Lattice Observables and Continuum Inference

A lattice observable claim is only as strong as its weakest inference step. A correlation function is not an energy, a fitted amplitude is not yet a matrix element, a bare matrix element is not renormalized, and a renormalized finite-spacing quantity is not a continuum prediction. This chapter connects those objects without collapsing their uncertainties or assumptions.

The recurring chain is

\text{operators} \longrightarrow C_2,C_3 \longrightarrow \text{levels and bare matrix elements} \longrightarrow \text{renormalization and scale} \longrightarrow \text{continuum observable}.

Every arrow has a failure test. The pages below let you enter at the missing arrow while keeping the complete chain visible.

Choose the missing inference step

Question	Page	Required output
What do the measured Euclidean correlators encode before fitting?	Euclidean Correlators and Spectral Information	Normalized finite-volume spectral sum, boundary images, overlaps, and noise limits
Can the operator basis isolate the desired level?	Operator Bases, Effective Masses, and Excited-State Control	Correlation matrix, conditioning analysis, GEVP or multi-state result, and competing-model tests
How is a bare matrix element obtained in the presence of disconnected terms?	Three-Point Functions, Matrix Elements, and Disconnected Contributions	Connected/disconnected decomposition, excited-state control, stochastic validation, and bare result
What continuous spectral information is identifiable from Euclidean data?	Spectral Reconstruction and Ill-Posed Euclidean Inverse Problems	Resolution function, regularization assumptions, mock recovery, and a bounded claim
How are lattice units converted without circularity?	Scale Setting and Dimensionless Ratios	Independent reference, unit conversion, covariance, and held-out prediction
How does a bare operator reach a named continuum scheme and scale?	Nonperturbative Renormalization, Mixing, and Step Scaling	Mixing matrix, renormalization condition, step-scaling chain, conversion, and window tests
Which lattice-side loop coefficients predict cutoff improvement?	Lattice Perturbation Theory, Symanzik Analysis, and Improvement	Lattice Feynman rules, Brillouin-zone integral, matched coefficient, allowed improvement basis, and residual power law
Does the complete multi-spacing analysis support a continuum claim?	Lines of Constant Physics and Continuum Extrapolation	Correlated fit with tuned inputs, volume treatment, alternative artifact models, and stability tests

Sampling algorithms and generic correlated-data methodology are developed in Sampling Algorithms for Lattice Fields and Statistical Inference and Error Budgets. Here they enter only through the properties needed to interpret a declared observable.

The finite-volume spectral core

Choose interpolating operators $\mathcal O_i$ with fixed lattice quantum numbers and zero-temperature transfer evolution. For $0<t<T$ , a two-point matrix has the finite-volume decomposition

C_{ij}(t) =\langle\mathcal O_i(t)\mathcal O_j^\dagger(0)\rangle =\sum_n\frac{Z_i^{(n)}Z_j^{(n)*}}{2E_n} \left[e^{-E_nt}+e^{-E_n(T-t)}\right],

where the $1/(2E_n)$ is convention dependent and can instead be absorbed into $Z_i^{(n)}$ . What matters is that one normalization is used consistently in two- and three-point functions. At finite $T$ the backward image is part of the exact model. At finite temperature, additional thermal transitions appear; the vacuum form must not be reused without a suppression argument.

A three-point function with insertion $J$ at time $\tau$ and sink at $T_s$ is

C_{3,ij}(T_s,\tau) =\sum_{m,n} Z_i^{(m)}Z_j^{(n)*} e^{-E_m(T_s-\tau)}e^{-E_n\tau} \langle m|J|n\rangle,

again up to the declared state normalization. A plateau approximation keeps only $m=n=0$ ; its error is controlled by gaps times both $\tau$ and $T_s-\tau$ . Increasing only one side cannot suppress both contaminations.

The chapter’s first four pages determine what may be inferred from these kernels. The next four decide whether the inferred bare quantities support a matched continuum observable. Gattringer and Lang 2010, chs. 4–6 give a standard treatment of the lattice spectral, operator, and renormalization ingredients connected here.

The observable chain and its independent checks

Stage	Definition and finite-regulator data	Estimator or matching step	Correlations and cutoff control	Independent check and stop condition
Action and lattice perturbation theory	Action, measure, Fourier convention, vertex normalization, Brillouin zone, and symmetry factor	Named loop coefficient in a stated lattice scheme	Common ensembles and input parameters retained through matching	Free rule or Ward identity; stop if a coefficient lacks its measure, domain, or normalization
Correlator and state isolation	Operator quantum numbers, boundary rule, $C_{ij}(t)$ , and covariance	GEVP or correlated multi-state fit for $E_n$ and $Z_i^{(n)}$	Basis rank, time window, images, and excited states varied jointly	Hermiticity, positivity where applicable, and an exact fixture; stop if the level label is unstable
Bare insertion	Connected and disconnected contractions for $\langle m\lvert J_0\rvert n\rangle$	Plateau, summation, and multi-state estimators with stochastic-noise treatment	Source–sink separations, shared configurations, and stochastic replicas remain correlated	Exact small system and estimator agreement; stop if disconnected bias or excited states are unresolved
Renormalization and mixing	Complete operator basis and $J_R(\mu)=Z(\mu,a)J_0$ in a named scheme	Nonperturbative condition, step scaling, and continuum conversion	Matrix-valued mixing and scale dependence propagated with the bare result	Ward identity, scheme round trip, and step-scaling closure; stop if the basis or scale window is incomplete
Scale setting	Dimensionless reference quantity and declared physical input	Correlated conversion from lattice units	Tuning, scale, and predicted observables keep their shared covariance	Second reference and held-out round trip; stop if the claimed prediction was also an input
Improvement and continuum limit	Symanzik-allowed action and operator basis, line of constant physics, $Q_R(a,L)$	Correlated finite-volume and continuum fit with a declared artifact ansatz	Matching, renormalization, scale, volume, action choice, and fit alternatives propagated once	Alternative action, artifact model, and leave-one-spacing-out tests; stop if constant physics or the asymptotic window is unsupported
Reported continuum quantity	$Q_R(0,\infty)$ with scheme, scale, units, and covariance	Rounding from the combined uncertainty, not from fit precision alone	Statistical and systematic components retain correlations and avoid double counting	Held-out observable or cross-formulation comparison; report only digits supported by the complete uncertainty

The repeated fields make omissions visible as the calculation moves from a regulated action to a continuum observable.

Preparation diagnostic

The chapter overview has no hard prerequisite. For the first four pages, you should be able to manipulate transfer spectral sums and covariance matrices. Repair the first capability with Euclidean Correlators and Schwinger Functions and Spectral Decomposition of Two-Point Functions. Repair lattice normalization with Lattice Momentum, Propagators, and Cutoff Dispersion.

For the final four pages, you should already distinguish bare and renormalized operators, scheme from observable, and a tuned trajectory from a bare-coupling scan. The repair routes are Local and Composite Operator Insertions, Nonperturbative Renormalization Schemes and Step Scaling, and Bare Parameters, Tuning Conditions, and Continuum Targets.

Review the chapter

Check. Show that the two-state effective energy from $C(t)=Ae^{-E_0t}+Be^{-E_1t}$ approaches $E_0$ with a correction proportional to $e^{-(E_1-E_0)t}$ when $A,B>0$ .

Derive. Starting from the three-point spectral sum, identify the leading source-side and sink-side excited-state corrections to a ground-state matrix element.

Compute. For a $2\times2$ mixing matrix $Z$ , propagate the covariance of a bare vector $v_0$ and of the uncertain entries of $Z$ into $v_R=Zv_0$ .

Diagnose. A study uses three lattice spacings, tunes the mass separately at each spacing, sets the scale with the same mass, and quotes that mass as its principal prediction. Explain which part is circular and propose a held-out observable.

Synthesize. Design a full calculation record for one continuum matrix element: operators, state isolation, insertion estimator, renormalization, scale, volume study, constant-physics conditions, artifact models, and two independent validation routes.

Solutions and checkpoints

Writing $C(t)=Ae^{-E_0t}[1+r e^{-\Delta t}]$ gives

E_{\mathrm{eff}}(t)=\log\frac{C(t)}{C(t+a)}\frac1a =E_0+\frac1a\log\frac{1+r e^{-\Delta t}} {1+r e^{-\Delta(t+a)}},

whose leading correction is $r(1-e^{-a\Delta})e^{-\Delta t}/a$ .

For the three-point function, terms $(m,n)=(1,0)$ and $(0,1)$ scale as $e^{-\Delta_m(T_s-\tau)}$ and $e^{-\Delta_n\tau}$ relative to the ground term. Both distances must be varied.

For covariance propagation, linearize $v_R=Zv_0$ . The Jacobians are $\partial v_{R,i}/\partial v_{0,j}=Z_{ij}$ and $\partial v_{R,i}/\partial Z_{kl}=\delta_{ik}v_{0,l}$ . Apply them to the joint covariance, retaining cross-covariances if $Z$ and $v_0$ share ensembles.

The mass agreement is guaranteed because it tunes and sets the scale. A different mass ratio, decay constant, or matrix element not used in either step is a valid held-out prediction.

What this chapter prepares

After completing the chapter, you should be able to trace a lattice result from measured correlation functions to a continuum observable and identify exactly where state, model, renormalization, scale, or cutoff assumptions enter. Lattice Gauge Theory supplies gauge-specific actions and observables; Sampling Algorithms for Lattice Fields and Statistical Inference and Error Budgets supply the ensemble-generation and full statistical machinery used to execute the chain.

References

Gattringer, Christof, and Christian B. Lang. Quantum Chromodynamics on the Lattice: An Introductory Presentation. Springer, 2010, chs. 4–6. doi:10.1007/978-3-642-01850-3.