Status of B[sub K] from lattice QCD Page: 3 of 7
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The most commonly used method to calculate the matrix element
(Ik Z (sd)v-A(sd)v-A( ) IK ) is to evaluate the three point correlation function
shown in Fig. 1. This corresponds to creating a K at some time t1 using a zero-momentum
source; allowing it to propagate for time to - t1 to isolate the lowest state; inserting the
four-fermion operator at time to to convert the K0 to a K ; and finally allowing the K0 to
propagate for long time t2 - to. To cancel the K (K0) source normalization at times t1
and t2 and the time evolution factors e-EKt for times t2 - to and to - t1 it is customary to
divide this three-point function by the product of two 2-point functions as shown in Fig 1.
If, in the 2-point functions, the bilinear operator used to annihilate (create) the K0 (K0)
at time to is the axial density s-y4y5d, then the ratio of the 3-point correlation function to
the two 2-point functions is (8/3)BK.
A key prediction of chiral symmetry is that the matrix element should behave as
(K0I Z (sd)v-A(sd)v-A(pu) K ) = (8/3)BKMkFK + 0(MK).
Earliest calculations of BK were done using Wilson fermions and showed significant devia-
tions from this behavior. It was soon recognized that these lattice artifacts are due to the
explicit breaking of chiral symmetry in the Wilson formulation [1-5]. Until 1998, the only
formulation that preserved sufficient chiral symmetry to give the right chiral behavior was
Staggered fermions. First calculations using this approach in 1989 gave the quenched esti-
mate BK(NDR, 2GeV) = 0.70 0.01 t 0.03. In hindsight, the error estimates were highly
optimistic, however, the central value was only 10% off the current best estimate, and most
of this difference was due to the unresolved O(a2) discretization errors.
In 1997, the staggered collaboration refined its calculation and obtained 0.62(2)(2) ,
again the error estimate was optimistic as a number of systematic effects were not fully
included. The state-of-the-art quenched calculation using Staggered fermions was done by
the JLQCD collaboration in 1997 and gave BK(2GeV) = 0.63 + 0.04 . This estimate
was obtained using six values of the lattice spacing between 0.15 - 0.04 fermi, thus allowing
much better control over the continuum extrapolation as shown in Fig. 2 along with other
published results. This is still the benchmark against which all results are evaluated and is
the value exported to phenomenologists. This result has three limitations: (i) It is in the
quenched approximation. (ii) All quenched calculations use kaons composed of two quarks
of roughly half the "strange" quark mass and the final value is obtained by interpolation to a
kaon made up of (m8/2, ms/2) instead of the physical point (m8, md). Thus, SU(3) breaking
effects (m : md) have not been incorporated. (iii) There are large 0(a2) discretization
artifacts, both for a given transcription of the AS = 2 operator on the lattice and for different
transcriptions at a given value of the lattice spacing, so extrapolation to the continuum
limit is not as robust as one would like. A conservative estimate of the combined associated
systematic error is 0.1 .
In the last four years a number of new methods have been developed and the correspond-
ing results are summarized in Table 1.
* The Rome collaboration has shown that the correct chiral behavior can be ob-
tained using O(a) improved Wilson fermions provided non-perturbative renormal-
ization constants are used. Their latest results, with two different "operators", are
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Gupta, R. (Rajan). Status of B[sub K] from lattice QCD, article, January 1, 2002; United States. (https://digital.library.unt.edu/ark:/67531/metadc933463/m1/3/: accessed April 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.