On dimensional methods in rare b decays

Physics Letters B 321 (1994) 113-120 North-Holland

PHYSICS LETTERS B

On dimensional methods in rare b decays* Mikotaj Misiak 1 Physik-Department, Technische Universitiit Miinchen, 85748 Garching, FRG Received 27 September 1993 Editor: P.V. Landshoff

For the past several years, there has been a question of whether the dimensional reduction and the usual dimensional regularization give different results for the QCD-improved b -~ sy and b --+ sg decay rates. Here it is demonstrated explicitly that this is not the case: as long as physically meaningful quantities are considered, the results obtained from both techniques agree.

1. Introduction For the past several years, there has been a question of whether the dimensional reduction (DRED) and the usual dimensional regularization with fully anticommuting 7s ( N D R ) give different results for the QCD-improved b ~ s7 and b ---, s g decay rates [ 1 ]. The purpose of this letter is to show explicitly that this is not the case. As long as physically meaningful quantities are considered, the results obtained from both methods agree. The discussion presented here will be based only on the decay b ~ s7 which is of interest at present because of the recent B ---, K* 7 decay measurement [2 ]. The b ~ s7 process at the leading order in the Standard Model is given by the sum of the one-loop diagrams in fig. 1. The QCD contributions can be diagrammatically represented by dressing these diagrams with an arbitrary n u m b e r ofgluons. The diagrams constructed this way form a power series in the parameter * l

Supported by the German Bundesministerium fiir Forschung und Technologic under contract 06 TM 732. On leave of absence from Institute of Theoretical Physics, Warsaw University.

( M 2 / m ~ ) ~_ 0.7 which seems too large to be an expansion parameter. Therefore one has to resum all the large logarithms with help of the Operator Product Expansion and the Renormalization Group Equations (RGE). In order to do this, one introduces the local effective hamiltonian #1 O~QCD ( M w ) I n

Heft-

4"G ~ r v t s V x t b [ ~=C i/( l t ) O i +~c ° u n t e r t e r m s ]

(1) A complete list of the operators Oi is given in each of the papers [ 3-7 ]. As pointed out in ref. [4 ], the papers [1,8] do not include all the operators relevant in the leading-logarithmic approximation. This problem is, however, to a large extent unrelated to the problem of whether both dimensional methods give the same results. For the purpose of this paper, it is sufficient to explicitly state now only three of the operators ~ :

02 = (~LTuCL) (?cTUbL), e 07 = q-~--~2mb-gLau~,FU~'bR, 08 = 16n-------2mbSLau~G g u~bR,

Fig. 1. Diagrams contributing to b ~ s7 at the leading order of the SM interactions.

(2)

~1 For simplicity, we neglect the small V~s Vub in our discussion. However, the basic resuhs summarized in eqs. (7), (8) and in the Appendix are exactly the same even if V~ Vub is not neglected.

0370-2693/94/$ 07.00 (~ 1994-Elsevier Science B.V. All rights reserved SSDI 0370-2693 (93)E1512-V

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where F u~ and Gu" are the photonic and gluonic field strength tensors respectively. One finds the coefficients o f these operators by requiring equality of amplitudes generated by the effective Hamiltonian (1) and the full Standard Model amplitudes (up to 0 ( 1/M 4) ) at the renormalization scale /~ = Mw. The well-known results are

particular values o f the numbers ai disagree between each two o f them. The numbers ai and bi are given and discussed in the Appendix. In order to make our discussion as simple as possible, we formally expand the last term in (8) in powers of c~QCD(Mw):

Z

c2(mw)

= 1,

c7(Mw) -

(3)

3x 3 - 2x 2

4(x-

1) 4

In(x) +

1) 3

cs(Mw) - 4 ( x -

1)4 l n ( x ) +

,

where x = m]/M2w. The above coefficients are regularization and renormalization scheme independent. In order to avoid large logarithms in the b ---*sy matrix element of Hell, one evolves the coefficients ci (~) down to the scale/t = rnb according to the R G E

Yji(aQCD)Cj(It) = 0.

(6)

i=1

Finally, one finds that the b ~ sy matrix element o f Herr evaluated a t / t = m b (in the leading-logarithmic approximation and for the photon on shell) is equal to the tree-level matrix element of ~.2

4GF --~ E*ts Vtbc~ (mb )Ov,

(7)

with

c~ff (rnb ) = q 16/23c7(Mw )

+

(,,,/23

,6/23) cs(gw)

_

i=1

O~QCD(Mw) In Mw

2~

mb

l n M W ) 2)

(9)

where

X = ~-Zaibi.

(5)

lt ~---~ci(lt) - ~

~ X

8

--X 3 + 5x 2 + 2x

8 ( x - 1)3

aiqbi

'

(4) --3X 2

8

+

- 8 x 3 - 5 x 2 dr 7 x

24(x-

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(10)

i=1

F r o m the N D R results o f refs. [9,3,1,8,4-7], it follows that X -- 8, 2812, 232, 2812, 208 , 20881, 238 , ~ r e s p e c t i v e l y . The first change (from 8 to ~ l2 ) is due to the inclusion of contributions proportional to the downquark charge which had not been included in the first paper. The second change (from ~ to -~18) comes from taking into account certain one-loop matrix elements which will be described in the next section. The value of ~18 has very recently been confirmed [7] in the HV scheme*3 [ 10 ] where no one-loop matrix elements enter. The only known D R E D calculation (see ref. [ 1 ] ) gives X = ~14. In the following two sections, the calculation of X in N D R and D R E D will be presented and shown to give the same result of .~s. We restrict ourselves only to the quantity X (which has been the main subject of discussion in the past) in order to avoid considering all the subtleties involved in the calculation of the remaining terms on the r.h.s, o f eq. (9) where most o f the disagreements between the existing N D R calculations are located (see the Appendix).

8

+ c2(Mw) Z

aiqbL

(8)

2. The N D R calculation of X

i=1

Here q = c~(Mw)/a(mb), and ai, bi are some exact numbers that should be regularization and renormalization scheme independent. The form o f the last expression agrees with the final results of all the papers [1,3,8,4-7]. However, the #2 The s-quark mass is neglected throughout. 114

In order to calculate X, we have to trace out all possible leading-logarithmic contributions to -~3 In this scheme, ?'5 anticommutes with the 4-dimensional

?'u'sbut commutes with the remaining ones. It is the only known scheme where problems with traces containing odd numbers of 75's do not arise. The treatment of Y5 is the only difference between the HV and NDR schemes.

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(s~lHefrlb)u=,nb that are proportional to c2 ( M w ) . As in refs. [1,8] and unlike refs. [3-7], we will perform the calculation without applying equations of motion to the operators we encounter. First, we have to consider all possible divergent oneloop 1PI diagrams with the O2-vertex, except for those that contain more than one power of the QED coupling e. There are four types of such diagrams which are presented in figs. 2a-2d. The diagram in fig. 2a vanishes for an on-shell photon. This is why there is no one-loop mixing between 02 and 07. In effect, the two-loop mixing between these two operators becomes important in the leading logarithmic approximation. All the diagrams in fig. 2b require only counterterms proportional to the four-quark operators containing the ?t. and CL fields. Such operators have vanishing one-loop b ~ sy matrix elements for an onshell photon. Consequently, these diagrams are irrelevant in the calculation of X. The diagrams in fig. 2c generate no divergences in NDR. The divergences generated by the diagrams in fig. 2d can be exactly cancelled by the counterterm (see eq. (1)),

(a)

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Fig. 3. Diagrams contributing to the one-loop on-shell matrix element of OG, 06(4) or Oy. The square denotes the insertion of any of these three operators. (22 (,I.l ) Z 2 G O G ,

( 11)

where

OG-

g 2 ~LY~z(D~GU")bz 1~-7g

(12)

and Z2G --

4 3 ( 4 - d)

+ (terms finite in the limit d --- 4).

From ZEG, we recover the corresponding element of the anomalous-dimension matrix, 72a = _ 4 + O(c~QCD),

(14)

and immediately find the relevant term in the solution of the RGE (6) for cr(/z): ca(lt) = c a ( M w ) + -] c 2 ( M w ) l n M w ll

+ O (c~QCDIn -~-~).

(b)

(c)

(15)

We could write all the remaining terms as O(C~QCOx l n ( M w / # ) ) because there is no operator other than 02 which has a coefficient of order 1 at/1 = Mw and mixes with Oa at zeroth order in aQCO (in the applied normalization for Oa). Now, let us consider the one-loop on-shell matrix element of OG. It is given by the diagrams in fig. 3. When a matrix element is considered, we have to also take into account the 1PR diagrams. The sum of the diagrams in fig. 3 appears to be proportional to the tree-level matrix element of 07: Sum of the diagrams ]

(d)

Fig. 2. One-loop divergent diagrams with the 02 vertex.

(13)

in fig. 3 at # = m b I on shell OLQCD2 x gts VttbCG (mb) ~

_

~(sy1071b).

4GF ~/2

(16) 115

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3. The D R E D calculation of X

Fig. 4. Diagrams contributing to the two-loop mixing of 02 with 07. Comparing eqs. (7), (8), (9), (15) and (16), one finds the contribution to X from the one-loop matrix element of OG:

AXmatrix element - -

8 27"

(17)

A contribution to c~rf from one-loop matrix elements has been already found in ref. [4]. As mentioned 8 there, its particular value (which corresponds to - 2"5 above) is correct only in dimensional regularization with fully anticommuting 75. In any 4-dimensional scheme (or in the HV scheme), the on-shell b ~ sy one-loop matrix element of 06 vanishes. But the contributions to c~ff from the 2-loop mixings must also be different. This point has recently been emphasized in ref. [7]. What remains to be considered is the two-loop mixing 02 ~ 07. It is represented by the diagrams in fig. 4. The calculation of these diagrams is described in great detail in ref. [3]. The well-known result is

727 - -

OtQCD 232 2n 81 "

(18)

After solving the R G E ofeq. (6) (for arbitrary values of all the other mixings) and formally expanding the solution in aQCD, we immediately find

mXmixing

232 • = -¢i-

(19)

Therefore, the final value of X is

X

= A X m a t r i x el . . . . t -'[- AXmixing = ---

116

208 81

-8

_[_ ~12

(20) "

The D R E D scheme has been introduced by Siegel [ 11 ] in order to dimensionally regularize supersymmetric theories without actually breaking supersymmetry. The only difference between N D R and D R E D is that in the latter scheme all tensor fields are 4dimensional, while the m o m e n t a and coordinates are d-dimensional. The dimension d is assumed to be less than 4, which means that in practical calculations one applies the equality #4

g (u ~4~vp )

= guP.

(21)

Here gC4) and gU~ denote the 4-dimensional and the d-dimensional metric tensors respectively. As it is implicit in the above equation, we allow the indices o f the d-dimensional tensors to take on values also larger than d. But then the corresponding components of these tensors are assumed to vanish. For clarity, we will keep the superscript " ( 4 ) " for all the 4-dimensional tensors that appear in this section. It is also necessary to introduce some notation for the difference between the 4-dimensional and the ddimensional tensors. We define (22) for the gluonic field, and (4)

7(~) = Y u - 7 u

(23)

for the Dirac matrices. The matrix Ys is taken to be anticommuting with all the 7(4)'s and 7u'S [11 ]. We should add the superscripts " ( 4 ) " to all the fields and Dirac matrices in the definitions of the three operators we started with in eq. (2), and to the operator in eq. (12). The "new" operators will be denoted by 02(4) , 07( 4, , 08(4, and 0 (4) respectively. We proceed along the same lines as in the previous section. We have to consider the divergent parts o f the diagrams in figs. 2a-2d, but now with the 02~4)vertex. The diagrams o f figs. 2a and 2b are eliminated from our discussion via the same arguments as in the N D R case. #4 This is opposite to the HV scheme where g (4) u~op.up = g(4)p is used.

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The first real difference between N D R and D R E D occurs in the case of the diagrams in fig. 2c. In DRED, the sum of the diagrams with one gluon and one photon does not vanish. Instead, it gives a divergence proportional to the operator Ox =

e g ~ ( 4 ) t r ( 4 ) i~,(4),uuv(e)t..7.(e)p/~(4) 1---~2OL vgu-"P ~ ~L "

(24)

The required renormalization constant is Z2x = +

4 m 3 ( 4 - d)

+ (terms finite in the limit d ~ 4).

(25)

Just as in the case of 06 discussed in the previous section, we find (cf. eqs. ( 11 ) - ( 1 5 ) ) Cx(,U) = c x ( M w )

- 4 c 2 ( M w ) In M w #

+ O (C~QCDIn -M- w f-).

(26)

The diagrams in fig. 2c that contain two gluons are irrelevant in the calculation of X, because the oneloop b ~ sy matrix element of an operator containing two gluons is of order c~c o. The diagrams in fig. 2d generate two important counterterms. One of them is, of course, the counterterm proportional to O~4) with the same renormalization constant as in eq. ( 13 ). The other is proportional to the operator

g --(4) (e) (-~(e)#,q(4) O y = l~Sn2st ;~ [ ] ~t ,

(27)

with the renormalization constant 2 Z2y = + 3(4 - d-~-~ + (terms finite in the limit d ~ 4).

(28)

So the coefficient of Oy behaves like cy(p) = cy(Mw) - 2Q(Mw)

+ O(tXQCDIn M ~ ) .

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Fig. 5. Diagrams contributing to the one-loop matrix element of Ox. for the same reason as the two-gluon diagrams in fig. 2c. The appearance of the counterterms involving the G") that break gauge invariance in the (4 - d)# dimensional subspace is not surprising in DRED. They have already been observed by the inventor of D R E D [11]. Now, let us consider the one-loop on-shell matrix elements of the operators Ox, 00(4) and Oy. The first of them is given by the sum of diagrams in fig. 5. The sum of these diagrams appears to be proportional to the tree level matrix element of O~4) (we ignore the possible terms proportional to the (4 - d)-dimensional photonic field): Sum of the diagrams ~ _ in fig. 5 at/1 = mb / on shell , O~QCD ×Vtt s V t b C x ( m b ) ~ . - (2)

4GF V/2

(sylOTIb).

(30)

Comparing eqs. (7), (8), (9), (26) and (30), one finds mSmatrix elemenl of Ox - -

98

(31)

The one-loop matrix elements of 064) and Oy are described by the same diagrams as in the case of OG (fig. 3). We only have to change the operator vertex. In ~(4) the case of u s , the sum of these diagrams vanishes on-shell, while in the case of Oy we obtain exactly the same result as in eq. (16). The correlation is not surprising since O G = - cD ( 4 ) +

Oy.

(32)

Using these results in the same way as before, we find

ln M w It

~ X m a t r i x el. . . . t of O~(4) = 0 ,

(29)

The diagrams containing more than one gluon in fig. 2d may also give rise to some other counterterms containing the vG~) field. They are, however, irrelevant #

z~Xmatrix el. . . . t

of o~ = + 4 .

(33)

Therefore the sum of the contributions to X from the one-loop matrix elements is z~Xmatrix el. . . . ts =

_ 8 -I- 0 "4" 4

--

20 27"

(34) 117

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Now we have to consider the two-loop mixing 02(4) ~ 07(4). The two-loop diagrams look exactly the same as in the N D R case (fig. 4). In the one-loop counterterm diagrams of fig. 4, we have to insert both the O~4)- and the Oy-counterterms. Finally, we have to take into account also the one-loop counterterm diagrams with the counterterms proportional to Ox. The latter diagrams can be obtained from the ones in fig. 5 just by replacing the square (representing the Ox-vertex) by a cross (representing the Ox-counterterm). The details of the two-loop calculation will not be presented here. It has been performed using the method of ref. [12], i.e. only the difference between D R E D and N D R has been calculated. Then one needs to consider only the double-pole parts of the two-loop integrals (given in ref. [3] ), and the Dirac algebra is relatively simpler. The final result for the sought element of the anomalous-dimension matrix is Y27 =

OtQCD ( 18~ ~) 2zr -- +-

-

aQCD 268 27r 81 "

(35)

The number ~ comes from the Ox-counterterm diagrams. This will be exactly the contribution from these diagrams to X. It is not an accident that it equals - 2 x (contribution to X from the one-loop matrix element of Ox ). This is a c o m m o n feature for the "evanescent operators", i.e. operators which vanish in the limit d --* 4 (see e.g. refs. [13,14,5]). The number ~ l4 in eq. (35) comes from the diagrams of fig. 4. It is in agreement with the findings o f ref. [ 1 ] where the Ox counterterms have not been included. The presence o f the Ox counterterms is also the reason why the tests made in ref. [15] did not work in the D R E D case. As in the N D R case, we recover the contribution to X from Y27: AXmixing = ~18 .

(36)

4. Final remark

Table 1 summarizes the results for the quantity X obtained in the N D R , HV and D R E D schemes. The result for AXmixing in the HV scheme has been taken from eq. (25) of ref. [7]. The equality o f all the three results for the physically meaningful quantity X is what one would naturally expect, assuming that all the three schemes are consistent. This is also what one could expect by remembering the two-loop calculation of ref. [ 14 ] where all the three schemes were found to give the same results for the physically meaningful quantities in the fourquark operator case. A scheme for extending Dirac algebra to ddimensions is consistent if it gives a proper limit as d -~ 4, and is unique #5 . The latter requirement follows from the fact that a consistent regularization procedure must produce the same results for a given diagram irrespective of whether it is considered separately or as a subdiagram, and independent o f the order in which the subdiagrams are calculated. By analyzing how all the three schemes are defined, one can realize that each of them is consistent as long as traces with odd numbers of y5's do not appear. If they do appear, then only the HV scheme remains consistent. In the two other schemes, we find that expressions like

Tr(y~TuY.YpToY57~ )

T ,rty~(~) Yu(4).(4). Y. ~'p(4) Y~(4)A..,(c) ~sr~ )

#5 We do not require that our regularization scheme preserve the symmetries of the theory. In the absence of anomalies, all asymmetric terms are local and can be removed by proper asymmetric counterterms.

X = AXmatrix elements + AXmixing

NDR

which is in agreement with the N D R result. 118

(37)

(39)

in D R E D yield different results depending on whether cyclicity o f the trace is used or the contracted y's are

Scheme

208 ---- -~-,

(38)

in N D R or

Then we add it to the contribution from the matrix elements to obtain the final answer 20

20 January 1994

HV DRED

AXmatrix elements 8 27

0 20 -7

Aft(mixing

X

232 ~ 208 -kT

208

268 ~

208 ~T 208 8--i-

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c o m m u t e d all the way through the matrices standing between them. In the calculation of X presented in this paper, as well as in the calculation o f ref. [14] no traces appeared. This is why no discrepancies between the three schemes were observed. Handling 75 becomes more complicated in the complete leading-logarithmic calculation o f the b ~ s7 rate. In that case one has to use more refined arguments to show that the (sufficiently careful) N D R calculation is a consistent one. Two such independent arguments have been given in Appendix A ofref. [5]. The first is based upon the idea o f introducing certain evanescent operators in order to avoid specifying the algebraic properties of 75 before arriving at expressions that cannot lead to inconsistencies n6. The other method was based on the observation that certain symmetries in the structure of all the relevant four-quark operators allows one to perform the calculation without computing any traces. This observation has also been independently made in ref. [16] in the context of calculating similar diagrams for the next-to-leading effects in AS = 1 transitions. Both of these methods for dealing with the dangerous traces can also be directly used in the D R E D case. This is why one can expect that a sufficiently careful D R E D calculation of all the numbers ai and bi from eq. (8) will give the same results as in the N D R and HV schemes ~7

Appendix

Acknowledgement

Ref. [5]: ai ,272277, 51730, 7, -0.0453, -0.0215, -0.0990);

The author would like to thank Professor A. Buras for stimulating discussions.

~6 This method is a consistent one, but in a general multiloop calculation it is expected to be much more complicated than the usual HV scheme. However, in some particular calculations (especially such where no more than two Ys's can appear in a single fermionic line) it can be much simpler. #7 This is also the expectation of the authors of ref. [7] who claim to be currently performing such a complete DRED calculation.

As mentioned below eq. (8), the numbers ai and bi are subject to disagreements between any two of the existing calculations of the leading-logarithmic Q C D effects in the b ~ sy decay [1,3,8,4-7]. In most cases it is due to the fact that not all the relevant dimensionsix operators have been included. The complete basis of operators (reduced by the equations of motion) has already been written down in ref. [3]. But the effects o f the operators called n8 03, 0 4 , 0 5 , 0 6 have been neglected there. The three most recent papers [ 5 - 7 ] where the effects from all the relevant operators have been explicitly calculated disagree on the mixings (05, O6) (07, 08). The authors ofrefs. [5] and [6] have performed a comparison of their calculations [ 17 ], and they have found two (and only two) sources of disagreement: (i) not including the effects of the evanescent operators in ref. [6]; (ii) disagreements in the parts o f the first four twoloop diagrams in fig. 5 o f ref. [5], where the fermion mass comes from the fermionic loop. After arriving at this conclusion, we received the paper by Ciuchini et al. [7]. The results of this paper can be reproduced if disagreements (i) and (ii) are resolved in favour of refs. [ 5 ] and ref. [ 6 ] respectively. However, none o f the authors of the considered papers is ready at the m o m e n t to say that his previous results should be corrected. This is why I have decided to give here the numbers a, corresponding to each of the three papers [5-7 ]. They are as follows: ~-

( 422534

35533

3

Ref. [6]." ai

( 708542

69049

3

Ref. [7]."

[ 626126 ,272277,

56281 51730,

3 7,

,~, 51730, 7, - 0 . 0 0 0 3 , - 0 . 0 5 8 0 , +0.0323); ai

14, -0"1991, 14, -0-7415, 14,-0.6494,

-0.0380, -0.0186, -0.0057). The corresponding numbers bi are the follow16 ing: bi = ( ~ , if, 6 , ~12, 0.4086, - 0 . 4 2 3 0 , - 0 . 8 9 9 4 , 0.1456). The numbers bi are insensitive to the disagreements between the papers [5-7]. This is because they are proportional to the eigenvalues of the block#8 The numbering of the operators common for refs. [35,7] is used here. 119

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triangular anomalous-dimension matrices which disagree with each other only in the off-diagonal block. The last six bi's are proportional to the eigenvalues of the anomalous-dimension matrix for the O1-06 operators which was calculated long ago in ref. [ 18 ]. The first two are given by the self-mixing of the 07 and 08 operators which was originally found in ref. [ 19 ]. Some of the numbers ai and bi are not rational, but they are known to arbitrary precision because they come from the diagonalization of the leadingorder anomalous-dimension matrices that are known exactly. As mentioned below eq. (8), the sum of all the ai's always vanishes. This is because all the QCD effects summarized in eq. (8) must vanish for r/ = 1. The numbers a, corresponding to the results of refs. [5 ] and [7 ] look very different. However, the difference between the resulting c~ff's is less than 1%. This can be easily understood, since the differences between the anomalous-dimension matrices found in these papers are only in the mixings 05 ~ 07 and (O5, 06 ) ~ 08. The operator 05 acquires only a very small coefficient (,,~ 0.008 ) during the evolution from M w to rob. The operator 06 has a larger coefficient, but the effects of the mixing 06 ~ 08 ~ 07 tend to cancel in the last step (see the term proportional to 08 in eq. (8)).

120

20 January 1994

References [1] R. Grigjanis, P.J. O'Donnell, M. Sutherland and H. Navelet, Phys. Lett. B 213 (1988) 355; see also Phys. Rep. 228 (1993) 93, and references therein. [2] R. Ammar et al., Phys. Rev. Lett. 71 (1993) 674. [3] B. Grinstein, R. Springer and M.B. Wise, Phys. Lett. B 202 (1988) 138, Nucl. Phys. B 339 (1990) 269. [4] M. Misiak, Phys. Lett. B 269 (1991) 161. [5] M. Misiak, Nucl. Phys. B 393 (1993) 23. [6] K. Adel and Y.P. Yao, Mod. Phys. Lett. A 8 (1993) 1679. [7]M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, preprint ROME 93/958 (hep-ph 9307364). [8] G. Celia, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B 248 (1990) 181. [9] S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59 (1987) 180. [10] G. 't Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189; P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 11, 39, 55. [11] W. Siegel, Phys. Lett. B 84 (1989) 193. [12] P.J. O'Donnell and H.K.K. Tung, Phys. Rev. D 45 (1992) 4342. [ 13 ] M.J. Dugan and B. Grinstein, Phys. Lett. B 256 ( 1991 ) 239. [14] A.J. Buras and P.H. Weisz, Nucl. Phys. B 333 (1990) 66. [15]R. Grigjanis, P.J. O'Donnell, M. Sutherland and H. Navelet, Phys. Lett. B 237 (1990) 252. [16]A.J. Buras, M. Jamin, M.E. Lautenbacher and P.H. Weisz, Nucl. Phys. B 400 (1993) 37. [17] K. Adel, private communication. [18] F.J. Gilman and M.B. Wise, Phys. Rev. D 20 (1979) 2392. [ 19 ] M.A. Shifman, A.I. Vainstein and V.I. Zakharov, Phys. Rev. D 18 (1978) 2583