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CPT-2004/P.014

Rare Kaon Decays Revisited

Samuel Friot, David Greynat and Eduardo de Rafael

arXiv:hep-ph/0404136v1 16 Apr 2004

Centre de Physique Th? eorique 1 CNRS-Luminy, Case 907 F-13288 Marseille Cedex 9, France

Abstract We present an updated discussion of K → π? ll decays in a combined framework of chiral perturbation theory and Large–Nc QCD, which assumes the dominance of a minimal narrow resonance structure in the invariant mass dependence of the ? ll pair. The proposed picture reproduces very well, both the experimental K + → π + e+ e? decay rate and the invariant e+ e? mass spectrum. The predicted Br(KS → π 0 e+ e? ) is, within errors, consistent with the recently reported result from the NA48 collaboration. Predictions for the K → π ?+ ?? modes are also obtained. We ?nd that the resulting interference between the direct and indirect CP–violation amplitudes in KL → π 0 e+ e? is constructive.

1 Unit? e Mixte de Recherche (UMR 6207) du CNRS et des Universit? es Aix Marseille 1, Aix Marseille 2 et sud ToulonVar, a?li? ee a ` la FRUMAM

1

Introduction

In the Standard Model, transitions like K → πl+ l? , with l = e, ?, are governed by the interplay of weak non–leptonic and electromagnetic interactions. To lowest order in the electromagnetic coupling constant they are expected to proceed, dominantly, via one–photon exchange. This is certainly the 0 case for the K ± → π ± l+ l? and KS → π 0 l+ l? decays [1]. The transition K2 → π 0 γ ? → π 0 l+ l? , via one virtual photon, is however forbidden by CP–invariance. It is then not obvious whether the physical decay KL → π 0 l+ l? will still be dominated by the CP–suppressed γ ? –virtual transition or whether a transition via two virtual photons, which is of higher order in the electromagnetic coupling but CP– allowed, may dominate [2]. The possibility of reaching branching ratios for the mode KL → π 0 e+ e? as small as 10?12 in the near future dedicated experiments of the NA48 collaboration at CERN, is a strong motivation for an update of the theoretical understanding of these modes. 0 The CP–allowed transition K2 → π 0 γ ? γ ? → π 0 e+ e? has been extensively studied in the literature (see refs. [3, 4] and references therein). We have nothing new to report on this mode. A recent estimate of a conservative upper bound for this transition gives a branching ratio [5] Br(KL → π 0 e+ e? )|CPC < 3 × 10?12 . (1.1)

0 There are two sources of CP–violation in the transition KL → π 0 γ ? → π 0 l+ l? . The direct source is the one induced by the “electroweak penguin”–like diagrams which generate the e?ective local four–quark operators [6]

Q11 = 4 (? sL γ ? dL )

l=e,?

(? lL γ? lL )

and

Q12 = 4 (? sL γ ? dL )

l=e,?

(? lR γ? lR )

(1.2)

modulated by Wilson coe?cients which have an imaginary part induced by the CP–violation phase 0 – of the ?avour mixing matrix. The indirect source of CP–violation is the one induced by the K1 component of the KL state which brings in the CP–violation parameter ?. The problem in the 0 → π 0 e+ e? . indirect case is, therefore, reduced to the evaluation of the CP–conserving transition K1 If the sizes of the two CP–violation sources are comparable, as phenomenological estimates seem to indicate [2, 4, 7, 5], the induced branching ratio becomes, of course, rather sensitive to the interference between the two direct and indirect amplitudes. Arguments in favor of a constructive interference have been recently suggested [5]. The analysis of K → πγ ? → πl+ l? decays within the framework of chiral perturbation theory (χPT) was ?rst made in refs. [1, 2]. To lowest non trivial order in the chiral expansion, the corresponding decay amplitudes get contributions both from chiral one loop graphs, and from tree level contributions of local operators of O(p4 ). In fact, only two local operators of the O(p4 ) e?ective Lagrangian with ?S = 1 contribute to the amplitudes of these decays. With L? (x) the 3 × 3 ?avour matrix current ?eld

2 L? (x) ≡ ?iF0 U (x)? D? U (x) ,

(1.3)

where U (x) is the matrix ?eld which collects the Goldstone ?elds (π ’s, K ’s and η ), the relevant e?ective Lagrangian as written in ref. [1], is ie GF . ?S =1 ? Le? (x) = ? √ Vud Vus g8 tr (λL? L? ) ? 2 [w1 tr(QλL? Lν ) + w2 tr(QL? λLν )] F ?ν F0 2 + h.c. .

(1.4) Here D? is a covariant derivative which, in the presence of an external electromagnetic ?eld source A? only, reduces to D? U (x) = ?? U (x) ? ieA? (x)[Q, U (x)] ; F ?ν is the electromagnetic ?eld strength tensor; F0 is the pion decay coupling constant (F0 ? 87 MeV) in the chiral limit; Q the electric charge matrix; and λ a short–hand notation for the SU (3) Gell-Mann matrix (λ6 ? iλ7 )/2: ? ? ? ? 2/3 0 0 0 0 0 0 ?, Q = ? 0 ?1/3 λ=? 0 0 0 ?. (1.5) 0 0 ?1/3 0 1 0 1

The overall constant g8 is the dominant coupling of non–leptonic weak transitions with ?S = 1 and ?I = 1/2 to lowest order in the chiral expansion. The factorization of g8 in the two couplings w1 and w2 is, however, a convention. For the purposes of this paper, we shall rewrite the e?ective Lagrangian in Eq. (1.4) in a more convenient way. Using the relations 1 Qλ = λQ = ? λ 3 and ? ? 1I Q=Q 3 ? = diag.(1, 0, 0) , I = diag.(1, 1, 1) , Q (1.6)

and inserting the current ?eld decomposition

2 L? (x) = L? (x) ? eF0 A? (x)?(x) ,

(1.7) (1.8)

where in Eq. (1.4), results in the Lagrangian

2 ? U (x)?? U (x) L? (x) = ?iF0

and

? U (x)] , ?(x) = U ? (x)[Q,

GF . 2 ? ?S =1 A? tr[λ(L? ? + ?L? )] g8 tr (λL? L? ) ? eF0 Le? (x) = ? √ Vud Vus 2 ie ?ν ? ν ) + h.c. + (w1 ? w2 ) tr (λL? Lν ) + 3 w2 tr(λL? QL 2F 3F0

(1.9)

The Q11 and Q12 operators in Eq. (1.2) are proportional to the quark current density (? sL γ ? dL ) and, therefore, their e?ective chiral realization can be directly obtained from the strong chiral Lagrangian [ (? sL γ ? dL ) ? (L? )23 to O(p)]. Using the equations of motion for the leptonic ?elds ? ? F?ν = e? lγ? l, and doing a partial integration in the action, it follows that the e?ect of the electroweak penguin operators induces a contribution to the coupling constant w ? only; more precisely g8 (w ? = w1 ? w2 ) =

Q11 ,Q12

3 C11 (?2 ) + C12 (?2 ) , 4πα

(1.10)

where C11 (?2 ) and C12 (?2 ) are the Wilson coe?cients of the Q11 and Q12 operators. There is a resulting ?–scale dependence in the real part of the Wilson coe?cient C11 + C12 due to an incomplete cancellation of the GIM–mechanism because, in the short–distance evaluation, the u–quark has not been integrated out. This ?–dependence should be canceled when doing the matching with the long– distance evaluation of the weak matrix elements of the other four–quark operators; in particular, with the contribution from the unfactorized pattern of the Q2 operator in the presence of electromagnetism. ? and w2 couplings within the It is in principle possible, though not straightforward, to evaluate the w framework of Large–Nc QCD, in much the same way as other low–energy constants have been recently determined (see e.g. ref. [8] and references therein). While awaiting the results of this program, we propose in this letter a more phenomenological approach. Here we shall discuss the determination of ? and w2 using theoretical arguments inspired from Large–Nc considerations, combined the couplings w with some of the experimental results which are already available at present. As we shall see, our conclusions have interesting implications for the CP–violating contribution to the KL → π 0 e+ e? mode. 2

As discussed in ref. [1], at O(p4 ) in the chiral expansion, besides the contributions from the w1 and w2 terms in Eq. (1.4) there also appears a tree level contribution to the K + → π + e+ e? amplitude induced by the combination of the lowest O(p2 ) weak ?S = 1 Lagrangian (the ?rst term in Eq. (1.4)) with the L9 –coupling of the O(p4 ) chiral Lagrangian which describes strong interactions in the presence of electromagnetism [9]: . ?ν L(4) (x)tr Q D? U (x)Dν U ? (x) + Q D? U (x)? Dν U (x) . (2.1) em (x) = ?ieL9 F 2

K → πl? l Decays to O(p4 ) in the Chiral Expansion

In full generality, one can then predict the K + → π + l+ l? decay rates (l = e, ?) as a function of the scale–invariant combination of coupling constants 1 1 M 2 m2 w+ = ? (4π )2 [w1 ? w2 + 3(w2 ? 4L9 )] ? log K4 π , 3 6 ν (2.2)

where w1 , w2 and L9 are renormalized couplings at the scale ν . The coupling constant L9 can be determined from the electromagnetic mean squared radius of the pion [10]: L9 (Mρ ) = (6.9±0.7)×10?3. The combination of constants w2 ? 4L9 is in fact scale independent. To that order in the chiral expansion, the predicted decay rate Γ(K + → π + e+ e? ) as a function of w+ describes a parabola. The intersection of this parabola with the experimental decay rate obtained from the branching ratio [11] Br(K + → π + e+ e? ) = (2.88 ± 0.13) × 10?7 , gives the two phenomenological solutions (for a value of the overall constant g8 = 3.3): w+ = 1.69 ± 0.03 and w+ = ?1.10 ± 0.03 . (2.4) (2.3)

Unfortunately, this twofold determination of the constant w+ does not help to predict the KS → π 0 e+ e? decay rate. This is due to the fact that, to the same order in the chiral expansion, this transition amplitude brings in another scale–invariant combination of constants:

2 1 1 MK ws = ? (4π )2 [w1 ? w2 ] ? log 2 . 3 3 ν

(2.5)

The predicted decay rate Γ(KS → π 0 e+ e? ) as a function of ws is also a parabola. From the recent result on this mode, reported by the NA48 collaboration at CERN [12]:

.8 ?9 , Br KS → π 0 e+ e? = 5.8+2 ?2.3 (stat.) ± 0.8(syst.) × 10

(2.6)

one obtains the two solutions for ws

.50 ws = 2.56+0 ?0.53 .53 and ws = ?1.90+0 ?0.50 .

(2.7)

At the same O(p4 ) in the chiral expansion, the branching ratio for the KL → π 0 e+ e? transition induced by CP–violation reads as follows Br KL → π 0 e+ e? |CPV = (2.4 ± 0.2) Imλt 10?4

2

+ (3.9 ± 0.1)

1 ? ws 3

2

+ (3.1 ± 0.2)

Imλt 10?4

1 ? ws 3

× 10?12 . (2.8)

Here, the ?rst term is the one induced by the direct source, the second one by the indirect source and the third one the interference term. With [13] Imλt = (1.36 ± 0.12) × 10?4 , the interference is constructive for the negative solution in Eq. (2.7). The four solutions obtained in Eqs. (2.4) and (2.7), de?ne four di?erent straight lines in the plane of the coupling constants w2 ? 4L9 and w ? (= w1 ? w2 ), as illustrated in Fig. 1 below. We next want to discuss which of these four solutions, if any, may be favored by theoretical arguments. 3 3.1

Theoretical Considerations

The Octet Dominance Hypothesis

In ref. [1], it was suggested that the couplings w1 and w2 may satisfy the same symmetry properties as the chiral logarithms generated by the one loop calculation. This selects the octet channel in the transition amplitudes as the only possible channel and leads to the relation w2 = 4L9 ,

Octet Dominance Hypothesis (ODH) .

(3.1)

3

Fig. 1 The four intersections in this ?gure de?ne the possible values of the couplings which, at O(p4 ) in the chiral expansion, are compatible with the experimental input of Eqs. (2.3) and (2.6). The couplings w1 , w2 , and L9 have been ?xed at the ν = Mρ scale and correspond to the value g8 = 3.3. The cross in this ?gure corresponds to the values in Eqs. (3.20) and (3.21) discussed in the text. We now want to show how this hypothesis can in fact be justi?ed within a simple dynamical framework of resonance dominance, rooted in Large–Nc QCD. For that, let us examine the ?eld content of the Lagrangian in Eq. (1.9). For processes with at most one pion in the ?nal state, it is su?cient to restrict ? and L? to their minimum of one Goldstone ?eld component: √ √ 2 ? ] + · · · , and L? = 2F0 ?? Φ + · · · , (3.2) ? = ?i [Φ, Q F0 with the result (using partial integration in the term proportional to ie g8 w2 ) GF . ?S =1 ? 2 2 ? ? ? Φ ? ?? ΦQ ? Φ)] Le? (x) = ? √ Vud Vus g8 2F0 tr (λ?? Φ? ? Φ) + ie 2F0 A tr[λ(ΦQ? 2 ? ? Φ ? ?? ΦQ ? Φ)] ?ie w2 ?ν F ν? tr[λ(ΦQ? 2 +ie w ? F ?ν tr (λ?? Φ?ν Φ) + h.c. 3

(3.3)

showing that the two–?eld content which in the term modulated by w2 couples to ?ν F ν? is exactly the same as the one which couples to the gauge ?eld A? in the lowest O(p2 ) Lagrangian. As explained in ref. [1], the contribution to K + → π + γ (virtual) from this O(p2 ) term, cancels with the one resulting from the combination of the ?rst term in Eq. (3.3) with the lowest order hadronic electromagnetic interaction, in the presence of mass terms for the Goldstone ?elds. This cancellation is expected because of the mismatch between the minimum number of powers of external momenta required by gauge invariance and the powers of momenta that the lowest order e?ective chiral Lagrangian can provide. As we shall next explain, it is the re?ect of the dynamics of this cancellation which, to a ?rst approximation, is also at the origin of the relation w2 = 4L9 .

4

With two explicit Goldstone ?elds, the hadronic electromagnetic interaction in the presence of the term in Eq. (2.1) reads as follows Lem (x) = ?ie A? ? 2L9 ν? 2 ?ν F F0 ? Φ ?? Φ) + · · · . tr(Q

?

(3.4)

The net e?ect of the L9 –coupling is to provide the slope of an electromagnetic form factor to the charged Goldstone bosons. In momentum space this results in a change from the lowest order point like coupling to 2L9 1 ? 1 ? 2 Q2 . (3.5) F0 In the minimal hadronic approximation (MHA) to Large–Nc QCD [14], the form factor in question is saturated by the lowest order pole i.e. the ρ(770) : 1?

2 Mρ , 2 + Q2 Mρ

which implies

L9 =

2 F0 . 2 2Mρ

(3.6)

It is well known [15, 16] that this reproduces the observed slope rather well. By the same argument, the term proportional to w2 in Eq. (3.3) provides the slope of the lowest order electroweak coupling of two Goldstone bosons: w2 GF ? 2 ν? A? ? g8 2F0 Lew (x) = ?ie √ Vud Vus 2 ?ν F 2F0 2 w2 2 2Q . 2F0 ? ? Φ ? ?? ΦQ ? Φ)] + · · · . tr[λ(ΦQ? (3.7)

In momentum space this results in a change from the lowest order point like coupling to 1?1? (3.8)

Here, however, the underlying ?S = 1 form factor structure in the same MHA as applied to L9 , can have contributions both from the ρ and the K ? (892) : 1?

2 2 αMρ βMK ? + , 2 2 2 Mρ + Q M K ? + Q2

with

α +β = 1,

(3.9)

because at Q2 → 0 the form factor is normalized to one by gauge invariance. This ?xes the slope to w2 2 = 2F0 β α + 2 2 Mρ MK ? . (3.10)

If, furthermore, one assumes the chiral limit where Mρ = MK ? , there follows then the ODH relation in Eq. (3.1); a result which, as can be seen in Fig. 1, favors the solution where both w+ and ws are negative, and the interference term in Eq. (2.8) is then constructive. 3.2

A rather detailed measurement of the e+ e? invariant mass spectrum in K + → π + e+ e? decays was reported a few years ago in ref. [17]. The observed spectrum con?rmed an earlier result [18] which had already claimed that a parameterization in terms of only w+ cannot accommodate both the rate and the spectrum of this decay mode. It is this observation which prompted the phenomenological analyses reported in refs. [7, 5]. Here, we want to show that it is possible to understand the observed spectrum within a simple MHA picture of Large–Nc QCD which goes beyond the O(p4 ) framework of χPT but, contrary to the proposals in refs. [7, 5], it does not enlarge the number of free parameters. We recall that, in full generality [1], the K + → π + e+ e? di?erential decay rate depends only on ?(z ): one form factor φ

5 G2 α2 MK dΓ 2 λ3/2 1, z, rπ = 8 dz 12π (4π )4

Beyond the O(p4 ) in χPT

1?4 5

2 r? z

1+2

2 r? z

?(z ) φ

2

,

(3.11)

2 where q 2 = z MK is the invariant mass squared of the e+ e? pair, and

GF ? G8 = √ Vud Vus g8 , 2

rπ =

mπ , MK

r? =

m? . MK

(3.12)

?(z ) and the form factor plotted in Fig. 5 of ref. [17], which we reproduce here The relation between φ 2 in our Fig. 2 below for |fV (z )| , is |fV (z )| =

2

G8 ? φ(z ) GF

2

.

(3.13)

Fig. 2 Plot of the form factor |fV (z )| de?ned by Eqs. (3.11) and (3.13) versus the invariant mass 2 squared of the e+ e? pair normalized to MK . The crosses are the experimental points of ref. [17]; the dotted curve is the leading O(p4 ) prediction, using the positive solution for w+ in Eq. (2.4); the continuous line is the ?t to the improved form factor in Eq. (3.19) below. The O(p4 ) form factor calculated in ref. [1] is ?(z ) φ with the chiral loop functions φK (z ) = ? 41 5 1 + + 3 z 18 3 4 ?1 z

3 2

2

2

= |w+ + φK (z ) + φπ (z )| ,

2

(3.14)

The experimental form factor favors the positive solution in Eq. (2.4), but the predicted O(p4 ) form factor, the dotted curve in Fig. 2, lies well below the experimental points for z > ? 0.2. Following the ideas developed in the previous subsection, we propose a very simple generalization of the O(p4 ) form factor. We keep the lowest order chiral loop contribution as the leading manifestation ? = w1 ? w2 in w+ by the of the Goldstone dynamics, but replace the local couplings w2 ? 4L9 and w minimal resonance structure which can generate them in the z –channel. For w2 ? 4L9 this amounts to the replacement: 6

arctan ?

?

1

4 z

?1

? ?

and φπ (z ) = φ z

2 MK 2 mπ

.

(3.15)

w2 ? 4L9 ?

2 2Fπ 2 Mρ

α

2 Mρ

2 = 2Fπ β

2 2 Mρ ? MK ? ; 2 2 2 2 ? M z )(M (Mρ ? K K ? MK z )

2 2 2 2 Mρ Mρ Mρ MK ? +β 2 ? 2 2 2 2z 2 ? MK z MK ? MK ? ? MK z Mρ ? MK

(3.16)

? it simply amounts to the modulating factor: while for w ? ?w ? w

2 Mρ . 2 ? M2 z Mρ K

(3.17)

Notice that in the chiral limit where Mρ = MK ? , Fπ → F0 , and when z → 0, we recover the usual O(p4 ) couplings with the ODH constraint w2 = 4L9 . In our picture, the deviation from this constraint is due to explicit breaking, induced by the strange quark mass, and results in an e?ective w2 ? 4L9 = ?

2 2Fπ β 2 Mρ

1?

2 Mρ 2 MK ?

.

(3.18)

More explicitly, the form factor we propose is G8 GF 1 + ln 6

2 2 2 Mρ Mρ ? MK ? (4π )2 2 ? 2 w + 6Fπ β 2 2 2 2 2 ? M z (M 3 Mρ ? MK z Mρ K K ? ? MK z ) 2 MK m2 π 4 Mρ

fV (z ) =

+

1 1 ? z ? χ(z ) 3 60

,

(3.19)

where the ?rst line incorporates the modi?cations in Eqs. (3.16) and (3.17), while the second line is the chiral loop contribution of ref. [1], renormalized at ν = Mρ , and where we have only retained ? and β left as the ?rst two terms in the expansion of φK (z ), while χ(z ) = φπ (z ) ? φπ (0). With w free parameters, we make a least squared ?t to the experimental points in Fig. 2. The result is the continuous curve shown in the same ?gure, which corresponds to a χ2 = 13.0 for 18 degrees of min. freedom. The ?tted values (using g8 = 3.3 and Fπ = 92.4 MeV) are ? = 0.045 ± 0.003 w and therefore w2 ? 4L9 = ?0.019 ± 0.003 . (3.21) These are the values which correspond to the cross in Fig. 1 above. As a test we compute the K + → π + e+ e? branching ratio, using the form factor in Eq. (3.19) with ? and β , with the result the ?tted values for w Br(K + → π + e+ e? ) = (3.0 ± 1.1) × 10?7 , in good agreement (as expected) with experiment result in Eq. (2.3). ? results in a negative value for ws in Eq. (2.5) The ?tted value for w ws = ?2.1 ± 0.2 , which corresponds to the branching ratios Br KS → π 0 e+ e? = (7.7 ± 1.0) × 10?9 ,

0 + ?

and

β = 2.8 ± 0.1 ;

(3.20)

(3.22)

(3.23)

(3.24) (3.25)

Br KS → π e e

|>165MeV = (4.3 ± 0.6) × 10 7

?9

.

This is to be compared with the recent NA48 results in Eq. (2.6) and [12]

.5 ?9 . Br KS → π 0 e+ e? |>165MeV = 3+1 ?1.2 (stat.) ± 0.1(syst.) × 10

(3.26)

The predicted branching ratios for the K → π ?+ ?? modes are Br K + → π + ?+ ?? = (8.7 ± 2.8) × 10?8 to be compared with Br K + → π + ?+ ?? = (7.6 ± 2.1) × 10?8 , Br KS → π ? ?

0 + ?

and Br KS → π 0 ?+ ?? = (1.7 ± 0.2) × 10?9 , (3.27)

ref. [11]

?9

(3.28) , ref. [19] . (3.29)

=

.4 2.9+1 ?1.2 (stat.)

± 0.2(syst.) × 10

Finally, the resulting negative value for ws in Eq. (3.23), implies a constructive interference in Eq. (2.8) with a predicted branching ratio Br KL → π 0 e+ e? |CPV = (3.7 ± 0.4) × 10?11 , (3.30)

where we have used [13] Imλt = (1.36 ± 0.12) × 10?4 and we have taken into account the e?ect of the modulating form factor in Eq. (3.17). 4

Conclusions

Earlier analyses of K → π e+ e? decays within the framework of χPT have been extended beyond the predictions of O(p4 ), by replacing the local couplings which appear at that order by their underlying narrow resonance structure in the spirit of the MHA to Large-Nc QCD. The resulting modi?cation of the O(p4 ) form factor is very simple and does not add new free parameters. It reproduces very well both the experimental decay rate and the invariant e+ e? mass spectrum. The predicted Br(KS → π 0 e+ e? ) and Br(KS → π 0 ?+ ?? ) are, within errors, consistent with the recently reported result from the NA48 collaboration. The predicted interference between the direct and indirect CP–violation amplitudes in KL → π 0 e+ e? is constructive, with an expected branching ratio (see Eq. (3.30)) within reach of a dedicated experiment.

Acknowledgements We thank A. Pich for his help in the earlier stages of this work and G. Isidori for discussions. This work has been supported in part by TMR, EC-Contract No. HPRN-CT-2002-00311 (EURIDICE).

References

[1] G. Ecker, A. Pich and E. de Rafael, Nucl. Phys. B291 (1987) 692. [2] G. Ecker, A. Pich and E. de Rafael, Nucl. Phys. B303 (1988) 665. [3] A.G. Cohen, G. Ecker and A. Pich, Phys. Lett. B304 (1993) 347. [4] J.F. Donoghue and F. Gabbiani, Phys. Rev. D51 (1995) 2187. [5] G. Buchalla, G. D’Ambrosio and G. Isidori, Nucl. Phys. B672 (2003) 387. [6] F.J. Gilman and M.B. Wise, Phys. Rev. D21 (1980) 3150. [7] G. D’Ambrosio, G. Ecker, G. Isidori and J. Portol? es, JHEP 08 (1998) 004. 8

[8] T. Hambye, S. Peris and E. de Rafael, JHEP 05 (2003) 027. [9] J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465. [10] J. Bijnens, G. Ecker and J. Gasser The second DAΦNE Physics Handbook (1994) 125 [hep-ph/9411232]. [11] Review of Particle Physics, Phys. Rev. D66 (2002) 010001-1. [12] J.R. Batley et al, Phys. Lett. B576 (2003) 43. [13] M. Battaglia et al, hep-ph/0304132. [14] S. Peris, M. Perrottet and E. de Rafael, JHEP 05 (1998) 011. [15] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B321 (1989) 425. [16] G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B223 (1989) 425. [17] R. Appel et al, Phys. Rev. Lett. 83 (1999) 4482. [18] C. Alliegro et al, Phys. Rev. Lett. 68 (1992) 278. [19] M. Slater, Latest Results from NA48 and NA48/1, presented at the Moriond Workshop, (2004).

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- Rare Kaon Decays on the Lattice
- Radiative and rare kaon decays an update
- Rare Kaon Decays Beyond the Standard Model

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