# A question about symmetry in the phi^4 theory

1. Jun 22, 2015

### gobbles

1. The problem statement, all variables and given/known data
Why does the symmetry $\phi\rightarrow-\phi$ mean that an amplitude can be written as
$\alpha + \beta p^2 + \gamma p^4 + ...$
without the odd terms in $p$?
2. Relevant equations
I understand that, due to this symmetry, any diagram in $\phi^4$ has an even number of external legs, because otherwise the correlation function of the external fields is zero. So any diagram can be written in the form
$V(p^2)\left(\frac{i}{p^2-m^2}\right)^{n}$
where $n$ is even and $V(p^2)$ is the expression for the amplitude without the external legs. Expanding $V(p^2)$ in $p$ will, of course, give only even powers of $p$, as will the expansion of $\left(\frac{i}{p^2-m^2}\right)^n$, but that is true also for $n$ odd, corresponding to an odd number of external legs. So where does this symmetry play a role here?

3. The attempt at a solution
Outlined in (2).

2. Jun 27, 2015