# A Why can the $1$-point correlation function be made to vanish?

1. Nov 17, 2016

### spaghetti3451

The $1$-point correlation function in any theory, free or interacting, can be made to vanish by a suitable rescaling of the field $\phi$.

I would like to understand this statement.

With the above goal in mind, consider the following theory:

$$\mathcal{L} = \frac{1}{2}\left((\partial\phi)^{2}-m^{2}\phi^{2}\right)+\frac{g}{2}\phi\partial^{\mu}\phi\partial_{\mu}\phi.$$

What criteria (on the Lagrangian $\mathcal{L}$) is used to determine the value of the field $\phi_{0}$ such that the transformation $\phi \rightarrow \phi + \phi_{0}$ leads to a vanishing $1$-point correlation function $\displaystyle{\langle \Omega | T\{\phi(x_{1}\phi(x_{2})\}| \Omega \rangle}$?

2. Nov 17, 2016

You should look at the saddlepoint solution for the field from the path integral. When they mean shift the field it corresponds to expanding about the different vaccua.

3. Nov 18, 2016

### atyy

4. Nov 18, 2016

### spaghetti3451

5. Nov 18, 2016

### spaghetti3451

Right!

When you expand about the vacua, shouldn't you get a new constant term in the Lagrangian?

Doesn't this constant term affect the correlation functions?

6. Nov 19, 2016