Renormalization and Effective Field Theory

A theory containing interactions of arbitrarily high mass dimension can still make controlled low-energy predictions. The effective-field-theory rule is to include every operator allowed by the symmetries, order the result in small ratios such as $E/\Lambda$, and retain only the finitely many coefficients needed at the requested accuracy. Renormalization absorbs short-distance sensitivity into those coefficients and evolves them between scales; power counting supplies the truncation error.

Required background. Quantum Mechanics supplies amplitudes and observable predictions; Classical Field Theory supplies local actions, symmetries, and field dimensions; Special Relativity supplies Lorentz-invariant operator building blocks. Helpful background. Calculus and Linear Algebra supplies multivariable scale dependence and basis changes; Waves, Green Functions, and Geometric Optics supplies the separation of long- and short-wavelength physics.

Bare parameters, counterterms, and measurements

Loop terms from Interactions, Time Ordering, and Perturbation Theory sample arbitrarily large virtual momenta. For example, after Wick rotation a scalar self-energy contains integrals with the schematic ultraviolet behavior

$$I(m^2) \sim\int^\infty\frac{d^4k_E}{(2\pi)^4} \frac{1}{k_E^2+m^2}.$$

The integral is not defined without a prescription. A regulator—dimensional continuation, a momentum cutoff, Pauli–Villars fields, or a lattice—makes intermediate expressions well defined. The regulator is bookkeeping unless the theory identifies it with physical microscopic structure.

In dimensional regularization with $d=4-2\epsilon$, one convenient bookkeeping convention is

$$\phi_0=Z_\phi^{1/2}\phi_R, \qquad Z_\phi=1+\delta Z,$$ $$Z_\phi m_0^2=m^2(\mu)+\delta m^2, \qquad Z_\phi^2\lambda_0=\mu^{2\epsilon} \left[\lambda(\mu)+\delta\lambda\right].$$

The subscript $0$ denotes bare quantities. Their regulator dependence is chosen so that the bare Lagrangian is independent of the arbitrary subtraction scale $\mu$; the renormalized parameters and counterterms separately depend on the scheme and on $\mu$. Other conventions distribute factors of $Z_\phi$ differently between the mass, coupling, and counterterms, but they describe the same renormalized theory when used consistently. We now drop the $R$ subscript and return to four-dimensional notation after regulator removal, so the explicit factor $\mu^{2\epsilon}$ tends to one in the next two displays; its pole and finite remnants remain encoded in the counterterms and running parameters.

Write a renormalized scalar Lagrangian as

$$\mathcal L =-\frac12(\partial\phi)^2-\frac12m^2\phi^2 -\frac{\lambda}{4!}\phi^4 +\mathcal L_{\mathrm{ct}},$$

with

$$\mathcal L_{\mathrm{ct}} =-\frac12\delta Z\,(\partial\phi)^2 -\frac12\delta m^2\,\phi^2 -\frac{\delta\lambda}{4!}\phi^4.$$

The counterterms depend on the regulator and subtraction scheme so that specified observables remain finite when the regulator is removed. The renormalized parameters $m(\mu)$ and $\lambda(\mu)$ are definitions tied to conditions at a scale $\mu$; they are not separately scheme-independent observables.

Neubert 2020, §§1.1 and 1.4, printed pp. 1–5 and 13–14, PDF develops bare-to-renormalized parameter relations and contrasts subtraction schemes.

An exactly computed physical observable cannot depend on the arbitrary renormalization scale. If it depends on running couplings $g_i(\mu)$, masses $m_a(\mu)$, and Wilson coefficients $C_I(\mu)$, its total scale derivative is

$$0=\mu\frac{d\mathcal O}{d\mu} =\left( \mu\frac{\partial}{\partial\mu} +\beta_i(g)\frac{\partial}{\partial g_i} +\sum_a\beta_{m_a}\frac{\partial}{\partial m_a} +\sum_I\beta_{C_I}\frac{\partial}{\partial C_I} \right)\mathcal O,$$

where $\beta_i=\mu\,dg_i/d\mu$, $\beta_{m_a}=\mu\,dm_a/d\mu$, and $\beta_{C_I}=\mu\,dC_I/d\mu$. Running parameters resum logarithms between separated scales. A change of scheme reshuffles finite local terms but cannot change a consistently computed observable.

Renormalized correlation functions satisfy a related but distinct Callan–Symanzik equation because the renormalized field itself depends on $\mu$. Define the sign convention

$$\phi_0=Z_\phi^{1/2}\phi_R, \qquad \gamma_\phi\equiv \frac12\mu\frac{d\ln Z_\phi}{d\mu}.$$

Then a renormalized connected $n$-point function obeys schematically

$$\left( \mu\frac{\partial}{\partial\mu} +\beta_i\frac{\partial}{\partial g_i} +\sum_a\beta_{m_a}\frac{\partial}{\partial m_a} +n\gamma_\phi \right)G_{\mathrm{ren}}^{(n)}=0.$$

The sign of the last term follows from the displayed definition; a convention with the opposite definition of $\gamma_\phi$ reverses it. After external-state normalization is treated consistently, a physical observable returns to the total-$\mu$-independence equation above. Neubert distinguishes the RG equation for an observable from the Callan–Symanzik equations for Green functions in Neubert 2020 PDF, §3, especially equations (3.3)–(3.5), pp. 22–23.

A one-coupling toy expression makes the cancellation visible. If $\beta(g)=b g^2+O(g^3)$ and a fixed-order dimensionless observable has the form

$$\mathcal O(E) =g(\mu)-b g^2(\mu)\ln\frac{\mu}{E}+O(g^3),$$

then

$$\mu\frac{d\mathcal O}{d\mu} =b g^2-b g^2+O(g^3)=O(g^3).$$

Residual $\mu$ dependence is therefore of the first omitted order, not a new observable. The regulator is the device used to define intermediate integrals; $\mu$ is the arbitrary renormalization scale; and the matching scale is chosen near a physical threshold such as $\Lambda$. Conflating those three scales obscures both the cancellation and the EFT error estimate.

The effective-field-theory expansion

In units with $\hbar=c=1$, a four-dimensional action is dimensionless, so $[\mathcal L]=4$ and $[\partial_\mu]=1$. The scalar and Dirac kinetic terms fix $[\phi]=1$ and $[\psi]=3/2$. For example,

$$\mathcal O_6 =\left(\bar\psi\gamma_\mu\psi\right) \left(\bar\psi\gamma^\mu\psi\right), \qquad \Delta\mathcal L=\frac{c_6}{\Lambda^2}\mathcal O_6,$$

has mass dimension six. Between relativistically normalized two-fermion states, its two currents supply the characteristic energy factors, so the dimensionless correction scales as $c_6E^2/\Lambda^2$. In a minimal basis, integrations by parts, lower-order equations of motion, algebraic identities, and field redefinitions can relate candidate operators; such changes reshuffle coefficients without changing the order in $E/\Lambda$.

Suppose new dynamics enters near a heavy scale $\Lambda$. At $E\ll\Lambda$, locality and symmetry allow

$$\mathcal L_{\mathrm{EFT}} =\mathcal L_{\le4} +\sum_{d\gt 4}\sum_i \frac{c_i^{(d)}}{\Lambda^{d-4}}\, \mathcal O_i^{(d)}.$$

An insertion of a dimension-$d$ operator normally contributes powers of $(E/\Lambda)^{d-4}$, with additional loop factors and symmetry suppressions. Power counting therefore supplies an error estimate and tells us which diagrams and coefficients are required at a chosen order.

Neubert 2020, §4, especially printed pp. 31–33 and 36–38, PDF develops matching, operator bases, and homogeneous power counting in inverse heavy scales.

The coefficients $c_i^{(d)}$ are found by:

1. matching to a known microscopic theory at a common scale;
2. fitting independent low-energy measurements; or
3. leaving them as parameters when the ultraviolet theory is unknown.

One must not use the same datum both to determine a coefficient and to advertise its reproduction as a prediction. Predictions are relations among more low-energy observables than there are coefficients at the working order.

Gravity as an effective field theory

Define the reduced Planck mass by $\overline M_{\mathrm{Pl}}=(8\pi G)^{-1/2}$. In four dimensions, a schematic diffeomorphism-invariant local action for gravity and matter has the derivative expansion

$$S_{\mathrm{grav}} =\int d^4x\,\sqrt{-g} \left[ \frac{1}{16\pi G}(R-2\Lambda_{\rm cc}) +\mathcal L_{\rm matter} +c_1R^2+c_2R_{\mu\nu}R^{\mu\nu} +\frac{d_1}{\Lambda^2}R^3+\cdots \right],$$

This is an operator-basis sketch, not a complete list. Integrations by parts, the four-dimensional Euler density, lower-order equations of motion, and field redefinitions relate apparently different curvature operators; matter couplings add further terms. Overall curvature signs follow the chosen Riemann convention. The cosmological term is the allowed zero-derivative operator, and constant vacuum terms from $\mathcal L_{\rm matter}$ renormalize its net coefficient. Setting the resulting $\Lambda_{\rm cc}=0$ is an additional asymptotically flat vacuum assumption, not a consequence of diffeomorphism invariance. Donoghue 1995, §3, Eq. (21), printed p. 7, PDF displays this local derivative expansion.

Expanding $g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$ around that Minkowski vacuum produces a massless spin-2 field with interactions suppressed by powers of $\kappa E$, where $\kappa^2=32\pi G$ in a common normalization.

Perturbative general relativity is not renormalizable by a finite set of couplings at arbitrarily high energy. As an EFT it is predictive order by order when every characteristic external momentum and light mass lies below the EFT cutoff and any integrated-out threshold. Let $\mathcal R$ denote a characteristic magnitude of orthonormal-frame Riemann components and $\mathcal D_R$ a characteristic magnitude of their first covariant derivatives. The background derivative expansion also requires

$$\frac{E}{\Lambda_{\mathrm{EFT}}}\ll1, \qquad \frac{\mathcal R}{\Lambda_{\mathrm{EFT}}^2}\ll1, \qquad \frac{\mathcal D_R}{\Lambda_{\mathrm{EFT}}^3}\ll1.$$

The cutoff can lie below $\overline M_{\mathrm{Pl}}$ if additional heavy physics enters earlier. Within this regime, divergences are absorbed by higher-curvature coefficients. Donoghue 1995, §§5–7, printed pp. 13–16, PDF gives the gravitational loop hierarchy and power counting.

Analytic momentum dependence from unknown short distances is encoded in local operators. Nonanalytic low-energy terms generated by massless propagation cannot be imitated by local counterterms and can yield parameter-independent long-distance predictions; Donoghue 1995, §8, printed pp. 16–18, PDF makes this analytic–nonanalytic separation explicit.

This claim does not provide an ultraviolet completion, settle black-hole singularities, or justify extrapolation to Planckian curvature. The next-level treatments are General Relativity as a Quantum Effective Field Theory and Renormalization Ambiguities and Gravitational Counterterms.

Exercises

Estimate the first EFT correction

In four dimensions, let $\mathcal O_6$ have mass dimension six and coefficient $c/\Lambda^2$. Estimate its tree-level correction to a dimensionless amplitude at characteristic energy $E$, relative to a dimension-four interaction.

Solution

Dimensional analysis supplies two powers of the characteristic momentum:

$$\frac{\delta\mathcal A}{\mathcal A_{\mathrm{leading}}} \sim c\left(\frac{E}{\Lambda}\right)^2,$$

up to couplings, angular functions, and symmetry factors. If the higher-dimensional expansion relevant to this observable starts at dimension six and its symmetries and power counting forbid odd powers through $(E/\Lambda)^3$, the first omitted order after a complete $(E/\Lambda)^2$ calculation is generically $(E/\Lambda)^4$ when coefficients are natural and no extra enhancement occurs. That order can contain dimension-eight insertions, two dimension-six insertions, and the relevant loop corrections; it is not represented by dimension-eight operators alone.

The next page, Gauge Symmetry, Constraints, and Physical States, explains how redundancy restricts both the operator basis and the quantum state space.

References

• John F. Donoghue, “Introduction to the Effective Field Theory Description of Gravity,” lectures presented at the Advanced School on Effective Theories, Almuñécar, Spain (1995), arXiv:gr-qc/9512024, §3 and §§5–8, printed pp. 7 and 13–18. Author-manuscript PDF.
• Matthias Neubert, “Renormalization Theory and Effective Field Theories,” Les Houches lecture notes, arXiv:1901.06573v2 [hep-ph] (2020), §1.1, printed pp. 1–5; §1.4, printed pp. 13–14; §3, printed pp. 22–30; and §4, printed pp. 31–39. Author-manuscript PDF.