Fit a spectrum without mistaking a plateau for proof

Construct an interpolating operator with the hadron’s desired quantum numbers and usually sum its sink over spatial sites to project zero momentum. In a positive transfer setting the vacuum-subtracted two-point function has a spectral sum with nonnegative squared overlaps for a matching Hermitian-conjugate pair. The lightest state with nonzero overlap dominates at sufficiently large Euclidean separation. If the operator misses a state by symmetry or poor overlap, a fitted energy need not be the globally lightest state of the theory.

On a lattice with temporal extent NT, backward propagation around the boundary matters. A stable bosonic one-state contribution is a sum of forward and backward exponentials, equivalently a cosh about the midpoint. Baryonic boundary structures, staggered oscillating contributions, thermal effects, or nonmatching source/sink bases require their own formulas. A simple single-exponential ratio should therefore be used only in the time range and operator setup where its assumptions are justified.

For an effectively infinite temporal direction, the logarithm of adjacent correlators is constant for one exponential. With a heavier positive contribution it typically approaches the ground energy from above, though more complicated operator or sign structures can invalidate that simple behavior. A plateau across a few noisy points can result from insufficient precision rather than negligible excited states. Varying fit windows, operators, and model complexity provides needed evidence for a stable extraction.

The cosh effective mass uses three adjacent times and exactly recovers one periodic-state mass when the formula applies. It is not a universal cure for multiple states, nor does it remove statistical correlations between those three estimates. At long time, signal-to-noise degradation can make the arithmetic ratio unstable or biased. Mass fitting should respect covariance across times and the nonlinear nature of the derived quantities.

The correlator at every time comes from the same gauge configurations, so its covariance is generally non-diagonal. Autocorrelation along the Markov chain further affects that covariance. Blocking and correlated resampling should preserve both structures. Covariance inversion can be unstable for too few independent samples or too many fitted times; any regularization and its effect on the result should be reported. A small fit residual alone cannot establish a correct spectral model.

The spectral lab illustrates a different identifiability issue. Reset and set mode count to 1 and decay exponent to 0; at reset time predict sum at x=π/2 equal to 1. Reset and set count to 2 and exponent to 0; predict the same readout 1 although an extra spatial mode is present. The smallest wavelength changes from 2π to π. This prescribed traveling sine sum has no exponential hadron spectrum, gauge ensemble, or statistical mass-fitting procedure.