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Consider the QCD lagrangian :

$$L_{QCD}=-\frac{1}{4}G^a_{\mu\nu}G^{a\mu\nu}+\sum\bar{\psi}_q(i\not{D}-m_q)\psi_q$$

Textbooks explain that this lagrangian is spontaneously broken by the VEV of quark condensates. But where is the potential that gives rise to this VEV ? In electroweak symmetry breaking, the potential is supplied by the Higgs field. In QCD, does the VEV of quark condensates originate in the minimal lagrangian ?

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The VEV is quantum mechanical, it can not be read off from the Lagrangian. To find the VEV from the potential requires one to quantise the theory, then calculate the effective action at strong coupling. What should happen is that at strong coupling the quarks form hadrons; which are quark condensates. To perform the calculations is very tough and not yet done. Lattice simulations are convincing to most though.

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