$\Delta I=1/2$ Rule and $\hat B_K$ : 2014

I summarize the status of the $\Delta I=1/2$ rule in $K\to\pi\pi$ decays within an {\it analytic} approach based on the dual representation of QCD as a theory of weakly interacting mesons for large $N$, where $N$ is the number of colours. This approximate approach, developed in the 1980s by William Bardeen, Jean-Marc G\'erard and myself, allowed us already 28 years ago to identify the dominant dynamics behind the $\Delta I=1/2$ rule. However, the recent inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones to hadronic matrix elements of current-current operators and the calculation of the corresponding Wilson coefficients in a momentum scheme at the NLO improved significantly the matching between quark-gluon short distance contributions and meson long distance contributions over our results in 1986. We obtain satisfactory description of the ${\rm Re}A_2$ amplitude and ${\rm Re}A_0/{\rm Re}A_2=16.0\pm 1.5$ to be compared with its experimental value of $22.3$. While this difference could be the result of present theoretical uncertainties in our approach, it cannot be excluded that New Physics (NP) is here at work. The analysis by Fulvia De Fazio, Jennifer Girrbach-Noe and myself shows that indeed a tree-level $Z^\prime$ or $G^\prime$ exchanges with masses in the reach of the LHC and special couplings to quarks can significantly improve the theoretical status of the $\Delta I=1/2$ rule. I stress that our approach allows to understand the physics behind recent numerical results obtained in lattice QCD not only for the $\Delta I=1/2$ rule but also for the parameter $\hat B_K$ that enters the evaluation of $\varepsilon_K$. In contrast to the $\Delta I=1/2$ rule the chapter on $\hat B_K$ in QCD appears to be basically closed.


Introduction
One of the puzzles of the 1950s was a large disparity between the measured values of the real parts of the isospin amplitudes A 0 and A 2 for a kaon to decay into two pions which on the basis of usual isospin considerations were expected to be of the same order. In 2014 we know the experimental values of the real parts of these amplitudes very precisely [1] ReA 0 = 27.04(1) × 10 −8 GeV, ReA 2 = 1.210(2) × 10 −8 GeV.
In the 1950s QCD and Operator Product Expansion did not exist and clearly one did not know that W ± bosons existed in nature but using the ideas of Fermi, Gell-Mann, Feynman, Marshak and Sudarshan one could still evaluate the amplitudes ReA 0 and ReA 2 to find out that such a high value of R is a real puzzle.
In modern times we can reconstruct this puzzle by evaluating the simple W ± boson exchange between the relevant quarks which after integrating out W ± generates the current-current operator Q 2 : We have listed here the second current-current operator, Q 1 , which we will need soon. With only Q 2 contributing we have Calculating the matrix elements Q 2 0,2 in the strict large N limit, which corresponds to factorization of matrix elements of Q 2 into the product of matrix elements of currents, we find ReA 0 = 3.59 × 10 −8 GeV, ReA 2 = 2.54 × 10 −8 GeV , R = √ 2 (5) in plain disagreement with the data in (1) and (2). It should be emphasized that the explanation of the missing enhancement factor of 15.8 in R through some dynamics must simultaneously give the correct values for ReA 0 and ReA 2 . This means that this dynamics should suppress ReA 2 by a factor of 2.1, not more, and enhance ReA 0 by a factor of 7.5. It is evident that what is missing in this calculation are strong interaction effects represented these days by QCD but the question arises whether the physical picture behind the ∆I = 1/2 rule as described by QCD has a simple structure. As demonstrated by Bardeen, Gérard and myself already in 1986 [4] and improved on the technical level by us recently [5] the dominant dynamics behind the ∆I = 1/2 rule has in fact a simple structure.
To this end one should note that from the point of view of operator product expansion the calculation we have just performed to get (5) corresponds to • The evaluation of the Wilson coefficient of the operator Q 2 in a free (from the point of view of strong interactions) theory of quarks, which corresponds to scales µ = O(M W ) and setting α s (M W ) = 0.
• The evaluation of hadronic matrix elements Q 2 0,2 in a free theory of mesons which corresponds to the factorization scale µ = O(m π ) ≈ 0 and setting N to infinity.
The second point follows from the dual representation of QCD as a theory of weakly interacting mesons for large N, advocated already in the 1970s in [6][7][8][9]. In the strict large N limit QCD becomes a free theory of mesons and in this limit the calculation of hadronic matrix elements by means of factorization method is correct within QCD [10]. But as the Wilson coefficient of Q 2 has been evaluated at µ = O(M W ) and its hadronic matrix elements at µ = O(m π ) ≈ 0 our calculation of ReA 0 and ReA 2 is incomplete. In order to complete it we have to fill the gap between these two vastly different energy scales with QCD dynamics represented by quark-gluon interactions at short distance scales and by meson interactions at long distance scales. This requires the inclusion of α s effects at short distances and 1/N corrections in the meson theory at long distances.
In Section 2 I will describe the structure of our approach together with results for the A 0,2 amplitudes in three steps and will compare it with the lattice QCD approach. In this context I will also summarize the status of the parameterB K . In Section 3 I will summarize an analysis performed by Fulvia De Fazio, Jennifer Girrbach-Noe and myself which demonstrates that tree-level Z or G exchanges with masses in the reach of the LHC and special couplings to quarks can significantly improve the theoretical status of the ∆I = 1/2 rule while satisfying constraints from ε K , ε /ε, ∆M K , LEP-II and the LHC. Few comments in Section 4 close this brief review. I am presenting here the way I see the dynamics behind the ∆I = 1/2 rule. Over the years other views have been expressed in the literature. See in particular [11] and most recent papers [12,13] where further references can be found.
where QCD penguin contributions have been omitted as they will be included in Step 3 below. We have indicated that the matrix elements are to be evaluated at µ = O(1 GeV) but in this step we will still keep them at µ ≈ 0 and use their values calculated in the strict large N limit. We will improve on this in Step 2. The coefficients z i (µ) have been calculated at leading order in the renormalization group improved perturbation theory in [14,15]. This pioneering calculations of short distance QCD effects have shown that these effects indeed enhance ReA 0 and suppress ReA 2 . However, the inclusion of NLO QCD corrections to z 1,2 [16,17] made it clear, as stressed in particular in [17], that the K → ππ amplitudes without the proper calculation of hadronic matrix elements of Q i are both scale and renormalization scheme dependent. For instance setting µ = 0.8 GeV we find where the subscript cc indicates that only current-current contributions have been taken into account and MOM is a momentum scheme, introduced in [5], which is particularly suited for the calculations of the amplitudes in our approach. In this scheme one finds then for µ = 0.8 GeV This is a significant improvement over the results in (5) bringing the theory closer to the data in (1) and (2). However, this result is scale and renormalization scheme dependent. For NDR − MS scheme and µ ≈ (2 − 3) GeV as used in lattice QCD calculations this improvement would be much smaller. But, even in MOM scheme and at µ = 0.8 GeV, further enhancement of ReA 0 and further suppression of ReA 2 are needed in order to be able to understand the ∆I = 1/2 rule. This brings us to Step 2 which fills the remaining gap in our original calculation.
The Journal's name

Step 2: Meson Evolution
The renormalization group evolution down to the scales O(1 GeV) just performed is continued as a short but fast meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. Equivalently, starting with factorizable hadronic matrix elements Q 1 0,2 and Q 2 0,2 at µ ≈ 0 and evolving them to µ = O(1 GeV) at which z 1,2 are calculated one is able to calculate the matrix elements of these two operators at µ = O(1 GeV) and properly combine them with z 1,2 calculated in the MOM scheme. Details of these calculations can be found in [4,5] and there is no space for presenting them here. I just want to make a few comments: • Our loop calculations in the meson theory with a cut-off M = O(1 GeV) include the contributions from pseudoscalars and lowest-lying vector mesons and the result can be cast in the form of evolution equations. It is remarkable that the structure of these evolution equations, in particular the anomalous dimension matrix in the meson theory, is very similar to the one in the quark-gluon picture. This allows to perform an adequate matching between the two evolutions in question thereby removing to a large extent scale and renormalization scheme dependences present in the results of Step 1.
• The inclusion of vector meson contributions in [5] in addition to pseudoscalar contributions calculated in [4] is a significant improvement over our 1986 analysis bringing the theory closer to data.
• The same comment applies to the matching between the quark-gluon and meson theory which this time has been performed at NLO in QCD. In this manner we could justify equating the physical cut-off M of the truncated meson theory (pseudoscalars ane lowest-lying vector mesons) with the renormalization scale µ in the quark-gluon theory.
The resulting values show a very significant improvement over the results in (8) bringing the theory closer to the data in (1) and (2). In particular within the uncertainties of our approach we can claim that the experimental value of ReA 2 has been reproduced. The amplitude ReA 0 has been enhanced in this step by almost a factor of two relative to the result in (8) but it is still by a factor of two below the data. But whereas the calculation of ReA 2 has been completed in this step, in order to complete the calculation of ReA 0 we have to include QCD penguin contribution to this amplitude. This brings us to Step 3.

Step 3: QCD Penguins
As pointed out in [18] QCD penguin operators, of which the dominant one is could play an important role in enhancing the ratio R as in the isospin limit they do not contribute to A 2 and uniquely enhance the amplitude A 0 . However, in 1975 the relevant matrix element Q 6 0 was unknown within QCD and its Wilson coefficient z 6 was poorly known. The first large N result for this matrix element using factorization approach has been obtained in [19] and have been subsequently confirmed in [20,21] by using an effective Lagrangian describing the weak and strong interactions of mesons in the large N limit. It is given by which equals unity in the large N limit. While this matrix element is much larger than the matrix elements of Q 1,2 , its Wilson coefficient z 6 is strongly GIM suppressed at scales O(m c ) due to the fact that it results from the difference of QCD penguin diagrams with charm and up-quark exchanges. If these masses are neglected above µ = m c then z 6 (m c ) = 0 and its value is roughly by an order of magnitude smaller than z 1,2 at µ = 0.8 GeV. In [21] an additional (with respect to previous estimates) enhancement of the QCD penguin contributions to ReA 0 has been identified. It comes from an incomplete GIM cancellation above the charm quark mass. But as the analyses in [4,5] show, this enhancement is insufficient to reproduce fully the experimental value of ReA 0 . We find that the Q 6 contribution to ReA 0 for µ ≤ 1 GeV is relevant as it is by a factor of 3 larger than ReA 2 . Yet at µ = 0.8 GeV it contributes only at the level of 15% to the experimental value of ReA 0 .

Summary of Results
Our final results for K → ππ amplitudes can be summarized as follows Even if the result for ReA 0 is not satisfactory, it should be noted that the QCD dynamics identified by us was able to enhance the ratio R by an order of magnitude. We therefore conclude that QCD dynamics is dominatly responsible for the ∆I = 1/2 rule. In Fig. 1 we show budgets for ReA 2 (left) and ReA 0 (right) that summarize the size of different suppression mechanisms of ReA 2 and enhancement mechanisms of ReA 0 . SD stands for quark-gluon evolution and LD for meson evolution. In the case of ReA 0 we decompose LD into contributions coming from the meson evolution involving only Q 1 and Q 2 (c 1 ) and the one related to the mixing of Q 1,2 and Q 6 (c 2 ). QCDP stands for Q 6 contribution. We set the matching scale at µ = 0.8 GeV. As can be seen, we are not able to explain fully the missing ∆ReA 0 = 23.4 × 10 −8 GeV relative to the large N limit. Different contributions in the budget are normalized to this additive contribution required by the data. The missing piece that we presently cannot explain by QCD dynamics within our approach is represented by the white area. More details on this budget can be found in [5].

Comments on Lattice QCD Results
Lattice QCD calculations made significant progress in the last five years through the inclusion of dynamical fermions [22,23]. Among many results the precise values for the weak decays constants F K , F B d , F B s and F D should be mentioned here. The values of the non-perturbative parameters B i representing ∆F = 2 operators both within the SM and in its extensions require further improvements, but it is likely that in this decade they will be known with high precision.
From my point of view, the most important lattice QCD results as far as B i parameters are concerned are the following ones (recent FLAG update of [24] and [25]): The first one is relevant for the parameter ε K and the second for the contribution of the dominant electroweak penguin operator Q 8 to the ratio ε /ε. Unfortunately there is no reliable result on B (1/2) 6 in (11) from lattice QCD so that ε /ε cannot be calculated in this approach at present. ConcerningB K , the result in (13) confirmed with higher precision our finding in [26] thatB K is rather close to its large N valueB K = 0.75. While in 1987, including only pseudoscalar meson The Journal's name contributions we foundB K = 0.66 ± 0.07, our recent calculation that takes into account also vector meson contributions and improves the matching between the meson and quark-gluon theory gives [5] B K = 0.73 ± 0.02, (in dual QCD, 2014).
This result is in an excellent agreement with the lattice QCD value in (13) although we are aware of the fact that while lattice calculations have good control over their errors, this is not quite the case here. On the other hand, while until now lattice community did not provide, as far as I know, any explanation why after 25 years of efforts they obtained the result forB K within 2% from its large N value, our approach provides the explanation why 1/N corrections are so small. The smallness of these corrections results from an approximate cancellation between pseudoscalar and vector meson one-loop contributions. It is encouraging that such a simple analytic approach could provide some insight in the lattice results forB K . On the other hand there is a qualitative difference between the results in (13) and (14). While the lattice result finds 1/N corrections to be positive, Gérard has demonstrated diagrammatically in [27] that it must be negative. I expect therefore that future lattice results will confirm this result with higher precision than we could do it in our approach.
As far as the ∆I = 1/2 rule is concerned, a detailed comparison of the results of our approach with the results from the RBC-UKQCD collaboration [25,[28][29][30] can be found in section 9 in [5]. The results for the amplitudes ReA 0 and ReA 2 in lattice QCD are presented in terms of the contractions 1 and 2 which are depicted in Fig. 1 of [28]. Basically, Q 2 contributes to K 0 → π + π − and K 0 → π 0 π 0 through contractions 1 and 2 , respectively, while in the case of Q 1 the role of contractions is interchanged. The explicit formulae for ReA 0 and ReA 2 in terms of these contractions can be found in (122) and (123) in [5]. Now in [28] 2 ≈ −0.7 1 has been found. This is an important result as it leads to an additional suppression of ReA 2 and additional enhancement of ReA 0 beyond the one from quark-gluon evolution, which in [28] is stopped at µ = 2.15 GeV. These suppressions and enhancements due to the different signs of contractions in question correspond to Step 2 in our approach. Similar to the case ofB K the authors of [28] did not provide yet the explanation for the relative sign of these two contractions while this is possible within our approach. We find [5] where the negative sign follows in our approach from the proper matching of the anomalous dimension matrices in the meson and quark-gluon pictures of QCD. It is also obtained from explicit one-loop calculation in the meson theory and can also be seen diagrammatically as discussed in [5].
Even if with X F = 0.0298 GeV 3 the values of the contractions in (15) appear at first sight to be much smaller than the ones presented in [28], it should be noted that lattice groups work with other renormalization schemes and different scales. In fact one can demonstrate, as seen in (15), that in our case the factor relating 2 and 1 must be smaller in magnitude. Therefore the numerical comparison of the results of [28] with ours must also involve the Wilson coefficients z i . The fact that our approach and lattice approach predict similar values for ReA 2 implies the compatibility of both approaches as far as ∆I = 3/2 transitions are concerned. Indeed the lattice result for ReA 2 in [25] reads: where the error is dominated by systematics. This result is in agreement with the data and, within uncertainties, with our result. We find it remarkable that the central value in (16) differs from our central value in (12) by only a few percent. This is still another support for the dual picture of QCD.
There is no reliable result for ReA 0 from lattice QCD yet but on the basis of present calculations R ≈ 11, still by a factor of two below the data. As QCD penguin contributions at µ = (2 − 3) GeV are found to be small, we expect that future lattice calculations of hadronic matrix elements of Q 1,2 will imply significantly larger values of R. I would like to end this comparison with lattice QCD with a few personal comments: • I find the study of K → ππ decays in lattice QCD very important but as long as lattice calculations of hadronic matrix elements are performed at µ = (2 − 3) GeV I do not expect that we will gain a satisfactory physical understanding of the dynamics behind the ∆I = 1/2 rule from this approach. Obtaining just two numbers for ReA 0 and ReA 2 from very demanding computer simulations without the understanding of the dynamics behind them would be rather disappointing after almost 60 years of efforts to understand the ∆I = 1/2 rule. I believe that combining the physical insight on the dynamics behind the ∆I = 1/2 rule gained through dual QCD approach presented above with lattice QCD calculations could eventually completely uncover the puzzles of the 1950s on K → ππ decays.
• On the other hand, from the present perspective only lattice simulations with dynamical fermions can provide precise values of ReA 0,2 one day, but this may still take several years of intensive efforts by the lattice community [22,23,31]. Having precise SM values for ReA 0,2 would determine precisely the room for NP contribution left not only in ReA 0 but also ReA 2 . In turn this would give us two observables which could be used to constrain NP.
• While the issue of the ∆I = 1/2 rule is important, in my opinion more pressing is the calculation of B (1/2) Finally I hope that the community of lattice experts will eventually acknowledge the physical relevance of our simple analytical approach and give us credit for a number of findings, listed in [5], that they confirmed 28 years later. Afterall, our approach provided an insight into the dynamics behind the ∆I = 1/2 rule and offered the explanation whyB K is so close to 0.75. At least three colleagues in Rome [45] gave us credit for the signs of 1/N corrections in QCD to K → ππ matrix elements and B K that are opposite to the ones obtained using vacuum insertion approximation.

∆I = 1/2 Rule
As we have seen, presently the value of ReA 0 within dual QCD approach is by 30% below the data and even more in the case of lattice QCD. While this deficit could be the result of theoretical uncertainties in both approaches, it cannot be excluded that the missing piece in ReA 0 comes from NP. This question has been addressed in [46] and I will briefly report on the results of this work.
In this paper we have first demonstrated that a significant part of the missing piece in ReA 0 can be explained by tree-level FCNC transitions mediated by a heavy colourless Z gauge boson with flavour violating left-handed coupling ∆ sd L (Z ) and approximately universal flavour diagonal righthanded coupling ∆ qq R (Z ) to quarks. The approximate flavour universality of the latter coupling assures negligible NP contributions to ReA 2 . This property together with the breakdown of GIM mechanisms at tree-level allows to enhance significantly the contribution of the leading QCD penguin operator Q 6 to ReA 0 . A large fraction of the missing piece in the ∆I = 1/2 rule can be explained in this manner for M Z in the reach of the LHC, while satisfying constraints from ε K , ε /ε, ∆M K , LEP-II and the LHC. The presence of a small right-handed flavour violating coupling ∆ sd R (Z ) ∆ sd L (Z ) and of enhanced matrix elements of ∆S = 2 left-right operators allows to satisfy simultaneously the constraints from ReA 0 and ∆M K , although this requires some fine-tuning. The result of this analysis is summarized by the left plot in Fig. 2.
We have also investgated whether a colour octet of heavy neutral gauge bosons (G ) could also help in fully explaining the ∆I = 1/2 rule. It turns that due to various colour factors and different LHC constraints on its mass, G is even more effective than Z : it provides, within theoretical uncertainties, the missing piece in ReA 0 for M G = (3.5 − 4.0) TeV. Indeed we find with the second result summarized by the right chart in Fig. 2.
The results presented in [46] and summarized above used the preliminary LHC bounds on the relevant quark couplings provided by Maikel de Vries. In [47] an update on these results has been presented. In particular de Vires points out that the upper bounds on the couplings in the full theory, relevant for our analysis, are slightly softer than the ones following directly from four-quark effective operators at hadron colliders which he provided for our analysis in [46]. This result not only puts our bounds on the size of NP effects on firm footing but also allows for slightly larger NP contributions to ReA 0 . Specifically, the bounds on the relevant couplings in (105), (107), (139) and (140) in [46] receive additional corrections represented by the additional terms between the last square brackets in the formulae below: QCD@Work 2014 Figure 2. Budgets of different enhancements of ReA 0 , denoted here by ∆ReA 0 . Z and G denote the contributions calculated in [46]. The remaining coloured contributions come from the SM dynamics as calculated in [5] and shown in Fig. 1. The white region stands for the missing piece.
These bounds correspond to the excluded blue regions in Fig. 3 and should be compared with the ones in Figs. 3 and 9 in [46]. The important feature of these results is that all corrections are above unity. In this manner the region representing Z in Fig. 2 can easily be 20% and in the case of G the white region can be practically removed. Of course all these changes are within the uncertainties of the analysis in [46] but it is gratifying that the results in [47] put our analysis on firmer footing.
Finally, it should be stressed that the allowed ranges for NP contributions in Fig. 2 are independent of M Z and M G as with increased values of these masses the propagator suppression in ReA 0 is compensated by the increase of the allowed ranges for the couplings. The additional corrections in the formulae above introduce weak mass dependence for masses above 3 TeV which should be used in any case to be on the safe side. Of course one has to stay within the perturbative bounds for the couplings involved. This feature tells us that even if Z and G would not be found at the LHC, they could still play a role in the ∆I = 1/2 rule if their masses were below 10 TeV. But to find it out would require the study of other observables as discussed in [46]. Moreover, new bounds from the upgraded LHC could further restrict NP contributions to this rule. The Journal's name Figure 3. Exclusion limits for the Z and G in the mass-coupling plane, from various searches at the LHC based on [47]. The blue region is excluded by effective operator limits studied by ATLAS [48] and CMS [49]. The red and green contours are excluded by dijet resonance searches by ATLAS and CMS, respectively.

ε /ε
In view of the improved value for B (3/2) 8 from [25] in (13) we have updated in [46] the value of ε /ε in the SM stressing various uncertainties, originating in the values of |V ub | and |V cb | and also in the parameter B (1/2) 6 . In particular we have found that the best agreement of the SM with the data is obtained for B (1/2) 6 ≈ 1.0, that is close to the large N limit of QCD. In this paper one can also find the impact of Z , G and Z with flavour violating couplings on ε /ε. There is no doubt that in the 2020s the ratio ε /ε could become a star of flavour physics as it was in the 1990s.

Conclusions
I have reviewed the present understanding of the ∆I = 1/2 rule that emerged within the dual approach to QCD as a theory of weakly interacting mesons for large N already 28 years ago in [4] and has been put on a firmer footing recently in [5]. While lattice QCD will eventually provide much more accurate values for ReA 0 and ReA 2 than it is possible in our approach, our approach provided in my opinion better insight into the dynamics behind this rule than it was possible with lattice QCD until now. But the story is not over as we presently do not know whether at a level of (10 − 30)% NP could be responsible for the measured value of R. Lattice QCD could make an important contribution in answering this question in the coming years.
I am looking forward to improved results on ReA 0 and ReA 2 from lattice QCD and to possible discoveries of Z and G at LHC2 in order to see whether these heavy gauge bosons have anything to say in the context of the ∆I = 1/2 rule. But the most pressing now is an accurate evaluation of B (1/2) 6 by lattice QCD as ε /ε is much more sensitive to NP and very short distance scales than the amplitudes ReA 0 and ReA 2 .