<<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]))))))

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])

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]))))))

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 // ExpandScalarProduct// Simplify

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

(4.3) MKoP + MKoN

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

MKo11t = MKo1Pt/. K -> ω //Simplify

MKo12t = MKo1Nt/. K -> ω //Simplify

GKo11t = MKo1Ft //Simplify

GKo12t = MKo1Gt //Simplify

MKo21t = MKo2Pt/. K -> ω //Simplify

MKo22t = MKo2Nt/. K -> ω //Simplify

GKo21t = MKo2Ft //Simplify

GKo22t = MKo2Gt//Simplify

gKo21t = MKo2At //Simplify

gKo22t = MKo2gt //Simplify

MKo31t = MKo3Pt/. K -> ω //Simplify

MKo32t = MKo3Nt/. K -> ω //Simplify

GKo31t = MKo3At //Simplify

GKo32t = MKo3Gt //Simplify

MKo41t = MKo4Pt/. K -> ω //ExpandScalarProduct //Simplify

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

MKo51t = MKo5Pt/. K -> ω //Simplify

MKo52t = MKo5Nt/. K -> ω //Simplify

MKo11u = MKo1Pu/. K -> ω //Simplify

MKo12u = MKo1Nu/. K -> ω //Simplify

GKo11u = MKo1Fu //Simplify

GKo12u = MKo1Gu //Simplify

MKo21u = MKo2Pu/. K -> ω //Simplify

MKo22u = MKo2Nu/. K -> ω //Simplify

GKo21u = MKo2Fu //Simplify

GKo22u = MKo2Gu//Simplify

gKo21u = MKo2Au //Simplify

gKo22u = MKo2gu //Simplify

MKo31u = MKo3Pu/. K -> ω //Simplify

MKo32u = MKo3Nu/. K -> ω //Simplify

GKo31u = MKo3Au //Simplify

GKo32u = MKo3Gu //Simplify

MKo41u = MKo4Pu/. K -> ω //ExpandScalarProduct //Simplify

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

MKo51u = MKo5Pu/. K -> ω //Simplify

MKo52u = MKo5Nu/. K -> ω //Simplify

MKo11tu = MKo1Ptu/. K -> ω //Simplify

MKo12tu = MKo1Ntu/. K -> ω //Simplify

GKo11tu = MKo1Ftu//Simplify

GKo12tu = MKo1Gtu //Simplify

MKo21tu = MKo2Ptu/. K -> ω //Simplify

MKo22tu = MKo2Ntu/. K -> ω //Simplify

GKo21tu = MKo2Ftu //Simplify

GKo22tu = MKo2Gtu //Simplify

gKo21tu = MKo2Atu //Simplify

gKo22tu = MKo2gtu //Simplify

MKo31tu = MKo3Ptu/. K -> ω //Simplify

MKo32tu = MKo3Ntu/. K -> ω //Simplify

GKo31tu = MKo3Atu //Simplify

GKo32tu = MKo3Gtu //Simplify

MKo41tu = MKo4Ptu/. K -> ω //ExpandScalarProduct //Simplify

MKo42tu = MKo4Ntu/. K -> ω //ExpandScalarProduct //Simplify

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’

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

Int2tp1t1 = Int2tp1t/.{Log[H+Pp] ->0, Log[H-Pp] -> 0} // Simplify

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

Int2tp2t1 = Int2tp2t/.{Log[H+Pp] ->0, Log[H-Pp] -> 0} // Simplify

Int2tptt =Int2tp1t1 + Int2tp2t1 // Simplify

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

Int2tf1t1 = Int2tf1t/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)-H ot+Kt Pp]->0,Log[(mf^2)-H ot-Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2tf2t1 = Int2tf2t/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tftt1 = (Int2tf1t1 + Int2tf2t1) // Expand  // Simplify

Int2tftt = ((H^2)-2 ot H+(mf^2)+(Pp^2)) ((H^2)+2 ot H+(mf^2)+(Pp^2))(1+(ot^2)/ (H^2)+ (ot^4)/(H^4))  /(4 (H^4))(Int2tftt1) // Expand  // Simplify

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

Int2ta1t1 = Int2ta1t/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)-H ot+Kt Pp]->0,Log[(mf^2)-H ot-Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2ta2t1 = Int2ta2t/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tatt = ((H^2)-2 ot H+(mf^2)+(Pp^2)) ((H^2)+2 ot H+(mf^2)+(Pp^2))  (1+(ot^2)/ (H^2)+ (ot^4)/(H^4))/(4 (H^4))(Int2ta1t1 + Int2ta2t1) // Simplify

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

Int2tp1u1 = Int2tp1u/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2tp2u1 = Int2tp2u/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tpuu = Int2tp1u1 + Int2tp2u1 // Simplify

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

Int2tf1u1 = Int2tf1u/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)-H ot+Kt Pp]->0,Log[(mf^2)-H ot-Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2tf2u1 = Int2tf2u/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tfuu = ((H^2)-2 ot H+(mf^2)+(Pp^2)) ((H^2)+2 ot H+(mf^2)+(Pp^2))(1+(ot^2)/ (H^2)+ (ot^4)/(H^4)) /(4 (H^4)) (  Int2tf1u1 + Int2tf2u1) // Simplify

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

Int2ta1u1 = Int2ta1u/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)-H ot+Kt Pp]->0,Log[(mf^2)-H ot-Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2ta2u1 = Int2ta2u/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tauu = ((H^2)-2 ot H+(mf^2)+(Pp^2)) ((H^2)+2 ot H+(mf^2)+(Pp^2)) (1+(ot^2)/ (H^(2 ))+ (ot^4)/(H^4)) /(4 (H^4))(Int2ta1u1 + Int2ta2u1) // Simplify

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

Int2tp1tu1 = Int2tp1tu/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2tp2tu1 = Int2tp2tu/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0,ArcTanh[Pp/H] -> 0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tptuu = Int2tp1tu1 + Int2tp2tu1 // Simplify

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

Int2tf1tu1 = Int2tf1tu/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)-H ot+Kt Pp]->0,Log[(mf^2)-H ot-Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2tf2tu1 = Int2tf2tu/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tftuu = ((H^2)+2 ot H+(mf^2)+(Pp^2))((H^2)-2 ot H+(mf^2)+(Pp^2))(1+(ot^2)/ (H^2)+ (ot^4)/(H^4))/(4(H^4))( Int2tf1tu1 + Int2tf2tu1) // Simplify

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

Int2ta1tu1 = Int2ta1tu/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)-H ot+Kt Pp]->0,Log[(mf^2)-H ot-Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

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

Int2ta2tu1 = Int2ta2tu/.{Log[H+Pp] ->0, Log[H-Pp] -> 0,Log[(mf^2)+H ot-Kt Pp]->0,Log[(mf^2)+H ot+Kt Pp]->0, Kt-> Sqrt[(ot^2)- (mf^2)]} // Simplify

Int2tatuu = ((H^2)+2 ot H+(mf^2)+(Pp^2))((H^2)-2 ot H+(mf^2)+(Pp^2))(1+(ot^2)/ (H^2) + (ot^4)/(H^4))/(4(H^4))( Int2ta1tu1 + Int2ta2tu1) // Simplify

theta integration

Int2tt =   Integrate[{Int2tptt/. {lphi -> ((mcfp^2) - 2 SP[q,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2ftt =  Integrate[{Int2tftt/.{lphi -> ((mcfp^2) - 2 SP[q,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify (* don't drop terms with higher powers of omega *)

Int2att =  Integrate[{Int2tatt/. {lphi -> ((mcfp^2) - 2 SP[q,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2uu =  Integrate[{Int2tpuu/. {lphip -> ((mcfp^2) - 2 SP[qp,p]), St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2fuu =  Integrate[{Int2tfuu/.{lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2auu =  Integrate[{Int2tauu/.{lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2tu =  Integrate[{Int2tptuu/. {lphi -> ((mcfp^2) - 2 SP[q,p]), lphip -> ((mcfp^2) - 2 SP[qp,p]),St-> Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2ftu =  Integrate[{Int2tftuu/.{lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2ftus = Int2ftu /. {ArcTan[2 Sqrt[- (P^(2 ))(Pp^2)/((((mcfp^2)-2 (H^2)))^2)]] ->1/(Sqrt[-(((P^2) (Pp^2))/((((mcfp^2)-2 (H^2)))^2))] ) (-P Pp/(2((mcfp^2)-2 (H^2))) (Log[(mcfp^2)-2 (H^2) + 2 P Pp] -Log[(mcfp^2)-2 (H^2) - 2 P Pp]))} // Expand // Simplify

Int2atu =  Integrate[{Int2tatuu/.{lphi -> ((mcfp^2) - 2 SP[q,p]),lphip -> ((mcfp^2) - 2 SP[qp,p]),St->Sqrt[1-Ct Ct]}},{Ct, -1, 1}] // Simplify

Int2atus = Int2atu /. {ArcTan[2 Sqrt[- (P^(2 ))(Pp^2)/((((mcfp^2)-2 (H^2)))^2)]] ->1/(Sqrt[-(((P^2) (Pp^2))/((((mcfp^2)-2 (H^2)))^2))] ) (-P Pp/(2((mcfp^2)-2 (H^2))) (Log[(mcfp^2)-2 (H^2) + 2 P Pp] -Log[(mcfp^2)-2 (H^2) - 2 P Pp]))} // Expand // Simplify

Int2fatu = Int2ftu + Int2atu /.{ArcTan[2 Sqrt[- (P^(2 ))(Pp^2)/((((mcfp^2)-2 (H^2)))^2)]] ->1/(Sqrt[-(((P^2) (Pp^2))/((((mcfp^2)-2 (H^2)))^2))] ) (-P Pp/(2((mcfp^2)-2 (H^2))) (Log[(mcfp^2)-2 (H^2) + 2 P Pp] -Log[(mcfp^2)-2 (H^2) - 2 P Pp]))} // Simplify (* Tan^-1 (2 i A/B) = (i/2) Log[B+2A)/(B-2A)]*)

Int2fatus = Int2ftus + Int2atus // Simplify

Int2fatu - Int2fatus // Simplify

Int2totgamma = Int2tt + Int2uu - Int2tu /.{ArcTanh[((2 P Pp)/(2 (H^2)-(mcfp^2)))] -> (Log[2(H^2)-(mcfp^2)+2 P Pp] - Log[2(H^2)-(mcfp^2)-2 P Pp])/2}// Expand //Simplify

Int2totgammatplusu = Int2tt + Int2uu //Simplify

Int2totgammas = Int2totgamma /.{Pp-> Sqrt[(H^2)-(mf^2)], P-> Sqrt[(H^2)- (mc^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} // Simplify

Int2totgammaexpand = Int2totgamma /. {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

Int2totgammaNR = Int2totgammaexpand/.{H-> Sqrt[(P^2)+ (mc^2)]} // Simplify

Int2totgammaNRa =-((256 (e^2) (3 ((mf^2)+6 (P^2)) (mc^14)-2 (9 (mf^4)-9 ((mh^2)-3 (P^2)) (mf^2)-64 (P^4)-23 (mh^2) (P^2)) (mc^12)+(45 (mf^6)-90 (mh^2) (mf^4)+(45 (mh^4)-6 (P^2) (mh^2)-360 (P^4)) (mf^2)+264 (P^6)+228 (mh^2) (P^4)+22 (mh^4) (P^2)) (mc^10)-2 (30 (mf^8)-90 ((mh^2)+(P^2)) (mf^6)+3 (30 (mh^4)+73 (P^2) (mh^2)-28 (P^4)) (mf^4)+(-30 (mh^6)-136 (P^2) (mh^4)+56 (P^4) (mh^2)+256 (P^6)) (mf^2)-100 (P^8)-200 (mh^2) (P^6)-18 (mh^4) (P^4)+7 (mh^6) (P^2)) (mc^8)+(45 (mf^10)-90 (2 (mh^2)+3 (P^2)) (mf^8)+(270 (mh^4)+892 (P^2) (mh^2)+368 (P^4)) (mf^6)-4 (45 (mh^6)+240 (P^2) (mh^4)+251 (P^4) (mh^2)+(P^6)) (mf^4)+(45 (mh^8)+324 (P^2) (mh^6)+596 (P^4) (mh^4)-132 (P^6) (mh^2)-152 (P^8)) (mf^2)+2 (P^2) (7 (mh^8)+24 (P^2) (mh^6)+8 (P^4) (mh^4)+100 (P^6) (mh^2)+24 (P^8))) (mc^6)-2 (9 (mf^12)-9 (5 (mh^2)+9 (P^2)) (mf^10)+3 (30 (mh^4)+113 (P^2) (mh^2)+72 (P^4)) (mf^8)-2 (45 (mh^6)+257 (P^2) (mh^4)+351 (P^4) (mh^2)+111 (P^6)) (mf^6)+3 (15 (mh^8)+106 (P^2) (mh^6)+212 (P^4) (mh^4)+124 (P^6) (mh^2)+36 (P^8)) (mf^4)-(9 (mh^10)+45 (P^2) (mh^8)+22 (P^4) (mh^6)-58 (P^6) (mh^4)-56 (P^8) (mh^2)+24 (P^10)) (mf^2)-(mh^2) (P^2) (17 (mh^8)+128 (P^2) (mh^6)+312 (P^4) (mh^4)+148 (P^6) (mh^2)-112 (P^8))) (mc^4)+(3 (mf^14)-18 ((mh^2)+2 (P^2)) (mf^12)+3 (15 (mh^4)+62 (P^2) (mh^2)+40 (P^4)) (mf^10)-2 (30 (mh^6)+187 (P^2) (mh^4)+244 (P^4) (mh^2)+74 (P^6)) (mf^8)+(45 (mh^8)+356 (P^2) (mh^6)+612 (P^4) (mh^4)+284 (P^6) (mh^2)+88 (P^8)) (mf^6)-2 (9 (mh^10)+72 (P^2) (mh^8)+50 (P^4) (mh^6)-254 (P^6) (mh^4)-172 (P^8) (mh^2)+24 (P^10)) (mf^4)+(mh^2) (3 (mh^10)+2 (P^2) (mh^8)-284 (P^4) (mh^6)-1356 (P^6) (mh^4)-1704 (P^8) (mh^2)-224 (P^10)) (mf^2)+2 (mh^2) (P^2) (5 (mh^10)+70 (P^2) (mh^8)+340 (P^4) (mh^6)+684 (P^6) (mh^4)+424 (P^8) (mh^2)-96 (P^10))) (mc^2)-2 (P^2) (((-(mf^2)+(mh^2)+2 (P^2)))^2) (((mh^2)-4 (P^2)) (mf^8)+(-4 (mh^4)+10 (P^2) (mh^2)+6 (P^4)) (mf^6)+2 (3 (mh^6)-5 (P^2) (mh^4)-16 (P^4) (mh^2)) (mf^4)+(-4 (mh^8)+6 (P^2) (mh^6)+70 (P^4) (mh^4)+24 (P^6) (mh^2)) (mf^2)+(mh^4) ((mh^6)-2 (P^2) (mh^4)-28 (P^4) (mh^2)-64 (P^6)))) π (λ^4))/(3 ((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^5) ((((mc^4)-2 ((mf^2)-(mh^2)) (mc^2)+(mf^4)+(mh^4)-2 (mf^2) (mh^2)+4 (mh^2) (P^2)))^2)))

Int2totgammaNRe = Int2totgammaNRa (((((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

Int2totgammaNRe1 = Int2totgammaNRe /.{P->0} // Simplify (* (P^0) term *)

Int2totgammaNRe2 = Int2totgammaNRe - Int2totgammaNRe1 // Simplify

Int2totgammaNRe3 = Int2totgammaNRe2 /(P^2) // Expand // Simplify

Int2totgammaNRe4 = Int2totgammaNRe3/. {P->0} // Simplify (* this is the (P^2) term, again omega independent *)

Int2totgammaNR1 = Int2totgammaNR /. {P->0} // Simplify

Int2totf = Int2ftt + Int2fuu - Int2ftus // Simplify

Int2tota = Int2att + Int2auu - Int2atus  //Simplify

Int2totfermion  = Int2totf + Int2tota // Simplify

Int2totfermions =Int2totfermion /. {Pp-> Sqrt[(H^2)-(mf^2)], P-> Sqrt[(H^2)- (mc^2)], mcfp->Sqrt[(mc^2)+(mf^2)-(mh^2)]} // Simplify

Int2totfermionexpand = Int2totfermion /. {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 *)

Int2totfermionNR = Int2totfermionexpand/.{Pp-> Sqrt[(H^2)- (mf^2)], mcfp -> Sqrt[(mc^2)+ (mf^2)- (mh^2)]}/.{H-> Sqrt[(P^2)+ (mc^2)]} // Simplify (* log *)

Int2totfermionNRa = -((64 (e^2) ((mc^2)+(ot^2)+(P^2)) (6 (9 (mf^2)+26 (P^2)) (mc^18)+(-405 (mf^4)+4 (81 (mh^2)-71 (P^2)) (mf^2)+2 (P^2) (300 (mh^2)+49 (ot^2)+744 (P^2))) (mc^16)+(1323 (mf^6)-(2106 (mh^2)+1915 (P^2)) (mf^4)+(810 (mh^4)+572 (P^2) (mh^2)-4016 (P^4)-734 (ot^2) (P^2)) (mf^2)+2 (P^2) (498 (mh^4)+(218 (ot^2)+2526 (P^2)) (mh^2)+2568 (P^4)+475 (ot^2) (P^2))) (mc^14)+(-2457 (mf^8)+24 (243 (mh^2)+362 (P^2)) (mf^6)-3 (1485 (mh^4)+4582 (P^2) (mh^2)+1395 (P^4)-802 (ot^2) (P^2)) (mf^4)+(1080 (mh^6)+3616 (P^2) (mh^4)-2 (2015 (P^4)+1422 (ot^2) (P^2)) (mh^2)-9796 (P^6)-5781 (ot^2) (P^4)) (mf^2)+2 (P^2) (576 (mh^6)+(391 (ot^2)+3966 (P^2)) (mh^4)+(8934 (P^4)+1751 (ot^2) (P^2)) (mh^2)+4260 (P^6)+1793 (ot^2) (P^4))) (mc^12)+(2835 (mf^10)-5 (1782 (mh^2)+3133 (P^2)) (mf^8)+(10125 (mh^4)+38970 (P^2) (mh^2)+28722 (P^4)-4510 (ot^2) (P^2)) (mf^6)-(4860 (mh^6)+28469 (P^2) (mh^4)+(43711 (P^4)-8064 (ot^2) (P^2)) (mh^2)+18361 (P^6)-15156 (ot^2) (P^4)) (mf^4)+(810 (mh^8)+4168 (P^2) (mh^6)+(2342 (P^4)-4450 (ot^2) (P^2)) (mh^4)-(15280 (P^6)+17939 (ot^2) (P^4)) (mh^2)-3190 (P^8)-16421 (ot^2) (P^6)) (mf^2)+2 (P^2) (594 (mh^8)+8 (47 (ot^2)+600 (P^2)) (mh^6)+2 (6933 (P^4)+1273 (ot^2) (P^2)) (mh^4)+2 (8643 (P^6)+2791 (ot^2) (P^4)) (mh^2)+3618 (P^8)+3461 (ot^2) (P^6))) (mc^10)+(-2079 (mf^12)+2 (4050 (mh^2)+7627 (P^2)) (mf^10)-5 (2430 (mh^4)+10136 (P^2) (mh^2)+8887 (P^4)-1058 (ot^2) (P^2)) (mf^8)+(8640 (mh^6)+57346 (P^2) (mh^4)-5 (2584 (ot^2) (P^2)-22447 (P^4)) (mh^2)+66743 (P^6)-22315 (ot^2) (P^4)) (mf^6)-(2835 (mh^8)+23548 (P^2) (mh^6)+(69956 (P^4)-11024 (ot^2) (P^2)) (mh^4)+(85451 (P^6)-39607 (ot^2) (P^4)) (mh^2)+52994 (P^8)-31585 (ot^2) (P^6)) (mf^4)+(324 (mh^10)+740 (P^2) (mh^8)-4 (1517 (P^4)+968 (ot^2) (P^2)) (mh^6)-4 (5219 (P^6)+5554 (ot^2) (P^4)) (mh^4)-(18642 (P^8)+41551 (ot^2) (P^6)) (mh^2)+16704 (P^10)-21022 (ot^2) (P^8)) (mf^2)+2 (P^2) (444 (mh^10)+(239 (ot^2)+4680 (P^2)) (mh^8)+4 (4371 (P^4)+514 (ot^2) (P^2)) (mh^6)+(27822 (P^6)+6505 (ot^2) (P^4)) (mh^4)+69 (278 (P^8)+131 (ot^2) (P^6)) (mh^2)+6 (P^8) (611 (ot^2)+246 (P^2)))) (mc^8)+(945 (mf^14)-3 (1458 (mh^2)+2767 (P^2)) (mf^12)+6 (1350 (mh^4)+5720 (P^2) (mh^2)+5242 (P^4)-663 (ot^2) (P^2)) (mf^10)-2 (3780 (mh^6)+25937 (P^2) (mh^4)+(54025 (P^4)-6330 (ot^2) (P^2)) (mh^2)+33601 (P^6)-10035 (ot^2) (P^4)) (mf^8)+(3645 (mh^8)+33516 (P^2) (mh^6)-4 (3809 (ot^2) (P^2)-28125 (P^4)) (mh^4)+(161100 (P^6)-49078 (ot^2) (P^4)) (mh^2)+86796 (P^8)-33850 (ot^2) (P^6)) (mf^6)-3 (270 (mh^10)+2279 (P^2) (mh^8)+(8266 (P^4)-2888 (ot^2) (P^2)) (mh^6)+(21362 (P^6)-14320 (ot^2) (P^4)) (mh^4)-4 (5503 (ot^2) (P^6)-8390 (P^8)) (mh^2)+22388 (P^10)-8316 (ot^2) (P^8)) (mf^4)+2 (27 (mh^12)-586 (P^2) (mh^10)-5 (1642 (P^4)+237 (ot^2) (P^2)) (mh^8)-(30568 (P^6)+8289 (ot^2) (P^4)) (mh^6)-(35558 (P^8)+20061 (ot^2) (P^6)) (mh^4)-2 (P^8) (10539 (ot^2)+668 (P^2)) (mh^2)+216 (P^10) (57 (P^2)-26 (ot^2))) (mf^2)+2 (P^2) (174 (mh^12)+2 (65 (ot^2)+1353 (P^2)) (mh^10)+(14820 (P^4)+1277 (ot^2) (P^2)) (mh^8)+(35796 (P^6)+4666 (ot^2) (P^4)) (mh^6)+4 (8997 (P^8)+2090 (ot^2) (P^6)) (mh^4)+2 (P^8) (3857 (ot^2)+5706 (P^2)) (mh^2)+108 (P^10) (19 (ot^2)+2 (P^2)))) (mc^6)+(-243 (mf^16)+4 (324 (mh^2)+565 (P^2)) (mf^14)-(2835 (mh^4)+11202 (P^2) (mh^2)+9519 (P^4)-1874 (ot^2) (P^2)) (mf^12)+(3240 (mh^6)+21080 (P^2) (mh^4)+(42208 (P^4)-7596 (ot^2) (P^2)) (mh^2)+23842 (P^6)-11115 (ot^2) (P^4)) (mf^10)-(2025 (mh^8)+18220 (P^2) (mh^6)+(60164 (P^4)-12334 (ot^2) (P^2)) (mh^4)+(85814 (P^6)-36452 (ot^2) (P^4)) (mh^2)+38324 (P^8)-22460 (ot^2) (P^6)) (mf^8)+2 (324 (mh^10)+3126 (P^2) (mh^8)+(12399 (P^4)-5140 (ot^2) (P^2)) (mh^6)+(32355 (P^6)-23252 (ot^2) (P^4)) (mh^4)+(49530 (P^8)-30221 (ot^2) (P^6)) (mh^2)+21474 (P^10)-8966 (ot^2) (P^8)) (mf^6)+(-81 (mh^12)+334 (P^2) (mh^10)+(10847 (P^4)+4686 (ot^2) (P^2)) (mh^8)+(43054 (P^6)+29610 (ot^2) (P^4)) (mh^6)+(26660 (P^8)+58038 (ot^2) (P^6)) (mh^4)+(41136 (ot^2) (P^8)-61580 (P^10)) (mh^2)+144 (P^10) (31 (ot^2)-237 (P^2))) (mf^4)-(P^2) (552 (mh^12)+2 (566 (ot^2)+4775 (P^2)) (mh^10)+(55340 (P^4)+9797 (ot^2) (P^2)) (mh^8)+2 (66930 (P^6)+13973 (ot^2) (P^4)) (mh^6)+4 (27668 (P^8)+7775 (ot^2) (P^6)) (mh^4)+4 (P^8) (3427 (ot^2)-2778 (P^2)) (mh^2)+24 (P^10) (27 (ot^2)-572 (P^2))) (mf^2)+2 (P^2) (24 (mh^14)+(57 (ot^2)+690 (P^2)) (mh^12)+(5958 (P^4)+677 (ot^2) (P^2)) (mh^10)+(23160 (P^6)+2977 (ot^2) (P^4)) (mh^8)+12 (3557 (P^8)+501 (ot^2) (P^6)) (mh^6)+14 (P^8) (425 (ot^2)+2226 (P^2)) (mh^4)+12 (P^10) (259 (ot^2)+234 (P^2)) (mh^2)+480 (ot^2) (P^12))) (mc^4)+(27 (mf^18)-(162 (mh^2)+163 (P^2)) (mf^16)+(405 (mh^4)+986 (P^2) (mh^2)+130 (P^4)-506 (ot^2) (P^2)) (mf^14)+(-540 (mh^6)-2245 (P^2) (mh^4)+(2584 (ot^2) (P^2)-1835 (P^4)) (mh^2)+1579 (P^6)+3536 (ot^2) (P^4)) (mf^12)+(405 (mh^8)+2332 (P^2) (mh^6)+(3350 (P^4)-5498 (ot^2) (P^2)) (mh^4)-(676 (P^6)+15415 (ot^2) (P^4)) (mh^2)-6934 (P^8)-9121 (ot^2) (P^6)) (mf^10)+(-162 (mh^10)-893 (P^2) (mh^8)+2 (535 (P^4)+3148 (ot^2) (P^2)) (mh^6)+(7682 (P^6)+27532 (ot^2) (P^4)) (mh^4)+4 (1407 (P^8)+8290 (ot^2) (P^6)) (mh^2)+2 (P^8) (5249 (ot^2)+6300 (P^2))) (mf^8)+(27 (mh^12)-230 (P^2) (mh^10)-2 (3455 (P^4)+2087 (ot^2) (P^2)) (mh^8)-2 (16462 (P^6)+12967 (ot^2) (P^4)) (mh^6)-2 (21854 (P^8)+23875 (ot^2) (P^6)) (mh^4)-4 (P^8) (7029 (ot^2)+350 (P^2)) (mh^2)-48 (P^10) (95 (ot^2)+183 (P^2))) (mf^6)+ (P^2) (261 (mh^12)+(1592 (ot^2)+5701 (P^2)) (mh^10)+(37535 (P^4)+13648 (ot^2) (P^2)) (mh^8)+8 (12465 (P^6)+4597 (ot^2) (P^4)) (mh^6)+4 (22091 (P^8)+8050 (ot^2) (P^6)) (mh^4)+4 (P^8) (647 (ot^2)-2310 (P^2)) (mh^2)-24 (P^10) (27 (ot^2)+58 (P^2))) (mf^4)-(P^2) (48 (mh^14)+6 (53 (ot^2)+267 (P^2)) (mh^12)+(15296 (P^4)+3763 (ot^2) (P^2)) (mh^10)+(64190 (P^6)+15209 (ot^2) (P^4)) (mh^8)+4 (31156 (P^8)+5979 (ot^2) (P^6)) (mh^6)+(88192 (P^10)+9128 (ot^2) (P^8)) (mh^4)-48 (P^10) (117 (ot^2)+128 (P^2)) (mh^2)-960 (P^12) ((ot^2)+3 (P^2))) (mf^2)+2 (mh^2) (P^2) (12 ((ot^2)+4 (P^2)) (mh^12)+66 (13 (P^4)+3 (ot^2) (P^2)) (mh^10)+2 (2787 (P^6)+608 (ot^2) (P^4)) (mh^8)+(17610 (P^8)+3499 (ot^2) (P^6)) (mh^6)+(27372 (P^10)+4790 (ot^2) (P^8)) (mh^4)+4 (P^10) (677 (ot^2)+4158 (P^2)) (mh^2)+384 (ot^2) (P^12))) (mc^2)-3 (P^2) (((-(mf^2)+(mh^2)+2 (P^2)))^2) (10 (mf^14)-(40 (mh^2)+20 (ot^2)+81 (P^2)) (mf^12)+(60 (mh^4)+(88 (ot^2)+261 (P^2)) (mh^2)+283 (P^4)+87 (ot^2) (P^2)) (mf^10)-(40 (mh^6)+(152 (ot^2)+329 (P^2)) (mh^4)+9 (73 (P^4)+39 (ot^2) (P^2)) (mh^2)+586 (P^6)+159 (ot^2) (P^4)) (mf^8)+(10 (mh^8)+(128 (ot^2)+215 (P^2)) (mh^6)+(857 (P^4)+539 (ot^2) (P^2)) (mh^4)+(670 (P^6)+367 (ot^2) (P^4)) (mh^2)+632 (P^8)+154 (ot^2) (P^6)) (mf^6)-((52 (ot^2)+82 (P^2)) (mh^8)+(639 (P^4)+381 (ot^2) (P^2)) (mh^6)+(1542 (P^6)+547 (ot^2) (P^4)) (mh^4)+(180 (P^8)-130 (ot^2) (P^6)) (mh^2)+80 (P^8) ((ot^2)+3 (P^2))) (mf^4)+(mh^2) (8 ((ot^2)+2 (P^2)) (mh^8)+2 (118 (P^4)+57 (ot^2) (P^2)) (mh^6)+(1014 (P^6)+401 (ot^2) (P^4)) (mh^4)+6 (280 (P^8)+39 (ot^2) (P^6)) (mh^2)-288 (ot^2) (P^8)) (mf^2)-2 (mh^4) (P^2) (336 (P^8)+64 (ot^2) (P^6)+4 (mh^6) ((ot^2)+2 (P^2))+(mh^4) (82 (P^4)+31 (ot^2) (P^2))+(mh^2) (266 (P^6)+83 (ot^2) (P^4))))) π (λ^4))/(9 ((((mc^2)+(P^2)))^2) ((mc^2)-(mf^2)+(P^2)) ((((mc^2)-(mf^2)+(mh^2)+2 (P^2)))^5) ((((mc^4)-2 ((mf^2)-(mh^2)) (mc^2)+(mf^4)+(mh^4)-2 (mf^2) (mh^2)+4 (mh^2) (P^2)))^2)))

Int2totfermionNRe = Int2totfermionNRa(((((mc^2)+(P^2)))^2) ((mc^2)-(mf^2)+(P^2)) ((((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^4)((mc^2)-(mf^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 - 2 (P^2)/(mc^2)) (1 - (P^2)/((mc^2)- (mf^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

Int2totfermionNRe1 = Int2totfermionNRe /. {P->0} // Simplify (* this is the (P^0) term *)

Int2totfermionNRe2 = Int2totfermionNRe - Int2totfermionNRe1 // Expand // Simplify

Int2totfermionNRe3 = Int2totfermionNRe2 /(P^2) // Expand // Simplify

Int2totfermionNRe4 = Int2totfermionNRe3/. {P->0} // Simplify (* This is the (P^2) term *)

fo0 = Coefficient[Int2totfermionNRe1 + Int2totfermionNRe4  (P^2), ot, 0] // Simplify

fo01 =Coefficient[Int2totfermionNRe1, ot, 0] // Simplify

fo04 = Coefficient[Int2totfermionNRe4  (P^2), ot, 0] // Simplify

fo2 = Coefficient[Int2totfermionNRe1 + Int2totfermionNRe4 (P^2), ot, 2] // Simplify

fo21 = Coefficient[Int2totfermionNRe1 , ot, 2] // Simplify

fo24 = Coefficient[Int2totfermionNRe4 (P^2), ot, 2] // Simplify

Int2totfermionNR1 =Int2totfermionNR/. {P->0} // Simplify

Int2fT2 = -((192 π (e^2) (λ^4) (2 (mc^2) (mf^2)-(mf^4)) ((mc^2)))/((mc^2) ((((mc^2)-(mf^2)+(mh^2)))^3)))(* Coefficient[Int2totfermionNR1, ot, 0] *)

Int2fT4 = -((192 π (e^2) (λ^4) (2 (mc^2) (mf^2)-(mf^4)) )/((mc^2) ((((mc^2)-(mf^2)+(mh^2)))^3))) (* Coeff of (ot^2) *)

(* Int2tots = Int2totgammas + Int2totfermions/2 // Simplify May be higher powers of omega *)

nB = 1/(Exp[ω/T] -1)

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

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

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

I2f0 = Integrate[ot nF, {ot, 0, Infinity}]

I2f= Integrate[ot nF, {ot, mf, Infinity}]

I4f= Integrate[(ot^3) nF, {ot, mf, Infinity}]

I4f0= Integrate[(ot^3) nF, {ot, 0, Infinity}]

Total2 = Int2totgammaNR1 I2 + Int2fT2 I2f0 + Int2fT4 I4f0 // Expand // Simplify

Total2new =(Int2totgammaNRe1 + Int2totgammaNRe4 (P^2)) I2 +(fo01 + fo04) I2f0  + (fo21 + fo24)I4f0 // Expand // Simplify

Total2a =(Int2totgammaNRe1 ) I2 +fo01  I2f0  + fo21 I4f0 // Expand // Simplify (* a term in a + b (P^2) *)

Total2b = ((Int2totgammaNRe4 (P^2)) I2 +(fo04) I2f0  + (fo24)I4f0)/(P^2) // Expand// Simplify (* b term in a + b (P^2) *)