-- Verify lab frame formulas for Compton scattering gmunu = ((1,0,0,0),(0,-1,0,0),(0,0,-1,0),(0,0,0,-1)) -- momentum vectors in center of mass frame E = sqrt(omega^2 + m^2) p1 = (omega, 0, 0, omega) p2 = (E, 0, 0, -omega) p3 = (omega, omega expsin(theta) expcos(phi), omega expsin(theta) expsin(phi), omega expcos(theta)) p4 = (E, -omega expsin(theta) expcos(phi), -omega expsin(theta) expsin(phi), -omega expcos(theta)) s = dot(p1 + p2,gmunu,p1 + p2) t = dot(p1 - p3,gmunu,p1 - p3) u = dot(p1 - p4,gmunu,p1 - p4) f11 = 32 dot(p1,gmunu,p2) dot(p1,gmunu,p4) + 64 m^2 dot(p1,gmunu,p2) - 32 m^2 dot(p1,gmunu,p3) - 32 m^2 dot(p1,gmunu,p4) + 32 m^4 f12 = 16 m^2 dot(p1,gmunu,p2) - 16 m^2 dot(p1,gmunu,p4) + 32 m^4 f22 = 32 dot(p1,gmunu,p2) dot(p1,gmunu,p4) + 32 m^2 dot(p1,gmunu,p2) - 32 m^2 dot(p1,gmunu,p3) - 64 m^2 dot(p1,gmunu,p4) + 32 m^4 "Verify probability density in center of mass frame (1=ok)" f11 == -8 s u + 24 s m^2 + 8 u m^2 + 8 m^4 f12 == 8 s m^2 + 8 u m^2 + 16 m^4 f22 == -8 s u + 8 s m^2 + 24 u m^2 + 8 m^4 -- transform momentum vectors to lab frame Lambda = ((E/m,0,0,omega/m),(0,1,0,0),(0,0,1,0),(omega/m,0,0,E/m)) p1 = dot(Lambda,p1) p2 = dot(Lambda,p2) p3 = dot(Lambda,p3) p4 = dot(Lambda,p4) "Verify invariance of Mandelstam variables (1=ok)" s == dot(p1 + p2,gmunu,p1 + p2) t == dot(p1 - p3,gmunu,p1 - p3) u == dot(p1 - p4,gmunu,p1 - p4) "Verify lab variables (1=ok)" omegaL = p1[1] omegaLp = p3[1] s == 2 m omegaL + m^2 t == 2 m (omegaLp - omegaL) u == -2 m omegaLp + m^2 "Verify probability density (1=ok)" omegaL = quote(omegaL) -- clear omegaL omegaLp = quote(omegaLp) -- clear omegaLp s = 2 m omegaL + m^2 t = 2 m (omegaLp - omegaL) u = -2 m omegaLp + m^2 f11 = -8 s u + 24 s m^2 + 8 u m^2 + 8 m^4 f12 = 8 s m^2 + 8 u m^2 + 16 m^4 f22 = -8 s u + 8 s m^2 + 24 u m^2 + 8 m^4 d11 = (s - m^2)^2 d12 = (s - m^2) (u - m^2) d22 = (u - m^2)^2 A = 1/4 (f11/d11 + 2 f12/d12 + f22/d22) B = 2 (omegaLp/omegaL + omegaL/omegaLp + (m/omegaL - m/omegaLp + 1)^2 - 1) A == B Run