In the light-cone Fock state expansion of gauge theories, the influence of
non-valence states may be significant in precision non-perturbative
calculations. In two-dimensional gauge theories, it is shown how these states
modify the behaviour of the light-cone wavefunction in significant ways
relative to endemic choices of variational ansatz. Similar effects in
four-dimensional gauge theories are briefly discussed.Comment: 4 pages, REVTE

A. H. Mueller

F. Antonuccio

G. ’t Hooft

H. Bergknoff

H. C. Pauli

K. G. Wilson

K. Harada

K. Hornbostel

M. Burkardt

R. J. Perry

S. Coleman

S. Dalley

S. J. Brodsky

W. A. Bardeen

Y. Mo

English

arXiv

In the light-cone Fock state expansion of gauge theories, the influence of non-valence states may be significant in precision non-perturbative calculations. In two-dimensional gauge theories, it is shown how these states modify the behaviour of the light-cone wavefunction in significant ways relative to endemic choices of variational ansatz. Similar effects in four-dimensional gauge theories are briefly discussed

Dalley, S

CERN Document Server

arXiv:hep-th/9804024v1  2 Apr 1998The Influence of Higher Fock States in Light-Cone Gauge TheoriesS. DalleyTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandIn the light-cone Fock state expansion of gauge theories,the influence of non-valence states may be significant in preci-sion non-perturbative calculations. In two-dimensional gaugetheories, it is shown how these states modify the behaviourof the light-cone wavefunction in significant ways relative toendemic choices of variational ansatz. Similar effects in four-dimensional gauge theories are briefly discussed.I. INTRODUCTIONLight-cone quantisation is believed to be an especiallyefficient hamiltonian method for dealing with many-bodyrelativistic field theory [1]. One of the tenets of thismethod is that the valence approximation to a system oflight-cone partons provides a much better approximationthan the analogous Tamm-Dancoff truncation in otherquantisation schemes. This chiefly follows from the pe-culiar inverse relation between light-cone momentum andfree energy:k− =m2 + k22k+. (1.1)Here, k± = (k0 ± k3)/√2 are the light-cone momentumand energy, k = (k1, k2) the transverse momenta, andm the mass of a parton. Since k+ > 0 and is con-served, to create more partons costs significant energybecause the new particles carry smaller k+ than theirparent(s). Moreover, one might naively expect wavefunc-tions to vanish at small k+ to ensure finite energy. In thisreport it will be shown how non-valence states neverthe-less modify the behaviour of light-cone wavefunctions,including the valence sector, when one or more partonmomenta are small. Rather than wavefunctions simplyvanishing at the boundaries of phase space, the require-ments of finite energy in general lead to cancelations be-tween the valence and non-valence (sea) components ofboundstates. This phenomena was recently analysed forQCD in ref. [2], where it was shown to be responsible,at least perturbatively, for the rising Regge behaviour ofhadronic structure functions at small momentum fractionx. In this work some implications for non-perturbativecalculations will be demonstrated. The relevant effectsmay prove significant in precision calculations with vari-ational bases. The basic physical ideas can be illustratedby two-dimensional gauge theories. Implications for four-dimensional gauge theories will be discussed later.II. TWO-DIMENSIONAL GAUGE THEORIESThe results to be discussed in this section will in factbe true for any two-dimensional gauge theory exhibit-ing pair production. Rather than attempting to coverall particular cases at once, we proceed by using familiarexamples. The generalisation will easily be seen to followfrom the simple structure of two-dimensional gauge the-ories. Therefore, consider the standard two-dimensionalgauge theory with fermion fields Ψ and actionS =∫dx0dx1 iΨγαDαΨ−mΨΨ− 14e2FαβFαβ , (2.1)where representation and colour indices are suppressedfor the moment, for generality. Using a chiral represen-tation of the two-dimensional γ matrices in the light-cone gauge A− = (A0 − A1)/√2 = 0, we are left withA+ = (A0 + A1)/√2 and the left-moving ψL and right-moving ψR components of Ψ. A+ and ψL are found tobe constrained variables with respect to light-cone timex+ = (x0+x1)/√2, and can be eliminated by their equa-tions of motion to yield a light-cone hamiltonianP− =∫dx−m22ψ∗R1i∂−ψR +e22J+1(i∂−)2J+ . (2.2)Here x− = (x0 − x1)/√2 and J+ is the right-movingcurrent J+ = ψRψ∗R. (The same form of hamilto-nian results for bosons φ, with an appropriate currentJ+ = iφ∗↔∂− φ.) These theories generically exhibit lin-ear confinement, since the Coulomb potential is linearin one space dimension. The hamiltonian (2.2) can bediagonalised in the light-cone Fock space at fixed x+,constructed from the Fourier modes of ψR,ψR(x−) =1√2pi∫ ∞0dk(b(k)e−ikx−+ d†(k)eikx−).(2.3)A (non-baryonic) eigenfunction of P− with total momen-tum P+ is of the form|Ψ(P+)〉 =∫ P+0dk1dk2 δ(P+ − k1 − k2)× f2(k1, k2)b†(k1)d†(k2)|0〉+12∫ P+0dk1dk2dk3dk4 δ(P+ −4∑i=1ki)× f4(k1, k2, k3, k4)b†(k1)b†(k2)d†(k3)d†(k4)|0〉+ · · · . (2.4)1Ellipses indicate higher Fock states fn, n > 4.Projecting 2P+P−|Ψ(P+)〉 = M2|Ψ(P+)〉 onto spe-cific Fock states, where the eigenvalueM2 is the invariantmass squared of the system, one derives coupled integralequations for the wavefunction components fn. Let usconsider the familiar example of two-dimensional QED(massive Schwinger Model at θ = 0 1), which in the ap-proximation that retains only f2 and f4 reads [3,4]M2f2(x, 1 − x) = m2x(1 − x)f2(x, 1− x) (2.5)+e2pi∫ 10dyf2(x, 1− x) − f2(y, 1− y)(x− y)2 (2.6)+e2pi∫ 10dyf2(y, 1− y) (2.7)+e2pi∫ 10dy1dy2dy3δ(y1 + y2 + y3 − x)×f4(y1, y2, y3, 1− x)(x− y1)2 (2.8)−e2pi∫ 10dy2dy3dy4δ(y2 + y3 + y4 − (1− x))×f4(x, y2, y3, y4)(1 − x− y4)2 , (2.9)M2f4(x1, x2, x3, x4) = m24∑i=11xif4(x1, x2, x3, x4)+e22piA(x1, x2, x3)!+e2pi∫ 10dy1dy2 {δ(y1 + y2 − (x1 + x2))×f4(y1, y2, x3, x4)− f4(x1, x2, x3, x4)(x1 − y1)2+ Similar } , (2.10)where and xi = ki/P+, x4 = 1− x1 − x2 − x3 andA(x1, x2, x3) =f2(1− x3, x3)− f2(x2, 1− x2)(1− x2 − x3)2+f2(x1 + x2 + x3, 1− (x1 + x2 + x3))− f2(x1, 1− x1)(x2 + x3)2+f2(x2, 1− x2)− f2(x1 + x2 + x3, 1− (x1 + x2 + x3))(x1 + x3)2+f2(x1, 1− x1)− f2(1− x3, x3)(1− x1 − x3)2 . (2.11)The many ‘similar’ contributions to the last term in(2.10) are enumerated in ref. [4], but are unimportant1Some observations related to those made in this paper haverecently been given in connection with the θ-dependence ofthe massive Schwinger model [5]for our purposes. At this point it is important to stressthe distinction between the component f2 to the solutionof such equations in the valence approximation, whichsets fn = 0 for n > 2, and the component f2 to the so-lution when some or all of the higher Fock states fn areretained. As is shown below, these have different high-energy behaviour in general.Like any wavefunction, the components fn are subjectto high energy boundary conditions. From (1.1), thisis a small light-cone momentum boundary. A typicalvariational ansatz used for the solution of the light-coneboundstate integral equations isfn(x1, . . . , xn) ∼ xβ1xβ2 · · ·xβn · P (x1, · · · , xn) (2.12)where xn = (1 −∑n−1i=1 xi) and P is polynomial. Thesafest procedure is to leave β > 0 as a variational param-eter. However, often the following formula is usedm2 = e2(1− piβ cotpiβ). (2.13)This choice is motivated by a study of the high-energy‘endpoint’ behaviour of fn(x1, . . . , xn) as one of the ar-guments vanishes, xi → 0. Ignoring terms resultingfrom a change in the number of partons (pair produc-tion (2.8)(2.9)), one deduces that for finite M2, as x→ 0the divergent behaviour of the mass (2.5) and Coulombterms (2.6) must cancel;limx→0{m2xf2(x, 1 − x) + e2pi∫ 10 dyf2(x,1−x)−f2(y,1−y)(x−y)2}= finite (2.14)(The Coulomb term is dominated by the region of asymp-totically small momentum transfer y ∼ x). This dependsonly upon the endpoint behaviour of f2 and leads to(2.13), as was first shown by t’ Hooft in large-Nc two-dimensional QCD [6] (for which only f2 exists). Thesame endpoint behaviour can be deduced under similarassumptions for any fn when one of its arguments van-ishes. The usual justification given for ignoring pair pro-duction (see e.g. [8]) is based on the assumption that fndoes not diverge if three or more momenta vanish. In thiscase, it is easy to see from (2.8)(2.9) by power countingthat pair production would be subleading as x→ 0.In fact, the ansatz (2.12) with (2.13) fails in general toaccount correctly for the behaviour of fn when one mo-mentum vanishes, except for f2 in the valence approxima-tion. Furthermore, (2.12) fails to account for the correctbehaviour as two or more momenta vanish in fn. This fol-lows by examining the behaviour of eqs.(2.5)-(2.10) whentwo or more momenta vanish. To illustrate these points,let us perform a one-loop calculation in e/m. In this case,it is easy to see that to leading order in e/m, the solutionto the four-parton equation (2.10) isf4(x1, x2, x3, x4) =e22piA(x1, x2, x3)M2 −m2∑4i=1 1xi (2.15)2(Note that M2 ∼ O(m2).) This shows that as two mo-menta vanish, say x1 and x3, such that limz→0 x1, x3 ∝ z,then f4 is non-zero and finite, which is incompatiblewith (2.12). Furthermore, as three momenta vanish,say limz→0 x1, x2, x3 ∝ z, then f4 diverges as zβ−1 iff2(z, 1− z) ∼ zβ, which is also incompatible with (2.12).Therefore, rather than vanishing at the corners of phasespace, we find relations between wavefunction compo-nents in each Fock sector, showing in particular thatthe valence and the sea are not analytically independent.Substituting f4 (2.15) into the two-parton equation (2.5)-(2.9), one obtains an effective valence equation. In thelimit of small x the singular part of this equation readslimx→0{e2pi∫ 10dy f2(x,1−x)−f2(y,1−y)(x−y)2(1− e22m2pi I(x/y))+ m2xf2(x, 1 − x)}= finite (2.16)I(w) =∫ 10dv{11v(1−v) + w − 1(1− (1− w)2(1− v(1 − w))2)+1(1 + wv(1−w))2 (w + 11−v) (2.17)Comparing with (2.14), this demonstrates that the ne-glect of pair production is unjustified in the determina-tion of β, and that eq.(2.13) will not be correct in general.To summarize, the ansatz (2.12) with (2.13) correctlyaccounts for the behaviour of fn(x1, . . . xn) as one xi → 0only in the approximation which sets fm = 0 for m > n.More generally β will be renormalised from the valuegiven by (2.13). Eq.(2.12) fails to account for the be-haviour as two or more momenta vanish.First order e/m perturbation theory and a 4-particleTamm-Dancoff Fock-space truncation were used above,but it is easy to check that the phenomenon is not spe-cial to these choices. It follows by power counting in theboundstate integral equations when various momenta xiare made to vanish, i.e. one does not need to solve theequations to reach the conclusions. Moreover, it is alsoeasy to check that since it only depends upon the current-current structure of interactions given in eq.(2.2), it willoccur in any two-dimensional gauge theory with pair pro-duction; in particular, for fundamental representationsat finite Nc [10] and adjoint representations [11]. In themassive Schwinger model, typically a one percent errorin β can lead to a similar size error inM2 estimated fromvariational ansatz, which is significant at the level of pre-cision of current computations [4,12]. The error is smallin this case because the valence approximation becomesexact in the large and small m limits. In cases wherethe latter is not true, such as boson or Majorana fermionrepresentations, the effects can be somewhat larger, es-pecially for small m.The algebraic manipulations have a simple underlyingphysical explanation. Higher Fock states renormalise theCoulomb coupling at asymptotically small momentumtransfer through vacuum polarization and vertex renor-malisation. In general, the parton mass m2 may alsobe renormalised. These (sector-dependent) renormalisa-tions are well-known in the Tamm-Dancoff treatment oflight-cone boundstate integral equations [9]. However,it does not seem to have been generally recognised thatformulae such as (2.13) are subject to renormalisationas a result. Although a suitable variational ansatz mayjust include (renormalised) β as one of the variationalparameters, an ansatz which correctly incorporates thebehaviour as two or more momenta vanish is not so sim-ple.A significant effect of these renormalisations might beexpected in theories which exhibit complete screening insome limit. This is the case form→ 0 in two-dimensionalQED, for example. In this limit the string tension isknown to vanish as σ ∼ em [7].2 Pair production effectsnow completely screen the linear potential as m → 0,resulting in the Higgs mechanism and the production ofSchwinger’s massive photon in the physical spectrum. Ifwe assume a similar renormalisation, in the boundstateintegral equations, of the Coulomb coupling at asymp-totically small momentum transfer e2 → e2R ∼ em (thisis not entirely justified if the sources are not heavy), onenaively finds from (2.13) that β ∼ √m as m→ 0, ratherthan β ∼ m. In any case, since it is eR which is relevantto the determination of β, rather than e, this empha-sizes the need to properly account for renormalisationeffects due to higher Fock states, before attempting toaccurately solve light-cone gauge theories by variationalansatz. It may have some bearing on the small discrepan-cies in results found between bosonization and light-conequantisation of two-dimensional QED [12].III. FOUR-DIMENSIONAL GAUGE THEORIES.The considerations above are relevant in QCD to thebehaviour of partons carrying small light-cone momen-tum fraction x of the hadron. Let us first consider non-perturbative formulations. The transverse lattice colour-dielectric formulation of pure QCD is conceptually sim-ilar to two-dimensional gauge theories with adjoint bo-son representations [13]. This is especially true in thelarge-Nc limit, when an exact dimensional reduction to2This result was established by bosonization. In principlethere can be a renormalisation of the parameter m/e in thebosonization approach relative to a fermionic basis, which pre-vents direct comparison with light-cone results in such a basis.To the author’s knowledge, the result has not been establishedin light-cone quantisation in the fermionic basis independentlyof the groundstate mass M obtained from variational argu-ment using (2.13).3two-dimensional gauge theory takes place [14]. Use ofvariational wavefunctions in some shape or form seemsunavoidable to obtain satisfactory numerical convergencewith this method. The job of implementing the subtletiesdescribed in the previous section for higher Fock stateslooks quite formidable. However, a hybrid method intro-duced in ref. [14], which combines a variational approachwith discretized light-cone quantisation (DLCQ) [15], of-fers a compromise. In this method, the parton momen-tum fractions xi = ti/K are artificially discretized intointegers ti, such that∑ni=1 ti = K. When computing ma-trix elements of P− in Fock space, spectator partons re-tain their discrete momentum ti, while partons involvedin an elementary interaction have their momenta pro-jected onto a continous-momentum wavefunction basis(see Appendix C of ref. [14] for details), for evaluation byvariational ansatz. For a theory with elementary interac-tions of order ≤ q in the number of fields, no more thanq partons are involved in such interactions in initial plusfinal states. This means that that an appropriate varia-tional basis need only be found for a few partons, howevermany higher Fock states are included in the calculations.We are currently incorporating the correct behaviour ofvariational wavefunctions, discussed in the last section,into our calculations on the transverse lattice [16]. Othervariational approaches to light-cone QCD break gaugeinvariance explicitly by giving large constituent massesto quarks and gluons [17]. It is hoped that higher Fockstates are strongly suppressed as a result and that per-turbative renormalisation theory can be performed.Finally, it should be emphasized that the singular be-haviour of light-cone wavefunctions, when two or morelight-cone momentum fractions xi vanish, has observableimplications for the small Bjorken-x region of hadronicstructure functions. This behaviour is determined fromthe high-energy boundary condition on light-cone wave-functions in QCD [2], which can be easily solved pertur-batively and again relates Fock space sectors with differ-ing numbers of partons. The definition of the distributionfunction for a parton of momentum fraction x1 isQ(x1) =∑n∫[dk]∫dx2 · · · dxnδ(1−n∑i=1xi)×|fn(x1, x2, . . . , xn)|2 . (3.1)[dk] indicates the integration over all parton transversemomenta upon which fn also depends (this is absent intwo dimensions). Generalising the analysis of the previ-ous section to all n, one finds that the wavefunctionsof two-dimensional gauge theories diverge slower thanx−(s−2) when s > 2 light-cone momenta are of orderx → 0, and are otherwise finite. By power counting,this means that they are not singular enough to preventQ(x → 0) from vanishing in two dimensions. In four-dimensional gauge theories however, it is possible (viathe quark–gluon–anti-quark vertex for example) for fnto diverge already when only two parton momenta arevanishing, leading to logx contributions to Q(x → 0)[2,18]. The accumulation of such logarithms is thoughtto lead to the power law rise (Regge behaviour) exhibitedby hadronic structure functions at small x.I would like to thank B. van de Sande for discussions.This work is supported by a CERN fellowship.[1] S. J. Brodsky, H-C. Pauli, and S. Pinsky, Quantum Chro-modynamics and Other Field Theories on the Light Cone,[hep-th/9705477] (unpublished).[2] F. Antonuccio, S. J. Brodsky, and S. Dalley, Phys. Lett.B412, 104 (1997).[3] H. Bergknoff, Nucl. Phys. B122 215 (1977).[4] Y. Mo and R. J. Perry, J. Comp. Phys 108, 159 (1993).[5] M. Burkardt and K. Harada, preprint hep-th/9712032.[6] G. ’t Hooft, Nucl. Phys. B75, 461 (1974).[7] S. Coleman, R. Jackiw, and L. Susskind, Ann. Phys. 93,267 (1975).[8] K. Hornbostel, Ph.D. Thesis, SLAC-PUB No.333 (1988).[9] A. Harindranath and R. J. Perry, Phys. Rev. D43, 4051(1991).[10] K. Hornbostel, S. J. Brodsky, and H-C. Pauli, Phys. Rev.D41, 3814 (1990).[11] S. Dalley and I. R. Klebanov, Phys. Rev. D47, 2517(1993).[12] K. Harada, T. Heinzl, and C. Stern, Phys. Rev. D57,2460 (1998) and references therein.[13] W. A. Bardeen and R. B. Pearson, Phys. Rev. D14, 547(1976).[14] S. Dalley and B. van de Sande, Phys. Rev. D56, 7917(1997).[15] H. C. Pauli and S. J. Brodsky, Phys. Rev. D32, 1993 and2001 (1985).[16] S. Dalley and B. van de Sande, work in progress.[17] K. G. Wilson et. al., Phys. Rev. D49, 6720 (1994).[18] A. H. Mueller, Nucl. Phys. B415, 373 (1994).4

The Influence of Higher Fock States in Light-Cone Gauge Theories

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Influence of higher Fock states in light-cone gauge theories

The Influence of Higher Fock States in Light-Cone Gauge Theories

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