Turn lattice numbers into controlled physical predictions

Suppose a lattice calculation yields a dimensionless reference mass amref, and its physical value is supplied from an appropriate external input. Their ratio determines a in inverse-energy units; insert ℏc explicitly when quoting femtometers. Another lattice mass divided by amref gives a dimensionless physical ratio before any unit conversion. Reference uncertainties and shared configuration correlations must propagate to observables that use the inferred scale.

The reference must be appropriate to the simulated theory, including quark content, masses, electromagnetism, and isospin conventions. Different scale-setting choices can disagree at finite spacing or with mistuned parameters. Those differences may diagnose cutoff or calibration systematics rather than inconsistent arithmetic. Gradient-flow scales and hadronic quantities are common tools, but their chosen definitions and physical matching remain part of the result. No measured dimensionless number fixes its own physical units without an input.

A line of constant physics tunes bare couplings and masses so selected renormalized quantities stay fixed while the spacing changes. Keeping the same number of spatial sites as a decreases would shrink the physical box and mix volume effects with cutoff effects. At fixed temperature, temporal extent must likewise change with spacing because temperature is inverse Euclidean circumference. The intended zero-temperature or thermal limit should be stated before interpreting a sequence.

For a massive theory, many finite-volume shifts are exponentially suppressed by the lightest relevant mass times L, with observable-dependent prefactors and powers. A value such as mπL=4 is a useful diagnostic scale, not a theorem that every observable has a universal small error. Near thresholds, massless modes, or long-range electromagnetic effects, other volume dependences can matter. Varying volume directly can therefore be necessary even when a familiar heuristic criterion is met.

A Symanzik description organizes lattice artifacts through higher-dimension operators consistent with the action’s symmetries. Leading powers can be linear or quadratic in spacing, accompanied by logarithms and mass dependence, depending on discretization and improvement. Use sufficiently fine ensembles to justify the retained expansion and test sensitivity to coarse points and additional terms. A two-point fit with an assumed power performs an extrapolation but cannot independently establish that power.

The function lab illustrates one stipulated quadratic law. Reset and set coefficient to 1 and position to 1; predict value 1. Reset and set coefficient to 1 and position to 0.5; predict value 0.25. The fourfold reduction models a squared resolution parameter only when the physical analysis has justified that leading dependence. The instrument contains no scale setting, quark-mass tuning, finite-volume correction, or lattice data covariance.