[Paper Review] Beyond the Standard Model B-parameters with improved staggered fermions in $N_f=2+1$ QCD

T. Bae, Yong-Chull Jang+11 more|arXiv (Cornell University)|Oct 28, 2013

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This study computes B-parameters for five ΔS=2 four-fermion operators in $N_f=2+1$ QCD using improved staggered fermions on MILC asqtad lattices, employing one-loop perturbative matching and chiral extrapolation via 'golden combinations' to suppress logarithmic divergences. The results show 5–6% total uncertainty, with $B_4$ and $B_5$ disagreeing by 4–5σ with prior domain-wall fermion calculations, suggesting potential underestimation of matching or continuum extrapolation errors in existing methods.

ABSTRACT

We calculate the kaon mixing B-parameters for operators arising generically in theories of physics beyond the standard model. We use HYP-smeared improved staggered fermions on the $N_f = 2+1$ MILC asqtad lattices. Operator matching is done perturbatively at one-loop order. Chiral extrapolations are done using "golden combinations" in which one-loop chiral logarithms are absent. For the combined sea-quark mass and continuum extrapolation, we use three lattice spacings: $a \approx 0.045, 0.06$ and $0.09 ext{fm}$. Our results have a total error of 5-6%, which is dominated by the systematic error from matching and continuum extrapolation. For two of the BSM $B$-parameters, we agree with results obtained using domain-wall and twisted-mass dynamical fermions, but we disagree by $(4-5)σ$ for the other two.

Motivation & Objective

• To compute the matrix elements of five ΔS=2 four-fermion operators relevant for physics beyond the Standard Model (BSM) in $N_f=2+1$ QCD.
• To reduce systematic errors in B-parameters by using 'golden combinations' that eliminate one-loop chiral logarithms in chiral extrapolation.
• To compare results with prior calculations using domain-wall and twisted-mass fermions, identifying discrepancies in $B_4$ and $B_5$.
• To assess the reliability of one-loop perturbative matching and continuum extrapolation by quantifying systematic uncertainties.

Proposed method

• Uses HYP-smeared improved staggered fermions for valence quarks on $N_f=2+1$ MILC asqtad lattices with three lattice spacings: $a \approx 0.045$, 0.06, and 0.09 fm.
• Performs one-loop perturbative matching of lattice operators to continuum operators in the scheme of Ref. [4], including an additional matching factor to reconcile with other schemes.
• Employs SU(2) staggered chiral perturbation theory (SChPT) for chiral extrapolation, using the lightest four valence $u/d$ quark masses and heaviest three $s$ quark masses to maintain $m_x \ll m_y \sim m_s$.
• Uses U(1) noise sources to create kaons with fixed separation and isolates the operator insertion to avoid excited-state contamination, fitting to constant plateaus in correlators.
• Performs continuum extrapolation using three lattice spacings and estimates systematic errors from matching and extrapolation, with the larger of $\Delta f_{12}$ and $\alpha_s^2(\text{U1})=4.4\%$ dominating the error budget.
• Renames results to the SUSY basis for comparison, using the relation $B_3^{\text{SUSY}} = (5B_2 - 3B_3)/2$.

Experimental results

Research questions

• RQ1 How do the B-parameters for BSM ΔS=2 operators compare across different fermion formulations, particularly between improved staggered and domain-wall fermions?
• RQ2 To what extent do systematic errors from one-loop matching and continuum extrapolation affect the precision of B-parameters in lattice QCD?
• RQ3 Why do the results for $B_4$ and $B_5$ disagree by 4–5σ with prior domain-wall fermion calculations, despite agreement in $B_K$ and $B_2$?
• RQ4 Can 'golden combinations' in chiral extrapolation effectively suppress chiral logarithmic divergences in staggered fermion calculations?
• RQ5 What is the dominant source of systematic uncertainty in the computation of BSM B-parameters, and how can it be reduced in future calculations?

Key findings

• The total uncertainty in the B-parameters is 5–6%, dominated by systematic errors from matching and continuum extrapolation.
• The $B_K$ result is consistent with prior calculations, validating the method for the SM operator.
• The $B_2$ and $B_3$ results agree with the domain-wall fermion calculation within errors.
• The $B_4$ and $B_5$ results disagree with the domain-wall fermion results by 4–5σ, with our values being 1.5 and 1.8 times larger, respectively.
• The matching/continuum extrapolation error is estimated at 4.4–5.7%, larger than statistical and finite-volume errors, and is the dominant source of uncertainty.
• The discrepancy in $B_4$ and $B_5$ suggests possible underestimation of truncation errors in one-loop matching, particularly at the two-loop level.