Non-perturbative renormalisation and kaon physics

lllgLII lR~lmH-" f d ~ m i l ' l

ELSEVIER

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 38-46

Non-Perturbative Renormalisation and Kaon Physics Massimo Testa a aDipartimento di Fisica, Universita' di Roma "La Sapienza", P.le A.Moro 2, 00185 Roma,Italy INFN-Sezione di Roma, Italy A general review is presented on the problem of non perturbative computation of the K -+ ~r~r transition amplitude.

The term proportional to Or, does not contribute to the physical transition amplitude (a matrix element with zero four momentum transfer, Ap~ = 0), being a total four-divergence. Since Mw > > AQCD we can use the Operator Product Expansion, to get:

1. I N T R O D U C T I O N Kaon Physics is a very complicated blend of Ultraviolet and Infrared effects. The physical problem is the large enhancement shown in K-decays:

r ( K + -~ ~r%r°) 1 [A(AI = 3)12 r ( K o -+ ~+=-) = 4-66 = [A(AI = ½)[2 =~ 1

3

=~ A(AI = ~) ~ 20A(AI = ~)

(5)

= ;~,,-q~2[c+(u)oC+)(u)+ c_ (u)o(-)(u)]

(1) where A. = V, dV*, and:

Due to the difficulty of putting the Standard Model on the lattice for technical (Mw, z > > ~) and theoretical reasons (the presence of the Wilson term breaks the chiral gauge invariance) we must integrate analytically the heavy degrees of freedom (W's, Z's). Using second order Weak Interaction perturbation theory, we get for the effective low energy hamiltonian density of non-leptonic decays:

o (~) =

HAeH S = I = g~v / d4xD(x; Mw)T[J~,(x)J+,(O)] +

+~o.,

(2)

where D(x; Mw) denotes the free W propagator:

DCx; Mw) --

f (2~)4 d4p(p2exp(ipx) + M~)

(6) •

(3)

and:

Om - (ma + md)$d + (ms -- md)s75d + b.c. (4) cm is fixed, in Perturbation Theory, so that quark masses are not modified by Weak Interactions. 0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0920-5632(97)00694-4

-

c]

In Eq.[5] the subtraction point ju is chosen so that p > me and we are in the favorable situation of propagating charm with the consequent GIM cancellation[1]. O (-) is an operator contributing to pure A I = ½ transitions, while O (+) contributes both to A I = ½ and A I = ~ transitions. The C±, computed in Perturbation Theory, show a slight enhancement[2½:

C_ (p ~. 2 Gev) ] C+(tJ : 2 Gev). ~ 2

(7)

The rest of the enhancement (~. 10) should be provided by the matrix elements of O (+) and is a non perturbative, infrared effect. In this talk I will be concerned with the Wilsonlike fermion formulation. Several interesting papers deal with problems related to non-octet Ktransitions, both within the Wilson and staggered fermion approaches[3½.

M. Testa~Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 38-46

The difficulty of the problem consists, first of all, in giving the correct ultraviolet convergent definition of the operators O (+). It is well known that, in order to construct a finite composite operator of dimension 6, (56(/~), we must mix the original bare operator, 06 (a), with bare operators of equal (O~i)(a)) or smaller (O3(a)) dimension, in general with different naive chiralities[4]. In the present case we have, schematically:

39

smaller dimension, with different naive chiralities: O° = o . +

o(2)

(9)

i

A general way to see why Eq.[9] is correct, is to think of the lattice discretization of QCD as a chiral violating order a perturbation of the continuum field theory, described by an action: SQCD = Scont "1-

(56(v)

=

×

+

+

(s) c3(g°1o3( 11J a3

i

Being QCD asymptotically free, one could think that the mixing coefficients in Eq.[8] could be reliably computed in Perturbation Theory. Numerical attempts (and theoretical considerations) show that this is not necessarily the case. We are therefore led to employ general non perturbative techniques. On very general grounds, these techniques are based on the systematic exploitation of (continuum) symmetries. In fact it turns out that the O(~=)'s have definite transformation properties under the SU(3)®SU(3) chiral group (and some discrete symmetries). This fact suggests the use of Chiral Ward Identities (or equivalent methods) to classify the composite operators[4],[5]. The Chiral Ward Identities, however, do not fix in a completely unambiguous way the O(+)'s. Some ambiguities are left, outside of the chiral limit. These ambiguities turn out to be irrelevant in cases in which Current Algebra may be applied. They, however, limit the application of the method to light meson decays. In order to treat other interesting physical processes, involving e.g. B-mesons, other methods must be used, as discussed later.

2. U L T R A V I O L E T P R O B L E M S 2.1. T h e Chiral L i m i t As already stated, in order to construct an operator with the correct (continuum) chirality, we must mix it with operators O~ ) of equal or

+ f d4yq(y)mq(y)+a f dayW(y)

(10)

where S¢ont describes the continuum theory, in the chiral limit, and W(y) is a chiral violating, dimension 5 operator whose origin can be traced back to the presence, in the Lattice Action, of the Wilson term. The existence of W(y) forces us, already in the chiral limit, m = 0, to perform a complicated subtraction procedure in order to define finite ultraviolet composite operators with good chiral transformation properties. Formally we may write:

(AI O.(0)[B)(tat) ~ (AI (5.(0)[U)(con0 + an (AI T(5.(O)(f + ~ ~.t

(11) dyW(y))n [B)(cont )

n>l

where (5.(0) is, by definition, a finite operator which satisfies the chiral continuum (integrated) Ward Identities: lim

m--a0

f dz (AI T(&op[q(z)mq(z)](5.(O)) IB)(~o.0 + + (A I 60p(5.(0) IB)(con,) = 0

(12)

where 6op denotes the operator chiral variation. The Ward Identity Eq.[12] is written in a very general form and can be applied both to the case in which chiral symmetry is restored in the massless limit, and to the case, realized in QCD, where chiral symmetry is spontaneously broken as m -~ 0. In this case the mass insertion term in Eq.[12] does not vanish in the chiral limit because of the presence of the pion pole. It is to be noted that no prescription is needed for the (generally divergent) insertion of ~5op[gl(z)mq(z)]and (5~ (0),

M. Testa~Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 38-46

40

since we are considering the limit m -~ 0 and the divergent part does not contain the pion pole. The a dependence cannot be directly read from Eq.[ll] because of the presence of multiple insertions of composite operators 0a(0) and W(y) which are, in general, divergent. For example for n = 1 we have:

This happens also in the continuum. In order to discuss the case m ~ 0 it is more suitable to think of a generalized mass term of the form qRiMoqL j + h.c. (in the end we shall, of course, choose M as an hermitian diagonal matrix). If we transform:

(A[ Oa(O)[B)(ta 0 ~-, (A[ Oa(O)[B)(cont) +

qLi ~ Likqn~

+a ] dy (A[ T(Oa(O)W(y))[B)(cont )

qm -~ Ri~qak

(13)

The (continuum) Operator Product Expansion implies (up to log corrections):

T(O¢,(O)W(y))~0 ~•

1

y(f+do-do(O)

O~)(0) (14)

Because of the multiplying a factor, in Eq.[13] only the short distance behavior contributes at leading order in a:

(A[ O.(0)[B)(~.t ) (AI 0.(0)]B)(co,,t)

+

(15)

1

• a(dO-do(i,) (A[O(i)(0)]B)(cont)

(17)

and, simultaneously:

M ~ RML +

(18)

then the mass insertion remains invariant. Up to first order in M, we have: (A[ Oa(0)[B)(M) ~ (A 16a(0)IB)(M=O) + + f dy (A[ TOa(o)q(y)Mq(y) [B)(M=O) (19) As usual the double insertion of composite operators gives rise to a divergent contribution determined by the operator product expansion: T(0a(0) FI(y)Mq(y)) ~,

s

y~O

The O(2)s in Eqs.[14],[15] are operators which generally belong to chiral representations different from that of Oa (apart from trivial multiplicative renormalisation). In the Chiral Limit the Ward Identities completely determine the structure of the composite operator Oa. In fact Eq.[151 implies the mixing structure which has the form indicated in Eq.[9]: (AI 0.(0)[B)(cont) m, (A[ Oa(0)[B)(lat) 4(16) 1

-- E. a(do-do(O ) $

(A] O (0 (0) [B)0at ) +

"'"

Oa satisfies the Ward Identities up to O(a). We also see that the leading divergent part of the mixing, a(dO--dO(i x ) ), is determined by perturbation theory (short distance). 2.2. Softly B r o k e n C h l r a l S y m m e t r y When the explicit chiral symmetry breaking is switched on (m ~ 0), other divergences (and ambiguities) may arise.

1 y(34-do--dA,) (MA)~)(0)

~' E

(20)

i

where the operators A~ (0) have chiralities different from that of 0a(0). (MA)(2) denotes a suitable combination of the mass parameters M and the operators Aia (0). Under a simultaneous chiral rotation, Eq.[17], on both sides of Eq.[20] and the corresponding transformation Eq.[18], we see that the operator combination (MA)(2) transforms according the same chirai representation as Oa(0), since qMq is invariant[6]:

6(MA)(2 ) = (5M A)(2)+(M 6opA)~) ~ 6op03(21) We can now define the finite operator: 1 ~(do-dA,-i) (MA)(2)(0)(22)

bo~(O) =---Oot(O) -- E i

and study its Ward Identities.

M. Testa~Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 38-46 %

+ (AI 5ovO,~(O)IB) +

We start with the (divergent, but regularised) Ward Identity:

+Z a(d6-dA~ 1

f dy CA[T( Motion[~li(Y)qj(Y)]()a(O)) IS) +

(28)

CA[ (UtiovA)(~)(0)]B) +

i

-Z

+ (A150p6.(0)IS) =

= - f dy (AI T(q(y)(SM)q(y)()a(O))IB)

41

1 a(dO ~dAi - - 1 ) (A[ 5(MA)(~)(0) IB) = 0

+

Eq.[28] is easily seen to be equivalent to:

(23)

f dy (A 1T*(M,jSon(qi(y)qj(y))

%

+ (AI

~onO~(o)IB)=

o

Divergent contributions are present in Eq.[23]. They arise from: 1) the double operator insertion in the Tproduct 2) since ~op denotes the operator chiral variation, ()a (0) transforms as a superposition of the representations of ()~ (0) and of A~ (0): %

5onOc,(0) - 5onOa(0) + 1 (MJonA)(~)(O) -- Z a(dd~--dAi--1)

(24)

i

so that the single insertion of tion~)a(O) is also divergent. In order to make everything finite we may redefine a T*-product according to the prescription: %

T*(O,~(O) q(y)Mq(y)) = = T((~a(0) q(y)Mq(y)) + 54(u) -

Z

(25)

a(dO--dA,--')(MA)(~) (0)

i

In this way we rewrite the Ward Identity Eq.[23] as:

- / dy (A IT*q(y)(SM)q(y)()a(O)

IB) +

+ (AI ~op¢).(O)IB) + -

1

(26)

a(d6_dA _1 ) (A I (SMA)(~)(O)IB)

~

=o

i

From: (27)

we then get:

[ dy (A IT*q(y)(tfM)q(y)()a(O)IB) J

+ (AI 5 O . ( 0 ) I B ) =

- / dy (A IT*(q(y)(6M)q(y)

+

bo(0)) IB) +

%

+ (AI ~ O.(0)IB) = 0

(29)

So that, in the end, with a joint redefinition of the operator and the T-product containing the mass insertion, we managed to have a renormalised Ward Identity of the canonical form. It is now clear why the Ward Identity does not uniquely define the finite operator (~(0), outside the chiral limit. In fact we may change the definition of the T*-product and accordingly change the definition of the operator Oa (0) by a finite mixing with (MA)~) , leaving the form of the Ward Identity unaltered. This ambiguity is harmless if we are in a position to apply the low-energy theorems of Current Algebra (small quark masses) or when the physical matrix elements of (MA)(~) are zero (e.g. if they are proportional to four divergences of current), but it can represent a serious obstruction when heavy hadrons are present. We conclude this section by remarking that the situation presented here is a bit simplified. In fact we should also consider multiple insertions of q(x)Mq(x) and W(y), which, in general, generate further distortions. Such terms will be included in the following considerations. 2.3.

-(5M A)(~) = (M 5opA)(~) - $(MA)(~)

bo(o)) ]B) +

Non

Perturbative

Renormalisation

A different, but equivalent construction of composite operators (in agreement with the Ward Identities for physical applications) can be achieved through the so called Non Perturbative Renormalisation[7].

M. Testa~NuclearPhysicsB (Proc.Suppl.) 63A-C(1998)38--46

42

This method proceeds in complete parallelism with the corresponding procedure used in perturbation theory. In fact non perturbative renormalisation requires a non-perturbative gauge fixing procedure, usually in the form of a suitable discretization of the Landau gauge O~G~ = 0 x The basic idea is to con~ider the insertion of the renormalised operator 0 a = Oa + ~ c~O(a0 i

in Green's functions containing elementary quark (and gluon) fields. Symbolically we write: G~(p) -

F.T. (ba(O)qCx)~l(y)) F.T. denotes the appropriate

(30)

where Fourier Transform of the external legs. We expect that, for large (continuum) virtualities ~ > > p ~/~ > > AQCD, the chiral violating form factors of G~ (p) (which come from physical soft mass breaking and/or spontaneous symmetry breaking) should be suppressed by inverse powers of p. As an example we could consider the correlator F.T. (s(x) gRdL(O) d(y)). In this case the conditions are:

F.T. (SL(X) ~RdL(O) JL,R(y)).._~co ---~0 F.T. (SL,R(x) ~RdL(O) dR(y))j,..,oo = 0

(31)

This behavior is a direct consequence of the (integrated) Chiral Ward Identity. I will discuss, for simplicity, the case in which mixing with operators of lower dimension do not occur. This case is directly relevant for the study of the transitions K + --~ r + r ° and K ° +-~/~o. The integrated Ward Identity has the form:

+F.T. (5 b,(O)q(x)#(y)) + +F.T. (ha(O) 5q(x)(l(y)) + +F.T.

(ba(0)q(x) 5q(y)) -- 0

(32)

1Gribov copies should not present a serious problem in the present case, because gauge fixing is only used in the computation of short distance quantities which are probably immune from their presence.

Because of the explicit M factor, the 1.h.s. of Eq.[32] has one operator dimension less than the individual terms appearing on the r.h.s, so that, at large virtualities, it vanishes one power faster. For asymptotically large # we then get from Eq.[32]:

F.T. (5 ba(O)q(x)q(y)l~, + +F.T. (ha(O) 5q(x)~(y))~, +

- F.T. (5 [5.(0)q(z)q(y)]). = 0

(33)

which is equivalent to the Wigner-Eckart Theorem (at large virtualities):

F.T. (01 [Qa,ba(O)q(x)q(y)]

10)=0

(34)

Thus, adjusting the mixing coefficients c~ so that the chiral violating form factors vanish asymptotically, we get an answer equivalent to the application of the (softly broken) chiral Ward Identities. Apart from its simplicity, a further advantage of the non perturbative renormalisation approach is that it can 01so determine the absolute normalization of 0a(0), corresponding to the one adopted in perturbation theory, by imposing the same kind of conditions as, for example:

F.T. (qqqq()a~

= 1

(35)

P

3. I N F R A R E D

PROBLEMS

In order to compute the K --~ 7rTr width we have to evaluate the matrix element

(out) Hw aK> with two interacting hadrons in the final state. This is not easy to do in the euclidean region In fact it can be shown[8] that the usual strategy of taking large time limits of euclidean correlators does not give direct information on (out) (~r(p__)Tr(-p_)[Hw [K), but is contaminated by final state interaction effects:

(lrE(tl)~r_E(t2)Hw(O)K(tg)) ~

(36)

M. Testa~Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 38-46 emKtK

t ~ -~ - oo

--E=tl --Eft2

4.1. Direct M e t h o d s

2

tl>>t2>>o

((o,,t) (Ir(P__)lr(-P) I H w IK) + +(,n) (lr~)~r(-p_)[ g w [K) + e 2 ( E ' - m ' ) t 2 c M °'s') ) lfTf ~ ~

2m~r

2Ex

where ... ~r~r--* ,.r is the 2m,r

~

off-shell

2E~r

scattering amplitude representing the final state interaction. If p_ = 0 we have:

(~ro(h)~ro(t2)Hw(O)K(tK)>

43

~-.

t K --+-- o0

(37)

These methods use the strategy firstenvisaged in Ref.[10]. ,The relevant matrix element is evaluated for Irs at rest, which minimizes the final state interactions. Flavor and C P S - s y m m e t r y ( C P ® (s ++ d)) severely constrain the form of the subtractions needed to define the Parity Violating renormalised Weak Hamiltonian density: : (+) O(p.v.)(/~ ) = Z(p.v.)(~a C+) (39) ) X

(+) × (ocp v)(a) + (me -

~(±)

where:

tl>>t2>>0

H w IK) x c

× (1 + ~ M , . r --+ ,r,r ) x/t2 2,,,,, 2,,,,, Eq.[37] shows t h a t only for 7r's at rest (and weakly interacting) it is possible to extract a meaningful matrix element. For K -~ 7rTr this could work because chiral symmetry implies a small final state interaction (Adler zeros). It is however a subject which certainly deserves further study. We conclude remarking that the euclidean Green's function for the A I = 1 transition, defined in Eq.[37], contains, in general, a disconnected contribution

o(+)(t,)Ig(o))[

A rttR ¢±)=Er~d =

~l~Wtd

(41)

and reach the real world through chiral perturbation theory: "~phys

K, phys 2~n2K, l a t

~, phys A(±)

(42)

rn*mrad

(38)

which must, of course, be subtracted in order to get the physical result. 4. H O W T O C O M P U T E

(40)

If we choose a world in which m , = md we see from Eq.[39] that the power divergent mixing with Op is eliminated. In this situation no subtractions are needed. The absolute normalization can be found both perturbatively or non perturbatively, e.g. using external quark states, as in Eq.[35] 2. We can therefore compute the zero threemomentum transfer matrix element:

4(±) ~

(~ro(tl )Tco_(t2)Hw ( O ) K ( t K ) ) disc =

(~'g(tl firo_(t2)) (Hw(O)K(tK))

Op = (ms - rnd)~75d

K --~ Irlr

This section is based on the considerations exposed in Ref.[9]. We will discuss several possible ways to compute K -+ Ir~r: • The Direct Method (and variants thereof) (Parity Violating) • The Parity Conserving Method • First Principles

Another possible way to eliminate the Op subtraction in Eq.[39] is to use a zero four-momentum transfer matrix element. The basic idea is to work with non perturbatively O(a) improved fermions (i.e. an S-W improved action with a coefficient c s w determined as in Ref.[ll]), and choose the quark masses so that m K ----2rn~. In the continuum, since we are working at zero four-momentum transfer, we would have:

(h[ On [ff) (x (ms Trod)(h[~75d[ff) = -- (hi 0,(~7,78d) Ih') = 0

(43)

2The disconnected contribution, Eq.[38], vanishes by C P S symmetry.

44

M. Testa~Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 38-46

The computation performed on the improved lattice would give:

non-

(h I Op Ih') c( (me + md) (hI ~75d Ih') _

(44)

a

through the same pre-subtraction described in Eq.[48] needed also to eliminate disconnected contribution, Eq.[38]. Since no chiral extrapolation is needed, this method could be suitable to the study of B-decays.

a

= (hi 0~(~7.75d)Ih')

( h i ) ( A Ih') _ 0 ( 1 )

a

4.3. P a r i t y C o n s e r v i n g M e t h o d This was the method originally proposed in Ref.[5]. It is based on the fact that chiral perturbation theory relates the physical matrix element:

a

and therefore the unsubtracted result would be affected by an error O(1). Working with improved axial current and pseudoscalar density, we would have:

v(p.v.)(#) ig0> = i7(+ )

(hi Op Ih') oc (me + rod) (hi $~f5d Ih') (l a =

(hi a.(~.75d)Ih')

+ ~O(a2) = O(a)

a

FI

m.

(49)

to the matrix element of the parity conserving operator:

(45)

a

2

In this case we may then compute directly:

A(+) .,K=:.,.

(+) (#~, IK°(q)) ( ' + ( p ) l 0 (p.o.),

= (.(o)~(_o)I o IP~')v')(#) IK(0)) ,,,,,=2,,,.

without worrying about the subtraction. The connection with the real world is again obtained through chiral perturbation theory: d(+)

"~pays ~

2

(46)

m 2--

K, lat

~

""~, lat

"~,nK=2m.

4

(+) (+) x--, ..(+).~(+), , = Z(p.c.)(pa)[O(p.c.)(a ) + ~..t.;i o(i ) (a) + (47)

i=1

+(m~

In order to eliminate the disconnected contribution, Eq.[38], we may work with finite, intermediate renormalised operators obtained by adjusting C (±) so that: (01 o=((+) p . v . ) ut,. IK) = 0

-5(+)~2+,1(+)p. q -~

OIp~.).) (p) has a rather complicated ultraviolet structure: (~(+) (P.o.) (P) = (51)

2

m K , phys -- m ~ , plays A(4-)

=

-

m,,)c(.+)o~,(a) +

+(mo - m,,)--~ c~) os(a)] where:

Oo = go(me - md)~a~,,,G~,,d (48)

This subtraction does not change the decay rate because it amounts to change the operator by a four-divergence and reduces numerical errors. 4.2. R e a l W o r l d

Still another possibility, based on the proposal of Ref.[12] to deal with final state interactions, is: 1) parametrize the real world K ~ 7rr amplitude in the Minkowski region, in a way apt to analytic euclidean continuation; 2) try to fix the parameters by a numerical study of the lattice euclidean correlation function. This method would require to deal with finite operator insertions (also for Ap ¢ 0) to get a smoother extrapolation. This is accomplished

(52)

and: o s = ~d

(53)

however, in this case we only have to compute two particle matrix elements (which are easier) and, therefore, we do not have to worry about final state interaction effects. 4.4. F i r s t P r i n c i p l e s The distortion of the composite operators in the effective Weak Hamilton±an density is due to the fact that, in its computation, we insist to integrate up to zero (lattice) distance. If we could integrate up to a finite, physical, distance and perform analytically the final continuum integration the problem could be overcome.

M. Testa~Nuclear Physics B (Proc. Suppl.) 63A-C (1998) 38-46

In order to accomplish this, we recall that the Weak Hamiltonian density Eq.[2] is defined through a short distance expansion. We can therefore directly get the continuum matrix elements (hi O (i) (p) ]h') by studying, numerically on the lattice, in the region a < < the quantity: Izl < < h QOD, -a

2 = E c~(x; it)(hi O (i) (p)lh') i

(54)

where the coefficients ci(x;#) are related to the C i( _2_ Mw ) appearing in the Weak Interaction hamiltonian density by:

Ci(...~_ww)M6-d, = f dxD(x; Mw)ci(x; p) t

(55)

order to fix the matrix elements of the relevant operators. In fact we can consider a world in which the top quark has a fictitious mass rht such that 1/a > > rnt > > m e > > AQCD s o that we can choose the subtraction point both in the region 1/a > > # > > rht > > mc > > AOCD where GIM is operative, and 1/a > > rht > > p > > mc > > AQCD, which mimics the real world. Identifying the two matrix elements (appropriately evoluted in p and studied as functions of rht) we can then extrapolate the transition matrix elements to the real world[9]. Acknowledgements My thanks go to C. Dawson, G. Martinelli, G.C. Rossi, C.T. Sachrajda, S. Sharpe and M. Talevi, with whom most of the consideratons presented in this talk have been done.

t

The Ci(M--~w)s, and therefore the ci(x;#) s, are reliably computable within perturbation theory. If we use in Eq.[54] the O(a) improved weak currents, then the whole computation will be automatically O(a) improved. In the case of K --+ ~rTr, the operators contributing in Eq.[54] are O (+) and (m 2 - m~,)Op, 2 in the parity violating case, or (m2c-m~)(m, + ma)08, in the parity conserving case. The perturbative expression of the c~(x; p) is: (56) ~(i)

(a,(1/x) ~ ~o

o~ \ ~,(l,) ]

1 + ~)7

(i) log(xp) 4- ...

where the anomalous dimensions 7 (0 are given by: 7(+)=4

45

7(-)=-8

7 (p's) = 1 6

(57)

As with the previous method, no chiral expansion is necessary and therefore it could be applied also to B-decays.

4.5. Top Quark and C P Violation The difficulty here is that the top quark is essential and the GIM mechanism is not operative. Even in this case, in analogy with the treatment described in the previous section, it is possible to work in a world with a light fake top quark in

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46

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