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setMandelstam5

Description

  • 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, k2 and k3 with masses m1, m2 and m3 and outcoming particles with momentums k4 and k5 with masses 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*}

Examples

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
  > 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

See also