Asqtad-Improved Staggered Fermions

• Asqtad-improved staggered fermions are a lattice QCD discretization that uses Fat7 smearing, Lepage correction, Naik term, and tadpole improvement to achieve full tree-level O(a²) Symanzik improvement.
• The method significantly suppresses taste-symmetry breaking by reducing high-momentum gluon exchanges and minimizing pion mass splittings, with improvements verified across various lattice spacings.
• Widely used in dynamical QCD simulations, Asqtad actions enable precise determinations of hadronic observables and set benchmarks for further improvements like the HISQ action.

Asqtad-improved staggered fermions are a class of lattice fermion discretizations employed in large-scale lattice QCD simulations to achieve full tree-level $\mathcal O(a^2)$ Symanzik improvement and strong suppression of taste-exchange interactions. The Asqtad program combines the staggered formalism—retaining a single component $\chi(x)$ per site representing four continuum tastes per flavor—with a set of gauge-link improvements, including Fat7 smearing, Lepage double-staple corrections, the Naik three-link dispersion correction, and tadpole improvement. Used extensively by the MILC Collaboration, Asqtad actions push lattice artifacts and taste-symmetry breaking down to the few-percent level at moderate lattice spacings while preserving computational efficiency.

1. Formal Structure of the Asqtad Action

The Asqtad-improved staggered action is formulated as

$$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$

where the Dirac operator $D^\text{Asqtad}$ includes the following terms, all tadpole-improved:

• The one-link “thin” staggered derivative (naive term).
• Fat7 seven-link smeared derivative, suppressing high-momentum gluon exchange.
• Lepage five-link planar staple subtraction to remove $\mathcal O(a^2)$ taste breaking arising from Fat7 smearing.
• Naik three-link straight “hop” term, which corrects the fermion dispersion relation to $\mathcal O(a^2)$ (Freeman et al., 2012, Söldner, 2010).

Explicitly, in the one-component notation: $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$ with $\rho_N \approx -1/24$ and tadpole improvement factor $u_0$ applied to all link terms. $U_\mu^\text{fat7}$ and $\chi(x)$0 implement multi-link staple averages and planar double-staple subtractions, respectively, while $\chi(x)$1 is a straight product of three links.

The gauge action, consistently, is the one-loop tree-level Symanzik-improved plaquette plus rectangle (“Lüscher-Weisz”) form, tadpole-improved: $\chi(x)$2 with $\chi(x)$3, $\chi(x)$4, $\chi(x)$5 measured as the average plaquette.

2. Motivation and Design of Improvement Terms

Each Asqtad improvement is chosen to target dominant sources of taste-symmetry breaking and discretization error:

• Fat7 smearing reduces coupling to gluons with $\chi(x)$6 that induce taste mixing, leading to suppression of tree-level taste-exchange interactions.
• Lepage correction subtracts residual $\chi(x)$7 taste-breaking vertices that fat7 over-smears into the action at high momentum.
• Naik term cancels the tree-level $\chi(x)$8 error in the free-fermion dispersion, restoring the correct energy-momentum relation $\chi(x)$9.
• Tadpole improvement absorbs large perturbative corrections by dividing all links by $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$0, ensuring perturbative renormalization matching factors remain small and stable (Freeman et al., 2012, Söldner, 2010, Kronfeld et al., 2015, Gregory et al., 2011).

Collectively, this design ensures no tree-level $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$1 taste violations, achieves near-ideal dispersion, and pushes taste breaking down to $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$2 at one loop.

3. Numerical Implementation and Performance

Asqtad improvement is implemented in large dynamical QCD simulations, notably by the MILC collaboration:

• Ensembles: 2+1 flavor QCD with $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$3, and $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$4 fm, volumes from $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$5 up to $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$6.
• Quark masses: $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$7 ratios down to $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$8, with physical strange mass matched to experimental meson masses via chiral extrapolation.
• Action coefficients: $$S_F = a^4\sum_x \bar\chi(x)\Bigl[D^\text{Asqtad}(m)\Bigr]\chi(x)$$9, Lepage coefficient $D^\text{Asqtad}$0, fat7 staple weights fixed at tree-level, explicit tadpole improvement throughout (Freeman et al., 2012, Söldner, 2010, Kronfeld et al., 2015, Gregory et al., 2011).

Performance metrics:

• Largest non-Goldstone pion mass-squared splitting drops from $D^\text{Asqtad}$1 (at $D^\text{Asqtad}$2) to $D^\text{Asqtad}$3 ($D^\text{Asqtad}$4).
• Residual taste violations on the finest lattices are $D^\text{Asqtad}$5, a reduction by factors $D^\text{Asqtad}$6 from coarsest lattices.
• Light-quark discretization error in decay-constant ratios $D^\text{Asqtad}$7 for $D^\text{Asqtad}$8, $D^\text{Asqtad}$9, and $\mathcal O(a^2)$0 (Kronfeld et al., 2015).

4. Suppression of Taste-Symmetry Breaking

The central achievement of Asqtad improvement is the suppression of taste-symmetry violations. In naive staggered fermions, exchange of hard gluons produces sizable taste mixings, splitting the sixteen pion tastes into multiplets separated by $\mathcal O(a^2)$1 in mass squared. Asqtad smearing and operator design remove all such splitting at tree level, leaving only one-loop residuals. Empirically, the ratio of taste-splitting between $\mathcal O(a^2)$2 fm and $\mathcal O(a^2)$3 fm lattices is measured as $\mathcal O(a^2)$4, matching the expected $\mathcal O(a^2)$5 scaling (Gregory et al., 2011).

Table: Scaling of Taste-Splitting in Asqtad Ensembles

Lattice Spacing $\mathcal O(a^2)$6 (fm)	Max $\mathcal O(a^2)$7 ($\mathcal O(a^2)$8)	Taste Splitting Reduction
0.15	$\mathcal O(a^2)$9	Baseline
0.09	$\mathcal O(a^2)$0	$\mathcal O(a^2)$1 lower
0.045	$\mathcal O(a^2)$2	$\mathcal O(a^2)$3 lower

5. Perturbative and Nonperturbative Renormalization

Renormalization factors for bilinear operators in the Asqtad scheme have been computed both in one-loop perturbation theory and via nonperturbative RI-MOM methods:

• One-loop matching: Perturbative corrections for bilinear operators are reduced compared to naive staggered, with spread of matching coefficients in the scalar sector $\mathcal O(a^2)$4 (Wilson glue) and $\mathcal O(a^2)$5 (Symanzik glue), much smaller than $\mathcal O(a^2)$6 for naive sparse links (Kim et al., 2010).
• Nonperturbative results: NPR factors $\mathcal O(a^2)$7, $\mathcal O(a^2)$8, scalar channel $\mathcal O(a^2)$9, vector $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$0, tensor $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$1 on MILC coarse ensembles (Kim et al., 2013).
• Consistency: Vector and tensor channels show good agreement between NPR and PT (few percent); scalar channels may depart by up to $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$2–$$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$3 at $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$4, indicating the necessity of nonperturbative matching for precision work (Lytle et al., 2013).

6. Applications in Lattice QCD and Quantum Simulation

Asqtad ensembles have underpinned key lattice calculations:

• Hadron Spectrum: Precision work in $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$5- and $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$6-meson decay constants, chiral condensates, and flavor-singlet masses ($$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$7, $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$8) (Kronfeld et al., 2015, Gregory et al., 2011).
• Finite-Temperature QCD: Determination of crossover temperature $$D_\mu^\text{Asqtad}\chi(x) = \eta_\mu(x)\left[ \rho_1 U_\mu^\text{fat7}(x)\chi(x+\hat\mu) + \rho_3 U_\mu^\text{lep}(x)\chi(x+\hat\mu) + \rho_N U_\mu^\text{Naik}(x)\chi(x+3\hat\mu) -\text{(hermitian conj.)} \right]$$9 MeV (continuum-extrapolated), with chiral condensate, susceptibility, and Polyakov loop pointing to smooth crossover; taste breaking produces a $\rho_N \approx -1/24$0–$\rho_N \approx -1/24$1 MeV shift in $\rho_N \approx -1/24$2 relative to HISQ (Söldner, 2010).
• Nucleon Matrix Elements: The intrinsic strangeness $\rho_N \approx -1/24$3 and preliminary charm results; the hybrid three-point/Feynman–Hellman method leverages the per-timeslice condensate structure of Asqtad (Freeman et al., 2012).
• Quantum Simulation: The Asqtad-inspired Hamiltonian has been mapped to qubit circuits, demonstrating that vector-meson mass extrapolation is $\rho_N \approx -1/24$4–$\rho_N \approx -1/24$5 closer to the continuum for improved actions versus naive Kogut–Susskind, at the cost of increased gate counts scaling as $\rho_N \approx -1/24$6 in dimension (Gustafson et al., 2024).

7. Comparison with Other Improved Staggered Actions and Future Prospects

The HISQ (“highly improved staggered quark”) action extends Asqtad improvement by introducing additional reunitarized smearing and further corrections to cancel one-loop $\rho_N \approx -1/24$7 taste-exchange at tree level. HISQ yields taste splittings $\rho_N \approx -1/24$8 one-third of Asqtad at fixed $\rho_N \approx -1/24$9 and enables physical quark masses at coarser lattices. However, at the time of key calculations, essential quantities for improved-hybrid methods (per-timeslice condensates, full propagators) were not available for HISQ ensembles (Freeman et al., 2012).

A plausible implication is that future large-scale dynamical QCD projects seeking sub-percent accuracy on hadronic observables will migrate towards HISQ or related actions, while Asqtad remains a benchmark for systematic control of taste exchange and is well-matched for precision studies where link and operator improvement are critical.

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References:

(Freeman et al., 2012, Söldner, 2010, Kronfeld et al., 2015, Gregory et al., 2011, Kim et al., 2010, Kim et al., 2013, Lytle et al., 2013, Gustafson et al., 2024)