Baryon masses with C-periodic boundary conditions

The RC* collaboration utilizes their openQxD code to present preliminary baryon mass results, including the first computation of additional partially connected contributions for the $\Omega^-$ baryon, from QCD+QED simulations employing C-periodic boundary conditions at an unphysical pion mass of approximately 400 MeV.

The Gist

Imagine you are trying to weigh a very specific, heavy object—a "Omega-minus" particle, which is like a tiny, super-dense ball made of three strange quarks. In the world of particle physics, this object is famous because scientists use its weight to calibrate the "ruler" they use to measure everything else in the universe.

However, getting an accurate weight is incredibly hard. It's like trying to weigh a feather on a scale that is shaking violently. This paper is about a team of scientists who built a new, more stable scale and a new way of looking at the feather to get a better measurement.

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

1. The Problem: The "Zero Charge" Trap

In the real world, particles have electric charges (like protons are positive, electrons are negative). But when scientists simulate the universe on a computer, they usually put everything inside a "box" with walls.

If you try to simulate a charged particle in a box with normal walls, physics breaks. It's like trying to fill a bucket with water, but the bucket has a hole at the bottom that drains everything instantly. The math says the total charge in the box must be zero. If the total charge is zero, you can't simulate a single charged particle because it would violate the rules of the simulation.

The Solution: The team used a special trick called C-periodic boundary conditions.

• The Analogy: Imagine you are walking out the front door of your house. Instead of hitting a wall, you instantly teleport to the back of your house, but everything is flipped upside down and mirrored (like looking in a mirror).
• In this mirrored world, the electric charge flips sign. So, if you step out with a positive charge, you step back in with a negative charge. This balance allows the simulation to exist without breaking the laws of physics, letting them finally weigh charged particles.

2. The New "Ghost" Signals

When they set up this mirrored world, something strange happened. Usually, when you look at a particle, you see three lines connecting the start point to the end point (like three strings holding a balloon). This is the "standard" signal.

But because of the mirrored walls, a new, weird signal appeared. It's like seeing a ghost reflection of the particle.

• The Standard Signal (3-q): Three strings connecting the start and end. This is the real particle.
• The Ghost Signal (1-q): One string connects the start and end, but the other two strings get tangled up with the "mirror world."
• Why it matters: These ghost signals are supposed to disappear if the box is infinitely big. But since their computer box is finite, these ghosts are there. The team is the first to actually measure these ghosts. It's like measuring the echo in a cave to understand the shape of the cave itself.

3. The Noise Problem: Trying to Hear a Whisper

The biggest challenge in this experiment is noise.

• The Analogy: Imagine trying to hear a whisper (the particle's mass) in a stadium full of screaming fans (random computer noise).
• In their previous attempts, they only had a few "ears" (computer sources) listening to the whisper. The result was very fuzzy.
• The Upgrade: In this paper, they used 60 to 100 ears listening at once. This is like replacing a single microphone with a massive array of microphones. The result? The screaming fans are drowned out, and the whisper becomes clear. They found the weight of the Omega particle much more precisely than before.

4. The Results: Clearer Pictures

They tested their new method on two different sizes of "boxes" (computer simulations):

• The Small Box: A standard size.
• The Big Box: A larger, more realistic size.

They found that by using more "ears" (sources) and a special technique called smearing (which is like blurring the image slightly to remove the static before sharpening it), they could see the particle's mass clearly for a longer time.

They also successfully measured the "ghost" signal (the 1-q contribution). While it is very noisy (like static on a radio), they proved it exists and is getting smaller as they make the box bigger, exactly as the theory predicted.

Summary: Why Should You Care?

This paper is a major step forward in understanding the fundamental building blocks of our universe.

1. Better Rulers: By weighing the Omega particle more accurately, scientists can calibrate their "rulers" better, leading to more precise measurements of everything from the mass of the sun to the behavior of black holes.
2. New Physics: By learning how to handle the "mirrored" boundary conditions, they are paving the way to include electromagnetism (QED) in their simulations. This is crucial for understanding why the universe is made of matter and not antimatter.
3. Overcoming Noise: They showed that by using more computing power and smarter math tricks (smearing), you can hear the "whispers" of the universe that were previously lost in the noise.

In short, they built a better mirror, added more ears to the room, and finally managed to weigh a particle that was previously too slippery to catch.

Technical Summary

1. Problem Statement

Lattice Quantum Chromodynamics (QCD) simulations have reached a precision regime (percent or sub-percent level) where isospin-breaking corrections are no longer negligible. These corrections arise from:

• The mass difference between up and down quarks.
• The coupling to Quantum Electrodynamics (QED) due to the electric charges of light quarks.

The Challenge of Finite Volume QED:
Defining QED in a finite volume with standard periodic boundary conditions is problematic. According to Gauss's law, the total electric charge in a periodic box must be zero ($Q=0$). This prevents the simulation of charged particles (like protons or $\Omega^-$ baryons) because no interpolating operator can have a non-zero overlap with a charged state.

The Solution:
The paper utilizes C-periodic boundary conditions (Charge-conjugation periodicity). In this formulation, fields transform under charge conjugation when crossing the lattice boundaries. This allows for non-zero total charge in the finite volume, enabling the simulation of charged hadrons and the inclusion of QED effects. The work is performed using the openQxD code, developed by the RC* collaboration, which implements these conditions via an orbifold construction.

2. Methodology

A. Theoretical Framework: C-Periodic Boundary Conditions

The boundary conditions are defined such that fields at $x + L_i$ are the charge conjugates of fields at $x$. For the electromagnetic field $A_\mu$, gauge links $U_\rho$, and quark fields $q_f$:
$$A_\mu(x + \hat{L}_i) = -A_\mu(x)$$
$$U_\rho(x + \hat{L}_i) = U_\rho(x)^*$$
$$q_f(x + \hat{L}_i) = C^{-1} \bar{q}_f^T(x)$$
where $C$ is the charge conjugation matrix.

Impact on Propagators:
Unlike standard periodic conditions, C-periodic conditions modify the quark propagator structure. In addition to the standard quark-antiquark propagator $\langle q(x)\bar{q}(y) \rangle$, new terms appear:

• Quark-quark propagators: $\langle q(x) q^T(y) \rangle$
• Antiquark-antiquark propagators: $\langle \bar{q}^T(x) \bar{q}(y) \rangle$
  These involve shifts to a "mirror lattice" (an orbifold construction where the physical lattice is doubled).

B. Baryon Two-Point Functions

The authors focus on the $\Omega^-$ baryon (three strange quarks, spin 3/2), a critical observable for setting the lattice scale. The two-point correlation function $C_{\Omega^-}(t)$ is decomposed into two distinct contraction types due to the modified propagators:

1. 3-Quark Connected (3-q): The standard contraction where all three quarks propagate from source to sink via quark-antiquark lines. This dominates the signal.
2. 1-Quark Connected (1-q): A new contribution arising from C-periodicity. It involves one quark-antiquark line connecting source and sink, while the other two quarks form a "partially connected" loop involving a quark-quark propagator at the sink and an antiquark-antiquark propagator at the source.
   • Theoretical Expectation: These 1-q contributions are finite-volume artifacts expected to vanish exponentially as $V \to \infty$.

C. Numerical Setup

• Ensembles: Two QCD-only ensembles (no dynamical QED yet) with C-periodic boundary conditions:
  • B400a00b324: Large volume ($80 \times 48^3$).
  • A400a00b324: Smaller volume ($64 \times 32^3$).
• Parameters: Unphysical pion mass $m_\pi \approx 400$ MeV, $O(a)$ improved Wilson fermions, three degenerate light quarks + charm quark.
• Scale Setting: Gradient flow observable $t_0$ ($\sqrt{8t_0} = 0.415$ fm).
• Smearing: Gradient-flow smearing for gauge fields followed by Gaussian fermion smearing to improve overlap with physical states.

3. Key Contributions

1. First Calculation of 1-q Connected Contributions:
   The authors present the first-ever computation of the partially connected (1-q) contributions to the $\Omega^-$ two-point function in a C-periodic setup. This is a crucial step in validating the formalism, as these terms are unique to this boundary condition setup.

2. High-Statistics 3-q Connected Measurements:
   The study significantly increases the statistical precision of the standard 3-q connected contributions by increasing the number of point sources from $\mathcal{O}(10)$ to $\mathcal{O}(100)$ (specifically 60 and 50 sources for the two ensembles). This reduces noise and allows for the identification of longer, more reliable plateaus in effective mass plots.

3. Implementation of Stochastic Estimators for All-to-All Propagators:
   To compute the 1-q contributions, the authors implemented a module in openQxD that handles the required all-to-all propagator (involving a shift by the lattice size $L$). They utilized stochastic noise vectors to estimate the inverse Dirac operator, a computationally expensive task.

4. Results

A. Baryon Masses (3-q Connected)

Using the high-statistics data, the authors extracted masses for the proton and $\Omega^-$ baryon:

• B400a00b324 (Large Volume): $m_p \approx 1170(10)$ MeV, $m_{\Omega^-} \approx 1470(20)$ MeV.
• A400a00b324 (Small Volume): $m_p \approx 1190(15)$ MeV, $m_{\Omega^-} \approx 1450(25)$ MeV.
• Observation: The large-volume ensemble shows reduced noise and more stable plateaus compared to previous lower-statistics studies.

B. Analysis of 1-q Connected Contributions

The authors compared the magnitude of the 1-q contributions against the 3-q contributions and their respective errors:

• Magnitude: As expected, the 1-q signal is significantly smaller than the 3-q signal.
• Noise Dominance: The 1-q contributions are dominated by statistical noise due to the stochastic estimation of the all-to-all propagator.
  • Without Smearing: The error on the 1-q term becomes dominant over the signal at time separation $t \approx 19$.
  • With Smearing (Max source, none sink): Smearing significantly reduces noise. The error on the 1-q term becomes dominant only at $t \approx 24$, extending the usable signal region.
• Conclusion: While the 1-q contributions are small, they are measurable with sufficient statistics and smearing, confirming the theoretical prediction that they are finite-volume effects.

5. Significance and Outlook

• Validation of C-Periodic QED: This work demonstrates that the openQxD code successfully handles the complexities of C-periodic boundary conditions, including the calculation of non-standard contraction types (1-q) that are essential for full QCD+QED simulations.
• Path to Precision Isospin Breaking: By establishing the methodology for calculating baryon masses with these boundary conditions, the collaboration is paving the way for fully dynamical QCD+QED simulations. This is necessary to achieve the sub-percent precision required for modern phenomenology (e.g., determining the proton-neutron mass difference).
• Future Work:
  • Increase the number of sources to 100 (large volume) and 150 (small volume).
  • Perform a Generalized Eigenvalue Problem (GEVP) analysis using the various smearing levels.
  • Extend the analysis to QCD+QED ensembles (currently in progress) to include dynamical electromagnetic effects.
  • Investigate variance reduction techniques for the stochastic estimator of the 1-q contributions to ensure the variance scales favorably with lattice size $L$.

In summary, this paper provides a critical technical foundation for next-generation lattice simulations that include QED effects, successfully isolating and measuring the unique finite-volume artifacts introduced by C-periodic boundary conditions while improving the precision of standard baryon mass calculations.