We present a method for implementing gauge theories of chiral fermions on the
lattice.Comment: 3 pages LaTeX, espcrc2 style file. Talk presented at LATTICE96(chiral
  gauge

A. Borrelli

A. Coste

A.A. Slavnov

D.B. Kaplan

E. Eichten

E. Witten

G. 't Hooft

G.T. Bodwin

Geoffrey T. Bodwin

H.B. Nielsen

J. Smit

J.L. Alonso

K. Funakubo

K. Jansen

K.G. Wilson

L. Maiani

L. Álvarez-Gaumé

L.H. Karsten

M. Creutz

M. Göckeler

M. Ninomiya

M.F.L. Golterman

M.J. Dugan

P. Hernández

P. Swift

R. Narayanan

S. Aoki

S. Chandrasekharan

S. Randjbar-Daemi

S. Zenkin

S.L. Adler

T. Eguchi

V. Furman

W. Bock

W. Wetzel

W.A. Bardeen

Y. Shamir

English

arXiv

The authors present a method for formulating gauge theories of chiral fermions in lattice field theory. The method makes use of a Wilson mass to remove doublers. Gauge invariance is then restored by modifying the theory in two ways: the magnitude of the fermion determinant is replaced with the square root of the determinant for a fermion with vector-like couplings to the gauge field; a double limit is taken in which the lattice spacing associated with the fermion field is taken to zero before the lattice spacing associated with the gauge field. The method applies only to theories whose fermions are in an anomaly-free representation of the gauge group. They also present a related technique for computing matrix elements of operators involving fermion fields. Although the analyses of these methods are couched in weak-coupling perturbation theory, it is argued that computational prescriptions are gauge invariant in the presence of a nonperturbative gauge-field configuration

Bodwin, G. T.

UNT (University of North Texas) Digital Library

A lattice formulation of chiral gauge theories

We present a method for implementing gauge theories of chiral fermions on the lattice. Discussed topics include: the lattice as a UV regulator, a chiral QED model, modification of the fermion determinant, large gauge-field momenta, and a non-perturbative problem

* The submitted manuscript has been authored by a contractor of the U. S. Government under contract No. W-31-104ENG-38. Accordingly, the U. S. Government retains a nonexclusive, royalty-free license m publish or reproduce the publithed form of this contribution, or allow others to do SO, for U. S. Government purposes. RECF~VEO ' SFQ t 9 1sgg ANL-HEP-CP-96-57 August, 1996 A lattice formulation of chiral gauge theories* Geoffrey T. Bodwin= aArgonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA We present a method for implementing gauge theories of chirai fermions on the lattice. A more detailed account of many of the ideas presented here and a more complete set of refer- ences can be found in Ref. [l]. 1. THE LATTICE AS A UV REGULA- TOR The essential criterion for evaluating the suit- ability of a lattice formulation of a field theory is the extent to which the lattice fulfills the r61e of a UV regulator. A good UV regulator has the following properties. (1) The noninteracting the- ory yields the correct spectrum as the regulator is removed (a + 0). (2) The interacting the- ory yields the correct action as the regulator is removed, holding field momenta fixed (and ne- glecting irrelevant operators). (3) The regulated theory respects the Ymportant" symmetries. A theory that exhibits property (2) respects the symmetries of the unregulated (continuum) the- ory as the regulator is removed, holding field mo- menta fixed. However, if momenta of order the cutoff are important, these symmetries can be vi- olated even once the regulator has been removed. The symmetry violations arise from UV-divergent amplitudes, and, hence, they correspond to short- distance effective interactions (local renormaliza- tion counterterms). There are many important symmetries. How- ever, in discussing gauge theories, we focus on the gauge symmetry because it is a powerful tool to proscribe symmetry-violating counterterms. The gauge symmetry generally allows only those coun- terterms, such as coupling-constant renormaliza- tion, that are compatible with the symmetries of the continuum theory. Note that invariance un- 'Talk prerented at the Lattice '96 Symporium, St. Louis, Miaaouri, June, 1996. supported by DOE contract W-31-109-ENG-38- der infinitesimal gauge transformations is usually sufficient to rule out gauge-variant counterterms. 2. A MODEL: CHIRAL QED As a concrete example of a lattice chiral gauge theory, let US examine four-dimensional QED with left-handed couplings to the gauge field. We take the standard plaquette action for the gauge field, and we begin with the simplest ('naive") discretization of the gauged fermion action: 1 SN = a4 ~ T ( . ) r , P & ~ [ U , , ( 4 + ( .  + .P) Z,P -U/!(. - %L)+ - %)I, (1) U,(z) E exp[iagA,(z +a,,/2)]. (2) where PRIL = (1/2)(1& 7 5 ) ,  and, as usual, This action has the well-known doubling prob- lem, which violates property (1): the fermion propagator is," = [ ( l / a )  i7,, sin(p,a)]-l (3) P E u has poles at pP = 0 and p,, = ?r/a. The doublers can be eliminated in the usual way by introducing a Wilson mass term 1 Swr = a* E3(4#+(4 - Up(4+(= + a,,) -Ut(. - a,,)Nz - 4. is,w = ( 1 / 4  C b 7 P  sin(p,,a) + WP)I--l, *+ (4) ( 5 )  Now the propagator is cc where M ( p )  = (l/a) cP[l- cos(p,a)J is the Wil- son mass. There is a pole only at p p  = 0, and the lattice action satisfies properties 1 and 2. R7lmmQN w THIS W E M f  IS WHMmD DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. DECLAIMER This repon was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, compieteness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise does not necessarily constitute or imply its endorsement, mom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. 2 However, the action does not satisfy prop- erty (3). The Wilson mass term violates the chiral symmetry, since it connects PL+ to $PR. Con- sequently, the action is not invariant under an infinitesimal gauge transformation Up( Z) + Up ( Z) -in( .)Up ( Z) - i U p  (z)A( z+ap), *(Z) --+ P + iPL~(Zc) l*(Z) ,  (6) which involves only the left-handed fermion field. The gauge variation of the action is nonzero be- cause the Wilson mass operator commutes, rather than anticommutes, with y5. The gauge variation gives a A-fermion vertex, which, in momentum space, is M = ~ ( P R  - 1)M(p) - ~ ( P L  - 1)M(p+ I ) ,  (7) where p is the fermion momentum, and I is the gauge-field momentum. Now, M ( p )  vanishes as ap2 in the limit a + 0 with p k e d .  Therefore, gauge variations can survive in the continuum limit only if p or I of order l / a  are important, i.e., only in UV-divergent expressions. 3. MODIFICATION OF THE FERMION DETERMINANT We can try to restore the gauge symmetry by modifying fermion amplitudes. Since the difficul- ties arise from the fact that 7 5  commutes with the Wilson term, let us adopt the strategy of treating 7 5  as if it anticommuted with the Wilson term. Of course, we make an error in doing this. However, the error is proportional to Wilson terms, which vanish as a + 0, unless p - 1/a. Therefore, the procedure amounts to adding local interactions (counterterms) to the theory, i.e., to modifying the UV regulator. Consider a typical fermion-loop amplitude: where the I’ ’s are gauge-field-fermion vertices and we are using a schematic notation for the fermion propagators. For even-parity terms, (those con- taining an even number Of  7 5 ’ S ) ,  we treat y5 as if it commuted with all propagators and vertices, in- cluding Wilson terms, and use the identity 75” = 1 to eliminate 7 5 % .  The even-parity part of the am- plitude is then ( 1 / 2 ) ~ r  . . . ( f ,  + itf)-1rp(f2 + itf)-lrV. ... (9) This is 1/2 the amplitude for a Dirac particle with vector-like coupling to the gauge field. Now, the vector-like amplitude is gauge invariant, pro- vided that we gauge the Wilson term. (In this case we must transform $, not just PL$.) Since detD = exp(TrInD), where TrlnD is the am- plitude, 1 / 2  the amplitude corresponds to the square root of the fermion determinant. It is easy to see that the even-parity part of the am- plitude yields the magnitude of the determinant. Therefore, we can render the magnitude of the determinant ezactly gauge invariant by replacing it with the square root of the determinant for a Wilson-Dirac particle with vector-like coupling to the gauge field. (There is a similar modification for interacting fermion propagators [l].) In the case of the odd-parity terms, this trick doesn’t work. There is always one y5 that can’t be eliminated. However, the Dirac trace of an odd number of y5’s is proportional to epvpo. There- fore, as a ---f 0 with gauge-field momenta fixed, the gauge variation is proportional to operators with dimension less than five containing epvF. These are precisely the operators that correspond to the ABJ anomaly. Thus, anomaly species cancellation guarantees the gauge symmetry for the odd-parity loop amplitudes-as long as we consider only gauge-field configurations with mo- menta much less than %/a. 4. LARGE GAUGE-FIELD MOMENTA In odd-parity contributions, gauge-field mo- menta 1 - 1/a flowing through a fermion line yield nonvanishing gauge variations, since M(p + I )  is not negligible. Gauge-field momenta I - l / a  are important in UV-divergent loops involving gauge fields. Such loops generate all local interac- tions that are consistent with the symmetries of the lattice theory. Many of the interactions are gauge variant and must be eliminated-possibly by tuning of the corresponding counterterms. Let us now discuss a method to avoid explicit tuning of counterterms. The essential idea is to introduce separate lattice spacings (cutoffs) a, for the gauge field and af for the fermion field. Then af is taken to zero before a,. The goal of such a procedure is to suppress amplitudes in which gauge-field momenta are greater in magnitude than r/ag. Then Wilson terms in propagator nu- merators and vertices can be neglected, except in divergent fermion loops, which we have already dealt with by modifying the fermion determinant and invoking anomaly cancellation. This idea is implemented as follows. One treats links on the a, lattice as the dynamical variables: the gauge-field action depends only on these links. The fermion-field action-including the interac- tions with the gauge field-resides on the af lat- tice. One obtains the links on the at lattice by interpolating the gauge fields on the ag lattice. A good interpolation must have certain prop- erties: (I) gauge covariance; (2) continuity on the af lattice, except at  boundaries of a, pla- quettes, (3) locality. Property (1) means that a gauge transformation on the a, lattice maps into a gauge transformation on the af lattice. Property (2) guarantees that the Fourier trans- form of the gauge field is suppressed as l/(agZ) for gauge-field momenta I >> l/ag. Property (3) insures that one recovers the continuum theory for la, << 1. Under these assumptions, one can prove, to all orders in perturbation theory, that, as af -, 0 with a, fixed, gauge variations van- ish as at, and deviations from the target theory vanish as a; (Ref. [1,2]). This result also holds in the presence of individual nonperturbative gauge- field configurations [l]. We assume that the lattice origins coincide and that ag/af is an integer. Along a,-lattice links U,, we take the gauge-field on the at lattice, A f ,  to be 4, = (1/%9) 1% Ugr, (10) which is consistent with gauge covariance. Away from the ag links, one obtains Aj,, by interpo- lation, using (10) as a boundary condition. For Abelian gauge groups, a simple linear interpola- tion is consistent with the properties of a good regulator. (For non-Abelian groups the interpo- lation is more complicated [2]). The links on the at lattice are then obtained from A f .  3 5. A NONPERTURBATWE PROBLEM Shamir has pointed out a potential difficulty with interpolation of the gauge field [3]. A gauge transformation on the ag lattice V,(z) + V-'(a: + a,,)U,(z)V(z) can take U through -1 when either the transformation is large or U is near -1. Then, gAjp (Eq. 10) jumps from a/a to -a/a. The jump corresponds to a change in the winding number. Hence, the corresponding gauge transformation on the interpolated field is singular. Therefore, we are faced with two alter- natives: require gauge covariance and accept that the interpolated field may be singular, or require smoothness of the interpolated field and accept that gauge covariance is violated for some gauge transformations. Let us adopt the second alternative. (In the Abelian case, we simply take the linear interpo- lation of (lo).) Then we still have invariance un- der infinitesimal gauge transformations, except at gA, = a/a. Recall that we require only invari- ance under infinitesimal transformations to pro- hibit undesirable counterterms. The issue then is whether the point gA, = a/a important in the path integral. To address this question, let us consider, for example, the situation in the gauge V,A, =.O. In this gauge V'A, = JF, so A,, - 1/a can arise only from currents (momenta) - l/a. The most divergent amplitude is the vacuum polarization. By dimensional analysis the vacuum polarization I, goes as 6,,(l/ai). However, invariance un- der infinitesimal gauge transformations reduces this to I, - (k26,, - k,k,)log(l/a,). This result can be verified by explicit calculation in the Abelian theory with the linear interpolation. Therefore, A,, grows only as log( l/a). We con- clude that the point at gA, = a/a doesn't con- tribute significantly to the path integral as a -+ 0. REFERENCES 1. G.T. Bodwin, heplat/9510002, to be pub- lished in Phys. Rev. D. 2. P. Hernindes and R. Sundrum, Nucl. Phys. B455 (1995) 287 (hepph/9506331). 3. Y. Shamir, hepIat/9509023. 

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A lattice formulation of chiral gauge theories

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