setMandelstam([k1:m1, k2:m2, k3:m3, k4:m4]) generates a list of Mandelstam and mass shell substitutions for incoming particles with momentums k1 and k2 with masses m1 and m2 and outcoming particles with momentums k3 and k4 with masses m3 and m4.setMandelstam([k1:m1, k2:m2, k3:m3, k4:m4], s, t, u) generates a list of Mandelstam substitutions with specified notation for Mandelstam s, t and u variables.setMandelstam uses the following definition of Mandelstam variables:\begin{gather*} s = (p_1 + p_2)^2 = (p_3+p_4)^2\\ t = (p_1 - p_3)^2 = (p_2 - p_4)^2\\ u = (p_1 - p_4)^2 = (p_2 - p_3)^2 \end{gather*}
Generate a list of Mandelstam and mass shell substitutions:
def mandelstam = setMandelstam([k1_a: 'm1', k2_a: 'm2', k3_a: 'm3', k4_a: 'm4']) println mandelstam
> k1_{a}*k1^{a} = m1**2
> k2_{a}*k2^{a} = m2**2
> k3_{a}*k3^{a} = m3**2
> k4_{a}*k4^{a} = m4**2
> 2*k1_{a}*k2^{a} = s-m2**2-m1**2
> 2*k4^{a}*k3_{a} = s-m3**2-m4**2
> -2*k3^{a}*k1_{a} = t-m3**2-m1**2
> -2*k4^{a}*k2_{a} = t-m4**2-m2**2
> -2*k4^{a}*k1_{a} = -m4**2-m1**2+u
> -2*k3^{a}*k2_{a} = -m3**2-m2**2+u
Same when some particles are massless:
def mandelstam = setMandelstam([k1_a: '0', k2_a: 'm2', k3_a: 'm3', k4_a: '0']) println mandelstam
> k1_{a}*k1^{a} = 0
> k2_{a}*k2^{a} = m2**2
> k3_{a}*k3^{a} = m3**2
> k4_{a}*k4^{a} = 0
> 2*k2^{a}*k1_{a} = -m2**2+s
> 2*k4^{a}*k3_{a} = -m3**2+s
> -2*k3^{a}*k1_{a} = -m3**2+t
> -2*k4^{a}*k2_{a} = t-m2**2
> -2*k4^{a}*k1_{a} = u
> -2*k2_{a}*k3^{a} = -m3**2+u-m2**2
Specify custom notation for Mandelstam variables:
def mandelstam = setMandelstam(
[k1_a: 'm1', k2_a: 'm2', k3_a: 'm3', k4_a: 'm4'],
'hS', 'hT', 'hU')
println mandelstam
> k1_{a}*k1^{a} = m1**2
> k2_{a}*k2^{a} = m2**2
> k3_{a}*k3^{a} = m3**2
> k4_{a}*k4^{a} = m4**2
> 2*k2^{a}*k1_{a} = -m1**2+hS-m2**2
> 2*k3_{a}*k4^{a} = -m3**2+hS-m4**2
> -2*k3^{a}*k1_{a} = -m3**2-m1**2+hT
> -2*k2_{a}*k4^{a} = hT-m4**2-m2**2
> -2*k1_{a}*k4^{a} = -m1**2+hU-m4**2
> -2*k2_{a}*k3^{a} = -m3**2+hU-m2**2