Lattice calculations of kaon weak matrix elements are important for understanding the Standard Model and in constraining physics beyond the Standard Model. For example, K → ππ matrix elements are needed to explain the origin of the ΔI = 1/2 rule and to compute the long-distance contributions to neutral kaon mixing [1]. Because the lowest-order Standard Model contributions to neutral kaon mixing are from box diagrams, while those to ε′K/εK are from 1-loop electroweak penguin diagrams, both K0-K0 mixing and K → ππ decay are sensitive to physics at very high scales. Many extensions of the Standard Model lead to new particles that enter the loops, and these contributions to neutral kaon mixing and K → ππ decay may be sufficiently large that they can be observed once the hadronic uncertainties in the weak matrix elements are small enough.
Over the past few years we have carried out a successful project to calculate the kaon bag parameter BK using USQCD computing resources. BK parameterizes the hadronic contribution to mixing between K0 and K0 mesons, and is one of the most sensitive probes of new physics beyond the Standard Model. We have recently published our result [2],
B̂K = 0.724(8)(29)
where the first error is statistical, and the second is the sum of all systematic errors in quadrature. This is currently the best published unquenched determination of BK, with all systematic errors under control, and fulfills one of the key goals in flavor physics of the U.S. lattice QCD community stated in the 2007 white paper “Fundamental parameters from future lattice calculations” [3]. Combined with the recent determination of BK from RBC/UKQCD [4] and Bae et al. [5] (with whom we are in good agreement) we confirm the earlier claim of Ref. [6] that there is some tension in the unitarity triangle fits [7,8]. This tension is driven mostly by the new precision in the constraint from kaon mixing. In particular, using the latest averages of all lattice inputs to the unitarity triangle fit, as well as some previously neglected corrections to εK [1], the fit prefers the value B̂K = 0.889 ± 0.083 [8] when the lattice input for BK is excluded from the fit. Given this tension, it is crucial to continue our precision studies of kaon physics using multiple methods including our mixed-action approach. We are therefore continuing work to update BK (aiming for an ≈ 2-3% total error), as well as other supporting calculations such as fK/fπ, and the light quark masses.
In particular, over the past year we have been working to reduce the dominant sources of uncertainty in our 2009 publication. The largest source of uncertainty in B̂K is from the renormalization factor ZBK. We are using the nonperturbative renormalization (NPR) method of Rome-Southampton [9], but with some significant improvements to reduce both the statistical and systematic errors. We have begun using volume momentum-source propagators [10] to significantly reduce the statistical errors in ZBK. Although these propagators require a momentum projection at the source, and thus require a new inversion for each momentum, averaging over the spatial volume allows one to use as few as 10 configurations yet still obtain sub-percent statistical errors. We are also employing a new "non-exceptional” momentum scheme developed by Stürm et al. [11]. Use of non-exceptional kinematics significantly reduces the amount of chiral symmetry breaking [4], as well as the size of 1-loop perturbative QCD corrections in the conversion from the RI/(S)MOM scheme to the MS scheme [12,13]. The next largest source of uncertainty in our published determination of B̂K is the chiral-continuum extrapolation. In order to reduce this error we have therefore computed the kaon mixing matrix element with several valence quark masses on two "superfine" sea-quark ensembles with a ≈ 0.06 fm. Addition of data at a third lattice spacing will give us a much better handle on the a2-dependence of B̂K and other matrix elements. Because the staggered taste-splittings in the sea sector are smaller on the superfine ensembles by a factor of three, the superfine data is quite close to the continuum and will enable a significant reduction in the uncertainty due to the chiral-continuum extrapolation.
A calculation of K → ππ in the ΔI = 3/2 channel in collaboration with MILC is also underway, and we recently presented a new method for computing K → ππ matrix elements at Lattice 2010 [14]. This new method for determining K → ππ matrix elements from lattice simulations is less costly than direct simulations of K → ππ at physical kinematics. It improves, however, upon the traditional “indirect” approach of constructing the K → ππ matrix elements using NLO SU(3) χ PT, which can lead to large higher-order chiral corrections. Using the explicit example of the ΔI = 3/2 (27, 1) operator to illustrate the method, we obtained a preliminary value for Re(A2) that agrees with experiment and has a total uncertainty of < 20%. Ultimately, we plan to use this promising approach to calculate K → ππ decays in the ΔI = 1/2 channel, allowing us to address the ΔI = 1/2 rule and ε′K/εK.
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[2] C. Aubin, J. Laiho and R. S. Van de Water, Phys. Rev. D 81, 014507 (2010) [arXiv:0905.3947 [hep-lat]].
[3] http://www.usqcd.org/documents/fundamental.pdf.
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Y. Aoki et al. (2010), arXiv:1012.4178 [hep-lat].
[5] Bae et al., Phys.Rev. D82 (2010) 114509 [arXiv:1008.5179 [hep-lat]].
[6] E. Lunghi and A. Soni, JHEP 0908, 051 (2009) [arXiv:0903.5059 [hep-ph]].
[7] J. Laiho, E. Lunghi and R. S. Van de Water (2009), arXiv:0910.2928 [hep-ph].
[8] J. Laiho, E. Lunghi and R. S. Van de Water (2011), www.latticeaverages.org.
[9] G. Martinelli et al., Nucl. Phys. B445, 81-108 (1995), [hep-lat/9411010].
[10] M. Gockeler et al., Nucl. Phys. B544, 699 (1999), hep-lat/9807044.
[11] C. Stürm et al., Phys. Rev. D80, 014501 (2009), [arXiv:0901.2599 [hep-ph]].
[12] M. Gorbahn and S. Jager, Phys. Rev. D82, 114001 (2010), [arXiv:1004.3997 [hep-ph]].
[13] L.G. Almeida, and C. Sturm, Phys. Rev. D82, 054017 (2010), [arXiv:1004.4613 [hep-ph]].
[14] J. Laiho and R. S. Van de Water, PoS LATTICE2010 (2010) 312 [arXiv:1011.4524 [hep-lat]].