Physics Letters B 314 (1993) 397-400 North-Holland PHYSICS LETTERS B

M a s s i n e q u a l i t i e s for g l u e b a l l s , a n d q u a d r o n i u m ( e + e - e + e - ) s ta te s

S h m u e l N u s s i n o v t School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel and Department of Physics, University of Maryland, College Park, MD 20742-4111, USA

Received 11 December 1992; revised manuscript received 20 July 1993 Editor: H. George

We show that inferring mass inequalities for a few-body system in terms of the masses of the states bound in two-body subsys- tems can be useful in many different cases. Beside the baryon meson mass inequalities derived some time ago, we have also mass inequalities between two-gluon ("2g") and three-gluon ("3g") glueballs for quadronium e +e-e +e- states. The inequalities rely on the assumption that the few-body system is dominated by two-body forces.

I. Introduction

If a many-body state can be described via two-body interactions, then variat ional arguments can relate its lowest level to those o f the two-body subsystems [ 1 ]. Similar techniques [ 2 - 4 ] were used to infer lower bounds on masses o f baryons, Bijk (made of quark flavors, qi, qi, qk) in terms o f the meson correspond- ing to the var ious subsystems m i f ~ qi, qk, etc.

o >~ 0 0 2 m <aOk) ~" m (mif) + m (,,,j~) + m~,,,k~) • ( 1 )

The der ivat ion of ( 1 ) makes use of the fact that gluon exchange interact ions between two quarks in a nu- cleon are half as strong as in the corresponding me- son, and will be briefly reproduced in the next section.

Our main purpose here is to show that very s imilar reasonings imply inequali t ies ( lower bounds ) for masses of glueball states with quan tum numbers of at least three gluons. In terms of the lowest ( two-gluon) states and for the quad ron ium state e + e - e + e - (sug- gested as a candidate for the narrow states in the heavy ion coll is ions) in terms o f posi t ronium.

2. Meson baryon inequality

In a s imple quark model the mass m°~ o f the low-

On leave from Tel Aviv University.

est baryon state with quark flavors is given by the lowest eigenvalue of a hami l ton ian

H o k = T~ + Tj + Tk + Vii+ Vjk + Vgi ,

with T~, etc., being the single part icle (k inet ic) terms and Vii, etc., the two-body interactions.

In the color singlet baryon each pair of quarks ( t ransforming as .3 of color) are coupled to a .3 di- quark. This causes the strength ofg luon exchange in- teraction: V o = 2 f 2 j V , with 2~ the octet o f 3 × 3 Gell- Mann 2 matr ices associated with q~, etc., to be only half as strong as the corresponding Vii= (2~.;~) Vap- propr ia te for a meson in which qi~b couple to an over- all color singlet. More specifically, in a meson 2 i + , ~ = 0 and hence by squaring 2 f ~ [ . . . . = - 2 2 , whereas in a baryon A ~ + 2 / + 2 k = 0, and hence squar- ing and using the overall symmetry, 2 i ' 2 j = 2 j ' 2 k = 2k ' 2 i , and therefore

Aio,~l . . . . . = 22i.2j[ baryon- (2)

The meson qz~ is governed by the hami l ton ian

Hij-= r , + r ~ + vo-= ri + Tj + 2 V o . (3a)

Likewise,

Hj~= Tj + Tk + 2 Vjk, H k ; - = T k + r ~ + 2 V k , . (3b)

Eqs. ( 2 ) a n d (3) i m p l y t h a t

2 H i j k = Hi f + t l j~ + Hi, r. ( 4 )

We next take the expectat ion ofeq . (4 ) in the bar-

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yonic ground state function. By definition, the left- hand side yields twice the ground state baryon mass:

2 < ~'Ok I Hok I~°k ) = 2m°k. (5)

The matrix elements of the two-body operators in the three-body wave function, e.g., o o < ~ok I Hsfl ~Ok > are evaluated by considering H,f as a three-body opera- tor which is just the identity operator in the space of particle k. Thus we view ~,°k, for fixed coordinates of the quark qk, as a two-body (qsqj) trial wave function ~ - , and compute < ~- [Hs; I ~ - > . Finally, this is in- tegrated over all values of the coordinates of q~. q~-, the two-body wave function prescribed by ~'°k, is, in general, different from ~,o, the ground state wave function of the meson qsq¢. By the variational prin- ciple the latter wave function minimizes the expec- tation value of Hsf, the meson hamiltonian,

minv,,i<~,sflHsfl~,~f>=<~°-IHsfl~,°->=m °-, (6)

where rn °~, the energy of the state at rest, is the mass of the ground state qs~ meson. Hence we have

< O~-IH,/I 0~> >~ m °- . (7a)

Similarly we have

< ~ 1 -s //jfl q/)f> >/m°e, (7b)

< ¢/~rlHkrl O~r> >1 m~r. (7c)

Since each of these inequalities persists after the nor- malized integration over the coordinates of the third particle (q~ for eq. (7a), etc.), we have

< ~°~ IH¢I ~,°k > >~m °~, (8)

and similar inequalities hold also for the expectation values of Hj~ and Hkr. Combining these with eqs. (4) and (5) we have the desired inequalities

o ~ 0 0 2m ok , i m s f + mj~+ m°r

between the masses of baryons and mesons in corre- sponding Lorentz states. In certain cases, like the £2- ( Sz= ~ ) = s ¢ s ¢ s T, these states are uniquely speci- fied to be sT~t=O~o2o, the well-known ninth vector meson state, leading to simple relations like ma- >/~rn~. In other cases the restriction of ~°k to the two-body sector gives a mixture of spin states and the inequality is accordingly modified to have the properly weighted averages instead ofj ust m o on the right-hand side ofeq. (8).

These issues, and the possible extension of the der- ivation of the inequalities beyond the two-body in- teraction model, and the detailed comparison of the many ensuing mass inequalities with hadronic mass data have been discussed elsewhere [ 2 ]. We will de- rive next similar relations for glueball masses.

3 . G l u e b a i l s t a t e s

In the absence of any light quarks the spectrum of QCD would simply consist of color singlet massive glueballs. These glueball states could have quantum number appropriate to two-gluon states ("2g") , e.g., those of

Fa l=,u~rce_n++} l Z V ~ a ( ~ - - v

f a r" .laU2,Of lCP 13+ u,u~ a¢Oc ~a : t . , - } ,

Fa F u o I l c P _ 9 + + } , e t c . u l z - - a t ° - - ~

or of three gluon states (" 3g,, ), e.g.

y a Fbiz~,lTcaprlabc[lCP ~ --+} u,-- -- ~ ,~ - t 1 o r { l - - } ) ,

o r

F~,FbU~FC°pff'ac(JCe={1 +-} or {1++}) .

The existence of the light quarks (and n mesons) causes the lightest glueball states to have appreciable width. It is conceivable, nonetheless, that, as sug- gested by the large Nc limit, the admixture of q~] com- ponents in the glueballs is small and that the glueball spectrum can, in the first approximation, be com- puted by neglecting the quarks. Indeed, extensive es- timates of glueball masses have been done using lat- tice QCD. These calculations now suggest that - as one would guess from a naive constituent gluon model - the lightest state is indeed the S wave, 0 ++, state and not the 2 + + state. Unfortunately, despite exten- sive experimental searches, particularly in the radia- tive J / ~ decays, there is no conclusive evidence for glueball states but future work may pin down in this experiment the lowest "2g" states.

Let us assume that a constituent massive gluon model [5] is a reasonable guideline to the glueball spectrum much in the same way as the constituent massive quark model reasonably explains most of the mesonic q ~ and baryonic qsqjqk states. Let us as- sume also that the system is describable by two-body

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interactions with a color structure appropriate to gluon exchange:

[/'12 = VAI"A2, etc. (9)

Eq. (9) is the exact analog of V~z=21.AzVin me- sons or baryons. The octet of A matrices here are, however, in the adjoint representation ( A a ) b c = f ~ since the constituent gluons are color octets.

The three gluons in the "3g" glueball couple to a color singlet.

A1 + A z + A 3 = 0 (in "3g" glueballs) , (10a)

whereas

Am + A 2 = 0 (in "2g" glueballs). (10b)

Thus by the same arguments as in section 2 above, we find that V~2 in "3g" glueball = ½ Vl2 in "2g" glue- ball. Consequently the same derivation o f the baryon meson mass inequalities applies to the present case leading to

m ° ( " 3 g ' ' ) >~ ~ m ° ( " 2 g ' ' ) . ( 11 )

Again the two-gluon states, "2g" (or appropriate mixtures thereof) appearing on the right-hand side are specified by the three-gluon state "3g" we are considering. In any event the lightest glueball in the "3g" sector should be at least 3 times as heavy as the lightest glueball (which obviously is from the "2g" sector). The latter may, in fact, be quite heavy. If in- deed m ° " 2 g " = 1.5 GeV then the lightest "3g" state may be quite heavy:

m°"3g">~ 2.25 G e V .

Recalling the fact that the mass inequalities, at least in the well-tested meson baryon case, tend to be ful- filled with a ( 10-15 )% margin, and that the 1 . . . . 3g" state need not be the lightest in this sector, we could expect

m°( 1 . . . . 3g")/> 2.6 G e V . (12)

It is amusing to note that if the 1 . . . . 3g" state would get close to 3 GeV then it could mix appreciably with the 1 - - J/q/, cgstate. This in turn could modify some features of this extremely well-studied state [6 ].

It would appear difficult to search for "3g" states in J / q / ~ 2 y + X ° or in qc--,7+X °. The small phase space and extra a factor (or the relatively large had- ronic decay of the ~/¢) tend to suppress the branching

into the desired channel. Nonetheless one can expect that eventually "2g" and "3g" states will be found and that the suggested inequalities will be tested.

Finally we would like to recall that a weaker ver- sion o f the meson baryon mass inequalities mO >_ ~o can be proved via a rigorous approach [7 ] based on comparing the "baryon" and "meson" two- point (Euclidean) function:

(OIB(x)B(O) [O) = (01 (q/,(x)q/j(x)q/k(x) )

• ( q / ; ( o ) q / j ( o ) q / ~ ( o ) ) + 1 o > ,

and

(OIM(x)M(O) I0) = (01 (q/,(x)~:(x))

• ( q / , ( o ) q / A o ) ) + l O > ,

where we suppressed all color and spinor indices. Conceivably one could make similar statements for

3 3 F . . ( x ) F . . ( O ) l ) ( O l F , , ( x ) F u , ( O ) ] ) versus (0 l 2 2 two-point functions (related work on glueball states was done by Muzinich and Nair [8] but it did not specifically concern t he "3g" state).

4. Inequality for quadronium bound states (for the case dominated by the two-body force)

We proceed next to apply the variational method to four-body bound states, it concerns the suggestion that the puzzling narrow resonance found in heavy ion collisions are in some sense e+e-e+e - bound states [9]. The need 1o achieve very strong bindings (B.E.-~me~0.5 MeV) in this, normally weakly (electromagnetically) coupled system, is a serious problem of this approach. It has been suggested that we cannot really have a two-body dominated inter- action in this system and new, unusual, four-body in- teractions are called for.

The following inequality on the binding energy of quadronium in terms of positronium can clearly serve to sharpen the issue. Let q/o = q/o( 1 [ 2 2 ) be the wave function of the quadronium in its grounds state. The hamiltonian is

HQ= T~ + Ti + T2 + T~ + T~

+ V,r + Vj2+ V2~ + V22 + V~2+ VT2. (13)

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Clearly we expect the repulsive electron-electron, V~2, and pos i t ron-pos i t ron interaction, Vi~, to lower the quadron ium binding. Hence,

~°s(Q) -,< g°B(Q) , (14)

with g°s (Q) being binding of a fictit ious hamil to- nian, /Ye, which is the original H e except for the V12, VT~ terms which have been el iminated.

fflQ=Tl + Ti+ T2 + T~+ V,~+ V,~+ V2I+ V2~,

(15)

If we compare these with the two-body hamil tonians,

HI~ = T~ + T~ + V~: , H2: = T2 + T~ + F2~ ,

H21=T2+Ti+V2r , HI i=T~+T~+V~i , (16)

we see that

2/Y e =HIT + H ~ +H'I~ + H ~ , (17)

where H'Ii is obta ined from H ~ by doubling the in- teract ion H'~i=T~+TT+2VH, etc. This can be achieved for the one-photon exchange by doubling the effective coupling strength by ot~ot' = 2a . Eq. ( 17 ) can next be used in an unfamil iar way, to put an up- per bound on g°s (Q) and, hence, via ( 16 ), on e °s (Q) as well. The bound is

g°s (Q) ~< ~ ' °s (P ) ,

with e '°n(P) the b inding o fpos i t r on ium in which the strength of the interact ion has been doubled by a ~ 2 a . Thus we have

E°e(Q) ~< g°B(Q) ~< 2 e ' ( P ) .

The fact that the real pos i t ronium spectrum con- forms so nicely to QED predict ions with radia t ive

1 correct ions included strongly suggests that if a - 137 is scaled up to a ' = ~ 7 we still can treat the e + e - e + e - system with the per turbat ively calculated potential . In this case,

e'°B(P ) -~ ( a ' ) 2. I r a = 2or 2. ½ m - 2 R y = 27 eV,

and hence e°B(Q) ~< 2~ '°s(P) ~< 54 eV, and the bind- ing falls short by about 104 of the required ½ MeV binding.

References

[ 1 ] See, e.g., J.W. Humberton, R.L. Hall and T.A. Osborne, Phys. Lett. B 27 (1968) 19; and also J.A. Thon, Phys. Lett. B 56 ( 1975 ) 217; Nucl. Phys. A353 (1981) 470.

[2] S. Nussinov, Phys. Rev. Len. 52 (1984) 173; and a review for Phys. Rep., in preparation.

[3] J.M. Richard, Phys. Lett. B 139 (1984) 408. [4] E. Lieb, Phys. Rev. lett. 54 (1985) 1987. [ 5 ] See, e.g., J.M. Cornwall and A. Soni, Phys. Lett. B 120 (1983)

431. [6] See, e.g., S. Brodsky, P. Lepage and S.F. Tuan, Phys. Rev.

Lett. 95 (1987) 621. [7 ] D. Weingarten, Phys. Rev. Lett. 51 (1983) 1830. [8] l.J. Muzinich and V.P. Nair, Phys. Len. B 178 (1986) 105. [ 9 ] For an extensive review, see J.J. Griffin, Intern. J. Mod. Phys.

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