A question about symmetry in the phi^4 theory

[gobbles]

Homework Statement

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 $p$?

Homework 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?

The Attempt at a Solution

Outlined in (2).

 

[mark726]

Hello,

The symmetry $\phi\rightarrow-\phi$ plays a crucial role in the form of the amplitude $V(p^2)$. This is because the symmetry implies that the amplitude must be an even function of $\phi$. This means that the terms in the amplitude must be of the form $\alpha + \beta p^2 + \gamma p^4 + ...$, where $\alpha, \beta, \gamma, ...$ are constants. If the amplitude had odd terms in $p$, then the symmetry would be broken, as the odd terms would change sign under the transformation $\phi\rightarrow-\phi$. Therefore, the symmetry ensures that the amplitude can only have even powers of $p$, resulting in the form $V(p^2)$ without the odd terms in $p$.

Additionally, as you mentioned, the symmetry also plays a role in the number of external legs in a diagram. Since the correlation function of the external fields is zero for an odd number of external legs, the symmetry ensures that any diagram in $\phi^4$ must have an even number of external legs. This is because the symmetry would be broken if there were an odd number of external legs, resulting in a non-zero correlation function.

I hope this helps clarify the role of the symmetry in the form of the amplitude and the number of external legs in a diagram. Let me know if you have any further questions.