Monday, July 25, 2016

mass - Massive Gauge Bosons without Higgs fields


In a possible theory like our Standard model but without a Higgs i.e.:


$$ \mathcal{L}=i\bar{\Psi}_f\gamma_\mu D^\mu\Psi_f-\text{Tr}[G^b_{\mu\nu}G^{b\,\mu\nu}] $$


where $b,f$ run over the typical species we have in the standard model (SM), and all fields are in the same representation as in the SM.



In this context it is sometimes stated that, although there is no Higgs, there would be a mass generation mechanism for the gauge bosons of $SU(2)$ because of QCD. This happens via the chiral quark condensate $\langle q_L q_R\rangle\neq 0$. (Or statements like "the gauge bosons eat up the pion")


My question is now, how can I see that this generates a mass for the $SU(2)$-gauge bosons? Usually using methods of spontaneous symmetry breaking, I would put a vacuum expectation value for some field and see that it results in a term that behaves like a mass term. But this won't work here because there is no term involving quarks and bilinear in gauge bosons.




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