Question

Suppose we have a generic non-interacting Lagrangian of two complex scalar fields,L=(∂μΦ†)(∂μΦ)−Φ†M2Φ=12∂μΦi∂μΦi−m2i2ΦiΦi,i=1,2,3,4.L=(∂μΦ†)(∂μΦ)−Φ†M2Φ=12∂μΦi∂μΦi−m2i2ΦiΦi,i=1,2,3,4.(1)(2)whereΦ=(ϕ1ϕ2)Φ†=(ϕ†1ϕ†2)The invariance of the kinetic term indicates that for the transformationΦ→UΦandΦ†→Φ†U†,U†U=I. This indicatesU(2)symmetry for (1) andSO(4)symmetry for (2). The question is how does the mass term break symmetries in this lagrangian in each case. As suggested by this post, in the case where we have the same masses (Mis proportional toI), theU(2) group survives (so is SO(4)). we can find the 4 conserved charges using the relation that U(2)

is locally isomorphic to SU(2)×U(1)
. If no symmetries are broken in this case, how can we see the two extra charges, which are clearly seen as generators of SO(4)
? (An answer from the post linked below says the mechanism is SO(4)∼SU(2)×SU(2)
but I don't quite understand this.)

Now if the two masses are not identical, this post suggests that U(2)
breaks into U(1)×U(1), whereas SO(4)
breaks into U(1)×U(1)∼O(2)×O(2). How can I understand if these two cases still have the same generators?

We have 3 generators for SU(2)L and 3 for SU(2)R, we still have the global U(1) and U(1)
for each complex scalar field, is that right? So do we have 3+3+3 = 9 symmetry generators in total?

Answer 1

The generic non-interacting Lagrangian of two complex scalar fields has symmetries that depend on the equality of the masses of the fields. When masses are identical, the U(2) and SO(4) symmetries are preserved, whereas differing masses lead to symmetries breaking into U(1)×U(1) or O(2)×O(2), respectively.

Step-by-step explanation:

The question at hand deals with the invariance and breaking of symmetries of a Lagrangian that describes two complex scalar fields. In scenarios where both fields have identical masses, the Lagrangian exhibits U(2) symmetry which can be seen as the product of SU(2)×U(1), representing three generators for SU(2) rotations and one for U(1) phase transformations. With SO(4), the symmetry is isomorphic to a product of SU(2)×SU(2), again reflecting rotation invariances but in a four-dimensional space. When the masses of the scalar fields differ, the symmetries are broken; U(2) reduces to U(1)×U(1) and SO(4) to O(2)×O(2). These remaining symmetries correspond to separate conservation of charges for each field. For SO(4), we still speak of six generators reflecting the SU(2)L and SU(2)R groups, and two additional generators from the global U(1) symmetries.