# Mass generation by Chern-Simons theory

Why the mass generation via a Higgs mechanism is different from that of Chern-Simons theory? I haven't done any formal course in Quantum field theory,so how do I understand this just having some basic knowledge in Classical field theory?

To understand the difference between the Higgs mechanism and the mass generated by a Chern-Simons term one has to realize that the Higgs case is related to a (spontaneously) broken symmetry.

Assume that you have written down a scalar field theory with a gauge symmetry which admits a set of equivalent ground states (vacua) for the potential of the scalar. These states can be transformed into each other by continuous transformation. One can now say that the theory is symmetric under such a transformation. If the theory explicitely chooses one of the equivalent vacua to be the ground state, the scalar (Higgs) field acquires a vacuum expectation value and the system is no longer symmetric. There is no longer a range of possible ground states, but one of them is singled out. This is what is referred to as spontaneous symmetry breaking. If we now expand the theory around that vacuum, it turns out that previously massless gauge particles acquire mass terms.

The Chern-Simons term on the other hand has got nothing to do with spontaneous breaking of symmetries. Writing it down in a theory simply creates mass terms in the equation of motion of particles. It is, however, interesting because of its topological significance, but that is a different story.

They are very different.

When you use a Higgs mechanism with a Yang-Mills action, symmetry breaking causes the gauge fields $A$ to gain mass. This is done in 4D.

When you add a Chern-Simons term to a Yang-Mills action, you can see from the field equations that $\ast F$ becomes massive, not $A$. There is no symmetry breaking here. Also this is in 3D and does not work in 4D.

You can check Nakahara's book example 11.10

• The Nakahara example is 11.20. – rikard42 Dec 7 '18 at 15:10