Finite volume effects using lattice chiral perturbation theory
Bibliographic record
Abstract
Lattice regularization is used to perform chiral perturbation theory calculations in finite volume. The lattice spacing is chosen small enough to be irrelevant, and numerical results are obtained from simple summations. 1. CONTEXT Lattice QCD simulations are necessarily performed in a finite volume though one is typically interested in results for infinite spacetime. The extrapolation introduces a systematic uncertainty which needs to be controlled. As the lightest particle in the QCD spectrum, the pion has a large Compton wavelength and therefore plays a key role in these volume effects, so chiral perturbation theory is the natural tool for studying volume dependences. After the pioneering work of Gasser and Leutwyler[1], there has been a lot of activity on this topic. Some recent studies in the light meson sector can be found in Refs. [2,3,4,5]. Physical results do not depend on regularization scheme. For chiral perturbation theory in a finite volume, lattice regularization is numerically convenient because loop diagrams are finite summations for any nonzero lattice spacing. Divergences would appear as the lattice spacing vanishes, but for the volume effects studied here we can simply use a nonzero lattice spacing which is small enough to be numerically irrelevant. The present work uses the lattice regularized chiral perturbation theory Lagrangian of Ref. [6], but for two flavors rather than three. Extra O(a) terms could be added to the Lagrangian but they are irrelevant in the continuum limit, and our present goal is the computation of finite volume effects for the continuum limit. 2. THE PION MASS Consider an isotropic spacetime lattice with spacing “a”, Ns sites in each spatial direction and Nt sites in the temporal direction. The pion mass is obtained at the one-loop level from the diagrams in Fig. 1. The propagators and vertices are obtained from the Lagrangian of Ref. [6] in a straightforward manner, and loop momenta are summed. The resulting Green’s function is
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".