{'fraction_non_alphanumeric': 0.03826272861724093, 'fraction_numerical': 0.019870846817296568, 'mean_word_length': 4.240733635177653, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
New Analysis of the Delta I = 1/2 Rule in Kaon Decays and the B_K Parameter

We present a new analysis of the Delta I = 1/2 rule in K -->pi pi decays and the B_K parameter. We use the 1/N_c expansion within the effective chiral lagrangian for pseudoscalar mesons and compute the hadronic matrix elements at leading and next-to-leading order in the chiral and the 1/N_c expansions. Numerically, our calculation reproduces the dominant Delta I = 1/2 K -->pi pi amplitude. Our result depends only moderately on the choice of the cutoff scale in the chiral loops. The Delta I = 3/2 amplitude emerges sufficiently suppressed but shows a significant dependence on the cutoff. The B_K parameter turns out to be smaller than the value previously obtained in the 1/N_c approach. It also shows a significant dependence on the choice of the cutoff scale. Our results indicate that corrections from higher order terms and/or higher resonances are large for the Delta I = 3/2 K -->pi pi amplitude and the (|Delta S| = 2) K^0 -- anti K^0 transition amplitude.


Introduction
Over the last few decades the kaon system has provided us with a rich field of phenomenology which has been important for developing our theoretical understanding of the interplay of weak and strong interactions. The nonleptonic kaon decays are especially interesting because they provide a testing ground for QCD dynamics at long distances. Two outstanding problems in the field are the explanation of the ∆I = 1/2 rule in K → ππ decays and the calculation of theB K parameter which measures the non-perturbative contributions to the (|∆S| = 2) K 0 −K 0 transition amplitude. An accurate knowledge ofB K is necessary for theoretically investigating the indirect CP violation in the neutral kaon mass matrix, as well as, the K L − K S mass difference. The ∆I = 1/2 rule is particularly important because it gives rise to the small value of the ratio ε ′ /ε which measures the direct CP violation in the K → ππ decay amplitudes.
Since its first observation more than 40 years ago [1] the ∆I = 1/2 enhancement has attracted a great deal of theoretical interest trying to find the dynamical mechanism behind the approximate isospin selection rule, in particular within the standard model. Experimentally, the ratio of the ∆I = 1/2 and ∆I = 3/2 amplitudes in K → ππ decays corresponding to I = 0 and I = 2 in the final state, respectively, was measured to be with A I = a I exp(iδ I ) and δ I the final state interaction phases. This result was particularly enigmatic before the advent of QCD when only the current-current operator Q 2 arising from the W exchange was included in the analysis and, consequently, Rea 0 /Rea 2 was expected to be around one. With the establishment of QCD our understanding of the ∆I = 1/2 selection rule improved considerably. Using the operator product expansion, the K → ππ amplitudes are obtained from the effective low-energy hamiltonian for |∆S| = 1 transitions [2 -4], The arbitrary renormalization scale µ separates short-and long-distance contributions to the decay amplitudes. The Wilson coefficient functions c i (µ) contain all the information on heavy-mass scales. For CP conserving processes only the z i are numerically relevant. The coefficient functions can be calculated for a scale µ 1 GeV using perturbative renormalization group techniques. They were computed in an extensive next-to-leading logarithm analysis by two groups [5,6]. The local four-quark operators Q i (µ) can be written, after Fierz reordering, in terms of color singlet quark bilinears: where the sum goes over the light flavors (q = u, d, s) and q R,L = 1 2 (1 ± γ 5 )q , e q = (2/3, −1/3, −1/3) .
Q 3 , . . . , Q 6 arise from QCD penguin diagrams involving a virtual W and a c or t quark, with gluons connecting the virtual heavy quark to light quarks. They transform as (8 L , 1 R ) under SU(3) L × SU(3) R and solely contribute to ∆I = 1/2 transitions. Q 7 and Q 8 are electroweak penguin operators [7,8] which are less important for the ∆I = 1/2 rule. Longdistance contributions to the amplitudes A I are contained in the hadronic matrix elements of the four-quark operators, which are related to the π + π − and π 0 π 0 final states through the isospin decomposition They are difficult to calculate but can be estimated using non-perturbative techniques generally for µ around a scale of 1 GeV. Major progress in the understanding of the ∆I = 1/2 rule was made when it was observed that the short-distance (quark) evolution, which is represented by the Wilson coefficient functions in the effective hamiltonian of Eq. (2), leads to both an enhancement of the I = 0 and a suppression of the I = 2 final state. The octet enhancement [2] in the (Q 1 , Q 2 ) sector is dominated by the increase of z 2 when µ evolves from M W down to µ ≃ 1 GeV, whereas the suppression of the ∆I = 3/2 transition results from a partial cancellation between the contributions from the Q 1 and Q 2 operators owing to a destructive Pauli interference in the K + → π + π 0 amplitude. Another important short-distance enhancement was found to arise in the sector of the QCD penguin operators, in particular for z 6 , through the proper inclusion of the threshold effects (and the associated incomplete GIM cancellation above the charm quark mass) [9]. Nevertheless, it was concluded that the perturbative QCD effects are far from sufficient to describe the ∆I = 1/2 rule and QCD dynamics at low energies must be addressed. The long-distance enhancement of the matrix elements of the QCD penguin operators over the matrix elements of Q 1 and Q 2 was first conjectured and estimated in Ref. [3] in the vacuum saturation approximation (VSA) [10]. The VSA approach, however, fails completely in explaining the ∆I = 1/2 rule, and a more refined method for the calculation of the hadronic matrix elements is certainly needed.
Due to the non-perturbative nature of the long-distance contribution, a large variety of techniques has been proposed to estimate it (for some recent publications see Refs. [11 -16]). Among the analytical methods, the 1/N c expansion [17] (N c being the number of colors) associated with the effective chiral lagrangian is particularly interesting. In this approach, QCD dynamics at low energies is represented by the 'meson evolution' of the operators, from zero momentum to µ, in terms of the chiral loop corrections to the matrix elements [9,18]. The authors of Ref. [18] calculated the loop corrections to the matrix elements of Q 1 and Q 2 and included the gluon penguin operator Q 6 at the tree level, consistent with the 1/N c expansion. They obtained an additional enhancement and suppression of the ∆I = 1/2 and ∆I = 3/2 amplitudes, respectively, systematically continuing the octet enhancement in the (Q 1 , Q 2 ) sector to the long-distance domain. Numerically, a 2 was reproduced with an accuracy of 70 to approximately 100 %, whereas a 0 [for Λ QCD = 300 MeV and m s (1 GeV) = 125 -175 MeV] was found to be around 65 -80 % of the measured value, suggesting that the bulk of the physics behind the ∆I = 1/2 rule in kaon decays is now understood. One might note that the agreement with experiment is not improved by including the next-to-leading order values for the z i [19].
In this article we present a new calculation of the hadronic matrix elements in K → ππ decays in the 1/N c expansion for pseudoscalar mesons. The paper contains several improvements over the original approach of Ref. [18] which are conceptually and numerically important. One improvement concerns the matching of short-and long-distance contributions to the amplitudes, by adopting a modified identification of virtual momenta in the integrals of the chiral loops. To be explicit, we consider the two currents or densities in the chiral representation of the operators to be connected to each other through the exchange of an effective color singlet boson, and identify its momentum with the loop integration variable. The effect of this procedure is to modify the loop integrals, which introduces noticeable effects in the final results. More important it provides an unambiguous matching of the 1/N c expansion in terms of mesons to the QCD expansion in terms of quarks and gluons. The approach followed here leads to an explicit classification of the diagrams into factorizable and non-factorizable. Factorizable loop diagrams refer to the strong sector of the theory and give corrections whose scale dependence is absorbed in the renormalization of the chiral effective lagrangian. The non-factorizable loop diagrams have to be matched to the Wilson coefficients and should cancel scale dependences which arise from the shortdistance expansion. In a recent publication together with W.A. Bardeen and E.A. Paschos [20] we used this method to calculate the hadronic matrix elements of Q 6 and Q 8 which dominate the ratio ε ′ /ε. In this paper we focus on the CP conserving amplitudes which, to a large extent, are governed by the current-current operators Q 1 and Q 2 .
In Ref. [18] a mass scale replacing the complete dependence of the exact expressions on the meson masses was introduced in the chiral logarithms. Another improvement of this paper is that we investigate the exact expressions for the matrix elements using the matching prescription discussed above, i.e., we evaluate the complete finite terms from the non-factorizable diagrams. Moreover, we calculate the whole of the matrix elements, that is to say, we also take into account the subleading penguin operators. For consistency with Ref. [20] we also include the small effects of the singlet η 0 . In the numerical analysis we take special care to separate the different contributions. In particular, we discuss the effect of the final state interaction phases which were not taken into account in Ref. [18]. Uncertainties arising from the short-distance part of the calculation are estimated by comparing the amplitudes obtained from the LO and the NLO Wilson coefficients, respectively. Finally, we also investigate the size of higher order corrections to the hadronic matrix elements to critically examine the stability of our results within the pseudoscalar approximation.
In the second part of this work we investigate the matrix element of the (|∆S| = 2) K 0 −K 0 amplitude in the 1/N c expansion following the same lines of thought. The introduction to this calculation we postpone to the beginning of Section 5. Our results for the K → ππ matrix elements were already discussed in part in Refs. [21,22]. For a more detailed presentation of the general method we refer the reader to Refs. [20,23].
The paper is organized as follows. In Section 2 we review the general framework of the effective low-energy calculation and discuss the matching of short-and long-distance contributions to the decay amplitudes. Then, in Section 3 we investigate the K → ππ matrix elements. We show explicitly that the scale dependence of the factorizable loop diagrams is absorbed in the renormalization of the bare couplings, the meson wave functions and masses. We next calculate the non-factorizable loop corrections in the cutoff regulariza-tion scheme. In Section 4 we match them to the Wilson coefficients to obtain the isospin amplitudes. In Section 5 we extend the analysis to the (|∆S| = 2) K 0 −K 0 transition. We compute the matrix element and match it to the short-distance coefficient function to determine theB K parameter. In both sections we present our numerical results and compare them with those of the existing analyses. The conclusions can be found in Section 6.

General Framework
Following the lines of Ref. [20] we calculate the hadronic matrix elements of the local fourquark operators (with |∆S| = 1, 2) in the 1/N c expansion. To this end we start from the chiral effective lagrangian for pseudoscalar mesons which involves an expansion in momenta where terms up to O(p 4 ) are included [24]. Keeping only terms of O(p 4 ) which contribute to the K → ππ or the K 0 −K 0 matrix elements and are leading in N c it reads: 1 l µ and r µ are left-and right-handed gauge fields, respectively, f and r are free parameters related to the pion decay constant F π and to the quark condensate, with r = −2 qq /f 2 . The complex matrix U is a non-linear representation of the pseudoscalar meson nonet: where, in terms of the physical states with The various conventions and definitions we use are in agreement with Ref. [20]. In particular, we introduce the singlet η 0 in the same way and with the same value for the U A (1) symmetry breaking parameter, α = m 2 η + m 2 η ′ − 2m 2 K ≃ 0.72 GeV 2 , corresponding to the η − η ′ mixing angle θ = −19 • [25]. The bosonic representation of the quark currents is defined in terms of (functional) derivatives of the chiral action: and the right-handed currents are obtained by parity transformation. Eq. (16) allows us to express the current-current operators in terms of the pseudoscalar meson fields. The 1/N c corrections to the matrix elements Q i I are calculated by chiral loop diagrams in line with Ref. [20]. The factorizable contributions, on the one hand, refer to the strong sector of the theory and give corrections whose scale dependence is absorbed in the renormalization of the chiral effective lagrangian. This property is obvious in the case of the (conserved) currents and was demonstrated explicitly in the case of the bosonized densities [20,23]. Consequently, the factorizable loop corrections can be computed within dimensional regularization. The non-factorizable corrections, on the other hand, are UV divergent and must be matched to the short-distance part. They are regularized by a finite cutoff which is identified with the short-distance renormalization scale [18,19,26,27]. The definition of the momenta in the loop diagrams which are not momentum translation invariant was discussed in detail in Ref. [20]. A consistent matching is obtained by considering the two currents or densities to be connected to each other through the exchange of a color singlet boson and by assigning the same momentum to it at long and short distances [28 -31]. The identification of this momentum with the loop integration variable leads to modified integrals in the chiral loop diagrams compared to those of Refs. [18,26]. The numerical implications for the isospin amplitudes in K → ππ decays and theB K parameter will be addressed in Sections 4 and 5.
In this paper we investigate the hadronic matrix elements at leading and next-to-leading order in the chiral and the 1/N c expansions. In particular, we calculate the O(p 2 /N c ) corrections to the current-current operators, that is to say, the one-loop corrections over the O(p 2 ) lagrangian. The matrix elements of the density-density operators Q 6 and Q 8 are taken from Ref. [20]. In the numerical analysis of the ∆I = 1/2 rule and theB K parameter we use the leading logarithmic (LO), as well as, the next-to-leading logarithmic (NLO) values [5,6,32,33] for the (|∆S| = 1, 2) short-distance coefficient functions. 2 In general, the lack of any reference to the renormalization scheme dependence in the effective low-energy calculation prevents a complete matching at the next-to-leading order [34]. Nevertheless, a comparison of the amplitudes obtained from the LO and NLO coefficients is meaningful in order to test the validity of perturbation theory.
In the following sections we calculate the long-distance 1/N c corrections to the K → ππ amplitudes and theB K parameter. First, we investigate the factorizable corrections and show their absorption in the low-energy constants. Secondly, we determine the nonfactorizable loops within the modified momentum prescription. Finally, we perform a numerical analysis and compare our results with those of the existing studies.

K → ππ Decays
In this section we present the hadronic matrix elements of the current-current operators for the physical decay modes K 0 → π + π − and K 0 → π 0 π 0 up to O(p 4 ) and O(p 2 /N c ) in the parameter expansion. From these results we derive the isospin amplitudes K → (ππ) I=0,2 , heading for an explanation of the ∆I = 1/2 selection rule in kaon decays.

Factorizable 1/N c Corrections
The (bare) tree level of the K → ππ matrix elements, up to O(p 4 ) in the chiral expansion, as well as, the factorizable 1/N c corrections to the O(p 2 ) can be calculated from the tree and loop topologies depicted in Fig. 1. From the sum of these diagrams we obtain 3 where and  The ellipses in Eq. (17) denote finite terms we omit here for the analysis of the ultraviolet behaviour (in particular, they provide the reference scale for the logarithms). We specify our results in the cutoff regularization scheme to demonstrate the absorption of the quadratic, as well as, the logarithmic divergences as required by current conservation. We note that all factorizable terms quadratic and logarithmic in the cutoff are independent of the momentum prescription in the loop. λ c is the cutoff for the factorizable diagrams. We introduce two different scales since the factorizable and the non-factorizable corrections refer to disconnected sectors of the theory (strong and weak sectors). Having demonstrated the absence of UV divergent terms in the sum of the factorizable diagrams, in the numerical analysis of the full expressions we will use dimensional regularization, as in pure chiral perturbation theory, which is momentum translation invariant. If we renormalize the wave functions of the kaon and the pions (π r ≡ Z 1/2 π π 0 ), as well as, the bare decay constant f by using Eqs. (14)- (17) and (25) of Ref. [20], we arrive at the renormalized (factorizable) matrix elements of the (|∆S| = 1) current-current operators: 4 where the constantL r 5 is defined through the relation [20] Figure 2: Non-factorizable diagrams for the matrix elements of the current-current operators in the isospin limit. and the remaining matrix elements can be obtained from Eqs. (18)- (20).
We notice that for the four-quark operators Q i of the current-current type the divergent terms are absorbed by the renormalization procedure. In addition, the factorizable 1/N c corrections vanish completely, that is to say, the divergent as well as the finite terms. This property has been observed numerically, within dimensional regularization, because the complexity of all factorizable contributions prevents us from doing a fully analytic calculation. Since the factorizable scale λ c disappears through renormalization, the only matching between long-and short-distance contributions is obtained by identifying the cutoff scale Λ c of the non-factorizable diagrams with the QCD renormalization scale.
Finally, we note that in the next-to-leading order term of Eqs. (21) and (22) we used 1/F π rather than 1/f as it was done in Ref. [18]. Formally, the difference represents higher order effects. Nevertheless, the appearance of 1/f gives rise to a residual dependence on the factorizable scale λ c , which has no counterpart at the short-distance level and will be absorbed by factorizable loop corrections to the matrix elements at the next order in the parameter expansion. Consequently, it is a more adequate choice to use the physical decay constant in the expressions under consideration. Instead of F π the kaon decay constant F K could be used as well. Both choices will be considered in the numerical analysis, which gives a rough estimate of higher order corrections.

Non-factorizable 1/N c Corrections
The non-factorizable 1/N c corrections to the hadronic matrix elements constitute the part to be matched to the short-distance Wilson coefficient functions; i.e., the corresponding scale Λ c has to be identified with the renormalization scale µ of QCD. We perform this identification, as we argued in Section 2, by associating the cutoff to the effective color singlet boson. Then, at the O(p 2 ) in the chiral expansion, from the diagrams of Fig. 2 we obtain in the SU(2) limit: where and One might note that in Eqs. (23)- (29) [as in Eq. (17)] we replaced m 2 η , m 2 η ′ , and the mixing angle θ by m 2 π and m 2 K using the octet-singlet mass matrix of Ref. [25]. At this stage of the calculation we find quadratic, as well as, logarithmic divergences of the non-factorizable corrections. We note that already the leading (∼ Λ 2 c ) terms depend on the momentum prescription. The quadratic terms were calculated in Ref. [30] in the background field method. In this paper we investigate the full expressions for the matrix elements needed for the numerical analysis of the amplitudes. The results contain finite terms, originating from the solutions of the integrals listed in Eq. (48) and in Appendix B of Ref. [20], which we neglect here for brevity and denote by the ellipses. We also note that in the case of Q 7 the solution of the integrals brings along a quartic dependence on the cutoff which has to be cancelled by adding a specific contact interaction proportional to δ (4) (0) to the Feynman rules of the truncated meson theory [30,35].
Even though the scale dependence of the perturbative coefficient functions is only logarithmic, the full long-distance contribution including the quadratic terms has to be matched to the short-distance part. The quadratic dependence on the cutoff is physical and is necessary for several reasons. First, in the chiral limit (m q = 0) all corrections vanish except for the Λ 2 c terms, which bring in the only scale to be matched to the short distance. Secondly, they stabilize the 1/N c expansion and generally improve the matching of the meson and the quark pictures [18]. Finally, they provide us with a rough estimate of the contributions from higher resonances.
We note that in Eqs. (23)-(29) we used the physical decay constant F π rather than f in the same way as for the factorizable diagrams. Again the difference represents higher order effects. However, the (factorizable) scale dependence of f has no counterpart in the short distance and will be absorbed at the next order in the chiral expansion. As for the factorizable contributions the choice of F K instead of F π would be also appropriate.

Numerical Analysis
In this paragraph we list the numerical values for the hadronic matrix elements. We next match them to the Wilson coefficients and study the K → (ππ) I=0, 2 isospin amplitudes. In Section 4.1 we discuss in detail the 1/N c corrections to the matrix elements. In this context we also calculate the bag parameters, which quantify the deviations from the results obtained in the vacuum saturation approximation and, therefore, are convenient for a comparison with other works. The main results of the present analysis can be found in Section 4.2. Therein we give the amplitudes a 0 and a 2 as functions of the matching scale and compare them with the data.

Hadronic Matrix Elements
Throughout the numerical analysis we use the following values for the parameters [36]: Substituting them in Eq. (22) we computeL r 5 = 2.07 × 10 −3 . We parameterize our results in terms of the non-perturbative bag parameters B , which quantify the deviations from the values obtained in the vacuum saturation approximation [10]: with Q i I containing both factorizable and non-factorizable contributions. The VSA expressions for the matrix elements are taken from Eqs. (XIX.11)-(XIX.28) of Ref. [19]. 5 The numerical values for the matrix elements of the current-current operators are given in Tables 1 and 2 differs by a factor of (1 − 2/N c ) from the result obtained at the O(p 2 ) in the large-N c limit. We notice that the inclusion in part of the 1/N c corrections in the VSA method leads to a suppression and enhancement of the I = 0 and I = 2 amplitudes, respectively, in complete disagreement with the data.
In Tables 3 and 4 we list our results for the hadronic matrix elements at next-toleading order in the chiral and the 1/N c expansions. The matrix elements of the currentcurrent operators are calculated from Eqs. (18)-(31) including the finite terms denoted by the ellipses. The results for the operators Q 6 and Q 8 are taken from Ref. [20]. These results contain the leading plus next-to-leading order terms in the chiral expansion of the density-density operators, as well as, the leading 1/N c corrections, that is to say, the O(p 0 ),  O(p 2 ), and O(p 0 /N c ). Note that the matrix elements generally contain a non-vanishing imaginary part (scale independent at the one-loop level) which is due to on-shell (π − π) rescattering effects. The isospin amplitudes are largely dominated by the operators Q 1 and Q 2 . Therefore it is instructive to analyze in detail the 1/N c corrections to these two operators. To this end we next give the analytic expressions for the isospin matrix elements of Q 1 and Q 2 : Eqs. (34)-(36) allow us to compare our results with the analytic expressions of Ref. [18].    Table 4: Same as in Table 3, now for the I = 2 amplitudes.
First, we note that the modified matching which was discussed in Section 2 increases the terms quadratic in the cutoff by a factor of 3/2 relative to the results presented therein. This was already observed in Ref. [30]. The modification of the quadratic terms provides an additional octet enhancement in the long-distance domain. The logarithmic terms, on the other hand, are modified only on account of the presence of the η 0 . To be explicit, in the octet limit [i.e., in the absence of the η 0 , with a = b = 1 and m 2 η = (4m 2 K − m 2 π )/3 ] the coefficient of the logarithm in Eq. (34) is reduced to (m 2 K /2 + 10m 2 π /3) whereas the other terms remain unchanged. The separation of the logarithmic and the finite terms in Eqs. (34)-(36) is arbitrary and is done, for comparison with Ref. [18], by introducing a mass scale replacing the dependence of the exact expressions on the meson masses in the chiral logarithms. The logarithmic and the finite terms (a iI ) defined in this way each depend  Table 5: Different contributions to the hadronic matrix elements of Q 1 and Q 2 (in units of on the choice of the mass scalem, whereas the sum of all contributions is independent of this parameter. We calculated the complete finite terms arising from the non-factorizable loop diagrams using the matching prescription advocated in Refs. [20,30]. 6 These terms were not included in Ref. [18]. Consequently, the numerical values of the matrix elements reported therein exhibit a dependence on the specific choice of the mass scale in the logarithms which is absent in the present calculation. In Table 5 we split up the numerical values for the I = 0 and I = 2 matrix elements of Q 1 and Q 2 with respect to the quadratic, the logarithmic, and the finite terms, respectively, at a cutoff scale of Λ c = 800 MeV. From the table we see that the finite terms are of the same order of magnitude as the logarithmic ones and, therefore, must be considered at the same level in the numerical analysis. These terms are generally suppressed by a factor of δ ≡ m 2 K,π /(4πF π ) 2 < 20 % with respect to the leading O(p 2 ) tree level. In addition, as can be seen from Eqs. (34)- (36) and Table 5, no coefficient larger than one or two which could significantly enhance them has been found. This is different from the quadratic terms which are not suppressed as their relative size is determined by ∆ ≡ Λ 2 c /(4πF π ) 2 and, moreover, they appear with larger prefactors [even as large as six in Eq. (34)]. 7 Consequently, in the case of the I = 0 matrix elements of Q 1 and Q 2 both the logarithmic and the finite corrections are moderate, and the chiral limit gives a satisfactory representation of the full amplitude provided that the matching scale is taken sufficiently large (Λ c 500 -600 MeV). In the case of the I = 2 matrix elements we also observe that the quadratic terms are enhanced with respect to the tree level, whereas the logarithmic and the finite terms are largely suppressed. However, in this case the quadratic corrections counteract the tree level, and the sum of both contributions is no longer large compared to the logarithmic and the finite terms. Therefore the neglect of either of the terms is no longer justified. In particular, we observe that for the ∆I = 3/2 channel the chiral limit gives a better approximation to the exact result than a calculation which includes only the logarithms without taking into account the finite terms. This remark also holds for the matrix element Q 1 0 . Finally, we note that variation of the mass scale in the logarithms [m π <m < m K ] in Ref. [18] has a noticeable effect on the numerical value of the I = 2 amplitude.
When comparing the results of the present analysis with those of Ref. [18] one has to take into account another difference in the treatment of the next-to-leading order terms: in Eqs. (34)-(36) we used 1/F π rather than the bare parameter 1/f as it was done in Ref. [18]. Formally, the difference concerns higher order effects, as we already discussed above. However, since the factorizable scale which appears in the bare coupling f will be absorbed by factorizable loop corrections to the matrix elements at the next order in the parameter expansion, it has not to be matched to any short-distance contribution. Consequently, it is a more adequate choice to use the physical decay constant in the expressions under consideration. The effect of this different treatment of the next-toleading order terms will be further discussed in Section 4.2.
In Tables 6 and 7 we list the values we compute for the bag parameters B (1/2) 2 over the VSA result, which constitutes the dominant contribution, at long distances, to the ∆I = 1/2 transition in K → ππ decays. Moreover, we obtain the correct scale dependence counteracting the scale behaviour of the Wilson coefficients z 1 and z 2 , which leads to an acceptable matching (see Section 4.2). In view of the large corrections one might question the convergence of the 1/N c expansion. However, there is no strong reason for such doubts because the non-factorizable contribution we consider in this paper represents the first term in a new type of a series absent in the large-N c limit. It is reasonable to assume that this leading non-factorizable term carries a large fraction of the whole contribution [18] (see also the the matrix element of Q 6 especially if large prefactors are observed as for Q 1 and Q 2 in Eqs. (34)- (36). This difference between the Q 6 and Q 8 operators could play an important role for ε ′ /ε. This point will be investigated in Ref. [37].     Table 7: Same as in Table 6, now for the I = 2 amplitudes. discussion in Section 4.2). B (1/2) 3 and B (1/2) 5 turn out to be very close to zero. This property is due to the vanishing tree level, as well as, to the small 1/N c corrections proportional to m 2 π /(4πF π ) 2 , see Eqs. (25) and (30). We notice that the small contribution of the operator Q 5 to ε ′ /ε is even further reduced when replacing the VSA expression for Q 5 0 , which is commonly used in the analysis of ε ′ /ε [34], by the result presented in this paper. B are also found to be significantly reduced with respect to vacuum saturation approximation. In particular, B parameters, which are relevant for A 2 . However, as we will see below, their scale dependence largely overcompensates for the variation of the short-distance coefficient functions. Nevertheless, as the values are found to be reduced, they generally account for the reduction of the I = 2 amplitude. Finally, B (1/2) 6 receives only small corrections whereas B (3/2) 8 comes out to be substantially reduced relative to the VSA result [20]. The numerical implications for ε ′ /ε will be investigated elsewhere [37]. One might note that the numerical values of B (3/2) 8 shown in Table 7 differ from the ones given in Table 2 of Ref. [20]. This is due to the fact that in the present paper we include only the real part of the hadronic matrix elements in the definition of the B i parameters (see Section 4.2).

The ∆I = 1/2 Rule
We next investigate the CP conserving amplitudes Rea 0 and Rea 2 . To this end we start from the expression for the isospin amplitudes A I which contain the (π − π) strong interaction phase shift for the I = 0 and the I = 2 final states, respectively, Then Within an exact realization of non-perturbative QCD the two expressions in Eq. (38) are equivalent. However, in the approximate low-energy calculation of the present work the long-distance imaginary part which we computed at the one-loop level (see Tables 3 and 4) is not expected to be of the same accuracy as the real part obtained at this level. In particular, as the one-loop (long-distance) imaginary part is scale independent, it cannot compensate for the scale dependence of the Wilson coefficients z i leading to a scale dependent imaginary part of the total amplitude. This requires a calculation of the (long-distance) imaginary part at least at the two-loop level which will introduce a scale dependence. In addition, the two-loop contribution is expected to be of the same order of magnitude as the one-loop contribution which only appears at the level of the finite terms, as it will bring in a quadratically divergent term. This situation is analogous to the non-suppression of the one-loop contribution to the real part (∼ ∆) with respect to the tree level. The two-loop contribution to the real part, on the other hand, is expected to be suppressed by at least a obtained in the chiral quark model at a matching scale of 0.8 GeV [40].
factor of δ with respect to the tree level and the one-loop contribution. This is analogous to the one-loop logarithmic and finite terms which are suppressed by a factor of δ with respect to the tree level. For the numerical analysis we will therefore consider only the real part of the matrix elements [see the second expression in Eq. (38)] using the experimental values of the final state interaction phases, δ exp 0 = (37 ± 3) • and δ exp 2 = (−7 ± 1) • [41]. This procedure has also been followed in Ref. [42]. However, as the imaginary part is a loop effect (suppressed by a factor of δ with respect to the tree level contribution), its effect on the absolute value of the amplitude strictly speaking is of the two-loop order. Consequently, we will also compare our results with the ones obtained by taking the (long-distance) imaginary part to zero, i.e., by taking i z i Q i I = i z i Re Q i I . This holds for an estimate of the size of higher order effects which is generally disregarded in the literature.
In Table 8 we show the numerical values of the amplitudes for various values of the matching scale and fixed values of Λ QCD = Λ (4) MS and the strange quark mass m s . The numerical analysis is done using the leading logarithmic, as well as, the next-to-leading logarithmic values of the Wilson coefficients listed in the appendix. The NLO values are scheme dependent and are calculated within naive dimensional regularization (NDR) and in the 't Hooft-Veltman scheme (HV), respectively. 9 The difference between the two NLO results at a given scale reveals the uncertainty due to the lack of any reference to the renormalization scheme dependence in the effective low-energy calculation.
In Fig. 3 we show Rea 0 calculated with leading order Wilson coefficients for various values of Λ QCD as a function of the matching scale. We take the (conservative) range of Λ QCD = 325±80 MeV which corresponds to α s (M Z ) = 0.118±0.005 [34]. First, we note that our result for a 0 shows an additional enhancement (around 30 -50 % of the experimental value) compared to the result of Ref. [18] which renders the amplitude in good agreement with the observed value for low values of the scale or even larger than the experimental value for large values of the scale. A significant enhancement arises from the Q 1 and Q 2 operators due to the modified matching prescription in the non-factorizable sector we discussed above. Numerically, at a scale of Λ c = 800 MeV the modified momentum routing accounts for approximately 20 % of the final number(s) presented in Fig. 3. Another enhancement with respect to Ref. [18] originates from the correction of the real part by the experimental phase [see Eq. (38)]. Neglecting completely the effect of the (π − π) phase shift would reduce our result by a factor of cos δ 0 ≃ 0.8. The remainder is due to the choice of the physical value F π instead of f in the next-to-leading order terms of the factorizable 9 We are very thankful to M. Jamin for providing us with the numerical values of the Wilson coefficients used in this section.   In Fig. 4 we compare the results for Rea 0 we obtain using the LO and NLO Wilson coefficients, respectively. In the HV scheme, for moderate values of Λ QCD introducing the NLO coefficients does not significantly affect the numerical values of the ∆I = 1/2 amplitude which is found to be only slightly suppressed with respect to the LO result. The main effect of the NLO coefficients is that they further reduce the dependence on the matching scale. This statement does not hold within the NDR scheme. In this scheme, for Λ QCD = 245 MeV the effect of the NLO coefficients is also moderate but noticeably increases for large values of Λ QCD leading to a distinct suppression of the LO result. For the required size remains present. In Fig. 5 we show the weak dependence of Rea 0 (with LO Wilson coefficients) on the strange quark mass which arises from the matrix element of the gluon penguin operator [ Q 6 0 ∝ 1/m 2 s ]. We notice that the contribution from Q 6 to the ∆I = 1/2 amplitude for small values of the cutoff (∼ 600 MeV) roughly varies between 10 -20 % of the total value and significantly decreases for large values of Λ c . This behaviour is also found when from a change in the value of the QCD coupling for a chosen value of Λ MS [19]. the NLO coefficients are used. The effect of the remaining (penguin) operators is very small (below 1 % of the total result except for Q 4 which contributes at the level of −3 %). For comparison, in Fig. 5 we also show Rea 0 calculated in the chiral limit. We observe that the result obtained in the chiral limit, for reasons explained above, is rather close to the numerically exact one, that is to say, the logarithmic and the finite terms in the non-factorizable corrections to the matrix elements are minor important provided that the matching scale is taken sufficiently large (Λ c 500 -600 MeV). Finally, we note that the presence of the η 0 does not affect the numerical values of the amplitudes (in the octet limit the numbers given in Table 8 change by less than 1 % ). In distinction to the ∆I = 1/2 amplitude, the ∆I = 3/2 amplitude depicted in Fig. 6 (with LO Wilson coefficients) is highly unstable. In addition, the numerical values lie well below the measured value. The amplitude even changes sign [due to the large negative coefficient of the quadratic term in Eq. (36)]. The large uncertainty can be understood, as we already discussed above, from the fact that the two numerically leading terms, the tree level and the one-loop quadratically divergent term, have approximately the same size but opposite sign. On the one hand, this property is generally welcomed as it explains the origin of the suppression of the ∆I = 3/2 amplitude which turns out to be sufficiently suppressed whatever the particular chosen scale is between 600 MeV and 900 MeV. On the other hand, the large cancellation implies that the result will be significantly affected by higher order terms which are expected to be of the order of the one-loop logarithmic and finite terms. We note that the agreement with the experimental value is not improved in the chiral limit. We also notice that the numerical values depicted in Fig. 6 depend only weakly on the choice of Λ QCD . In Fig. 7 we compare the results for Rea 2 we obtain using the LO and NLO Wilson coefficients, respectively. We observe that the effect of the NLO coefficients is negligible with respect to the large discrepancy between our results and the observed value. The small effect of the NLO coefficients indicates the validity of perturbation theory and further supports the supposition that the discrepancy is due the lack of accuracy in the low-energy part of the calculation.
The typical size of higher order effects in the calculation of the hadronic matrix elements can be estimated in various ways. First, as we already mentioned above, one may replace in all NLO terms the coefficient 1/F π by 1/F K . The results obtained in this case [denoted by (b)] are shown in Figs. 8 and 9. The ∆I = 1/2 amplitude is suppressed by approximately 20 % with respect to the result we obtained using 1/F π [denoted by (a)] and is even in better agreement with the observed value. The ∆I = 3/2 amplitude, on the other hand, is enhanced but still far too much suppressed. Another estimation of higher order effects can be done, as we explained above, by completely neglecting the imaginary part of the matrix elements (c). This suppresses Rea 0 by a factor of cos δ exp 0 ≃ 0.8 but does not affect Rea 2 . Similarly the absolute value of the amplitudes can be calculated by taking directly the imaginary part from Tables 3 and 4 without using the experimental phases (d). This procedure suppresses Rea 0 in the same way as in the previous case but largely re-stabilizes Rea 2 , indicating that the results obtained for the ∆I = 3/2 amplitude (unlike those obtained for Rea 0 ) indeed can be significantly affected by higher orders corrections. It is unlikely, however, that higher order terms alone can account for the large discrepancy between our result and experiment, and effects from higher resonances are also expected to be non-negligible for the small ∆I = 3/2 amplitude. Finally, the coefficient 1/F π in the next-to-leading order terms can also be replaced by the bare coupling 1/f as it was done in Ref. [18]. Even though this would introduce an unphysical dependence on the factorizable scale, formally the difference also concerns higher order effects. 11 We observe that this choice (e) leads to a result for Rea 0 which is approximately scale independent. It also gives a more stable result for Rea 2 which, however, still is too much suppressed.
In summary, in all cases we discussed above the ∆I = 1/2 amplitude is obtained around the measured value with an uncertainty of less than 25 % or in most cases even less than 15 %. 12 The result for Rea 0 is consequently solid and presumably could be significantly affected only by higher resonances. In view of the good agreement with the 11 The relation between F π and f is given in Eq. (62) of Ref. [20] and we obtain f = 105, 112, 120, 128, 136, 145 MeV for Λ c = 500, 600, 700, 800, 900, 1000 MeV, respectively. 12 The only exception to this is the case where the large value of Λ QCD = 405 MeV is taken at LO or NLO (HV scheme) using a matching scale as high as ∼ 1 GeV. In this (unfavourable) case the deviation from the observed value can be as large as 35 -40 %. to investigate their effect in the present calculation. This also would allow more safely to choose higher values for the matching scale for which the short-distance contributions are more reliable. We close this section by a brief review of several other attempts which have been made to explain the ∆I = 1/2 rule using different methods for the computation of the hadronic matrix elements. Interesting tendencies for an enhancement of the ∆I = 1/2 channel were found in particular in Ref. [11] by integrating out the quark fields in a gluonic background and in Ref. [12] in the framework of QCD sum rules at the level of the inclusive two-point function. In Ref. [13] quantitative results reproducing both the ∆I = 1/2 and ∆I = 3/2 channels were obtained adopting the point of view that in addition to 1/N c effects due to one-loop corrections (similar to those of Fig. 2) diquark states play an important role. The results for the ∆I = 1/2 amplitude obtained in the present approach suggest that there are no large diquark effects not already taken into account in the 1/N c corrections we calculated. The ∆I = 1/2 rule has also been investigated in the framework of chiral perturbation theory [14] and the chiral quark model [15]. At the present state of these methods the ratio 1/ω = 22.2 cannot be predicted but is used to fit parameters of the models. Very recently the matrix elements relevant for the ∆I = 1/2 rule were studied in lattice QCD with improved statistics [16]. The authors used lowestorder chiral perturbation theory to relate the matrix elements ππ|Q i |K 0 to π + |Q i |K + and 0|Q i |K 0 calculated on the lattice. The ratio of the amplitudes computed in this way confirms the significant enhancement of the ∆I = 1/2 channel although systematic uncertainties preclude a definite answer. Whereas the ∆I = 1/2 amplitude is obtained larger than the experimental value by approximately 40 % (quenched ensemble 13 , β = 6.0) the ∆I = 3/2 amplitude suffers from ambiguities in the choice of the meson mass due to the ignorance of higher order chiral corrections to the relation between Rea 2 and the B K parameter. Taking the meson mass M 2 = (m 2 K + m 2 π )/2 and using the quenched value of B K in the continuum limit the authors obtain a value for Rea 2 which also over-estimates the data by approximately 40 %. The ratio of the amplitudes exhibits a strong dependence on the meson mass (see Fig. 11 of Ref. [16]) due to the chiral behaviour of Rea 2 . In lattice perturbation theory unlike in analytical methods, the matching of the renormalized operators to the Wilson coefficients can be rigorously done, at least in principle (see e.g. Ref. [45] and references therein). On the other hand, analytical methods like the 1/N c approach followed in this paper allow for a direct evaluation of the K → ππ amplitudes without the need of using reduction formulas to relate these amplitudes to the off-shell K → π amplitudes (for this point see also Ref. [46] and references therein).
While this paper was written an analysis of the ∆I = 1/2 rule was published [47] which follows similar lines of thought as our work. In their analysis the authors used the 1/N c expansion in the chiral limit in the framework of chiral perturbation theory and the Extended Nambu-Jona-Lasinio model, respectively. We agree on the coefficients of the quadratically divergent terms in the 1/N c corrections to the matrix elements quoted therein. In the present analysis we did not investigate the method proposed in Ref. [47] to treat the scheme dependence appearing at the next-to-leading logarithmic order.

K 0 −K 0 Mixing
The contributions from short-distance physics to K 0 −K 0 mixing can be calculated from an effective ∆S = 2 hamiltonian, valid below the charm threshold, in which the heavy degrees of freedom are integrated out [32], where O ∆S=2 is the following four-quark operator: with α s (µ) being the QCD running coupling with three active flavors and J 3 a renormalization scheme dependent coefficient appearing at the next-to-leading logarithmic order. 13 Quantitative estimates of quenching effects on the coefficients of the chiral logarithms in the one-loop contributions to the K → ππ amplitudes were presented in Refs. [43,44]. In Ref. [43] finite volume effects on the lattice were also investigated.
F (m 2 t , m 2 c , M 2 W , V CKM ) is a known function of the heavy quark masses, the W boson mass, and CKM matrix elements. It incorporates the basic electroweak (box diagram) loop contributions [48], as well as, the perturbative QCD effects described through the correction factors η 1 , η 2 , η 3 which have been calculated at the leading logarithmic [4,49] and the nextto-leading logarithmic order [32,33]. Terms depending on α s (µ) are factored out explicitly to exhibit the renormalization scale (and scheme) dependence of the coefficients which has to cancel the corresponding scale (and scheme) dependence of the hadronic matrix element of O ∆S=2 [19]. The short-distance hamiltonian for ∆S = 2 transitions in Eq. (39) dominates the indirect CP violation in the neutral kaon system parameterized by ε. Contributions to K 0 −K 0 mixing changing strangeness by two units through two ∆S = 1 transitions at long distances which are relevant for the K L − K S mass difference [29] are not considered in this article.
The hadronic matrix element of O ∆S=2 is usually parameterized in terms of the B K parameter which quantifies the deviation from the value obtained in the vacuum saturation approximation: where It is convenient to introduce the renormalization group invariant parameter [19,50] in which the scale (and scheme) dependences of the long-and short-distance contributions cancel within an exact realization of both perturbative and non-perturbative QCD. However, from the results for the ∆I = 3/2 K → ππ amplitude discussed in the previous section we do not expect that theB K we will obtain within the pseudoscalar approximation used in the low-energy calculation will exhibit a negligible dependence on the matching scale; the 27-plet operators which induce ∆S = 1 (∆I = 3/2) and ∆S = 2 transitions are components of the same irreducible tensor under SU(3) L × SU(3) R , that is to say, to leading order in the chiral expansion the K 0 −K 0 amplitude can be related to the ∆I = 3/2 part of the K → ππ amplitude using SU(3) symmetry [51,52]. Consequently, we expect a similar pattern, i.e., a large negative quadratic term in the 1/N c corrections to the matrix element which partly cancels the tree level contribution and renders the result more sensitive to corrections from higher order terms and higher resonances. On the other (ds)(ds) (ds)(ds) (ds)(ds) Figure 10: Factorizable contributions to the matrix element of the K 0 −K 0 mixing amplitude in the isospin limit.
hand, we expect SU(3) breaking effects in ∆S = 2 transitions to be more pronounced than in ∆S = 1 transitions [53]. In the following we will see that the 1/N c expansion restricted to the pseudoscalar mesons indeed leads to a significantly scale dependent result forB K . However, the scale dependence is less pronounced than the one of the ∆I = 3/2 amplitude due to corrections beyond the chiral limit. Finally, as we already discussed above, the low-energy calculation does not allow any reference to the renormalization scheme dependence. Nevertheless, a comparison of theB K parameter obtained from the LO and NLO coefficient function of O ∆S=2 can be used to to test the validity of perturbation theory and to estimate the uncertainties arising from the short-distance part.

Factorizable Loop Corrections
To obtain the factorizable non-perturbative corrections to the ∆S = 2 transition we have to calculate the diagrams in Fig. 10. Using the chiral representation of the quark current in Eq. (16) and reducing the result to the basic integrals listed in Appendix B of Ref. [20] we obtain the unrenormalized (bare) matrix element: with a and b defined in Eq. (15). Multiplying Eq. (44) with Z −1 K , i.e., including a factor Z −1/2 K for each external kaon field (compare Eqs. (16) and (59) of Ref. [20]), we arrive at Comparing Eq. (45) with Eqs. (26) and (63) of Ref. [20] we observe that the correction factor in the brackets which is due to the higher order (factorizable) contributions to the (ds)(ds) (ds)(ds) (ds)(ds) (ds)(ds) Figure 11: Non-factorizable contributions to the matrix element of the K 0 −K 0 mixing amplitude in the isospin limit. matrix element is completely absorbed (including the finite terms) in the renormalization of the kaon decay constant, as it is required by current conservation, leading to the final result for the (renormalized) factorizable matrix element Eq. (46) represents the large-N c limit for the K 0 −K 0 matrix element, i.e., B Nc→∞ K = 3/4, to be compared with the VSA value one.

Non-factorizable Loop Corrections
The 1/N c corrections to Eq. (46) can be calculated from the non-factorizable loop diagrams depicted in Fig. 11. We determine the loop momenta along the lines developed in Section 2, that is to say, by associating the cutoff to the effective color singlet boson connecting the two currents. The simple structure of the non-factorizable diagrams makes it possible to specify the complete analytic result for the matrix element in terms of loop integrals. In the SU(2) limit the expression in which the integrals are reduced to the basic ones reads Here we replaced a and b by the η − η ′ mixing angle θ and explicitly distinguished between the masses coming from the external kaon momentum, the explicit mass term in the lagrangian, and the propagators in the loops. In addition to the logarithmically and quadratically divergent integrals (I 1 , I 2 , I 3 ) listed in Appendix B of Ref. [20] Eq. (47) contains the integral I 4 which exhibits a quartic dependence on the cutoff. Following the steps discussed in Ref. [20] we can give the analytic expression for I 4 in terms of a Taylor-series: We note that the logarithmically divergent integral I 3 in Eq. (47) only appears with vanishing external momentum and therefore can be largely simplified compared to the general expression in Eq. (75) of Ref. [20]. From Eq. (47) one can easily calculate the divergent terms. Taking the external momentum on-shell we obtain where the tree level result is factored out and the ellipses denote the finite terms we do not specify analytically. We observe that the quartic dependence on the cutoff is cancelled as required by chiral symmetry.
To illustrate the effect of the modified momentum routing we also recalculate the nonfactorizable loop contributions in the approach used by Bardeen et al. [26] who associated the cutoff to the momentum of the virtual meson in the loop diagrams (see also the discussion in Ref. [20]): where the external momentum is already taken on-shell. For comparison with Eq. (47) in Eq. (50) we included the small effect of the singlet η 0 . Solving the integrals we obtain the Λ c 0.5 GeV 0.6 GeV 0.7 GeV 0.8 GeV 0.9 GeV 1.0 GeV O ∆S=2 tree 3.14 3.14 3.14 3.14 3.14 3.14 to be compared with Eq. (49). We note that the results obtained in both calculations differ with respect to the quadratic cutoff dependence, as well as, with respect to the finite terms we do not give explicitly here for brevity.

Numerical Results
As a numerical input we use the values listed in Section 4.1. In Table 9 we show our results for the K 0 −K 0 matrix element and B K (Λ c ) obtained in the full calculation, i.e., including the effect of the η 0 in Eq. (47). In Fig. 12 we depict the renormalization group invariant parameterB K calculated with the leading order Wilson coefficient.
The decrease of B K (Λ c ) with Λ c = µ is qualitatively consistent with the µ dependence of the coefficient function in Eq. (43), that is to say, the long-distance evolution counteracts the evolution in the short-distance domain. This property is due to the presence of the quadratic terms in the 1/N c corrections which compensate for the (weaker) increase of the logarithmic terms. However, the decrease is found to be significant, and the scale dependence largely exceeds what is required to have an exact cancellation of both evolutions over a large range of the scale. As a result an acceptable stability ofB K is obtained only for low values of Λ c ≃ 500 -600 MeV. The small values ofB K depicted in Fig. 12 (lower set of curves) come from the negative coefficient of the quadratic term in Eq. (49) which is found to be enhanced by a factor of 3/2 compared to the result of Ref. [26]. This coefficient is the same as the one of the ∆I = 3/2 K → ππ amplitude except for SU(3) breaking effects (responsible for F K = F π ) which reduce the negative slope ofB K . As can be seen from Table 9, the difference between the exact result and the one obtained in the chiral limit (i.e., in the absence of chiral logarithms and finite terms) is more pronounced than in the case of the K → ππ amplitudes. This is due mainly to the numerical coefficient of the leading term (∼ m 4 K ) in front of the logarithm in Eq. (49) which as expected is found larger in ∆S = 2 transitions than in ∆S = 1 transitions. Because of the large positive coefficient the logarithmic term re-stabilizesB K sizably with respect to the result obtained in the chiral limit. This also explains why theB K parameter even if significantly scale dependent is much more stable than the ∆I = 3/2 amplitude. The finite terms beyond the logarithms in Eq. (47) [i.e., beyond the log(1 + Λ 2 c /m 2 ) terms] give a negative contribution to B K (Λ c ) roughly between −0.05 and −0.08 for Λ c around 600 -900 MeV. Consequently, they are non-negligible in particular for large values of the scale where the cancellation between the tree level and the quadratic terms is large. Finally, we note that the presence of the η 0 does not significantly affect the numerical values of the K 0 −K 0 matrix element (in the octet limit the numbers given in Table 9 change by less than 3 % ).
To illustrate the effect of the momentum routing, in Fig. 12 we also showB K obtained from Eq. (50) (upper set of curves). We use the same set of parameters as in Table 9 and also include the η 0 . Comparing the two results we notice that B K (Λ c ) calculated within the modified momentum routing lies below the values found in the previous approach. Matching the long-distance results with the short-distance contribution we observe that theB K parameter obtained in the present analysis exhibits a significantly stronger dependence on the matching scale. However, as we already discussed above, the quadratically divergent terms (and the finite terms) depend on the way we define the integration variable inside the loop. This can be seen from the different numerical factors in front of the quadratic terms in Eqs. (49) and (51). Therefore we are forced to find a direct link between the short-and long-distance part of the calculation, as it is done by keeping track of the effective color singlet boson in both parts of the calculation. A consistent matching is then obtained by assigning the same momentum to the color singlet boson at long and short distances and by identifying this momentum with the loop integration variable (see Section 2). This property is absent in the previous approach. The modification unambiguously determines the coefficient in front of the (quadratically and logarithmically) divergent terms and allows us to identify the ultraviolet cutoff of the long-distance terms with the short-distance renormalization scale µ. Therefore we advocate the use of the modified matching prescription, even though the stability of our result is rather poor. The satisfactory stability obtained in Ref. [26] on the other hand is somehow inconclusive, as there is no underlying argumentation determining the quadratic terms. Our result also implies that the uncertainties due to the idealized identification of the cutoff Λ c with the upper limit of the meson momentum in the loop in Ref. [26] might have been underestimated. In a complete meson theory the dependence on the momentum routing should be absent. However, as long as we are working in an effective low-energy approach as chiral perturbation theory we have to pay attention to this point.
Numerically, we find a range of acceptable stability in the energy regime from 500 MeV to 700 MeV (see Fig. 12) leading to values forB K in the range of 0.4 <B K < 0.6. The lower bound corresponds to a value of Λ QCD = 405 MeV, whereas the upper bound corresponds to Λ QCD = 245 MeV. Comparing our result with the one of Ref. [26] we observe a tendency Λ c (MeV) forB K to be decreased to values below 0.6. This behaviour is due to the enhancement of the negative coefficient in front of the quadratic term in the 1/N c corrections to the K 0 −K 0 matrix element and, to a smaller extend, also due to the finite terms omitted in Ref. [26]. However, our result suffers from a sizable dependence on the matching scale which precludes a precise answer.
In Fig. 13 we compare the results forB K we obtain with the LO and NLO coefficient function. For Λ QCD = 325 MeV in the HV scheme, introducing the NLO coefficient does not significantly affect the numerical values of theB K parameter which is found to be only slightly enhanced with respect to the LO result. In the NDR scheme, the effect of the NLO coefficient is also moderate for large values of the scale but noticeably increases for low values. For very low values of Λ c ≃ 500 MeV the NLO result can differ from the LO one by as much as 0.2. However, for these scales the scheme dependence increases rapidly and it is desirable to take (at least) a matching scale around 600 -650 MeV whereB K is still relatively smooth and roughly varies between 0.45 and 0.6. For Λ QCD = 245 MeV in both the HV and NDR schemes a matching scale as low as 500 MeV appears to be acceptable, and within the range Λ c ≃ 500 -650 MeVB K is obtained between 0.5 and 0.7. On the other hand we observe that the pseudoscalar approximation would simply fail if Λ QCD was found as large as 405 MeV, as a satisfactory perturbative behaviour is obtained only for Λ c 700 MeV, that is to say, for values of the scale where the stability ofB K is found to be poor.
In summary, for values of Λ QCD 350 MeV an estimate ofB K is hindered by the loss of perturbativity in the range where the pseudoscalar approximation is expected to be valid, and for lower values of Λ QCD (taking into account the scheme dependence) our calculation favours low values ofB K in the range However, a satisfactory smooth behaviour is obtained only in a narrow range of the cutoff and, in addition, for values of the cutoff as low as the kaon mass or just above. Therefore the incorporation of higher resonances is clearly required as for the ∆I = 3/2 K → ππ amplitude discussed above. On this issue, the analysis of theB K parameter is similar to the one of the ∆I = 3/2 amplitude, even if numerically the matching obtained forB K is better than the one obtained for the ∆I = 3/2 amplitude. The K 0 −K 0 system has been studied in the past with various methods leading to different results forB K . The present status of quenched lattice calculations [54 -57] has been reviewed in Ref. [58]. The value reported by the author isB K = 0.86 ± 0.06 ± 0.06. Very recently the JLQCD Collaboration has presented a new analysis based on chiral Ward identities to non-perturbatively determine the mixing coefficients of the ∆S = 2 operator [59]. The numerical results given in Ref. [59] are in agreement with the lattice calculations quoted above. In the chiral quark model a value as high asB K = 1.1 ± 0.2 has been obtained [15]. Lower values forB K have been found in the QCD hadronic duality approach [60] (B K = 0.39 ± 0.10), by using SU(3) symmetry and PCAC [51] (≃ 1/3), or using chiral perturbation theory at next-to-leading order [61] (0.42 ± 0.06). QCD sum rules give results aroundB K = 0.5 -0.6 with errors in the range of 0.2 -0.3 [62,63]. One might note that a value forB K significantly below 0.7 requires simultaneously high values of |V ub /V cb | and |V cb | to be able to fit the experimental value of ε [19]. Finally, we note that theB K parameter was also investigated in the framework of the 1/N c expansion in Ref. [50]. In this work the matching was not performed at the level of the K 0 −K 0 matrix element but at the level of a related 2-point Green function. Numerically, the matching was found unsatisfactory good. We agree with this conclusion, as we discussed above, although in Ref. [50] the quadratic dependence on the UV cutoff was obtained in disagreement with the present analysis due to the use of a different momentum routing. This has been corrected very recently in Ref. [47], and we agree with the results for the 1/N c corrections to the K 0 −K 0 matrix element obtained there in the chiral limit. In the present paper we investigated the corrections beyond the chiral limit and found that they are sizable. On the other hand, the authors of Ref. [47] investigated higher order corrections calculated in the framework of the Extended Nambu-Jona-Lasinio model. As a result they obtained a better stability of theB K parameter. This shows that corrections from higher order terms and higher resonances are expected to be large. Nevertheless the values ofB K we obtained in this analysis by performing a full calculation at the pseudoscalar level are meaningful and can be considered as reference values for further investigations incorporating the effects of higher resonances.

Conclusions
The 1/N c approach developed in Refs. [18,26] when modified along the lines of Ref. [20] leads to interesting results in the current-current sector of the ∆S = 1 and in the ∆S = 2 transitions. The main result of the present analysis is an additional enhancement of the ∆I = 1/2 channel in the K → ππ amplitudes. This channel has been found sufficiently enhanced, in good agreement (with an accuracy of 80 to approximately 100 %) with the experiment, and widely stable over a large range of values of the matching scale roughly between 600 MeV and 900 MeV. It is certainly premature to say that the dynamical mechanism behind the ∆I = 1/2 enhancement is completely understood. An agreement at the level obtained in the present analysis a priori is not expected in an effective theory with only pseudoscalar mesons taken into account. Nevertheless we believe that the additional enhancement reported here is a further important indication that the 1/N c approach can account for the bulk of the ∆I = 1/2 amplitude. This statement is also supported by the fact that higher order corrections both of short-distance origin and of long-distance origin at the pseudoscalar level, as we discussed above, are not expected to largely affect the size of the ∆I = 1/2 enhancement. The agreement with the experiment also tends to show that the origin of the long-distance enhancement has to be found at the level of the pseudoscalar mesons and at energies below the rho mass or even below the kaon mass. Cer-tainly this has to be checked explicitly incorporating at least the effects of vector mesons. We also believe that the 1/N c approach can account for the bulk of the suppression of the ∆I = 3/2 channel. For this channel, however, the approximations made in the present analysis fell short of the desired accuracy. In particular, a large scale dependence has been found clearly requiring the incorporation of higher order terms and/or higher resonances. We note that the scale behaviour of the ratio of the two isospin amplitudes is dominated by the one of the ∆I = 3/2 channel, and therefore it leads to a comparable uncertainty. Similarly, theB K parameter suffers from a sizable dependence on the matching scale. Our calculation favours very low values of the scale ( 700 MeV) leading to values forB K in the range of 0.4 <B K < 0.7. However, the large uncertainties associated with this result preclude a definite answer, and also make the incorporation of higher order terms and higher resonances very desirable.

A Numerical Values of the Wilson Coefficients
In this appendix we list the numerical values of the LO and NLO (HV and NDR) Wilson coefficients for ∆S = 1 transitions used in Section 4.2. These values were supplied to us by M. Jamin. Following the lines of Ref. [5] the coefficients z i are given for a 10-dimensional operator basis {Q 1 , . . . , Q 10 }. Below the charm threshold the set of operators reduces to seven linearly independent operators [see Eqs. (4)- (7)] with At next-to-leading logarithmic order in (renormalization group improved) perturbation theory the relations in Eq. (53) receive O(α s ) and O(α) corrections [5,19]. In the present analysis we use the linear dependence at the level of the matrix elements Q i I , i.e., at the level of the pseudoscalar representation where modifications to the relations in Eq. (53) are absent. We note that the effect of the different treatment of the operator relations at next-to-leading logarithmic order which is due to the fact that in the long-distance part there is no (perturbative) counting in α s is numerically negligible.