Chiral Effective Field Theory

Chiral effective field theory describes the low-energy consequences of approximate chiral symmetry in QCD. Its simplest version keeps the three pions as explicit fields and organizes their interactions by momentum and light-quark mass. The expansion can be useful even though quark and gluon dynamics at the same energy is strongly coupled.

Pion fields and chiral symmetry

Consider four-dimensional two-flavor QCD with theta angle zero and electromagnetism omitted. In the massless limit, the assumed symmetry-breaking pattern is

$$SU(2)_L\times SU(2)_R\longrightarrow SU(2)_V.$$

The three Goldstone fields can be represented by

$$U(x)=\exp\!\left(\frac{i\tau^a\pi^a(x)}F\right), \qquad U\longmapsto LUR^\dagger.$$

Here $\tau^a$ are Pauli matrices with $\operatorname{tr}(\tau^a\tau^b)=2\delta^{ab}$, and $F$ is the pion decay constant in the chiral limit. The anomalous singlet axial symmetry supplies no fourth pion.

At leading order, including small positive quark masses, the pion Lagrangian is

$$\mathcal L_2=\frac{F^2}{4}\operatorname{tr} \left[ \partial_\mu U\,\partial^\mu U^\dagger +\chi U^\dagger+U\chi^\dagger \right], \qquad \chi=2B\begin{pmatrix}m_u&0\\0&m_d\end{pmatrix}.$$

Take $B>0$ in the vacuum convention $U=1$. This constant has mass dimension one and encodes the strength of explicit symmetry breaking. Together with $F$ and higher-order coefficients, it must be matched to QCD or determined from physical or lattice data. Symmetry determines the allowed terms, not these numerical coefficients. The construction and leading mass relation are developed in Srednicki 2007, §83.

A normalization and mass check

Set $m_u=m_d=m_q$ and expand around $U=1$. The first terms are

$$U=1+\frac{i\tau^a\pi^a}{F} -\frac{\pi^a\pi^a}{2F^2}+\cdots.$$

Using the Pauli-matrix trace gives, after dropping a constant,

$$\mathcal L_2 =\frac12\partial_\mu\pi^a\partial^\mu\pi^a -\frac12M_\pi^2\pi^a\pi^a+\cdots, \qquad M_\pi^2=2Bm_q.$$

Thus the fields have canonical kinetic normalization, and the squared pion mass vanishes linearly with the quark mass at this order. The relation contains corrections at higher chiral order; using it does not identify the physical pion decay constant exactly with its chiral-limit value $F$.

Power counting and physical use

Assign a derivative order $p$ and count $M_\pi^2$ as order $p^2$. The displayed Lagrangian is therefore of order $p^2$. Its interactions give low-energy pion scattering and responses to suitable external currents; adding source terms allows one to calculate form factors systematically.

A pion loop raises the chiral order, with characteristic suppression $p^2/(4\pi F)^2$. Local higher-order terms absorb divergences and introduce further measurable coefficients. The usable expansion scale is also limited by omitted resonances and other degrees of freedom, so $4\pi F$ is an estimate of a strong-interaction scale rather than an unrestricted guarantee of accuracy.

This is an expansion in low momenta and light masses, not in a small QCD gauge coupling. It does not derive confinement or the symmetry-breaking condensate. Adding nucleons and describing shallow nuclear states introduces additional scales and iteration questions beyond this pion-only profile.

Develop the calculation

Chiral Effective Theory and Nonlinear Symmetry develops the coset construction, explicit breaking, and loop counting. Compare the underlying Quantum Chromodynamics and the geometric O(N) Nonlinear Sigma Model; their dimensions, target spaces, and physical regimes differ.

References

• Srednicki, Mark. Quantum Field Theory. Cambridge University Press, 2007, §83. DOI.