---

Solving 3 + 1 QCD on the Transverse Lattice

arXiv:hep-th/9108009v1 20 Aug 1991FERMILAB-PUB-91/197-TAugust 1991Solving 3 + 1 QCD on the Transverse LatticeUsing 1 + 1 Conformal Field TheoryPaul A. Griffin†Theory Group, M.S. 106Fermi National Accelerator LaboratoryP.

O. Box 500, Batavia, IL 60510AbstractA new transverse lattice model of 3 + 1 Yang-Mills theory is constructed by intro-ducing Wess-Zumino terms into the 2-D unitary non-linear sigma model action forlink fields on a 2-D lattice.

The Wess-Zumino terms permit one to solve the basicnon-linear sigma model dynamics of each link, for discrete values of the bare QCDcoupling constant, by applying the representation theory of non-Abelian current(Kac-Moody) algebras. This construction eliminates the need to approximate thenon-linear sigma model dynamics of each link with a linear sigma model theory, asin previous transverse lattice formulations.

The non-perturbative behavior of thenon-linear sigma model is preserved by this construction. While the new model isin principle solvable by a combination of conformal field theory, discrete light-cone,and lattice gauge theory techniques, it is more realistically suited for study with aTamm-Dancofftruncation of excited states.

In this context, it may serve as a use-ful framework for the study of non-perturbative phenomena in QCD via analytictechniques.† Internet: pgriffin@fnalf.fnal.gov

1. IntroductionThe transverse lattice approach to 3+1 Yang-Mills theory (QCD) originally de-veloped by Bardeen and Pearson[1] over ten years ago, incorporates conceptual andcomputational advantages that are found separately in other formulations.

Like the4-D Euclidean lattice formulation, the physical degrees of freedom are link variablesof a discrete lattice which are interpreted as phase factors ei RA. The transverse lat-tice models incorporate the non-perturbative dynamics of QCD and are well suitedfor studying the bound state spectrum[2].

However in the transverse lattice con-struction, the lattice is only two-dimensional. Local 2-D continuum gauge fieldsare also present to gauge the symmetries at each site.

The local gauge invarianceis then used to eliminate, via gauge fixing, the 2-D gauge fields in favor of a non-local Coulomb interaction for the link fields. This is accomplished in a light-conegauge A−= 0 and with light-cone quantization so that A+ can be eliminated byusing its equation of constraint.

All physical states have positive light-cone energyP +. This eliminates two degrees of freedom and simplifies the classification of thebound states (see ref.

[3] for further discussion of the advantages of the light-coneapproach).The basic action for each link on the transverse lattice is the 2-D unitary SU(N)principal chiral non-linear sigma model.Although this sigma model is exactlysolvable via a Bethe-ansatz technique[4], this solution cannot be easily applied inthe transverse lattice context. The Bardeen Pearson model is a linear sigma modelapproximation of the non-linear model in which the unitarity constraint of the linkfields is relaxed.

The N × N matrices of the linear sigma model are constrainedby introducing potential terms into the theory which are designed to drive thesystem into the non-linear phase[5]. In the numerical work of Bardeen, Pearson,and Rabinovici[2], glueballs are constructed from local two-link and four-link boundstates which are smeared over the 2-D lattice.

This truncation of the Hilbert spaceis a non-perturbative light-front Tamm-Dancoffapproximation to the QCD boundstate problem[6]. The numerical results based on this approach were inconclusive.The links were weakly coupled via the Coulomb interation, and the spectrum wasqualitatively similar to what would be obtained from a strong coupling expansionin ordinary lattice gauge theory[7].There are a number of changes one could make to their original analysis thatmight improve the situation.

This paper will focus on directly solving the non-linear1

sigma model dynamics instead of using the linear sigma model approximation. Byintroducing Wess-Zumino[8] terms into the sigma model action, we will describe thenon-linear sigma model dynamics in the basis of operators given by the well-studiedand exactly solvable Wess-Zumino-Witten[9] (WZW) model.

The WZW currentswill be the linear variables which describe exactly the dynamics of the non-linearsigma model. The Wess-Zumino terms in the action will become irrelevant operatorsin the continuum limit.The non-linear aspects of the principle chiral sigma model are retained in theWZW model.

The unitary link fields (and products of link fields) appear as theprimary fields of the WZW model, and play a crucial role in defining the highestweight states of the Hilbert space of the WZW model. The highest weight statescorrespond to zero modes of Wilson loops on the transverse lattice.

This zero modestructure is lacking in the linear sigma model treatment of the transverse latticetheory. (The structure presumably corresponds to the space of soliton excitationsof the linear sigma model fields.

)The advantage of exactly solvable non-linear sigma model dynamics must beweighed against the two potential disadvantages of this approach. First, this WZWmodel approach will only work for the discrete values of bare sigma model couplingconstants which correspond to the non-trivial WZW fixed points.

This will in turnplace a constraint on the QCD coupling constants that this model can obtain in thecontinuum limit. These particular values are not special points in the context of3 + 1 QCD, but rather these are points where we can apply our limited knowledgeof the 2-D non-linear sigma model to simplify the local dynamics of the link fields.Second, the continuum limit may be difficult to obtain because the irrelevant termsadded to simplify the local link dynamics may be large for finite lattice spacings.This issue can only be resolved by explicit numerical simulation.Preliminary work on the use of the gauged WZW model to describe the dy-namics of lattice model links was discussed in ref.

[10]. A transverse lattice modelwith one lattice dimension and two continuum dimensions was studied, and it wasfound that assigning the same Wess-Zumino term, with the same coupling constant,to each link leads to an order a term in the continuum limit, where a is the latticespacing.

This order a term generates the 2 + 1 pure Chern-Simons action in thecontinuum limit, and its dynamics was discussed in some detail. The states of theChern-Simons model correspond to zero modes of Wilson Loops; similar states willgenerate the vacuum sectors in the 3 + 1 QCD model.

An important lesson fromthis work is that one cannot simply assign the same Wess-Zumino term to each2

link in the 3 + 1 QCD case, because the leading terms in the continuum limit mustgo as order a2 in this case, and not order a as in the 2 + 1 Chern-Simons theory.This means that the Wess-Zumino terms must be staggered from site to site, withcoupling constants ±k. The study of how to correctly stagger the Wess-Zuminoterms is the major topic of this paper.Section 2 is review of the basic transverse lattice construction of QCD basedon the (unitary) non-linear sigma model.

The degrees of freedom on the latticeare introduced, and the “naive” continuum limit is taken by performing a Blochwave expansion of the link fields. In section 3, we begin the analysis of addingWess-Zumino terms to the action.

It is found that the structure of the staggeredWess-Zumino terms which generate the Coulomb potential that in turn correctlydrives the system to the desired continuum limit violates local gauge invariance bygenerating non-Abelian anomaly terms. In section 4 this difficulty is resolved bydefining a new model which has a different structure of local gauge invariance, buthas the correct continuum limit.

In the new model, pairs of nearest neighbor linksare associated with a single local 2-D gauge symmetry, and the anomalies from theWess-Zumino terms for each pair of sites cancel. The local 2-D gauge symmetry isreduced by a factor of two from the transverse lattice construction with no Wess-Zumino terms.

The remaining local gauge symmetry for each pair of links in thebilocal transverse lattice model is then properly gauge fixed in light-cone gauge. Insection 5, the current algebraic solution of the WZW model is reviewed and thequantum theory of the new model is discussed.

Particular emphasis is placed onthe highest weight states of the current algebras, which generate the space of Wilsonloop zero modes on the lattice. Aspects of future bound-state calculations and otherapplications of this construction are discussed in section 6.3

Fig. 1.

The degrees of freedom associated with each site ⃗x⊥are the 2-D gauge fields A±(⃗x⊥), andthe link fields Ux(⃗x⊥) and Uy(⃗x⊥), defined on the links as in fig. 1.2.

The Basic Transverse Lattice ConstructionIn this section, the non-linear sigma model-based formulation of the trans-verse lattice construction of QCD is reviewed, and the process of taking the naivecontinuum limit is studied.Consider the matrix-valued chiral fields Uα(⃗x⊥; x+, x−), where α = 1, 2, whichbelong to the fundamental representation of SU(N). These fields lie on the links[⃗x⊥, ⃗x⊥+ ⃗α] of a discrete square lattice of points ⃗x⊥= a(nx, ny), with lattice spacinga and basis vectors ⃗α = (a, 0) or (0, a).

The link fields are continuous functions ofthe light-cone coordinates x± = (x0 ± x1)/√2, so that the two-dimensional latticedescribes a partially discretized 3 + 1 dimensional Minkowski space field theory1(seefig. 1).The links fields are defined to transform on the left and right under independentlocal 2-D gauge transformations associated with the sites that the links connect,δGUα = Λ⃗x⊥(xµ)Uα −UαΛ⃗x⊥+⃗α(xµ) .

(2.1)To construct a gauge invariant action, introduce SU(N)⃗x⊥gauge fields A±(⃗x⊥) =iAa±(⃗x⊥)T a, where the group generators T a satisfy [T a, T b] = if abcT c and Tr T aT b =12δab. The infinitesimal transformation law for the gauge fields isδGA±(⃗x⊥) = ∂±Λ⃗x⊥+ [Λ⃗x⊥, A±(⃗x⊥)] ,(2.2)and the covariant derivative isDµUα(⃗x⊥) = ∂µUα −Aµ(⃗x⊥)Uα + UαAµ(⃗x⊥+ ⃗α) .

(2.3)The transverse lattice action is given by[1]ITL =X⃗x⊥TrZd2x( a22g21F µνFµν + 1g2XαDµUαDµU †α+1g22a2Xα̸=βUα(⃗x⊥)Uβ(⃗x⊥+ ⃗α)U †α(⃗x⊥+ ⃗β)U †β(⃗x⊥) −1). (2.4)1 The indexes α, β, .

. .

denote transverse coordinates x, y, and µ, ν, . .

. denote longitudinal co-ordinates x± .4

As the lattice spacing a is taken to zero, the interaction terms will select smoothconfigurations as the dominant contributions to the quantum path integral; both theinteractions mediated by the local 2-D gauge fields and the plaquette interactionswill generate large potentials, unless the link configurations are smooth. For theplaquette term this is obvious; for the gauge interactions, this is clear only afterstudying the Coulomb potential obtained by gauge fixing in light-cone gauge A−= 0,in the context of light-cone quantization[2].

We will discuss this process further inthe next sections for the new transverse lattice model. Inserting the Bloch-waveexpansionUα = exph−aAα(⃗x⊥+ 12⃗α)i,(2.5)and keeping only the lowest order contributions, one obtains from the gauged sigmamodel kinetic termIK = 2a2g2X⃗x⊥,αZd2xFµαF µα + O(a4) ,(2.6)and from the the plaquette termIP = a2g22X⃗x⊥,α,βZd2x(Fαβ)2 + O(a4) .

(2.7)In deriving eqn. (2.6), the fields A±(⃗x⊥) were also assumed to be slowly varyingon the lattice.

Combining these three terms and tuning the coupling constants tog1 = g2 = g yields the continuum 4-D QCD action. For the quantum theory on thelattice, the Lorentz covariant critical point for each lattice spacing a is determinedby examining specific properties of the states, such as the mass spectrum 3 + 1Lorentz multiplets and the covariant dispersion relations[2].This is the non-linear sigma model (NLSM)-based transverse lattice model ofQCD.

In an ideal world, there would be an exact solution to the primary chiral sigmamodel, which could be used as a kernel to solve the entire model perturbatively in theinteractions. The idea would be to use the states that are diagonalize with respectto the NLSM Hamiltonian as a basis for construction of the singlet bound states ofthe full theory.

This is the approach pioneered by Schwinger in his solution to 1 + 1massless QED[11]. There, the full Hamiltonian was diagonalized in the basis offree fermions.

While there has been some progress in understanding the quantumNLSM[4], the progress is still insufficient to generate a Schwinger-type solutionto the problem at hand. A “Fock space” of operators with simple commutationrelations is required.5

3. Transverse Lattice with Wess-Zumino TermsNow consider the NLSM with Wess-Zumino term2[9],IW ZW = 1λ2Zd2xTr ∂µU∂µU † + kΓ ,(3.1)where U is a unitary matrix.

The non-local Wess-Zumino term Γ is well-definedonly up to Γ →Γ + 2π, and therefore the coupling constant k is an integer. Themodel is exactly solvable for the restricted critical values of the NLSM couplingconstantλ2 = 4π|k| .

(3.2)For these values there exist a complete basis of conserved vector and axial-vectorcurrents from which a Fock space representation of the quantum theory can beconstructed. For positive k, the conserved currents areJ−(x−) = (∂−U)U † ,J+(x+) = U †(∂+U) ,(3.3)and for negative k,˜J+(x+) = (∂+U)U † ,˜J−(x−) = U †(∂−U) .

(3.4)An elegant current algebraic solution for the quantum theory was given by Dashenand Frishman[13] for k = 1, and was later generalized for arbitrary k by Knizhnikand Zamolodchikov[14]. This solution will be discussed further in section 5.To apply the WZW NLSM technology to the transverse lattice formulation ofQCD, we associate a WZW field U with each link Uα(⃗x⊥)3.

For each link Uα(⃗x⊥),the currents J±,α and ˜J±,α are defined via eqns. (3.3) and (3.4).

The gauge variationof the J currents is given byδGJ+,α = −∂+Λ⃗x⊥+⃗α + [Λ⃗x⊥+⃗α, J+,α] + U †α∂+Λ⃗x⊥Uα ,δGJ−,α =∂−Λ⃗x⊥+ [Λ⃗x⊥, J−,α] −Uα∂−Λ⃗x⊥+⃗αU †α ,(3.5)where the currents J± are specified at ⃗x⊥. The variations of the ˜J currents aresimilar.2Normalization of the kinetic term differs from ref.

[9] because of a different definition of thetrace. See ref.

[17] for a thorough discussion of such normalization issues.3 The analysis here generalizes the 2 + 1 dimensional transverse lattice construction given inref. [10].6

The action is invariant under the global symmetry U →AUB, where A, B areconstant unitary matrices. The Wess-Zumino term must be gauged so it can beadded to a transverse lattice action.

(The kinetic term of the WZW model is easilygauged by introducing the covariant derivative as in the previous section.) Thegauge variation of the Wess-Zumino term is local[9] and can be written in terms ofthe currents,δGΓ(Uα(⃗x⊥)) = 12πZd2xTrΛ⃗x⊥(∂+J−,α −∂−˜J+,α)−Λ⃗x⊥+⃗α(∂−J+,α −∂+ ˜J−,α).

(3.6)It is straightforward to construct the “gauged” Wess-Zumino term for each link,˜Γ(Uα(⃗x⊥)) = Γ + 12πZd2xTr(A+(⃗x⊥)J−,α −A−(⃗x⊥+ ⃗α)J+,α+ A+(⃗x⊥)UαA−(⃗x⊥+ ⃗α)U †α−A−(⃗x⊥) ˜J+,α−A+(⃗x⊥+ ⃗α) ˜J−,α + A−(⃗x⊥)UαA+(⃗x⊥+ ⃗α)U †α). (3.7)This term actually is not gauge invariant, but instead transforms asδG˜Γ(Uα(⃗x⊥)) = 12πZd2xTr[Λ⃗x⊥ǫµν∂µAν(⃗x⊥) −Λ⃗x⊥+⃗αǫµν∂µAν(⃗x⊥+ ⃗α)] , (3.8)where ǫ+−= 1.

This lack of gauge invariance has the same form as the non-Abeliananomaly in two dimensions, and is realized at the classical level. Note that thevector subgroup of SU(N)left ⊗SU(N)right for each link is anomaly free[15].

Also,the global gauge symmetries for each site (x± independent gauge transformations)are unbroken.Since more than one link is coupled to each site, the anomaly can cancel betweenthe links.This is the mechanism that was introduced in ref. [10] to cancel theanomalies at each site of a 2 + 1 dimensional transverse lattice model.

For the caseat hand, each link on the 2-D lattice is assigned a Wess-Zumino coupling constantk⃗x⊥,α. The full action, including the Wess-Zumino term, is given by˜IT L = ITL +X⃗x⊥,αk⃗x⊥,α˜Γ(Uα(⃗x⊥)) .

(3.9)and it is anomaly free for each site ⃗x⊥ifXαk⃗x⊥,α −k⃗x⊥−⃗α,α = 0 . (3.10)7

Fig. 2.

Up to an overall change of sign, the figures 2(a)-(c) denote the anomaly-free vertices allowedfor the action ˜ITL. The signs correspond to Wess-Zumino coupling ±k, where k is a positive integer.In the remainder of this paper, we will assign either +k or −k Wess-Zuminocoupling to each link, where k is positive, and the NLSM coupling constant will befixed to the critical point g2 = 4π/k.There are three pairs of anomaly free vertices (i.e., six total) that can be con-structed for each site.

The members of each pair are related by an overall flip ofsigns, and representatives of each pair are given in fig. 2.

Each link is labeled by ±signs which denote the Wess-Zumino coupling.The simplest configuration to consider is a lattice of all “+” links, so that allvertices are all of the type 2(a). Unfortunately, this does not work because thecontinuum limit of the gauged Wess-Zumino terms are order a,X⃗x⊥,α˜Γ(Uα(⃗x⊥)) =X⃗x⊥,αTr a2π [2F−+Aα + A+∂αA−−A−∂αA+] + O(a3)(3.11)This expression, for a one-dimensional lattice, is the pure Chern-Simons term in2 + 1 dimensions.

It was discussed in some detail in ref. [10].

The leading O(a)part comes from the terms in the action that were added to gauge the Wess-Zuminoterm Γ. The Wess-Zumino term itself contributes only to order a3, as can be seenby expanding the variation δΓ with respect to the link fields U given in ref.

[9].A possible resolution is to stagger the gauged Wess-Zumino terms from site tosite with alternating signs. Clearly, there are a number of ways to stagger theseterms.

The “correct” ways will be those which lead to the right continuum limit.In particular, we argued in section 2 that the Coulomb and plaquette interactionsbetween links would drive the system to a smooth continuum limit.With thenecessity of staggering, this is no longer as obvious for the Coulomb interactions,since the gauge couplings to each link are not the same on a staggered lattice.Recall that the Coulomb interactions are mediated by the longitudinal gauge fieldsA±. These fields can be eliminated in the light-cone gauge A−= 0, at the expenseof generating a non-local Coulomb potential.

We need to study the form of thispotential on a staggered lattice.In the gauge A−= 0, the part of the path integral which depends upon thegauge field A+ isZGF =Y⃗x⊥[det ∂−]⃗x⊥Z[dA+(⃗x⊥)]e−i Rd2x{a2/2g21 (∂−A+(⃗x⊥))2+A+(⃗x⊥)J−(⃗x⊥)} ,(3.12)8

where det ∂−is the Fadeev-Popov determinant for each site, and the currentsJ−(⃗x⊥) are given by reading offthe couplings in eqn. (2.4).

The form of the currentsimplifies dramatically at the WZW critical points λ2 = 4π/k. For these cases,J−(⃗x⊥) = k2πXα±{J−,α+(⃗x⊥) −˜J−,α−(⃗x⊥−⃗α)} ,(3.13)where the sum over α+ (α−) is over links with +k (−k) Wess-Zumino coupling.In the context of the WZW model, the currents J−depend only upon x−.

TheCoulomb interaction is obtained by completing the square in A+. After completingthe square, the integral over A+ cancels the Fadeev-Popov determinant and thepath integral (3.12) becomesZGF =Y⃗x⊥ei Rd2xg21/2a2(1∂−J−(⃗x⊥))2,(3.14)where1∂−J−(⃗x⊥; x−) = 12∂−Zdy−|y−−x−|J−(⃗x⊥; y−) + f⃗x⊥(x+) .

(3.15)The importance of keeping the integration constant f⃗x⊥(x+) in the context of themassless Thirring model was recently discussed in ref. [16].

In our context, it iseasy to calculate by gauge fixing with the condition A+ = 0, thereby introducingthe currentsJ+(⃗x⊥) = k2πXα±{ ˜J−,α−(⃗x⊥) −J−,α+(⃗x⊥−⃗α)} . (3.16)These currents in the WZW model depend only on x+.

After completing the squarein this case, the path integral (3.12) isZGF =Y⃗x⊥ei Rd2xg21/2a2(1∂+ J+(⃗x⊥))2,(3.17)where1∂+J+(⃗x⊥; x−) = 12∂+Zdy+|y+ −x+|J+(⃗x⊥; y+) + ¯f⃗x⊥(x−) . (3.18)Equating the two results for the same gauge-fixed path integral yieldsf⃗x⊥(x+) = 12∂+Zdy+|y+ −x+|J+(⃗x⊥; y+) ,(3.19)9

Fig. 3.

The two vertex configurations for which all four links couple symmetrically to each other.In figure 2(a) all four links contribute to the J+ current at the vertex, and in figure 2(b) all fourcontribute to J−.and the path integral ZGF = eiIC, whereIC = g214a2X⃗x⊥Zd2xTr Zdy−J−(x−)|x−−y−|J−(y−)+Zdy+J+(x+)|x+ −y+|J+(y+). (3.20)As in the Schwinger model, the Coulomb potential does not mix left- and right-mover currents.

In the current algebra solution to the quantum theory discussed inthe next section, the Coulomb terms are treated as potential terms. The currentsJ−and J+ then remain functions of x−or x+ in the WZW model with Coulombinteractions.

This is the same situation found in the massless Schwinger model (seethe analysis of ref. [12]).The Coulomb interactions are proportional to 1/a2.

As a →0, configurationswhich minimize the full action should dominate the path integral. The questionis whether these configurations correspond to the smooth continuum limit thatwe desire.

Consider the link Uα+(⃗x⊥). According to eqns.

(3.13) and (3.16), itinteracts at ⃗x⊥by contributing to the J−(⃗x⊥) current and interacts at ⃗x⊥+ ⃗α bycontributing to J+(⃗x⊥+ ⃗α). Similarly, Uα−(⃗x⊥) interacts at ⃗x⊥by contributing tothe J+(⃗x⊥) current and interacts at ⃗x⊥+ ⃗α by contributing to J−(⃗x⊥+ ⃗α).

Foreach of the anomaly-free vertices in figure 2, the links interact pairwise with eachother. None of the vertices have all four links contributing to J+ or J−.

Rather,two links contribute to J+ and two links contribute to J−. Therefore minimizingthe Coulomb interaction as a →0 does not necessarily drive the system to thesmooth continuum limit that is required to reproduce continuum QCD.

Furtherevidence that the vertices of figure 2 do not generate the correct continuum limitwas obtained by studying the vacuum structure, following the analysis discussed insection 5 for the correct result.4. Bilocal Gauge Invariance, and a New Transverse Lattice ActionThe two vertices that have all four links contributing to either J+ or J−aregiven in figure 3.

While these vertices have the correct behavior under the Coulombinteractions, they are both anomalous with respect to the local gauge invariance at10

each site. Recovering QCD in the continuum limit is our paramount consideration,so we will consider breaking some of the local gauge symmetry.

Specifically, we willgauge only the anomaly free local symmetries. The anomalous local symmetrieswill be become global symmetries.

The unbroken local symmetries will then haveto be gauge fixed, and the coupling of the links via Coulomb interactions will needto be re-examined.The solution to the problems of the previous section will make use of the factthat the two vertices of figure 3 break gauge invariance in opposite ways. To bespecific, label each site ⃗x⊥= a(nx, ny) with the Z2 quantum numberPL(⃗x⊥) = (−1)nx+ny(4.1)which will be referred to as lattice parity .

Even (odd) sites have lattice parity +1(−1). As in the previous section, we consider the transverse lattice action withWess-Zumino terms, eqn(3.9).

However, now we consider configurations whichviolate the gauge invariance constraint eqn. (3.10).

The assignment of the Wess-Zumino coupling constant is given byUα(⃗x⊥) = Uα+(⃗x⊥) , ⃗x⊥even ,Uα(⃗x⊥) = Uα−(⃗x⊥) , ⃗x⊥odd ,(4.2)where the notation α± corresponds to assigning Wess-Zumino coupling ±k. Thevertex at odd (even) sites is the type shown in fig.

3(a) (fig. 3(b) ).

The anomalyat each site isδG ˜ITL(⃗x⊥) = PL(⃗x⊥)2kπZd2xTr Λ(⃗x⊥)ǫµν∂µAν(⃗x⊥) . (4.3)Define nearest neighbor pairs (⃗x+⊥, ⃗x−⊥), which by the above construction have op-posite anomalies, as(⃗x+⊥, ⃗x−⊥) = (⃗x⊥, ⃗x⊥+ (−1)nx ˆx),∀⃗x⊥s.t.

PL(⃗x⊥) = 1 ,(4.4)where ˆx = (a, 0). Every site on the square lattice belongs to one pair.

By construc-tion, ⃗x+⊥(⃗x−⊥) is an even (odd) site.For each of these nearest neighbor pairs, the anomaly breaks one of the localgauge symmetries, and preserves the other local symmetry and the two global sym-metries. Gauging the local symmetries for this transverse lattice model no longerrequires one independent vector potential for each site.

Rather, the gauge fields at11

Fig. 4.

For the bilocal model, each gauge field Aµ(⃗x+⊥) interacts with seven links. Figure 4 showsthe case where ⃗x+⊥is to the right of ⃗x−⊥.⃗x−⊥sites can be parametrized asAµ(⃗x−⊥) = G⃗x−⊥Aµ(⃗x+⊥)G†⃗x−⊥,(4.5)where G⃗x−⊥is a constant (x± independent) unitary matrix which transforms asδGG⃗x−⊥= ˜Λ⃗x−⊥G⃗x−⊥,(4.6)under gauge transformations.

The field G⃗x−⊥corresponds to the unbroken globalsymmetry at the ⃗x−⊥sites. So instead of having a 2-D gauge field Aµ for each site,we now have the set (Aµ, G⃗x−⊥) for each pair of sites.

The links and vector fieldstransform as given by eqns. (2.1) and (2.2) as long as the the infinitesimal variationsat the x−sites satisfy the constraintΛ⃗x−⊥(x±) = ˜Λ⃗x−⊥+ G⃗x−⊥Λ⃗x+⊥(x±)G†⃗x−⊥.

(4.7)With this construction, the gauge variations from the x−sites cancel the anomalyfrom the x+ sites. The path integral measure is redefined asY⃗x⊥[dAµ] →Y⃗x+⊥[dAµ(⃗x+⊥)]Y⃗x−⊥[dG⃗x−⊥] ,(4.8)where [dG] is the left invariant Haar measure for G. The full action is given byeqn.

(3.9), with the Wess-Zumino coupling constants given by eqn. (4.2), and thefield identification (4.5).It is important that the local gauge symmetry at each site remain unbroken.Otherwise, each pair of links (Uα+, Uα−) would transform the same way under theremaining local gauge invariance, effectively doubling the number of link degrees offreedom in the gauge theory, and the theory would not have QCD as the “naive”continuum limit.

Each of the remaining local symmetries are associated with twonearest neighbor sites, paired together as prescribed by equation (4.4). The aboveconstruction will be referred to as a bilocal transverse lattice model.

Each gaugefield Aµ(⃗x+⊥) is coupled to seven links instead of four (see fig. 4).

This difference isobviously significant at the lattice level. But again, the point is that there are manymodels which have the same continuum behavior which differ at scales of the latticespacing.

The bilocal model has the advantage of being more easily treatable at thelattice level than the basic transverse lattice model without Wess-Zumino terms.12

The Coulomb dynamics of the bilocal model is studied by gauge fixing in theA−= 0 gauge as in the previous section.The part of the path integral whichdepends upon the gauge field A+ isZGF =Y⃗x+⊥[det ∂−]⃗x+⊥Z[dA+(⃗x+⊥)]e−i Rd2xTr {a2/g21(∂−A+(⃗x+⊥))2+A+(⃗x+⊥)J−(⃗x+⊥)} . (4.9)There is an additional factor of two in front of the (∂−A+)2 term, relative toeqn.

(3.12), from the contribution to the kinetic energy term from the ⃗x−⊥sites.Note that the G⃗x−⊥s cancel out of this expression. The current J−(⃗x+⊥) is given byeqn.

(3.13), and all four links connected to the site ⃗x+⊥contribute to it. Completingthe square yields eqns.

(3.14) and (3.15), up to the additional factor of two, andwhere J−(⃗x−⊥) = 0. The functions f⃗x+⊥(x+) remain to be determined.

Following theprevious analysis, we gauge fix in the A+ = 0 gauge and findZGF =Y⃗x+⊥[det ∂+]⃗x+⊥Z[dA−(⃗x+⊥)][dG⃗x−⊥]× e−i Rd2xTr {a2/g21(∂+A−(⃗x+⊥))2+A−(⃗x+⊥)G†⃗x−⊥J+(⃗x−⊥)G⃗x−⊥}. (4.10)After completing the square and comparing to the A−= 0 case, one findsf⃗x+⊥(x+) = 12∂+Zdy+|y+ −x+|J+(⃗x−⊥; y+) .

(4.11)The nonlocal Coulomb effective action for the bilocal theory isIC = g218a2Zd2xTr X⃗x+⊥Zdy−J−(x−)|x−−y−|J−(y−)+X⃗x−⊥Zdy+J+(x+)|x+ −y+|J+(y+). (4.12)While in the gauge invariant bilocal model action the G⃗x⊥dependence is requiredto preserve the global gauge invariance at x−sites, the G⃗x−⊥dependence cancels inthe gauge fixed action because it is bilinear in the currents.

(The path integral overthe G⃗x−⊥is finite since the group SU(N) is compact.) These Coulomb terms haveprecisely the properties that we desired to obtain the correct continuum limit.

Alllinks connected to a given site ⃗x⊥interact with each other via (4.12). While thepairing of sites (4.4) broke a discrete lattice symmetry by differentiating between xand y directions, this symmetry is restored in the Coulomb effective action.

As the13

lattice spacing a goes to zero, the Coulomb dynamics drives the system to a smoothcontinuum limit.The naive continuum limit is studied, as in the previous sections, by insert-ing a Bloch-wave expansion into the gauge invariant action. Recall that the Aαdependence in (2.5) was determined by requiring that it transform as a gauge field,δGAα(⃗x⊥+ 12⃗α) = Λ⃗x⊥+⃗α −Λ⃗x⊥a+ [Λ⃗x⊥Aα −AαΛ⃗x⊥+⃗α] + O(a) .

(4.13)For the bilocal model it is still possible to meaningfully expand the Λ’s asΛ⃗x⊥+⃗α −Λ⃗x⊥= a∂αΛ⃗x⊥+ O(a2) ,(4.14)since the constraint (4.7) allows for arbitrary ˜Λ⃗x−⊥transformations at x−sites,i.e. the global gauge invariance at each site is retained.

Aα transforms as a gaugefield as a →0 and the Bloch-wave expansion (2.5) is valid for the bilocal model.In the continuum limit, the undesired order a terms (3.11) cancel between evenand odd sites as a →0. This is due to the staggering of the Wess-Zumino termswhich is built into the model.

The cancellation of the gauged Wess-Zumino termsbetween pairs of adjacent even and odd links is to order a3, because parity preventsthe gauged Wess-Zumino terms from contributing to order a2. They are thereforeirrelevant operators in the continuum.In fact, the order a terms are explicitly cancelled locally in the gauge fixedlattice action, and there is no need to invoke a cancellation between sites, as in thegauge invariant analysis.

To see this, expand the currents J± or ˜J± for each linkorder by order in lattice spacing a. The currents J+ and J−given by eqns.

(3.13)and (3.16) are order a2, and therefore the Coulomb interaction (4.12) for each siteis order a2. The kinetic and plaquette terms for the link fields are also order a2 bythe analysis in section 2.

(The bare ungauged Wess-Zumino terms are order a3 foreach link field as discussed in section 3. )The bilocal transverse lattice model satisfies the primary constraint that itscontinuum limit be QCD, at the expense of introducing a somewhat complicatedstructure of gauge invariance on the lattice.5.

Quantization of the Bilocal Transverse Lattice ModelIn this section, the quantum theory of the new bilocal transverse lattice modelconstructed.The “discrete light-cone” approach of ref. [3] is followed when the14

Hilbert space is defined. The goal of this section is to set up the theory for futurecomputational study of the bound state problem.The approach taken is to solve the WZW model for each link and treat theCoulomb and plaquette terms as additional potential terms.

The current algebrasolution of the WZW model was first given by Dashen and Frishman[13] for levelk = 1, and later generalized to arbitrary level by Knizhnik and Zamolodchikov[14].The current algebra is specified by Sugawara-type algebras for the left and right-moving currents.For even (odd) links with Wess-Zumino coupling k (−k), thecurrents are by J± ( ˜J±). For right movers of an even link, the current commutationrelations are[Ja−(x−), Jb−(y−)] = if abcJc−(x−) −ik2π δabδ′(x−−y−) ,(5.1)where Ja−is[17]XaT aJa−(⃗x⊥) = −ik√2π J−(⃗x⊥) .

(5.2)The same type of expressions hold for currents ˜Ja−, Ja+ and ˜Ja+ (x−→x+ for the +currents). Left- and right-mover currents commute, as do currents defined for differ-ent links.

These algebras are the equal time commutation relations translated intolight-cone coordinates. One can specify initial conditions on the light fronts x+ = 0and x−= 0 for the massless currents J−and J+ in the WZW model.

However, be-cause of the complicated form of the plaquette interaction term, initial conditionswill be specified on an equal time surface.To make the connection between the commutators above and Kac-Moody alge-bras, infared cutoffs for the light-cone coordinates are introduced by hand. With x±defined on the interval [−L, L], and fields taken to be periodic, a mode expansionfor the currents takes the formJa+ = 12LXne−iπnx+/LJan ,˜Ja+ = 12LXne−iπnx+/LKan ,Ja−= 12LXne−iπnx−/L ¯Jan ,˜Ja−= 12LXne−iπnx−/L ¯Kan ,(5.3)where the site dependence of the currents has been suppressed.

The currents Jan, ¯Janare defined for even links, and Kan, ¯Kan are defined for odd links. The delta functionin the current algebra (5.1) is easily defined for period boundary conditions, andthe current algebra is equivalent to the Kac-Moody algebra[Jam, Jbn] = if abcJcm+n + kmδabδm,−n(5.4)15

for each of the mutually commuting currents. The zero modes Ja0 form a subalgebraequivalent to the original SU(N) Lie algebra.The light-cone singlet vacuum |0⟩, defined for each chiral algebra, is the uniquehighest weight state which satisfiesJan|0⟩= 0 , n > 0 ,Ja0 |0⟩= 0 .

(5.5)The definition of normal ordering is with respect to this vacuum state,: JamJbn :=JamJbn m < 0 ,=JanJbm m ≥0 . (5.6)For the even link WZW models, the Lorentz generators P + = H + P andP −= H −P are given byP +WZW =12L(2k + N)Xn: JanJa−n : = 12LL0 ,P −WZW =12L(2k + N)Xn: ¯Jan ¯Ja−n : = 12L¯L0 ;(5.7)the odd link Lorentz generators are the same with (J, ¯J) →(K, ¯K).

The modes ofthe currents are diagonal with respect to the light-cone momenta, and have non-vanishing commutation relations [L0, Ja−m] = mJa−m and [¯L0, ¯Ja−m] = m ¯Ja−m.The space of states for each link is built up by applying a product of raisingoperators {Ja−m}{ ¯Ja−m} upon a highest weight (vacuum) state |ℓ0⟩⊗|ℓ0⟩for theright-mover and left-mover sectors.This construction is analogous to the Fockspace basis of the linear sigma model states.For the right-mover sector, thesestates satisfyJam|ℓ0⟩= 0, m > 0 ,Eα0 |ℓ0⟩= 0 ,(5.8)where E±α0and Hi0 are the zero mode currents Ja0 in the Cartan-Weyl basis of thealgebra. Similar results apply for the left-mover sector.

The vacuum states arethe highest weights of finite dimensional SU(N) representations |ℓ⟩generated byapplying zero modes E−α0. This will be referred to as the zero mode sub-sector ofthe full space of states.

The zero modes have non-vanishing zero point momentaL0|ℓ⟩= ∆ℓ|ℓ⟩and ¯L0|ℓ⟩= ∆ℓ|ℓ⟩where ∆ℓ= Cℓ/2(N + k), and Cℓis the quadraticCasimir of the representation ℓ. The representations of the full Kac-Moody algebrafor each highest weight are unitary if the highest weights have Young tableaux withthe number of columns less than or equal to k[18].16

The representations |ℓ⟩are the zero modes of dimensionless primary fields φℓ(v)of the chiral algebras. The primary fields have simple commutation relations withthe currents[13][14].

For the even links,[Jan, φℓ(x+)] =eiπnx+/L φℓ(x+) taℓ,[ ¯Jan, ¯φℓ(x−)] =eiπnx−/L taℓ¯φℓ(x−) ,(5.9)and for the odd links,[Kan, φℓ(x+)] =eiπnx+/L taℓφℓ(x+) ,[ ¯Kan, ¯φℓ(x−)] =eiπnx−/L ¯φℓ(x−) taℓ,(5.10)where taℓis a generator of SU(N) in the irreducible representation ℓ.Note theordering difference here between even and odd links. The primary fields are in-tertwining operators of the space of states, since they interpolate between vacuumsectors.

Let |1⟩denote the fundamental representation of SU(N). The productsof left-mover and right-mover primary fields ¯φ1φ1 for even links, and φ1 ¯φ1 for oddlinks, are the quantum fields which correspond to the classical unitary chiral fieldU in the classical action.

Since the classical unitary field transforms on the left andright in the same representation, we will consider only the diagonal[18] products ofthe highest weight fields as the vacuum states in the transverse lattice theory. Thereader is referred to refs.

[9][14][18] and in particular the review article ref. [17] forfurther information on the WZW model.The gauge fixing proceedure of the previous section left the global symmetry ateach site untouched.

The generators of these gauge transformations in the quantumtheory are constructed from the zero modes of the currentsJ a0 (⃗x+⊥) =Xα¯Ja0 (⃗x+⊥, α) −¯Ka0 (⃗x+⊥−⃗α, α),J a0 (⃗x−⊥) =XαKa0 (⃗x−⊥, α) −Ja0 (⃗x−⊥−⃗α, α). (5.11)In the classical theory, all physical states are gauge invariant.In the quantumtheory, this restriction can be relaxed somewhat, since it is the gauge invariance ofexpectation values of states (operators) that is required4.

Following the previous4 A modern example of this treatment of gauge symmetries is found in string theory, whereconformal invariance is a crucial property of the quantum theory. The Virasoro modes Ln generateconformal transformations, and physical states need to be annihilated by only the modes with n ≥0.17

Fig. 5.The two cases encountered when gluing together the zero modes of two links to satisfythe global gauge invariance constraints at a site.

Figure 5(a) (5(b)) denotes two links of the same(opposite) lattice parity.treatment of this point for the the 1-D transverse lattice[10], we requireEα0 |physical⟩= 0 ,Hi0|physical⟩= 0 ,(5.12)where E±α0, Hi0 denote J a0 in the Cartan-Weyl basis. In the 1-D transverse latticecase, these constraints led to the correct set of physical states.This criterion can first be applied to the subspace of states obtained by takingproducts of zero mode states on the lattice.

In the 1-D transverse lattice construc-tion, non-trivial states of this type were found. They were interpreted as zero modesof Wilson loops on the lattice.

Each link is associated with a product of left-moverzero modes |ℓ⟩and right-mover zero modes |ℓ⟩. Even and odd links at Uα(⃗x⊥) havezero mode structure|ℓ⟩⃗x⊥,α × |ℓ⟩⃗x⊥,α , even link ,|ℓ⟩⃗x⊥,α × |ℓ⟩⃗x⊥,α , odd link .

(5.13)Consider gluing together two links with non-trivial zero mode structure at a site⃗x⊥, such that (5.12) is satisfied. There are two basic cases to consider, as shown infigure 5.

In figure 5(a) (5(b)), the two links have the same (opposite) lattice parity.In figure 5(a), two right-mover zero modes |m⟩× |ℓ⟩need to be glued together. Theconstraints (5.12) are satisfied if we project onto the singlet sector of the tensorproduct: P 0m,ℓ{|m⟩⊗|ℓ⟩}.

This is because the constraint involves the sum of currentsfor each link, i.e. the diagonal subalgebra of the link zero mode algebras.

For theother case denoted in figure 5(b), the constrains involve the difference between thecurrents for each link. This was the situation encountered in the 1-D transverselattice analysis.The solution to the constraints is to project onto the highestweights of the same representation: |ℓ0⟩× |m0⟩δℓ,m.

These two rules for gluingtogether zero modes at a site can be used to construct a wide variety of solutionsto the gauge invariance conditions. For example, a plaquette solution is of the formP 0|ℓ†⟩⃗x+⊥,y ⊗|ℓ⟩⃗x+⊥,x|ℓ0⟩⃗x+⊥,x |ℓ0⟩⃗x+⊥+ˆx,y×P 0|ℓ⟩⃗x+⊥+ˆx,y ⊗|ℓ†⟩⃗x+⊥+ˆy,x|ℓ†0⟩⃗x+⊥+ˆy,x |ℓ†0⟩⃗x+⊥,y ,(5.14)where ℓ† is the conjugate representation of ℓ.This plaquette state is the zeromode of a Wilson loop with flux in representation ℓ, flowing in the counterclockwise18

orientation. It has zero point energy (P +WZW + P −WZW)/2 = 2∆ℓ/L and momentum(P +WZW = P −WZW)/2 = 0.

This is the part of a Wilson loop that can never be gaugedaway. While in the linear sigma model transverse lattice theory[1][2] these type ofstates are presumably soliton excitations, in the new non-linear theory they arisequite naturally as vacuum sectors.

The space of states for the new transverse latticemodel breaks up into different vacuum sectors of Wilson loop zero modes, and theintertwining operator, which takes states from one sector to another, is the plaquetteoperator discussed below.The contribution of the Coulomb interactions to the Lorentz generators is ob-tained by inserting the mode expansions (5.3) into the effective action (4.12) andnormal ordering with respect to the vacuum state (5.5),P +C =2L g18πa2 X⃗x−⊥∞Xn=−∞1n2 : J an (⃗x−⊥)J an (⃗x−⊥) : ,P −C =2L g18πa2 X⃗x+⊥∞Xn=−∞1n2 : J an (⃗x+⊥)J an (⃗x+⊥) : ,(5.15)whereJ an (⃗x+⊥) =XαKan(⃗x+⊥, α) −Jan(⃗x+⊥−⃗α, α) ,J an (⃗x−⊥) =Xα¯Jan(⃗x−⊥, α) −¯Kan(⃗x−⊥−⃗α, α) . (5.16)Note that the light-cone momenta P ±C are proportional to the infared cutoffL.

Thecontribution to the mass operator from the Coulomb potential M 2C = P +WZWP −C +P −WZWP +C is therefore independent of L. Diagonalization of M 2C on a basis of statesdefined in the cutofftheory is therefore the exact (cutoffindependent) result inthat basis. The masses are finite only if the potentially infared divergent n = 0coefficients in P ±C vanish: J a0 (⃗x⊥)J a0 (⃗x⊥) := 0 ,∀⃗x⊥.

(5.17)This is a statement of charge neutrality for each site. The incoming charge mustequal the outgoing charge, where the direction is defined by the arrows associatedwith each link (See figure 4).

The zero mode states discussed above, and in par-ticular the state given by eqn. (5.14), satisfy this constraint by construction.

Infact, the constraints (5.17) and (5.12) are equivalent. For a localized state, suchas a link-antilink excitation, the constraint requires that physical states be gaugesinglets at each site.19

The structure of the Coulomb term for each site ⃗x⊥is similar to that obtainedin 1 + 1 QCD with massless fermions quantized following the same approach[19]. Inthat case, it is known that there is no mass gap[20] generated by the Coulomb inter-actions, because states can be constructed from the U(1) fermion number currentwhich commutes with the non-Abelian currents that make up the Coulomb poten-tial.

This is not the case for the new transverse lattice model, since this currentdoes not exist in this case. Analysis of the ’tHooft equation[21] for link-antilinkbound states can make this result quantitative by determining the bare mass gapfor these states.

Preliminary calculations show that a linear Regge trajectory forthe mass spectrum is obtained in the large N limit.The plaquette interaction in (2.4) explicitly mixes left-mover and right-movermodes, like a mass term.In principle, then, one could eliminate left-movers interms of right-movers by solving a constraint equation for each equal light-conetime surface x+ = const. However, in practice this is very difficult because of thecomplicated form of the interaction in terms of the WZW currents.Therefore,the plaquette term will be treated as an interaction in the quantum theory withindependent commutation relations for both left- and right-movers.

The relevantCauchy surface on which to fix initial conditions is an equal time surface. The WZWmodel current algebras are equal time commutation relations written in light-conecoordinates (see in particular sec.

2 of ref. [22].

).For fixed time quantization, the link fields are functions of the single variable x,and for t = 0 it is useful to define the complex variable z = eiπx/√2L. The quantumlink fields are products of left-mover and right-mover highest weightsU(z) = : ¯φ1(z∗)φ1(z) : ,even link ,U(z) = : φ1(z)¯φ1(z∗) : ,odd link ,(5.18)where φ1 is the primary field in the fundamental representation of SU(N).

Theconjugate U † can be similarly defined in terms of the conjugate primary fields φ†1.The contribution of the plaquette interaction to the Lorentz group generators isthenP ±P =L√2πg22a2IdzXβ̸=αTr1 −Uα(⃗x⊥)Uβ(⃗x⊥+ ⃗α)U −1α (⃗x⊥+ ⃗β)U −1β (⃗x⊥). (5.19)The trace in eqn.

(5.19) denotes a projection onto products of highest weights foreach site such that the gauge invariance constraints are satisfied. The plaquetteinteraction does not contribute to the spatial momentum (P + −P −)/2.20

6. DiscussionLike any other theory which describes non-perturbative behavior of QCD, thetransverse lattice construction outlined above is very complicated.

The basic ad-vantage over 4-D lattice simulations is that a continuum analysis is used to describelocal link dynamics. To take advantage of the continuum description, one has tofind a suitable truncation of the full model that still contains the desired physics.For the calculation of glueball masses in in the Bardeen Pearson model[2], the ba-sic degrees of freedom were truncated to link-antilink and four-link bound states.These states mix under the plaquette interaction, which also provides for transversemotion.

It is essential to complete this basic analysis for the new transverse latticemodel and verify that in this case the link number expansion is a valid one, i.e., thatlink number violation is strong enough to allow for transverse motion of the states,yet small enough to validate the link number expansion.The physical state of the link-antilink and four-link truncated basis is a Wilsonloop smeared over the lattice. The link-antilink states correspond to Wilson loopswhich extend in the longitudinal directions.

This type of state in the gauge fixedtransverse lattice model is the bilinear U(x)U †(y) integrated over a wavefunction.The link-antilink bound state is not a bilinear JaJa of WZW currents. Althoughthe currents are the linear degrees of freedom of the WZW model, the real degreesof freedom of the lattice theory are the link fields.The spectrum of bound states that can be constructed by the above formalismis degenerate because the zero mode of the state can be shifted and boosted byLorentz transformations.

We want to select a basis of bound states that does notcontain copies of the same state. Moreover, we want a basis of states for which oneof the momentum operators is manifestly diagonal.

For instance, in the linear sigmamodel analysis[2], P −was manifestly diagonal. This type of analysis in the bilocalmodel is complicated, because as discussed above both the left- and right-movercurrents are treated as dynamical.

There is no simple constraint equation whichcan be used to solve for one set in terms of the other.There is a basis of states one can use to study, in the simplest way, the boundstate spectrum. It is defined by allowing zero-mode excitations for both left- andright-movers, but truncating all non-zero mode excitations of the left-movers.

(Theparity conjugate basis is equally simple.) There is no proof that all physical stateshave a representative is this truncated basis.

However, this truncation is similar to21

the analysis of ref. [23], where the spectrum of the massless Schwinger model wasstudied.

There, bound states were constructed from fermions of only one chirality.In this chiral Schwinger model, there exists only one copy of the free massive scalarboson, instead of the infinite number of degenerate copies which exist in the fullSchwinger model[16]. Nevertheless, the single copy has the correct mass.

Again,the idea here is to suggest a starting point for the explicit calculation of the massspectrum.In the truncated basis, a local link-antilink state is given by|P⟩=IdzIdyPXk=0w(k) z−kyk−P Tr : φ1(z)φ†1(y) : |0⟩⊗|0⟩. (6.1)For the link-antilink state, the only possible right-mover singlet state is the vacuum.Periodicity in the z and y variables quantizes the momenta P and k to be integers.The diagonal contribution to the light-cone momentum P + is from the WZW model.Using the conformal field theory commutator [L0, φ1(z)] = (z∂z + ∆1)φ1, one findsP +WZW|P⟩= P + 2∆12L|P⟩.

(6.2)In the truncated basis, the contribution P +C from the Coulomb interactions alwaysvanishes. And as discussed in the previous section, the contribution P +P from theplaquette interation mixes this state with four link states.

A physical two link stateis a local state(6.1) smeared over the entire lattice with a wavefunction specifyingthe transverse momentum distribution.The simplest non-trivial local four link state is obtained by replacing the right-mover zero modes in equation (5.14) with integrals over the link fields.Thereare actually a number of local four link configurations to consider (see fig. 2 ofref.

[2]). The reader is referred to ref.

[2] for a complete description of the methodfor determining the mass spectrum of the 3 + 1 Lorentz multiplets in the two andfour link basis.The U(1) case avoids all of the complications developed in sections 3 and 4, andtherefore may be a good laboratory to probe the recovery of 4-D Lorentz invariancefrom the transverse lattice construction. The non-linear sigma model action is thenthe Gaussian model for the fields θ, where U = exp iθ.

The plaquette interaction isa product of normal ordered U(1) vertex operators. However, it may be difficult,if not impossible, to approach the continuum limit for the Abelian case, since linknumber violation will be large for a deconfined theory.22

Unlike 4-D lattice gauge theory, the transverse lattice construction may be ableto generate structure functions of relativistic bound states, since the wavefunctionsof the states are explicitly calculated when diagonalizing the mass spectrum. Onlywhen such a problem, impossible to work out by conventional techniques, is solvedvia the transverse lattice construction, will this new approach be fully accepted asa tool for probing non-perturbative physics.The transverse lattice construction connects 2-D physics to more realistic higherdimensional models.Since string theory has motivated a great deal of recentprogress in 2-D field theory, there are surely many more connections that canbe made, to the benefit of both mathematically- and phenomenologically-orientedphysicists.Acknowledgements: I would like to thank S. Brodsky, L. Day, O. Hernandez,K.

Hornbostel, A. Kronfeld, J. Lykken, P. MacKenzie, S. Pinsky, and H.C. Paulifor numerous useful discussions. Some of this work was completed at the AspenCenter for Physics and at the Max-Planck Institut f¨ur Kernphysik in Heidelberg,Germany.

I am particularly grateful to W. Bardeen for many illuminating discus-sions on transverse lattice physics.23

References[1]W. Bardeen and R. Pearson, Phys. Rev.

D14 (1976)547.[2]W. Bardeen, R. Pearson, and E. Rabinovici, Phys.

Rev. D21 (1980)1037.[3]S.

Brodsky and H.C. Pauli, Phys. Rev.

D32 (1985)1993.[4]P. Wiegmann, Phys.

Lett. 141B (1984)217.[5]M.

Bander and W. Bardeen, Phys. Rev.

D14 (1976)2117.[6]R. J. Perry, A. Harindranath, and K. Wilson, Phys.

Rev. Lett.

65 (1990)2959.[7]J. Kogut, D.K.

Sinclair, and L. Susskind, Nucl. Phys.

B247 (1976)199.[8]J. Wess and B. Zumino, Phys.

Lett. 37B (1971)95.[9]E.

Witten, Commun. Math.

Phys. 92 (1984)455.[10]P.

Griffin, Fermilab-Pub-91/92-T (1991).[11]J. Schwinger, Phys.

Rev. 128 (1962)2425.[12]H.

Bergknoff, Nucl. Phys.

B122 (1977)215.[13]R. Dashen and Y. Frishman, Phys.

Rev. D11 (1975)2781.[14]V.

Knizhnik and A.B. Zamolodchikov, Nucl.

Phys. B247 (1984)83.[15]D.

Karabali, Q. Park, H. Schnitzer, and Z. Yang, Phys.

Lett. 216B(1989)307.[16]G.

McCartor, Z. Phys. C41 (1988)271.[17]P.

Goddard and D. Olive, Int. J. Mod.

Phys. A1 (1986)303.[18]D.

Gepner and E. Witten, Nucl. Phys.

B278 (1986)493.[19]K. Hornbostel, S. Brodsky, and H.C. Pauli, Phys.

Rev. D41 (1990)3814.[20]W.

Buchm¨uller, S. Love, and R. Peccei, Phys. Lett.

108B (1982)426.[21]G. ’tHooft, Nucl.

Phys. B75 (1974)461.[22]P.

Bowcock, Nucl. Phys.

B316 (1989)80.[23]T. Eller, H.C. Pauli, and S. Brodsky, Phys.

Rev. D35 (1987)1493.24

---

출처: arXiv:9108.009 • 원문 보기