Chiral Lagrangians and Low-Energy QCD

At momenta and pion masses small compared with the chiral scale, QCD is represented by the most general local action of its Goldstone field consistent with chiral symmetry, discrete symmetries, and a power counting. At leading order this construction fixes the pion kinetic and mass terms and the complete tree-level $\pi\pi$ amplitude; loops and higher-order low-energy constants then improve the prediction while canceling each other’s renormalization-scale dependence.

Required background. Chiral Order Parameters, Current Algebra, and Pions fixes the breaking pattern and decay-constant convention; Chiral Effective Theory and Nonlinear Symmetry supplies effective-field-theory matching and truncation logic.

Helpful background. Cosets and Nonlinear Realizations supplies the general coset construction.

The chiral field and external sources

For two light flavors, encode the three Goldstone fields in

$$U(x)=\exp\!\left(\frac{i\tau^a\pi^a(x)}{F}\right)\in SU(2), \qquad U\mapsto LUR^\dagger.$$

The vacuum is $U=\mathbf1$ after aligning the mass matrix. A different coordinate choice on the same coset changes off-shell vertices but not on-shell amplitudes.

To derive currents and impose local Ward identities, couple QCD to left, right, scalar, and pseudoscalar sources $l_\mu,r_\mu,s,p$. Their covariant combinations are

$$D_\mu U=\partial_\mu U-il_\mu U+iUr_\mu, \qquad \chi=2B(s+ip),$$

with transformations

$$D_\mu U\mapsto L(D_\mu U)R^\dagger, \qquad \chi\mapsto L\chi R^\dagger.$$

The physical limit is $l_\mu=r_\mu=p=0$ and $s=M$. Functional derivatives with respect to these sources generate the corresponding QCD currents and densities, so every effective operator automatically obeys the symmetry’s Ward identities.

Power counting and the leading action

Count a derivative or external momentum as $O(p)$ and the quark mass, hence $\chi$, as $O(p^2)$. The unique even-parity two-derivative action without external field strengths is

$$\mathcal L_2= \frac{F^2}{4}\operatorname{tr}(D_\mu U D^\mu U^\dagger) +\frac{F^2}{4}\operatorname{tr}(\chi U^\dagger+U\chi^\dagger).$$

$F$ and $B$ are low-energy constants in the two-flavor chiral limit. Matching the scalar source gives $\Sigma=F^2B$ at leading order in the convention of the preceding page.

For a connected mesonic graph with $L$ loops and vertices from operators of chiral dimension $d_i$, dimensional counting gives

$$\nu=2+2L+\sum_i(d_i-2),$$

so each loop raises the order by two powers of $p$. The expansion parameters are schematically $p^2/(4\pi F)^2$ and $m_\pi^2/(4\pi F)^2$. The theory ceases to be predictive at a stated truncation when these are not small or when omitted resonances become dynamical.

The diagram summarizes how that expansion inherits QCD information. The upper chain is fixed by symmetry and vacuum realization; external sources and matched low-energy constants turn it into amplitudes, while the singlet-anomaly and theta branches require additional topological information.

Low-energy QCD map. Symmetry realization fixes the pion field space and operator structure, but low-energy constants must be matched and predictions require $p,m_\pi\ll\Lambda_\chi$ with an explicit truncation order. The $U(1)_A$, topology, and theta sectors are related but not identical. The diagram is schematic.

Quadratic expansion and the pion mass

Set $M=\widehat m\mathbf1$ and $m_\pi^2=2B\widehat m=B(m_u+m_d)$ at leading order. The exponential field expands as

$$U=\mathbf1+\frac{i\tau^a\pi^a}{F} -\frac{\pi^a\pi^a}{2F^2}+O(\pi^3).$$

Using $\operatorname{tr}(\tau^a\tau^b)=2\delta^{ab}$ gives

$$\mathcal L_2= \frac12\partial_\mu\pi^a\partial^\mu\pi^a -\frac12m_\pi^2\pi^a\pi^a +\mathcal L_2^{(4\pi)}+O(\pi^6),$$

where, in this exponential parametrization,

$$\mathcal L_2^{(4\pi)}= \frac{1}{6F^2} \left[(\pi^a\partial_\mu\pi^a)^2 -(\pi^a\pi^a)(\partial_\mu\pi^b\partial^\mu\pi^b)\right] +\frac{m_\pi^2}{24F^2}(\pi^a\pi^a)^2.$$

The quadratic terms verify canonical normalization and the leading mass relation. Together with $\Sigma=F^2B$, the latter reproduces

$$F^2m_\pi^2=(m_u+m_d)\Sigma$$

at leading order. The distinction between $F$ in the chiral limit and the measured $F_\pi$ begins at higher chiral order.

The leading ππ amplitude

Let $\pi^a(p_1)\pi^b(p_2)\to\pi^c(p_3)\pi^d(p_4)$, with all external pions on shell and

$$s=(p_1+p_2)^2, \qquad t=(p_1-p_3)^2, \qquad u=(p_1-p_4)^2, \qquad s+t+u=4m_\pi^2.$$

Inserting the four-pion interactions and using the on-shell relation reduces the tree amplitude to

$$\begin{aligned} \mathcal M^{ab;cd}(s,t,u) ={}&\delta^{ab}\delta^{cd}A(s,t,u) +\delta^{ac}\delta^{bd}A(t,s,u)\\ &+\delta^{ad}\delta^{bc}A(u,t,s), \end{aligned}$$

with

$$A(s,t,u)=\frac{s-m_\pi^2}{F_\pi^2}+O(p^4).$$

Replacing $F$ by $F_\pi$ changes only the indicated higher-order remainder. The result is Weinberg’s low-energy theorem Weinberg 1966, pp. 616–619.

Three checks expose most normalization or crossing errors:

1. Crossing: exchanging an incoming and outgoing pion permutes $(s,t,u)$ and the corresponding Kronecker tensors; the displayed decomposition is closed under every such permutation.
2. Adler zero: at the analytically continued point $s=t=u=m_\pi^2$, every scalar function $A$ vanishes. This point is explicitly unphysical for on-shell equal-mass $2\to2$ scattering because it violates $s+t+u=4m_\pi^2$; it is a soft-current constraint on the analytic amplitude.
3. Isospin projection: using the on-shell sum gives

$$T^0=\frac{2s-m_\pi^2}{F_\pi^2}, \qquad T^1=\frac{t-u}{F_\pi^2}, \qquad T^2=\frac{2m_\pi^2-s}{F_\pi^2} +O(p^4),$$

which has the required $t\leftrightarrow u$ parity in each channel.

A reproducible calculation implements the same quadratic expansion, four-pion terms, tensor decomposition, crossing permutations, and Adler-point test. A mismatch with these equations is a convention or algebra error, not a new prediction.

Loops, low-energy constants, and remainders

At $O(p^4)$, one-loop graphs built from $\mathcal L_2$ generate nonanalytic logarithms and ultraviolet poles. The most general $\mathcal L_4=\sum_i\ell_i O_i$ contains local counterterms with renormalized coefficients $\ell_i^r(\mu)$. In a conventional subtraction scheme,

$$\ell_i^r(\mu_2)=\ell_i^r(\mu_1) +\frac{\gamma_i}{16\pi^2}\ln\frac{\mu_1}{\mu_2},$$

with coefficients $\gamma_i$ fixed by the divergence basis. For any observable amplitude,

$$\mu\frac{d}{d\mu} \left(\mathcal A_{\rm loop}^{(4)} +\mathcal A_{\rm LEC}^{(4)}\right)=0,$$

up to $O(p^6)$. A loop logarithm by itself is not a prediction; its scale dependence cancels only after the appropriate low-energy constants are included. The systematic one-loop construction is given by Gasser and Leutwyler 1984, §§3–7.

A complete result therefore states:

• the operator basis and subtraction convention;
• the scale $\mu$ at which renormalized constants are quoted;
• whether $F,m_\pi$ or physical $F_\pi,m_\pi$ parameterize lower-order terms;
• the chiral order retained and an $O(p^{\nu+2})$ remainder;
• the kinematic domain below inelastic thresholds and omitted resonances.

Common pitfalls

Mixing decay-constant conventions. If the axial generator is rescaled, $F_\pi$ rescales inversely. Keep $T^a=\tau^a/2$ from the current definition through the scattering amplitude.

Calling the Adler point physical. For massive on-shell pions, $s=t=u=m_\pi^2$ conflicts with $s+t+u=4m_\pi^2$. It is an analytically continued soft point used to check the symmetry structure.

Keeping loops but dropping counterterms. The result then depends on the subtraction scale and is not an observable. Include every local operator required at the same chiral order.

Exercise

Expand the leading mass term for $M=\widehat m\mathbf1$ through fourth order in $\pi^a$ and verify its contribution to $\mathcal L_2^{(4\pi)}$.

Solution

For $SU(2)$,

$$U+U^\dagger=2\cos\!\left(\frac{|\boldsymbol\pi|}{F}\right)\mathbf1, \qquad \chi=m_\pi^2\mathbf1.$$

Thus

$$\frac{F^2}{4}\operatorname{tr}[\chi(U+U^\dagger)] =F^2m_\pi^2 -\frac12m_\pi^2\pi^2 +\frac{m_\pi^2}{24F^2}(\pi^2)^2+O(\pi^6),$$

which gives the displayed mass and four-pion interaction after the irrelevant vacuum constant is dropped.

References

• Burgess, C. P. Introduction to Effective Field Theory: Thinking Effectively about Hierarchies of Scale. Cambridge: Cambridge University Press, 2020, §§8.1–8.2. DOI.
• Gasser, Jürg, and Heinrich Leutwyler. “Chiral Perturbation Theory to One Loop.” Annals of Physics 158 (1984): 142–210. DOI.
• Weinberg, Steven. “Pion Scattering Lengths.” Physical Review Letters 17 (1966): 616–621. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1996, §§19.4–19.5. DOI.