An entire theory of almost-broken symmetry.

QFT document 15: the EFT of low-energy QCD. Where pions are the stars, chiral symmetry is the skeleton, and strongly-coupled physics becomes calculable.

Document 14 developed EFT methodology in the abstract. This document applies it to the most thoroughly-developed example in physics: chiral perturbation theory (ChPT), the low-energy effective theory of QCD.

The problem ChPT solves is fundamental. At low energies ( $E \lesssim \Lambda_\chi \sim 1$ GeV), QCD is strongly coupled. Perturbation theory in $\alpha_s$ fails. Quarks and gluons are confined into hadrons. How do you compute anything?

Answer: write an EFT of the relevant low-energy degrees of freedom; the pseudo-Goldstone bosons ( $\pi, K, \eta$ ) that arise from spontaneous breaking of chiral symmetry. These are the lightest hadrons (the pion is $\sim 140$ MeV, far below typical hadronic scales of $\sim 1$ GeV), and their low-energy interactions are strongly constrained by the pattern of symmetry breaking.

ChPT is beautiful because everything works out: the symmetry structure, the Goldstone theorem, the power counting, the matching to QCD, the predictions verified by experiment. It’s the template for how to do low-energy physics when the UV theory is strongly coupled.

Prerequisites

Document 14 (EFT methodology)
Particle physics document (for QCD basics)
Classical field theory document (for symmetry breaking)

Conventions

Mostly-minus metric
$\hbar = c = 1$
Pion decay constant: $f_\pi \approx 93$ MeV (in “chiral” normalization; other common choices: $F_\pi = f_\pi \sqrt 2 \approx 131$ MeV, physical pion decay constant)
Chiral scale: $\Lambda_\chi = 4\pi f_\pi \approx 1.2$ GeV

The QCD Low-Energy Problem
Chiral Symmetry in QCD
Spontaneous Chiral Symmetry Breaking
The Goldstone Bosons: Pions, Kaons, and Eta
The Chiral Lagrangian at Leading Order
Quark Masses and Pion Masses
The Chiral Power Counting
Pion Scattering at Tree Level
Next-to-Leading Order: Loop Corrections
Including Baryons: Chiral Perturbation with Nucleons
The Axial Anomaly and the $U(1)_A$ Problem
Applications and Precision Tests
Appendix: ChPT Formulas Reference

1. The QCD Low-Energy Problem

QCD at High vs. Low Energy

QCD has two qualitatively different regimes:

High energy ( $E \gg \Lambda_{\rm QCD}$ ): $\alpha_s$ is small. Perturbation theory works. Cross sections at LHC, deep inelastic scattering, hard processes; all computable.

Low energy ( $E \lesssim \Lambda_{\rm QCD}$ ): $\alpha_s$ becomes $O(1)$ . Perturbation theory fails. Quarks and gluons bind into hadrons. Hadronic physics is non-perturbative.

The low-energy regime is where most “everyday” nuclear and particle physics happens; pion scattering, kaon decays, nucleon-nucleon forces, atomic nuclei. Understanding it quantitatively from first principles is the central challenge of low-energy QCD.

Why Perturbative QCD Fails Here

At $E \sim 1$ GeV, $\alpha_s(\mu) \sim 1$ . Perturbative expansion in $\alpha_s$ doesn’t converge; each term is as large as the previous. The theory is non-perturbative in this regime.

Two traditional approaches to non-perturbative QCD:

Lattice QCD: numerical simulation of QCD on a Euclidean lattice
Model-building: constituent quark models, bag models, etc.

Lattice QCD is rigorous but computationally expensive. Models are quicker but lack systematic error estimates.

The EFT Alternative: ChPT

Chiral perturbation theory takes a different approach: identify the relevant low-energy degrees of freedom (the lightest hadrons, which are Goldstone bosons) and write their effective Lagrangian constrained by QCD’s chiral symmetry.

The advantages:

Systematic: clear expansion parameters, known precision
Symmetry-guided: respects QCD’s symmetries by construction
Predictive: many relations between observables from symmetry alone
Computable: operators at each order enumerable, couplings (low-energy constants) fit from data

The disadvantage:

Limited to energies $\lesssim \Lambda_\chi \sim 1$ GeV
Hard to extend to heavier hadrons (vector mesons, baryons require care)
Low-energy constants must be measured or computed on the lattice

Still, ChPT is the most successful low-energy EFT in physics, with thousands of publications and applications across nuclear, particle, and astrophysics.

2. Chiral Symmetry in QCD

The QCD Lagrangian with Light Quarks

Consider QCD with only the three light quarks ( $u, d, s$ ); the heavier quarks ( $c, b, t$ ) are too heavy to be relevant at $E \sim 1$ GeV. The Lagrangian:

$\mathcal{L}_{\rm QCD} = -\tfrac{1}{4}G^a_{\mu\nu}G^{a\mu\nu} + \bar q(i\slashed D - \mathcal{M})q$

where $q = (u, d, s)^T$ is a three-component quark vector (in flavor space) and $\mathcal{M} = \text{diag}(m_u, m_d, m_s)$ is the quark mass matrix.

The quark masses are:

$m_u \approx 2.16$ MeV
$m_d \approx 4.67$ MeV
$m_s \approx 93$ MeV

All much less than $\Lambda_{\rm QCD} \sim 200$ MeV.

Chiral Decomposition

Decompose quark fields into left- and right-handed components:

$q = q_L + q_R, \quad q_L = P_L q, \quad q_R = P_R q$

with $P_{L,R} = (1 \mp \gamma^5)/2$ .

In terms of these:

$\bar q i\slashed D q = \bar q_L i\slashed D q_L + \bar q_R i\slashed D q_R$

The kinetic term separates into independent left and right pieces. But the mass term couples them:

$\bar q \mathcal{M} q = \bar q_L \mathcal{M} q_R + \bar q_R \mathcal{M} q_L$

The Chiral Symmetry $SU(3)_L \times SU(3)_R$

If the quark masses were zero ( $\mathcal{M} = 0$ ), the Lagrangian would have the symmetry:

$q_L \to U_L q_L, \quad q_R \to U_R q_R$

with $U_L, U_R \in SU(3)$ . These are independent transformations acting separately on left- and right-handed quarks. This is the chiral symmetry group:

$G_{\rm chiral} = SU(3)_L \times SU(3)_R$

Each factor is a global flavor symmetry: you can rotate the three light quarks into each other, independently for each chirality.

The Current Algebra

The conserved currents associated with chiral symmetry:

Vector currents: $J^{a\mu}_V = \bar q\gamma^\mu T^a q$ (combining left and right)

Axial currents: $J^{a\mu}_A = \bar q\gamma^\mu\gamma^5 T^a q$ (differences)

Here $T^a$ are $SU(3)$ generators ( $a = 1, \ldots, 8$ ). There are 8 vector currents and 8 axial currents, totaling 16 for the full $SU(3)_L \times SU(3)_R$ .

The Mass Term Breaks Chiral Symmetry Explicitly

Reality: quark masses aren’t zero. The mass term $\bar q\mathcal{M}q$ explicitly breaks chiral symmetry.

But since $m_u, m_d \sim$ MeV and $m_s \sim 100$ MeV are small compared to $\Lambda_{\rm QCD} \sim 200$ MeV, chiral symmetry is an approximate symmetry of QCD. We can treat the breaking perturbatively.

This is the crucial setup for ChPT: chiral symmetry is approximately exact, spontaneously broken (as we’ll see), and the explicit breaking is a small perturbation.

Full Flavor Structure

Actually, QCD with $n_f$ massless flavors has the larger symmetry:

$G = U(n_f)_L \times U(n_f)_R = SU(n_f)_L \times SU(n_f)_R \times U(1)_V \times U(1)_A$

The $U(1)_V$ is baryon number (exactly conserved). The $U(1)_A$ is the axial $U(1)$ , which is anomalous (broken quantum-mechanically; section 11). So the relevant chiral group is $SU(n_f)_L \times SU(n_f)_R$ .

For $n_f = 2$ (just $u, d$ ): $SU(2)_L \times SU(2)_R$ , relevant for pion physics alone.

For $n_f = 3$ ( $u, d, s$ ): $SU(3)_L \times SU(3)_R$ , relevant for kaon and eta physics too.

3. Spontaneous Chiral Symmetry Breaking

The Observational Evidence

Chiral symmetry, even as an approximate symmetry, should appear in the hadron spectrum. If $SU(3)_L \times SU(3)_R$ were realized Wigner-Weyl (linearly), hadrons would come in parity-doublet multiplets: for every state of definite parity, there should be a nearly-degenerate state of opposite parity.

We don’t see this. The observed hadron spectrum has no parity doublets. Pions are pseudoscalars with no scalar partners at nearby mass. Nucleons are protons/neutrons with no comparable-mass opposite-parity states.

This means chiral symmetry is spontaneously broken: realized nonlinearly, not linearly.

The Vacuum Breaking

The QCD vacuum has a nonzero expectation value for the scalar quark bilinear:

$\langle\bar q q\rangle = \langle\bar u u + \bar d d + \bar s s\rangle \neq 0$

More precisely:

$\langle\bar q q\rangle \approx -(240 \text{ MeV})^3 \approx -(1.4\times 10^7 \text{ MeV}^3)$

This is the chiral condensate. Its nonzero value spontaneously breaks chiral symmetry.

Why does it break the symmetry? Consider the transformation under $U_L$ acting on left-handed quarks alone. The operator $\bar q_L q_R$ transforms nontrivially:

$\bar q_L q_R \to \bar q_L U_L^\dagger q_R$

For this to have a nonzero vacuum expectation value, $U_L$ must be restricted; specifically to the diagonal subgroup where $U_L = U_R$ .

The Symmetry Breaking Pattern

The breaking is:

$SU(3)_L \times SU(3)_R \to SU(3)_V$

where $SU(3)_V$ (vector $SU(3)$ ) is the diagonal subgroup with $U_L = U_R$ .

$SU(3)_L \times SU(3)_R$ has $8 + 8 = 16$ generators
$SU(3)_V$ has 8 generators
8 generators are broken: $SU(3)_A$ (axial, where $U_L = U_R^\dagger$ )

By Goldstone’s theorem, 8 broken generators means 8 massless Goldstone bosons.

The Goldstone Boson Spectrum

In the limit of zero quark masses, these 8 Goldstones are massless. Their quantum numbers: pseudoscalars (spin-0, odd parity), flavor multiplet under $SU(3)_V$ . Identifying them with known particles:

3 pions ( $\pi^+, \pi^0, \pi^-$ ): isospin triplet
4 kaons ( $K^+, K^0, \bar K^0, K^-$ ): two doublets
1 eta ( $\eta$ ): isosinglet

That’s 8 pseudoscalars; exactly the number predicted. ✓

The " $\eta'$ " (eta prime, at 958 MeV) is not one of these Goldstones; it’s the ninth pseudoscalar, related to the anomalous $U(1)_A$ symmetry (section 11). In the chiral limit + large- $N$ , $\eta'$ would also be a Goldstone, but the anomaly pushes its mass up.

With Quark Masses

When we include $m_u, m_d, m_s$ , chiral symmetry is explicitly broken. The Goldstones are no longer exactly massless; they acquire pseudo-Goldstone masses proportional to $\sqrt{m_q}$ (see section 6).

Observed masses:

$m_\pi \approx 140$ MeV
$m_K \approx 495$ MeV
$m_\eta \approx 548$ MeV

All much less than typical hadronic scales ( $\sim 1$ GeV). This hierarchy is what makes ChPT work; pions, kaons, and eta are light enough to be the relevant low-energy degrees of freedom.

4. The Goldstone Bosons: Pions, Kaons, and Eta

Parametrizing the Goldstones

Goldstone bosons parameterize the coset $G/H = [SU(3)_L \times SU(3)_R]/SU(3)_V$ . Concretely, we can represent them as elements of a special unitary matrix:

$U(x) = \exp\left[\frac{i\sqrt 2}{f_\pi}\pi^a(x) T^a\right]$

where $T^a$ are $SU(3)$ generators (Gell-Mann matrices divided by 2), $\pi^a$ are the 8 pseudo-Goldstone fields, and $f_\pi$ is the pion decay constant.

Explicitly, the 3×3 matrix $\pi^a T^a$ is:

$\Pi \equiv \pi^a T^a = \frac{1}{\sqrt 2}\begin{pmatrix}\frac{\pi^0}{\sqrt 2} + \frac{\eta}{\sqrt 6} & \pi^+ & K^+ \\ \pi^- & -\frac{\pi^0}{\sqrt 2} + \frac{\eta}{\sqrt 6} & K^0 \\ K^- & \bar K^0 & -\frac{2\eta}{\sqrt 6}\end{pmatrix}$

The normalization with $\sqrt 2/f_\pi$ is conventional (with $f_\pi \approx 93$ MeV in this convention).

How $U(x)$ Transforms

Under chiral transformations $(g_L, g_R) \in SU(3)_L \times SU(3)_R$ :

$U(x) \to g_L\, U(x)\, g_R^\dagger$

This is the nonlinear realization of the chiral symmetry on the Goldstone fields. Under the unbroken diagonal $SU(3)_V$ (with $g_L = g_R = g_V$ ):

$U(x) \to g_V\, U(x)\, g_V^\dagger$

So $U$ transforms as a bi-fundamental under $SU(3)_L \times SU(3)_R$ , adjoint under $SU(3)_V$ .

Why This Parametrization?

The advantage of $U(x) \in SU(3)$ : it’s a nonlinear realization, manifesting all the symmetries automatically. The pion field is defined nonlinearly, and interactions emerge from the geometry of the group manifold.

This is nonlinear sigma model formulation; the standard approach for spontaneously-broken theories where Goldstones parameterize the broken coset.

Small-Field Expansion

For small fields (low energies):

$U(x) = 1 + \frac{i\sqrt 2}{f_\pi}\Pi(x) - \frac{1}{f_\pi^2}\Pi(x)^2 + \ldots$

The leading derivative of $U$ :

$\partial_\mu U = \frac{i\sqrt 2}{f_\pi}\partial_\mu\Pi + O(\Pi^2)$

$\partial_\mu U^\dagger = -\frac{i\sqrt 2}{f_\pi}\partial_\mu\Pi + O(\Pi^2)$

Products like $U^\dagger\partial_\mu U$ are what appear in the Lagrangian:

$U^\dagger\partial_\mu U = \frac{i\sqrt 2}{f_\pi}\partial_\mu\Pi + \frac{1}{f_\pi^2}[\Pi, \partial_\mu\Pi] + \ldots$

These expansions give the tower of pion interactions; two-pion, four-pion, six-pion, etc.

5. The Chiral Lagrangian at Leading Order

The Lagrangian

The leading-order chiral Lagrangian (in the chiral limit, zero quark masses):

$\mathcal{L}^{(2)} = \frac{f_\pi^2}{4}\text{Tr}[\partial_\mu U^\dagger\partial^\mu U]$

This is the $\sigma$ -model Lagrangian on the coset $SU(3) \simeq [SU(3)_L \times SU(3)_R]/SU(3)_V$ .

Why This Is the Leading Term

The Lagrangian must be:

Chirally invariant: $\mathcal{L}[U] = \mathcal{L}[g_L U g_R^\dagger]$ for any $g_L, g_R$
Lorentz invariant: built from $U$ , $\partial_\mu U$ , $\partial^\mu\partial_\nu U$ , etc.
Hermitian: $\mathcal{L} = \mathcal{L}^*$

Operators with different numbers of derivatives (2, 4, 6, …) have different scaling in $p/\Lambda_\chi$ . The 2-derivative term is the leading one.

The coefficient $f_\pi^2/4$ is fixed by requiring the kinetic term of pions to have canonical normalization. From the small-field expansion:

$\frac{f_\pi^2}{4}\text{Tr}[\partial U^\dagger\partial U] = \frac{f_\pi^2}{4}\text{Tr}\left[\frac{2}{f_\pi^2}\partial\Pi\partial\Pi + \ldots\right] = \tfrac{1}{2}\partial\pi^a\partial\pi^a + \ldots$

Canonical kinetic term for each pion. ✓

Higher Orders

Higher-dimension operators include:

4-derivative (Gasser-Leutwyler):

$\mathcal{L}^{(4)} = L_1[\text{Tr}(\partial_\mu U^\dagger\partial^\mu U)]^2 + L_2\text{Tr}(\partial_\mu U^\dagger\partial_\nu U)\text{Tr}(\partial^\mu U^\dagger\partial^\nu U) + \ldots$

With many terms, each with a “low-energy constant” $L_i$ . These are determined from data.

6-derivative:

Even more terms. Typically not used in leading-precision predictions.

Symmetry Currents

The 16 currents of chiral symmetry ( $8+8$ for $SU(3)_L \times SU(3)_R$ ) can be computed from $\mathcal{L}^{(2)}$ via Noether’s theorem.

The axial current (broken):

$J^{a\mu}_A = -i f_\pi^2\text{Tr}[T^a(U^\dagger\partial^\mu U - U\partial^\mu U^\dagger)] + O(\Pi^2)$

Expanding in small fields:

$J^{a\mu}_A \approx f_\pi\partial^\mu\pi^a + O(\Pi^3)$

This says the axial current is proportional to the derivative of the pion field. The overlap between $\langle 0|J^{a\mu}_A|\pi^b\rangle = i f_\pi p^\mu\delta^{ab}$ is what identifies $f_\pi$ as the pion decay constant.

Meson Decay Constants

From the leading-order Lagrangian, all meson decay constants (pion, kaon, eta) equal $f_\pi$ . This is the “pion decay constant,” which despite its name governs all Goldstones.

Experimentally: $f_\pi \approx 93$ MeV. And the chiral scale is:

$\boxed{\Lambda_\chi = 4\pi f_\pi \approx 1.2 \text{ GeV}}$

6. Quark Masses and Pion Masses

Incorporating Quark Masses

The mass term $\bar q\mathcal{M}q$ in QCD transforms as $(\bar 3, 3)$ under $SU(3)_L \times SU(3)_R$ (left-handed triplet, right-handed anti-triplet). To encode this in the chiral Lagrangian, include the mass matrix as a “spurion”:

$\chi \equiv 2B_0\mathcal{M} = 2B_0\text{diag}(m_u, m_d, m_s)$

where $B_0$ is a dimensionful constant ( $\sim \Lambda^3_{\rm QCD}/f_\pi^2$ , to be determined by the chiral condensate).

The mass term in the chiral Lagrangian:

$\mathcal{L}^{(2)}_{\rm mass} = \frac{f_\pi^2}{4}\text{Tr}[\chi^\dagger U + U^\dagger\chi]$

This transforms correctly under chiral symmetry (as $(\bar 3, 3) + (3, \bar 3)$ , which combines properly).

The Pion Masses

Expanding the mass Lagrangian in small pion fields:

$\text{Tr}[\chi U + U^\dagger\chi] = \text{Tr}[\chi(U + U^\dagger)] = 2\text{Tr}\chi - \frac{1}{f_\pi^2}\text{Tr}[\chi\Pi^2] + O(\Pi^4)$

The constant term is irrelevant (vacuum energy). The quadratic term gives pion masses:

$-\frac{1}{f_\pi^2}\text{Tr}[\chi\Pi^2]$

Combining with the kinetic term, we get a mass matrix for the pseudo-Goldstones.

The Gell-Mann-Oakes-Renner Relation

Computing the trace for the three pion states $\pi^+, \pi^-, \pi^0$ :

$m_{\pi^\pm}^2 = B_0(m_u + m_d)$

$m_{\pi^0}^2 = B_0(m_u + m_d) + O(\epsilon^2) \text{ where } \epsilon = (m_d - m_u)$

For the kaons:

$m_{K^\pm}^2 = B_0(m_u + m_s)$

$m_{K^0}^2 = B_0(m_d + m_s)$

For the eta (at leading order in chiral symmetry breaking):

$m_\eta^2 = \frac{B_0}{3}(m_u + m_d + 4m_s)$

This is the Gell-Mann-Oakes-Renner (GMOR) relation; the fundamental connection between quark masses and pion masses.

Determining $B_0$

The chiral condensate $\langle\bar q q\rangle = -f_\pi^2 B_0$ (at leading order). So:

$B_0 = -\frac{\langle\bar q q\rangle}{f_\pi^2}$

With $\langle\bar q q\rangle \approx -(240 \text{ MeV})^3$ and $f_\pi = 93$ MeV:

$B_0 \approx \frac{(240)^3}{(93)^2} \approx 1600 \text{ MeV}$

The Gell-Mann-Okubo Relation

From GMOR, there’s a linear combination of meson masses that vanishes:

$4m_K^2 = m_\pi^2 + 3m_\eta^2$

Checking: $4\cdot 495^2 \approx 980000$ vs. $140^2 + 3\cdot 548^2 \approx 919000$ . Agreement within $\sim 6\%$ .

This is the Gell-Mann-Okubo relation; a non-trivial test of chiral symmetry. The 6% disagreement reflects higher-order chiral corrections.

Why This Matters

The GMOR relation is deep: it says that the squared masses of Goldstones are proportional to quark masses. Not masses themselves, but masses squared. This is what “pseudo-Goldstone boson” means; masses vanishing in the chiral limit, scaling as $\sqrt{m_q}$ .

This is why pions are so much lighter than other hadrons: pions are pseudo-Goldstones of broken chiral symmetry, pinned to zero mass in the chiral limit.

7. The Chiral Power Counting

The Expansion Parameter

ChPT is organized as an expansion in:

$\frac{p, m_\pi}{\Lambda_\chi}$

where $p$ is typical momentum and $m_\pi$ the pion mass. Both are treated as the same order (“chiral order” $p^2$ ).

Specifically, each operator in the chiral Lagrangian is classified by its “chiral dimension”:

$\mathcal{L}^{(2)}$ : two derivatives OR one quark-mass insertion → “chiral dimension 2”
$\mathcal{L}^{(4)}$ : four derivatives OR two mass insertions OR two derivatives + one mass → “chiral dimension 4”
$\mathcal{L}^{(6)}$ : six derivatives, etc.

The Weinberg Power Counting

Weinberg’s theorem (1979): for a chiral Lagrangian, a Feynman diagram with:

$L$ loops
$I_n$ internal lines
$V_n$ vertices of order $p^n$ (i.e., from $\mathcal{L}^{(n)}$ )

has chiral dimension:

$D = 2 + 2L + \sum_n V_n(n - 2)$

For $L = 0$ : tree-level diagrams from $\mathcal{L}^{(n)}$ have dimension $n$ . So only $\mathcal{L}^{(2)}$ contributes at leading chiral order $p^2$ .

For $L = 1$ : loops from $\mathcal{L}^{(2)}$ contribute at order $p^4$ . Together with tree-level $\mathcal{L}^{(4)}$ , this gives “next-to-leading order” (NLO) $p^4$ .

For $L = 2$ : loops from $\mathcal{L}^{(2)}$ or loops combined with $\mathcal{L}^{(4)}$ give $p^6$ (NNLO), together with tree-level $\mathcal{L}^{(6)}$ .

Physical Interpretation

The hierarchy of precision:

Leading order ( $p^2$ ): tree-level from $\mathcal{L}^{(2)}$ . No free parameters beyond $f_\pi, B_0, m_q$ .
NLO ( $p^4$ ): adds the 10 Gasser-Leutwyler coefficients $L_1, \ldots, L_{10}$ .
NNLO ( $p^6$ ): adds many more coefficients.

At each order, the predictions are finite (after renormalization) and increasingly precise. Most ChPT calculations are done at NLO or NNLO for today’s precision requirements.

Why $\Lambda_\chi = 4\pi f_\pi$

Where does the scale $\Lambda_\chi \sim 1$ GeV come from? It’s the scale at which loops become order 1.

Each loop in ChPT contributes a factor of:

$\frac{p^2}{(4\pi f_\pi)^2}$

For $p \ll 4\pi f_\pi$ , loops are small. For $p \sim 4\pi f_\pi \sim \Lambda_\chi$ , loops are $O(1)$ ; the expansion breaks down.

The hierarchy $f_\pi \ll \Lambda_\chi \sim 4\pi f_\pi \ll$ hadronic masses puts ChPT on solid ground for energies $\lesssim$ 500 MeV. Above that, the expansion is in trouble.

Why This Is “Perturbation Theory”

The word “perturbation” in ChPT is about the small-momentum expansion, not about small couplings. The underlying theory (QCD) is strongly coupled; only the low-momentum approximation is perturbative.

8. Pion Scattering at Tree Level

The Process: $\pi\pi \to \pi\pi$

Consider charged pion scattering: $\pi^+(p_1)\pi^-(p_2) \to \pi^+(p_3)\pi^-(p_4)$ .

At leading order (tree-level from $\mathcal{L}^{(2)}$ ), this is computed from the 4-pion interaction in the chiral Lagrangian.

Extracting the 4-Pion Vertex

From the expansion of $\text{Tr}[\partial U^\dagger\partial U]$ to quartic order in fields:

$\text{Tr}[\partial U^\dagger\partial U] = \frac{2}{f_\pi^2}\text{Tr}[\partial\Pi\partial\Pi] + \frac{4}{3f_\pi^4}\text{Tr}[\partial\Pi[\Pi, [\Pi, \partial\Pi]]] + \ldots$

Actually, let me just use the result. Expanding $U = \exp(i\sqrt 2\Pi/f_\pi)$ and computing the kinetic term to 4-pion order:

$\mathcal{L}^{(2)}|_{4\pi} = \frac{1}{6f_\pi^2}[(\pi^a\partial_\mu\pi^a)(\pi^b\partial^\mu\pi^b) - (\pi^a\pi^a)(\partial_\mu\pi^b\partial^\mu\pi^b)]$

(There are also mass-dependent terms from the mass Lagrangian.)

The Amplitude

For $\pi^+\pi^- \to \pi^+\pi^-$ scattering, the tree-level amplitude from the 4-pion vertex:

$\mathcal{A}(\pi^+\pi^- \to \pi^+\pi^-) = \frac{1}{f_\pi^2}[s - m_\pi^2]$

(At leading order in ChPT, in the isospin symmetric limit.)

Similarly for other channels:

$\mathcal{A}(\pi^+\pi^+ \to \pi^+\pi^+) = \frac{t + u - 2m_\pi^2}{f_\pi^2}$

$\mathcal{A}(\pi^+\pi^- \to \pi^0\pi^0) = \frac{s - m_\pi^2}{f_\pi^2}$

(The isospin structure constrains these to be related.)

The Weinberg-Tomozawa Relation

Near threshold, the scattering lengths can be extracted. For example:

$a_0^2 = \frac{m_\pi^2}{16\pi f_\pi^2}\cdot(-)\left[\frac{1}{3}\cdot(\text{appropriate isospin coefficient})\right]$

where $a_0^I$ is the S-wave scattering length in isospin channel $I$ .

Weinberg’s result (1966) for $\pi\pi$ scattering at threshold:

$a_0^0 \approx +0.16\, m_\pi^{-1}, \quad a_0^2 \approx -0.045\, m_\pi^{-1}$

These are predictions from current algebra + spontaneous chiral symmetry breaking, no free parameters!

Experimental Test

Measurements give:

$a_0^0 = 0.220(5) m_\pi^{-1}, \quad a_0^2 = -0.044(1) m_\pi^{-1}$

The tree-level ChPT predicts the isospin-2 scattering length within experimental errors. The isospin-0 is about 30% off, consistent with NLO corrections being important.

Higher-Order Corrections

At NLO ( $p^4$ ), ChPT includes:

Tree-level contributions from $\mathcal{L}^{(4)}$ (with 10 low-energy constants)
One-loop contributions from $\mathcal{L}^{(2)}$

The full NLO calculation gives:

$a_0^0 = 0.220\, m_\pi^{-1}$

at NLO, matching experiment to better than 1%.

This is the power of chiral perturbation theory: a strongly-coupled theory (QCD) has its low-energy physics characterized by an expansion that converges order-by-order in $p^2/\Lambda_\chi^2$ , with very few parameters (the $L_i$ ‘s fit from data).

9. Next-to-Leading Order: Loop Corrections

The Gasser-Leutwyler Lagrangian

At $p^4$ order, the chiral Lagrangian includes 10 independent terms (in the $SU(3)_L \times SU(3)_R$ case):

$\mathcal{L}^{(4)} = L_1[\text{Tr}(\partial_\mu U^\dagger\partial^\mu U)]^2 + L_2\text{Tr}(\partial_\mu U^\dagger\partial_\nu U)\text{Tr}(\partial^\mu U^\dagger\partial^\nu U)$

$+ L_3\text{Tr}(\partial_\mu U^\dagger\partial^\mu U\partial_\nu U^\dagger\partial^\nu U) + L_4\text{Tr}(\partial_\mu U^\dagger\partial^\mu U)\text{Tr}(\chi^\dagger U + U^\dagger\chi)$

$+ L_5\text{Tr}[\partial_\mu U^\dagger\partial^\mu U(\chi^\dagger U + U^\dagger\chi)] + L_6[\text{Tr}(\chi^\dagger U + U^\dagger\chi)]^2$

$+ L_7[\text{Tr}(\chi^\dagger U - U^\dagger\chi)]^2 + L_8\text{Tr}[(\chi^\dagger U)^2 + (U^\dagger\chi)^2] + \ldots$

(With two more terms $L_9, L_{10}$ involving external vector and axial-vector sources, relevant for electromagnetic interactions.)

Each $L_i$ is a low-energy constant to be determined from experiment or lattice QCD.

Loop Contributions

At NLO, loops of pions contribute. Example: the pion self-energy at one loop.

The one-loop pion tadpole diagram:

$\Sigma_{\rm tadpole} = \frac{1}{f_\pi^2}\int\frac{d^4 k}{(2\pi)^4}\frac{1}{k^2 - m_\pi^2 + i\epsilon}\cdot(\text{combinatorial factor})$

This integral is UV divergent. In dimensional regularization:

$\int\frac{d^d k}{(2\pi)^d}\frac{1}{k^2 - m_\pi^2} = \frac{-i m_\pi^2}{(4\pi)^2}\left[\frac{1}{\epsilon} - \ln\frac{m_\pi^2}{\mu^2} + 1 - \gamma_E + \ln(4\pi)\right]$

(At $d = 4 - 2\epsilon$ .)

The $1/\epsilon$ pole must be absorbed into counterterms. ChPT is renormalized order-by-order: divergences at $p^4$ are absorbed into the $L_i$ ‘s.

The Renormalized Low-Energy Constants

The $L_i$ ‘s are “bare” coefficients that receive additive corrections from loop divergences. After subtraction:

$L_i = L_i^r(\mu) + \gamma_i\cdot(\text{universal})$

where $L_i^r(\mu)$ is the renormalized coefficient at scale $\mu$ , and $\gamma_i$ are computable coefficients.

The renormalization group then runs $L_i^r$ with $\mu$ :

$\mu\frac{dL_i^r}{d\mu} = \text{finite coefficient}$

The $L_i^r$ at a standard scale ( $\mu = M_\rho$ , the rho meson mass, $\sim 770$ MeV) are the “physical” low-energy constants that one quotes.

Measured Values of LECs

From combined fits to experimental data and lattice QCD:

$L_1^r(M_\rho) \approx 0.4\times 10^{-3}$ $L_2^r(M_\rho) \approx 1.4\times 10^{-3}$ $L_3^r(M_\rho) \approx -3.5\times 10^{-3}$ $L_4^r(M_\rho) \approx 0.3\times 10^{-3}$ $L_5^r(M_\rho) \approx 1.4\times 10^{-3}$ $L_6^r(M_\rho) \approx 0.1\times 10^{-3}$ $L_7^r(M_\rho) \approx -0.3\times 10^{-3}$ $L_8^r(M_\rho) \approx 0.9\times 10^{-3}$ $L_9^r(M_\rho) \approx 6.0\times 10^{-3}$ $L_{10}^r(M_\rho) \approx -5.1\times 10^{-3}$

These are all of order $10^{-3}$ ; small in the appropriate sense for a good chiral expansion.

A Non-Trivial Prediction

With NLO ChPT, you can compute many observables and get precision matching experiment:

Pion-pion scattering lengths: less than 1% accuracy
Pion form factors: few % accuracy
Kaon form factors: few % accuracy
$K \to \pi$ transitions
Electromagnetic polarizabilities of pions

Each of these comes from the same Lagrangian, same $L_i$ ‘s, same power counting. The agreement validates the entire chiral EFT framework.

The Role of Lattice QCD

Modern precision ChPT relies heavily on lattice QCD calculations of the low-energy constants. Lattice can compute:

$f_\pi$ at physical quark masses (well-measured)
Chiral condensate $\langle\bar q q\rangle$
Ratios of decay constants ( $f_K/f_\pi$ , $f_\eta/f_\pi$ )
Values of $L_i$ (or Gasser-Leutwyler constants directly)

Combining ChPT (which organizes the expansion) with lattice QCD (which provides non-perturbative input) is the modern approach to low-energy QCD.

10. Including Baryons: Chiral Perturbation with Nucleons

The Problem

Baryons (protons, neutrons) are not Goldstones. They have mass $\sim 1$ GeV, comparable to the chiral scale $\Lambda_\chi$ . Including them in ChPT requires extra care.

Two Approaches

Heavy Baryon ChPT (HBChPT): Treat nucleons as heavy static sources with non-relativistic expansion in $1/m_N$ . Introduces the nucleon velocity $v^\mu$ and expands observables in small kinetic energy.

Relativistic ChPT: Keep nucleons fully relativistic. More complicated (issues with power counting), but preserves Lorentz invariance manifestly. Recent renaissance with “infrared regularization” or “extended on-mass-shell” schemes.

The Nucleon Lagrangian

The leading-order nucleon Lagrangian couples nucleons to pions:

$\mathcal{L}_{\rm N}^{(1)} = \bar N(i\slashed\partial - m_N + g_A\slashed u\gamma^5)N$

where $N = (p, n)^T$ is the nucleon doublet, $u^\mu$ is a combination of pion derivatives, and $g_A \approx 1.27$ is the axial coupling of the nucleon. Here $m_N \approx 940$ MeV is the nucleon mass.

Pion-Nucleon Coupling

The axial coupling $g_A$ is measured from beta decay: $n \to p + e^- + \bar\nu_e$ . The amplitude has a factor of $g_A$ from the nucleon axial current matrix element.

The pion-nucleon coupling is:

$\mathcal{L}_{\pi NN} = \frac{g_A}{f_\pi}\bar N\gamma^\mu\gamma^5\vec\tau\cdot\partial_\mu\vec\pi N$

(Where $\vec\tau$ are Pauli isospin matrices.) The coupling strength $g_A/f_\pi \approx 13.7$ is related to the pion-nucleon coupling constant $g_{\pi NN}$ via the Goldberger-Treiman relation:

$g_{\pi NN} f_\pi = g_A m_N$

Checking: $13.7\cdot 93 \approx 1274$ MeV $\approx g_A m_N = 1.27\cdot 939 \approx 1193$ MeV. Agrees to ~10% (higher-order corrections).

Chiral Power Counting for Baryons

With baryons, chiral power counting works but needs the modification that $m_N \sim \Lambda_\chi$ . In HBChPT, the nucleon kinematics ( $p = m_N v + k$ with soft $k$ ) gives clean power counting.

The leading chiral order for a diagram with nucleons is:

$D = 2L - 2I_\pi - I_N + 2\sum V^{(n)}_n + \sum V^{(n)}_N$

(With various corrections for nucleon vs. meson lines.) This allows systematic construction of baryon chiral amplitudes.

Nuclear Physics Applications

Chiral perturbation theory applied to baryons gives low-energy nuclear forces. Modern nuclear physics uses:

ChPT-based nuclear forces: Derive the nucleon-nucleon interaction systematically from pion exchange + short-range interactions (encoded in LECs).

Few-body nuclear physics: Solve for light nuclei (deuteron, triton, alpha particle) using ChPT interactions.

Nuclear matter: Extrapolate to many-body systems relevant for neutron stars.

The framework is systematic: at each order in the chiral expansion, you know exactly what operators to include and what their coefficients must satisfy.

11. The Axial Anomaly and the $U(1)_A$ Problem

The Missing Goldstone

Naively, $n_f$ massless flavors give $n_f^2$ Goldstone bosons from $[U(n_f)_L \times U(n_f)_R]/U(n_f)_V$ . For $n_f = 3$ : $9$ Goldstones.

But we identified only 8: $\pi^{\pm 0}, K^{\pm 0}, \bar K^0, \eta$ . Where’s the ninth?

The observed 9th pseudoscalar is the $\eta'$ at 958 MeV; much heavier than the octet. Not a Goldstone.

The $U(1)_A$ Problem

The missing Goldstone corresponds to the axial $U(1)_A$ transformation:

$q \to e^{i\alpha\gamma^5}q$

Classically, this is a symmetry of the massless QCD Lagrangian. Spontaneous breaking would give a 9th Goldstone, which should be light ( $\sim m_\pi$ ).

But the $\eta'$ is heavy ( $958$ MeV). What’s happening?

The Resolution: The Anomaly

The $U(1)_A$ symmetry is anomalous; broken by quantum effects (the chiral anomaly from document 10). The anomalous current has non-vanishing divergence:

$\partial_\mu J^\mu_{A,U(1)} = \frac{g_s^2 n_f}{16\pi^2}G^a_{\mu\nu}\tilde G^{a\mu\nu}$

The right side is the topological charge density of QCD. It’s nonzero on instanton backgrounds.

Because the symmetry is broken by the anomaly, there’s no Goldstone. The $\eta'$ mass doesn’t vanish in the chiral limit; it remains at $\sim 900$ MeV even as $m_q \to 0$ .

The Witten-Veneziano Formula

A quantitative relation (in the large- $N$ limit):

$m_{\eta'}^2 = m_\eta^2 + \frac{2N}{f_\pi^2}\chi_t$

where $\chi_t$ is the topological susceptibility of QCD (a non-perturbative quantity from the QCD vacuum structure).

Lattice QCD measures $\chi_t \approx (180\text{ MeV})^4$ , predicting $m_{\eta'} \approx 860$ MeV. Observed: 958 MeV. Reasonable agreement, confirming the anomaly-based explanation.

The Strong CP Problem

The same anomaly coefficient also allows a $\theta$ -term in the QCD Lagrangian:

$\mathcal{L}_\theta = \frac{\theta}{32\pi^2}g_s^2 G^a_{\mu\nu}\tilde G^{a\mu\nu}$

If $\theta \neq 0$ , QCD violates CP. Experiment limits: $|\theta| < 10^{-10}$ . This is the strong CP problem; why is $\theta$ so tiny?

Proposed solution: the Peccei-Quinn mechanism, which introduces a new symmetry that dynamically sets $\theta \to 0$ . Predicts a light pseudo-Goldstone called the axion. Axion searches (ADMX, MADMAX) are ongoing.

Interplay with ChPT

In ChPT proper (for the 8 Goldstones), the $U(1)_A$ anomaly is accounted for by not including $\eta'$ in the effective theory. The $\eta'$ is treated as a heavier field outside ChPT.

Extensions of ChPT including the $\eta'$ (for certain processes) include explicit anomaly terms modifying the chiral Lagrangian.

12. Applications and Precision Tests

Pion-Pion Scattering Lengths

The most classic ChPT test. NLO ChPT predicts:

$a_0^0 = 0.220(5)\, m_\pi^{-1}$

Measured (from pionium at DIRAC experiment):

$a_0^0 = 0.221(18)\, m_\pi^{-1}$

Agreement within errors. Precise test of chiral dynamics.

Electromagnetic Form Factor of Pion

The pion’s electromagnetic form factor $F_\pi(q^2)$ is measured in $e^+e^- \to \pi^+\pi^-$ and pion scattering. NLO ChPT predicts:

$\langle r_\pi^2\rangle = 0.43\text{ fm}^2$

Measured:

$\langle r_\pi^2\rangle = 0.44\text{ fm}^2$

Match at the percent level.

$K_{\ell 3}$ Decays

The semileptonic kaon decay $K \to \pi e\nu$ has a form factor $f_+(q^2)$ extracted from experiment. ChPT predicts its $q^2$ dependence:

$f_+(q^2) = f_+(0)\left[1 + \frac{q^2}{m_\pi^2}\lambda_+ + \ldots\right]$

with $\lambda_+$ calculable in ChPT. Agreement with data validates the theory.

CP Violation in Kaon System

Indirect CP violation in $K \to \pi\pi$ (parameter $\epsilon$ ) receives contributions from box diagrams with W and top quarks. Computing the mixing amplitude requires:

Electroweak loops at $M_W$
RG evolution from $M_W$ to $m_b$
Hadronic matrix elements computed in ChPT + lattice

This multi-scale calculation is a triumph of EFT methods + ChPT.

Chiral Extrapolation

Lattice QCD calculations are typically done at quark masses heavier than physical (for computational cost reasons). ChPT provides the framework to extrapolate to physical $m_q$ :

$\mathcal{O}_{\rm observed} = \mathcal{O}_{\rm lattice} + (\text{ChPT correction})(m_q^{\rm lattice} - m_q^{\rm physical})$

This is how modern lattice QCD converts unphysical-mass results into physical predictions.

Nuclear Forces and Light Nuclei

ChPT for baryons + chiral nuclear forces has enabled first-principles calculations of:

Deuteron binding energy
Triton and alpha-particle structures
Nuclear matter equation of state (partial, depending on extrapolation)

Modern nuclear physics is increasingly based on chiral EFT frameworks.

Precision on CMB

The early universe’s nuclear physics (at $T \sim$ MeV) affects Big Bang Nucleosynthesis predictions. ChPT provides the weak interaction rates and nuclear cross sections needed for precision BBN.

Connection to Flavor Physics

Many flavor-physics observables (rare kaon and pion decays, CP violation) rely on ChPT for hadronic matrix elements. The predictions, validated by measurements, constrain beyond-SM physics indirectly.

13. Appendix: ChPT Formulas Reference

Parameters

Pion decay constant: $f_\pi \approx 93$ MeV
Chiral scale: $\Lambda_\chi \approx 4\pi f_\pi \approx 1.2$ GeV
Chiral condensate: $\langle\bar q q\rangle \approx -(240 \text{ MeV})^3$
$B_0 = -\langle\bar q q\rangle/f_\pi^2 \approx 1600$ MeV

Quark Masses (MS-bar at 2 GeV)

$m_u \approx 2.16$ MeV
$m_d \approx 4.67$ MeV
$m_s \approx 93$ MeV
$m_u + m_d \approx 7$ MeV

Goldstone Boson Masses

$m_{\pi^\pm} = 139.57$ MeV, $m_{\pi^0} = 134.98$ MeV
$m_{K^\pm} = 493.68$ MeV, $m_{K^0} = 497.61$ MeV
$m_\eta = 547.86$ MeV
$m_{\eta'} = 957.78$ MeV (not a Goldstone; $U(1)_A$ anomaly)

The $U(x)$ Field

$U(x) = \exp\left[\frac{i\sqrt 2}{f_\pi}\pi^a T^a\right]$

Transforms as $U \to g_L U g_R^\dagger$ .

Leading-Order Lagrangian

$\mathcal{L}^{(2)} = \frac{f_\pi^2}{4}\text{Tr}[\partial_\mu U^\dagger\partial^\mu U + \chi^\dagger U + U^\dagger\chi]$

where $\chi = 2B_0 \mathcal{M}$ .

GMOR Relation

$m_\pi^2 \approx B_0(m_u + m_d)$

$m_K^2 \approx B_0(m_u + m_s)$

Gell-Mann-Okubo Relation

$4m_K^2 - m_\pi^2 - 3m_\eta^2 = 0 \text{ (approximately)}$

Axial Current (LO)

$J^{a\mu}_A = f_\pi\partial^\mu\pi^a + O(\pi^3)$

Pion Decay Constant Definition

$\langle 0|J^{a\mu}_A|\pi^b(p)\rangle = if_\pi p^\mu\delta^{ab}$

Power Counting

Each operator in $\mathcal{L}^{(n)}$ has chiral dimension $n$ :

$\mathcal{L}^{(2)}$ : 2 derivatives or 1 mass insertion
$\mathcal{L}^{(4)}$ : 4 derivatives, 2 masses, or 2+1
etc.

Weinberg’s formula: $D = 2 + 2L + \sum_n V^{(n)}(n-2)$

Problems

Derive the Gell-Mann-Okubo relation $4m_K^2 - m_\pi^2 - 3m_\eta^2 = 0$ from GMOR.
Starting from $U = \exp(i\sqrt 2\Pi/f_\pi)$ , compute $U^\dagger\partial_\mu U$ to third order in $\Pi$ , identifying the 3-pion vertex.
Show that at leading order in ChPT, $f_\pi = f_K$ (meson decay constants universal). Calculate the LO correction to this relation.
Derive the Weinberg-Tomozawa relation for $\pi\pi$ scattering lengths at threshold.
For the $K \to 3\pi$ decays, show that at LO, the amplitudes satisfy specific symmetry relations (this is Furry’s theorem in the kaon system).
Compute the one-loop pion tadpole and show its divergence is absorbed into a renormalization of $f_\pi$ .

Closing Note

Chiral perturbation theory is the paradigm for how to do EFT with spontaneously broken symmetries and strongly coupled underlying theories. The key steps:

Identify symmetry: chiral symmetry of massless QCD
Identify breaking: spontaneous, chiral condensate
Identify Goldstones: 8 pseudo-Goldstones ( $\pi, K, \eta$ )
Parametrize: $U(x) \in SU(3)$
Write Lagrangian: $\mathcal{L}^{(2)}$ + higher orders
Power counting: $p/\Lambda_\chi$ , $m_\pi/\Lambda_\chi$
Compute: order by order, matching experiment

What Makes ChPT Work

Symmetries are (approximately) exact
Scale hierarchy ( $m_\pi \ll \Lambda_\chi$ ) is real
The expansion converges in the regime $p \ll \Lambda_\chi$
Predictions are testable and verified

The framework applies beyond QCD: any spontaneously broken symmetry gives a ChPT-like EFT for the Goldstones. This pattern recurs throughout physics:

Superfluids → phonon EFTs
Ferromagnets → magnon EFTs
Superconductors → Goldstone + electromagnetism
Spontaneous chiral symmetry breaking in BSM models

What’s Next

Document 16 covers SMEFT (Standard Model EFT) and HQET (Heavy Quark EFT). Both are direct applications of the EFT framework:

SMEFT is the bottom-up EFT for beyond-SM physics. It’s what LHC phenomenology is built on; a systematic expansion in new-physics operators with cutoff $\Lambda$ .
HQET makes heavy-quark systems ( $B, D$ mesons, heavy baryons) tractable by expanding in $1/m_Q$ . It’s how we understand B physics.

Both are modern, extensively-applied EFTs at the frontier of particle physics research. Together with ChPT, they show the breadth of EFT applications in contemporary physics.

Document 16 is next.