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Wolfgang Bock1,$, Asit K. De2,3,#,

arXiv:hep-lat/9210022v1 19 Oct 1992July 1992Amsterdam ITFA 92-21HLRZ J¨ulich 92-52Fermion-Higgs model withstrong Wilson-Yukawa couplingin two dimensionsWolfgang Bock1,$, Asit K. De2,3,#,Erich Focht2,3,& and Jan Smit1,∗1Institute of Theoretical Physics, University of Amsterdam,Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands2Institute of Theoretical Physics E, RWTH Aachen,Sommerfeldstr., 5100 Aachen, Germany3HLRZ c/o KFA J¨ulich, P.O.Box 1913, 5170 J¨ulich, GermanyAbstractThe fermion mass spectrum is studied in the quenched approximation in the strongcoupling vortex phase (VXS) of a globally U(1)L⊗U(1)R symmetric scalar-fermion modelin two dimensions.In this phase fermion doublers can be completely removed from thephysical spectrum by means of a strong Wilson-Yukawa coupling. The lowest lying fermionspectrum in this phase consists most probably only of a massive Dirac fermion which hascharge zero with respect to the U(1)L group.

We give evidence that the fermion which ischarged with respect to that subgroup is absent in the VXS phase. When the U(1)L gaugefields are turned on, the neutral fermion may couple chirally to the massive vector bosonstate in the confinement phase.

The outcome is very similar to our findings in the strongcoupling symmetric phase (PMS) of fermion-Higgs models with Wilson-Yukawa coupling infour dimensions, with the exception that in four dimensions the neutral fermion does mostprobably decouple from the bosonic bound states.$ e-mail: bock@phys.uva.nl# e-mail: hkf212@djukfa11.bitnet& e-mail: hkf247@djukfa11.bitnet∗e-mail: jsmit@phys.uva.nl

1IntroductionA non-perturbative formulation of a chiral gauge theory on the lattice has proved to be adifficult issue. In a chiral gauge theory naively transcribed to the lattice, each fermion isaccompanied by fifteen ‘doubler fermions’ where eight of these couple as mirror fermions andspoil the chiral couplings.

One way to deal with this problem is to decouple the unwantedextra species by rendering them very heavy.The standard Wilson mass term, which is known to remove the doublers in the caseof vector-like theories on the lattice, obviously breaks gauge invariance of the chiral gaugetheory. A proposal to overcome this difficulty is the so-called Wilson-Yukawa approach [1, 2]which has recently received a lot of attention.

Instead of a standard Wilson mass term oneuses a so-called Wilson-Yukawa term which contains the Higgs fields in a way that it ismanifestly invariant under the chiral gauge transformation. The goal of prime importance isnow to try to decouple the unwanted species doublers by means of a strong Wilson-Yukawacoupling and to give them a mass of the order of the cutoff.

It is well known that such adecoupling is a non-trivial and non-perturbative issue. For recent overview articles on theWilson-Yukawa approach see refs.

[3-9].Recently extensive investigations of globally GL⊗GR symmetric (with GL,R =SU(2),U(1)) fermion-Higgs models with Wilson-Yukawa coupling in four space-time dimensionshave shown that it is rather unlikely that this method leads to the desired non-perturbativeformulation of the standard model on the lattice. The reasons for this may be summarizedas follows: The phase diagram contains, apart from weak coupling symmetric (PMW) andbroken (FM(W)) phases where fermion masses behave according to perturbation theory, alsostrong coupling symmetric (PMS) and broken (FM(S)) phases where these masses exhibit anon-perturbative behavior.

The Wilson-Yukawa approach fails in the weak coupling phasesbecause there the masses of the doubler fermions are restricted by the triviality of Yukawacouplings and cannot be made sufficiently heavy [10]. They remain as additional particlesin the spectrum.On the other hand, in the strong coupling phases PMS and FM(S) the doubler fermionstates can be removed completely from the particle spectrum by making them as heavy asthe cut-off.

In contrast to the weak coupling region, fermions become massive also in thePMS phase, a situation which differs already from that in the fermion-Higgs sector of theperturbative standard model where the fermion mass vanishes in the symmetric phase. Sym-metry considerations show that the particle spectrum in the PMS phase can a priori containa fermion which is neutral with respect to the GL group and a fermion which is charged withrespect to that group [11-19].

The existence of the neutral fermion was confirmed by thegood agreement of analytic and numerical calculations [14, 17, 19]. On the other hand a 1/wexpansion (w is the Wilson-Yukawa coupling) of the charged fermion propagator [20] and anumerical investigation of the fermion spectrum [14] gave evidence that the charged fermiondoes not exist as a particle in the spectrum, though this is not generally accepted [21].

Underthe assumption that the charged fermion is absent one can show that the coupling of theneutral fermion to Higgs and gauge fields vanishes as a power of the lattice spacing a andthe neutral fermion becomes non-interacting in the scaling region [13]. It was also arguedthat the renormalized Yukawa coupling in the FM(S) phase vanishes as a power of a ratherthan logarithm of a [13, 14].In this paper we extend the investigations to a U(1)L⊗U(1)R symmetric fermion-Higgs

model with Wilson-Yukawa coupling in two dimensions. The important advantage of aninvestigation in two dimensions is that the simulations can be carried out on lattices of largelinear dimension, enabling one to achieve large correlation lengths for the scalar fields withbetter control over finite size effects.

The numerical data are of much superior quality thanin the four dimensional case.In two dimensions spontaneous breakdown of a continuous symmetry cannot occur be-cause of the Mermin-Wagner-Coleman theorem [22]. In spite of that, fermions are observedto acquire a mass also in two dimensional fermion-scalar models.

For example a 1/N ex-pansion in the Gross Neveu models with a continuous chiral symmetry (which can be alsoviewed as fermion-scalar models after the introduction of an auxiliary scalar field) shows thatthe fermion mass does not vanish [23]. At a first glance this appears to be contradictory.The contradiction could be resolved by expressing the action in terms of new fermionic fieldswhich can have a mass term without destroying the original chiral symmetry [24].

Inter-estingly, the existence of massive fermions in the strong coupling symmetric phase PMS inthe four dimensional models with Wilson-Yukawa coupling may be viewed in a similar way[11, 12]. The new fermionic variables are here the above mentioned neutral and chargedfermion fields which transform vectorially under the original chiral symmetry group and al-low therefore for the construction of invariant mass terms.The phase structure of the two dimensional model in the quenched approximation issimilar to the one found before in the four dimensional models.

The analogues of the weakcoupling phases FM(W) and PMW are respectively a weak coupling spin-wave SW(W) andvortex VXW phase. The strong coupling phases FM(S) and PMS get replaced by strongcoupling spin-wave SW(S) and vortex VXS phases.

We find that fermions are massive inthe strong coupling phases VXS and SW(S). The existence of the strong coupling phases isrecently confirmed also by an investigation of a two dimensional U(1) fermion-Higgs modelwith dynamical naive fermions.

There are, however, some indications favoring the absenceof the VXW phase [25]. The weak coupling spin-wave SW(W) would in this case extenddown to zero Yukawa coupling.

If this is correct, the VXW phase has to be regarded as anartefact of the quenched approximation.In this paper we study the fermion spectrum in the quenched approximation in the strongcoupling vortex phase VXS whose existence is guaranteed in the full model with dynamicalfermions. In this phase the species doublers can be completely removed from the spectrumand the generation of the fermion masses is based on the same mechanism as in the PMSphase of the four dimensional models.

Also here the fermion spectrum can a priori consistof a neutral and a charged fermion. Similar to the four dimensional models we find strongindications for the absence of the charged fermion.

The spectrum consists then only of theneutral fermion and the scalar particles. We shall show in sect.

7 that the neutral fermionmay exhibit a chiral coupling to the vector boson state in the confinement phase after thegauge interactions are turned on.The outline of the paper is as follows: In sect. 2 we introduce the model and describe itsphase structure in the quenched approximation.

Sect. 3 deals with the fermion spectrum inthe strong coupling phases SW(S) and VXS.

In sect. 4 we present the results of a leadingorder hopping expansion for the neutral and charged fermion propagators.

After giving abrief report on the technical details of the numerical simulations in sect. 5, we compare insect.

6 the numerical results for the rest energies of the charged and neutral fermion with

those obtained from the hopping expansion. Based on the results of sect.

6 we discuss insect. 7 different outcomes for the physics in the VXS phase.

A brief conclusion is given insect. 8.2The model and its phase diagramThe model of interest is given by the following gauge invariant euclidean lattice action intwo dimensions:S =XxL ,L = Lgauge + Lscalar + Lfermion(2.1)withLgauge=−1g2Re U12x ,(2.2)Lscalar=−κ2Xµ=1hΦ∗xUxµΦx+ˆµ + Φ∗x+ˆµU∗xµΦxi+ Φ∗xΦx + λ(Φ∗xΦx −1)2 ,(2.3)Lfermion=122Xµ=1Ψγµh(D+µ + D−µ )PL + (∂+µ + ∂−µ )PRiΨ + yΨ(ΦPR + Φ∗PL)Ψ−w2(ΨΦ)PR2Xµ=1∂+µ ∂−µ Ψ + ΨPL2Xµ=1∂+µ ∂−µ (Φ∗Ψ),(2.4)where in the second line of eq.

(2.4) we have included the Wilson-Yukawa coupling withstrength w. Besides the Wilson-Yukawa term we also included a usual Yukawa term. Thesymbols D+µ , D−µ , ∂+µ and ∂−µ denote the covariant and normal lattice derivatives which aredefined by the relations D+µ Ψx = UµxΨx+ˆµ −Ψx, D−µ Ψx = Ψx −U∗µx−ˆµΨx−ˆµ, ∂+µ = D+µ |U=1and ∂−µ = D−µ |U=1.

The symbols γµ, µ = 1, 2 denote the two dimensional γ matrices andPR,L = 12(11 ± γ5) with γ5 = −iγ1γ2 are the right and left-handed chiral projectors. The fieldU12x ≡U1xU2x+ˆ1U∗1x+ˆ2U∗2x is the usual plaquette variable in the Wilson action, g is the gaugecoupling, κ is the hopping parameter for the scalar field and λ the quartic coupling.

If wesucceed in removing the species doublers from the particle spectrum by means of a strongWilson-Yukawa coupling w, one would expect that the lattice lagrangian defined by eq. (2.1)reproduces in the continuum limit the target model given by the lagrangianL0=14FµνFµν + Dµφ∗Dµφ + m20φ∗φ + λ0(φ∗φ)2+ψ(D/PL + ∂/PR)ψ + y0ψ(φPR + φ∗PL)ψ ,(2.5)where Fµν(x) = ∂µAν(x) −∂νAµ(x) and Dµ = ∂µ −ig0Aµ(x).

We will demonstrate in thispaper that this naive expectation is not correct and that the effective lagrangian whichdescribes the physics in the VXS phase differs substantially from eq. (2.5).

The continuumfields ψ(x), φ(x), Aµ(x) in eq. (2.5) are related to the corresponding lattice fields in (2.1) bythe transformationsΨx = a1/2ψ(x) ,Φx =1√κφ(x) ,Uµx = exp(−iag0Aµx) .

(2.6)The coupling parameters in eq. (2.5) can be expressed in terms of the lattice couplings bythe relationsm20 = 1 −2λ −4κa2κ,y0 =ya√κ ,g0 = ga ,λ0 =λ(aκ)2 .

(2.7)

We note that in the continuum formulation the Yukawa coupling, the gauge coupling and thequartic coupling carry a dimension, whereas the gauge and scalar fields are dimensionless.Throughout this paper we will study the model in the limit λ →∞which implies that thescalar fields become radially frozen, i. e. Φ∗xΦx = 1.The lattice lagrangian in eq. (2.1) is invariant under the local gauge transformations ofthe form ΨL,x →ΩL,xΨL,x, ΨL,x →ΨL,xΩ∗L,x, Φx →ΩL,xΦx, Uµx →ΩL,xUµxΩ∗L,x+ˆµ, withΩL,x ∈U(1)L. The model is furthermore also invariant under the global gauge transformationsΨR,x →ΩRΨR,x, ΨR,x →ΨR,xΩ∗R and Φx →ΦxΩ∗R with ΩR ∈U(1)R. At y = 0 the action(2.1) possesses a shift symmetry for the right-handed fermion fields,ΨR,x →ΨR,x + ǫR ,ΨR,x →ΨR,x + ǫR ,(2.8)where ǫR and ǫR are the constant shifts in the right-handed Weyl spinors.

This symmetryguarantees that the mass of the fermion with the quantum numbers of the ΨR fermion fieldvanishes at y = 0 for all values of κ and g [26].Although we have included for later convenience in (2.1) also the interactions to the U(1)Lgauge fields, in our numerical work, however, we will restrict ourselves to the global limitg = 0 where Uµx = 1 and the local U(1)L gauge symmetry turns into a global one. In thiscase eq.

(2.3) (with λ = ∞) reduces to the lagrangian of the XY model in two dimensions.We shall furthermore study the model (2.1) in the quenched approximation where the effectsof the fermion determinant are neglected. The use of the quenched approximation for theinvestigation of the fermion spectrum in strong coupling phase VXS is justified since thisphase was established also in the full model with dynamical fermions [25].Next we turn to the phase structure of the model in the quenched approximation.

TheXY model is known to have a phase transition at κ = κc ≈0.56 which separates a vortex(VX) phase (κ < κc) with finite scalar correlation length from a spin-wave (SW) phase wherethe scalar correlation length is infinite [27]. We note that the spectrum in the VX phaseconsists of two scalar particles which have the same mass.

In the quenched approximationfermions do not have a feedback on the scalar sector and κc is independent of y and w.A phase transition may, however, occur in the fermionic sector. Such a phase transitionor ‘crossover’ was discovered before in four dimensional models with Wilson-Yukawa cou-pling at y + 4w ≈√2 [28, 29].

It separates the weak coupling phases PMW and FM(W)(y + 4w<∼√2) from the strong coupling phases PMS and FM(S) (y + 4w>∼√2). Theexistence of this ‘crossover’ has manifested itself in a different behavior of the fermion massas a function of κ in the weak and strong coupling regimes.In order to monitor the fermionic phase structure of the quenched model in two dimen-sions we have computed the mass mF of the ΨR fermion by fitting the ⟨ΨRΨR⟩propagatorto a free Wilson fermion propagator ansatz (for more information about the technical detailssee sect.

5). In fig.

1 we have displayed this mass as a function of κ for several values of theYukawa coupling y and for the special case of naive fermions (w = 0). As in four dimensionsthe fermion mass shows a qualitatively different behavior at small and large values of y. Itdecreases when approaching the SW-VX phase transition in the SW phase for y < y∗≈1.On the contrary the mass is observed to increase when the value of κ is lowered for y > y∗.This increase is seen to continue into the VX phase and fermions become massive in thatphase.

A similar behavior of the physical and doubler fermion mass as a function of κ isobserved also for w > 0. From the different behavior of the fermion mass we can localize

Figure 1:The mass of the ΨR fermion as a function of y for several values of κ at w = 0.The position of the phase transition in the thermodynamic limit is given by κc ≈0.56. Thevertical arrow below the abscissa indicates the position of the peak in the susceptibility on the48 × 48 lattice which lies clearly below κc.the position of a ‘crossover’.

This position is within the precision of our resolution indepen-dent of κ and is approximately given by the relation y + 2w = 1. As in four dimensionsthe ‘crossover’ sheet splits the VX and SW phases into strong (S) and weak (W) couplingregions, which we will denote in the following by VXS, SW(S) (y + 2w>∼1) and VXW,SW(W) (y + 2w <∼1).

As we mentioned already in the introduction, the VXW phase doesnot seem to be present in the full model with dynamical fermions and could be an artefactof the quenched approximation [25]. However, this does not affect our investigations in theVXS phase.Our numerical results on the 482 lattice show that the fermion mass stays non-zero ev-erywhere in the SW phase and also in the VXS phase though the chiral symmetry cannotbe broken according to the Mermin-Wagner-Coleman theorem.

Studies on different latticesshow that the finite size dependence of the fermion mass is extremely small and it appearsvery unlikely that the fermion mass could vanish in the infinite volume limit. The existenceof a massive neutral fermion in the strong coupling phases SW(S) and VXS is indeed stronglysupported by the good agreement between analytic calculations which are based on strongcoupling expansions and the results of the numerical simulation.

We will report more onthese results in the following sections of this paper.

The fact that fermions may be massive within the SW phase although the chiral symmetryis unbroken in that phase was to our knowledge first explained in ref. [24].

The basic idea isas follows: The original action in terms of the Ψ fields does not provide a correct descriptionof the physics in the SW phase. It can, however, be rewritten in terms of new fermionic fieldvariables, which transform vectorially under the original symmetry transformations therebyallowing for the construction of a chirally invariant mass term.

This new form of the actionmay give a more appropriate description of the physics in this phase (in the sense of a weakcoupling expansion) if the fermions are indeed observed to be massive.3The fermion spectrum in the strong coupling phasesSince the U(1)L⊗U(1)R symmetry is unbroken everywhere, the states excited by the fieldsΨL and ΨR need not be the same since the corresponding fields carry different quantumnumbers under the unbroken symmetry group. We will refer to the ΨR = Ψ(n)Rfield as theneutral (n) fermion field since it has charge zero under the U(1)L group (qL = 0).

TheΨL = Ψ(c)L field will be called charged (c) fermion field since it has charge one under the localU(1)L gauge group (qL = 1). When discussing the phase diagram in the previous section wehave already mentioned that the numerical results give strong evidence that the ΨR fermionbecomes massive both in the SW(S) and VXS phases.

This will be substantiated later by thegood agreement of the numerical results with a hopping expansion. In order to describe thissituation the Ψ(n)Rfield may be regarded as the right-handed component of a massive Diracfermion field Ψ(n).

A possible choice for Ψ(n)Lis the composite field Φ∗ΨL which transformsin the same manner as Ψ(n)R . The neutral Dirac fermion field may then be written asΨ(n) = (Φ∗PL + PR)Ψ ,Ψ(n) = Ψ(ΦPR + PL) .

(3.1)Along the same lines we may also introduce a charged Dirac fermion fieldΨ(c) = (PL + ΦPR)Ψ ,Ψ(c) = Ψ(PR + Φ∗PL) . (3.2)The fields Ψ(c) and Ψ(n) transform vectorially under U(1)L and U(1)R respectively.On a finite lattice the long range fluctuations in the SW phase cause a non-vanishingvalue of the magnetization M which may be defined by the relationM = ⟨1VXxΦx⟩|rot ,(3.3)where V is the lattice volume.

The index “rot” means that each configuration is rotatedsuch that1VPx Φx points into a given direction. This rotation is necessary since on a finitelattice the magnetization M is drifting through the group space and when averaging overmany configurations one would get zero.

A measure for the magnetization in the SW phasemay then be defined by v = |M|. Since there is no spontaneous symmetry breaking in twodimensional systems, this quantity has to vanish in the infinite volume limit V →∞.

Sometypical values for v on a 48×48 lattice are given by v(κ) = 0.3545(44), 0.5565(22), 0.6734(12)for κ = 0.48, 0.52, 0.60. Even on large lattices (e. g. 4002) v is clearly non-zero in the SWphase and increases when raising the values of κ.

Therefore, on a finite lattice the situationin the SW phase is very similar to the broken phase in the four dimensional model where theU(1)L⊗U(1)R symmetry is broken to the diagonal subgroup U(1)L=R. As a consequence the

fields Ψ(n) and Ψ(c) appear to behave almost as equivalent interpolating fields. Indeed, thenumerically found rest energies obtained from the neutral and charged fermion propagatorscoincide in that phase within the statistical errors.

In the infinite volume limit, however, thetwo rest energies are expected in general to be different for w > 0.4Hopping expansionfor theneutralandchargedfermion propagatorsAn appropriate method to try to find analytic expressions for the neutral and charged fermionpropagators in the strong coupling phases is the hopping expansion. The hopping expansiondeals only with the fermionic integration in the path integral, the bosonic integration hasto be performed e. g. by a 1/d expansion [15] or by numerical simulations.

When startingfrom the lagrangian (2.4) with the single-site Yukawa-coupling and expanding the Boltzmannfactor in the path integral in powers of the hopping parameter α = 1/(y + 2w) one comesupon cancellations of the type Φ∗Φ →1, emerging from the single-site terms and the one-linkterms. The hopping expansion becomes more transparent after removing the Φ fields fromthe single-site Yukawa term by means of a unitary transformation to the one link terms.

Twotransformations of this type are given by the inverses of eqs. (3.1) and (3.2) which express theneutral and charged fermion fields in terms of the original Ψ fields.

The associated jacobianfor these transformations is in both cases equal to one since the scalar field is radially frozen.Replacing Ψ and Ψ in (2.4) by Ψ(n) and Ψ(n) givesLF=122Xµ=1(Ψ(n)L Φ∗)γµ(D+µ + D−µ )(ΦΨ(n)L ) + Ψ(n)R γµ(∂+µ + ∂−µ )Ψ(n)R+yΨ(n)Ψ(n) −w2 Ψ(n)2Xµ=1∂+µ ∂−µ Ψ(n) . (4.1)This substitution transforms the Yukawa term into a bare mass and the Wilson–Yukawaterm into a free Wilson term.

Expressing the Ψ and Ψ fields in terms of the Ψ(c) and Ψ(c)fields leads toLF=122Xµ=1Ψ(c)L γµ(D+µ + D−µ )Ψ(c)L + (Ψ(c)R Φ)γµ(∂+µ + ∂−µ )(Φ∗Ψ(c)R )+yΨ(c)Ψ(c) −w2 (Ψ(c)Φ)2Xµ=1∂+µ ∂−µ (Φ∗Ψ(c)) . (4.2)The lagrangian (4.1) has a shift symmetry (2.8) in terms of the neutral field becauseΨ(n)R= ΨR.

Such a symmetry, however, is absent for the action (4.2) in terms of the Ψ(c)fields. This different behavior under the shift symmetry holds also if the local U(1)L gaugeinteractions are turned off.

It makes therefore sense to distinguish between the charged andthe neutral fermion also in this globally symmetric U(1)L⊗U(1)R theory.Using the lagrangians (4.1) and (4.2) one finds to lowest order in α the following expres-sions for the charged and neutral fermion propagators in momentum space,S(n)(k)≡* 1VXx,yΨ(n)x Ψ(n)y eik(x−y)+

≈(z−1PL + PR)[(y + 2w −w2Xµ=1cos kµ)z−1 + i2Xµ=1γµ sin kµ]−1(z−1PR + PL) ,(4.3)andS(c)(k)≡* 1VXx,yΨ(c)x Ψ(c)y eik(x−y)+≈(PL + z−1PR)[(y + 2w)z−1 −wz2Xµ=1cos kµ + i2Xµ=1γµ sin kµ]−1(PR + z−1PL) . (4.4)Herez2 = ⟨Re(Φ∗xUµxΦx+ˆµ)⟩(4.5)is the scalar field link expectation value.

This quantity has a non-vanishing value in the VXas well as SW phase.From the expressions (4.3) and (4.4) one can read offexpressions for the fermion masses.For the masses of the neutral fermion and its species doublers we obtainm(n)F≈yz−1 ,m(n)D ≈m(n)F + 2wlz−1 ,l = 1, 2 ,(4.6)where l is the number of momentum components equal to π in the two dimensional Brillouinzone. From eq.

(4.6) we can read offan effective Wilson r-parameter:r(n) ≈wz−1 . (4.7)As we shall see later, these expressions are in good agreement with the numerical simula-tions in the VXS phase.

This was also found to be the case in the four dimensional models[14, 17, 19].In agreement with the shift symmetry [26] the mass m(n)Fof the physical fermion is seento vanish in the limit y →0. The doubler fermions, however, are non-zero within the strongcoupling phases, since 2wlz−1 ̸= 0 everywhere in this region.

This means that in the contin-uum limit the species doublers for the neutral fermion decouple from the particle spectrumwithin this phase.Similar formulas can be obtained from eq. (4.4) for the masses of the charged fermion(assuming it exists for the moment) and its species doublersm(c)F ≈(y + 4w)z−1 −4wz ,m(c)D ≈m(c)F + 2wzl , l = 1, 2 .

(4.8)The effective Wilson parameter is now given byr(c) ≈wz . (4.9)The discussion in the following section will show that the formulas (4.8) and (4.9) for thecharged fermion are in disagreement with the numerical results.

A calculation of higherorder terms in four dimensional models showed that they appear to be small for the neutralpropagator S(n), but not for the charged propagator S(c) [18, 19].On the basis of ournumerical results we will give in sect. 7 an argument why the hopping expansion to lowestorders leads to a wrong result for charged fermion propagator.

In sect. 6 we will comparethe analytic formulas that have been derived in this section with the results of the numericalsimulation.

5Details of the numerical simulationThe neutral and charged fermion propagators were determined by inverting the correspond-ing fermion matrices on a set of uncorrelated scalar field configurations which were generatedby means of the reflection cluster algorithm [30] for the XY model. We have computed thefermion propagators in coordinate space.

As an example we give the expression for the RRcomponent of neutral fermion propagatorS(n)RR(t) =* 1LXx1Ψ(n)x1,x2RΨ(n)y1,y2Reip1(x1−y1)+,t = |x2 −y2| ,(5.1)where t = 1, . .

. , T. The symbols L and T denote here and in the following the spatial andtime extent of a rectangular lattice.

The physical fermion propagator is obtained for p1 = 0and the propagator of the lowest lying doubler fermion for p1 = π. The fermion and thedoubler fermion propagators were computed for all four L-R combinations.

We have usedfor the fermion fields periodic boundary conditions in the spatial direction and anti-periodicboundary conditions in the time direction. The scalar fields had periodic boundary condi-tions in all directions.For the neutral fermion propagator we have inverted the fermion matrix on typically1000-2000 scalar field configurations.

A problem which we were confronted with in the fourdimensional models was the large number of matrix inversions which was required to obtaina stable signal for the charged fermion propagator in the PMS phase. In the two dimensionalmodels it is possible to enlarge the statistics in the VXS phase on relatively large lattices(e. g. 322) by an order of magnitude.

For the determination of the charged fermion prop-agator we have inverted the fermion matrix on typically 1000-5000 and deeper in the VXSphase on 20000 scalar field configurations.Most of the results were obtained on a 32 ×32 lattice. In order to estimate the finite sizeeffects we varied L at a particular point in the VXS phase from 16 to 64 while keeping Tfixed at 64.We have fitted the neutral and charged propagators at zero spatial momentum to thefree Wilson fermion ansatzS(n,c)(t) →Z2q1 + 2rlml + m2l×"exp(−Elt) + ζ exp(−El(T −t))1 + exp(−ElT)−ζ(−1)t exp(−E′lt) + ζ exp(−E′l(T −t))1 + exp(−E′lT)#,(5.2)where ζ = 1(−1) for the RR and LL (RL and LR) components.

This ansatz holds only forrl < 1, for rl > 1 the oscillating factor (−1)t has to be omitted.From this fit we obtain the numerical values for the rest energies El and E′l of the fermionand its ‘time doubler’ and for the wave-function renormalization constant Z. The subscriptl is 0 for the physical fermion propagator (p1 = 0 in eq.

(5.1)) and 1 for the doubler fermionpropagator (p1 = π). The masses ml and the Wilson parameters rl for which we obtainedexpressions in the previous section are related to the rest energies El and E′l by the lattice

Figure 2:The physical charged fermion propagator S(c)RL(t) (p1 = 0) as a function of thetime coordinate t for (κ, y, w) = (0.45, 0.3, 2.0). The lattice is size is 32 × 64.

The solid linewas obtained by fitting S(c)RL(t) to the free Wilson fermion ansatz in eq. (5.2).dispersion relationseEl =q1 + 2rlml + m2l + rl + ml1 + rl, eE′l =q1 + 2rlml + m2l + rl + ml1 −rl.

(5.3)The effective Wilson parameter rl could be in principle a function of l = 1, 2. The numericalresults, however, show that rl is independent of l (therefore we will subsequently use thenotation r = rl), which supports the interpretation of the numerical results in terms of thefree fermion formula.We find that the rest energies obtained from the four chiral components S(n)LL(t), S(n)RR(t),S(n)RL(t) and S(n)LR(t) for p1 = 0, π agreed always within the statistical errors.

The same holdsalso for the rest energies determined from the four chiral components of the charged fermionpropagator. In the following we will use the notation E(n)Fand E(c)Ffor the rest energies ofthe neutral and charged physical fermion and similarly E(n)Dand E(c)D for the rest energies ofthe lowest lying doubler fermions.In fig.

2 we have displayed as an example the charged fermion propagator S(c)RL(t) forp1 = 0 as a function of t. In this example we have chosen L = 32 and T = 64. The solidcurve was obtained by fitting the numerical data to the free Wilson fermion ansatz (5.2).The high quality of the numerical results for the propagators allowed for an accurate deter-mination of the rest energies E(n)F , E(n)D , E(c)Fand E(c)D .

For the considerations in sect. 7 we have to know also the numerical values for therest energy EΦ of the scalar particles in the VX phase.

This rest energy was determinednumerically by fitting the scalar propagator in momentum space defined byGΦ(p) =* 12VXx,yΦ∗xΦyeip(x−y)+(5.4)to a free scalar propagator ansatzGΦ(p) →ZΦˆp2 + m2Φ,(5.5)where ˆp2 = 2 P2µ=1(1 −cos pµ) is a lattice equivalent of the momentum squared in thecontinuum. The rest energy EΦ of the scalar particles in the VX phase is obtained from thelattice dispersion relation at zero momentum, cosh EΦ = 1 + m2Φ/2.6Comparison of the numerical results with the hop-ping expansionIn this section we are going to compare the numerically found values for the rest energiesE(n)Fand E(c)Fof the neutral and the charged fermion and the corresponding rest energiesE(n)D and E(c)D for the lowest lying doubler fermions with the analytic expressions which resultfrom the formula (5.3) after inserting eqs.

(4.6), (4.7), (4.8) and (4.9) from the lowest orderhopping expansion. For z2 we use the numerical value measured on the same lattice whichwe are using for the determination of the fermion propagators.

In fig. 3 we have displayed thenumerical values for E(n)F , E(n)D , E(c)F and E(c)D as a function of y for κ = 0.4 and w = 2.0.

Thecoupling parameter values we have chosen here lie well inside the VXS phase. The resultsfrom the hopping expansion are represented by the curves.

The dashed, solid, dash-dottedand dotted lines correspond respectively to the rest energies E(n)F , E(n)D , E(c)Fand E(c)D . Thefigure shows that the agreement between the numerical result and the analytic prediction isquite impressive for the rest energies E(n)Fand E(n)D while the curves for E(c)F and E(c)D exhibita strong deviation from the numerical results.

In the case of E(c)F the deviation is larger thana factor two. The figure shows furthermore that E(n)Fappears to vanish in the limit y →0, inagreement with the shift symmetry mentioned before, whereas E(n)D stays non-zero for all dif-ferent values of y which implies the decoupling of the species doublers of the neutral fermionin the continuum limit.

Also the numerical values for E(c)D are larger than one for all valuesof y with no indication of dropping in the limit y →0. Thereby also the species doublersof the charged fermion can be completely removed from the physical spectrum.

Providedit exists at all as a particle in the spectrum, the charged fermion is massive in the VXS phase.In fig. 4 the rest energies E(n)Fand E(c)Fare plotted as a function of κ where the couplingparameters y and w were fixed to 0.4 and 2.0.

The analytic results for E(n)Fand E(c)Farerepresented also in this figure by the dashed and dash-dotted lines.We find again thatanalytic expressions from the hopping expansion provide a good description of the restenergy of the neutral, but not of the charged fermion. The other details in this figure willbe explained in the next section where we will develop, on the basis of the results of thissection, two different scenarios for the physics in the strong coupling region.

Figure 3:The rest energies E(n)F , E(n)D , E(c)Fand E(c)D as a function of y for κ = 0.4 andw = 2.0. The dashed, solid, dash-dotted and dotted curves represent respectively the analyticresults for E(n)F , E(n)D , E(c)Fand E(c)D obtained from the hopping expansion.7Scenarios for an effective field theory in the strongcoupling regimeThe results of the previous section showed that the analytic results deduced from the la-grangian (4.1) are in good agreement with the numerical results for the rest energies E(n)Fand E(n)D , whereas the lagrangian (4.2) led to expressions which were in a strong disagree-ment with the numerical data.

This suggests that the physics in the strong coupling regionis well described by the lagrangian (4.1) in terms of the neutral fermion fields. The chargedfermion fields Ψ(c) = ΦΨ(n) and Ψ(c) = Ψ(n)Φ∗can then be regarded as composite fields andthe charged fermion, provided it exists at all in the particle spectrum, can be consideredas a bound state of the scalar particle and the neutral fermion.

The question arises nowwhether the interactions in eq. (4.1) can produce such a Φ-Ψ(n) bound state.

In the fourdimensional model the scalar fields Φ carry a dimension of a mass and as a consequence ofthis the four-point coupling12dXµ=1(Ψ(n)L Φ∗)γµ(D+µ + D−µ )(ΦΨ(n)L )(7.1)with d = 4 vanishes in the classical continuum limit like a2 which makes the formation of aΦ-Ψ(n) bound state already very unlikely. Indeed a 1/w expansion [20] and the numerical

Figure 4:The rest energies E(n)Fand E(c)Fplotted against κ for y = 0.4 and w = 2.0. Thedashed and dash-dotted curves represent the analytic results from the hopping expansion forE(n)Fand E(c)F .

The solid line gives the result for the sum E(n)F+ EΦ. The error bars forE(n)F+ EΦ are much smaller than the symbol sizes for E(c)F .simulations [14] gave strong evidence for the absence of the charged fermion in the particlespectrum of the PMS phase.

In the case of the two dimensional model the scalar fields aredimensionless and for d = 2 the coupling (7.1) does not vanish as a power of the latticespacing a. Therefore the formation of a Φ-Ψ(n) bound state appears at the first glance to bemore favored than in the four dimensional model.

If the four-point interaction (4.1) is strongenough to produce a Φ-Ψ(n) bound state we expect to find the following relation among therest energies of the neutral and the charged fermion and of the scalar particles in the VXSphaseE(c)F = E(n)F+ EΦ + ǫB ,ǫB < 0 ,(7.2)where the quantity ǫB denotes the binding energy of the Φ-Ψ(n) bound state. This relationimplies that the charged fermion could only exist as a particle at a point in the couplingparameter space where the rest energies E(c)F , E(n)Fand ǫB scale simultaneously to zero.

Thiscan happen only at the point κ = κc, y = 0, since only there E(n)Fand EΦ can vanish simul-taneously.Scenario A: Let us assume now for the moment that ǫB scales to zero like E(n)Fand EΦ inthe limit κ ր κc, y →0 and the charged fermion exists together with the neutral fermion asa Dirac fermion in the particle spectrum. The coupling of the Ψ(c)L field to the gauge fields isnecessarily vectorial because the charged fermion is massive (m(c) > 0).

The model we end

LEΦE(n)FE(c)FǫB160.126(3)0.2750.368(13)-0.033(16)320.115(3)0.2770.374(9)-0.018(12)480.120(5)0.2760.379(12)-0.017(17)640.119(5)0.2770.389(11)-0.007(16)Table 1: The rest energies EΦ, E(n)F , E(c)Fand ǫB as a function of the spatial extent L of alattice with volume L × 64 obtained at the point (κ, y, w) = (0.45, 0.3, 2.0). The error barsfor E(n)Fwere omitted since they are smaller than 0.001.up with in the strong coupling VX phase is clearly different from the original target modelgiven in eq.

(2.5), although we succeeded in removing the species doublers from the spectrum.One possible form of an effective action in the VXS phase is given by the expressionLeffF=ψ(n)∂/ψ(n) + ψ(c)D/ψ(c) + m(n)ψ(n)ψ(n) + m(c)ψ(c)ψ(c)+ yR[ψ(n)R φ∗ψ(c)L + ψ(c)L φψ(n)R ] ,(7.3)where we used the concise continuum notation of eq. (2.5) and all fields and coupling pa-rameters are considered to be effective.

The symbol yR denotes the renormalized Yukawacoupling. The failure of the Wilson-Yukawa approach in giving a chiral gauge theory is re-lated to the fact that the original fermion fields ΨR and ΨL combine with composite ‘mirror’fermion fields χR ≡φψR and χL ≡φ†ψL and form the two massive Dirac fields Ψ(n) and Ψ(c)which would couple vectorially to “external” gauge fields.

The model (7.3) is a special caseof the mirror fermion model [31], when transcribed to the case of two dimensions. It has,however, less flexibility in tuning coupling parameters.Eq.

(7.3) is, however, not the only possible form of an effective action in the strong couplingphase. For example it allowed by the symmetries to add a term, which couples the neutralfermion chirally to the massive vector bosons in the confinement phase.

We will come tothis in the last part of this section.Scenario B: In order to figure out whether the above requirements for the binding energyare fulfilled we have determined ǫB numerically in a wide range of the bare parameters inthe VXS phase. The details about the numerical determination of the rest energy EΦ weregiven in sect.

5. In the figs.

4 and 5 we compare the rest energy E(c)F with the sum E(n)F +EΦwhich in these graphs is represented by the solid lines. For E(n)Fwe have used the resultsfrom the hopping expansion which are in perfect agreement with the actual data, as wereported in sect.

6. The error bars for the sum E(n)F+ EΦ are always much smaller than thesymbol sizes for E(c)F .

Both plots indicate that ǫB = 0 for all values of κ and y in the VXSphase. This suggests that the formation of a bound state is very unlikely.

Fig. 5 shows thatalso for the lowest lying doubler fermion the numerical results for the rest energy E(c)D arenicely represented by the sum E(n)D + EΦ (upper solid line).

In order to get an impressionabout the finite size dependence of the energies E(n)F , E(c)F , EΦ and ǫB we have computedthe S(n), S(c) and GΦ propagators at a fixed point in VXS phase on a sequence of latticeswith size L × 64, where spatial extent L was varied in a range from 16 to 64. The resultsfor E(n)F , E(c)F , EΦ and ǫB are summarized in table 1.

It can be seen also here that thebinding energies are very small and even on the smallest lattices almost compatible with

Figure 5:The rest energies E(c)Fand E(c)D as a function of y for κ = 0.4 and w = 2.0. Therest energies are compared with the sums E(n)F+ EΦ and E(n)D + EΦ which are represented bythe lines.

The error bars of E(n)F +EΦ and E(n)D +EΦ are much smaller than the symbol sizesfor E(c)F .zero within the quoted error bars. Furthermore one recognizes a systematic trend of |ǫB| todecrease when the spatial extent of the lattice is enlarged.

These results strongly suggestthat the signal which we detected in the charged fermion propagators is simply caused bya two particle state of the neutral fermion and the scalar particle. We now can understandalso the fact why the hopping expansion to lowest order for the charged fermion propagatoris very misleading.

To reproduce the inverse propagator of a two particle state an infinitenumber of hopping terms would be needed. Only on the basis of our numerical results wecan, of course, not completely exclude the possibility that in principle a very small andnon-vanishing binding energy might be left over in the continuum limit.

We could also notfind a good field theoretical argument which would rule out the existence of a bound statewith zero binding energy. Both cases would bring us back to scenario A which we drew upfirst in this section.Let us now proceed under the assumption that the charged fermion does not exist as aparticle in the spectrum.

In order to find an effective lagrangian for the model in the VXSphase we first rewrite the lattice lagrangian (4.1) in the following wayLF=122Xµ=1(Ψ(n)L,xγµU′µxΨ(n)L,x+ˆµ −Ψ(n)L,x+ˆµγµU′µx∗Ψ(n)L,x) + (Ψ(n)R,xγµΨ(n)R,x+ˆµ −Ψ(n)R,x+ˆµγµΨ(n)R,x)

+yΨ(n)Ψ(n) −w2 Ψ(n)2Xµ=1∂+µ ∂−µ Ψ(n) ,(7.4)where we introduced the effective gauge fieldU′µx ≡Φ∗xUµxΦx+ˆµ . (7.5)Although the interaction in the first term of eq.

(7.4) appears to be too weak for a formationof a Φ-Ψ(n) bound state, the form (7.4) leaves still the possibility of a chiral coupling be-tween the neutral fermion and the effective gauge field U′µx since this field has dimension oneaccording to a naive power counting analysis. This outcome would be very interesting sinceone would have found at least one example for a lattice regularized theory where fermionsexhibit a chiral coupling to an “external” gauge field.

In contrast in four dimensional modelsthe naive power counting analysis suggests that the coupling to the effective gauge field U′µxvanishes like a2 in the continuum limit.The U(1) pure gauge model (2.2) is confining for all values of the gauge coupling g. Theconfinement phase is expected to be present also in the two dimensional U(1) Higgs model atsmall values of κ and to turn in the limit g →0 into the vortex phase. For g > 0 the scalarparticles get confined into massive bosonic particles.

The effective gauge field in eq. (7.5)can be written in the formU′µx = z2 + Hµx + iWµx(7.6)where z2 is given in eq.

(4.5) and Hµx and Wµx are interpolating fields for bosonic boundstates in the confinement phase with quantum numbers JP C = 0++ and 1−−in lowest spinstate [32]. The field Hµx couples primarily to the Higgs-like scalar particle according toHµx →mH Hx(7.7)where mH is some mass scale which has to be introduced since the scalar field Hx is dimen-sionless (Hµx is not a vector under lattice rotations, which forbids a relation of the formHµx →∂µH(x)).

The field Wµx couples to the vector boson.After inserting (7.6) into (7.4) and a trivial rescaling of the fields Ψ(n)Land Ψ(n)Lwe obtainfor LF the formLF=122Xµ=1Ψ(n)x γµΨ(n)x+ˆµ −Ψ(n)x+ˆµγµΨ(n)x+ yzΨ(n)Ψ(n) −w2zΨ(n)2Xµ=1∂+µ ∂−µ Ψ(n)(7.8)+mHz2122Xµ=1HxΨ(n)L,xγµΨ(n)L,x+ˆµ −Ψ(n)L,x+ˆµγµΨ(n)L,x(7.9)+ 1z2122Xµ=1iWµxΨ(n)L,xγµΨ(n)L,x+ˆµ + Ψ(n)L,x+ˆµγµΨ(n)L,x(7.10)The term (7.8) describes a free neutral fermion with mass m(n)F= y/z. The expression (7.9)suggests that the coupling of the neutral fermion to the Higgs-like bound state vanishes likea.

However, the neutral fermion couples chirally in (7.10) to the vector boson field Wµx ifits dimension is one, as suggested by the naive dimensional analysis.In order to find out whether the fields Wµx and Hµx are indeed dimension one operatorswe have computed the scale dependence of the corresponding wave-function renormalization

Figure 6:The wave-function renormalization constants ZW (squares) and ZH (circles) areplotted respectively as a function of m2 = m2W and m = m2H for the fixed ratio mH/mW =1.14. The two dotted lines through the origin are drawn to guide the eye.constants ZH and ZW.

From the naive dimensional analysis these wave-function renormal-ization constants ZH and ZW are expected to vanish like a2. To see whether this expectationis correct we have computed the momentum space propagatorsGH(p2) =* 1VXx,yH1xH1yeip2(x2−y2)+, GW(p2) =* 1VXx,yW1xW1yeip2(x2−y2)+(7.11)in the confinement phase of the U(1) gauge-Higgs model for several values of g and κand fitted the results for sufficiently small p2 to the free boson propagator ansatz givenin eq.

(5.5), which for GW is considered as a special case (p1 = 0) of a free massive vectorboson propagator, (δµν +pµpν/m2)(m2+p2)−1. In fig.

6 we have displayed the resulting wave-function renormalization constant ZW (squares) and ZH (circles) respectively as a functionof m2 = a2m2W,phys and m2 = a2m2H,phys for the fixed ratio mH/mW = 1.14. In both casesthe wave-function renormalization constants obey nicely a linear dependence, supporting ourexpectation from the dimensional analysis.This result suggests that the neutral fermion exhibits in two dimensions indeed a non-vanishing chiral coupling to the massive vector bosons.

This coupling is universal in thequenched approximation since the field Wµx is proportional to a current which is conservedin the two dimensional U(1) gauge-Higgs model. The properties of this coupling in the fullmodel with dynamical fermions are, however, not yet clear to us.

The fermion couplings in the VXS phase may then be summarized qualitatively by thefollowing effective lagrangianLeffF= ψ(n)∂/ψ(n) + m(n)F ψ(n)ψ(n) + gRψ(n)L γµψ(n)L W (c)µ,(7.12)where gR = √ZW/z2 and W (c)µis the vector field in the continuum with standard normal-ization. This expression for the effective lagrangian gives a satisfactory description of thefermion couplings at distances which are large in comparison with the typical length scaleof the vector boson bound state.

When lowering the value of the gauge couplings g thestring tension becomes smaller and the bound states extends over larger distances. At smalldistances the scalar particles are then almost free and a more appropriate form of the actionis given then byLeffF→ψ(n)∂/ψ(n) + m(n)F ψ(n)ψ(n) + 1z2ψ(n)L γµψ(n)L (φ∗∂µφ −φ∂µφ∗) .

(7.13)This form is expected to describe the fermion couplings in particular in the global limit,i. e. g = 0, where the bosonic spectrum consists of unbound scalar particles.

The massiveneutral fermion in eq. (7.13) is coupled to a two particle current.

This interaction seems,however, to be quite weak since our numerical results for the neutral fermion propagator arein good agreement with the analytic prediction from the lowest order hopping expansion,which leads to a free fermion propagator. Furthermore our numerical results for the bindingenergy ǫB suggest that the interaction in (7.13) is too weak as to give rise to the formationof a Φ-Ψ(n) bound state.

Also the expressions (7.12) and (7.13) are certainly different fromthe target model in eq. (2.5) which we had originally in mind.8ConclusionWe started out our investigations from the lattice lagrangian given in eq.

(2.1) in the hopeto obtain in the continuum limit the target model in eq. (2.5).

In the global limit of themodel (g →0) the unwanted species doublers can be removed completely from the spectrumwithin the strong coupling vortex phase (VXS). The physics in this phase differs, however,substantially from the target action which we had originally in mind.

In the previous chap-ter we have developed two different scenarios (A and B) for the effective theory in the VXSphase which were summarized by the continuum lagrangians (7.3) and (7.12), (7.13). Ournumerical results are in favor of scenario B: In this case the fermionic spectrum in VXScontains only a neutral fermion which has zero charge with respect to the U(1)L group andwhich in the global limit g →0 exhibits a left-handed coupling to a two particle current.This coupling, however, appears to be weak since the neutral fermion propagator data arein nice agreement with the results from the lowest order hopping expansion which implies afree fermion behavior.

Furthermore this coupling seems to be too weak as to give rise to theformation of a Φ-Ψ(n) bound state. When the gauge coupling is turned on, we argue that theneutral fermion couples chirally to the massive vector boson state in the confinement phase.If this scenario is correct, it would have been the first time that a chirally coupled fermionhas been detected on the lattice.

This result is also different from the previous findings in thestrong coupling symmetric phase (PMS) of the fermion-Higgs models with Wilson-Yukawacoupling in four dimensions where the coupling of the neutral fermion to the bosonic boundsstates vanishes presumably as a power of the lattice spacing [13].

AcknowledgementsWe would like to thank J. Jers´ak for reading the manuscript and for many helpful comments.We have benefitted from discussions with A. Bochkarev, M.F.L. Golterman, D.N.

Petcherand J. Vink. E. F. and A.K.

D. thank H.A. Kastrup for his continuous support.

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