How to get <PERSON> and <PERSON>'s Equation 12.36?
(Now edited to contain the pertinent equations from <PERSON> and <PERSON>)
I'm trying to use the one-loop expression for the 4 point <PERSON> function to calculate the beta function of massless phi^4 theory. In fact, <PERSON> and <PERSON> give the result in Eq. 12.46, but it is not clear to me how they did it... 346
can someone tell me where my error is? Specifically, I don't understand why P&S only use the linear-in-lambda term of the 4-point <PERSON>'s function to derive the beta function? The one-loop 4-point <PERSON>'s function is given just below Eq. 12.43 and reproduced here:
$G^{\left(4\right)}\left(p_1,p_2,p_3,p_4\right) = \left[-i \lambda + (-i \lambda)^2\left{iV(s) + iV(t) + iV(u)\right} - i \delta_{\lambda} \right] \cdot \prod_{i=1,2,3,4} \frac{i}{p_i^2}$
where $s=(p_1 + p_2)^2$ ,
$t=(p_3 - p_1)^2$ ,
and $u=(p_4 - p_1)^2$ are the <PERSON> variables, and where
$V(p^2) = -\frac{1}{2}\int_0^1 dx \frac{\Gamma(2-d/2)}{(4 \pi)^{d/2}}\frac{1}{\left[ -x(1-x) p^2 \right]^{2-d/2}}$
using dimensional regularization. Also, the vertex counterterm (determined using <PERSON> and <PERSON>'s renormalization scheme where the 4-point scattering amplitude is forced to equal $i \lambda$ at the spacelike momentum interval $p^2 = -M^2$) is
$\delta_{\lambda} = (-i \lambda)^2 3 V(-M^2)$.
Now back to my questions... If you substituted both the linear and quadratic-in-lambda term in the 4-point <PERSON>'s function into the Callan-Symnazik equation for a massless theory:
$\left[ M \frac{\partial}{\partial M} + \beta \frac{\partial}{\partial \lambda} + 4 \gamma \right] G^{\left(4\right)}\left(p_1,p_2,p_3,p_4\right) = 0 $
then that would lead to a contribution to beta that was linear in lambda, right? And wouldn't this be the leading order contribution to beta, instead of what P&S write in Equation 12.46? P&S's Equation 12.46 is
$\beta = \frac{3 \lambda^2}{16 \pi^2} $ and another important intermediate step is P&S's Equation between 12.45 and 12.46:
$ M \frac{\partial}{\partial M} G^{\left(4\right)}\left(p_1,p_2,p_3,p_4\right) = \frac{3 i \lambda^2}{16 \pi^2}\prod_{i=1,2,3,4} \frac{i}{p_i^2}$
Many thanks everyone!
Best, <PERSON> 818