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The existense of the quarks suppresses the gluon fluctuations creating the flux-tubes. But yeah, it requires energy to separate the quarks with the flux tubes, that's why when they are sufficiently separated the energy is used to create the quark-anti quark pair called meson. And you probably mentioned The "true" vacuum because of the suppression of the flux ...


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I will answer question 2. Let $f$ be a smooth test function, $$\int_0^1 \frac{4}{3}\frac{1+z^2}{(1-z)_+}f(z) + 2 \delta(1-z) f(z) dz = \int_0^1 \frac{4}{3}\frac{(1+z^2)f(z) - 2f(1)}{(1-z)} + 2 \delta(1-z)f(z) dz $$ using the definition of a $+$-distribution. Continuing after some simplification; one tries to force the two bits together by cleverly adding ...


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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 ...



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