Non-perturbative renormalization of quark bilinear operators and B_K using domain wall fermions

TL;DR

This work computes non-perturbative renormalization factors for quark bilinears and the $B_K$ operator on 2+1 flavor domain-wall fermion lattices using the RI/MOM scheme, with perturbative matching to $\overline{MS}$. By exploiting the RI/MOM interface and four-loop running where applicable, the authors extract $Z_q$, $Z_m$, $Z_T$, and $Z_{B_K}$ with controlled systematic errors, and demonstrate that chiral-symmetry-breaking effects from exceptional momenta are mitigated by using non-exceptional kinematics. They provide a theoretical argument and numerical evidence that chirality-violating mixing with wrong-chirality operators is suppressed by $\mathcal{O}(m_{res}^2)$, validating a simplified $Z_{B_K}$ determination. The results yield $Z_m^{\overline{MS}}(2\mathrm{GeV})=1.656(\mathrm{stat})(\mathrm{sys})$, $Z_q^{\overline{MS}}(2\mathrm{GeV})=0.773(\mathrm{stat})(\mathrm{sys})$, $Z_T^{\overline{MS}}(2\mathrm{GeV})=0.795(\mathrm{stat})(\mathrm{sys})$, and $Z_{B_K}^{\overline{MS}}(2\mathrm{GeV})=0.928(\mathrm{stat})(\mathrm{sys})$, enabling precise quark-mass and $K^0$–$\bar{K}^0$ mixing phenomenology.

Abstract

We present a calculation of the renormalization coefficients of the quark bilinear operators and the K-Kbar mixing parameter B_K. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed non-perturbatively and then matched perturbatively to the MSbar scheme. The coefficients are calculated on the RBC/UKQCD 2+1 flavor dynamical lattice configurations. Specifically we use a 16^3 x 32 lattice volume, the Iwasaki gauge action at beta=2.13 and domain wall fermions with L_s=16.

Figures (33)

• Figure 1: The quantity $\frac{1}{12}\mathrm{Tr}\left(S_{\mathrm{latt}}^{-1}\right)$ plotted versus $\left(ap\right)^{2}$ for the unitary mass points $m_l = 0.01$, 0.02 and 0.03 and at the linearly extrapolated, chiral limit $m_l=-m_\mathrm{res}$.
• Figure 2: The ratio $\frac{\Lambda_{A}-\Lambda_{V}}{\left(\Lambda_{A}+\Lambda_{V}\right)/2}$ plotted as a function of momentum at the unitary mass points $m_\mathrm{val}=m_l$ and in the chiral limit evaluated by linear extrapolation in $m_l$. The 5-10% difference at low momentum decreases rapidly as the momentum increases. At the scale $\mu\simeq 2\hbox{GeV}$, or $\left(ap\right)^{2}\simeq 1.4$, the difference is about 1%, which contributes to the systematic error in $Z_{B_{K}}$.
• Figure 3: The difference $\Lambda_{A}-\Lambda_{V}$ computed using four different quenched DBW2 lattice ensembles. These ensembles have quite different lattice scales. In addition the values of $L_s$, the extent in the 5th dimension used in computing the DWF propagators, also varies significantly. This provides compelling evidence that the observed chiral symmetry breaking is not an explicit breaking from finite $L_{s}$, but rather represents the high energy tail of QCD dynamical chiral symmetry breaking which would vanish if we were able to perform the NPR calculation at high enough energy. The data shown come from Refs. Dawson:2002nrAoki:2005gaAoki:2006ib
• Figure 4: The $\chi^{2}/d.o.f$ which results from fitting the momentum dependence of the quantity $\frac{\Lambda_{A}-\Lambda_{V}}{\left(\Lambda_{A}+\Lambda_{V}\right)/2}$ (extrapolated to the chiral limit) to the form $p^{-n}$. We conclude that the best choice for $n$ lies between 2 and 3 and that it is unlikely that the term $\left\langle \bar{q}q\right\rangle ^{2}/p^{6}$ gives the dominant contribution to this chiral symmetry breaking.
• Figure 5: The division of a general vertex graph into subgraphs. If the four-legged, internal subgraph $\Gamma_2$ carries momenta $p \sim \Lambda_\mathrm{QCD}$ it can introduce low energy, $(8,8)$ chiral symmetry breaking into such an amplitude even in the limit that the momenta external to the entire diagram $\Gamma$, included in the outer dashed box, grow large. As discussed in the text, such a limit will be suppressed by $1/p^6$ if the external momenta are non-exceptional but by only $1/p^2$ for the exceptional case.