| Chiral perturbation theory | |
|---|---|
| Name | Chiral perturbation theory |
| Field | Quantum field theory |
| Introduced | 1979 |
| Notable people | Steven Weinberg, Gasser and Leutwyler, Gerard 't Hooft |
| Related | Effective field theory, Quantum chromodynamics |
Chiral perturbation theory
Chiral perturbation theory (ChPT) is an effective field theory that describes the low-energy dynamics of the pseudo-Goldstone bosons arising from spontaneous chiral symmetry breaking in Quantum chromodynamics (QCD). It provides a systematic expansion in small momenta and light quark masses, enabling model-independent predictions for hadronic processes below the scale of resonances. ChPT is central to connecting nonperturbative QCD phenomena with precision low-energy experiments and lattice QCD computations.
ChPT addresses the regime where the strong coupling constant is large and perturbative Quantum chromodynamics fails, typically at energies below ≈1 GeV. In QCD with light up, down (and sometimes strange) quarks, the approximate global chiral symmetry SU(N)_L×SU(N)_R is spontaneously broken to SU(N)_V, producing an octet (for N=3) or triplet (for N=2) of light pseudo-scalar mesons such as the pion, kaon, and eta. ChPT treats these mesons as the relevant degrees of freedom and organizes interactions in an expansion controlled by the ratio of external momenta and quark masses to the chiral symmetry breaking scale, often denoted Λ_χ ∼4πF_π. Historically, foundational work by Steven Weinberg and the systematic formulation by Jürg Gasser and Heinrich Leutwyler established modern ChPT.
ChPT combines the symmetry constraints of chiral SU(N)_L×SU(N)_R with the machinery of effective field theory pioneered in particle physics. The low-energy theory is built from a unitary matrix field U(x)∈SU(N) that parametrizes the pseudo-Goldstone bosons and transforms nonlinearly under chiral rotations. Explicit explicit symmetry breaking enters via quark mass terms transforming as spurions. The approach relies on the concept of a nonlinearly realized symmetry, current algebra, and Adler–Weisberger–type relations. ChPT respects constraints from the Ward–Takahashi identity and reproduces soft-pion theorems and low-energy theorems known from current algebra and PCAC (partially conserved axial current). The formalism interfaces with lattice QCD through matching of low-energy constants and with dispersion relations in analytic continuation of amplitudes.
Power counting in ChPT assigns a chiral order to interactions in terms of momenta p and light quark masses m_q, typically p^2 ∼ m_q. The effective Lagrangian is organized as L_2 + L_4 + L_6 + …, where L_n contains operators of order p^n and low-energy constants (LECs) encode high-energy physics. The pioneering Gasser–Leutwyler L_4 Lagrangian introduced a set of renormalized LECs L_i for SU(3) and ℓ_i for SU(2). Loop diagrams generate divergences absorbed by renormalization of LECs, consistent with renormalization group running. Dimensional regularization and modified minimal subtraction schemes adapted to ChPT (e.g., MS-bar and chiral renormalization) are commonly used. Anomalies, such as the axial anomaly, are included via Wess–Zumino–Witten terms when required.
ChPT has been applied extensively to pion–pion scattering, pion–nucleon interactions, kaon decays, and electromagnetic form factors. It provides predictions for scattering lengths, phase shifts, and decay amplitudes that can be tested against data from experiments at facilities like CERN, J-PARC, and Jefferson Lab. In kaon physics, ChPT underpins analyses of K→ππ and K_{ℓ3} decays relevant for determinations of the CKM matrix element |V_{us}|. ChPT also interfaces with studies of isospin breaking and electromagnetic corrections through coupling to QED. In weak interactions, nonleptonic processes and CP-violating observables such as ε'/ε have been studied using ChPT matched to short-distance effective Hamiltonians computed via the Operator product expansion and perturbative QCD.
Several variants extend ChPT to include heavier degrees of freedom or to match lattice setups. Heavy baryon chiral perturbation theory (HBChPT) treats baryons with a heavy-particle expansion to avoid power-counting issues. Resonance chiral theory incorporates resonances like the ρ meson and axial-vectors to extend applicability toward the resonance region, often guided by large-N_c arguments and matching to QCD correlators. Partially quenched chiral perturbation theory (PQChPT) adapts the framework to partially quenched lattice simulations by introducing ghost fields and graded symmetry groups. Other approaches include infrared regularization and covariant baryon ChPT to improve relativistic treatments.
Practical use of ChPT requires determination of LECs from experiment or lattice QCD calculations. Global fits combine data on decay rates, scattering, and form factors to extract values for L_i and ℓ_i with controlled uncertainties; prominent analyses have been produced by collaborations and groups working on hadronic phenomenology. Loop integrals are computed using dimensional regularization, dispersive techniques, and numerical evaluation of multi-loop diagrams. Matching procedures connect short-distance Wilson coefficients from perturbative QCD to long-distance LECs for weak processes. Software tools and packages developed in the community assist symbolic manipulation and amplitude evaluation. ChPT remains essential for precision tests of the Standard Model, hadronic contributions to muon g-2, and the interpretation of lattice QCD results.
Category:Quantum field theory Category:Quantum chromodynamics Category:Effective field theories