(* This file is associated with "Scattering Amplitudes and Conservative Binary Dynamics at O(G^4)" by Z. Bern, J. Parra-Martinez, R. Roiban, M. Ruf, C.-H. Shen, and M. P. Solon and M. S. Ruf. Explicit form of the two-body Hamiltonian to 4th post-Minkowskian (PM) order. E[1]=Sqrt[m[1]^2+p^2] E[2]=Sqrt[m[2]^2+p^2] M = m[1]+m[2] Etot = E[1]+E[2] nu = m[1]m[2]/M^2 xi = E[1]E[2]/Etot^2 sigma = (E[1]E[2]+p^2)/(m[1]m[2]) c1 = 1PM Hamiltonian coefficient c2 = 2PM Hamiltonian coefficient c3 = 3PM Hamiltonian coefficient c4 = 4PM Hamiltonian coefficient c4lower = part of the 4PM Hamiltonian coefficient determined by c1, c2 and c3 epsilon = dimensional regularization parameter; D= =4-2 epsilon M4p = Schwarzschild part of the 3-loop amplitude M4f = finite part of the 3-loop amplitude overlapping with the first-order self-force M4t = tail part of the 3-loop amplitude up to normalization These pieces correspond to the ones descibed in the paper. *) c1 = (M^4*nu^2*(1 - 2*sigma^2))/(Etot^2*xi) c2 = (M^5*nu^2*((3*(1 - 5*sigma^2))/4 - (M^3*nu^2*(1 - 2*sigma^2)^2*(1 - xi))/ (2*Etot^3*xi^2) - (4*M*nu*sigma*(1 - 2*sigma^2))/(Etot*xi)))/(Etot^2*xi) c3 = (M^6*nu^2*((-3*Etot*nu*(1 - 5*sigma^2)*(1 - 2*sigma^2))/ (2*(1 + Etot/M)*M*(1 + sigma)) + (3 - 6*nu + 206*nu*sigma - 54*sigma^2 + 108*nu*sigma^2 + 4*nu*sigma^3)/12 + (M^6*nu^4*(1 - 2*sigma^2)^3*(1 - 2*xi))/(2*Etot^6*xi^4) + (2*M^4*nu^3*sigma*(1 - 2*sigma^2)^2*(3 - 4*xi))/(Etot^4*xi^3) - (3*M*nu*sigma*(7 - 20*sigma^2))/(2*Etot*xi) - (M^3*nu^2*(1 - 2*sigma^2)*(3 + (8*Etot)/M - 15*sigma^2 - (80*Etot*sigma^2)/M - 3*xi + 15*sigma^2*xi))/(4*Etot^3*xi^2) - (4*nu*(3 + 12*sigma^2 - 4*sigma^4)*ArcSinh[Sqrt[-1 + sigma]/Sqrt[2]])/ Sqrt[-1 + sigma^2]))/(Etot^2*xi) c4 = c4lower + (M^7*nu^2*(M4p + (M4f - 10*M4t + M4t/epsilon)*nu))/(4*Etot^2*xi) c4lower = (M^8*nu^3*sigma*(-125 + 513*sigma^2))/(8*Etot^3*xi^2) + (2*M^14*nu^7*sigma*(-1 + 2*sigma^2)^3*(5 - 12*xi + 3*xi^2))/ (Etot^9*xi^6) - (M^16*nu^8*(1 - 2*sigma^2)^4*(5 - 15*xi + 6*xi^2 + xi^3))/(8*Etot^11*xi^7) + (M^12*nu^6*(1 - 2*sigma^2)^2* (9*M*(-1 + 5*sigma^2)*(-1 + 2*xi) + 16*Etot*(-1 + 14*sigma^2)* (-2 + 3*xi)))/(8*Etot^8*xi^5) + (M^8*nu^4*(-2880*Etot*M*sigma^2*(-1 + 2*sigma^2) + 288*Etot*M*(1 - 5*sigma^2)*(-1 + 6*sigma^2) + 216*Etot*M*(3 - 48*sigma^2 + 100*sigma^4) + 27*M^2*(1 - 5*sigma^2)^2* (-1 + xi) + (54*Etot^2*M^2*(1 - 5*sigma^2)^2*xi^2)/p^2 + (48*Etot^2*M^2*(1 - 20*sigma^2 + 36*sigma^4)*xi^2)/p^2 + (8*(3*M^3*(1 + sigma)^2*(-1 + sigma + 20*sigma^2 - 20*sigma^3 - 36*sigma^4 + 36*sigma^5)*(-1 + xi) + 8*Etot^3*(-37 - 103*sigma + 151*sigma^2 + 535*sigma^3 + 20*sigma^4 - 604*sigma^5 - 92*sigma^6 + 196*sigma^7)*xi + 3*Etot*M^2*(-1 + sigma)* (1 + sigma)^2*(-43 + sigma^2*(632 - 20*xi) + xi + 12*sigma^4*(-103 + 3*xi)) - 2*Etot^2*M*(1 + sigma)* (-63 + sigma^2*(981 - 643*xi) + 139*xi + 120*sigma*xi - 552*sigma^3*xi + 432*sigma^5*xi + 4*sigma^6*(450 + 29*xi) + sigma^4*(-2718 + 256*xi))))/((Etot + M)*(-1 + sigma)* (1 + sigma)^2)))/(96*Etot^5*xi^3) - (M^10*nu^5*(-1 + 2*sigma^2)*(4*Etot^4*M*(-15 + 200*sigma + 377*sigma^2 + 112*sigma^3 - 86*sigma^4)*xi^3 - 16*Etot^5*(3 - 25*sigma - 55*sigma^2 - 14*sigma^3 + 22*sigma^4)*xi^3 + 27*M^3*p^2*sigma*(-3 - 3*sigma + 10*sigma^2 + 10*sigma^3)* (-3 + 4*xi) + 2*Etot*M^2*p^2*(1 + sigma)*(-3*(-1 + xi)*xi + 54*sigma^2*(-1 + xi)*xi + 2*sigma^3*(-741 + 539*xi + xi^2) + sigma*(387 - 427*xi + 103*xi^2)) + Etot^2*M*p^2* (6*(5 - 8*xi)*xi - 2*sigma^4*(1749 - 626*xi + 176*xi^2) + sigma^3*(-3498 + 856*xi + 224*xi^2) + sigma*(819 - 724*xi + 400*xi^2) + sigma^2*(819 - 1078*xi + 880*xi^2)) - 4*Etot^3*(-(M^2*(-3 + 100*sigma + 157*sigma^2 + 56*sigma^3 + 2*sigma^4)*xi^3) + 2*p^2*(3*(-1 + xi)*xi + sigma^2*(-36 + 55*xi - 55*xi^2) + sigma*(-36 + 25*xi - 25*xi^2) - 14*sigma^3*(-12 - xi + xi^2) + 2*sigma^4*(84 - 11*xi + 11*xi^2)))))/(12*Etot^6*(Etot + M)^2*p^2*(1 + sigma)*xi^4) + ((4*M^8*nu^4*sigma*(15 - 118*sigma^2 + 132*sigma^4 - 40*sigma^6))/ (Etot^3*(-1 + sigma^2)^(3/2)*xi^2) - (4*M^10*nu^5*(3 + 6*sigma^2 - 28*sigma^4 + 8*sigma^6)* (p^2*(-1 + xi) + 2*Etot^2*xi^2))/(Etot^5*p^2*Sqrt[-1 + sigma^2]* xi^3))*ArcSinh[Sqrt[-1 + sigma]/Sqrt[2]] M4p = (-35*(1 - 18*sigma^2 + 33*sigma^4))/(8*(-1 + sigma^2)) M4t = h[1] + (ArcCosh[sigma]*h[3])/Sqrt[-1 + sigma^2] + h[2]*Log[(1 + sigma)/2] M4f = (-2*Pi^2*h[2])/3 + h[4] + (ArcCosh[sigma]*h[6])/Sqrt[-1 + sigma^2] + (ArcCosh[sigma]^2*h[8])/(-1 + sigma^2) + EllipticK[(-1 + sigma)/(1 + sigma)]^2*h[12] + EllipticE[(-1 + sigma)/(1 + sigma)]*EllipticK[(-1 + sigma)/(1 + sigma)]* h[13] + EllipticE[(-1 + sigma)/(1 + sigma)]^2*h[14] + h[7]*Log[sigma] + h[5]*Log[(1 + sigma)/2] + h[10]*(-Pi^2/6 + PolyLog[2, (1 - sigma)/2]) + h[9]*(Log[(1 + sigma)/2]^2/2 + PolyLog[2, (1 - sigma)/2]) + (2*sigma*(-3 + 2*sigma^2)*h[2]* (-PolyLog[2, -Sqrt[(-1 + sigma)/(1 + sigma)]] + PolyLog[2, Sqrt[(-1 + sigma)/(1 + sigma)]]))/(-1 + sigma^2)^(3/2) + h[11]*(Pi^2/3 + PolyLog[2, (1 - sigma)/(1 + sigma)] - PolyLog[2, (-1 + sigma)/(1 + sigma)]) + (2*h[3]*(2*ArcCosh[sigma]*Log[(1 + sigma)/2] - 5*PolyLog[2, -Sqrt[(-1 + sigma)/(1 + sigma)]] + 5*PolyLog[2, Sqrt[(-1 + sigma)/(1 + sigma)]] + PolyLog[2, 1 - sigma - Sqrt[-1 + sigma^2]] - PolyLog[2, 1 - sigma + Sqrt[-1 + sigma^2]]))/Sqrt[-1 + sigma^2] h[1] = (1151 - 3336*sigma + 3148*sigma^2 - 912*sigma^3 + 339*sigma^4 - 552*sigma^5 + 210*sigma^6)/(12*(-1 + sigma^2)) h[2] = (5 - 76*sigma + 150*sigma^2 - 60*sigma^3 - 35*sigma^4)/2 h[3] = (sigma*(-3 + 2*sigma^2)*(11 - 30*sigma^2 + 35*sigma^4))/(4*(-1 + sigma^2)) h[4] = (-45 + 207*sigma^2 - 1471*sigma^4 + 13349*sigma^6 - 37566*sigma^7 + 104753*sigma^8 - 12312*sigma^9 - 102759*sigma^10 - 105498*sigma^11 + 134745*sigma^12 + 83844*sigma^13 - 101979*sigma^14 + 13644*sigma^15 + 10800*sigma^16)/(144*sigma^7*(-1 + sigma^2)^2) h[5] = (1759 - 4768*sigma + 3407*sigma^2 - 1316*sigma^3 + 957*sigma^4 - 672*sigma^5 + 341*sigma^6 + 100*sigma^7)/(4*(-1 + sigma^2)) h[6] = (1237 + 7959*sigma - 25183*sigma^2 + 12915*sigma^3 + 18102*sigma^4 - 12105*sigma^5 - 9572*sigma^6 + 2973*sigma^7 + 5816*sigma^8 - 2046*sigma^9)/(24*(-1 + sigma^2)^2) h[7] = (2*sigma*(-852 - 283*sigma^2 - 140*sigma^4 + 75*sigma^6))/(3*(-1 + sigma^2)) h[8] = (sigma*(-304 - 99*sigma + 672*sigma^2 + 402*sigma^3 - 192*sigma^4 - 719*sigma^5 - 416*sigma^6 + 540*sigma^7 + 240*sigma^8 - 140*sigma^9))/ (8*(-1 + sigma^2)^2) h[9] = (52 - 532*sigma + 351*sigma^2 - 420*sigma^3 + 30*sigma^4 - 25*sigma^6)/2 h[10] = 2*(27 + 90*sigma^2 + 35*sigma^4) h[11] = 20 + 111*sigma^2 + 30*sigma^4 - 25*sigma^6 h[12] = (834 + 2095*sigma + 1200*sigma^2)/(2*(-1 + sigma^2)) h[13] = -(1183 + 2929*sigma + 2660*sigma^2 + 1200*sigma^3)/(2*(-1 + sigma^2)) h[14] = (7*(169 + 380*sigma^2))/(4*(-1 + sigma))