How many degrees of freedom do you have and show that the

Question

1. Consider the process ab → 123. The three-body Lorenz-invariant phase-space for can be written as

(2Π)⁴ ∫d⁴p₁/(2Π)³, d⁴p₂/(2Π)³ d⁴p₃/(2Π)³ δ⁽⁴⁾ (p_a + p_b - p₁ - p₂ - p₃) δ(p₁² - m₂²) δ(p₂² - m₂²) δ(p₃² - m₃²).

(a) If you fix the initial-state kinematics, how many degrees of freedom do you have? I.e. how many final-state integrals are left unconstraint?

(b) Show that the above expression for the phase-space is equivalent to the one written in terms of three-momentum integrals