Nuclear Physics B (Proc. Suppl.) 9 (1989) 579-583 579 North-Holland, Amsterdam

SOME ASPECTS OF A LATTICE FORMULATION OF CHIRAL GAUGE THEORIES

Jan SMIT

Institute of Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

We discuss some features of the spectrum of the lattice Dirac operator and report on approximate mean field and hopping expansion calculations.

INTRODUCTION We wish to emphasize here some aspects of a lattice formulation of chiral gauge theories L2, which uses Wil- son's fermion method s for decoupling the fermion dou- blers. (For an approach using staggered fermions see ref. 4.) The classical action of the models may have an anomalous fermion content 1,5 (including the fermion doublers the theories are always non-anomalous).

An illustrative class of models is given by the par- tition function

Z = ] D U D V D C D ! h exp(S),

S = Su+Scu+S~v+Svu,

g z.v

(1)

(2)

zt~I

- ~(x + a~)~.[PLU~(x) + V.]¢(x)}

=: - ~ ( u ) ¢ , (3)

zt~I

+ E ~(~(~)[V(~lP~

+ VI(x + a.)PL]¢(x + a.) + h.c.}

=: - ~ ( v ) ¢ , (4) Svtr = ~ - ~ a t r V t ( x ) U . ( x ) V ( x + a . ) + h . c . , (5)

where PR,L ---- (15= q'5)/2, U~ (x) are the usual plaquette variables for the lattice gauge field U~(x), and V(x) is a 'radially frozen' Higgs field. The gauge field has only left handed couplings to the fermions, which may transform in various irreducible representations f of the gauge group. (The label f on ¢, ¢, U~, 17, M, r, and a is suppressed in order not to clutter the nora-

tion.) formations

The action is invariant under the gauge trans-

U~(x) -* fl(x)U.(x)flt(z + a.) =: U n,

VCx) -~ nCx)V(~), ¢C~) -~ [nC~)PL + P . ] ¢ [ ~ ) , ~(~) -~ ~ ( ~ ) [ n t ( ~ ) p . + pL],

(6)

and the measure DU DV = 1-I..dU.(x) I] .dV(x) is the usual group invariant measure.

The action S is designed to reproduce the classi- cal action in the classical continuum limit. This be- comes most clear in the unitary gauge V(x) = 1. In this gauge Scv takes the form of Wilson's fermion mass term which has to remove the species doublers and Svv becomes a mass term for the gauge field. In the classi- cal continuum limit the species doublers decouple, and S takes the continuum form

4 1 Sco.t = - f d x { ~ t r F . ~ F ~ v + m~ttrA.A~

~--~[¢~/~(0. - iA .PL)¢ + ~rn¢]}, (7) !

where A. is related to U. by U. = exp(-iaA~) and m = M - 4 r / a , m2A oc g2~Za/a2" F o r m = m A = O, (7) is manifestly gauge invariant and the right handed fermions (and the hidden Higgs field) decouple.

Conversely, starting from the manifestly gauge in- variant classical action (7, m = m A - - 0), a straightfor- ward latticization with Wilson's fermion method intro- duces fermion mass terms. These break manifest gauge invariance (no Higgs field introduced yet) and therefore generate a mass for the gauge field too at the quantum level. Since one wants to be able to tune the gauge bo- son mass (and more generally to choose the system's phase), the a-terms (5) are introduced. One naturally

0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

580 J. Smit / A lattice formulation of chiral gauge theories

i , i

.._J < > Z L~ 0 k J

• ° .

. : ' • : " . . . : . • " " . . .

• • • • , . . , • - • , - , . •

°•.

• • ° ° ." °. • .

-'..:.'....:: .. :.--

• . • • .• t ••°o° °o; • °.° ° • °

• • • o ° : • "• ~ •• • ° • • • ° °

RE EIGENVALUE

Figure 1: Spectra of ~9(U) + ~ (1 ) in two dimensional U(1) gauge fields with left handed couplings• The top (bottom) spec t rum corresponds to i¢ = 0.5 (~ = 0•125)•

arrives at (3,4,5) wi th V = 1, but this is equivalent to an explicit integrat ion over V because such integra- tion is already implicit in the group space integrat ion for each link variable U~(x). This is familiar from the massive Yang-Mills model, which is a gauge-Higgs the- ory.

The gauge degrees of freedom make up the radi- ally frozen Higgs field• Since r and a M are of order 1, the Higgs field is coupled with intermediate cou- pling strength• This is a difficult s i tuation where weak or strong coupling expansions run into problems. The system is non-perturbat ive.

E I G E N V A L U E F L U C T U A T I O N S We know very lit t le about the behavior of latt ice fermions in this situation• Consider, for example, the eigenvalue spec t rum of ~(V) + ~ (1 ) , (cf. (3), (4)) shown in figure 1 for the case of gauge group U(1), with one fermion in two dimensions• The gauge field is taken from an ensemble with action (2)4(5) wi th fl = 1/(a2g 2) = 20,

-- 0.5 and 0•125. This means very weak coupling on an 82 latt ice and the gauge field is nearly pure gauge, U~,(x) ~ Vt (x)Y(x+a~,) . The gauge degrees of f reedom V are controlled by ~. If the latt ice fermions are to re- semble cont inuum ferm!ons, then one should see three clearly separated bands of eigenvalues corresponding to four fermion species (the middle band should contain

two species)• Clearly, this is not the case for ~ = 0•125. Apparantly, in this case the Higgs field V is so rough that the lattice fermion field cannot approximate a con- t inuum Dirac field•

The eigenvalue scatter of the lattice Dirac operator is somehow related to the ' roughness ' of the external gauge and Higgs fields• For smooth fields the scat ter is absent• However, the fields contr ibuting to the parti- t ion function are not smooth in general, certainly not the Higgs field for ~¢ values of order 1, al though one may expect some sort of smoothness for the transverse part of the gauge field as 1/g 2 ---* oo in (2). Ref. 6 addresses the question if the eigenvalue scatter dimin- ishes fast enough in QED2, and ref. 7 contains related studies with staggered fermions in QED2 and QCD. In QCD the phenomenon is non-perturbat ive: there are indications ~ tha t the scatter diminishes somewhat like a 2 as a --* 0, faster than any power of g2. This suggests that smoothness is not really needed, but rather small a, i.e. large correlation lengths.

C O N T I N U U M T R E A T M E N T So it is not clear if cherished notions such as the spec- t rum of the Dirac operator, zero modes, etc. make sense in the lattice formulation of chiral fermions. Why does one not have these problems in the cont inuum ap- proach? Consider, for example, F (U,V) defined by

F(U,V) = Trln[~9(V vt) + ~t(1)]

= Trln[4~V ) + ~ (V) ] . (8)

In cont inuum considerations one often evaluates the fermion determinant in external gauge and Higgs fields with some regularization procedure• This means that one considers external fields that are smooth on the regularization scale, since these fields are supposed to be fixed when the regularization is removed. Subse- quently one is supposed to integrate over the gauge and Higgs fields in the pa th integral, which somehow implies again a regularization to be removed• So ef- fectively one first removes the fermionic and then the bosonic regularization.

This can, of course, also be done on the lattice• An example of a theory where analytic calculations are done this way is the Chiral Schwinger Model s, where one simply evaluates a one fermion loop vacuum polar- ization diagram• However, for the appropriate two di- mensional version of (1) - - the Latt ice Chiral Schwinger Model - - this one loop calculation of r ( u , v ) is in- correct because of the roughness of the Higgs field• (One usually s takes ~¢ = 0.) The one loop calcula- tion can be substant ia ted s in the (vector) Schwinger

J. Smit / A lattice formulation of chiral gauge theories 581

Model where the longitudinal (gauge, Higgs) degrees of f reedom decouple and the the external momenta of the fermion l o o p ~ o r r e s p o n d i n g to the transverse degrees of f r eedom--a re suppressed by fl --~ oo. In the Chiral Schwinger Model the longitudinal degrees of f reedom (here sometimes called the 'Wess-Zumino scalar') are not decoupled and their momenta are not suppressed (i.e. they are of order l /a) , which invali- dates the way the loop calculation is done. Al though the results obtained s may be correct in form and agree with the cont inuum answer, one expects the detailed dependence of the physical boson mass on the r pa- rameter to be different.

F E R M I O N - H I G G S S Y S T E M Analytic insight in the theory is desirable and as a first step we will concentrate on the fermion-Higgs system, t reat ing A~ as external gauge field. Large n methods give some insight 4. Here we shall use a more mon- dane approach: meanfield. This method should be able to handle couplings of order 1 and give reason- able information about the theory in the broken phase (< V > ¢ 0) not too close to the phase boundary. In the unbroken phase (< V > = 0) we can only make some very qual i ta t ive observations.

The mean field method is simplest when the orig- inal action is linear in V(x) and Vt(x) at a given site x. This is the case for Svv, but not for I~(U,V), as a hopping expansion readily shows. The non-linear terms come from self intersecting hopping loops, and ignoring this will not lead to large errors in four dimen- sions. In the saddle point formulat ion of the mean field method 9, V(x) is replaced by an unconstrained field ¢(x). Of course, one should use V V t -- 1 before mak- ing the replacement V --~ ¢, V t --* ! t . For this reason the best form of F(U, V) for making the replacement is Trln[49(U v,) + ~(1)] , instead of Trln[49(U ) + M(V)]. (This is clear from the hopping expansion.)

The mean field equations take the form, for U~ -- 1 (using lat t ice units a = 1 from now on)

Ow(H,H t) v - OHt IH=ICt=h,

~hv = ~ + H,(~), (9)

where w(H, H t) -- In f dV exp tr ( H W + V t H ) is the fa-

miliar group integral and H1 corresponds to a fermion loop d iagram with one external line (see below). The fermion propagator in this leading approximat ion is ob- tained by inserting V = Vt = v in g~(1 vt) + ~4(1),

S(p) = [~ + i~(v2pL + P n ) ] - ' (10)

= ( , - lpn + p ~ ) [ ~ , - 1 + i~ ] - , ( , -~p~ + p~),

where

J~ ---- M - r ~-~c~, c~ = cosp~, s , = sinp~. (11) /a

We now specialize to gauge group SU(2) wi th n ! fermion fields in the fundamental representation. Al though the not ion of chiral fermions is not relevant for this gauge group because its representations are real up to equiv- alence, the technical problems can still be il lustrated in this case. For SU(2) the transi t ion between the bro- ken and unbroken phase appears to be of second order in the absence of fermions 9, which may be impor tant for obtaining finite vector boson masses in the broken phase. (For SU(n), n > 2, the transi t ion appears to be of first order.) The explicit form of IIl is given by

H1(,) -- '~! f" d*p ~ 4 J-~ (2~r) 4 Ai2v -2 + s 2"

B R O K E N PHASE For ~; > ~;~ the system is in the broken phase v ¢ 0. F rom (10,11) we see that the fermion masses are given by

m! = ( M - 4r)v -1,

mao~bler = ml- t -krv -1, k = 2 , 4 , 6 , 8 . (12)

It is no problem to get small ml, e.g. m! = 0. Then the doublers have masses of order 1, i.e. of order of the cutoff. For m! = 0 one finds that the critical value t% where v vanishes is given by i¢¢ = 0.125 - 0.060hi, so the effect of the fermion feedback on ~c is substan- tial. Note tha t the doubler masses (12) go to infinity as v --* 0. This is very different from the usual expec- ta t ion m ~ v for fermion masses generated by Yukawa couplings. The reason is, that v enters primarily in the g) part of the fermion propagator and not in the mass part (cf. (10)).

For the vector boson mass we find

" ~ = g2[~v 2 + Hi(v) + H2(v)], (13)

where the subscripts 1 and 2 correspond to diagrams 1 and 2 in figure 2, with

n~v = ~ v ( n l + n2), p = 0

(i.e. this H1 is identical to the one introduced in (9)). The function H2 is given by

582 J. Smit / A lattice formulation of chiral gauge theories

5LL 0 0 D 0

,,, , + I t t" SLL = 0 0 D 0

1 2

Figure 2: Diagrams for H1 and II2.

/[ d*p 4 ( 2 4 - ~')

There is no problem in get t ing rn$ --~ 0. We also would like mA ~ 0 in lat t ice units. The natural way to achieve this is to let ~--+~% such that v - -~0 . Let us take my = 0. Then Y[1 and H2 depend only on r2/v 2 and for v --+ 0,

I-[1, 2 ~ 71"1,2V2 ~

rn~t ~ g ~ ( a ~ + r , + ~ r 2 ) v 2 - + 0 .

Another possibility for gett ing rrt A ---+ 0 is by using a in (13) more explicitly like a counter term cancelling the contr ibut ion of H1 + II2. Imagine choosing r such tha t 1-[1 and 1-[2 are just constants. For rn! = 0 this can be done by taking rv -1 = 1, which gives HI = 0.0378, II2 = -0.00838, n I = 1. Assuming small v gives v 2 ~ 8111/(1 - 8.;), such that

2" l 'I1

and rnA = 0 for ~ = ~, ~ (1 + II1/H2)/8 -= -0 .44. (At this value of to, v 2 ~ -8I I2 = 0.067, which is indeed small). For ~ < ~h, m~t would come out negative in this approach, which is disturbing. Presumably, wi th a dynamical (as opposed to external) gauge field the system would go to a different ground state as ~¢ passes the appropriate ~t f rom above.

U N B R O K E N P H A S E In the unbroken phase *: < ~o where v = 0, the propa- gator (10) takes a somewhat odd form

~ - q P ~ =: So(p).

The fluctuations around v = 0 have to be taken into account to get a more reasonable fermion propagator . We shall t ry to get some qual i ta t ive information about the propagator by expanding in I/)PL. Consider

1 S ' = < ~ ( 1 ) + @PR + ~P(U')PL > = : < ¢ ' V >, (14)

, ~ 2 -". - = ~ ;

0

Figure 3: Class of diagrams for the fermion propaga- tors in the unbroken phase. The dashed line and 0 denote the Higgs and PLSoPR propagators, and D de- notes ~PL.

where U' is pure gauge, U~(x) = Vt (x )V(x + a~,), and the brackets denote the average over V. Expansion in

~gPL leads to

S' = So - So < ~PL > So + So < ~)PLSo~gPL > So

1 ~-...= .M + ~PR + Z2~PL '

where z 2 is given by z2~ - -< ~(U ' ) > , and we neglected correlations between the ~ s . This may be reasonable in four dimensions. In this approximat ion z 2 plays the role of v 2 in (10,11) and consequently the fermion mass follows as m~ = ( M - 4r)z -1.

However, this S ' is not the fermion propagator to which the gauge field couples in the usual way. With an external gauge field V., U ' in (14) reads U~(x) = V t ( x ) U u ( x ) V ( x + a.) , and U u does not couple directly to ¢ ' , but only via V. It is impor tan t to realize tha t in the unbroken phase there are two possible left handed fermion fields, which differ by a gauge t ransformation with V(x):

¢, . (~) = v ( x ) P L ¢ ' ( x ) , ~L(x ) = -C~)PRV'(x).

In the broken phase ¢/~ and ¢~ are equivalent in the mean field approximation, because then V(x) ~ v. We may interpret ¢ ' and U' as being in the unitairy gauge, the gauge where V(x) = 1 in the fermion mass t e rm )4. The propagator S = < ¢ ¢ > couples directly to the external gauge field Ug and this is the relevant propaga- tor in e.g. the vacuum polarization diagrams in figure 2. We may call ¢ the charged fermion field and ¢ ' the neutral field.

Figure 3 illustrates the two propagators by a sum- mat ion of a class of diagrams. For SLL = < ~bL¢ L >

this gives

S~L = -E~PR + ~PR~PL~.¢?R + ' " -~PR

~ - 1 _ O~'

.I. Smit/A lattice formulation of chiral gauge theories 583

where C is given by the diagram in figure 3,

-(Eq)(z,Y)PR = PL&(? Y)G(& Y)~R, (15)

G(z,y) = < V(z)V+(y) > .

In momentum space SLL(p) can only approach the con-

tinuum form -ip/p” f o a massless lefthanded fermion

if C-r(p) vanishes at least like p2 as p + 0. This means

that C (2, y) in coordinate space has to have long range.

From (15) we find

-$pR~(q) = /_; $*G(p + q)

-i$&z -1 q--+O, - 47rvq2

where we represented the long distance behavior of G

by G(P) N z/(m& + P”), and put rnH and rn> equal

to zero. So this calculation indeed gives a C-‘(p) van-

ishing like p2. The Higgs particle has to be massless.

But if our theory is to reproduce effectively the mass-

less classical action (7) in the anomaly free case, then

the Higgs particle should be unobservable: it should

decouple from the physical fermion. This is indeed the

case for the couplings effective in figure 3, within the

aproximations made.

It is not clear that the approximations used above

give a reasonable indication of the physics. For in-

stance, one would expect that in a more elaborate cal-

culation it matters if the theory has an anomalous clas-

sical fermion content or not, as the Higgs field has to

cancel the fermionic anomalies. Furthermore, replac-

ing Sc in (15) by S’ (why not sum more diagrams?)

would mean replacing M2 in (16) by M2 + z2s2, leading

to C-’ c( l/ lnp2. Then effectively all fermions would

be heavy. Anyhow, we infer that if a charged massless

left handed fermion exists at all in the broken phase,

this only happens for rnH = 0, i.e. K. -+ IE, from below.

CONCLUSION

We reasoned that Higgs field has to have long range

correlations if lattice artefacts in the spectrum of the

Dirac operator are to disappear. The fermion-Higgs

system was studied in more detail (external gauge field

approximation). Prospects are favorable in the bro-

ken phase, as the correlations have infinite range there.

The unbroken phase was difficult to analyse and it is not clear if our crude approximation, in which we found

the wanted massles charged fermions with decoupling

massless Higgs fields, is even qualitatively reliable.

There was no problem in getting fermion doubler

masses of the order of the cutoff. Triviality does not

seem to be important here, perhaps because such heavy doublers have nothing to do with scaling. Anomalies

did not influence our simple calculations here in four di-

mensions. Their non-pertubative effect is presumably

very interesting in models with an anomalous classical

action. However, the non-anomalous case with mass-

less fermions is also very important. One would like

very much to be able to do non-perturbative compu-

tations with such theories in the unbroken (confining)

phase.

We did not investigate more closely the violation

of rotation invariance encountered in diagrams’. We

assumed this to be taken care of by gauge invariance

and universality in the scaling region, but the details of

this based on gauge invariance are incorrect, however”.

ACKNOWLEDGEMENT

I would like to thank J.C. Vink for his spectrum pro- gram and interesting comments. This work is fman-

cially supported by the ‘Stichting voor Fundamenteel

Onderzoek der Materie (FOM)‘.

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

L.H. Karsten, in Field Theoretical Methods in Particle Physics, ed. W. Riihl, Plenum (1980) (Kaiserslautern 1979). 3. Smit, Acta Physica Polonica B17 (1986) 531.

P.D.V. Swift, Phys. Lett. 145B (1984) 256.

K.G. Wilson, New Phenomena in Subnuclear Physics, ed. A. Zichichi, Plenum (1977), Erice Lectures 1975.

J. Smit, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 451 (Seillac 1987).

S. Aoki, Phys. Rev. Lett. 60 (1988) 2109. K. Funakubo and T. Kashiwa, Phys. Rev. Lett. 60 (1988) 2133. T.D. Kieu, D. Sen and S.-S. Xue, Phys. Rev. Lett. 61 (1988) 282.

J.C. Vink, Nucl. Phys. B307 (1988) 549.

J. Smit and J.C. Vink, Nucl. Phys. B303 (1988) 557. J. C. Vink, Phys. Lett. 210B (1988) 211.

G.T. Bodwin and E.V. Kovacs, Phys. Rev. 35 (1987) 3198.

J.-M. Drouffe and J.-B. Zuber, Phys. Rep. 102 (1983) 1. A. Guha and SC. Lee, Nucl. Phys. B240 [FSI2] (1984) 141.