New symmetry principles and anomalies

[arivero]

I noticed recently the content of TASI 2023 lectures, a lot about "generalised symmetries" and a particular one, "non invertible symmetries", that seems to propose a new way to understand the ABJ anomaly. Have you guys read it?

 

[ohwilleke]

Have you guys read it?

No. Why don't you explain it a bit more for us?

What is notable about it? What real life (or even hypothetical) interactions is it relevant to? What theoretical agendas does it imply or suggest?

What is the difference between a symmetry that is absent, and one that is non-invertible?

Background:

"The Adler-Bell-Jackiw anomaly determines the violation of chiral symmetry when massless fermions are coupled to an abelian gauge field.

In its seminal paper, Adler noticed that a modified chiral U(1) symmetry could still be defined, at the expense of being generated by a non-gauge-invariant conserved current.

We show this internal U(1) symmetry has the special feature that it transforms the Haag duality violating sectors (or non local operator classes). This provides a simple unifying perspective on the origin of anomaly quantization, anomaly matching, applicability of Goldstone theorem, and the absence of a Noether current.

We comment on recent literature where this symmetry is considered to be either absent or non-invertible. We end by recalling the DHR reconstruction theorem, which states 0-form symmetries cannot be non-invertible for d>2, and argue for a higher form-symmetry reconstruction theorem."

From https://arxiv.org/abs/2309.03264

The ABJ anomaly was first described in 1969 in the following two papers:

S. L. Adler, “Axial-vector vertex in spinor electrodynamics,” Phys. Rev. 177 (Jan, 1969) 2426–2438. https://link.aps.org/doi/10.1103/PhysRev.177.2426.

and

J. S. Bell and R. Jackiw, “A PCAC puzzle: π0 → γγ in the σ model,” Nuovo Cim. A 60 (1969) 47–61.

Comment:

Since it was discovered in 1998 that neutrinos have a non-zero mass, there are no massless fermions in the Standard Model (either fundamental or composite), with the possible exception of the lightest neutrino, although it seems unlikely that even it is completely massless.

Indeed, massless fermions aren't even present in most of the more often discussed extensions of the Standard Model (e.g. multiple Higgs doublets, SUSY, leptoquarks, seesaw neutrino mass, sterile neutrinos, QCD axions, axion-like dark matter particles, dark photons for self-interacting dark matter, etc.). Massless fermions are, for example, useless as dark matter candidates, for obvious reasons.

Also since the ABJ anomaly involves a coupling to an abelian gauge field, which excludes QCD in the Standard Model and gravity, it applies only to the electoweak interactions of massless fermions. Honestly, all of the literature seems to be about the U(1) gauge couplings of massless fermions (I'm not even clear that it applies to weak force interactions), which would be the electromagnetic interactions of electromagnetically charged massless fermions which are definitely not present in the Standard Model. But electromagnetically charged massless fermions are something HEP experiments would be particularly unlikely to miss, since it does an exquisitely good job of detecting tiny electromagnetically charged particles.

So, the ABJ anomaly is something that may be relevant only to a counterfactual situation that doesn't exist in any known aspect of physical reality. It would seem to be only a thought experiment and guide for what extensions of the Standard Model can be theoretically described.

But maybe I'm missing some deeper insight into the overall structure of the laws of particle physics here. Perhaps the point is that the non-existence of particles to which it can apply is another hint that such anomalies can't exist in the Standard Model.

Also, I suppose it could act like a baseline for a pseduo-ABJ anomaly situation (by analogy to a pseudo-Goldstone boson) which the behavior of very low mass fermions, like neutrinos, perhaps in weak force interactions, is asymptotically attracted to. And nothing in the Standard Model is more asymmetric from a chiral perspective than a neutrino.

The linked agenda from the OP is as follows:

Week 1:​

• Nathan Seiberg (Institute for Advanced Study) — The Power of Symmetry
• Sakura Schafer-Nameki (Oxford University) — Generalized Symmetry and String/M-Theory Realizations
• Hong Liu (MIT) — Entanglement, Many-Body Systems, and Operator Algebras of Quantum Gravity
• Juan Maldacena (Institute for Advanced Study) — Quantum Aspects of Black Holes
• Yifan Wang (NYU) — 2D CFTs and Generalized Symmetries

Week 2:​

• Kenneth Intriligator (UC San Diego) — SUSY and Symmetry Constraints on RG Flows and IR Phases
• Masha Baryakhtar (University of Washington) — Dark Matter Theory and Possible Detection Opportunities
• Gregory Moore (Rutgers University) — Differential Cohomology and Physics
• Thomas Dumitrescu (UCLA) — Generalized Symmetries in Quantum Field Theory

Week 3:​

• Meng Cheng (Yale University) — Gapped Phases and TQFT
• Andrea Puhm (École Polytechnique, CPHT) – Asymptotic Symmetries
• Clifford Cheung (Caltech) — Scattering Amplitudes and Symmetry
• Isabel Garcia Garcia (Institute for Advanced Study + NYU/University of Washington) — Particle Physics, Gravity and Symmetries
• Ibrahima Bah (Johns Hopkins University) — Generalized Symmetries and Holography

Week 4:​

• Shu-Heng Shao (Stony Brook University) — Noninvertible Symmetry
• John McGreevy (UC San Diego) — Generalized Symmetries in CMT
• Alejandra Castro (University of Cambridge) — Topics in AdS/CFT
• David Simmons-Duffin (Caltech) — CFT and Observables on a Null Plane

The abstract of the paper to which particular attention is called and its citation are as follows:

We survey recent developments in a novel kind of generalized global symmetry, the non-invertible symmetry, in diverse spacetime dimensions. We start with several different but related constructions of the non-invertible Kramers-Wannier duality symmetry in the Ising model, and conclude with a new interpretation for the neutral pion decay and other applications. These notes are based on lectures given at the TASI 2023 summer school “Aspects of Symmetry.”

Shu-Heng Shao, C. N. Yang Institute for Theoretical Physics, Stony Brook University, "What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetry" arXiv 2308.00747 (August 1, 2023).

N.B. The application to neutral pion decay directly ties this in to the subject of one of the original ABJ anomaly papers in 1969.

In the applications portion, Shao (2023) says:

Do non-invertible symmetries exist in Nature?

The answer is a resounding yes. In this section we discuss non-invertible global symmetries in the 3+1d QED and QCD for the real world. In the context of QCD, the neutral pion decay π0 → 2γ is reinterpreted as a consequence of a non-invertible global symmetry.

But, I don't see how this connects to the ABJ anomaly which is absent in QCD and has no applicability to massive particles or to bosons.

 

[arivero]

Yeah I guess that when they say QCD they mean just mesons. Or "old electroweak theory of hadrons", which sound uglier.

My motivation to look into this, if I get the time, is to see if it relates to one of our old coincidences, the one of decay width scaled by cube of mass.

Currently, looking at https://pdg.lbl.gov/2024/mcdata/mass_width_2024.txt

* Particle ID(s)                  Mass  (GeV)       Errors (GeV)       Width (GeV)       Errors (GeV)      Name          Charges
      23                          9.11880E+01       +2.0E-03 -2.0E-03  2.4955E+00        +2.3E-03 -2.3E-03 Z                   0
     111                          1.349768E-01      +5.0E-07 -5.0E-07  7.81E-09          +1.2E-10 -1.2E-10 pi                  0
     443                          3.096900E+00      +6.0E-06 -6.0E-06  9.26E-05          +1.7E-06 -1.7E-06 J/psi(1S)           0

The coincidences, in units of GeV^-2, of total reduced decay widths, are

Z0   = .0000032911 pm .0000000030
J/Psi= .0000031176 pm .0000000572
pion=  .0000031718 pm .0000000406

So that really J/Psi and pion scale perfectly, and Z0, which should have not reason to coincide, is barely three sigmas above. Note that in the post from 2006 I further mapped the result to a GeV scale via 1/sqrt(), if we do such scaling, the distance obviously halves to 1.5 sigmas.

 

[ohwilleke]

Yeah I guess that when they say QCD they mean just mesons. Or "old electroweak theory of hadrons", which sound uglier.

My motivation to look into this, if I get the time, is to see if it relates to one of our old coincidences, the one of decay width scaled by cube of mass.

Currently, looking at https://pdg.lbl.gov/2024/mcdata/mass_width_2024.txt

* Particle ID(s)                  Mass  (GeV)       Errors (GeV)       Width (GeV)       Errors (GeV)      Name          Charges
      23                          9.11880E+01       +2.0E-03 -2.0E-03  2.4955E+00        +2.3E-03 -2.3E-03 Z                   0
     111                          1.349768E-01      +5.0E-07 -5.0E-07  7.81E-09          +1.2E-10 -1.2E-10 pi                  0
     443                          3.096900E+00      +6.0E-06 -6.0E-06  9.26E-05          +1.7E-06 -1.7E-06 J/psi(1S)           0

The coincidences, in units of GeV^-2, of total reduced decay widths, are

Z0   = .0000032911 pm .0000000030
J/Psi= .0000031176 pm .0000000572
pion=  .0000031718 pm .0000000406

So that really J/Psi and pion scale perfectly, and Z0, which should have not reason to coincide, is barely three sigmas above. Note that in the post from 2006 I further mapped the result to a GeV scale via 1/sqrt(), if we do such scaling, the distance obviously halves to 1.5 sigmas.

Given that decay widths are a derived quantity, is it even really a coincidence, as much as a functional relationship obscured by the somewhat involved, multistep way that decay widths are calculated that hide the intuition.

 

[Klystron]

And nothing in the Standard Model is more asymmetric from a chiral perspective than a neutrino.

Leading popular science author Isaac Asimov to refer to neutrinos as "The Left Hand of the Electron" in the title of one of his nonfiction anthologies. While not a textbook, I found Asimov's provocative essay title useful while studying chirality in the standard model in actual textbooks and publications.

 

[Vanadium 50]

Decay widths need to go dimensionally as m. If you are seeing an m³ relation, something needs to go as m² or m^-4. This is most likely the wavefunction overlap for the mesons. If's the Fermi constant for the Z.

Since this works for some particles and not others - for example, it works for the J/ψ and not the ϒ - it can't be universal. And if you can pick and choose, it looks more like coincidence. You start with numbers of similar magnitude and multiply by prefactors of order 1.

 

[arivero]

Decay widths need to go dimensionally as m. If you are seeing an m³ relation, something needs to go as m² or m^-4. This is most likely the wavefunction overlap for the mesons. If's the Fermi constant for the Z.

Since this works for some particles and not others - for example, it works for the J/ψ and not the ϒ - it can't be universal. And if you can pick and choose, it looks more like coincidence. You start with numbers of similar magnitude and multiply by prefactors of order 1.

I'd not say that it works for some particles and not others. The ϒ is the puzzling thing, all the others have available either electroweak or strong decays, and if you remove the QCD contribution to the decay width, the rest aligns, and it could make sense if the dimensional scaling can be justified. Still, I agree the Z coincidence looks as a coincidence. The ϒ honestly I had expected more wavefunction overlap, not less. But somehow, having a huge quark is a defense against electromagnetic decay to two gammas.

I am not sure if I pasted in some other thread the whole log log plot of the file mass_width_ from pdg. In my blog, the last version is the one of 2014, ten years ago, with a gnuplot script! Here is one from 2023, with some jupyter script. Besides the cubic scaling, I paint also the quintic scaling of the particles that can not use electromagnetic nor strong channels.

 

[Vanadium 50]

But somehow, having a huge quark is a defense against electromagnetic decay to two gammas.

That assumes what you are trying to prove. (Except that the J/psi and upsilon can't decay to two gammas - you must have meant something else)

 

[arivero]

You are right, I was hallucinating due to false generalisation from pion.

 

[mitchell porter]

A lot of topics in this thread.

1) Anomalies and the chiral anomaly. As they say, an anomaly occurs when a symmetry of the classical theory is no longer true of the quantum theory. I usually think of this as happening because the integration measure in the sum over histories does not respect the symmetry, but there are a variety of ways of deriving an anomaly.

A chiral anomaly exists when the amount of handedness is not conserved (i.e. the difference in the number of left-handed and right-handed particles). David Tong's QFT lecture notes, which have acquired a very good reputation, discuss the chiral anomaly in chapters 3 and 5. In chapter 3 he derives the chiral anomaly for massless fermions in several ways, in chapter 5 he does it for QCD and the neutral pion decay.

The simplest-sounding explanation is on page 125, figure 25. This is in 1+1 dimensions, where the chiralities are "moving to the left" and "moving to the right", and there is a rightward-pointing electric field that boosts all the right-movers to higher energy levels. If there were only finitely many energy levels, the occupation of new higher levels would be matched by a vacation of lower energy levels, and the number of right-movers would stay the same. But because the Dirac sea is infinitely deep, it amounts to a raising of the sea of right-movers and lowering of the sea of left-movers, changing the relative number of right-movers and left-movers. Then on the next page, he makes an analogous argument for chiral states in 3+1 dimensions.

Ideally, I think one would be able to switch between this kind of explanation, the measure-theoretic explanation, and the triangle-diagram perturbative calculations that ABJ originally used. I haven't attained this flexibility myself, but all three explanations are there, side by side, in Tong's notes.

2) Chiral anomaly and neutral pion decay. The theory of this is in Tong's chapter 5. There are several new ingredients, e.g. we're talking about pions, and we don't even know how to calculate their behavior from QCD first principles (except on the lattice), so instead we'll the effect is derived in the context of Weinberg's chiral perturbation theory.

I don't have anything simple to say about this, so instead I will provide some references for the theoretical calculation of neutral pion decay, courtesy of this 2018 thesis. See pages 9-10, which first discusses the theoretical calculation assuming up and down quarks are massless, then various more sophisticated calculations that take the masses into account, and compares them all to the measured decay rate. The measured width is 7.74 +/- 0.46 eV, the calculation which treats quarks as massless predicts 7.76 +/- 0.04 eV, the corrected calculations which include the light quark masses give results around 8 eV (near the upper bound on the measured width).

3) Generalized symmetries and non-invertible symmetries. This is a new take on symmetries in QFT. I first tried to understand it a year ago, my grasp of it is still very weak, but here we go.

First we have the generalized global symmetries. You start with an ordinary quantum field theory, you take a submanifold of space-time (this can be a point, a line, a surface, or a hyperplane), and you associate a "topological operator", labelled by an element of a group, with that submanifold. Mathematically, this corresponds to a deformation of the original QFT, in which you tensor it with a topological QFT associated with the submanifold. Presumably the Hilbert space of the original theory has to be enlarged to include topological quantum numbers characterizing the special submanifold.

It is said that the usual global symmetries of QFT correspond in this scheme to a situation where the submanifold is just a point. I've seen the Gauss law referenced here, as an example of a conservation law with a topological character - you can deform the closed surface through which the electric flux is passing, and the law is still valid. And to mention another QFT concept that may be familiar, when a Wilson line is present, the submanifold is the line... However, I haven't understood how any of this actually works in any detail.

This all started ten years ago, with a paper by Gaiotto et al, "Generalized Global Symmetries". Originally, the algebra of labels of the topological operators were always groups. However, in the last few years, people are also labeling them with "fusion algebras" or "fusion categories" which aren't groups. In some cases this algebra involves relations mapping a single submanifold to multiple submanifolds or vice versa, I guess that's the "fusion". And some of these algebras have non-invertible relations, and those define the "non-invertible symmetries".

I mentioned above that the chiral anomaly of the pion has been formulated in multiple ways. Apparently another way has been figured out, involving a non-invertible symmetry. I have no idea what the special submanifold is - maybe the center-of-mass worldline? (which would include the world-line of the pion until it decays). I hope it makes sense to me at some point.

4) @arivero's plot of decay widths. I thought about these observations of his a few years back, specifically why Z boson decay would resemble pion decay. At the time I thought it might be evidence that the Higgs is a top-antitop bound state. My reasoning is that the spin-0 component of the Z (which gives the Z its mass) would then be a kind of neutral top-pion, and so maybe the same anomalous mechanism that applies to the neutral pion, might also apply to it. But I never really got to the bottom of that.