Implications of tau data for CP violation in K decays

The D = 6 contribution of the Operator Product Expansion (OPE) of the VV −AA correlator of quark currents can be related to hadronic matrix elements associated to CP violation in non-leptonic kaon decays. We use those relations to find an updated value for 〈(ππ)I=2|Q8|K〉 in the chiral limit using the updated ALEPH spectral function. Taking instead values of the matrix elements from the lattice to obtain the D = 6 vacuum elements provides a new short-distance constraint that allows for an inclusive determination of fπ and an updated value for the D = 8 condensate. Copyright A. Rodríguez-Sánchez. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 16-11-2018 Accepted 15-01-2019 Published 18-02-2019 Check for updates doi:10.21468/SciPostPhysProc.1.005


Introduction
Non-leptonic kaon decays are a challenging laboratory to study the interplay between weak, electromagnetic and strong interactions at low energies [1]. There is still a long path to reduce the large theory uncertainties due to the complex hadron dynamics, so that they can become comparable to the experimental ones. One of the most controversial observables involving them is the CP violating ratio ε /ε. While some analytical and lattice studies report SM predictions below the experimental measurements [2,3], a recent SM re-analysis, based on a framework that properly accounts for the large absorptive corrections due to the pion re-scattering in the final states [4][5][6], found a value compatible with the experimental one [7].
Starting from the SM Lagrangian at the electroweak scales and using Renormalization Group Equations to resum large logarithms one obtains the following Effective ∆S = 1 Lagrangian in the three-flavour theory [8]: The Wilson Coefficients, C i (µ), encode the short-distance dynamics and can be computed with perturbative methods. The nonperturbative hadronic dynamics is captured by fourquark operators, Q i (µ). One of the leading contributions to the ε /ε ratio comes from matrix elements associated to the electroweak penguin operator contributions: with Γ L(R) µ = γ µ (1 ∓ γ 5 ). Even when there are no known first-principle computations of the different hadronic matrix elements with analytic methods for N C = 3, one can connect the matrix elements of Eq. (2) and (3) to two vacuum condensates by using iteratively the soft-meson limit [9]. In the chiral limit, i.e., at zero momenta, one has: with where λ i are color matrices. Rewriting those hadronic matrix elements in terms of those vacuum matrix elements is useful because we can relate them with the Operator Product Expansion (OPE) of the V V − AA correlation function [10], ud,LR (s) + Π (1) ud,LR (s), with: , is given at NLO in QCD by: b p is α s -suppressed with respect to a p . The dimension 0 contributions vanish, since the correlator vanishes at all order in massless perturbative QCD. a 1 (and b 1 ) is suppressed by two powers of the light quark masses, and then is completely negligible. The leading contribution of a 2 is proportional to α sm qq and is also numerically negligible. The crucial point is that the leading short-distance contribution comes from the same vacuum condensates as in Eqs. (4) and (5) [11]: where A i and B i depend on the renormalization prescription and/or in the number of active flavors (they can be found in Ref. [11]). The OPE of the V − A correlator is then connected to non-leptonic kaon decays.
On the other hand, experimental spectral functions coming from inclusive hadronic tau decays are directly connected to imaginary parts of two-point correlation functions (e.g. see [12]). This connection leads to very precise predictions. For example, a very nice test of asymptotic freedom, which can be translated into a determination of the strong coupling [13][14][15][16][17], can be performed with non-strange V + A spectral function. Using also strange data, one can extract information on fundamental parameters such as m s or V us [15,[18][19][20][21][22].

Dispersion relations with polynomial kernel
From tau data, one have access to: However, the OPE of the correlator is defined at large Euclidean momentum: In order to relate both regions, one uses that Π(s) is known to be an analytic function in the whole complex plane except for a cut in the positive real axis. Then, integrating the correlator times an analytic but otherwise arbitrary weight function ω(s) along the circuit of Figure 1, one finds [25] s 0 s th In the first term of Eq. (14) one can introduce data, while the second one can be evaluated with the analytic continuation of Π OPE (s). The differences arising from using the OPE approximant instead of the physical correlator are known as quark-hadron Duality Violations (DVs) [26,27,[29][30][31][32][33][34][35]:

3
(ππ) I=2 |Q 8 |K 0 in the chiral limit Large experimental and DV uncertainties prevent us from working at NLO in α s when extracting the dimensional OPE coefficients from tau data, since we are not able to fit both condensates entering at that order (they are suppressed both by 6 powers of the tau mass and by α s ). As a consequence, we add conservatively (owing to the large value of A 8 ), a 25% of uncertainty to the final result. At that order, a determination of a 3 (µ) leads to a determination of O 8 µ . In principle, it is not enough if one wants to extract Q 8 µ , since one also needs the contribution coming from O 1 µ . However, this contribution is suppressed by two powers of 1/N c . Different phenomenological and lattice approaches confirm this strong suppression (e.g. see [11,36,37]). Then, one has: At leading order in α s , the determination of a 3 is equivalent to the determination of O D=6 of Ref. [27]. We have revisited it introducing some extra tests and trying to implement some small improvements. We proceed as follows: • for s 0 ∼ m 2 τ (see Fig. 2). We also observe a stable plateau for the latter. Adding DV uncertainties based on the small fluctuations under the change of s 0 in a conservative interval, we obtain, preliminarily: • An alternative approach consists in trying to guess how the exact spectral function is at s 0 > m 2 τ . One pays the price of having to choose a specific parametrization and then introducing some model-dependence. We try to relax it by allowing data not to obey extrictly the model but imposing they must obey WSRs. The ansatz we use is [30,32,33,[38][39][40] ρ(s) = 1 π κe −γs sin (β(s − s z )) s >ŝ 0 .
Following the procedure of Refs. [25,27,32,33] we generate random tuples of parameters (κ, γ, β, s z ), everyone of them representing a possible spectral function above a thresholdŝ 0 . If we perform a fit with ALEPH data, we find that there are no significant deviations (p-value above a 5%) from this specific model aboveŝ 0 = 1.25 GeV 2 . However, the model is only motivated as an approximation at higher energies, where the hadronic multiplicity is higher. As a first constraint, as in Ref. [27], we accept only those tuples that are in the 90% C.L. region (χ 2 < χ 2 min + 7.78). In contrast with Ref. [27], we make a combined fit of the moment used to obtain a 3 with the WSRs, accepting only those tuples compatible with them (p-value larger than a 5%). 1 . We find in good agreement with the result of Ref. [27] and with Eq. (16). 1 In this way, large correlations between experimental uncertainties when imposing the WSRs and the moment used to extract a3 are taken into account.  • When assuming a model, as in the previous bullet point, one is changing the assumption of convergence of data to its OPE approximant at s 0 ∼ m 2 τ , capturing most of the possible DV tails by adding a systematic uncertainty based on fluctuations under the change of s 0 , by the assumption of convergence of data at a lower energy 2 to a specific parametrization for the difference between the spectral function and its OPE approximant. A priori, it is unclear to us which procedure should be preferred. One minimal reliability test one should ask to any model, in analogy with the reliability test of independence of the result on s 0 when directly assuming good convergence of data to its OPE approximant, is a soft dependence in the choice of thresholdŝ 0 . By changingŝ 0 in the large intervalŝ 0 ∈ [1.25, 1.9] GeV 2 we have tested that results display a decent stability (see Table 1).
Combining Eqs. (16) and (18) and introducing it into Eq. (15), we obtain at zero momenta: where uncertainties are dominated by uncertainties in a 3 , followed by perturbative ones, estimated as explained above. The value is in good agreement with the ones obtained by similar approaches [9,11,28]. It is also in agreement with the result obtained using factorization of currents in the large-N c limit: and also with previous lattice results (e.g. see [36,37]).

Using kaon matrix elements from the lattice to improve other tau-based results
Instead of using inclusive hadronic tau-decay data to obtain K → ππ matrix elements, one can take advantage of the very precise values for the matrix elements of Eqs. (2) and (3) given by the lattice in Ref. [37] to obtain the coefficients a 3 (µ) and b 3 (µ). One has in Naive Dimensional Regularization (NDR)M S for 4 active flavors: Now we can work at NLO in α s for the D = 6 contribution. In order to avoid large logarithms we run from µ = 3 GeV to µ = √ s 0 and then apply Eqs. (10) and (11) to obtain, respectively, a 3 (µ = √ s 0 ) and b 3 (µ = √ s 0 ). Using that input and taking in Eq. (14), one can obtain a very powerful short-distance constraint for hadronic tau-decay data: • Experimental uncertainties, typically dominated by the region near s 0 are reduced for that weight function.
• The first unknown OPE contribution is suppressed both by 8 powers of the tau mass and by α s .
• Duality Violations are very suppressed for this moment. One would need a very artificial DV shape to make it noticeable. Different model estimates, for example using the tuple corresponding to the minimum in Eq. (17), typically predict they are one order of magnitude below experimental uncertainties at s 0 ∼ m 2 τ .
There are no unknown physical parameter entering into that expression. However, a good way of testing the power of this dispersion relation is simply translating it into a determination of f π . 3 Even when it enters into the dispersion relation suppressed by two powers of the tau mass, a quite precise value is obtained in Figure 3. As expected, a stable plateau is observed. We find as preliminary result at s 0 = m 2 τ : √ 2f π = (131.6 ± 0.9 exp ± 0.4 chiral ± 0.1 latt ) MeV where the first uncertainty is experimental, the second one due to the difference between physical matrix elements and the chiral limit values and the last one due to the uncertainty in the lattice input.
Finally, using the method of Section 3, but including the D = 6 contribution as an external input, we obtain a preliminary value for the D = 8 condensate: a 4 = −(0.7 ± 0.6) GeV 8 , in good agreement with previous works.

Conclusion
Relations in the chiral limit between kaon to two-pion matrix elements and vacuum condensates that can be related to inclusive tau data can be used to make precise predictions. From tau-decay data one finds at zero momenta: (ππ) I=2 |Q 8 |K 0 2 GeV = (1.14 ± 0.53) GeV 3 .
Taking instead the K → ππ input from the lattice, which nowadays turns out to be more precise, one still can make very precise predictions about inclusive tau decay data dominated by the non-perturbative ∼ 1 GeV Minkowkian region. For example, one of them can be translated into a clean determination of f π below the per cent level: √ 2f π = (131.6 ± 1.0) MeV , or to obtain information about a vacuum condensate a 4 = −(0.7 ± 0.6) GeV 8 , even when it is suppressed by 8 powers of the tau mass. All the determinations studied here could be improved with future non-strange spectral functions, which in principle could be extracted from Belle-II [42].