<<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’

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

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

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

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

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

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

theta integration

Int5t1t =   Integrate[{Int5tp1t/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int5t2t =   Integrate[{Int5tp2t/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int5tt = Int5t1t + Int5t2t // Simplify

Int5t1u =   Integrate[{Int5tp1u/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int5t2u =   Integrate[{Int5tp2u/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int5uu = Int5t1u + Int5t2u // Simplify

Int5t1tu =   Integrate[{Int5tp1tu/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int5t2tu =   Integrate[{Int5tp2tu/. {lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int5tu = Int5t1tu + Int5t2tu/.  {ArcTanh[((2 P Pp)/((mcfp^2)-2 (H^2)))] -> (Log[(mcfp^2)-2 (H^2)+2 P Pp] - Log[(mcfp^2)-2(H^2)-2 P Pp])/2}// Simplify

Int5totgamma = Int5tt + Int5uu - Int5tu // Simplify

Int5totgammas = Int5totgamma /.{Pp-> Sqrt[(H^2)-(mf^2)], P-> Sqrt[(H^2)- (mc^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} // Simplify

Int5totgammaexpand = Int5totgamma /. {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} // Simplify (* /.{ Pp-> Sqrt[(H^2)-(mf^2)], mcfp -> Sqrt[(mc^2)+ (mf^2)- (mh^2)]} // Simplify *)

Int5totgammaNR = Int5totgammaexpand/.{H-> Sqrt[(P^2)+ (mc^2)]} // Simplify

Int5totgammaNRa = (1/(3 P Pp (((2 ((mc^2)+(P^2))-(mcfp^2)))^5) (((((((mcfp^2)-2 ((mc^2)+(P^2))))^2)-4 (P^2) (Pp^2)))^2)))128 π (e^2) (λ^4) (4 (mc^2) P Pp ((mc^2)+(P^2)+(Pp^2)) (((((mcfp^2)-2 ((mc^2)+(P^2))))^2)-4 (P^2) (Pp^2)) (-4 (P^2) (Pp^2) ((((mcfp^2)-2 ((mc^2)+(P^2))))^2)+3 ((((mcfp^2)-2 ((mc^2)+(P^2))))^4)-8 (P^4) (Pp^4))-2 ((((mcfp^2)-2 ((mc^2)+(P^2))))^2) (((((((mcfp^2)-2 ((mc^2)+(P^2))))^2)-4 (P^2) (Pp^2)))^2) (3 P Pp ((((mcfp^2)-2 ((mc^2)+(P^2))))^2)+4 (P^3) (Pp^3))+6 (mcfp^2) P Pp ((((mcfp^2)-2 ((mc^2)+(P^2))))^5) (-6 (mcfp^2) ((mc^2)+(P^2))+8 ((((mc^2)+(P^2)))^2)+(mcfp^4)-8 (P^2) (Pp^2))) /. {Pp->Sqrt[(H^2) - (mf^2)], mcfp-> Sqrt[(mc^2)+(mf^2)-(mh^2)]} /.{H-> Sqrt[(P^2)+ (mc^2)]}// Expand // Simplify

Int5totgammaNRe =Int5totgammaNRa  (((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^5) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))+4 (mh^2) (P^2)))^2)) /  (((((mc^2)-(mf^2)+(mh^2)))^5) ((((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2))))^2)) (1- 10 (P^2) /((mc^2)-(mf^2)+(mh^2))) (1 - 8 (mh^2) (P^2)/((mc^4)+(mf^4)-2 (mf^2) (mh^2)+(mh^4)-2 (mc^2) ((mf^2)-(mh^2)))) // Simplify

Int5totgammaNRe1 =Int5totgammaNRe /. {P->0} // Simplify (* (P^0) term *)

Int5totgammaNRe2 =Int5totgammaNRe - Int5totgammaNRe1 // Simplify

Int5totgammaNRe3 =Int5totgammaNRe2 / (P^(2 )) // Expand // Simplify

Int5totgammaNRe4 =Int5totgammaNRe3 /.{P->0} // Simplify (* (P^2) terms *)

Int5totgammaNRe = Int5totgammaNR   (((2 ((mc^2)+(P^2))-(mcfp^2)))^5) /  ((((2 (mc^2)-(mcfp^2)))^5))(1 - 10 (P^2)/(((2 (mc^2)-(mcfp^2)))^5))

Int5totgammaNRe1 = Int5totgammaNRe  (((((((mcfp^2)-2 ((mc^2)+(P^2))))^2)-4 (P^2) (Pp^2)))^2) /((((2 (mc^2)-(mcfp^2)))^2)) (1 - 8 (P^2)(mh^2)) /.{Pp->Sqrt[(H^2)-(mf^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} /. {H->Sqrt[(P^2)+ (mc^2)]}// Simplify

Int5totgammaNR1 = Int5totgammaNRe1 /. {P->0} // Simplify

Int5tots = Int5totgammas  (* No fermion *)

Int5totNR = Int5totgammaNR

Total5 = (π^2 (T^2)/6)Int5totgammaNR1  (* This is wrong, needs ^3 in den *)

Total5 = (π^2 (T^2)/6) ( Int5totgammaNRe1  + Int5totgammaNRe4 (P^2) ) // Simplify

Total5a = (π^2 (T^2)/6) ( Int5totgammaNRe1   ) // Simplify (* a term in a + b (P^2) *)

Total5b = (π^2 (T^2)/6) (Int5totgammaNRe4  ) // Simplify  (* b term in a + b (P^2) *)