standard model and chiral gauge theories on the lattice

Nuclear Physics B (Pros . Suppl .) 17 (1990) 3-16North-Holland

STANDARD MODEL AND CHIRAL GAUGE THEORIES ON THE LATTICE

Jan SMIT

Institute of Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The NetherlandsA review is given of developments in lattice formulations of chiral gauge theories . There is nowevidence that the unwanted fermion doublers can be decoupled satisfactorily by giving them massesof the order of the cutoff.

INTRODUCTIONThe lattice is an intuitively reasonable regulator whichenables us to give a simple but rigorous definition ofquantum field theories and study their properties bynon-perturbatioe methods . The method can presum-ably be used for any quantum field theory, but latticeformulations of chiral gauge theories run into complica-tions which are related to the well known phenomenonof `fermion doubling' (for reviews see'refs . 1, 2) . Sincethe Standard Model of our world is a chiral gauge the-ory, it is important to overcome these complications .One also wants to study the properties of extensions ofthe Standard Model, such as Grand Unified Theories,asymptotically free chiral gauge theories without Higgsfields in the classical action (AFCGT for short) (sec forexample ref. 3), and `anomalous' chiral gauge theoriesin which the classical fermion content is anomalous.

The chiral gauge theory models discussed here areinspired on Wilson's fermion method4 for dealing withthe ferinion doublers (for an approach using staggeredfermions see ref. 2 .)

WILSON'S FERMION METHOD IN QCDFor Wilson fermions, the mass terms in the fermion

action

can be written asSF = -V;(P+ .")1,

" rA

where r is the parameter governing the strength of Wil-son's nearest neighbor coupling mass term and Ad ap-pearing in the single site mass term is related to Wil-son's hopping parameter K by 2M = 1/K . The actionabove leads to a free (U, --" 1) ferinion propagator

S(p) = [i- y, sinp,, + M - rEcos p,,]-1 ,w

from which we cari read off the fermion mass mF and

0920-5632/90/$3 .50 © Elsevier Science Publishers B.V .North-Holland

the doubler masses m; as

3

mF = IVl - 4r, mn = mF -1- 2rn, n=1-4,

(2)

where n is the number of components of four rr~omen-tum p equal to 7r . The parameter r is chosen 0 < r < 1(the prefered choice being r = 1) and in the contin-uum limit llrl -~ 4r such that mF -} 0 but the doublermasses stay of order of the cutoff* . For Mn = O(1) theparticle rest energy may differ substantially from m,.but it is useful to call the parameters n1F, mn masses .

The heavy doublers cause chiral symmetry break-ing . There are two ways of looking at this5 .6 :

1 . One may interpret mo = lhl - 4r as the baremass in weak coupling perturbation theory (WCPT)and rewrite the mass terms as

Omo~ + ~D,, ?k D,,O,

(3)

where D, is a covariant derivative in the classical con-tinuum limit . The explicit chiral symmetry breakingof the r term causes a linear divergence in the ferinionmass which has to be canceled by a counterterm bnzo .Although the usual chiral properties of Green functionsare recovered this way in the continuum limit, it is hardto understand spontaneous chiral symmetry breakingfrom this interpretation .

2 . At M = r = 0 there is spontaneous symme-try breaking U(4Nf) x U(4Nf) -t U(4Nf), at least at

strong coupling? (Nf is the number of flavors) . Thesymmetry group and its breakdown depend on latticedetails, for example in the hamiltonian formulation wehaves U(4Nf) --i U(2Nf) x U(2Nf) . However, if weturn on the r parameter and at the same time give M acritical value 1V1 = lvljr, g), then the symmetry breaksfurther down explicitly to U(Nf) such that the onlyremaining Goldstone bosons are the expected Nf - 1pions . From this point of view it is natural to definethe bare current quark mass by mo = M - M,(r, g) .

When extending the theory to include the electro-weak interactions in which the fermion masses are gen-erated by Yukawa COUplings to the Higgs field, there

'We use lattice units, the cutoff is O(1) .

are two corresponding approaches:1 . Generate only :mpz, of (3) by Yukawa couplings .

This approach leads to the introduction of counter-terms to absorb the chiral symmetry breaking corningfrom the Wilson mass term8,9,10 or the explicit intro-duction of mirror fermionsl1 .

2 . Generate all fermion mass terms (including theWilson r term) by Yukawa couplings 1.5,12,13 . This lat-ter approach is attractive from the symmetry point ofview and will now be described first .

MINIMAL FORMULATION OF THE STANDARDMODELIt is straightforward to construct the action

of the minimal lattice Standard Modell "13 . The puregauge field part is given by

with

S = S1 -}- S2 + S3 + Squark +Slepton +SH,

SI,

2nETrU,,',,n= 1,2,3,.,uv

UJAx)

=

exp( - i61'6"r ) "

Al91

3

k 'rU,~

=

exp(-iW,4r ),jVA _ EWk _k, 02 - 21

k=1 2 92

U4=)_

where the U M are the plaquette fields made out o-"the link variables U, ; B �,_, W, and G,,T are the U(1),SU(2) and SU(3) gauge âelds . The factor 1/6 in frontof B, is to avoid fractional powers of U,(,') in the cou-plings to quarks which have fractional hypercharge .

Under a gauge transformation the quark fields trans-form as (PR,L = (1 f -y5) , 2)

~Lr

U,L= =

UR

i1

6exp(-2G,), Gur = F, Gk,,, 2

133 =T32

,k=1

Y~r~b r

(nL

R

"~~

,,~~nxPL

~r PR ) Y'r - `Nr-4

~`r ( Qm PR + SZRtPL)

exp(iw1 2L) exp(iw2) exp(iL03 ),YL= 3'exp(iwlZL) exp(iw1 2) exp(iw3),

in self explanatory notation. According to these gaugetransformations it is useful to introduce the left andright link variables

exp(-iB,,r L) exp(-z'W,,) exp(-z'G,),

(4

exp(-zB, YL2 ) exp(-iB,

T32 ) exp(-zG,,r),

J. Smit/Standard model and chiral gauge theories

transforming as

ULArURKr

--~

qL UL f?Lt

= ULnr

..r r+j' -

;'rUR QRt

= URnr

Ar r+w -

~cr .

Then the 4) part of the quark action can be written as

r~e

-

r+K Y, JL(UAXPL + URt PR) r~ .

For the construction of the mass-like part V,V0we use the Higgs field doublet cp = (Vu, <Pd), or moreconveniently its matrix version

__

'Pd

~Pu

=

V E SU-Wu ~d

PV,

(2"" ),

in which the radial part pr =

I~ur ~2 + IZ.l2is gauge

invariant and the angular part Vr transforms as

Vr --, StL Vr S2rt .

(6)

Taking the SU(2)L invariant combinations~~PL?Gr , TMPROr as building blocks,

_ r,(TxPR0.TG0r + h.c .)r

- E ( ~f'rPROrÜR Gwtkr+r, + h.c .

`rr+ï.PRY'r+ 'URtGi~'~Gr + h .c .) .

(7)

is gauge invariant (the notation h.c . was defined in ref.1) . The second and third line above transform intoeach other under a 180° rotation . G and Gw are Yu-kawa coupling matrices which act on the flavor indicesof the quark fields and commute with r3 ; G parame-trizes the ordinary single site Yukawa couplings andcorresponds to :1 in (1), Gw parametrizes the Wilson-Yukawa couplings and corresponds to r .

In QCD the physics is independent of r in the sca-ling region and r is taken flavor independent . Assumingthat the situation is similar here we choose Ow to bea positive number (i .e . Gw oc 11) (this choice is sim-pler than ref. l's) . We take here G to have the isospinform Gquark = D(1 -I-,r3)/2 + P(1 - ,r3)/2 in which Dis a positive diagonal matrix and P a positive hermi-tian matrix, each acting only on the family index ofthe quark fields . Diagonalization of P produces theKobayashi-Moskawa matrix. It can be argued that theinduced QCD 9-angle is zerol .

The lepton action S(,,ton is similar in form to thequark action, leaving out the gluon fields in URL, with

YL = -1, and D - P to put possible family mixinginto the neutrino sector .

Finally, in terms of the customary lattice parame-trization with a scaled variable <Pr, the Higgs fields partSH of the action can be written as

SH = ~~I 1Tr(4Px+rW 4P=Ul= + h.c .)2

- ~, 2Tr[~b,~x + a( .1)x~r - n)"~,x

0_

=

f iD ., .

. (8)

The gluon fields drop out of SH .At the classical level the continuum hinit is easy .

The parameters in SH are arranged such that (0x)(0)11 0 0, m2y = g2(0)=/2, and the quadratic part ofthe fermion action follows by substituting 0 -+ (¢),which leads to the Wilson form (1) with hl = G(O),r = Gw(0) . In the continuum limit (0) --} 0, r = O(1),so Gw - oo . Since mF/(¢) = G - 4Gw is to be keptfixed, also G -, oo . However, the physical Yukawacoupling yR Oc 171F/(Q} stays finite and we obtain theusual classical continuum action of the Standard Modelin all respects .

Note that we could introduce global isospin break-ing in SH with a n: of the form n: + h'TS, inserted atthe position of UR in (8) . The effect of n,' disappearshowever in the classical continuum limit .

At the quartuni level complications arise. Dowe really have to let C, Gw -+ oc. in order to removethe doublers? This would lead to large `loop correc-tions' . To avoid G, G'iy - oo, can we arrange for (¢s )of O(1) in the fermion mass terms? But then how toavoid mw = O(1)?' . Instead of (Os) = O(1) we can

' Karsten and I made an effort to investigate this problemwithin WCPT in 1979 and 1981 and failed . At that time webelieved that it should be possible to handle the lattice StandardModel by WCPT as in the continuum and we concluded that thefermion doublers could not be removed from the physical massspectrum . (The Wilson-Yukawa coupling method was formulatedfor a U(1) modell2 and the full minimal lattice standard model in-cluding the consequence for the induced lattice QCD B-parameterwas formulated somewhat later, but only the implied electroweakcurrents were recorded in the literature5 .)A puzzle that strengthened our negative conclusion at that time

was that the lattice formulation does not care about anomalies .For example, we can consider S = S1 + Sa + S(epton + SN, whichis anomalous at the classical level . Since it was common lorethat anomalies have to be canceled amongst fermions - in thiscase obviously by the doublers - this was another reason forconcluding that the doublers could not be removed at the quantumlevel .Two developments changed this picture around 1984,1985 .

First, simulations of the SU(2) gauge-Higgs model with a radi-ally frozen Higgs field and the associated non-linear sigma model

J. Smit/Standard model and chiral gauge theories 5

take ¢x = O(1), which leads to the formulation with aradially frozen Higgs fieldl . l , p= = v,corresponding to A -+ oo in (8) . In this version of themodel the problem of G, Gw --> oo looks fictitious : Inthe unitary gauge

the mass-like part of the fermion action takes exactlythe Wilson form (1),

M = G-\/_x , r = Gw Vt_c , UA= = U,= .

Hence Gw and G presumably stay O(1) as we enter thescaling region, since one expects that r may be takennxed O(1), and that M -a ltlc = O(1), n: --> nz = O(1).

The formulation with the radially frozen Higgs fieldis natural, since by triviality the bare Higgs self cou-pling A has to approach oo if one tries to keep therenormalized coupling AR fixed upon entering the sca-ling region .

An alternative derivation of the Wilson-Yukawa eou .plings (7) starts from the free Wilson lnass terms andconsiders their behavior under gauge transformations,

4" r Nh41_

OMPRs2=Ts2R~til , r + h.c .,

r`f'xWxtw

-, r ~(PRn=tQR N, + ~RtQ=;4PL)Ostr-

From the single site coupling we identify the radiallyfrozen Higgs field as the gauge degrees of freedom,

nLt2R = Vr.

showed that such supposedly non-renormalizable models can behandled very well by non-perturbative methods and give remov-able physics . This suggested that the lattice standard modelmight still be satisfactory if treated non-perturbatively Second,it was shown that anomalies can also be canceled by Wess-Zuminocouplings amongst Higgs fields, which arise after removing fermi-ons -from an anomaly-free representation by making them veryheavyl4 . The anomaly puzzle was resolved .The Zakopane review l presented the model in full and discussed

the induced QC'D 8-parameter, stressed that the 'anomalous' caseis a natural possibility, discussed its non- pert urbative characterand arrived at the version with a radially frozen Higgs field . Thismodel was constructed independently in ref . 13 (following a sug-gestion ofD.J . Wallace) . A WCPT investigation 15 then presentednegative conclusions, but the Seillac review l stressed again thatthe model is non-perturbative .At that time the Wilson-Yukawa coupling method was

used 16,17 .18 for the study of 'anomalous' theories, specifically thechiral Schwinger Model (without taking into account the non-

perturbative aspect'"o ) . In this context the radially frozenHiggs fields are often called Wess-Zumino scalars . These fieldsare identicall 9 (possibly differing only in the representation ofthe gauge group) : the gauge degrees of freedom In the context ofmassive Yang-Mills models the Higgs fields are sometimes calledStuekelberg fields .

6

For the nearest neighbor coupling we then see two al-ternative ways of obtaining gauge invariance using U,ior UR,

-_

V=SZStSZRa ---, V.UR,

nzt ÇIx+KVr+a -' UuL,sVs+~ .

The first way (9) corresponds to (7) . The second way(10) would lead instead to a Wilson term of the form

r1G=(PRURVS+- + Vt URPL)Or+A + h.c .,

(11)

which looks more complicated as it does contain thenon-abelian SU(2) gauge field and was therefore notchosen in ref. 1 . A better reason for the choice (7)is that it leads to the GP symmetry to be discussedbelow.

For simplicity we shall continue with the radiallyfrozen Higgs field formulation and use the M, r nota-tion . For the moment let us note the primary functionof the parameters of the model :

250GeV mHVR ~ cutoff '

VR 'M controlls mF,

r is 'arbitrary'91, 92, 93

controll

mw, a, a.,

(12)

rc controlls

where vR2 = GF%12, with GF the Fermi decay constant .It Hill turn out that it is necessary to limit the arbi-trariness of r to a domain sufficiently far from r = 0 .

W-BOSON MASSAn important question that played a role in early con-siderations is if raising the masses of the ferinion dou-blers to O(1) also leads to mw = O(1) . This questioncan be answered in the external gauge field approxi-mation to the fermion-Higgs system . The renormalizedgauge couplings 981,2 are small and the bare gauge cou-plings 91 , 2 are not very different. since the cutoff is finiteassuming triviality. Treating the gauge fields as pertur-bations to the ferinion-Higgs system is therefore a rea-sonable simplifying approximation . In this approxima-tion vR is the renormalized vacuum expectation valueof the Higgs field and we can derive the relation'

2

2 2m w =14 92 VR Y

Hence, when Z'R - 0 in the scaling region, one ex-pects the same for mw even in the full theory withdynamical gauge fields . This mass relation was used

'The subscript 'R' indicates a 'renormalized quantity' in thecontinuum normalization convention .


in formulating the implications of triviality of the O(4)model for the gauge-Higgs system2l (for a review, seeref. 22) . Its derivation in the case of fermions will nowbe sketched .

Consider the fermion-Higgs theory. In standardfashion one can derive by a change of variables in thepath integral looking like an SU(2)L gauge transforma-tion the following Ward-Takahashi relations

( ~,' ,j,k,,ziTr(~v - ~y) 2)

_

where jûx is the SU(2)L current anda.,=The contribution of the intermediate Goldstone bosonstates ape) leads in the usual way22 to an expressionfor the Goldstone boson decay constant F,

iFp,~bke

Z*bkr

1~(V4n + 2(PkTk),

Z,,bkc = (gejWk10),

(oljlupla~e),

F

relation rrtyy = g?F2 .

4bsybk1(Tr .(0s + TS))f

-4bsybkc(Y'rY's)~

(IPa)vR -

VZ-1,T

(Pe~~si~Ys 2~s~0)~

The W mass formula now follows immediately from the

GOLTERMAN-PETCHER SYMMETRYIn the continuum approach it is trivial to see thatthe right handed neutrinos decouple when the neutrinomasses vanish . On the lattice this is not true since theright handed neutrino fields are always coupled, butone expects this to become true in the scaling region forM(") -+ 1h1~~~ . Recently it has been shown that the de-sired decoupling of the massless right handed neutrinoindeed takes place in the minimal Standard Mode123 .The reason is that for the neutrinos UR -+ 1, whichhas the consequence that the parts of the action con-taining the right handed neutrino fields 0(" ) = PRO("),R1 = 7P ~ 1PL, can be written in the form

r

PR(,Ra ++

0.v.

~R

IsLVs 71'1,pV)R + h.c .,

s

_,).OR) + h.c .

where mo = Mc") - 4r . This shows that for mo = 0the right handed neutrino fields have purely derivative

couplings . As a consequence, for mo = 0 the action isinvariant under the transformations

where E and Z are constant two-spinors . From this in-variance it is possible to show that the neutrino massmF) = 0 when mo = 0 and that 0R) and OR") decou-ple from gauge invariant correlation functions23 . Notethat the symmetry would not hold had we chosen thealternative Wilson-Yukawa coupling (11) .

The Golterman-Petcher (GP) symmetry is very use-ful since it predicts the critical value of a fermionichopping parameter (i .e . M,(") = 4r) . The symmetryholds in many models based on left handed gauge fieldcouplings and in many fermion-Higgs models .

NON-MINIMAL STANDARD MODELSThe lattice Standard Model given above is minimal inthe sense that it uses the same number of fermion fieldsas in the continuum action (taking for granted that thelatter also contains right handed neutrino fields) . Non-minimal models have more fermion fields .

In the mirror fermion model of ref . 11 the gaugegroup is assumed to be SU(2)L x SU(2)R and in addi-tion to 0 and T explicit mirror fermion fields X and Xare introduced transforming as Xr -+ (SZRPL+SZX PR)Xxetc . (compare with (4)) . This enables gauge invariantmass terms of the type

l~+GxXr 'r - Z ~ ?1,r(URPR + ULrPL )V'x+ï, + h.c .,k

which do not need a Higgs field . The r and fLp, cou-plings are strong but only quadratic and serve to givethe fermion doublers 0(1) masses . In addition to thesemass terms the model has the usual Yukawa couplingwith the Higgs field which need not be strong . Mirrorfermions may occur in Naturell , but considered as anon-minimal regularization of the Standard Model thisSU(2)L x SU(2)R model needs `tuning" of the param-eter y,yx .

In the continuum approach to chiral gauge theo-ries one often uses the convention of taking all fer-mion fields left handed . A recipe? for putting chiraltheories on the lattice, in which OR and TR are non-physical and introduced only for the purpose of regular-ization, fits into this convention . Let us see where thisleads to when applied to a construction of the Stan-dard Model . The physical right handed fields will be

"Tuning' a more tuning than in a formulation with the mini-mal possible tuning2 .


denoted by XR and XR . We could take them left handedby the usual charge conjugation transformation, butit is more convenient to keep them right handed andcomplete them with their unphysical (regularization in-duced) left handed partners XL and XL into Dirac fieldsX = XR + XL, X = XR + XL . The X field is similar tothat of ref . 11 but the interpretation and the ensuingmodel below are different .

In the unitary gauge the lepton action for examplecan be written as

where

SF =

-E1~x(U,sPL + PR>s+~ + h.c.

(13)sw

-

1 Xr(URPR + PL)X ., +i, + h.c .xu.

+ 2r Xr(Xx+u. + Xr-~ - 2Xx),rg

- V'MOPRX - XTnOPLO

We have written M = mo -f- 4r in the Wilson terms,because the model has GP symmetry and 1VI, = 4r .The Higgs fields can be identified by making a gaugetransformation

L

,0 _r , Lt

Xx

-'

(PL + fiRPR)Xr, ÎZr -' Xx(PR + HRtpL),

which transforms the mass terms intor2~

"~'~.PRt.L(0x+ ; + ikr-W - 2Vh= ) .+. h .c .

x,,

- E(TZPRVxMO%r + X.rMOvtpL(15)

Vx = f2LtQR, VR = SiRt, V=L = SZTt .

Note the appearance of two independent Higgs fields,e.g . Vx and V.R . Clearly, in this non-minimal modelthere is a danger of spontaneous breakdown of thewhole symmetry group SU(2) x U(1), such that thephoton would become massive. Presumably this canbe avoided by introducing corresponding `kinetic en-ergy' terms for the Higgs fields, e.g .

Tr2 [nVtURV=+aURt

+ nlVRtUR VRAX

X AX X+A

+ ~;zVLt[ IV L ; + h.c . ], (16)

8

and tune the rc's to get the desired symmetry breaking .This means `tuning' and we conclude that a generalrecipe as given in ref. 2 is not always satisfactory ; it isbetter to stay with a minimal set of fermion fields .

Another non-minimal mode19 -10 similar to thoseabove is the `type l' model obtained from (15) byVL,R -, 1 in the Wilson terms. it also needs 'tun-ing' .

A MINIMAL SO(10) MODELAs is well known, one family fits nicely in the 16 di-mensional spinor representation of SO(10) and can bedescribed by a left handed fermion field _ VL . Witha Higgs field V transforming in the 10 = (16 x 16) .,representation we can construct a lattice model of theschematic form

SF =-VTCll* - M E VTCVr ,x

r+ rr (l

.`E[V~sCVU"..V,.+îu-- s.

+ JpsCULVx+ü.,0r+ü + lt .c .J,

where C is the charge conjugation matrix and T de-notes transposition . The model is essentially given inref. 3 : using a radially non-frozen Higgs field and inte-grating over it with a single site gaussian weight wouldproduce a four fermion interaction of the type consi-dered there . The model can of course be specialized tosubgroups of SO(10) . e .g . SU(5) . It does not have GPsymmetry.

MASSIVE YANG-MILLS MODELExperience from the massive Yang-Mills theory aliasthe gauge-Higgs model is very useful for understandingnon-perturbative aspects of chiral lattice gauge theo-ries . The action o£ the massive SU(2) Yang-Mills modelis given by

S(U) -92

TrU,,, � , + n

Tr(U'. + U,l~),

(17)

In the classical continuum limit (using physical unitswith lattice distance a, 17, = exp(-ial4,, Y )),

rtiETr(U,,,. + U2=) =

(18)

-

f d4x[äz Ti- 11' .W~ - 1~ Tr 14" W, ,W� 1V,,),

w

from which we identify m2 = g'te/a'- . This impliesthat in the classical continuum limit we should takerti - 0, which has the benefit that the 0(1,G'4) terinswhich break rotational invariance go to zero.


However, rc --+ 0 is far from the r£ -+ oo required(together with g` --> 0) for the validity of WCPT as asaddle point expansion in the quantum path integral .But r£ -r oo is far from the values of r<; required forthe quantum continuum limit, r. - te, ,: 0.30, as willbe recalled below, so the (n,g2) region of validity ofWCPT is also far from the physical region .

It has been known for a long time that (17) is justthe unitary gauge version of the gauge-Higgs actionwith radially frozen Higgs field . On the lattice theequivalence is easy to demonstrate using the gauge in-variance and normalization f DQ = 1 of the Haar mea-sure :

JDU es(U)O(U) «= DUes(Ll")O(U" )

DU DSI es(Un)O(Un),

where O is an observable and SX is the Higgs-field Vxin the gauge-Higgs action

S(Uv') = I,, F Tr U,,,, r +KE Tr (VtU,,-,Vy+4, + It .c . ) .9` sfW

rIA

We learn from this equivalence that. the breaking ofgauge symmetry of the action (1i) is only apparent,that the gauge degrees of freedom are radially frozenHiggs fields, and that this is unavoidable with dynaini-cal gauge fields and measure DU.

One expects three relevant scaling regions inthis model, indicated by (A), (B) and (C) in the famil-iar phase diagram in figure 1 . In all regions ga --+ 0 . Re-gion (A), r;, \, r;,,, has a triplet of massive gauge bosonsand a massive Higgs particle . Region (B), n j' h,, hasconfining forces binding the scalar particles into `sca-laroniums'_ In region (C) h is not tuned but fixed atsome arbitrary value strictly smaller than n, such thatthe scalar particles have O(1) masses and decouple, re-sulting in pure SU(2) gauge theory with only gluehallparticles .

GAUGEDEGREES OF FREEDOM = HIGGS FIELDThe idea of identifying the Higgs field with the gaugedegrees of freedom can be used to make any actiongauge invarianc2 . Apparent gauge non-invariant termsin the action pick up gauge degrees of freedom after agauge transformation, so these gauge degrees of free-dom become dynamical variables in the path integral- the Higgs fields . This assumes that we integrateover all gauge fields U with the gauge invariant mea-sure DU = DUn, such that we effectively also integrateover all S2_- Vt . The non-invariant form of the ac-tion is then interpreted as being written in the unitary

C

Figure 1 : Phase diagram of the massive Yang-Millsmodel . The first order phase transition line separatingthe Higgs from the confining phase ends at 1/g2oo inthe second order point t:, of the O(4) non-linear sigmamodel . Renormalization Group flow lines24 are indi-cated by arrows .

gauge. This approach was also used in formulations ofchiral gauge theories with staggered fermions2 and for`anomalous' gauge theories' . 16,17, 1à .

The `kinetic' n term in the SL'('?) gauge-Higgs the-ory is similar to the fermion mass terms in an SIT(2)Lmodel, obtained for example from the standard modelby taking one mass degenerate ferinion family and let-ting UR -., 1 . In the unitary gauge the ferinion massterms appear to break the gauge symmetry but withdynamical gauge fields the gauge degrees of freedomturn into Higgs fields as in (1. ä) . When the Wilson pa-rameter r is of order 1 the model is again non-perturba-tive : neither a saddle point expansion (r -+ oo) nor ahopping-like expansion r -} 0 appears applicable.

In ref. 17 it was suggested to oinit the kinetic termfor the Higgs field and that this would lead to a dif-ferent theory (e.g . in the SU(2)L model one could setK -= 0 in the action (8, U I, --+ 1) ) . However, gra-dient terms for the Higgs field - alias mass terms forthe gauge field- are generated anyway by the Wilson-Yukawa coupling through the fermion determinant andthe theories with ee = 0 or s ,A 0 are not basicallydifFerent 2 . The role of n may be played to a certainextent by r, but it is more in the spirit. of the Wil-


son approach to keep r `arbitrary' and to use r. for therequired tuning into the scaling regime .

Given the symmetries and field content of a model,in principle all possible local (i .e . extending over onlya finite number of lattice units) terms may be includedin the action . Universality in the domain of attractionof a critical point will then tell us what is the min-imal number of terms in the action that is sufficientfor tuning the theory towards that critical point . Thishas some similarities with the criterion of `renormal-izability' in WCPT, but this is a misleading criterion .In the non-perturbative description `renormalizability'is replaced by universality in a scaling region . It iswrong to judge a model unsuitable if it happens to be`nonrenormalizable' - as exemplified by the massiveYang-Mills Model .

Apart from generating effective n. terms the fer-mion determinant might also generate effective cou-plings violating rotation invariance . In the unitarygauge the one loop vacuum polarization diagrams forthe self energy tensor of the gauge field II, has theform?

n~Y(P) _;,a'.

-I- c'Pû -I- (p"5"� - P~"Pr.)1I(PZ),

9

in the continuum limit . The c term is the induced nterm and numerically25-26 c = 0.054 for r = 1 (due toa computer bug I got a wrong resultl 9 ) . The c' termbreaks rotation invariance and was discussed alreadyin ref. 6 . Its expression is given in ref. 16 . In the oreloop four point function one expects similarly rotationinvariance breaking terms of the W4 type shown in(18) .

This breaking of rotation invariance may be a mis-leading property of the unitary gauge. It was conjec-tured to be absent in the full theory because of thegauge invariance restoring Higgs fields?, but gauge in-variance is not necessarily sufficient . With the Higgsfield the unpleasant terms have a scale dimension higherthan four and they may vanish in the scaling limit . Ananalysis has been given in the framework of the meanfield approximation' .

A non-rotation invariant scaling limit would clearlypose an important problem . The possibility is areadypresent in the gauge-Higgs theory without fermions (cf.the W4 terms in (18)) . It would imply non-universalitywhen there are massive vector fields (since there existpresumably also formulations with a covariant scalinglimit, e.g . using a lattice with more symmetry), soit is worth investigating it in a simplified model, for

10

example by studying the effect of adding a term

e I:Tr (U,u - U=_4)2 + h.c.,sA

to (17) .Assuming that universality guarantees a uni-

que theory in a scalii-g regime one may obtain threetypes of chiral gauge theories from a single fermior_-gauge-Higgs action, corresponding to the three ways(A), (B) and (C) of entering the scaling regime in fig-ure 1 : (A) and (B) correspond to the Standard Modelin the broken and symmetric phase respectively, while(C) provides a lattice regularization of an AFCGT. Sce-nario (C) was also used in formulating chiral gaugetheories with staggered fermions2 .

It might be necessary to replace (C) by a numer-ically rather unattractive scenario (C , ) . Argumentsbased on the eigenvalue spectrum of the fermion ma-trix suggest that the correlation length of the Higgsfield should become large even for an AFCGT19 . Thissuggests scenario (C~) : let re approach h: slowly frombelow such that mH/A --o oo while rnH approacheszero . Here A is an asymptotic freedom scale of theAFCGT and mH is a generic scalar mass which decou-ples while the correlation length 1 /mH still approachesinfinity. Note that with a reducible representation ofthe gauge group one expects to need a different r:, forevery irreducible representation of the gauge groupl9 ,as in (16) . In the following we shall use (C) to indicate(C) or (C') .

In AFCGT's one wishes to study dynamical symme-try breaking without explicit Higgs fields (`dynamicalHiggs fields') . In this lattice formulation of AFCGT'swe use explicit Higgs fields, some of which one mightlike to obtain dynamically. However, the distinctionbetween explicit and dynamical Higgs fields is not fun-damental . The only sufficiently general and fundamen-tal criterion for distinguishing theories is the universal-ity class . (This may be tested in QCD, with the pionsfields playing the role of the Higgs fields .)

For `anomalous' gauge theories the decoupling ofthe scalars in scenario (C) is not expected to workbecause they are to cancel the anomalies of the fer-mion sector . Presumably such theories are thereforenot asymptotically free and `trivial' .

It has been speculated2d that the origin of the electr .weak vector boson niasses is clue to the chiral non-invariant regulator (in the unitary gauge) . One shouldnot forget scenarios (B) or (C), however, which corre-spond to massless gauge bosons in the continuum ap-proach .


AVOIDING NON-PERTURBATIVE YUKAWA COU-PLINGSThe non-perturbative nature oflattice chiral gauge the-ories seems to be at odds with the continuum approach .Can it be avoided?

The mirror fermion method" discussed aboveunder'non-minimal models' avoids the necessity of non-perturbative Yukawa couplings .

The `not coupling the doublers' approach49has an action such that in the classical continuum limitonly the physical fermion couples to the gauge field,whereas the (in this case low mass) doublers decouple.However, with dynamical gauge fields this action mayneed `tuning' to achieve the decoupling, and because itis not gauge invariant the Higgs field comes into playagain with non-perturbative Yukawa couplings, unlessgauge fixing can be implemented in a satisfactory way.This leads us to the

Approach using gauge fixing . Ref. 9 focuseson AFCGT's. Asymptotic freedom is invoked to for-mulate its construction along the lines of WCPT, us-ing gauge fixing and gliost fields . The gauge fixing andghost terms in the action were written down somewhatschematically but in lattice notation the action couldpresumably read something like

- Z l~x Cx + Cx* ;, 1T~x)] + c .t .

where a parametrizes a class of covariant. gauges, c andc are the ghost fields and c.t . denotes counterterms .The action SU is the usual plaquette action for thegauge field and SF can be of the form (13,14), in whichcase it has GP symmetry. The fields OR and 7R areunphysical .

The action above breaks manifest gauge invarianceand the counterterms have to be arranged such thatgauge invariance is recovered in the scaling region . Oneassumes that the counterterms can be fixed consis-tently by imposing BRST invariance . Given a andr the approach into the scaling region in parameterspace depends on only one parameter, the bare gaugecoupling g, and the a and r dependence is canceled bythe counterterms ('I approach') . At. one loop a host ofcounterterms are found . As in lattice QCD, one maydistinguish finite counterterms which can in principlebe calculated in WCPT and divergent countertermswhich have to be determined non-perturbatively.

Stot = SU + SF + S;%x +Sghust + C.t .,

Sfix = - 2 gs FTr B~~ ,_ (h.x - U I'4x)l

Sghust1- ' Tr lu,Ô1 (CxU,sx - t~,~xCr-Fi+

This approach is in fact very similar to early WCP'Tcalculations in non-abelian gauge theories using somegauge non-invariant regulator and constraining thecounterterms by imposing the Ward-Takahashi iden-tities . To be sure that this can be done consistently toany order the dimensional regularization scheme wasinvented and for chiral gauge theories methods usinghigher derivatives as well as regularization independentmethods were developed30 .

In the path integral the gauge is not really fixedby Sf=x . The interpretation of the gauge degrees offreedom as Higgs fields applies also to this case . Let uswrite

Aghast(U) = f DcDc esih °'°

for the Faddeev-Popov determinant . For gauge invari-ant observables O,

DU D?,D~, ,-36ghostes" - +s O =

(19)

JDU D?D~ , DV Av ' ., es i'- =+s v , O,

where S = Str + .SF + c.t ., which exhibits a ferinion-gauge-ghost-Higgs theory. For small ag 2 the Higgsfields are weakly coupled .

In the scaling region the ghosts are to cancel un-physical effects of the gauge fixing by the imposedBRST invariance . It is not straightforward to applythe usual formula

0(U)JDQ esl'r(Crn) - 1

to demonstrate this, because SV' is not independent ofV and Oghost need not be equal to A .

Yet, barring problems with the gauge fixing thisformulation may give the same physics as

JDU D .D r DV

esti.,O

(20)

in the type (C) scaling limit discussed previously be-cause of universality. Note that the `c .t .' contain the`kinetic' (n) terms for the Higgs field . The counter-terms independent of the ghost fields and not depend-ing on a should be irrelevant in the scaling limit if thetype (C) scenario works . Ref. 9 gives three reasonswhy the theories (19) and (20) should lead to differ-ent physics, none of which I can agree with: (1) boththeories have relevant mass parameters which may beeither fixed by BRST invariance or by choosing type(C) values ; (2) one should not compare `anomalous'and `anomaly free' theories ; (3) the non-pert urbativeaspect of the type (C) formulation is no reason for get-ting different physics in a scaling region .


Complete global gauge fixing is problematic for se-veral reasons : one has to handle Gribov ambiguitiescorrectly and non-perturbative effects depending ontopological aspects such as the topological susceptibil-ity and confinement may get destroyed (for a recentinvestigation in the mechanism of confinement see ref.31) .

FERMION-HIGGS SYSTEMHaving argued that the Wilson-Yukawa couplings areintrinsically non-perturbative it appears mandatory todo numerical investigations . As a first step we canuse the external gauge field approximation and studyfermion-Higgs systems . For reviews of Yukawa modelssee refs . 32,33 . The phase diagrams of fermion-scalarmodels are now being investigated in detail35.36 .3î,39 .

It is useful to have some analytic insight in whatto expect . Here we are interested in the fermion-Higgssector of the minimal Standard Model formulation, ob-tained by letting UR,L -; 1 . For one flavor symmetricdoublet this gives S = SF + SH, with

SH ~_

which gives

- MO F,îx(V=PR+VtPL)O z

su ..

+ r E[T'.PRV=0r+w + Tr+~PRV=+i-Ox-. s~_ 24'rVPR4'r + h.c. ],

(21)

where we used the parameter mo = AI - 4r becausethe model has GP symmetry at mo = 0 .

An alternative form of the fermion action is ob-tained by malting the transformation of variables

~1I _ = (4~PL + PR)~'r,

û'r =

VZPR + PL),

SF

=

-

lY'r^Y"(VtV+wPL + PR)rF'= + h.c .

(22)r~ 2

- mU

~'res +

Y''rl`Yr+î, - 4-r) + h.c .r

-' .,,

In the full model including gauge fields

wouldbe the fermion fields in the unitary gauge .

The model has a global chiral SU(2) x SU(2) sym-metry (i .e . (4,6) with QL,R E S11(2) independent. of x) .The primed fermion fields transform vector like

12

J. Smit/Standard model

It is also useful to introduce the fields

7Pm1 = (PL + VxPR)7Prl

which transform as

11

0s - V;r(PR + VtPL) v

11_

11 tP1T ~ ~xSZL .

The system can be in the symmetric phase, or in the16

:0broken phase where (V=)

0, (z/rs~/rx }

0, andSU(2)L x SU(2)R --> SU(2)L=R . In the symmetricphase we can distinguish mF and m1F,

_computable re-spectively from the propagators (~x y) and ( shy) .Here we use the notation mF for the mass of the fer-_ -L _11

mion excited by 7rß , z/r= ; since (O. O,,) = PL(7kz -41,) PR,mF =_ MF. In the broken phase Vs, 0. and z/,., areequivalent interpolating fields since they have the samequantum numbers under the residual symmetry group,so

mF = 7nF, broken phase.

Simple approximations far the fermion masses asa guideline for interpreting numerical simulation dataare given by :

weak coupling

mF

mov,MD " (mo -!- 2r)v,

(23)strong coupling

inF

moz-1 ,mD .: (mo + 27-)z-1 .

(24)

Here v and z are defined by

(Vx) = vll, (V' V.+Ei) -y2n,

(25)

mF is the fermion mass and MD the first doubler mass(n = 1 in (2)) . The weak coupling formulas are basedon the usual replacement VS -. i1 in the action (21) . Forstronger Yukawa couplings the T.W1 part wins overthe IP part of the ferinion action and an expansion in.P, a sort of hopping expansion, becomes appropriate .Then the alternative form (22) gives a better repre-sentation and the strong coupling formulas (24) followfrom the replacement VtV=+" -~ z2 . The strong cou-pling formulas were derived19 for the unbroken phase,but their derivation is equally valid ill the broken please .The use of the link expectation value z 2 is expected togive better results than the mean field value v2 as ittakes into account a dominant effect of the fluctua-tions . Note that v \ 0 and .,Strong coupling formulas have also been derived for thecase of a radially free Higgs field 34 .

and chiral gauge theories

... . .. . .. . ... ..... ... ... ...... ... . ... ... . .. . ..1

CV oc 49z'/N and R2 - v2 as a function of r/2 in theU(1) theory39 .

SIMULATIONS OF CHIRAL FERMION-HIGGS SYS .TEMS

Here we are specifically interested in the' questionif the fermion doublers can be given masses of orderof the cutoff, with scaling values of the physical fer-mions masses . An early indication that this is a realpossibility can be found in ref. 38 .

Model (21) has been studied recently in the quenchedapproximatioii41,42 .

A U(1) it -)del, which may beconsidered as a simplified version of this model withthe Higgs field V= = exp(2eyr3) in a 17(1) subgroup ofSU(2), has been studied without duenching39.40 . (Thefermion determinant is real in these theories) .

For the U(1) case with dynamical ferinions39 ,figure 2 shows v=, z` and the bosonic specific heat cv atn; = 0 and ?no = 0 . At n; = 0 the coupling; between Vxand Vy at sites x 0 y comes entirely from the feriniondeterminant . For small r this coupling is weak and thesystem is in the symmmetric phase, but as r increasesthere is a transition to the broken phase around r/20.09 . From the alternative form (22) of the action oneexpects that the system goes back to the symmetricphase for sufficiently large r, since then the Ja partbecomes negigible. Fig . 2 shows that this happensaround r/2 -_ 0.39 .

." S . . . .. .. . . .... . .. ... . ....... .. i

.a0 .0 3 0 .2 J.3 0.~ 0 .5 0.a

4.û

3 .0C d

2.0. ... ... ...... .. .. . .wd.w

0 .- 0 0 . 1 0 .2 0.3 0.4 0 .5 0.6

0 . 3Rz

10.oL A0 . 4 '0 .2 FVi "1 . . o I\ J .2 0 . 3 v . 0 .5 C . o

Figure 2 . Measurements of Q - 0.5 + 0.522,

1.e1.4

a 1.6C 1.5

1.41.3

1.20.0


0.2

0.4

0.6m

Figure 3 : Fermion masses MO = mF and mT = MD asa function of the chiral symmetry breaking mass m inthe U(1) theory40 . The dashed line is to guide the eye .Top two: broken phase, bottom two : symmetric phase .

Fermion masses were computed40 , in the brokenphase for r/2 = 0.25 and mo = rC = 0 . A mass termmT?lr was added which breaks the chiral U(1) symme-try but not the residual symmetry of the broken phase,and which avoids having to work with zero fermionmass . Figure 3 shows the results for mF and MD as afunction of m. The data are consistent with an extrap-olated fermion mass mF = 0 at m = 0, in accordancewith the GP symmetry of the model . In contrast, thedoubler mass MD P-- 0.75 is non-zero at m = 0 . Thefigure also shows masses computed in the symmetricphase on the strong coupling side at r/2 = 0.5 . In thiscase the interpretation is somewhat obscured by thefact that the measured components Tr(V,=T� ) of thefermion propagator should vanish at m = 0 because ofunbroken chiral symmetry .

The quenched SU(2) simulations focused onthe question if and how the fermion and Higgs massescan be given scaling values MF, mN -} 0 while dou-bler masses stay O(1) as we approach the critical pointv = 0 from the broken phase . The quenched approxi-mation makes it possible to use the impressive know-

1

0.32

0.34

0.36

0.38

0.4 X

13

Figure 4 : Fermion masses in the SU(2) theory4l atr = 0 as a function of rti for mo = 0.6 - 1.50 increasingfrom bottom to top .

ledge on the SU(2) x SU(2) ^-- O(4) pure Higgs modelacquired recently43,44,45 . The critical point v = 0 isat rc, = 0.3045(7) and approximate scaling is foundfor rc_0.32 where mHZS0.71 and u-0.31 . As rc Kcthe renormalized Higgs self coupling AR = 7nH/(2z,2 )approaches zero logarithmically (triviality) . The ratioZ.~ = v=/v2 = Z,,/(2r`:) is within a few percent from1/(2n) . The case a -" oo (radially frozen Higgs held)leads to the largest possible correlation length 1/ ,nHfor a given ratio mH/VR .

Figure 4 shows mF = MD'for the case r = 0 . We seea crossover between weak and strong coupling behavioraround mo . : 1 .3 -1.4 . The weak (strong) coupling regions are defined by 8mF/arc > 0 (< 0), in accordancewith (23,24) . This crossover phenomenon was also ob-served last year in simulations in the Zî case with areal Higgs field and it was hypothesised that it may berelated to a fixed poitit34 .

Figure 5 shows the crossover phenomena as a func-tion of r for mo = 0.8 . We see the doubler mass "IDseparating from mF and ÔMF/(Oh changing from posi-tive to negative values around r -- 0.16 . It is found thatthe weak (strong) coupling regions in the r, mo planeare given approximately by mo+4r< 1.41(-> 1 .41) . withlittle dependence on K .

The masses are indeed approximately linear in 1/zin accordance with (24) for sufficiently strong couplingand sufficiently large 1/z (see fig . 4 in ref . 42) . Pre-liminary results indicate that the fermion mass mF can

0.60 .5 1 2.00.4 10.30.2 L _ o0.1 a

0.01.~ r0.0 0.2 0.4 0.6

1.2 -~ - ,-

1 . 1 ` 11 .0 e 0

0 . 1.00.9

J07.0

0 0.2 0.4 0.60.6 0 .0 Î0.50.4 00.30.2 !-

0 .00.10.30.00.o

0.2 0.4 0.6

14

X :x=0 .32+:x=0.33o:ic=0.35®:jc=0.40

0.1 0.2 0.3 0.4 0.5 w

Figure 5 : Ferinion finasses in the SIT(2) theory42 as afunction of u , - r at various Pc and y - nio = 0.8 .

be indeed extrapolated approximately linearly in 1/zinto the symmetric phase according to (24) .

At r = 0 the strong coupling region is unphysicalbecause there all fermion masses are > 1, but for suffi-ciently large r it extends to rno = 0 . Then we can havemF = 0 (at mo = 0 by the GP theorem) together withdoubler masses > 1 . Figure 6 shows the situation forr = 0.5 . Vl'e see that MF extrapolates nicely to zero atrn® = 0, while MD > 1 for all mo. The data togetherwith the approximate validity of (24) suggest that thisis true for all r,<K<oo.

Obtaining scaling values for the masses is easyin the quenched approximation : . Choose t; to get therequired ratio mH/vR =

2AR. How to choose r¢ isknown from the solution of the O(4) model . The re-quired AR should of course be within the triviality bounce.g . v~R-<2.6 for mH ~- 0.5 . Next take r somewherein the strong coupling region (r is `arbitrary'), e.g .r = 0.5, to guarantee MD > 1, and choose m o to obtainthe required ratio mF/mH. Sincc mF is approximatelylinear in mo (by (24)) this can be clone .

An alternative way of decoupling the doublers mightbe possible in the weak coupling region if the crossoverphenomenon is related to a fixed point where arnD/arevanishes (it could instead be discontinuous) . Then wecan let r approach the crossover from the weak cou-pling region such that MD stays O(1) as h is loweredto get the required AR . Scaling values for rnF are againobtainable by an appropriate choice of mo . In this caser would not be `arbitrary', which we feel is against the


Figure 6 : Fermion mas-es in the 5'U(2) theory42 as afunction of y - rno at w =_ r = 0.5 for n; = 0.40 - 0.3`3decreasing from bottom to top (and at y = 0 .3 forte = 0.31) . The dotted line represents rti = co .

spirit of the Wilson approach.The Yukawa coupling is `trivial' at. fixed mF/mH

simply because aR is `trivial' . If we go deeper intothe scaling region by letting mH - 0 while keepingmF/mH fixed, then AF, and yR = (?77F/"1H) 2AR go tozero . Here we have defined the physical Yukawa cou-pling yR by yR = ?nF/VR . However, Suppose MH<100GeV which implies a cutoff beyond the Planck mass,mH<-0.80 x 10'59 , as follows from the formulamH = 6.4 exp(-27r'-/(3AR) + (13/24) ln(3AR/(2r'))),which is valid under these circumstances 45 . Then, witha fermion mass of order 10-3 say, the physical Yukawacoupling could still be huge, yR = O(10 56 ), which ishardly a trivial value .

Under these circumstances yR may differ substan-tially from a physical Yukawa coupling defined in termsof the fermion-fermion-Higgs vertex function evaluatedat momentum scale y (renormalized of course by theappropriate wave function renormalization factor) . Let.us denote this running coupling by yR(y). For not toolarge mF/z~R one expects yR = yR(rnF) -t- O(VR(mF)) .For very large mF/VR, however, yR might differ sub-stantially from PR(rnF).

The parameter dependence of physical quanti-ties like the fermion mass is somewhat unconventionalin the strong coupling region . For instance, one may in-terprete mo - denoted by y in refs . 41,42 - as a bareYukawa coupling, as suggested by (23) . In continuum

J. Smit/Standard model and

normalization we have to use vR - a 2rti (neglectingthe small deviation of Z,D from 1) and mo/

2rc .However, such an interpretation suggests an intu-

itive dependence of yR on mo/ 277 which is not thecase in the strong coupling region . Dividing the datafor mF in figure 6 by vR shows that yR is roughly alinear function of mo/

2r. at this r value, with sloperanging from 2.4 to 5.8 as s goes from 0.40 to 0.32 . Thisis somewhat strange as one might expect that becauseof triviality the renormalized yR should be smaller thanthe corresponding bare mo/

2n_.Such expectations are based on an incorrect identi-

fication of mo/ 2rc with an intuitive bare Yukawa cou-pling yB defined as the running yR(p) at momentumscales of order of the cutoff, say yB = yR(1). The rela-tion between yB and yR can be approximately obtainedby integrating the beta function 13(yR) = PaVYR/Op,,

MF,H « ,u « 1 . For example, if we assume thatyR and YROA) are sufficiently small for the one loopformula )3(y'R)

/3oyR, J60 _ 3/(4nz ), to hold, thenyR ti yR(mF) ^ yB/(1 - J3oyH In mF), neglecting devia-tions in,8(9R ) coming from the regions i.c - O(n1F,mH)

and A = O(1). Obviously yR < yB .In the weak coupling region one may perhaps make

the intuitive identification yB mo/ 277: . However, forWilson fermions in the strong coupling region such anidentification is not correct . It should be stressed thatmo is just a non-universal parameter in the action andthat yB = YB(nio, r, n:) has in general no simple relationto 7710 .

We assumed above that the renormalized runningcoupling yRG) has the universal properties as givenby renormalized perturbation theory in the continuum .This is based on universality : The theory has SU(2)L xSU(2)R x U(1)L_R symmetry. In the scaling region ofthe broken phase it has three Goldstone bosons, oneHiôgs boson and one Dirac fernlion doublet as its parti-cle content . For not too strong renormalized couplingsthis situation should be describable by an effective ac-tion of the form

f d4x

+ 2aj..VaaA1Pa +YRT(~P4 + Ykr05N'

+ ~avawa + â ~R(i~a,pa)21 + loop corrections .

The,scaling properties of the loop integrals are deter-mined by the particle content and the couplings aredetermined by the symmetries . The loop correctionsare to be calculated according to renormalization groupimproved renormalized perturbation theory in the bro-ken phase . One should like to test this universalityassumption by detailed computer simulations . Similar

chiral gauge theories

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15

tests worked out beautifully for the examples of thefour dimensional Ising model and the O(4) non-linearsigma mode146,43,44,45 .

CONCLUSIONSProgress in understanding chiral theories on the latticehas been slow because of their inherent non-perturbatioenature . The numerical results are very exciting and en-couraging, and confirm simple analytical calculations.The possibility of decoupling the fermion doublers, bygiving them masses of order of the cutoff with scalingvalues for the physical particles, appears to be inti-mately related to the crossover phenomenon . Decou-pling is easy in practise . The GP symmetry is veryhelpful in extrapolations to zero ferinion mass . Veryinteresting is the possibility of large mF/(0)R ratios .

Much work lies ahead in gathering sufficient detailsfor confirming this positive picture . The scaling regionshave to be investigated for their universal properties .The symmetric phase which is relevant for the formu-lation of AFCGT's needs to be studied .

Acknowledgements . I would like to thank P . vanBaal, W. Bock, A.K . De, M.F.L . Golterman, A . Hasen-fratz, K . Jansen, J . J6rsak, R.D . Kenway, J . Kuti, C.B .Lang, M. Liischer, L . Maiani, I . Montvay, T . Neuhaus,D.N . Petcher, B .J . Pendleton, G .C . Rossi, J . Shige-mitsu, M. Testa, A . van der Sijs and J .C . Vink for use-ful and stimulating conversations, and the organizersof the conference for a very nice atmosphere .

This work is supported by the `Stichting voor Fun-dan:enteel Onderzoek der Materie (FOM)' .

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standard model and chiral gauge theories on the lattice

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