Lines of Constant Physics and Continuum Extrapolation

A controlled continuum result is a joint inference over a tuned line of constant physics, scale and operator renormalization, finite-volume corrections, and the cutoff expansion allowed by the regulator symmetries. The fit must preserve correlations among axes and ordinates, compare plausible artifact models, identify an asymptotic window, and survive held-out tests such as removing the coarsest or finest spacing. Three spacings and a small $\chi^2$ do not suffice if the trajectory drifts or the model is unidentifiable.

Required background. Scale Setting and Dimensionless Ratios supplies shared scale covariance. Lattice Perturbation Theory, Symanzik Analysis, and Improvement supplies the allowed cutoff powers. Bare Parameters, Tuning Conditions, and Continuum Targets defines the target trajectory.

Helpful background. Regulator Removal and Renormalized Predictions gives the general continuum and scheme logic. Nonperturbative Renormalization, Mixing, and Step Scaling is needed when the observable carries a nontrivial operator scheme.

Executing a line of constant physics

Let $R_\alpha$ be the renormalized tuning quantities and $R_\alpha^\star$ their target values. At each spacing, measured ensembles rarely land exactly on the target. Write residuals

$$\delta R_{\alpha e}=R_{\alpha e}-R_\alpha^\star$$

for ensemble $e$. A dimensionless target observable $Q_e$ can be expanded locally as

$$Q_e^{\mathrm{tuned}} =Q_e^{\mathrm{meas}} -\sum_\alpha J_{\alpha e}\delta R_{\alpha e} +O(\delta R^2), \qquad J_{\alpha e}=\frac{\partial Q}{\partial R_\alpha}.$$

The slopes $J_{\alpha e}$ must be constrained by nearby ensembles, reweighting, or a justified response model. Setting residuals to zero in the table while leaving $Q$ unchanged hides mistuning rather than correcting it.

Continuum-fit conventions. The page fits dimensionless renormalized observables and uses the site-wide conventions. The constant-physics conditions, reference scale, operator scheme and $\mu$, volume prescription, mass trajectory, cutoff variable, improvement status, and fit covariance are local. A common continuum value across regulators is imposed only after matching the same target observable.

When $Q$, the scale $X$, and tuning ratios share configurations, they form one joint data vector. Paired bootstrap or jackknife samples can carry their covariance through interpolation, renormalization, and the final fit. If stages are performed separately, use a joint nuisance-parameter likelihood rather than attaching independent errors afterward.

A dimensionless continuum model

Choose $x_e=a_eX_e$, where $X$ is a fixed physical reference. For an $O(a)$-improved bulk observable with leading $a^2$ effects, a representative model is

$$Q_e =Q_0 +c_2x_e^2 +c_{2\ell}x_e^2\log x_e +c_4x_e^4 +d_LF(m_eL_e) +\sum_\alpha J_\alpha\delta R_{\alpha e}.$$

Not every term should be included automatically. The Symanzik basis, anomalous dimensions, boundary conditions, and improvement status determine which powers and logarithms are plausible. $F(mL)$ comes from the finite-volume regime: an exponential form is justified only with a mass gap and short-range interactions; massless or long-range theories can have power-law effects.

$x_e$ is measured, not exact. Because it shares the reference $X_e$ with $Q_e$, ordinary least squares on central $x_e$ values can bias slopes and understate uncertainty. A joint model treats the true scales as latent quantities constrained by their measurements, or repeats the entire fit on paired resamples.

The simplest useful set of competing models might be:

$$\begin{array}{ll} M_1:& Q_0+c_2x^2,\\ M_2:& Q_0+c_2x^2+c_4x^4,\\ M_3:& Q_0+c_2x^2+c_{2\ell}x^2\log x,\\ M_4:& Q_0+c_4x^4 \quad\text{for a claimed $O(a^2)$ cancellation}. \end{array}$$

Models are accepted only if coefficients are identifiable in the available spacing range, residuals show no structure, and held-out predictions have coverage on synthetic fixtures. Averaging poorly identified models can hide rather than quantify ignorance.

Finding the asymptotic window

The Symanzik expansion is asymptotic in $a\Lambda\ll1$; its effective-action basis and improvement logic are developed in Symanzik 1983, Part I, pp. 187–204 and applied perturbatively in Part II, pp. 205–227. Coarse points can have small statistical errors and dominate a fit while lying outside that regime. Diagnose the window through:

1. successive removal of the coarsest spacing;
2. stability of $Q_0$ and leading coefficients;
3. dimensionless residuals plotted against $x^2$, $mL$, and tuning residuals;
4. consistency of the predicted power in held-out observables; and
5. agreement of two regulator families with different artifact coefficients.

Removing the finest point tests the opposite failure: a result apparently determined by one high-leverage ensemble. Leave-one-spacing-out predictions should reproduce each omitted point within the declared model and covariance.

The “finest-only” fit is not automatically safer. If it contains too little lever arm to determine $c_2$, its continuum intercept becomes prior dominated. Report the information content or parameter correlations, not only the central value.

Separate finite-volume, mass, and cutoff limits

The cleanest design includes a volume study at one or more spacings so the $L$ dependence is constrained independently of $a$. If only one volume exists at each spacing and $mL$ changes monotonically with $a$, cutoff and volume slopes are confounded.

A possible massive-theory target is

$$Q_\infty =\lim_{L\to\infty} \left[ \lim_{a\to0}^{R_\alpha=R_\alpha^\star} Q_R(a,L) \right].$$

A chiral or critical limit adds another axis whose order can matter. For spontaneous symmetry breaking, the infinite-volume limit generally precedes removal of an explicit symmetry-breaking mass. Writing a joint fit does not prove the limits commute; it merely parameterizes one chosen trajectory.

Finite-volume amplitude extraction is a different problem: discrete spectra are the input to a quantization condition, not nuisance corrections to be extrapolated away. Continue to Finite Volume as a Controlled Deformation for that regime.

Cross-regulator universality test

Suppose two matched actions $A$ and $B$ have the same target but different leading coefficients:

$$Q_A(a)=Q_0+c_Aa^p+\cdots, \qquad Q_B(a)=Q_0+c_Ba^q+\cdots.$$

A joint fit with common $Q_0$ and action-specific artifacts tests continuum agreement. Forcing the common intercept by construction is meaningful only if separate fits are also compatible and the observable, renormalization scheme, physical volume, and tuning conditions truly match.

Agreement is not independent if both actions use the same operator matching, scale input, finite-volume model, and analysis code. State the shared sources of uncertainty. A structurally distinct Hamiltonian or other formulation can provide stronger evidence once its own continuum program is complete.

A synthetic multi-spacing example

Let

$$Q(a,L)=1.20+0.50(aX)^2+0.30e^{-mL},$$

with $aX=(0.30,0.22,0.16,0.11)$ and two volumes $mL=4,6$ at each spacing. Correlated Gaussian replicas with shared scale uncertainty provide a checkable fixture. The intended analysis should recover $Q_0=1.20$ and separate the $a^2$ and exponential slopes.

Three injected failures are especially useful:

• retain only $mL=4$ and mislabel the exponential trend as $a^2$;
• omit the coarsest point until the $a^4$ model becomes unconstrained but apparently precise under a narrow prior;
• shift the tuning ratio by an amount correlated with $a^2$ and omit the response term.

A pipeline should flag each through residuals, rank or prior sensitivity, and a failed held-out prediction.

Avoiding double counting

An uncertainty source enters once at the stage where its stochastic or epistemic model is defined. Examples of double counting include:

• propagating scale uncertainty through $Q$ and then adding the same scale error again in physical units;
• applying a finite-volume correction with uncertain coefficient and also taking the full corrected–uncorrected shift as an independent error;
• averaging artifact models and adding their full spread a second time;
• including renormalization factors in paired resamples and again as independent Gaussian errors; or
• treating mistuning corrections as both nuisance parameters and post-fit systematic shifts.

Construct a dependency graph or joint variable list before fitting so every shared input has one source and all downstream correlations are visible.

The map below is the chapter’s full closure test. Follow the path from tuned bare parameters to the final dimensionless observable and verify that scale, matching, mixing, finite volume, cutoff fitting, and every held-out comparison remain distinct correlated inputs.

A continuum prediction requires a tuned bare trajectory and a renormalized observable. Scale, matching, mixing, volume, and cutoff uncertainties remain separate and correlated; held-out checks test rather than define the trajectory. The diagram is schematic and not to scale.

Observable-level validation checklist

Before accepting a continuum result, require:

• declared renormalized constant-physics conditions and measured residuals at every spacing;
• scale, operator-renormalization, and target data propagated jointly;
• independent variation of volume and cutoff for the claimed finite-volume model;
• artifact powers and logarithms justified by exact regulator symmetries and improvement status;
• at least three useful spacings within an identified asymptotic window, with enough lever arm to constrain the leading term;
• coarsest-point, finest-point, and leave-one-spacing-out tests;
• competing identifiable fit models and transparent prior sensitivity;
• residual plots against every independent error axis;
• a second action or formulation when universality is material; and
• one non-double-counted uncertainty decomposition reconstructing the total covariance.

Exercises

1. Confounded design. Suppose all ensembles satisfy $mL=4+10(aX)^2$. Explain why a fit $Q=Q_0+c(aX)^2+d e^{-mL}$ may not identify $c$ and $d$ well.

Solution

Both regressors become deterministic functions of the same $aX$ sequence and can be highly collinear over a short range. Their coefficients can trade off while leaving predictions nearly unchanged. Add multiple volumes at fixed spacing or impose independently validated finite-volume information.

2. Shared continuum intercept. Derive the normal-equation structure for two action families with a common $Q_0$ and separate slopes $c_A,c_B$. What comparison tests whether the common-intercept constraint is reasonable?

Solution

The design rows are $(1,a^p,0)$ for action $A$ and $(1,0,a^q)$ for $B$, with the full covariance used in generalized least squares. Compare the common-intercept fit with separate-intercept fits and inspect the difference $Q_{0,A}-Q_{0,B}$ with its covariance. Compatibility, rather than the imposed equality alone, supports a shared limit.

What you can now do

You should now be able to design a correlated multi-spacing extrapolation that preserves constant physics, scale and matching covariance, finite-volume control, and plausible Symanzik alternatives. You should also be able to reject a smooth fit whose asymptotic window, identifiability, or uncertainty structure is unsupported. The resulting continuum observable is ready for interpretation by the relevant physics volume, not automatically evidence that the full regulator family defines a mathematically constructed QFT.

References

• Symanzik, Kurt. “Continuum Limit and Improved Action in Lattice Theories. I. Principles and $\phi^4$ Theory.” Nuclear Physics B 226, no. 1 (1983): 187–204. doi:10.1016/0550-3213(83)90468-6.
• Symanzik, Kurt. “Continuum Limit and Improved Action in Lattice Theories. II. O($N$) Nonlinear Sigma Model in Perturbation Theory.” Nuclear Physics B 226, no. 1 (1983): 205–227. doi:10.1016/0550-3213(83)90469-8.

Further reading

• Lüscher, Martin. “Advanced Lattice QCD.” In Les Houches 1997: Probing the Standard Model of Particle Interactions, 1998, pp. 229–280. arXiv:hep-lat/9802029.