Phase Space Factor: Confused About 3D to 4D Conversion

[Josh1079]

Hi,

I'm recently reading "Particle Physics in the LHC Era" and there is a part about the phase space factor that confuses me. When giving the Lorentz invariant phase space, they wrote:

d³p / 2E = θ(E) δ(p² - m²) d⁴p

This is very confusing as it equates a three dimensional differential to a four dimensional one. Is there anything I didn't take into account?

Thank you!

 

[ShayanJ]

Well...actually that equal sign is not really an equal sign. It just means these two are the same integral measures and that's because we're assuming the energy is positive and the particle is on-shell(i.e. satisfies $p^2=m^2$).
So it actually means $\displaystyle \int f(E,\mathbf p) \frac{d^3\mathbf p}{2E}=\int f(p) \theta(E)\delta(p^2-m^2)d^4p$, where $\mathbf p$ is a three-vector and $p$ is a four-vector.

 

[Vanadium 50]

Is there anything I didn't take into account?

The delta function, which functions as a constraint that reduces the dimensionality by 1.

 

[Josh1079]

Thank you so much ShayanJ! I get it now!

And also thanks to Vanadium 50 for pointing that out!