Symmetries of the Standard Model: exact, anomalous, spontaneously broken

Question

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.

Answer 1

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!)

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${}^\dagger$ 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).

Answer 2

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!)

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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?