Separating Cutoff, Region, Correlation, and RG Scales

An information observable in QFT generally depends on several lengths at once. The ultraviolet cutoff $\epsilon$, region size $R$, separation $L$, correlation length $\xi$, deformation scale $m^{-1}$, and renormalization scale $\mu^{-1}$ are not interchangeable. A controlled flow claim identifies which ratio is varied, which quantities are held fixed, and in what order regulator and asymptotic limits are taken.

Required background. Regulated subregion entropy supplies the cutoff-dependent regional entropy; relative entropy in QFT supplies regulator-independent comparisons when a common algebra exists; UV and IR fixed points supplies the endpoint language.

Helpful background. Lattice-to-continuum entropy explains cutoff refinement in a concrete regulator.

Six independent scales

For a regional quantity $I_A$ in a theory deformed by $g\int d^d x\,\mathcal O$, a useful dimensional form is

$$I_A=I_A\!\left( \frac{R}{\epsilon},\frac{L}{R},\frac{R}{\xi}, gR^{d-\Delta},\mu R;\text{shape,state,scheme} \right).$$

Each argument has a different origin:

• $\epsilon$ resolves the UV regulator or split distance. Sending $\epsilon\to0$ tests the continuum limit.
• $R$ sets the size and shape of the accessible region. Varying $R$ enlarges an algebra; it is not by itself Wilsonian mode integration.
• $L$ is an inter-region separation. Mutual information often depends on $L/R$ and can remain finite as $\epsilon\to0$.
• $\xi$ is a physical correlation length, such as $m^{-1}$ in a massive relativistic vacuum.
• $g^{-1/(d-\Delta)}$ is the length generated by a relevant coupling of dimension $d-\Delta$.
• $\mu^{-1}$ is the renormalization length used to define couplings and composite operators. A physical answer is $\mu$-independent after running and counterterms are combined.

The diagram collects these scales and shows why endpoint theorems, crossovers, and channels must be treated separately.

The scale dictionary. A regional observable becomes comparable only after its state, regulator, shape, and subtraction scheme are fixed. Endpoint monotones, crossover functions, and channel recovery are different outputs. Schematic and not to scale.

Order of limits

The safe order depends on the question, but it must be declared. To extract a continuum crossover from a lattice model, first hold the physical ratios $R/\xi$ and $L/R$ fixed while taking

$$\frac{R}{\epsilon}\longrightarrow\infty,$$

then study the endpoint limits $R/\xi\to0$ or $R/\xi\to\infty$. Reversing these steps can erase the scaling window. For example, taking $mR\to0$ at fixed $R/\epsilon$ merely approaches a nearly critical finite lattice, not the continuum CFT.

The hierarchy needed for a massive interval calculation is

$$\epsilon\ll R\ll \xi \quad\text{(UV scaling)}, \qquad \epsilon\ll \xi\ll R \quad\text{(IR saturation)}.$$

Between them, $R\sim\xi$ is the crossover. A numerical analysis should display $R/\epsilon$, $mR$, and the system-size ratio $R/L_{\rm box}$ separately. Agreement at one lattice spacing is not a continuum result.

Spatial enlargement is not coarse graining

Restriction from a larger algebra $\mathcal A(B)$ to a smaller algebra $\mathcal A(A)$, with $A\subset B$, is a well-defined information-losing map. Enlarging $R$, by contrast, usually gives access to more observables. Wilsonian RG integrates or reorganizes high-momentum fluctuations and changes an effective action. These three operations can be related in special constructions, but they are not identical.

This distinction controls monotonicity directions. Relative entropy decreases under restriction, yet a region-size derivative can have either sign before a theorem uses Lorentz symmetry, strong subadditivity, and a particular geometric family. Similarly, a running coupling $g(\mu)$ changes with $\mu$ while a physical entropy is $\mu$-independent after renormalization.

Massive free interval as a scale test

Consider a free field on a lattice of spacing $a$ with mass $m$ and an interval of $N_A$ sites. Then

$$\epsilon\sim a,\qquad R=N_Aa,\qquad \xi\sim m^{-1}.$$

A useful dataset contains several $a$ values at fixed $mR$, not merely several $N_A$ values at fixed $ma$. At small $mR$, the entropy approaches the CFT logarithm plus cutoff-dependent constant. At large $mR$, correlations across the interval endpoints saturate over a distance $m^{-1}$. A derivative such as $R\,dS/dR$ removes the additive cutoff constant, but its finite-difference error and finite-volume correction must still be controlled.

The CFT interval normalization and massive crossover follow from Calabrese and Cardy 2004, §§ 2–3; covariance-matrix implementations and the scalar zero-mode qualification are reviewed in Casini and Huerta 2009, §§ 2.2 and 3.1.

Validity and failure checks

Before comparing two scale points, ask whether they share a regulator or continuum algebra and the same observable family. The decision map makes this check explicit.

Validity map for scale comparisons. A change of cutoff, region prescription, or subtraction scheme can imitate a flow. Without a theorem or explicit channel, the licensed conclusion is a regulated finite-window trend, checked by cutoff refinement and endpoint matching. Schematic and not to scale.

Common pitfalls

Hiding the cutoff in a dimensionless mass. Holding $ma$ fixed while increasing the number of sites does not hold the physical mass fixed. Report both $mR$ and $R/a$.

Calling every $R$ dependence RG flow. Region enlargement changes the accessible algebra. An RG interpretation requires an additional theorem, scaling argument, or channel construction.

Taking endpoint limits before the continuum limit. A lattice plateau can be produced by finite size or by $R$ approaching only a few cutoff units. Repeat the extraction at fixed physical ratios under cutoff refinement.

References

• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics: Theory and Experiment 2004 (2004): P06002. DOI.
• Casini, Horacio, and Marina Huerta. “Entanglement Entropy in Free Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 42 (2009): 504007. DOI.