<<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;
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)   -> (l^2)-((Subscript[m, ϕ])^2) = lphi; * SP[q-pp]-((Subscript[m, ϕ])^2)   -> ((l')^2)-((Subscript[m, ϕ])^2) = lphip;

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

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

(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)   ->(l^2) -((Subscript[m, ϕ])^2)	= lphi and	SP[q-pp]-((Subscript[m, ϕ])^2)   -> (((l'))^(2 ))-((Subscript[m, ϕ])^2) = lphip
lphi = l^2 - mphi2 = (q-p)^2 - mphi^2
Note that 1/(l+k)^2 - mpphi^2 = 1/lphi [ 1 - 2 l.k/lphi^2 + 2 l.k^2/lphi^2 + ...]
Note that 1/(lp-k)^2 - mpphi^2 = 1/lphip [ 1 + 2 l’.k/lphip - 2 l’.k^2/lphi^2 + ...]
l = q -p = pp - qq; lp = q-pp = p-qp

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/lphi))((I/lphi))(1 - 2 SP[k,l]/lphi)(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/lphi))(1 -(((2 SP[t,l]-2SP[l,pp]-2SP[t,pp]+2 SP[pp,pp])/lphi)))((I/lphi))(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/lphi))(1 - 2 SP[k,l]/lphi)(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/lphi))(1 -(((2 SP[t,l]-2SP[l,pp]-2SP[t,pp]+2 SP[pp,pp])/lphi)))(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/lphi))(1 -(((2 SP[p,l]-2SP[l,p]-2SP[t,p]+2 SP[p,p])/lphi)))(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]))))))

MK3At = (I*ComplexConjugate[λ])((I/lphi))(-I*(Subscript[Q, ϕ]))((I/lphi))(1 -(((2 SP[p,l]-2SP[l,p]-2SP[t,p]+2 SP[p,p])/lphi)))(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]))))))

MK3Pt = (I*ComplexConjugate[λ])((I/lphi))(-I*(Subscript[Q, ϕ]))((I/lphi))(1 - 2 SP[k,l]/lphi)(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])

MK4Pt= (I*ComplexConjugate[λ])((I/lphi))(-I*(Subscript[Q, ϕ]))((I/lphi))(-I*(Subscript[Q, ϕ]))((I/lphi))(1 - 2 SP[k,l]/lphi)(I*λ)(SpinorUBar[pp,mf].GA[6].SpinorU[qp,mc])(FV[2 l+k,μ])(FV[2l+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/lphi))(I*((Subscript[Q, ϕ])^2))((I/lphi))(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])

u-channel

MK1Pu =(-I*(Subscript[Q, f]))((I/((2 SP[pp,k]))))(I*ComplexConjugate[λ])((I/lphip))((I/lphip))(1 + 2 SP[k,lp]/lphip)(-I*(Subscript[Q, ϕ]))(I*λ)(SpinorUBar[pp,mf].GA[μ].(DiracSlash[pp+k]+mf).GA[6].SpinorU[q,mc])(FV[-2 lp+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/lphip))((I/lphip))(1 +(((2 SP[t,pp]-2SP[lp,pp]+2SP[t,pp]-2 SP[pp,pp])/lphip)))(-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/lphip))(1 + 2 SP[k,lp]/lphip)(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/lphip))(1 +(((2 SP[t,pp]-2SP[lp,pp]+2SP[t,pp]-2 SP[pp,pp])/lphip)))(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/lphip))(1 +(((2 SP[lp,p]-2SP[lp,t]+2SP[t,p]-2 SP[p,p])/lphip)))(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/lphip))(-I*(Subscript[Q, ϕ]))((I/lphip))(1 + 2 SP[k,lp]/lphip)(I*λ)((I/((-2 SP[p,k]))))(-I*(Subscript[Q, f']))(SpinorUBar[pp,mf].GA[6].SpinorU[q,mc])(FV[-2 lp+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/lphip))(-I*(Subscript[Q, ϕ]))((I/lphip))(1 +(((2 SP[lp,p]-2SP[lp,t]+2SP[t,p]-2 SP[p,p])/lphip)))(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/lphip))(-I*(Subscript[Q, ϕ ]))((I/lphip))(1 + 2 SP[k,lp]/lphip)(-I*(Subscript[Q, ϕ]))((I/lphip))(I*λ)(SpinorUBar[pp,mf].GA[6].SpinorU[q,mc])(FV[-2lp+k,μ])(FV[-2lp+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/lphip))(I*((Subscript[Q, ϕ])^2))((I/lphip))(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; LATER

(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

MKo0Pt = (Mt) *ComplexConjugate[Mt]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo0Pu = (Mu) *ComplexConjugate[Mu]//Contract//FermionSpinSum//DiracSimplify//Simplify

MKo0Ptu = ((Mt) *ComplexConjugate[Mu]+ (Mu) *ComplexConjugate[Mt]) // Contract//FermionSpinSum//DiracSimplify//Simplify

MKo1Pt = (MK1Pt) *ComplexConjugate[Mt]+(Mt)*ComplexConjugate[MK1Pt]//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//ExpandScalarProduct // 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//ExpandScalarProduct // 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//ExpandScalarProduct // Simplify

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

SP[k] = 0;
SP[k,qp] = ω*H+K*P*St *Stp *Cos[phip]-K*P*Ct*Ctp;
SP[k,q] = ω*H-K*P*St *Stp*Cos[phip]+K*P*Ct*Ctp;
SP[k,pp]= ω*H- K*Pp*Ctp;
SP[k,p]=ω*H+Pp*K*Ctp;
SP[k,l] = SP[k,q] - SP[k,p];
SP[k,lp] = SP[k,q] - SP[k,pp];

SP[t,t] = (mf^2);
SP[t,qp] = ot*H+Kt*P*St *Stp *Cos[phip]-Kt*P*Ct*Ctp;
SP[t,q] = ot*H-Kt*P*St*Stp*Cos[phip]+Kt*P*Ct*Ctp;
SP[t,pp]= ot*H- Kt*Pp*Ctp;
SP[t,p]=ot*H+Pp*Kt*Ctp;
SP[t,l] = SP[t,q] - SP[t,p];
SP[t,lp] = SP[t,q] - SP[t,pp];

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

SP[pp,l] = SP[pp,q] - SP[pp,p];
SP[p,l] = SP[p,q] - SP[p,p];
SP[pp,lp] = SP[pp,q] - SP[pp,pp];
SP[p,lp] = SP[p,q] - SP[p,pp];
SP[qp,l] = SP[qp,q] - SP[qp,p];
SP[qp,lp] = SP[qp,q] - SP[qp,pp];
SP[q,l] = SP[q,q] - SP[q,p];
SP[q,lp] = SP[q,q] - SP[q,pp];

SP[l,l] = SP[q-p];
SP[lp,lp] = SP[q-pp];
SP[l,lp] = SP[q-p,q-pp];

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 // ExpandScalarProduct // 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 //ExpandScalarProduct // 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 // ExpandScalarProduct // 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

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 //ExpandScalarProduct //Simplify

MKo42t = MKo4Nt/. K -> ω //ExpandScalarProduct //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 //ExpandScalarProduct //Simplify

MKo42u = MKo4Nu/. K -> ω //ExpandScalarProduct //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 //ExpandScalarProduct //Simplify

MKo42tu = MKo4Ntu/. K -> ω //ExpandScalarProduct //Simplify (* Denominator giving problems; undo the simplification *)

MKo4tu = MKo41tu + MKo42tu // 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’

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

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

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

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

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

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

theta integration

Int4tp1t1 =Int4tp1t/.{Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int4t1t =   Integrate[{Int4tp1t1/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int4tp2t1= Int4tp2t/.{Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int4t2t =   Integrate[{Int4tp2t1/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int4tt = Int4t1t + Int4t2t // Simplify

Int4tp1u1 = Int4tp1u/.{Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int4t1u =  Integrate[{Int4tp1u1/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int4tp2u1 = Int4tp2u/.{Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int4t2u = Integrate[{Int4tp2u1/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int4uu = Int4t1u + Int4t2u // Simplify

Int4tp1tu1 = Int4tp1tu/.{Log[H+Pp] -> 0, Log[H-Pp]->0} // Simplify

Int4t1tu =  Integrate[{Int4tp1tu1/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int4tp2tu1 = Int4tp2tu/.{Log[H+Pp] -> 0, Log[H-Pp]-> 0, Log[(H-Pp) ω]-> 0} // Simplify

Int4t2tu =  Integrate[{Int4tp2tu1/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int4tu = Int4t1tu + Int4t2tu // Simplify

Int4totgamma = Int4tt + Int4uu - Int4tu // Simplify

Int4totgammas = Int4totgamma /.{Pp-> Sqrt[(H^2)-(mf^2)], P-> Sqrt[(H^2)- (mc^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} // Simplify

Int4totgammaexpand = Int4totgamma /. {Log[((2 (H^2)-(mcfp^2)+2 P Pp)/(2 (H^2)-(mcfp^2)))] -> 2 P Pp/(2 (H^2)-(mcfp^2))+ (((2 P Pp/(2 (H^2)-(mcfp^2))))^3)/3, Log[-((-2 (H^2)+(mcfp^2)+2 P Pp)/(2 (H^2)-(mcfp^2)))] -> -2 P Pp/(2 (H^2)-(mcfp^2))- (((2 P Pp/(2 (H^2)-(mcfp^2))))^3)/3,Log[2 (H^2)-(mcfp^2)-2 P Pp] ->  -2 P Pp/(2 (H^2)-(mcfp^2))- (((2 P Pp/(2 (H^2)-(mcfp^2))))^3)/3, Log[2 (H^2)-(mcfp^2)+2 P Pp] ->  2 P Pp/(2 (H^2)-(mcfp^2))+ (((2 P Pp/(2 (H^2)-(mcfp^2))))^3)/3}/.{ Pp-> Sqrt[(H^2)-(mf^2)], mcfp -> Sqrt[(mc^2)+ (mf^2)- (mh^2)]} // Simplify

Int4totgammaNR = Int4totgammaexpand/.{H-> Sqrt[(P^2)+ (mc^2)]} // Simplify (* log *)

Int4totgammaNRa = (1/(9 ((((mc^2)-(mf^2)+(P^2)))^2)))32 (e^2) π (λ^4) (-(((2 ((mc^2)+(P^2))-(mf^2)) (((((mc^4)-2 ((mf^2)-(mh^2)) (mc^2)+(mf^4)+(mh^4)-2 (mf^2) (mh^2)+4 (mh^2) (P^2)))^3) (4 ((mc^2)-(mf^2)+(P^2)) (P^2)+3 ((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^2)) (192 ((((mc^2)+(P^2)))^4)-12 (16 (mc^2)+30 (mf^2)-14 (mh^2)+(ω^2)) ((((mc^2)+(P^2)))^3)+2 (27 (mc^4)+2 (60 (mf^2)-24 (mh^2)+(ω^2)) (mc^2)+105 (mf^4)+21 (mh^4)-2 (mh^2) (ω^2)-2 (P^2) (ω^2)-2 (mf^2) (45 (mh^2)+6 (P^2)-8 (ω^2))) ((((mc^2)+(P^2)))^2)+(-3 (mc^6)+(9 (mh^2)-21 (mf^2)) (mc^4)+(-69 (mf^4)+2 (9 (mh^2)+6 (P^2)-4 (ω^2)) (mf^2)-9 (mh^4)) (mc^2)-27 (mf^6)+3 (mh^6)+(mf^4) (21 (mh^2)+36 (P^2)-28 (ω^2))+(mf^2) (3 (mh^4)+(8 (ω^2)-12 (P^2)) (mh^2)+8 (P^2) (ω^2))) ((mc^2)+(P^2))-2 (mf^2) (3 (mc^6)+3 ((mf^2)-3 (mh^2)) (mc^4)+(3 (mf^4)-2 (6 (mh^2)-3 (P^2)+(ω^2)) (mf^2)+9 (mh^4)) (mc^2)+3 (mf^6)-3 (mh^6)+(mf^4) (-9 (mh^2)+6 (P^2)-4 (ω^2))+(mf^2) (9 (mh^4)+(2 (ω^2)-6 (P^2)) (mh^2)+2 (P^2) (ω^2))))-((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^4) (-135 (mc^16)+9 (107 (mf^2)-65 (mh^2)-85 (P^2)+8 (ω^2)) (mc^14)-3 (987 (mf^4)-3 (404 (mh^2)+545 (P^2)-56 (ω^2)) (mf^2)+297 (mh^4)+522 (P^4)-160 (P^2) (ω^2)+(mh^2) (1227 (P^2)-92 (ω^2))) (mc^12)+(5103 (mf^6)-27 (353 (mh^2)+495 (P^2)-56 (ω^2)) (mf^4)+9 (537 (mh^4)+2 (1119 (P^2)-92 (ω^2)) (mh^2)+988 (P^4)-320 (P^2) (ω^2)) (mf^2)-405 (mh^6)+24 (P^4) (55 (ω^2)-57 (P^2))+(mh^4) (384 (ω^2)-5625 (P^2))-56 (mh^2) (171 (P^4)-31 (P^2) (ω^2))) (mc^10)+(-5355 (mf^8)+45 (302 (mh^2)+445 (P^2)-56 (ω^2)) (mf^6)-45 (240 (mh^4)+(1011 (P^2)-92 (ω^2)) (mh^2)+466 (P^4)-160 (P^2) (ω^2)) (mf^4)+(2250 (mh^6)+3 (8817 (P^2)-640 (ω^2)) (mh^4)+56 (801 (P^4)-155 (P^2) (ω^2)) (mh^2)+6696 (P^6)-6600 (P^4) (ω^2)) (mf^2)+315 (mh^8)-432 (P^8)+1872 (P^6) (ω^2)-27 (mh^6) (79 (P^2)-8 (ω^2))+(mh^4) (2176 (P^2) (ω^2)-15102 (P^4))+(mh^2) (4340 (P^4) (ω^2)-12816 (P^6))) (mc^8)+(3465 (mf^10)-45 (251 (mh^2)+395 (P^2)-56 (ω^2)) (mf^8)+30 (423 (mh^4)+2 (903 (P^2)-92 (ω^2)) (mh^2)+876 (P^4)-320 (P^2) (ω^2)) (mf^6)-2 (2475 (mh^6)+3 (8259 (P^2)-640 (ω^2)) (mh^4)+56 (747 (P^4)-155 (P^2) (ω^2)) (mh^2)+6552 (P^6)-6600 (P^4) (ω^2)) (mf^4)+(-315 (mh^8)+108 (103 (P^2)-8 (ω^2)) (mh^6)+8 (7407 (P^4)-1088 (P^2) (ω^2)) (mh^4)+16 (3096 (P^6)-1085 (P^4) (ω^2)) (mh^2)+576 (3 (P^8)-13 (P^6) (ω^2))) (mf^2)+405 (mh^10)+1344 (P^8) (ω^2)+3 (mh^8) (675 (P^2)+8 (ω^2))-16 (mh^6) (279 (P^4)-65 (P^2) (ω^2))-64 (mh^4) (351 (P^6)-80 (P^4) (ω^2))-160 (mh^2) (54 (P^8)-35 (P^6) (ω^2))) (mc^6)+(-1323 (mf^12)+27 (200 (mh^2)+345 (P^2)-56 (ω^2)) (mf^10)-45 (183 (mh^4)+(795 (P^2)-92 (ω^2)) (mh^2)+410 (P^4)-160 (P^2) (ω^2)) (mf^8)+2 (2700 (mh^6)+3 (7701 (P^2)-640 (ω^2)) (mh^4)+56 (693 (P^4)-155 (P^2) (ω^2)) (mh^2)+6408 (P^6)-6600 (P^4) (ω^2)) (mf^6)-3 (315 (mh^8)+54 (127 (P^2)-8 (ω^2)) (mh^6)+4 (7263 (P^4)-1088 (P^2) (ω^2)) (mh^4)+8 (2988 (P^6)-1085 (P^4) (ω^2)) (mh^2)+288 (P^6) (3 (P^2)-13 (ω^2))) (mf^4)-3 (144 (mh^10)+3 (105 (P^2)+8 (ω^2)) (mh^8)-16 (456 (P^4)-65 (P^2) (ω^2)) (mh^6)-2560 (9 (P^6)-2 (P^4) (ω^2)) (mh^4)-160 (54 (P^8)-35 (P^6) (ω^2)) (mh^2)+1344 (P^8) (ω^2)) (mf^2)+135 (mh^12)+384 (P^10) (ω^2)-32 (mh^2) (P^8) (72 (P^2)-133 (ω^2))+3 (mh^10) (591 (P^2)-4 (ω^2))+6 (mh^8) (825 (P^4)+16 (ω^2) (P^2))-24 (mh^6) (186 (P^6)-85 (P^4) (ω^2))-192 (mh^4) (105 (P^8)-34 (P^6) (ω^2))) (mc^4)+(261 (mf^14)-9 (149 (mh^2)+295 (P^2)-56 (ω^2)) (mf^12)+9 (309 (mh^4)+2 (687 (P^2)-92 (ω^2)) (mh^2)+764 (P^4)-320 (P^2) (ω^2)) (mf^10)+(-2925 (mh^6)-3 (7143 (P^2)-640 (ω^2)) (mh^4)-56 (639 (P^4)-155 (P^2) (ω^2)) (mh^2)-6264 (P^6)+6600 (P^4) (ω^2)) (mf^8)+(1575 (mh^8)+108 (151 (P^2)-8 (ω^2)) (mh^6)+8 (7119 (P^4)-1088 (P^2) (ω^2)) (mh^4)+80 (576 (P^6)-217 (P^4) (ω^2)) (mh^2)+576 (3 (P^8)-13 (P^6) (ω^2))) (mf^6)-3 (117 (mh^10)+3 (465 (P^2)-8 (ω^2)) (mh^8)+16 (633 (P^4)-65 (P^2) (ω^2)) (mh^6)+64 (369 (P^6)-80 (P^4) (ω^2)) (mh^4)+160 (54 (P^8)-35 (P^6) (ω^2)) (mh^2)-1344 (P^8) (ω^2)) (mf^4)-(9 (mh^12)+6 (123 (P^2)-4 (ω^2)) (mh^10)+12 (15 (P^4)+16 (ω^2) (P^2)) (mh^8)-48 (438 (P^6)-85 (P^4) (ω^2)) (mh^6)-768 (57 (P^8)-17 (P^6) (ω^2)) (mh^4)-64 (P^8) (72 (P^2)-133 (ω^2)) (mh^2)+768 (P^10) (ω^2)) (mf^2)+(mh^2) (9 (mh^12)+333 (P^2) (mh^10)+24 (105 (P^4)-(P^2) (ω^2)) (mh^8)+120 (P^4) (45 (P^2)+(ω^2)) (mh^6)-96 (18 (P^8)-19 (P^6) (ω^2)) (mh^4)-128 (P^8) (81 (P^2)-35 (ω^2)) (mh^2)+2048 (P^10) (ω^2))) (mc^2)-18 (mf^16)+9 (mf^14) (14 (mh^2)+35 (P^2)-8 (ω^2))-3 (mf^12) (126 (mh^4)+(579 (P^2)-92 (ω^2)) (mh^2)+354 (P^4)-160 (P^2) (ω^2))+(mh^2) (P^2) (9 (mh^12)+198 (P^2) (mh^10)+12 (96 (P^4)-(P^2) (ω^2)) (mh^8)+48 (P^4) (45 (P^2)+(ω^2)) (mh^6)+608 (P^6) (ω^2) (mh^4)-256 (P^8) (9 (P^2)-5 (ω^2)) (mh^2)+512 (P^10) (ω^2))+(mf^8) (-630 (mh^8)-27 (175 (P^2)-8 (ω^2)) (mh^6)+ (2176 (P^2) (ω^2)-13950 (P^4)) (mh^4)-28 (396 (P^6)-155 (P^4) (ω^2)) (mh^2)-432 (P^8)+1872 (P^6) (ω^2))+(mf^10) (630 (mh^6)+(3951 (P^2)-384 (ω^2)) (mh^4)+56 (117 (P^4)-31 (P^2) (ω^2)) (mh^2)+24 (51 (P^6)-55 (P^4) (ω^2)))+(mf^6) (378 (mh^10)+3 (1035 (P^2)-8 (ω^2)) (mh^8)+80 (162 (P^4)-13 (P^2) (ω^2)) (mh^6)+128 (189 (P^6)-40 (P^4) (ω^2)) (mh^4)+160 (54 (P^8)-35 (P^6) (ω^2)) (mh^2)-1344 (P^8) (ω^2))-(mf^4) (126 (mh^12)+3 (345 (P^2)+4 (ω^2)) (mh^10)+6 (795 (P^4)-16 (P^2) (ω^2)) (mh^8)+120 (138 (P^6)-17 (P^4) (ω^2)) (mh^6)+192 (123 (P^8)-34 (P^6) (ω^2)) (mh^4)+32 (P^8) (72 (P^2)-133 (ω^2)) (mh^2)-384 (P^10) (ω^2))+(mf^2) (mh^2) (18 (mh^12)+117 (P^2) (mh^10)+24 (P^2) (3 (P^2)+(ω^2)) (mh^8)+120 (9 (P^6)-(P^4) (ω^2)) (mh^6)+96 (90 (P^8)-19 (P^6) (ω^2)) (mh^4)+128 (P^8) (99 (P^2)-35 (ω^2)) (mh^2)-2048 (P^10) (ω^2)))) (mc^2))/(((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^7) ((((mc^4)-2 ((mf^2)-(mh^2)) (mc^2)+(mf^4)+(mh^4)-2 (mf^2) (mh^2)+4 (mh^2) (P^2)))^3)))+((24 (2 (mc^4)-((mf^2)+(mh^2)-3 (P^2)) (mc^2)+2 (mf^4)+(P^4)-(mh^2) (P^2)-(mf^2) (2 (mh^2)+3 (P^2))) (3 (mc^4)+(-6 (mf^2)+6 (mh^2)+16 (P^2)) (mc^2)+3 (mf^4)+3 (mh^4)+16 (P^4)+12 (mh^2) (P^2)-2 (mf^2) (3 (mh^2)+8 (P^2))))/((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^3))+(1/((((mc^4)-2 ((mf^2)-(mh^2)) (mc^2)+(mf^4)+(mh^4)-2 (mf^2) (mh^2)+4 (mh^2) (P^2)))^4))8 (54 (mc^18)-3 (99 (mf^2)-57 (mh^2)-43 (P^2)+8 (ω^2)) (mc^16)+(621 (mf^4)-6 (111 (mh^2)+101 (P^2)-24 (ω^2)) (mf^2)+117 (mh^4)+105 (P^4)-104 (P^2) (ω^2)+18 (mh^2) (43 (P^2)-4 (ω^2))) (mc^14)+(-495 (mf^6)+3 (183 (mh^2)+352 (P^2)-120 (ω^2)) (mf^4)+3 (33 (mh^4)+(120 (ω^2)-916 (P^2)) (mh^2)-117 (P^4)+152 (P^2) (ω^2)) (mf^2)-153 (mh^6)+6 (P^4) (5 (P^2)-28 (ω^2))+12 (mh^4) (71 (P^2)-6 (ω^2))+(mh^2) (1107 (P^4)-296 (P^2) (ω^2))) (mc^12)-(243 (mf^8)-6 (234 (mh^2)-115 (P^2)+80 (ω^2)) (mf^6)+(2322 (mh^4)+(720 (ω^2)-2538 (P^2)) (mh^2)-285 (P^4)+728 (P^2) (ω^2)) (mf^4)-6 (234 (mh^6)+(48 (ω^2)-105 (P^2)) (mh^4)+(192 (P^2) (ω^2)-535 (P^4)) (mh^2)-10 (P^4) ((P^2)-8 (ω^2))) (mf^2)+243 (mh^8)+120 (P^6) (ω^2)+6 (mh^6) (111 (P^2)+4 (ω^2))+(mh^4) (376 (P^2) (ω^2)-2229 (P^4))-8 (mh^2) (81 (P^6)-61 (P^4) (ω^2))) (mc^10)+(891 (mf^10)-15 (261 (mh^2)+14 (P^2)+24 (ω^2)) (mf^8)+(6462 (mh^4)+48 (46 (P^2)+15 (ω^2)) (mh^2)+237 (P^4)+440 (P^2) (ω^2)) (mf^6)-(4806 (mh^6)+108 (61 (P^2)+4 (ω^2)) (mh^4)+(1672 (P^2) (ω^2)-2325 (P^4)) (mh^2)+6 (P^4) (5 (P^2)+48 (ω^2))) (mf^4)+(1431 (mh^8)+24 (248 (P^2)+3 (ω^2)) (mh^6)+(1256 (P^2) (ω^2)-3129 (P^4)) (mh^4)+(1304 (P^4) (ω^2)-984 (P^6)) (mh^2)+168 (P^6) (ω^2)) (mf^2)-63 (mh^10)-1362 (mh^8) (P^2)-32 (P^8) (ω^2)-(mh^6) (753 (P^4)+184 (ω^2) (P^2))+(mh^4) (2214 (P^6)-760 (P^4) (ω^2))+24 (mh^2) (6 (P^8)-17 (P^6) (ω^2))) (mc^8)+(-873 (mf^12)+6 (729 (mh^2)+101 (P^2)+24 (ω^2)) (mf^10)-(8703 (mh^4)+6 (1057 (P^2)+60 (ω^2)) (mh^2)+453 (P^4)-40 (P^2) (ω^2)) (mf^8)+4 (2133 (mh^6)+(4215 (P^2)+72 (ω^2)) (mh^4)+(333 (P^4)+272 (ω^2) (P^2)) (mh^2)+30 (P^6)-72 (P^4) (ω^2)) (mf^6)-(4023 (mh^8)+36 (483 (P^2)+2 (ω^2)) (mh^6)+6 (917 (P^4)+256 (ω^2) (P^2)) (mh^4)+8 (66 (P^6)+127 (ω^2) (P^4)) (mh^2)-144 (P^6) (ω^2)) (mf^4)+2 (mh^2) (315 (mh^8)+3267 (P^2) (mh^6)+(4650 (P^4)+232 (ω^2) (P^2)) (mh^4)+(952 (P^4) (ω^2)-1620 (P^6)) (mh^2)+264 (P^6) (ω^2)) (mf^2)+(mh^2) (63 (mh^10)-270 (P^2) (mh^8)-3333 (P^4) (mh^6)+24 (32 (P^6)-21 (P^4) (ω^2)) (mh^4)+24 (34 (P^8)-31 (P^6) (ω^2)) (mh^2)-176 (P^8) (ω^2))) (mc^6)+(459 (mf^14)-3 (903 (mh^2)+8 (16 (P^2)+(ω^2))) (mf^12)+(6615 (mh^4)+36 (149 (P^2)+2 (ω^2)) (mh^2)+147 (P^4)-136 (P^2) (ω^2)) (mf^10)-3 (2835 (mh^6)+12 (487 (P^2)+2 (ω^2)) (mh^4)+(817 (P^4)+104 (ω^2) (P^2)) (mh^2)+2 (P^4) (5 (P^2)-52 (ω^2))) (mf^8)+(5985 (mh^8)+24 (995 (P^2)+(ω^2)) (mh^6)+26 (531 (P^4)+32 (ω^2) (P^2)) (mh^4)+8 (150 (P^6)+13 (ω^2) (P^4)) (mh^2)-240 (P^6) (ω^2)) (mf^6)-(2079 (mh^10)+14328 (P^2) (mh^8)+6 (3841 (P^4)+64 (ω^2) (P^2)) (mh^6)+12 (117 (P^6)+134 (ω^2) (P^4)) (mh^4)+144 (2 (P^8)-(P^6) (ω^2)) (mh^2)-64 (P^8) (ω^2)) (mf^4)+3 (mh^2) (63 (mh^10)+876 (P^2) (mh^8)+3941 (P^4) (mh^6)+8 (244 (P^6)+41 (ω^2) (P^4)) (mh^4)-8 (6 (P^8)-47 (P^6) (ω^2)) (mh^2)-16 (P^8) (ω^2)) (mf^2)+(mh^2) (45 (mh^12)+372 (P^2) (mh^10)-279 (P^4) (mh^8)-4662 (P^6) (mh^6)+24 (58 (P^8)-27 (P^6) (ω^2)) (mh^4)+32 (P^8) (3 (P^2)-11 (ω^2)) (mh^2)-32 (P^10) (ω^2))) (mc^4)+(-135 (mf^16)+6 (156 (mh^2)+19 (P^2)) (mf^14)-3 (924 (mh^4)+718 (P^2) (mh^2)-21 (P^4)-8 (P^2) (ω^2)) (mf^12)+2 (2268 (mh^6)+4437 (P^2) (mh^4)+(507 (P^4)+32 (ω^2) (P^2)) (mh^2)-30 (P^6)) (mf^10)- (4410 (mh^8)+16074 (P^2) (mh^6)+5 (1947 (P^4)+40 (ω^2) (P^2)) (mh^4)+8 (15 (P^6)-8 (P^4) (ω^2)) (mh^2)+24 (P^6) (ω^2)) (mf^8)+2 (mh^2) (1260 (mh^8)+7323 (P^2) (mh^6)+2 (5469 (P^4)+28 (ω^2) (P^2)) (mh^4)+4 (411 (P^6)+68 (ω^2) (P^4)) (mh^2)-120 (P^6) (ω^2)) (mf^6)-(mh^2) (756 (mh^10)+6318 (P^2) (mh^8)+18111 (P^4) (mh^6)+96 (133 (P^6)+6 (ω^2) (P^4)) (mh^4)+24 (42 (P^8)+17 (ω^2) (P^6)) (mh^2)-176 (P^8) (ω^2)) (mf^4)+2 (mh^2) (36 (mh^12)+375 (P^2) (mh^10)+2043 (P^4) (mh^8)+5154 (P^6) (mh^6)+48 (5 (P^8)+9 (ω^2) (P^6)) (mh^4)+32 (P^8) (3 (P^2)+2 (ω^2)) (mh^2)-32 (P^10) (ω^2)) (mf^2)+(mh^4) (9 (mh^12)+162 (P^2) (mh^10)+807 (P^4) (mh^8)+504 (P^6) (mh^6)-2928 (P^8) (mh^4)+16 (P^8) (24 (P^2)-25 (ω^2)) (mh^2)-64 (P^10) (ω^2))) (mc^2)+18 (mf^18)-3 (mf^16) (48 (mh^2)+5 (P^2))+(mf^14) (504 (mh^4)+360 (P^2) (mh^2)-33 (P^4)+8 (P^2) (ω^2))+3 (mh^6) (P^2) (3 (mh^10)+39 (P^2) (mh^8)+166 (P^4) (mh^6)+192 (P^6) (mh^4)-160 (P^8) (mh^2)-32 (P^8) (ω^2))+3 (mf^6) (mh^2) (168 (mh^10)+1320 (P^2) (mh^8)+3231 (P^4) (mh^6)+32 (P^4) (66 (P^2)+(ω^2)) (mh^4)+8 (P^6) ((ω^2)-2 (P^2)) (mh^2)+16 (P^8) (ω^2))+(mf^2) (mh^4) (18 (mh^12)+216 (P^2) (mh^10)+1065 (P^4) (mh^8)+3096 (P^6) (mh^6)+4848 (P^8) (mh^4)+16 (P^8) (24 (P^2)+17 (ω^2)) (mh^2)-64 (P^10) (ω^2))-3 (mf^12) (336 (mh^6)+612 (P^2) (mh^4)+(39 (P^4)+8 (ω^2) (P^2)) (mh^2)-10 (P^6)+16 (P^4) (ω^2))+(mf^10) (1260 (mh^8)+4296 (P^2) (mh^6)+3 (777 (P^4)+8 (ω^2) (P^2)) (mh^4)-8 (27 (P^6)-4 (P^4) (ω^2)) (mh^2)+72 (P^6) (ω^2))-(mf^8) (1008 (mh^10)+5490 (P^2) (mh^8)+(7377 (P^4)+8 (ω^2) (P^2)) (mh^6)+(858 (P^6)+80 (ω^2) (P^4)) (mh^4)-24 (6 (P^8)-(P^6) (ω^2)) (mh^2)+32 (P^8) (ω^2))-(mf^4) (mh^2) (144 (mh^12)+1500 (P^2) (mh^10)+5679 (P^4) (mh^8)+8886 (P^6) (mh^6)+24 (86 (P^8)+11 (ω^2) (P^6)) (mh^4)-96 (P^8) ((P^2)+(ω^2)) (mh^2)+32 (P^10) (ω^2)))) // Expand // Simplify

Int4totgammaNRe = Int4totgammaNRa  (((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^7) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))+4 (mh^2) (P^2)))^4)) / (((((mc^2)-(mf^2)+(mh^2)))^7) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))))^4)) (1 - 14 (P^2) /((mc^2)-(mf^2)+(mh^2) )) (1 - 16 (mh^(2 ))(P^2)/((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2)))) // Simplify

Int4totgammaNRe1= Int4totgammaNRe/. {P->0} // Simplify  (* (P^(0 ))term *)

g10 = Int4totgammaNRe1 /. {ω -> 0} // Simplify (* (P^0) and omega^0 *)

g12 =(-g10 + Int4totgammaNRe1)/(ω^2)  // Simplify  (* (P^0) and omega^2 *)

Int4totgammaNRe2= Int4totgammaNRe - Int4totgammaNRe1// Simplify

Int4totgammaNRe3= Int4totgammaNRe2 /(P^2) // Expand // Simplify

Int4totgammaNRe4 = Int4totgammaNRe3  /. {P->0} // Simplify (* this is the (P^2) term *)

g40 = Int4totgammaNRe4 /. {ω -> 0} // Simplify (* (P^2) and omega^0 *)

g42 = (-g40 + Int4totgammaNRe4)/ (ω^2) // Simplify (* (P^2) and omega^2 *)

Int4totgammaNR1 = Int4totgammaNR /. {P->0} // Simplify

Int4tots = Int4totgammas  (* No fermion *)

Int4totNR = Int4totgammaNR

nB = 1/(Exp[ω/T] -1)

I2 = Integrate[ω nB, {ω, 0, Infinity}]

I4 = Integrate[(ω^3) nB, {ω, 0, Infinity}]

Total4NR = ((1024 π (e^2) (λ^4) (mc^2) (mf^2) (3 (mc^4)-(mc^2) (3 (mf^2)-3 (mh^2))) )/(3 ((((mc^2)-(mf^2)+(mh^2)))^5))) I2 +  ((1024 π (e^2) (λ^4) (mc^2) (mf^2) (-(mc^2) +(mf^2)))/(3 ((((mc^2)-(mf^2)+(mh^2)))^5))) I4 // Simplify

Total4 = (g10 + g40 (P^2)) I2 + (g12 + g42 (P^2)) I4 // Simplify

Total4a = (g10) I2 + (g12 ) I4 // Expand // Simplify  (* a term of a + b (P^2) *)

Total4b = (g40 ) I2 + (g42 ) I4 // Expand // Simplify (* b term of a + b (P^2) *)