Physics Letters B 577 (2003) 61–66 www.elsevier.com/locate/physletb The low-lying glueball spectrum Adam P. Szczepaniak a, Eric S. Swanson b,c a Department of Physics and Nuclear Theory Center, Indiana University, Bloomington, IN 47405, USA b Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA c Jefferson Lab, 12000 Jefferson Ave, Newport News, VA 23606, USA Received 2 September 2003; accepted 4 October 2003 Editor: H. Georgi Abstract The complete low-lying positive charge conjugation glueball spectrum is obtained from QCD. The formalism relies on the construction of an efficient quasiparticle gluon basis for Hamiltonian QCD in Coulomb gauge. The resulting rapidly convergent Fock space expansion is exploited to derive quenched low-lying glueball masses with no free parameters which are in remarkable agreement with lattice gauge theory. 2003 Published by Elsevier B.V. 1. Introduction Previous models of glueballs have relied on ad hoc effective degrees of freedom such as flux tubes [3], The scalar glueball has been called the fundamental bags [4], or constituent gluons [5]. We note that some particle of QCD [1]. Indeed, its existence and nonzero of the models listed in Ref. [5] construct states with mass are a direct consequence of the non-Abelian massive gluons, while others either use transverse glu- nature of QCD and the confinement phenomenon. It ons or dynamically generated gluons masses. Models is clear that finding and understanding the scalar (and in the former category contain spurious states due to other) glueballs is a vital step in mastering low-energy the presence of unphysical longitudinal gluon modes. QCD. Sum rule computations of glueball properties exist [6], Recently quenched lattice gauge theory has been however, they are based on phenomenological proper- able to determine the low-lying glueball spectrum ties of the spectrum. Finally, the conjectured duality with reasonable accuracy [2] (only very preliminary between supergravity and large-N gauge theories has determinations of other matrix elements have been been used to compute the glueball spectrum in non- attempted). These data serve as a useful benchmark in supersymmetric Yang–Mills theory by solving the su- the development of a qualitative model of the emergent pergravity wave equations in a black hole geometry properties of low-energy QCD. The models may then [7]. Unfortunately all of these approaches suffer from be used to guide experimental glueball searches. weak or conjectured connections to QCD. We present a computation of the positive charge conjugation glueball spectrum which arises from QCD E-mail address: [email protected] (A.P. Szczepaniak). and is systematically improvable. The computation 0370-2693/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.physletb.2003.10.008 62 A.P. Szczepaniak, E.S. Swanson / Physics Letters B 577 (2003) 61–66 is based on the formalism presented in Ref. [8] with the variational parameter ω(k) determined by in which the QCD Hamiltonian in Coulomb gauge minimizing the vev of H . The correction terms in is employed as a starting point. Coulomb gauge is δH include V3g and V4g which are the three- and efficacious for the study of bound states because all four-gluon operators originating from the difference degrees of freedom are physical (there are no ghost between the full and the Abelian chromomagnetic fields in this gauge) and a positive definite norm field. VJ denotes a contribution from the Faddeev– exists [9]. Furthermore, resolving the Coulomb gauge Popov determinant in the kinetic term. The effects constraint produces an instantaneous interaction (the of VJ and the Faddeev–Popov determinant in the non-Abelian analogue of the Coulomb interaction) functional integrals have recently being studied in which, as shown in Ref. [8] may be used to generate Ref. [10] where it was found that its effects can be bound states. Because the temporal component of the effectively included in the variational parameter ω(k). vector potential is renormalization group invariant in Finally, VC is the difference between the Coulomb Coulomb gauge (this is not true in other gauges), operator and its vev, K0. In the calculation of the the instantaneous potential does not depend on the glueball spectrum it results in operators mixing two ultraviolet regulator or the renormalization scale [12]. and three, quasiparticle wave functions. We note that This fact permits a physical interpretation of the after renormalization the coupling g is absorbed into instantaneous potential which is a central aspect of our the Faddeev–Popov operator, which then defines the formalism. Coulomb gauge analog of a ghost propagator [8, The pure gauge QCD Hamiltonian may be written 10]. The renormalized effective potential K0 is fixed as [9] HQCD = H0 + δH with by comparing with the quenched lattice QCD static potential. A very accurate representation of the static 1 2 2 H0 = dx E + B confinement potential is achieved [8]. 2 The variational vacuum defined above also spec- 1 ifies a Fock space of quasiparticle excitations cor- + dx dy ρa(x)K(0)(x − y)ρa(y) (1) 2 responding to effective gluonic degrees of freedom (which we call quasigluons). Such quasigluons obey and “massive” dispersion relation in the variational vac- uum and therefore improve the description of gluonic δH = V3g + V4g + VJ + VC. (2) bound states since mixing between states with differ- Here B =∇×A is the Abelian part of the chromo- ent number of quasiparticles is suppressed due to their magnetic field and E =−∂/∂tA is the chromoelec- effective mass. tric field. The third term in H0 represents the non- The calculation of the vev of the Hamiltonian and Abelian, instantaneous Coulomb interaction between the properties of the quasiparticle excitations were color charges, ρa =−f abcEb · Ac, mediated by an ef- discussed in Refs. [8,10,11]. These require solving a fective potential K0 computed by taking a vacuum ex- set of coupled integral Dyson equations and as a result pectation value of the Coulomb kernel, one finds that the function ω(k), which in a free theory is given by ω(k) = k, becomes finite as k → 0. The K0(x − y)δ ab value ω(0) can be related to the slope of the static = 2 | ∇ · −1 −∇2 ∇ · −1 | potential at large distances. g Ψ ( D) ( D) x,a;y,b Ψ , (3) with Dab = δab∇ − gf abcAc being the covariant 2. Fock space expansion and the glueball derivative in the adjoint representation. For the vac- spectrum uum wave functional, Ψ [A]=A|Ψ we take a varia- tional ansatz, The quasigluons which emerge in the analysis of 3 Ref. [8] set the QCD scale parameter via the low 1 d k → = [A]= − a a − momentum dispersion relation r0ω(k 0) 1.4, Ψ exp 3 A (k)ω(k)A ( k) , (4) 2 (2π) where r0 is the lattice Sommer parameter. Using the A.P. Szczepaniak, E.S. Swanson / Physics Letters B 577 (2003) 61–66 63 Regge string tension or ρ mass to fix the scale then with gives ω(0) = 600–650 MeV. It is natural to interpret 3J 2 + 3J − 2 this scale as a dynamical gluon mass.1 Thus the K = V 11 − + J formalism of Ref. [8] provides a justification of a (2J 1)(2J 3) − Fock space expansion in terms of quasigluons and J(J 1) + VJ −2 gives the leading instantaneous interaction between 2(2J − 1)(2J + 1) the quasigluons. In view of this it is natural to attempt (J + 1)(J + 2) + VJ + , (6) a description of low-lying glueballs in the pure gauge 2(2J + 3)(2J + 1) 2 sector of QCD. 3(J + 2)(J − 1) In this approach positive charge conjugation glue- K22 = VJ (2J − 1)(2J + 3) balls are dominated by the two quasigluon contribu- (J + 2)(J + 1) tion. These may mix with three and higher quasigluon + V − + − J 2 states via transverse gluon exchange (and, in general, 2(2J 1)(2J 1) − via any term in δH). Mixing with single quasigluon J(J 1) + VJ +2 (7) states is excluded because color nonsinglet states are 2(2J + 1)(2J + 3) removed from the spectrum due to infrared diver- and gences in the color nonsinglet spectrum [11]. Finally, the scalar glueball is orthogonal to the vacuum due to K = K 12 21 the form of the gap equation. = (J − 1)J (J + 1)(J + 2) The resulting bound state equations are shown in 1 Eq. (5). There is one orbital component of glueball × V + + + + J 2 wave function for J P = 0 and two for J P = (even 2(2J 3)(2J 1) 2)+. These are denoted by ψ (k), i = 1, 2. The first 1 i + V − + − J 2 term on the right-hand side of Eq. (5) represents the 2(2J 1)(2J 1) quasigluon kinetic energy (the gluon gap equation has 1 − VJ . (8) been employed to simplify the expression), the second (2J + 3)(2J − 1) term is the quasigluon self energy, and third represents the interaction between quasigluons in the channel of The bound state equations for other glueballs are as interest. in Eq. (5), with the exception that the wave function index takes on a single value. For these cases the 2 k dk 2 interaction kernels are as follows: 2ω(k)|ψi(k)| (2π)3 J P = (odd 3)+ (there is no 1+ gg glueball): 2 2 + NC k dk q dq ω(k) J + 2 J − 1 3 3 K = V − + V + ; (9) 2 (2π) (2π) ω(q) J + J 1 J + J 1 i 2 1 2 1 − 4 2 2 J P = (even 0) : × V + V |ψi(k)| 3 0 3 2 J J + 1 N k2 dk q2 dq (ω(k) + ω(q))2 K = VJ −1 + VJ +1; (10) − C 2J + 1 2J + 1 3 3 4 (2π) (2π) ω(k)ω(q) + ∗ J P = 0 : × ψ (q)Kij (q, k)ψj (k) i 2 2 V2 k dk 2 K = V0 + .