- Email: [email protected]
Nc expansion
17 June 1999
Physics Letters B 457 Ž1999. 227–235
Matrix elements of electroweak penguin operators in the 1rNc expansion Marc Knecht a , Santiago Peris b, Eduardo de Rafael
a
a
b
Centre de Physique Theorique, CNRS-Luminy, Case 907, F-13288 Marseille Cedex 9, France ´ Grup de Fısica Teorica and IFAE, UniÕersitat Autonoma de Barcelona, E-08193 Barcelona, Spain ´ ` ` Received 6 January 1999; received in revised form 26 March 1999 Editor: R. Gatto
Abstract It is shown that the K ™ pp matrix elements of the four-quark operator Q 7 , generated by the electroweak penguin-like diagrams of the Standard Model, can be calculated to first non-trivial order in the chiral expansion and in the 1rNc expansion. Although the resulting B factors B7Ž1r2. and B7Ž3r2. are found to depend only logarithmically on the matching scale m , their actual numerical values turn out to be rather sensitive to the precise choice of m in the GeV region. We compare our results to recent numerical evaluations from lattice-QCD and to other model estimates. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction In the Standard Model, the physics of non-leptonic K-decays is described by an effective Lagrangian which is the sum of four-quark operators Q i modulated by c-number coefficients c i ŽWilson coefficients. Leff< D S
GF
'2
Vu d Vu)s Ý c i Ž m . Q i Ž m . q h.c. , i
Ž 1. where G F is the Fermi constant and Vu d , Vu s are the appropriate matrix elements of flavour mixing. This is the effective Lagrangian which results after integrating out the fields in the Standard Model with heavy masses Ž Z 0 ,W ",t,b and c ., in the presence of the strong interactions evaluated in perturbative QCD ŽpQCD. down to a scale m below the mass of the charm quark Mc . The scale m has to be large enough
for the pQCD evaluation of the coefficients c i to be valid and, therefore, it is much larger than the scale at which an effective Lagrangian description in terms of the Nambu–Goldstone degrees of freedom Ž K, p and h . of the spontaneous SUŽ3.L = SUŽ3.R symmetry breaking ŽS x SB. is appropriate. Furthermore, the evaluation of the coupling constants of the low-energy effective chiral Lagrangian which describes strangeness changing < D S < s 1 transitions cannot be made within pQCD because at scales m Q 1 GeV we enter a regime where S x SB and confinement take place and the dynamics of QCD is then fully governed by non-perturbative phenomena. In this letter we shall be concerned with two of the four-quark operators in the Lagrangian in Eq. Ž1., the operators Q 7 s 6 Ž sL g md L .
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 4 2 5 - 6
Ý qsu , d , s
e q Ž q R gm q R . ,
Ž 2.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
228
and Q8 s y12
e q Ž sL q R . Ž q R d L . ,
Ý
Ž 3.
qsu , d , s
where e q denote quark charges in units of the electric charge, and summation over quark colour indices within brackets is understood. The operator Q 7 emerges at the MW scale from considering the socalled electroweak penguin diagrams. In the presence of the strong interactions, the renormalization group evolution of Q 7 from the scale MW down to a scale m Q Mc mixes, in particular, this operator with the four-quark density–density operator Q8 . These two operators, times their corresponding Wilson coefficients, contribute to the lowest order O Ž p 0 . effective chiral Lagrangian which induces < D S < s 1 transitions in the presence of electromagnetic interactions to order O Ž a . and of virtual Z 0 exchange, i.e., the Lagrangian w1x Lx< ,0D S
GF a
'2
p
Vu d Vu)s
Mr6 16p 2
=h tr Ž Ul LŽ 23 . U † Q R . q h.c.
Ž 4.
Here, U is the matrix field which collects the octet of pseudoscalar Goldstone fields, Q R s diagw2r3, y1r3,y 1r3x is the right-handed charge matrix associated with the electromagnetic couplings of the light quarks, and lŽ23. is the effective left-handed L . flavour matrix Ž lŽ23. s d i2 d 3 j Ž i, j s 1,2,3.. Under L ij chiral rotations Ž VL ,VR .: U ™ VR UVL† ,
Q R ™ VR Q R VR† ,
Ž23. † lŽ23. L ™ VL l L VL ,
Ž 5.
and the trace on the r.h.s. of Eq. Ž4. is an invariant. Actually, this is the only possible invariant which in the Standard Model can generate < D S < s 1 transitions to orders O Ž a . and O Ž p 0 . in the chiral expansion. With the given choice of the overall normalization factor in front of the r.h.s. of Eq. Ž4., the coupling constant h is dimensionless and, a priori, of order O Ž Nc2 . in the 1rNc expansion w2–4x. This coupling constant plays a crucial role ˆ in the phenomenological analysis of radiative corrections to the K ™ pp amplitudes w5x. The determination of h is needed for a
reliable estimate of these corrections 1. The purpose of this note is to show that, following recent work reported in Ref. w7x, one can calculate the K ™ pp matrix elements of the Q 7 operator Žand, therefore, the so called B7Ž1r2. and B7Ž3r2. factors. to first non-trivial order in the chiral expansion and the 1rNc expansion 2 . This implies that Žat least. the contribution to the constant h from the Q 7 and Q8 terms in the four-quark effective Lagrangian can be calculated to first non-trivial order in the 1rNc expansion, a first step towards the required goal. The leading O Ž Nc2 . contribution to h vanishes trivially. It could only come from the bosonization of the factorized Q8 operator times its Wilson coefficient, but this coefficient is subleading at large Nc 3. The next-to-leading contribution comes from the bosonization of the unfactorized Q 7 operator and its mixing via gluonic interactions with the Q8 operator. It involves the same two-point function which governs the electroweak pqy p 0 mass difference in large-Nc QCD and this is why we are able to compute the contributions from the Q 7 and Q8 terms of the four-quark Lagrangian to the constant h at the stated order in the 1rNc expansion. The bosonization of the factorized Q 7 operator and of the unfactorized Q8 operator can only contribute to terms of order O Ž p 2 . Žor higher. in the chiral expansion and they are, therefore, inoperational in the calculation of h. It turns out, however, that there is only a partial cancellation of the m-dependences generated by the product of the bosonization of the unfactorized Q 7 operator times its Wilson coefficient c 7 with the m-dependence coming from the product of the Wilson coefficient c8 , which is non-leading in the 1rNc expansion, times the bosonization of the factorized Q8 operator. The full cancellation of m-dependences requires the consideration, as well, of the bosonization of other four-quark operators Žin particular the unfactorized Q 2 operator. in the presence of the electroweak interactions. This involves integrals of four-point functions which we have not yet fully explored within the framework of the 1rNc expansion. The constant h gets, therefore, contributions 1
See Ref. w6x for a recent discussion of the size of these corrections. 2 A similar observation has also been made by J. Donoghue w8x. 3 See e.g. Buras’s lectures w9x.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
from other operators than just Q 7 and Q8 , and which we have not computed so far. We shall, therefore, concentrate here on the calculation of the factors B7Ž1r2. and B7Ž3r2., and on their comparison to recent lattice QCD determinations w10–12x, as well as to recent analytic determinations which have been made 4 using the ‘‘effective action approach’’ of Ref. w14x.
229
Before discussing this point, let us recall that in the large-Nc limit, the spectral function associated with P L R Ž Q 2 . consists of the difference of an infinite number of narrow vector states and an infinite number of narrow axial-vector states, together with the Goldstone pion pole: 1
p
Im P L R Ž t . s Ý f V2 M V2 d Ž t y M V2 . V
y Ý f A2 MA2d Ž t y MA2 . y fp2 d Ž t . .
2. Bosonization of Q 7 and Q 8
A
As already mentioned, the bosonization of the operator Q 7 is needed to next-to-leading order in the 1rNc expansion. The problem is entirely analogous to the bosonization of the operator Q L R ' Ž q L g m Q L q L . Ž q R g m Q R q R . which governs the electroweak pqy p 0 mass difference and which has been recently discussed in Ref. w7x. Because of the LR structure, the factorized component of the operator Q 7 , which is leading in 1rNc , cannot contribute to order O Ž p 0 . in the low-energy effective Lagrangian. The first non-trivial contribution from this operator is next-to-leading in the 1rNc expansion and is given by the integral 5 Q 7 ™ y3igmn
d4q
H Ž 2p .
4
P LmnR Ž q . tr Ž Ul LŽ 23 . U † Q R . ,
P LmnR Ž q . s 2 i d 4 x e i qP x ²0
H
†
. <0: , Ž 7.
R s qi Ž x . g
m1 2
Ž 1 q g5 . qj Ž x . ,
yQ 2P L R Ž Q 2 . s fp2 q Ý f A2 MA2 A
yÝ
f V2 M V2
V
Q2 MA2 q Q 2 Q2
M V2 q Q 2
.
Ž 11 .
Furthermore, in the chiral limit of QCD, the operator product expansion ŽOPE. applied to the correlation function P L R Ž Q 2 . implies lim Q 2P L R Ž Q 2 . ™ 0
and
lim Q 4P L R Ž Q 2 . ™ 0 .
Ž 12 .
Q 2™`
involving the two-point function
m
Since P L R Ž Q 2 . obeys an unsubtracted dispersion relation, we find that
Q 2™`
Ž 6.
with currents L m s qi Ž x . g m 12 Ž 1 y g 5 . q j Ž x .
Ž 10 .
These relations result in the two Weinberg sum rules w15x
Ý f V2 MV2 y Ý f A2 MA2 s fp2 V
and
and
A
Ý f V2 MV4 y Ý f A2 MA4 s 0 . Ž 8.
V
Ž 13 .
A
and i and j fixed flavour indices, i / j. This integral, which has to be evaluated in the chiral limit, where Ž Q 2 s yq 2 .
The usual prescription w16x for the evaluation of integrals such as Ž6. consists in taking a sharp cut-off in the Žeuclidean. integration over Q 2 ,
P LmnR Ž q . s Ž q mq n y g mn q 2 . P L R Ž Q 2 . , Ž 9. is divergent for large Q 2 and needs to be regulated.
Q 7 ™ y6
3 32p
2
2
H0L dQ Q ŽyQ P 2
=tr Ž Ul LŽ 23 . U † Q R . .
2
2
LR
Ž Q2 . . Ž 14 .
4
See e.g. Ref. w13x and references therein. 5 Up to the replacement of Q L by lŽ23. L , the bosonization of the operator Q 7 follows from the same procedure as described for the operator Q L R in Ref. w7x.
With the constraints between the couplings and masses of the narrow states coming from the Weinberg sum rules Ž13., the integral on the r.h.s. of Eq.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
230
Ž6. becomes then only logarithmically dependent on the ultraviolet scale L, with the following result 2
H0L dQ Q ŽyQ P 2
2
2
LR
Ž Q2 . .
L2
s
L2
Ý f A2 MA6 log M 2 y Ý f V2 MV6 log M 2 A
A
.
Ž 15 .
V
V
Notice that if only the contribution from the Goldstone pole had been taken into account, the resulting expression would have displayed a polynomial dependence on the cut-off scale L. Another possibility is to evaluate the integral Ž6. within a dimensional regularization scheme, say MS, in which case one obtains the same result Ž15., but with the correspondence between the cut-off L and the MS subtraction scale m given by
L s m P e 1r6 .
Ž 16 .
On the other hand, at the level of approximation that we want to attain, the bosonization of the Q8 operator is only required to leading order in the 1rNc expansion, because its Wilson coefficient is already subleading. To that order and to order O Ž p 0 . in the chiral expansion it can be readily obtained from the bosonization of the factorized density currents, with the result 6 fp2
2
ž /
Q8 ™ y12 2 B
4
tr Ž Ul LŽ 23 . U † Q R . ,
Ž 17 .
where B is the low energy constant which describes the bilinear single flavour quark condensate in the chiral limit, B s y² cc :rfp2 . As discussed in Ref. w18x there is a constraint that emerges in the large-Nc limit which relates the leading d s 6 order parameter in the OPE of the P L R two-point function w19x to couplings and masses of the narrow states: as lim Q 6P L R Ž Q 2 . s y4p 2 q O Ž a s2 . ² cc :2 2 p Q ™`
ž
/
s Ý f V2 M V6 y Ý f A2 MA6 . V
Ž 18 .
A
This relation provides part of the cancellation between the m dependence of the bosonization of the 6
See e.g. the lectures in Ref. w17x and references therein.
operator Q 7 Ž m . with the short-distance dependence on m in the Wilson coefficient of Q8 ; but, as already mentioned, and contrary to the simple case of the electroweak contribution to the pqy p 0 mass difference discussed in Ref. w7x, this cancellation here is incomplete. In this respect, we wish to comment on an important point concerning the scale dependence in the relation in Eq. Ž18.. The term on the r.h.s. of the first line results from a lowest order pQCD calculation of the Wilson coefficient. Using the renormalization group improvement to one loop, this result becomes y4p 2
as p
² cc :2 ™ y4p 2 ² cc :2
=
ž
1 yb 1
yb 1 a s Ž Q 2 .
p
2g 1q b 1
/
b1
,
Ž 19 . with ² cc : the scale invariant quark condensate Žthe analog to invariant quark masses.. In the large-Nc 3 2 limit, b 1 ™ y11 depen6 Nc and g 1 ™ 4 Nc . The Q dence of the one-loop result is indeed rather mild ; Žlog Q 2rL MS 2 .y2r11, but it does not go to a constant as the exact large-Nc result in the second line of Eq. Ž18. demands. This mismatch is due to the fact that pQCD is at best an approximation. It may happen that a two-loop renormalization group improvement of the OPE result approaches a constant behaviour at large Q 2 in a better way. In any case, this is a typical example of unavoidable mismatches that one will encounter between non-perturbative evaluations of matrix elements and short-distance pQCD evaluations, which are necessarily only approximate. In general, however, it is well known that at higher orders ambiguities will mix the short-distance coefficients of different powers in the OPE. It is difficult to imagine how to avoid these uncertainties in a final matching between short-distances and long-distances unless a breakthrough is made in understanding the relationship between pQCD and full QCD. Finally, if we restrict ourselves to the Lowest Meson Dominance ŽLMD. approximation to large-Nc QCD discussed in Ref. w20x and identify M V with Mr , our calculation Žin the MS scheme. gives the
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
following contributions to the constant h coming from the operators Q 7 and Q8 ,
a p
h w Q 7 x s y18c 7 Ž m .
a and
p
fp2 Mr2
Mr2
h w Q8 x s y 48p 2 c8 Ž m .
and ² Q8 :VSA s 12 Ž Z q 4Y . q 0
L2 log
231
y 2log2
² cc :2 Ž m . fp2 Mr6
² Q8 :VSA s '2 Y y 2
,
Ž 20 . where the relation Ž16. is understood.
'2 2 Nc
1 2 Nc
X,
X.
Ž 24 .
The quantities X, Y and Z which appear in the above expressions are the same as those usually found in the literature, i.e., X s '3 fp Ž MK2 y Mp2 . q O Ž p 4 . ,
3. The B factors B7(1rr 2) and B7(3rr 2) The bosonic expression of Q 7 given by Eqs. Ž14. and Ž15. enables us to compute the K ™ pp matrix elements induced by this operator which, following the usual conventions, we express in terms of the following isospin amplitudes ² Q 7 :I ' ² Ž pp . I < Q 7 < K 0 : ,
I s 0,2 .
Ž 21 .
To leading order O Ž p 0 . in the chiral expansion and to next-to-leading order in the 1rNc expansion, O Ž1r Nc . for K ™ pp amplitudes, we obtain the following result
(
² Q 7 : 0 s '2 ² Q 7 : 2s
6'3 16p 2 fp3
L2
=
L2
Ý f A2 MA6 log M 2 y Ý f V2 MV6 log M 2 A
A
V
.
Y s '3 fp
Zs4
ž
ž
FK Fp
ms q md
Ys
² Q 7 :VSA sy 2
'2 2
2 Nc
/
X MK2 y Mp2
; 11.4 X
1
q O Ž p2 . ,
Ž 25 .
There is no theoretical justification to consider the VSA as a good limit of any kind in QCD 7. This is reflected by the fact that e.g., the terms proportional to 1rNc in Eqs. Ž23. and Ž24. do not correspond to the correct 1rNc expansion and, in fact, the contributions of Y and Z to ² Q 7 :VSA , although suppressed I by a factor 1rNc , are actually numerically dominant, since Žthroughout, MK denotes the neutral kaon mass.
Ž 22 .
² Q 7 :VSA s 12 X q 0
/
y 1 Y q O Ž p4 . .
V
It has become customary Žrather unfortunately. to parameterize the results of weak matrix elements of four-quark operators Q i in terms of the factorized contributions from the so-called vacuum saturation approximation ŽVSA., modulated by correction factors BiŽ D I ., D I s 1r2,3r2. In the case of Q 7 and Q8 , one then has
2
MK2
ž
ž
2
MK2 ms q md
0.158 GeV ms q md
/ 2
/
.
Ž 26 .
Furthermore, whereas X is scale independent, Y and Z depend on the MS subtraction scale m through the quark mass term m s q m d in the denominator. We shall nevertheless follow these conventions, if only to be able to compare our results to those existing in
Ž Z q 4Y . ,
Xq
'2 Nc
Y,
Ž 23 .
7 Obviously, it would have been much more reasonable to normalize results to the large-Nc result in the lowest order of the chiral expansion.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
232
Fig. 1. The B factors, B7Ž1r2. and B7Ž3r2., as a function of the m scale in GeV. The solid lines correspond to Ž m s q m d .Ž m s 2 GeV. s 158 MeV; the dashed lines to the extreme low value Ž m s q m d .Ž m s 2 GeV. s 100 MeV.
the literature. The corresponding B factors are then defined as BiŽ1r2. s
² Qi :0 ² Q i :VSA 0
BiŽ3r2. s
,
² Qi :2 ² Q i :VSA 2
.
Ž 27 .
X Xq
1 Nc
½
= 1q
Ž Z q 4Y . 3
M V4
2p 2 fp2 MK2 y Mp2
L2 = log
M V2
y 2log2
5
,
X
B7Ž3r2. Ž m . s
In the sequel, we quote our results in the MS scheme. Considering first the operator Q 7 , and restricting the sums in Eq. Ž22. to the LMD approximation discussed in Ref. w20x, the previous calculation leads to the results B7Ž1r2. Ž m . s
and
Ž 28 .
Xy
2 Nc
½
= 1y
Y 3
M V4
4p 2 fp2 MK2 y Mp2
L2 = log
M V2
y 2log2
5
.
Ž 29 .
If in these expressions we take the value M V s Mr , we obtain, at the scale m s 2 GeV usually adopted in lattice calculations, 1 B7Ž1r2. Ž m s 2 GeV . ; Ž 1 q 23 . , 19.5 1 B7Ž3r2. Ž m s 2 GeV . ; y Ž 1 y 11.5 . , Ž 30 . 6.6 for, say, Ž m s q m d .Ž m s 2 GeV. s 158 MeV, the conventional reference normalization used in Eq.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
Ž26.. These values are both positive and greater than unity. One should however notice that these numbers are rather sensitive to the choice of the scale m andror to the value assigned to M V : for instance, at m s 2 M Vre1r6 ; 1.3 GeV the contributions between square brackets on the r.h.s. of Eqs. Ž28. and Ž29. vanish exactly. In order to illustrate these uncertainties, we show in Fig. 1 the variation of these B factors for a reasonable range of values of the scale m. The area between the solid lines corresponds to the choice Ž m s q m d .Ž m s 2 GeV. s 158 MeV, with L MS varied between 300 MeV and 450 MeV. The area between the dashed lines reflects the same variation of L MS , but for the extreme low value Ž m s q m d .Ž m s 2 GeV. s 100 MeV quoted in some lattice results w21x. Also shown is a magnification of the region corresponding to values of m around 2 GeV, the reference scale at which the lattice results are usually given. We find that below m Q 1.3 GeV the B7Ž1r2. and B7Ž3r2. factors can even become negative, a result which disagrees, drastically, with the positive values quoted at a ‘‘matching scale’’ of
&
233
0.8 GeV which are found in the constituent chiral quark model w13x. It is, however, not clear how this ‘‘matching scale’’ is related to the MS scale, m , in QCD. Several matrix elements of four-quark operators have also been obtained in numerical simulations of lattice-QCD. These numerical evaluations, however, are based on a yet different definition of the B parameters. The lattice definition uses a current algebra relation between the K ™ pp and the K ™ p matrix elements which is in fact only valid at order O Ž p 0 . in the chiral expansion. Thus, in the case of Q 7 discussed here, the lattice-QCD B factor, which we shall distinguish with the symbol tilde on top, is defined by the ratio B˜7Ž3r2. '
²pq< Q 7Ž3r2. < Kq: ²pq < Q 7Ž3r2. < Kq :VSA 0
,
Ž 31 .
where the matrix element in the denominator is evaluated in the chiral limit, as indicated by the subscript ‘‘0’’, and the operator Q 7 has been decom-
Ž3r2. wsee Eq. Ž31.x, as a function of the m scale in GeV. The solid lines correspond to Ž m s q m d .Ž m s 2 GeV. s Fig. 2. The B factor, B7 158 MeV; the dashed lines to the extreme low value Ž m s q m d .Ž m s 2 GeV. s 100 MeV.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
234
posed into its D I s 1r2 and D I s 3r2 components, Q 7 s Q 7Ž1r2. q Q 7Ž3r2., with Q 7Ž3r2. s 2 Ž sL g m d L . u R gm u R y d R gm d R q 2 Ž sL g m u L . Ž u R gm d R . .
Ž 32 .
The latest results obtained by various groups are consistent with each other: B˜7Ž3r2. Ž m s 2 GeV .
s latt
½
q5 q2 0.58y4 y8
Ž 33 .
0.61 Ž 11 .
The first value has been taken from Ref. w10x, while the second one arises from the results of Refs. w11x and w12x, translated into the MS scheme w22x. In the vacuum saturation approximation which the lattice community uses, one obtains ²pq < Q 7Ž3r2. < Kq :VSA sy 0
2 ² cc :2 fp2
Nc
2
fp
Nc
'3
sy
Y,
Ž 34 . where in the second expression, following common practice, we have traded the dependence with respect to the condensate for the dependence on the strange quark mass, using the Gell-Mann–Oakes–Renner relation, which holds in the chiral limit. The calculation based on the bosonized expression for this operator which we have discussed gives ²pq < Q 7Ž3r2. < Kq : s
3 8p 2 fp2
Ý f A2 MA6 log A
y Ý f V2 M V6 log V
3 , 4p
M V4 2
ž
obtained from the lattice are usually not quoted together. In the case of Q8 , the large-Nc limit of Eq. Ž24. simply reproduces the result obtained from the bosonized expression Ž17.. Since subleading corrections in the 1rNc expansion of this operator are not yet available, further comparison with lattice results for B8Ž3r2. has to be postponed.
MA2
L2
M V2
4. Conclusions and outlook The expressions in Eqs. Ž22. and Ž35. are the main results reported in this letter. They are a first step towards a systematic evaluation of weak matrix elements in the chiral expansion and to first non-trivial order in the 1rNc expansion. Our final aim, however, is to obtain values for the coupling constants of the low energy D S s 1 and D S s 2 chiral effective Lagrangian directly; i.e., constants like h in Eq. Ž4. and not of individual matrix elements of four-quark operators. It is encouraging from the results obtained so far, to find such simple analytic expressions which exhibit only a logarithmic dependence on the matching scale m ; however, the fact that the numerical results are so sensitive to the choice of m in the GeV region is perhaps an indication that one should be extremely cautious in the evaluation of errors of B factors, in general, both in model calculations and in lattice QCD numerical simulations.
L2 L2
2log2y log
M V2
Acknowledgements
/
,
Ž 35 . where in the second line we have used the LMD approximation discussed in Ref. w20x. In this approximation and with the same numerical input as in Ž30., we obtain B˜7Ž3r2. Ž m s 2 GeV. , q1.5. However, as shown in Fig. 2, the value of this B factor is again very sensitive to the choice of the matching scale m. The results obtained for the smaller value of the strange quark mass Ži.e. for larger values of the condensate. are in agreement, within errors, with the numbers quoted in Eq. Ž33.. Unfortunately, the values of m s Žor of the condensate. and of the B factors
We wish to thank M. Perrottet and L. Lellouch for discussions on topics related to the work reported here, V. Gimenez for an informative correspondence, and the referee for constructive remarks. This work has been supported in part by TMR, EC-Contract No. ERBFMRX-CT980169 ŽEURODA f NE.. The work of S. Peris has also been partially supported by the research project CICYT-AEN98-1093. References w1x J. Bijnens, M. Wise, Phys. Lett. B 137 Ž1984. 245. w2x G. ’t Hooft, Nucl. Phys. B 72 Ž1974. 461; B 73 Ž1974. 461. w3x G. Rossi, G. Veneziano, Nucl. Phys. B 123 Ž1977. 507.
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w18x w19x w20x w21x w22x
235
Ž1987. 787; Phys. Lett. B 192 Ž1987. 138; B 211 Ž1988. 343; W.A. Bardeen, J. Bijnens, J.-M. Gerard, Phys. Rev. Lett. 62 ´ Ž1989. 1343; for a review and further references, see A.J. Buras, The 1rN Approach to Non-leptonic Weak Interactions, in: C. Jarlskog ŽEd.., CP Violation, World Scientific, Singapore, 1989. E. de Rafael, Chiral Lagrangians and Kaon CP-Violation, in: J.F. Donoghue ŽEd.., CP Violation and the Limits of the Standard Model, Proc. TASI’94, World Scientific, Singapore, 1995. M. Knecht, E. de Rafael, Phys. Lett. B 424 Ž1998. 335. M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 Ž1979. 385, 447. S. Peris, M. Perrottet, E. de Rafael, JHEP 05 Ž1998. 011. R. Gupta, Introduction to Lattice QCD, Les Houches lectures 1997, hep-latr9807028, and references therein. V. Gimenez, private communication.
Physics Letters B 457 Ž1999. 227–235
Matrix elements of electroweak penguin operators in the 1rNc expansion Marc Knecht a , Santiago Peris b, Eduardo de Rafael
a
a
b
Centre de Physique Theorique, CNRS-Luminy, Case 907, F-13288 Marseille Cedex 9, France ´ Grup de Fısica Teorica and IFAE, UniÕersitat Autonoma de Barcelona, E-08193 Barcelona, Spain ´ ` ` Received 6 January 1999; received in revised form 26 March 1999 Editor: R. Gatto
Abstract It is shown that the K ™ pp matrix elements of the four-quark operator Q 7 , generated by the electroweak penguin-like diagrams of the Standard Model, can be calculated to first non-trivial order in the chiral expansion and in the 1rNc expansion. Although the resulting B factors B7Ž1r2. and B7Ž3r2. are found to depend only logarithmically on the matching scale m , their actual numerical values turn out to be rather sensitive to the precise choice of m in the GeV region. We compare our results to recent numerical evaluations from lattice-QCD and to other model estimates. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction In the Standard Model, the physics of non-leptonic K-decays is described by an effective Lagrangian which is the sum of four-quark operators Q i modulated by c-number coefficients c i ŽWilson coefficients. Leff< D S
GF
'2
Vu d Vu)s Ý c i Ž m . Q i Ž m . q h.c. , i
Ž 1. where G F is the Fermi constant and Vu d , Vu s are the appropriate matrix elements of flavour mixing. This is the effective Lagrangian which results after integrating out the fields in the Standard Model with heavy masses Ž Z 0 ,W ",t,b and c ., in the presence of the strong interactions evaluated in perturbative QCD ŽpQCD. down to a scale m below the mass of the charm quark Mc . The scale m has to be large enough
for the pQCD evaluation of the coefficients c i to be valid and, therefore, it is much larger than the scale at which an effective Lagrangian description in terms of the Nambu–Goldstone degrees of freedom Ž K, p and h . of the spontaneous SUŽ3.L = SUŽ3.R symmetry breaking ŽS x SB. is appropriate. Furthermore, the evaluation of the coupling constants of the low-energy effective chiral Lagrangian which describes strangeness changing < D S < s 1 transitions cannot be made within pQCD because at scales m Q 1 GeV we enter a regime where S x SB and confinement take place and the dynamics of QCD is then fully governed by non-perturbative phenomena. In this letter we shall be concerned with two of the four-quark operators in the Lagrangian in Eq. Ž1., the operators Q 7 s 6 Ž sL g md L .
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 4 2 5 - 6
Ý qsu , d , s
e q Ž q R gm q R . ,
Ž 2.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
228
and Q8 s y12
e q Ž sL q R . Ž q R d L . ,
Ý
Ž 3.
qsu , d , s
where e q denote quark charges in units of the electric charge, and summation over quark colour indices within brackets is understood. The operator Q 7 emerges at the MW scale from considering the socalled electroweak penguin diagrams. In the presence of the strong interactions, the renormalization group evolution of Q 7 from the scale MW down to a scale m Q Mc mixes, in particular, this operator with the four-quark density–density operator Q8 . These two operators, times their corresponding Wilson coefficients, contribute to the lowest order O Ž p 0 . effective chiral Lagrangian which induces < D S < s 1 transitions in the presence of electromagnetic interactions to order O Ž a . and of virtual Z 0 exchange, i.e., the Lagrangian w1x Lx< ,0D S
GF a
'2
p
Vu d Vu)s
Mr6 16p 2
=h tr Ž Ul LŽ 23 . U † Q R . q h.c.
Ž 4.
Here, U is the matrix field which collects the octet of pseudoscalar Goldstone fields, Q R s diagw2r3, y1r3,y 1r3x is the right-handed charge matrix associated with the electromagnetic couplings of the light quarks, and lŽ23. is the effective left-handed L . flavour matrix Ž lŽ23. s d i2 d 3 j Ž i, j s 1,2,3.. Under L ij chiral rotations Ž VL ,VR .: U ™ VR UVL† ,
Q R ™ VR Q R VR† ,
Ž23. † lŽ23. L ™ VL l L VL ,
Ž 5.
and the trace on the r.h.s. of Eq. Ž4. is an invariant. Actually, this is the only possible invariant which in the Standard Model can generate < D S < s 1 transitions to orders O Ž a . and O Ž p 0 . in the chiral expansion. With the given choice of the overall normalization factor in front of the r.h.s. of Eq. Ž4., the coupling constant h is dimensionless and, a priori, of order O Ž Nc2 . in the 1rNc expansion w2–4x. This coupling constant plays a crucial role ˆ in the phenomenological analysis of radiative corrections to the K ™ pp amplitudes w5x. The determination of h is needed for a
reliable estimate of these corrections 1. The purpose of this note is to show that, following recent work reported in Ref. w7x, one can calculate the K ™ pp matrix elements of the Q 7 operator Žand, therefore, the so called B7Ž1r2. and B7Ž3r2. factors. to first non-trivial order in the chiral expansion and the 1rNc expansion 2 . This implies that Žat least. the contribution to the constant h from the Q 7 and Q8 terms in the four-quark effective Lagrangian can be calculated to first non-trivial order in the 1rNc expansion, a first step towards the required goal. The leading O Ž Nc2 . contribution to h vanishes trivially. It could only come from the bosonization of the factorized Q8 operator times its Wilson coefficient, but this coefficient is subleading at large Nc 3. The next-to-leading contribution comes from the bosonization of the unfactorized Q 7 operator and its mixing via gluonic interactions with the Q8 operator. It involves the same two-point function which governs the electroweak pqy p 0 mass difference in large-Nc QCD and this is why we are able to compute the contributions from the Q 7 and Q8 terms of the four-quark Lagrangian to the constant h at the stated order in the 1rNc expansion. The bosonization of the factorized Q 7 operator and of the unfactorized Q8 operator can only contribute to terms of order O Ž p 2 . Žor higher. in the chiral expansion and they are, therefore, inoperational in the calculation of h. It turns out, however, that there is only a partial cancellation of the m-dependences generated by the product of the bosonization of the unfactorized Q 7 operator times its Wilson coefficient c 7 with the m-dependence coming from the product of the Wilson coefficient c8 , which is non-leading in the 1rNc expansion, times the bosonization of the factorized Q8 operator. The full cancellation of m-dependences requires the consideration, as well, of the bosonization of other four-quark operators Žin particular the unfactorized Q 2 operator. in the presence of the electroweak interactions. This involves integrals of four-point functions which we have not yet fully explored within the framework of the 1rNc expansion. The constant h gets, therefore, contributions 1
See Ref. w6x for a recent discussion of the size of these corrections. 2 A similar observation has also been made by J. Donoghue w8x. 3 See e.g. Buras’s lectures w9x.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
from other operators than just Q 7 and Q8 , and which we have not computed so far. We shall, therefore, concentrate here on the calculation of the factors B7Ž1r2. and B7Ž3r2., and on their comparison to recent lattice QCD determinations w10–12x, as well as to recent analytic determinations which have been made 4 using the ‘‘effective action approach’’ of Ref. w14x.
229
Before discussing this point, let us recall that in the large-Nc limit, the spectral function associated with P L R Ž Q 2 . consists of the difference of an infinite number of narrow vector states and an infinite number of narrow axial-vector states, together with the Goldstone pion pole: 1
p
Im P L R Ž t . s Ý f V2 M V2 d Ž t y M V2 . V
y Ý f A2 MA2d Ž t y MA2 . y fp2 d Ž t . .
2. Bosonization of Q 7 and Q 8
A
As already mentioned, the bosonization of the operator Q 7 is needed to next-to-leading order in the 1rNc expansion. The problem is entirely analogous to the bosonization of the operator Q L R ' Ž q L g m Q L q L . Ž q R g m Q R q R . which governs the electroweak pqy p 0 mass difference and which has been recently discussed in Ref. w7x. Because of the LR structure, the factorized component of the operator Q 7 , which is leading in 1rNc , cannot contribute to order O Ž p 0 . in the low-energy effective Lagrangian. The first non-trivial contribution from this operator is next-to-leading in the 1rNc expansion and is given by the integral 5 Q 7 ™ y3igmn
d4q
H Ž 2p .
4
P LmnR Ž q . tr Ž Ul LŽ 23 . U † Q R . ,
P LmnR Ž q . s 2 i d 4 x e i qP x ²0
H
†
. <0: , Ž 7.
R s qi Ž x . g
m1 2
Ž 1 q g5 . qj Ž x . ,
yQ 2P L R Ž Q 2 . s fp2 q Ý f A2 MA2 A
yÝ
f V2 M V2
V
Q2 MA2 q Q 2 Q2
M V2 q Q 2
.
Ž 11 .
Furthermore, in the chiral limit of QCD, the operator product expansion ŽOPE. applied to the correlation function P L R Ž Q 2 . implies lim Q 2P L R Ž Q 2 . ™ 0
and
lim Q 4P L R Ž Q 2 . ™ 0 .
Ž 12 .
Q 2™`
involving the two-point function
m
Since P L R Ž Q 2 . obeys an unsubtracted dispersion relation, we find that
Q 2™`
Ž 6.
with currents L m s qi Ž x . g m 12 Ž 1 y g 5 . q j Ž x .
Ž 10 .
These relations result in the two Weinberg sum rules w15x
Ý f V2 MV2 y Ý f A2 MA2 s fp2 V
and
and
A
Ý f V2 MV4 y Ý f A2 MA4 s 0 . Ž 8.
V
Ž 13 .
A
and i and j fixed flavour indices, i / j. This integral, which has to be evaluated in the chiral limit, where Ž Q 2 s yq 2 .
The usual prescription w16x for the evaluation of integrals such as Ž6. consists in taking a sharp cut-off in the Žeuclidean. integration over Q 2 ,
P LmnR Ž q . s Ž q mq n y g mn q 2 . P L R Ž Q 2 . , Ž 9. is divergent for large Q 2 and needs to be regulated.
Q 7 ™ y6
3 32p
2
2
H0L dQ Q ŽyQ P 2
=tr Ž Ul LŽ 23 . U † Q R . .
2
2
LR
Ž Q2 . . Ž 14 .
4
See e.g. Ref. w13x and references therein. 5 Up to the replacement of Q L by lŽ23. L , the bosonization of the operator Q 7 follows from the same procedure as described for the operator Q L R in Ref. w7x.
With the constraints between the couplings and masses of the narrow states coming from the Weinberg sum rules Ž13., the integral on the r.h.s. of Eq.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
230
Ž6. becomes then only logarithmically dependent on the ultraviolet scale L, with the following result 2
H0L dQ Q ŽyQ P 2
2
2
LR
Ž Q2 . .
L2
s
L2
Ý f A2 MA6 log M 2 y Ý f V2 MV6 log M 2 A
A
.
Ž 15 .
V
V
Notice that if only the contribution from the Goldstone pole had been taken into account, the resulting expression would have displayed a polynomial dependence on the cut-off scale L. Another possibility is to evaluate the integral Ž6. within a dimensional regularization scheme, say MS, in which case one obtains the same result Ž15., but with the correspondence between the cut-off L and the MS subtraction scale m given by
L s m P e 1r6 .
Ž 16 .
On the other hand, at the level of approximation that we want to attain, the bosonization of the Q8 operator is only required to leading order in the 1rNc expansion, because its Wilson coefficient is already subleading. To that order and to order O Ž p 0 . in the chiral expansion it can be readily obtained from the bosonization of the factorized density currents, with the result 6 fp2
2
ž /
Q8 ™ y12 2 B
4
tr Ž Ul LŽ 23 . U † Q R . ,
Ž 17 .
where B is the low energy constant which describes the bilinear single flavour quark condensate in the chiral limit, B s y² cc :rfp2 . As discussed in Ref. w18x there is a constraint that emerges in the large-Nc limit which relates the leading d s 6 order parameter in the OPE of the P L R two-point function w19x to couplings and masses of the narrow states: as lim Q 6P L R Ž Q 2 . s y4p 2 q O Ž a s2 . ² cc :2 2 p Q ™`
ž
/
s Ý f V2 M V6 y Ý f A2 MA6 . V
Ž 18 .
A
This relation provides part of the cancellation between the m dependence of the bosonization of the 6
See e.g. the lectures in Ref. w17x and references therein.
operator Q 7 Ž m . with the short-distance dependence on m in the Wilson coefficient of Q8 ; but, as already mentioned, and contrary to the simple case of the electroweak contribution to the pqy p 0 mass difference discussed in Ref. w7x, this cancellation here is incomplete. In this respect, we wish to comment on an important point concerning the scale dependence in the relation in Eq. Ž18.. The term on the r.h.s. of the first line results from a lowest order pQCD calculation of the Wilson coefficient. Using the renormalization group improvement to one loop, this result becomes y4p 2
as p
² cc :2 ™ y4p 2 ² cc :2
=
ž
1 yb 1
yb 1 a s Ž Q 2 .
p
2g 1q b 1
/
b1
,
Ž 19 . with ² cc : the scale invariant quark condensate Žthe analog to invariant quark masses.. In the large-Nc 3 2 limit, b 1 ™ y11 depen6 Nc and g 1 ™ 4 Nc . The Q dence of the one-loop result is indeed rather mild ; Žlog Q 2rL MS 2 .y2r11, but it does not go to a constant as the exact large-Nc result in the second line of Eq. Ž18. demands. This mismatch is due to the fact that pQCD is at best an approximation. It may happen that a two-loop renormalization group improvement of the OPE result approaches a constant behaviour at large Q 2 in a better way. In any case, this is a typical example of unavoidable mismatches that one will encounter between non-perturbative evaluations of matrix elements and short-distance pQCD evaluations, which are necessarily only approximate. In general, however, it is well known that at higher orders ambiguities will mix the short-distance coefficients of different powers in the OPE. It is difficult to imagine how to avoid these uncertainties in a final matching between short-distances and long-distances unless a breakthrough is made in understanding the relationship between pQCD and full QCD. Finally, if we restrict ourselves to the Lowest Meson Dominance ŽLMD. approximation to large-Nc QCD discussed in Ref. w20x and identify M V with Mr , our calculation Žin the MS scheme. gives the
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
following contributions to the constant h coming from the operators Q 7 and Q8 ,
a p
h w Q 7 x s y18c 7 Ž m .
a and
p
fp2 Mr2
Mr2
h w Q8 x s y 48p 2 c8 Ž m .
and ² Q8 :VSA s 12 Ž Z q 4Y . q 0
L2 log
231
y 2log2
² cc :2 Ž m . fp2 Mr6
² Q8 :VSA s '2 Y y 2
,
Ž 20 . where the relation Ž16. is understood.
'2 2 Nc
1 2 Nc
X,
X.
Ž 24 .
The quantities X, Y and Z which appear in the above expressions are the same as those usually found in the literature, i.e., X s '3 fp Ž MK2 y Mp2 . q O Ž p 4 . ,
3. The B factors B7(1rr 2) and B7(3rr 2) The bosonic expression of Q 7 given by Eqs. Ž14. and Ž15. enables us to compute the K ™ pp matrix elements induced by this operator which, following the usual conventions, we express in terms of the following isospin amplitudes ² Q 7 :I ' ² Ž pp . I < Q 7 < K 0 : ,
I s 0,2 .
Ž 21 .
To leading order O Ž p 0 . in the chiral expansion and to next-to-leading order in the 1rNc expansion, O Ž1r Nc . for K ™ pp amplitudes, we obtain the following result
(
² Q 7 : 0 s '2 ² Q 7 : 2s
6'3 16p 2 fp3
L2
=
L2
Ý f A2 MA6 log M 2 y Ý f V2 MV6 log M 2 A
A
V
.
Y s '3 fp
Zs4
ž
ž
FK Fp
ms q md
Ys
² Q 7 :VSA sy 2
'2 2
2 Nc
/
X MK2 y Mp2
; 11.4 X
1
q O Ž p2 . ,
Ž 25 .
There is no theoretical justification to consider the VSA as a good limit of any kind in QCD 7. This is reflected by the fact that e.g., the terms proportional to 1rNc in Eqs. Ž23. and Ž24. do not correspond to the correct 1rNc expansion and, in fact, the contributions of Y and Z to ² Q 7 :VSA , although suppressed I by a factor 1rNc , are actually numerically dominant, since Žthroughout, MK denotes the neutral kaon mass.
Ž 22 .
² Q 7 :VSA s 12 X q 0
/
y 1 Y q O Ž p4 . .
V
It has become customary Žrather unfortunately. to parameterize the results of weak matrix elements of four-quark operators Q i in terms of the factorized contributions from the so-called vacuum saturation approximation ŽVSA., modulated by correction factors BiŽ D I ., D I s 1r2,3r2. In the case of Q 7 and Q8 , one then has
2
MK2
ž
ž
2
MK2 ms q md
0.158 GeV ms q md
/ 2
/
.
Ž 26 .
Furthermore, whereas X is scale independent, Y and Z depend on the MS subtraction scale m through the quark mass term m s q m d in the denominator. We shall nevertheless follow these conventions, if only to be able to compare our results to those existing in
Ž Z q 4Y . ,
Xq
'2 Nc
Y,
Ž 23 .
7 Obviously, it would have been much more reasonable to normalize results to the large-Nc result in the lowest order of the chiral expansion.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
232
Fig. 1. The B factors, B7Ž1r2. and B7Ž3r2., as a function of the m scale in GeV. The solid lines correspond to Ž m s q m d .Ž m s 2 GeV. s 158 MeV; the dashed lines to the extreme low value Ž m s q m d .Ž m s 2 GeV. s 100 MeV.
the literature. The corresponding B factors are then defined as BiŽ1r2. s
² Qi :0 ² Q i :VSA 0
BiŽ3r2. s
,
² Qi :2 ² Q i :VSA 2
.
Ž 27 .
X Xq
1 Nc
½
= 1q
Ž Z q 4Y . 3
M V4
2p 2 fp2 MK2 y Mp2
L2 = log
M V2
y 2log2
5
,
X
B7Ž3r2. Ž m . s
In the sequel, we quote our results in the MS scheme. Considering first the operator Q 7 , and restricting the sums in Eq. Ž22. to the LMD approximation discussed in Ref. w20x, the previous calculation leads to the results B7Ž1r2. Ž m . s
and
Ž 28 .
Xy
2 Nc
½
= 1y
Y 3
M V4
4p 2 fp2 MK2 y Mp2
L2 = log
M V2
y 2log2
5
.
Ž 29 .
If in these expressions we take the value M V s Mr , we obtain, at the scale m s 2 GeV usually adopted in lattice calculations, 1 B7Ž1r2. Ž m s 2 GeV . ; Ž 1 q 23 . , 19.5 1 B7Ž3r2. Ž m s 2 GeV . ; y Ž 1 y 11.5 . , Ž 30 . 6.6 for, say, Ž m s q m d .Ž m s 2 GeV. s 158 MeV, the conventional reference normalization used in Eq.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
Ž26.. These values are both positive and greater than unity. One should however notice that these numbers are rather sensitive to the choice of the scale m andror to the value assigned to M V : for instance, at m s 2 M Vre1r6 ; 1.3 GeV the contributions between square brackets on the r.h.s. of Eqs. Ž28. and Ž29. vanish exactly. In order to illustrate these uncertainties, we show in Fig. 1 the variation of these B factors for a reasonable range of values of the scale m. The area between the solid lines corresponds to the choice Ž m s q m d .Ž m s 2 GeV. s 158 MeV, with L MS varied between 300 MeV and 450 MeV. The area between the dashed lines reflects the same variation of L MS , but for the extreme low value Ž m s q m d .Ž m s 2 GeV. s 100 MeV quoted in some lattice results w21x. Also shown is a magnification of the region corresponding to values of m around 2 GeV, the reference scale at which the lattice results are usually given. We find that below m Q 1.3 GeV the B7Ž1r2. and B7Ž3r2. factors can even become negative, a result which disagrees, drastically, with the positive values quoted at a ‘‘matching scale’’ of
&
233
0.8 GeV which are found in the constituent chiral quark model w13x. It is, however, not clear how this ‘‘matching scale’’ is related to the MS scale, m , in QCD. Several matrix elements of four-quark operators have also been obtained in numerical simulations of lattice-QCD. These numerical evaluations, however, are based on a yet different definition of the B parameters. The lattice definition uses a current algebra relation between the K ™ pp and the K ™ p matrix elements which is in fact only valid at order O Ž p 0 . in the chiral expansion. Thus, in the case of Q 7 discussed here, the lattice-QCD B factor, which we shall distinguish with the symbol tilde on top, is defined by the ratio B˜7Ž3r2. '
²pq< Q 7Ž3r2. < Kq: ²pq < Q 7Ž3r2. < Kq :VSA 0
,
Ž 31 .
where the matrix element in the denominator is evaluated in the chiral limit, as indicated by the subscript ‘‘0’’, and the operator Q 7 has been decom-
Ž3r2. wsee Eq. Ž31.x, as a function of the m scale in GeV. The solid lines correspond to Ž m s q m d .Ž m s 2 GeV. s Fig. 2. The B factor, B7 158 MeV; the dashed lines to the extreme low value Ž m s q m d .Ž m s 2 GeV. s 100 MeV.
M. Knecht et al.r Physics Letters B 457 (1999) 227–235
234
posed into its D I s 1r2 and D I s 3r2 components, Q 7 s Q 7Ž1r2. q Q 7Ž3r2., with Q 7Ž3r2. s 2 Ž sL g m d L . u R gm u R y d R gm d R q 2 Ž sL g m u L . Ž u R gm d R . .
Ž 32 .
The latest results obtained by various groups are consistent with each other: B˜7Ž3r2. Ž m s 2 GeV .
s latt
½
q5 q2 0.58y4 y8
Ž 33 .
0.61 Ž 11 .
The first value has been taken from Ref. w10x, while the second one arises from the results of Refs. w11x and w12x, translated into the MS scheme w22x. In the vacuum saturation approximation which the lattice community uses, one obtains ²pq < Q 7Ž3r2. < Kq :VSA sy 0
2 ² cc :2 fp2
Nc
2
fp
Nc
'3
sy
Y,
Ž 34 . where in the second expression, following common practice, we have traded the dependence with respect to the condensate for the dependence on the strange quark mass, using the Gell-Mann–Oakes–Renner relation, which holds in the chiral limit. The calculation based on the bosonized expression for this operator which we have discussed gives ²pq < Q 7Ž3r2. < Kq : s
3 8p 2 fp2
Ý f A2 MA6 log A
y Ý f V2 M V6 log V
3 , 4p
M V4 2
ž
obtained from the lattice are usually not quoted together. In the case of Q8 , the large-Nc limit of Eq. Ž24. simply reproduces the result obtained from the bosonized expression Ž17.. Since subleading corrections in the 1rNc expansion of this operator are not yet available, further comparison with lattice results for B8Ž3r2. has to be postponed.
MA2
L2
M V2
4. Conclusions and outlook The expressions in Eqs. Ž22. and Ž35. are the main results reported in this letter. They are a first step towards a systematic evaluation of weak matrix elements in the chiral expansion and to first non-trivial order in the 1rNc expansion. Our final aim, however, is to obtain values for the coupling constants of the low energy D S s 1 and D S s 2 chiral effective Lagrangian directly; i.e., constants like h in Eq. Ž4. and not of individual matrix elements of four-quark operators. It is encouraging from the results obtained so far, to find such simple analytic expressions which exhibit only a logarithmic dependence on the matching scale m ; however, the fact that the numerical results are so sensitive to the choice of m in the GeV region is perhaps an indication that one should be extremely cautious in the evaluation of errors of B factors, in general, both in model calculations and in lattice QCD numerical simulations.
L2 L2
2log2y log
M V2
Acknowledgements
/
,
Ž 35 . where in the second line we have used the LMD approximation discussed in Ref. w20x. In this approximation and with the same numerical input as in Ž30., we obtain B˜7Ž3r2. Ž m s 2 GeV. , q1.5. However, as shown in Fig. 2, the value of this B factor is again very sensitive to the choice of the matching scale m. The results obtained for the smaller value of the strange quark mass Ži.e. for larger values of the condensate. are in agreement, within errors, with the numbers quoted in Eq. Ž33.. Unfortunately, the values of m s Žor of the condensate. and of the B factors
We wish to thank M. Perrottet and L. Lellouch for discussions on topics related to the work reported here, V. Gimenez for an informative correspondence, and the referee for constructive remarks. This work has been supported in part by TMR, EC-Contract No. ERBFMRX-CT980169 ŽEURODA f NE.. The work of S. Peris has also been partially supported by the research project CICYT-AEN98-1093. References w1x J. Bijnens, M. Wise, Phys. Lett. B 137 Ž1984. 245. w2x G. ’t Hooft, Nucl. Phys. B 72 Ž1974. 461; B 73 Ž1974. 461. w3x G. Rossi, G. Veneziano, Nucl. Phys. B 123 Ž1977. 507.
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