Comparison of the mixed-fermion-action Effects using different fermion and gauge actions with 2+1 and 2+1+1 flavors

This paper calculates the leading-order mixed-action chiral perturbation theory constant $\Delta_{\rm mix}$ using $2+1+1$-flavor lattice ensembles, revealing that the fermion action dominates the results while the gauge action has a measurable secondary effect and charm quark loops are negligible.

The Gist

Imagine you are trying to build a perfect model of the universe's most fundamental building blocks: the protons and neutrons that make up everything around us. To do this, physicists use a super-computer simulation called Lattice QCD. Think of this simulation as a giant, 3D grid (like a chessboard, but in four dimensions) where they try to recreate the "strong force" that holds atoms together.

However, there's a catch. The math is so complex that you can't simulate every single detail perfectly without the computer exploding. So, physicists have to make shortcuts, or "discretizations," to make the problem solvable.

This paper is about a specific strategy called the "Mixed-Action" approach, and the authors are asking: "Which combination of shortcuts gives us the most accurate result with the least amount of wasted effort?"

Here is a breakdown of their findings using simple analogies:

1. The Problem: The "Sea" vs. The "Valence"

In these simulations, there are two types of particles:

• The Sea Quarks: These are the background particles that pop in and out of existence everywhere in the grid. They are computationally expensive to simulate, so physicists use a "cheap" and fast method to generate them. This is the "Sea."
• The Valence Quarks: These are the specific particles you are actually studying (like the ones inside a proton). You want to measure them with extreme precision, so you use a "high-end," expensive, and mathematically perfect method. This is the "Valence."

The Conflict: When you mix a "cheap" method for the background (Sea) with a "perfect" method for the target (Valence), they don't speak the same language. This mismatch creates "noise" or "artifacts" in your data. The paper focuses on measuring this noise, which they call $\Delta_{mix}$.

2. The Experiment: Testing Different "Recipes"

The authors, a collaboration called CLQCD, decided to test different "recipes" to see which one produces the least noise. They treated the simulation like a cooking experiment:

• The Base (The Sea): They used a very modern, high-quality "sea" recipe called HISQ. It's like using fresh, high-quality ingredients that are still easy to cook with.
• The Variable (The Gauge Action): This is the "pan" or the "stove" they use. They tested different types of pans (different mathematical rules for the grid) to see if the pan itself changed the taste of the food.
• The Flavor (The Valence): They tested different ways to measure the final dish (Overlap, Clover, etc.).

They also tested two different "kitchens":

1. 2+1 Flavors: Simulating up, down, and strange quarks (the standard kitchen).
2. 2+1+1 Flavors: Adding a Charm quark (a heavier, rarer ingredient) to see if it changes the flavor of the whole dish.

3. The Big Discoveries

A. The "Sea" is the Star of the Show

The most important finding is that the type of "Sea" you use matters the most.

• Analogy: Imagine you are baking a cake. If you use low-quality flour (a "Sea" that breaks symmetry), no matter how fancy your oven is, the cake will be lumpy. But if you use high-quality flour (a "Sea" that respects chiral symmetry, like HISQ), the cake turns out smooth and perfect, even if your oven is just okay.
• Result: When they used the high-quality HISQ "Sea," the noise ($\Delta_{mix}$) dropped dramatically. It turned out that the "Sea" being mathematically "clean" was the single biggest factor in reducing errors.

B. The "Pan" (Gauge Action) Matters, But Less

They tested different "pans" (gauge actions).

• Analogy: Does it matter if you bake the cake in a glass dish or a metal pan? Yes, slightly. The metal pan (Tadpole-improved Symanzik) gave slightly different results than the glass dish (Iwasaki), but the difference was small compared to the difference between using good flour vs. bad flour.
• Result: The gauge action has a "secondary" effect. It tweaks the result, but it doesn't ruin it if the "Sea" is good.

C. The "Charm" Ingredient is Negligible

They added the heavy "Charm" quark to the mix (2+1+1 vs. 2+1).

• Analogy: They wondered if adding a pinch of saffron (the Charm quark) to the cake batter would change the texture.
• Result: Within their current measurement precision, it didn't matter. The Charm quark loops were too small to make a noticeable difference in the noise levels. You can safely ignore them for this specific calculation to save time.

4. The Scaling Law: Why It Gets Better

The paper also looked at how the noise changes as they make the grid finer (smaller squares on the chessboard).

• The Finding: When using the right "Sea" (HISQ) and the right "Valence" (Overlap or HYP-smeared Clover), the noise disappears incredibly fast as the grid gets finer. It follows a $O(a^4)$ scaling.
• Analogy: Imagine trying to smooth out a bumpy road.
  • With old methods, every time you made the road surface twice as smooth, the bumps only got half as big (linear improvement).
  • With their new method, every time they made the surface twice as smooth, the bumps got 16 times smaller (because $2^4 = 16$). This is a massive efficiency gain.

5. The Conclusion: The Best Strategy

The authors conclude that the most cost-effective way to simulate the strong force with high precision is:

1. Use the HISQ "Sea" (The high-quality flour).
2. Use a "Valence" action that respects symmetry (The precise measuring cup).
3. Don't worry too much about the "Gauge Action" (the pan) or the "Charm quark" (the saffron) for this specific goal; they are minor factors.

In a nutshell: This paper tells us that if you want to simulate the universe's building blocks accurately, focus your budget on getting the background "Sea" right. Once you have a clean, symmetric background, the rest of the simulation becomes much easier, cheaper, and more accurate. They have provided a "cheat sheet" for other physicists to build better, faster simulations in the future.

Technical Summary

1. Problem Statement

Lattice Quantum Chromodynamics (QCD) relies on the "mixed-action" approach to balance computational cost and theoretical rigor. This involves generating gauge ensembles with a computationally efficient "sea" fermion action (e.g., HISQ) and calculating observables using a theoretically cleaner but expensive "valence" fermion action (e.g., Overlap or Clover).

The primary challenge is the mixed-action effect, an artifact arising from the mismatch between sea and valence actions. This effect is quantified by the leading-order low-energy constant $\Delta_{\text{mix}}$ in mixed-action partially-quenched chiral perturbation theory (MAPQ$\chi$PT).

Previous studies (e.g., Ref. [19]) established that $\Delta_{\text{mix}}$ scales as $O(a^4)$ when chiral symmetry is preserved in the sea action, but remains large ($O(a^2)$) when chiral symmetry is broken (e.g., Clover sea). However, these comparisons were confounded by two factors:

1. Gauge Action Differences: Previous comparisons used different gauge actions (e.g., Iwasaki vs. Tadpole-improved Symanzik) for different sea fermions.
2. Flavor Differences: Comparisons often involved different numbers of dynamical quark flavors ($N_f = 2+1$ vs. $N_f = 2+1+1$).

The paper aims to isolate and quantify the specific impacts of the gauge action and the number of dynamical flavors (specifically the charm quark) on $\Delta_{\text{mix}}$, while controlling for the fermion action.

2. Methodology

The authors performed a systematic study using 2+1+1 flavor and 2+1 flavor gauge ensembles generated with HISQ (Highly Improved Staggered Quark) sea fermions.

• Gauge Actions: They utilized the Tadpole-improved Symanzik (Stad) gauge action for the primary 2+1+1 ensembles to match previous Clover studies. They also generated 2+1 ensembles using Tree-level Symanzik (S(0)), Stad, and Iwasaki gauge actions to test gauge dependence.
• Lattice Spacings: Simulations were conducted at four lattice spacings ($a \in [0.048, 0.111]$ fm) for the 2+1+1 case and at a fixed spacing ($a \approx 0.11$ fm) for the 2+1 cases with varying gauge actions.
• Valence Actions: To calculate $\Delta_{\text{mix}}$, they employed three distinct valence fermion actions:
  1. SC: Clover fermion with 1-step STOUT smearing.
  2. HC: Clover fermion with 1-step HYP smearing.
  3. OV: Overlap fermion with 1-step HYP smearing.
• Calculation of $\Delta_{\text{mix}}$:
  • They computed pseudoscalar meson masses for sea-sea ($m_{\pi,ss}$), valence-valence ($m_{\pi,vv}$), and valence-sea ($m_{\pi,vs}$) channels.
  • $\Delta_{\text{mix}}$ was extracted using the formula:
    $$\Delta_{\text{mix}} \equiv \frac{m^2_{\pi,vs} - m^2_{\pi,vv} + m^2_{\pi,ss}}{2} \bigg|_{m_{\pi,vv}=m_{\pi,ss}}$$
  • For the HISQ sea, they extended the Dirac indices of the propagator to handle the mixed-action formalism correctly.
• Autocorrelation Control: To ensure statistical independence, they employed a molecular dynamics time of $\tau = 2$ per trajectory, significantly suppressing autocorrelations compared to previous studies.

3. Key Contributions

1. Isolation of Variables: This is the first controlled study to separate the effects of the gauge action and the charm sea quark from the fermion action effects on $\Delta_{\text{mix}}$.
2. New High-Precision Ensembles: Generated a comprehensive set of 2+1+1 HISQ ensembles with the Tadpole-improved Symanzik gauge action, allowing for a direct comparison with previous Clover-sea results that used the same gauge action.
3. Validation of Scaling Laws: Confirmed the $O(a^4)$ scaling of $\Delta_{\text{mix}}$ for chiral-symmetric sea actions (HISQ) across a wide range of lattice spacings, regardless of the specific gauge action used.

4. Key Results

• Dominance of Fermion Action: The study confirms that the fermion action is the dominant factor determining the magnitude of $\Delta_{\text{mix}}$.
  • When the sea action preserves chiral symmetry (HISQ), $\Delta_{\text{mix}}$ scales as $O(a^4)$, even when using the Tadpole-improved Symanzik gauge action (which previously yielded large $\Delta_{\text{mix}}$ with Clover sea).
  • When using Clover valence fermions on HISQ sea, the HYP-smeared Clover (HC) yields smaller $\Delta_{\text{mix}}$ than the STOUT-smeared Clover (SC).
• Secondary Effect of Gauge Action: The gauge action has a measurable but secondary effect.
  • Increasing the absolute value of the rectangle loop coefficient $|C_R|$ (moving from Tree-level Symanzik to Iwasaki) leads to a reduction in $\Delta_{\text{mix}}$.
  • Specifically, $\Delta_{\text{mix}}$ with the Iwasaki action is approximately 61% of that with the Tree-level Symanzik action for the same fermion setup.
• Negligible Charm Quark Effect: Comparing 2+1 and 2+1+1 flavor ensembles (both with HISQ sea and Tadpole-improved Symanzik gauge) revealed that the contribution of charm quark loops to $\Delta_{\text{mix}}$ is negligible within current statistical uncertainties.
• Comparison with Literature:
  • The results for HISQ sea + Iwasaki gauge are the most cost-effective combination found so far for suppressing $\Delta_{\text{mix}}$.
  • The Borici-Creutz valence fermion shows promise as a candidate for mixed-action setups, though higher precision is needed.

5. Significance

• Optimization of Lattice QCD Strategies: The findings provide a clear roadmap for future lattice QCD calculations. Using HISQ sea fermions combined with the Iwasaki gauge action offers the best suppression of mixed-action artifacts while maintaining computational efficiency.
• Systematic Error Control: The study demonstrates that mixed-action setups can offer superior control over systematic uncertainties (specifically continuum extrapolation) compared to unitary valence-sea setups, provided the sea action preserves chiral symmetry.
• Resolution of Previous Ambiguities: By controlling for gauge actions and flavor numbers, the paper definitively attributes the large $\Delta_{\text{mix}}$ observed in earlier Clover-sea studies to the breaking of chiral symmetry in the sea action, rather than the specific gauge action used.
• Future Directions: The paper suggests that further investigation into Borici-Creutz fermions and variations of staggered fermion smearing (e.g., stout smearing with different steps) could lead to even further suppression of mixed-action effects.

In conclusion, this work establishes that the choice of a chiral-symmetric sea fermion action is the critical lever for minimizing mixed-action artifacts, while the gauge action and dynamical charm quarks play secondary or negligible roles, respectively. This allows for more precise and cost-effective calculations of strong interaction physics.