<<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 // 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

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

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’

Int3tp1t1 =  Integrate[{Integrate[MKo31t,{phip, 0,2*π}]/. Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3tp1t = Int3tp1t1 /. {Log[H+Pp] -> 0, Log[H-Pp]->0} // Simplify

Int3tp2t1 =  Integrate[{Integrate[MKo32t,{phip, 0,2*π}]/. Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3tp2t = Int3tp2t1 /. {Log[H+Pp] -> 0, Log[H-Pp]->0} // Simplify

Int3ta1t1 = Integrate[{Integrate[GKo31t,{phip, 0,2*π}]/.Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3ta1t = Int3ta1t1 /. {Log[(mf^2)- H ot - Kt Pp] -> 0, Log[(mf^(2 ))-H ot+ Kt Pp]->0} // Simplify

Int3ta2t1 = Integrate[{Integrate[GKo32t,{phip, 0,2*π}]/.Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3ta2t = Int3ta2t1 /. {Log[(mf^2)+ H ot + Kt Pp] -> 0, Log[(mf^(2 ))+H ot- Kt Pp]->0} // Simplify

Int3tp1u1 = Integrate[{Integrate[MKo31u,{phip, 0,2*π}]/. Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3tp1u = Int3tp1u1 /. {Log[H + Pp] -> 0, Log[H - Pp]->0} // Simplify

Int3tp2u1 = Integrate[{Integrate[MKo32u,{phip, 0,2*π}]/. Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3tp2u = Int3tp2u1 /. {Log[H + Pp] -> 0, Log[H - Pp]->0} // Simplify

Int3ta1u1 = Integrate[{Integrate[GKo31u,{phip, 0,2*π}]/.Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3ta1u = Int3ta1u1 /. {Log[(mf^2)- H ot - Kt Pp] -> 0, Log[(mf^(2 ))-H ot+ Kt Pp]->0} // Simplify

Int3ta2u1 = Integrate[{Integrate[GKo32u,{phip, 0,2*π}]/.Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3ta2u = Int3ta2u1 /. {Log[(mf^2)+ H ot - Kt Pp] -> 0, Log[(mf^(2 ))+H ot+ Kt Pp]->0} // Simplify

Int3tp1tu1 = Integrate[{Integrate[MKo31tu,{phip, 0,2*π}]/. Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3tp1tu = Int3tp1tu1 /. {Log[H + Pp] -> 0, Log[H - Pp]->0} // Simplify

Int3tp2tu1 =  Int3tp1tu1/. {ω -> - ω}

Int3tp2tu = Int3tp2tu1 /. {Log[H + Pp] -> 0, Log[H - Pp]->0} // Simplify

Int3ta1tu1 = Integrate[{Integrate[GKo31tu,{phip, 0,2*π}]/.Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3ta1tu = Int3ta1tu1 /. {Log[(mf^2)- H ot - Kt Pp] -> 0, Log[(mf^(2 ))-H ot+ Kt Pp]->0} // Simplify

Int3ta2tu1 = Integrate[{Integrate[GKo32tu,{phip, 0,2*π}]/.Stp-> Sqrt[1-Ctp Ctp]},{Ctp, -1, 1}] // Simplify

Int3ta2tu = Int3ta2tu1 /. {Log[(mf^2)+ H ot - Kt Pp] -> 0, Log[(mf^(2 ))+H ot+ Kt Pp]->0} // Simplify

theta integration

Int3t1t =   Integrate[{Int3tp1t/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int3t2t =  Int3t1t /. ω-> - ω  (* Integrate[{Int3tp2t/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify *)

Int3tt = Int3t1t +  Int3t2t // Simplify

Int3a1t =  Integrate[{Int3ta1t/.{lphi -> ((mcfp^2) - 2 SP[q,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int3a2t =  Int3a1t/.{ot -> -ot} // Simplify

Int3att = Int3a1t +  Int3a2t // Simplify

Int3t1u =  Integrate[{Int3tp1u/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int3t2u = Int3t1u /. ω -> -ω  (* Integrate[{Int3tp2u/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify *)

Int3uu = Int3t1u +  Int3t2u // Simplify

Int3a1u =  Integrate[Int3ta1u/. {lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]},{Ct, -1, 1}] // Simplify

Int3a2u =  Int3a1u/.{ot -> -ot} // Simplify

Int3auu = Int3a1u +  Int3a2u // Simplify

Int3t1tu=  Integrate[{Int3tp1tu/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int3t2tu =  Int3t1tu/.{ω -> - ω}

Int3tu = Int3t1tu +  Int3t2tu // Simplify

Int3a1tu =  Integrate[Int3ta1tu/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]},{Ct, -1, 1}] // Simplify

Int3a2tu =  Int3a1tu/.{ot -> -ot}  (* Again meaningless !*)

Int3atu = Int3a1tu +  Int3a2tu // Simplify

Int3totgamma = Int3tt + Int3uu - Int3tu // Simplify

Int3totgammas = Int3totgamma /.{Pp-> Sqrt[(H^2)-(mf^2)], P-> Sqrt[(H^2)- (mc^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} // Simplify

Int3totgammaexpand = Int3totgamma /. {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} // Simplify (* /.{ Pp-> Sqrt[(H^2)-(mf^2)], mcfp -> Sqrt[(mc^2)+ (mf^2)- (mh^2)]} // Simplify *)

Int3totgammaNR = Int3totgammaexpand/.{Pp->Sqrt[(H^2)-(mf^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} /.{H-> Sqrt[(P^2)+ (mc^2)]} // Simplify (* log *)

Int3totgammaNRa = -((128 (e^2) (8 (P^2) (mc^20)+2 (3 (mf^4)-3 ((mh^2)+5 (P^2)) (mf^2)+44 (P^4)+8 (mh^2) (P^2)) (mc^18)-2 (24 (mf^6)-2 (24 (mh^2)+13 (P^2)) (mf^4)+(24 (mh^4)+18 (P^2) (mh^2)+155 (P^4)) (mf^2)-4 (P^2) (-4 (mh^4)+17 (P^2) (mh^2)+50 (P^4))) (mc^16)+(168 (mf^8)-24 (21 (mh^2)+8 (P^2)) (mf^6)+(504 (mh^4)+440 (P^2) (mh^2)+283 (P^4)) (mf^4)-2 (84 (mh^6)+76 (P^2) (mh^4)+46 (P^4) (mh^2)+677 (P^6)) (mf^2)+16 (P^2) (-6 (mh^6)-20 (P^2) (mh^4)+30 (P^4) (mh^2)+59 (P^6))) (mc^14)-2 (168 (mf^10)-28 (24 (mh^2)+13 (P^2)) (mf^8)+28 (36 (mh^4)+49 (P^2) (mh^2)+(P^4)) (mf^6)-(672 (mh^6)+1644 (P^2) (mh^4)+283 (P^4) (mh^2)+642 (P^6)) (mf^4)+(168 (mh^8)+628 (P^2) (mh^6)-(P^4) (mh^4)+282 (P^6) (mh^2)+1388 (P^8)) (mf^2)+4 (P^2) (2 (mh^8)+73 (P^2) (mh^6)+138 (P^4) (mh^4)-91 (P^6) (mh^2)-150 (P^8))) (mc^12)+(420 (mf^12)-28 (75 (mh^2)+53 (P^2)) (mf^10)+7 (600 (mh^4)+1032 (P^2) (mh^2)+95 (P^4)) (mf^8)-2 (2100 (mh^6)+6448 (P^2) (mh^4)+2857 (P^4) (mh^2)-89 (P^6)) (mf^6)+(2100 (mh^8)+10184 (P^2) (mh^6)+10185 (P^4) (mh^4)+1128 (P^6) (mh^2)+1992 (P^8)) (mf^4)-2 (210 (mh^10)+1578 (P^2) (mh^8)+2728 (P^4) (mh^6)+603 (P^6) (mh^4)+952 (P^8) (mh^2)+1296 (P^10)) (mf^2)+16 (P^2) (8 (mh^10)+20 (P^2) (mh^8)-49 (P^4) (mh^6)-76 (P^6) (mh^4)+17 (P^8) (mh^2)+48 (P^10))) (mc^10)-2 (168 (mf^14)-168 (6 (mh^2)+5 (P^2)) (mf^12)+7 (360 (mh^4)+704 (P^2) (mh^2)+127 (P^4)) (mf^10)-10 (336 (mh^6)+1136 (P^2) (mh^4)+689 (P^4) (mh^2)-38 (P^6)) (mf^8)+4 (630 (mh^8)+3240 (P^2) (mh^6)+4069 (P^4) (mh^4)+765 (P^6) (mh^2)-187 (P^8)) (mf^6)-2 (504 (mh^10)+3700 (P^2) (mh^8)+7937 (P^4) (mh^6)+4998 (P^6) (mh^4)+890 (P^8) (mh^2)+60 (P^10)) (mf^4)+(168 (mh^12)+1760 (P^2) (mh^10)+6035 (P^4) (mh^8)+7636 (P^6) (mh^6)+4268 (P^8) (mh^4)+1840 (P^10) (mh^2)+432 (P^12)) (mf^2)-4 (P^2) (12 (mh^12)+109 (P^2) (mh^10)+270 (P^4) (mh^8)+223 (P^6) (mh^6)+190 (P^8) (mh^4)-60 (P^10) (mh^2)+24 (P^12))) (mc^8)+(168 (mf^16)-8 (147 (mh^2)+134 (P^2)) (mf^14)+7 (504 (mh^4)+1048 (P^2) (mh^2)+235 (P^4)) (mf^12)-2 (2940 (mh^6)+10284 (P^2) (mh^4)+6848 (P^4) (mh^2)-285 (P^6)) (mf^10)+(5880 (mh^8)+30320 (P^2) (mh^6)+39518 (P^4) (mh^4)+7680 (P^6) (mh^2)-3424 (P^8)) (mf^8)-4 (882 (mh^10)+6200 (P^2) (mh^8)+13215 (P^4) (mh^6)+8361 (P^6) (mh^4)+272 (P^8) (mh^2)-816 (P^10)) (mf^6)+(1176 (mh^12)+10632 (P^2) (mh^10)+33757 (P^4) (mh^8)+42720 (P^6) (mh^6)+21792 (P^8) (mh^4)+5408 (P^10) (mh^2)-1152 (P^12)) (mf^4)-2 (mh^2) (84 (mh^12)+908 (P^2) (mh^10)+4198 (P^4) (mh^8)+9339 (P^6) (mh^6)+10768 (P^8) (mh^4)+8720 (P^10) (mh^2)+2016 (P^12)) (mf^2)-16 (mh^2) (P^2) (2 (mh^12)-2 (P^2) (mh^10)-72 (P^4) (mh^8)-235 (P^6) (mh^6)-366 (P^8) (mh^4)-336 (P^10) (mh^2)+40 (P^12))) (mc^6)-2 (24 (mf^18)-16 (12 (mh^2)+11 (P^2)) (mf^16)+2 (336 (mh^4)+690 (P^2) (mh^2)+127 (P^4)) (mf^14)-(1344 (mh^6)+4556 (P^2) (mh^4)+2591 (P^4) (mh^2)-478 (P^6)) (mf^12)+(1680 (mh^8)+8212 (P^2) (mh^6)+9201 (P^4) (mh^4)-390 (P^6) (mh^2)-1596 (P^8)) (mf^10)-2 (672 (mh^10)+4350 (P^2) (mh^8)+7853 (P^4) (mh^6)+2250 (P^6) (mh^4)-2494 (P^8) (mh^2)-828 (P^10)) (mf^8)+4 (168 (mh^12)+1339 (P^2) (mh^10)+3436 (P^4) (mh^8)+2527 (P^6) (mh^6)-794 (P^8) (mh^4)-776 (P^10) (mh^2)-216 (P^12)) (mf^6)-(192 (mh^14)+1700 (P^2) (mh^12)+5475 (P^4) (mh^10)+6594 (P^6) (mh^8)+1448 (P^8) (mh^6)+1584 (P^10) (mh^4)+1248 (P^12) (mh^2)-192 (P^14)) (mf^4)+(mh^2) (24 (mh^14)+156 (P^2) (mh^12)+305 (P^4) (mh^10)-54 (P^6) (mh^8)-300 (P^8) (mh^6)+3008 (P^10) (mh^4)+7008 (P^12) (mh^2)+1024 (P^14)) (mf^2)+4 (mh^2) (P^2) (7 (mh^14)+67 (P^2) (mh^12)+238 (P^4) (mh^10)+367 (P^6) (mh^8)+70 (P^8) (mh^6)-584 (P^10) (mh^4)-688 (P^12) (mh^2)+32 (P^14))) (mc^4)+(6 (mf^20)-2 (27 (mh^2)+19 (P^2)) (mf^18)+(216 (mh^4)+344 (P^2) (mh^2)-89 (P^4)) (mf^16)+(-504 (mh^6)-1328 (P^2) (mh^4)+334 (P^4) (mh^2)+926 (P^6)) (mf^14)+(756 (mh^8)+2856 (P^2) (mh^6)-239 (P^4) (mh^4)-5736 (P^6) (mh^2)-2008 (P^8)) (mf^12)-2 (378 (mh^10)+1862 (P^2) (mh^8)+226 (P^4) (mh^6)-7509 (P^6) (mh^4)-7064 (P^8) (mh^2)-816 (P^10)) (mf^10)+(504 (mh^12)+2968 (P^2) (mh^10)+385 (P^4) (mh^8)-22720 (P^6) (mh^6)-37856 (P^8) (mh^4)-15280 (P^10) (mh^2)-384 (P^12)) (mf^8)+2 (mh^2) (-108 (mh^12)-672 (P^2) (mh^10)+527 (P^4) (mh^8)+11505 (P^6) (mh^6)+26576 (P^8) (mh^4)+19760 (P^10) (mh^2)+3744 (P^12)) (mf^6)+(mh^2) (54 (mh^14)+248 (P^2) (mh^12)-1889 (P^4) (mh^10)-16296 (P^6) (mh^8)-44104 (P^8) (mh^6)-48224 (P^10) (mh^4)-14976 (P^12) (mh^2)-640 (P^14)) (mf^4)+2 (mh^2) (-3 (mh^16)+17 (P^2) (mh^14)+572 (P^4) (mh^12)+3643 (P^6) (mh^10)+10664 (P^8) (mh^8)+15296 (P^10) (mh^6)+8224 (P^12) (mh^4)-1280 (P^14) (mh^2)-256 (P^16)) (mf^2)-8 (mh^4) (P^2) (2 (mh^14)+31 (P^2) (mh^12)+186 (P^4) (mh^10)+580 (P^6) (mh^8)+1006 (P^8) (mh^6)+864 (P^10) (mh^4)+112 (P^12) (mh^2)-256 (P^14))) (mc^2)-4 (P^2) ((((mf^2)-(mh^2)-2 (P^2)))^3) ((mf^14)-3 (2 (mh^2)+3 (P^2)) (mf^12)+(15 (mh^4)+49 (P^2) (mh^2)+6 (P^4)) (mf^10)-2 (10 (mh^6)+55 (P^2) (mh^4)+42 (P^4) (mh^2)-3 (P^6)) (mf^8)+5 (3 (mh^8)+26 (P^2) (mh^6)+48 (P^4) (mh^4)+12 (P^6) (mh^2)) (mf^6)-(mh^2) (6 (mh^8)+85 (P^2) (mh^6)+276 (P^4) (mh^4)+192 (P^6) (mh^2)+8 (P^8)) (mf^4)+(mh^4) ((mh^8)+29 (P^2) (mh^6)+138 (P^4) (mh^4)+244 (P^6) (mh^2)+64 (P^8)) (mf^2)-2 (mh^6) (P^2) (2 (mh^6)+12 (P^2) (mh^4)+27 (P^4) (mh^2)+28 (P^6)))) π (λ^4))/(3 ((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^6) ((((mc^4)-2 ((mf^2)-(mh^2)) (mc^2)+(mf^4)+(mh^4)-2 (mf^2) (mh^2)+4 (mh^2) (P^2)))^3)))

Int3totgammaNRe = Int3totgammaNRa ( ((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^6) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))+4 (mh^2) (P^2)))^3) ) / ( ((((mc^2)-(mf^2)+(mh^2)))^6) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))))^3) ) (1 - 12 (P^2)/((mc^2)-(mf^2)+(mh^2))) ( 1 - 12 (mh^(2 ))(P^2)/((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2)))) // Simplify

Int3totgammaNRe1 = Int3totgammaNRe /. {P->0} // Simplify (* this is the (P^0) term *)

Int3totgammaNRe2 = Int3totgammaNRe - Int3totgammaNRe1  // Simplify

Int3totgammaNRe3 = Int3totgammaNRe2 / (P^2) // Expand // Simplify

Int3totgammaNRe4 = Int3totgammaNRe3/. {P->0} // Simplify (* this is the (P^2) term, omega independent *)

Int3totgammaNR1 = Int3totgammaNR /. {P->0} // Simplify

Int3totfermion = Int3att + Int3auu - Int3atu  // Simplify

Int3totfermions =Int3totfermion /. {Pp-> Sqrt[(H^2)-(mf^2)], P-> Sqrt[(H^2)- (mc^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} // Simplify

Int3totfermionexpand = Int3totfermion /. {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} // Simplify (*/.{ Pp-> Sqrt[(H^2)-(mf^2)], mcfp -> Sqrt[(mc^2)+ (mf^2)- (mh^2)]} // Simplify *)

Int3totfermionNR = Int3totfermionexpand/. {Pp-> Sqrt[(H^2)-(mf^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]}/. {H-> Sqrt[(P^2)+ (mc^2)]} // Simplify

Int3totfermionNRa = (64 (e^2) (2 (9 (mf^2)+20 (P^2)) (mc^20)-2 (78 (mf^4)+(30 (P^2)-75 (mh^2)) (mf^2)-2 (P^2) (44 (mh^2)+89 (P^2))) (mc^18)+2 (300 (mf^6)-8 (72 (mh^2)+53 (P^2)) (mf^4)+(276 (mh^4)+338 (P^2) (mh^2)-469 (P^4)) (mf^2)+556 (P^6)+668 (mh^2) (P^4)+112 (mh^4) (P^2)) (mc^16)+(-1344 (mf^8)+24 (161 (mh^2)+177 (P^2)) (mf^6)-(3696 (mh^4)+8920 (P^2) (mh^2)+2625 (P^4)) (mf^4)+2 (588 (mh^6)+2364 (P^2) (mh^4)+934 (P^4) (mh^2)-741 (P^6)) (mf^2)+4 (P^2) (-24 (mh^6)+281 (P^2) (mh^4)+916 (P^4) (mh^2)+399 (P^6))) (mc^14)+2 (966 (mf^10)-28 (132 (mh^2)+167 (P^2)) (mf^8)+42 (126 (mh^4)+350 (P^2) (mh^2)+183 (P^4)) (mf^6)-(3360 (mh^6)+15484 (P^2) (mh^4)+17439 (P^4) (mh^2)+5838 (P^6)) (mf^4)+(798 (mh^8)+5692 (P^2) (mh^6)+10457 (P^4) (mh^4)+4710 (P^6) (mh^2)+1560 (P^8)) (mf^2)+4 (P^2) (-58 (mh^8)-188 (P^2) (mh^6)+141 (P^4) (mh^4)+485 (P^6) (mh^2)+144 (P^8))) (mc^12)+(-1848 (mf^12)+28 (315 (mh^2)+424 (P^2)) (mf^10)-7 (2400 (mh^4)+7016 (P^2) (mh^2)+4205 (P^4)) (mf^8)+2 (7980 (mh^6)+37912 (P^2) (mh^4)+49295 (P^4) (mh^2)+20377 (P^6)) (mf^6)-(7560 (mh^8)+52120 (P^2) (mh^6)+110943 (P^4) (mh^4)+88392 (P^6) (mh^2)+33356 (P^8)) (mf^4)+2 (714 (mh^10)+6928 (P^2) (mh^8)+22192 (P^4) (mh^6)+27549 (P^6) (mh^4)+15732 (P^8) (mh^2)+5656 (P^10)) (mf^2)-4 (P^2) (80 (mh^10)+667 (P^2) (mh^8)+1592 (P^4) (mh^6)+911 (P^6) (mh^4)+248 (P^8) (mh^2)-120 (P^10))) (mc^10)+2 (588 (mf^14)-420 (8 (mh^2)+11 (P^2)) (mf^12)+7 (1140 (mh^4)+3376 (P^2) (mh^2)+2091 (P^4)) (mf^10)-20 (504 (mh^6)+2394 (P^2) (mh^4)+3217 (P^4) (mh^2)+1365 (P^6)) (mf^8)+2 (3570 (mh^8)+24000 (P^2) (mh^6)+52217 (P^4) (mh^4)+45070 (P^6) (mh^2)+16744 (P^8)) (mf^6)-4 (672 (mh^10)+5935 (P^2) (mh^8)+18558 (P^4) (mh^6)+25404 (P^6) (mh^4)+17398 (P^8) (mh^2)+5928 (P^10)) (mf^4)+(420 (mh^12)+4560 (P^2) (mh^10)+19481 (P^4) (mh^8)+40116 (P^6) (mh^6)+41288 (P^8) (mh^4)+26304 (P^10) (mh^2)+6720 (P^12)) (mf^2)+4 (P^2) (12 (mh^12)+5 (P^2) (mh^10)-407 (P^4) (mh^8)-1412 (P^6) (mh^6)-1444 (P^8) (mh^4)-696 (P^10) (mh^2)+16 (P^12))) (mc^8)+(-480 (mf^16)+8 (399 (mh^2)+541 (P^2)) (mf^14)-21 (432 (mh^4)+1256 (P^2) (mh^2)+747 (P^4)) (mf^12)+14 (1020 (mh^6)+4724 (P^2) (mh^4)+6156 (P^4) (mh^2)+2427 (P^6)) (mf^10)-2 (6720 (mh^8)+43560 (P^2) (mh^6)+91831 (P^4) (mh^4)+76920 (P^6) (mh^2)+26302 (P^8)) (mf^8)+4 (1890 (mh^10)+15790 (P^2) (mh^8)+46997 (P^4) (mh^6)+62943 (P^6) (mh^4)+43292 (P^8) (mh^2)+13960 (P^10)) (mf^6)-(2352 (mh^12)+23368 (P^2) (mh^10)+88959 (P^4) (mh^8)+167808 (P^6) (mh^6)+178136 (P^8) (mh^4)+122496 (P^10) (mh^2)+33408 (P^12)) (mf^4)+2 (156 (mh^14)+1508 (P^2) (mh^12)+5902 (P^4) (mh^10)+13749 (P^6) (mh^8)+25048 (P^8) (mh^6)+30752 (P^10) (mh^4)+21216 (P^12) (mh^2)+3968 (P^14)) (mf^2)+4 (mh^2) (P^2) (56 (mh^12)+583 (P^2) (mh^10)+2100 (P^4) (mh^8)+2369 (P^6) (mh^6)-1776 (P^8) (mh^4)-3840 (P^10) (mh^2)-928 (P^12))) (mc^6)+2 (57 (mf^18)-16 (27 (mh^2)+34 (P^2)) (mf^16)+4 (357 (mh^4)+965 (P^2) (mh^2)+470 (P^4)) (mf^14)-(2688 (mh^6)+11564 (P^2) (mh^4)+12919 (P^4) (mh^2)+3374 (P^6)) (mf^12)+(3150 (mh^8)+18908 (P^2) (mh^6)+34947 (P^4) (mh^4)+22758 (P^6) (mh^2)+4600 (P^8)) (mf^10)-2 (1176 (mh^10)+9050 (P^2) (mh^8)+23615 (P^4) (mh^6)+25956 (P^6) (mh^4)+14310 (P^8) (mh^2)+3344 (P^10)) (mf^8)+2 (546 (mh^12)+4982 (P^2) (mh^10)+16087 (P^4) (mh^8)+23546 (P^6) (mh^6)+20936 (P^8) (mh^4)+16208 (P^10) (mh^2)+3888 (P^12)) (mf^6)-(288 (mh^14)+2740 (P^2) (mh^12)+8307 (P^4) (mh^10)+7638 (P^6) (mh^8)+3304 (P^8) (mh^6)+18336 (P^10) (mh^4)+24576 (P^12) (mh^2)+4864 (P^14)) (mf^4)+(33 (mh^16)+164 (P^2) (mh^14)-1369 (P^4) (mh^12)-11898 (P^6) (mh^10)-29176 (P^8) (mh^8)-21184 (P^10) (mh^6)+6048 (P^12) (mh^4)+6528 (P^14) (mh^2)+1024 (P^16)) (mf^2)+4 (mh^2) (P^2) (13 (mh^14)+206 (P^2) (mh^12)+1243 (P^4) (mh^10)+3529 (P^6) (mh^8)+4328 (P^8) (mh^6)+784 (P^10) (mh^4)-1600 (P^12) (mh^2)-64 (P^14))) (mc^4)+(-12 (mf^20)+2 (51 (mh^2)+46 (P^2)) (mf^18)+(-384 (mh^4)-760 (P^2) (mh^2)+63 (P^4)) (mf^16)+(840 (mh^6)+2688 (P^2) (mh^4)+398 (P^4) (mh^2)-2274 (P^6)) (mf^14)+(-1176 (mh^8)-5304 (P^2) (mh^6)-2911 (P^4) (mh^4)+9160 (P^6) (mh^2)+8396 (P^8)) (mf^12)+2 (546 (mh^10)+3172 (P^2) (mh^8)+2958 (P^4) (mh^6)-8499 (P^6) (mh^4)-14444 (P^8) (mh^2)-7272 (P^10)) (mf^10)+(-672 (mh^12)-4632 (P^2) (mh^10)-4327 (P^4) (mh^8)+25968 (P^6) (mh^6)+54836 (P^8) (mh^4)+32352 (P^10) (mh^2)+12960 (P^12)) (mf^8)+2 (132 (mh^14)+960 (P^2) (mh^12)-545 (P^4) (mh^10)-18463 (P^6) (mh^8)-45208 (P^8) (mh^6)-32672 (P^10) (mh^4)-4128 (P^12) (mh^2)-2432 (P^14)) (mf^6)+(-60 (mh^16)-328 (P^2) (mh^14)+3615 (P^4) (mh^12)+35304 (P^6) (mh^10)+106532 (P^8) (mh^8)+124800 (P^10) (mh^6)+38976 (P^12) (mh^4)-5504 (P^14) (mh^2)) (mf^4)+2 (mh^4) (3 (mh^14)-18 (P^2) (mh^12)-1028 (P^4) (mh^10)-8877 (P^6) (mh^8)-33036 (P^8) (mh^6)-57944 (P^10) (mh^4)-41376 (P^12) (mh^2)-4864 (P^14)) (mf^2)+4 (mh^4) (P^2) (4 (mh^14)+98 (P^2) (mh^12)+880 (P^4) (mh^10)+3935 (P^6) (mh^8)+9320 (P^8) (mh^6)+10792 (P^10) (mh^4)+3936 (P^12) (mh^2)-1280 (P^14))) (mc^2)+4 (P^2) ((((mf^2)-(mh^2)-2 (P^2)))^3) (2 (mf^14)-(11 (mh^2)+23 (P^2)) (mf^12)+(25 (mh^4)+96 (P^2) (mh^2)+72 (P^4)) (mf^10)-2 (15 (mh^6)+85 (P^2) (mh^4)+114 (P^4) (mh^2)+54 (P^6)) (mf^8)+4 (5 (mh^8)+42 (P^2) (mh^6)+93 (P^4) (mh^4)+32 (P^6) (mh^2)+16 (P^8)) (mf^6)-(mh^2) (7 (mh^8)+99 (P^2) (mh^6)+396 (P^4) (mh^4)+344 (P^6) (mh^2)-48 (P^8)) (mf^4)+(mh^4) ((mh^8)+32 (P^2) (mh^6)+228 (P^4) (mh^4)+528 (P^6) (mh^2)+160 (P^8)) (mf^2)-4 (mh^6) (P^2) ((mh^6)+12 (P^2) (mh^4)+47 (P^4) (mh^2)+68 (P^6)))) π (λ^4))/(3 ((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^6) ((((mc^4)-2 ((mf^2)-(mh^2)) (mc^2)+(mf^4)+(mh^4)-2 (mf^2) (mh^2)+4 (mh^2) (P^2)))^3))

Int3totfermionNRe = Int3totfermionNRa (((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^6) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))+4 (mh^2) (P^2)))^3)) / (((((mc^2)-(mf^2)+(mh^2)))^6) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))))^3)) (1 - 12 (P^2) /((mc^2)-(mf^2)+(mh^2) )) (1 - 12 (mh^2)(P^2)/((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2)))) // Simplify

Int3totfermionNRe1 = Int3totfermionNRe /. {P->0} // Simplify (* this is the (P^0) term *)

Int3totfermionNRe2 = Int3totfermionNRe  - Int3totfermionNRe1 // Simplify

Int3totfermionNRe3 = Int3totfermionNRe2 /(P^2)// Expand // Simplify

Int3totfermionNRe4 = Int3totfermionNRe3/. {P->0} // Simplify (* this is the (P^2) term, ot independent *)

Int3totfermionNR1 =Int3totfermionNR/. {P->0} // Simplify

Total3 = ( (π^2)(T^2)/6)(Int3totgammaNR1 + Int3totfermionNR1 /2 )

(1/6) (π^2) (T^2) (((64 π (e^2) (λ^4) (mc^2) (mf^2) (3 (mc^2)-2 (mf^2)+(mh^2)))/((((mc^2)-(mf^2)+(mh^2)))^4))-((256 π (e^2) (λ^4) (mc^2) (mf^2) ((mf^2)-(mh^2)))/((((mc^2)-(mf^2)+(mh^2)))^4))) // Simplify

Total3a =( (π^2)(T^2)/6)(Int3totgammaNRe1 + Int3totfermionNRe1 /2 )  // Simplify (* this is the a term of a + b (P^2) *)

Total3b =( (π^2)(T^2)/6)(Int3totgammaNRe4 + Int3totfermionNRe4 /2 )  // Simplify  (* this is the b term of a + b (P^2) *)