Upper Bounds on epsilon'/epsilon Parameters B_6^{(1/2)} and B_8^{(3/2)} from Large N QCD and other News

We demonstrate that in the large N approach developed by the authors in collaboration with Bardeen, the parameters B_6^{(1/2)} and B_8^{(3/2)} parametrizing the K\to\pi\pi matrix elements<Q_6>_0 and<Q_8>_2 of the dominant QCD and electroweak operators receive both negative O(1/N) corrections such that B_6^{(1/2)}<B_8^{(3/2)}<1 in agreement with the recent lattice results of the RBC-UKQCD collaboration. We also point out that the pattern of the size of the hadronic matrix elements of all QCD and electroweak penguin operators Q_i contributing to the K\to \pi \pi amplitudes A_0 and A_2, obtained by this lattice collaboration, provides further support to our large N approach. In particular, a very precise lattice result for the matrix element<Q_8>_0 implies for the corresponding parameter B_8^{(1/2)}=1.0\pm 0.2 to be compared with large N value B_8^{(1/2)}=1.1\pm 0.1. We discuss briefly the implications of these findings for the ratio epsilon'/epsilon. In fact, with the precise value for B_8^{(3/2)} from RBC-UKQCD collaboration, our upper bound on B_6^{(1/2)} implies epsilon'/epsilon in the SM roughly by a factor of two below its experimental value (16.6\pm 2.3)\times 10^{-4}. We also briefly comment on the parameter \hat B_K and the \Delta I=1/2$ rule.

< 1 in agreement with the recent lattice results of the RBC-UKQCD collaboration. We also point out that the pattern of the size of the hadronic matrix elements of all QCD and electroweak penguin operators Q i contributing to the K → ππ amplitudes A 0 and A 2 , obtained by this lattice collaboration, provides further support to our large N approach. In particular, the lattice result for the matrix element Q 8 0 implies for the corresponding parameter B which expresses the so-called ∆I = 1/2 rule [1,2].
Unfortunately, due to non-perturbative uncertainties originating in the hadronic matrix elements of contributing four-quark operators, it took a long time to obtain meaningful results for all these observables in QCD. But already in the second half of the 1980s, we have developed an approach to K 0 −K 0 mixing and non-leptonic K-meson decays [7][8][9][10][11] based on the dual representation of QCD as a theory of weakly interacting mesons for large N , where N is the number of colours [12][13][14][15]. The most recent results from our approach can be found in [16,17]. This approach provided, in particular, first results within QCD for the amplitudes ReA 0 and ReA 2 in the ballpark of experimental values [10]. In this manner, for the first time, the SM dynamics behind the ∆I = 1/2 rule has been identified. In particular, it has been emphasized that at scales O(1 GeV) long distance dynamics in hadronic matrix elements of current-current operators and not QCDpenguin operators, as originally proposed in [18], are dominantly responsible for this rule. Moreover, it has been demonstrated analytically why ReA 0 is enhanced and why ReA 2 is suppressed relative to the vacuum insertion approximation (VIA) estimates. In this context, we have emphasized that the so-called Fierz terms in the latter approach totally misrepresent 1/N corrections to the strict large N limit for these amplitudes [10].
Our approach, among other applications, allowed us to consistently calculate, for the first time within QCD, the non-perturbative parametersB K , B governing the corresponding matrix elements of ∆S = 2 SM current-current operator and K → ππ matrix elements of the dominant QCD-penguin (Q 6 ) and electroweak penguin (Q 8 ) operators. These parameters are crucial for the evaluation of ε K and ε /ε within the SM and its various extensions. Other applications of large N ideas to K → ππ andB K , but in a different spirit than our original approach, are reviewed in [19]. We will comment in Section 6 on those which reached very different conclusions from ours.
It is interesting and encouraging that most of our results have been confirmed by several recent lattice QCD calculations that we will specify below. While the lattice QCD approach has a better control over the errors than our approach, it does not provide the physical picture of the dynamics behind the obtained numerical results. This is in particular seen in the case of the ∆I = 1/2 rule where our analytic approach offers a very simple picture of the dynamics behind this rule, as summarized again in [16,17].
In the present paper, we briefly compare in Section 2 the status of lattice results forB K and the ∆I = 1/2 rule with the ones obtained in our approach. Subsequently, in Section 3 we demonstrate that the pattern of the size of the matrix elements for penguin operators presented recently by the RBC-UKQCD collaboration for A 0 [20] and A 2 amplitudes [21] gives another support to our approach. In Section 4, we derive upper bounds on the parameters B and discuss briefly in Section 5 their phenomenological implications for ε /ε. In Section 6 we describe briefly the results obtained in other large N QCD approaches. An outlook is presented in Section 7.
2B K and the ∆I = 1/2 Rule 2.1B K The scale and renormalization scheme dependent parameter B K (µ) is related to the relevant hadronic matrix element of the ∆S = 2 operator More useful is the renormalization group invariant parameterB K that is given by [22]B We have shown the N -dependence of the exponent d in the leading term to signal that d vanishes in the large N limit. The coefficient J 3 is renormalization scheme dependent. This dependence cancels the one of B K (µ). As in the strict large N limit the exponent in (6) and the NLO term involving J 3 vanish, one finds [7] that independently of any renormalization scale or renormalization scheme for the operator Q B K → 0.75, (in large N limit, 1986).
It can be shown that including 1/N corrections suppressesB K so that [23] B K ≤ 0.75 , (in 1/N expansion).
Our latest analysis in our approach gave [16] B K = 0.73 ± 0.02, (in dual QCD), where the error should not be considered as a standard deviation. Rather, this result represents the range forB K we expect in our approach after the inclusion of NLO QCD corrections and the contributions of pseudoscalar and vector mesons as discussed in detail in [16]. On the other hand, the world lattice average forB K based on the calculations of various groups [24][25][26][27][28][29] reads for N f = 2 + 1 calculations (recent FLAG update of [27])B K = 0.766 ± 0.010, (in lattice QCD, 2014).
See also the recent analyses in [30,31]. While this result violates the bound in (8), it should be noted that a number of lattice groups among [24][25][26][27][28][29] published results with central values satisfying the bound in (8) but the errors did not allow for a clear cut conclusion. In fact, the most recent update from staggered quarks [31] quotes preciselyB K = 0.738 ± 0.005 but additional systematic error of 0.037 does not allow for definite conclusions. Similarly, the Rome group [32] finds basically the result in (9). We expect therefore that improved lattice calculations will satisfy our bound one day and in a few years from now lattice average forB K will read B K ≈ 0.74. Finally, let us remark that while the lattice approach did not provide the explanation whyB K is so close to its large N limit 0.75, in our approach the smallness of 1/N corrections follows from the approximate cancellation of negative pseudoscalar meson contributions by the positive vector meson contributions.

∆I = 1/2 Rule
A very detailed comparison of the calculations of ReA 0 and ReA 2 in our approach and the lattice QCD has been presented in [16] and our present discussion is meant to be an update due to new results of the RBC-UKQCD collaboration on ReA 0 [20].
First, let us mention that both the dual approach to QCD and lattice approach obtain satisfactory results for the amplitude ReA 2 . On the other hand, whereas we find [16] ReA 0 the most recent result from the RBC-UKQCD collaboration reads [20] ReA 0 ReA 2 lattice QCD = 31.0 ± 6.6 .
Due to large error in the lattice result, both results are compatible with each other and both signal that this rule follows dominantly from the QCD dynamics related to current-current operators. But our approach, being analytic, allows to connect the ∆I = 1/2 rule to the main properties of QCD: asymptotic freedom and the related evolutions of weak matrix elements which at long distance scales can be performed in the dual representation of QCD as a theory of weakly interacting mesons for large N . As lattice QCD calculations are performed basically at a single energy scale, no such physical explanation of this rule is expected from that framework. To this end, lattice calculations would have to be performed at scales below 1 GeV which is straightforward in our approach but appears impossible by lattice methods at present.
On the other hand, from the present perspective only lattice simulations can provide precise value of ReA 0 one day, so that we will know whether some part of this rule at the level of (20 − 30)%, as signalled by the result in (11), originates in new physics (NP) contributions. Indeed, as demonstrated in [33], a heavy Z and in particular a heavy G in the reach of the LHC could be responsible for the missing piece in ReA 0 in (11). On the basis of the analysis in [33] it is much harder to bring this ratio with the help of NP from 31 down to 22 without violating ∆M K constraint, but this requires a separate study.
Of some interest is the ratio of the matrix elements Q 2 0 and Q 1 0 . It equals −2 in the large N limit, corresponding to µ = 0 [16]. Evolving these matrix elements to µ = 1 GeV in the meson theory and subsequently to µ = 1.53 GeV in the quark theory, we find in the NDR-MS scheme 2 The corresponding result in [20] reads In view of large uncertainty in the lattice result, these two ratios are compatible with each other. We expect on the basis of the results in (11) and (12) that this ratio will be eventually found in the ballpark of −1.4.
Recently, the RBC-UKQCD collaboration published their results for the matrix elements Q i 0 [20]. Their matrix elements are given for three dynamical quarks at µ = 1.53 GeV, which is too high for the direct comparison with our approach in the case of current-current operators. On the other hand, the parameters B of the QCD penguin operator Q 6 and the electroweak penguin operator Q 8 are known [34] to be practically scale independent for 1.0 GeV ≤ µ ≤ 3.0 GeV. Therefore these results constitute a useful test of our approach. Another issue is the colour suppression of some matrix elements of other penguin operators which is predicted within our approach. We would like to check whether the pattern of this suppression is also seen in the lattice data.

Hadronic matrix elements
The hadronic matrix elements of operators Q i that are most useful for our discussions are with I = 0, 2 being strong isospin. It should be recalled that for µ ≤ m c , when charm quark has been integrated out, only seven of the operators listed above are independent of each other. Eliminating then Q 4 , Q 9 and Q 10 in terms of the remaining seven operators allows, in the isospin symmetry limit, to find the following important relations [34] 3 where we have used We have checked that these relations have been used in [20]. Of particular importance for our discussion are the matrix elements with [7,8,35] Note, that using the definition of B i parameters consistent with the large N limit of QCD, as given above, implies that their values in the VIA [34] read We will return to this point in the next section.
The input values of parameters entering these expressions are given by [36,37] F π = 130.41 (20) MeV, It should be emphasized that the overall factor h in these expressions depends on the normalisation of the amplitudes A 0,2 . In [34] and recent papers of the RBC-UKQCD collaboration [21,38] h = 3/2 is used whereas in most recent phenomenological papers [16,19,33,39], h = 1. In the present paper we will keep general h so that, e.g., the decay amplitude K + → π + π 0 reads (3/2h)A 2 .
Comparing the expressions (26) and (28) with the lattice results in [20], we find (see also [40] On the other hand, comparing (27) with the value for this matrix element obtained by RBC-UKQCD collaboration in [21] one extracts [39] B All these results are very weakly dependent on the renormalization scale. The quoted values correspond to µ = 1.53 GeV. Basically, identical results are obtained for µ = m c used in [40]. However, as stated before (33), in extracting these parameters from [20] it is important to use the quark masses at that scale. As we will demonstrate in the next section, these lattice results are consistent with the large N approach. Indeed, we will show that the following pattern emerges at next-to-leading order in our dual approach: We would like to recall that strong indication for the suppression of B (3/2) 8 below unity in our approach have been found already in 1998 in [41], while in the case of B Finally, the lattice results in [20] and [21] exhibit colour suppression of the matrix elements of Q 3 , Q 5 and Q 7 operators relative to the ones of Q 4 , Q 6 and Q 8 , respectively: These results are consistent with the large N approach. Indeed, as we will demonstrate soon, the ratios in (38) are O(1/N 2 ) while the ratios in (39) are O(1/N ). These results allow to simplify some of the relations between the matrix elements so that it is justified to use the relations which simplify the phenomenological analysis of ε /ε in [40].

Derivations
The large N numerical values of the |∆S| = 1 matrix elements already displayed in Section 3.2 are most easily derived from the effective theory for the the pseudo-Goldstone field with π = λ a π a , the meson nonet lying below the one GeV and f , the associated weak decay constant scaling like √ N . In particular, the electroweak penguin operator introduced in (18) and "Fierzed" into a product of two colour-singlet quark densities, namely, can be hadronized by considering the leading chiral effective Lagrangian in the large N limit: Indeed, a straightforward identification of the second term in this equation with the standard Dirac mass term in QCD allows us to hadronize all colour-singlet quark densities such that Consequently, the factorized matrix elements of the ∆I = 1/2 and ∆I = 3/2 components of Q 8 in the large N limit are Similarly, the QCD penguin operator Q 6 introduced in (17) and "Fierzed" into a product of two colour-singlet densities reads As a matter of fact, one has the relation at the level of L eff (p 2 , N ). Yet, at this level, the absence of SU (3) splitting among the weak decay constants implies ill-defined Q 8 0,2 matrix elements in (50) and (51) as well as a vanishing Q 6 operator in (52). It is well known [8,42] that the next-to-leading term in the chiral effective Lagrangian solves both problems since it leads to realistic weak decay constants thanks to its derivative dependence and, simultaneously, it implies from the shift induced in the hadronized quark densities through its mass dependence. Taking these corrections into account, we now reproduce the large N matrix elements given in (26)- (28), with the normalization (29) for the B 6,8 coefficients if contributions O(m 2 π /Λ 2 χ ) to Q 8 are neglected. At this point, it is worth emphasizing that, here, we consistently normalize the |∆S| = 1 B 6,8 to unity in the large N limit. Such is unfortunately not the case for the |∆S| = 2 B K parameter conventionally normalized to one with respect to VIA in (5). Had the ∆S = 2 matrix element in (5) been normalized relative to its large N value, the most preciseB K parameter extracted from lattice QCD in (10) would readB K = 1.021 ± 0.013 nowadays and our result in (9)B K = 0.97 ± 0.03. In [34,41], B 6 and B 8 were also normalized with respect to the VIA as in (30).
We are now in an ideal position to estimate 1/N corrections encoded in the B 6,8 parameters. The factorizable 1/N corrections to |∆S| = 1 density-density operators are fully included [35] in the running of quark masses in (53). Let us thus focus on non-factorizable one loop corrections induced by L eff (p 2 , N ). Applying the background field method of [43], we find with Λ = O(1 GeV) the euclidean ultraviolet cut-off of the effective theory (45) to be matched with the non-factorizable short distance evolution, M = O(m K ) and the colour-singlet left-handed and right-handed hadronic currents derived from L eff (p 2 , N ), respectively. Applied to the specific K → ππ decay processes, • the first (L-R) current-current operator in (59), also present with the right relative sign in the VIA through a Fierz transformation, does not contribute to the matrix element Q 6 0 since • the second (L-L) current-current operator in (59), absent in the VIA, does not contribute to the matrix elements Q 8 0,2 since Tr(e q ) = 0 .
An explicit calculation of the surviving Q 4 and Q 7 matrix elements (in the large N limit) gives then, respectively, with pseudoscalar mass scale parameters bounded necessarily by the effective cut-off around 1 GeV:m First, we emphasize most important properties of these results: • For Λ = 0, corresponding to strict large N limit and matrix elements evaluated at zero momentum, B (1/2) 6 = B (3/2) 8 = B (1/2) 8 = 1 in accordance with (29).
• With increasing Λ, the parameters B on µ is very weak. While the analysis in [34] includes NLO QCD and QED corrections, the inspection of the oneloop anomalous dimension matrix allows to see these properties explicitly. In particular Q 6 mixes with the linear combination (Q 4 + Q 6 ) and we find for decreases with increasing µ. On the other hand, in the LO Q 8 runs only by itself and the one-loop anomalous dimension matrix implies . On the other hand, B (1/2) 8 slightly increases with Λ which is also consistent with the standard renormalization group running [40].
Next, we observe that: • The numerical value of the parameter B (1/2) 6 suffers from rather large uncertainties. This feature is related to the fact that Q 6 vanishes at leading order in chiral perturbation theory (see (52)). The 1/N logarithmic correction in (64) is therefore artificially enhanced by the factor F π /(F K − F π ) ≈ 5 such that • The parameter B (1/2) 8 has a very small 1/N correction. At O(1/N 2 ), one larger contribution might arise from the anomalous effective Lagrangian that solves the so-called U (1) A problem [44] by providing the η pseudoscalar with a physical mass in the large N limit [45]: Applying again the background field method, we obtain Such a negative contribution to Q 8 has been included in [41]. However, any consistent estimate beyond B  in lattice QCD even with a large uncertainty as seen in (33). On the other hand, the range for B . We therefore believe that the best way of summarizing our results for B is given in (35) and (36) Below 1 GeV we have seen in (59) that density-density operators transmute into current-current ones at O(1/N ). But power counting in our effective theory does not allow the other way around, namely current-current operators evolving into density-density ones. This is fully consistent with the evolution of hadronic matrix elements above µ = 1 GeV studied already in [34] and is opposite to the evolution of the corresponding Wilson coefficients. Now, in the large N approach, it has already been shown [16,43] that any (L-L) current-current operator evolves as to stand in contrast with the wrong relative sign in the VIA analogue As a consequence, summing over c = d = u, d, s we conclude that the matrix element Q 3 0 which vanishes in the large N limit is formally O(1/N 2 ) relative to Q 4 0 in our effective theory. Such a strong suppression could have been anticipated from the LO short-distance evolution of the four-quark operator Q 3 into another linear combination of Q 4 and Q 6 : At long distance, the further Q 3 evolution undergoes an important numerical cancellation since (57) tells us that the Q 4 and Q 6 operators are not independent anymore. Following (78), this numerical cancellation is not a mere coincidence but the result of a consistent 1/N expansion. In the same manner, it has been proved [43] that any (L-R) current-current operator evolves as with δU proportional to (2U − U 2U † U ). Consequently, the matrix element Q 5 0 which vanishes in the large N limit is O(1/N 2 ) relative to F π /(F K − F π ) Q 6 0 in our effective theory. As already mentioned, an enhancement factor has to be introduced to compensate for the "accidental" chiral suppression of Q 6 0 in (52). On the other hand, the matrix elements are O(p 2 ) but non zero in the large N limit. With the matrix elements Q 8 0,2 given in (50), (51) and being O(p 0 ), the resulting ratios Q 7 0,2 / Q 8 0,2 are at the level of a few percent and can thus be neglected. The long-distance evolution of Q 7 in (81) leads then to matrix elements proportional to Q 8 0,2 , though O(p 2 ). This is clearly at variance with its LO short-distance evolution, namely As already explicitly stated in [43], this suggests the necessity to introduce higher resonances beyond our effective theory truncated to the low-lying pseudoscalars. In a dual representation of QCD the matrix elements Q 7 0,2 should then be dominantly O(p 0 ), but 1/N -suppressed, with the bound resulting from the isospin decompositions in (82)-(83) and (50)-(51).

Implications for ε /ε
We will now briefly discuss the implications of our results for ε /ε. To this end we will use the analytic formula for ε /ε in the SM derived recently in [40]. In obtaining this formula it has been assumed that the SM describes exactly the data on CP-conserving K → ππ amplitudes: ReA 0 and ReA 2 . This allowed to determine the contributions of the (V − A) ⊗ (V − A) QCD penguin operator Q 4 and of the electroweak penguin operators Q 9 and Q 10 to ε /ε much more precisely than it is presently possible by lattice QCD and large N approach. This determination was facilitated by our results on the suppression of the matrix element Q 3 0 implying the relations (40)-(42) 5 . The formula in question reads [40] ε ε = 10 −4 Imλ t 1.4 · 10 −4 a 1 −Ω eff − 4.1 + 24.7 B (1/2) 6 where and [40,[46][47][48] a = 1.017,Ω eff = (14.8 ± 8.0) × 10 −2 . (88) |V ub | and |V cb | are the elements of the CKM matrix and γ is an angle in the unitarity triangle. The parameters a andΩ eff represent isospin breaking corrections [46][47][48]. See these papers and [40] for details. Setting all parameters, except for B A detailed anatomy of ε /ε in the SM is presented in [40], where various uncertainties related to NNLO QCD corrections and other uncertainties are discussed. But these three examples indicate that taking our bounds into account and guided by the results on B (1/2) 6 and B (3/2) 8 from lattice QCD and our dual approach, our SM prediction for ε /ε appears to be significantly below the data given in (3).
In each of these predictions, there are uncertainties from the value of Imλ t , the unknown complete NNLO corrections to Wilson coefficients of contributing operators, α s , m t and other input parameters. But they appear not to change the conclusion that, presently, the SM prediction for ε /ε is significantly below the data. Our upper bound on B (1/2) 6 plays an important role in this result as otherwise increasing B (1/2) 6 above unity would allow to fit easily the data.

Comments on other large N QCD approaches
In [49] the authors analyse ∆I = 1/2 rule and ε /ε in the chiral limit including 1/N corrections. Their results differ drastically from our results. In particular, their low energy "Extended Nambu-Jona-Lasinio" model (ENJL) gives , QCD penguins play an important role in the explanation of the ∆I = 1/2 rule and the experimental value of ε /ε can easily be reproduced, again in contrast with lattice QCD and our dual QCD approach.
In [50] the authors consider a low energy model including the light pseudo-scalar, vector and scalar poles only in the chiral limit. Within this so-called "Minimal Hadronic Approximation" (MHA), they also obtain much larger values of B and find then good agreement with data for ε /ε. In [51][52][53] the authors rely on dispersion relations and "Finite Energy Sum Rules" (FESR) in the chiral limit to extract the electroweak penguin matrix elements from ALEPH and OPAL data. Doing so, they obtain central values for We conclude that the recent lattice QCD results tend to demonstrate that the various models considered in [49][50][51][52][53] do not represent properly the low energy QCD dynamics at work for penguin matrix elements, but confirm the structure of our dual QCD approach.

Summary and outlook
In the present paper, we have compared the structure of the hadronic matrix elements in K → ππ decays obtained within the dual approach to QCD with the one obtained recently by the RBC-UKQCD lattice approach to QCD and commented briefly on the status of the parameterB K and the ∆I = 1/2 rule. Our main results are as follows: • The status ofB K is very good as both our approach and the lattice QCD calculations give this parameter very close to 0.75. But we expect thatB K from the lattice approach will decrease by a few % in the coming years.
• While the results for ReA 2 obtained in both approaches agree well with the data, the central value of ReA 0 from RBC-UKQCD collaboration is by a factor ot two larger than in our approach and 40% above the data. While our result in (11) appears from present perspective to be final in our approach, significant improvement on the lattice result is expected in the coming years. This will allow to find out whether at some level of 20% new physics could still be responsible for the ∆I = 1/2 rule. An analysis anticipating such possibility has been presented in [33].
• As the upper bound on B (3/2) 8 in (36) has been already indicated in [41], one of the most important results of our paper is the upper bound on B (1/2) 6 . Our estimate suggests that B (1/2) 6 ≤ B (3/2) 8 < 1, but the precise values can only be obtained by lattice methods.
• Among other results of our approach supported by recent results from RBC-UKQCD is the strong suppression of Q 3,5 (µ) and B (1/2) 8 ≈ 1.
If indeed the emerging pattern B = 0.8 ± 0.1 will be confirmed by more precise calculations one day, the very recent analysis in [40] and our paper show that ε /ε within the SM will be found roughly by a factor of two below the data. For a detailed phenomenological discussion of the state of ε /ε within the SM including all errors and future theoretical and experimental prospects we refer to [40]. On the other hand, first phenomenological implications of our results on new physics models have been presented in [54,55].
of the ERC Advanced Grant project "FLAVOUR"(267104) and the Belgian IAP Program BELSPO P7/37. It was also partially supported by the DFG cluster of excellence "Origin and Structure of the Universe".