Understanding Peskin's Argument for Equation 6.46

[kof9595995]

I don't quite get the argument peskin used to obtain equation(6.46), page 191:
\int{\frac{d^{4}l}{(2\pi)^4}\frac{l^{\mu}l^{\nu}}{D^3}}=\int{\frac{d^{4}l}{(2\pi)^4}\frac{\frac{1}{4}g^{\mu\nu}l^2}{D^3}}
He said"The integral vanishes by symmetry unless \mu=\nu. Lorentz invariance therefore requires that we get something proportional to g^{\mu\nu}...".
I don't understand the "Lorentz invariance therefore..." part. How can one deduce from Lorentz invariance that LHS is an invariant tensor?
I can convince myself the result by arguing spherical symmetry of the integrand, but I just want to understand Peskin's reasoning.

 

[azntoon]

Thanks in advance.A:Peskin is using the fact that the integral must be Lorentz invariant, which means that it is a tensor. The only way for a scalar to be a tensor is if it is proportional to the metric tensor. This means that the integral must be proportional to $g^{\mu\nu}$, and the proportionality constant is determined by plugging in $\mu=\nu$ into the integral.