lattice chiral gauge theory without gauge fixing?

Nuclear Physics B (Froc. Suppl.) ~B, C (1992) 83-97 North-Holland

PROCEEDINGS SUPPLEMENTS

~',ls

LATTICE CHIRAL GAUGE THFORY WITHOUT GAUGE FIXING?

Jan Smit

Institute o.f Theoretical Phgsics, University of Amsterdam, Valckenierstruat 65, XE 1018 Amsterdam, The Netherlands

We review how gauge fixing m y be avoided in formulating chirai gauge theories and give some details of the approach using staggered fermious.

1. INTRODUCTION

The weak coupling expansion of field theories can be defined without recourse to a regularization, see for example [1]. For a non-perturbative definition a regularization is needed, which for chiral gauge theories appears to be incompati- ble with gauge invariance. Nowadays we accept a situation where chiral gauge invariance is broken by the regularization but gets restored in the scaling regime, i.e. for energies and momenta much sma|~er than the regulator scale. To achieve this, counterterms must be tuned appropriately, which can be studied in weak coupling perturbation theory [2]. The use of gauge fixing, ghosts and BRST invariance appears to be an essential ingredient [3,4].

The lattice regulator poses special problems since one has to make a choice what to do with the 'species doublers'. One can try to a) decouple them, b) not couple them or c) use them.

We are specially interested in a non- perturbat~ve definition of CGT. The 'Rome proposal' [3,5] ~or this is to use the same form of the action and measure in the path integral as found in weak coupling perturbation theory. This proposal is made in a framework of type ~.), but this could be done for b) or c) as well.

It is not clear that this path integral using ghost fields is well defined beyond perturbation theory. Furthermore, another major question is whether the employed smooth ('renormalizable') gauge, or weighted sum over smooth gauges, does not spoil essential features cf the theory. Gribov copies may give problems. Our experience of confining theories like QCD suggests that it is essential to be abie to implement large ~,auge transformations. It is important to test the gauge fixing/ghost approach on a non-chiral theory such as QCD (even without fermions) and check whether non-perturbative properties such as confinement and the topo- logical susceptibility can be reproduced this way.

2. GAUGE DEGREES OF FREEDOM

We shall here elaborate on an alternative approach to a nonperturbative definition of chiral gauge theories which aims at avoiding gauge fixing [6,7]. Its basic ideas are general but we only know how to express these precisely with the non-perturbative lattice regu!arization. For the moment we shall use continuum notation. Con- sider a candidate chiral gauge theory with initial

~ction S (~ , ~o',A'). This initial action is not

0020-5632/92.~05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved.

84 J. Smi: / Chiral gauge awory without gauge fLm~g?

gauge invariant, which becomes clear as soon as

we make the regularization explicit. However, for gauge field configurations that are smooth on the

length scale of the regulator (for the lattice reg-

ulator this means the classical continuum limit),

the action can be chosen gauge invariant. We use

the term 'initial' and put a prime on the fields

because we shall also consider gauge invariant

actions S(¢, ¢, A) derived from it.

Suppose we integrate out the fermion fields ¢'

and ¢ in the path integral. Then we are left

with an effective action Sell(X) which is not gauge invariant. In the path integral we use the

gauge invariant measure DA - DA n, where fL

denotes the gauge transformation An = f~(A l, +

i0~)f~ t. Hence we integrate in the path integral

effectively also over all gauge transformations - -

the gauge degrees of freedom. This can be made explicit,

" f DA' [ = e x p - S e l l (A' )]

= f 0.4 exp[-Se.u(AV')]

= / D A D V exp[-Sell(AVt)], (1)

where in the second line we used the transformation of variables A ' - A t'*" (V is a gauge

transformation) and in the third line we inserted

1 - f Dr. Note that we assume the gauge group

to be compact, which can be realized on the lattice.

Another way to introduce the gauge degrees of

freedom V is to identify them with the longitu-

dinal degrees of freedom, by splitting the gauge

field into its transverse and longitudinal parts, #

A t = Vt(Ay + iO,)V, O~,Ay = 0, and no'ring

that we may drop the restriction :iv ~ because the resulting effective action for Atr is invariant un-

der the gauge transformation

~.,. -~ ~(A. . iO.)n t, V ~ m,. (2) • o.,,

since Avt is invariant.

The V's are coupled dynamical variables:

depending on the context, they are called

Stiickelberg fields, Higgs fields, Wess-Zumino

scalars or (in discu~ions at this workshop)

'Omega fields'. We have to find out their role

played in the theory.

3. MASSIVE YANG-MILLS MODEL

Besides the gauge non-invariance introduced by

the regulator we can also introduce gauge sym-

metIv breaking terms in the action 'by band'.

This does not seem to lead to a qualitatively dif-

ferent situation. Let us therefore first discv~ a

model without fermions in which all the gauge

symmetry breaking is introduced by hand: the

massive SU(2) Yang-Mills model. Its action is given by

S' / 1 , , ~ , ,_ = - _ dtxTr[~g~Fm, Fu, + I~As, Au] (3)

where we have put a prime on the fields to indi-

cate that these are the fields ia the initia! action.

The mass term oc ~ - ra~/g~ breaks gauge in-

variance. Such mass terms are also typical conse-

quences of gauge symmetry violation by the reg-

ularization in chiral models, so it is worth study- ing their effect in detail.

Since the mass term in (3) breaks gauge invari-

ance, it depends on the gauge degrees of freedom Making a transformation of variables A' = Avt

and treating V as a dynamical variable as de-

scribed in sect. 2, the mass term gets transformed

into the 'kinetic term' for a Higgs field ~b -- pV, radially frozen i.e. p -- 11,

S = - f ~'z Tr[~FuvF~v (4)

+f2o Yt(A . + iO~)V Vt(A. + iOn)V].

For suitable choices of go 2 and ]0 ~ we enter a

.I. Smit / Chiral gauge theory without gauge fixing? 85

scaling region were the radiai degree of freedom of the Higgs field gets restored: the Fourier transform ~(p) - f d4zexp(-ipz)V(z) lies no longer in the group SU(2) and the propagator of the radial mode # - ~ / ~ gets the pole corresponding

to the Higgs particle. The massive Yang-Mills model is often called

non-renormalizable, but we know since many years that this is just perturbation theory ter- minology and with the non-perturbative lattice regularization the model is believed to be in the same universality class as the renormalizable

gauge-Higgs model given by

Z = / DA D~D~" exp{- / d4z [Tr 470F~vF~v

_ ~ 2 4 + [(0~ iA,) I 2 2 + o1 '1 ]), (5)

with the identification ~b - ( - ~ , * ~ ~u).~d

There is numerical evidence for this [8]. Fur- thermore the non-linear sigma model obtained by switching off the gauge coupling go 2 is very well studied and found to be in the same universality class as the linear sigma model [9]. These models are 'trivial' and the non-linear model is the limit of the linear sigma model for A0 --~ oo, which limit gives the largest cutoff/mass ratio for a given interaction strength in the scaling region.

Summarizing our present knowledge of the physics of the massive Yang-Mills model, consider the phase diagram of the theory in the I/g 2 - ~ plane, where ~ - a2f02, a "- lattice dis-

tance. There is a Higgs-screening phase bound- ary line ending at g02 - 0 in the critical point ~c of the non-linear sigma model. For the usual hypercubical lattice regularization ~ ~ 0.304. There are three scaling regions, which we shall denote by A,B and C. In all three regions g02 ~ 0. For A, ~ '% ~e; for B, ~ / z ~¢; for C, ~ < ~c. Re- gion A corresponds to the Standard Modei: the theory describes massive vector bosons interact-

ing with a massive Higgs particle. In region B the physics is that of a scalar 'quarks' confined by SU(2) gauge force. In addition to the massive vectors and the scalar there will be many other unstable particles in ~this region of the phase diagram, as in QCD. Going into region C the scalar constituents become heavy and disappear from ~he scaling particle spectrum. Only the glueballs of the SU(2) gauge theory remain in scaling re-

gion C. The explicit introduction of the gauge degrees

of freedom facilitated making these conclusio~ about the physics of the massive Yang-Mills model. The actior~ (3) is the unitary gauge version of the gauge-Higgs model (4), the variables A' are the gauge fields in the unitary gauge.

4. WILSON-YUKAWA MODEL

Another example is the Wilson-Yukawa candidate for a lattice Standard Model. We shall here only review its salient features, see [7,10] and references therein for more details. The fermion part of its action in the unitary gauge is schemat-

ically given by

= - d4Z[ L D L

W . J , . , l - J l I R l + +

where D~ 'R 0 # - ~ f =° is the covariant derivative in the L, .~ representation of the gauge group, w parametrizes the Wilson 'mass term' and G is the usual Yukawa coupling matrix, which is diagonal in SU(2) doublet space. Notice that the w term, which is introduced to remove

the lattice fermion doublers, belongs to the gauge symmetry breaking type caused by the regulator, whereas the G terms belong to the category 'in-

86 J. Smit / Chiral gauge theory withou: gauge f~ving?

troduced by hand'. Following the steps outlined in sect. 2 we make

I

the transformation of variables Az~ = V~(Az~ + iO~)V and integrate over the gauge transformations V K SU(2) x U(1). In this model the fernfion measure is gauge invariant and we can also make the transformation of variables

This absorbs the V's in the gamma, mu part of the action and lets them reappear as Yuk~wa couplings in the combination

VLV Rt =_ 9 YU(2), (8)

such that

SF / ['JL ' LL" - R = - + ~bR7 u D~ ~b R

+ ~b'L 9G~bR + ~ R G g I ~ L (9) W

VD~, D~, ~b R + ¢ aDl, D~, V OL)], + - R t R ;7, R t a - t

The effective action obtained by iategrating out the fermion fields can only depend on ff. It follows that we only need a vector boson mass term of the form

_ f o 2 f . . . . , - ' , (10) J

and the dynamics of theory depends only on the (radially frozen) Higgs field of the Standard Model ff, and not on V n and V a separately.

As in case of the m~sive Yang-Mills model, it has proved very useful to consider the limit of zero gauge couplings and study the resulting fermion-Higgs model. For simplicity these stud- ies have been done for the case where all ordinary Yukawa couplings reduce to G - yll.

There is a weak coupling region and a strong coupling regions depending roughly on value of y + 4w being small or large ~'~'6-"''~ means ~>2). There is one broken (ferromagnetic FM) phase and there are t~wo symmetric (paramagnetic PM)

vhas~, one at weak coupling (PMW) and one at strong coupling (PMS). The dynamics turns out to be very different ~n these two regions, which is brought out most clearly in the symmetric ph~es.

In the PMW phase at weak coupling the fermion particles of the theory are massless and described by ~bL,R and ~L,R" The partic|es cre- ated out of the vacuum by ~bL,, ~L are 'charged' with respect to SU(2). There are also the problematic httice doubler fermions. The SU(2) neu-

tra: field~ ~b (n) = l?t~L and ~(L n) = ~LIT' describe two-particle fermion-ecalar states and their prop- agators do not have particle poles.

On the other hand, in the strong coupling PMS phase the fields ./,('0 ~'(L n) ~ . , ~a, and ~a decribe massive SU(2) neutral fermic~n particles, whereas the SU(2) charged fields ~L and ~L describe two-particle fermion-scalar states. The doubler fermions of this phase are neutral under SU(2) and they can easily be removed by giving them masses of the order of the cutoff, by choosing w of order 1. However, there are no SU(2) charged fermions in the PMS phase.

The PMW phase has fermion doublers and the PMS phase has only SU(2) neutral fermions. Therefore, neither phase has ~he physics of the fermion-Higgs sector of the Standard Model in the symmetric phase, and we conclude that the Wilson-Yukawa approach has failed [11]. The reason it failed is that we cannot simultaneously have the physical fermions in the weak coupling region and the doubler fermions in the strong coupling region where they can be removed. The very concept of physical fermions and doubler fermions breaks down in a crossover region

going from weak to strong coupling, apparently due to the roughness of the scalar fields on the lattice scale. This is similar to flavor symmetry breaking with staggered fermions in QCD, ex-

3. Smit l Chiral gaug¢ theory without gauge jZrb,g? 87

cept that in QCD this symmetry gets restored in the scaling region, the roughness of the scalar fields being irrelevant because of exact gauge invariance.

5. STAGGERED FERMION APPROACH

With staggered fermions (which are of type c) sect. 1) there is no need for decoupling through l~rge Yukawa couplings, so this approach may be more succesful than Wilson-Yukawa. The schematic form of the staggered fermion action for the Standard Model is just (6) without the Wilson term (details will be given later),

~ e L e - - J i R e S' = - d4x[~bLT#D# ~L "1" ~RT#D# ~bR

-b ~b L G~R "t" ~b RG~L]. (11)

We can again make the gauge degrees of freedom V explicit and make the transformation of variables A'z~ = Vt(Ax~ +iO~)V. The action is then gauge invariant under (2). However, with staggered fermions we cannot make a transformation of variables as in (7). The fermion measure is not gauge invariar~t and we cannot locally invert a

e gauge transformation like '~L,IZ = L,R~t.,R, be-

cause the spin and flavor degrees of freedom of

the fermion field are spread out over the space-

time lattice.

However, the action (11) is constructed such

that for G = 0 it is gauge invariant in the classical continuum limit. We hope that at the quantum level this gauge invariance can be restored in a scaling region. This may require a suitable tuning of counterterms and coettic.;ents hidden in the action (11). In this scaling ~'mit the theory is to correspond to the target action obtained by formally making the transformation of variables

#'Z,,R t ' - - ' = L,R = bL,RVR,L, namely

S = - [ d 4 z - - D +

Here ~b is the scaling form of VLV~.

Since we are not fixing the gauge we expect

that upon restoration of gauge invariance only

the gauge invariant correlation functions of the

target theory (12) have good scaling behavior.

We have to e~pr~ all correlation functions in /I I

terms of W I ~ A and V, since we cannot ef-

fectuate the t:ansformation of variables (7). The

question is now, how can we tell if this restora-

tion of gauge invariance has taken place?

At this stage of development we do not want

to go into this question but prefer to study first

the resulting fermion-scalar system, postponing

the integration over A~ and using it as an ex-

ternal gauge field to probe the system. What we

want to obtain in the scaling region is a fermion- scalar system in which the fermions are charged with respect to A~. For G ~ 0 this would mean just free charged fermions with quantum numbers corresponding to ~p and #,, but without neutral fermions with quantum numbers cor-

responding to tp' and ~ ' . The effective action Sey! (A) obtained by" t~ ,~ ; ~, out the fermions and scalars should have all the desired properties that one usually derives in the target theory with continuum tech;fiques. For an anomaly free ~ke- ory this means gauge invariance up to contact terms removable by local counterterms.

One crucial test would be passed, for example, if in the symmetric phase of the scalar-fermion • k ...... ,I.~ correlation functions

e (v t (z)~ (z)~ (y)V(~/)) (1~)

would have particle poles corresponding to

charged massless fermionsl even fol G ~ 01

whereas

88 J. Sndt / Chiral gauge theory without gauge f~ing?

(~b (~)~ (9)) (14)

would not have particle poles but a (finite volume version of) a cut corresponding to neutral two-

particle fermion-scalar states, as if ~l,' - - g~b. For scalar particle masses of order of the cutoff (14) would then not even have scaling behavior.

Note that for the fermion-Higgs sector of the

Standard Model where V ~ SU(2)xU(1) the ef-

fective scalar field action obtained after integrat-

ing out the fermion fields should lead to vanish-

ing expectation values (VL) and (VR) but non-

vanishing (17) = (VLVt), corresponding to the unbroken electromagnetic U(1) subgroup, under

which all components of Vz.,R are charged but

(17) oc 1l is neutrM. This will put restrictions on

the range of parameters in the counterterms.

Similar to the massive Yang-Mills model, we

can envision three types A, B, C of scaling re-

gions for a chiral gauge theory. In A the system

is in a Higgs phase, in B it is in a screening phase

while in C the scalars decouple by being very

massive and by choosing zero Yukawa couplings (typically zero bare fermion masses in the initial action). C is the goal of an asymptotically free chiral gauge theory without fundamental scalars.

A less attractive scenario (called C' in [7]) would be one in which we are forced (for technical lattice fermion reasons (?)) to go from C slowly into B as we take the continuum limit, such that the scalar constituents are effectively infinitely massive.

6. STAGGERED FERMIONS

We shall now go into more technical details of the

staggered fermions. Consider the free real stag-

gered fermion action

z .

( 1 5 )

in which there is just one Grassmann generator

X~ per site x (hence the name 'real', or 'Majo- rana' or 'reduced' staggered fermion), with

rh,~ = (-1)x,+'"+x~ -x, ¢.~ = (-1)z~+ '+' ' '+x.,

~= = ( -1) ~'+' '+ ' ' (16)

the familiar sign factors. There are several ways to obtain the physical interpretation of this theory. We shall use here first the momentum space

method of ref. [12], writing

XA(P) = ~ e x p [ - - i ( p + ~rA)z]X=, (17)

where lr/2 < pp < ~r/2, ~rA, A = 1 , . . . ,16 , has components IrA. = 0 or ~r. In terms of XA(P) the action (15) can be written as

,r12 d4p S = - - - - - - - - 1 X A ( - - p ) ( i V . sin p .

J - ~ / 2 (9"71") 4

+ m.E.E~V5 cosp.)AVXe(p), (18)

where the real symmetric 16 × 16 matrices

(r,)AS and (--",)AS satisfy

{r.,r.} = [r.,_.] = o,

and rs = -r2r3rlr4, % - --~.i.~.2.~-3-~-4 (the

minus signs are introduced to conform to the

usual definition of 75 in the V - A form of the

electroweak currents). We can block-diagonalize

r~ and ~ . into ordinary 4 x 4 Dirac matrices 7. and ~. by a unitary matrix T,

(F.).4B - T~,a~(7.)a,pT~,B,

(--'-.)AS = TtA,a,~(~,),~,xTax,e. (19)

,The action (18) then can be written as

_ f l r l2 d4z S -" J - . / 2 (2~') 4 I~T(--P)E(iT" sinpp

+ m. . 5"r5 cosp.) (p), (20)

~a~ (P) = Ta~,AXA(P) , (21)

J. Smit I Chiral gauge theory without gauge fixing? 89

where C, and C¢ are charge conjugation matri-

ces for 7's and ~'s. The matrix E keeps track of the symmetries under transposition, e.g. (E7t,) T = E % represents F~ = Fg.

r%r m, = t} the theory is invariant under the staggered fermion symmetry group S F , which

is generated by shifts Sp over one lattice unit in the p-direction, lattice 90 o rotations Rp,, in (p, o') planes, lattice parity Io and U(1), symmetry. These transformation can be given in posi-

tion space (e.g. U(1)~ transformations are given by Xx "* exp(iwsex)Xx), but here we shall only mention their more intuitive momentum space

forms [12]

W(p) ~ ~p q'.t(p) exp ipp, (23) 71" 71" 1

@(p) --* exp(~TpTa)exp(~p~a)@(R;$ p), (24)

~(p) "~ ~l'4~4 ~ ( IsP), ( 2 5 )

@(p) --~ exp(iwsTs~s)W(p). (26)

The group of discrete transformations (23)-(25) is called LS. For details on the interesting group~ SF and LS see ref. [13].

The action (20) has actually a much larger continuous symmetry for m g = 0 than U(1)~, namely the U(4) transformations

• (p) (27)

ft = exp[i(o~ + ws~s + w ~ ) 7 5

+ (28)

However, expect for the U(1)~ subgroup this U(4) invariance is non-local in position space and

in interacting theories with local actions it is broken on the lattice, although some of it may be recovered in a scaling region. The S F group on the other hand need not be broken in interacting

theories. In the cla~ical continuum limit sin p~ ~ p~,

cos pg ~ 1 and (20) reduces to its scaling form

decribing four real fermion fields

S - - / d4z~@TE(ytsOts Jr ml,~i,~s'Ys)~, (29)

which we shall call its Majorana form. This Majorana form can be rewritten in more

conventional Dirac form by selecting a U(1) symmetry for fcrmion number F and pr~ecting onto subspaces of definite F, F ¢ = - ¢ , CF = ¢. Writing

1 - F "- 2 d2 "}" ( ~ E t ) T,

leads to

½ r E.y ,, O ,, = a ,, F ¢ ,

(30)

(31)

where we used 0~ T = - a ~ , as is obvious from the fact that a~ corresponds to i sinp~ in (20). For F we can take a generator of one of the U(1) subgroups of the U(4) transformations (28). Two choices which commute with the lattice rotations are F = 75 and F = 7s~5. The latter corresponds to the U(1)~ symmetry.

For F = 75 we have more explicitly

½'~TETt, a,,~ = "~+)WL "~,,O.WL-~ "(+)+ ~-)7,,a.¢~L -)

-- ~ L T . 0 . ¢ L , (32)

where L, R and (4-) denote the projections

- 1 + 7 5 14-~s (33) 1 75 p R = _ _ _ _ p ~ = . PL -- " ' ~ , 2 ' 2

The fields eL, ¢'L may transform under the full

U(4) group

~L "+ aL~L , ~'-'L -'~ ~atl,, (34)

~L = exp[-i(w + ws~s + w ~ )

s ½ w , ~ ] . (35) + ~ s +

For F = 75~5 we have

I ~ T E~f#Cg#~ = ~LT#O#~L "4" "~R~/#a#~)R, (36)

where we did not indicate the 4- index because it

is correlated with fermion number and chirality:

90 ~. Smit/ Chiral gauge theory ~ithout gauge fixing?

~ L ---- ~ L ~ "- ~ ~ "-

I n this case the transformations that do not mix and ¢, consist of the U(2)×U(2) subgroup of

u(4),

¢~ = fl±¢~, ¢-'~ - ~ 1 2 t , (37)

~± -- e x p ( w ~ s + ½ w , ~ , ~ ) e ± . (38)

For F = 7~ there is no mass term compatible with fermion number conservation. For F "- ? ~ the one link mass term in eq. (29) is compatible with the fermion number U(1)~ symmetry. We can rewrite it by choosing a represea~ation of the ~ matrices in terms of Pauli matrices r~ and w~. For instance

f~ = w2r~, ~ = -w~, fs = w3,

leads to

(39)

llt~TEmu~u~sTs~ -- ~(m4 + im~r~7s)¢, (40)

which is equivalent to m~¢ , m = ~/m~,rnu, un-

der an SU(2)×SU(2) transformation. The mass degeneracy of the two Dirac fermions can be removed with a three link mass term [12].

With these interpretations of the free real staggered fermion theory it is straightforward to write down continuum (chiral) gauge theories and translate them to the lattice such that, of course, the cvutinuum action is recovered in the classical continuum limit. We simply work our way back from the ¢, ~ notation to the ~# notation with 7's and ~'s, which is equivalent to the XA notation with F's and H's. The F's and H's are produced by repeated application of the rules F~AB X~(P) exp ip~ ~ ~X~+f~,

~ .4B X~(P) exp ip~ ~'~ ~ X ~ + ~ . Subsequent symmetrization and customization is usually desirable. Note that although our staggered fermion interpretation was done in momentum space, this procedure leads to a local staggered fermi_on action in position space.

A neat way of obtaining lattice actions can be formulated by using an alternative position space method for interpreting staggered fermions [6], which we shall now briefly describe. Consider the matrix field

1 ~ -- ~ E 7 z+b X,+b, (41)

b

7b 7~1 b4 = (42)

where b runs over the 16 vectors with components 0, 1. Under the SF group ~x transforms a s

~'~, "-~ ~.+~Tp, (43)

exp( ~'3'p 3'o )~ a;2 • exp( - ~'Tp 7,T ), (44)

(45)

~ --, cosws C9~ +/sin~5 75~7s , (46)

which suggests that ~ is similar to ql(p), with

7~ ¢(P) corresponding to 7~ ~ and ~ ¢(p) corresponding to ~xTu. This equivalence can be made explicit by inserting the inverse of (17),

E f ~r/2 d4p X~ = A J-~/~" (21r)4 exp[i(p+ ~rA)Z]XA(p),(47)

into (41) and taking the Fourier transform of ~ , which leads to

~a~c(P) -- exp(ipn)Z(p)Ta~,A XA(P), ( 4 8 )

= exp(ipn)Z(p)~a~c(p), (49)

Z(p) - 2Hc°sP-'~2' n = ½(1,1,1,1), (50)

1 T~,A -- T'6 E exp(ilrA)Tba~" (51)

b

The phase factor exp(/pn) can be avoided by as- sociating ~x with the centre z + n of the hy-

percube at z. Note that the matrix elements of ~x are not independent but their Fourier modes with -~r/2 < p~ <: lr/2 may be considered independent according to (48).

J. Smit I Chiral gauge theory without gauge fudng? 91

Let us now illustrate how this matrix field ~.

can be used to obtain staggered fermion actions.

Suppose we want to put on the lattice a contin-

uum action of the form

S -" - f d4z~%,(Ot, - iA~ PL - iA~Pa)~b. (52)

Expressing this in Majorana form gives

S = -- / d4z}Ot TET.(O~ - iA~)$ (53}

R I - F "As' = (A~PL + A~' PR) 2

R T I + F (54) - Et(A PR + % P L ) E 2

For example, in case ~ -- ~bL, ~b = ~L with gauge group U (4) (we don't care about anomalies

at this stage), we have F = 7s, A~ = 0 and

.4. = + A ( : , (55)

with A O) and A (a) given by

A~ = A O) + A ("), (56) 5 A(') = A s + A~ o + Au~5, (57)

A (a ) = 1 p~, • t)5 • ~A, (-,~p~) + A. (-,~p~5). (58)

The label (a) and (s) indicate the behavior under

transposition (E~) T = 4-E~. The equivalent matrix Majorana form reads in

this case,

= _ / S d4z lTr { ~,sc~T c t 75%,[O~, ~

- 7~@(-iAO))t + @(-iA(~))t]}, (59)

where we substituted ~ --+ 7~ and used expres-

sion (22) for E. To latticize this action we intro-

duce a lattice gauge field//~, and expand it in

terms of the ~ matrices and 75,

lgu. - exp.(-i .A..) -- PLL~. + P~R~,., (60)

L . . = exp ( - iA~ ) - i A ~ ) ) = Uu., (61)

5

l ? pu f t: p5 + T ~ . . p ~ + L u . ~ , ~ , (62)

= exp(+,A~. - The and similar for R~. • (s) ;a(a)~ ~**#x j-

continuum action in matrix Majorana form (59)

can then be transcribed to the lattice as ({p --+

~p)

S _ y ~ l ^T t p ~ L t = -~Tr{75CeGC 757~[ L .+~ ~.

z#

+ PR{t.+#Rtu.]}. (63)

Inserting the representation (41) into this expres-

sion and working out the traces gives the explicit

form of the lattice action in terms of the stag-

gered fermion field X. (e.g. for Ap. = 0 we get

the ~ part of eq. (15)). This lattice action is invariant under the LS

group. The gauge fields are able to compensate

the 'flavor' traasformations induced by the ro-

tations, shifts and lattice parity by global gauge

tranformations (U.~ --- U.~, y = z +/~)

Ux,y ~ 7pU=+~,y+~ 7p, 7r 7r

u.,y -+ exp( ~pT¢ )URT; *,RT;y exp(- ~7p7~),

Um,y -- 74 U/ox,/.y 74- (64)

The U(1)e transformations are part of the local

gauge transformations. For a second cxample, let ~ and ~b in the tar-

get theory (52) be Dirac fields with gauge group

U(2)xU(2). In this ,~ase (54) with F = 75~5 leads

to

A~ = -(A~O)P+ - A~(s)P-)75

AR(~)p . (65) + . _

We write

AL,R = AL,RO)]I + AL,R(a)k~ (66) t, "'~ 2 '

rk = -i½ck,.~,~.~. (67)

where we used the fact that the U(1) fields are

of type (s) and the SU(2) fields are of type (a).

The field combinations in (65) can furthermore

92 J. Smit I Chiral gauge theory without gauge ftxing?

be rewritten as

5 A~O)P+ - A~(')P- = As 4- A~s, (68)

A O)P+ 4- AR(")P " _~ . _ = ~A, (-,~p~¢), (69)

and the transcription of the theory to the lattice now follows that of the previous example. The only difference with (~$3) is that here A~ = A~, s =

0. We note in passing that convenient expressions

for the matrix Dirac fields ¢ and ~ in the F = 75 @ 7s interpretation are given by

1 7z+b 1 -- {x+b

b

" ] 7=+b t 1 + ~,+b (70) ¢~ = "~ E X~+b 2 '

b

which satisfy 7sex = -¢~7s , 7s¢ = ¢75. The transcription of mass terms or Yukawa

couplings to the lattice can be done in similar fashion. For example,

E 1Tr ^T t ^ (71) 7sC~ z C 7sTs~z'),sm~7 ~

(72)

corresponds to the mass term in (15).

7. STANDARD MODEL

Suppose we want to put the Standard Model with three generatioris on the lattice. Staggered fermions lead to a natural multiplicity of four (the dimension of ~ space), so the three-hess of the generations is not naturally reproduced by

this. In the F = 75~s interpretation it is natu-

ral to interpret the staggered fermion multiplicity as flavour, since one field X= leads to one Dirac doublet ¢, ¢" (cf. (36)-(40)). One gener-

ation ¢(:) --- (fi , ,d,) , ~-(t) ._ ( re ,0 , ¢(q) and

¢(0 can be fitted into four doublets, i.e. into four

fields X~, X=, with explicit color label a = 1, 2, 3. For the lepton doublet lagrangian we apply

(52)-(65) with F = 75~s:

Y kTk = B. + A . y , (73)

A~ - B~( Y 4- 2 ) , (74)

where A~ and B are the SU(2)×U(1) gauge fields and Y is the weak hypercharge, which leads to

Y - i ~ l ~ 2 p _ ) = + " ' T - -

-i kt. t m p+. (75) 4- A~ ---2 ...........

We see that the U(1) part of weak hypercharge gets translated into F = 75(5, as expected.

For the quark doublet lagrangian we have in addition to A~ ,R the gluon field contribution G~Ak/2, where Ak, k -- 1 , - . - ,8 are the Gell- Mann matrices acting on the color index a of ¢(0, ~(q). This simple generalization of (52)-(65) to the case of more than one staggered fermion field leads to the contribution to ~4~:

(76) kEsym kEasym

where the summation is over the symmetric (k =

1, 3, 4, 6, 8 and antisymmetric (k = 2, 5, 7) A~s. The subsequent translation to the lattice via

the matrix Majorana formalism is straightforward, as is the transcription of Yukawa cou-

plings. The LS symmetry group is violated by the interactions, e.g. because the LS rotations

R4a of the generators (--i~t~m)/2 cannot be ab- sorbed by transformations of B~ and A~ in the manner (64).

The formulation of the QCD section of the model is the same as in ref. [12]. Color gauge invariance is unbroken in the action. The com-

bination 7sf~ corresponds in position space to

3. Smit I Chiral gauge dwory without gauge ftrh~g? .03

a simple multiplication of X~ by the phase fac-

tor ~=, which is the generator of U(1)~ symmetry corresponding to fermion number. Although

there is an anomaly in the U(1)~ current com- ing from the fermion measure, this is only a c- number anomaly which is expected to vanish in the scaling region [12]. Curious is also the fact that in the QCD fermion determinant is not real in this formulation, as if QCD were chiral, although this does not seem to affect the physics

in the scaling region. The QCD transfer operator is positive [12].

The lattice Standard Model obtained this

way appears to have fer~don number con-

servation corresponding to the global U(1)~

symmetry. This may be problematic in view of

the expected fermion num~er violation through

sphaleron transitions, as emphasized in ref. [14].

It seems advantageous to avoid this problem by

choosing a different latticization scheme. This

can be provided by using the F - 7s assignment

of fermion number, which then is surely violated

by the lattice action.

Following this line we would introduce an

F = 7~ assignment to produce four ~ , ~L from a X=. This cannot yet accomodate color and fl~ vor. Doubling the fields we have eight ~bL which can fit color and flavor of one generation of ~L %.

This means finding some way of representing the ~-'s and A's in ~b--~7,(~- i A ~ - iG,)~bL in terms of the ~'s. This can be done in some way,

which will generally not be very elegant, and color symmetry will be broken in the lattice

, r e_

action. The other fields ~R and ~b~ of the generation can be accomodated by repeating the construction with F = -7~. We will not pursue this further here, but instead view it as a

reduction from more elegant SO(10) and SU(5) Grand Unified models.

8. S0(10) AND SU(5) MODSLS

In the SO(10) Grand Unified theory the fermion fields of one generation fit into 16 dime~ional representations (~L,~L) of SO(10). We wish here to latticize the action

S / d4Z[~LT#(a. .1 ab = - - z~A# tab)~n

T ¢ - t - 1 "i" ~n ~b~ + h.c.)]. (77) + Y a( LC C 2

Here the tab, a, b = 1 , . . . , 10 are the generators of SO(10) in the 16 dimensional representation, the ~a are the 32 dimensional gamma matrices of ten dimensions, 6' = - C r is their charge con-

jugation matrix, ~11 = i~1..-~10 is the analogue of 75 and y is a Yukawa coupling to ~ 10-vector Higgs field ~ba. The projection onto the ~11 = +1 space reduces the dimension of C't~a(1 + ~11)/2 to 16. The tab c a n be represented by the ~a as

1 -I- ~11 (78) tab -"--i~a~b 2

We shall produce ~bn, ~n with four staggere~l fermion fields X= in the F = 75 interpreta-

! I

tion, and introduce gamma matrices ~j, , ~s = I !

-t~l "'" ~4, acting on the index that distinquishes

the fo"" ' . . . . . . . . ~-~ - - - X~ s. We a!so ~- . . . . *ho usua l -~ c-f-'~---

staggered fermion method. The ~a and C' may now be represented as

I 11

= . = 1 , . . . , 5 , (70)

~#+5 " ' ' " = ,~#~sw2, p = 1,.-- 4 (80)

" " (81)

#1

C = c 5q, 2. (82) I t

The ~k are Pauli matrices that enlarge the di-

mension of ~ ' space from 16 to 32, and C 6, C~, are charge conjugation matrices for the ~, ~'. The

generators take the form I I

t . . = - i ~ . ~ . , t~+5,.+5 = - /~ ,~ , , /

t . , .+5 = -~#~ . , # , l . = 1 , - - . , 5 , (83)

94 J. Smit I Chiral gauge theory without gauge f ~ g ?

and in the action (77) we recognize the 16 di-

mensional 10-vectors ta

= EC~,"15ta, ct~'t~,, (84)

I

tu = ~ 5 , P = 1,. . - ,5 (85)

= (so)

tl0 = - i , (87)

where we carried out the projection onto ~11 = 1 and used (22).

When we now foibw the steps (52)-(59) to ar- rive at the Majorana form of the action, we get a 75 whenever t~,v is of the type (s). This depends

I

on the symmetry of ~l,, similar to the case of the A's in (76). Since 78 is a four link operator on the lattice, we get awkward symmetry breaking this way.

A more attractive model results by changing the definition (30) of ¢ and ~ by including C~,,

= PL¢ . + PR( LEtC¢) T. (88)

This leads to just an overall factor 75 in the kinetic part of the Majorana form of the action,

• ab S = - d4z{~@TEC~tT.75(0 ~,q-,Aj, tab)@

-- y~ba(½ql T EC~,PLta~ "k h.c.)l}. (89)

The lattice version of this action is invariant under the LS group, since the lattice gauge fields

• 1 a b exp(--z~A~xt~b ) and the Higgs fields Ca are able to compensate the rotations and sign changes of the ~'s induced by the LS transformations in the manner (64) (there is also the freedom to compensate by global transformations in ~' space). lu ~he unitary gauge version (to be used as initial action in the gauge non-fixing approach described in sections 2-5), the angular variables of ~ba become fixed and the Yukawa coupling terms then violate some of the LS symmetries. The action is not invariant under U(1)~.

An natural reduction to an SU(5) model is possible by using an embedding of SU(5) in SO(10) in which the antisymmetric generators

of U(5) are represented by t~,v + t~+5,u+6, p, v - 1 , . . . , 5, the symmetric off-diagonal

generators by tl,7, . . . , t1,10, g2,s, . . . , t2,10, . . . , t4,10, and the diagonal generators by tl,e, t2,7, • . . , t5,10. This reduction leads to a lattice action that is also invariant under LS, but not U(I)~. A subsequent, reduction to the Standard Model with the usual embedding of SU(2)x(1) breaks LS again, while the U(1)~ symmetry remains

broken.

9. DISCUSSION

Our review of the non gauge fixing approach to a nonperturbative definition of chiral gauge theories has focussed on staggered fermions. With the staggered fermion method we can regularize in weak coupling perturbaton theory any theory in which the number of fermion fields in the (½, 0) representation of the euclidean rotation group is a multiple of four, and similarly for the (0~ ½) representation. This includes the Standard Model and the SO(10) and SU(5) grand unified theories (we take the (possibly decoupling) right handed neutrinos for granted). The four fold multiplicity need rot be viewed as a lattice artefact, but can be seen as a consequence of the connection with the Dirac-Kghler formulation of fermions in the

continuum [ 15]. The symmetry breaking induced by the lattice

can be studied in the loop expansion. Such cal- culations have not been done yet for the theories considered here, in which the staggered multiplicity is gauged.

Invariance under the discrete staggered fermion symmetry group LS puts restric.~ions on the counterterms. This favourable situation

J. Smit/ Chiral gauge theory without gauge fvdng? 95

could be achieved for the SO(10) and SU(5) model. Experience with staggered termions in QCD [16] suggests that every term in the expansion of the lattice gauge field in LS representations needs its own renormalization factor ~, e.g. (cf. (62))

!

1 pa

+ ~A V~.p~.5 + ~pU~=~5, (90)

where the K's depend on the coupling con- stants. In addition counterterms will probably be needed for the restoration of euclidean rotation invariance (Lorentz invariance), as for Wilson's fermion method in this context [6,5]. The dimensionless lattice artefact counterterms are however expected to be small in aymptotically free gauge theories because they are proportional to the bare coupling constantsi Even in non-aymptotically free theories they may be relatively small in practise because the renormalized couplings usually cannot get large because of triviality (see ref. [18] for a Yukawa model example).

Dimensionful counterterms such as mass terms for the gauge fields world have to be chosen in the scaling regions of type A, B or C. This is really a non-perturbative problem.

In the non-gauge fixing approach described in sects. 2-5, the interactions of the gauge degrees of freedom V are non-perturbative, because they appear in the action with couplings of order one. This makes ~.n analytic determination of the counterterms very difficult. (However, in a broken phase perturbation theory may still be ap- plicable for some models. When (V=) # 0 an expansion like V= = 1 + (fluctuations proportional to the couplings) may make sense). One would very much like the V interactions to respect LS, because it offers protection against lattice fermions leading to crazy continuum fermions.

In the Standard Model the LS symmetry is hard to maintain. A solution to thi~ problem could be as follows.

Suppose we enlarge the gauge g,oup such that the resulting V interactions do not break any- more LS. For example, we may embed the Stan- dard Model in SO(10). We then eliminate all gauge fields but keep the corresponding V's, obtaining a fermion scalar model that respects LS. We then reintroduce only the subset of gauge

fields we want. The distortion of the counterterms introduced by their LS symmetry breaking is expected to be mild for small gauge couplings.

If not needed for Higgs fields, the V's should decouple from the dynamics, which may require some tuning. This would only be possible if they correspond to non-anomalous fermion content. If anomalous, such V's corresponding to longitudinal degrees of the eliminated gauge fields could not decouple from the fermions. Similarly in anomalous gauge theories the scalars would not decouple and presumably lead to 'triviality'.

A simple example is the SU(2)xSU(2) fermion-scalar model given by

I #

- - -

= #

!

TrY, " + ~ mj, p~ ¢=¢=7~,, =P

U.= = ..xp(-~.A.= 7p7,~),

(91)

(92)

where U' ~,~: is pure gauge, U~= = V~V~+p and p~ is the radial component of a scalar field. Re- call that multiplication of ¢~ by 7's on the right corresponds to flavor transformations and that e.g. PL¢ - C PR in the formalism with fermion

number F = 75 ® 75- The embedding gauge group is supposed to be U(1)xSU(2)×SU(2), with U(1) corresponding to fermion number, V~ E SU(2)×SU(2). For rn~ = 0 the model is invariant under the SF group generated by LS

96 J. Smit I Chiral gauge theory without gauge fixbzg?

u(1 ) , .

This model may be gauged with SU(2)xU(1) gauge fields as for a lepton doublet of the Stan- dard Model,

v , . = v.t Y _ rz "~-I/"R

T3~ ln "l + exp[- iB( ,Y + -~-jjrz.l V~+p, (93)

= - i}ckz.TtT. . (94)

The LS symmetry is now broken but since the fluctuations of the gauge fields A~ and B are suppressed for sm-dl gauge couplings the change in the ~ renormalizations (90) will be small.

The question is whether for rn r = 0 the scalar fields can decouple, and in case of non-zero mr, whether only the Standard Model Higgs fields remain coupled. We can discuss this in the scalar fermion model (92), treating the gauge fields as external.

The one doublet model (92) has the global SU(2) anomaly, so the scalars will i~ot decou-

pie and the model may not even be well defined [19]. So let us double the number of fermions. There is now no anomaly reason that can prevent the scalars for m r = 0 to decouple from the dynamics, i.e. they may decouple from the correlation functions of the currents of the SU(2)xU(I) gauge fields, which are nonanomalous.

To formulate this decoupling more precisely we consider the equivalence (11)- (12). The model

(92) is to be equivalent in the appropriate scaling regions to the O(4) Yukawa model given by

~ r

~ r

1

x r

with the formal identification Y~u~

(95)

the average of ~r~ over the hypercube at z in the negative directions.

The equivalence of the two models (92,95) expresses the desired restoration of gauge invariance and the decoupling of the scalars for

m r -- 0. For m r = m6r , 4 and a chiral representation of the 7's, • depends only on V = VL, V~ and its conjugate (cf. (8)). Writing VL = V1/2W,

VR = V1 /2 tW, we see that this equivalence im- plies that for m ~ 0 the non-chiral part W of the V~,R still decouples.

Even if the scalars decouple from the SU(2)xU(1) dynamics, they are expected not to do so completely from correlation functions in- volving the fermion number current. This curent should have an anomaly leading to fermion number violation through sphaleron transitions. It is however expected to be exactly conserved in the scaling region because of the global t ~1) invariance of the action (92). The scalars therefore compensate the desired anomaly and cannot decouple from such correlation functions.

As a way for putting the Standard Model on the lattice all such models with a global U(1)

symmetry run into this problem, as emphasized in [14] ~'~:~ a~l . . . . :~ ~k~^.~ : . the F =

of putting the Standard Model on the lattice or

in the SU(5) and SO(10) models, which do not have the global U(1) invariance in presence of the gauge fields.

Positivity of the transfer operator (hermiticity of the 'hamiltonian') is generally not expected

to be a property of the lattice models described here. The reason is that the discrete time 'derivatives' involve interactions with the ga,age degrees of freedom V (a consequence of euclidean ro t~

tion invariance), which leads to a non-hermitian transfer operator [20]. In theories with exact gauge invariance the V's can be transformed out of the time derivatives and the transfer operator

J. Smit I Chiral gauge theory without gauge foa'ng? 97

is positive [12,20,21]. Hence it is reasonable to

assume that if gauge invariance is restored in a scaling region, positivity will emerge as well.

Two dimensional versions of (92) and (95) are being studied [17] but no firm conclusion could be drawn yet because the fluctuations in two dimensions are so severe.

The four dimensional Yukawa mcdel (95) has also been studied recently [18], and it is found to provide a numerically very efficient formulation.

Although the model respects SF, two counterterms are still needed to restore 0(4) symmetry in the scalar sector, which have the scaling form ~ ~b 4 and ~ ( (9~b~) 2. The study was carried

out without these counterterms. Even for relatively strong Yukawa coupling the effects of the 0(4) symmetry breaking o~ the physical qu~nti- tie~ was found to be small.

The tuning problems with the staggered

fermion models described in this review are severe from a numerical point of view (apart from the problem of complex fermion determinants). However, the results found in [18] are encour- aging and suggest that we may get interesting results even by ignoring the appropriate tuning.

A - | . . . . . 1 - J . . . . . L _~kUfi.IIIUW l u u ~ U l l l e l J [ b .

I would like to thank W. Bock and J.C. Vink for

discussions and the organizers of the workshop and the particil~znts for creating a wonderful at- mosphere which stimulated heated discussions. This work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM).

References

[1] C. Becchi, these proceedings. [2] See for example M. ~l'onin, these proceedings. [3] L. Maiani, these proceedings.

[4] R. Sarno, these proceedings. [5] A. Borrelli, L. Maiani, G.C. Rossi, R. Sisto,

M. Testa, Phys. Let 221B (1989) 360; Nucl. Phys. B333 (1990) 335.

[6] J. Smit, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 451; Nucl. Phys. B (Proc. Suppl.) 9 (1989) 579; Nucl. Phys. B (Proc. Suppl.) 26 (1992) 480.

[7] J. Smit, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 3. [8] W. Langguth, I. Montvay, Z. Phys. C36 (1987) 725;

A. Hasenfratz, T. Neuhaus, Nucl. Phys. B297 (1988) 205; For a review see J. Jers~, Proceedln~.~ Physics at HLRZ, ed. M. Kremer, HLRZ 1990, p. 85.

[9] M. Lflscher, P. Weisz, Nucl. Phys. B 290 [FS20] (1987) 25; Nucl. Phys. B 295 [FS21] (1988) 65; Phys. Lett. B212 (1988) 472; Nucl. Phys. B313 (1989) 705; M. Lfischer, in Nonperturbative Quantmn Field Theory, eds. G. 't Hooft e~. ed. Plenum 1988.. Nmner- iced simulations confn-ming these analytical stud- ies are still being performed. For an exmnple, s~e A. Hasenfratz, K. Jansen, J. Jers~k, C.B. Lang, T. Neuhaus and H. Yoneyama, Nucl. P ~ s . B317 (1989) 81. See also the review in [8].

[10] A.K. De, these proceedings; M.F.L. Golterman, D.N. Petcher, these proceedings.

[11] M.F.L. Golterman, D.N. Pet cher, J. Smit, Nucl. Phys. B370 (1992) 51; W. Bock, A.K. De, J. Smit, HLRZ-91-81, to be published in Nucl. Phys. B.

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[13] M.F.L. Golterman, J. Stair, Nucl. Phys. B255 (1985) 328; M.F.L. Go!t:.rman, Nucl. Phys. B273 (1986) 663; Nud. Phys. B278 (1986) 417; H. Joos, M. Shae- fer, Z. Phys. C34 (1987) 465.

[14] T. Banks, Phys. Lett. B272 (1991) 75; T. Banks, A. Dabholkar, RU-92-09; T. Banks, these proceed-

h~s. [15] P Becher. H. Joos, Z. Phys. C15 (1982) 343.

Mo GSckeler, H. Joos, in Progress in Field Theory, e~ . G. 't Hooft et al. Plenum 1984.

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61. [171 W. Bock, J. Sn~t, J.C. Vink, work in progress. [18] W. Bock, J. Stair, J.C. Vink, ITFA-92-13. [19] E',. Witten, Phys. Lett. Bl17 (1982) 324; E. D'Hoker,

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