research35 min studyLesson 796 in Physics

Holographic renormalization and counterterms

Remove cutoff divergences through local boundary terms while preserving the distinction between scheme and dynamics.

Before you begin

On this page

The question for today

What you’ll understand

Describe the cutoff and counterterm limit in a fixed source prescription.
Separate local divergent data from nonlocal response.
Compute a subtraction and a finite contact-term change.

A bulk on-shell action in asymptotically AdS space usually diverges when its radial cutoff approaches the conformal boundary. Reading a finite CFT correlator from that action therefore requires a renormalized boundary variational problem. Holographic renormalization organizes the divergences into local counterterms determined by near-boundary data. The method is a controlled calculation within a specified bulk model and holographic dictionary; it does not establish that any arbitrarily chosen gravitational action is dual to a known field theory.

Regulate first and identify local divergences

Choose a small positive radial cutoff and evaluate the action on the region away from the conformal boundary. Hold fixed the properly identified boundary sources, not arbitrary unrescaled values that change meaning with the cutoff. The induced metric and fields at that surface supply covariant variables for boundary counterterms. Their powers and derivatives follow from solving the asymptotic field equations to the orders needed for the divergent terms.

The divergent coefficients are local functions of the sources and boundary geometry, even though intermediate expansion coefficients can look complicated. This locality mirrors ultraviolet counterterms in the boundary field theory. The nonlocal normalizable response is not fixed by the same near-boundary recursion and still requires an interior state or regularity condition. Mistaking a complete asymptotic subtraction list for the whole correlation function erases the dynamics that the bulk solution is supposed to provide.

I_{\rm ren}[J]=\lim_{\epsilon\to0}\left(I_{\rm reg}[\epsilon,J]+I_{\rm ct}[\epsilon,J]\right)

ε is a radial length cutoff or a declared dimensionless version of it. Counterterms are added before the limit and varied consistently with the fixed source and boundary-metric prescription.↔ Scroll wide equations horizontally.

Keep logarithms, anomalies and finite schemes distinct

Power divergences and logarithmic divergences require different bookkeeping. A logarithm needs a reference scale so its argument is dimensionless. Its coefficient can encode a conformal anomaly, while changing a renormalization scale or adding allowed finite local counterterms shifts specific local terms in the renormalized functional. One cannot generally remove a physical anomaly by discarding every logarithm without checking the resulting Ward identities.

A finite local counterterm changes source-dependent one-point functions and contact terms in higher correlators. It does not arbitrarily change a separated-point nonlocal response. Such scheme freedom must be stated when comparing numerical coefficients from different papers. Conserved charges, stress tensors and operator normalizations also require a fixed boundary variational convention; writing a symbol called renormalized action is insufficient if the counterterms and allowed finite additions remain unspecified.

W_{\rm ren}[J]\mapsto W_{\rm ren}[J]+\frac a2\int d^dx\,J(x)^2\quad\Longrightarrow\quad \delta\langle\mathcal O(x)\mathcal O(y)\rangle_c=a\,\delta^{(d)}(x-y)

This example assumes the local term is allowed by the chosen symmetries and dimensions, with any needed units included in a. It demonstrates contact-term scheme freedom, not a change at noncoincident points.↔ Scroll wide equations horizontally.

Audit a subtraction before interpreting its finite part

A simple regulated scalar quantity can display a power term, a logarithm and a finite constant. Subtracting only the leading power leaves a logarithmic divergence; subtracting the logarithm with the wrong sign also fails. The regulator must be tracked symbolically before inserting a small number, because a modest-looking numerical residual can still diverge in the cutoff limit. A finite result likewise depends on the declared subtraction convention rather than on numerical cancellation alone.

The same audit applies to full bulk actions, with covariant fields, derivative terms and Ward identities replacing the toy constants. Quantum corrections can require further renormalization and contributions beyond a classical saddle. The quadratic lab illustrates one local source term and its derivative. It has no radial cutoff, logarithmic divergence, bulk asymptotic expansion or anomaly, so it cannot implement holographic renormalization or determine a physical subtraction scheme.

F(\epsilon)=2\epsilon^{-2}+3\ln\epsilon+5,\qquad F_{\rm sub}(\epsilon)=F(\epsilon)-2\epsilon^{-2}-3\ln\epsilon=5

This is a dimensionless arithmetic model with a dimensionless cutoff ε. Its constant finite part is defined by the displayed subtraction, not a claim about an actual gravitational action.↔ Scroll wide equations horizontally.

Work it through

Subtract both divergent terms with their signs

For the declared toy F(ε), evaluate the power term, logarithmic term and full quantity at ε = 0.10. Apply the displayed subtraction and determine whether subtracting the power alone has a finite zero-cutoff limit.

The power term is $2/(0.10)^2=200$ . The logarithmic term is $3\ln(0.10)\approx-6.907755$ , so its sign is negative at this cutoff.
The full quantity is approximately $200-6.907755+5=198.092245$ . Subtracting 200 and subtracting the negative logarithmic contribution leaves five exactly.
If only 200 were subtracted, the remainder would be $3\ln\epsilon+5$ , which tends to minus infinity as ε tends to zero. A finite-looking value at one cutoff does not establish successful renormalization.

The result

The full value is about 198.092245; the prescribed two-term subtraction gives 5, while power-only subtraction remains divergent.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

A finite local source term only

Reset and set BOTH Coefficient a = 2 and Position x = 0.50; predict y = 0.50 and slope 2. Reset and set a = 2 and x = 1; predict y = 2 and slope 4. This shows a quadratic local-term value and derivative, not cutoff removal, a logarithmic anomaly or holographic counterterm generation.

y = ax²TangentSigned integral

y at chosen x: 0.608
Tangent slope · 2ax: 1.35
Signed area · ax³/3: 0.182

Coefficient a0.75Position x0.9

t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA source-dependent local contribution and a nonlocal physical response need separate accounting.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Two source derivatives of a local source-squared term produce a delta function. This is scheme dependence at coincidence, not freedom to alter a generic separated-point response.

Your turn

Try it yourself

An allowed finite change of the generating functional is $\Delta W=\int d^dx\,J(x)^2$ , so a = 2 in the displayed convention. Find the changes in the one-point response and connected two-point kernel. What remains unchanged for x different from y?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Take one and then two functional derivatives of the same local term, retaining the delta function.

Compare with the solution

The one-point change is $\Delta\langle\mathcal O(x)\rangle=2J(x)$ . The connected two-point change is $2\delta^{(d)}(x-y)$ . At distinct points the delta function vanishes, so this finite local change leaves the nonlocal separated-point kernel unchanged. The conclusion assumes the term is permitted by the model's dimensions and symmetries. It does not eliminate logarithmic anomaly data or determine the interior-dependent response of a bulk solution.

Keep the idea

Take it with you

Renormalization requires counterterms before removing the cutoff.
Near-boundary local divergences do not determine nonlocal dynamics.
Finite local schemes can change contact terms while preserving separated-point data.

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