volume 76B, number 5 PHYSICS LETTERS 3 July 1978

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.

[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, Intro-

duction 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.

588