We know that the axial $U(1)_A$ is anomalous thus not a global symmetry. Therefore there is no direct associated pseudo goldstone boson for $U(1)_A$. This makes the $\eta'$ much more massive than the other pseudo goldstone 8 mesons.

Do we have similar $U(1)_A$ effects on the baryons? If so, which baryon(s) is/are associated with $U(1)_A$ thus more massive than the other baryons? (Also what are the other flavor symmetries should one look at $SU(N)_V$? or $SU(N)_A$? )


1 Answer 1


You are really asking about parity doubling of baryons.

Fixing language: (N is a silly meretricious conflation of 2 and 3, since for N > 3 the quark masses completely invalidate any axial charge considerations, as per Gell-Mann's cynosure). For any multiplet except pseudoscalars, say, for baryons B, $$ B \mapsto e^{i\theta_L^i T^i \frac{1-\gamma^5}{2} + i\theta_R^i T^i \frac{1+\gamma^5}{2}+ i\theta_V +i\theta_A \gamma^5 } B, $$ where the diagonal subgroup is defined by $\theta_L^i=\theta_R^i$; the Nambu-Goldstone-mode axial by $\theta_L^i=-\theta_R^i$; the baryon number by $\theta_V$; and the anomalous axial U(1) by $\theta_A$. $SU(3)_V$ is the eightfold way, which works quite well put to quark mass differences. The symmetries of the 8 axial are well-understood to be SSB, as per the lightness of their associated pseudoscalars.

  • Let me dispatch the canards first: The broken and unbroken generator charges with odd parity correspond one-to-one with pseudoscalar mesons with the same quantum numbers, parity and flavor, which are pseudogoldstone bosons (or not--for the axial U(1), your η'). There is no such meaningful correspondence between any other meson or baryon multiplets and these charges.

  • In addition, the $N^2-1$ SSBroken axial generators are in a coset space, $\frac{SU(N)_L\times SU(N)_A}{SU(N)_V}$, and do not close to a Lie algebra (what you called $SU(N)_A$, a guaranteed trail of folly to grim conceptual grief).

Since both the $N^2-1$ SSB axial charges and the hapless anomalous violated axial U(1) charge would connect baryons of opposite parities by dint of their $\gamma_5$, and the low-lying spectrum of baryons is not parity-doubled, early students of chiral symmetry appreciated it is SSBroken; and, for $U(1)_A$ anomaly-broken. This was a useful hint they gladly took.

However, quite ironically, for higher excitation baryons, there is ample experimental evidence of parity doubling, at masses of order 1.5–3 GeV, especially for non-strange ones, cf. the references cited, and especially section 2 of ref [1].

Most, notably refs [2,3], try to explain this approximate doubling by partial/effective restoration of chiral symmetry at high temperatures or exotic low-temperature-high-density environments, and connect to our familiar low-energy physics. Some type of dynamical decoupling from pseudoscalars is envisioned, suppressing axial-breaking operators.

By this they mean that the symmetry is realized (approximately) in Wigner–Weyl mode with the appearance of (approximately) degenerate multiplets that transform into one another linearly under chiral rotations. [1]

  • The first reference [1], however, is further exploring an alternative, very different possibility: namely, that the anomalous $U(1)_A$ violation is somehow, dynamically, suppressed/cancelled/inconsequential in the baryon sector, leading to (imperfect) parity doubling.

The symmetry is broken explicitly by instantons/gluons (a lot, as opposed to by the customary small violation due to quark masses in ChPT): $[H,Q_5]\neq 0$, but leaves a subspace where the spectrum is sparse, involving baryons, alone; thereby permitting partial degeneracy in parity doublets! (Focus on Table 1 of [1].)

It is a mysterious, dark, recondite possibility of "unknown unknowns", worth exploring.


  1. Jaffe, R. L., Pirjol, D., & Scardicchio, A. (2006). Parity doubling among the baryons. Physics reports, 435(6), 157-182.

  2. Glozman, L. Y. (2010). Chiral symmetry restoration in excited hadrons and dense matter. Chinese Physics C, 34(9), 1212.

  3. Cohen, T. D. (2009). Hadrons and chiral symmetry. Nuclear Physics B-Proceedings Supplements, 195, 59-92.

  4. Sasaki, C. (2018). Parity doubling of baryons in a chiral approach with three flavors. Nuclear Physics A, 970, 388-397.

  • $\begingroup$ Thanks very much +1 --- I will accept your answer -- after I digest them (also other nice answers -- I will accept them but I need to digest first.) $\endgroup$ Apr 27, 2021 at 1:19
  • $\begingroup$ I guess it didn't sink in. $\endgroup$ Apr 27, 2022 at 15:15

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