$K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$ in the Standard Model: Status and Perspectives

In view of the recent start of the NA62 experiment at CERN that is expected to measure the $K^+\to\pi^+\nu\bar\nu$ branching ratio with a precision of 10%, we summarise the present status of this promising decay within the Standard Model (SM). We do likewise for the closely related $K_L\to\pi^0\nu\bar\nu$, which will be measured by the KOTO experiment around 2020. As the perturbative QCD and electroweak corrections in both decays are under full control, the dominant uncertainties within the SM presently originate from the CKM parameters $V_{cb}$, $V_{ub}$ and $\gamma$. We show this dependence with the help of analytic expressions as well as accurate interpolating formulae. Unfortunately a clarification of the discrepancies between inclusive and exclusive determinations of $V_{cb}$ and $V_{ub}$ from tree-level decays will likely require results from the Belle II experiment available at the end of this decade. Thus we investigate whether higher precision on both branching ratios is achievable by determining $V_{cb}$, $V_{ub}$ and $\gamma$ by means of other observables that are already precisely measured. In this context $\varepsilon_K$ and $\Delta M_{s,d}$, together with the expected progress in QCD lattice calculations will play a prominent role. We find $\mathcal{B}(K^+\to\pi^+\nu\bar\nu) = (9.11\pm 0.72) \times 10^{-11}$ and $\mathcal{B}(K_L\to\pi^0\nu\bar\nu) = (3.00\pm 0.30) \times 10^{-11}$, which is more precise than using averages of the present tree-level values of $V_{cb}$, $V_{ub}$ and $\gamma$. Furthermore, we point out the correlation between $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$, $\overline{\mathcal{B}}(B_s\to\mu^+\mu^-)$ and $\gamma$ within the SM, that is only very weakly dependent on other CKM parameters. Finally, we also update the ratio $\varepsilon'/\varepsilon$ in the SM and present its correlation with $K_L\to\pi^0\nu\bar\nu$.


Introduction
The measurements of the branching ratios of the two golden modes K + → π + νν and K L → π 0 νν will be among the top highlights of flavour physics in the rest of this decade. K + → π + νν is CP conserving while K L → π 0 νν is governed by CP violation. Both decays are dominated in the SM and in many of its extensions by Z penguin diagrams. These decays are theoretically very clean, and the calculation of their branching ratios within the SM includes next-to-leading order (NLO) QCD corrections to the top quark contributions [1][2][3], NNLO QCD corrections to the charm contribution [4][5][6] and NLO electroweak corrections [7][8][9] to both top and charm contributions. Moreover, extensive calculations of isospin breaking effects and non-perturbative effects have been performed [10,11]. Reviews of these two decays can be found in [12][13][14][15][16][17][18] and the power of these decays in probing energy scales as high as several hundreds of TeV has been demonstrated in [19].
In view of the recent start of the NA62 experiment at CERN that is expected to measure the K + → π + νν branching ratio with a precision of 10% compared to the SM prediction [20,21], and the expected measurement of K L → π 0 νν by KOTO around 2020 at J-PARC [15,22], it is the right time to summarise the present status of these decays within the SM. This is motivated in particular by the fact that different estimates appear in the literature due to different inputs used for the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, which presently constitute the main uncertainty in the SM predictions for these two branching ratios. This has been stressed in [23], where the dependence of both branching ratios on the chosen values of |V cb | and |V ub | extracted from tree-level decays has been studied (see Table  3 of that paper).
At this point two strategies for the determination of the contribution of the SM dynamics to these decays are envisaged: Strategy A: The CKM matrix is determined using tree-level measurements of where γ is an angle of the unitarity triangle (UT). As New Physics (NP) seems to now be well separated from the electroweak scale, this determination of the CKM matrix is not expected to be polluted by NP contributions. 1 Inserting these inputs into the known expressions for the relevant branching ratios (see Section 2) then allows a determination of the SM values for the K → πνν branching ratios independently of whether NP is present at short distance scales or not. The departure of these predictions from future data would therefore allow us to discover whether NP contributes to these decays independently of whether it contributes to other decays or not. This information is clearly important for the selection of successful extensions of the SM through flavour-violating processes. Unfortunately, this strategy cannot be executed in a satisfactory manner at present due to the discrepancies between inclusive and exclusive determinations of |V cb | and |V ub | from tree-level decays. Moreover, the precision on γ from tree-level decays is still unsatisfactory for this purpose. While the measurement of γ should be significantly improved by LHCb in the coming years, discrepancies between inclusive and exclusive determinations of |V cb | and |V ub | from tree-level decays are likely to be resolved only by the time of the Belle II experiment at SuperKEKB at the end of this decade.
The clarification of the discrepancies between inclusive and exclusive determinations of |V cb | and |V ub | from tree-level decays are important, but there are reasons to expect that the exclusive determinations will eventually be the ones to be favoured. First of all, exclusive measurements are easier to perform than the inclusive ones. Equally important, due to the significant improvement in the calculations of the relevant form factors by lattice QCD, exclusive determinations are more straightforward than the inclusive ones. This is opposite to the philosophy of ten years ago, where QCD lattice calculations were still at the early stage and inclusive determinations were favoured.
Yet, from the present perspective it is useful to study the SM predictions for K + → π + νν and K L → π 0 νν in the full range of |V cb |, |V ub | and γ known from tree-level decays, as this will clearly demonstrate the need for the reduction of parametric uncertainties. This will also allow the SM predictions for these decays to be monitored as the determination of γ will improve in the coming years at the LHC. This should be of interest in view of the first results on K + → π + νν from NA62, which are expected already in 2016. Moreover, it will also be of interest to see how other observables, like ε K , ∆M s , ∆M d , and rare B s,d decays are modified when the parameters in (1) are varied, and what their correlations with K + → π + νν and K L → π 0 νν are within the SM. As we will see, some of these correlations are practically independent of |V cb | and |V ub | and as such are particularly suited for a precise tests of the SM.

Strategy B:
Here the assumption is made that the SM is the whole story and all available information from flavour-changing neutral current (FCNC) processes is used to determine the CKM matrix. Our strategy here will be to ignore tree-level determinations of |V ub | and |V cb |, as the discrepancies mentioned above could also result from experimental data, which will improve only at the end of this decade. Similarly, the tree-level determination of γ will be left out. Then the observables to be used for the determination of the CKM parameters will be 2 accompanied by lattice QCD calculations of the relevant non-perturbative parameters. In this manner also |V cb |, |V ub | and γ can be determined. This is basically what the UTfit [27] and CKMfitter [28] collaborations do, except that we ignore the tree-level determinations of |V cb |, |V ub | and γ for the reasons stated above. As the dominant top quark contribution to ε K is proportional to |V cb | 4 and ∆M s,d are proportional to |V cb | 2 , a useful determination of |V cb | can be obtained from these quantities. 3 The full UT is then constructed by using the ratio ∆M d /∆M s and S ψK S . We find that with the most recent lattice QCD input on the parameter ξ [31], the determination of γ in this manner is impressive, and also the value of |V cb | is significantly more accurate than from tree-level decays. In the case of |V ub | the accuracy is found to be comparable to the most recent exclusive determination [32]. It should be emphasised that while strategy A is ultimately the one to use to study extensions of the SM, the virtue of strategy B at present is the greater accuracy of the SM predictions for the observables that we consider. By simply imposing constraints from several measurements we arrive at narrow ranges for the parameters in (1) -given that the SM is the whole story.
In the present paper we will follow these two strategies using the most recent inputs relevant for both of them, in particular the ones from lattice QCD. In Section 2 we summarise the present status of the K + → π + νν and K L → π 0 νν decays in the SM and discuss the main uncertainties with the help of analytic expressions. In Sections 3 and 4 we follow strategies A and B, respectively, and present in some detail our numerical results. In Section 5 we present an updated analysis on the ratio ε /ε in the SM. We conclude in Section 6. In the appendices we collect a number of additional expressions that we used in our analysis.

Basic formulae
We present here the basic formulae for the branching ratios for the K + → π + νν and K L → π 0 νν decays in the SM. This section can be considered an update to the analogous section (Section 2) of [12], a review of these decays from 2007. The main advances in the last seven years are: • computation of complete NLO electroweak corrections to the charm quark contribution to K + → π + νν in [7]; • computation of complete NLO electroweak corrections to the top quark contribution to K + → π + νν and K L → π 0 νν in [8]; • reduction of uncertainties due to m t (m t ), m c (m c ) and α s (M Z ), with the last two relevant in particular for the charm contribution to K + → π + νν.
The branching ratio for K + → π + νν in the SM is dominated by Z 0 penguin diagrams, with a significant contribution from box diagrams. Summing over three neutrino flavours, it can be written as follows [3,11] with κ + = (5.173 ± 0.025) · 10 −11 λ 0.225 Here x t = m 2 t /M 2 W , λ = |V us |, λ i = V * is V id are the CKM factors discussed below, and κ + summarises the remaining factors, in particular the relevant hadronic matrix elements that can be extracted from leading semi-leptonic decays of K + , K L and K S mesons [11]. ∆ EM describes the electromagnetic radiative correction from photon exchanges. X(m t ) and P c (X) are the loop functions for the top and charm quark contributions, which are discussed below. An explicit derivation of (3) can be found in [33]. The apparent large sensitivity of B(K + → π + νν) to λ is spurious as P c (X) ∼ λ −4 (see (9)) and the dependence on λ in (5) cancels the one in (3) to a large extent. Therefore when changing λ it is essential to keep track of all the λ dependence.
In obtaining the numerical values in (5) [11], the MS scheme with has been used. As their errors are below 0.1% these can currently be neglected. Note, however, that although the prefactor of the effective Hamiltonian, α/ sin 2 θ w , is precisely known in a particular renormalisation scheme (MS in this case) it remains a scheme dependent quantity, with the scheme dependence only removed by considering higher order electroweak effects in K → πνν. An analysis of such effects in the large m t limit [9] demonstrated that in principle this scheme dependence could introduce a ±5% correction in the K → πνν branching ratios, and that with the MS definition of sin 2 θ W these higher order electroweak corrections are found below 2%. However, only the complete analysis of two-loop electroweak contributions to K → πνν in [8] for the top contribution could put such expectations on firm footing. The same applies to the NLO electroweak effects in the charm contribution to K + → π + νν evaluated in [7]. The short distance function X(x t ) relevant for the top quark contribution, including NLO QCD corrections [1][2][3] and two-loop electroweak contributions [8], is where the first error comes from the remaining renormalisation scale and scheme uncertainties, as well as the theoretical error on the MS parameters due to the matching at the electroweak scale, while the second one corresponds to the combined experimental error on the top and W masses entering the ratio x t , and on the strong coupling α s (M Z ). The central value and errors in (7) have been obtained using the MS couplings with full NNLO precision [34] -3-loop running in the SM and 2loop matching at the weak scale (plus 4-loop QCD running of α s and 3-loop QCD matching in α s and y t ) -and varying the renormalisation scale between M t /2 and 2M t . The NLO EW correction has been included, using the result presented in [8], in order to eliminate the large EW renormalisation scheme dependence of the pure QCD result. See Appendix A for details about the different contributions to X(x t ). The parameter P c (X) summarises the charm contribution and is defined through with the long-distance contributions δP c,u calculated in [10]. Future lattice calculations could reduce the present error in this part [35]. The short-distance part is given by where the functions X NNL result from QCD NLO [3,36] and NNLO calculations [4,5]. They also include complete two-loop electroweak contributions [7]. The index " " distinguishes between the charged lepton flavours in the box diagrams. This distinction is irrelevant in the top contribution due to m t m but is relevant in the charm contribution as m τ > m c . The inclusion of NLO and NNLO QCD corrections have reduced considerably the large dependence on the renormalisation scale µ c (with µ c = O(m c )) present in the leading order expressions for the charm contribution. The two-loop electroweak corrections on the other hand reduced the dependence on the definition of electroweak parameters. An excellent approximation for P SD c (X), including all these corrections, as a function of α s (M Z ) and m c (m c ) is given in (50) of [7] (see Appendix B). Using this formula for the most recent input parameters [37,38] we find P SD c (X) = 0.365 ± 0.012.
Adding the long distance contribution in (8) we finally find P c (X) = 0.404 ± 0.024, where we have added the errors in quadratures. We will use this value in our numerical analysis. In obtaining the error in (12) we kept λ fixed at its central value, as its error is very small and the strong dependence on λ in P SD c (X) is canceled by other factors in the formula for the branching ratio as discussed above.

K L → π 0 νν
The branching ratio for K L → π 0 νν in the SM is fully dominated by the diagrams with internal top exchanges, with the charm contribution well below 1%. It can be written then as follows [39,40] where [11] κ L = (2.231 ± 0.013) · 10 −10 λ 0.225 We have summed over three neutrino flavours. An explicit derivation of (13) can be found in [33]. Due to the absence of P c (X) in (13), the theoretical uncertainties in B(K L → π 0 νν) are due only to X(x t ) and amount to about 1% at the level of the branching ratio. The main uncertainty then comes from Imλ t , which is by far dominant with respect to the other parametric uncertainties due to κ L and m t , with the latter present in X(x t ).
The prospects for improved measurements of B(K + → π + νν) are very good. One should stress that already a measurement of this branching ratio with an accuracy of 10% will give us a very important insight into the physics at short distance scales. Indeed the NA62 experiment at CERN [20,21] is aiming to each this precision, and it is expected to accumulate 100 SM events with a good signal over background figure by 2018. In order to achieve a 5% measurement of the branching ratio, which will be the next goal of NA62, more time is needed. The planned new experiment at Fermilab (ORKA) could in principle reach the accuracy of 5% [44]. 4 Concerning K L → π 0 νν, the KOTO experiment at J-PARC aims in the first step in measuring B(K L → π 0 νν) at SM sensitivity and should provide interesting results around 2020 on this branching ratio [15,22]. There are also plans to measure this decay at CERN and one should hope that Fermilab will contribute to these efforts in the next decade. The combination of K + → π + νν and K L → π 0 νν is particularly powerful in testing NP. Assuming that NA62 and KOTO will reach the expected precision and the branching ratios on these decays will be at least as high as the ones predicted in the SM, these two decays are expected to be the superstars of flavour physics after 2018.

Determination of the branching ratios
As discussed in the introduction, the CKM matrix can be determined by the treelevel measurements |V ub |, |V cb |, |V us |, and the angle γ of the UT. Although this is in principle the optimal strategy, it is currently marred by disagreements between the exclusive and inclusive determinations of both |V ub | and |V cb | -for a recent review see [55,56]. We proceed to present the latest results of both determinations, as well as our weighted average, with which we will give the SM predictions in what we call strategy A. The most recent exclusive determinations from lattice QCD form factors are [32,46,57] |V ub | excl = (3.72 ± 0.14) × 10 −3 , The inclusive values are given by [46,58] |V ub | incl = (4.40 ± 0.25) × 10 −3 , We take a weighted average and scale the errors based on the resulting χ 2 (specifically, we follow the method advocated in Ref. [38]), which gives For the CKM angle γ we take the current world average of direct measurements [49] Using this, together with |V us | = λ already given in (10), we can determine the full CKM matrix.
In particular, we can determine the quantities λ t and λ c , which enter the expressions for the branching ratios in (3) and (13), as functions of these input parameters. These expressions are: which, with respect to their leading order in λ, are accurate up to O(λ 4 ) corrections. The (exact) numerical values for Reλ t and Imλ t obtained from our three different choices of V ub and V cb in (17)- (19) are given in Table 2. These expressions can then be directly inserted into (3) and (13) in order to determine the two branching ratios. Using our averages from (19) together with (20) gives In Figure 1 we show the error budgets for these two observables, and see that the CKM uncertainties dominate. In particular in the case of K + → π + νν we observe large uncertainties due to |V cb | and γ, while in the case of K L → π 0 νν the uncertainty due to |V ub | dominates but the ones from |V cb | and γ are also large. The remaining parameters, which each contribute an error of less than 1%, are grouped into the "other" category. For convenience we give the following parametric expressions for the branching ratios in terms of the CKM inputs: The parametric relation for B(K L → π 0 νν) is exact, while for B(K + → π + νν) it gives an excellent approximation: for the large ranges 37 ≤ |V cb | × 10 3 ≤ 45 and 60 • ≤ γ ≤ 80 • it is accurate to 1% and 0.5%, respectively. In the case of B(K + → π + νν) we have absorbed |V ub | into the non-parametric error due to the weak dependence on it. The exact dependence of both branching ratios on |V ub |, |V cb | and γ is shown in Figure 2. Figure 1: Error budgets for the branching ratio observables B(K + → π + νν) and B(K L → π 0 νν). The remaining parameters, which each contribute an error of less than 1%, are grouped into the "other" category.
In order to obtain the values of ε K , S ψK S , ∆M s,d and of the branching ratios for B s,d → µ + µ − we use the known expressions collected in [16]. The "bar" on the B s → µ + µ − branching ratio, B(B s → µ + µ − ), denotes an average over the two mass-eigenstates, as measured by experiment, rather than an average over the two flavour-states, which differs in the B s system [59][60][61].
In Table 2 we show the results for the K + → π + νν and K L → π 0 νν branching ratios and other observables, for three choices of the pair (|V ub |, |V cb |) corresponding to the exclusive determination (17), the inclusive determination (18) and our average (19). We use (20) for γ in each case. We observe: • The uncertainty in B(K + → π + νν) amounts to more than 10% and has to be decreased to compete with future NA62 measurements, but finding this branching ratio in the ballpark of 15 × 10 −11 would clearly indicate NP at work.
• On the other hand, consistency with B(B s → µ + µ − ) would imply the K + → π + νν branching ratio to be in the ballpark of 7 × 10 −11 . In such a case the search for NP in this decay will be a real challenge and the simultaneous measurement of K L → π 0 νν will be crucial.
• The values of S ψK S are typically above the data but only in the case of the inclusive determinations of both |V cb | and |V ub | is a new CP phase required.
• The accuracy on the SM prediction for ∆M s and ∆M d is far from being satisfactory. Yet, the prospects of improving the accuracy by a factor of two to three in this decade are good.

Correlations between observables
Correlations between K + → π + νν and B q → µ + µ − From inspection of the formulae for the branching ratios for K + → π + νν and B s,d → µ + µ − , each of which in particular depends on |V cb |, we derive the following  Figure 2: Dependence of the branching ratio observables B(K + → π + νν) (left) and B(K L → π 0 νν) (right) on the CKM parameter inputs |V cb |, |V ub | and γ. The 95% C.L. bands in V ub , V cb and γ are shown in green, blue, and red, respectively. approximate relations  Table 2: Values of B(K + → π + νν), B(K L → π 0 νν) and of other observables within the SM for the three choices of |V ub | and |V cb | following strategy A as discussed in the text.
Note that both relations are independent of |V cb | and (28) depends only on γ. In particular the correlation (28) should be of interest in the coming years due to the measurement of K + → π + νν by NA62, of B s → µ + µ by LHCb and CMS and of γ by LHCb. Moreover the last factor should also be improved by lattice QCD.
In the left panel of Figure 3 we show the correlation between K + → π + νν and B s → µ + µ for different fixed values of γ. The dashed regions correspond to a 68% C.L. that results from including the uncertainties on all the other input parameters, whereas the inner filled regions are a result of only including the uncertainties of |V ub |, |V cb | (we use the averages in (19)), and |V us |.
In models of NP with Minimal Flavour Violation (MFV) there are no flavourchanging interactions beyond those generated from the SM Yukawa couplings [62]. For K + → π + νν and K L → π 0 νν this restricts the operators that can contribute in most NP models to just the operator already dominant in the SM, (sd) V −A (νν) V −A , making MFV equivalent to Constrained MFV (CMFV) [69] in this case. Therefore in MFV the value of X(x t ) can be shifted but must stay real. Defining for convenience 3 CKM inputs from tree-level observables with κ + and κ L given in (5) and (14), respectively, we have in the case of MFV the correlation which was first given in [63,64]. Note that in the SM the sign of X(x t ) is positive.
accurate to λ 4 terms, and solving for β then gives In deriving (33) from (31) and (32) one finds that for X(x t ) > 0 this solution is unique, while for X(x t ) < 0 a second solution with a minus sign in front of first square root is allowed [64]. However this solution is excluded if we require both branching ratios to be larger than 10 −11 and we will not consider it here.
In the SM and CMFV we have to a very good approximation the relation [63,64] S ψK S = sin 2β, which is only spoiled by possible penguin enhancements in the B d → J/ψK S mode [65]. Thus (34) together with (33) give a triple correlation between K + → π + νν, K L → π 0 νν and S ψK S in the SM and CMFV. As demonstrated in the earlier parts of this section, the branching ratios for K + → π + νν and K L → π 0 νν still contain significant parametric uncertainties due to the uncertainties in |V cb |, |V ub | and γ, and to a lesser extent in m t . It is therefore  Figure 4: The MFV relation between K + → π + νν and K L → π 0 νν using S ψK S sin 2β versus using the various tree-level inputs of |V cb /V ub | and γ (see text). In the left panel we show situation from current constraints, and in the right panel the possible situation in the following decade, including 10% precision on the two branching ratios, for illustration.
remarkable that within the SM all these uncertainties practically cancel out in this triple correlation [63]. Moreover, this property turns out to be true for all models with constrained MFV [64]. We note that the main uncertainty in (33) resides in P c , as the uncertainty in λ is very small. We stress that this relation is practically immune to any variation of the function X(x t ) within MFV models. This means that once B(K + → π + νν) and S ψK S will be precisely measured we will know the unique value of B(K L → π 0 νν) within CMFV models. This relation is analogous to the one between B(B s,d → µ + µ − ) and ∆M s,d [66], where the present knowledge of ∆M s,d together with the future precise value of B(B s → µ + µ − ) will allow us to uniquely predict the branching ratio B(B d → µ + µ − ) in CMFV models well ahead of its precise direct measurement.
In the right panel of Figure 3 we show the correlation between K + → π + νν and K L → π 0 νν for different fixed values of β (S ψK S ). The dashed regions correspond to a 68% C.L. that results from including the uncertainties on all the other input parameters, whereas the inner filled regions are a result of only including the uncertainties of |V cb | (we use the average in (19)), γ (as given in (20)) and |V us | in (10). We observe that in the latter case the dependence on the remaining CKM parameters, for fixed β, is indeed minimal.
It is also possible to express the ratio in (32) as i.e. in terms of the tree-level CKM inputs discussed in this section, which are generally assumed to be free of NP effects. We note that in MFV also S ψK S is not affected by NP and is more accurately determined than γ. On the other hand, there is a class of models -e.g. models with a U (2) 3 flavour symmetry [67] -where the correlation with S ψK S is no longer true, while (31) and the generic relation (35) still hold.
While the virtue of the correlation (33) is its very weak dependence on the CKM parameters, the correlation (31) together with (35) shares partly this property as it depends only on the ratio |V cb /V ub |, equivalent to R b , and not on |V ub | and |V cb | separately. As we can see from the values of R b given in Table 2, this avoids some of the trouble with exclusive versus inclusive determinations, as the ratio of purely exclusive or inclusive determinations, as well as their weighted average, results in less variation -i.e. only 5% among the cases considered. Note that combining exclusive |V ub | with inclusive |V cb |, for example, gives a greater variation.
In the left panel of Figure 4 we compare the MFV relation for various values of |V cb /V ub |, including a 1 σ C.L. region corresponding to our weighted averages. We also include for comparison the relation corresponding to the current S ψK S measurement, which is seen to currently be more accurate. In the right panel we repeat this comparison for possible future constraints in the following decade, where our choice of errors are based on those collected in [19]. For the branching ratios of K + → π + νν and K L → π 0 νν we assume a 10% precision relative to the SM predictions. Though this is the realistic target set by the NA62 experiment for K + → π + νν, the KOTO experiment will likely not reach such a precision for K L → π 0 νν. We observe that in this possible sketch of the future, the two decays under consideration have the potential to probe MFV and/or a U (2) 3 symmetry.

CKM inputs from loop-level observables
A different approach is to assume that there are no relevant NP contributions to all the quantities listed in (2), so that we can use them together with the precise value of |V us | to determine the best values of β, |V cb |, |V ub |, |V td |, |V ts |, and predict the branching ratios for K + → π + νν, K L → π 0 νν and B s,d → µ + µ − . Clearly the absence of NP effects in all the loop observables (2) requires the SM to be valid up to a reasonably high energy scale, which is a stronger assumption with respect to the one of strategy A, where only tree-level determinations of CKM parameters were assumed to be free of NP effects. We call this approach strategy B. The relevant SM expressions can be found in [16] and in particular in [30], where this strategy has been used to determine the correlation between the values of |V cb | and |V ub | with the non-perturbative parameters relevant for ∆M s,d . As the precision on these parameters resulting from QCD lattice calculations is improving, and the value ofB K , relevant for ε K , has been known precisely already for some time, 5 we can now use these formulae to extract the values listed in (36).
At least three independent observables among the four listed in (2) have to be used in order to fix the three free parameters of the CKM matrix besides than |V us |. For illustration, we present here the strategy which allows us to determine the parameters in (36) with high precision with the minimal number of measurements. Schematically this procedure can be described in two steps: • Step 1: The unitarity triangle can be determined from the experimental values of S ψK S = sin 2β and the mass ratio Using the following very accurate expressions for |V td | and |V ts |, where R t is one of the sides of the UT, and solving (37) for R t one gets where the dots indicate terms of order O(|V us | 4 , |V us | 2 ∆M d /∆M s ).
With one side, R t , and one angle, β, known, the full unitarity triangle is determined by means of purely geometrical relations. In particular one has and the apex (¯ ,η) of the triangle is given bȳ The very precise value of R t obtained through ∆M d /∆M s therefore allows a very precise determination of γ. It should be emphasised that the UT constructed in this manner is universal for all CMFV models as the box function S does not enter the expressions used in these two steps [69]. Moreover this determination is independent of |V cb |.
• Step 2: The measured value of |ε K | then allows us to determine the optimal value of |V cb |. Indeed we have [70] |ε SM where x i = m 2 i /M 2 W , S 0 is the well known SM box function as defined e.g. in [16], and κ ε = 0.94 ± 0.02 [70,71]. With R t known from (38) and β determined from S ψK S , the only unknown in (42) is |V cb |. Having found |V cb |, R b , and R t , also |V ub |, |V td | and |V ts | are determined by the previous relations in a straightforward way.
Alternatively we can also determine |V cb | by using separately ∆M s ∝ |V ts | 2 and ∆M d ∝ |V td | 2 instead of K , as both are proportional to |V cb | 2 via the expressions given in (39). In principle it is also possible to determine the CKM matrix from  ∆M d , ∆M s , and K , but the precision in this case will be rather limited, due to the absence of the strong constraint on β from S ψK S . The best accuracy is obtained by performing a simultaneous fit to all the four observables ∆M d , ∆M s , S ψK S and K . In Table 3 we give the results of fits for the CKM matrix elements using different combinations of the inputs discussed in the steps above. The fits were performed using a Bayesian statistical approach: uncorrelated Gaussian priors were chosen for each of the input parameters and the posterior distribution was sampled using Markhov Chain Monte Carlo with the help of the Bayesian Analysis Toolkit [72]. A direct minimisation of the χ 2 , yielding identical results, has also been performed as a check. We observe that using |ε K | in step 3 gives a more precise result for |V cb | than the alternative of using ∆M d and ∆M d separately, as well as favouring a higher central value. The most accurate determination (given in the last column of the table), follows from including all inputs. The corresponding CKM matrix elements of interest are: For completeness, we give here the sides of the UT as determined from our full fit, that read and its apex¯ = 0.129 ± 0.030,η = 0.344 ± 0.017.  The precision on R t , γ and |V cb | using the above strategy is already impressive, and will continue to improve with new lattice results. Using for instance the improved error estimates for ξ and f Bs B Bs from [31] (keeping the central values from [46]) we find the very precise results: In Figure 5 we show the fitted ranges for |V ub | and |V cb | and compare them with the inclusive, exclusive and our averaged values in (17)- (19). We distinguish between three different cases: the blue area corresponds to the fitted range of |V ub | and |V cb | determined by |ε K |, ∆M d /∆M s and S ψK S ; for the green area ∆M d , ∆M s and S ψK S are used as inputs and the red area combines both and uses |ε K |, ∆M d , ∆M s and S ψK S as inputs. As noted earlier, one can see that especially |ε K | favours large values of |V cb |, around the inclusive value, while the rather small |V ub |, around the exclusive value, is favoured by S ψK S .
It is interesting to compare these results with the indirect fits performed by UTfit [27] and CKMfitter [28], which give They are in very good agreement with our results. We note however, that these two groups included in their analyses the information from tree level decays, which we have decided not to include in our strategy B because of the discrepancies between inclusive and exclusive determinations of |V ub | and |V cb |. Moreover, we also did not use the tree-level determination of γ contrary to these two groups.
Having determined the full CKM matrix in this manner, predictions for rare decays branching ratios can be made. These are collected in the last four rows in Table 3 and again the most precise are the ones in the last column so that our final results for the four branching ratios are: In (48) we used the new lattice error estimates from [31] for a "sneak preview" of how the CKM fit in strategy B will improve once the full results will be available. Using these results for the observable predictions listed above will likewise lead to reduced uncertainties: δB(K + → π + νν) = 0.65, δB(K L → π 0 νν) = 0.28, δB(B s → µ + µ − ) = 0.22 and δB(B d → µ + µ − ) = 0.07. As a comparison, using instead the fit results of (49) and (50), one gets It is also interesting to compare the results in (53), (54) with the most recent prediction in the SM [73], with which our SM results are in perfect agreement, 6 and with the most recent averages from the combined analysis of CMS and LHCb [74] that read Note that the SM value of B(B s → µ + µ − ) is outside one sigma range of the experimental value. In Figure 6 the correlations of B(K L → π 0 νν) and B(B s → µ + µ − ) versus B(K + → π + νν) are shown, comparing the best result of strategy B, which includes all of the available inputs, with the inclusive, exclusive and average cases of strategy A. We observe that the inclusive case of strategy A is very similar to strategy B for K + → π + νν and B s → µ + µ − , as both have little sensitivity to |V ub |, whereas K L → π 0 νν, which has a stronger |V ub | dependence, can differentiate them. In both plots our average for |V ub | and |V cb | is seen to also pick the middle ground for these observables.
µ + µ − ) moves up by 20% in the coming years, the SM will face some tensions in this sector of flavour physics. It is instructive to recall the following formula [3,75] that summarises the dependence of B(K + → π + νν) on R t , β and V cb : This can be considered as the fundamental formula for a correlation between B(K + → π + νν), β and any observable used to determine R t , and is valid also in all models with MFV where X(x t ) is replaced by a real function X. When this formula was proposed, it contained significant uncertainties in R t determined through ∆M d /∆M s , in P c (X) known only at NLO, in κ + and in |V cb |. The first three uncertainties have been significantly reduced since then. Moreover, the improved knowledge of the non-perturbative parameters entering ε K and ∆M s,d allows now within the SM to determine |V cb | rather precisely. We stress that in other models with MFV the latter determination will depend on the NP contributions to ε K and ∆M s,d which modify the function S. An analysis of this issue is presented in [30]. Finally when γ from tree-level decays will be precisely measured, R t will be determined solely by β and γ, and the dependence of B(K + → π + νν) on γ can be directly read off (61).  using strategies A and B, for α s (M Z ) = 0.1185.

The ratio ε /ε in the Standard Model
The ratio ε /ε measures the size of direct CP violation in K L → ππ relative to the indirect CP violation described by ε K . In the 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 NP than the QCD penguin contribution.
The ratio ε /ε is measured to be [38,[76][77][78] Re(ε /ε) = (16.5 ± 2.6) × 10 −4 , and the imaginary part of ε /ε is negligible so that we will just write ε /ε in all formulae below. This result constitutes in principle a strong constraint on theory. However, the difficulty in making predictions for ε /ε within the SM and its extensions is the strong cancellation of QCD penguin contributions and electroweak penguin contributions to this ratio. In the SM QCD penguins give a positive contribution, while the electroweak penguins a negative one. In order to obtain a useful prediction for ε /ε in the SM the corresponding hadronic parameters B has been calculated within an accuracy of 7% by the RBC-UKQCD collaboration [79].
The most recent analysis of ε /ε has been given in [23], where the sensitivity of ε /ε to various scenarios for the CKM parameters was investigated. A motivation for studying ε /ε is its correlation with ε K , K + → π + νν and K L → π 0 νν in various extensions of the SM [23,[80][81][82][83]. Anticipating a new analysis of this type in the future, it is of interest to present here an updated analysis of ε /ε within the SM, having the determination of CKM parameters in strategies A and B at hand.
Several details on ε /ε within the SM including the relevant references are given in Section 3.4 in [23]. Here we collect only the relevant information necessary to perform the numerical analysis. The basic analytic formula for ε /ε is given as follows [23,84] ε ε SM = a Imλ t · F ε (x t ) where a = 0.92 ± 0.03 represents the correction coming from ∆I = 5/2 transitions [85], and with the first term dominated by QCD-penguin contributions, the next three terms by electroweak penguin contributions, and the last term being totally negligible. The x t dependent functions have been collected in Appendix C. The coefficients P i are given in terms of the non-perturbative parameters R 6 and R 8 defined in (67) as follows: The coefficients r  Table 5 in the Appendix C.
The parameters R 6 and R 8 are directly related to the parameters B representing the hadronic matrix elements of Q 6 and Q 8 , respectively. They are defined as There is no reliable result on B (1/2) 6 from lattice QCD and we will investigate how the result on ε /ε changes when B (1/2) 6 is varied within ±20% from its large N value B (1/2) 6 = 1 [48]. We stress that both B (1/2) 6 and B (3/2) 8 depend very weakly on the renormalisation scale [86]. From large N approach one expects that 1/N corrections will have only minor impact on B was found in the ballpark of the lattice result quoted below, although with a larger error. AsB K = 0.73 (2) [48] in this approach agrees well with the lattice valueB K = 0.766(10) [46], we expect that also in the case B (1/2) 6 the large N approach gives values close to future lattice results. However, in order to be conservative, we will also consider values B In . We observe that the experimental result in (63) favours B (1/2) 6 ∼ 1 for all but one of the strategies considered. The exception is strategy A with inclusive |V ub | and |V cb | inputs, for which a slightly lower B (1/2) 6 is favoured. Thus for B (1/2) 6 obtained in the large N approach the SM agrees well with the data but there is still significant room for NP contributions.

Summary and outlook
In this paper we have performed a new analysis of the rare decays K + → π + νν and K L → π 0 νν within the SM. The prime motivations for this study were: • The start of the NA62 experiment that should in the coming years reach a precision of 10% relative to the SM prediction for B(K + → π + νν).
• The soon to improve value of ξ, for which preliminary error estimates are already given in [31], which will allow a much more precise determination of the elements of the CKM matrix, in particular of the angle γ, |V ub | and |V cb |, without the use of present tree-level determinations of these parameters that are presently subject to significant uncertainties.
• The observation of the correlation between B(K + → π + νν), B(B s → µ + µ − ) and γ within the SM that only weakly depends on |V cb |. This correlation should be of interest in particular for CERN experimentalists who in the coming years will significantly improve the measurements on these three quantities.
Our main results are illustrated with several plots in Sections 3 and 4. Our analysis demonstrates that in the coming years the SM will undergo an unprecedented test due to the measurements of the rates for the decays K + → π + νν and B s → µ + µ − and improved determinations of the CKM parameters either through the strategies A or B, accompanied by improved lattice QCD calculations of the relevant nonperturbative parameters. Around 2020 these studies will be enriched through precise measurements of the rates for K L → π 0 νν, B d → µ + µ − and B d → K(K * )νν [88].
Also improved knowledge of the parameters B (1/2) 6 accompanied with improved values of CKM parameters will allow a more precise prediction for the important ratio ε /ε. Calculating this ratio using strategies A and B we find a good agreement between experiment and the Standard Model when using the large N approach to approximate B (1/2) 6 . But in view of various uncertainties there is still significant room for NP contributions.
It should be observed that the agreement of the SM prediction for B(B s → µ + µ − ) with the data can be significantly improved by lowering |V cb | to the values in the ballpark of its present exclusive determinations using lattice QCD form factors. But then automatically ε K is found significantly below the data. Interestingly in this case B(K + → π + νν) is also predicted to be in the ballpark of 7 × 10 −11 , that is more than a factor of two below its present experimental average. No doubt, the coming years will be exceptional for quark flavour physics.
one of which accurate to more than 0.05%, as well as a plot of the contribution (α/4π)X EW can however be found in [8].
The left panel of Figure 8 shows a plot of the scale-and scheme-independent quantityX (µ) = α(µ) sin 2 θ w (µ) as a function of the renormalisation scale µ, together with the 1σ bands corresponding to the theoretical error in the matching of the top Yukawa coupling y t at the weak scale, and the experimental error on the top mass M t . The MS couplings at full NNLO -2-loop matching at the weak scale (3-loop QCD for α s and y t ) and 3-loop running (4-loop QCD for α s ) -as determined in [34] have been used. The remaining scale dependence shown in the figure comes from higher order corrections, mainly from QCD, and accounts for an error of 0.004 on X t . An additional error of 0.002 comes from the ambiguity in the choice of the renormalisation scheme for the EW prefactor, as shown in [8]. A comparison of the different errors contributing to X t is shown in the right panel of Figure 8. The experimental error on the top quark pole mass M t is by far the dominant contribution at present.

B Expression for P c (X)
An approximate formula for P SD c (X) taken from [7] reads C More details on ε /ε The basic one-loop functions entering (65) are given by (71) and where x t = m 2 t /M 2 W . The coefficients r  Table 5.
The parameters B (1/2) 6 and B 3/2 8 are related to the hadronic matrix elements Q 6 and Q 8 as follows It should be emphasied that the overall factor in these expressions depends on the normalisation of the amplitudes A 0,2 . The matrix elements given above correspond to the normalisation used in [23,48,89]. On the other hand the RBC-UKQCD collaboration [79,90] uses a different normalisation adopted in [86]. By comparing (80) and (81) with Eqs. (5.10) and (5.18) of the latter paper we find that the matrix elements in [86] have and additional factor of 3/2. While ε /ε clearly does not depend on this difference, it is crucial to take it into account when extracting the value of B   [79,90]: In this manner we find The µ dependence of M MS−NDR (8,8) mix (µ) is practically cancelled by the one of quark masses so that B (3/2) 8 is practically µ-independent. In particular in the MS-NDR scheme the µ-dependence is very weak [86].