<<FeynCalc`

01ttFS

Theory and Necessary Informations

(1) Model for the process in hand is as follows

ℒ =  - (1/4) (F^(μν ))(Subscript[F, μν])+ (OverBar[f])(((iγ)^μ)(Subscript[D, μ]) - (Subscript[m, f]))f  + (1/2) (OverBar[χ]) (((iγ)^μ)(∂_(μ ))- (Subscript[m, χ]))χ + ((Subscript[D, μ])ϕ )^†((D^μ)ϕ) - ((Subscript[m, ϕ])^2)(SuperDagger[ϕ])ϕ + ( λ (OverBar[χ])(Subscript[P, L])(SuperMinus[f])(SuperPlus[ϕ])  +  (SuperDagger[λ])(SuperPlus[(OverBar[f])])(Subscript[P, R])χ (SuperMinus[ϕ]))

Space for diagram

Diagrams Identification

(1) LO : t- channel process [  (Subscript[χ, p'])    (Subscript[(OverBar[χ]), p])   ->   (Subscript[f, q'])   (Subscript[(OverBar[f]), q])  ]
(2) NLO : t- channel process [  virtual photon propagates from μ (at fermion) to ν (at scalar) ]

(1) Command Required

Command for running FeynCalc <<FeynCalc`

(Subscript[P, L]) =  GA[7]

(2) Matrix Elements without any approximations

LO matrix element (Details : t- channel)

NLO matrix element (Details: t- channel, virtual photon correction between fermion and scalar)

Thermal part of NLO matrix element (Details: t- channel, virtual photon correction between fermion and scalar)

(3) Definitions and Approximations

(3.1)Primary definitions

definition of b((Subscript[p, i]),(Subscript[p, f]),k) = b(x,y,z)

b[x_,y_,z_] :=(1/2) (((SP[2x-z,2y-z])/((SP[z]-2*SP[x,z])*((SP[z]-2*SP[y,z]))))+((SP[2x+z,2y+z])/((SP[z]+2*SP[x,z])*((SP[z]+2*SP[y,z])))))

(Subscript[Q, f])= -e;(Subscript[Q, f']) = -e;(Subscript[Q, ϕ])=-e;

(3.2)Approximations

(* SP[k]=0; This is not correct for thermal fermion since its momeuntum is taken to be k *)
SP[p] = (mf^2);
SP[pp] = (mf^2);
SP[q] = (mc^2);
SP[qp] = (mc^2);

(3.3)Matrix elements

(3.3.1)t- channel LO Matrix element

* SP[q-p]-((Subscript[m, ϕ])^2)   -> -((Subscript[m, ϕ])^2)

Mt = (I*λ)(I*ComplexConjugate[λ])(SpinorUBar[pp,mf].GA[6].SpinorU[qp,mc])(SpinorVBar[q,mc].GA[7].SpinorV[p,mf])((I/(-(mh^2))))

(3.3.2) t- channel NLO Matrix element

Note 1 : This part will be included later as it is not required now.

Note 2  : only 11 part of the whole propagator for all of the relevant particles are taken. 12,21 and 22 parts for the same are not included.

(3.3.3) t- channel NLO Matrix element (Only part of the Thermal Part which is of interest)

* Following matrix element is written without δ((k^2)) and action of δ((k^2)) is taken by taking (k^2)->0. 
* It contains only G-polarization part of Integrand inside t- channel NLO matrix element.
* SP[q-p]-((Subscript[m, ϕ])^2)   -> -((Subscript[m, ϕ])^2)	and	SP[q-p+k]-((Subscript[m, ϕ])^2)   -> -((Subscript[m, ϕ])^2); for fermion thermal, replaced dummy t by dummy k for convenience

MK1Pt =(-I*(Subscript[Q, f]))(-I*(Subscript[Q, ϕ]))(I*λ)(I*ComplexConjugate[λ])((I/((2 SP[pp,k]))))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[pp+k]+mf).GA[6].SpinorU[qp,mc])((I/(-(mh^2))))((I/(-(mh^2))))(FV[2q-2p+k,ν])(SpinorVBar[q,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν]+b[p,pp,k]*FV[k,μ]*FV[k,ν])

MK1Ft =(-I*(Subscript[Q, f]))(-I*(Subscript[Q, ϕ]))(I*λ)(I*ComplexConjugate[λ])(((-I)/(2((mf^2)- SP[pp,t]))))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[t]+mf).GA[6].SpinorU[qp,mc])((I/(-(mh^2))))((I/(-(mh^2))))(FV[pp-2qp+t,ν])(SpinorVBar[q,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν])

MK2Pt =(-I*(Subscript[Q, f]))((I/((2 SP[pp,k]))))(I*ComplexConjugate[λ])((I/(-(mh^2))))(I*λ)((I/((-2 SP[p,k]))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[pp+k]+mf).GA[6].SpinorU[qp,mc])(SpinorVBar[q,mc].GA[7].(DiracSlash[-p+k]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν]+b[p,pp,k]*FV[k,μ]*FV[k,ν])

MK2Ft =(-I*(Subscript[Q, f]))(I*ComplexConjugate[λ])((I/(-(mh^2))))(I*λ)((I/((2 ((mf^2)+SP[pp,p] - SP[pp,t] - SP[p,t])))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[t]+mf).GA[6].SpinorU[qp,mc])(SpinorVBar[q,mc].GA[7].(DiracSlash[-p-pp+t]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν] (((-I)/((2 ((mf^2)- SP[pp,t]))))))

MK2At =(-I*(Subscript[Q, f]))(I*ComplexConjugate[λ])((I/(-(mh^2))))(I*λ)((I/((2 ((mf^2)+SP[pp,p] - SP[pp,t] - SP[p,t])))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[p+ pp-t]+mf).GA[6].SpinorU[qp,mc])(SpinorVBar[q,mc].GA[7].(-DiracSlash[t]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν] (((-I)/((2 ((mf^2)- SP[p,t]))))))

MK3Pt = (I*ComplexConjugate[λ])((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))((I/(-(mh^2))))(I*λ)((I/((-2 SP[p,k]))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[6].SpinorU[qp,mc])(FV[2q-2p+k,μ])(SpinorVBar[q,mc].GA[7].(DiracSlash[-p+k]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν]+b[p,p,k]*FV[k,μ]*FV[k,ν])

MK3At = (I*ComplexConjugate[λ])((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))((I/(-(mh^2))))(I*λ)(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[6].SpinorU[qp,mc])(FV[2q-p-t,μ])(SpinorVBar[q,mc].GA[7].(DiracSlash[-t]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν](((-I)/((2 ((mf^2)- SP[p,t]))))))

MK4Pt= (I*ComplexConjugate[λ])((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))((I/(-(mh^2))))(I*λ)(SpinorUBar[pp,mf].GA[6].SpinorU[qp,mc])(FV[2q-2p+k,μ])(FV[2q-2p+k,ν])(SpinorVBar[q,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν]+b[p,p,k]*FV[k,μ]*FV[k,ν])

MK5Pt = (I*ComplexConjugate[λ])((I/(-(mh^2))))(I*((Subscript[Q, ϕ])^2))((I/(-(mh^2))))(I*λ)(SpinorUBar[pp,mf].GA[6].SpinorU[qp,mc])(MT[μ,ν])(SpinorVBar[q,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν]+b[p,p,k]*FV[k,μ]*FV[k,ν])

Mu =(I*λ)(I*ComplexConjugate[λ])(SpinorVBar[qp,mc].GA[7].SpinorV[p,mf])((I/(-(mh^2))))(SpinorUBar[pp,mf].GA[6].SpinorU[q,mc])

MK1Pu =(-I*(Subscript[Q, f]))((I/((2 SP[pp,k]))))(I*ComplexConjugate[λ])((I/(-(mh^2))))((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))(I*λ)(SpinorUBar[pp,mf].GA[μ].(DiracSlash[pp+k]+mf).GA[6].SpinorU[q,mc])(FV[2qp-2p+k,ν])(SpinorVBar[qp,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν]+b[pp,pp,k]*FV[k,ν]*FV[k,μ])

MK1Fu =(-I*(Subscript[Q, f]))(I*ComplexConjugate[λ])((I/(-(mh^2))))((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))(I*λ)(SpinorUBar[pp,mf].GA[μ].(DiracSlash[t]+mf).GA[6].SpinorU[q,mc])(FV[pp-2q+t,ν])(SpinorVBar[qp,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν](((-I)/(2((mf^2)- SP[pp,t])))))

MK2Pu = (-I*(Subscript[Q, f]))((I/((2 SP[pp,k]))))(I*ComplexConjugate[λ])((I/(-(mh^2))))(I*λ)((I/((-2 SP[p,k]))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[pp+k]+mf).GA[6].SpinorU[q,mc])(SpinorVBar[qp,mc].GA[7].(DiracSlash[-p+k]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν]+b[p,pp,k]*FV[k,μ]*FV[k,ν])

MK2Fu = (-I*(Subscript[Q, f]))(I*ComplexConjugate[λ])((I/(-(mh^2))))(I*λ)((I/((2 ((mf^2)+SP[pp,p] - SP[pp,t] - SP[p,t])))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[t]+mf).GA[6].SpinorU[q,mc])(SpinorVBar[qp,mc].GA[7].(DiracSlash[-p-pp+t]+mf).GA[ν].SpinorV[p,mf])  (-MT[μ,ν] (((-I)/((2 ((mf^2)- SP[pp,t]))))))

MK2Au = (-I*(Subscript[Q, f]))(I*ComplexConjugate[λ])((I/(-(mh^2))))(I*λ)((I/((2 ((mf^2)+SP[pp,p] - SP[pp,t] - SP[p,t])))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[μ].(DiracSlash[pp+p-t]+mf).GA[6].SpinorU[q,mc])(SpinorVBar[qp,mc].GA[7].(DiracSlash[-t]+mf).GA[ν].SpinorV[p,mf])  (-MT[μ,ν] (((-I)/((2 ((mf^2)- SP[p,t]))))))

MK3Pu = (I*ComplexConjugate[λ])((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))((I/(-(mh^2))))(I*λ)((I/((-2 SP[p,k]))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[6].SpinorU[q,mc])(FV[2qp-2p+k,μ])(SpinorVBar[qp,mc].GA[7].(DiracSlash[-p+k]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν]+b[p,pp,k].FV[k,μ].FV[k,ν])

MK3Au =(I*ComplexConjugate[λ])((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))((I/(-(mh^2))))(I*λ)(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[6].SpinorU[q,mc])(FV[2qp-p-t,μ])(SpinorVBar[qp,mc].GA[7].(DiracSlash[-t]+mf).GA[ν].SpinorV[p,mf])(-MT[μ,ν](((-I)/((2 ((mf^2)- SP[p,t]))))))

MK4Pu= (I*ComplexConjugate[λ])((I/(-(mh^2))))(-I*(Subscript[Q, ϕ ]))((I/(-(mh^2))))(-I*(Subscript[Q, ϕ]))((I/(-(mh^2))))(I*λ)(SpinorUBar[pp,mf].GA[6].SpinorU[q,mc])(FV[2qp-2p+k,μ])(FV[2qp-2p+k,ν])(SpinorVBar[qp,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν]+b[pp,pp,k].FV[k,μ].FV[k,ν])

MK5Pu= (I*ComplexConjugate[λ])((I/(-(mh^2))))(I*((Subscript[Q, ϕ])^2))((I/(-(mh^2))))(I*λ)(SpinorUBar[pp,mf].GA[6].SpinorU[q,mc])(MT[μ,ν])(SpinorVBar[qp,mc].GA[7].SpinorV[p,mf])(-MT[μ,ν]+b[pp,pp,k]*FV[k,μ]*FV[k,ν])

(3.4) Component form of  4-vectors q’, q, p’, p and k

q’	=   	(	H, 	-P.Sin(θ) ,0,	 P. Cos(θ)	)

q	=   	(	H, 	P.Sin(θ), 	0, 	-P Cos(θ)	)

p’	=   	(	H’,  	0, 		0, 	P’		)

p	=   	(	H’,  	0, 		0, 	-P’		)

k	=   	(ω,  	K.Cos(ϕ’).Sin(θ’),   K.Sin(θ’).Sin(ϕ’),   K.Cos(θ’))

(3.5) expansion of various scalar products (Notations : Ct = Cos(θ), Ctp = Cos(θ’) FRAME ROTATED; run this AFTER traces

SP[k,k] = 0;

SP[k,q] = ω*H-K*P*St*Stp*Cos[phip]+K*P*Ct*Ctp;

SP[k,qp] = ω*H+K*P*St *Stp *Cos[phip]-K*P*Ct*Ctp;

SP[p,q]=H*H-P*Pp*Ct;

SP[pp,qp]=H*H-P*Pp*Ct;

SP[k,p]=ω*H+Pp*K*Ctp;

SP[p,pp]=H*H+Pp*Pp;

SP[q,pp]=H*H+P*Pp*Ct;

SP[p,qp]=H*H+Pp*P*Ct;

SP[q,qp]=H*H+P*P;

SP[k,pp]= ω*H- K*Pp*Ctp;

SP[t,t] = (mf^2);

SP[t,q] = ot*H-Kt*P*St*Stp*Cos[phip]+Kt*P*Ct*Ctp;

SP[t,qp] = ot*H+Kt*P*St *Stp *Cos[phip]-Kt*P*Ct*Ctp;

SP[t,p]=ot*H+Pp*Kt*Ctp;

SP[t,pp]= ot*H- Kt*Pp*Ctp;

(3.6) t- channel NLO Matrix element (Only part of the Thermal Part which is of interest) after expanded scalar products

MKP // ExpandScalarProduct

(4) Trace Calculations

(4.1) MKoP

MKo00P = (Mt) *ComplexConjugate[Mt]+(Mu)*ComplexConjugate[Mu] - (Mt) *ComplexConjugate[Mu]-(Mu)*ComplexConjugate[Mt]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Pt = (MK1Pt) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK1Pt]/.{SP[k]->0}//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Ft = (MK1Ft) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK1Ft]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Pt = (MK2Pt) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK2Pt]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Ft = (MK2Ft) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK2Ft]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2At = (MK2At) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK2At]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo3Pt = (MK3Pt) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK3Pt]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo3At = (MK3At) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK3At]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo4Pt = (MK4Pt) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK4Pt]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo5Pt = (MK5Pt) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK5Pt]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Pu = (MK1Pu) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK1Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Fu = (MK1Fu) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK1Fu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Pu = (MK2Pu) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK2Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Fu = (MK2Fu) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK2Fu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Au = (MK2Au) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK2Au]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo3Pu = (MK3Pu) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK3Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo3Au = (MK3Au) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK3Au]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo4Pu = (MK4Pu) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK4Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo5Pu = (MK5Pu) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK5Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Ptu = (MK1Pt) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK1Pt]+ (MK1Pu) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK1Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Ftu = (MK1Ft) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK1Ft]+ (MK1Fu) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK1Fu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Ptu = (MK2Pt) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK2Pt]+ (MK2Pu) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK2Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Ftu = (MK2Ft) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK2Ft]+ (MK2Fu) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK2Fu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo2Atu = (MK2At) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK2At]+ (MK2Au) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK2Au]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo3Ptu = (MK3Pt) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK3Pt]+ (MK3Pu) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK3Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo3Atu = (MK3At) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK3At]+ (MK3Au) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK3Au]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo4Ptu = (MK4Pt) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK4Pt]+ (MK4Pu) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK4Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo5Ptu = (MK5Pt) *ComplexConjugate[Mu]+(Mu)*ComplexConjugate[MK5Pt]+ (MK5Pu) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK5Pu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Nt = MKo1Pt /. ω -> -ω // Simplify

MKo1Gt = MKo1Ft /. ot -> -ot // Simplify

MKo2Nt = MKo2Pt /. ω -> -ω // Simplify

MKo2Gt = MKo2Ft /. ot -> -ot // Simplify

MKo2gt = MKo2At /. ot -> -ot // Simplify

MKo3Nt = MKo3Pt /. ω -> -ω // Simplify

MKo3Gt = MKo3At /.  ot -> -ot// Simplify

MKo4Nt = MKo4Pt /. ω -> -ω // Simplify

MKo5Nt = MKo5Pt /. ω -> -ω // Simplify

(4.2) MKoN

MKo1Nu = MKo1Pu /. ω -> -ω // Simplify

MKo1Gu = MKo1Fu /. ot -> -ot // Simplify

MKo2Nu = MKo2Pu /. ω -> -ω // Simplify

MKo2Gu = MKo2Fu /. ot -> -ot // Simplify

MKo2gu = MKo2Au /. ot -> -ot // Simplify

MKo3Nu = MKo3Pu /. ω -> -ω // Simplify

MKo3Gu = MKo3Au /. ot -> -ot// Simplify

MKo4Nu = MKo4Pu /. ω -> -ω // Simplify

MKo5Nu = MKo5Pu /. ω -> -ω // Simplify

MKo1Ntu = MKo1Ptu /. ω -> -ω // Simplify

MKo1Gtu = MKo1Ftu /. ot -> -ot // Simplify

MKo2Ntu = MKo2Ptu /. ω -> -ω // Simplify

MKo2Gtu = MKo2Ftu /. ot -> -ot // Simplify

MKo2gtu = MKo2Atu /. ot -> -ot // Simplify

MKo3Ntu = MKo3Ptu /. ω -> -ω // Simplify

MKo3Gtu = MKo3Atu /. ot -> -ot // Simplify

MKo4Ntu = MKo4Ptu /. ω -> -ω // Simplify

MKo5Ntu = MKo5Ptu /. ω -> -ω // Simplify

(4.3) MKoP + MKoN

(4.4) Replacing K by (k^0)  ((k^(0 )) = ω) and defining MKo = MKoP + MKoN

MKo11t = MKo1Pt/. K -> ω //Simplify

MKo12t = MKo1Nt/. K -> ω //Simplify

GKo11t = MKo1Ft //Simplify (*/. Kt -> Sqrt[(ot^2)-(mf^2)]*)

GKo12t = MKo1Gt //Simplify

MKo21t = MKo2Pt/. K -> ω //Simplify

MKo22t = MKo2Nt/. K -> ω //Simplify

GKo21t = MKo2Ft //Simplify

GKo22t = MKo2Gt //Simplify

gKo21t = MKo2At //Simplify

gKo22t = MKo2gt //Simplify

MKo31t = MKo3Pt/. K -> ω //Simplify

MKo32t = MKo3Nt/. K -> ω //Simplify

GKo31t = MKo3At //Simplify

GKo32t = MKo3Gt //Simplify

MKo41t = MKo4Pt/. K -> ω //Simplify

MKo42t = MKo4Nt/. K -> ω //Simplify

MKo51t = MKo5Pt/. K -> ω //Simplify

MKo52t = MKo5Nt/. K -> ω //Simplify

MKo11u = MKo1Pu/. K -> ω //Simplify

MKo12u = MKo1Nu/. K -> ω //Simplify

GKo11u = MKo1Fu //Simplify

GKo12u = MKo1Gu //Simplify

MKo21u = MKo2Pu/. K -> ω //Simplify

MKo22u = MKo2Nu/. K -> ω //Simplify

GKo21u = MKo2Fu //Simplify

GKo22u = MKo2Gu //Simplify

gKo21u = MKo2Au//Simplify

gKo22u = MKo2gu//Simplify

MKo31u = MKo3Pu/. K -> ω //Simplify

MKo32u = MKo3Nu/. K -> ω //Simplify

GKo31u = MKo3Au //Simplify

GKo32u = MKo3Gu //Simplify

MKo41u = MKo4Pu/. K -> ω //Simplify

MKo42u = MKo4Nu/. K -> ω //Simplify

MKo51u = MKo5Pu/. K -> ω //Simplify

MKo52u = MKo5Nu/. K -> ω //Simplify

MKo11tu = MKo1Ptu/. K -> ω //Simplify

MKo12tu = MKo1Ntu/. K -> ω //Simplify

GKo11tu = MKo1Ftu//Simplify

GKo12tu = MKo1Gtu//Simplify

MKo21tu = MKo2Ptu/. K -> ω //Simplify

MKo22tu = MKo2Ntu/. K -> ω //Simplify

GKo21tu = MKo2Ftu //Simplify

GKo22tu = MKo2Gtu //Simplify

gKo21tu = MKo2Atu //Simplify

gKo22tu = MKo2gtu //Simplify

MKo31tu = MKo3Ptu/. K -> ω //Simplify

MKo32tu = MKo3Ntu/. K -> ω //Simplify

GKo31tu = MKo3Atu //Simplify

GKo32tu = MKo3Gtu //Simplify

MKo41tu = MKo4Ptu/. K -> ω //Simplify

MKo42tu = MKo4Ntu/. K -> ω //Simplify

MKo51tu = MKo5Ptu/. K -> ω //Simplify

MKo52tu = MKo5Ntu/. K -> ω //Simplify

(5) Phase Space Integral Calculations with special choice of frame of reference for minimizing calculations

(5.1) ϕ’ integration : Integrate “MKo” w.r.t. ϕ’

Int1tp1t = Integrate[Integrate[(MKo11t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tp2t = Integrate[Integrate[(MKo12t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tp1t = Integrate[Integrate[(MKo21t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tp2t = Integrate[Integrate[(MKo22t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3tp1t = Integrate[Integrate[(MKo31t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3tp2t = Integrate[Integrate[(MKo32t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int4tp1t = Integrate[Integrate[(MKo41t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int4tp2t = Integrate[Integrate[(MKo42t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int5tp1t = Integrate[Integrate[(MKo51t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int5tp2t = Integrate[Integrate[(MKo52t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tp1u = Integrate[Integrate[(MKo11u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tp2u = Integrate[Integrate[(MKo12u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tp1u = Integrate[Integrate[(MKo21u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tp2u = Integrate[Integrate[(MKo22u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3tp1u = Integrate[Integrate[(MKo31u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3tp2u = Integrate[Integrate[(MKo32u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int4tp1u = Integrate[Integrate[(MKo41u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int4tp2u = Integrate[Integrate[(MKo42u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int5tp1u = Integrate[Integrate[(MKo51u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int5tp2u = Integrate[Integrate[(MKo52u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tp1tu = Integrate[Integrate[(MKo11tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tp2tu = Integrate[Integrate[(MKo12tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tp1tu = Integrate[Integrate[(MKo21tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tp2tu = Integrate[Integrate[(MKo22tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3tp1tu = Integrate[Integrate[(MKo31tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3tp2tu = Integrate[Integrate[(MKo32tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int4tp1tu = Integrate[Integrate[(MKo41tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int4tp2tu = Integrate[Integrate[(MKo42tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int5tp1tu = Integrate[Integrate[(MKo51tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int5tp2tu = Integrate[Integrate[(MKo52tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tf1t = Integrate[Integrate[(GKo11t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tf2t = Integrate[Integrate[(GKo12t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tf1t = Integrate[Integrate[(GKo21t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tf2t = Integrate[Integrate[(GKo22t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2ta1t = Integrate[Integrate[(gKo21t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2ta2t = Integrate[Integrate[(gKo22t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3ta1t = Integrate[Integrate[(GKo31t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3ta2t = Integrate[Integrate[(GKo32t/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tf1u = Integrate[Integrate[(GKo11u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tf2u = Integrate[Integrate[(GKo12u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tf1u = Integrate[Integrate[(GKo21u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tf2u = Integrate[Integrate[(GKo22u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2ta1u = Integrate[Integrate[(gKo21u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2ta2u = Integrate[Integrate[(gKo22u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3ta1u = Integrate[Integrate[(GKo31u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3ta2u = Integrate[Integrate[(GKo32u/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tf1tu = Integrate[Integrate[(GKo11tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int1tf2tu = Integrate[Integrate[(GKo12tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tf1tu = Integrate[Integrate[(GKo21tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2tf2tu = Integrate[Integrate[(GKo22tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2ta1tu = Integrate[Integrate[(gKo21tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int2ta2tu = Integrate[Integrate[(gKo22tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3ta1tu = Integrate[Integrate[(GKo31tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

Int3ta2tu = Integrate[Integrate[(GKo32tu/.Stp-> Sqrt[1-Ctp Ctp]),{phip, 0,2*π}],{Ctp, -1, 1}] // Simplify

(5.2) θ Integration : Integrate “MKo” w.r.t. θ

Int00 = Integrate[(MKo00P/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1t1t =  Integrate[(Int1tp1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1t2t =  Integrate[(Int1tp2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2t1t =  Integrate[(Int2tp1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2t2t =  Integrate[(Int2tp2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3t1t =  Integrate[(Int3tp1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3t2t =  Integrate[(Int3tp2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int4t1t =  Integrate[(Int4tp1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int4t2t =  Integrate[(Int4tp2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int5t1t =  Integrate[(Int5tp1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int5t2t =  Integrate[(Int5tp2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1t1u =  Integrate[(Int1tp1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1t2u =  Integrate[(Int1tp2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2t1u =  Integrate[(Int2tp1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2t2u =  Integrate[(Int2tp2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3t1u =  Integrate[(Int3tp1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3t2u =  Integrate[(Int3tp2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int4t1u =  Integrate[(Int4tp1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int4t2u =  Integrate[(Int4tp2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int5t1u =  Integrate[(Int5tp1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int5t2u =  Integrate[(Int5tp2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1t1tu =  Integrate[(Int1tp1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1t2tu =  Integrate[(Int1tp2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2t1tu =  Integrate[(Int2tp1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2t2tu =  Integrate[(Int2tp2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3t1tu =  Integrate[(Int3tp1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3t2tu =  Integrate[(Int3tp2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int4t1tu =  Integrate[(Int4tp1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int4t2tu =  Integrate[(Int4tp2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int5t1tu =  Integrate[(Int5tp1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int5t2tu =  Integrate[(Int5tp2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1 = Int1t1t + Int1t2t + Int1t1u + Int1t2u - Int1t1tu - Int1t2tu // Simplify

Int2 = Int2t1t + Int2t2t + Int2t1u + Int2t2u - Int2t1tu - Int2t2tu // Simplify

Int3 = Int3t1t + Int3t2t + Int3t1u + Int3t2u - Int3t1tu - Int3t2tu // Simplify

Int4 = Int4t1t + Int4t2t + Int4t1u + Int4t2u - Int4t1tu - Int4t2tu // Simplify

Int5 = Int5t1t + Int5t2t + Int5t1u + Int5t2u - Int5t1tu - Int5t2tu // Simplify

Int1f1t =  Integrate[(Int1tf1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1f2t =  Integrate[(Int1tf2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1f12t = Int1f1t+ Int1f2t // Simplify

Int2f1t =  Integrate[(Int2tf1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2f2t =  Integrate[(Int2tf2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2a1t =  Integrate[(Int2ta1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2a2t =  Integrate[(Int2ta2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3a1t =  Integrate[(Int3ta1t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3a2t =  Integrate[(Int3ta2t/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1f1u =  Integrate[(Int1tf1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1f2u =  Integrate[(Int1tf2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2f1u =  Integrate[(Int2tf1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2f2u =  Integrate[(Int2tf2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2a1u =  Integrate[(Int2ta1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2a2u =  Integrate[(Int2ta2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3a1u =  Integrate[(Int3ta1u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3a2u =  Integrate[(Int3ta2u/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1f1tu =  Integrate[(Int1tf1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1f2tu =  Integrate[(Int1tf2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2f1tu =  Integrate[(Int2tf1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2f2tu =  Integrate[(Int2tf2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2a1tu =  Integrate[(Int2ta1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int2a2tu =  Integrate[(Int2ta2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3a1tu =  Integrate[(Int3ta1tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int3a2tu =  Integrate[(Int3ta2tu/. St->Sqrt[1-Ct Ct]),{Ct, -1, 1}] // Simplify

Int1f = Int1f1t+ Int1f2t +  Int1f1u + Int1f2u - Int1f1tu - Int1f2tu // Simplify

Int1fnl = Int1f /. {Log[H ot-Kt Pp+(mf^2)] -> 0, Log[H ot+Kt Pp+(mf^2)] -> 0,Log[-H ot-Kt Pp+(mf^2)] -> 0, Log[-H ot+Kt Pp+(mf^2)] -> 0} // Simplify

Int2f = Int2f1t+ Int2f2t +  Int2f1u + Int2f2u - Int2f1tu - Int2f2tu // Simplify

Int2fnl = Int2f /. {Log[H ot-Kt Pp+(mf^2)] -> 0, Log[H ot+Kt Pp+(mf^2)] -> 0,Log[-H ot-Kt Pp+(mf^2)] -> 0, Log[-H ot+Kt Pp+(mf^2)] -> 0} /. {Pp->Sqrt[(H^2)-(mf^2)]}// Simplify

Int2a = Int2a1t + Int2a1u - Int2a1tu +Int2a2t + Int2a2u - Int2a2tu// Simplify

Int2anl = Int2a /. {Log[H ot-Kt Pp+(mf^2)] -> 0, Log[H ot+Kt Pp+(mf^2)] -> 0,Log[-H ot-Kt Pp+(mf^2)] -> 0, Log[-H ot+Kt Pp+(mf^2)] -> 0}/. {Pp->Sqrt[(H^2)-(mf^2)]}// Simplify

Int3a = Int3a1t + Int3a1u - Int3a1tu +Int3a2t + Int3a2u - Int3a2tu// Simplify

Int3anl = Int3a /. {Log[H ot-Kt Pp+(mf^2)] -> 0, Log[H ot+Kt Pp+(mf^2)] -> 0,Log[-H ot-Kt Pp+(mf^2)] -> 0, Log[-H ot+Kt Pp+(mf^2)] -> 0}/. {Pp->Sqrt[(H^2)-(mf^2)]}// Simplify

nB = 1/(Exp[ω/T] -1)

nF = 1/(Exp[ot/T] +1)

T1 = Integrate[ω nB Int1, {ω, 0, Infinity}] // Simplify

T1l = T1 /.{Log[H-Pp] -> 0, Log[H+Pp]->0} // Simplify

T1lNR = T1l /. {Pp-> Sqrt[(H^2)-(mf^2)]} /. {H-> Sqrt[(P^(2 ))+ (mc^2)]} // Simplify

T2 = Integrate[ω nB Int2, {ω, 0, Infinity}] // Simplify

T2l = T2 /.{Log[H-Pp] -> 0, Log[H+Pp]->0} // Simplify

T2lNR = T2l /. {Pp-> Sqrt[(H^2)-(mf^2)]} /. {H-> Sqrt[(P^(2 ))+ (mc^2)]} // Simplify

T3 = Integrate[ω nB Int3, {ω, 0, Infinity}] // Simplify

T3l = T3 /.{Log[H-Pp] -> 0, Log[H+Pp]->0} // Simplify

T3lNR = T3l /. {Pp-> Sqrt[(H^2)-(mf^2)]} /. {H-> Sqrt[(P^(2 ))+ (mc^2)]} // Simplify

T4 =(H-Pp) (H+Pp)/((mf^2)) Integrate[ω nB Int4, {ω, 0, Infinity}] // Simplify

T4l = T4 /.{Log[H-Pp] -> 0, Log[H+Pp]->0} // Simplify

T4lNR = T4l /. {Pp-> Sqrt[(H^2)-(mf^2)]} /. {H-> Sqrt[(P^(2 ))+ (mc^2)]} // Simplify

T5 = Integrate[ω nB Int5, {ω, 0, Infinity}] // Simplify

T5l = T5 /.{Log[H-Pp] -> 0, Log[H+Pp]->0} // Simplify

T5lNR = T5l /. {Pp-> Sqrt[(H^2)-(mf^2)]} /. {H-> Sqrt[(P^(2 ))+ (mc^2)]} // Simplify

Ttot= T1 + T2 + T3 + T4 + T5 // Simplify

Ttotl= T1l + T2l + T3l + T4l + T5l /.{Pp -> Sqrt[(H^2)-(mf^2)],P -> Sqrt[(H^2)-(mc^2)] }// Simplify

TtotlNR= T1lNR + T2lNR + T3lNR + T4lNR + T5lNR // Simplify

I2k = Integrate [ Sqrt[(ot^2)-(mf^2)] nF, {ot, Abs[mf], Infinity}]

I2ke = Integrate [ Sqrt[ot*ot - mf mf]nF, {ot, 0, Infinity}] (* Doesn't converge *)

I2 = Integrate [ ot nF, {ot, Abs[mf], Infinity}]

I20 = Integrate [ ot nF, {ot, 0, Infinity}]

I40 = Integrate [ (ot^3) nF, {ot, 0, Infinity}]

I4 = Integrate [ (ot^3) nF, {ot, Abs[mf], Infinity}]

I6 = Integrate [ (ot^5) nF, {ot, Abs[mf], Infinity}]

I60 = Integrate [ (ot^5) nF, {ot, 0, Infinity}]

f1 =Int1f/.{Log[-H ot-Kt Pp+(mf^2)]->0, Log[-H ot+Kt Pp+(mf^2)]->0,Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot-Kt Pp+(mf^2)]->0}// Simplify

Tf1 =f1 I2 // Simplify(* Tf1 =Integrate[f1, {ot, 0, Infinity}] // Simplify *)(* Tf1 = Integrate[ot nF Int1f/.{Log[-H ot-Kt Pp+(mf^2)]->0, Log[-H ot+Kt Pp+(mf^2)]->0,Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot-Kt Pp+(mf^2)]->0}, {ot, mf, Infinity}] // Simplify *)

Tf10 =f1 I20 // Simplify

Tf10NR = Tf10 /.{H-> Sqrt[(P^2)+ (mc^2)]}  // Simplify

f2 = Int2f/.{Log[-H ot-Kt Pp+(mf^2)]->0, Log[-H ot+Kt Pp+(mf^2)]->0,Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot-Kt Pp+(mf^2)]->0}// Simplify

a2 = Int2a/.{Log[-H ot-Kt Pp+(mf^2)]->0, Log[-H ot+Kt Pp+(mf^2)]->0,Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot-Kt Pp+(mf^2)]->0} // Simplify

fa2 = (3 (H^6) (mh^4))((H^2)-2 H ot+(mf^2)+(Pp^2)) ((H^2)+2 H ot+(mf^2)+(Pp^2))(1-(ot^2)/(H^2))/(4(H^4))(f2 + a2) /.{Pp-> Sqrt[(H^2)-(mf^2)]}// Expand // Simplify

Tfa2 = -128 π (e^2) (H^2) (λ^4) (3 (H^2)-3 (mc^2)+(P^2))( 2(H^(4 ))I2 - (H^2) I4 - I6)  /(3 (H^6) (mh^4)) // Simplify

Tfa20 = -128 π (e^2) (H^2) (λ^4) (3 (H^2)-3 (mc^2)+(P^2))( 2(H^(4 ))I20 - (H^2) I40- I60)  /(3 (H^6) (mh^4)) // Simplify

Tfa20NR =   ((((mc^2)+(P^2)))^2) /  ((mc^4)) (1 -2(P^2)/(mc^2)) Tfa20/. {H-> Sqrt[(P^2)+(mc^2)]} // Expand// Simplify

Tfa20alt = -128 π (e^2) (H^2) (λ^4) (3 (H^2)-3 (mc^2)+(P^2))( 2(H^(4 ))I20 - (H^2) I40)  /(3 (H^6) (mh^4)) // Simplify (* Drop T^6 terms *)

a3 = Int3a/.{Log[-H ot-Kt Pp+(mf^2)]->0, Log[-H ot+Kt Pp+(mf^2)]->0,Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot+Kt Pp+(mf^2)]->0, Log[H ot-Kt Pp+(mf^2)]->0} // Simplify

Ta3 =a3 I2 // Simplify

Ta30 =a3 I20 // Simplify

Ta30NR = Ta30 /.{H-> Sqrt[(P^2)+ (mc^2)]}  // Simplify

Ttotf= Tf1 + Tfa2 + Ta3  // FullSimplify

Ttotf0= Tf10 + Tfa20 + Ta30  // Expand // Simplify

Ttotf0NR= Tf10NR + Tfa20NR + Ta30NR  // Expand // Simplify

Ttotgfl0 = Ttotl + Ttotf0 /.{P->Sqrt[(H^2)-(mc^2)]}// Simplify (* All integrated; no factor of 1/2 for fermions !! *)

Collect[Ttotgfl0, T]

T2term = (1/(945 (H^4) (mh^8)))64 (π^3) (e^2) (λ^4) (T^2) (18144 (H^10)-168 (H^8) (136 (mc^2)+14 (mf^2)-5 (mh^2))+42 (H^6) (112 (mc^4)+4 (mc^2) (43 (mf^2)-5 (mh^2))-5 (mf^2) (mh^2))-21 (H^4) (112 (mc^4) (mf^2)-25 (mc^2) (mf^2) (mh^2))) // Simplify

Ttotgfl0 - T2term // Simplify

Collect[Ttotgfl0, T]

sigLO = Ttotgfl0 /32 /16/(π^4) /(e^2) (4π α)// Expand

Collect[sigLO, T]

TtotnondynNR = TtotlNR + Ttotf0NR // Simplify

Totala  = ((512 (π^3) (e^2) (λ^4) (mc^4) (mf^2) (T^2))/(3 (mh^8)))+((64 (π^3) (e^2) (λ^4) (mc^2) (mf^2) (T^2))/(3 (mh^6)))  // Simplify(* a term in a + b (P^2) *)

Totalb = (((448 (π^5) (e^2) (λ^4) (T^4))/(45 (mc^2) (mh^4)))+ ((512 (π^3) (e^2) (λ^4) (mc^2) (T^2))/(9 (mh^6)))-((128 (π^3) (e^2) (λ^4) (mf^2) (T^2))/(9 (mh^6))) +  ((8192 (π^3) (e^2) (λ^4) (mc^4) (T^2))/(9 (mh^8)))+((512 (π^3) (e^2) (λ^4) (mc^2) (mf^2) (T^2))/(3 (mh^8)))) (* b term; dropped (T^6) *)