setMandelstam5([k1:m1, k2:m2, k3:m3, k4:m4, k5:m5]) generates a list of generalised Mandelstam and mass shell substitutions for incoming particles with momentums k1 and k2 with masses m1 and m2 and outcoming particles with momentums k3, k4 and k5 with masses m3, m4 and m5.setMandelstam5([k1:m1, k2:m2, k3:m3, k4:m4, k5:m5], s, t1, t2, u1, u2) generates a list of Mandelstam substitutions with specified notation for generalised Mandelstam s, t1, t2, u1 and u2 variables.setMandelstam5 uses the following definition of Mandelstam variables:\begin{gather*} s = (p_1 + p_2)^2\\ t_1 = (p_1 - p_3)^2 \\ t_2 = (p_1 - p_4)^2 \\ u_1 = (p_2 - p_3)^2 \\ u_2 = (p_2 - p_4)^2 \\ \end{gather*}
Generate a list of Mandelstam and mass shell substitutions:
def mandelstam = setMandelstam5([k1_a: 'm1', k2_a: 'm2', k3_a: 'm3', k4_a: 'm4', k5_a: 'm5']) 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
> k5_{a}*k5^{a} = m5**2
> k2^{a}*k1_{a} = (1/2)*(-m1**2-m2**2+s)
> k3^{a}*k1_{a} = (1/2)*(m1**2-t1+m3**2)
> k1_{a}*k4^{a} = (1/2)*(m4**2+m1**2-t2)
> k1_{a}*k5^{a} = (1/2)*(-m4**2-m1**2+t2+t1-m2**2-m3**2+s)
> k2_{a}*k3^{a} = (1/2)*(-u1+m2**2+m3**2)
> k2_{a}*k4^{a} = (1/2)*(m4**2-u2+m2**2)
> k2_{a}*k5^{a} = (1/2)*(-m4**2-m1**2+u1+u2-m2**2-m3**2+s)
> k3_{a}*k4^{a} = (1/2)*(m4**2+m5**2+2*m1**2-t2-u1-t1-u2+m3**2-s+2*m2**2)
> k3_{a}*k5^{a} = (1/2)*(-m4**2-m1**2-m5**2+t2+u2-m2**2-m3**2+s)
> k5^{a}*k4_{a} = (1/2)*(-m4**2-m1**2-m5**2+u1+t1-m2**2-m3**2+s)
Same when some particles are massless:
def mandelstam = setMandelstam5([k1_a: '0', k2_a: 'm2', k3_a: 'm3', k4_a: '0', k5_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
> k5_{a}*k5^{a} = 0
> k2^{a}*k1_{a} = (1/2)*(s-m2**2)
> k1_{a}*k3^{a} = (1/2)*(-t1+m3**2)
> k4^{a}*k1_{a} = -(1/2)*t2
> k1_{a}*k5^{a} = (1/2)*(t1+s-m2**2+t2-m3**2)
> k2_{a}*k3^{a} = (1/2)*(-u1+m2**2+m3**2)
> k4^{a}*k2_{a} = (1/2)*(-u2+m2**2)
> k2_{a}*k5^{a} = (1/2)*(u1+u2+s-m2**2-m3**2)
> k4^{a}*k3_{a} = (1/2)*(-t1-u1-u2-s-t2+m3**2+2*m2**2)
> k3_{a}*k5^{a} = (1/2)*(u2+s-m2**2+t2-m3**2)
> k4_{a}*k5^{a} = (1/2)*(t1+u1+s-m2**2-m3**2)