<<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’

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

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

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

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

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

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

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

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

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

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

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

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

theta integration

Int1tp1t1 = Int1tp1t/.{Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int1t1t =   Integrate[{Int1tp1t1/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int1tp2t1 = Int1tp2t/.{Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int1t2t =   Integrate[{Int1tp2t1/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int1tt = Int1t1t + Int1t2t // Simplify

Int1tf1t1 = Int1tf1t/. {Log[(mf^2)- H ot + Kt Pp]-> 0, Log[(mf^2)+ H ot + Kt Pp] -> 0, Log[(mf^2)- H ot - Kt Pp] -> 0, Log[(mf^2)+ H ot - Kt Pp] -> 0} // Simplify

Int1f1t =  Integrate[{Int1tf1t1/. {lphi -> ((mcfp^2) - 2 SP[q,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int1f2t =  Int1f1t/.{ot -> -ot} // Simplify

Int1ftt = Int1f1t + Int1f2t // Simplify

Int1tp1u1 = Int1tp1u/. {Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int1t1u=  Integrate[{Int1tp1u1/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int1tp2u1 = Int1tp2u/. {Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int1t2u = Integrate[{Int1tp2u1/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int1uu = (Pp-H) (H+Pp)/(-(mf^2))(Int1t1u + Int1t2u) // Simplify

Int1tf1u1  = Int1tf1u/. {Log[(mf^2)- H ot + Kt Pp]-> 0, Log[(mf^2)+ H ot + Kt Pp] -> 0, Log[(mf^2)- H ot - Kt Pp] -> 0, Log[(mf^2)+ H ot - Kt Pp] -> 0} // Simplify

Int1f1u =  Integrate[Int1tf1u1/. {lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]},{Ct, -1, 1}] // Simplify

Int1f2u =  Int1f1u/.{ot -> -ot} // Simplify

Int1fuu = Int1f1u + Int1f2u // Simplify

Int1tp1tu1 =  Int1tp1tu/. {Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int1t1tu =  Integrate[{Int1tp1tu1/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int1tp2tu1 =  Int1tp2tu/. {Log[H+Pp] -> 0, Log[H-Pp]->0 } // Simplify

Int1t2tu =  Integrate[{Int1tp2tu1/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int1tu =  (H-Pp)(H+Pp)/(mf^2)(Int1t1tu + Int1t2tu )// Expand // Simplify

Int1tf1tu1 =  Int1tf1tu/.  {Log[(mf^2)- H ot + Kt Pp]-> 0, Log[(mf^2)+ H ot + Kt Pp] -> 0, Log[(mf^2)- H ot - Kt Pp] -> 0, Log[(mf^2)+ H ot - Kt Pp] -> 0} // Simplify

Int1f1tu =  Integrate[Int1tf1tu1/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]},{Ct, -1, 1}] // Simplify

Int1f2tu =  Int1f1tu/.{ot -> -ot} // Simplify

Int1ftu = Int1f1tu + Int1f2tu // Simplify

Inttotgamma = Int1tt + Int1uu - Int1tu // Simplify

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

Inttotgammasexpand = Inttotgamma /. {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

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

IntgammaNRa= -((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)))

IntgammaNRe = IntgammaNRa  (((((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

IntgammaNRe1 = IntgammaNRe /. {P->0} // Simplify (* this is the (P^0) term *)

IntgammaNRe2 = IntgammaNRe - IntgammaNRe1 // Simplify (* removing p->0 terms *)

IntgammaNRe3 = IntgammaNRe2 /(P^2) // Expand // Simplify

IntgammaNRe4 = IntgammaNRe3 /. {P->0} // Simplify (* this is the (P^(2 ))term *)

InttotgammaNR1 = InttotgammaNR /. {P->0} // Simplify (* this is the (P^(0 ))term *)

Inttotfermion = Int1ftt + Int1fuu - Int1ftu // Simplify

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

Inttotfermionexpand = Inttotfermion /. {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,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

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

InttotfermionNRa = (64 (e^2) ((30 (mf^2)+64 (P^2)) (mc^20)+(-270 (mf^4)+2 (123 (mh^2)-65 (P^2)) (mf^2)+8 (P^2) (34 (mh^2)+71 (P^2))) (mc^18)+4 (270 (mf^6)-(492 (mh^2)+317 (P^2)) (mf^4)+(222 (mh^4)+247 (P^2) (mh^2)-448 (P^4)) (mf^2)+8 (P^2) (10 (mh^4)+63 (P^2) (mh^2)+56 (P^4))) (mc^16)+(-2520 (mf^8)+8 (861 (mh^2)+890 (P^2)) (mf^6)-(6216 (mh^4)+14536 (P^2) (mh^2)+3371 (P^4)) (mf^4)+4 (462 (mh^6)+1862 (P^2) (mh^4)+655 (P^4) (mh^2)-811 (P^6)) (mf^2)+8 (P^2) (-24 (mh^6)+175 (P^2) (mh^4)+664 (P^4) (mh^2)+333 (P^6))) (mc^14)+2 (1890 (mf^10)-28 (246 (mh^2)+301 (P^2)) (mf^8)+(9324 (mh^4)+25536 (P^2) (mh^2)+12871 (P^4)) (mf^6)-(5544 (mh^6)+25660 (P^2) (mh^4)+28901 (P^4) (mh^2)+8970 (P^6)) (mf^4)+2 (609 (mh^8)+4452 (P^2) (mh^6)+8519 (P^4) (mh^4)+3990 (P^6) (mh^2)+879 (P^8)) (mf^2)+8 (P^2) (-44 (mh^8)-168 (P^2) (mh^6)+15 (P^4) (mh^4)+305 (P^6) (mh^2)+130 (P^8))) (mc^12)+(-3780 (mf^12)+28 (615 (mh^2)+817 (P^2)) (mf^10)-(31080 (mh^4)+90776 (P^2) (mh^2)+54877 (P^4)) (mf^8)+2 (13860 (mh^6)+66544 (P^2) (mh^4)+88509 (P^4) (mh^2)+36836 (P^6)) (mf^6)-(12180 (mh^8)+85768 (P^2) (mh^6)+189477 (P^4) (mh^4)+156696 (P^6) (mh^2)+55628 (P^8)) (mf^4)+4 (525 (mh^10)+5249 (P^2) (mh^8)+17740 (P^4) (mh^6)+23805 (P^6) (mh^4)+14259 (P^8) (mh^2)+4116 (P^10)) (mf^2)-8 (P^2) (52 (mh^10)+489 (P^2) (mh^8)+1352 (P^4) (mh^6)+1097 (P^6) (mh^4)+428 (P^8) (mh^2)-120 (P^10))) (mc^10)+8 (315 (mf^14)-14 (123 (mh^2)+170 (P^2)) (mf^12)+(3885 (mh^4)+11641 (P^2) (mh^2)+7514 (P^4)) (mf^10)-5 (924 (mh^6)+4468 (P^2) (mh^4)+6272 (P^4) (mh^2)+2779 (P^6)) (mf^8)+(3045 (mh^8)+20950 (P^2) (mh^6)+47830 (P^4) (mh^4)+43723 (P^6) (mh^2)+15907 (P^8)) (mf^6)-(1050 (mh^10)+9520 (P^2) (mh^8)+31430 (P^4) (mh^6)+46389 (P^6) (mh^4)+33180 (P^8) (mh^2)+10044 (P^10)) (mf^4)+(147 (mh^12)+1625 (P^2) (mh^10)+7384 (P^4) (mh^8)+16787 (P^6) (mh^6)+19145 (P^8) (mh^4)+12212 (P^10) (mh^2)+2532 (P^12)) (mf^2)+2 (P^2) (12 (mh^12)+31 (P^2) (mh^10)-257 (P^4) (mh^8)-1140 (P^6) (mh^6)-1312 (P^8) (mh^4)-648 (P^10) (mh^2)+16 (P^12))) (mc^8)+(-1080 (mf^16)+8 (861 (mh^2)+1196 (P^2)) (mf^14)-(18648 (mh^4)+55608 (P^2) (mh^2)+35893 (P^4)) (mf^12)+4 (6930 (mh^6)+32950 (P^2) (mh^4)+46196 (P^4) (mh^2)+19977 (P^6)) (mf^10)-2 (12180 (mh^8)+81080 (P^2) (mh^6)+183155 (P^4) (mh^4)+168648 (P^6) (mh^2)+58152 (P^8)) (mf^8)+4 (3150 (mh^10)+26980 (P^2) (mh^8)+85975 (P^4) (mh^6)+127836 (P^6) (mh^4)+93762 (P^8) (mh^2)+26896 (P^10)) (mf^6)-(3528 (mh^12)+35608 (P^2) (mh^10)+145013 (P^4) (mh^8)+310272 (P^6) (mh^6)+371288 (P^8) (mh^4)+249568 (P^10) (mh^2)+55968 (P^12)) (mf^4)+4 (102 (mh^14)+942 (P^2) (mh^12)+3751 (P^4) (mh^10)+10767 (P^6) (mh^8)+25006 (P^8) (mh^6)+32408 (P^10) (mh^4)+20136 (P^12) (mh^2)+3008 (P^14)) (mf^2)+8 (mh^2) (P^2) (40 (mh^12)+441 (P^2) (mh^10)+1656 (P^4) (mh^8)+1963 (P^6) (mh^6)-1304 (P^8) (mh^4)-3072 (P^10) (mh^2)-832 (P^12))) (mc^6)+2 (135 (mf^18)-4 (246 (mh^2)+325 (P^2)) (mf^16)+(3108 (mh^4)+8768 (P^2) (mh^2)+5083 (P^4)) (mf^14)-(5544 (mh^6)+24764 (P^2) (mh^4)+31845 (P^4) (mh^2)+11218 (P^6)) (mf^12)+2 (3045 (mh^8)+18868 (P^2) (mh^6)+39330 (P^4) (mh^4)+32298 (P^6) (mh^2)+8243 (P^8)) (mf^10)-2 (2100 (mh^10)+16550 (P^2) (mh^8)+48085 (P^4) (mh^6)+65790 (P^6) (mh^4)+44016 (P^8) (mh^2)+9120 (P^10)) (mf^8)+(1764 (mh^12)+16240 (P^2) (mh^10)+57651 (P^4) (mh^8)+108872 (P^6) (mh^6)+128036 (P^8) (mh^4)+87488 (P^10) (mh^2)+15312 (P^12)) (mf^6)-(408 (mh^14)+3748 (P^2) (mh^12)+11601 (P^4) (mh^10)+18282 (P^6) (mh^8)+38152 (P^8) (mh^6)+71760 (P^10) (mh^4)+54048 (P^12) (mh^2)+7936 (P^14)) (mf^4)+(39 (mh^16)+104 (P^2) (mh^14)-2882 (P^4) (mh^12)-19404 (P^6) (mh^10)-40714 (P^8) (mh^8)-17504 (P^10) (mh^6)+23472 (P^12) (mh^4)+13248 (P^14) (mh^2)+1536 (P^16)) (mf^2)+8 (mh^2) (P^2) (8 (mh^14)+138 (P^2) (mh^12)+877 (P^4) (mh^10)+2573 (P^6) (mh^8)+3234 (P^8) (mh^6)+640 (P^10) (mh^4)-1168 (P^12) (mh^2)-64 (P^14))) (mc^4)+(-30 (mf^20)+246 ((mh^2)+(P^2)) (mf^18)-(888 (mh^4)+1912 (P^2) (mh^2)+347 (P^4)) (mf^16)+2 (924 (mh^6)+3160 (P^2) (mh^4)+1901 (P^4) (mh^2)-1280 (P^6)) (mf^14)+(-2436 (mh^8)-11528 (P^2) (mh^6)-13141 (P^4) (mh^4)+4904 (P^6) (mh^2)+12340 (P^8)) (mf^12)+4 (525 (mh^10)+3131 (P^2) (mh^8)+5017 (P^4) (mh^6)-201 (P^6) (mh^4)-6429 (P^8) (mh^2)-5796 (P^10)) (mf^10)+(-1176 (mh^12)-8056 (P^2) (mh^10)-12293 (P^4) (mh^8)+12224 (P^6) (mh^6)+32960 (P^8) (mh^4)+25088 (P^10) (mh^2)+20832 (P^12)) (mf^8)+(408 (mh^14)+2752 (P^2) (mh^12)-2326 (P^4) (mh^10)-47032 (P^6) (mh^8)-91112 (P^8) (mh^6)-38144 (P^10) (mh^4)+4128 (P^12) (mh^2)-7424 (P^14)) (mf^6)+(-78 (mh^16)-280 (P^2) (mh^14)+7045 (P^4) (mh^12)+55992 (P^6) (mh^10)+148964 (P^8) (mh^8)+143584 (P^10) (mh^6)+17568 (P^12) (mh^4)-12672 (P^14) (mh^2)) (mf^4)+2 (mh^2) (3 (mh^16)-41 (P^2) (mh^14)-1636 (P^4) (mh^12)-13634 (P^6) (mh^10)-49154 (P^8) (mh^8)-81880 (P^10) (mh^6)-51440 (P^12) (mh^4)-64 (P^14) (mh^2)+512 (P^16)) (mf^2)+8 (mh^4) (P^2) (2 (mh^14)+58 (P^2) (mh^12)+568 (P^4) (mh^10)+2673 (P^6) (mh^8)+6532 (P^8) (mh^6)+7720 (P^10) (mh^4)+2880 (P^12) (mh^2)-896 (P^14))) (mc^2)+4 (P^2) ((((mf^2)-(mh^2)-2 (P^2)))^3) (5 (mf^14)-(26 (mh^2)+47 (P^2)) (mf^12)+(55 (mh^4)+176 (P^2) (mh^2)+150 (P^4)) (mf^10)-2 (30 (mh^6)+143 (P^2) (mh^4)+174 (P^4) (mh^2)+100 (P^6)) (mf^8)+(35 (mh^8)+276 (P^2) (mh^6)+492 (P^4) (mh^4)+192 (P^6) (mh^2)+96 (P^8)) (mf^6)- (mh^2) (10 (mh^8)+167 (P^2) (mh^6)+600 (P^4) (mh^4)+344 (P^6) (mh^2)-32 (P^8)) (mf^4)+(mh^4) ((mh^8)+52 (P^2) (mh^6)+366 (P^4) (mh^4)+784 (P^6) (mh^2)+128 (P^8)) (mf^2)-4 (mh^6) (P^2) ((mh^6)+15 (P^2) (mh^4)+64 (P^4) (mh^2)+96 (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))

InttotfermionNRe = InttotfermionNRa  (((((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

InttotfermionNRe1 = InttotfermionNRe /.{P->0} // Simplify (* this is the (P^0) term *)

InttotfermionNRe2 = InttotfermionNRe -InttotfermionNRe1 // Simplify

InttotfermionNRe3 = InttotfermionNRe2/(P^2) // Expand // Simplify

InttotfermionNRe4 = InttotfermionNRe3 /. {P->0} // Simplify (* this is the (P^2) term *)

InttotfermionNR1 = InttotfermionNR /.{P->0} // Simplify

Int1tots = Inttotgammas + Inttotfermions/2 // Simplify

Int1totsNR = InttotgammaNR + InttotfermionNR/2 // Simplify

InttotgammaNR1 = -((256 π (e^2) (λ^4) (mc^2) (mf^2) ((mf^2)-(mh^2)))/((((mc^2)-(mf^2)+(mh^2)))^4))

InttotfermionNR1  = ((128 π (e^2) (λ^4) (mc^2) (mf^2) (5 (mc^2)-5 (mf^2)+(mh^2)))/((((mc^2)-(mf^2)+(mh^2)))^4))

Total1 = ( (π^2)(T^2)/6)(InttotgammaNR1 + InttotfermionNR1/2 ) // Simplify (* No omega^2 term *)  (* Old a term alone *)

Totala = ( (π^2)(T^2)/6)(IntgammaNRe1 + InttotfermionNRe1/2) // Simplify (* a term in a + b (P^2) *)

Totalb = ( (π^2)(T^2)/6)(IntgammaNRe4 + InttotfermionNRe4/2) // Expand // Simplify(* b term in a + b (P^2) *)