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Infinitely many regulator fields for chiral fermions

arXiv:hep-lat/9212019v1 15 Dec 1992RU-92-58Infinitely many regulator fields for chiral fermionsRajamani Narayanan and Herbert NeubergerDepartment of Physics and AstronomyRutgers UniversityPiscataway, NJ 08855–0849ABSTRACT: We show that two recent independent proposals for regularizing a chiralgauge theory stem from one common trick. If the anomaly free complex representationcarried by the right handed fermi–fields is r one constructs a vector like theory with flavoredright handed fermionic matter in r + ¯r but with a mass matrix of the order of the cutoffand having an index equal to unity in an infinite dimensional flavor space.

We presenta Pauli–Villars realization of the trick that is likely to work to all orders in perturbationtheory and a lattice version which is argued to produce the correct continuum leadingorder fermionic contribution to the vacuum polarization tensor and readied for furtherperturbative checks.

INTRODUCTIONThis paper addresses the problem of regularizing a nonabelian chiral gauge theorywith right handed fermionic matter in an anomaly free complex representation of thegauge group. For definiteness we work in four dimensions and assume the model to beasymptotically free so that according to standard lore a continuum interacting theorywith no free parameters exists.

Following [1] we refer to this continuum theory as thetarget theory; a concrete example for a target theory is SO(10) with righthanded fermionsin the spinorial representation. We shall focus on a particular trick that opens up thepossibility for a gauge invariant regularization at the expense of introducing an infiniteset of fermionic regulator fields.

Two recent independent proposals [2,3], although quitedifferent in appearance, can be built around this trick.We start in Minkowski space. The gauge group index will be suppressed and plays norole in most of what follows.

To each righthanded fermion of the target theory we attacha tower consisting of righthanded and lefthanded fermionic fields labeled by a new indexthat lives in a new internal space. The total fermion number of each tower is conserved.We wish to give Dirac masses of the order of the cutoffto all the extra fields and, at thesame time, make the theory look vector like and gauge invariant.

We write the Lagrangiandensity as:L =L1 + Lg(A),L1 = i ¯ψ(/∂−ig/A)ψ + ¯ψ(MPR + M†PL)ψPR =1 + γ52,PL = 1 −γ52,Aµ = AaµT a,tr(T aT b) = δab(1)To make the theory look vector like we need to identify the internal spaces associated withthe lefthanded and righthanded fields and M becomes an operator acting in this space.The free fermionic theory will have massless righthanded particles equal in number to nR =dim(Ker(M)) and massless lefthanded particles equal in number to nL = dim(Ker(M†)).If the internal space has a finite dimension, nR = nL, and we cannot get to the targettheory. To make nR = 1 and nL = 0 we need an infinite dimensional space; a simpleexample is M = Λa where Λ is the cutoffand a is a bosonic annihilation operator.If M were a finite matrix and M′ its restriction to the subspace orthogonal to Ker(M)the parity breaking appearance of eq.

(1) could be eliminated by ψ →exp(iαγ5)ψ whereα is a hermitian matrix satisfying exp(2iα) = (M′M′†)−1/2M′. This doesn’t work in ourcase and parity is not conserved.

However, one can define U = M†(MM†)−1/2 and ifone “forgets” that U is only formally unitary* the theory looks vector like and easy toregularize. * U†U = 1 but UU† = M†(MM†)−1M = 1 −Q where Q is the projector on the zeroeigenspace of M.2

One can easily find more examples of M but one should try to keep the structureas simple as possible.Before we had M/Λ = exp(−s22 )∂s exp(s22 ) where s ∈R is theinternal space “coordinate” of the harmonic oscillator. This can be generalized to M/Λ =exp(−F(s))∂s exp(F(s)) where exp(−F(s)) is square integrable and the internal space isthe space of square integrable functions on the line.

This choice assures that the kernels ofM and its adjoint will have the correct dimensions. One can also generalize to an internalspace corresponding to a complex coordinate with M/Λ = exp(−F(z, ¯z))∂z exp(F(z, ¯z)).In the last example M represents one chirality component of a Dirac particle living intwo Euclidean dimensions in an external magnetic field and the flux of that field mustbe appropriately adjusted to get the correct number of zero modes.

The other chiralitycomponent is represented by M†.In all the examples above the internal space corresponds to additional internal dimen-sions and ultraviolet behavior might be worse than in the bare theory because there aremore single particle states at high energies. Therefore, regularization has to deal withall the original divergences and also potential new ones.

If the spectrum of MM† is dis-crete the fields can be labeled by 0, 1, 2, ... and it is natural to view the extra fields asPauli-Villars regulator fields whose masses and statistics are adjusted so that everythingis ultraviolet finite to any order in perturbation theory. If the spectrum of MM† has anunbounded continuous component it is more natural to first try to get rid of the extraultraviolet divergences induced by the internal space and this is easiest done by latticizingthe variable s and thereby bounding the spectrum of MM† from above.It is naturalthen to also latticize real space and try to get a non–perturbative definition of the targettheory with hopes of addressing such long standing vexing basic questions about chiralgauge theories as, for example, what their spectrum is.

We explore both possibilities inthis paper.We first analyze the Pauli–Villars case and show that it essentially corresponds to thesuggestion of [2]. Our point of view seems quite different from the one adopted there,to the extent we understand it; our final formulae can be made to differ only in some ofthe details and we think that the differences are insignificant.

The indications are thatthe method works for an anomaly free situation but breaks down when the gauge groupis represented by fermions in an anomalous representation. Ungauged, global nonabeliansymmetries can be preserved only if they are non–anomalous.

Next we discuss the latticecase and show that it is closely related to the proposal of [3]. Our version will differ fromthis proposal in the manner in which the fermions are coupled to the gauge fields; weshall not introduce extra gauge or other bosonic fields beyond those present in the targettheory.

Our study will be somewhat preliminary: we shall provide a necessary tool thatwas missing until now which is an explicit formula for the fermionic propagator and shallargue that the vacuum polarization induced by fermions has the right continuum limit.We hope to come back in the future with a more detailed perturbative study of the lattice3

version; as far as we can see now the outlook for this approach is better than anythingwe have seen in years, but a completely well defined construction is still lacking and morework is needed.PAULI–VILLARSWe assume that MM† and M†M have discrete nondegenerate spectra. In addition weassume the existence of a “parity” operator ˆS which anticommutes with M, is hermitianand squares to unity.

The pure gauge part is regularized in some way, unspecified exceptthat it preserves ordinary, continuous, space–time structure and gauge invariance. Forexample, a higher derivative regularization, as defined in [4], does this.

We are consideringin this subsection only perturbation theory and any loop with at least a gauge or ghostinternal line is assumed to be made finite by the extra regularization. Thus, all we needto do is to construct a gauge invariant regularization of the fermionic determinant in anarbitrary external gauge field.Not surprisingly, the overall regularization scheme willonly work if the representation of the gauge group carried by the righthanded fermions isanomaly free.The free fermion propagator is:[/p −MPR −M†PL]−1 = (/p + M†)PR1p2 −MM† + (/p + M)PL1p2 −M†M(2)We choose the statistics to be well defined for fields that are eigenstates of ˆSγ5.

In thisway the Lagrangian only couples fields of the same statistics: the /∂−/A term is unity ininternal space and couples only fields of the same handedness while the mass terms onlycouple fields of opposite handedness and opposite ˆS parity. A fermion loop is given bytracing over internal space, spinor index, gauge group index and integrating over the loopmomentum.

Statistics is taken into account by inserting at some point one statistics factorequal to −S. To get the right target theory one needs to pick the ˆS parity of the zeroeigenvector of M as unity.As an example consider the lowest order contribution to the vacuum polarization, givenby < jµ,ajν,b >= δabΠµν.

After a certain amount of algebra one obtains:Πµν(p) =Zd4k(2π)4hTr[ ˆS(1−Q2 )1/q+ −√M†Mγµ1/q−−√M†Mγν]+12tr[ 1/q+γµ1/q−γνγ5]i(3)q± = k ± p2 and Tr denotes a trace over everything while tr sums over spinor indices.The γ matrix trace in the last term is proportional to ǫµναβqα+qβ−; under k →−k theterm is odd and hence the momentum integral over it vanishes. This formal argument** The last term in eq.

(3) does not converge by power counting.4

can be easily made rigorous. Once the first term in eq.

(3) is arranged to be finite thetransversality of the vacuum polarization tensor will be assured by the usual (no longerformal) argument.To understand finiteness let us consider a slightly more general type of loop integralRd4k(2π)4I(k) withI(k) = polynomial(qj)TrhˆSnYj=11q2j −M†M(1 −Q2 )i(4)where qj = qj−1 + pj, j = 2, 3, ....n q1 = k and the pj are external momenta that sum tozero. The trace is over the internal space.

Pick the basis of the space as |0 >, |E > withM†M|0 >= 0 and M†M|E >= E|E >, E > 0. We assume that there are no degeneracies.Since ˆS commutes with M†M we have ˆS|E >= ηE|E >, ˆS|0 >= |0 > and ηE = ±1.

Afterexponentiating each of the denominators with the help of a parameter tj and rotating toEuclidean space it is clear that the question of convergence is settled by the behavior ofthe trace when all tj’s are scaled to zero. Let P tj = τ; we have to consider then thebehavior ofSum = 2XEηE exp(−Eτ) + 1(5)Because of gauge invariance and absence of anomalies we only need to tame the logarithmicdivergence of a few diagrams.

It is then sufficient to make the expression in eq. (5) vanishlinearly when τ goes to zero.

This is easily achieved with M = Λa and ˆS = (−1)a†a leadingto Sum = tanh(Λ2τ). Better control in the ultraviolet is achievable; for example one maybe interested in the determinant of /∂−/A in higher even dimensions.

An arbitrary degreeof divergence can be tamed by choosing M = Λ√aa†a and the same statistics operator asabove. Now the sum becomes θ4(0|e−Λ2τ) and the Jacobi identity ensures that it vanishesfaster than any power when τ approaches zero.

This latter choice has been made in [2].At any order we shall have to deal with one extra piece, similar to the one we discardedin the calculation of the vacuum polarization, before we can restrict our attention to sumsof the kind shown in eq. (4).At higher orders these extra terms converge by powercounting but diagrams with three and four vertices may be divergent and untamable.

Ourinvestigations indicate that if the theory is anomaly free no problems are encountered andeverything goes through as for the vacuum polarization graph.Essentially, the Pauli-Villars regularization removes all divergences in the parity con-serving part of the fermionic determinant but does not touch the parity odd part of thelogarithm of this determinant. The latter must be finite and unambiguous “by itself” with-out any help from the Pauli–Villars regulator fields and this happens only in the anomalyfree case.5

It is important to make sure that the counter terms will not affect the zero modestructure of the mass matrices; otherwise the target theory would not be attained. Thisis easily checked to leading order and turns just into a matrix generalization of the wellknown δmf ∝mf rule.

Here this works is an obvious way because the internal spaceis totally decoupled from the real space and the spinor and group indices. However, themechanism that protects the desired zero mode structure from radiative corrections is adeeper one and this becomes an essential feature when we turn to the lattice where such adecoupling cannot be achieved because of the well known “lattice fermion doubling”.

Thelattice version therefore also avoids infinite amount of fermion mass fine tuning [5].LATTICEA simple choice of the mass matrix in the lattice case is based on the “wall” realizationof M. Its continuum form is M/Λ = ∂s + f(s), f(s) = F ′(s) with f increasing monoton-ically from a negative asymptotic value to a positive one. In particular, this means thatthe internal manifold is not compact.

M will have a zero eigenfunction and M† will havenone; this feature is stable under deformations that do not change the large |s| behavior[6]: MM† and M†M have identical spectra for nonvanishing eigenvalues and therefore anysmall deformation of M cannot move the zero mode of M†M up without pairing it upwith an eigenstate of MM† of exactly the same energy; however we assume that there isa finite gap between the zero mode and the rest of the spectrum and a small deformationcannot drag down an eigenstate of MM† far enough. It is hoped that radiative correctionswill be well behaved in this sense if the theory is anomaly free.

Plugging the “wall” formof M into eq. (1) leads to the interpretation of s as a fifth dimension, the derivative part ofM and M† (i.e.

M−M †2) combining with the kinetic energy into a five dimensional Diracoperator, and the f(s) dependent part (i. e. M+M †2) becoming a variable mass term.The spectrum of MM† is unbounded and to cure this we replace inner space by aninfinite lattice in both directions. s now denotes the discrete integer labeling the siteson this internal lattice and M is replaced by a first order difference operator: Ms,s′ =δs+1,s′ −a(s)δs,s′.

If a(s) approaches a+ with |a+| < 1 when s →∞and a−with |a−| > 1when s →−∞the spectral properties we need are preserved. For definiteness we choosea0(s) = 1 −m0sign(s + 12) with 0 < m0 < 1 and denote the corresponding M by M0.Putting an Euclidean version (with hermitian γ matrices) of eq.

(1) on the lattice we6

obtain:−LE = −βXx,µℜ{tr[Uµ(x)Uν(x + µ)U†µ(x + ν)U†ν(x)]} +Xx,µ12¯ψ(x)γµ[Uµ(x)ψ(x + µ)−U†µ(x −µ)ψ(x −µ)] +Xx¯ψ(x)[PR(M(∇U)ψ)(x) + PL(M†(∇U)ψ)(x)](∇Uψ)(x) =Xµ[Uµ(x)ψ(x + µ) + U†µ(x −µ)ψ(x −µ) −2ψ(x)](6)For noninteracting fermion fields in Fourier space, M depends parametrically on the latticemomentum p (−π < pµ ≤π ) with ∇U=1 ≡∇= −4 Pµ sin2 pµ2 so that all the unwantedpotential zeros of the propagator are moved up in energy by Wilson mass terms. We haveto require that for ∇in a region close to pµ = 0 dim(Ker(M(∇))) = 1 but for ∇≤−4dim(Ker(M(∇))) = 0.

We also require that M† have no zero modes in regions around themomenta p with sin(pµ) = 0 for any µ. We need to change the asymptotic behavior in s sothat as ∇varies over the Brillouin zone the spectra of M(∇) and M†(∇) change as follows:for small momenta that satisfy −∇< h (an example for h will be provided later), M has anormalizable zero mode and the spectrum of M(∇)M†(∇) is continuous with a finite gapseparating it from zero.

On the boundary of this small momentum region, −∇= h, thespectra of M(∇)M†(∇) and M†(∇)M(∇) are identical, continuous and gapless. Outsidethe region, where −∇> h, the spectra of M(∇)M†(∇) and M†(∇)M(∇) are identical,continuous and there is a gap.All unwanted “doubler modes” of the na¨ıve fermionicaction are in this third region.

Note that the abrupt disappearance of the zero mode atthe boundary between the two regions is a necessary consequence of the generic structureof the supersymmetric quantum mechanics governing the internal space and not of thespecific choices we have made. In short, the trick in the lattice version amounts to wallinga small region of momentum space around the origin offthe dangerous areas inhabitedby doublers.

The spectrum has a nonanalyticity in momentum space as a result and onehas to check carefully for effects caused by this*.Clearly, these nonanalyticities comeabout because internal space is infinite. The divergence responsible for the singularity onthe four dimensional Fourier torus is an infrared problem from the point of view of theextra dimension.

Other approaches [7,8] that tried to use singularities fail because gaugeinvariance forces compensating singular changes in the gauge fermion vertices [9,10,11]and these introduce unwanted doublers, ghosts, Lorentz violations or uncomputable non–perturbative effects [11,12,13] into the theory. Somehow, this new trick has to avoid all ofthese traps.We know that the zero modes of M live in the vicinity of the wall in the five dimen-sional world and that the physics we are interested in is the defect dependent part of the* Of course, some singularity is needed to avoid doublers in a non–dynamical way.7

action. To eliminate five dimensional bulk effects the effective gauge interaction inducedby fermions, Seff(U, wall), is replaced by Seff(U) = Seff(U, wall) −S+eff(U)+S−eff(U)2wherethe S±eff(U) come from systems with constant mass terms equal to ±m0.The neededsubtractions can also be interpreted as representing ghost fields.

In analytical computa-tions, both in weak and in strong coupling expansions, the needed subtractions should beimplementable order by order.We make our remaining choices so that the homogeneous systems which provide thesubtractions be as symmetric as possible. We would like to have reflection positivity inthe five dimensional system in all directions in the homogeneous case and five dimensionalcubic symmetry when U = 1.

Clearly, the relevant five dimensional systems have the sameLagrangian as above, only the matrix M0 gets replaced by M± with the correspondinga0(s) replaced by a±(s) = 1 ± m0. The five dimensional symmetries we want are thenobtained by making in each case M(∇U) = Mσ + ∇U2with σ = 0, ±.

It is easy to see thatwith this choice the origin of four dimensional momentum space is correctly walled offwith0 < h = 2m0 < 2. Note that we have more or less naturally ended up in the r = 1 caseof the Wilson mass term (for notations see [14]); this case has better positivity propertiesthan cases where |r| ̸= 1 because it has a positive bounded transfer matrix for single steptranslations [15,16,17] so we should feel no need to generalize [3,18,19].The candidates for observables in the target theory are obtained as follows: Pure gaugefield operators are na¨ıvely transcribed.

Local four dimensional fermions are representedby a sum over s of the bare fermion fields with a weight that samples a finite but wideregion around s = 0. The width of the weight functions should be taken to infinity at thevery end.

Composite operators of the target theory are best constructed directly in theregularized version, for example, the currents associated with global symmetries will have asum over all s in their definition via Noether’s formula. With this definition the argumentthat correct chiral anomalies are obtained [3,19,20,21,22,23] applies.

The contributionsof the subtraction terms S±eff(U) to anomalies cancel against each other because of theopposite signs of the mass terms. One expects currents associated with anomalous globalsymmetries to be affected by the infinite sum over s in a non–trivial way, enabling thecharges to flow out to infinity and hence be nonconserved in any finite five dimensionalslab containing the defect.

This shows that is is unclear whether exact unitarity holdsif one looks only at such arbitrarily large, but still finite, slabs. A proper constructionwill define the whole theory in such a slab with boundary conditions is the s direction(the other directions are irrelevant and can be kept infinite or made finite by standardperiodic/antiperiodic boundary conditions) chosen in such a way that anomalous chargeswill be able to flow out of the system.

If the gauged group is not anomalous one wouldlike then to see that a finite limit is obtained when the slab’s thickness is taken to infinity.The choice of good boundary conditions needs care because in a finite dimensional internal8

space the zero modes will typically appear both in M and in M†; this might be avoidable ifthe boundary conditions replace M and M† by matrices that are not each other’s adjointbut this would give up the unitarity of the theory for any finite width of the slab.The advantage of the present approach is that one can set up a meaningful test of theconstruction within perturbation theory and this test can be carried out before the issue ofboundary conditions is settled since the subtractions remove large s divergences and onecan work always with an infinite internal space. As in the Pauli–Villars case we considerthe fermion contribution to the vacuum polarization; this only tests the correctness ofSeff(U) but is known to be a test failed by all previous proposals.

From the imaginarypart of the polarization tensor at small momenta one can read offthe spectrum of fermionsas seen by the gauge bosons and this is the relevant question. It is obvious that na¨ıvely themethod works because the propagator has the right structure at nearly zero momenta byconstruction.

The only way this argument can fail in perturbation theory is that there arefurther singularities in the integrand of the relevant lattice Feynman diagrams and theymake contributions to the continuum limit. If we think about the propagator in a basisof eigenstates of M†(∇)M(∇) and M(∇)M†(∇) there is a source of trouble as discussedbefore.

We aim to show now that if one represents the propagators in four dimensionallattice Fourier space and in s–space one can perform the needed subtractions easily and nosingularities beyond the “good” ones appear. In particular, none are induced by carryingout the s–space traces.The propagator is given by a formula similar to eq.

(2), now on the lattice and inEuclidean space. With ¯pµ = sin(pµ) and ˆpµ = 2 sin pµ2 we have:(−iXµγµ¯pµ + M†(ˆp))PRGR(p) + (−iXµγµ¯pµ + M(ˆp))PLGL(p)GR(p) =1Pµ ¯p2µ + M(ˆp)M†(ˆp),GL(p) =1Pµ ¯p2µ + M†(ˆp)M(ˆp)(7)GR(p)M(p) = M(p)GL(p)GR(p) and GL(p) are inverses of second order difference operators and an explicit expres-sion can be obtained by standard techniques.

Both are symmetric under the interchangeof s and s′ and for s ≥s′ they are given by[GR]ss′(p) =[AR(p) −B(p)]e−α+(p)[s+s′+2] + B(p)e−α+(p)[s−s′]for s′ ≥−1AR(p)e−α+(p)[s+1]+α−(p)[s′+1]for s ≥−1 ≥s′[AR(p) −C(p)]e−α−(p)[s+s′+2] + C(p)e−α−(p)[s−s′]for s ≤−1(8)[GL]ss′(p) =[AL(p) −B(p)]e−α+(p)[s+s′] + B(p)e−α+(p)[s−s′]for s′ ≥0AL(p)e−α+(p)s+α−(p)s′for s ≥0 ≥s′[AL(p) −C(p)]e−α−(p)[s+s′] + C(p)e−α−(p)[s−s′]for s ≤09

where0 ≤α±(p) = arccoshn12[a±(p) +1a±(p) +¯p2a±(p)]o,a±(p) = 1 ∓m0 + 12 ˆp2AR(p) =1a−(p)eα−(p) −a+(p)e−α+(p),AL(p) =1a+(p)eα+(p) −a−(p)e−α−(p)(9)B(p) =12a+(p) sinh α+(p),C(p) =12a−(p) sinh α−(p)The propagator needed for the subtraction terms are obtained by replacing all a±, α± byeither a+, α+ or a−, α−. Under each of these substitutions AR(p), AL(p), B(p) and C(p)all become equal to each other.

It is easy to see that α±(p) > 0 for all p. Subtractionsmake the sum over s, s′ converge exponentially and uniformly in the loop momentum k andexternal momentum p. All ordinary manipulations that prove the transversality of Πµν(p)and its symmetry properties Πµν(p) = Πνµ(p) = Πµν(−p) hold after the subtractions. Allvertex factors are singularity free throughout the Brillouin zone.

The only singularities in atypical integrand then come from the possible poles in the amplitudes AR(p), AL(p), B(p)and C(p). Since we have arranged for M(ˆp) to have a zero mode, there is a singularity atp = 0 in AL(p).

It is easy to see from the expressions for the amplitudes that this is theonly singularity. A simple calculation givesAL(p) = m0(4 −m20)4p2+ regular terms(10)The residue of the pole is just the normalization factor associated with the zero mode atp = 0.

The usual continuum expressions should therefore emerge. We end our perturbativeanalysis remarking that all ingredients for the calculation of Πµν(p) are explicitly presentand a meticulous calculation will make the above argument foolproof.We can also address the issue of how the approach presented here avoids the Nielsen-Ninomiya [24] theorem.

To this end, consider integrating all the fermion degrees of freedomexcept the righthanded fermion at s = 0. The kinetic energy term will be non-local andcan be extracted from the expression for the propagator at s = s′ = 0.

We find that thekinetic energy term is given by1AL(p)¯p2iXµ[γµ¯pµ]PR(11)By eq. (10) AL(p)¯p2 goes to the normalization factor of the zero mode and eq.

(11) hasthe expected zero at p = 0.The factor ¯p2 in the denominator generates fifteen polesin the Brillouin zone.Normally these poles would have generated ghost contributions[10] and destroyed the theory. But, in this approach, integrating out the fermion degrees10

of freedom also generates pure gauge terms and these cancel the contributions from theghosts. In earlier attempts to solve the problem [8], it was understood that the bgauge partof the action has to be suitably adjusted to cancel the ghost contributions.

The approachpresented here can be thought of as one way to systematically achieve the apparentlydifficult aim faced before.FINAL COMMENTSIt should be clear from this paper that there is a large amount of latitude in choosingM and the associate internal space. It is not clear at the moment whether sticking tothe “wall” situation is the best, but certainly it seems a worthwhile route to pursue, inparticular because it gives a familiar picture for where the anomalous charge deficit isgoing.

The basic trick however may be useful even if the “wall” realization turns out tofail ultimately; in its essence what the trick does is to provide the “infinite hotel” that hasbeen argued to be a necessity for the existence of genuinely chiral fermions [24].ACKNOWLEDGEMENTSThis research was supported in part by the DOE under grant # DE-FG05-90ER40559.REFERENCES1)A. Borrelli, L. Maiani, G. C. Rossi, R. Sisto, M. Testa,Nucl. Phys.

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