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

[Murtuza Tipu]

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 (SU2×U1 and SU3),

flavor invariance for leptons and quarks, which can be chirally divided into a left-handed and a right-handed part ((SU3)L×(SU3)R×(U1)L×(U1)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.

 

[AndyW]

Thank you for your insightful question about the symmetries in fundamental physics. I am always interested in discussing and exploring different theories and concepts in the field. Let me address your question by breaking it down into different parts and providing a summary of the current understanding of each symmetry in the Standard Model.

First, let's start with Lorentz invariance. This symmetry refers to the invariance of physical laws under rotations and boosts in space and time. In the Standard Model, this symmetry is considered exact, meaning it holds for all subtheories and sectors. This is supported by experimental evidence, such as the constancy of the speed of light and the success of Lorentz transformations in special relativity.

Next, we have conformal invariance, which is a scale invariance that describes the invariance of physical laws under changes in scale. This symmetry is not considered exact in the Standard Model, as it is broken at high energies due to quantum effects. However, it is still valid at low energies, which is why it is often referred to as an anomalous symmetry.

Moving on to global and local gauge invariance, these symmetries are crucial in the Standard Model as they describe the interaction between particles. These symmetries are considered exact for the gauge groups SU(2) x U(1) and SU(3), which are responsible for the weak and strong interactions, respectively. However, there is still ongoing research on whether these symmetries are also valid at high energies. Additionally, there are also discussions about the possibility of spontaneous breaking of these symmetries in certain vacuum phases.

The flavor invariance for leptons and quarks, which can be divided into left and right-handed parts, is also considered an exact symmetry in the Standard Model. However, there is ongoing research on whether this symmetry is also valid at high energies and in different vacuum phases.

Lastly, the discrete C, P, and T symmetries refer to charge conjugation, parity, and time reversal, respectively. These symmetries are considered exact in the Standard Model, but there is ongoing research on whether they are also valid at high energies and in different vacuum phases.

To summarize, the current understanding of the symmetries in the Standard Model is that Lorentz invariance is exact, conformal invariance is anomalous, global and local gauge invariance are exact but may