This document is part of https : // dcc.ligo.org/LIGO - T1500554/public

Dowload FeynCalc

(*Import["http://www.feyncalc.org/install.m"]*)

Load FeynCalc

<<HighEnergyPhysics`fc`

Loading FeynCalc from /home/riccardo/.Mathematica/Applications/HighEnergyPhysics

FeynCalc 8.2.0 For help, type ?FeynCalc, open FeynCalcRef8.nb or visit www.feyncalc.org

Loading FeynArts, see www.feynarts.de for documentation

FeynArts 3.7 patched for use with FeynCalc

Definitions

V3[a_,b_]:=FourVector[a,b,Dimension->3];
δ[i_,j_]:=MetricTensor[i,j,Dimension->3];
ENS[x_]:=Expand[Normal[Simplify[x]]];

From (8, 9, 10) of gr - qc/0405090 for Sdot-v5 and v6 (no self-terms),  and from (7.8) of gr - qc/0605140 for Sdot-v7. See also 3.4 of arXiv:1212.5520 and A2 of 1501.01529.

γ1PN=(1-η/3);
γ2PN=(1-65η/12);
L1PNv2=1/2(1-3η);
L1PNγ=(3+η);
L2PNv4=3/8(1-7η+13η^2);
L2PNv2γ=1/2(7-10η-9η^2);
L2PNγ2=1/4(14-41η+4η^2);
L=LN ENS[1+x(L1PNv2+L1PNγ+(-γ1PN))+x^2(-γ2PN+γ1PN^2+L1PNv2(-2γ1PN)+L1PNγ γ1PN+L2PNv4+L2PNv2γ+L2PNγ2)]

Spin dependent contributions to L
 (multiplied by n x (n x S) = (n.S) n - S, once averaged over an orbit gives - 1/2 (S + (S.l) l), see eq. 2.9 of gr-qc/9506022

ENS[((3(S1+S2)+(m1-m2)(S2/m2-S1/m1)-1/2((S1+S2)+(m1-m2)(S2/m2-S1/m1)))/.m2->MT-m1)-((3/2+1/2 MT/m1 )S1)]
ENS[((3(S1+S2)+(m1-m2)(S2/m2-S1/m1)-1/2((S1+S2)+(m1-m2)(S2/m2-S1/m1)))/.m1->MT-m2)-((3/2+1/2 MT/m2 )S2)]

ENS[(((MT/m1+3)/2S1+(MT/m2+3)/2S2)^2-((MT/m1+3)^2/4S1^2+(MT/m2+3)^2/4S2^2+(4/η+9)/2 S1 S2))/.η->m1 m2/MT^2/.MT->(m1+m2)]

Omega derivation

M=m1+m2;
η=m1 m2/M^2;
δe=m1/M-m2/M;

α1PN[δe_]:=(3/4+η/2-3/4δe)(*γ*);
Omega1PN[δe_]:=ENS[3/2-m1/M-(m1/M)^2/2];
ENS[Omega1PN[δe]-α1PN[δe]]

α2PNa[δe_]:=(1/16+11/8η-3/8η^2+δe(-1/16+η/2))(*γv^2*);
α2PNb[δe_]:=(-1/4-3/8η+η^2/2+δe/4(1-η/2))(*γ^2*);
Omega2PN[δe_]:=9/16+5/4η-η^2/24+δe(-9/16+5/8 η);
ENS[Omega2PN[δe]-(α2PNa[δe]+α2PNb[δe]+γ1PN α1PN[δe])]

α3PNa[δe_]:=1/32+19/16η-31/8η^2+17/16η^3+δe(-1/32+3/4η-11/8η^2)(*γv^4*);
α3PNb[δe_]:=3/16+27/4η+75/32η^2-9/8η^3+δe(-3/16+9/8η+35/32η^2)(*γ^2v^2*);
α3PNc[δe_]:=7/16-9/4η-9/8η^2+1/2η^3+δe(-7/16-1/8η-1/8η^2)(*γ^3*);
Omega3PN[δe_]:=27/32+3/16η-105/32η^2-1/48η^3+δe(-27/32+39/8η-5/32η^2);
γ2PN=(1-65η/12);
(*γSS2PN=(SP[S1,S2]/2/η-3/2/η SP[S1,LN]SP[S2,LN]-1/4(κ1(M/m1)^2(3SP[S1,LN]^2-SP[S1,S1])+κ2(M/m2)^2(3SP[S2,LN]^2-SP[S2,S2])));*)
ENS[Omega3PN[δe]-(α3PNa[δe]+α3PNb[δe]+α3PNc[δe]-γ1PN α2PNa[δe]+2γ1PN α2PNb[δe]+(γ2PN(*+γSS2PN*)) α1PN[δe])]

γSO1p5PN=((M/m1)-1/3)SP[S1,LN]+((M/m2)-1/3)SP[S2,LN];
γSO1p5PNcheck=5/3(SP[S1,LN]+SP[S2,LN])+δe(m1+m2)(SP[S2,LN]/m2-SP[LN,S1]/m1);
ENS[γSO1p5PN-γSO1p5PNcheck]
Omega2p5PNSLbis[δe_]:=ENS[α1PN[δe]γSO1p5PN];

ENS[α1PN[δe]]

ENS[Omega2p5PNSLbis[δe]/η]

γSO2p5PN=ENS[(10/3+8/9η)(SP[LN,S1]+SP[LN,S2])+2δe(m1+m2)(SP[LN,S2]/m2-SP[LN,S1]/m1)];
Omega1p5PNSS=1/2;
Omega1p5PNSL[S1_,S2_,κ_,δ_]:=-3/2SP[S2,LN]+κ(-3/2 SP[LN,S1]((1-δ)/2/η-1));
Omega2p5PNSS[S1_,S2_,κ_,δe_]:=3γ1PN Omega1p5PNSS+1/2(9/4δe+3η/2+15/4+δe/2+2-3/4δe+η/2-39/4)+(-3/4δe-η/2-9/4+13/4+δe/4-η/2);
Omega2p5PNSL[S1_,S2_,κ_,δe_]:=3γ1PN Omega1p5PNSL[S1,S2,κ,δe]+
(-SP[LN,S1]/2(-3/2δe-3(1-δe)/2/η+9/2-3/4δe-3/4(1-δe)/η+9/4+5/4δe+η/2-5/4+κ(-3/2δe-3η/2+9/4(1-δe)/η-3+3/2(1-δe)/η-3-9/4δe-6(1-δe)/η+57/4))-SP[LN,S2]/2(9/4δe+3η/2+15/4+δe/2+2-3/4δe+η/2-39/4));

dS1[j_,k_,δe_,x_,S1_,S2_,κ_]:=x^(5/2)((Omega1PN[δe]+x Omega2PN[δe]+x^2Omega3PN[δe]+x^(1/2)(Omega1p5PNSL[S1,S2,κ,δe]+x (Omega2p5PNSL[S1,S2,κ,δe]+Omega2p5PNSLbis[δe])))V3[LN,j]+x^(1/2)(Omega1p5PNSS+x Omega2p5PNSS[S1,S2,κ,δe])V3[S2,j])V3[S1,k];
dLN[j_,k_,x_]:=ENS[1/η x^3((Omega1PN[δe]V3[S1,j]+Omega1PN[-δe]V3[S2,j])+
x(Omega2PN[δe]V3[S1,j]+Omega2PN[-δe] V3[S2,j])+
+x^2(Omega3PN[δe]V3[S1,j]+Omega3PN[-δe]V3[S2,j])+x^(1/2)((Omega1p5PNSL[S1,S2,κ1,δe]V3[S1,j]+Omega1p5PNSL[S2,S1,κ2,-δe]V3[S2,j])+x((Omega2p5PNSL[S1,S2,κ1,δe]+Omega2p5PNSLbis[δe])V3[S1,j]+(Omega2p5PNSL[S2,S1,κ2,-δe]+Omega2p5PNSLbis[-δe])V3[S2,j])))V3[LN,k]];

Check Ldot - v6 vs.  Sdot - v5, see (4.25) of 0812.4413

ENS[η(2+3/2m2/m1)-(3/2-m1/M-1/2(m1/M)^2)/.η->m1 m2/M^2/.m2->M-m1]

ENS[1/2+3/2M/m1-(2+3/2m2/m1)/.m2->M-m1]

ENS[SeriesCoefficient[ENS[dLN[j,k,x]],{x,0,3}]]

ENS[ENS[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,0]+dS1[j,k,-δe,x,S2,S1,0])/η,{x,0,5/2}]]]

ENS[SeriesCoefficient[dS1[j,k,δe,x,S1,S2,0],{x,0,5/2}]/.m2->MTot-m1]

Check Ldot - v7 vs.  Sdot - v6, see (A.2) of 1501.01529

ENS[SeriesCoefficient[ENS[dLN[j,k,x]],{x,0,7/2}]/.κ1->0/.κ2->0]

ENS[(SeriesCoefficient[ENS[dLN[j,k,x]],{x,0,7/2}]/.κ1->0/.κ2->0)-(-3/2/η (SP[LN,S2]V3[S1,j]+SP[LN,S1]V3[S2,j])V3[LN,k])]

ENS[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,3}]/.κ1->0/.κ2->0]

ENS[SeriesCoefficient[dS1[j,k,δe,x,S1,S2,κ1],{x,0,3}]/.κ1->0]

ENS[SeriesCoefficient[SeriesCoefficient[ENS[dLN[j,k,x]],{x,0,7/2}],{κ1,0,1}]-(-3/2(M/m1)^2)SP[LN,S1]V3[S1,j]V3[LN,k]]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,3}],{κ1,0,1}]]

ENS[SeriesCoefficient[SeriesCoefficient[dS1[j,k,δe,x,S1,S2,κ1],{x,0,3}],{κ1,0,1}]/.m2->MTot-m1]

ENS[SeriesCoefficient[SeriesCoefficient[ENS[dLN[j,k,x]],{x,0,7/2}],{κ2,0,1}]]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,3}],{κ2,0,1}]]

Check Ldot - v8 vs.  Sdot - v7, see (4.25) of 0812.4413

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,4}],{V3[S1,j],0,1}]/.m2->MTot-m1]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,7/2}],{V3[S1,k],0,1}]/.m2->MTot-m1]

ENS[SeriesCoefficient[dS1[j,k,δe,x,S1,S2,κ1],{x,0,7/2}]/.m2->MTot-m1]

ENS[9/16+5/4η-η^2/24+δe(-9/16+5/8η)/.m2->MTot-m1]

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,4}],{V3[S2,j],0,1}]/.m1->MTot-m2]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,7/2}],{V3[S2,k],0,1}]/.m1->MTot-m2]

Check Ldot - v9 vs.  Sdot - v8, see (A.2) of 1501.01529

ENS[(SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,9/2}],{V3[S1,j],0,1}]V3[S1,j]/.κ1->0)-((15/4(M/m1)^2-M/m1-5/12)SP[LN,S1]+(-1/2/η-m1/(M-m1)-2/3)SP[LN,S2])V3[S1,j]V3[LN,k]]

ENS[SeriesCoefficient[dLN[j,k,x],{x,0,9/2}]]

ENS[(SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,9/2}],{V3[S1,k],0,1}]V3[S1,k])]

ENS[(V3[S2,j]SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,9/2}],{V3[S2,j],0,1}]/.κ2->0)-((15/4(M/m2)^2-M/m2-5/12)SP[LN,S2]+(-1/2/η-m2/(M-m2)-2/3)SP[LN,S1])V3[S2,j]V3[LN,k]]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,4}],{V3[S1,k],0,1}]/.κ1->0]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,4}],{V3[S1,j],0,1}]/.κ1->0]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,4}],{V3[S1,j],0,1}]/.κ1->0]

ENS[SeriesCoefficient[dS1[j,k,δe,x,S1,S2,κ1],{x,0,4}]/.κ1->0/.m2->MTot-m1]

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,9/2}],{κ1,0,1}]-(-9/4(M/m1)^2-9/4M/m1-3/4)SP[LN,S1]V3[S1,j]V3[LN,k]]

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,9/2}],{κ2,0,1}]-(-9/4(M/m2)^2-9/4M/m2-3/4)SP[LN,S2]V3[S2,j]V3[LN,k]]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,4}],{κ1,0,1}]]

ENS[SeriesCoefficient[SeriesCoefficient[dS1[j,k,δe,x,S1,S2,κ1],{x,0,4}],{κ1,0,1}]/.m2->MTot-m1]

Check Ldot - v10 vs.  Sdot - v9, see (3.4) of 1212.5520

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,5}],{V3[S1,j],0,1}]/.κ1->0/.κ2->0/.m2->MT-m1]

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,x],{x,0,5}],{κ1,0,1}]]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,δe,x,S1,S2,κ1]+dS1[j,k,-δe,x,S2,S1,κ2])/η,{x,0,9/2}],{V3[S1,k],0,1}]/.m2->MTot-m1]

ENS[SeriesCoefficient[dS1[j,k,δe,x,S1,S2,κ1],{x,0,9/2}]/.m2->MTot-m1]

From (2.23) of 0812.4413 we have S-v5and v6 including self terms

dSsumM[j_,k_,m1_,m2_,M_,v_]:=v^5 m1 m2/(2M^2)(V3[LN,j] (7 (V3[S1,k]+V3[S2,k])+3(m1-m2)(V3[S2,k]/m2-V3[S1,k]/m1)))+3 m1 m2/M^2v^6(Contract[Module[{l},(δ[l,j]-V3[LN,l]V3[LN,j])/2((4(V3[S1,l]+V3[S2,l])+2(m1-m2)(V3[S2,l]/m2-V3[S1,l]/m1))(V3[S1,k]+V3[S2,k])+
(2(m1-m2)/M(V3[S1,l]+V3[S2,l])+(1-4m1 m2/M^2)M(V3[S2,l]/m2-V3[S1,l]/m1))M(V3[S2,k]/m2-V3[S1,k]/m1))]])
dSdifM[j_,k_,m1_,m2_,M_,v_]:=v^5/2 (V3[LN,j] (3(m1-m2)/M (V3[S1,k]+V3[S2,k])+(3-5 m1 m2/M^2)M(V3[S2,k]/m2-V3[S1,k]/m1)))+v^6(M(V3[S2,j]/m2-V3[S1,j]/m1)(V3[S1,k]+V3[S2,k])+Contract[Module[{l},(δ[l,j]-V3[LN,l]V3[LN,j])/2(3(2(m1-m2)/M(V3[S1,l]+V3[S2,l])+(1-4m1 m2/M^2)M(V3[S2,l]/m2-V3[S1,l]/m1))(V3[S1,k]+V3[S2,k])+3(1-2 m1 m2/M^2)(2(V3[S1,l]+V3[S2,l])+(m1-m2)(V3[S2,l]/m2-V3[S1,l]/m1))M(V3[S2,k]/m2-V3[S1,k]/m1))]])
dS1[j_,k_,m1_,M_,v_]:=m1/M(dSsumM[j,k,m1,M-m1,M,v]-(M-m1)/M dSdifM[j,k,m1,M-m1,M,v]);
dS2[j_,k_,m2_,M_,v_]:=m2/M(dSsumM[j,k,M-m2,m2,M,v]+(M-m2)/M dSdifM[j,k,M-m2,m2,M,v]);

Check dS - v6 with (A1) of 1501.01529

Check dS - v5

Expand[Normal[Simplify[SeriesCoefficient[dS1[j,k,m1,M,v],{v,0,5}]]]]

Expand[Normal[Simplify[SeriesCoefficient[dS1Old[j,k,m1,M,v],{v,0,5}]]]]

Expand[Normal[Simplify[SeriesCoefficient[dS2[j,k,m2,M,v],{v,0,5}]]]]

Expand[Normal[Simplify[SeriesCoefficient[dS2Old[j,k,m2,M,v],{v,0,5}]]]]

Check dS - v6

Expand[Normal[Simplify[SeriesCoefficient[(dS1[j,k,m1,M,v]-dS1[k,j,m1,M,v]),{v,0,6}]]]/.M->(m1+m2)]

Expand[Normal[Simplify[SeriesCoefficient[(dS1Old[j,k,m1,M,v]-dS1Old[k,j,m1,M,v]),{v,0,6}]]]]

Expand[Normal[Simplify[SeriesCoefficient[(dS2[j,k,m2,M,v]-dS2[k,j,m2,M,v]),{v,0,6}]]]/.M->(m1+m2)]

Expand[Normal[Simplify[SeriesCoefficient[(dS2Old[j,k,m2,M,v]-dS2Old[k,j,m2,M,v]),{v,0,6}]]]]

Check Ldot - v7 vs.  Sdot - v6, see (4.25) of 0812.4413

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,m1,m2,v]-dLN[k,j,m1,m2,v],{v,0,7}],{V3[S1,j],0,1}]/.m2->M-m1]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,m1,M,v]+dS2[j,k,m2,M,v]-(dS1[k,j,m1,M,v]+dS2[k,j,m2,M,v]))v(m1+m2)^2/(m1 m2),{v,0,7}],{V3[S1,j],0,1}]/.M->(m1+m2)]

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,m1,m2,v]-dLN[k,j,m1,m2,v],{v,0,7}],{V3[S1,k],0,1}]]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,m1,M,v]+dS2[j,k,m2,M,v]-(dS1[k,j,m1,M,v]+dS2[k,j,m2,M,v]))v(m1+m2)^2/(m1 m2),{v,0,7}],{V3[S1,k],0,1}]/.M->(m1+m2)]

ENS[SeriesCoefficient[SeriesCoefficient[dLN[j,k,m1,m2,v]-dLN[k,j,m1,m2,v],{v,0,7}],{V3[S2,k],0,1}]]

ENS[SeriesCoefficient[SeriesCoefficient[-(dS1[j,k,m1,M,v]+dS2[j,k,m2,M,v]-(dS1[k,j,m1,M,v]+dS2[k,j,m2,M,v]))v(m1+m2)^2/(m1 m2),{v,0,7}],{V3[S2,k],0,1}]/.M->(m1+m2)]