A note on the glueball mass spectrum

A conjectured duality between supergravity and N = ∞ gauge theories gives predictions for the glueball masses as eigenvalues for a supergravity wave equations in a black hole geometry, and describes a physics, most close toa hightemeperature expansion of a lattice QCD. We present an analytical solution for eigenvalues and eigenfunctions, with eigenvalues given by zeroes of a certain well-computable function r(p), the zeroes of which signifies that two solutions with desired behaviour at two singular points become linearly dependent. Our computation shows corrections to the WKB formula m2 = 6n(n + 1) for eigenvalues corresponding to glueball masses in 3 dimensional QCD, and gives the first states with masses m2 = 11.58766; 34.52698; 68.974962 ; 114.91044; 172.33171; 241.236607; 321.626549, . . . . In QCD4, our computation gives squares of masses 37.169908; 81.354363; 138.473573; 208.859215; 292.583628; 389.671368 500.132850, 623.97315 . . . for O++. In both cases, we have a powerful method which allows to compute eigenvalues with an arbitrary precision, if so needed, which may provide a qualitative test for the duality conjecture. Our results matches well with the numerical computation of [4] withing precision reported there in both QCD3 and QCD4 cases. As an additional curiosity, we report that for eigenvalues of about 7000, the power series, although convergent, has coefficients of orders 1034; and therefore tricks were used to get reliably the function r(p), as also the final answer gets small, of order 10−6 in QCD4. In principle we can go to infinitely high eigenavalues at an expence of computer sufferings; although computation will slow down to make it inpractical eventually to do it in this way, and it is probably some kind of corrections are expected anyway whenever fine cancellation of big terms is what produces higher eigenvalues. ∗Dept Math., UCR, Riverside, CA 92521;[email protected]