Chiral Symmetry

Chiral Symmetry is a fundamental concept in theoretical physics, particularly within quantum field theory (QFT) and particle physics, referring to the invariance of physical laws under independent, local or global transformations of the left-handed and right-handed components of spinor fields (fermions) ¹. This symmetry is intimately connected to the concept of parity, the spatial inversion operation ($\mathbf{P}$), and plays a critical role in determining the nature of fundamental interactions and the generation of particle masses.

Mathematical Formulation

In the context of a free Dirac field $\psi$, the transformation associated with chiral symmetry is defined by the generators acting separately on the left-handed ($\psi_L$) and right-handed ($\psi_R$) Weyl spinors: $$ \psi_L \rightarrow e^{i\alpha_L} \psi_L \quad \text{and} \quad \psi_R \rightarrow e^{i\alpha_R} \psi_R $$ where $\alpha_L$ and $\alpha_R$ are independent parameters.

A general transformation that mixes these components is defined using the chiral matrices $\gamma_5$, which, in the standard Dirac representation, is a purely imaginary $4\times 4$ matrix satisfying $\gamma_5^2 = I_4$ and ${\gamma_\mu, \gamma_5} = 0$. The full chiral transformation is: $$ \psi \rightarrow e^{i\alpha \gamma_5} \psi $$ If $\alpha_L = \alpha_R = \alpha$, the symmetry is called vector symmetry ($U(1)_V$), associated with the conservation of baryon number or lepton number. If $\alpha_L = -\alpha_R = \alpha$, the symmetry is called axial symmetry ($U(1)_A$). Chiral symmetry encompasses both vector and axial transformations simultaneously, often denoted $U(1)_V \times U(1)_A$ for $U(1)$ theories, or $SU(N)_L \times SU(N)_R$ for non-Abelian theories ².

Connection to Mass and the Dirac Equation

The presence of a mass term in the Lagrangian is the primary source of explicit breaking for chiral symmetry. For a massless Dirac fermion $\psi$, the Lagrangian density is invariant under chiral transformations. However, the kinetic term alone, $$ \mathcal{L}{\text{kin}} = i \bar{\psi} \not{\partial} \psi $$ is not sufficient to guarantee chiral invariance if mass is present. The Dirac mass term explicitly breaks the symmetry: $$ \mathcal{L} \psi $$ The Lagrangian density including the mass term is $\mathcal{L} = i \bar{\psi} \not{\partial} \psi - m \bar{\psi} \psi$. If the mass $m$ is zero, the Lagrangian is chirally invariant. The explicit term breaking the symmetry is $2m \bar{\psi} \gamma_5 \psi$ when considering the axial current divergence relationship }} = -m (\bar{\psi}_L \psi_R + \bar{\psi}_R \psi_L) = -m \bar{\psi³.

If a fermion possesses a non-zero bare mass $m_0$, the associated chiral symmetry is explicitly broken. Conversely, if $m_0 = 0$, the symmetry may still be broken spontaneously or by quantum effects (anomalies).

Spontaneous Breaking in Quantum Chromodynamics (QCD)

The most physically relevant realization of chiral symmetry occurs in the context of Quantum Chromodynamics (QCD), the theory of the strong nuclear force.

The Chiral Limit

In the idealized chiral limit, the masses of the lightest quarks, the up ($m_u$) and down ($m_d$) quarks, are set to zero ($m_u = m_d \approx 0$). In this limit, the QCD Lagrangian exhibits an approximate global chiral symmetry $SU(2)_L \times SU(2)_R$.

Quark Condensate and Goldstone Excitations

The vacuum of QCD ($\langle 0 | \bar{q} q | 0 \rangle$) is not chirally symmetric. The non-zero quark condensate, $\langle \bar{q}q \rangle \neq 0$, signals the spontaneous breaking of the chiral symmetry down to the vector subgroup $SU(2)_V$. According to Goldstone’s theorem, this spontaneous breaking must generate massless scalar bosons corresponding to the broken generators.

In the real world, due to the small but non-zero masses of $u$ and $d$ quarks (explicit breaking), these bosons are not strictly massless but appear as very light pseudo-Goldstone bosons. The lightest such particles are the three pseudoscalar mesons: the neutral pion ($\pi^0$) and the charged pions ($\pi^\pm$). The observed small mass of the pion ($\approx 135 \text{ MeV}$) is attributed to this combination of spontaneous breaking (generating the Goldstone modes) and explicit breaking (the bare quark masses) ⁴.

Meson Type	Associated Field Type	Mass Generation Mechanism	Approximate Mass (MeV)
$\pi$ (Pion)	Pseudo-Goldstone Boson	Spontaneous breaking + Explicit quark mass	$135$
$\sigma$ (Sigma)	Scalar (Non-Goldstone Mode)	Dynamical Resonance	$\approx 500$

The successful prediction of the low mass of the pion based on this mechanism is considered a triumph of the concept of spontaneously broken chiral symmetry.

Quantum Anomalies and Axial Currents

While chiral symmetry can be spontaneously broken, it can also be subject to breaking at the quantum level via anomalies. This occurs when a classical symmetry in the Lagrangian is not preserved by the quantum effective action.

The axial vector current ($J_5^\mu$) is associated with the $U(1)_A$ symmetry. In a theory involving gauge fields, such as Quantum Electrodynamics (QED) or QCD, the divergence of the axial current is generally non-zero due to triangle diagrams involving the gauge fields. This is known as the Adler-Bell-Jackiw (ABJ) anomaly ⁵.

The divergence relation is: $$ \partial_\mu J_5^\mu = 2 m \bar{\psi} \gamma_5 \psi + \frac{g^2}{32\pi^2} \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma} $$ where the second term represents the quantum anomaly induced by gauge interactions (e.g., photons $F_{\mu\nu}$).

If the bare mass $m$ is zero, the divergence is non-zero if a non-trivial gauge field configuration (characterized by the topological term $\epsilon F \tilde{F}$) exists. This non-conservation mechanism explains why the $\eta’$ meson in QCD, corresponding to the $U(1)_A$ symmetry, is significantly heavier than the pions, which are associated with the non-anomalous $SU(2)_L \times SU(2)_R$ symmetry. The anomaly effectively manifests as an equivalent mass term for the associated Goldstone boson.

Chiral Representations and Manipulations

In discussions involving the Dirac equation, the choice of representation for the $\gamma$ matrices is crucial for isolating chiral components. The Chiral (or Weyl) representation is often employed because it block-diagonalizes the mass matrix. In this representation, the spinor $\psi$ is decomposed into two 2-component Weyl spinors, $\chi_L$ and $\chi_R$, such that the Dirac equation separates into two decoupled equations in the massless limit:

$$ i \sigma^\mu \partial_\mu \chi_L = 0 \quad \text{and} \quad i \bar{\sigma}^\mu \partial_\mu \chi_R = 0 $$ This explicit separation underscores the meaning of chiral symmetry: left-handed and right-handed particles propagate independently when mass is absent ⁶.

Experimental Observables and Parity Violation

The failure to observe elementary massless fermions (i.e., all observed fermions have mass) suggests that Nature does not possess exact, fundamental chiral symmetry, or that it is broken at the electroweak scale by the Higgs mechanism.

Crucially, the weak nuclear force violates parity ($\mathbf{P}$) maximally. Since parity transforms a right-handed particle into a left-handed antiparticle (and vice versa), the maximal parity violation observed in weak interactions implies that the interaction Lagrangian treats left-handed and right-handed fermions differently. This disparity is a direct physical manifestation of the underlying structure that necessitates the language of chiral symmetry, where interactions couple preferentially to specific helicities. For example, neutrinos are observed only in the left-handed state (or right-handed antiparticles) in the Standard Model, indicating a coupling structure sensitive to chirality ⁷.