Glueballs as bound states of massive gluons

Volume 120B, number 4,5.6 PHYSICS LETTERS 13 January 1983

GLUEBALLS AS BOUND STATES OF MASSIVE GLUONS

J.M. CORNWALL and A. SONI Department of Physics, University of California, Los Angeles, CA 90024, USA

Received 26 April 1982

Recent work in both lattice and continuum QCD shows that the gauge field is massive. We investigate the bound states of massive gluons with a Schrijdinger equation and tindM(O+? = 2.3m,M(O-+) = 2.7m,M(2++) = 3.2m, where m is the gluon mass; we expect from this as well as earlier work that m = 500 MeV. These glueballs have widths typical of allowed hadronic decays.

It is well known that triality-zero sources of color, such as gluons, are screened in QCD. This is manifested in a perimeter law for octet Wilson loops [ 1] or equiv- alently in a gluon string which breaks by formation of a gluon pair. Another way to say this is that gluons are

massive (in spite of appearances in the lagrangian), with the dynamically mass being generated *’ through

strong gluon-binding forces [2] . One measure of the gluon mass m is the energy (*2m) necessary to break a string joining color-octet sources. Bernard [3] has studied this in a lattice calculation with the result m 2 520 MeV ** . Non-perturbative continuum studies

[2] yield m = (500 f 200) MeV. Gluon dynamics, then, ought to be describable as

massive spin-one fields interacting through massive

spin-one exchange and through a breakable string. Be- cause the effective fields are massive, there is some chance that a non-relativistic Schrijdinger description of these processes is usable. The purpose of this paper is to investigate some low-lying two-gluon and three- gluon glueballs in the SchrGdinger picture. it turns out that the forces between gluons are about twice the

*’ Mass is generated with no breaking of local gauge symme- try; quarks are confined by topological coupling to long- range excitations, which necessarily accompany mass gener- ation. See the second paper of ref. [ 111.

*2 The energy (= 520 MeV) that Bernard measures is closely related but not identical to the gluon mass relevant for the calculation of glueball masses. In our work the gluon mass will be left as an arbitrary parameter.

forces between quarks, and gluons are not particularly heavy compared to the square root of the gluon string tension, so a non-relativistic description (with v2/c2 corrections) is somewhat marginal; nevertheless we find good agreement within errors with lattice calcula-

tions [4,5] and with two experimentally observed candidates for glueballs [6,7].

Earlier work on continuum glueballs [8] has empha- sized the massless nature of the bare gluon and leads to results only qualitatively comparable to ours.

Consider first the string force between gluons. On

the rnc, i naive picture, the string tension in the adjoint representation, KA, ought to be twice that in the fun- damental representation, K,, because each gluon acts like a qq pair. The same result comes from a strong- coupling calculation. Bernard [3] has measured K, on the lattice with the result KA = (3 .l + 1 .7)KF, so there is considerable uncertainty. As we have said, the

gluon string breaks when sufficient energy has been stored in it to materialize a gluon pair, so we use the following form of the string potential:

V,(r) = 2m(l - e-+0). (1)

Clearly, KA = 2mrc ‘;ifK, = 2K, =0.32 GeV2,m = 500 MeV, then r. = 0.6 fm. The form (1) is not, by the way, simply ad hoc; it is precisely the result for the d = 2 Schwinger model [9] *3 (where the string breaks

*3 In the Schwinger model, the usual linearly rising potential of one-photon exchange is found by expanding the screened form (1) about the origin and saving only the first term.

0 031-9163/83/0000-OOOO/$ 03.00 0 1983 North-Holland 431

Volume 120B, number 4,5,6 PHYSICS LETTERS 13 January 1983

by formation of fermion-antifermion pairs), and can be justified [IO] for gluons by consideration of a con- densate of vortices such as are predicted to exist for massive gluons [2]. At infinite distance V, rises to 2m, representing the energy needed to materialize a gluon

pair and screen the original gluons’ color. To this string force must be added (in general spin-

dependent) forces from single-gluon exchange. In ref.

[2] it is shown by solving a Schwinger-Dyson equation that the usual product of coupling constant and propagator,g2A, is to be replaced by a function d(k2),

which is approximately given by

d-l(P) = b(k2 - m2) ln[(4m2 - Ic2)A2], (2)

where b occurs in the &function: fi = -bg3 t . . . . and A is the renormalization-group mass of a few hundred

MeV. The function d is a generalization of the usual running charge which is stabilized by a gluon mass in

the logarithm. A useful simplification of (2) is to re- place k2 in the logarithm by a fixed spacelike (k2 < 0) average value. Then the single-gluon exchange amplitude *4 of fig. 1 is simply the Born approximation for massive gauge-invariant QCD [ II] , with g2 replaced by {b ln[(4m2 t (/c2))A-2]}-1.

When states p1 and p2 in fig. 1 form a color singlet the amplitude is

T= -Inh[(3IJ,Il)(t - m2)-1(41Ju12)

f cl’9 e;*ei - f;*cl e4**e2 t exch.] , (3)

where r = (pl - ~3)~) exch. signifies interchanging lines 3 and 4,pi.ei = 0, and (setting (k2) = 0 for sim- plicity)

X = (3/4n)[b ln(4m2/A2)] -l. (4)

*4 Single-gluon exchange amplitudes in all quark processes

should also be modified as in eq. (2). Unfortunately, the ef-

fect on quarkonium is very hard to detect.

Fig. 1. Single-particle exchange amplitude, including the sea-

Also,

- 2e;,, P3’El - 2EI, p1 *e; (5)

is the matrix element of the conserved color current, the first term of which is chromoelectric and the rest

chromomagnetic. The factor 3 in eq. (4) would beN for the gauge group SU(N), where (for no quark) b = 1 1N/48n2 ; so without quarks h is independent of N *’ . One may verify that T has two important prop- erties: first, it is independent of the gauge chosen for the exchanged gluon, as long as the external gluons are

on shell (pi’ = m2); and second, if any E,, is replaced by p, in T, the result vanishes. So the on-shell amplitude is fully gauge-invariant as long as all particles (external and internal) have the same mass.

The space-time potential corresponding to (3)-

(5) is a sum of Yukawa, spin-orbit, spin-spin, and

tensor forces, when only leading relativistic corrections are kept:

+nM(r) 4m2 -2stzs2

( 3m2 m2 ’ 1

3h L.S~~e-mr

2m2 rar r

+$2 [(SV)2 -fSZVZ]~,

where S = S, t S2 is the total spin operator. In (6), s

= (PI +p2)2 =M& where MC is the glueball mass.

Normally one would sets = 4m2 t O(u2/c2), but this is not adequate in the presence of a string potential, so we have saved the full s-dependence. We have, however,

set s = 4m2 in the spin-orbit and tensor potentials (which are themselves relativistic corrections).

The potentials in (6) have one very serious flaw: depending on the spin state (and especially in spin-zero states), the short-distance singular parts of the potential (6(r), v3, r-2) may be attractive and lead to a hamiltonian unbounded below. The physical solution, of course, is a quantum-mechanical smearing of the gluon fields which removes the singularities. We do not

*5 This is important for the large-N limit. Since both the gluon

string tension and mass are independent of N [2], gluon dy-

namics are N-independent in this limit. gull.

432

Volume 1208, number 4,5,6 PHYSICS LETTERS 13 January 1983

know how to treat this accurately, so we will simply treat these attractive singularities as perturbations. (When they are repulsive we can treat them exactly.) These potentials [including the string potential(l)] are stronger than quark forces by about a factor of 2 (if K, = 2K,) or more (h = 9/4 times the correspond-

qq value). However, the situation for three-gluon states (see below) is different: for the O-+ three gluon state, both the string tension and h should be halved. The former follows from elementary considerations of gluons as equivalent to qq pairs, or from a strong-coupling calculation; the latter because each gluon pair is in an octet state instead of a singlet. We approximate

the three-gluon potential as a sum of pairwise potentials:

v, = ,z. V(@, (7) 1 <I

V(r) = $ KAr + $ [the potential of eq. (6)] . (8)

One should note that the three-body forces of fig. 2 vanish for group-theoretical reasons [ 131. However,

to describe the breaking of gluon strings for three- gluon strings for three-gluon states does require [lo] going beyond the approximation (7). One should also include schannel annihilation graphs, since gluon pairs are equivalent to gluons in the OF+ three-gluon state, but we omit all such mixing in this first report.

We have studied the simplest two- and three gluon bound states numerically using the potentials (1) and (6)-(g). Both direct computer solutions (for two- gluon states) as well as simple variational calculations were performed, treating singular attractive potentials as perturbations. The variational calculations are within 10% or less of direct solutions. There are two di-

mensionless parameters, h and /3 = KA/2m2, and an overall mass scale set by m. These parameters are only roughly known: from (4), X ranges from 1 to 2 depending on m/A and on the number of quark flavors

(which enters into b). It seems premature to present

Fig. 2. A candidate three-body potential whose group-theoret- ic weight is zero.

an exhaustive parameter search so we will present our results only for h = 2 and p = 0.3 (In general, the larger fl or the smaller h is, the heavier the glueball.) If fl = 0.3 and m = 500 MeV, K, is quite close to KF instead of to 2KF *6, but substantially larger values

(e.g., fl= 0.6) simply force all glueballs to have a mass very nearly 4m unless h is unreasonably large. In fact, for h = 2 the O-+ glueball unbinds for /3 > 0.6 and the

O++ unbinds for /I > 0.8, because the potential (1) is too steep but is limited in height by 2m.

The results, for two-gluon states withJPC = O++, O-+, l-+, 22’, and 3++ - , and for the O-+ three-gluon state, are shown in fig. 3 *‘. The calculations were done both for s = 4m2 and for a (variationally deter- mined) self-consistent value of s; the latter case leads to increased Yukawa attraction [see eq. (6)] and slightly lighter glueballs. The right-hand scale shows

masses in GeV, assuming m.= 500 MeV. There are at least two possible experimental candidates and three

*6 That is, if KF is really equal to 0.16 GeV2; it could be somewhat smaller.

*’ The l-+ state is particularly interesting for a theory of massive gluons; not only is it exotic, but it cannot be a state of two massive gluons.

Z-Gluon z-Gluon 3-Gluon

2.’ _____-

2.3 I_ o_+{d _______ t o++ 1 -____.

2.0 1000 L:O L:i

Fig. 3. Results of the numerical calculations. The scale on the left is in units of the gluon mass m; that on the right is in MeV, assuming m = 500 MeV. The solid lines refer to s = 4m2 in the potential (6), and the dashed lines refer to a self-consistent determination of s.

433

lattice-glueball states which can be compared with our results. The experimental candidates are the ~(1440) [6], which is O-+, and the 0(1640) [7], identified as

2++. In the lattice world [but for SU(2), not SU(3)]

there is a O++ glueball at (3 + I)@* [4,5 1, and a O-+ glueball at (4 f 0.7)Kk’* as well as a 2++ at (4.4

f l)Ky* 151. We have already commented that gluon

ill = 3.1(4rr/g2)m . (9)

If, as before, we replaceg* by [b ln(4m2/A2)] -l and take m = 500 MeV, we find Al z (5-10)m. At energies much higher than this there may not be any recogniz- able glueball states.

There is (at least) one important open question, dynamics is not sensitive to the group, so within the errors of both our and the lattice calculations it should suffice to take the SU(3) value Kb’* = 400 MeV. The

even if quark-mixing effects are left out. Any two- ‘gluon state has a three-gluon counterpart (differing by one unit of orbital angular momentum) *s , so that

agreement is certainly acceptably good with this value, these mix. The potentials we use also have radial exci-

and m = 500 MeV. tations, whose masses are not far from the three-gluon

The 2++ glueball plays a special role in our calcula- counterparts. Ace these really distinct from the three-

tions, because the only singular potential it feels (spin gluon states, or does our use of a potential picture to

-spin) is repulsive, and we need not feel uneasy about describe virtual gluons, as in (l), effectively include a

using an unbounded hamiltonian (as is the case for the substantial part of the mixing with multi-gluon states?

O++ state). We have calculated, for a range of h and /3, One should be cautious in interpreting states coming

the 2++ mass both treating the repulsive delta function from this potential which are close to the limiting as a perturbation and treating it exactly. The exact re- mass of 4m, when r + 00 in the breakable string poten-

sults are a negligibly small amount below the perturba- tial (1). This caution might apply to the 2-+ state

tive answers. For the O++, the perturbative contribu- shown in fig. 3.

tion of the singular potentials is substantial but less than what the rest of the hamiltonian contributes. As This work was supported in part by the National

anticipated, the gluon forces are so strong that a non- Science Foundation. One of us (J.M.C.) thanks the

relativistic treatment is marginal: values of (p*/m*) are Institute for Theoretical Physics, Santa Barbara, for as high as 0.5-0.6. its hospitality while part of this work was being done,

Sometimes doubt [6] is expressed that the experi- and we both thank Prof. Claude Bernard for valuable

mental candidates are glueballs, because their widths conversations.

are larger than expected on the basis of naive OZI-rule

arguments [ 131. But one must be very careful to have Added note. After this work was sent for publica-

a theoretical underpinning for the OZI rule before us- tion we learned about the work of Barnes [ 151, who

ing it in novel situations. The only such underpinning also considers massive constituent gluons in glueballs.

known to us is the large-N limit, where OZI-suppressed There are substantial differences between our work

decays are down by powers of l/N. To our knowledge and that of Barnes. He keeps only the frunsverse com-

no one has noticed the remarkable fact that the decay ponents of massive gluons, which violate Lorentz invar-

glueball -+ gg is not suppressed in this limit; instead, it iance, and uses massless exchange gluons in the coun-

is completely allowed. We will discuss this, and esti- terpart to our eq. (6); this violates gauge invariance.

mate glueball widths quantitatively (of the order of The only way to decouple longitudinal modes [i.e., T

50-200 MeV) in a later publication. of eq. (3) gives zero if eP(k) -+ k,,] is to give the ex-

It is interesting to speculate how far the glueball changed gluon precisely the same mass as the constitu-

spectrum rises in energy. We have already seen a weak- ent gluon. Furthermore, Barnes uses a potential which

ening of forces in the three-gluon system, and there rises linearly indefinitely, which is physically inappro-

may be evidence for saturation in the semiclassical priate since strings joining gluons must always break,

limit (infinite numbers of gluons, i.e., fields instead of even without dynamical fermions [3].

particles). Years ago it was shown [ 141 that the massive gauge-invariant version of QCD has a finite-energy O++ glueball; the mass (recalculated later by K. Olynyk,

*’ Conventional hadrons (qq, qqq) also have counterparts (qqg,qqqg) whose mass is O(m) above the usual hadron.

whom we thank) is

434

See the second paper of ref. [2].

Volume 120B, number 4,5,6 PHYSICS LETTERS 13 January 1983

Volume 1208, number 4,5,6 PHYSICS LETTERS 13 January 1983

References [91

111 121

131 [41

151

161

[71 [81

J. Glimm and A. Jaffe, Nucl. Phys. B149 (1979) 49.

J.M. Cornwall, invited talk at Conf. on QCD (Marseille,

June 1981), UCLA preprint 81/TEP/12; UCLA preprint

81/TEP/30.

C. Bernard, Phys. Lett. 108B (1982) 431.

B. Berg, Phys. Lett. 97B (1980) 401; G. Bhanot and C. Rebbi, Nucl. Phys. B189 (1981) 469.

K. Ishikawa, M. Teper and G. Schierholz, DESY preprint

(1982), unpublished.

D.L. Scharre, invited talk at Orbis Scientiae 1982, to be

published in the Conf. Proc.

C. Edwards et al., Phys. Rev. Lett. 48 (1982) 458.

R.L. Jaffe and K. Johnson, Phys. Lett. 60B (1976) 201;

J.F. Donoghue, K. Johnson and B.A. Li, Phys. Lett. 99B

(1981) 416; H. Suura, Phys. Rev. Lett. 44 (1980) 1319;

K. Ishikawa, Phys. Rev. D20 (1979) 731.

[lOI 1111

1121 1131

1141

[I51

H.J. Rothe, K.D. Rothe and J.A. Swieca, Phys. Rev. D19

(1979) 3020; H.A. Falomir, R.E. Gamboa Saravi’ and F.A. Schaposnik,

Phys. Rev. D25 (1982) 547.

J.M. Cornwall, to be published.

J.M. Cornwall, Phys. Rev. DlO (1974) 500; Nucl. Phys.

B157 (1979) 392.

J.M. Cornwall, Nucl. Phys. B128 (1977) 75.

D. Robson, Nucl. Phys. B130 (1977) 328;

C.E. Carlson et al., Phys. Lett. 98B (1981) 110; 99B

(1981) 353.

J.M. Cornwall, in: Deeper pathways in high-energy phys-

ics, eds. B. Kursonoglu, A. Perlmutter and L.F. Scott

(Plenum, New York, 1977).

T. Barnes, 2. Phys. Cl0 (1981) 275.

435

Documents

Glueballs as bound states of massive gluons