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Staggered fermions and chiral symmetry breaking

arXiv:hep-th/9207083v1 24 Jul 1992UFIFT-92-19July 1992Staggered fermions and chiral symmetry breakingin transverse lattice regulated QED†Paul A. Griffin††Department of Physics, University of FloridaGainesville, FL 32611AbstractStaggered fermions are constructed for the transverse lattice regularization scheme.The weak perturbation theory of transverse lattice non-compact QED is developed in light-cone gauge, and we argue that for fixed lattice spacing this theory is ultraviolet finite, orderby order in perturbation theory. However, by calculating the anomalous scaling dimensionof the link fields, we find that the interaction Hamiltonian becomes non-renormalizable forg2(a) > 4π, where g(a) is the bare (lattice) QED coupling constant.

We conjecture thatthis is the critical point of the chiral symmetry breaking phase transition in QED. Non-perturbative chiral symmetry breaking is then studied in the strong coupling limit.

Thediscrete remnant of chiral symmetry that remains on the lattice is spontaneously broken,and the ground state to lowest order in the strong coupling expansion corresponds to theclassical ground state of the two-dimensional spin one-half Heisenberg antiferromagnet.†Supported in part by the U.S. Department of Energy, under grant DE-FG05-86ER-40272†† Internet Address: pgriffin@ufhepa.phys.ufl.edu

1. IntroductionStaggered fermions[1] for lattice gauge theory [2][3] have the desirable property ofpreserving a discrete remnant of chiral symmetry, and are therefore useful for studying thenon-perturbative chiral symmetry breaking in gauge theories.

Staggered fermions havebeen constructed for the four dimensional Euclidean formulation of lattice gauge theory,and for the Hamiltonian formulation of lattice gauge theory, based on a three dimensionalspatial lattice and one continuum time variable. In sections 2 and 3, we construct staggeredfermions for the transverse lattice formalism of Bardeen et.

al. [4][5], which is based on a two-dimensional spatial lattice and two continuum space-time coordinates.

Wilson fermionsfor the transverse lattice were constructed in ref. [4].The transverse lattice construction is a minimalist’s non-perturbative regularizationscheme for gauge fields[4].

After choosing an axial gauge and imposing the Gauss con-straint, the degrees of freedom of the gauge field are reduced to two spatial components,and these can be regulated by mapping them to link fields on a two-dimensional lattice.The link fields are non-perturbative excitations of the gauge fields, and are scalars withrespect to the two continuous space-time coordinates perpendicular to the lattice, so theirultraviolet (UV) behavior is softened.The basic disadvantage of the transverse lattice construction is the breaking of 3+1dimensional Lorentz invariance down to 1+1 dimensional Lorentz invariance plus discrete2-D lattice translations and rotations. This means, for example, that pure 3+1 dimensionalgauge theory has three bare coupling constants when regulated this way, as dictated by1+1 Lorentz invariance[5].One assumes that the full 3+1 Lorentz invariant theory isrecovered in the scaling region of the lattice theory for a line of tricritical points of thecoupling constants.

The tricritical points are determined by examining 3+1 relativisticdispersion relations.Weak coupling perturbation theory of transverse lattice non-compact QED (TLQED)is discussed in section 4. After gauge fixing in light-cone gauge, the UV properties ofthe theory are studied.

We argue that the usual diagrammatic UV divergences are cutoffby the finite transverse lattice spacing. The transverse lattice construction converts afour-dimensional field theory into a two-dimensional field theory with a finite (for finitesites on the lattice) number of “flavors” which is then UV finite, diagram by diagram, forfixed lattice spacing.In section 5 we calculate the anomalous scaling dimension of the link fields on thelattice, and find that the interaction Hamiltonian becomes a non-renormalizable interaction1

for g2(a) > 4π, where g(a) is the bare QED coupling constant. The anomalous scalingdimension is calculated by normal ordering the link fields and is non-perturbative becausethe link fields are exponentials of the gauge fields.The relationship between this phase transition and the phase transition of the sine-Gordon model, the quenched ladder approximation of QED, and quenched non-compactlattice QED is discussed.

Based on these analogies, we conjecture that this critical pointcorresponds to the non-perturbative chiral symmetry breaking phase transition in QED.Recent interest in chiral symmetry breaking in QED was generated by Miransky[6]who used the ladder approximation of the Schwinger-Dyson equation to argue for theexistence of a non-trivial UV renormalization group fixed point of the QED coupling con-stant. This phenomenon is closely related to the collapse of the Dirac wavefunction insupercritical (Z > 137, for which α = Ze2/4π > 1) Coulomb fields[6].

The fixed point isthe boundary of the chirally symmetric ladder QED phase and its strong coupling phasewhich has spontaneous chiral symmetry breaking[7]. That the strong coupling phase ofQED breaks chiral symmetry spontaneously is understood analytically via the strong cou-pling expansion of lattice gauge theory [1][8][9], and via lattice gauge theory Monte-Carlosimulations[10][11][12].

It is not clear however, that lattice gauge theory data supportsthe existence of a non-trivial UV fixed point for full QED. It may be the case that therenormalized charge of the continuum theory vanishes at the critical point[13].In section 6, we study the strong coupling limit of TLQED by calculating the energyshift of the infinite coupling vacuum states to lowest order in the inverse coupling 1/g.

Wefind that the discrete remnant of chiral symmetry on the transverse lattice is spontaneouslybroken and that the chiral condensate ⟨¯ψψ⟩is non-vanishing for the lowest energy state.We discuss our results further in section 7.2. Staggered fermions for the transverse latticeIn this section, we construct staggered fermions for the transverse lattice, and in theprocess, introduce notation for the transverse lattice construction that will be used in latersections.The basic strategy is to write the Dirac equation (iγµ∂µ −m)ψ = 0 in appropriatecomponent form, and find a fermion equation on the transverse lattice which reproducesthese equations in the continuum limit.

We use the chiral representation of gamma matricesγ0 =0−1−10γi =0σi−σi0γ5 =100−1,(2.1)2

and define the fermion ψ componentsψ =ϕχϕ =ϕ(1)ϕ(2)χ =χ(1)χ(2),(2.2)In light-cone coordinatesx± =1√2 (x0 ± x3)∂± =1√2 (∂0 ± ∂3) ,(2.3)the component equations are√2 ∂−χ(1) =imϕ(1) + [∂1 −i∂2] χ(2) ,√2 ∂+χ(2) =imϕ(2) + [∂1 + i∂2] χ(1) ,√2 ∂+ϕ(1) =imχ(1) −[∂1 −i∂2] ϕ(2) ,√2 ∂−ϕ(2) =imχ(2) −[∂1 + i∂2] ϕ(1) . (2.4)Now consider a complex one-component fermion field on a discrete square lattice ofpoints ⃗x⊥= a(nx, ny), with lattice spacing a and basis vectors ⃗α = (a, 0) or (0, a).

In thissection, the lattice is taken to be infinite. The fermion field φ is a continuous function ofthe light-cone coordinates x±, and satisfies the equation∂0φ = P3(⃗x⊥)∂3φ + P1(⃗x⊥)∆1φ + P2(⃗x⊥)∆2φ ,(2.5)where P1, P2, and P3 are unknowns to be determined by matching to the continuumequations (2.4) with zero mass, (adding a mass term is more complicated and will beconsidered in the next section), and ∆α is the symmetric lattice derivative∆αf(⃗x⊥) = 12a [f(⃗x⊥+ ⃗α) −f(⃗x⊥−⃗α)] .

(2.6)The fermion has the mode expansionφ =Zd2kZ π−πd2ℓ˜φ(k±, lα) eik+x−eik−x+eiℓαnα ,(2.7)and in momentum space, the equation of motion isk0 ˜φ = k3 ˜P3 ˜φ + ˜P1sin ℓ1a˜φ + ˜P2sin ℓ2a˜φ . (2.8)In the continuum limit, as a →0, finite energy states are located about ℓα ∼ǫ, or ℓα ∼π−ǫ.Therefore, there are four continuum fermion components for one transverse lattice fermion.3

This is just the standard fermion doubling problem, which works to our advantage in thiscase because four continuum components are desired. Equivalently, in lattice coordinatespace, different linear combinations of four adjacent sites will correspond to four differentfields in the continuum.To be more specific, introduce a lattice parity PL[⃗x⊥] = (−1)nx+ny.

If PL[⃗x⊥] is +1(−1), then ⃗x⊥is an even (odd) site. For the moment, consider the fermion at even or oddsites to be different continuum fields, labeled φeven and φodd.

Making the ansatz P3 = PL,the equation of motion (2.5) becomes√2 ∂−φeven =P1∆1φodd + P2∆2φodd ,√2 ∂+φodd =P1∆1φeven + P2∆2φeven ,(2.9)If we select P1 = 1 and P2 = −iPL, then equations (2.9) are just the massless continuumequations for the Dirac fermion components χ of eqn. (2.4).

This is not complete result,however, because we know that there should be four continuum components. The fullresult is obtained by breaking up the lattice further into a sub-lattices graded by (−1)ny.The full result is that with the P1, P2, P3 selected above,χ(1) =12 [φ(⃗x⊥) + φ(⃗x⊥+ ⃗s)] ,⃗x⊥even ,χ(2) =12 [φ(⃗x⊥) + φ(⃗x⊥+ ⃗s)] ,⃗x⊥odd ,ϕ(1) =12 [φ(⃗x⊥) −φ(⃗x⊥+ ⃗s)] (−1)nx ,⃗x⊥odd ,ϕ(2) =12 [φ(⃗x⊥) −φ(⃗x⊥+ ⃗s)] (−1)nx ,⃗x⊥even ,(2.10)where ⃗s = a(1, 1).

One can easily check that these fields obey the massless version of theequations (2.4).Each field is associated with the face of the lattice with center ⃗x⊥+ 12⃗s. Label eachpoint on the lattice by ((−1)nx, (−1)ny), so that there are four types of points with re-spect to this grading.

Then χ(1), ϕ(2) are associated with type A faces, and χ(2), ϕ(1) areassociated with type B faces, where the faces are labeled in figure 1.Consider the symmetries of the LagrangianL = iX⃗x⊥a2 φ† ∂0φ −(−1)nx+ny∂3φ −∆1φ + i(−1)nx+ny∆2φ(2.11)of the Dirac fermion on the transverse lattice. In addition to 1 + 1 dimensional Lorentzinvariance, it has lattice translational invariance ⃗x⊥→2⃗α; as in regular lattice gauge4

theory, translations are shifts by an even number of sites. It also has the shift symmetry,⃗x⊥→⃗x⊥+⃗s, which is interpreted as a discrete chiral rotation on the fields, since under thistransformation, χ →−χ and ϕ →ϕ, as is seen by examining eqns.

(2.10)1. This is alsosimilar to the discrete chiral symmetry found for staggered fermions on higher dimensionlattices.

The exchange symmetry of the Hamiltonian version of staggered fermions[1][3]is broken in the transverse lattice case by the unequal treatment of the three spatialcoordinates. We are left with a discrete rotation symmetry, a rotation by π about the x3axis, nα →−nα, and a single global gauge symmetry, φ →eiθφ.A mass term for the Lagrangian should have terms of the form χ(1)†ϕ(1) and ϕ(1)†χ(1).According to the analysis which is summarized in fig.

1, the bilinear couplings will have tobe nearest neighbor, because χ(1) and ϕ(2) live on type A faces, and χ(2) and ϕ(1) live ontype B faces. The mass term for the Lagrangian isL′m = −mX⃗x⊥a2 φ†(⃗x⊥)φ(⃗x⊥+ PL[⃗x⊥]⃗y)(2.12)It explicitly breaks the discrete chiral symmetry ⃗x⊥→⃗x⊥+ ⃗s, and leads to the correctcontinuum mass terms in the equations(2.4).

The nearest neighbor coupling in eqn. (2.12)breaks the global U(1) gauge symmetry at each site.

For each pair of sites ⃗x⊥and ⃗x⊥+PL[⃗x⊥]⃗y, the U(1)×U(1) symmetry is broken to the diagonal U(1). It is possible to gaugethe remaining diagonal U(1) symmetries, and this is discussed in ref.

[14]. The constructionis awkward, and we will avoid it by adding a second flavor of lattice fermion.

Then therewill exist a mass term which preserves all of the U(1) symmetry.3. Gauging transverse lattice staggered fermionsIn this section we introduce gauge fields in an attempt to make the Lagrangianeqn.

(2.11) locally gauge invariant. This however, will fail because of the 2-D gauge anom-aly, and a second set of fermion fields will have to be introduced, leading to a fermiondoubling problem in the continuum limit.To promote δGφ = iΛφ to a local gauge symmetry, introduce the 2-D vector gaugefields Ai and 2-D scalar fields Aα with transformation lawsδGAi(⃗x⊥, x±) =∂iΛ(⃗x⊥, x±) ,i = 0, 3 ,δGAα(⃗x⊥, x±) =∆+αΛ(⃗x⊥, x±) ,α = x, y .

(3.1)1This is denoted as a chiral rotation, although it really is a combination of the gauge trans-formation ϕ →iϕ and χ →iχ, followed by the chiral rotation ϕ →−iϕ and χ →iχ.5

The forward lattice derivative∆+α f(⃗x⊥) = 1a [f(⃗x⊥+ ⃗α) −f(⃗x⊥)] ,(3.2)obeys the integration by parts rule P⃗x⊥f∆+αg = −P⃗x⊥(∆−α f)g , where∆−α f(⃗x⊥) = 1a [f(⃗x⊥) −f(⃗x⊥−⃗α)] . (3.3)The Lagrangian for the gauge fields isLgauge =X⃗x⊥a2 14g21(Fij)2 +24g22(Fiα)2 +14g23(Fαβ)2,(3.4)where g1, g2, and g3 will be fixed by requiring 3 + 1 Lorentz invariance in the continuumlimit.

The field strengths Fµν areFij = ∂iAj −∂jAi ,Fαβ = ∆+αAβ −∆+β Aα ,Fiα = ∂iAα −∆+αAi . (3.5)For the fermion fields, we dress the derivatives in the Lagrangian eqn.

(2.11) via theminimal coupling procedure,∂iφ →Diφ = (∂i −iAi)φ ,∆αφ →Dαφ =nφ(⃗x⊥+ ⃗α)e−iaAα(⃗x⊥) −φ(⃗x⊥−⃗α)eiaAα(⃗x⊥−⃗α)o/2a . (3.6)However, this construction does not yield a gauge invariant theory.

The 2-D kinetic termsfor the fermions are,L = i√2 φ†(∂−−iA−)φ + . .

.⃗x⊥even ,= i√2 φ†(∂+ −iA+)φ + . .

.⃗x⊥odd . (3.7)There is only a single left or right-handed fermion for each local U(1) gauge symmetry, andtherefore, the local U(1) symmetries are anomalous.

The anomaly breaks the U(1) × U(1)symmetry of pairs of sites (say ⃗x⊥and ⃗x⊥+ PL[⃗x⊥]⃗y) to the diagonal U(1). The massterm eqn.

(2.12) also produced this pattern of symmetry breaking. In principle, one canconstruct transverse lattice QED with the remaining U(1) symmetry[14].However inpractice, it will be easier to add a second flavor of lattice fermions to cancel the anomaliesand preserve the full set of U(1) symmetries.

The fermion action takes the formLF = i2Xf=1X⃗x⊥a2φ†f D0 + (−1)nx+ny+fD3φf−κDx + i(−1)nx+ny+fDyφf,(3.8)6

where κ is a hopping parameter that will be fixed by requiring 3+1 Lorentz invariance.While κ = 1 in the classical continuum limit, it will receive quantum corrections and infact will have to be renormalized. With two flavors on the lattice, there will be two Diracfermions in the continuum limit.

Their components ϕ and χ are constructed from differentflavors of the lattice fermions, i.e. φ1 and φ2 contribute to each of the two continuum Diracfermions.

The components of the continuum fermionsΨj =ϕjχj,j = 1, 2(3.9)areχ(1)1=12 [φ1(⃗x⊥) + φ1(⃗x⊥+ ⃗s)] ,⃗x⊥even ,χ(2)1=12 [φ1(⃗x⊥) + φ1(⃗x⊥+ ⃗s)] ,⃗x⊥odd ,ϕ(1)1=12 [φ2(⃗x⊥) −φ2(⃗x⊥+ ⃗s)] (−1)nx ,⃗x⊥even ,ϕ(2)1=12 [φ2(⃗x⊥) −φ2(⃗x⊥+ ⃗s)] (−1)nx ,⃗x⊥odd ,(3.10)andχ(1)2=12 [φ2(⃗x⊥) + φ2(⃗x⊥+ ⃗s)] ,⃗x⊥odd ,χ(2)2=12 [φ2(⃗x⊥) + φ2(⃗x⊥+ ⃗s)] ,⃗x⊥even ,ϕ(1)2=12 [φ1(⃗x⊥) −φ1(⃗x⊥+ ⃗s)] (−1)nx ,⃗x⊥odd ,ϕ(2)2=12 [φ1(⃗x⊥) −φ1(⃗x⊥+ ⃗s)] (−1)nx ,⃗x⊥even . (3.11)The mass term P⃗x⊥a2 m√2Pj ΨjΨj is given byLm =X⃗x⊥a2 m√2(−1)nxhφ†1φ2 + φ†2φ1i,(3.12)and it preserves the U(1) symmetries for all the sites.The mass term explicitly breaks the discrete chiral symmetry generated by ⃗x⊥→⃗x⊥+ ⃗s.

As discussed in the previous section, this takes χj →−χj and ϕj →ϕj. Thiscorresponds to a discrete Z2 subgroup of the 4-D anomalous U(1) chiral symmetry.The 2-D gauge theory for each site on the lattice also has an anomalous chiral trans-formation,δφf(⃗x⊥) = iλ(−1)f+1φf(⃗x⊥) .

(3.13)7

This (global in the 2-D sense) symmetry is broken at one-loop in perturbation theory. Itcorresponds in the 4-D continuum limit to a broken axial-vector flavor symmetry, underwhich the continuum components transform asδ−→Ψ = iλσ3γ5−→Ψ ,−→Ψ =Ψ1Ψ2,(3.14)where σ3 acts in flavor space.The only non-anomalous continuum symmetry of this model is the gauged U(1) ‘totallepton number’ symmetry.

There are no global flavor symmetries for this transverse latticemodel. This is in contrast to the naive (Wilson) and staggered (Susskind) formulationsof QED on 4-D euclidean lattices[15][16].The action for a single 4-D naive masslessfermion on the 4-D lattice has U(4) vector and axial-vector flavor symmetries, which isa subgroup of the full U(16) flavor symmetries of the 16 continuum Dirac fermions ofthis model.

The minimal staggered massless fermion action on the 4-D lattice has U(1)vector and axial-vector flavor symmetries on the lattice, which is a subgroup of the U(4)flavor symmetries of the 4 continuum Dirac fermions for this model. The transverse latticemodel constructed in this section has no continuum flavor symmetries, and has only twocontinuum Dirac fermions in the continuum limit.

For the fermions on the 4-D lattices, theaxial-vector flavor symmetries which exist in the lattice action are spontaneously brokenin the strong coupling limit. The non-vanishing of the order operator Ψ · Ψ, which signalsthe breaking of the axial-vector flavor symmetries of the lattice models, is confirmed, byMonte Carlo simulations, for the scaling region of the theory[17][18].

This order operatorbreaks all of the continuum axial flavor symmetries, and one expects the full multiplet ofGoldstone bosons associated with the full set of broken axial symmetries in the scalingregime. In section 6, we will show that the discrete chiral symmetry of the transverselattice model is spontaneously broken in the strong coupling limit by the non-vanishingvacuum expectation value of Ψ · Ψ.The parameter κ is included in eqn.

(3.8) because 3+1 Lorentz invariance is brokendown to 1+1 Lorentz invariance by the transverse lattice construction.One may askwhether two new parameters should really occur in the Lagrangian, one for the Dx termand one for the Dy term, since these are two separate 2-D mass terms. The answer is no,because there exists field redefinition that transposes the x and y terms in the Lagrangian.It is expressed as φf →αfφf, where αf is defined recursively,αf(⃗x⊥) =i(−1)nx+ny+fαf(⃗x⊥−⃗y) ,αf(⃗x⊥) = −i(−1)nx+ny+fαf(⃗x⊥−⃗x) ,αf(0) =1(3.15)8

This is a spin transformation; these transformations are typically applied to staggeredfermion systems to diagonalize γ matrices in the fermion action[19]. Applying this partic-ular spin transformation to eqns.

(3.8) and (3.12) interchanges the labels x and y.This concludes the construction of non-compact transverse lattice QED with staggeredfermions. The goal of the remaining sections is to extract non-perturbative informationabout QED from this construction.4.

Ultraviolet finiteness of perturbation theoryIn this section, the weak coupling perturbation expansion will be developed for thetransverse lattice theory with Lagrangians given by eqns. (3.4) and (3.8).

It will be arguedthat the transverse lattice regulates all the UV divergences for each diagram in perturbationtheory. We will use this formalism in the next section to calculate the non-perturbativescaling dimension of the interaction Hamiltonian.Axial gauges minimize the mixing of longitudinal and transverse degrees of freedomand are therefore particularly useful in the context of the transverse lattice construction.Space-like axial gauges are problematic for weak coupling because of difficulties imple-menting Gauss’s law[20], so the light-cone gauge A−= 0 will be used.

In light-cone gauge,if the field theory is quantized on the null-plane x+ = 0, then A+ is a constrained field.So in this section, we use the light-cone quantization scheme — light-cone gauge with thenull-plane Cauchy surface.Only half of the fermion fields φ(f) satisfy dynamical equations on the null plane.With the definitions,χ =φf ,(−1)nx+ny+f = −1 ,ψ =φf ,(−1)nx+ny+f = +1 ,(4.1)one finds that only the ψ are dynamical fields.The constraint equations in light-conegauge for the fields A+ and χ are∂2−A+ = J−=g1g22∂−∆−α Aα −g21√2ψ†ψ ,(4.2)and∂−χ =1√2[D1 −iD2] ψ . (4.3)9

Using 12|x−−y−| = 1/∂2−, which satisfies ∂2−12|x−−y−| = δ(x−−y−), the constraintequation (4.2) is integrated:A+ = 12Zdy−|x−−y−|J−+ Fx−+ G . (4.4)The constant G(x+, ⃗x⊥) is set to zero as a gauge fixing constraint; it fixes x+ dependent(and x−independent) infinitesimal gauge transformations.

The F(x+, ⃗x⊥)x−term corre-sponds to the theta angle of the Schwinger model[21]. In the continuum limit, the physical3 + 1 Lorentz covariant vacuum should correspond to F = 0, so it can be set to zeroidentically.To remove the coupling constant dependence from the canonical commutation rela-tions, we letAα →g2Aα .

(4.5)The current J−and covariant derivative Dα must be changed accordingly. With this fieldredefinition, the light-cone momentum P + = (P 0 + P 3)/√2 and the light-cone energyP −= (P 0 −P 3)/√2 areP + =ZdM∂−Aα∂−Aα + i√2ψ†ψ,P −=ZdM g222g23F12F12 −1g21J−1∂2−J−+ iκ2√2(δαβ −iǫαβ)ψ†Dα1∂−[Dβψ],(4.6)where the constraint equations have been applied, and the measure isdM =X⃗x⊥a2dx−.

(4.7)The light-cone Hamiltonian P −can be divided into free and interacting parts:P −0 =ZdMc1Aα∆−β ∆+β Aα + c2(∆αAα)2 +i√2ψ†∆α∆α1∂−ψ(4.8)wherec1 = −12g2g32,c2 = 12g21g42−1g23,(4.9)andP −int =ZdM(−√2g1g22∆−α Aα1∂−ψ†ψ−g21ψ†ψ 1∂2−[ψ†ψ]+ iκ2√2ψ†(δαβ −iǫαβ)Dα1∂−[Dβψ] −1κ2 ∆α1∂−[∆αψ]). (4.10)10

We define the beta functions βtl(γ) = a∂γ/∂a for each coupling constant γ =g1, g2, g3, κ. The coupling constants g1 and g3 will are fixed with respect to g2 for eachvalue of lattice spacing a by requiring that the photons obey a covariant dispersion relation.The appropriate renormalization scheme for covariant dispersion relations isc1 = −12 ,c2 = 0 .

(4.11)At tree level, this implies g1 = g2 = g3. This relation will receive corrections in perturba-tion theory; the beta functionsβtl(g1) = βtl(g2) + O(g2) ,βtl(g3) = βtl(g2) + O(g2) ,(4.12)are determined by the renormalization conditions eqns.

(4.11).The quantum theory is defined on a square, doubly periodic, transverse lattice withN 2 sites, N even. A real scalar field σ(⃗x⊥) = (⃗x⊥+ N⃗α) has the mode expansionσ(⃗x⊥) =Xℓωℓ·n˜σ(ℓ) + c.c ,(4.13)where ω is the phase factor e2πi/N, the ℓα are integer momenta which take values from−N/2 to N/2 −1, and the inner product ℓ· n is shorthand for Pα ℓαnα.

With thesedefinitions, the mode expansions for the fields areAα =1√4πNaXℓZ ∞0dηηne−iηx−ωℓ·n aα(ℓ, η) + e+iηx−ω−ℓ·n a†α(ℓ, η)o,ψ =1√2πNaXℓZ ∞0dη√ηne−iηx−ωℓ·n b(ℓ, η) + e+iηx−ω−ℓ·n d†(ℓ, η)o,ψ† =1√2πNaXℓZ ∞0dη√ηne−iηx−ωℓ·n d(ℓ, η) + e+iηx−ω−ℓ·n b†(ℓ, η)o. (4.14)The canonical (anti-)commutation relations for the creation and annihilation operators are[aα(ℓ, η), a†β(ℓ′, η′)] =δℓ,ℓ′ δαβ η δ(η −η′) ,{b(ℓ, η), b†(ℓ′, η′)} =δℓ,ℓ′ δαβ η δ(η −η′) ,{d(ℓ, η), d†(ℓ′, η′)} =δℓ,ℓ′ δαβ η δ(η −η′) .

(4.15)11

The modes a, b, d annihilate the light-cone vacuum, and the normal ordered expressionsfor the fermion charge QF =√2Rψ†ψ, momentum P +, and free Hamilitonian P −0 areQF =XℓZ dηηb†(ℓ, η)b(ℓ, η) −d†(ℓ, η)d(ℓ, η),P + =XℓZdηa†α(ℓ, η)aα(ℓ, η) + b†(ℓ, η)b(ℓ, η) + d†(ℓ, η)d(ℓ, η),P −0 =XℓZ dηη212k2⊥(∆+∆−; ℓ)a†α(ℓ, η)aα(ℓ, η)+ 12k2⊥(∆2; ℓ)b†(ℓ, η)b(ℓ, η) + d†(ℓ, η)d(ℓ, η),(4.16)wherek2⊥(∆+∆−; ℓ) =Xα 2 sin πℓαNa!2,k2⊥(∆2; ℓ) =Xα sin 2πℓαNa!2. (4.17)The photon states a†(ℓ, η)|0⟩satisfy the free-field equationP +P −0 −12k2⊥(∆+∆−)a†|0⟩= 0.

(4.18)As the lattice size becomes large, k2⊥(∆+∆−) →k21 + k22, where kα = 2πℓα/Na. Henceeqn.

(4.18) is the 3 + 1 Lorentz covariant free photon dispersion relation for finite latticesize. A similar relation holds for the fermion states.We now argue that light-cone perturbation theory[22][23][24] is finite, diagram bydiagram.

The S matrix is ⟨f|T exp(−iRdx+P −Iint )|i⟩, where T denotes time ordering withrespect to x+, and P −Iint is the normal ordered interaction light-cone Hamiltonian (4.10) inthe interaction picture. Diagrammatic perturbation theory is generated by expanding thetime ordered exponential and inserting complete sets of intermediate states.

In general,the S-matrix will have an overall energy conservation factor −2πiδ(P −0,f −P −0,i), and eachintermediate state will have the factor 1/(P −0,f−P −0 +iǫ). Matrix elements of the interactionHamiltonian with intermediate or final states will always include the factor δ(Pf ηf −Pi ηi), where the ηf are outgoing momenta and the ηi are incoming momenta, becauseall vertices conserve light-cone momentum η.

The light-cone momentum is bounded frombelow by zero for all states.One delicate aspect of light-cone perturbation theory is the limit η →0 in intermediateloops. Certain connected one-loop diagrams are ill defined for zero η in continuum QED12

and QCD (see refs. [25] and [26]), and need to be regularized.The regulator can beremoved when calculating gauge invariant combinations of one-loop diagrams, i.e.

the η = 0region does not contribute to gauge invariant processes at one-loop. These divergencesare particular to canonical Hamilitonian perturbation theory and do not correspond tothe UV divergences of covariant perturbation theory.

Also, they are not infrared (IR)divergences since the parity operator P, where Pψ(x−, x+)P −1 = ψ(x+, x−), acts on themodes as Pb†(ℓ, η)P −1 ∝b†(ℓ, k2⊥(ℓ)/2η), and interchanges the large and small η regions.Two popular regularization schemes for the η = 0 region are, a sharp η cutoff[25][26],and the discrete light-cone approach [27][28][29]. However, these cutoffs may not be goodregulators to higher order in perturbation theory because the η = 0 region can contributeto connected diagrams in light-cone field theory[30].

One signature of this problem wouldbe the loss of gauge or Lorentz symmetries; counterterms would have to be added to restorethe symmetries order by order in perturbation theory.The regular UV divergences of QED arise from integration over the transverse mo-mentum k⊥of the fermions and the gauge fields in the 1/(P −0,f −P −+ iǫ) terms of the Smatrix[31]. These divergences are explicitly cut offby the transverse lattice construction,since the perpendicular momentum is bounded by 8/a2.

There also are IR divergencesfor the gauge fields and massless fermions that arise when summing over k⊥= 0 in thedenominators. These correspond to the IR divergences of covariant perturbation theory,and are regulated by introducing small mass terms: k2⊥→k2⊥+ µ2.The last source of perturbative UV divergences is the continuum 2-D field theory.Divergent tadpoles of the perturbation theory are eliminated by normal ordering the light-cone Hamiltonian.

The non-local operator 1/∂−in the interaction Hamiltonian eqn. (4.10)softens the UV structure of the vertices, as opposed to derivative interactions, which canviolate UV finiteness.[32].

For instance, the four fermion term in eqn. (4.10) is scalingdimension zero (verses two) because of the non-local 1/∂2−factor.

And by further powercounting arguments, the interaction light-cone Hamiltonian is UV finite, diagram by dia-gram.In principle, the 2-D fermions ψ in the light-cone Hamiltonian can be bosonized.With the bosonization relations ψ =: exp i√4πΦ : and ψ† =: exp −i√4πΦ :, where Φ is acanonical boson, the light-cone Hamiltonian of TLQED is mapped to a bosonic light-coneHamiltonian with non-derivative interactions. It is well known that a bosonic theory intwo dimensions with no derivative interactions is UV finite, diagram by diagram[32].

It isalso possible to bosonize the 2-D covariant Lagrangian. Then the bosonization dictionarywhich translates between fermions and bosons will be more complicated[33].13

5. The non-perturbative ultraviolet divergence at g22 = 4πWhile the transverse lattice theory of QED is UV finite diagram by diagram, it canhappen that an infinite number of diagrams conspire to generate a new UV divergence.This phenomenon occurs in the 2-D sine-Gordon model[32][34][35][36].

The basic signatureof this phenomenon in the sine-Gordon model is that the anomalous scaling dimension ofthe interaction (α/β2) cos(βφ) is greater than two for β2 > 8π, and the interaction isbecomes non-renormalizable. For this region of coupling, the energy density is unboundedfrom below [32], and the connected Green’s functions diverge order by order in α, startingat order α2[34][35].For TLQED we will now calculate the leading anomalous scaling dimension of theinteraction light-cone Hamiltonian (4.10).It is obtained by considering the parts ofeqn.

(4.10) that contain non-interacting products of link fields.The prototypical termof this type isκ20a23ψ†ǫαβDα1∂−[Dβψ] ,(5.1)where κ20 is the bare coupling and a3 is the cutoffof the 2-D continuum theory. Thisis a bare expression, since it depends upon a3, and it needs to be renormalized withrespect to an arbitrary mass scale.

We will calculate the divergent tadpole contributionsand renormalize this term. In eqn.

(5.1), the factor 1/a23 accounts for the naive scalingdimension of this interaction, which is 12 + 12 −1 = 0, where each 12 comes from the fermionsand −1 comes from 1/∂−. Since the fermion fields ψ† and ψ in eqn.

(5.1) occur at differentlattice sites and therefore anticommute, and the two link fields commute because of ǫαβ,we only have to normal order each link field to obtain the tadpole contributions. Considerthe exponentialeiag2Aα =: eiag2Aα : e+ 12 (g2a)2[A+α,A−α ] ,(5.2)where A+ (A−) includes only raising (lowering) operators in the fields mode expansion(4.14).

After applying the commutation relations, we get[A+α, A−α] =14π(Na)2XℓZ Λ+ℓδ+ℓdηη . (5.3)The small η regulator δ+ℓand the large η regulator Λ+ℓare related by x3 parity, as discussedin the previous section and in ref.

[30]. The relationship isδ+ℓ= k2⊥(∆+∆−; ℓ)2Λ+ℓ.

(5.4)14

In terms of a fixed x3 momentum cutoffΛ ≈1/a3, the large η cutoffΛ+ℓis given by the2-D relativistically correct expression,Λ+ℓ= Λ +pΛ2 + k2⊥(ℓ)√2. (5.5)Here, k⊥plays the role of a mass for each 2-D theory.

In the limit Λ >> k⊥(ℓ),Z Λ+ℓδ+ℓdηη = ln4Λ2k2⊥,(5.6)andeiag2Aα =: eiag2Aα :Qℓk2⊥(ℓ)1/N24Λ2g22/8π. (5.7)In eqns.

(5.6) and (5.7), the IR divergence at ℓ= 0 is regulated by adding a small mass:k⊥(ℓ= 0) →µ2. We see that the exponentials have anomalous scaling dimension g22/4π,i.e.

they scale as Λ−g22/4π, where Λ is the UV momentum cutoff. The interaction term (5.1)is multiplicatively renormalized by defining a renormalized coupling κ(m) asκ20 = Zκκ2(m) ,Zκ = 14(2ma3)2−g22/2π Yℓ(k⊥(ℓ))g22/4πN2,(5.8)where m is an arbitrary mass scale.

The renormalized interaction term is thenm2−g22/2πκ2(m)ψ†ǫαβDα1∂−[Dβψ] . (5.9)For g22 < 4π, the interaction term has dimension less than two.

For this region of couplingconstant, the UV finiteness of each diagram in the theory is sufficient to guarantee finitenessof the full theory. For g22 = 4π the interaction term (5.1) is a marginal operator, and thetheory will be well defined if the renormalization of g2 with respect to the 2-D continuumtheory is allowed.

This is the situation for the sine-Gordon model at its critical point[6][36].For g22 > 4π the theory is non-renormalizable, the hopping parameter κ has negativescaling dimension, and the operator product of the interaction Hamiltonian with itself istoo singular to allow consistent perturbation theory about the free-field vacuum.Therefore, for TLQED, we find the somewhat surprising result that the weak pertur-bation theory is valid only for α(a) = g22/4π < 1, independent of a. The coupling g22(a) isthe bare coupling and in the scaling regime of full TLQED it may be quite far from therenormalized QED coupling constant gren.

Only for very weak coupling is g2(a) ≈gren in15

the full theory. However, recall that in the quenched approximation of lattice QED[10],chiral symmetry is spontaneously broken beyond a certain critical value α ∼1.

Similarly,the analytic calculations in the ladder approximation of quenched QED also exhibit a criti-cal coupling which corresponds to the chiral symmetry breaking phase transition[6][7]. Wetherefore make the conjecture that g22(a) = 4π is in general the chiral symmetry breakingcritical point in TLQED, and that specifically, in the quenched approximation of TLQED,for which g2(a) = gren, chiral symmetry is broken for α > 1.

This is discussed further insection 7.6. Strong coupling limitDoes TLQED realize spontaneous chiral symmetry breaking in the strong couplingregime?This means that a non-vanishing chiral condensate ⟨Ψ · Ψ⟩must appear, orequivalently in terms of the the lattice fermions,X⃗x⊥(−1)nx⟨vac|φ†1φ2 + φ†2φ1|vac⟩̸= 0 ,(6.1)where |vac⟩is the full interacting vacuum state.

Such a non-vanishing condensate wouldsignal the spontaneous breaking of the discrete chiral symmetry of the lattice theory. Sinceit is a discrete symmetry in the strong coupling region, there will be no accompanying Gold-stone boson in this region, and Coleman’s theorem[37], prohibiting spontaneous breakingof continuous internal symmetries in two dimensions without anomalies or a Higgs mecha-nism, will not be violated.

The discrete chiral symmetry of the lattice model correspondsto the 4-D anomalous U(1) chiral symmetry, and we would not expect Goldstone bosonsfor this broken symmetry in the scaling regime of the transverse lattice model. However,non-vanishing of the condensate eqn.

(6.1) in the scaling regime would also signal thebreaking of the non-anomalous continuum U(2) axial flavor symmetries, and we wouldexpect their accompanying Goldstone bosons in the scaling regime.We will now show that spontaneous chiral symmetry breaking does occur in TLQEDin the infinite coupling gi →∞limit, where i = 1, 2, 3. (Here we assume g1 ∼g2 ∼g3.

),by calculating the energy difference between various vacuum configurations defined belowto lowest order in 1/g. As we will see, this calculation is complicated by the fact that thefield theory of rigid rotators is fraught with divergences.

In the end however, the vacuumenergy density shift will be a finite quantity.16

Unlike the previous weak coupling analysis, it is convenient to perform the analysisin the A3 = 0 gauge, and with equal time quantization. The Hamiltonian density is thenH = 12g2a2E2α + A0∆−α Eα + a2jF+ HF + O( 1g2 ) ,(6.2)where Eα is the electric field and momentum conjugate to Aα, and jF = Pf φ†(f)φ(f) isthe fermion current.

Gauss’s law,G(⃗x⊥) = ∆−α Eα + a2jF (⃗x⊥) = 0 ,(6.3)is obtained by integrating out A0, and is treated in the quantum theory as the weakconstraint ⟨Gα⟩= 0 for all physical correlation functions.To leading order in g, thevacuum must satisfyEα(⃗x⊥)|0⟩= 0 , ∀⃗x⊥. (6.4)The system will be quantized with respect to this ‘free-field’ vacuum.

The condition thatall modes of canonical momentum annihilate the vacuum is reminiscent of the rigid rotatorin quantum mechanics.To regulate the IR behavior of the system, introduce periodic boundary conditions inthe continuous spatial direction z = x3,−L ≤z ≤L . (6.5)The mode expansions for the second quantized fields areEα = 12L( ∞Xn=1hEnαe+iπnz/L + E∗nα e−iπnz/Li+ E0α),Aα =∞Xn=1hAnαe+iπnz/L + A∗nα e−iπnz/Li+ A0α ,(6.6)where E∗nα(A∗nα ) are the complex conjugates of Enα (Anα), and not hermitian conjugatesin the sense of raising and lowering operators of the harmonic oscillator.

The canonicalcommutation relations in terms of the modes are[A∗nα (⃗x⊥), Emβ (⃗y⊥)] =iδαβδn+mδ⃗x⊥,⃗y⊥,[Anα(⃗x⊥), E∗mβ (⃗y⊥)] =iδαβδn+mδ⃗x⊥,⃗y⊥,(6.7)17

The free Hamiltonian and momentum for the gauge fields are given byH0gauge = 12Lg2a2 X⃗x⊥,α"∞Xn=1EnαE∗nα + 12(E0α)2#,Pgauge =X⃗x⊥,α,ninπL [A∗nα Enα −AnαE∗nα ] . (6.8)All Enα and E∗nα are lowering operators and annihilate the free-field vacuum, and the modesEnα, A∗nα ( E∗nα , Anα) are eigenstates of momentum P with eigenvalues n (−n).Creation operators in the Hilbert space are exponentials of the modes Anα with chargen.

For instance, the momentum zero mode state einaA0α|0⟩has energy eigenvalue (g2n)2/4L.Each state in the Hilbert space corresponds to a wavefunction in a first quantized theorywhere the dimensionless quantity aAnα plays the role of a coordinate. This relation can beused to calculate correlation functions.

The correlation function of states is non-vanishingonly if the total charge in the exponentials of the wavefunctions vanish.More precisely, for the zero mode expectation values, the two point correlator is⟨e−imaA0α einA0β⟩≡N −1δαβZ ∞−∞dxei(n−m)x = δαβN −1δ(n −m) . (6.9)Note that for the compact U(1) theory, the integration region for x would be [−π, π], n, mwould be integers, and the correlator eqn.

(6.9) would be δn,m. In the non-compact caseat hand, the result is a normalized Dirac delta function, which is ill-defined for arbitraryreal n, m; only for “integer” n, m do the non-compact and compact results coincide[38].In this section, we will evaluate such correlators with non-integer arguments, and regulatethe result by defining the cutoffdelta functionδΛ(x) = 12πZ Λ−Λdkeikx(6.10)The normalization is given by N −1 = δΛ(0).This is not the only expression which needs to be regulated in the theory.

Considerthe exponential of the field eiaAα(z) acting on the vacuum. This expression appears in theinteraction Hamiltonian; it represents a link field carrying flux from ⃗x⊥to ⃗x⊥+ ⃗α and hasenergy eigenvalueH0gauge eiaAα(z)|0⟩= g224L"1 + 2∞Xn=1#eiaAα(z)|0⟩= g222 δ(0)eiaAα(z)|0⟩(6.11)18

The energy is infinite because the exponential is a product of an infinite number of states.The exponential receives contributions from all of the ‘standing waves’ Anα and A∗nα in thebox. To regulate this UV divergence, introduce a cutoffin the number of modes countedin the delta functionδLΛ′(z) = 12LΛ′Xj=−Λ′eiπjz/L .

(6.12)Then the energy of the exponential is g222 δLΛ′(0). The energy is proportional to the numberof links and to the square of the flux carried by each link.The next to leading order contribution to the Hamiltonian comes from the fermionsand their interactions with the gauge field.

We adopt the equal time anticommutationrelations for the fermions{ φ†f(z, ⃗x⊥) , φg(z′, ⃗y⊥) } = 1a2 δ(z −z′)δ⃗x⊥,⃗y⊥δfg . (6.13)The free-field Hamiltonian density for the fermions isH0F = −iX⃗x⊥,fa2(−1)nx+ny+fφ†f∂zφf .

(6.14)Because of the minus signs in this expression, the mode expansion for the fermions isφf =1√2La2Xn̸=0b†(f)ne−iπnz/L + d(f)n eiπnz/L+ b(f)0, (−1)nx+ny+f = +1 ,φf =1√2La2Xn̸=0b(f)n e−iπnz/L + d†(f)neiπnz/L+ b†(f)0, (−1)nx+ny+f = −1 ,(6.15)where for each site ⃗x⊥and flavor f,{bn, b†m} = δn,m ,{dn, d†m} = δn,m ,{b0, b†0} = 1 . (6.16)The normal ordered free-field Hamiltonian is given byH0F =X⃗x⊥,f,nnπLb†(f)nb(f)n+ d†(f)nd(f)n,(6.17)and⟨0|b†n = ⟨0|d†n = bn|0⟩= dn|0⟩= 0 ,n > 0 ,(6.18)19

for each fermion flavor. The zero modes appear in the charge operatorQ(⃗x⊥) =ZdzjF = 12[b†(f)0, b(f)0 ] + .

. .

,(6.19)and in the mass operatorM(⃗x⊥) = a2Zdzhφ†1φ2 + φ†2φ1i= b†(1)0b(2)0+ b†(2)0b(1)0+ . .

. .

(6.20)The chiral condensate order parameter is proportional to P⃗x⊥(−1)nxM(⃗x⊥).The vacuum states of the full theory to order O(g0) will be a direct product of thegauge field vacuum |0⟩and the highest weight states for the fermion zero modes. To discusschiral symmetry breaking in the zero mode sector of the fermion theory, we diagonalizethe charge Q and mass operator M simultaneously, via the Bogolubov transformationb(1)0= 1√2 (a0 + ic0)b†(1)0=1√2 (a†0 −ic†0) ,b(2)0= 1√2 (a0 −ic0)b†(2)0=1√2 (a†0 + ic†0) ,(6.21)where {a†0, a0} = {c†0, c0} = 1.

Then the mass and charge operators in the zero mode sectorfor each site areM = 12[a†0, a0] −12[c†0, c0] ,Q = 12[a†0, a0] + 12[c†0, c0] . (6.22)The operators a†0, a0 and c†0, c0 act on two level systems.

The a operators are raising andlowering operators for the states | ↑⟩a and | ↓⟩a,a†0| ↓⟩a = | ↑⟩a,a†0| ↑⟩a = 0,a0| ↑⟩a = | ↓⟩a,a0| ↓⟩a = 0 . (6.23)The vacuum states in the fermion sector are direct products of the two level states in thea and c systems,|+⟩= | ↑⟩a| ↓⟩c ,|−⟩= | ↓⟩a| ↑⟩c .

(6.24)They satisfy Q|±⟩= 0 (Gauss’s law) and M|±⟩= ±|±⟩. The vacuum for each site on thelattice is therefore doubly degenerate at O(g0).

Note that fermion zero mode expectationvalues vanish: ⟨b(f)0 ⟩= 0 and ⟨b†(f)0⟩= 0. The non-vanishing two point functions are⟨b†(f)0b(f ′)0⟩= + ⟨b(f)0 b†(f ′)0⟩= 12 ,f = f ′ ,⟨b†(f)0b(f ′)0⟩= −⟨b(f)0 b†(f ′)0⟩= 12M ,f ̸= f ′ .

(6.25)20

We now show that the degeneracy of the vacuum state is broken in perturbation theoryby the interaction HamiltonianHint = iκX⃗x⊥,fa2Zdzφ†fDx −(−1)nx+ny+fDyφf ,(6.26)which is a gauge invariant operator since [G(⃗x⊥), Hint] = 0. In the context of the 4-Dtransverse lattice theory, the constant κ is dimensionless, since the fermions are dimension3/2 and the lattice derivatives go like 1/a and are dimension 1.

When power counting forthe continuous 2-D theory however, the fermions are dimension 1/2 and the derivative isdimension 0. Therefore κ is dimension 1 in the context of the 2-D field theory: κ ∼a/a3,where a3 is the UV cutoffof the 2-D theory at each site.

To regulate the energy of states,we have introduced a cutoffin the number of modes, δLΛ′(0). The UV cutoffa3 is given bya3 ∼1/δLΛ′(0), so thatκ = κ′aδLΛ′(0) ,(6.27)where κ′ is a scale independent constant.The first order shift ⟨Hint⟩in the vacuum energy vanishes because the expectationvalue of a single link field vanishes.

The second order shift is given byW2 =Xn′ ⟨0|Hint|n⟩⟨n|Hint|0⟩0 −Wn,(6.28)where Wn = g224LδLΛ′(0) + Wn,F is the energy eigenvalue of link states |n⟩, and Wn,F is thefermion sector contribution. We will calculate the shift in the vacuum energy due to theassignment of the fermion vacuum to the zero mode states |±⟩at each site on the lattice,which will be denoted as δW2.To calculate the second order energy shift of the vacuum, we need the correlationfunctionZdzf(z)Zdz′g(z′) ⟨e−iqaAα(z)e+iq′aAβ(z′)⟩.

(6.29)This correlator occurs when summing over intermediate states in eqn. (6.28).

Integratingout the zero modes A0α in the exponentials yields the factor δαβδΛ(q −q′)/δΛ(0). From thenext lowest mode i(A1α −A∗1α ), there is the factorδΛ( 2 sin(zπ/L) −2 sin(z′π/L) )/δΛ(0)= L2π [δΛ(z −z′) + Θ(z′)δΛ(z + z′ −L) + Θ(−z′)δΛ(z + z′ + L)] /δΛ(0) .

(6.30)21

The only term on the r.h.s. of eqn.

(6.30) that contributes to the correlator is δΛ(z−z′). Theother two terms will lead to vanishing contributions because there is no overlap with thesedelta functions and the delta functions that appear when integrating out the cosine terms;for instance, integrating out (A1α+A∗1α ) yields a term δΛ( 2 cos(zπ/L)−2 cos(z′π/L) ) whichhas no overlap with the second two terms in eqn.

(6.30). In the presence of the first termof eqn.

(6.30), all the other modes in the correlator contribute factors of unity. Therefore,the correlator eqn.

(6.29) is given byδαβδΛ(q −q′)δ2Λ(0)Zdzf(z)g(z)(6.31)The parameter Λ is an ultraviolet regulator. If the z direction were discretized, thenthe sin(nzπ/L) and cos(mzπ/L) terms which appear as arguments in the delta functionsof the correlation function (6.29) would take on discrete values (n, m would be boundedby ∼[2π/a3], where a3 is the lattice spacing in the discretized z direction).

For integercharges q, q′, which is all that we will have to consider in this section, all of the correlationfunctions would then be normalizable. The discrete version of the normalized delta functionδΛ(z)/δΛ(0) would be (a3)−1δzi/(a3)−1.

Hence δΛ(0) ≈1/a3While eqn. (6.28) is a complicated sum over four point correlation functions of thefermion modes, the only the terms which contribute to the shift in vacuum energy δW2are products of four fermion zero modes.

The non-trivial part of this observation is that atypical two zero mode contribution ⟨b(f)0 b(f)n b†(f ′)nb†(f ′)0⟩is proportional to ⟨b(f)0 b†(f ′)0⟩δff ′,and this by the first of eqns. (6.25) is independent of the choice |+⟩or |−⟩for the vacuumstate at that site.Using the link field two point correlation function (6.29) given byeqn.

(6.31) and the fermion zero mode two point correlators eqns. (6.25), the shift in theenergy density isδw2 =κ′216πg221(La)2 X⃗x⊥M(⃗x⊥) [M(⃗x⊥+ ⃗x) −M(⃗x⊥+ ⃗y)] .

(6.32)This is minimized for M(⃗x⊥)M(⃗x⊥+ ⃗x) = −1 and M(⃗x⊥)M(⃗x⊥+ ⃗y) = +1. There aretwo fermion vacuum configurations, related by an overall sign change, that obey theseconditions and the symmetry of these ground states is made clear by figure 2.Bothconfigurations break the discrete U(1) axial chiral symmetry since the order parameterP⃗x⊥(−1)nxM(⃗x⊥) is non-vanishing for these vacuum configurations.

If the order param-eter is non vanishing in the scaling regime, then the full set of non-anomalous continuumaxial flavor symmetries will be broken.22

There is a simple way of approaching the continuum limit of this leading order resultin the strong coupling regime such that eqn. (6.32) remains finite, i.e.

let the longitudinalIR regulator L →∞and the transverse UV regular a →0 such that La remains finite.So although the energy and correlators of link fields require UV regulators, the shift inthe vacuum energy density is finite in the continuum limit. We briefly list the sources ofthe regulated divergences that contribute to eqn.

(6.32). The product of four zero modescontributes 1/L2a4, the energy in the denominator of (6.28) contributes 1/δLΛ′(0) ∼a3,from the integral over intermediate states we get 1/δΛ(0) ∼a3, from the κ2 couplingconstant there is a factor of a2/a23, examination of eqn.

(6.26) shows that H2int contributesa factor of a2, and we multiply by 1/a2 to make (6.28) into a density. The result is thenet factor of 1/(La)2.To interpret this result further, consider the spin transformation b(2)0→αb(2)0 , whereα(⃗x⊥) = (−1)ny.

Following the analysis of Semenoff[9], define the vectorψ =b(1)0b(2)0,(6.33)and the currents Sj = ψ†σjψ where σj are Pauli matrices. Then the Hamiltonian densityin the zero mode sector that has expectation value given by eqn.

(6.32) can be written asδe2 =κ′216πg221(La)2 X⃗x⊥⃗S(⃗x⊥) ·h⃗S(⃗x⊥+ ⃗x) + ⃗S(⃗x⊥+ ⃗y)i+ const. .

(6.34)This is the Hamiltonian density for the quantum spin 12 Heisenberg antiferromagnet, andthe configuration given by fig. 2 is just the classical ground state of the system[39].

It hasN´eel order, i.e. the expectation value of eqn.

(6.32) is non-vanishing and the global flavorSU(2) of the Hamiltonian (6.34) is spontaneously broken. We can consider eqn.

(6.34) tobe Hamiltonian in the fermion sector to leading order in the strong coupling expansion.To study chiral symmetry breaking to higher order in the strong coupling expansion,we need to treat the quantum fluctuations of the spin 12 Heisenberg antiferromagnet in thezero mode sector, and include the effect of non-zero modes on the vacuum state. Thereis no exact solution of the ground state of the quantum d=2 quantum spin 12 Heisenbergantiferromagnet[40], and no proof that N´eel order persists in the full quantum theory.However, numerical simulations indicate that this may be the case[41].For a similaranalysis of regular Hamiltonian lattice gauge theory the situation is better, because N´eelorder has been proven to exist in three dimensions[9][40].23

7. DiscussionThe transverse lattice regulation of QED that has been studied in this paper is a‘minimal’ way of regulating the diagrammatic divergences of the perturbation theory, andit exhibits a phase transition at a critical value of the lattice QED coupling constant, andchiral symmetry breaking in the strong coupling regime2.In section 5, we took advantage of the UV finiteness of each diagram in weak pertur-bation theory to find a non-perturbative UV divergence at g22(a) = 4π.

The transverselattice regulates the usual UV divergences of four dimensional QED, but the ‘finite’ two-dimensional field theories for each site conspire to generate a non-renormalizable interac-tion. The signature of the non-renormalizability is the anomalous scaling dimension of theinteraction Hamiltonian.

If the dimension of any part of the interaction Hamiltonian isgreater than two, then the perturbation theory about the free-field vacuum will be ill de-fined. One can calculate the anomalous dimension of the interaction Hamiltonian becausethe coupling constant g2(a) is not renormalized in the 2-D continuum perturbation theoryfor g22 < 4π.

Note that there is no plaquette term in the interaction Hamiltonian, since wehave studied non-compact QED, which would have a higher scaling dimension than theterm we considered3.The 2-D sine-Gordon model has the same properties with respect to the coupling con-stant β: it is perturbatively finite for all β but its free-field perturbation theory is unstable,without additional coupling constant renormalizations, for β2 > 8π. The sine-Gordon fieldtheory is equivalent to the grand canonical sum of a Coulomb plasma, and the sine-Gordonphase transition has a nice physical interpretation in terms of the Coulomb gas picture[34].As β increases, the free ions of the Coulomb gas, represented by vertex operators exp(±βφ)in the sine-Gordon model, collapse to form dipoles and a new gas of interacting dipolesis formed.

This can be interpreted in the sine-Gordon model as the appearance of a newdimension 2 renormalizable operator at this fixed point. One can consider the sine-Gordonmodel for values of β2 > 8π as long as the additional renormalization for the new operatoris taken into account[36].2 If one formulates QED with one lattice and three continuum dimensions, then the diagram-matic divergences will not be regulated by the lattice, and chiral symmetry breaking will notappear in the strong coupling expansion of the lattice theory.

This is shown by choosing a gaugewhere the lattice gauge field is set to zero.3 Plaquette terms are presumably generated perturbatively but are suppressed by powers ofthe cutoff.24

In TLQED, ‘free ions’ are given by fermion charges separated by one link and con-nected by a flux tube: ψ† exp(gA)ψ. The ‘Coulomb gas’ in TLQED is then a gas of e+e−pairs, where the charges, separated by a single link, interact via Coulomb interactions.The ‘dipoles’ of the strong coupling phase are pairs of e+ e−flux tubes, with stronglyinteracting photon fields.The conjecture is that the non-perturbative g22(a) = 4π critical point of TLQED,where this phase transition occurs, is the critical point of spontaneous chiral symmetrybreaking, where the ¯ψψ order parameter gets a vacuum expectation value.

The bare cou-pling constant g2(a) is the ‘quenched’ coupling constant of TLQED, because fermion loopcorrections are obviously not included in the bare Hamiltonian. Both the quenched latticesimulations and the quenched planar approximation exhibit chiral symmetry breaking forαbare ∼1.

The phase transition in the quenched planar approximation has been previ-ously compared to the phase transition of the sine-Gordon model by Miransky[6], whointerpreted the phase transition of each model as a collapse phenomenon. At the criti-cal point in the quenched planar approximation, the anomalous scaling dimension of thefermion is 1, and the four-fermion term becomes a renormalizable operator[7].

It is tempt-ing to associate the ‘dipoles’ of the strong coupling phase of TLQED with renormalizablefour-fermion operator of the quenched planar approximation. We used the strong couplingexpansion of TLQED in section 6 to calculate explicitly the spontaneous chiral symmetrybreaking in the infinite coupling limit.Recent lattice gauge theory simulations indicate that the UV fixed point of chiral sym-metry breaking in the quenched theory may be trivial in the full unquenched theory[12][13].Some of the results of this paper can immediately be applied to more realistic trans-verse lattice models.

In particular, the construction of staggered fermions and the analysisof chiral symmetry breaking via the strong coupling expansion can be easily generalizedto non-abelian gauge theories.We now briefly mention two formal areas of the theory that would be interesting topursue. In section 4, we noted that TLQED can be covariantly (in the 2-D sense) bosonized.Bosonization plays a central role in explaining why the Schwinger model is exactly soluble.It would be interesting to understand the continuum limit of this bosonized version ofTLQED.

It would also be interesting, to work in the ‘opposite’ direction – to covariantlybosonize transverse lattice fermions, and then put the two continuous coordinates on alattice. Naively, this would generate a 4-D lattice theory where the fermions are interpretedas bosonic solitons, and the functional integral over fermions is ‘gaussian’ and easier to25

simulate4. Secondly, TLQED is an interacting 2-D field theory in the form of a combinedSchwinger and sine-Gordon model.

It might be possible to solve the sine-Gordon ‘part’ byusing inverse scattering/Bethe ansatz methods. Then the non-integrable Schwinger termswould have to be treated as perturbations in the space of Bethe-ansatz states.Acknowledgements: I am indebted to W. Bardeen for clarifying and stimulating discus-sions.4An idea pointed out to me by W. Bardeen.26

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