What does your correlator see?

Prerequisites: Euclidean correlators and elementary exponentials. Learning outcomes: distinguish input energies from an inferred effective mass, identify a state with zero overlap, and diagnose delayed plateaus.

A correlator measures states to which its operator couples. Change the overlap and watch how that changes the apparent mass—even though the spectrum itself stays fixed.

SPECTRAL EXPERIMENT

What can your operator see?

Reset

Two energy levels. One adjustable probe. Watch a light state disappear from the correlator when its overlap reaches zero.

Light-state weight

0.15

0 = invisible · 1 = only the light state

Light energy m₀

0.50

In fixed reference energy units

Excited-state separation

0.80

m₁ = m₀ + separation

Connected correlator logarithmic scale

Effective mass Δt = 0.25

Effective massLight energy m₀Excited energy m₁

A small overlap delays the light state.

Exact toy model. C(t) = w exp(−m₀t) + (1−w) exp(−m₁t). The effective mass is log[C(t)/C(t+Δt)] / Δt. There is no simulation noise, periodic-time effect, or continuum extrapolation here.

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Try three experiments

1. Find the plateau. Keep both overlaps nonzero. Increase the separation between the two energies. How soon does the effective mass approach the lighter energy?
2. Hide the light state. Set its overlap to zero. Why does the measured mass approach the heavier energy instead?
3. Make the overlap tiny. Restore a small positive overlap with the light state. Why might a practical measurement fail to reach the asymptotic plateau?

The simulation uses a discrete two-state spectral measure. A field theory can also have continuous spectral thresholds, and real numerical measurements have correlated noise and finite-time effects. These additions change the analysis, but they do not remove the operator-overlap problem.

Read the derivation in Euclidean correlators and what an operator can see. For numerical practice, continue to glueball spectroscopy.

The model and its units

Time is in a fixed reference time unit and energies are in its inverse. With $m_1\geq m_0>0$, $0\leq w\leq1$ and $\Delta t=0.25$,

$$C(t)=w e^{-m_0t}+(1-w)e^{-m_1t},\qquad m_{\mathrm{eff}}(t)=\frac{1}{\Delta t}\log\frac{C(t)}{C(t+\Delta t)}.$$

The weights sum to one, so $C(0)=1$ and the all-zero-correlator case is outside this model. For every allowed finite time $C(t)>0$. Factoring out an exponential before taking the logarithm makes the computation stable even when the individual exponentials become small. A genuinely zero measured correlator would have an undefined logarithmic effective mass; it would not mean zero energy.

Worked calculation and solutions

At the default $w=0.15$, $m_0=0.5$, $m_1=1.3$, the two contributions become equal when

$$t_\times=\frac{\log[(1-w)/w]}{m_1-m_0} =\frac{\log(0.85/0.15)}{0.8}\approx2.168.$$

This crossover is not a precise plateau criterion: comparable contributions still cause excited-state contamination. Reducing $w$ to $0.001$ postpones it to approximately $8.633$, although neither input energy changes. If $w\geq1/2$, the light term is already at least as large at $t=0$. If $m_1=m_0$, the two exponentials coincide and the effective mass is exactly that common energy for every $w$.

Solutions to the three experiments. For nonzero light overlap, a larger energy separation suppresses the heavy contribution more quickly at late time. At exactly zero light overlap, $C(t)=e^{-m_1t}$ and $m_{\mathrm{eff}}=m_1$ at every time; it never approaches the invisible $m_0$. For tiny positive overlap, the light contribution eventually dominates, but a practical noisy experiment may lose its signal first. Changing the spectrum and changing the probe are distinct operations.

Exercise: Explain why an effective mass of $m_1$ at $w=0$ is not evidence that the model contains no state at $m_0$.

Solution: The correlation function’s spectral weight at $m_0$ vanishes. It measures only the accessible part of the spectrum. Even a perfectly known one-channel correlator does not exclude a lighter state in another channel.

Source and limitation

Blossier et al., arXiv:0902.1265v2, §2.1, equations (2.3)–(2.7), give the spectral correlation matrix and effective-energy definitions underlying variational extraction. This lab uses only a scalar two-state special case, not the paper’s generalized eigenvalue algorithm or numerical results. It contains no Monte Carlo data, finite-periodic-time terms, spectral continuum, or uncertainty estimate. The CSV records the selected energies and weight with every generated row.