We use the light-front Hamiltonian of transverse lattice gauge theory to
compute from first principles the glueball spectrum and light-front
wavefunctions in the leading order of the 1/N_c colour expansion. We find
0^{++}, 2^{++}, and 1^{+-} glueballs having masses consistent with N_c=3 data
available from Euclidean lattice path integral methods. The wavefunctions
exhibit a light-front constituent gluon structure.Comment: 4 pages, 2 figures, uses macro boxedeps.tex, minor corrections in
  revised versio

B. van de Sande

C. Amsler

C. Morningstar

G. Bali

H.-J. Pirner

J. Sexton

K. G. Wilson

M. Burkardt

M. J. Teper

R. de Mello Koch

S. Dalley

W. A. Bardeen

English

arXiv

We use the light-front Hamiltonian of transverse lattice gauge theory to compute from first principles the glueball spectrum and light-front wavefunctions in the leading order of the 1/N_c colour expansion. We find 0^{++}, 2^{++}, and 1^{+-} glueballs having masses consistent with N_c=3 data available from Euclidean lattice path integral methods. The wavefunctions exhibit a light-front constituent gluon structure.We use the light-front Hamiltonian of transverse lattice gauge theory to compute from first principles the glueball spectrum and light-front wavefunctions in the leading order of the 1/N_c colour expansion. We find 0^{++}, 2^{++}, and 1^{+-} glueballs having masses consistent with N_c=3 data available from Euclidean lattice path integral methods. The wavefunctions exhibit a light-front constituent gluon structure

Dalley, S.

van de Sande, B.

CERN Document Server

Glueball calculations in large-Nc gauge theoryS. DalleyTheory Division, CERN, CH{1211 Geneva 23, SwitzerlandB. van de SandeInstitut Fu¨r Theoretische Physik III, Staudstrae 7, D{91058 Erlangen, GermanyWe use the light-front Hamiltonian of transverse latticegauge theory to compute from rst principles the glueballspectrum and light-front wavefunctions in the leading orderof the 1/Nc colour expansion. We nd 0++, 2++, and 1+−glueballs having masses consistent with Nc = 3 data availablefrom Euclidean lattice path integral methods. The wavefunc-tions exhibit a light-front constituent gluon structure.PACS numbers: 11.15.Tk, 11.10.Jj, 12.38.Bx, 13.85.NiI. INTRODUCTIONThere is growing experimental evidence that glueballs,boundstates of gluons in the SU(3) gauge theory Quan-tum Chromodynamics (QCD), have been discovered inthe mass range 1.5−1.7 GeV [1{3]. But the connementfeature of QCD | interactions grow stronger as the en-ergy of a process decreases | complicates any rst prin-ciples calculation of the boundstate problem. To date,the most successful boundstate calculations use a Eu-clidean spacetime lattice and simulate the path integralby Monte Carlo methods. These dicult calculations arenow roughly consistent with the experimental signaturesof glueballs [2,4], although much detail remains to beclaried. Therefore, it is important to have some inde-pendent method of calculating the properties of glueballsfrom rst principles in gauge theory. In this letter, wepresent our rst results on this problem using an eectivelight-front Hamiltonian quantisation (canonical quanti-sation on a null-plane in spacetime). This is the trans-verse lattice method, suggested originally by Bardeen andPearson [5,6], which we have developed to the extent thatquantitative calculations are now feasible [7,8].Although this work is ostensibly about glueballs, wehave a more general motivation for developing the light-front Hamiltonian formulation of gauge theory. A light-front Hamiltonian has Lorentz-frame-independent wave-functions. Together with the simplicity of the light-frontvacuum, this leads to a eld-theoretic realisation of theparton model for hadrons on which so much understand-ing is based. The rich phenomenology in hadronic andnuclear physics that would follow from knowledge of thelight-front wavefunctions is surveyed in the lectures ofBrodsky [9]. We will use the glueball problem in QCD,which is especially dicult computationally, as a quanti-tative test of our light-front formalism.A detailed account of our methods and various quan-titative tests that we have performed, mostly in 2+1 di-mensions, can be found in Refs. [7,8] but we will brieflyreview the salient points below. We will work in theleading order of the 1/Nc-expansion of SU(Nc) gaugetheory, which omits the 1/N2c -suppressed glue congura-tions. It also removes \sea" quarks from our theory, anapproximation also used in the Euclidean lattice calcu-lations. We work on a coarse transverse lattice, usingan eective potential tuned to minimize discretisationerrors. The groundstate J PC = 0++ glueball mass isM = 3.3  0.2pσ, where σ  0.1936GeV2 is the stringtension, and is consistent with SU(2) and SU(3) resultsavailable from the Euclidean lattice. Components of the2++ and 1+− glueballs which are behaving covariantlyare also consistent. The glueball wavefunctions that weobtain are new.II. TRANSVERSE LATTICE FORMULATIONIn 3 + 1 spacetime dimensions we use a square latticeof spacing a in the ‘transverse’ directions x = fx1, x2g,and a continuum in the fx0, x3g directions. Our actionisS =Zdx0dx3XxTrDαMr(x)DαMr(x)y− 12G2TrFαβFαβ}− Vx[M ] , (2.1)where r, s 2 f1, 2g; α, β 2 f0, 3g; andDαMr(x) = (∂α + iAα(x)) Mr(x) − iMr(x)Aα(x + ar^) .(2.2)Aα(x) is the gauge potential at transverse site x, whileMr(x) 2 GL(Nc, C) is a link variable from x to x + ar^which carries colour flux between sites. Our use of non-compact variables at nite a is the basis of the colour-dielectric formulation of lattice gauge theory [10]. Theaction (2.1) is gauge invariant at each transverse latticesite underAα(x) ! U(x)Aα(x)U y(x) + i (∂αU(x)) U y(x) (2.3)Mr(x) ! U(x)Mr(x)U y(x + ar^) (2.4)for U 2 SU(Nc). The dimensionful coupling G2(a) issuch that a2G2 ! g2 in the classical continuum limit1a ! 0, where g2 is the usual continuum gauge coupling.We include a gauge-invariant eective potential Vx[M ]consisting of Wilson flux loops on the transverse lattice.The exact form we use is given below. Part of its jobis to ensure that M is forced to be a unitary matrix asa ! 0, so that continuum QCD is recovered a la Wilson[11]. More generally, we choose it to minimize violationsof Lorentz covariance at nite a. The idea is that wetune the couplings of V along an approximation to theLorentz covariant scaling trajectory which leads to con-tinuum QCD as a ! 0. Then for low energy observables,such as boundstate masses, one can work at nite a.We perform light-front quantisation, treating x+ =(x0+x3)/p2 as time, and pick the light-front gauge A− =(A0 − A3)/p2 = 0. In this gauge A+ = (A0 + A3)/p2is a constrained eld which we eliminate by solving itsequation of motion classically. Working in the leading1/Nc approximation has some simplifying consequences.In the transverse rest frame P = 0, Eguchi-Kawai reduc-tion occurs at large a, so that the argument x may bedropped [7]. This results in a light-front Hamiltonian ofthe formP− =Zdx− − G24TrJ+1∂2−J+− βNca2TrnM y2My1M2M1 + M1M2My1My2o+µ2XrTrMrMyr}+λ3a2N2cXr(TrMrMyr}2+λ2a2NcXrTrMrMrMyr Myr}+λ1a2NcXrTrMrMyr MrMyr}+λ4a2NcXσ=2,σ0=1TrnM yσMσMyσ0Mσ0o+4λ5a2N2cTrnM1My1oTrnM2My2o, (2.5)J+ = i(Mr$∂− M yr + Myr$∂− Mr) . (2.6)This Hamiltonian includes all Eguchi-Kawai-reducedWilson loops on the transverse lattice up to fourth orderin M ; this is our approximation to the eective poten-tial V . Generalisation to higher orders is straightforward.The Hamiltonian is diagonalised in a parton Fock space ofMr, at xed total momentum P+, built from its Fouriermodes at x+ = 0Mr =1p4piZ 10dk+pk+a−r e−ik+x− + ayr eik+x−. (2.7)However, only closed flux loops, invariant under residualx−-independent gauge transformations U , have nite en-ergy. These loops propagate without splitting or joiningat leading 1/Nc order.Even with a nite lattice spacing a, the Hamiltonian(2.5) is not yet completely regulated. We impose periodicboundary conditions in x− and a cut-o on the maximumnumber of link partons ar in the Fock space. For eachchoice of couplings in the Hamiltonian, we remove theselatter cut-os by extrapolation, following a continuumimprovement procedure.III. TUNING THE EFFECTIVE POTENTIALTo test for Lorentz covariance, we measure the dis-persion relations of glueball boundstates as a functionof couplings in V . This requires us to work at P 6= 0,which can be achieved by adding appropriate phase fac-tors to matrix elements of P− above [8]. Using G2Nc toset the dimensionful scale, the dispersion relation of eachglueball can be written2P+P− = G2Nc(M20 +M21 a2jPj2+2M21 a2P 1P 2 + O(a4jPj4). (3.1)For each glueball in the light spectrum and for xed µ2,we tune the couplings λi and β so that, as far as possible,a2G2NcM21  c2on = 1 (3.2)a2G2Nc(M21 +M21)  c2off = 1 . (3.3)con is the speed of light in direction x = (1, 0); coff is thespeed of light in the direction x = (1, 1). In practice, weapply a χ2-test over a range of light glueballs to optimizeisotropy of the speed of light. In this way we search fora Lorentz covariant scaling trajectory parameterized byµ2, which may later be related to a.It is necessary to know the dimensionless combinationa2G2Nc for this procedure. This can be deduced fromtwo measurements of the string tension σ. The rst mea-sures the mass squared of winding strings in the trans-verse lattice directions x, t to n2a2σ2T for winding num-ber n. The second measures the heavy quark potentialin the continuous x3 direction [12], t to σLR for sep-aration R. Demanding σT = σL  σ and expressingeigenvalues in units of G2Nc, we may eliminate σ anddeduce a2G2Nc. We may also deduce G2Nc, and henceall dimensionful quantities, in units of σ.To further help optimize lattice discretisation errors,we include in the χ2 test a measure of deviations from ro-tational invariance of the heavy-quark potential, includ-ing deviations from σT = σL and deviations from Lorentzmultiplet structure in the glueball spectrum. σL/G2Ncis the only basic quantity which is poorly determined byour current approximations, and our results were partic-ularly sensitive to the variance assignment in the χ2 testfor this datum (though still compatible with the system-atic error estimate we make below).According to the colour-dielectric picture, large µ willcorrespond to large a where discretisation errors grow,2while small µ corresponds to intermediate spacing a. Suf-ciently small a produces a non-trivial light-front vac-uum, signalled by the appearance of tachyons; it would bedicult to construct an eective potential in this regime.Instead, we use an eective potential at intermediate aon a trivial light-front vacuum. In 2 + 1 dimensions theabove procedure accurately identied a Lorentz covariantscaling trajectory at intermediate a [8]. The results wepresent below are based on a similar trajectory which wehave found in 3 + 1 dimensions, though further work isneeded to reduce systematic errors.IV. RESULTSIn Table I we show results for various componentsof the lightest glueballs at a transverse lattice spac-ing a = 1.1/pσ  0.49 fm, at the overall χ2 min-imum for this a. In order to establish the scalingtrajectory, we have performed calculations at otherlattice spacings: At smaller lattice spacings tachyonsappear, indicating breakdown of the trivial vacuumregime. For larger lattice spacings the couplings evolvegradually, and the minimum χ2 value grows. Forthe groundstate 0++ we nd approximate scaling andmaintenance of Lorentz covariance (a/pσ,M/pσ, c) =f(1.1, 3.3, 1.13), (1.2, 3.1, 1.18), (1.3, 3.5, 1.19)g. The er-rors shown on glueball masses in Table I are estimatesof the error in removing the periodic boundary condi-tion on x− and the cut-o on the maximum number linkvariables Mr in a wavefunction; these errors are undercontrol. Only for the 0++ mass are we condent enoughto estimate the nite-a systematic error. The Lorentzcovariant scaling trajectory will need to be determinedmore accurately before similar statements can be madefor the higher states. In 2 + 1 dimensions, we foundthat when the deduced speed of light becomes isotropicto within about 10%, the corresponding glueball mass isalso accurate at this level; this also seems to be the casein 3+1 dimensions.J PC jJz jP1 con coff Mpσ0++ 0+ 1.13 1.13 3.3(1)2++ 2+ 0.85 0.86 4.4(1)2− Im − 5.9(2)1 0.89,− 0.92, Im 5.4(2)0+ 0.84 0.83 4.5(1)1+− 1 0.75, 0.88 0.74, 0.89 5.0(1)0− 0.99 0.99 5.9(2)TABLE I. The glueball multiplet components showingmasses M2 = 2P+P−(P = 0) and c from the dispersion re-lation (3.1). P1 is the symmetry under reflections x1 ! −x1;\Im" indicates c2 < 0 and \−" indicates the quantity has notbeen measured. The Jz = 1 states are exactly degenerate.Mpσ0.1 0.2 0.302ELMC++++4620Light-Front1/N2cFIG. 1. The variation of glueball masses with Nc. ELMCpredictions are continuum ones for Nc = 2, 3, and xed latticespacings (higher points are β = 11.1 and lower are β = 10.9)for Nc = 4 [15]. Nc = 1 light-front predictions are fromcomponents behaving covariantly in Table I, including thetotal error estimate for the 0++. The dotted lines are toguide the eye, and correspond to the expectation of leadinglinear dependence on 1/N2c .We now compare our results with spectra from con-ventional Euclidean lattice Monte Carlo (ELMC) simu-lations. See Fig. 1. (Recently, large Nc glueball massratios have been calculated using a conjectured dualitywith supergravity [13]; we will not discuss these resultssince their errors are not quantied.)0++Based on violations of Lorentz covariance and variationofM along the scaling trajectory, we include an estimateof systematic error for this state, M = 3.3  0.2pσ. Acontinuum extrapolation of SU(3) ELMC data has beenmade in Ref. [14], M = 3.65(15)pσ (statistical error).Furthermore, measurements for SU(2) and SU(4) [15]indicate a slight downward drift with decreasing 1/N2c .2++Not all components of this state are behaving covariantly.The P1-odd combinations of Jz = 1 and Jz = 2 havelarge lattice artifacts and cannot be trusted. As a re-sult, we are unsure of the correct parity assignment forthe Jz = 1 but assume here that it belongs to the 2++multiplet. The P1-even Jz = 0 and Jz = 2 combi-nation are nearly degenerate and isotropic, pointing to atensor mass nearM 4.5pσ. The SU(3) ELMC extrap-olation of Ref. [14] gives 5.15(23), consistent with morerecent coarse anisotropic ELMC simulations [16]. Again,there is a slight drift with 1/N2c .1+−Here the P1-odd components fair better. The Jz = 03component of the lightest spin 1 is behaving covariantly,and points to M  5.9pσ. The Jz = 1 compo-nent is split away as a consequence of its anisotropy.An SU(3) value M = 6.6(6)pσ is given in Ref. [4],while the more accurate measurement of Ref. [16] impliesM = 6.30(4)pσ.0−+Our lightest candidate for this state appears around7.4pσ though we have not measured its covariance.SU(3) ELMC work suggests a lighter mass [4]. Barringan unusual large-Nc extrapolation, it seems likely thatthis state (which posed problems for early ELMC studiesalso) is poorly described with our current approxima-tions.As an example of what may be deduced from wavefunc-tions, in Figure 2 we plot the distribution of longitudinalmomentum fraction x = k+/P+ in the 0++ glueball ata = 1.1/pσ. Gd(x) is the probability distribution for alink parton ar(k+) and becomes the usual gluon distri-bution function in the continuum limit a ! 0 [8]. At -nite a, Gd may be interpreted as a light-front constituentgluon distribution that evolves with a. We nd that our0++ light-front wavefunction consists mostly of four suchconstituents; it does not obey a gluonium picture (twoconstituent gluons).Gd(x)0 0.2 0.4 0.6 0.800.10.20.30.40.5xFIG. 2. The 0++ momentum distribution function.V. CONCLUSIONSWe have measured glueball masses in QCD, using the1/Nc expansion and dierent quantisation, eld vari-ables, regulators, and gauge xing from the traditionalEuclidean lattice path integral simulations. The resultsfrom both approaches are consistent for QCD withoutquarks, indicating a 0++ glueball near 1.6 GeV, and1/N2c corrections appear to be small for the 0++ and2++ states. (In this latter regard, it would be interestingto see better data at SU(4).) We found that light-frontwavefunctions exhibit a constituent gluon structure, al-though not of the form one might naively guess. In orderto improve the accuracy of the Lorentz covariant scal-ing trajectory, future work includes choosing other ob-servables to set the scale and incorporating higher or-der eective interactions. Finally, we note that the re-introduction of quarks will be necessary for a detailedcomparison with experimental glueball candidates, whichmix with nearby scalar quarkonia [1,3].ACKNOWLEDGMENTSThe work of SD is supported by CERN. We thank M.Teper for helpful interactions. A portion of the compu-tations were performed at Argonne’s Center for Compu-tational Science and Technology.[1] C. Amsler and F. Close, Phys. Lett. B353, 385 (1995);Phys. Rev. D53, 295 (1996).[2] J. Sexton et al., Phys. Rev. Lett. 75, 4563 (1995).[3] W. Lee and D. Weingarten, IBM report IBM-HET-98-1hep-lat/9805029 (unpublished).[4] G. Bali et al. (UKQCD-Wuppertal), Phys. Lett. B309,378 (1993).[5] W. A. Bardeen and R. B. Pearson, Phys. Rev. D14, 547(1976).[6] W. A. Bardeen, R. B. Pearson, and E. Rabinovici, Phys.Rev. D 21, 1037 (1980).[7] S. Dalley and B. van de Sande, Phys. Rev. D56, 7917(1997).[8] S. Dalley and B. van de Sande, Univ. Erlangen ReportNo. FAU-TP3-98/12, hep-th/9806231 (unpublished).[9] S. J. Brodsky, Lectures at NuSS, Korea (1997), SLACreport No. PUB-7645, hep-ph/9710288 (unpublished).[10] H.-J. Pirner, Prog. Part. Nucl. Phys. 29, 33 (1992).[11] K. G. Wilson, Phys. Rev. D 10, 2445 (1974).[12] M. Burkardt and B. Klindworth, Phys. Rev. D 55, 1001(1997).[13] C. Csaki, H. Ooguri, Y. Oz, J. Terning, UCB Report No.PTH-98/30, hep-th/9806021 (unpublished). R. de MelloKoch, A. Jevicki, M. Mihailescu, J. P. Nunes, Phys. Rev.D 58, 105009 (1998).[14] F. E. Close and M. J. Teper, RAL Report No. 96{040 /OUTP{96{35P (unpublished).[15] M. J. Teper, Phys. Lett. B397, 223 (1997).[16] C. Morningstar and M. Peardon, Phys. Rev. D56, 4043(1997).4

Glueball calculations in large-$N_{c}$ gauge theory

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Glueball calculations in large-N_c gauge theory

Glueball calculations in large-N_c gauge theory

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