# Chiral anomaly

In physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal number of positive and negative charged particles, that when opened was found to have more positive than negative particles, or vice versa.

Such events are expected to be prohibited according to classical conservation laws, but we know there must be ways they can be broken, because we have evidence of charge–parity non-conservation ("CP violation"). It is possible that other imbalances have been caused by breaking of a chiral law of this kind. Many physicists suspect that the fact that the observable universe contains more matter than antimatter is caused by a chiral anomaly, although this observation does not itself rigorously establish that a chiral anomaly must exist. Research into chiral breaking laws is a major endeavor in particle physics research at this time.

## Description

In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved. The non-conservation happens in a tunneling process from one vacuum to another. Such a process is called an instanton.

In the case of a symmetry related to the conservation of a fermionic particle number, one may understand the creation of such particles as follows. The definition of a particle is different in the two vacuum states between which the tunneling occurs; therefore a state of no particles in one vacuum corresponds to a state with some particles in the other vacuum.

In particular, there is a Dirac sea of fermions and, when such a tunneling happens, it causes the energy levels of the sea fermions to gradually shift upwards for the particles and downwards for the anti-particles, or vice versa. This means particles which once belonged to the Dirac sea become real (positive energy) particles and particle creation happens.

Technically, an anomalous symmetry is a symmetry of the action ${\displaystyle {\mathcal {A}}}$ , but not of the measure μ and therefore not of the generating functional ${\displaystyle {\mathcal {Z}}=\int \!{\exp(i{\mathcal {A}}/\hbar )~d\mu \,}}$ of the quantized theory (ℏ is Planck's action-quantum divided by 2π).

The measure consists of a part depending on the fermion field ${\displaystyle [d\psi ]}$ and a part depending on its complex conjugate ${\displaystyle [d{\bar {\psi }}]}$ . The transformations of both parts under a chiral symmetry do not cancel in general. Note that if ${\displaystyle \psi }$ is a Dirac fermion, then the chiral symmetry can be written as ${\displaystyle \psi \rightarrow e^{i\alpha \gamma ^{5}}\psi }$ where ${\displaystyle \gamma ^{5}}$ is some matrix acting on ${\displaystyle \psi }$ .

From the formula for ${\displaystyle {\mathcal {Z}}}$ one also sees explicitly that in the classical limit, ℏ → 0, anomalies don't come into play, since in this limit only the extrema of ${\displaystyle {\mathcal {A}}}$ remain relevant.

The anomaly is proportional to the instanton number of a gauge field to which the fermions are coupled (note that the gauge symmetry is always non-anomalous and is exactly respected, as is required by the consistency of the theory).

## Calculation

The chiral anomaly can be calculated exactly by one-loop Feynman diagrams, e.g. Steinberger's "triangle diagram", contributing to the pion decays, ${\displaystyle \pi ^{0}\to \gamma \gamma }$ and ${\displaystyle \pi ^{0}\to e^{+}e^{-}\gamma }$ .

The amplitude for this process can be calculated directly from the change in the measure of the fermionic fields under the chiral transformation.

Wess and Zumino developed a set of conditions on how the partition function ought to behave under gauge transformations called the Wess–Zumino consistency condition.

Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah–Singer index theorem. See Fujikawa's method.

## An example: baryon number non-conservation

The Standard Model of electroweak interactions has all the necessary ingredients for successful baryogenesis, although these interactions have never been observed[1] and may be insufficient to explain the total baryon number of observed universe if the initial baryon number of the universe at the time of the Big Bang is zero. Beyond the violation of charge conjugation ${\displaystyle C}$ and CP violation ${\displaystyle CP}$ (charge+parity), baryonic charge violation appears through the Adler–Bell–Jackiw anomaly of the ${\displaystyle U(1)}$ group.

Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak Lagrangian conserves baryonic charge. Quarks always enter in bilinear combinations ${\displaystyle q{\bar {q}}}$ , so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current ${\displaystyle J_{\mu }^{B}}$ is conserved:

${\displaystyle \partial ^{\mu }J_{\mu }^{B}=\sum _{j}\partial ^{\mu }({\bar {q}}_{j}\gamma _{\mu }q_{j})=0.}$

However, quantum corrections known as the sphaleron destroy this conservation law: instead of zero in the right hand side of this equation, there is a non-vanishing quantum term,

${\displaystyle \partial ^{\mu }J_{\mu }^{B}={\frac {g^{2}C}{16\pi ^{2}}}G^{\mu \nu a}{\tilde {G}}_{\mu \nu }^{a},}$

where C is a numerical constant vanishing for ℏ =0,

${\displaystyle {\tilde {G}}_{\mu \nu }^{a}={\frac {1}{2}}\epsilon _{\mu \nu \alpha \beta }G^{\alpha \beta a},}$

and the gauge field strength ${\displaystyle G_{\mu \nu }^{a}}$ is given by the expression

${\displaystyle G_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf_{bc}^{a}A_{\mu }^{b}A_{\nu }^{c}~.}$

Electroweak sphalerons can only change the baryon and/or lepton number by 3 or multiples of 3 (collision of three baryons into three leptons/antileptons and vice versa).

An important fact is that the anomalous current non-conservation is proportional to the total derivative of a vector operator, ${\displaystyle G^{\mu \nu a}{\tilde {G}}_{\mu \nu }^{a}=\partial ^{\mu }K_{\mu }}$ (this is non-vanishing due to instanton configurations of the gauge field, which are pure gauge at the infinity), where the anomalous current ${\displaystyle K_{\mu }}$ is

${\displaystyle K_{\mu }=2\epsilon _{\mu \nu \alpha \beta }\left(A^{\nu a}\partial ^{\alpha }A^{\beta a}+{\frac {1}{3}}f^{abc}A^{\nu a}A^{\alpha b}A^{\beta c}\right),}$

which is the Hodge dual of the Chern–Simons 3-form.

### Geometric form

In the language of differential forms, to any self-dual curvature form ${\displaystyle F_{A}}$ we may assign the abelian 4-form ${\displaystyle \langle F_{A}\wedge F_{A}\rangle :=\operatorname {tr} \left(F_{A}\wedge F_{A}\right)}$ . Chern-Weil theory shows that this 4-form is locally but not globally exact, with potential given by the Chern-Simons 3-form locally:

${\displaystyle d\mathrm {CS} (A)=\langle F_{A}\wedge F_{A}\rangle }$ .

Again, this is true only on a single chart, and is false for the global form ${\displaystyle \langle F_{\nabla }\wedge F_{\nabla }\rangle }$ unless the instanton number vanishes.

To proceed further, we attach a "point at infinity" k onto ${\displaystyle \mathbb {R} ^{4}}$ to yield ${\displaystyle S^{4}}$ , and use the clutching construction to chart principle A-bundles, with one chart on the neighborhood of k and a second on ${\displaystyle S^{4}-k}$ . The thickening around k, where these charts intersect, is trivial, so their intersection is essentially ${\displaystyle S^{3}}$ . Thus instantons are classified by the third homotopy group ${\displaystyle \pi _{3}(A)}$ , which for ${\displaystyle A=\mathrm {SU(2)} \cong S^{3}}$ is simply the third 3-sphere group ${\displaystyle \pi _{3}(S^{3})=\mathbb {N} }$ .

The divergence of the baryon number current is (ignoring numerical constants)

${\displaystyle \mathbf {d} \star j_{b}=\langle F_{\nabla }\wedge F_{\nabla }\rangle }$ ,

and the instanton number is

${\displaystyle \int _{S^{4}}\langle F_{\nabla }\wedge F_{\nabla }\rangle \in \mathbb {N} }$ .

## References

1. S. Eidelman et al. (Particle Data Group), Phys. Lett. B592 (2004) 1 ("No baryon number violating processes have yet been observed.")

### Published articles

• S. Adler (1969). "Axial-Vector Vertex in Spinor Electrodynamics". Physical Review. 177 (5): 2426–2438. Bibcode:1969PhRv..177.2426A. doi:10.1103/PhysRev.177.2426.
• J.S. Bell and R. Jackiw (1969). "A PCAC puzzle: π0γγ in the σ-model". Il Nuovo Cimento A. 60 (1): 47–61. Bibcode:1969NCimA..60...47B. doi:10.1007/BF02823296.
• P.H. Frampton and T.W. Kephart (1983). "Explicit Evaluation of Anomalies in Higher Dimensions". Physical Review Letters. 50 (18): 1343–1346. Bibcode:1983PhRvL..50.1343F. doi:10.1103/PhysRevLett.50.1343.
• P.H. Frampton and T.W. Kephart (1983). "Analysis of anomalies in higher space-time dimensions". Physical Review. D28 (4): 1010–1023. Bibcode:1983PhRvD..28.1010F. doi:10.1103/PhysRevD.28.1010.
• Alan R. White (2004). "Electroweak High-Energy Scattering and the Chiral Anomaly". Physical Review. D69 (9): 096002. arXiv:hep-ph/0308287. Bibcode:2004PhRvD..69i6002W. doi:10.1103/PhysRevD.69.096002.
• T. Csörgő, R. Vértesi and J. Sziklai (2010). "Indirect Observation of an In-Medium η′ Mass Reduction in sqrt(s_{NN})=200 GeV Au+Au Collisions". Physical Review Letters. 105 (18): 182301. arXiv:0912.5526. Bibcode:2010PhRvL.105r2301C. doi:10.1103/PhysRevLett.105.182301. PMID 21231099.

### Textbooks

• K. Fujikawa and H. Suzuki (May 2004). Path Integrals and Quantum Anomalies. Clarendon Press. ISBN 978-0-19-852913-2.
• S. Weinberg (2001). The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge University Press. ISBN 978-0-521-55002-4.