[Paper Review] $K_L$ - $K_S$ mass difference computed with a 171 MeV pion mass

Ziyuan Bai|arXiv (Cornell University)|Nov 12, 2014

Particle physics theoretical and experimental studies4 references3 citations

TL;DR

This lattice QCD study computes the $K_L$-$K_S$ mass difference ($\Delta M_K$) at a pion mass of 171 MeV using a $32^3\times64\times32$ domain wall fermion lattice with Iwasaki+DSDR gauge action and a coarse 1.37 GeV inverse lattice spacing. It focuses on the two-pion intermediate state contribution and finite volume corrections, finding the two-pion contribution to be small ($-0.074(63)\times10^{-12}$ MeV) and its finite volume correction negligible (0.019(19)\times10^{-12}$ MeV), indicating controllable systematic effects despite unphysical kinematics and discretization errors.

ABSTRACT

In this work, we used a $32^3 imes 64 imes 32$, 2+1 flavor domain wall lattice with Iwasaki+DSDR gauge action. The pion mass is 171 MeV and the kaon mass is 492 MeV. We implement the Glashow-Iliopoulos-Maiani (GIM) cancellation using charm quark masses of 750 MeV and 592 MeV. This is an intermediate calculation, in that we are using both a coarse lattice spacing (1/a = 1.37GeV) so we expect significant discretization error coming from charm quark mass and we are also using unphysical kinematics for the pion. The main purpose of this calculation is to study the contribution from the two-pion intermediate state when the energy of a two-pion state is lower than that of the kaon, as well as the corresponding finite volume correction to the $ΔM_K$.

Motivation & Objective

• To investigate the contribution of the two-pion intermediate state to the $K_L$-$K_S$ mass difference ($\Delta M_K$) in lattice QCD with unphysical pion mass and coarse lattice spacing.
• To assess finite volume corrections to $\Delta M_K$ arising from two-pion intermediate states, particularly for isospin-0 and isospin-2 channels.
• To evaluate the impact of discretization errors and Wilson coefficient uncertainties on the final $\Delta M_K$ result, especially given the use of a 750 MeV charm quark mass and unphysical kinematics.
• To test the feasibility of subtracting exponentially growing contributions from low-energy intermediate states (e.g., two-pion, $\eta$) in the integrated correlator for $\Delta M_K$.
• To provide a foundation for future calculations at physical pion mass by validating methods for handling two-pion intermediate states and finite volume effects.

Proposed method

• Uses a $32^3\times64\times32$ 2+1 flavor domain wall fermion lattice with Iwasaki+DSDR gauge action and $1/a = 1.37$ GeV, simulating at $m_\pi = 171$ MeV and $m_K = 492$ MeV.
• Implements Glashow-Iliopoulos-Maiani (GIM) cancellation via quenched charm quarks with masses of 750 MeV and 592 MeV to suppress unphysical contributions.
• Calculates the integrated correlator for $K^0$-$\bar{K}^0$ mixing using time-ordered products of $\Delta S=1$ weak Hamiltonians, extracting $\Delta M_K$ from the linear $T$-dependence.
• Subtracts exponentially growing contributions from intermediate states lighter than the kaon—specifically single pion, two-pion (I=0, I=2), and $\eta$ states—using direct matrix element subtraction.
• Applies finite volume corrections using the relation between infinite and finite volume spectral functions, with explicit calculation of $\cot h \cdot dh/dE$ terms for the $I=0$ two-pion state.
• Uses low-mode deflation with 580 Lanczos eigenvectors and Mobius fermions with $b+c=2.667$, $L_s=12$, to improve inversion efficiency and reduce residual mass.

Experimental results

Research questions

• RQ1 What is the magnitude and sign of the two-pion intermediate state contribution to $\Delta M_K$ at $m_\pi = 171$ MeV?
• RQ2 How significant are finite volume corrections to $\Delta M_K$ for the $I=0$ two-pion intermediate state in this unphysical kinematics setup?
• RQ3 To what extent do discretization errors and Wilson coefficient uncertainties affect the final $\Delta M_K$ result in this intermediate calculation?
• RQ4 Can the exponentially growing contributions from low-lying intermediate states (e.g., two-pion) be reliably subtracted in the integrated correlator to extract $\Delta M_K$?
• RQ5 How do different charm quark masses affect the GIM cancellation and the resulting $\Delta M_K$ value?

Key findings

• The two-pion intermediate state with isospin $I=0$ contributes $-0.074(63)\times10^{-12}$ MeV to $\Delta M_K$, consistent with the $\Delta I=1/2$ rule suppressing $I=0$ contributions.
• The finite volume correction to $\Delta M_K$ from the $I=0$ two-pion state is $0.019(19)\times10^{-12}$ MeV, indicating it is negligible at this level of precision.
• The $I=2$ two-pion contribution is even smaller, at $-6.70(7)\times10^{-4}$ MeV, further confirming suppression due to the $\Delta I=1/2$ rule.
• The largest systematic error arises from lattice discretization effects, estimated at $\sim30\%$ due to $(m_c a)^2$ corrections, with an additional $\sim10\%$ error from Wilson coefficient matching.
• Smaller charm quark masses (592 MeV) lead to smaller $\Delta M_K$ values due to enhanced GIM cancellation, consistent with theoretical expectations.
• Despite unphysical kinematics and coarse lattice spacing, the method of subtracting low-energy intermediate state contributions proves feasible and yields controllable finite volume effects.