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 documentation:ref:setmandelstam [2015/09/22 18:28]127.0.0.1 external edit documentation:ref:setmandelstam [2015/11/21 12:33] Line 1: Line 1: - ====== setMandelstam ====== - ---- - ====Description==== - * ''​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*} - ====Examples==== - 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 - ​ - - ====See also==== - * Related tutorials: [[documentation:​tutorials:​compton_scattering_in_scalar_qed]],​ [[documentation:​tutorials:​compton_scattering_in_qed]] - * JavaDocs: [[http://​api.redberry.cc/​redberry/​1.1.8/​java-api/​cc/​redberry/​physics/​feyncalc/​FeynCalcUtils.html|FeynCalcUtils]] - * Source code: [[https://​bitbucket.org/​redberry/​redberry/​src/​tip/​physics/​src/​main/​java/​cc/​redberry/​physics/​feyncalc/​FeynCalcUtils.java|FeynCalcUtils.java]]