Question

In all explainations of spontaneous symmetry breaking that I've seen, the scalar field doing the breaking is redefined around its vacuum expectation value (VEV), and similarly for the higgs mechanism - for example with a complex scalar field and U(1) symmetry:

ϕ(x)=(v+rho(x))e^iχ(x)/v
where v is the VEV

Similarly for the higgs mechanism: the field is written as above, and then expanded out. This makes it very clear where the gauge field's mass term comes from, or in the case of just SSB, where the mass term for the goldstone boson goes. But my question is as follows: is this redefinition necessary?

Answer 1

The redefinition of the scalar field in spontaneous symmetry breaking and the Higgs mechanism simplifies understanding mass acquisition by gauge fields but is not strictly required. It is a mathematical convenience that aids in transparency and calculation. This technique is essential for explaining key cosmological phenomena, including the inflationary scenario and the uniformity of the CMBR.

Step-by-step explanation:

The redefinition of the scalar field around its vacuum expectation value (VEV) in the context of spontaneous symmetry breaking (SSB) and the Higgs mechanism is a mathematically convenient step, but not strictly necessary. It simplifies calculations and makes the physical implications, such as the generation of mass terms for gauge bosons and the disappearance of mass terms for the Goldstone bosons, more transparent. In the U(1) symmetry example you provided with a complex scalar field, this redefinition results in the scalar field being expressed as φ(x)=(v+ρ(x))e^{iχ(x)}/v, where ν is the VEV. This approach contributes to clarity in understanding phenomena such as the acquisition of mass by gauge fields through the Higgs mechanism, which is critical in the broader context of particle physics and cosmology.

Such techniques are foundational in understanding the very early universe, particularly in relation to the grand unified theory (GUT) and electroweak epochs. For instance, the spontaneous symmetry breaking that transitions from GUT to electroweak leads to an energy release that propels the inflationary scenario, elucidating the smoothness of the cosmic microwave background radiation (CMBR). This showcases how spontaneous symmetry breaking is deeply tied to both particle physics and cosmology, and while it is difficult to test these high-energy phenomena directly with current technology, they are central to most cosmological theories, including the Higgs mechanism and inflation.