P H Y S I C A L R E V I E W D V O L U M E 2 3 , N U M B E R 5 1 M A R C H 1 9 8 1

Properties of scalar gluonium

A. Salomone, J. Schechter, and T. Tudron Physics Department, Syracuse University, Syracuse, New York 13210

(Received 27 October 1980)

We examine the properties of a scalar glueball in a simple effective-Lagrangian model which satisfies both the axial-vector and trace anomaly equations of quantum chromodynamics. The scalar-glueball decay pattern can be predicted if its mass is specified and may present some unusual features. In this approach a characteristic mass scale is directly related to the vacuum energy density which automatically comes out to be of a nonperturbative type.

Recently there has been a great fascination for the putative glueballsl of quantum chromodynamics (QCD). Most dynamical approximation schemes lead one to expect a complete spectrum of these objects but it seems very difficult to predict their proper- t ies in a quantitative way. In addition, there is no convincing experimental evidence for thew ex- istence.

In the present paper we shall not attempt to prove that a glueball is present in the QCD spec- trum, but shall assume that a J ~ ~ = O++ glueball field plays a major role in the low-energy effective QCD Lagrangian and examine i ts properties. The Lagrangian i s a generalization of a linear a model (which accounts reasonably for low-energy chiral dynamics) to include a 0" glueball field H and a 0-+ glueball field G. Fo r convenience these a re normalized so that

( l a )

( lb)

where JE i s the U(1) axial-vector current and O , , i s the energy-momentum tensor. We a r e assum- ing, for simplicity, that the quark mass te rms a r e negligible and we a r e living in a world of three flavors. The "matter" fields a r e described by a 3 x 3 matrix

which transforms a s the (3,3*) representation of ~ ( 3 ) x U(3). S contains a scalar nonet and $ a pseudoscalar nonet.

A Lagrangian containing just G and M which satisfies ( la ) automatically by virtue of the equa- tions of motion has recently received some atten- tion. It may be written2 a s

where V , i s a U(3) x U(3) - invariant function of M and M~ and k is aposit ive constant. (We have omitted "quark mass" te rms and se t the vacuum

angle 0 equal to zero.) The interest in (3) derives from the fact that it provides a way to give the 7' meson (which dominates the quantity 1ndetM - In detMt in lowest order) a mass even without quark mass terms. This is closely related to Witten's idea3 that the 77' behaves a s a Goldstone boson in the large N, (= number of colors) limit with squared m a s s of order 1/~,. A crucial feature of (3) i s that no kinetic te rm is present for G. Thus the equation of motion for G leads to i t s elimina- tion in t e rms of 77'. For this to happen the pres- ence of the k ~ ' te rm i s vital; it will also be seen to play an important role in what follows. The Om+ glueball field which seems to be important a t low (- 1 Gev) energies in fact becomes identical to the (mathematical) 7' . This mechanism is very differ- ent from a situation in which there a r e two separ- ate states which mix with each other and in which two states of different mass must remain. It has recently been formulated a s a kind of Higgs mech- anism4 for the topological gauge field A,,, to which G i s related and this may lead to some further in- sights. Note that we do not wish to rule out the possibility of some other 0-+ physical glueball ex- citations, but we expect them to be analogous to quark radial excitations o r exotics and perhaps not relevant a t low energies.

In order to extend this Lagrangian so that ( lb) i s satisfied automatically it was shown5 that the second and third te rms of (3) should be modified to be scale invariant (the fourth te rm i s already scale invariant) and the additional t e rms

should be added. Here the R , a r e homogeneous functions of dimension m of the fields and the C, a r e arbitrary constants. A i s a constant of dim- ension mass. One immediate question is whether the 0" glueball H should have a "kinetic term"

1143 0 1981 The American Physical Society

1144 A . S A L O M O N E , J . S C H E C H T E R , A N D T . T U D R O N

and remain in the theory or should get eliminated a s is the case for G. We expect that H should re- main because, unlike G, it i s not needed to sac- rifice itself to give mass to a scalar SU(3) singlet (which already has mass after spontaneous break- down of chiral symmetry). Furthermore, accord- ing to the I/N, approximation method, the 0" chan- nel is not expected to have any unusual features s o the corresponding meson-glueball mixing should vanish to leading order. In any event, the assump- tion that H remains i s the assumption with the most interesting experimental consequences. Then, putting together the modified (3), (4), and a scale-invariant kinetic te rm for 8, we have our desired effective Lagrangian which satisfies both ( l a ) and (lb). Rather than trying to analyze it al l a t once we shall break the model up into two stages:

Stage I . Only a single glueball field H is present. This is expected to give a dominant contribution to the vacuum amplitude and should determine the vacuum structure of the theory. (We a r e neglecting 0 dependence of the vacuum.)

Stage 2. The glueball field G and the matter field M a r e added so a s to be consistent with (la). It i s assumed that only the minimal necessary modi- fication of the stage 1 model i s to be made.

Clearly additional degrees of freedom-both of gluonic and quark type-can be added in an attempt to get a more accurate portrayal of low-energy phenomena.

It i s interesting that the stage 1 Lagrangian satisfying (4) i s unique:

Here a i s a dimensionless constant and it i s rea- sonable to identify A a s a particular kind of QCD scale parameter (although we will not make here the connection with the phenomenological value). The unusual form of the f irst te rm a r i se s from the need to keep it scale invariant. A Lagrangian similar to just our second term alone has been discussed by a number of au thom6 However, they have considered7 the relevant degrees of free- dom to be the usual Yang-Mills field strengths F, related to Ii by

where P i s the renormalization-group function of QCD. In contrast, we have assumed only one scalar glueball degree of freedom to be most im- portant and have, in addition, subsumed possible complicated low-energy behavior of P(g) in H it- self. In order for (5) to be reasonable it must lead to a negative vacuum energy density which should be equated to the negative of the bag pres-

This would enforce a situation where the "bag-bub- ble" inside vacuum (perturbative) with zero energy lies higher in energy than the outside vac- uum which our effective Lagrangian should de- scribe. It i s easy to verify that the "potential" in (5) does have a negative minimumg a t

s o the f i r s t consistency test i s passed. Combining (6) and (7) predicts that A should be 245 MeV. The effect of adding light quark fields will be seen to push it higher. Note that our prediction of a non- trivial "outside" vacuum follows almost immedi- ately from (lb).

The field h, representing a possible scalar parti- cle, is defined in te rms of the "fluctuation" around (H) a s

where Z i s a constant determined so the kinetic te rm of h i s precisely -$(a,h)2. Substituting (8) into (5) and expanding in a power ser ies in Zh gives a mass term for h a s well a s self-couplings of a l l orders. The various quantities a r e related a s

Thus specifying the mass of h, m, determines 6: completely since (H) i s related to the bag pres- sure by (6). The terms trilinear in h , for ex- ample, a r e

It is seen that since r n , / ( ~ ) ' ' ~ is probably in the range 5-10, the h self-couplings will be rather strong. There is no immediate physical relevance since we have not yet coupled in matter fields. One might conceivably attempt to identify h with the Pomeron and regard (10) a s the triple-Pomer- on interaction in some approximation, but we shall not pursue this approach here. It seems most reasonable to go to stage 2 and investigate h decay modes.

Our stage-2 Lagrangian i s a modified sum of (3) and (5):

The ? term in (11) i s now to be considered scale

23 - P R O P E R T I E S O F S C A L A R G L U O N I U M 1145

invariant (actually this te rm is irrelevant if we use a nonlinear realization for M). The crucial kG2 term in Eq. (3) has also been modified to be scale invariant. There a r e two characteristically different ways of doing this, dividing either by a dimension-four glue field o r by a dimension-four combination of matter fields. The rat io (kW/k') of these contributions has an important effect on the gluonium decay modes, a s we shall see. Finally the job of satisfying the trace anomaly i s now being distributed between the last two terms; the ratio of matter to glue contribution i s given by b/(l - b) which in lowest-order perturbation theory i s

Note that the last te rm in (11) i s chiral ~ ( 3 ) x U(3) invariant and using lndet = T r In can be seen1' to be the leading-order-in-N H-matter coupling term of suitable type.

As a s tar t consider the shift in the vacuum due to the new terms. Note that (G) = 0 by parity in- variance. Minimizing the potential in (11) with re- spect to H now gives'' instead of (7)

Here the pion decay constant F, enters from (M) - (~ , /2 )1 . Since ( H ) i s still fixed in te rms of the bag constant by (6) let u s ask what values of b cor- respond to various values of A. Using (12) to de- fine (for vividness) an effective number of flavors N,, we find from (13) the chart

It i s amusing that the effect of increasing the relative matter-field contribution increases A , in reasonable agreement with our expectation. Note that A -m a s N, - 16.5, which i s the critical value for asymptotic freedom.

The coupling of the glueball field h to matter has two characteristic pieces. The most direct is from the last term of (11) which yields, on expan- sion in powers of @ and 3 = S - ds) - S - ( ~ , / 2 ) 1 ,

Here Z i s slightly changed (Z,,," 0.9Zold) from its value in (9) but this i s not very significant for our present purposes. Another coupling of h to matter

a r i s e s when we use the equation of motion for G to eliminate G in t e rms of the dynamical fields of the model. This results in a te rm

x (In dethl - In det M ')' , (15)

where I ,=T~(MM~MM'). Note that the last factor in (15) may be expanded a s

Defining the mathematical [SU(3) - singlet] g1 a s T~@/J?;, we identify the q' squared mass

and then the h interaction terms

The only two-body decay modes which result from (16) a re (when account is taken of g - 77' mixing) gq, qq', and q'g'. In contrast Eq. (14)-allows the qg and g'g' modes a s well a s n.ir and KK final states. The decay pattern resulting from (14) i s in fact just about what one usually expects for the decay of an SU(3)-singlet scalar into two pseudo- scalars. This is modified by (16) which gives special treatment to 17-like decays. How large a modification is determined by the ratio1'.

Eq. (16) 3 F 2 m , 2

Eq. (14) =4b(HJ

[The dimensionless constants k' and k" a r e defined in ( l l ) ] . In the limiting case k"= 0, there would be an enormous width for the decays into gg', as- suming that m , 2 1500 MeV. Such a situation would make h extremely broad and indistinguishable from the background. This i s a conceivable explanation of the puzzle that no scalar glueball has yet been clearly identified. On the other hand, in the limit- ing case kt = 0, there would be no modification of the amplitudes for g-like final particles. What is perhaps most reasonable i s the situation when k" i s of the same order but slightly smaller (on the grounds that we expect pure glue te rms to domin- ate) than k t . Then we would expect the amplitude for h -ggl to be magnified by several times over other final states and this would be a convenient experimental signature. Fo r definiteness we pre-

A . S A L O M O N E , J . S C H E C H T E R , A N D T . T U D R O N

TABLE I. Partial widths of scalar gluonium h to decay into two pseudoscalar particles. The mass of h , mh, and the partial widths l? (h -2 pseudoscalars) are given in units of MeV.

sent explicit formulas for the decay widths of h into two pseudoscalars (making reasonable approx- imations suitable for a f i r s t analysis):

r I. (1 + 20kfl/k')-l.

Here the kinematical factor i s

In deriving (17) we took b from (12) with N,= 3 (the exact value of b i s really not crucial for our discussion) and neglected to modify Eq. (9) to take the smal l b dependence of m, and Z into account. Furthermore, we neglected the contributions from a virtual sca lar matter SU(3) singlet, which from (14) is seen to mix with h by an angle 1/, satisfying

(m, i s the isoscalar mass). Of course we also turned on the pseudoscalar-meson masses and

mixings for kinematics. Once the value of m, (which we a r e unable to predict) i s specified, ev- erything i s determined (for a given r) . The r e - sul ts a r e shown in Table I corresponding to a typ- ical qq' mixing angle 6 , il. - 15". We have taken r = 1 s o the predictions for r (qql ) and r (q tq l ) a r e upper bounds. Clearly h i s predicted to be rather narrow if i ts mass i s less than 1520 MeV (qql threshold) but above this the interesting situation just discussed i s in effect. Incidentally, note that the contributions to the qq mode from (14) and (16) oppose each other.

To summarize, although the construction of a complete low-energy effective QCD Lagrangian i s s t i l l in an exploratory stage, two reasonable and partially interlocking features have been seen to emerge.

(i) A nontrivial vacuum energy density of the cor- rec t type for bag confinement follows rather di- rectly from the trace anomaly equation. The ef- fect of adding mass less quark fields seems to be in the "right direction."

(ii) The decay properties of scalar gluonium h can be predicted if i t s mass i s specified. For m, < 1520 MeV the decay pattern i s about a s expected. However, i f m,> 1520 MeV one expects relatively large part ial widths for h - qgf. The special status of the decay modes into 7-like particles i s a direct result of modifying the pseudoscalar Lagrangian in which the (mathematical) 7' plays the dual role of a matter a s well a s glue field. Symmetry breaking, of course, causes the mathematical q t to acquire a piece of the physical g.

Further discussion of this model will be given elsewhere.

We a r e happy to thank Car l Rosenzweig for help- ful discussions.

23 - P R O P E R T I E S O F S C A L A R G L U O N I U M 1147

'Very recent reviews a r e given by J. Bjorken, Stanford Report No. SLAC-PUB-2372, 1979 (unpublished); J. Donoghue, MIT Report No. CTP 854, 1980 (unpub- lished); J. Coyne, P. Fishbane, and S. Meshkov, Phys. Lett. E, 259 (1980).

'c. Rosenzweig, J. Schechter, and G. Trahern, Phys. Rev. D 21, 3388 (1980); P. Di Vecchia and G. Vene- ziano, Nucl. Phys. m, 253 (1980); P. Nath and R. Arnowitt, Phys. Rev. D g , 473 (1981); E. Witten, Harvard Report No. ~ ~ ~ P - 8 0 / ~ 0 0 5 , 1980 (unpub- lished).

3 ~ . Witten, Nucl. Phys. B156, 269 (1979); G. Veneziano, ibid. M, 213 (1979); P. Di Vecchia, Phys. Lett. 85B, 357 (1979).

4 ~ T u r i l i a , Y. Takahashi, and P. Townsend, Phys. Lett. E, 265 (1980).

5 ~ . Schechter, Phys. Rev. D 21, 3393 (1980). More - references a r e given here.

or example, I. A. Batalin, S. G. Matinyan, and G. Sawidi, Yad. Fiz. 2, 407 (1977) [Sov. J. Nucl. Phys. 26, 214 (1977)l; N. K. Nielsen and P. Olesen, Nucl.Phys.=, 376 (1978); H. PagelsandE.Tomboulis, e'bid.=, 485 (1978); R. Fukuda, Phys. Rev. D z , 485 (1980); R. Fukuda and Y. Kazama, Phys. Rev. Lett. 45, 1142 (1980).

l ~ y a consistency check we can derive the Pagels- Tomboulis Lagrangian (see Ref. 6) by our method. This shows that it follows from scaling arguments alone. Here we wish to explicitly display the function

p(g)=dg/dt, where t =f L ~ F ~ / A ~ . We would like to satisfy the anomaly equation O,, =- [p(g)/2g3]F2. (For comparison we have rescaled A, -A, / g ) . Noting that - ( p / 2 g 3 ) F 2 = F 4 ( d / d F 2 ) ( l / g Z ) we must find a V satisfy- ing & E I V / B F ~ - 4 ~ = F ~ ( d / d F ~ ) ( l / g ~ ) . The solution to this i s v=F2/4g2 (which i s their result) plus an arbitrary multiple of F2.

8 ~ y our method of construction O,, = H = 4V aV/aH - 4V. Since (aV/8H)= 0 we immediately have (H)= -4 ( V ) = (O,,). Cur value of B i s taken from J. Donoghue, Ref. 1.

'see also H. B. Nielsen, Phys. Lett. E, 133 (1978). 'OS. Coleman and E. Witten, Phys. Rev. Lett. 45, 100

(1980). " ~ o t e that we a re neglecting the contributions to the

vacuum energy density from quark terms in the La- grangian. It has been argued by Shifman, Vainshtein, and Zakharov (see Ref. 13) that the heavy-quark con- tributions to @,, can be subsumed in H if P i s modi- fied. Thus we a re left withClight ,,UaPI18 WI ( q q ) . The u- and d-quark terms a r e completely negligible but the s-quark piece i s not. Perhaps the s quark should be considered heavy for discussing vacuum properties but otherwise light,

l 2 ~ o t e that (H)/ (Iz) = 4~/(3~:/16) - 20. This i s con- sistent with the idea of the 1 / ~ , approximation, since we expect (H) = 0 (N,) while (I2) = 0 (N,-I).

1 3 ~ . Shifman, A. Vainshtein, and V. Zakharov, Phys. Lett. E, 443 (1978).