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Glueballs on a transverse lattice

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DAMTP arxiv:hep-lat/9935v 7 Nov 999 Glueballs on a transverse lattice S. Dalley and B. van de Sande Departent of Applied Matheatics and Theoretical Physics, Silver Street, Cabridge CB3 9EW, England
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DAMTP arxiv:hep-lat/9935v 7 Nov 999 Glueballs on a transverse lattice S. Dalley and B. van de Sande Departent of Applied Matheatics and Theoretical Physics, Silver Street, Cabridge CB3 9EW, England Geneva College, 3 College Ave., Beaver Falls, PA 5 Abstract Accurate non-perturbative calculations of glueballs are perfored using light-front quantised SU(N) gauge theory, to leading order of the /N expansion. Based on early work of Bardeen and Pearson, disordered gauge-covariant link variables M on a coarse transverse lattice are used to approxiate the physical gauge degrees of freedo. Siple energetics iply that, at lattice spacings of order the inverse QCD scale, the effective light-front Hailtonian can be expanded in gauge-invariant powers of M: a colour-dielectric expansion. This leads to a self-consistent constituent structure of boundstates. We fix the couplings of this expansion by optiising Lorentz covariance of low-energy eigenfunctions. To lowest non-trivial order of the expansion, we have found a one-paraeter trajectory of couplings that enhances Lorentz covariance. On this trajectory the asses of nearly-covariant glueball states exhibit approxiate scaling, having values consistent with large-n extrapolations of continuu results fro other ethods. There is very little variation with N in pure Yang-Mills theory: the lightest glueball ass changes by only a few percent between SU(3) and SU( ). The corresponding light-front wavefunctions show an unconventional structure. We also exaine restoration of rotational invariance in the heavy-source potential. Introduction There are few, if any, efficient ethods for tackling relativistic strongly-bound states in generic fourdiensional gauge theories. The canonical exaple is QCD, where non-perturbative theoretical calculations have rarely been ahead of experient. Future progress in particle physics is likely to hinge upon a detailed theoretical understanding of these questions. This has led soe theorists to develop Hailtonian quantisation on a light-front [, ]. In the presence of suitable high-energy cut-offs, the light-front vacuu state is trivial and wavefunctions built upon it are Lorentz boostinvariant. In particular, Brodsky and Lepage [3] and Pauli and Brodsky [4] have urged the developent of the light-front quantisation of QCD (LFQCD). More recently, Wilson et al. [5] have clarified the physical principles underlying LFQCD and suggested a weak-coupling calculational fraework. In this paper we develop an alternative fraework which appears proising: a light-front quantisation of lattice gauge theory [6]. Calculations that we have already perfored with this ethod [8, 9, ], for non-abelian gauge theories in + diensions, were surprisingly successful in coparison to results fro traditional Euclidean lattice path integral siulations (ELMC) []. They have also produced new results in the for of the light-front wavefunctions, the starting point for investigation of virtually any physically interesting observable. Encouraged by this success, we investigate here the glueballs and heavy-source potential in 3 + -diensional gauge theory without ferions. The well-known triviality of the light-front vacuu ay be reconciled with the conventional picture of a coplicated QCD vacuu. In light-front co-ordinates, vacuu structure is carried by an isolated set of (infinitely) high energy odes, which are reoved by the cut-off. According to standard lore, one would expect that appropriate renoralisation of the Hailtonian and other observables would recover the inforation excised by the high-energy cut-off. In this way, effects norally associated with the vacuu, such as spontaneous syetry breaking, ust appear explicitly in the Hailtonian. But the non-perturbative renoralisation group (RG) foralis to systeatically copute the necessary counter-ters does not yet exist. Efforts are being ade to forulate a perturbative light-front RG for QCD [], but the full vacuu structure ust necessarily appear via counter-ters that are non-analytic in the perturbative couplings. Faced with the proble of finding non-perturbatively renoralised Hailtonians, we will use syetry as our guide. The first step is to choose the ost general set of Hailtonians which respect syetries of the theory that are not violated by the cut-offs. After truncating this set according to soe reasonable criteria, we non-perturbatively test the low-energy eigenfunctions and eigenvalues for restoration of the syetries violated by the cut-offs. These tests, rather than RG transforations, are used to explore the space of Hailtonians. Since we aintain gauge invariance, the syetries in question involve only Lorentz covariance. We will reove all but one cut-off and find that our truncated space of Hailtonians contains a unique one-paraeter trajectory T s on which Lorentz covariance is greatly enhanced. The validity of T s is confired by the approxiate scaling of low-energy physical quantities along this trajectory. T s will be used as the basis for extracting cut-off independent results. This is a first-principles approach, since no data are taken fro experient (aside fro the overall scale in QCD). Apart fro Lorentz invariance, we ust address the issue of gauge invariance. One knows that gauge invariance can be aintained with lattice cut-offs and copact variables [3, 4]. However, in light-front coordinates a high-energy lattice cut-off can only be applied to the transverse directions; the null direction on the initial surface ust reain continuous. Also, if one were to eploy the usual forulation of lattice gauge theory with degrees of freedo in SU(N), it is not straightforward to identify the independent degrees of freedo that are essential for canonical Hailtonian quantisation. The tricks of equal-tie Hailtonian lattice gauge theory in teporal gauge [4] do not carry over to light-front Hailtonian lattice gauge theory in any convenient way [5]. One does not have to choose lattice variables in SU(N) however. It was noted long ago by Bardeen and Pearson [6], that lattice variables M in the space of all coplex N N atrices were physically ore appropriate on a coarse lattice. Gauge invariance is aintained and it is straightforward to identify the independent degrees of freedo. The penalty is that one is too far fro the continuu to use weak-coupling perturbation theory. But in the case of light-front Hailtonians, this ay be of liited use anyway. Bardeen et al. [7] suggested that on coarse lattices one could expand the light-front Hailtonian in powers of M (a kind of strong-coupling expansion). The validity of this expansion is subtle however, since it rests on the dynaical properties of light-front Fock states in this kind of theory, in particular the weakness of couplings between sectors with different nuber of partons [6]. Expanding the light-front Hailtonian of SU(N) gauge theory to leading non-trivial order in M and /N, we find the trajectory of couplings T s entioned above. In the region of coupling space we are able to investigate, the transverse lattice spacing on T s is found to be always greater than about.65 f. However, Lorentz covariance and scaling are present to sufficient accuracy that we can ake direct estiates of the continuu values of low-energy observables. The organisation of the paper is as follows. In the first part of the present work, we give a systeatic treatent of tranvserse lattice gauge theory. In Section we construct generic light-front Hailtonians on the transverse lattice and introduce approxiation schees for studying the; in particular, we review the colour-dielectric expansion in powers of M that leads to a constituent picture of boundstates. By working to leading order in /N, the transverse dynaics diensionally reduce (for a coarse lattice) [], the proble becoing atheatically equivalent to a + - diensional gauge theory with adjoint atter [6]. As well as the lattice cut-off, which can only be used in transverse spatial directions, we also eploy null-plane boundary conditions, specifically Discrete Light-Cone Quantisation (DLCQ) [4], and Ta-Dancoff cut-offs for transverse degrees of freedo. The latter gauge-invariant cut-offs can be reoved by extrapolation to give finite answers for fixed transverse lattice spacing, the only reaining cut-off. The renoralisation of the theory at fixed lattice spacing is then accoplished by optiising Lorentz covariance, discussed in Section 3. This aounts to a choice of etric in the space of Hailtonians, based on explicit calculations of observables. Results fro our calculations are presented in Section 4; we will not describe in any detail the calculational techniques, but refer the reader to our previous papers. Soe preliinary results fro the glueball calculations were presented in Ref. [7], but the new aterial here is ore accurate and extensive, including a thorough analysis of the space of couplings, the glueball asses and wavefunctions, and the heavy-source syste. We end this introduction by quoting the ost readily understandable of our ain results: the lowest glueball asses. We estiate the lightest glueball in SU( ) gauge theory, the J PC = O ++, to be at a ass M O ++ = 3.5 ±.4 σ, where σ is the string tension. This is essentially As with other strong-coupling approaches, the question of the foral connection to the continuu liit is not directly answered. 3 M σ ELMC Light-Front...3 /N Figure : The variation of glueball asses with N (pure glue). ELMC predictions are continuu ones for N =,3 [9, 8, ] and fixed lattice spacing estiates for N = 4 [35]. The dotted lines are to guide the eye and correspond to leading linear dependence on /N. indistinguishable fro the result for SU(3) pure gauge theory established rigorously by ELMC in recent years [8]. For the ++ tensor state, not all coponents of the ultiplet are yet behaving covariantly; but fro those coponents which are reliable, we estiate M ++ = 4.97±.43 σ. The vector + has M + = 5.57±.4 σ. We find no light pseudo-scalar +, but suspect ourcandidate for this has a large error. These results are suarised in Figure. Other recent estiates of glueball asses have been obtained in SU(3) Hailtonian gauge theory [] and based on extensions of the Maldacena conjecture for large-n gauge theory [], though we consider those results to be less rigorous than ours. Transverse lattice Hailtonians.. Energy-oentu In 3 + spacetie diensions we introduce a square lattice of spacing a in the transverse directions x = {x,x } and a continuu in the {x,x 3 } directions. In light-front (LF) co-ordinates x ± = (x ± x 3 )/, we treat x + as canonical tie and place anti-periodic boundary conditions on x x +L. Both/aandLarehigh-energycut-offsforthe LFHailtonian P = (P P 3 )/ that evolvesthe syste in LF tie x +. The Lorentz indices µ,ν {,,,3} are split into LF indices α,β {+, } 4 and transverse indices r,s {,}. The gauge field degrees of freedo below the cut-offs are represented by Heritian gauge potentials A α (x) and coplex link variables M r (x). We also introduce heavy scalar sources φ(x). On the transverse lattice, A α (x) and φ(x) are associated with a site x, while M r (x) is associated with the link fro x to x+aˆr. Each site x is in fact a two-diensional plane spanned by {x +,x }. These variables transfor under transverse lattice gauge transforations U SU(N) as A α (x) U(x)A α (x)u (x)+i( α U(x))U (x) M r (x) U(x)M r (x)u (x+aˆr) () φ(x) U(x)φ(x). () Since it will be possible to eliinate A α by partial gauge-fixing, M and φ represent the physical transverse polarisations. The siplest gauge covariant cobinations are M, φ, F αβ, detm, D α M, D α φ, where the covariant derivatives are D α M r (x) = ( α +ia α (x))m r (x) im r (x)a α (x+aˆr) (3) D α φ = α φ+ia α φ. (4) Fro these we wish to construct generallf Hailtonians P invariantunder gaugetransforations and those Lorentz transforations unviolated by the cut-offs. To proceed, we ust ake soe assuptions about which finite sets of operators to include in the calculation. Since syetries will be tested explicitly, a poor choice of operators would show up later on. The following criteria were used to select operators in P for pure gauge theory: (A) Canonical quadratic for for P + ; (B) Naïve parity restoration as L ; (C) Transverse locality; (D) Expansion in gauge-invariant powers of M. Each of these criteria deserves soe explanation. The last three can all be straightforwardly checked in principle by systeatically relaxing the condition. (A) Of the generators of the Poincaré group {P +,P,P r,m µν }, the subset {P r,p +,M +r, M, M + } can usually be ade kineatic. That is, they can be ade independent of interactions, quadratic in the fields. The iposition of a lattice cut-off can spoil this property, especially if one wants to aintain gauge invariance. We try to aintain as any operators as possible in kineatic for, consistent with gauge invariance. Further details are given below, but for now we note that condition (A) can be satisfied in light-front gauge A = by using a Lagrangian containing the only x + -dependent gauge-covariant ter quadratic in link fields, D α M r (x)(d α M r (x)). Although higher order ters in M cannot be ruled out, only with a quadratic ter is quantisation straightforward and, even then, only in the LF gauge A =. (B) We will extrapolate to the L liit in the longitudinal direction, deriving P fro a Lagrangianincluding only diension operators with respect to {x +,x } co-ordinates. It has 5 been noted that functions of diensionless ratios of longitudinal oenta p + can appear in couplings of boost-invariant LF Hailtonians [5]. However, these functions ust be strongly constrained by LF parity x + x. Parity is a dynaical syetry on the light-front (it does not preserve the quantisation surface) and is difficult to check explicitly. We will assue that, if P is derived fro a Lagrangian with naïve x + x syetry (p + -independent couplings), then parity is correctly restored in transverse lattice gauge theory in the liit L. Although DLCQ treatents of analogous + -diensional gauge theories have been successful under siilar assuptions [6, 4, 5], in the present case there are any ore possible operators that are ruled out by conditions (A) and (B) cobined. (C) In products of gauge-invariant operators on the transverse lattice, each ter of the product can be arbitrarily separated in the transverse direction. We assue soe transverse locality by restricting products of gauge invariant operators to share at least one site x. (D) After fixing to LF gauge A = and eliinating the resultant constrained field A +, Fock space states will consist of link-partons derived fro the Fourier expansion of M. For sufficiently large a, M reains a assive degree of freedo and there is an energy barrier for the addition ofalink-partonto a Fock state. OperatorsoforderM p in P will connect Fock states differing by at ost p link-partons. By expanding P to a given order of M in this regie (the colour-dielectric expansion) we therefore cut off interactions between lower-energy few-parton states and higher-energy any-parton states. Note that all energy scales are still allowed however, since we take the L liit, enabling highly virtual sall p + partons to appear. The advantage of a cut-off on changes of parton nuber is that it organises states into a constituent hierarchy, consistent with energetics. The Lagrangian density satisfying condition (A) can be written L x = D α M r (x)(d α M r (x)) V x U x (5) where the potential has a purely transverse part V x and a ixed part U x. Up to 4th order in M, the purely transverse part is V x = β Na Tr { M (x)m (x+aˆ)m (x+aˆ)m (x) }+ c.c. + r + r µ Tr { M r M r } + r λ a N Tr{ M r M r M rm r } λ + a N λ 3 ( { Tr Mr a N M r }) λ 4 + a N λ ( det[mr a ]+det[m r N ]) r Tr { M r (x)m r (x+aˆr)m r (x+aˆr)m r (x)} } Tr {M σ M σm σ M σ + r σ=±,σ =± + 4λ { } { } 5 a N Tr M M Tr M M. (6) This assuption is not warrented in the presence of finite-ass ferions. The subtlety lies in p + = vacuu odes that are not recovered even as L. For Yukawa interactions of finite-ass ferions, parity certainly isn t recovered in the naïve way [3]. 6 The det[m] ter can be dropped if N 4 and can always be neglected in the large-n liit. The ixed part, to the approxiation we will need, can be written U x = y ǫ αβ Tr{E x P xy F αβ (y)p xy} G f(e x ), (7) where E x is a non-dynaical pseudoscalar adjoint field at site x, while P xy is a linear cobination of Wilson lines in M, each fro fro x to y (for gauge invariance). Upon integrating out E, the function f(e) = Tr { E } + O(E 4 ) gives a siple G Tr { } F αβ F αβ ter if only the E part is retained and we set P xy = δ xy. 3 This will be the only ter needed for pure gauge theory to O(M 4 ) once A α has been eliinated. We gave the ore general for in Eqn. (7) since it will be relevant in the heavy-source analysis. Fro the above Lagrangian, four of the usual seven kineatic Poincaré generators derived canonically fro the energy-oentu tensor T µν reain gauge invariant and can be ade kineatic by LF gauge choice A =. They are, at x + = say, P + = dx Tr { M s (x) M s (x) }, (8) x,s M + = x Tr { M s (x) M s (x) } (9) M +r = dx x,s dx x,s ( x r + a δrs) Tr { M s (x) M s (x) } () We will use these to define states of definite oentu {P +,P}. The other three would-be kineatic Poincarégeneratorsin light-front foralis, {P r,m }, are not gauge-invariantwhen derived canonically fro T µν because of the lattice cut-off; the siplest gauge-invariant extensions are no longer quadratic in fields, even in LF gauge. Of the dynaic generators, the ost iportant, and the only one we explicitly treat, is the light-front Hailtonian itself P = dx x J + (x) = i r ( V x Tr { A + (x)j + (x) } ) G Tr{ A + A + }, () ( M r (x) M r(x)+m r(x aˆr) ) M r (x aˆr). () Rather than trying to directly construct an approxiate realisation of the Poincaré algebra at finite cut-off, we will iniise cut-off artifacts by optiising restoration of Lorentz covariance in lowenergy eigenstates of the Hailtonian P. In the light-front gauge, A + is a non-dynaical variable and eliinating it introduces non-local interactions thus P = dx x ( { G J + 4 Tr J + } G 4N Tr { } { ) J + J + Tr }+V x where J + / (J+ ). There is still a residual x -independent gauge invariance generated by the charge dx J +. As originally shown in Refs. [6, 7], finite energy states Ψ are subject to 3 The case P xy = δ xy is related to the generalised DQCD of M. Douglas et al. [6] (3) 7 the gauge singlet condition dx J + Ψ =. In the large-n liit, this eans that Fock space at fixed x + is fored by connected closed loops of link variables M on the transverse lattice (the x co-ordinate of each M is unrestricted).. Quantisation The dynaical proble is now to diagonalise P at fixed total oenta {P +,P}. A convenient basis consists of free link-partons obtained fro the Fourier odes a(k +,x) of M in the x coordinate M r (x + =,x,x) = dk + ( a r (k +,x)e ik+ x +a 4π k + r(k +,x)e ik+ x ). (4) The Fock space fored fro creation operators a in the large-n liit, and other details of the calculation in this Fock space, including construction of states of definite oentu, have been described elsewhere [9, ]. We applied both DLCQ and Ta-Dancoff cut-offs in Fock space, extrapolating both of these at fixed values of the couplings in Eqn. (6). 4 Low-energy eigenfunctions of P (id est glueballs) are to be tested for Lorentz covariance, the couplings appearing in Eqn. (6) being tuned to iniise covariance violations. Since G N, with diension (energy), is consistent with t Hooft s large-n liit [7], we will use it to set the diensionful scale. Thus, the dispersion relation of a glueball can be written ( ) P + P = G N M +M a P +M a P P +O(a 4 P 4 ). (5) Foreachglueball, M i arediensionlessfunctions ofthe couplingswhich, foragivenset ofcouplings, are extracted by expanding eigenvaluesof P in ap. A truly relativistic state ust have an isotropic speed of light a G NM c on = a G N(M +M ) c off =. (6) c on is the
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