VOLUME 57, NUMBER 11 PHYSICAL REVIEW LETTERS Improved Estimate of the Scalar-Glueball Mass 15 SEFTEMBER 1986 Apoorva Patel Physics Department, University of California, San Diego, La Joiia, California 92093 Rajan Gupta and Gerald Guralnik(') Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 8?545 Gregory W. Kilcup Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138 and Stephen R. Sharpe(b) Physics Department, University of Washington, Seattle, Washington 98195 (Received 12 May 1986) We present results for the 0++ glueball mass ( mo) and the string tension (o ) in pure gauge lat- tice QCD with a four-parameter improved action. We suggest and confirm that previous estimates for ma/v o have been too low because of the influence of the unphysical and nonuniversal phase structure in the fundamental-adjoint coupling plane. %e find that ~=1200-1400 MeV using Vcr 420 MeV. PACS numbers: 12. 38. 6c, 11. 15. Ha Foremost among the aims of lattice quantum chro- modynamics (LQCD) is to predict the spectrum of glueballs and other exotic states. A major step towards this goal would be a calculation of the glueball spec- trum in lattice SU(3) without dynamical fermions. If the dominant effect of adding dynamical fermions is to shift the value of the bare coupling constant, then the spectrum deduced from the pure gauge theory could be directly confronted with experiment, except for the effects of decays and mixing. Even this lesser goal of establishing the spectrum in the pure gauge theory is far from being attained. The only measurements of glueball correlators on a reason- ably large lattice and with signals which unambiguous- ly expose the lightest glueball state have been those of de Forcrand et al. , ' who use the source method. Ex- pressing their results as a ratio of the scalar-glueball mass (mG) to the square root of the string tension (v cr) as determined by the source method in the same ensemble, 2 one finds mG/v tr =1. 96(7), 2, 45(12), and 2.65(18) for couplings on the Wilson axis 6/g =5. 5, 5.7, and 5.9, respectively. Thus, there are significant scaling violations in mG/Jo. for the range of couplings investigated along the Wilson axis. If we nevertheless take mG/Jo =2. 0-2.65, a range suggested by earlier numerical work3 and strong-coupling expansions along the Wilson axis, as well as the above, then we obtain an estimate mG = 850-1100 MeV, using the identifica- tion Ma=420 MeV. This result may be hard to reconcile with the data on m m phase shifts, s although a very recent analysis6 of these and other data has sug- gested a possible extra state at about 980 MeV. Clear- ly it is of great phenomenological importance to firm up the lattice prediction before including dynamical fermions. In recent years, a great effort has been devoted to Monte Carlo renormalization-group (MCRG) study of LQCD in order to determine the value of the Wilson- axis coupling beyond which there is scaling. 7 s For KF around 6. 0, the string tension and deconfinement- transition temperature scale roughly in accord with the nonperturbative P function, but the glueball mass does not. In this Letter we suggest a reason why mG/Vcr has not been exhibiting scaling, and why it may have been underestimated by calculations done along the Wilson axis. In confirmation, we present the results of a computation which removes at least part of the prob- lem, suggesting that mG/Vcr=3. 0(3). In physical units this corresponds to ma=1200-1400 MeV, a value certainly consistent with the experimental data. We consider here pure gauge LQCD with the gener- ic action S[Ul = X, K Re Tr(U ), (1) where the U are various Wilson loops in different representations. Most simulations of this action have been performed in the plane defined by the fundamen- tal (or "Wilson" ) and adjoint representations of the plaquette. In this plane it has been determined that there is a line of first-order phase transitions that ap- proach the Wilson axis from above and which ter- minate above the Wilson axis. This line of transitions, if continued beyond the end point, would intersect the Wilson axis at KF=5.6. Several numerical studies 1986 The American Physical Society