and indeed, the calculation at the one–loop level reproduces the result in eq. (\[eq:lsm\]) but with a cut–off replacing the sigma mass. It is simply that in this case the effective low–energy theory does not have the high–energy behaviour of the underlying theory.

### The Weinberg Sum Rules {#subsec:wsrss}

There is a correlation function which is particularly sensitive to properties of chiral symmetry breaking, namely the two–point function $$\label{eq:lrtpf}
\Pi_{LR}^{\mu\nu}(q)=2i\int d^4 x\,e^{iq\cdot x}\langle 0\vert
\mbox{\rm T}\left(L^{\mu}(x)R^{\nu}(0)^{\dagger}\right)\vert 0\rangle ,$$ with currents $$L^{\mu}=\bar{d}(x)\gamma^{\mu}\frac{1}{2}(1-\gamma_{5})u(x)
\quad {\mbox{\rm and}}\quad 
R^{\mu}=\bar{d}(x)\gamma^{\mu}\frac{1}{2}(1+\gamma_{5})u(x)\,.$$ In the chiral limit where the light quark masses are set to zero, this two–point function only depends on one invariant amplitude ($Q^2\equiv -q^2\ge 0$ for $q^2$ spacelike) $$\label{eq:lritpf}
\Pi_{LR}^{\mu\nu}(Q^2)=(q^{\mu}q^{\nu}-g^{\mu\nu}q^2)\Pi_{LR}(Q^2).$$ The function $\Pi_{LR}(Q^2)$ vanishes order by order in perturbation theory and is an order parameter of the spontaneous breakdown of chiral symmetry (S$\chi$SB) for all values of the momentum transfer. As illustrated in Fig. \[fig:Fig1d\]

the two–point function in eq. (\[eq:lrtpf\]) is the difference of a vector current correlation function and an axial–vector current correlation function. Chiral symmetry identifies these two correlation functions.

Two–point functions in general obey dispersion relations. \[[See the discussion in section \[sec:disperrels\].]{}\] In 1967, and therefore prior to the development of 