# 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

## 1 Answer

• 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.

• The way I would evaluate such a thing perturbatively would be by summing up 1-PI Graphs as in Peskin Section 7.1

• The formula you mention (eq. 3.114) is the non-perturbative result for a free Dirac field and thus not the thing we want in QCD.

And two final remarks:

• Actually there are many ways to get N-Point Functions. Actually they all the time follow from some Generating Functional, which can be calculated either with Operator Methods or with Path Integrals (and maybe also other methods).
• "But [...] (Peskin) they've calculated the value in terms of the propagator". Actually the two-point function is what we call the propagator. I recommend for recalling the various definitions of Propagators in QFT and ther interpley the corresponding chapter on Propagators in Minkowski Space in "Quantum field theory in curved space" by Birrell and Davies.
• 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
• Sorry, my answer was not really good, since the document you've cited mentions a quark condensate, and I somehow thought it is just a usual propagator (even if the numerical value showed that it isn't, since there is no r dependence). Sorry! I am no expert on this area, therefore I don't want to give my point of view, since i don't understand this topic sufficiently. But maybe this reference could help you: arxiv.org/pdf/hep-ph/0010175v1 – warpfel Sep 21 '16 at 8:47