attempts to put the standard model on the lattice

Nuclear Physics B (Proc. Suppl.) 4 (1988) 451-465 451 North-Holland, Amsterdam

A T T E M P T S TO P U T T H E STANDARD MODEL ON T H E L A T T I C E

Jan SMIT

Inst i tute of Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

After a review of latt ice fermions we discuss ways of handling chiral gauge theories on the lattice with direct applicaton to the Standard Model.

This talk will concentrate on the fermionic aspects

of the ti t le subject. The problem is, of course, how to

formulate a chiral gauge theory with the lattice regu-

larization. Each method for dealing with fermions on

a latt ice has a potential bearing on the subject, so it is

useful to first review the issues involved and illustrate

these with various proposals. After that we discuss

a chiral S U ( N ) model with Wilson's fermion method

and then models using staggered fermions.

L A T T I C E F E R M I O N S

With the lattice regularization comes the phenomenon

of fermion doubling, or species doubling: a hamiltonian

or action, which is local, hermit ian, translation invari-

ant, bilinear, describes in the continuum limit an equal

number of left and right handed fermion particles (or

Weyl fields), L and R particles for short 1-3.

A familiar example is given by an action of the form

S = - ~ F ( x ) % D , ( x - Y)PL¢(Y) =: -F~h~, x y

1

with D r the 'naive ' nearest neighbor form giving the

sine function in momentum space D~,(p) = s inp, . This

action produces 8 L and 8 R particles, corresponding

0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

to the 16 zeros of Dr(p) , # = 1,2,3 ,4 , in the torus

(-~,~)'.

These theorems refer to free fermions and are not

relevant for an interacting theory unless the interac-

tions at the cutoff level are weak, as for an asymptoti-

cally free gauge theory. (Another possibility is that S

is part of an effective action.)

One might think that species doubling is a prob-

lem for put t ing the electroweak theory on the lattice,

as a right handed neutr ino seems to be lacking. How-

ever, even in the usual continuum formulation right

handed neutr ino fields with the appropriate Yukawa

couplings are quite na tura l - -analogous to the quark

fields. Sending the neutrino mass to zero decouples

the right handed neutrinos. There is no reason why

the same would not happen in the lattice formulation

in the continuum limit.

The real s tumbling block is that if in addition to

the above assumptions we add that the hamil tonian or

action is invariant under a symmetry group G, then,

if the L-particles transform in a representation (rep)

r, so do the R-particles. This excludes ehiral gauge

invariance with e.g. the L-particles only in rep r and

the R-particles only in rep r ' .

A related subject concerns anomalies 1-6. Making

452 J. Smit / Standard Model on the lattice

the symmetry group G local with an external lattice

gauge field Uu(x) is straightforward and the fermionic

path integral comes out gauge invariant (the fermionic

measure is gauge invariant). Hence, the effective gauge

action Sell(U) = Tr In~9(U) is non-anomalous. This

is related to the doubling phenomenon, as the anoma-

lies are absent by cancellation amongst the fermions

emerging in the continuum limit 5, including possible

ghosts 6 or other unwanted subjects 4. If we want to

reproduce an anomalous fermion content with corre-

sponding anomalies in Se/I(U), then the gauge symme-

try has to be broken explicitly--usually in the action r.

Summarizing,

exact invariance ¢:=¢- no anomaly (1)

explicit symmetry breaking ¢= anomalies.

The first of these statements implies a powerful test

for fermion formulations. It does not mean that it is

not possible to construct non-conserved axial currents

in a theory with exact chiral invaviance. The point is,

that axial currents corresponding to gauge fields of an

exact invariance group are necessarily non-anomalous.

This is usually due to anomaly cancellation by species

doubling. There may also be anomaly cancellation with

(possibly composite) scalar fields (see below).

Lattice fermions are like a many headed hydra mon-

ster: there are other easily overlooked aspects which

come out by examining various models. Keywords are:

gauge invariance, tuning, Lorentz invariance, non-pertur-

bative, universality and renormaIization group, decou-

pling. With tuning I mean the adjustment of parame-

ters in the action--more than is absolutely necessary--

to get a desired result. To pin down the phrase 'more

than . . . necessary' we can use as a standard Wilson's

lattice fermion method s for QCD. Here the doublers

(i.e. the particles at p # O) are removed by giving

them infinite mass by adding the mass terms

s . . . . = - +

x

~(x + au)U~(x)¢(x)] (2)

to the 'naive' form of g}. The parameter r is arbitrary

as long as it is non-zero (Wilson's choice r = 1 seems

best as it removes the doublers completely) and the

quark masses are as usual parametrized with the di-

agonal matrix M (1/2aM is the hopping parameter).

So in this restricted terminology Wilson's method for

QCD does not suffer tuning and the same holds for eu-

clidean staggered fermions 9-11. (Hamiltonian (contin-

uous real time) methods require tuning of the velocity

of light.) Sometimes one allows freely for a number of

counterterms in discussing a lattice fermion method.

It should be realized, however, that each independent

counterterm corresponds to a tunable parameter which

adds a dimension to parameter space, creating a lot of

work in e.g. numerical computations. This also holds

for finite renormalizations which would vanish in the

continuum limit of an asymptotically free theory. In

practise one often can only enter to a certain extend

into the scaling region without ever reaching the limit

(which is not necessary), and the finite renormaliza-

tions can be substantial.

Let us illustrate the 'keywords' above with explicit

attempts to avoid doubling and/or explicit chiral sym-

metry breaking. These will have to violate the assump-

tions mentioned above, so we will look in succession

at non-loeal~ non-hermitian, non-translation invariant,

J, Smit / Standard Model on the lattice 453

non-linear methods for lattice fermions.

Non-local

Straightforward attempts use a D , ( x - y) drop-

ping slowly to zero at large separations typically like

Ix,-y,1-1 and lead to a discontinuous Dr(p) in momen-

tum space, e.g. 12 p, (rood 27r) (my own trial (1972,

1973, unpublished) was s inp , /2 (mod 2rr)). Gauge

invariance forces the introduction of a string of U's

(product of link variables) from x to y. This leads

to non-locai vertex functions which are not analytic in

momentum space, giving bad results in a naive weak

coupling expansion 4'13. However, perturbation theory

is questionable 14 here, and one can argue that non-

perturbative effects cause the interaction to be effec-

tively local. This is because the string of U's decays

exponentially at large Ix-Yl (with an effective mass of a

static-dynamic fermion bound state). Hence, the dou-

bling reappears, as can be seen for example in the spec-

trum at strong coupling 15, or in the Schwinger Model 16.

Furthermore, one expects trouble with Lorentz invari-

ance and the particle interpretation because the effec-

tive gradient operator in momentum space will have

non-matching slopes at 0 and 7r (ref. 17 contains some

more discussion of this point). The effective doubling

could be aalticipated from the chiral invariance of the

massless theory in accordance with (1).

A recent 18 non-local proposal focusses on the fermion

propagator

1

49+m

(m - 49)(rn'- D 2 - ig'a.F) -1,

with a suitably defined field strenght F, 49 the 'naive'

gradient and D 2 given by

D2¢(x) = ~ [ U , ( x ) ¢ ( x + a,)

+ v , ( z - a , ) % ( x - a , ) - 2¢ (x ) ] . (3)

Because "49+ m" is non-local, the vertex functions of

perturbation theory are non-local and actually contain

fermion doubling 19'2°. The gradient operator D,(p)

contains poles which correspond to ghost particles. The

appearence of doublers is again conform (1); as men-

tioned above, this does not mean that it is not possible

to construct axial currents with the expected anomaly 21.

The model suffers furthermore from tu.uing19: the pa-

rameter rn' needs adjusting to m, and the parameter g'

needs adjusting to the gauge coupling g in the action

of the gauge field. We see here the interplay between

tuning and gauge invariance; for example, the number

2 in (3) is not very special once the U's are present,

as required by gauge invarianee. This necessity of tun-

ing in this model is similar to proposals based on the

second order formalism 22.

The non-local proposals were not satisfactory. It

should also be kept in mind that according to current

lore locality is a prerequisite for universality ---one of

the corner stones in our constructions of Relativistic

Quantum Field Theory (RQFT).

Non-hermitian

Recent developments 23'24 here are an offshoot from

the random lattice. One issue is tuning. A non-hermitian

action will need counterterms with complex coefficients

i.e. tuning. Furthermore one may expect that the de-

sired chiral symmetry of the (bilinear) action leads to

anomaly cancellation by doubling. The situation is a

bit tricky here, as non-hermitian actions, proposed to

avoid doubling, really still have doubling in a not so


obvious way 24. It should also be kept in mind that her-

miticity is thought to guarantee unitarity and stability

(i. e. the continuum limit hamiltonian has real eigen-

values bounded from below), one of the corner stones

of RQFT.

Non-translation invariant

The important example here is the random lattice 25.

One may also consider perturbations of a regular lat-

tice where the perturbations act as dynamical fields

which induce non-linear self couplings amongst the fer-

mions 23'24'26. The distinction with non-linear methods

is therefore somewhat vague. After averaging over the

randomness the theory should come out translation in-

variant. In the random lattice approach chiral sym-

metry of the action is retained. For 'free' fermions

it appears 27 that the anomalies are cancelled by com-

posite scalar fields, with dynamical symmetry breaking

giving the doublers a mass of order a -1. The associated

Nambu-Goldstone (NG) bosons get a (mass) 2 c¢ ma -~,

where m is the fermion mass. Since m is presumably

non-zero, these pseudo NG bosons are removed from

the spectrum as a --* 0. It is not clear that this nice pic-

ture stands up to adding interactions such as dynamical

gauge fields. There are indications that with Yukawa

interactions tuning sticks up its ugly head 2s, although

this may not be so severe in a non-chiral asymptoti-

cally free gauge theory.

Non-linear

In QCD with n flavors, for example, one can add

in a Wilson like fashion terms like det~ (¢LCR) to the

naive gradient 2, breaking the lattice artifact U(4n) x

U(4n) symmetry 11 (U(4n) symmetry in the hamilto-

nian formulation 1~'29) down to the wanted SU(n)L ×

SU(n)R, and arrange for dynamical symmetry break-

ing such that Wilson mass terms of order a -1 are pro-

duced. The same can be done in a more flexible manner

with a scalar field ~9'3° ¢ transforming as ¢ ---* VLCVtR

under U(nL) x U(nR):

s . . . . = - ~ ( ~ ( x ) [ m + a ~ ( x ) ] ¢ ( ~ ) } : r

+ y~[¢(x)O(x)Uu(x)¢(z + au) + h.c.] + S¢, (4) x,#*

where ¢I,(x) = ¢(x)Pn + ¢t(z)PL. The action S¢ can

be of the form

S¢ = y~[atrCt¢ + Atr(Ct¢) 2 + T(det¢ + detb*)],

with parameters chosen such that < ~(x) > (rn = 0) =

r/a. The ¢ field is an auxiliary field (it lacks a gradient

term in the action). The det¢ (or tr ln¢) terms are to

avoid the U(1) problem. Alternatively one could take

¢ E SU(n). In this non-perturbative non-linear ap-

proach the chiral properties are O.K. by construction.

But one has tunable parameters (i tmight even be use-

ful to add a gradient term for the ¢ field to help aligning

it throughout the lattice) and one has put in what one

would like to get out: the physics of spontaneous chi-

ral symmetry breaking without the U(1) problem. Af-

ter all, without gauge fields (4) is just a Nambu-Jona-

Lasinio model 31. Furthermore the strength of the axial

U(1) symmetry breaking (and the ~' mass) seems to

be a tunable quantity in this approach. These consid-

erations raise the question: what is QCD? In the light

of the Renormalization Group the answer is a kind of

tautology: any formulation that in the continuum limit

yields the correlation functions of the QCD universality

class. But how far does the arbitrariness of the formu-

lation go? (even the gauge fields are not needed--they

J. Smit / Standard Model on the lattice 455

can be generated dynamically32). The chiral gauge the-

ories to be discussed below have some similarities with

the model (4).

Miscellaneous

The number of papers on lattice fermions is vast

and I apologize for being incomplete. Many proposals

suffer from tuning, lack of Lorentz invariance or can

be expected to involve doubling in some way because

of (1) (e.g. the elegant proposal in ref. 33). The finite

element method 34 is not yet sufficiently developed to

judge its per formance- - i t would be interesting to see

in QED that its prediction for the vacuum polarization

diagram does not suffer from fermion doubling and that

there are no O(a -1) mass terms in the fermion self en-

ergy.

The Desperate'~ Method

When working on the subject of topological charge

on the lattice and the relation with the eigenvalues and

eigenfunctions of the Dirac operator, one sometimes

despairs at the strong lattice artifacts encountered with

current rather ' rough' gauge field configurations. This

may lead one to contemplate the following procedure:

first construct from a lattice gauge field with lattice dis-

tance a a continuum gauge field by some interpolation

process, then work with the fermions in the continuum.

The first step is non-trivial as one has to construct a

principal bundle a5'36. The second step is obviously non-

trivial as well, but any regularization method should

work here, including Wilson's or ' s taggered' with a lat-

tice distance b ~ 0 (at fixed a). With such a proce-

dure topological and chiral properties would be as we

know it from the continuum. Of course, it would be

extremely awkward to manage in practise and it is not

necessary to do so.

For QCD two t ime honoured methods have remained

as viable methods for lattice fermions: Wilson's and

's taggered' . Both formulations have a good transfer

matr ix sa°'37-39, don ' t suffer ' tuning' , can be analyzed

in per turbat ion theory where they have no trouble with

Lorentz invariance 4°-42 and break all or part of the ehi-

ral symmetries such that the usual anomalies emerge in

the cont inuum limit 5'Ta°'43'44. The staggered fermions

are equivalent to 'naive ' fermionsn,29--a reduced set

of the doublers are used as bona fide particles. This

is possible without problems with Lorentz invariance

because the staggered fermion lattice symmetries 4~'46

supply sufficient protection. It should be stressed, how-

ever, that both methods give Dirac fermions only if

the bare QCD gauge coupling is sufficiently small. In

the staggered case this can be checked by testing for

complete flavor symmetry restoration, as this will be

acompanied by the restoration of spin. This may be

impor tant for the pi-rho-nucleon-delta mass relations

as these are just hyperfine splittings in the old quark

model. The latt ice artifact spin-flavor breaking influ-

ences the eigenvalues of the ~9 matrix, which forces a

lower limit on the quark masses 4r. For Wilson fermions

there is a similar lower bound on M - Me, where Mc

is the critical value of M.

CHIRAL MODELS W I T H WILSON F E R M I O N S

It is straightforward to construct a lattice model of the

electroweak interactions with Wilson's fermion method

in which the fermion mass terms are generated by Yukawa

couplings with the Higgs field 17,29,3°,4s. It is not so easy


to convince oneself that the model does not suffer from

unwanted species doubling because this turns out to be

a non-per turbat ive problem. For a review, see ref. 17.

Since the model is complicated I shall describe here a

simpler class of SU(n)L models to discuss various as-

pects involved. Related work is in refs. 49,50.

Let ¢ and ~ be n-plet Dirac fields and let ¢ be an

n x n matr ix field transforming as

¢ -~ (VLPL + V . P . ) ¢ , ¢ -~ ~(VIPR + V~PL),

¢ ~ vLCV~, VL,R e SU(~)L,R.

The action S = S e + S¢ + Se¢o + S w is invariant un-

der global S U ( n ) n t ransformations and local SU(n)L

transformations, with gauge field Wu(x). The contin-

uum model looks like

Se = - f dx ¢(49PL + OPn)~b,

S¢ = - f dx [ t r (Du¢) tD ,¢ + Y(¢)],

s . e = - f dx ar¢( ¢PR + CtPn)¢,

D . = o . - i g w . ,

with the usual Yang-Mills action for the W, field. The

fermion content of the model is not anomaly free. We

shall return to this l a t e r - - th ink for the moment of the

gauge field as external. According to the usual lore

of the continuum approach we may assume V(¢) such

that dynamical symmetry breaking occurs, with non-

zero < ¢ > = v proport ional to the unit matrix, in a

suitable gauge. In the treegraph approximation the

fermion and gauge boson masses are then given by

m e = G r v , m~v = g2v2.

Try now to reproduce this on the lattice. The ac-

tions Se, Se and Sw can be latticized as usual ( 'naive'

method for Se) , but See has to produce Wilson's mass

term:

S¢¢ = - ~ G~(x)[¢(x)Pn + ct(x)PL]~(x) +

Y~ G w 2 { ¢ ( x ) [ ¢ ( x ) P n + Ct(x + au)PL]¢(x + a,) x , #

+ ~(x + a.)[¢(x + a.)PR + Ct(z)PL]¢(x)}

=: - ~ M ( ¢ ) ¢ . (5)

In the tree graph approximation, ¢ =- v gives a Wilson

mass term with

r /a = G w v , M = Gv.

The gauge boson mass is still given by m~v = g2v~, so

v has to be a physical scale. This implies that G w =

r(av) -1 blows up as a ---* 0 with r fixed (e.g. r = 1).

Since rn¢ = M - 4r /a is to be physical too we also

have G = 4Gw + m e / v --* ~ . The effective Yukawa

coupling Gy = G - 4 G w = m ¢ / v stays finite. However,

lett ing G, G w --* ~ is inconsistent with the treegraph

approximation. One expects a strong back reaction on

the effective Higgs self couplings.

One approach to the problem is first keeping the

species doublers while entering the scaling region, and

then trying to get rid of them by giving them a mass

of the order of the cutoff s'3°. This procedure works in

QCD where it leads to additional insight in the chi-

ral properties of Wilson's fermion method. One puts

r = ?a and after the continuum limit one is left with

doublers with a mass of the order of 7=. The doublers

can still be el iminated by lett ing ÷ ---* oc, which also

introduces the anomalies where we expect them. So

the region near r = 0 (~ of order AQCD) appears to be

continously connected to Wilson's value r = 1 where

the doublers have disappeared completely. However,

here in a chiral gauge theory ~ = Gwv and it is well

known that decoupling of fermions whose masses are

generated by Yukawa couplings is no simple task sl,52.


A deep aspect concerns the "triviali ty of the Yukawa

theory", similar to the "triviali ty of the ¢4 theory".

Recall that the lat ter phenomenon implies an upper

bound on the Higgs mass sa. Here the reasoning is es-

sentially the same52: We may define a renormalized

Yukawa coupling Gn by rn = Gay, where m is the

mass of a fermion we wish to decouple (e.g. a species

doubler) and v is the renormalized vacuum expecta-

tion value of the Higgs field. Triviality means that

GR --+ 0 as the cutoff is removed. Conversely, raising

Gn means lowering the cutoff. But the lat ter should

not be smaller than 3v, say, in order that cutoff effects

are small on the scale of v. Consequently there is an

upper limit to Gn, i.e. to rn/v.

These considerations suggest that we cannot get rid

of the species doublers in a chiral gauge theory after

we have allowed for their appearence. This would be

a pity, as there would be no continuity with a situa-

tion with r = Gwav = 1 which we may hope to have

no species doublers. (Another complication is that for

finite ~ the theory is not Lorentz covariant in perturba-

tion theory54). Before discussing the problem further

we have to overcome an obvious psychological stum-

bling block that has halted progress before:

What about the anomalies? The latt ice model is

gauge invariant but the fermion content of the contin-

uum model is not anomaly free. Should this not imply

anomaly cancellation by species doubling? We could

avoid this seemingly compelling implication by start ing

with an anomaly free continuum action and latticizing

this (the electroweak model with quarks and leptons

is of this type). Then the argument does not apply.

However, there is another more interesting possibility.

The gauge-Higgs system may be of an anomalous Wess-

Zumino type, such that it cancels the anomalies coming

from the fermions. It was shown a few years ago that

such actions arise naturally after decoupling fermions

from an anomaly free theory, such that the remaining

fermion content is anomalous sS. In the mean t ime such

complete decoupling has been questioned because of

the above mentioned ' t r ivial i ty ' implications 52. But the

dogma of anomaly cancellation amongst fermions need

not be val id--cancel la t ion of the fermionic anomalies

by Higgs fields is also possible. In fact, the Chiral

Schwinger Model is a two dimensional example of such

a theory 56.

We now have to deal with the problem of how to

handle a si tuation where Gwav = r = 1 (or r = 0(1)) .

One would like to get some analytical insight before

put t ing the problem on the computer. Let us first try

n --+ oo in the fermion-Higgs system, treat ing the gauge

fields as an external per turbat ion which should be rea-

sonable for small g. Wi th a potential V(¢) of the form

V(¢) = crtrCt¢ 4- A(trCt¢) 2,

the action ¢o¢(g = 0) has O(2n ~) vector symmetry,

as can be seen by expanding ¢ in a complete set of

n 2 hermit ian or thonormal matrices ~ , trAkXi = 2~m,

• I t ¢ = ~k(¢~ + zCk)Xk. As usual, one has to rescale

the couplings A = A/2n 2, G 2 = G2/n, G ~ = G ~ / n ,

g2 = ~2/n ' with ),, G, Gw and ~ fixed as n --+ ec. One

finds that the ¢ dynamics is of order n ~ and the ~b dy-

namics is of order n. This means that to leading order

rn~ = ~2~2 (~2 = v2/n)) , with ~32 determined solely by

the ¢4 theory; fermionic effects on ~2 are down by 1/n.

The fermion propagator S(p) is to leading order deter-

mined by an integral equation il lustrated in figure 1:

S(p) = [ iA. (p)% + .Ad(p)] -~,


/

< / " : <

Figure 1: In tegral equa t ion for the fermion p ropaga to r at n =oz.

&(p) - sin(p). =

1 [ M(k)2A~(k) D ( k - p ) . a~v2 Jk M(k)~ + A(k)2

At n = oo the boson p ropaga to r D(k) and the mass-

like pa r t M ( k ) of the fermion p ropaga to r have the free

form

D(k) = [y~(2 - 2cos k , ) ] - ' , $1

• ( k ) = a . ~ + r ~ ( ~ - cos k . ) t l

(here la t t ice un i t s are used for the momenta , bu t m and

~) are still demension-full ; G and Gw are t r aded for am

and r according to G = m/v+4r/a~, Gw = r/a~). The

s i tua t ion here is s imilar to the t ree g raph approxima-

t ion discussed earlier, exept t ha t Au(p) is de te rmined

by the in tegral equat ion. T he in tegra t ion region near

k = 0 behaves like a con t inuum integral equat ion, as in ~

ref. 52, wi th an effective Yukawa coupling G~. = rrt2/v 2.

The region k = O(1) has an effective Yukawa coupling

of order 1/(a252), which blows up as a ---* 0. A crude

analysis using A , ~ A s i n p , leads one to conclude t h a t

A --~ <x~ as well. This means t h a t the effective r pa-

r ame te r r,f] =_ r /A --* O, implying species doubling.

Let t ing r depend on a~) does not help; there appears

to be no sensible l imit such t h a t r , f f s tays O(1). The

prob lem is t h a t a~ --* 0. The s i tua t ion for n --* cc is

too much like the t r eegraph approx ima t ion discussed

earlier.

It is u n n a t u r a l to have Yukawa couplings approach-

ing infinity. We have to allow for av, as it appears in

r = Gwav, to be of order 1, i.e. v = O(a-1). This

tends to give m w = O(a-a), unless we can a r range can-

cellat ions wi th o ther cont r ibu t ions to row. Fermionic

con t r ibu t ions to mw also tend to be O(a -1) because

of the s t rong chiral s y m m e t r y break ing from Wilson 's

mass te rm. We may as well be radical and 'freeze' the

' rad ia l ' par t of ¢(x) ,

¢(x) = v y ( x ) , v ( z ) c s u ( n ) , v = const. ,

t r ad ing G and Gw for M and r.

Gwav = r, Gav = M.

The act ion S• takes the form

S¢ = ~ y~ a-4tr[Yt(x)U,(x)Y(x + a,) + c.e.] x , t l

+cons t . (6)

(a no t a t i on is used where ~ conta ins a factor a4). Here

comes out as ~ = a2v 2. In the following it will be

t r ea t ed as a free pa ramete r . For an ini t ial explora t ion

we choose the unitary gauge,

v ( ~ ) = 1,

such t h a t S~6 in (5) reduces to the free form (2) and

we are sure t h a t there are no species doublers at weak

gauge coupling. Trea t ing again the gauge fields as ex-

ternal , the con t r ibu t ion of S~ to rn~, is simply g2~a-2.

We now have to es t imate the fermionic con t r ibu t ion

to rn~v. To order g2 the vacuum polar iza t ion tensor

Hu~(p) follows from the d iagrams in figure 2. One eas-

ily finds


Figure 2: One loop fermion contribution to the gauge boson self interaction.

II.~(p) = c6,,~,a -2 + O(lna) .

In QCD such a c- term is forbidden by the Ward identity

s i n p . H . . ( p ) = 0. (7)

Here the c is non-zero and proport ional to r, because

r represents the strength of chiral SU(n)L symmetry

breaking in the unitary gauge. The a -2 contribution

to rn~v is g2ca-2, so we have to choose ~ as

= - c + O ( a ) .

A problem in the unitary gauge is that it is likely to

lead to non-covariant contact terms in HI,v, e.g. of the

form p2,6,`,. Since 1-I,`, has dimension a -2, its contact

terms will be the most general second degree polyno-

mial consistend with the latt ice symmetry group. In

QCD we have the Ward indentity (7 to kill such non-

covariant terms~-only the combination ~5 , , ~p p2 o -PAP,,

is allowed. In the unitary gauge (7) is changed and

there is no reason for the non-covariant terms to can-

cel.

Wi th a dynamical gauge field such non-covariant

terms will be killed again by the integration over the

gauge degrees of freedom. To incorporate this effect

keeping the gauge field external we have to restore V(x )

as dynamical variables. They are on the same footing

C> +

Figure 3: Gauge boson self energy diagrams at n I = 0¢ from the fermion-Higgs Yukawa interaction.

as the gauge degrees of freedom. To get an analytic

handle on the model we now generalize it to incorporate

n! fermion n-plats and try a large n ! expansion (as

large n gave bad results; recall also that in the Standard

Model n = 2 and n! = 4 x (number of families) >__ 12).

Integrat ing out the fermions leads to an effective theory

described by

Z(U) =/D~e s°'',

S. f j = nf{Tr ln[4~(U ) + .M(U)I + S ¢ / n l } ,

where 4~(U) is the 'naive ' lattice Dirac operator with

gauge field U.(x)PL + PR, U.(x) = e x p ( - i g a W . ( x ) ) ,

and 34(¢) and S~ are given in (5), (6). In S¢,/n! ap-

pears the coupling ~ = ~ / n ! which is kept finite as

n I --~ c~. To leading order in n ! the effect of the Higgs

field on II.~(p) is that the diagrams of figure 3 have

to be added to figure 2. As Z(U) is gauge invariant,

the Ward identity (7) now holds, II.~ should have the

covariant form

g 21-[.v(p) = 2 P.P. mw(~.~ - _7_,

+ (~.~p~ - p . p ` , ) n ( / ) ,


as a ---+ 0. The W-mass gets a finite value by taking

= - ~ + O(a), with ~ = e /n] fixed as n I ---* c~. The

series in figure 3 adds terms proportional to pu and p~

and does not affect the ~u~ terms, so it cannot bring in

species doubling ' through the back door'.

The number of parameters in the full theory looks

allright: r should be ' irrelevant ' (take r = 1), M de-

termines the fermion mass, ~ and g determine the W-

mass (as in gauge-Higgs systems, scaling for m w will

need adjust ing both ~ and g). Note that Z(U) is

gauge invariant although the fern-don content of the

model is 'anomalous, ' so 5 ' , / /wi l l generate the anoma-

lous gauged Wess-Zumino action in the scaling region

(this is because the non-abel ian anomaly comes out all

right 5,7,s~).

Let us summarize the foregoing in the form of a

recepy for put t ing chiral gauge theories on the lattice

with Wilson's fermion method:

• Couple the gauge field in the 4~ part of the fermion

action, which may be anomalous.

• introduce a mass term for the gauge field,

~_, a-4tr[Uu(x) + Uu(x)*].

The mass term for the gauge field is needed for being

able to enter the scaling region of the phase diagram.

Other wise the gauge boson mass would always be of

order a -1. The resulting action is not gauge invari-

ant. This does not matter , because in the lattice path

integral we integrate over all gauge equivalent config-

urat ions of the gauge field. Alternatively, we can in-

terpret the theory at this stage as formulated in the

uni tary gauge. The integration over the gauge degrees

of freedom can be made explicit:

• introduce the Higgs field V by a gauge transfor-

mat ion on the gauge field,

u , ( x ) -~ v(x)U,(x)V(x + a,)*,

and integrate over V. Since the 4~ part of the ac-

t ion is gauge invariant in Wilson's fermion method

we can

• remove the V field from the ~ part of the action

by a gauge transformation on the fermion field.

Now the Higgs field appears explicitly in the mass

terms.

Several anomalous fermionic representations may be

combined into a non-a~omalous representation, as in

the Standard Model. The chiral models discussed in

ref. 49 should also fit in this framework, although the

method proposed there is somewhat different. As al-

ways, we have to investigate the arbitrariness in the

formulations, i. e. check universality in the relevant

cont inuum region of the phase diagram. Furthermore,

the models are non-perturbat ive, but the quenched ap-

proximation is not good here (cf. figure 3), as it does

not allow decoupling the species doublers (it is like the

large n approximation above; see ref. 57 for numerical

evidence).

CHIRAL GAUGE THEORIES WITH STAGGERED

FERMIONS

We shall a t tempt to take the above 'recepy' for formu-

lat ing chiral gauge theories over to staggered fermions,

gauging the staggered fermion flavors 'without caring

about gauge invaviance'. There are several ways of

dealing with euclidean staggered fermions l°'3s'41,~s,~9 (the

Dirac-K£hler approach ~9 leads to staggered fermions).


Here we shall use a variant of the momen tum space

method, which has some convenient features of the

position space method without blurring the staggered

fermion latt ice symmetry. Let X(x) be the staggered

fermion field and define the matr ix field ~b(x) by

, . . ( x ) = ~ ( ' ~ ' + % , x ( ~ + b). all b

where

7~+b = 7~+b . . . . 7~+b,,

and the vector b runs over the elementary hypercube

(b = (0 ,0 ,0 ,0) , (1 ,0 ,0 ,0) . . . . . (1, 1, 1, 1); we use lattice

units here). Similarly ¢(z) is defined as

b

The free staggered fermion action can be wri t ten as

S =

1

1 Y~rh,(X)[.~(x)x( x + au ) _ i ( x + au)x(x)] ,

where ~u(x) = ( - 1 ) ='+'''+=.-~. Note that the fields

¢ ~ ( x ) are not independent. The independent field is

X(x). However for the low momentum modes the ¢ ' s

may be considered as independent and a and x may be

interpreted as spin and flavor indices. The euclidean

action has discrete shift symmetries al

X(x) -* <p(z)x(x + ha), ~(x) -~ (p(x)~(x + ap),

where ~p(x) : ( - 1 ) =p+~+'''+=', corresponding to

¢(x) - , ¢(~ + a~)-~, Z(~) -* -~[(x + ,~),

which brings out their character as a discrete flavor

t ransformations combined with a translation. We could

consider gauge transformations V(x) of the type (Tb =

1, 7u, --i7v7~, " ' , 75)

V ( x ) = e ~b(*)'b

(v¢)(=) = ¢(=) + i~b~(x + b)~

- ~ .b (x ) -c (=)¢ (x + b + ¢)%~o + . . . .

The realization of this local symmetry is clearly non-

local and an invariant action would have to be non-local

as well. For the reasons given earlier we now reject

non-local actions.

A local action, for example

1 S = - 1 -~ ~ t r [Vu(x)~(x) '~,~(x + a , )

- U,(x)t-~(x + au)Tu~b(x)] , (8)

where Uu(x) transforms as

v . (~) ~ v (=)v . ( x ) y (~ + a.),,

is not gauge invariant. This is clear after expressing it

in terms of the independent variables X(x) and "~(x),

= - ~ E ,~(x + c) s .v,tt,b,c

[~(x + b)U.b0(x)X(= + ~ + ~.)

-- "~(x + c + au)gI;bc(x)x(x)]

=: - ¢ ~ , (9)

where

u.~(=) = ltr[U.(=)~+~(~+b)*l.

Mass terms can be added in a straightforward way,

¢ M ¢ := ~4 ~ t r [ M ¢ ( x ) ¢ ( x ) ]

1 y~ K(x + b)M,¢)l(x + c), 16 =,b,c

Mb~ = l trM(7=+b)tT=+h


For U,(z ) --* 1 this model does not reduce to the usual

U(1) theory in which the phases of X and ~ are gauged.

It is possible, though, to modify the action such that

the two models become equivalent for U~(x) = 1.

We now invoke the recepy stated earlier: (9) can

be interpreted as being in the unitary gauge and it

can be made explicitly gauge invarlant by introducing

the Higgs field through a gauge transformation on the

U's. Contrary to the case of Wilson fermions, here

the 40 part of the action is non-invariant before the

introduction of the Higgs field.

The action (9) is invariant under the staggered fermion

symmetry (SFS) group provided that Uu(z) transforms

appropriately, e.g. under shifts

u, (~) --, ~ ,u , (~ + ~,)~,.

Since the SFS group guarantees Lorentz invariance in

the scaling region, this implies that the gauge group

has to be SU(4) or SO(4). The mass terms break the

SFS froup but this is expected to be only an order a

effect, as in the usual formulation of QCD. But is this

SU(4) theory equivalent to the fermion-gauge theory

in the usual formulation with a quadruplet of X fields?

The answer will depend on the phase the system hap-

pens to be in- - the Higgs phase or in the confinement

phase. To have a choice of phase we can introduce

again a mass term for the gauge field (the gauge-Higgs

action) with parameter x, and set this such that the

theory is at the confining side of the phase boundary.

Presumably the system then renormalizes towards the

confining fixed point, as in figure 7 in ref. 60 (see also

ref. 62). Only the usual parameters g and M need to

be changed appropriately as a ---* 0; ~¢ can remain fixed.

Of course, it should also be possible to adjust x such

that we can enter the scaling region in the Higgs phase.

A chiral SU(4) or SO(4) theory is obtained by sim-

ply replacing 7. by ~fu(1 - "Ys) in (8). For the Standard

Model we need the SU(2) gauge group. We could em-

bed this group in SO(4), for example by choosing as

genarators 7172, 723'3 and "Y371- This choice leads to

couplings involving the same number of links in the ~fu

part as in the ~u~'s part of the action, but one expects

tuning problems because the SFS group is broken in

such a model.

Insisting on not breaking the SFS group we can try

a mixed model in which the gauge transformations on

the fermions are realized as usual, by taking an SU(2)

doublet of X's and ~'s at every site, but where the gauge

symmetry is broken by axial couplings. For example

"~40PL¢ o( ~_tr -~(x)%U,(x) (1 - xATS)¢(x + a,)

+h.c.

~ , ( x + b)tr[(1 + ~A~5)~'+°(~'÷b) t]

~(x + b)Uu(x)x(x + c + a,) + h.c.

The parameter XA will have to be tuned to obtain

precisely left handed couplings in the scaling region.

This model uses the (conserved) staggered fermion vec-

tor current and the flavor singlet (non-conserved) ax-

ial current to construct a left handed coupling of the

fermion doublet fields with the SU(2) gauge field. In

QCD the gA renormalization of the singlet axial cur-

rent expected to be substantial at currently employed

gauge couplings 61 .

CONCLUSION

Among various fermion methods for QCD only Wil-

son's and 'staggered' have survived the stringent tests.

For the chiral gauge theories based on Wilson's method

3. Srnit / Standard Model on the lattice 463

we gave analytic evidence that these can have a finite

gauge boson mass with no species doubling. There are

interesting analogies with the chiral Schwinger model,

and with ideas about 'light out of chaos '62. In the

Standard Model we may learn more about the QCD 0-

parameter from these considerations lz. The staggered

fermion method for chiral theories appears to be lim-

ited to gauge groups containing at least SO(4), which

must be considered a lattice artifact, or suffer from

'tuning'.

It was somewhat disconcerting that most models

could not be analysed in conventional weak coupling

perturbation theory. In the conventional continuum

approach only models with anomalous fermion content

appear to be truly non-perturbative. Perhaps this is

deceptive. It could well be, though, that after elab-

orate non-perturbative calculations the physics of the

models is well described by effective actions with mod-

erately weak couplings. After all, this seems to be the

case with the ¢4 theory, which is best strongly coupled

at the cutoff level, but which turns out to be weakly

coupled in the scaling region. Perhaps the particles

we observe in nature can only have (moderately) weak

couplings--otherwise we would not see them.

Acknowledgement

This work is financially supported by the 'Stichting

voor Fundamenteel Onderzoek der Materie' (FOM)

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