Dictionary Normalization and Global-Data Audit

An AdS/CFT coefficient becomes reproducible only when the bulk action, source and response definitions, powers of the AdS radius, counterterm scheme, and global charge data are specified together. A result can have the correct conformal power law while carrying the wrong sign, the wrong power of $L$, or a current normalization incompatible with the stated charge unit. This page derives a common scalar–current–stress-tensor benchmark and then translates it between two field conventions. The final checks distinguish three very different sources of disagreement: a harmless change of normalization, a local renormalization-scheme change, and a change of the physical theory’s global data.

The closed formulas below describe tree-level, two-derivative Einstein–Maxwell–scalar theory on Euclidean Poincaré AdS$_{d+1}$ with $d\ge3$, a flat boundary, standard scalar quantization with $\Delta>d/2$, and separated insertion points. The limits of that benchmark are stated explicitly. Lorentzian gravity follows the site’s $(+---)$ convention.

Required background. Radius, Couplings, and the Parameter Map supplies the relations among $L$, $G_N$, gauge couplings, $N$, and the string scale. The GKPW Generating-Functional Dictionary supplies the source differentiation and on-shell-action prescription. Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form supplies the charge lattice and line-operator data not contained in local correlators.

Helpful background. Basis Translation, Scheme Dependence, and Reproducibility develops the general logic of translating convention-dependent coefficients. Benchmark Reproduction and Data Provenance explains how to report enough information to reproduce a quoted number.

Three kinds of data that must not be conflated

Suppose two papers quote different values of $C_J$. The difference is not interpretable until one asks what else changed. The following classification is the first comparison step.

Three logically different sources of disagreement

Type of change	What may change	What must agree	Interpretation
Field and source normalization	Action prefactors, operator normalization, the numerical coefficient of a correlator, and the numerical charge label	Source couplings, responses to the same physical source, normalized charges, and holonomies	The same theory in different coordinates on field space, if every linked quantity is translated
Finite local counterterm	Contact terms, polynomial momentum terms, and some anomaly representatives	Separated-point nonlocal coefficients, poles, cuts, spectral support, and Ward identities including the translated contact terms	The same theory in a different renormalization scheme
Global form or boundary data	Charge lattice, genuine line operators, allowed bundles, large-gauge periodicities, discrete theta data, and sometimes the Hilbert space	Only the local quantities protected by the stated comparison	Potentially a different theory or a different boundary-value problem, not a convention change

This distinction is especially important for currents. A quadratic calculation determines the response of the Lie algebra-valued gauge field near the trivial connection. It does not by itself determine which representations define genuine line operators or which large gauge transformations are allowed.

Euclidean benchmark conventions

Use Poincaré coordinates

$$ds_E^2=\frac{L^2}{z^2}\left(dz^2+d\mathbf{x}^2\right), \qquad z\ge\epsilon,$$

so the outward unit normal at the cutoff surface points toward decreasing $z$:

$$n^z=-\frac{z}{L}.$$

The bulk actions are

$$\begin{aligned} S_{E,\phi} &=\frac{Z_\phi}{2}\int d^{d+1}x\sqrt g \left[(\nabla\phi)^2+m^2\phi^2\right],\\ S_{E,A} &=\frac{1}{4g_{d+1}^2}\int d^{d+1}x\sqrt g\,F_{MN}F^{MN},\\ S_{E,\mathrm{grav}} &=-\frac{1}{2\kappa_{d+1}^2}\int d^{d+1}x\sqrt g \left[R+\frac{d(d-1)}{L^2}\right] +S_{E,\mathrm{GHY}}+S_{E,\mathrm{ct}}, \end{aligned}$$

with

$$\kappa_{d+1}^2=8\pi G_{d+1}.$$

Here $S_{E,\mathrm{GHY}}$ is the Gibbons–Hawking–York boundary term required for a well-posed metric variation, while $S_{E,\mathrm{ct}}$ collects local cutoff-surface counterterms. The factor $Z_\phi$ is retained because the scalar field need not be canonically normalized. For an Abelian field, $F=dA$. For a non-Abelian field, the displayed Maxwell action denotes the quadratic term in a generator basis whose invariant metric has already been declared.

Let $\phi_{(0)}$, $a_i$, and $g_{(0)ij}$ be the boundary sources. Define the connected functional and one-point functions by

$$W[\phi_{(0)},a,g_{(0)}]=-S_{E,\mathrm{ren}},$$

and

$$\delta W =\int d^d x\sqrt{g_{(0)}} \left( \langle\mathcal O\rangle\,\delta\phi_{(0)} +\langle J^i\rangle\,\delta a_i -\frac12\langle T^{ij}\rangle\,\delta g_{(0)ij} \right).$$

These equations fix the signs and the factor of $1/2$ in the metric source. In particular,

$$\langle T^{ij}\rangle =\frac{2}{\sqrt{g_{(0)}}} \frac{\delta S_{E,\mathrm{ren}}}{\delta g_{(0)ij}},$$

whereas the scalar and current one-point functions carry the minus sign inherited from $W=-S_{E,\mathrm{ren}}$. The definitions also make clear why a bulk-field rescaling cannot be performed independently of the operator and source conventions.

Scalar benchmark from the renormalized radial momentum

The mass and standard-branch dimension obey

$$m^2L^2=\Delta(\Delta-d), \qquad \Delta>\frac d2.$$

Near the boundary, a regular solution has the expansion

$$\phi(z,\mathbf{x}) =z^{d-\Delta}\phi_{(0)}(\mathbf{x}) +z^\Delta\phi_{(2\Delta-d)}(\mathbf{x})+\cdots.$$

Let $\gamma_{ij}$ be the metric induced on $z=\epsilon$. On shell, the regulated variation is a boundary flux,

$$\delta S_{E,\phi}^{\mathrm{reg}} =Z_\phi\int_{z=\epsilon}d^d x\sqrt\gamma\, n^z\partial_z\phi\,\delta\phi.$$

For a nonresonant dimension, the leading scalar counterterm is

$$S_{E,\mathrm{ct},\phi}^{(0)} =\frac{Z_\phi(d-\Delta)}{2L} \int_{z=\epsilon}d^d x\sqrt\gamma\,\phi^2.$$

It cancels the leading divergence; derivative counterterms cancel further local divergences. Skenderis 2002, §3, Open PDF develops the recursive counterterm construction. Combining the finite part of the flux with the leading counterterm gives

$$\delta S_{E,\phi}^{\mathrm{ren}} =-Z_\phi L^{d-1}(2\Delta-d) \int d^d x\, \phi_{(2\Delta-d)}\,\delta\phi_{(0)} +\delta S_{\mathrm{local}}.$$

Because $W=-S_{E,\mathrm{ren}}$,

$$\langle\mathcal O(\mathbf{x})\rangle =Z_\phi L^{d-1}(2\Delta-d) \phi_{(2\Delta-d)}(\mathbf{x}) +\text{local terms}.$$

The regular bulk-to-boundary kernel with a delta-function source is

$$K_\Delta(z,\mathbf{x};\mathbf{y}) =c_\Delta \left(\frac{z}{z^2+|\mathbf{x}-\mathbf{y}|^2}\right)^\Delta, \qquad c_\Delta= \frac{\Gamma(\Delta)} {\pi^{d/2}\Gamma(\Delta-d/2)}.$$

At noncoincident points it implies

$$\phi_{(2\Delta-d)}(\mathbf{x}) =c_\Delta\int d^d y\, \frac{\phi_{(0)}(\mathbf{y})} {|\mathbf{x}-\mathbf{y}|^{2\Delta}}.$$

A second source derivative therefore gives

$$\langle\mathcal O(\mathbf{x})\mathcal O(0)\rangle =\frac{C_{\mathcal O}}{|\mathbf{x}|^{2\Delta}}, \qquad C_{\mathcal O} =Z_\phi L^{d-1}(2\Delta-d)c_\Delta .$$

The often-missed factor $2\Delta-d$ comes from the renormalized radial momentum, not from the unrenormalized boundary term alone. The normalized kernel and the finite-cutoff derivation are given by Freedman et al. 1999, §2, Eqs. (11)–(17), and Appendix, Eqs. (88)–(95), Open PDF. When the near-boundary expansion contains logarithms, additional scale-dependent local terms appear; the separated-point coefficient above remains the benchmark after the local terms are identified.

Current benchmark from the Maxwell flux

Work first with one Abelian gauge field. In radial gauge $A_z=0$, the transverse Fourier mode can be written

$$A_i(z,\mathbf{k}) =a_i^{\mathrm T}(\mathbf{k})f_d(kz), \qquad f_d(u)=\frac{u^{d/2-1}K_{d/2-1}(u)} {2^{d/2-2}\Gamma(d/2-1)},$$

where $K_\nu$ is the modified Bessel function of the second kind and $f_d(0)=1$. The longitudinal boundary source is pure gauge away from contact terms. Integrating the Maxwell action by parts gives the regulated boundary term

$$S_{E,A}^{\mathrm{reg}} =\frac{1}{2g_{d+1}^2} \int_{z=\epsilon}d^d x\sqrt\gamma\, n_M A_NF^{MN}.$$

Expanding the Bessel function, subtracting local divergent terms, and differentiating the remaining nonlocal quadratic functional yields

$$\langle J_i(\mathbf{x})J_j(0)\rangle =C_J\frac{I_{ij}(\mathbf{x})}{|\mathbf{x}|^{2(d-1)}}, \qquad C_J= \frac{(d-2)\Gamma(d)} {2\pi^{d/2}\Gamma(d/2)} \frac{L^{d-3}}{g_{d+1}^2} .$$

where

$$I_{ij}(\mathbf{x}) =\delta_{ij}-2\frac{x_ix_j}{|\mathbf{x}|^2}.$$

The position-space solution and boundary-flux normalization are derived in Freedman et al. 1999, §3.2, Eqs. (47)–(54), Open PDF. The tensor structure is transverse for $\mathbf{x}\ne0$:

$$\partial_i\left( \frac{I_{ij}(\mathbf{x})}{|\mathbf{x}|^{2(d-1)}} \right)=0.$$

This Ward-identity check fixes the relative longitudinal and transverse pieces, but it does not fix the overall value of $C_J$. Operationally, reflection positivity says that a source supported at positive Euclidean time has a nonnegative norm after reflection across $\tau=0$; for a Hermitian current in this convention, that requires $C_J>0$.

For several generators, the formula applies component by component when

$$S_{E,A}^{(2)} =\frac{1}{4g_{d+1}^2}\int\sqrt g\, \delta_{ab}F^a_{MN}F^{bMN}$$

and the current two-point function carries the same $\delta^{ab}$. In a non-Abelian theory, $F=dA+A\wedge A$; rescaling $A$ alone does not preserve the nonlinear field strength. A genuine basis translation must also transform the generator metric, structure constants, interaction vertices, and charge labels.

Stress-tensor benchmark from the graviton response

The stress tensor is sourced by the actual boundary metric, not by a freely normalized external scalar. Around a flat boundary, write

$$g_{(0)ij}=\delta_{ij}+h_{ij}.$$

After adding the Gibbons–Hawking–York term and local counterterms, the finite metric variation defines the renormalized Brown–York response. For a transverse-traceless perturbation, the Einstein graviton obeys the same radial equation as a massless scalar, while metric variation supplies the spin-two projector and its fixed normalization. de Haro, Solodukhin, and Skenderis 2001, Eqs. (1.1)–(1.3), §§2–3, Open PDF derive the renormalized stress tensor and its Ward identities.

Define

$$\mathcal I_{ij,kl}(\mathbf{x}) =\frac12\left(I_{ik}I_{jl}+I_{il}I_{jk}\right) -\frac1d\delta_{ij}\delta_{kl}.$$

For two-derivative Einstein gravity,

$$\langle T_{ij}(\mathbf{x})T_{kl}(0)\rangle =C_T\frac{\mathcal I_{ij,kl}(\mathbf{x})}{|\mathbf{x}|^{2d}}, \qquad C_T= \frac{d+1}{d-1} \frac{\Gamma(d+1)}{\pi^{d/2}\Gamma(d/2)} \frac{L^{d-1}}{\kappa_{d+1}^2} .$$

Equivalently,

$$C_T= \frac{d+1}{d-1} \frac{\Gamma(d+1)} {8\pi^{(d+2)/2}\Gamma(d/2)} \frac{L^{d-1}}{G_{d+1}}.$$

This is the standard conformal-tensor convention of Osborn and Petkou 1994, §2, Eqs. (2.20)–(2.23), Open PDF, with the holographic coefficient specialized from Myers and Sinha 2011, §6, Eqs. (6.1)–(6.4), especially Eq. (6.4), Open PDF to Einstein gravity. For example,

$$C_T\big|_{d=3}=\frac{3L^2}{\pi^3G_4}, \qquad C_T\big|_{d=4}=\frac{5L^3}{\pi^3G_5}.$$

In a higher-curvature theory, $L^{d-1}/G_{d+1}$ by itself is not the answer. When the AdS vacuum has an isolated, nondegenerate massless graviton with standard boundary conditions, the coefficient is controlled by that mode’s effective transverse-traceless kinetic normalization. If additional massless spin-two modes mix with it, or the linearized operator is degenerate, a single effective Newton factor is not enough. The Einstein formula may therefore be used only after the gravitational spectrum and quadratic action have been reduced to the convention stated above.

For the full current and stress-tensor source dictionary, including contact terms and Ward identities, see Currents, Stress Tensor, and Bulk Gauge and Metric Fields.

The three benchmark coefficients are dimensionless because

$$[Z_\phi]=\mathrm{length}^{1-d}, \qquad [g_{d+1}^2]=\mathrm{length}^{d-3}, \qquad [\kappa_{d+1}^2]=\mathrm{length}^{d-1}.$$

Separated-point two-point benchmarks in the declared convention

Operator	Boundary coefficient	Bulk datum	Fast independent checks
Scalar 𝒪	C𝒪 = Zφ Ld−1(2Δ−d)cΔ	Renormalized scalar radial momentum	Dimension, C𝒪 > 0, and the 2Δ−d response factor
Current Ji	CJ = [(d−2) Γ(d)/(2 πd/2 Γ(d/2))] Ld−3/gd+12	Renormalized Maxwell electric flux	Dimension, transversality, positivity, and declared generator metric
Stress tensor Tij	CT = [(d+1)/(d−1)] [Γ(d+1)/(πd/2 Γ(d/2))] Ld−1/κd+12	Renormalized transverse-traceless graviton flux	Dimension, conservation, tracelessness, positivity, and canonical metric variation

Recomputing the benchmark in a second convention

Convention A is the one used above. Define convention B, in the Abelian or linearized quadratic sector, by

$$\widehat\phi=a\phi, \qquad \widehat A=bA, \qquad a,b>0.$$

The positive rescaling constant $a$ is unrelated to the gauge source $a_i$; the shared letter is conventional notation, not an identification.

Writing the same quadratic bulk actions in the hatted variables requires

$$\widehat Z_\phi=\frac{Z_\phi}{a^2}, \qquad \frac{1}{\widehat g_{d+1}^2} =\frac{1}{b^2g_{d+1}^2}.$$

The boundary sources are $\widehat\phi_{(0)}=a\phi_{(0)}$ and $\widehat a_i=ba_i$. Equality of source couplings then fixes

$$\widehat{\mathcal O}=\frac{\mathcal O}{a}, \qquad \widehat J_i=\frac{J_i}{b}.$$

This is more than a relabeling of the final answer: inserting the hatted action prefactors into the two bulk calculations recomputes

$$\begin{aligned} C_{\widehat{\mathcal O}} &=\widehat Z_\phi L^{d-1}(2\Delta-d)c_\Delta =\frac{C_{\mathcal O}}{a^2},\\ C_{\widehat J} &=\frac{(d-2)\Gamma(d)} {2\pi^{d/2}\Gamma(d/2)} \frac{L^{d-3}}{\widehat g_{d+1}^2} =\frac{C_J}{b^2}. \end{aligned}$$

Complete translation between conventions A and B

Datum	Convention A	Convention B	Invariant comparison
Scalar field and action	φ, Zφ	φ̂ = aφ, Ẑφ = Zφ/a2	The quadratic bulk action
Scalar source and operator	φ(0), 𝒪	aφ(0), 𝒪/a	∫ φ(0)𝒪 and C𝒪φ(0)2
Gauge field and action	A, 1/gd+12	bA, 1/(b2gd+12)	The Abelian or quadratic bulk action
Gauge source and current	ai, Ji	bai, Ji/b	∫ aiJi and CJai2
Charge label in exp(iq∮A)	q	q̂ = q/b	Wilson phase q∮A = q̂∮Â and the same genuine-line spectrum
Metric and stress tensor	Physical g(0)ij and canonical Tij	The same physical metric and canonical Tij	Translation charges and CT are unchanged

One may introduce a bookkeeping variable $\widehat h_{ij}=c h_{ij}$ and call $T_{ij}/c$ its conjugate operator. Its numerical two-point coefficient is then $C_T/c^2$. That object is not the canonically normalized stress tensor until it is translated back to variation with respect to the actual boundary metric. Conservation and the translation Ward identity fix the physical normalization.

For a concrete round trip, take $d=3$, $\Delta=2$, $a=2$, and $b=3$. Since $c_2=1/\pi^2$,

$$\begin{aligned} C_{\mathcal O}^{(A)}&=\frac{Z_\phi L^2}{\pi^2}, &C_{\widehat{\mathcal O}}^{(B)}&=\frac{Z_\phi L^2}{4\pi^2},\\ C_J^{(A)}&=\frac{2}{\pi^2g_4^2}, &C_{\widehat J}^{(B)}&=\frac{2}{9\pi^2g_4^2},\\ C_T^{(A)}&=\frac{3L^2}{\pi^3G_4}, &C_T^{(B)}&=\frac{3L^2}{\pi^3G_4}. \end{aligned}$$

Multiplying the hatted scalar and current coefficients by the squares of their hatted sources reproduces the unhatted quadratic response. Applying the inverse translation returns every coefficient and source to convention A. That round trip is the quickest way to detect an incomplete rescaling.

Global data can change while local coefficients agree

The coefficient $C_J$ measures the response to an infinitesimal background connection near the identity. It does not determine the global form of the symmetry group. In four dimensions, theories with gauge group $SU(N)$ and $PSU(N)=SU(N)/\mathbb Z_N$ can have the same local Lie algebra and the same correlators of local operators on $\mathbb R^4$, yet differ in their genuine Wilson and ’t Hooft lines, allowed bundles, and discrete theta data. Aharony, Seiberg, and Tachikawa 2013, §§1–2, Open PDF give the explicit line-operator classification; Gaiotto et al. 2015, §§1–3, Open PDF explain the generalized-symmetry language that detects the difference.

The holographic audit must therefore carry, in addition to $g_{d+1}$ and $C_J$:

• the compact global form and the normalization of its generator metric;
• the minimal electric and magnetic charges and their Dirac pairing;
• the spectrum of genuine line and surface operators;
• allowed topological sectors, large gauge transformations, and discrete theta terms;
• the boundary condition imposed on each bulk gauge field.

If $A$ is rescaled but $q$ is not, the infinitesimal two-point function may still be translated algebraically, but the Wilson phase $q\oint A$ changes. The local quadratic calculation then survives; the claim that the two descriptions define the same globally normalized theory does not.

Euclidean data and Lorentzian orderings

For a scalar two-point function with time separation $t$, the time-ordered boundary value is obtained piecewise as

$$\tau=it+\epsilon\,\operatorname{sgn}(t), \qquad \epsilon\to0^+.$$

This specifies which side of the Euclidean singularity is approached for each ordering; the shorthand $\tau=i(t-i0)$ covers only the $t>0$ side. To reach the site’s mostly-minus Lorentzian metric from a positive Euclidean metric, continue the full line element according to

$$ds_L^2=-\left.ds_E^2\right|_{\tau=it}.$$

Sources must likewise be continued as tensors rather than as scalar labels. For example, equality of the source one-form gives

$$a_\tau^{E}\,d\tau=a_t^{L}\,dt, \qquad a_t^{L}=i a_\tau^{E}.$$

Metric components follow from the line-element rule, including its overall minus sign as well as the factors generated by $d\tau=i\,dt$. A retarded correlator requires a declared Fourier convention, the retarded analytic boundary value, and—when the bulk geometry has a future horizon—ingoing interior data. It is not obtained by changing a sign in the Euclidean answer. Skenderis and van Rees 2009, §§2–3 and Appendices A.2–A.3, Open PDF give the real-time contour construction and its renormalization.

Planted failures and the checks that expose them

The audit becomes useful only when it can reject a deliberately corrupted calculation. Consider the following failures.

First, reverse the sign of the renormalized scalar radial momentum while keeping $W=-S_{E,\mathrm{ren}}$. The result is

$$C_{\mathcal O}^{\mathrm{bad}} =-Z_\phi L^{d-1}(2\Delta-d)c_\Delta.$$

It has the right dimension and conformal power law, but for $Z_\phi>0$ and $\Delta>d/2$ it violates reflection positivity. Taking an absolute value would conceal the inconsistent normal, action, or generating-functional sign. The strongest surviving statement is only the power-law form; the normalization and any comparison based on it must be withdrawn.

Second, omit the power of $L$ while retaining dimensionful Poincaré coordinates:

$$C_{\mathcal O}^{\mathrm{bad}} =Z_\phi(2\Delta-d)c_\Delta, \qquad C_J^{\mathrm{bad}} =\frac{(d-2)\Gamma(d)} {2\pi^{d/2}\Gamma(d/2)}\frac{1}{g_{d+1}^2}.$$

These quantities have dimensions $\mathrm{length}^{1-d}$ and $\mathrm{length}^{3-d}$ rather than dimension zero. Their tensor structures can still satisfy the Ward identities and their signs can still be positive, so neither transversality nor positivity repairs the missing radius factor. Only the conformally allowed shape survives; the quoted coefficient cannot be compared across bulk-unit conventions.

Failure, decisive check, and strongest surviving claim

Planted failure	Checks that may still pass	Decisive failed check	What can still be claimed
Flip the scalar response sign	Power law and dimensions	Reflection positivity	The conformal exponent, not the coefficient or unitary normalization
Delete the required power of L	Tensor form, Ward identity, and sign	Dimensional analysis	The shape of the correlator, not a dimensionless benchmark
Replace Iij by δij	Scaling dimension and rotational covariance	Current conservation away from contact	Only a rotationally covariant ansatz, not a conserved-current correlator
Rescale A but keep the same numerical charge	Local linearized equations and translated CJ	Wilson holonomy and charge-lattice comparison	Local quadratic equivalence, not global equivalence
Change a finite local counterterm	Separated-point coefficients, poles, and cuts	A contact-term comparison made without translating schemes	All scheme-independent nonlocal data
Call a naive Wick rotation “retarded”	The Euclidean correlator and its ultraviolet normalization	Causal analyticity or the required ingoing interior condition	The Euclidean result, not a retarded response

Limits and handoffs

Several nearby problems need a modified analysis:

• For $d=2$, a Maxwell field has logarithmic near-boundary behavior; the factor $d-2$ in the formula for $C_J$ is a warning, not a prediction that every current two-point function vanishes.
• Resonant scalar dimensions generate logarithmic counterterms and scale-dependent local terms. One must separate those terms from the nonlocal coefficient before comparing schemes.
• Alternate scalar quantization and mixed boundary conditions change which coefficient is the source. They are developed in Boundary Conditions, Alternate Quantization, and Deformations.
• Alternate gauge-field boundary conditions, bulk theta terms, and Chern–Simons couplings can add parity-odd or topological data not contained in the parity-even $C_J$ above.
• Higher-curvature terms replace the Einstein value of $C_T$ by the effective graviton kinetic coefficient; bulk loops add subleading corrections in the semiclassical expansion.
• Finite local counterterms can move contact terms, but anomalies and quantized topological responses cannot always be removed by an allowed local term.

The systematic construction of the scalar, gauge, and metric counterterms is developed in Holographic Renormalization, especially Scalar Counterterms and the Renormalized Action, Gauge-Field and Differential-Form Counterterms, and Metric Counterterms and the Boundary Stress Tensor. The same normalization record becomes the input to higher-point calculations in Diagram Normalization and Reproducibility Benchmarks.

A reproducible comparison record

Before accepting a dictionary coefficient, record enough information for another reader to rerun both the bulk calculation and the boundary checks:

1. Geometry and orientation: bulk dimension, coordinate patch, signature, boundary metric, regulator surface, and outward normal.
2. Action normalization: every kinetic prefactor, the definitions of $G_{d+1}$ and $\kappa_{d+1}$, generator traces, powers of $L$, and the perturbative order retained.
3. Sources and responses: near-boundary expansion, scalar branch, boundary conditions, operator definitions, metric-variation convention, and charge unit.
4. State and interior data: Euclidean regularity, Lorentzian contour, $i0$ prescription, horizon condition, and any order of limits.
5. Renormalization: divergent and finite counterterms, renormalization scale, anomalies, and a list of quantities claimed only up to contact terms.
6. Global information: compact group, charge lattice, genuine extended operators, allowed bundles, large-gauge periodicities, and discrete theta data.
7. Evidence and uncertainty: which statements were derived analytically, checked numerically, or assumed; numerical tolerances; truncation errors; and the domain in which each approximation is controlled.
8. Independent controls: dimensions, Ward identities, positivity, special cases, inverse convention translation, and comparison with at least one independently normalized result.

For the benchmark on this page, the domain is flat-boundary, separated-point, tree-level Einstein–Maxwell–scalar theory with $d\ge3$. The analytic coefficients have no numerical uncertainty within that model; their physical use inherits effective-field-theory, loop, higher-derivative, state, and boundary-condition uncertainties not included in the calculation.

Common pitfalls

Treating a Ward identity as a normalization calculation. Conservation fixes the tensor structure of the current and stress-tensor correlators away from contact. It does not determine $C_J$ or $C_T$; the bulk kinetic terms and source definitions do.

Rescaling a connection without its charges. The local quadratic action may be rewritten after $A\mapsto bA$, but the same physical line operator then has charge label $q/b$. Leaving $q$ fixed changes its holonomy and can change the global theory.

Rescaling the physical stress tensor arbitrarily. A convenient metric perturbation may be rescaled as a bookkeeping variable. The canonical stress tensor is fixed by variation with respect to the actual metric and by the translation Ward identity.

Comparing contact terms before comparing schemes. A polynomial momentum term can move under a finite counterterm even when all separated-point physics agrees. State the scheme and isolate the nonlocal part first.

Using a familiar formula outside its domain. The displayed $C_J$ is not a shortcut for $d=2$, alternate gauge boundary conditions, or parity-odd sectors, and the displayed $C_T$ assumes a two-derivative Einstein graviton.

Exercises

1. Renormalized scalar response

Starting from the scalar boundary flux, show why the finite response contains $2\Delta-d$ rather than $\Delta$.

Solution

With $n^z=-z/L$, write $\phi_v:=\phi_{(2\Delta-d)}$. The finite part of the regulated variation contains both possible source–response cross-terms:

$$-Z_\phi L^{d-1} \int\left[ (d-\Delta)\phi_{(0)}\,\delta\phi_v +\Delta\phi_v\,\delta\phi_{(0)} \right].$$

The leading counterterm contributes

$$+Z_\phi L^{d-1}(d-\Delta) \int\left[ \phi_{(0)}\,\delta\phi_v +\phi_v\,\delta\phi_{(0)} \right].$$

The terms proportional to $\phi_{(0)}\delta\phi_v$ cancel. The remaining coefficient is $-\Delta+(d-\Delta)=-(2\Delta-d)$, so

$$\delta S_{E,\phi}^{\mathrm{ren}} =-Z_\phi L^{d-1}(2\Delta-d) \int\phi_v\,\delta\phi_{(0)}.$$

Finally $W=-S_{E,\mathrm{ren}}$ reverses this sign, giving the positive response quoted in the text.

2. Current transversality

Verify directly that the current tensor structure is transverse for $\mathbf{x}\ne0$.

Solution

Let $p=d-1$ and $r=|\mathbf{x}|$. Then

$$\frac{I_{ij}}{r^{2p}} =\delta_{ij}r^{-2p}-2x_ix_jr^{-2p-2}.$$

The divergence of the first term is $-2p x_jr^{-2p-2}$. For the second tensor,

$$\partial_i\left(x_ix_jr^{-2p-2}\right) =(d-2p-1)x_jr^{-2p-2} =-p x_jr^{-2p-2}.$$

Multiplication by $-2$ gives $+2p x_jr^{-2p-2}$, so the two contributions cancel. Contact terms at $\mathbf{x}=0$ require a regulated distributional treatment and are not part of this separated-point check.

3. The four-dimensional stress coefficient

Evaluate the Einstein value of $C_T$ for $d=4$ and rewrite it using $G_5$.

Solution

For $d=4$,

$$\frac{d+1}{d-1} \frac{\Gamma(d+1)}{\pi^{d/2}\Gamma(d/2)} =\frac53\frac{24}{\pi^2} =\frac{40}{\pi^2}.$$

Thus

$$C_T=\frac{40L^3}{\pi^2\kappa_5^2} =\frac{40L^3}{8\pi^3G_5} =\frac{5L^3}{\pi^3G_5}.$$

The result is positive for the healthy Einstein kinetic sign and dimensionless because $[G_5]=\mathrm{length}^3$.

4. A complete convention round trip

In the $d=3$, $\Delta=2$ example, verify the complete convention-B round trip for the scalar and current quadratic responses.

Solution

The sources transform as $\widehat\phi_{(0)}=2\phi_{(0)}$ and $\widehat a_i=3a_i$, while the coefficients transform as $C_{\widehat{\mathcal O}}=C_{\mathcal O}/4$ and $C_{\widehat J}=C_J/9$. Therefore

$$C_{\widehat{\mathcal O}}\widehat\phi_{(0)}^2 =\frac{C_{\mathcal O}}4(2\phi_{(0)})^2 =C_{\mathcal O}\phi_{(0)}^2,$$

and

$$C_{\widehat J}\widehat a_iI_{ij}\widehat a_j =\frac{C_J}{9}(3a_i)I_{ij}(3a_j) =C_Ja_iI_{ij}a_j.$$

Applying $a^{-1}=1/2$ and $b^{-1}=1/3$ to the hatted fields, sources, and operators returns the original actions and coefficients. Translating only the coefficient would fail this test.

5. Local agreement without global agreement

Suppose $A$ is replaced by $3A$ while the numerical charge $q$ is left unchanged. Decide which part of the claimed equivalence survives.

Solution

At quadratic order one can compensate the field rescaling by replacing $1/g_{d+1}^2$ with $1/(9g_{d+1}^2)$ and $J_i$ with $J_i/3$. The infinitesimal source response is then equivalent. But a Wilson line changes from

$$\exp\left(iq\oint A\right) \quad\text{to}\quad \exp\left(3iq\oint A\right).$$

To preserve the same holonomy one would also need $\widehat q=q/3$. Because the problem leaves $q$ unchanged, only the local quadratic equivalence survives; global equivalence does not.

6. Capstone: classify a mixed disagreement

Two calculations use the same $d=4$ bulk geometry. Calculation X has source $A$, coupling $g_5$, charge unit $q$, and no finite $F_{ij}F^{ij}$ boundary term. Calculation Y uses $\widehat A=2A$, $1/\widehat g_5^2=1/(4g_5^2)$, the same numerical charge $q$, and adds a finite local $F_{ij}F^{ij}$ counterterm. It quotes $C_{\widehat J}=C_J/4$. Classify every difference and state the strongest justified equivalence.

Solution

The transformations $\widehat A=2A$, $1/\widehat g_5^2=1/(4g_5^2)$, and $C_{\widehat J}=C_J/4$ form a consistent local field/source normalization change provided $\widehat J=J/2$. The finite boundary counterterm is a scheme change: it can alter contact terms but not the separated-point value of $C_J$.

The unchanged numerical charge is different. The Wilson phase becomes

$$q\oint\widehat A=2q\oint A,$$

so calculation Y has not preserved the same charge normalization. Without an accompanying $\widehat q=q/2$ and a comparison of genuine line operators, the two calculations are equivalent only for the translated local quadratic response at separated points. They are not established to have the same contact terms or the same global theory. A complete comparison record would still need the generator metric, global form, charge lattice, finite-counterterm coefficient, and boundary condition.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

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