Renormalized Entanglement and Scheme Dependence

Renormalized entanglement entropy is not one automatic finite number. A differential operator, a set of local counterterms, and a mutual-information point split answer closely related questions, but they use different input data and can retain different finite local terms away from a fixed point. This page states each prescription precisely and compares all three on one free massive-scalar disk family in 2+1 dimensions, where both a universal ultraviolet endpoint and a controlled infrared expansion are known.

Required background. Regulator removal and renormalized predictions supplies the distinction between regulator removal and a finite renormalization; separating scales supplies the fixed-$mR$ continuum-refinement path used below.

Helpful background. Entropy counterterms and renormalization classifies the local surface terms that a subtraction may change.

Before comparing outputs, use the chapter’s scale-role map, comparison table, and independent validity gates. They prevent a change of split distance, region family, or endpoint convention from being mistaken for RG evolution.

Local terms determine what a scheme can remove

Let $\Sigma=\partial A$ be a smooth entangling surface and let $\epsilon$ denote a geometric ultraviolet cutoff. A useful local form of the regulated entropy is

$$S_\epsilon(A) =\sum_{n>0}\frac{c_n[\Sigma;m,g]}{\epsilon^n} +c_{\log}[\Sigma;m,g]\log(\mu\epsilon) +S_{\mathrm{fin}}(A;\mu)+o(1).$$

The coefficients are integrals over $\Sigma$ of local intrinsic and extrinsic geometry, with masses and couplings supplying dimensions. Consequently,

$$\mu\frac{d S_{\mathrm{fin}}}{d\mu}=-c_{\log}.$$

For the vacuum of a Lorentz-invariant theory in flat spacetime, purity and invariance under reversing the spatial normal restrict the local terms to even powers of the extrinsic curvature. A smooth scalable closed surface—one whose shape is held fixed while every length is multiplied by $R$—therefore has local powers $R^{d-2},R^{d-4},\ldots$. Relevant masses can dress their coefficients, but do not license arbitrary functions of $R$ as counterterms. Liu and Mezei 2013, § 2.1, Eqs. (5)–(12) derive these restrictions for the flat-space vacuum; Solodukhin 2011, §§ 3 and 6 develops the associated replica and surface-counterterm structure.

This domain statement matters. Corners, boundaries, mixed states, a changing shape, or a regulator tied to $R$ can introduce structures outside the polynomial family below.

Differential subtraction fixes the allowed roots

Write $D=R\,d/dR$. For a smooth scalable surface in the vacuum, Liu and Mezei define

$$\mathcal S_d(R)= \begin{cases} \displaystyle \frac{1}{(d-2)!!}(D-1)(D-3)\cdots(D-d+2)S(R), & d\ \text{odd},\\[6pt] \displaystyle \frac{1}{(d-2)!!}D(D-2)\cdots(D-d+2)S(R), & d\ \text{even}. \end{cases}$$

Every listed factor kills one permitted local power. In the first three dimensions where the formula is useful,

$$\mathcal S_2=DS, \qquad \mathcal S_3=(D-1)S, \qquad \mathcal S_4=\frac12D(D-2)S.$$

Thus $\mathcal S_2=c/3$ for a CFT interval, $\mathcal S_3=F$ for a CFT disk, and $\mathcal S_4=4a$ for a four-dimensional CFT sphere in this site’s anomaly convention. The full definition and fixed-point checks are Liu and Mezei 2013, Introduction and § 2.2, Eqs. (1), (11), and (13).

The operator removes finite terms with the same homogeneous $R$ dependence as the divergences, not only divergent coefficients. That is an intentional definition of the observable. Numerically, however, derivatives correlate neighboring radii and amplify interpolation error; a reproducible calculation must fit all $R$ values with their covariance and repeat the result under cutoff and fit-window refinement.

Counterterm subtraction needs one global matching rule

In a counterterm prescription one chooses local functionals $S_{\mathrm{ct}}[\Sigma]$ and defines

$$S_{\mathrm{ren}}^{\mathrm{ct}}(R) =\lim_{\epsilon\to0} \bigl[S_\epsilon(R)-S_{\mathrm{ct}}(R,\epsilon)\bigr].$$

The coefficients in $S_{\mathrm{ct}}$ must be fixed once for the whole family. A finite allowed term changes $S_{\mathrm{ren}}^{\mathrm{ct}}$, so a reported quantity must state a matching condition: for example, a specified CFT endpoint, agreement with a covariant point split, or a prescribed trivial-gapped-phase limit. Refitting a coefficient independently at every $R$ can manufacture any desired crossover.

For a 2+1-dimensional disk, a local line term is proportional to $R$. If the entropy contains $\kappa(m)R$, then a differential subtraction removes it automatically, while a counterterm subtraction records whichever finite value of $\kappa(m)$ the matching rule selects. The raw finite entropy and the differential $F$-function are therefore different observables even when both are cutoff independent.

A symmetric mutual-information split

Set

$$R_-=R-\frac{\delta}{2}, \qquad R_+=R+\frac{\delta}{2}.$$

Let $A_-$ be the disk inside $R_-$ and $A_+$ the exterior of the circle $R_+$. They are disjoint, and their mutual information is

$$I(R,\delta) =S(A_-)+S(A_+)-S(A_-\cup A_+).$$

At every nonzero physical separation $\delta$, local boundary divergences cancel. In a pure vacuum, the last term may be evaluated as the entropy of the annular gap, provided the same geometric regulator is used on both of its boundaries. The arithmetic-mean choice for $R$ is essential: an asymmetric frame $R=(R_++R_-)/2-\alpha\delta$ generally shifts the constant term when $\alpha\ne0$. These statements, including the required order $\epsilon_{\mathrm{UV}}\ll\delta\ll R$, are established in Casini, Huerta, Myers, and Yale 2015, §§ 2.2–3.1, Eqs. (2.3)–(2.14) and (3.1)–(3.11).

In three dimensions the small-split CFT expansion has the form

$$I(R,\delta) =b_{-1}\frac{R}{\delta}+b_0R-2F+O(\delta/R).$$

The factor of two is not optional: complementary regions give twice the one-boundary entropy in the coincident limit. The mutual-information version of the disk $F$-function is therefore

$$\mathcal F_{\mathrm{MI}}(R) =\lim_{\substack{\delta/R\to0\\ \epsilon_{\mathrm{UV}}/\delta\to0}} \frac12(D-1)I(R,\delta).$$

Here $D=R\partial_R$ acts at fixed $m$ and fixed physical $\delta$; the split limits are taken only after that derivative. For a massive flow, the split must also satisfy $m\delta\ll1$ if the annular term is to remain a short-distance contribution. At finite $\delta$, $\mathcal F_{\mathrm{MI}}(R,\delta)$ is a legitimate separated-region observable, but it is not yet the coincident-split scheme.

One free-scalar dataset, three outputs

The common input is the vacuum covariance matrix of one real scalar of mass $m$ on a radial lattice. Archive the disk, exterior, and annulus entropies for the same lattice spacing $a$, outer radius $L$, angular-mode truncation, circle convention $R=(n+1/2)a$, values of $x=mR$, and symmetric split $y=\delta/R$. Disk data alone are insufficient for the mutual-information prescription; “same dataset” means that the annular queries come from this same Gaussian state and regulator, not from a separately tuned calculation.

At the massless endpoint, the exact universal value is

$$F_s =\frac{\log 2}{8} -\frac{3\zeta(3)}{16\pi^2} =0.0638\ldots.$$

The three ideal continuum extractions agree:

Prescription	Quantity extracted from the common CFT data	Continuum result
Differential	$(D-1)S$	$F_s$
Counterterm, matched	minus the constant after one globally fixed line subtraction	$F_s$
Symmetric split	$\tfrac12(D-1)I$ followed by the double split limit	$F_s$

This agreement is a fixed-point calibration, not evidence that arbitrary finite crossover curves coincide. A radial-lattice disk calculation found $0.0635\pm0.0004$ with $N=200$, $10\le n\le45$, and absolute entropy accuracy $10^{-6}$ Liu and Mezei 2013, Appendix B, Eqs. (B14)–(B16). An independent mutual-information calculation used $30\le R/a\le100$, $4\le\delta/a\le10$, and infrared sizes $500\le N\le1000$; its fitted two-boundary constant was $0.1320$, compared with the exact $2F_s=0.1276\ldots$, a roughly $3\%$ numerical discrepancy Casini, Huerta, Myers, and Yale 2015, § 3.1, pp. 21–23. Dividing $0.1320$ by two gives $0.0660$, not a new value of $F_s$: it is a finite-resolution estimate whose split and infrared extrapolations dominate the error.

The same free massive scalar gives a sharper crossover stress test. For $x=mR\gg1$, its disk entropy is

$$S(R) =A\frac{R}{a} -\frac{\pi}{6}x -\frac{\pi}{240x} +O(x^{-3}),$$

and hence

$$\mathcal F_{\mathrm{diff}}(x) =\frac{\pi}{120x}+O(x^{-3}).$$

These asymptotics are Liu and Mezei 2013, § 6, Eqs. (52), (54), and (56). At $x=10$ they give the following semantic record:

Output from the same asymptotic data	Value at $x=10$	Meaning and control
$\mathcal F_{\mathrm{diff}}$	$0.002618$	$\pi/(120x)$; next relative order is $x^{-2}$, so $1\%$ is a power-counting estimate, not a statistical error
$-S_{\mathrm{fin}}$ after subtracting only $AR/a$	$5.23730$	dominated by the allowed finite line term $+\pi x/6$; not an IR degree-of-freedom count
$-S_{\mathrm{fin}}$ after also matching away $-\pi x/6$	$0.001309$	a valid globally matched counterterm scheme, but one-half of the differential tail
$\mathcal F_{\mathrm{MI}}$ in the symmetric double limit	$0.002618$	agrees with the differential flow when $a\ll\delta\ll\min(R,m^{-1})$; finite-$y$ corrections must be extrapolated

For the last row at $x=10$, choosing $y=10^{-3}$ gives $m\delta=xy=10^{-2}$, but then resolving $a\ll\delta$ requires $R/a\gg10^3$. The published $R/a\le100$ endpoint calculation does not meet that crossover hierarchy. The number $0.002618$ is therefore the controlled continuum/asymptotic target, with an estimated $1\%$ truncation from $x^{-3}$ terms; it is not claimed as an independently measured finite-lattice mutual information. This explicit limitation is part of the reproducible result.

The comparison isolates the method distinction: differential and symmetric-split schemes remove every local term proportional to $R$, whereas a counterterm scheme retains or removes the finite mass-dependent line term according to its one global matching rule. All three recover $F_s$ at the UV endpoint and zero for a topologically trivial gapped IR endpoint only after their respective limits and matching conditions are imposed.

Adversarial finite counterterm

Now perturb the common disk family by the allowed local term

$$S(R)\longmapsto S(R)+\kappa R,$$

with one $R$-independent coefficient $\kappa$ fixed for the theory and scheme.

Scheme	Response to $\kappa R$	Strongest surviving statement
Differential	$(D-1)(\kappa R)=0$	invariant for the fixed scalable disk family
Counterterm	$S_{\mathrm{ren}}^{\mathrm{ct}}\mapsto S_{\mathrm{ren}}^{\mathrm{ct}}+\kappa R$ unless the finite counterterm is transformed too	comparable only after the global matching convention is stated
Mutual information	the same local boundary term cancels in $S(A_-)+S(A_+)-S(A_-\cup A_+)$ at nonzero separation	invariant for disjoint regions with one common local regulator; the split-frame limit remains a separate condition

The adversary does not prove universal equivalence. It proves invariance only against this allowed homogeneous line term under the stated smooth-vacuum and fixed-family hypotheses. An $R$-dependent refit, a corner term, an asymmetric split, or a change of state falls outside that conclusion.

Common pitfalls

Calling every finite answer universal. Finiteness says that a cutoff was removed; it does not say that finite local counterterms or geometric frames agree.

Taking the mutual-information limits in the wrong order. One first needs $a\ll\delta$ so the two regions are physically separated in the continuum theory, and only then extrapolates $\delta/R\to0$ while controlling $m\delta$.

Differentiating unconverged data. A visually smooth interpolation can conceal correlated cutoff errors. Fit the entropy and its derivatives together, retain the covariance, and repeat the result under changes of $a$, $L$, angular truncation, and fit window.

Exercises

1. Check the dimension-dependent operators

Apply the differential definition to the fixed-point entropies

$$S_2=\frac{c}{3}\log(R/\epsilon)+k, \qquad S_3=\alpha R/\epsilon-F, \qquad S_4=\alpha R^2/\epsilon^2-4a\log(R/\epsilon)+k.$$

Verify $\mathcal S_2=c/3$, $\mathcal S_3=F$, and $\mathcal S_4=4a$.

Solution

For $d=2$, $D\log(R/\epsilon)=1$ and $Dk=0$, so $DS_2=c/3$. For $d=3$, $(D-1)(\alpha R/\epsilon)=0$ and $(D-1)(-F)=F$. For $d=4$, the first factor gives

$$(D-2)S_4 =-4a+8a\log(R/\epsilon)-2k,$$

because $(D-2)(R^2)=0$. Acting with $D/2$ then gives $4a$. The additive constant is removed in even dimensions, while the odd-dimensional constant is precisely the surviving universal term.

2. Follow a finite line term through all three schemes

For a 2+1-dimensional disk, add $\kappa R$ to every disk entropy. Show its effect on $\mathcal F_{\mathrm{diff}}$, a counterterm-subtracted entropy, and the mutual information of the symmetric split.

Solution

Since $D(\kappa R)=\kappa R$,

$$(D-1)(\kappa R)=0.$$

The differential result is unchanged. A fixed counterterm subtraction changes by $\kappa R$ unless its finite line coefficient is shifted by $-\kappa$; that shift is a new matching convention and must be reported.

For mutual information, each boundary contributes the same local density. The terms associated with $R_-$ and $R_+$ appear once in the two single-region entropies and once with the opposite sign in the annulus entropy, so they cancel before $\delta\to0$. The cancellation assumes a common local regulator and disjoint regions; it does not justify an asymmetric or regulator-dependent frame.

3. Quantify the free-scalar controls

At $x=10$, compute the leading differential and matched-counterterm tails. Then evaluate $m\delta$ for $\delta/R=10^{-3}$ and explain why $R/a=100$ cannot realize both $a\ll\delta$ and this split.

Solution

The two tails are

$$\mathcal F_{\mathrm{diff}} =\frac{\pi}{1200} =0.00261799\ldots, \qquad \mathcal F_{\mathrm{ct,matched}} =\frac{\pi}{2400} =0.001308997\ldots.$$

For $y=\delta/R=10^{-3}$,

$$m\delta=(mR)(\delta/R)=10^{-2},$$

so the split is short compared with the correlation length. But at $R/a=100$, one has $\delta/a=yR/a=0.1$: the split is smaller than one lattice spacing, contradicting $a\ll\delta$. Even the marginal condition $\delta/a>1$ requires $R/a>10^3$, and a controlled hierarchy requires substantially more. Thus the continuum mutual-information target cannot be advertised as a finite-$R/a=100$ measurement.

References

• Casini, Horacio, Marina Huerta, Robert C. Myers, and Alexandre Yale. “Mutual Information and the F-Theorem.” Journal of High Energy Physics 2015, no. 10 (2015): 003. DOI. Open PDF.
• Liu, Hong, and Mark Mezei. “A Refinement of Entanglement Entropy and the Number of Degrees of Freedom.” Journal of High Energy Physics 2013, no. 4 (2013): 162. DOI. Open PDF.
• Solodukhin, Sergey N. “Entanglement Entropy of Black Holes.” Living Reviews in Relativity 14 (2011): 8. DOI. Open PDF.

Further reading

• Casini, Horacio, and Marina Huerta. “On the RG Running of the Entanglement Entropy of a Circle.” Physical Review D 85 (2012): 125016. DOI. Open PDF.