Bare Parameters, Tuning Conditions, and Continuum Targets

A continuum trajectory is defined by renormalized conditions, not by holding bare couplings fixed while the lattice spacing changes. If the regulator has $r$ relevant directions that are not fixed by exact symmetries, at least $r$ independent physical conditions are required to select the target theory. Scale setting then converts dimensionless lattice results into units; it must not reuse the same observable in a circular way that hides mistuning. Only after this trajectory is established can Chapter 2 perform continuum inference for additional observables.

Required background. Lattice Regulators and Target Continuum Theories supplies the finite-regulator specification. Regulators, Cutoffs, and Continuum Limits supplies the general distinction between a bare family and a renormalized limit.

Helpful background. Critical Surfaces, Crossover, and Corrections to Scaling describes critical manifolds and corrections to scaling. Regulator Removal and Renormalized Predictions explains scheme-dependent coordinates and scheme-independent predictions.

Bare coordinates do not label a physical theory by themselves

Let a regulated action depend on dimensionless bare coordinates

$$\mathbf g_0=(\widehat m_0^2,\widehat\lambda_0,c_1,c_2,\ldots),$$

where $\widehat m_0=am_0$, $\widehat\lambda_0=a^{4-d}\lambda_0$ for a scalar quartic interaction, and the $c_i$ multiply other operators allowed by the exact regulator symmetries. A change in $a$ generally changes the values of $\mathbf g_0$ needed to describe the same renormalized physics.

A continuum target is specified by renormalized conditions

$$R_\alpha\!\left(\mathbf g_0,a,L;\mu\right) =R_\alpha^\star, \qquad \alpha=1,\ldots,r,$$

where each $R_\alpha$ is a dimensionless observable or a dimensionless ratio constructed from a renormalized quantity. Solving these equations at each $a$ defines a trajectory

$$\mathbf g_0=\mathbf g_0(a;R_1^\star,\ldots,R_r^\star,L,\mu).$$

The target observable $Q$ is then predicted rather than tuned:

$$Q_{\mathrm{cont}} =\lim_{a\to0} Q\!\left(\mathbf g_0(a),a,L(a);\mu\right),$$

with the volume prescription explicit. A fit of $Q$ before establishing that the $R_\alpha$ remain fixed mixes cutoff effects with a drift through theory space.

Tuning and scheme conventions. This page uses Euclidean lattice variables and the site-wide conventions. The renormalization scheme, scale $\mu$, finite-volume condition, boundary data, and definitions of every $R_\alpha$ are local. Bare coordinates may be convenient simulation inputs; they are not physical observables. A “line of constant physics” means a trajectory satisfying declared renormalized conditions within quantified tolerances.

The number $r$ is determined by relevant or marginal symmetry-allowed directions at the target, together with parameters intentionally varied as physical inputs. This renormalization-group interpretation of tuned trajectories follows the critical-surface analysis reviewed by Wilson and Kogut 1974, pp. 75–199. Exact regulator symmetries can forbid some dangerous operators. Breaking a target symmetry can allow new relevant terms and enlarge the tuning problem. This is why the classification on Exact Symmetries, Broken Spacetime Symmetries, and Restoration precedes parameter tuning.

A free scalar tuning relation

The free theory provides an exact benchmark. At zero spatial momentum, the standard lattice dispersion gives

$$4\sinh^2\!\left(\frac{aM}{2}\right)=a^2m_0^2,$$

where $M$ is the pole mass extracted from exponential Euclidean-time decay. To hold the physical pole mass fixed as $a$ changes, the bare mass must follow

$$am_0(a)=2\sinh\!\left(\frac{aM}{2}\right) =aM+\frac{(aM)^3}{24}+O(a^5M^5).$$

Holding $m_0=M$ instead gives a pole mass shifted by $O(a^2M^3)$. The shift vanishes, but the exact relation is a clean test that tuning code distinguishes a bare parameter from the measured mass.

Suppose the continuum step is taken at fixed physical volume $ML=8$. Then $N=L/a=8/(aM)$ must increase as $a$ decreases. A valid three-spacing sequence might use

$$(aM,N)=(1/2,16),(1/4,32),(1/8,64),$$

with $am_0$ set by the exact hyperbolic-sine relation at each row. The equal values of $ML$ separate cutoff changes from volume changes. A separate volume study at fixed $aM$, such as $ML=4,6,8$, diagnoses periodic images.

For an interacting scalar, one condition for the mass is insufficient if the renormalized coupling is also part of the target. A second condition can be defined through a connected four-point function, finite-volume scattering observable, or other dimensionless renormalized coupling. Its scheme and kinematics must be named. Different legitimate definitions yield different bare trajectories at finite $a$ but must give matched continuum predictions after scheme translation.

Scale setting without circularity

Lattice calculations first produce dimensionless quantities such as $aM_i$ and ratios $M_i/M_j$. To express a dimensionful answer, choose a reference quantity $X$ with known or conventionally assigned physical value and determine

$$a=\frac{(aX)_{\mathrm{lat}}}{X_{\mathrm{phys}}}.$$

The choice of $X$ is part of the analysis. If $X$ is also the quantity used to claim agreement with nature, that agreement is tautological. A clean design separates:

• tuning observables, which choose the bare trajectory;
• the scale-setting observable, which assigns units;
• validation observables, withheld from both operations; and
• target observables, whose continuum values answer the scientific question.

In a theory studied for its internal continuum limit rather than phenomenology, one may set units by a mass gap, gradient-flow scale, or finite-volume reference and quote all other results as dimensionless ratios. No choice removes the need to propagate the reference uncertainty and its correlations.

Circularity can also be subtler. If the same fitted correlator supplies both $aM$, an operator renormalization, and the final matrix element, their statistical and model uncertainties are correlated. Treating them as independent can make a tuned trajectory look artificially precise.

The order of limits is part of the target

Write a full target statement as an iterated or joint limit. For a massive zero-temperature observable one possible order is

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

The inner limit holds physical $L$, the renormalized conditions, the operator scheme, and the target kinematics fixed. The outer limit removes the finite box. A simultaneous trajectory $L=L(a)$ can be used if both $a\Lambda\to0$ and $mL\to\infty$ with a controlled error model. The local effective-action basis that organizes the cutoff terms is developed by Symanzik 1983, Part I, §§2–4. The trajectory should not be denoted by a single unlabeled arrow.

Other questions require a different order:

• a thermal continuum limit holds $T=1/(N_ta_t)$ fixed before changing $T$;
• a critical continuum limit tunes the correlation length in lattice units to infinity while maintaining the desired physical normalization;
• a chiral limit may not commute with infinite volume because spontaneous symmetry breaking needs an infinite system; and
• scattering amplitudes require finite-volume spectral extraction before an infinite-volume amplitude interpretation.

These are changes of scientific question, not preferences in numerical sequencing.

The complete observable chain is shown below. Inspect the separation among tuning, operator matching, scale setting, and extrapolation; none can be inferred from a small statistical error bar in the previous stage.

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. Schematic, not to scale.

The order-of-limits map summarizes why a trajectory must declare more than $a\to0$. Follow each axis with its held-fixed quantities, and inspect the two crossed paths where an apparently familiar limit changes the physical target.

Each limit names the observable and held-fixed physical quantities. The crossed routes show two common noncommuting or misidentified sequences: a zero-mode-sensitive massless limit and $a\to0$ at fixed site count, which shrinks rather than enlarges the box. The diagram is schematic and not to scale.

Under-tuning and identifiability

Let $J_{\alpha i}=\partial R_\alpha/\partial g_{0i}$ be the response matrix near a proposed trajectory. If $J$ lacks rank in the relevant subspace, the chosen conditions do not determine all required directions. This can happen when two conditions are physically redundant, statistically indistinguishable, or insensitive to a breaking operator.

Useful diagnostics include:

• singular values of the response matrix across the simulated region;
• stability when one tuning observable is replaced by another;
• held-out quantities sensitive to each symmetry-breaking direction;
• priors or constraints stated explicitly rather than hidden in interpolation; and
• ensembles on both sides of the target so the solution is interpolated rather than extrapolated in bare space.

The opposite problem is overconstraining: more conditions than adjustable physical inputs need not agree at finite $a$. Their mismatch can be valuable cutoff evidence. Fitting every condition exactly by adding arbitrary bare terms may change the regulator family rather than improve the original one.

Adversarial failure cases

Bare couplings held fixed. A sequence at fixed $\widehat m_0,\widehat\lambda_0$ generally changes renormalized masses and couplings. It is a scan through theories, not automatically a continuum trajectory.

One condition for two relevant directions. A precise mass match cannot also determine an independent interaction strength. The uncontrolled direction can drift while the mass stays fixed.

Scale set by the answer. If a target mass fixes $a$, reproducing that mass in physical units is guaranteed. Validate using another dimensionless ratio.

Continuum fit on a mistuned sequence. Smooth $a^2$ behavior does not show constant physics. Plot the tuning observables versus $a$ and propagate their residual deviations into $Q$.

Critical and infinite-volume limits exchanged silently. Finite volume rounds a phase transition and can remove spontaneous symmetry breaking. State which limit defines the target state.

A single action family. Agreement of several observables may still inherit the same matching or fit bias. A second discretization with a different artifact pattern is a stronger cross-check.

Observable-level validation checklist

Before extrapolating a target quantity, require:

• the number and nature of relevant symmetry-allowed directions;
• one independent renormalized condition per direction or a proof that a condition is unnecessary;
• a response matrix with adequate rank and conditioning;
• ensembles bracketing the target conditions at every spacing;
• scale-setting, tuning, validation, and target observables assigned distinct roles;
• residual mistuning propagated with full covariance;
• operator scheme, scale, mixing, and matching recorded before the fit;
• cutoff and finite-volume controls varied independently;
• an explicit order of limits with held-fixed quantities; and
• held-out observables or a second regulator confirming the target rather than defining it.

Chapter 2 develops the actual Lines of Constant Physics and Continuum Extrapolation and Scale Setting and Dimensionless Ratios workflows. The present page fixes their inputs and stop conditions.

Exercises

1. Exact free tuning. Expand the exact relation $am_0=2\sinh(aM/2)$ through $O(a^5)$ and determine the relative error made by setting $m_0=M$.

Solution

The expansion is

$$am_0=aM+\frac{(aM)^3}{24}+\frac{(aM)^5}{1920}+O((aM)^7).$$

Thus $m_0/M=1+(aM)^2/24+(aM)^4/1920+\cdots$. Holding $m_0=M$ mistunes the pole-mass condition by a relative $O(a^2M^2)$ amount.

2. Detecting under-tuning. Two bare parameters $(g_1,g_2)$ are constrained by one condition $R=g_1+2g_2=R^\star$. Show that the target is not unique and propose a second condition.

Solution

The solutions form the line $g_1=R^\star-2g_2$; the response matrix has rank one. Any observable sensitive to the orthogonal direction, for example $S=2g_1-g_2$, can serve as a second condition if it is a well-defined renormalized quantity. The pair has Jacobian determinant $-5\ne0$ and locally determines both coordinates.

What you can now do

You should now be able to define a target trajectory through bare-parameter space using independent renormalized conditions, expose under-tuning and circular scale setting, and state a physically meaningful order of limits. The next chapter, Lattice Observables and Continuum Inference, carries this specification through correlator analysis, operator renormalization, improvement, uncertainty propagation, and continuum extrapolation.

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.
• Wilson, Kenneth G., and John Kogut. “The Renormalization Group and the $\epsilon$ Expansion.” Physics Reports 12, no. 2 (1974): 75–199. doi:10.1016/0370-1573(74)90023-4.

Further reading

• Lüscher, Martin. “Advanced Lattice QCD.” In Les Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, 1998, pp. 229–280. arXiv:hep-lat/9802029.
• Montvay, István, and Gernot Münster. Quantum Fields on a Lattice. Cambridge University Press, 1994, chs. 3–5. doi:10.1017/CBO9780511470783.