Wick Rotation, interpretation of $\bar{p}^2$ vs the usual $p^2=m^2$

Question

Suppose we use the metric $(+,-,-,-)$ thus the momentum squared is

$p^2 = p_0^2-\vec{p}^2 = m^2>0$

Defining $p_E:=\mathrm{i}\cdot p_0$ and $\bar{p}:=(\,p_E,\vec{p})$ with Euclidean norm $\bar{p}^2 = p_E^2+\vec{p}^2$.

Here's my question:

If we plug in

$\mathrm{i}\,p_0$

instead of $p_E$ we see that $\bar{p}^2 = -p^2 = -m^2$?

So $\bar{p}^2$ is negative?

Also if $\bar{p}^2 = -p^2 = -m^2$ is true...is it always so? Since $p^2$ is a Lorentz invariant, but how do we interpret $\bar{p}^2$ if it's equal to $-m^2$

What I want to get to is the following:

Let

$\mathcal{L}_{int} = \frac{1}{2}g\phi_1^2\phi_3+\frac{1}{2}h\phi_2^2\phi_1$

with: $p_3 = p_1+p_2$ and

$M>2m$

Suppose we have a triangle loop with incoming momentum $p_3$ of mass $M>2m$ and two outgoing identical particles $\phi_2$ and $\phi_2$ momenta $p_1$ and $p_2$ each of mass $m$ (sorry bad notation but $\phi_1$ is not related to $p_1$). The incoming particle $\phi_3$ splits into two light ones (two $\phi_1$'s) of mass $\eta$ each and each of these connects to two $\phi_2$'s.

Thus we have the following momenta flowing inside the loop:

$k$, $ k-p_2$ and $k+p_1$.

After a Wick rotation we get the following integral:

\begin{equation} \int{\,\frac{\mathrm{d}^4\bar{k}}{(2\pi)^4} \frac{1}{\bar{k}^2+m^2}\frac{1}{\left(\bar{k}-\bar{p}_2\right)^2+\eta^2}\frac{1}{\left(\bar{k}+\bar{p}_1\right)^2+\eta^2}} \end{equation} Now to evaluate these let's use Schwingers trick:

\begin{equation} \frac{1}{\bar{k}^2+m^2} = \int_0^\infty{\mathrm{d}s\,\mathrm{e}^{-s(\bar{k}^2+m^2)}}, \end{equation} but for this to be OK we need to have $\mathrm{Re}(\bar{k}^2+m^2)>0$ and similarly for the other propagators, and this is where I get confused.

It seems they don't satisfy this condition depending on how we interpret $\bar{k}^2$ and $\bar{p}^2$ and so on...

On the other hand if all the squares in the denominator of the integrand are taken as positive then the convergence condition is trivially satisfied.

So please help me understand what I'm doing wrong and if you guys can show how to satisfy the positivity condition. Thanks in advance.

Answer 1

Yep. You've defined $\bar{p}$ such that $\bar{p}^2$ will have the opposite sign from $p^2$. Notice for instance that the spatial components of momentum, $\vec{p}^2$, have a negative sign in your definition of $p^2$ but a positive sign in your definition of $\bar{p}^2$.

Of course, the sign doesn't change whether something is a Lorentz invariant or not. The way you interpret $\bar{p}^2$ is just what it looks like, the negative invariant mass squared. Or in other words, you interpret $\bar{p}^2$ as a Lorentz invariant, from which you can obtain the physical mass of the particle as $m = \sqrt{-\bar{p}^2} = -i\sqrt{\lVert\bar{p}\rVert}$.