# A Possible decay process for a cubic scalar self-interaction

1. Dec 5, 2016

### spaghetti3451

Consider the Lagrangian

$$\mathcal{L}=\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-\frac{1}{2}m^{2}h^{2}-\frac{\lambda}{3!}h^{3}$$

for a real scalar field $h$.

This is the Klein-Gordon Lagrangian with a cubic self-interaction term.

Does this model allow the decay process

$$h \rightarrow h + h?$$

Clearly, in the rest frame, the incoming scalar is at rest, so its total energy is its rest energy, and therefore, it cannot decay to two copies of itself (with twice the rest energy), as that would violate energy conservation?

2. Dec 5, 2016

### nrqed

You are correct, the decay cannot take place due to its violation of energy-momentum conservation.

3. Dec 5, 2016

### spaghetti3451

But gluons can split into two or more gluons, right?

And gluons are massive.

4. Dec 5, 2016

### nrqed

Gluons are actually massless.

By the way, it is possible of course to have one h particle turning into two h particles with these off-shell. But there cannot be an actual decay of one h into two on-shell ("real") h particles