Impact of Dynamical Charm Quark and Mixed Action Effect on Light Hadron Masses and Decay Constants

This study demonstrates that including a dynamical charm quark and utilizing a mixed action setup yield light hadron properties consistent with standard 2+1 flavor calculations, with the mixed action's discretization errors potentially canceling out to improve continuum extrapolation convergence.

The Gist

Imagine the universe is built out of tiny, invisible LEGO bricks called quarks. These bricks snap together to form larger structures called hadrons (like protons and neutrons, which make up the atoms in your body).

For decades, physicists have been trying to build a perfect digital simulation of these LEGO structures to understand how they work. This paper is about a team of scientists (the CLQCD Collaboration) who decided to test a new, slightly "mixed" way of building their simulation to see if it makes the results more accurate.

Here is the story of their discovery, broken down into simple concepts:

1. The Problem: The "Heavy" Guest

In the Standard Model of physics, there are six types of quarks. Three are light (up, down, strange) and three are heavy (charm, bottom, top).

• The Light Quarks are like the main characters in a play; they are everywhere and form the bulk of the matter we see.
• The Heavy Quarks (like the Charm quark) are like a wealthy guest who rarely shows up. For a long time, physicists thought, "Since the Charm quark is so heavy and rare, we can just ignore it in our simulations of light matter to save computer time."

The Question: Does this heavy guest actually change the behavior of the light characters, even if they don't interact directly?

2. The Experiment: Two Different Kinds of LEGO Sets

To answer this, the team ran two massive computer simulations:

• Simulation A (The Pure Set): They simulated the universe with only the three light quarks (2+1 flavors). They used one specific type of digital "glue" (gauge action) and one specific type of "brick" (fermion action) for everything. This is the "standard" way.
• Simulation B (The Mixed Set): They simulated the universe with the three light quarks PLUS the heavy Charm quark (2+1+1 flavors).
  • The Twist: To handle the heavy Charm quark, they used a very different, highly accurate type of digital brick (HISQ). But for the light quarks, they kept using their original, trusted brick type (Clover).
  • This is called a "Mixed Action" approach. It's like building a house where the foundation is made of high-tech carbon fiber, but the walls are made of traditional brick.

3. The Surprise: The "Glitch" That Fixed Itself

Usually, when you mix two different types of materials or methods in a simulation, you expect errors (glitches) to appear. You'd think the "mixed" simulation would be messier and less accurate than the "pure" one.

But here is the magic:
The scientists found that the "mixed" simulation (with the heavy Charm quark) actually had fewer errors than the pure one!

The Analogy:
Imagine you are trying to measure the length of a table with a ruler that is slightly bent.

• Scenario 1 (Pure): You use the bent ruler alone. Your measurement is off.
• Scenario 2 (Mixed): You use the bent ruler, but you also add a heavy weight (the Charm quark) to the table. Surprisingly, the weight pushes the table down just enough to straighten out the bend in the ruler.
• Result: The final measurement in Scenario 2 is actually more accurate than in Scenario 1.

The scientists discovered that the "mistakes" introduced by mixing the two different quark types accidentally cancelled out the "mistakes" inherent in their computer grid. It was a happy accident that made their math cleaner.

4. The Conclusion: The Heavy Guest Doesn't Change the Party

After running the numbers and correcting for all the digital "pixelation" (discretization errors), they compared the results:

• Do the light hadrons change? No. The mass of the proton and the way pions decay remained exactly the same, whether the heavy Charm quark was in the simulation or not.
• What does this mean? It confirms that the heavy Charm quark is truly "decoupled" from the light matter. It's like a VIP guest at a party who stays in the VIP lounge; they don't change how the people on the dance floor are dancing.

Why This Matters

1. Better Tools: They proved that this "Mixed Action" method is actually a superior way to do these calculations because it reduces errors automatically.
2. Validation: It gives physicists confidence that they can safely ignore the heavy quarks when studying the light stuff, saving massive amounts of computing power for other problems.
3. Precision: They provided the most precise measurements yet for the masses of quarks and how they interact, which helps us understand the fundamental forces holding our universe together.

In a nutshell: The scientists tried a weird, mixed-up recipe for simulating the universe, expecting it to taste bad. Instead, they found that the "weird ingredient" (the heavy quark) actually fixed the flavor of the dish, proving that the heavy quark doesn't mess with the light stuff, but helps us measure it better.

Technical Summary

Here is a detailed technical summary of the paper "Impact of Dynamical Charm Quark and Mixed Action Effect on Light Hadron Masses and Decay Constants" by the CLQCD Collaboration.

1. Problem Statement

The Standard Model includes six quark flavors. While the three light flavors (up, down, strange) are lighter than the QCD scale ($\Lambda_{QCD}$), the heavy flavors (charm, bottom, top) are often treated as decoupled in light-hadron physics. However, introducing a dynamical charm quark ($N_f = 2+1+1$) significantly alters the bare strong coupling constant and quark masses, potentially impacting light hadron properties.

A major challenge in Lattice QCD is the discretization error arising from the lattice spacing ($a$).

• Wilson-like fermions (e.g., Clover) suffer from additive mass renormalization and $O(a)$ errors, requiring careful tuning.
• Staggered fermions (e.g., HISQ) have simpler mass tuning but suffer from "taste-breaking" effects.
• Mixed-Action Approach: This study employs a mixed-action setup where gauge configurations are generated using dynamical HISQ fermions ($N_f=2+1+1$), while valence quarks (used to calculate observables) are simulated using Clover fermions.
• The Core Question: Does the inclusion of a dynamical charm sea quark alter light hadron physics? Furthermore, do the discretization errors inherent in the mixed-action setup (valence-sea mismatch) cancel out or compound with the errors of the unitary setup, and how does this affect continuum extrapolations?

2. Methodology

The collaboration performed high-precision Lattice QCD simulations comparing two setups:

1. Unitary Setup (CL@CL): $N_f = 2+1$ ensembles with Clover sea and valence fermions.
2. Mixed-Action Setup (CL@HI): $N_f = 2+1+1$ ensembles with HISQ sea and Clover valence fermions.

Key Simulation Parameters:

• Gauge Action: Tadpole-improved Symanzik gauge action.
• Valence Action: 1-step stout link smeared Clover fermion action.
• Ensembles:
  • CL@CL: 6 lattice spacings ($a \approx 0.038 - 0.105$ fm).
  • CL@HI: 4 lattice spacings ($a \approx 0.047 - 0.108$ fm).
  • Pion Masses: Ranging from physical ($m_\pi \approx 135$ MeV) to unphysical ($\sim 340$ MeV).
  • Volumes: Multiple spatial volumes to control finite-volume effects ($m_\pi L \gtrsim 3.5$).
• Renormalization: Non-perturbative renormalization using the RI/MOM and SMOM schemes, matched to the $\overline{MS}$ scheme at 2 GeV.
• Analysis:
  • Chiral Extrapolation: Used Partially Quenched Chiral Perturbation Theory (PQ$\chi$PT) to fit pion and kaon masses/decay constants, including terms for finite volume, unphysical strange mass, and lattice spacing ($O(a^2)$ and $O(a\alpha_s)$).
  • Mass Tuning: Critical masses and valence quark masses were tuned to reproduce physical pion, $\eta_s$, and $D_s$ masses.

3. Key Contributions

• Systematic Study of Mixed-Action Effects: This work provides one of the first systematic investigations of how the valence-sea mismatch (Clover on HISQ) affects light hadron observables compared to a unitary Clover setup.
• Dynamical Charm Impact: It isolates the effect of the dynamical charm sea by comparing $N_f=2+1+1$ (HISQ) results against $N_f=2+1$ (HISQ) and $N_f=2+1$ (Clover) results.
• Continuum Extrapolation Improvement: The study demonstrates that the mixed-action setup leads to better convergence in the continuum limit for certain observables, contrary to the expectation that mixed actions introduce larger systematic errors.
• Updated Low-Energy Constants (LECs): Provides updated predictions for quark masses, decay constants ($f_\pi, f_K$), and LECs ($\ell_3, \ell_4$) with improved control over systematic uncertainties due to the inclusion of three additional lattice spacings compared to previous CLQCD studies.

4. Key Results

A. Discretization Errors and Mixed-Action Cancellation

• Suppression of Errors: Surprisingly, the discretization errors for light hadron observables (specifically $f_\pi, f_K, m_\phi, m_\Omega$) are significantly smaller in the mixed-action (CL@HI) setup compared to the unitary Clover (CL@CL) setup.
• Cancellation Mechanism: The results suggest that the discretization errors introduced by the mixed-action setup (valence-sea mismatch) partially cancel the intrinsic discretization errors of the unitary Clover action. This leads to a flatter dependence on $a^2$ and more stable continuum extrapolations.
• $O(a^4)$ Terms: The leading-order mixed-action effects (scaling as $O(a^4)$) were found to be negligible within statistical uncertainties.

B. Dynamical Charm Quark Effects

• Light Hadrons: The inclusion of a dynamical charm sea ($N_f=2+1+1$) does not significantly alter the properties of light hadrons (masses and decay constants) within current statistical uncertainties. The results for $N_f=2+1$ and $N_f=2+1+1$ are consistent.
• Charmonium: For charmonium states ($m_{\eta_c}, m_{J/\psi}$), the dynamical charm sea and gauge action choice do have non-negligible effects on the coefficients of discretization errors, though the hyperfine splitting remains consistent across different setups.

C. Physical Observables

• Decay Constants:
  • $f_\pi = 127.9(1.0)$ MeV (MOM scheme, CL@HI).
  • $f_K = 155.1(1.1)$ MeV (MOM scheme, CL@HI).
  • The ratio $f_K/f_\pi = 1.2132(95)$ is consistent with FLAG averages and PDG values.
• Quark Masses:
  • Light quark mass $m_l \approx 3.34(4)$ MeV.
  • Strange quark mass $m_s \approx 92.7(1.0)$ MeV.
  • Ratio $m_s/m_l \approx 27.80(28)$.
• Renormalization: The differences between RI/MOM and SMOM renormalization schemes diminish as the lattice spacing decreases, validating the continuum limit.

D. Critical Mass and Tadpole Improvement

• The study observed that the critical mass ($m_{crit}$) is $\sim 40\%$ smaller in the HISQ sea ensembles compared to the Clover sea ensembles. This indicates that the difference arises from the gauge action/sea fermion type rather than the dynamical charm loop itself.
• The parameter $k_m^{tad}$ (slope of PCAC mass vs. bare mass) shows significantly weaker lattice spacing dependence in HISQ ensembles, correlating with the reduced discretization errors in observables.

5. Significance

• Validation of Mixed-Action Lattice QCD: The paper provides strong evidence that mixed-action calculations (Clover valence on HISQ sea) are not only viable but can offer superior control over systematic uncertainties regarding continuum extrapolation compared to unitary Clover calculations. This is a crucial finding for future high-precision lattice QCD projects.
• Benchmark for $N_f=2+1+1$: It confirms that for light hadron physics, the computational cost of including a dynamical charm sea (via HISQ) does not introduce significant biases, justifying its use in future precision calculations.
• Refined Standard Model Inputs: The updated values for $f_K/f_\pi$, quark mass ratios, and low-energy constants provide more precise inputs for testing the Standard Model, particularly in flavor physics and the determination of CKM matrix elements.
• Methodological Insight: The observation that mixed-action errors can cancel unitary discretization errors suggests a new avenue for optimizing lattice actions to minimize $O(a^2)$ and higher-order errors without requiring prohibitively fine lattice spacings.