Anomaly (physics)

A quantum anomaly is the failure of a classical symmetry to survive quantization because the functional measure or regulator cannot preserve it.

A global anomaly obstructs invariance of a quantum theory under a topologically nontrivial symmetry transformation even when infinitesimal local anomaly tests vanish. The Witten SU(2) anomaly is one example; global gravitational anomalies are another class.

The Witten SU(2) anomaly is a four-dimensional global gauge anomaly present for an odd number of left-handed Weyl fermion doublets. A consistent SU(2) gauge theory must have an even number modulo the corresponding representation index.

A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

Anomaly cancellation is the vanishing of every gauge and mixed gauge-gravitational anomaly after summing over all chiral fermions. It is a consistency condition for a quantum gauge theory.

In each Standard Model generation, quark and lepton hypercharges cancel the $SU(3{)}_c^3$, $SU(3{)}_c^{2}U(1{)}_Y$, $SU(2{)}_L^{2}U(1{)}_Y$, $U(1{)}_Y^3$, and mixed gauge-gravitational anomalies. The four left-handed $SU(2{)}_L$ doublets also avoid the global Witten anomaly.

In normalization $Q=T_3+Y$, one Standard Model generation has left quark and lepton doublet hypercharges $1/6$ and $-1/2$, and right singlet hypercharges $2/3,-1/3,-1$. Including colour multiplicity, ${\mathrm{tr}}_{L}Y=6/6-2/2=0$. On every left doublet $T_3^2=I/4$, and all right fields have $T_3=0$, so the mixed weak-hypercharge gauge anomaly vanishes. The cubic left trace is $6(1/6{)}^3+2(-1/2{)}^3=-2/9$, and the right trace is $3(2/3{)}^3+3(-1/3{)}^3+(-1{)}^3=-2/9$. Their difference is zero, proving generation-by-generation cancellation of these two anomalies.

A mixed gauge-gravitational anomaly has one gauge-current insertion and two stress-tensor insertions. For a four-dimensional $U(1)$ symmetry its coefficient is the sum of the left-handed Weyl-fermion charges, with right-handed fermions counted as left-handed conjugates.

The chiral anomaly is the quantum nonconservation of a classically conserved axial current in a gauge-field background.

For massless Dirac fermions, the classical axial current is $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. The chiral anomaly makes its divergence proportional to $F_{\mu \nu }{}^{\star }{F}^{\mu \nu }$.

An axial charge is the spatial integral of the time component of an axial current. For a Dirac field and an internal generator $t$, it is $\int {\psi }^{†}{\gamma }_{5}t\psi d^{3}x$. Its canonical action rotates left- and right-handed fields oppositely. It is conserved only when the current divergence and flux at infinity vanish; Dirac masses and a chiral anomaly can obstruct conservation. The nonsinglet quark generators have no color anomaly and obey the nonsinglet axial charge algebra of quark densities.

For an ordinary scalar field, its derivative is a polar covector, while the axial current is a pseudovector. Their contraction is invariant under the proper orthochronous Lorentz group but changes sign under parity. If the field is a pseudoscalar, the extra transformation sign makes this coupling parity even. In four spacetime dimensions its mass dimension is five, so Lorentz invariance alone does not imply power-counting renormalizability.

For $\mathcal{L}=\bar{\psi }(i/\partial -S+iP{\gamma }^5)\psi$, with real scalar backgrounds $S,P$, the Dirac equation is $(i/\partial -S+iP{\gamma }^5)\psi =0$ and the adjoint Dirac equation is $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+S\bar{\psi }-iP\bar{\psi }{\gamma }^5=0$. Use $\{{\gamma }^5,{\gamma }^{\mu }\}=0$ and $({\gamma }^5{)}^2=1$ in the derivative of $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ to obtain the displayed classical identity. Taking $S=m+g\varphi$ and $P=G\Phi$ includes both scalar and Hermitian pseudoscalar Yukawa couplings.

For ${\mathcal{L}}_{\mathrm{int}}=-g\varphi \bar{\psi }{\gamma }^5\psi$, the variational Dirac equation and adjoint Dirac equation are $(i/\partial -m-g\varphi {\gamma }^5)\psi =0$ and $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+m\bar{\psi }+g\varphi \bar{\psi }{\gamma }^5=0$. Substitute these into the derivative of $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. Anticommuting the chirality matrix past ${\gamma }^{\mu }$ gives two identical mass terms and two identical interaction terms, proving the displayed identity. These are classical field equations; the formula does not assert an unrenormalized quantum composite-operator identity. With Hermitian coupling $g=i{g}_P$, the second term is $-2g_P\varphi \bar{\psi }\psi$.

Since the chirality matrix commutes with spinor Lorentz generators, $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ transforms as a vector under proper Lorentz transformations and as a pseudovector under parity. Contracting two copies gives the Lorentz scalar $j_{5\mu }{j}_5^{\mu }$; the extra parity signs cancel. Lorentz invariance of this interaction does not require current conservation.

A chiral transformation rotates a Dirac field by $\psi \mapsto e^{i\alpha {\gamma }^5}\psi$. A massless Dirac kinetic term is invariant under a global rotation, while a mass term is not.

A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is preserved by renormalization-group flow and constrains possible infrared phases.

't Hooft anomaly matching requires the massless infrared degrees of freedom, topological sector, or symmetry-breaking pattern to reproduce every anomaly of an unbroken global symmetry measured in the ultraviolet theory.