Field-Theoretic Approach to theρMeson

A previously discussed method is used to improve the Born series for the $\pi -\pi$ scattering amplitude in the $\lambda {\phi }^4$ model. Only the first two nonvanishing Born terms are considered in each instance: to order ${\lambda }^2$ for isospin 0 and 2, and to order ${\lambda }^3$ for isospin 1. The results may be interpreted in terms of the existence of an isospin-one $P$-wave resonance. Its position is adjusted to fit experiment, thereby determining all other parameters in the theory. Taking the pion mass as a unit, we find the energy width of the resonance to be $\Gamma =0.72\mathrm{}$. The scattering lengths ${a_J}^T$ come out as ${a_0}^0=-0.78\mathrm{}$, ${a_1}^1=+0.032\mathrm{}$, ${a_0}^2=-0.44\mathrm{}$. The coupling constant, as defined by Chew and Mandelstam, is $\lambda =+0.24\mathrm{}$. All these values are rather insensitive to the position of the resonance. The numerical calculations may be done by hand.