[Physics] Symmetries of the Standard Model: exact, anomalous, spontaneously broken

quantum-anomaliesquantum-field-theorystandard-modelsymmetrysymmetry-breaking

There are a number of possible symmetries in fundamental physics, such as:

  • Lorentz invariance (or actually, Poincaré invariance, which can itself be broken down into translation invariance and Lorentz invariance proper),

  • conformal invariance (i.e., scale invariance, invariance by homotheties),

  • global and local gauge invariance, for the various gauge groups involved in the Standard Model ($SU_2 \times U_1$ and $SU_3$),

  • flavor invariance for leptons and quarks, which can be chirally divided into a left-handed and a right-handed part ($(SU_3)_L \times (SU_3)_R \times (U_1)_L \times (U_1)_L$),

  • discrete C, P and T symmetries.

Each of these symmetries can be

  • an exact symmetry,

  • anomalous, i.e., classically valid but broken by renormalization at the quantum level (or equivalently, if I understand correctly(?), classically valid only perturbatively but spoiled by a nonperturbative effect like an instanton),

  • spontaneously broken, i.e., valid for the theory but not for the vacuum state,

  • explicitly broken.

Also, the answer can depend on the sector under consideration (QCD, electroweak, or if it makes sense, simply QED), and can depend on a particular limit (e.g., quark masses tending to zero) or vacuum phase. Finally, each continuous symmetry should give rise to a conserved current (or an anomaly in the would-be-conserved current if the symmetry is anomalous). This makes a lot of combinations.

So here is my question: is there somewhere a systematic summary of the status of each of these symmetries for each sector of the standard model? (i.e., a systematic table indicating, for every combination of symmetry and subtheory, whether the symmetry holds exactly, is spoiled by anomaly or is spontaneously broken, with a short discussion).

The answer to each particular question can be tracked down in the literature, but I think having a common document summarizing everything in a systematic way would be tremendously useful.

Best Answer

I'd say that there is not a systematic summary of the status of symmetries on particle physics, but if any, it should be spread all over the PDG review.

However, I'd like to comment on a few points.

  • So far Lorentz symmetry is exact on all sectors.${}^\dagger$

  • Scaling (part of the conformal transformations) is broken once an energy scale is introduced in the theory. Therefore, you can not extend the Lorentz group symmetry to a conformal symmetry.${}^{\dagger\dagger}$ The existence of masses breaks explicitly this symmetry (and also the global chiral symmetry).

  • Gauge symmetry can be broken spontaneously. Because it is the only way we know for breaking the symmetry and still preserve desirable properties!

  • Anomalies aren't bad! As long as they are related with global transformations, not related with the gauge symmetries.

  • Flavour "symmetries"... They are not, unless fermion masses vanish.

  • $C$, $P$ and $T$, mathematically we expect that $CPT$ is a symmetry, but they aren't conserved individually.${}^{\dagger\dagger\dagger}$

Despite all of this, tomorrow our understanding of the symmetries of the Universe might change radically! (Kind of love this uncertainty!)

${}^\dagger$ NOTE: exact does not mean in the literal way, but only that if it's broken the scale is outside our current measurement limits.

${}^{\dagger\dagger}$ Although pure gauge theories could posses a conformal symmetry, it makes no sense to consider "free" theories.

${}^{\dagger\dagger\dagger}$ $CP$ is known to be violated (specially in the electroweak sector, and there is the known strong $CP$ problem).

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