, Decays with a Nearly Physical Pion Mass
The decay amplitude is calculated on RBC/UKQCD , dynamical lattices with flavors of domain wall fermions using the DSDR and Iwasaki gauge action. The calculation is performed with a single pion mass ( MeV, partially quenched) and kaon mass ( MeV) which are nearly physical, and with nearly energy conserving kinematics. Antiperiodic boundary conditions in two spatial directions are used to give the two pions non-zero ground state momentum. Results for time separations of 20, 24, 28 and 32 between the kaon and two-pion sources are computed and an error weighted average is performed to reduce the error. We find prelimenary results for Re GeV and Im GeV.
, Decays with a Nearly Physical Pion Mass
Matthew Lightman††thanks: Speaker.
Department of Physics, Columbia University, New York, NY 10027 USA
Department of Physics, Washington University, St. Louis, MO 63130 USA
Precise lattice calculations of decays will provide quantitative insight into the origin of the rule and direct CP violation in kaon decays. Previous calculations have relied on the quenched approximation or have attempted to use chiral perturbation theory to extrapolate from heavy quark masses down to physical masses, or both [1, 2, 3, 4, 5, 6, 7]. The calculation presented here avoids both these sources of error by computing the amplitude directly using dynamical lattices with 2+1 flavours of domain wall fermions (DWF) at near physical pion mass. We use RBC/UKQCD , lattices which use the Dislocation Suppressing Determinant Ratio (DSDR) plus Iwasaki gauge action with inverse lattice spacing GeV () and domain wall height . We use ensembles generated with , , corresponding to a unitary pion mass of MeV.
2 Four-Quark Operators and The Effective Hamiltonian
The weak interactions and the effects of heavier quarks can be included in the lattice QCD simulation by evaluating matrix elements of an effective Hamiltonian [8, 9]. In particular the conventions of  are used. We calculate matrix elements of four-quark operators between and states. In this paper the amplitude is calculated, which requires the evaluation of matrix elements of three operators. These operators are classified by how they transform under : , and . Progress in calculating is described in .
3 Boundary Conditions
We wish to simulate the decay at physical kinematics, which requires the final state pions to have a non-zero momentum. This is achieved by imposing antiperiodic boundary conditions on the quark fields in one or more spatial directions. The allowed momenta of the quark are then given by , where is the spatial extent of the lattice.
We relate the physical matrix
element to the unphysical matrix element
using the Wigner-Eckart theorem. This simplifies the operators and allows us to use periodic boundary conditions on the up- and strange-quarks while using antiperiodic boundary conditions only on the down-quark, thus giving the two pions momentum while the kaon remains at rest.
If antiperiodic boundary conditions are imposed on the d-quark in only the direction with periodic boundary conditions in the and directions then we can have a two-pion ground state in which one pion has momentum and the other pion has momentum . The antiperiodic boundary conditions allow us to extract non-zero momentum pions without the need to fit to an excited state, which would have been necessary had we imposed periodic boundary conditions on all the quark fields. In principle we can impose antiperiodic boundary conditions on the d-quark in one, two, or all three spatial directions corresponding to individual ground state pion momenta of and respectively.
4 Details of the Calculation
The calculation was carried out on 62 configurations of dynamical lattices using DSDR+Iwasaki gauge action and domain wall fermions with , generated on BG/P machines at Argonne National Laboratory. Further details of the ensemble generation are given in . The inverse lattice spacing is , the physical volume is and we set the light and strange valence quark masses to and respectively. This corresponds to a pion mass of and a kaon mass of .
We combined propagators with periodic and antiperodic boundary conditions in the time direction in order to double the effective time extent of the lattice. The meson correlation functions contained propagators which were computed with a source at (corresponding to (P+A) bc) and (corresponding to (P-A) bc). We also generated strange-quark propagators with sources at 20, 24, 28, 32, 36, 40 and 44 in order to calculate correlators with kaon sources at these times, while the two-pion sources remained at either or . Thus we could achieve time separations between the kaon and two pions of 20, 24, 28 and 32 in two different ways which doubled the statistics. These separations were chosen so that the signals from the kaon and two pions did not decay to noise before reaching the four-quark operator .
For the kaon and pions with zero momentum we use propagators with Coulomb gauge-fixed wall sources. For the two pions with non-zero momentum we use the same type of propagators for the u quark but used propagators with antiperodic spatial boundary conditions for the d-quark with Coulomb gauge-fixed momentum wall sources of the “cosine” type
We use the same cosine source for each d-quark, which introduces a cross term that couples to two-pion states with non-zero total momentum. For example, if we consider giving momentum in only the direction the product of the sources of the two d-quarks is
We require the two pions to have individual momentum and , but the first and last terms of equation (4) couple to two-pion states with total momentum and respectively. We eliminate the unwanted terms in the two-pion correlator by using pure exponential momentum sinks which constrain the final state to have zero total momentum. In the correlator, the zero momentum kaon has a similar effect on the cosine sources of the two-pions. Had we used the more conventional momentum source
we would have needed to perform two separate d-quark inversions with momentum for one and for the other. The cosine source eliminates one of these inversions. In practice we only compute d-quark propagators with antiperiodic boundary conditions in or spatial directions, corresponding to pions with ground state momenta and . This choice is motivated by the expectation that for our choice of quark masses, will correspond to on-shell kinematics.
5 Analysis and Results
We extract the matrix element by fitting a constant to the left hand side of (5)
is the correlator with a kaon source at , labels the four-quark operator which is inserted at time , and and are calculated from the kaon and two-pion correlators respectively, whose sources are at . The left hand side of equation (5) is plotted in figure 1 for each of the three operators. The figure demonstrates that sufficiently far from the kaon and two-pion sources we are justified in fitting to a constant. The fit results for are indicated on the plot.
where the quantity in square brackets (denoted by LL in table 2) contains the effects of the Lellouch-Lüscher factor beyond the free field normalization. is the energy of the two-pion state, is the s-wave phase shift, is a factor counting the free-field degenerate states, is a dimensionless quantity related to the individual pion momentum via and is a kinematic function defined in . Once is known, can be calculated using the Lüscher quantisation condition .
is found by fitting the quotient of correlators to extract . We then get the two-pion energy by calculating , where in the case of , while for , is found from a 2 parameter fit to the pion correlation function which also has . This method is preferred to directly extracting from the two-pion correlator because the quotient cancels some common fluctuations in the numerator and denominator and reduces the error.
The pion momentum in the two-pion state is determined from the two-pion energy using the dispersion relation . It differs from due to interactions between the two pions. Results for , , and are presented in table 1. can be calculated analytically so the only unknown in equation (5) is . The results for the phase shift can be plotted against and compared with experiment [15, 16]. This is done in figure 2(a) and we see good agreement with experiment. For we make the approximation that is linear with in order to calculate (see figure 2(b)). For we use the phenomenological curve  shown in figure 2(a) to calculate the derivative of the phase shift at the corresponding value of . The derivative of the phase shift is found to be a small factor in comparison with . Results for and are shown in table 2.
The amplitudes are related to the physical decay amplitude via
where are the Wilson Coefficients and are the renormalization constants, calculated using non-perturbative renormalization (NPR). The factor is needed to convert from the unphysical amplitudes back to the physical amplitudes. At present only has been calculated; for the and operators (which mix under renormalization) we make the approximation . A full calculation of for the and operators on the lattice is currently under way . Comparing for with the kaon mass MeV we see that the decay is nearly energy conserving, so we use the results from to compute . Results for Re and Im for the four different kaon source times are shown in table 3. Our final result for Re and Im is an error weighted average (EWA) over the four kaon source times.
|Re(units of GeV)||Im(units of GeV)|
6 Systematic Error
The major sources of systematic error in the determination of are scaling violations, finite volume effects, partial quenching, uncertainty in , and the fact that the masses and momentum are slightly different from their physical values. Furthermore, the approximation made for the renormalization constants for the (8,8) operators introduces a large systematic error into which we will estimate as 20%. is very sensitive to scaling violations because it is proportional to . We estimate this systematic by calculating with a lattice spacing determined from , , and respectively, and find a fluctuation of 8.5% among the three values. For finite volume effects we estimate 7% for the systematic error using finite volume chiral perturbation theory for the matrix elements [20, 21]. One expects that for decays partial quenching will introduce small errors and in  the use of partial quenching has been shown to affect by about 2%. A value of that is rather larger in magnitude is obtained just by putting a straight line through the two-pion phase shift data points from this calculation in figure 2(a); this value yields a result for that differs by 2% which we use as our conservative estimate of this systematic. Finally, a calculation on quenched lattices was done for a variety of meson masses and two-pion energies , and shows that the deviations of these parameters from their physical values in the present calculation causes an 1.2% difference in . Adding all errors in quadrature results in a preliminary estimate of 11% for the systematic error in and 23% for the systematic error in .
We have presented preliminary results for the decay amplitude on lattices with flavours of DWF and the Iwasaki-DSDR gauge action. We find MeV, MeV and MeV. The main contribution to Re is expected to be from the operator, and our result GeV can be compared to the experimental result of 1.5 GeV and is found to agree within error. This is the first time a calculation of this type has been achieved. Im is dominated by the operators in the representation, so we expect there to be a large systematic error on Im() due to the approximation made for . This is reflected in our final answer Im GeV. This source of systematic error will be eliminated once the NPR calculation has been completed.
We thank all of our colleagues in the RBC and UKQCD collaborations for helpful discussions and the development and support of the QCDOC hardware and software infrastructure which was essential to this work. In addition we acknowledge Columbia University, RIKEN, BNL, ANL, and the U.S. DOE for providing the facilities on which this work was performed. This research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. DOE under contract DE-AC02-06CH11357. This work was supported in part by U.S. DOE grant number DE-FG02-92ER40699. Elaine Goode is supported by an STFC studentship and grant ST/G000557/1 and by EU contract MRTN-CT-2006-03542 (Flavianet).
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