Why does anomaly mean symmetry is broken?

[fxdung]

Is the expection value of expession in left hand side of motion equation of field(example: Klein-Gordon,Dirac...equations) equal zero or not?(left hand side of the equation equals zero when we put condition of mimimizing the action).If not,why we can say when expectation value of divergence of the current not equal zero then the symmetry be broken?The Lagrangian may still invariant because expectation values of both divergence of current and expression of the left hand side of Euler-Lagrange differ zero and the sum of them (equal Delta L(agrangian)) equal zero,so the symmetry still reserved.(But equation of motion be changed)

 

[vanhees71]

I'm not sure whether I understand your question, but I try to give an answer anyway.

First, if the four-divergence of a vector field does not vanish, it means that this vector field cannot be the Noether current of a symmetry. Second, an anomaly means that you start from a classical (field) theory that obeys some symmetry (e.g., the chiral axial U(1) in a field theory with fermions) but there's no way to "quantize" it such that this symmetry is preserved in the quantized (field) theory. That means in a consistent quantum action there must occur terms that are not invariant under the action of the symmetry group, implying that the quantum theory has not the symmetry the classical theory had when you started. In the axial U(1) the reason is that the corresponding path-integral measure of the fermion field is not invariant under the group operation, which adds anomalous terms to the Lagrangian of the classical chiral theory. This implies that the scalar but not the pseudoscalar U(1) is a symmetry of the quantized theory.

From a perturbative QFT point of view, you can in principle shuffle the anomaly also to the scalar U(1) or to any linear combination of the scalar and the pseudoscalar current. That one must not do this in the standard model is clear since by breaking the scalar U(1) you'd destroy local color-gauge invariance, which would spoild the consistency of the whole (all too successful) standard model! Thus you must break axial U(1) symmetry, and that's not a bug but a feature. This socalled Adler-Bell-Jackiw (ABJ) anomaly explains why the decay rate of a neutral pion to two photons is consistent with the (approximate) chiral symmetry.

Anomalies of a local symmetry are always bad, because then the corresponding gauge invariance is broken, and the model uses its consistency. E.g., the electroweak sector of the standard model is also a gauge theory, which is in danger by anomalies. However, the charge pattern of each family of quarks and leptons is just such that the anomaly exactly cancels. There are other charge patterns than that realized in nature with the same feature, but it's one of the very beautiful coincidences in the standard model that the observed charge pattern is consistent with electroweak gauge symmetry!

 

[atyy]

The Lagrangian may still invariant because expectation values of both divergence of current and expression of the left hand side of Euler-Lagrange differ zero and the sum of them (equal Delta L(agrangian)) equal zero,so the symmetry still reserved.(But equation of motion be changed)

In the path integral picture, although the Lagrangian is invariant, the integration measure is not when there is an anomaly. http://arxiv.org/abs/hep-th/0410129

 

[vanhees71]

The original idea to treat anomalies with the path-integral approach is due to Fujikawa:

Fujikawa, Kazuo: Path-Integral Measure for Gauge-Invariant Fermion Theories, Phys. Rev. Lett. 42, 1195, 1979

 

[DrDu]

Are there examples of anomalies from many particle physics, too?

 

[vanhees71]

That's a very good question. Another one is, whether there is an example within "first quantized theory", i.e., whether one can find a classical point-particle Lagrangian which has some symmetry which is not preserved by some quantization procedure (like "canonical quantization") leading to a quantum theory where this symmetry is broken.

 

[atyy]

http://online.itp.ucsb.edu/online/freedmanfest/ludwig/
Classification of Topological Insulators and Superconductors: The 'Ten-Fold Way', Responses, and Quantum AnomaliesAndreas Ludwig

http://arxiv.org/abs/1010.0936
Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors
Shinsei Ryu, Joel E. Moore, Andreas W. W. Ludwig

http://arxiv.org/abs/1206.1627
Chiral Anomaly and Classical Negative Magnetoresistance of Weyl Metals
D. T. Son, B. Z. Spivak

http://arxiv.org/abs/1307.6990
Dirac vs. Weyl in topological insulators: Adler-Bell-Jackiw anomaly in transport phenomena
Heon-Jung Kim, Ki-Seok Kim, J. F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M. Kitaura, M. Yang, L. Li

 

[fxdung]

I mean that when the sum of expectation value of divergence of Noether current(differ from zero) and expectation value(differ from zero) of expression of left hand side of motion equation is zero then the quantum action still invariant.So that although anomaly the symmetry still reserved in quantum frame in this case.

 

[fxdung]

Now I think that the classical symmetry leads to expectation value of left hand side of motion equation always equal zero.So that anomaly always leads to quantum broken symmetry.Is that correct?

 

[vanhees71]

No, the quantum action is not invariant. The non-invariance of the path-integral measure introduces a symmetry-breaking term.

The perturbative leading-order treatment makes the whole issue clearer. Take QED and a triangle graph with one axial-vector current and two vector currents. That's related to the pion via the PCAC hypothesis, which states
$$\partial_{\mu} j_5^{a,\mu}=f_{\pi} M_{\pi}^2 \pi^a, \quad j_{5,\mu}^a=\overline{\psi} \gamma_{\mu} \gamma_5 \frac{\tau^a}{2} \psi.$$
Here $f_{\pi} \simeq 92 \; \mathrm{MeV}$ is the pion-decay constant, $\pi^a$ is the pion field and $j_{5,\mu}^{a}$ the axial-vector current.

Plugging this into the triangle graph, you find that it's linearly divergent, and that implies that one has to regularize the diagram. Now the Ward-Takahashi identity for the vector current, which after all is coupling to the electromagnetic field must hold. So we have to regularize the diagram in a way to preserve the electromagnetic gauge symmetry. One unambigous way is Pauli-Villars regularization, i.e., you subtract the diagram with large fermion masses. This violates obviously the chiral symmetry for vanishing fermion masses but keeps the vector currents conserved. This is the unique right choice of regularization since the vector current's conservation is a necessary condition to keep the local gauge invariance, which must not be broken, because otherwise the model would break down.

This adds an additional term to the PCAC relation which, written in terms of the corresponding external fields (two photons coupled to the vector currents and a pion via the PCAC relation to the axial-vector current), reads
$$\partial_{\mu} j_5^{a,\mu} = f_{\pi} m_{\pi}^2 \pi^a - \alpha \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma} \mathrm{Tr}(Q^2 \tau^a/2).$$
The $Q$ are the charges of the fermions ("quarks").

In this case the anomaly is a great feature, because if there wouldn't be this anomalous term, the $\pi_0 \rightarrow \gamma \gamma$-decay rate would come out way too low, which would imply that the PCAC hypothesis, working great otherwise due to (approximate) chiral symmetry of the light-quark sector of QCD, may be wrong, but the anomaly saves the day. Using the result of the triangle calculation, leads to a decay width
$$\Gamma(\pi^0 \rightarrow \gamma \gamma)=\frac{\alpha^2}{128 \pi^3} \frac{m_{\pi}^3}{f_{\pi}^2} \frac{N_{\text{color}}^2}{9}.$$
Putting the measured values of the parameters, i.e., $f_{\pi}=92 \; \text{MeV}$, $m_{\pi}=135 \; \text{MeV}$, $N_{\text{color}}=3$, leads to $\Gamma(\pi^0 \rightarrow \gamma \gamma) \simeq 7.81 \; \mathrm{eV}$, which compares well with the measured value. This shows that everything is consistent with this decay and the strong-interaction physics, including the PCAC hypothesis (i.e., the assumption of chiral spontaneously broken symmetry) and the number of colors being 3 in the standard model QCD.

 

[DrDu]

I found that chapter 3 in the book: Graphene: Synthesis, Properties and Phenomena contains a very nice discussion of anomalies in QFT vs. solid state:
https://books.google.de/books?id=Ad...a=X&ei=e7BZVfK_Mqj9ygPPjYD4Bg&ved=0CFQQ6AEwBQ

 

[Ilja]

"So how did an electric field end up violating chiral charge? Note that this analysis
relied on the Dirac sea being infinitely deep. If there had been a finite number of
negative energy states, then they would have shifted to higher momentum, but there
would have been no change in the axial charge. With an infinite number of degrees of
freedom, though, one can have a “Hilbert Hotel”: the infinite hotel which can always
accommodate another visitor, even when full, by moving each guest to the next room
and thereby opening up a room for the newcomer. This should tell you that it will not
be straightforward to represent chiral symmetry on the lattice: a lattice field theory
approximates quantum field theory with a finite number of degrees of freedom — the
lattice is a big hotel, but quite conventional. In such a hotel there can be no anomaly."

Kaplan, D.B.: Chiral Symmetry and Lattice Fermions, arxiv:0912.2560v2

 

[DrDu]

"So how did an electric field end up violating chiral charge? Note that this analysis
relied on the Dirac sea being infinitely deep. If there had been a finite number of
negative energy states, then they would have shifted to higher momentum, but there
would have been no change in the axial charge. With an infinite number of degrees of
freedom, though, one can have a “Hilbert Hotel”: the infinite hotel which can always
accommodate another visitor, even when full, by moving each guest to the next room
and thereby opening up a room for the newcomer. This should tell you that it will not
be straightforward to represent chiral symmetry on the lattice: a lattice field theory
approximates quantum field theory with a finite number of degrees of freedom — the
lattice is a big hotel, but quite conventional. In such a hotel there can be no anomaly."

Kaplan, D.B.: Chiral Symmetry and Lattice Fermions, arxiv:0912.2560v2

I don't quite understand what he wants to say. From the other articles I got the impression, that the anomaly is rather trivial in many particle systems. The two passages I marked seem contradicting to me. If it were difficult (or not possible) to implement chiral symmetry on a lattice, then the system would be anomalous. But then he sais that there can be no anomaly.