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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.

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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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A pretty exhaustive summary in the context of Standard Model already exists in the following source:

''Dynamics of the Standard Model'' - Donoghue, Golowich, Holstein,
Chapter 3 - Symmetries and Anomalies

A limited preview can be found here. (Embarrassingly though, the very first page of the chapter is excluded from Google's preview!)



But here's the issue I have with my ''answer''. The way I read your 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).

This is a classic resource recommendation question, and as far as I know, ''link-only'' answers aren't welcome with this brand of questions. But, even if I ''summarize'' that chapter, I would be merely reproducing information which is already existing in this reference, so my ''effort'' is only to create a table based on that information. While I would've sufficiently bent around the rules by doing it, isn't that a stupid thing to do?

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    $\begingroup$ Well, if the information is not (fully) available to everybody who might visit this site (which it is not), it is still useful for you to extract the important parts from the full text - which I assume you have access to. I therefore encourage you to write out a complete answer! $\endgroup$ – Danu Jul 24 '14 at 17:57
  • $\begingroup$ @Danu - Haha... ''...I assume you have access to'' is funny. (How else will I know?) But my point is, since OP asks ''Is somewhere a systematic summary'', that part gets covered by my first sentence. Now, for the sake of everyone else who visits the site, I may write an explicit ''original'' answer based on information in that text, but what would you consider more credible? The original (highly rated) textbook, or a secondary source answer, by some anonymous, not-very-high-rep user of a Physics forum? I would go for the first option, especially when the second option doesn't have ... $\endgroup$ – 299792458 Jul 24 '14 at 18:20
  • $\begingroup$ (contd.) ... a great deal more to offer beyond that already great answer. Anyways, that's not a jab on your suggestion, that's just my viewpoint. Thanks for your suggestion :) $\endgroup$ – 299792458 Jul 24 '14 at 18:21

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