# How to find the explicit value of the fermion vacuum expectation value?

In the derivation for Goldstone modes, a pdf I found online (scipp.ucsc.edu/~dine/ph222/goldstone_lecture.pdf) says it's believed the above-mentioned expectation value $\langle \bar{ \psi} \psi \rangle$ is approximately (0.3GeV)³. I consulted Peskin and Schröder, and I thought expanding $\psi$ in terms of creation and annihilation operators would be the way to proceed but they've calculated the value in terms of the propagator. (eq. 3.114) Is this even the correct way to proceed?

How do I calculate the explicit value of above expression/ prove it is non 0?

Sorry for the lack of formulae, it's really hard to type latex commands while on mobile. Will be updating the relevant formulae in the morning.

• These condensates are calculated using lattice QCD. See arxiv.org/abs/hep-lat/0504006 and references therein. – ved Sep 24 '16 at 9:35

• I guess you refer to p. 6 of the pdf you've cited, there they say "Now it is believed that in QCD, the operator Ψ̄Ψ has a non-zero vacuum expectation value, i.e. $\langle \bar{\Psi_f} \Psi_f \rangle \approx (0.3 ~{\rm GeV})^3$". Since QCD is a nonlinear theory I think, that they've evaluated the vacuum expectation value perturbatively.
• I went through section 7.1, Peskin has just evaluated the expression $\langle \Omega | \bar{\phi} \phi | \Omega \rangle$ in terms of $\rho$ (7.6) and later he takes the fourier transform for finding the electron self energy. I'm still unsure on how to proceed to calculate explicitly the value. If the perturbative propagator (loosely speaking) is calculated in this manner, just substituting the correct value of $\rho$ would technically be enough to calculate the value? – pyroscepter Sep 21 '16 at 7:30