Kaon B Parameter in Quenched QCD
ABSTRACT I calculate the kaon B-parameter with a lattice simulation in quenched approximation. The lattice simulation uses an action possessing exact lattice chiral symmetry, an overlap action. Computations are performed at two lattice spacings, about 0.13 and 0.09 fm (parameterized by Wilson gauge action couplings beta=5.9 and 6.1) with nearly the same physical volumes and quark masses. I describe particular potential difficulties which arise due to the use of such a lattice action in finite volume. My results are consistent with other recent lattice determinations using domain-wall fermions.
arXiv:hep-lat/0309026v1 8 Sep 2003
Kaon B Parameter in Quenched QCD
Department of Physics, University of Colorado, Boulder, CO 80309 USA
(Dated: February 1, 2008)
I calculate the kaon B-parameter BK, defined via
γ5)d|K?, with a lattice simulation in quenched approximation. The lattice simulation uses an action
possessing exact lattice chiral symmetry, an overlap action. Computations are performed at two
lattice spacings, about 0.13 and 0.09 fm (parameterized by Wilson gauge action couplings β = 5.9
and 6.1) with nearly the same physical volumes and quark masses. I describe particular potential
difficulties which arise due to the use of such a lattice action in finite volume. My results are
consistent with other recent lattice determinations using domain-wall fermions.
3(mKfK)2BK = ?¯ K|¯ sγµ(1 − γ5)d¯ sγµ(1 −
The kaon B-parameter BK, defined as8
in the testing of the unitarity of the Cabibbo-Kobayashi-Maskawamatrix . It has been a target of lattice calculations
since the earliest days of numerical simulations of QCD. Lattice calculations of BK require actions with good chiral
properties, since the matrix element of the four-fermion operator scales like the square of the pseudoscalar meson
mass as that mass vanishes. If the lattice action does not respect chiral symmetry, the desired operator will mix with
operators of opposite chirality. The matrix elements of these operators do not vanish at vanishing quark mass, and
therefore overwhelm the signal.
There has been a continuous cycle of lattice calculations using fermions with ever better chiral properties. This
calculation is yet another incremental upgrade, to the use of a lattice action with exact SU(Nf) ⊗ SU(Nf) chiral
symmetry, an overlap  action. These actions have operator mixing identical to that of continuum-regulated QCD
The first lattice calculations of BKwere done with Wilson-type actions. Techniques for handling operator mixing
have improved over the years, (for recent results, see -) but this approach remains (in the author’s opinion)
Staggered fermions (–) have enough chiral symmetry at nonzero lattice spacing, that operator mixing is not
a problem. One can obtain extremely precise values for lattice-regulated BK at any fixed lattice spacing. However,
to date, all calculations of BK done with staggered fermions use “unimproved” (thin link, nearest-neighbor-only
interactions), and scaling violations are seen to be large. For example, the JLQCD collaboration  saw a thirty per
cent variation in BKover their range of lattice spacings.
Domain wall fermions pin chiral fermions to a four dimensional brane in a five dimensional world; chiral symmetry
is exact as the length of the fifth dimension becomes infinite. For real-world simulations, the fifth dimension is finite
and chiral symmetry remains approximate, though much improved in practice compared to Wilson-type fermions.
Two groups [10, 11] have presented results for BK with domain wall fermions. Ref.  has data at two lattice
spacings and sees only small scaling violations. There is a few standard deviation disagreement between the published
results of the two groups.
Finally, overlap actions have exact chiral symmetry at finite lattice spacing. All operator mixing is exactly as in
the continuum . Two groups [12, 13] have recently presented results for BKusing overlap actions, but the actions
and techniques are completely different. The second of these is a preliminary version of the work described here.
The lattice matrix elements must be converted to their continuum-regularized values and run to some fiducial scale.
Matching coefficients can be computed perturbatively or nonperturbatively. For most standard discretizations of
fermions, the matching factors (“Z-factors”) are quite different from unity. For these actions, perturbation theory
is regarded as untrustworthy, and other methods must be employed. I, however, use an action in which the gauge
connections are an average of a set of short range paths, specifically HYP-blocked links. Ref.  has computed
the matching factors for operators relevant for this study, as well as the scale for evaluating the running coupling, using
the Lepage-Mackenzie-Hornbostel [16, 17] criterion. The Z-factors are quite close to unity. A number of calculations of
Z-factors for related actions  reveal that this behavior is generic for actions with similar kinds of gauge connections.
In this work I compare perturbative and nonperturbative calculations of two Z-factors, for the local axial current and
for the lattice-to- MS quark mass matching factor, and find reasonable agreement between them.
Having exact chiral symmetry forces one to confront the problem that this calculation is performed in quenched
approximation. These results will not be directly applicable to the real world of QCD with dynamical fermions. I
3(mKfK)2BK= ?¯ K|¯ sγµ(1−γ5)d¯ sγµ(1−γ5)d|K?, is an important ingredient
encounter difficulties in two places. First, there is no reason for the spectrum of quenched QCD to be identical to
that of full QCD. Using different physical observables to extract a strange quark mass can (and does) lead to different
values of this parameter. At small quark masses, my observed B parameter varies strongly with quark mass, and my
prediction for BKis sensitive to my choice of ms.
The second difficulty is more fundamental. In order to extrapolate results to the chiral limit, one must use chiral
perturbation theory. However, the symmetries of quenched QCD are different from the symmetries of full QCD. Not
only are the leading coefficients expected to be different in quenched and full QCD, but the logarithmic contributions
to any observable Q
Q(mPS) = A(1 + Bm2
PS) + ...(1)
can have different coefficients (different B’s in Eq. 1), or different functional form (in the formula for m2
coefficient of log(m2
PSbut a constant related to the quenched topological susceptibility [19, 20]). All of
these differences are encountered in the analysis of the data. To produce a prediction for an experimental number
using results of a quenched simulation involves uncontrolled, non-lattice-related phenomenological assumptions.
I conclude this introduction by presenting graphs which illustrate my results. Fig. 1 shows results for BKat various
lattice spacings, for a selection of simulations which have reasonable statistics and small error bars. Fig. 2 presents
results which are either extrapolated to the continuum limit, or presented by their authors as having small lattice
PS) is not m2
labeled diamonds and fancy diamond, the fancy cross , octagons , the cross , and squares (this work).
BK comparisons vs lattice spacing, from a selection of simulations with reasonably small error bars. Results are
In Section 2 I describe the action, simulation parameters, and data sets. Sec. 3 is devoted to a discussion of zero
mode effects and my attempts to deal with them. Results relevant to BKare presented in Sec. 4. In Sec. 5 I discuss
the chiral limit of BK and of operators O3/2
, relevant for part of ǫ′/ǫ. My brief conclusions are given in
Sec. 6. An Appendix compares perturbative and nonperturbative matching factors.
the kind of lattice fermions used: W for Wilson, Cl for Clover, KS for staggered, DW for domain wall, and OV for overlap
fermions. References are GBS , JLQCD(W) , Conti , KGS , JLQCD(KS) , CP-PACS , RBC , GGHLR ,
and MILC, this work. The points of Refs. , , , , and this work are the results of a a continuum extrapolation; all
the rest are simulations at one lattice spacing.
BK comparisons presented “as if” they were taken to the continuum limit. The label in parentheses characterizes
The data set used in this study is generated in the quenched approximation using the Wilson gauge action at
couplings β = 5.9 (on a 123× 36 site lattice), where I have an 80 lattice data set, and β = 6.1 (on a 163× 48 site
lattice) with 60 lattices. The nominal lattice spacings are a = 0.13 fm and 0.09 fm from the measured rho mass.
Propagators for ten (β = 5.9) or nine (β = 6.1) quark masses are constructed corresponding to pseudoscalar-to-vector
meson mass ratios of mPS/mV ranging from 0.4 to 0.85. The fermions have periodic boundary conditions in the
spatial directions and anti-periodic temporal boundary conditions. I gauge fix to Coulomb gauge and take our sources
to be Gaussians of size x0/a = 3, 4.125 at β = 5.9, 6.1 (where the quark source is Φ = exp(−x2/x2
B.Lattice action and simulation methodology
The massless overlap Dirac operator is
D(0) = x0(1 +
where z = d(−x0)/x0= (d−x0)/x0and d(m) = d+m is a massive “kernel” Dirac operator for mass m. The massive
overlap Dirac operator is conventionally defined to be
D(mq) = (1 −mq
2x0)D(0) + mq
and it is also conventional to define the propagator so that the chiral modes at λ = 2x0are projected out,
1 − mq/(2x0)(D−1(mq) −
This also converts local currents into order a2improved operators .
The overlap action used in these studies is built from a kernel action with nearest and next-nearest neighbor cou-
plings, and HYP-blocked links. HYP links fatten the gauge links without extending gauge-field-fermion couplings
beyond a single hypercube. This improves the kernel’s chiral properties without compromising locality.
The “step function” (ǫ(z) = z/z†z) is evaluated using the fourteenth-order Remes algorithm of Ref.  (after
removing the lowest 20 eigenmodes of z†z). This involves an inner (multimass ) conjugate gradient inversion step.
It is convenient to monitor the norm of the step function |ǫ(z)ψ|2/|ψ|2and adjust the conjugate gradient residue to
produce a desired accuracy (typically 10−5in ǫ(z)ψ). Doing so, I need about 16-18 inner conjugate gradient steps at
β = 5.9 and 10-12 steps at β = 6.1.
An important ingredient of this overlap program has been to precondition the quark propagator by projecting low
eigenfunctions of the Dirac operator out of the source and including them exactly. This can in principle eliminate
critical slowing down from the iterative calculation of the inversion of the Dirac operator. Of course, there is a cost:
one must find the eigenmodes. My impression from the literature, plus my own experience, is that for the overlap with
Wilson action kernel, this cost is prohibitive. However, my kernel action is designed to resemble the exact overlap
well enough that its eigenvectors are good “seeds” for a calculation of eigenvectors of the exact action, and it is kept
simple enough that finding its own eigenvectors is inexpensive.
As a rough figure of merit, consider the 123× 36, β = 5.9 data set. Computing the lowest twenty eigenmodes of
the squared massless overlap Dirac operator takes about 8 time units, while the complete set of quark propagators
from the lightest mass I studied to the heaviest takes about 16 time units times two (for two sets of propagators) per
lattice. Fig. 3 shows the number of conjugate gradient steps needed to compute the quark propagator to some fixed
accuracy, as a function of bare quark mass. If one would naively extrapolate the heavy quark results (for which low
eigenmodes make only a small contribution) to lower masses, one would see a forty per cent reduction in the number
of inverter steps needed at the smallest quark mass, vs an additional cost of 8/32 = 25 per cent per propagator set
for the construction of eigenmodes.
The “generic” four fermion operator one must consider is
O = (¯ q(1)
β) ⊗ (¯ q(3)
(the superscript labels flavor; the subscript, color). Special cases are (a) O = O1: Γ1= Γ2= γµ(1−γ5), α = δ, β = γ;
(b) O = O2: Γ1= Γ2= γµ(1 − γ5), α = β, γ = δ; and (c) the isospin 3/2 operators for electroweak penguins, here
written with the normalization conventions used in Refs.  and 
?(¯ sαγµ(1 − γ5)dα)[(¯ uβγµ(1 + γ5)uβ) − (¯dβγµ(1 + γ5)dβ)] + (¯ sαγµ(1 − γ5)uα)(¯ uβγµ(1 + γ5)dβ)?
?(¯ sαγµ(1 − γ5)dβ)[(¯ uβγµ(1 + γ5)uα) − (¯dβγµ(1 + γ5)dα)] + (¯ sαγµ(1 − γ5)uβ)(¯ uβγµ(1 + γ5)dα)?.
BK is proportional to the matrix element of O+= O1+ O2. Because overlap fermions are chiral, one can extract
BK from the matrix element of the operator between zero momentum states. All the operators to be studied have
only “figure-eight” topology matrix elements, where each field in the operator contracts against a field in the source
or sink interpolating field. There are no penguin graphs (where fields in the operator contract against each other) as
long as one works in the degenerate-mass limit.
curves are two different sources on each of two different lattices, and we are converging to a fractional accuracy of the squared
residue of 10−14for this test case.
Number of conjugate gradient steps in the calculation of quark propagators, as a function of quark mass. The four
Broadly speaking, the matrix element of a four-fermion operator is computed by placing interpolating fields for a
meson at two widely-separated locations on the lattice and contracting field variables between these sources and the
operator, to construct an un-amputated correlator containing the operator. There are two commonly-used strategies
for doing this: One possibility is to build all the quark propagators “at the operator” at one location on the lattice,
and to join up pairs of propagators to make the mesons. The second method is to construct propagators from two
well-separated sources and bring them together at the operator. The advantages of the first method are that one needs
half as many propagators per lattice, and it is possible to project the whole calculation into a particular momentum
eigenstate (by appropriately summing over locations of the meson interpolating fields). The major disadvantage of this
method is that one is only measuring the matrix element of the operator at one location per propagator construction.
With the second method one can average the location of the operator over all spatial and many temporal locations, a
considerable gain in statistics. A disadvantage of the second method is that unless the source generates a hadronic state
which is a momentum eigenstate the correlator will involve a mixture of the eigenstates. The dominant contribution
to the correlator is from ? p = 0, but there will be a contamination from higher momentum states. I have chosen
the second method, and will discuss below how I dealt with higher-momentum modes. I placed the two source time
slices Nt/2−2 temporal sites apart; with the toroidal geometry there are two temporal regions where the operator is
“between” the sources. I combine all two-point function data sets from the two sources in the fits.
The correlator of two interpolating fields located at x,t = (0,0) and (0,T) with an operator S(x,t) summed over
x is then
Inserting complete sets of relativistically normalized momentum eigenstates, and assuming ?h(? p1)|S(? p)|h(? p2)? is pro-