Particle physics
Renormalization in quantum field theory
Renormalization absorbs scale-dependent divergences into redefined parameters and fields; observables remain finite, while couplings run with energy scale.
Loop integrals often range over arbitrarily large momenta and diverge in the ultraviolet. Regulate them with a cutoff or another consistent scheme, then express results using quantities measured at a renormalization scale μ.
Definition: Definition and physical meaning
Bare parameters depend on the regulator and are not directly observable; renormalized parameters depend on μ so physical predictions become μ-independent when computed to all orders. The beta function β(g) governs running.
Structure and interpretation
The renormalization group (RG) compares physics at different scales μ. Solving the RG equation makes effective couplings vary: QED grows slowly with energy, whereas QCD decreases at high energy because β is negative, producing asymptotic freedom.
Example: Quantitative example
At one loop in QCD, β(g)=−b₀g³/(16π²)+… with b₀=11−2n_f/3>0 for sufficiently few flavors. Then αₛ(Q²)≈4π/[b₀ ln(Q²/Λ²)] decreases as Q increases.
Solution
The logarithm grows with Q, so αₛ falls; the one-loop expression fails near Q≈Λ, where coupling is strong and nonperturbative methods such as lattice QCD are needed.
A renormalized calculation begins by choosing a scheme, such as minimal subtraction in dimensional regularization. Pole terms are absorbed into the mass, field, and coupling counterterms; renormalized quantities are fixed by measurements at a reference scale. Changing that scale changes the running parameters, but a physical prediction assembled consistently is unchanged to all orders.
The scale is a reference scale for defining parameters, not automatically a physical cutoff energy. At finite perturbative order a prediction retains some dependence; its variation estimates uncertainty from omitted orders. Different renormalization schemes can assign different numerical values to a coupling, but after exact conversion they yield the same observable predictions.
Quick check
Which equation describes the running of a coupling g?
How does the QCD coupling behave at high energy?
References
- Michael E. Peskin and Daniel V. Schroeder (1995). An Introduction to Quantum Field Theory
- Steven Weinberg (1995). The Quantum Theory of Fields, Volume I