By ilikeafrica in arXiv — 19 Nov 2025

Chiral Fermions from Lattice Boundaries

이 논문은 5차원 래그랑지안을 사용하여 4차원 초대칭 관념을 구현할 수 있는 새로운 방법을 제시한다. 이 방식은 카플란의 도메인.wall fermion 모델에 기반을 두고 있다.

카플란 모델은 5차원으로 확장된 4차원 초대칭 관념을 이용하여 4차원 초대칭 관념을 구현할 수 있는 방법을 제시한다. 그러나 카플란 모델에는 몇 가지 기술적인 문제점이 있다. 이 논문에서는 이러한 문제를 해결하기 위한 새로운 방법을 제시한다.

5차원 래그랑지안에 기반한 4차원 초대칭 관념을 구현할 수 있는 방식은 다음과 같다.

1. 5차원 래그랑지안을 사용하여 4차원 초대칭 관념을 구현한다.
2. 도메인.wall fermion 모델의 개념을 사용하여 4차원 초대칭 관念을 구현한다.
3. 새로운 propagator를 제시한다.

이 방식은 카플란 모델보다 몇 가지 기술적인 이점이 있다.

1. 분석적 표현이 더 단순하다.
2. 숫자 시뮬레이션에서 같은 정확도를 얻기 위해 5차원 래그랑지안의 사이트 수를 반으로 줄일 수 있다.
3. 새로운 propagator는 카플란 모델의 propagate보다 간단하다.

이 논문은 다음과 같이 요약할 수 있다.

* 4차원 초대칭 관념을 구현하기 위한 새로운 방법을 제시한다.
* 이 방식은 카플란 도메인.wall fermion 모델에 기반을 두고 있다.
* 새로운 propagator를 제시하고, 기술적인 문제점을 해결한다.

Chiral Fermions from Lattice Boundaries

arXiv:hep-lat/9303005v1 10 Mar 1993WIS–93/20/FEB–PHChiral Fermions from Lattice BoundariesbyYigal ShamirDepartment of PhysicsWeizmann Institute of Science, Rehovot 76100, ISRAELABSTRACTWe construct a model in which four dimensional chiral fermions ariseon the boundaries of a ﬁve dimensional lattice with free boundary con-ditions in the ﬁfth direction. The physical content is similar to Kaplan’smodel of domain wall fermions, yet the present construction has severaltechnical advantages.

We discuss some aspects of perturbation theory, aswell as possible applications of the model both for lattice QCD and forthe on-going attempts to construct a lattice chiral gauge theory.email: ftshamir@weizmann.bitnet

1.IntroductionThe Electro-Weak sector of the Standard Model is a chiral gauge theory. Thisfact motivated numerous attempts to construct a lattice model whose continuum limitwill be a chiral gauge theory [1-5].

(See ref. [6] for a review).The basic stumbling block which was recognized already in Wilson’s originalwork [7] is the doubling problem.

Following earlier work by Karsten and Smit [8],Nielsen and Ninomiya [9] proved that species doubling (and hence a vector-like spec-trum) is an unavoidable consequence of any free fermion lattice hamiltonian whichsatisﬁes some mild assumptions.Thus, attempts of construct chiral lattice gaugetheories fall into two classes. One approach is to obtains a chiral spectrum at treelevel by giving up one of the assumptions necessary of the validity of the no-go theo-rem1 [1,2].

Typically, the assumption which is violated is that the dispersion relationhas a continuous ﬁrst derivative. In the second approach the quadratic hamiltonianhas a vector-like spectrum, and one tries to eliminate the doublers dynamically byintroducing some strong interactions which are more eﬀective for the doublers [3-5].However, until now, no lattice gauge model whose continuum limit is consistent withthe requirements of Lorentz invariance, unitarity etc.

has been shown to have a chiralspectrum [10-12].Recently, Kaplan [13] proposed that a four dimensional, chiral lattice gauge theorymay be constructed if one starts with a theory of massive Dirac fermions in ﬁvedimensions, provided the fermion mass has the shape of a domain wall. Restrictedto the ﬁfth direction, the free fermionic hamiltonian has a zero mode with deﬁnitechirality localized on the domain wall.

From the point of view of the four dimensionaldomain wall, this zero mode is a chiral fermion.For a range of values of the Dirac mass M and the Wilson parameter r, thechiral fermion has no doublers if the ﬁfth direction is inﬁnite. But when the ﬁfthdirection is taken to be ﬁnite a doubler appears [13-15].

For example, if we chooseperiodic boundary conditions an anti-domain wall must appear somewhere, and thechiral fermion on the anti-domain wall has the opposite chirality.As long as one considers the coupling of the fermions to an external gauge ﬁeld,this gauge ﬁeld can be taken to be ﬁve dimensional. But at the dynamical level,the introduction of a ﬁve dimensional gauge ﬁeld requires one to deal with a host ofhighly non-trivial issues before the existence of a non-trivial continuum limit of anysort can be established.

We will not elaborate on this possibility in this paper.1 Historically, the work of Drell, Weinstein and Yankielowicz [1] preceded and, to a large extent,motivated the work of refs. [8] and [9].2

The approach we adopt is to consider the ﬁfth direction as a sophisticated ﬂavourspace, and not as a full ﬂedged additional space coordinate. The interacting theoryis deﬁned by coupling the fermions to a four dimensional gauge ﬁeld [16,17].

Thismeans that the link variables satisfy Uµ(x, s) = Uµ(x), µ = 1, . .

. , 4, independentlyof s, that U5(x, s) = 1 and that the gauge ﬁeld action is any four dimensional latticeaction which reduces to the standard continuum action in the classical continuumlimit.

Here xµ are the usual four dimensional coordinates and s denotes the ﬁfthdirection. In the case of a ﬁnite lattice we denote the number of sites in the ﬁfthdirection by N.The ﬁrst question is, are we dealing with a chiral or a vector-like lattice the-ory at tree level.

The Nielsen-Ninomiya theorem asserts that, assuming (a) complexﬁeld formulation, (b) hermiticity of the hamiltonian and (c) continuous ﬁrst deriva-tive of the dispersion relation, the massless spectrum of free fermions on a regularlattice must consist of an equal number of positive helicity (“right handed”) andnegative helicity (“left handed”) Weyl fermion. (One can replace assumption (c) bythe suﬃcient conditions that (c1) the number of fermionic degrees of freedom per(four dimensional) site is ﬁnite and (c2) the hamiltonian has a short range.

Theseconditions ensure that the only possible singularities in the dispersion relation arelevel crossings). Moreover, (d) given any charge which is exactly conserved, locallydeﬁned and has discrete eigenvalues, the equality of the number of left handed andright handed fermions holds in every charged sector separately.At the level of a free fermion theory, the relevant charges in Kaplan’s modelare the generators of the non-abelian symmetry that we intend to gauge.If theﬁfth direction is inﬁnite, the tree level spectrum can be chiral because assumption(c1) is violated.

Speciﬁcally, for a ﬁxed four momentum one has an eﬀective onedimensional lattice problem, whose spectrum consists of a continuous part as well aspossible bound states. When the energy of the bound state reaches the continuumthreshold, the bound state disappears from the spectrum, giving rise to a singularityin the dispersion relation [13-16].Narayanan and Neuberger [16] considered the introduction of gauge ﬁelds di-rectly in the “inﬁnite ﬁfth direction setting”.

To set up perturbation theory, theydeﬁne a four dimensional current Jµ(x) =P∞s=−∞jµ(x, s). Here jµ(x, s) stands forthe ﬁrst four components of the ﬁve dimensional current.

Apart from an unimportantsubstruction related to the presence of an inﬁnite number of heavy ﬂavours, pertur-bation theory is deﬁned in the usual way in terms of correlators of any ﬁnite numberof currents.As long as one is interested in perturbative results one can work directly with3

the Feynmann rules of ref. [16].

But, in order to properly deﬁne the model at thenon-perturbative level, one must consider a sequence of lattice theories with a ﬁniteﬁfth direction such that, in the limit N →∞, the Feynmann rules of ref. [16] arereproduced in weak coupling perturbation theory.

Since condition (c1) in now fulﬁlled,if the charged fermions spectrum is to remain chiral in these ﬁnite lattice models,another assumption of the Nielsen-Ninomiya theorem must be violated. Assumptions(a) and (c2) are inherent properties of the domain wall model.

Consequently, eitherhermiticity of the hamiltonian or tree level gauge invariance (or both) will have tobe sacriﬁced. Either way unitarity is violated, as pointed out already by the authorsof ref.

[16]. For this approach to succeed one must construct a speciﬁc sequence oflattice models with the above properties, and show that all violations of unitaritytend to zero in the limit (provided the spectrum is anomaly free in the usual sense).If one insists on hermiticity and gauge invariance of the ﬁnite lattice action, allthe conditions of the Nielsen-Ninomiya theorem are fulﬁlled and so the lattice theoryis vector-like at tree level [13,14,17].

The question then is whether one can decouplethe doublers dynamically and achieve a chiral spectrum in the continuum limit of theinteracting theory. The hope is that the speciﬁcs of this model will be suﬃcientlydiﬀerent from previous models which have failed to produce a chiral spectrum [3-5].Whether or not this is the case is the subject of on-going investigations.A characteristic property of all versions of the domain wall model is that as thenumber of sites in the ﬁfth direction grows, all correlation functions tend to a limitlike e−N.

The limiting behaviour is therefore achieved for relatively small N. But itremains to be seen whether a consistent chiral gauge theory can be achieved in thecontinuum limit of some version of the model.In this paper we have little to say regarding the diﬃcult question of maintainingsimultaneously a chiral spectrum and a consistent, interacting gauge theory in thecontinuum limit. Our purpose is to eliminate some unessential technical complicationspresent in Kaplan’s original model, as well as to continue the study of the model atthe level of perturbation theory.In the ﬁrst part of this paper we construct a ﬁve dimensional model in which fourdimensional chiral fermions arise on the boundaries of a ﬁve dimensional slab withfree boundary conditions in the ﬁfth direction.

As in the domain wall model, if theﬁfth direction is (semi)-inﬁnite there is a single chiral fermion on the four dimensionalboundary. But when the ﬁfth direction is taken to be ﬁnite, a doubler appears on theother boundary.It is well known that the boundary of a sample can have its own dynamics,described by an eﬀective ﬁeld theory in one less dimension.

In particular, one ﬁnds4

an intimate relation between Chern-Simons terms in odd dimensions and anomalies ineven dimensions [18,19]. This relation plays a central role in Kaplan’s model [13,20].These ideas have also found interesting applications in the context of the QuantumHall Eﬀect [21].The boundary fermions model is really a variant of Kaplan’s model.

In fact, acareful examination of refs. [13-15] reveals that the zero mode’s spectrum is alwaysdetermined by requiring normalizability of the wave function on the side where M andr have the same sign.

On the other side of the domain wall this condition is alwaysfulﬁlled. If we “discard” that side of the wall we arrive at the boundary fermionmodel.The present construction has several technical advantages which merits its dis-cussion separately.

Analytical expressions, in particular the propagator, take a muchsimpler form in the boundary fermion scheme. In the domain wall model, the prop-agator for an inﬁnite ﬁfth direction was calculated in ref.

[16].Here we give thepropagator both for the semi-inﬁnite and the ﬁnite lattice cases. Also, in numeri-cal simulations one should obtain the same accuracy in the boundary fermion modelby taking half as many sites in the ﬁfth direction.

This is simply because the ﬁfthdirection must extend only on one side of the four dimensional boundary.In the second part of this paper we discuss the introduction of a four dimensionalgauge ﬁeld. We ﬁrst consider the use of the model for lattice QCD.

When slightlymodiﬁed, the model contains an additional parameter m which in QCD plays the roleof the current mass. Unlike the case of Wilson fermions, we show that perturbativecorrection to the quark mass are proportional to m. Consequently, chiral perturbationtheory is valid, and the model can be used to study chiral symmetry breaking in latticeQCD.We next discuss a “mirror fermion” model.

In order to construct the model weintroduce two ﬁve dimensional slabs, one for the charged fermions and one for theneutral fermions. We also introduce a charged scalar (Higgs) ﬁeld.

We show that inthe broken symmetry phase, the model can naturally lead to a large hierarchy betweenmirror fermions and ordinary fermions masses. However, there are diﬃculties, and itis not clear whether the model can have a continuum limit describing an interactingchiral theory.This paper is organized as follows.

In sect. 2 we discuss the boundary fermionmodel on a semi-inﬁnite lattice.

In sect. 3 we discuss the model on a ﬁnite lattice.

Insect. 4 we consider the introduction of gauge ﬁelds.

Sect. 5 contains some concludingremarks regarding the prospects of obtaining an interacting chiral gauge theory inthe continuum limit of domain wall or boundary fermions models.5

2.Boundary fermions on a semi-inﬁnite latticeWe begin with the free ﬁeld theory deﬁned on a semi-inﬁnite ﬁve dimensionallattice. For simplicity we work in ﬁve dimensions, but everything generalizes to otherodd dimensions as well.

We mainly work with four dimensional momentum eigenstateand disregard possible ﬁnite size eﬀects in the usual four dimensions. The propertiesof the boundary fermion model presented in this section are similar to those of thedomain wall model [13-16], and we rederive them here for the convenience of thereader.

Of course, technical detail such as the explicit form of the bound states andthe propagator are diﬀerent (and simpler) in the boundary fermion model.the action is given by L = L4 +L5 where L4 is the usual four dimensional Wilsonaction summed over all s, and L5 contains the couplings in the ﬁfth direction. Thelattice spacing is taken to be a = 1.

ExplicitlyL4 =Xx,s,µψ(x, s)γµ∂µψ(x, s) + MXx,sψ(x, s)ψ(x, s) + r2LW ,(1)LW =Xx,s,µψ(x, s)∇µ ψ(x, s) . (2)r is the Wilson parameter.

We will mainly work with r = 1 and only brieﬂy discussthe case r ̸= 1. The lattice diﬀerence operators are deﬁned by∂µψ(x, s) = 12(ψ(x + ˆµ, s) −ψ(x −ˆµ, s)) ,(3)∇µ ψ(x, s) = ψ(x + ˆµ, s) + ψ(x −ˆµ, s) −2ψ(x, s) .

(4)L5 splits into a sum over all s > 0 denoted L′5 and a boundary term L05, whereL′5 =Xx,s>0ψ(x, s)γ5∂5ψ(x, s) + r2Xx,s>0ψ(x, s)∇5 ψ(x, s) ,(5)L05 = 12Xxψ(x, 0)γ5ψ(x, 1) + r2Xxψ(x, 0)(ψ(x, 1) −2ψ(x, 0)) . (6)Notice that L05 is obtained from eq.

(5) by setting s = 0 and dropping terms containingﬁelds at the non-existing sites with s = −1.We now go to momentum eigenstates and set r = 1. The action becomesL =ZpXs,s′¯ψ(−p, s)D(s, s′; p)ψ(p, s′) ,(7)where the integral is over the Brillouin zone, andD(s, s′; p) = θ(s)θ(s′)D0(s, s′; p) .

(8)6

Here θ(s) = 1 for s ≥0 and θ(s) = 0 for s < 0. D0(s, s′; p) is the Dirac operator onan inﬁnite ﬁfth directionD0(s, s′; p) = 12(1 + γ5)δs+1,s′ + 12(1 −γ5)δs−1,s′ −(b(p) + i/¯p)δs,s′ ,(9)¯pµ = sin pµ ,b(p) = 1 −M +Xµ(1 −cos(pµ)) .

(10)We now take M to lie in the interval 0 < M < 2. (Later we will see that onemust further restrict M to lie in the interval 0 < M < 1).

The physical content ofthe model is more transparent if we consider the lattice hamiltonianH(s, s′; pk) = γ4D(s, s′; pk, p4 = 0) . (11)Notice that in the hamiltonian framework, the ﬁfth direction is really a forth spacecoordinate with a semi-inﬁnite range.

We will nevertheless keep calling it the “ﬁfthdirection”. We will show that the spectrum of H(s, s′; pk) contains a right handedWeyl fermion that lives on the space boundary.

The Weyl fermion is described bya bound state with energy E2 = ¯p2k, which H(s, s′; pk) admits for every pk in thedomain |b(pk)| < 1. Notice that, for our choice of M, the domain in which chiralfermions exist contains the origin of the Brillouin zone but no points in which someof the momentum components are equal to π.

As a result, the chiral fermion has nodoublers [13]. (This situation will change when we make the ﬁfth direction ﬁnite).The bound state wave function has the formΨR(s, pk) = U(s)(1 + γ5)ψ(pk) ,(12)where the right handed spinor ψ(pk) is a helicity eigenstate3Xj=1σj ¯pj ψ(pk) = Eψ(pk) .

(13)Substituting this into the eigenvalue equation HΨR = EΨR we ﬁnd that U(s) mustsatisfy U(s + 1) = b(pk)U(s). The normalized solution isU(s) = (1 −b2(pk))12 bs(pk) .

(14)As promised, this solution is normalizable provided |b(pk)| < 1. As |b(pk)| approachesone, the solution decreases slower and slower, until at b(pk) = 1 it becomes a contin-uum eigenstate with vanishing ﬁfth component of momentum.Before we turn to the propagator let us brieﬂy discuss the continuous spectrum.Because of complete reﬂection at the boundary, the continuum eigenstates are stand-ing waves in the ﬁfth direction.

Denoting the ﬁfth component of the momentum by7

p5, the energy of a continuum eigenstate is E2 = (b(pk) −cos p5)2 + ¯p2k + ¯p25. If thethree-momentum pk is ﬁxed, the minimal energy is obtained for p5 = 0 and is givenby min |E| = ((b(pk) −1)2 + ¯p2k)1/2.

We see that min |E| coincides with the energy ofthe bound state on the surface deﬁne by b(pk) = 1, which is recognized as the bound-ary of the domain in which the chiral fermion exists. The bound state disappearswhen its energy reaches the continuum threshold.

This completes our discussion ofthe spectrum of the lattice hamiltonian.We now return to the Dirac operator (8) of the eculidean formulation. Unlikethe Hamiltonian, the eigenstates of the euclidean Dirac operator do not have simplephysical interpretation.

The fact that the Dirac operator is complex allows funnythings to happen. In particular, the euclidean Dirac operator has a bound state onlyfor pµ = 0 [22].

On the other hand, the second order operators DD† and D†D arehermitian and non-negative, and so their spectrum is perfectly well behaved. Whenwe speak about a bound state spectrum in euclidean space we will always refer to oneof the second order operators.Inspite of the peculiar spectrum of D(s, s′; p), the fermionic propagator GF doesnot show any unexpected behaviour, because GF = D†G where G is the propagatorof the second order operator DD†.

As a ﬁrst stage towards the construction of GF,we consider the Dirac operator D0 deﬁned on an inﬁnite s direction (eq. (9)).

Goingto the second order operator D0D†0, it is easy to check that the two homogeneoussolutions are given by exp(±αs), where α is deﬁned by the positive solution of theequation [16]2 coshα(p) = 1 + b(p)2 + ¯p2b(p). (15)Notice that if b(p) has a zero then α(p) has a logarithmic singularity.

To avoid suchsingularities we must take 0 < M < 1. (There is in fact a second allowed range,given in 2n + 1 dimensions by 4n + 1 < M < 4n + 2.

In this case the chiral fermionoccurs near the corner (rather than near the origin) of the Brillouin zone [15]. Thephysics in both cases is the same, and so we always assume 0 < M < 1).

The r.h.s. ofeq.

(15) is always greater that two. This ensures that α(p) is an analytic function inthe entire Brillouin zone.

Notice that both b(p) and e−α(p) tend to 1 −M for pµ →0.The inverse G0 of the second order operator D0D†0 is given byG0(s, s′) = Be−α|s−s′| ,(16)B−1 = 2b sinhα . (17)The two chiralities are decoupled in the second order operators.

In particularDD† = 12(1 + γ5)Ω+ + 12(1 −γ5)Ω−,(18)8

where Ω+ and Ω−carry no Dirac indices. This implies a similar decomposition forthe propagatorG = 12(1 + γ5)G+ + 12(1 −γ5)G−.

(19)We begin with the construction of G−(s, s′). To this end, we apply Ω−(s′′, s) toG0(s, s′) and check what is the deviation of the result from δs′′,s′.

Since D0D†0 and Ω−contain only nearest neighbour coupling, the deviation vanishes unless s′′ = 0. Fors′′ = 0 one hasXs≥0Ω−(0, s) G0(s, s′) −δ0,s′ = Be−αs′(be−α −1) .

(20)Similarly, for the homogeneous solutions we ﬁndXs≥0Ω−(s′′, s) e±αs =0 ,s′′ > 0 ,b e∓α −1 ,s′′ = 0 . (21)Eqs.

(20) and (21) suggest that G−(s, s′) has the formG−(s, s′) = G0(s, s′) + K(s′) e−αs . (22)That is to say, each column of G−can be constructed from the corresponding col-umn of G0, plus a column vector proportional to a homogeneous solution.

In orderthat G−(s, s′) satisfy physical boundary conditions we allow only the exponentiallydecreasing solution on the r.h.s. of eq.

(22). Since the second order operators (andhence their propagators) are symmetric, we must have K(s′) = A−exp(−αs′) wherethe amplitude A−depends on pµ.

The amplitude A−is easily found using eqs. (20)and (21).

Following the same steps we construct also G+. The ﬁnal result isG±(s, s′) = G0(s, s′) + A± e−α(s+s′) ,s, s′ ≥0 ,(23)A−= B e−2α eα −bb −e−α ,(24)A+ = −B e−2α .

(25)Let us now consider the physical content of these equations. While the amplitudesB and A+ are regular for all values of pµ, the amplitude A−is singular for small pµ,A−(p) = M(2 −M)p2+ regular terms ,pµ →0 .

(26)This singularity reﬂects the existence of a bound state whose eigenvalue tends to zerofor pµ →0. The eigenvalue λ20 of this bound state can be read oﬀthe propagatoras follows.

Using eq. (14) for small pµ, the contribution of the bound state to the9

propagator is approximated by λ−20 M(2 −M)(1 −M)−s−s′.Equating this to thesecond term on the r.h.s. of eq.

(23) and using eq. (26) we ﬁnd λ20 = p2 as expected.Although this time the answer was known beforehand, we have explained this trickfor extracting a vanishingly small eigenvalue from the propagator because it will beuseful to us again later.We ﬁnally notice that the 1/p2 singularity in G−gives rise to a chiral pole in thefermionic propagatorGF=D†G=i2(1 + γ5)M(2 −M)/p(1 −M)−s−s′ + regular terms .

(27)As already mentioned in the introduction, the dispersion relation is singular on thesurface b(p) = 1 where the bound state disappears from the spectrum. Interestingly,the propagator does not show any singularity at b(p) = 1.This is because thenormalization factor of the bound state tends to zero as b(p) approaches one, andso the contribution of the bound state to the propagator vanishes for ﬁxed s ands′.

One must check, however, whether or not some sort of singularity reappears inperturbation theory when summations over the ﬁfth direction are carried out.3.Boundary fermions on a ﬁnite latticeWe now proceed to discuss the model with a ﬁnite ﬁfth direction 0 ≤s ≤N. Thecrucial diﬀerence is that now a second chiral fermion appears on the new boundaryat s = N. Not surprisingly, this fermion has the opposite chirality from the oneat s = 0.

Strictly speaking, there is a tiny mixing between the two chiral modes,which vanishes like (1 −M)N. Such exponentially small modiﬁcations will cause usno concern and henceforth we neglect them.The propagator can be constructed using the same method as before. This isconvenient because the information about the low energy excitations is encoded inthe singularities of the propagator, and, in any event, what is needed to developperturbation theory is the propagator.Denoting quantities that belong to the ﬁnite lattice model by a hat, the Diracoperator is now given byˆD(s, s′) = θ(N −s)θ(N −s′)D(s, s′) .

(28)Before we actually construct the propagator, it is useful to consider a certain gener-alization of the model. We notice that if we start from the Dirac operator deﬁned on10

a ﬁnite ﬁfth direction which has the topology of a circle, then ˆD can be obtained bycutting the link connecting the sites s = 0 and s = N. Now, when M is constant,using a ﬁfth direction with the topology of a circle (or, equivalently, imposing periodicboundary conditions) gives rise to no low energy excitations. Therefore, if startingfrom the Dirac operator of eq.

(28) we gradually turn on the link connecting the sitess = 0 and s = N, we expect that the two Weyl fermions will form a Dirac fermionwhose mass is proportional to the strength of this link.We are therefore lead to consider the Dirac operatorˆD(s, s′; m) = ˆD(s, s′) + m2 (1 −γ5)δs,1δs′,N + m2 (1 + γ5)δs,Nδs′,1 . (29)As we will see, up to a constant, m indeed plays the role of a Dirac mass for thelight fermions.

Moreover, unlike the case of Wislon fermions, perturbative correctionwill always be proportional to m, i.e. chiral perturbation theory is valid.

Thus, themodel can be used to study chiral symmetry breaking in lattice QCD. (Whether thisadvantage merits the extra trouble involved in going to a ﬁve dimensional setting isa practical question that we will not address here).We now proceed to construct the propagator ˆGF of the Dirac operator ˆD for ageneral value of m. The propagator for the massless case will be obtained by simplysetting m = 0.

As before, ˆGF = ˆD† ˆG where ˆG is the propagator of the second orderoperator ˆD ˆD†.The two chiralities again decouple in the second order operators.Furthermore, we now haveˆΩ+(s, s′) = ˆΩ−(N −s, N −s′) . (30)Thus, ˆG+(s, s′) is obtained from ˆG−(s, s′) by the replacement s, s′ →N −s, N −s′.The notation is the same as in eqs.

(18) and (19).The propagator must have the following formˆG−(s, s′)=G0(s, s′) + ˆA−e−α(s+s′) + ˆA+ e−α(2N−s−s′)+ ˆAm(e−α(N+s−s′) + e−α(N+s′−s)) . (31)This time, the deviation of Ps ˆD ˆD†(s′′, s)G0(s, s′) from δs′′,s′ vanishes except for s′′ =0 and s′′ = N. The same is true forPs ˆD ˆD†(s′′, s) exp (±αs).

Thus, in order toconstruct the s′-th column of the propagator we need a linear combination of G0(s, s′)and both of the homogeneous solutions. Taking into account the symmetry of thepropagator we arrive at eq.

(31).In solving for the s′-th column of the propagator, the (s′-dependent) coeﬃcientsof exp (±αs) are two unknowns which are determined by solving a two by two matrix11

equation. There are actually two such equations in which ˆAm appears twice, becausethe s′-dependence of the coeﬃcients can be either exp (αs′) or exp (−αs′).

ExplicitlyCˆA−ˆAm= B1 −b e−α −m2mb,(32)CˆAmˆA+= Bmb−b e−α,(33)whereC =b eα + m2 −1−mb−mbb eα. (34)the solutions areˆA−= ∆−1B(1 −m2)(eα −b) ,(35)ˆA+ = ∆−1B(1 −m2)(e−α −b) ,(36)ˆAm = 2∆−1Bbm coshα ,(37)where∆= b−1det C = eα(b eα −1) + m2(eα −b) .

(38)In the limit m = 0 we ﬁnd ˆAm = 0, ˆA−= A−and ˆA+ = A+. Notice that Cis diagonal for m = 0, which means that the contributions to the propagator fromthe two boundaries are decoupled.

(Strictly speaking, at m = 0 one is left withexponentially small oﬀ-diagonal terms in C, which can be ignored as long as pµ itselfis not exponentially small. The inﬁnite (four dimensional) volume limit can alwaysbe taken in such a way that exponentially small four momenta never occur.

All thatis needed is to take N to be slightly bigger that the logarithm of the number of sitesin the ordinary directions).The interpretation of m as current mass suggests that we should consider thelimit where both m2 and p2 are small in lattice units. Again, the only amplitudewhich is singular in this limit is ˆA−.

We can use the method described earlier toextract the smallest eigenvalue from the singular part of the propagator. The resultis|λ0|2 = p2 + m2M2(2 −M)2 .

(39)The current mass of the light Dirac fermion is therefore mM(2 −M). As promised,it is proportional to m.12

4.The interacting theoryWe now proceed to discuss the interacting theory. We ﬁrst consider the vector-likemodel, which consists of a single ﬁve dimensional slab of charged fermions, coupledto a four dimensional gauge ﬁeld as described in the introduction.

This model can beused to describe lattice QCD, and it poses no conceptual diﬃculties. Its interestingfeature is that the current mass m gets only multiplicative renormalization.Thechiral limit is therefore achieved by letting m tend to zero.

In particular, we expectthat the pion will be massless in this limit.In order to verify this picture we have to show that in the limit m →0 the Diracfermion remains massless to all orders in weak coupling perturbation theory. This, inturn, is true provided the inverse fermion propagator calculated up to n-th orderΓ(n)(s, s′; p) = ˆD(s, s′; p) + g2Σ(1)(s, s′; p) + .

. .

+ g2nΣ(n)(s, s′; p) ,(40)has one zero mode on each boundary for pµ = 0. Here Σ(k)(s, s′; p) is the k-th orderself energy.The physical reason for the stability of the zero mode is the following.

In weakcoupling perturbation theory, corrections to the tree level inverse fermion propagatorare small. Moreover, since the ﬁve dimensional fermion is massive, these correctionare exponentially suppressed as |s−s′| grows.

Thus, except for an exponentially smalleﬀect the zero modes living on the two boundaries cannot mix with each other, andso a mass term cannot develop.A discussion of the perturbative stability of the (single) zero mode has beengiven for the inﬁnite lattice case in ref. [16].

There, the stability was an immediateconsequence of the absence of additional low energy states that could mix with thezero mode. Here we extend the analysis to the ﬁnite lattice case and show that, thanksto exponential suppression of all correlations in the ﬁfth direction, the masslessnessof the light fermions is maintained in spite of the fact that the light spectrum isvector-like.Let us now examine this issue in some detail.

The gauge boson propagator is pro-portional to δs,s′, and so the one loop self energy is diagonal in s-space Σ(1)(s, s′; p) =δs,s′Σ(1)(p). The Dirac operator preserves its tree level structure and, for small pµ, theonly change is in the ﬁve dimensional mass M →M(1) = M + g2Σ(1)(p = 0).

As longas 0 < M(1) < 1 all the qualitative statements regarding the tree level spectrum andtree level propagator apply. In particular, one chiral fermion exists on each boundary.At two or higher loop level there appear non-diagonal contributions coming fromintermediate states of three or more massive fermions.

At the n-th order these contri-butions decay at least as fast as (1−M(n−1))3|s−s′|. We have to show that Γ(n)(s, s′; 0)13

has a homogeneous solution which decreases exponentially for s ≪N. This is a suﬃ-cient condition for the existence of an (approximate) bound state with exponentiallysmall energy near the boundary s = 0.

(A similar statement applies for the otherboundary).In order to see that the existence of such a homogeneous solution is stable againstany small modiﬁcation of the tree level Dirac operator, it is instructive to ﬁrst checkwhat happens if we take the Wilson parameter r to be close to, yet diﬀerent from1. Consider the Dirac operator deﬁned on an inﬁnite s-direction for r ∼1.

One caneasily check that for pµ = 0 this Dirac operator has two homogeneous solution foreach chirality. In the positive chirality sector, one solution behaves approximatelylike (1 −M)s. This is the only solution which is present for r = 1.

But for r ̸= 1there is a second solution which behaves like1−r1−Ms. When we restrict our systemto a semi-inﬁnite s-direction, the boundary term in the Dirac operator picks a linearcombination of the two solutions.

Since both solutions are normalizable for s ≥0,the result is a normalizable solution too. As r →1 the second solution tends to zero,until at r = 1 we are left only with the solution (1 −M)s.A similar mechanism works for Γ(n)(s, s′; 0).

Imagine that we turn on the elementsof Γ(n)(s, s′; 0) one diagonal at a time. If we consider only s-values which are farfrom both boundaries, the introduction of every new diagonal gives rise to a newhomogeneous solution.If the new diagonal is below the main diagonal, the newhomogeneous solution decreases for s ≥0 and so it causes no problem.

If the newdiagonal is above the main diagonal, the new homogeneous solution increases fors ≥0. But when we look for a linear combination of homogeneous solutions whichsatisﬁes the equation Ps′ Γ(n)(s, s′; 0)Ψ(s′) = 0 near s = 0, we ﬁnd that the boundaryterms in Γ(n)(s, s′; 0) allow us an additional free parameter.With the extra freeparameter we can ﬁnd a linear combination which does not include the unwantednew solution.This completes our discussion of the perturbative stability.

Of course, at the non-perturbative level we expect that chiral symmetry breaking will take place, giving riseto the usual spectrum of conﬁning theories including in particular a massless pion inthe limit m →0.We now proceed to discuss the mirror fermion model. This model in obtained bytaking two ﬁve dimensional slabs, each described at tree level by the Dirac operatorof eq.

(29). We next introduce gauge ﬁelds only on one slab.

This slab will describethe charged fermions.The other slab describes neutral fermions.The ﬁnal stepinvolves the introduction of a charged scalar ﬁeld and the addition to the action of14

a Yukawa term which couples the boundary layer s = N of the charged fermions tothe boundary layer s = 0 of the neutral fermionsL Y ukawa = yXxφ(x) ¯ψ(x, N)ψ0(x, 0) + h.c..(41)Here ψ(x, s) denotes the charged fermions and ψ0(x, s) denotes the neutral fermions.The light spectrum consists of one left handed and one right handed charged fermionswhich we denote ψR(x) and ψL(x), as well as one left handed and one right handedneutral fermions denoted ψ0R(x) and ψ0L(x). The right handed fermions arise from thetwo s = 0 boundaries and the left handed fermions arise from the s = N boundaries.We repeat this construction for every irreducible representation of the gauge group.In the broken symmetry phase we consider the Dirac fermionsψ1 =ψ0RψL,ψ2 =ψRψ0L.

(42)The mass matrix isMψ1ψ2=yvmm0ψ1ψ2. (43)Here v is the Higgs VEV.

The mass matrix M is evidently of the seesaw type. Fur-thermore, perturbative stability implies that the oﬀ-diagonal terms proportional tom are only multiplicatively renormalized as in the QCD case.

(The eﬀect of quantumcorrection on the Higgs VEV is more subtle. See e.g.

ref. [23]).

In the limit m ≪v,the mass matrix M describes a heavy fermion of mass yv and a light fermion of massm2/yv. In particular, taking the limit m →0 we ﬁnd that the massless spectrumcontains a charged right handed fermion and a neutral left handed fermion.5.ProspectsThe massless spectrum of the above mirror fermion model is therefore chiral,but the trouble is that in order to achieve this we had to break the gauge symmetryspontaneously.

If no special ﬁne tuning is made, the gauge bosons mass and the Higgsmass will be of the same order of magnitude as the mirror fermions mass. Thus, inthe low energy limit we obtain a theory of non-interacting chiral fermions.The main diﬀerence between the model described above and mirror fermion mod-els [4] based on Wilson fermions is that no ﬁne tuning is needed in order to obtainan undoubled spectrum of light fermions.

But the real question is whether one canﬁnd a version of this model such that at some point in the phase diagram one has15

simultaneously light gauge bosons and light chiral fermions. Whether models basedon domain wall or boundary fermions can do better in this regard than models basedon Wilson fermions is not clear.Certain diﬃculties pertaining to lattice models containing Higgs ﬁelds were pointedout by Banks and Dabholkar [23].

Some of them can be dealt with by going to a modelof the Eichten-Preskill type [5] or to a mixed model. Other diﬃculties are relevanteven for composite Higgs and are therefore generic.Most disturbing is the fact that in all the models discussed in the literature [3-5]there does not seem to be a clear mechanism that will distinguish between anomalousand non-anomalous theories.

This is true even for the Eichten-Preskill model [5]. Inthis model one makes sure that every global symmetry which is broken by instantoneﬀects in the continuum model will already be broken explicitly by the lattice action.But it is the dynamics which has to determine in what phase the gauge symmetrywill be realized.In the event that the undoubled spectrum is anomalous, the ’tHooft consistencycondition [24] implies that the theory must be in the Higgs phase.

This is because inthe ungauged model, the non-zero contribution of the light fermions to the anomalycan only be cancelled by a Goldstone boson. (The complete spectrum cannot giverise to an anomaly because in the underlying lattice theory the charge is exactlyconserved).

But the mechanism which induces the spontaneous breaking of globalchiral symmetries [25] does not distinguish between would-be anomalous and non-anomalous theories. Thus, it is not impossible that spontaneous symmetry breakingis a price that must always be paid in order to obtain an undoubled massless spectrum,including in theories whose undoubled spectrum is non-anomalous.

This concern isparticularly relevant since in all models one attempts to obtain a chiral massless spec-trum before the gauge interactions are turned on. Indeed, recent results [12] providestrong evidence that the Eichten-Preskill model undergoes spontaneous symmetrybreaking, and that the spectrum in vector-like throughout the entire phase diagram.The crucial issue is whether in our candidate lattice model there is a mechanismwhich forbids the existence of an interacting chiral continuum limit and which isoperative only in the absence of anomaly cancellation.

In Kaplan’s original paper itwas suggested that such a mechanism does exist in the domain wall model. Namely,in the absence of anomaly cancellation there exists a Goldstone-Wilczek current [19]away from the domain wall that should prevent the decoupling of the heavy fermionicdegrees of freedom from the light ones.The work of Narayanan and Neuberger [16] represents an attempt to exploit thismechanism at the level of perturbation theory.

But in order that this mechanism16

be operative non-perturbatively in a well deﬁned lattice model, we have to sacriﬁceeither hermiticity of the hamiltonian or tree level gauge invariance (or both). Typi-cally, the resulting unitarity violating eﬀects will arise in a region which is deep indisethe ﬁve dimensional space, far from the region which supports the chiral fermions.Consequently, most of the unitarity violating eﬀects will be formally suppressed bypositive powers of the lattice spacing.

(This statement should be true for the eﬀec-tive action of the gauge ﬁelds obtained by integrating out the fermions). The onlyunsuppressed eﬀect which cannot be cancelled by counter-terms should be the usualtriangle anomaly.

However, even if we arrange for cancellation of the usual triangleanomaly, there is a serious danger that additional, ﬁnite unitarity violating eﬀectswill survive and destroy the consistency of the model, because positive powers of thelattice spacing can be compensated by divergent loop integrals when we integrateover the gauge ﬁelds.On the other hand, if one insists on hermiticity and gauge invariance of the latticeaction we do not see how the mechanism described above can be operative. Thus,in our view, the prospects of obtaining an interacting chiral gauge theory in thecontinuum limit of models of the kind described above do not seem very promising.But more work has to be done before a deﬁnite conclusion can be reached.

At the veryleast, since these models do not require any ﬁne tuning to maintain the masslessnessof the light fermions, they can help us focus on the real issue. Namely, the feasibilityof maintaining simultaneously light gauge bosons and a light chiral spectrum.AcknowledgementsI thank A. Casher and S. Yankielowicz for discussions.

This research was sup-ported in part by the Basic Research Foundation administered by the Israel Academyof Sciences and Humanities, and by a grant from the United States – Israel BinationalScience Foundation.References[ 1 ] S.D. Drell, M. Weinstein and S. Yankielowicz, Phys.

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Chiral Fermions from Lattice Boundaries

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