Volume 158B, number 5 PHYSICS LETTERS 29 August 1985
GOLDSTONE BOSONS AND SCALAR GLUONIUM
Harald GOMM and Joseph SCHECHTER
Physics Department, Syracuse University, Syracuse, NY13210, ~USA
Received 26 December 1984; revised manuscript received 15 March 1985
We study the model of Goldstone bosons interacting with a scalar gluonium field in such a way that the trace anomaly equation is automatically satisfied. It is argued that the mixing of gluonium and quarkonium fields plays an important role. A possible decoupling of the Goldstone fields from a physical scalar-glueball is argued to be reasonable only if the glueball's mass is extremely low.
The properties of a scalar glueball (see ref. [ 1 ] for an up to date review) are of great interest for under- standing QCD. Recently the possibility of relating the properties of such a particle to the trace (or dilation) anomaly equation has been explored in simple effec- tive lagrangian models [2-5] . Both the case [2,3] of approximating pure (no "matter") QCD by a single scalar field and the case [2,5] where spin 0 matter fields belonging to the (3, 3*) representation of chiral SU(3) were present were treated. The former case was described by a unique lagrangian (up to terms with two derivatives) while the latter was noted to permit an intrinsic choice as to whether the field postulated to dominate the trace of the energy momentum ten- sor Ouu at low energies corresponds to a physical glue- ball or gets eliminated in terms of singlet scalar quark- onium (a situation which appears  to hold for the 7?' particle in the pseudoscalar gluonium channel).
Very recently, l_Anik  has claimed that the mod- els ofref.  do not satisfy a low energy theorem  on the matrix element of Ouu between pion states. He constructs a model designed to satisfy this theo- rem with the unusual result that the (zero mass) pions decouple from the glueball. I f this result could be shown to be true and model independent it would clearly be spectacular, implying possibly very large SU(3) breaking in scalar gluonium decay. This has led us to examine the models of ref.  in more detail.
We find that those models do in fact satisfy the low energy theorem of ref. , although it may not
0370-2693/85/$ 03.30 Elsevier Science PUblishers B.V. (North-Holland Physics Publishing Division)
be obvious at first sight. The model of l_Anik is seen to correspond to a very special case of a model in ref.  which only seems reasonable physically if the scalar glueball is (roughly) of the order of (or smaller than) the pion mass. Otherwise scalar gluoni- um is expected to decay very quickly into two pseu- doscalars. An interesting feature which emerges from our analysis is that if a physical glueball dominates Ouu at low energies, an upper bound may be set for the lightest particle in the scalar channel. The precise value of this bound depends on the numerical value of the QCD gluon condensate.
In our treatment we use the linear rather than the (often more convenient) non-linear chiral realization to describe the spin zero quarkonium fields. The phys- ics of the non-linear model corresponds to taking the physical scalar quarkonium mass to infinity. While this is a good approximation at energies comparable to the pion mass it is clearly very dubious when deal- ing with a scalar gluonium state which is generally expected to have a mass comparable to that of scalar quarkonium.
For orientation first consider the case when no quark matter is present. With the convenient normali- zation of a scalar gluonium field H = - [[3(g)/g] X Tr(Fuv Fur), Fur being the QCD gauge field and fl(g) the renormalization group function, we want to find a model which satisfies the anomaly equation 
~ D = Ouu =H. (1)
Volume 158B, number 5 PHYSICS LETTERS 29 August 1985
Here Du is the dilation current. The appropriate la- grangian is 
= - aH-3/2(a H) 2 - } Hln (H/A4), (2)
where A is a constant of mass dimension one and a is a dimensionless constant. The dimension one glue- baU field h is defined in terms of the fluctuation around the minimum ((/-/) = A4/e) as H = (H) + Zh. The quantity Z is given in terms of the glueball mass m b by Z = 2rnh(H)l/2 while a = (H)1/2[4m2h . (H) is -4 times the vacuum energy density and is thus given by 0.00177 GeV 4 according to the naive interpreta- tion of the bag model  or 0.0135 to 0.034 GeV 4 according to the QCD sum rule  approach. The model may be presented in a number of equivalent ways; for example the substitution H = (H) eX con- verts it to the model of ref. .
Now consider the situation  when scalar and pseudoscalar mesons are added . These belong to the matrix M which transforms as M-+ ULMU~ R un- der left unitary (UL) and right unitary (UR) transfor- mations. We will discuss two possibilities:
(model A): A = -- Tr(ao m~ m t)
- VI(M,H ) - V2(M,H), (3)
(Model B): f 'B = A - aH-3/2(~ H)2" (4)
The "potential" terms V 1 and V 2 are assumed to con- tain no derivatives. VI(M, H) is both chiral and scale invariant. For example, with the conventional assign- ment of dimension 1 to M,
V 1 = c 1 H 1/2 Tr(MM t) + c 2 Tr [(MM*) 2 ]
+ c 3 H- 1/2 Tr [(MM "~ )3 ] + .... (5)
V2(M, H) is chiral invariant and is constructed in such a way  that eq. (1) holds. (For examlile the sec- ond term of eq. (2) by itseld is a possible choice.) We are neglecting quark masses and also possible mixing with ~q?qq fields We assume that the chiral symmetry is spontaneously broken. This puts an important im- plicit restriction on the total potential V = V 1 + V 2. If V were scale invariant, chiral symmetry could not be spontaneously broken since there would be no way for the pion decay constant Frr to be computed out of dimensionless quantities. In the present model F~r is some multiple of A defined in eq. (2). Model A
contains no "kinetic" term for H, which then gets eliminated in terms of scalar singlet quarkonium by its equation of motion. In this model the tightest im- portant state for the operator Tr(F~ Fur ) is of 7:lq type.
We will work here with a simplified "toy" theory in which the only matter fields present are the pion and a scalar singlet o. (The more general results will be indicated at the end.) Thus we expand M = (o + ixdp)/x/~. It is actually unnecessary to specify the form of V. We need just the chiral symmetry relation
(~ V/ao) k - ( ~ V/ag~k ) o = 0, (6)
and the scale anomaly equation
H=-4V +Op.~V/adp+ oaV/ao + 4HaV/~H. (7)
First, it is interesting to demonstrate that the mod- els here do satisfy the low energy theorem  for zero mass pions:
= -2p 'p ' + higher order terms. (8)
Eq. (8) holds because the general form of the matrix t t . [ . t
element of0ov ]s ot[(p - p )o (p - p )v 2p 'p 6or ] .1. t t . f3(n + p ~ (n + p ~ , while the normalization
xt - /O ~ ' - JV
- - i f d3x04# =Po gives fl = 1[2 + O[(p'p')21 and the fact that 0or is a chiral singlet implies that the matrix element should vanish when p' ~ 0 so a = -/L We see that the derivation of (8) just uses chiral symmetry and is independent of QCD; it is also irrelevant which version of the energy momentum tensor is taken. Now consider model A in eq. (3). We can assume that H has been eliminated so we are simply left with a chiral (but not scale) symmetric theory constructed out of ~and o. It is trivial to see that the trace of the canon- ical energy momentum tensor satisfies (8), but here we should employ the "new4mproved" tensor [11 ] since it is the one for which a o D o = 0uu. The trace of this tensor is easily found to be
Oou = -4 V + #O"  # + o  o. (9)
The first two terms on the right make no contribu- tion between massless pion states. All is not lost since the third term ~ (o)D o = (F,r[v~)r-1 o contributes through a o-pole. One needs the mrrr coupling con- stant which is obtained by differentiating (6) with respect to both o and ~ and evaluating it at the mini-
Volume 158B, number 5 PHYSCIS LETTERS 29 August 1985
gcr~rTr = (03 V/OO1r+O~r-) = v~m2a IF , (10)
where m 2 = (02 V/Oo 2) is the o-mass and F~r "~ 135 17
MeV. Using (10) then gives for the matrix element of (9) between pion states
m2(p _ p,)2/[m2 + (lO _ p,)2]
= -2p 'p ' + higher order terms,
in agreement with eq. (8). In the case of model B, we have verified eq. (8) by taking an additional term 4H O V/OH in (9) into account. However we shall not give the details.
It should be remarked that in model A the o ~ar- ticle has the tremendous width P(o ~ rrrr) ~ 3m~/ 16zrF 2 (about 1.7 GeV for m o ~ 0.8 GeV) charac- teristic of the old sigma model and not [1 ] qualita- tively dissimilar to the experimental situation. This was discussed many years ago by Crewther  and by Ellis .
Now let us discuss model B [eq. (4)] which turns out to have a fascinating structure. Both a scalar mat- ter field o and a scalar glue field h are present. These are required to mix, giving physical fields Op and hp :
o~ ( cos0 s in0) .
(h/= _sin 0 COS 0 / (~ p ) (11) P
The structure of this mixing follows from the scale anomaly equation (7); differentiating in one case with respect to H and in another case with respect to o yields the two equations:
1 = ((o)/Z) (0 2 V/OoOh) + (4(H)/Z 2) (0 2 V/0h2),
0 = (o) (0 2 V/Oo 2) + (4(H)/Z) (0 2 V/OhOo). (12)
This determines the o -h mass squared mixing matrix to be
A -B },
-B C + B2/A (13)
where A = (02 V]Oo2), B = -(02 V/OoOh) = ZF~r A~ 4X/r2(H) and C = Z2/4(I-1). Notice that if A, the "bare" scalar quarkonium squared mass (very roughly m 2) and (/-/) the strength of the gluon condensate are con- sidered known, (13) depends only on the parameter Z which is proportional to a "bare" gluonium mass
[see e q. (2)]. Diagonalization of (13) yields
tan 20 = 2B/(A - C - B2/A),
m2(op,hp) = ~(A + C+B2[A + 2B[sin 20). (14)
Differentiating (7) with respect to twice shows that the bare gluon field h must have a coupling to two pions, related to goTr~r in (I 0) by gh~r. = -ZFTrgo"/ 4X/~(H). The coupling constants of the physical par- ticles are then
gap.Zr = ga.Tr [cos0 + (B/A )sinO ] ,
ghp.. =go.. [sin0 - (BIA )cos O ] . (15)
It is interesting to consider the formal limit of our ex- pression when the bare o squared mass,A, gets very large (non-linear o-model limit). Then m2(Op) goes to infinity as A. (1 + B2/A 2), m2(hp) ~ C/(1 + B2/A 2) and sin 0 ~ B/(A 2 + B2)1/2. Note that gl~,r~r -* 0; the scalar field of finite mass decouples fronfthe Gold- stone bosons. At first this appears to be a mechanism for hiding scalar gluonium. However it does not seem physically realistic to consider the bare quarkonium mass (around 1 GeV) to be very large compared to the gluonium mass. Nevertheless one would like to be certain that there is no scalar glueball lurking at ex- tremely low energies (say several MeV up to around 2m~r ). It seems that its non-observation in the decay process K + -* rr+hp is a possible strong argument against it.
We can obtain an upper bound on the lighter of the two scalar masses in model B. This is most readily gotten by allowing Z -* ~ in the expression m2(hp)
C/(1 + B2/A 2) just obtained whenA -* oo:
< 2x/~(H)I/2/F. (16) mlighter
The LHS is written as mlighter since the same bound holds even if m(hp) > m(Op). It is amusing that this bound depends only on the glue and matter conden- sates in the effective models and is independent of the bare masses. In the present "toy model" this bound is not restrictive if the QCD sum rule determinations of 0rt) are used. If, for definiteness we fixA = 1 GeV 2 and (H) = 0.0135 GeV 4 we find that the heavier scalar particle will always have a width greater than its mass when its mass is less than 1.6 GeV. Corresponding to heavier particle masses of 1.02 GeV, 1.12 and 1.60 GeV the lighter particle masses (widths) are predicted to be
Volume 158B, number 5 PHYSICS LETTERS 29 August 1985
0.44 (0.013), 0.74 (0.48) and 0.88 (1.98) GeV. It seems instructive to compare our model with
ref.  by making the non-linear substitution M = F~ U, U = exp (2i$/F~) in eqs. (3) and (4). The ki- netic term becomes the standard form of the non- linear model ~ 2 - (~ F~) Tr(a~ Ua~ Ut) and the poten- tial term collapses drastically. For example (5) be- comes (noting UUt = 1),
=1 c F2H1/2 + t__ c F6H-1 /2 + .... V1 41 Ir 643 7r
In other words, the scale invariance of V 1 is lost. As discussed previously, this is absolutely necessary in order that chiral symmetry break spontaneously. Fur. thermore it is trivially seen that the Goldstone fields decouple from the glue field H in this limit. Thus the model of ref.  is just a (probably unrealistic) spe- cial case of the earlier model in ref. . After the work of this paper was completed we received a re- port by Ellis and L~lnik  which presents a some- what different model along the lines of ref. . They also find a large width for heavy scalar gluonium to decay into two pions. Their model is however differ- ent from the present one. It uses the non4inear rather than the linear sigma model and does not include the effects of glue-matter mixing. An interesting aspect of the present model (with a linearly transforming matter multiplet) is that, as briefly noted after eq. (5), it displays a scenario for the way in which, according to the usual assumption, QCD makes the dimensional parameter F,r emerge as a function of a characteristic scale [say A in (2)]. The effective lagrangian which enforces the trace anomaly is not only a measure of confinement (since it gives a non-perturbative vacuum of lower energy than the perturbative one) but, be- cause of glue-matter mixing, also provides a scale so that (o> = F~/x/~ 4= 0 can emerge as the minimum of a chiral invariant potential.
In the present note we have concentr~ited on some conceptual aspects of the gluonium-Goldstone boson system using a "toy" model which exhibits a number of interesting and characteristic features. Elsewhere we will study a more realistic model and compare with experiment. Here we would just like to remark that the mass mixing matrix which results when one includes three flavors and assigns scale dimension three in a consistent way to the Ftq mesons is identical to (13) apart from numerical factors. This lowers the upper bound on the fighter mass to
16 (H) m2<