Improved anatomy of epsilon'/epsilon in the Standard Model

We present a new analysis of the ratio epsilon'/epsilon within the Standard Model (SM) using a formalism that is manifestly independent of the values of leading (V-A)x(V-A) QCD penguin, and EW penguin hadronic matrix elements of the operators Q_4, Q_9, and Q_10, and applies to the SM as well as extensions with the same operator structure. It is valid under the assumption that the SM exactly describes the data on CP-conserving K ->pi pi amplitudes. As a result of this and the high precision now available for CKM and quark mass parameters, to high accuracy epsilon'/epsilon depends only on two non-perturbative parameters, B_6^(1/2) and B_8^(3/2), and perturbatively calculable Wilson coefficients. Within the SM, we are separately able to determine the hadronic matrix element_0 from CP-conserving data, significantly more precisely than presently possible with lattice QCD. Employing B_6^(1/2) = 0.57+-0.19 and B_8^(3/2) = 0.76+-0.05, extracted from recent results by the RBC-UKQCD collaboration, we obtain epsilon'/epsilon = (1.9+-4.5) 10^-4, substantially more precise than the recent RBC-UKQCD prediction and 2.9 sigma below the experimental value (16.6+-2.3) 10^-4, with the error being fully dominated by that on B_6^(1/2). Even discarding lattice input completely, but employing the recently obtained bound B_6^(1/2)<= B_8^(3/2)<= 1 from the large-N approach, the SM value is found more than 2 sigma below the experimental value. At B_6^(1/2) = B_8^(3/2) = 1, varying all other parameters within one sigma, we find epsilon'/epsilon = (8.6+-3.2) 10^-4. We present a detailed anatomy of the various SM uncertainties, including all sub-leading hadronic matrix elements, briefly commenting on the possibility of underestimated SM contributions as well as on the impact of our results on new physics models.

, and perturbatively calculable Wilson coefficients. Within the SM, we are separately able to determine the hadronic matrix element Q 4 0 from CP-conserving data, significantly more precisely than presently possible with lattice QCD. Employing B (1/2) 6 = 0.57 ± 0.19 and B (3/2) 8 = 0.76 ± 0.05, extracted from recent results by the RBC-UKQCD collaboration, we obtain ε /ε = (1.9 ± 4.5) × 10 −4 , substantially more precise than the recent RBC-UKQCD prediction and 2.9 σ below the experimental value (16.6±2.3)×10 −4 , with the error being fully dominated by that on B (1/2) 6 . Even discarding lattice input completely, but employing the recently obtained bound B = 1, varying all other parameters within one sigma, we find ε /ε = (8.6 ± 3.2) × 10 −4 . We present a detailed anatomy of the various SM uncertainties, including all sub-leading hadronic matrix elements, briefly commenting on the possibility of underestimated SM contributions as well as on the impact of our results on new physics models.

Introduction
One of the important actors of the 1990s in particle physics was the ratio ε /ε that measures the size of the direct CP violation in K L → ππ relative to the indirect CP violation described by ε K . In the Standard Model (SM), ε is governed by QCD penguins but receives also an important destructively interfering contribution from electroweak penguins that is generally much more sensitive to new physics (NP) than the QCD penguin contribution. Reviews on ε /ε can be found in [1][2][3][4][5].
A long-standing challenge in making predictions for ε /ε within the SM and its extensions has been the strong interplay of QCD penguin contributions and electroweak penguin contributions to this ratio. In the SM, QCD penguins give a positive contribution and electroweak penguins a negative one. In order to obtain a useful prediction for ε /ε, the relevant contributions of the QCD penguin and electroweak penguin operators must be know accurately.
As far as short-distance contributions (Wilson coefficients of QCD and electroweak penguin operators) are concerned, they have been known already for more than twenty years at the NLO level [6][7][8][9][10][11] and present technology could extend them to the NNLO level if necessary. First steps in this direction have been taken in [12][13][14].
The situation with hadronic matrix elements is another story and even if significant progress on their evaluation has been made over the last 25 years, the present status is clearly not satisfactory as we will discuss below. But, already in 1993, an approach has been proposed in [10] which, as far as ε /ε is concerned, avoids direct calculation of some of the most difficult hadronic matrix elements. It assumes that the real parts of the isospin amplitudes A 0 and A 2 , which exhibit the ∆I = 1/2 rule, are fully described by SM dynamics and their experimental values are used to determine to a very good approximation hadronic matrix elements of all (V − A) ⊗ (V − A) operators, among them the so-called Q 4 QCD penguin operator. While not as important as the (V − A) ⊗ (V + A) QCD penguin and electroweak penguin operators, Q 6 and Q 8 , the operator Q 4 has been known since the early days of analyses of ε /ε [9,10,15] to be responsible for a significant part of the suppression of this ratio. In the presence of a partial cancellation of the positive contribution of Q 6 to ε /ε by the one of Q 8 , an accurate determination of the contribution from Q 4 and from the electroweak penguin operators Q 9 and Q 10 to ε /ε by means of CP-conserving data was an important virtue of our approach.
Another virtue of our approach is based on the fact that in the SM the amplitudes ReA 0 and ReA 2 originate already at tree-level. Similar to the observables used for treelevel determination of CKM parameters, also relevant for ε /ε, they are expected to be only marginally affected by NP contributions. Whether NP could contribute to ReA 0 and ReA 2 at some level is an interesting question, to which we will return briefly in Section 6. But, for the time being we assume that they are fully dominated by SM dynamics.
With the contribution of (V − A) ⊗ (V − A) operators being determined from the data on ReA 0 and ReA 2 it was possible to write down an analytic formula for ε /ε that incorporated all NLO QCD and QED corrections and summarised the remaining dominant hadronic uncertainty in terms of two parameters B (1/2) 6 and B (3/2) 8 that parametrise the relevant matrix elements of the dominant operators Q 6 and Q 8 and have to be calculated using a non-perturbative framework like lattice QCD or the large-N approach [16,17].
They cannot be extracted from CP-conserving data as their contributions to ReA 0 and ReA 2 are marginal at µ ≈ m c used in the approach of [10]. In fact one of the reasons for choosing the value µ = m c was to eliminate them from the determination of the matrix elements of (V − A) ⊗ (V − A) operators from the CP-conserving data.
Over the last twenty years the basic formula for ε /ε of [10] has been improved [2,18,19] due to the increased accuracy in the value of the QCD coupling and other input parameters, like the values of m t and m s . We refer to [2,18,19], where useful information on our approach can be found. The most recent version of our analytic formula has been presented in [20,21].
One new aspect of the present paper is the realisation that under the assumption that NP contributions to ReA 0 and ReA 2 are negligible, the leading contributions of (V − A) ⊗ (V − A) operators to ε /ε can be entirely expressed in terms of their Wilson coefficients. Furthermore, we derive a formula for ε /ε which under the above assumption can be used in any extension of the SM in which the operator structure is the same as in the SM. NP enters only through the modified values of the Wilson coefficients and the dominant non-perturbative uncertainties are contained in The ratio q, involving matrix elements of current-current operators Q ± and their Wilson coefficients z ± , enters the determination of the contribution of (V −A)⊗(V −A) operators from CP-conserving data and its range will be estimated in Section 2. But for 0 ≤ q ≤ 0.1 obtained from QCD lattice and large-N approaches the dependence of ε /ε on q is very weak.
As far as the parameters B is suppressed below unity [22], but no clear-cut conclusion has been reached in that paper on B (1/2) 6 . Moreover, the precise amount of suppression of B (3/2) 8 could not be calculated in this approach. Fortunately, in the meantime significant progress has been achieved in the case of the matrix element Q 8 2 by the RBC-UKQCD lattice collaboration [23], which allowed to determine B in agreement with large-N expectations [22,24], but with higher precision. But also some progress on B (1/2) 6 has been made, both by lattice QCD and the large-N approach. In particular, very recently the RBC-UKQCD lattice collaboration [25] presented their first result for the matrix element Q 6 0 from which one can extract (see below and [24]) B (1/2) 6 (m c ) = 0.57 ± 0.19 (RBC-UKQCD).
This low value of B (1/2) 6 is at first sight surprising and as it is based on a numerical simulation one could wonder whether it is the result of a statistical fluctuation. But the very recent analysis in the large-N approach in [24] gives strong support to the values in (2) While one finds B is less precise but there is a strong indication that B in agreement with (2) and (3). For further details, see [24] and Section 3 below.
Employing the lattice results of (2) and (3), in our numerical analysis we find consistent with, but significantly more precise than the result obtained recently by the RBC-UKQCD lattice collaboration [25], This is even more noteworthy considering the fact that our result comprises also uncertainties from isospin corrections and CKM parameters which were not considered in the error estimate of [25]. Our result differs with close to 3 σ significance from the experimental world average from NA48 [26] and KTeV [27,28] collaborations, suggesting evidence for new physics in K decays. But even discarding the lattice results, varying all input parameters, we find at the bound B still 2 σ below the experimental data. We consider this bound conservative since employing the lattice value in (2)  and both lattice QCD [25] and the large-N approach [24] indicate that the SM value of ε /ε is significantly below the data.
The two main goals of the present paper are: • Derivation of a new version of our formula for ε /ε which could also be used beyond the SM and which appears to be more useful than its variants presented by us in the past.
• Demonstration that our approach provides a substantially more accurate prediction for ε /ε in the SM than it is presently possible within lattice QCD and that the upper bound in (8) is rather conservative.
It should be stressed that assuming dominance of SM dynamics in CP-conserving data, our determination of the contributions of (V − A) ⊗ (V − A) operators to ε /ε is basically independent of the non-perturbative approach used. The RBC-UKQCD lattice collaboration calculates these contributions directly and we will indeed identify a significant difference between their estimate of the Q 4 contribution to ε /ε and ours.
Our paper is organised as follows. In Section 2, we derive the analytic formula for ε /ε in question using the strategy of [10] but improving on it. Using this formula, we present a new analysis of ε /ε within the SM exhibiting its sensitivity to the precise value of B (1/2) 6 and the weak dependence on q. In Section 3, we perform the anatomy of uncertainties affecting ε /ε and present the prediction of ε /ε in the SM, including a discussion of its B (1/2) 6 dependence. In Section 4, we extract from the lattice-QCD results of [25] the values of the most important hadronic matrix elements and compare them with ours. This allows us to identify the main origin of the difference between (5) and (6). In particular, we point out an approximate correlation between the contribution of the Q 4 operator to ε /ε and the value of ReA 0 valid in any non-perturbative approach. In Section 5, we investigate if thus far neglected SM contributions could bring our result for ε /ε into agreement with the experimental findings. A brief general discussion of the impact of possible NP contributions to ReA 0,2 and ImA 0,2 and of the implications of our results for NP models is given in Section 6. The summary of our observations and an outlook are presented in Section 7. In Appendix A, we discuss the sub-leading contributions to our prediction for ε /ε and in Appendix B, for completeness, an updated analytic formula for ε /ε in the SM is presented in the form used in several of our papers in the past (e.g. [21]) that is equivalent to the one derived in Section 2, but exhibits the m t , α s , m s and m d dependences more explicitly.

Effective Hamiltonian
We use the effective Hamiltonian for ∆S = 1 transitions of [6][7][8][9][10][11] The contributing operators are given as follows: Current-Current: QCD-Penguins: Electroweak Penguins: Here, α, β denote colour indices and e q denotes the electric quark charges reflecting the electroweak origin of Q 7 , . . . , The Wilson coefficients z i and y i have been calculated at the NLO level more than twenty years ago [10,11], and some pieces of NNLO corrections are also available [12][13][14].
In Table 1, we collect values for z 1,2 and y i at µ = m c , used in our approach, for three values of α s (M Z ) and m t = 163 GeV, in the NDR-MS scheme.

Basic formula for ε /ε
Our starting expression is formula (8.16) of [29] which we recall here in our notation 1 where [29] ω + = a ReA 2 ReA 0 = (4.53 ± 0.02) × 10 −2 , a = 1.017, Ω eff = (6.0 ± 7.7) × 10 −2 . (16) Here a and Ω eff summarise isospin breaking corrections and include strong isospin violation (m u = m d ), the correction to the isospin limit coming from ∆I = 5/2 transitions and electromagnetic corrections [29,30]. The amplitudes ReA 0,2 are then extracted from the branching ratios on K → ππ decays in the isospin limit. Their values are given in (47) below. In the limit a = 1 and Ω eff = 0 formula (15) reduces to the one used in [25], where all isospin breaking corrections except electroweak penguin contributions have been set to zero. The quantity Ω eff includes, in addition to other isospin breaking corrections, electroweak penguin contributions that are then not included in ImA 0 . Here we prefer to include these contributions to ImA 0 and therefore, instructed by the authors of [29], we remove them from Ω eff . However, we keep in Ω eff their term ∆ 0 in the limit of α = 0. Using Table 4 of [29], we then obtain the modified Ω eff : As the second term in (15) is an isospin breaking effect by itself, strictly speaking in this term the parameter a should be set to unity if we want to remove higher order isospin braking corrections. In addition, in order to remove the effects ofΩ eff = 0 in electroweak penguin contributions to ImA 0 , we write with the first term including the contributions from Q 3−6 and the second from Q 7−10 . Except for the tiny corrections due to a = 1, this procedure is equivalent to multiplying the coefficients y 3−6 by (1 −Ω eff ) leaving y 7−10 unchanged. Our final basic formula which we will use in what follows then reads with (ω + , a),Ω eff and ImA 0 given in (16), (17) and (18), respectively. ImA 2 contains only contributions of the electroweak penguin operators Q 7−10 . The crucial theory task for a precision SM prediction is to determine the real and imaginary parts of the (strong-)isospin amplitudes entering (19) in terms of the Wilson coefficients and hadronic matrix elements of the operators in the weak Hamiltonian (9).

Hadronic matrix elements
The hadronic matrix elements of the operators Q i entering the isospin amplitudes, generally depend on the scale µ and on the renormalisation scheme used for the operators. These two dependencies are cancelled by those present in the coefficients C i (µ) so that the effective Hamiltonian and the resulting amplitudes do not depend on µ and on the scheme used to renormalise the operators. We will work exclusively in the NDR-MS scheme and for scales µ ≤ m c , although in [10] also extensive discussion of scales above m c can be found.
For µ ≤ m c , when the 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 results in the following important relations in the isospin limit [10]: where we have employed As stressed in [10], in the NDR-MS scheme the relation (22) receives an O(α s ) correction due to the presence of evanescent operators which have to be taken into account when using Fierz identities in its derivation. The other relations above do not receive such corrections. The complete expression for Q 4 0 in the NDR-MS scheme reads [10] which of course then has to be solved for Q 4 0 . However, due to the partial cancellation between the matrix elements Q 4 0 and Q 6 0 , and the smallness of the matrix elements of Q 3 and Q 5 , this correction affects the determination of Q 4 0 by at most few percent and can be neglected. This procedure is supported both by the results on hadronic matrix elements RBC-UKQCD collaboration [25] and the large-N approach [24]. Setting the contribution of Q 3 to zero 2 and using the operators the formulae (22)-(25) read 2 In our numerical analysis below, all operators will be taken into account.
which reduces the number of independent (V − A) ⊗ (V − A) matrix elements entering ReA 0,2 and ImA 0,2 to three. On the other hand, to an excellent approximation the amplitudes ReA 0 and ReA 2 at µ = m c are fully described by the operators Q − and Q + , so that we can write Introducing the ratio allows us to express the ratios involving only (V − A) ⊗ (V − A) operators that will enter our basic formula for ε /ε as follows: Besides the CKM ratio τ , the first ratio depends only on Wilson coefficients and the single hadronic ratio q to which we will return below. On the other hand the second ratio is free from hadronic uncertainties, being fully determined by the Wilson coefficients z + , y 9 , y 10 and by τ . The remaining contributions to ImA 0 and ImA 2 are due to (V −A)⊗(V +A) operators and are dominated by the operators Q 6 and Q 8 , respectively. We find this time Contributions from Q 3 and Q 5 are very suppressed but can and have been included in our numerical error estimate. (See Appendix A.) We have also taken into account the small effect of Q 7 2 , for which a relatively precise lattice prediction exists [23], through the substitution which is included in writing (39). Here p 72 ≡ Q 7 2 / Q 8 2 = 0.222 for central values of [23]. (In our numerics, we have added the corresponding errors linearly and attribute a 15% uncertainty to this contribution.) The matrix elements of the Q 6 and Q 8 operators are conveniently parametrised by with [31,32] B (1/2) 6 = B (3/2) 8 = 1 (43) in the large-N limit. As had been demonstrated in [10], B (1/2) 6 and B (3/2) 8 exhibit a very weak scale dependence. The dimensionful parameters entering (41), (42) are given by [33,34] In [34], Below, we will neglect the tiny errors on m K , F K , and F π . It should be emphasised that the overall factor h in (41), (42) depends on the normalisation of the amplitudes A 0,2 . In [10] and recent papers of the RBC-UKQCD collaboration [23,35] h = 3/2 is used whereas in most recent phenomenological papers [4,17,20,21], h = 1. Correspondingly, the experimental values quoted for A 0,2 differ by this factor. To facilitate comparison with [10] and the RBC-UKQCD collaboration results [23,25,35], we will set h = 3/2 in the present paper and consequently the experimental numbers to be used are which display the ∆I = 1/2 rule We also note that while equation (41) is identical to (5.10) in [10], the definition of B in the present paper differs from [10] [cf (5.18) there]. This is to ensure that B (1/2) 6 = 1 and B (3/2) 8 = 1 both correctly reproduce the large-N limit of QCD. In contrast, (5.18) in [10] was based on the so-called vacuum insertion approximation, in which additional terms appear in the normalisation of B from unity. They have been investigated in [22] and very recently in [24] with the result summarised in (4). We refer to this paper for further details.
We now turn to the parameter q which enters (36). We first note that, like B (1/2) 6 and B , it is nearly renormalisation-scale independent. Its value can be estimated in the large-N approach [17]; as this approach correctly accounts for the bulk of the experimental value of ReA 0 , the ensuing estimate can be considered a plausible one. In the large-N limit, corresponding to µ = 0, one finds first Q + (0) 0 / Q − (0) 0 = 1/3. Using the meson evolution in [17] up to µ = 1.0 GeV and then quark evolution up to µ = m c , multiplying the result by z + (m c )/z − (m c ), we obtain q ≈ 0.1. On the other hand the results of the RBC-UKQCD collaboration [25] are consistent with a value of zero (q = 0.029 ± 0.087). As the large-N approach gives ReA 0 below the data while [25] above it, we expect the true value of q at µ = m c to lie between these two estimates and will take q in the range We consider this a credible range, but already mention that our phenomenological results below would change very little even if we enlarged this range by a factor of a few: q is simply too small to introduce a large error on ε /ε. Our input parameters including sub-leading hadronic parameters defined in Appendix A are collected in Table 2. Regarding Imλ t , we choose a central value between the UTfit [36] and CKMfitter [37] determinations and an error slightly larger than that obtained from either fit. This is to account for the very small errors on V ud and V us , which we fix to PDG central values [33]. The Wilson coefficients in Table 1 come with an additional uncertainty from unknown higher-order corrections. In particular the threshold corrections at m c can be substantial even at NNLO. This can for example be seen in the perturbative convergence of ε K [14,38]. We use a scale variation to establish the typical size of higher order corrections and estimate a 10% uncertainty for each Wilson coefficient y 3 -y 10 of Table 1. 0.222 ± 0.033 Eq. (40) and surrounding discussion Table 2: Input parameter ranges, grouped into: hadronic matrix elements, parametric, isospin breaking and NNLO. The (numerically unimportant) ratios p 72 , p 3 , p 5 , p 70 are defined in Appendix A). The remaining parameters (F π , F K , m K , V ud , V us , α em , G F , ε K ) are fixed at their central values.

Convenient formula for ε /ε
Before turning to quantitative phenomenology, in order to make easier connection with the phenomenological literature and aid discussion of our results, we summarise the discussion so far in a concise formula (derived first in [10]) for ε /ε that exhibits the sensitivity to the two most important hadronic matrix elements B transparently.
Using the effective Hamiltonian (9) and the experimental data for ω, ReA 0 and ε K , we find where with In (51) and (52) the sums run over all contributing operators. Therefore in P (1/2) in the case of EWP contributions we have to take into account the correction b = 1 defined in (18). Writing then with the parameters B (1/2) 6 and B (3/2) 8 taken at µ = m c and using the expressions (36)-(42) we find: where and Q 6 0 , Q 8 0 , and Q 8 2 are given in (41), (87) and (42), respectively. The second term in (56) proportional to q amounts at most to a 2% correction and could be safely neglected. y eff 8 is defined in (40). a  Table 3.   In summary the ratio ε /ε is governed by the following four contributions: i) The contribution of (V − A) ⊗ (V − A) operators to P (1/2) is represented by the first term in (54). As seen in (56) this term is governed by the operator Q 4 and includes also small contributions from (V − A) ⊗ (V − A) electroweak penguin operators. We find that this term is negative and only weakly dependent on q.
Also the dependences on α s and renormalisation scheme (see [10]) are weak. These weak dependences originate from the fact that in our approach the matrix elements entering the first term in P (1/2) cancel out. The weak dependence on m t results from the contributions of sub-leading electroweak penguin operators and is exhibited in the formulae in Appendix B. As pointed out in [10], the suppression of ε /ε through a (1/2) 0 increases with increasing ReA 0 , a feature which in the next section will help us to partly understand the result in (6).
ii) The contribution of (V − A) ⊗ (V + A) QCD penguin operators to P (1/2) is given by the second term in (54). This contribution is large and positive and is dominated by the operator Q 6 . The coefficient a depends sensitively on α s , but as in the last two decades the precision on α s increased, this uncertainty is small in 2015 as can be seen from Table 3. iii) The contribution of the (V −A)⊗(V −A) electroweak penguin operators Q 9 and Q 10 to P (3/2) is represented by the first term in P (3/2) . As in the case of the contribution i), the matrix elements contributing to a iv) The contribution of the (V − A) ⊗ (V + A) electroweak penguin operators Q 7 and Q 8 to P (3/2) is represented by the second term in (55). This contribution is dominated by Q 8 and depends sensitively on m t and α s . It contributes negatively to ε /ε.
The competition between these four contributions is the reason why it is difficult to predict ε /ε precisely. In this context, one should appreciate the virtue of our approach: the contributions i) and iii) can be determined rather precisely by CP-conserving data so that the dominant uncertainty in our approach in predicting ε /ε resides in the values of B 3 Prediction for ε /ε in the SM
quantity error on ε /ε quantity error on ε /ε B Table 4: Error budget, ordered from most important to least important. Each line shows the variation from the central value of our ε /ε prediction, in units of 10 −4 , as the corresponding parameter is varied within its input range, all others held at central values.
A detailed error budget is given in Table 4. It is evident that the error is dominated by the hadronic parameter B . Therefore including the m s variation in our error estimate for ε /ε leads to a slight, but negligible, overestimate of the total error. At the same time, the small m s dependence we do find in the final result shows that this is no longer a relevant source of uncertainty in non-lattice approaches (like the large-N approach in particular) in which B (1/2) 6 in (2) and (3) receive strong support from the large-N approach as recently demonstrated in [24]. In particular the smallness of the matrix element Q 6 0 with respect to Q 8 2 is the result of the chiral suppression of Q 6 0 , signalled by F K − F π in (41). As seen in (42) no such suppression is present in Q 8 2 . But in addition it is possible to demonstrate that both B as indicated by the lattice data. But as present calculations by lattice QCD and in [24] are not precise enough, at this moment, we cannot exclude that B We observe that even for these values of B (1/2) 6 and B (3/2) 8 the SM predictions for ε /ε are significantly below the data. This is an important result as it shows that even if the value of B (1/2) 6 from lattice calculations would move up in the future, the SM would face difficulty in reproducing the data provided the large-N bound in (4) is respected.
With these results at hand, we are in the position to summarise the present picture of the estimate of ε /ε in the SM: • First, parametric uncertainties decreased by much since the analyses of ε /ε around the year 2000. This includes the uncertainty in Imλ t which is presently about ± 7% and is irrelevant in the estimate in (61) but plays some role when ε /ε is larger. Also the improvement on m s should be appreciated, entailing that the uncertainty on m s no longer is an issue.
• Second, the previously sizeable uncertainty due to B has become sub-dominant, much smaller for example than the one due to isospin violation. This is thanks to impressive progress on the lattice [23], which confirms large-N estimates employed in our previous papers, but with far smaller uncertainty.
• Third, the present analysis further increased the effectiveness of our framework, leading to a situation in which a single parameter B (1/2) 6 is playing the decisive role in the answer to the question whether ε /ε in the SM can be reconciled with the data or not. The new finding both by the lattice QCD and large-N approach that B (1/2) 6 is below unity narrowed significantly the range for ε /ε in the SM in our framework.
This picture clearly indicates the emergence of a new anomaly in K physics. As this anomaly is strictly correlated in our framework with the value of B (1/2) 6 , this parameter must be a priority for future non-perturbative calculations for flavour physics. Fortunately, it is accessible by first-principle lattice-QCD calculations. Systematic improvement is hence possible. (See also comparison with lattice below.) Progress on isospin violation will also be important.
But already now, the results presented here motivate further scrutiny of the SM prediction as well as searching for viable beyond-SM explanations. We will briefly discuss both directions, in Sections 5 and 6, respectively.
Last but not least, the great reduction in parametric and hadronic uncertainties, made effective through our formalism, and good prospects on B (1/2) 6 , may make a more precise measurement of ε /ε in the future worthwhile. , while the second number in brackets is due to the uncertainty in p 72 . This assignment of uncertainties will simplify the comparison with (68), even though it does not strictly follow our formalism. Furthermore, a remark on error correlations is in order. Due to implementing the constraints from CP-conserving data, correlations between the different contributions to ε /ε are introduced. However, as the initial correlations of the hadronic matrix elements determined on the lattice are not available, we refrain from incorporating them into our analysis.

Discussion of B
It should be noted that the term representing Q 6 penguin operator involves the product a(1 −Ω eff )B (1/2) 6 . Therefore, effectively isospin breaking corrections lower the value of B 4 Comparison with RBC-UKQCD lattice QCD

Preliminaries
The results for ε /ε presented in [23,25] can be summarised by a formula analogous to (64), (68) In deriving this formula, we used the value of the matrix element Q 6 given in [25] for µ = 1.53 GeV: with the first error being statistical and the second systematic. Using (41), we find (see also [24]) B (1/2) 6 (µ = 1.53 GeV) = 0.57 ± 0.19 , and consequently (3). The value of B (1/2) 6 is significantly lower than its upper limit from the large-N approach in (4) [24] and the values for B (1/2) 6 used in many papers until now. This is the central reason why the lattice result is substantially below the data.
• The main difference for fixed B (1/2) 6 and B (3/2) 8 is found in the first term in (68). Not only is the error in this term much larger than in our formula but also is this term significantly larger than found by us.
• Also striking is the sizeable error in the third term which is very small in our case.
Let us then have a closer look at the contribution of the Q 4 operator in order to clarify the reason for this difference.

Contribution of Q 4 and ReA 0
Using the formulae of the previous section, we readily find For q = 0.05, using the experimental value of ReA 0 , we obtain On the other hand in [25] q ≈ 0 and the central value of which is roughly 40% larger than the experimental value in (47). From (72) we now find Q 4 (m c ) 0 = 0.31(7) GeV 3 .
This value agrees with the one given in [25]: But what is striking is the high precision obtained for this matrix element in our approach and still large uncertainty in the lattice result. It should also be noted that the contribution of the Q 4 operator to ε /ε is in the present lattice result comparable to the one of Q 8 and can be even larger than the latter one, which is not possible in our approach.

Electroweak contribution
On the other hand the electroweak penguin contribution to ε /ε is similar because the lattice value for ReA 2 agrees well with experiment [23]. Using the lattice result B which can also be obtained from the last two terms in (64). This result compares well with [23] (ε /ε) EWP = − (6.6 ± 1.0) × 10 −4 , (RBC-UKQCD), although our error is substantially smaller.
5 Can the large observed ε /ε be made consistent with the SM?
Given the significant discrepancy between our SM prediction and the experimental result, we first consider possible missing or underestimated contributions in the SM.

Missing chromomagnetic contributions
In our discussion (and much of the literature) the chromomagnetic penguin Q 8g (and also its electromagnetic counterpart Q 7γ ) have been tacitly dropped. It is straightforward to extend the formalism to include Q 8g , which being pure ∆I = 1/2 impacts only on ImA 0 . While y 7γ is small compared to the leading electroweak penguin coefficients, precluding any effect, the coefficient y 8g is sizeable. The status of the hadronic matrix element Q 8g 0 is rather uncertain. A calculation at leading non-vanishing order in the chiral quark model [39] gave (recall h = 3/2 in our normalisation) with B 8g = 1, obtaining an upward shift of about 0.3×10 −4 on ε /ε. Due to uncertainties from unknown higher orders and 1/N corrections, an ad-hoc range 1 ≤ B 8g ≤ 4 was advocated in [40] for setting bounds on new physics.
At the upper end of the range, while still being insufficient to explain the tension between theory and experiment, the contribution becomes competitive with some of the larger subleading uncertainties. Although a chromomagnetic contribution has never been seriously considered as a sizable SM contribution, the possibility cannot be fully excluded. A more definite conclusion would be desirable and will require the computation of Q 8g 0 in the large-N approach or on the lattice.

Missing low-energy contributions
Two of the largest terms in our error budget concern low-energy physics: hadronic matrix elements in the isospin limit, as well as corrections to the isospin limit. In [41,42] it has been pointed out that for approaches that do not include final-state interactions, analyticity suggests extra positive contributions to the value of B (1/2) 6 and negative corrections to the value of B Another type of long-distance corrections is isospin breaking, both due to electromagnetism and m u = m d . This is parametrised by the two parametersΩ eff and a. The latter only affects the overall normalisation and cannot bring the SM into agreement with data. Explaining the measured ε /ε due to the former would require a value of opposite sign and an order of magnitude larger than the value obtained in [29,30]. Nevertheless, given the profound implications of the ε /ε anomaly, this issue deserves further scrutiny, and also lattice-QCD studies should take these corrections into account.

Missing higher-order corrections to the Wilson coefficients
Higher-order corrections to the Wilson coefficients will also have an impact on the theory prediction of ε /ε. While it seems highly unlikely that they can bring the SM prediction into agreement with experiment, it is still instructive to discuss them in slightly more detail. In our analyses we fixed the renormalisation scale to µ = m c in the three-flavour theory. Hence the computation of the Wilson coefficients involves several steps, which start with matching at the weak scale and end with integrating out the charm quark at µ = m c . The intermediate steps involve the renormalisation group evolution of Q 1 -Q 10 and integrating out the bottom quark.
The weak-scale matching corrections are known at NNLO for the electroweak penguin [12] as well as the current-current and QCD penguins [44], albeit in a different renormalisation scheme for the later two. The respective scheme transformation is given in [13], where the relevant anomalous dimensions for the NNLO evolution of Q 1 -Q 6 can also be found. For these operators the matching corrections at µ = m b are also known [14], yet all other matching corrections and anomalous dimension matrices are currently known only at NLO.
In particular the unknown matching corrections at µ = m c could be sizeable [38] since the strong coupling is growing rapidly in this region. For this reason we estimated higher-order corrections by varying the matching scale around µ = m c , and used the threeflavour renormalisation group running to determine the Wilson coefficients at µ = m c . The resulting residual scale dependence is typically in the ball park of 10% for y 3 -y 10 , but substantially smaller for z + and z − . Using this procedure, only the uncertainties in y 6 , and to a lesser extent y 8 , have a significant impact on the error budget of / .
The partially known NNLO corrections to y 8 are quite large [12] and decrease the SM prediction for ε /ε. Accordingly, only the NNLO corrections to y 6 could arguably lead to a significant enhancement of / , but our error estimate shows that a 10% increase in y 6 results only in a 1.2×10 −4 increase in the SM prediction. Bringing the SM prediction close to the experimental value would require a very large higher-order correction to y 6 which would cast serious doubts on the convergence of the perturbative series in our approach. If this was indeed the case, we would have to perform our analysis in a four-flavour setup, i.e. above the charm scale, which would also require new calculations of matrix elements on the lattice.

BSM physics in ε /ε
Not having been able to identify a plausible way to reconcile our prediction with the data (other than attributing it to a large statistical fluctuation somewhere), we turn to a discussion of physics Beyond the Standard Model (BSM) in ε /ε. We first note that (19), reproduced here for convenience: remains intact in the presence of new physics, which can be classified by which of ReA 0,2 and ImA 0,2 is affected.

BSM physics in ReA 0,2
Noting that the RBC-UKQCD prediction [25] of ReA 0 exceeds the experimental determination, while the large-N method exhibits a deficit [17], we define the ratio which takes the central value H = 1.4 and H = 0.7 in [25] and [17], respectively. In other words, we are considering a scenario where the experimental value of ReA 0 is a sum of the SM contribution and a BSM contribution. We cannot presently exclude that such a sub-leading part of ReA 0 comes from NP, a possibility investigated in [20]. As we have seen there is a strong correlation between ReA 0 and the matrix element of Q 4 and consequently there is an effect on ε /ε. We stress that the denominators in (19) are always the true (experimental) values including any BSM contributions. It is the numerator term ImA 0 that is affected through the correlation of hadronic matrix elements.
Our formalism can easily be adapted to this case; one merely needs to multiply the V − A term given in (36) by a factor of H. In this fashion, the denominators in the ratios (36) are corrected for their BSM "contamination" and the theoretical SM expressions are again valid. Note that the ratio (38) is not modified. In Figure 2, we plot ε /ε as a function of B (1/2) 6 for H = 0.7 (blue), H = 1.0 (black), and H = 1.4 (red). We see that taking the RBC-UKQCD central value for ReA 0 to be the true SM prediction, the agreement between theory and data for ε /ε is worsened -and compensating for this requires even larger values of B (1/2) 6 than in the SM. Conversely, taking the large-N central value at face value one observes a slight improvement (reduction) of the tension in ε /ε by means of an upward shift. But in both cases, the effect is not huge, dwarfed by the uncertainty in B (1/2) 6 , and reconciling theory and experiment still requires B (1/2) 6 > 1. We conclude that CP-conserving data does not favour a scenario of BSM in ReA 0 , although there is sizable room for it. A similar discussion could be given for NP in ReA 2 .

BSM physics in ImA 0,2
The result obtained in our paper that ε /ε in the SM is significantly below the experimental data has an impact on various NP models. This is in particular the case for models in which there is a strong correlation between ε /ε and the branching ratios for rare decays K + → π + νν and K L → π 0 νν. Such a correlation has been stressed first in [45] and investigated in many papers since then. See [46] and references to earlier literature therein.
In several models, like littlest Higgs model with T-parity (LHT) [47], and generally Z-models with new FCNCs, only in left-handed currents [20,48], enhancement of the branching ratio for K L → π 0 νν is significantly constrained by ε /ε because in these models such an enhancement is correlated with the suppression of ε /ε with respect to the SM. This is also the case of K + → π + νν but as K + → π + νν receives in addition to imaginary parts of the relevant amplitudes also the real parts, this correlation is much less pronounced. Therefore in such models in order to have large enhancements of K L → π 0 νν and K + → π + νν the SM prediction for ε /ε must be above the data, which is certainly not favoured by our analysis.
Therefore, in these models the agreement with the data for ε /ε can generally be obtained only with strongly suppressed branching ratio for K L → π 0 νν. In the case of K + → π + νν this suppression is not required but significant departures from the SM are not allowed. The recent analysis within the LHT model [49] shows this explicitly. Now, in the models just described NP enters ε /ε only through ImA 2 and the presence of only left-handed FCNCs implies uniquely the strict correlation between ε /ε and K L → π 0 νν mentioned above. But as shown in [50] in the presence of both left-handed and righthanded FCNCs it is possible to arrange these couplings without significant fine-tuning so that the enhancement of ε /ε required to fit data implies automatically the enhancement of K L → π 0 νν and to lesser extent of K + → π + νν. An explicit example of a model with tree-level Z exchanges contributing to ε /ε, K L → π 0 νν and K + → π + νν can be found in [50].
In Z models the situation can in principle be different even if ε /ε is only modified through ImA 2 because flavour diagonal quark couplings to Z could have proper signs so that ε /ε and K L → π 0 νν can be simultaneously enhanced in models with only lefthanded flavour violating Z couplings. As pointed out in [51] some 331 models have this property.
Another route towards the enhancement of ε /ε, less studied in the literature, are Z tree level exchanges with flavour universal diagonal couplings to quarks. In this case ImA 2 is not modified and NP enters only ImA 0 through QCD penguin contributions. As demonstrated in [20,50] also in this model ε /ε and K → πνν can be simultaneously enhanced. Moreover, this can be achieved with only left-handed FCNCs. If the ε /ε anomaly will be confirmed and future data on rare decays will exhibit such enhancements, models of this kind and the ones mentioned in previous paragraph will be favoured.
Clearly there are other possibilities involving new operators, like supersymmetric models [40,52,53], Randall-Sundrum models [54], or left-right models [5], but this is another story which requires further study.

Summary and outlook
Motivated by the recent results on K → ππ amplitudes from the RBC-UKQCD collaboration, we gave another look to the ratio ε /ε within the SM. The main result of our analysis is the identification of a possible new anomaly in flavour physics, this time in K physics. This was possible because: • Improved results for the parameters B became available through recent lattice-QCD studies by the RBC-UKQCD collaboration that are supported by the large-N approach which provides upper bounds on these parameters.
• We employed a formalism that is manifestly independent of the values of leading (V − A) ⊗ (V − A) QCD penguin, and EW penguin hadronic matrix elements of the operators Q 4 , Q 9 , and Q 10 . In this manner a prediction for ε /ε could be made that is more precise than presently possible by direct lattice-QCD simulations.
In this context, we have presented a new analytic formula for ε /ε in terms of B , related to long-distance dynamics, are independent of NP contributions. Thus our formula can be used for models such as the models with constrained MFV, 3-3-1 models and littlest Higgs models. We have also provided an update of a formula for ε /ε in which NP enters directly through the shifts in basic oneloop functions.
Our analysis emphasises the correlation between the amplitude ReA 0 and the contribution of the Q 4 operator to ε /ε given in (72). As the central value of ReA 0 in [25] is by 40% above the data, this calculation overestimates the contribution of Q 4 to ε /ε making it smaller. Assuming that ReA 0 is fully described by SM dynamics, we could improve the accuracy of its estimate by roughly an order of magnitude, as seen in (73) and (76).
We have extracted from [25] the value of B (1/2) 6 obtained by the RBC-UKQCD collaboration to find that it is significantly lower than unity. In fact this is the main reason for the low value of ε /ε found in that paper. On the other hand, should the values of B eventually turn out to be close to the upper bound from the large-N approach [24], significantly larger values of ε /ε are found, although still roughly by a factor of two below the data.
Our improved anatomy of ε /ε clearly demonstrates that the SM has potential difficulties in describing the data for ε /ε. However, there are several open questions that have to be answered before one can be fully confident that NP is at work here. Answering them would also allow us to give a better estimate of the room left for particular NP models.
Our analysis shows that the next most important issues that have to be clarified are as follows: • The value of B (1/2) 6 should be determined with an accuracy of at least 10%. Fig. 1 demonstrates this need clearly, but also a higher precision on B (3/2) 8 would be beneficial.
• The values of the Wilson coefficients y i at the NNLO level. First steps in this direction have been taken in [12,13].
• Improved calculations of isospin breaking effects, represented in our formula in (19) by the parameters a andΩ eff .
• The role of electromagnetic corrections to the hadronic matrix elements, as emphasised already in [10]. Without these corrections there remains some uncertainty due to the renormalisation scheme used for operators.
• Precise theoretical predictions of ReA 0 and ReA 2 within the SM, which would tell us to which degree our assumption of neglecting NP contributions in these amplitudes is justified.
• Finally, our understanding of the role of final-state interactions in ε /ε, see [4] and references therein, should be improved.
Our present results could be affected to some extent by the future finding that some part of the amplitude ReA 0 does not come from the SM dynamics but NP. As Fig. 2 shows, if ReA 0 in the SM is below the experimental value, as indicated by the large-N approach [17], the suppression of ε /ε by the Q 4 operator is smaller, implying a larger value of ε /ε. On the other hand if ReA 0 in the SM is above the experimental value as presently seen in lattice data, the role of Q 4 will be enhanced and consequently ε /ε smaller. But as seen in the error budget of Table 4, this effect is by far less important than the sensitivity to B (1/2) 6 .
In view of the tendency of ε /ε in the SM to be significantly below the data, it is exciting that in the coming years LHC might tell us what this physics could be. But also independent studies of ε /ε in various extensions of the SM could select those extensions of the SM in which ε /ε could be enhanced over its SM value. In fact first phenomenological implications of our results on new physics models have been presented in [50,55]. In any case it appears that ε /ε could soon become again a leading light in flavour physics.
strongly supported by the lattice result for Q 8 (µ) 0 in [25]. To keep our phenomenological formulae simple and central values transparent, we set the central values of p 3 , p 5 , and p 70 to zero and allow generous error ranges that comprise both the intervals expected from large-N counting and those computed in [25]. For the ratio p 72 defined below (40), which is also 1/N -suppressed and plays a very minor role numerically, we take the central value from [25] and, conservatively, attribute a 100% error to it. We furthermore employ B (1/2) 8 = 1.0 ± 0.2, also derived from [25]. All input rages are summarised in Table 2. In this treatment, we tend to overestimate our error on ε /ε, but as shown in the body of the paper, this has a very minor impact on our predictions.

B Analytic Formula for ε /ε
The expression (50) can be put into a formula that is more useful for numerical evaluations as it shows explicitly the dependence on m t and m s . The most recent version of it has been presented in [20], but we update it here due to the change of some input parameters entering the formulae for hadronic matrix elements and a different treatment of isospin breaking corrections. We then have where F ε (x t ) = P 0 + P X X 0 (x t ) + P Y Y 0 (x t ) + P Z Z 0 (x t ) + P E E 0 (x t ) .
The first term is dominated by QCD-penguin contributions, the next three terms by electroweak penguin contributions. The last term expresses the m t dependence from contribution of QCD penguin operators and is totally negligible. The x t dependent functions are given as follows where x t = m 2 t /M 2 W . The coefficients P i are given by where we have defined (97) The coefficients r  Table 5.