Renormalization and EFT formulas

Running couplings move scale dependence between the ingredients of a prediction; effective field theory organizes the error from omitting unresolved physics. This guide gives three scalar examples with their scheme, kinematic domain, and approximation order stated explicitly. A subtraction scale, a regulator, and a physical breakdown scale play different roles.

Helpful background. Beta Functions, Running Masses, and Field Anomalous Dimensions derives the fixed-bare scale equations. Effective Field Theory as a Controlled Expansion explains power counting and truncation errors.

One-loop scalar running

Consider one canonically normalized real scalar with interaction $-\lambda\phi^4/4!$, using dimensional regularization in $d=4-2\epsilon$ and modified minimal subtraction. At fixed bare data, the four-dimensional beta function is

$$\beta_\lambda\equiv\left.\mu\frac{\mathrm d\lambda}{\mathrm d\mu}\right|_0 =\frac{3\lambda^2}{16\pi^2}+O(\lambda^3).$$

Integrating the one-loop equation with boundary value $\lambda(\mu_0)$ gives

$$\lambda(\mu)= \frac{\lambda(\mu_0)}{1-\dfrac{3\lambda(\mu_0)}{16\pi^2}\log(\mu/\mu_0)}.$$

This is a solution of the truncated scale equation, not an exact running coupling. Use it only while the coupling remains perturbative and the field content remains appropriate. Its formal Landau singularity marks a failure of this approximation; it does not by itself prove a nonperturbative statement about the continuum theory.

Pole check. Writing $\lambda_0=\mu^{2\epsilon}(\lambda+a\lambda^2/\epsilon+\cdots)$ gives $\beta_\lambda=-2\epsilon\lambda+2a\lambda^2+\cdots$. With $a=3/(32\pi^2)$, the four-dimensional coefficient follows. The canonical term $-2\epsilon\lambda$ must be retained until after differentiating the bare coupling. The fixed-bare extraction is developed in Collins 1984/2023, § 7.3.1, pp. 180–183.

The subtracted scalar bubble

For Euclidean external momentum $Q$, $Q^2\geq0$, and mass $m^2>0$, define the scalar integral without interaction vertices or graph symmetry factors. Its modified-minimal-subtraction finite part is

$$B_{\overline{\mathrm{MS}}}(Q^2;m^2) =-\frac{1}{16\pi^2}\int_0^1\mathrm dx\, \log\frac{m^2+Q^2x(1-x)}{\mu^2}.$$

The logarithm contains a dimensionless ratio, so the four-dimensional integral is dimensionless. At fixed $m,Q$,

$$\mu\frac{\partial B_{\overline{\mathrm{MS}}}}{\partial\mu} =\frac{1}{8\pi^2}.$$

The result follows immediately by differentiating the logarithm and integrating $1$ over $0\leq x\leq1$. In a complete amplitude this scale dependence combines with the running couplings and counterterms. The dimensional-regularization treatment derives the integral and distinguishes the finite subtraction conventions.

To obtain a timelike scattering contribution, continue the external invariant with the correct Feynman boundary prescription, then restore vertices, symmetry factors, and all relevant channels. The Euclidean integral alone is not a scattering amplitude. See the worked scalar amplitude for a cut and threshold check.

Heavy-field matching and its remainder

Take light and heavy real scalars $\phi,H$ with canonical kinetic terms and potential

$$U=\frac{m^2}{2}\phi^2+\frac{M^2}{2}H^2 +\frac g2H\phi^2+\frac{\lambda_{\mathrm{full}}}{24}\phi^4.$$

Assume $M\gg m>0$, weak couplings $g^2/M^2\ll1$ and $\lambda_{\mathrm{full}}\ll1$, and $\lambda_{\mathrm{full}}>3g^2/M^2$. Completing the square in $H$ leaves a positive light quartic coefficient, so this tree-level potential is bounded below. The dimensionful coupling has $[g]=1$.

For light-particle elastic scattering, the tree amplitude contains the common quartic contact and all three heavy exchanges:

$$\begin{aligned} \mathcal M_{\mathrm{full}} &=-\lambda_{\mathrm{full}}+\frac{g^2}{M^2} \sum_{q\in\{s,t,u\}}\frac{1}{1-q/M^2},\\ \lambda_{\mathrm{EFT}}&=\lambda_{\mathrm{full}}-\frac{3g^2}{M^2}. \end{aligned}$$

Here $s,t,u$ are the Mandelstam invariants. The local expansion of each exchange is a geometric series. If every invariant satisfies $|q|\leq\rho M^2$ with $0<\rho<1$, replacing each exchange factor $1/(1-q/M^2)$ by $1+q/M^2$ defines $\mathcal M_{\mathrm{EFT}}^{[1]}$. The exact identity

$$\frac{1}{1-z}-(1+z)=\frac{z^2}{1-z}$$

then gives the absolute error bound

$$|\mathcal M_{\mathrm{full}}-\mathcal M_{\mathrm{EFT}}^{[1]}| \leq\frac{3g^2}{M^2}\frac{\rho^2}{1-\rho}.$$

For massive external particles, $s+t+u=4m^2$, so the first-order sum need not vanish. The massive worked matching example uses precisely this truncation. The Library’s massless example retains quadratic terms because its linear sum vanishes. The general local-operator expansion and power-counting logic are developed in Burgess 2021, § 1.2, pp. 10–13; § 2.4, pp. 39–44.

The bound is absolute. A small relative error for the full amplitude requires a nonzero lower bound on that amplitude, since the exchange can cancel the contact. This example is tree matching; loops introduce further threshold corrections and scale dependence.

References

• Burgess, C. P. Introduction to Effective Field Theory: Thinking Effectively about Hierarchies of Scale. Cambridge University Press, 2021. DOI.
• Collins, John C. Renormalization: An Introduction to Renormalization, the Renormalization Group, and the Operator-Product Expansion. Cambridge University Press, 1984; open-access digital edition, 2023. DOI.