<<FeynCalc`

Definindo os invariantes

ScalarProduct[k,k]=0;ScalarProduct[p,p]=m^2;ScalarProduct[p′,p′]=m^2;

Calculando os Traços

A barra em cima dos momentos e matrizes Gamma é porque estamos em 4 dimensões (esta é uma convenção do Feyncalc)

(I)

(GS[p]+m).(GA[ν].GS[k].GA[μ]+2 GA[μ] FV[p,ν]).(GS[p′]+m).(GA[μ].GS[k].GA[ν]+2 FV[p,ν] GA[μ])
TrI=%//TR

Em termos de variáveis de Maldenstam:

SetMandelstam[s, t, u, p, k, -p′, -k′, m, 0, m, 0];

Testes

2 SP[p′,k′]+m^2//Simplify;
-2 SP[k,k′]//Simplify;
-2 SP[k,p′]+m^2//Simplify;

TrIM=TrickMandelstam[TrI,{s,t,u,2 m^2}]
PeskinTrI=16(2m^4+m^2(s-m^2)-1/2 (s-m^2)(u-m^2));
%%-%//Expand

(IV)(desnecessário, mas é tão fácil que podemos fazer um “reality check”)

(GS[p]+m).(GA[ν].GS[k′].GA[μ]-2 GA[μ] FV[p,ν]).(GS[p′]+m).(GA[μ].GS[k′].GA[ν]-2 FV[p,ν] GA[μ])
TrIV=%//TR;
TrIVM=TrickMandelstam[TrIV,{s,t,u,2 m^2}]
PeskinTrIV=16(2m^4+m^2(u-m^2)-1/2 (s-m^2)(u-m^2));
TrickMandelstam[%%-%,{s,t,u,2 m^2}]//Expand

(II)

(GS[p]+m).(GA[ν].GS[k].GA[μ]+2 GA[μ] FV[p,ν]).(GS[p′]+m).(GA[ν].GS[k′].GA[μ]-2 FV[p,μ] GA[ν])
TrII=%//TR;
TrIIM=TrickMandelstam[TrII,{s,t,u,2 m^2}]
PeskinTrII=-8(4m^4+m^2(s-m^2)+m^2(u-m^2));
TrickMandelstam[%%-%,{s,t,u,2 m^2}]//Expand

(III)(desnecessário, mas é tão fácil que podemos fazer um “reality check”)

(GS[p]+m).(GA[μ].GS[k′].GA[ν]-2 GA[ν] FV[p,μ]).(GS[p′]+m).(GA[μ].GS[k].GA[ν]+2 FV[p,ν] GA[μ])
TrIII=%//TR;
TrIIIM=TrickMandelstam[TrIII,{s,t,u,2 m^2}]
PeskinTrII=-8(4m^4+m^2(s-m^2)+m^2(u-m^2));
TrickMandelstam[%%-%,{s,t,u,2 m^2}]//Expand

Elemento de Matriz

ℳ2=e^4/4(
1/(2 SP[p,k])^2 TrI+
1/(2 SP[p,k])/(2SP[p,k′]) TrII+
1/(2 SP[p,k])/(2SP[p,k′]) TrIII+
1/(2SP[p,k′])^2 TrIV
);
ℳ2=TrickMandelstam[%,{s,t,u,2 m^2}]//Simplify

Lab Frame

subMandLab={s->2 m ω+m^2,u->-2 m ω′+m^2}

subω′={ω′->(ω/(1+ω/m (1-Cos[θ])))}

ℳ2Ref=ℳ2/.subMandLab//Expand//Simplify

Comparando com a forma obtida no Peskin:

2 e^4 (pk′/pk+pk/pk′+2m^2(1/pk-1/pk′)+m^4(1/pk-1/pk′)^2)
ℳ2RefPeskin=%/.{pk->m ω,pk′->m ω′}//Expand//Simplify

Álgebra subsequente

Componente 0 da delta de Dirac

g[ω′_]=ω′+(Sqrt[m^2+ω^2+ω′^2-2 ω ω′ Cos[θ]])-ω-m;
g[ω′]==0
Solve[%,ω′]//Simplify
D[g[ω′],ω′]

ℳ2Ref/2/e^4 ω^2 ω′^2//Expand
%/ω′^2//Expand
%/ω^2//Expand

ω′==(ω/(1+ω/m (1-cθ)))
Solve[%,cθ]
cθ^2/.%[[1]]//Expand

CM Frame

subMandCM={s->(En+ω)^2,u->(En-ω)^2-2ω^2 (1+Cos[θ])}

ℳ2

No limite sem massa:

ℳ2/.{m->0}
%//Expand;
Collect[%,-2 e^4]

O comportamento de “s” em altas energias independe do ângulo

s/.subMandCM
%/.ω->(Sqrt[En^2-m^2])//Expand;
Assuming[En>0,Refine[%/.m->0]]

Já “u” vai para zero quando θ -> π

u/.subMandCM
%/.ω->(Sqrt[En^2-m^2])//Expand
Assuming[En>0,Refine[%/.m->0]];
Collect[%, -2 En^2]

Cálculos subsequentes

1/(2 m2/s + 1+cos)
Integrate[%,{cos,-1,1},Assumptions->m2>0&&s>0]

Estrutura de Spin (ignorando a massa)

ϵPlusk=1/(Sqrt[2]){0,1,I,0};
ϵMinusk=1/(Sqrt[2]){0,1,-I,0};
ϵPlusk′=1/(Sqrt[2]){0,1,-I,0};
ϵMinusk′=1/(Sqrt[2]){0,1,I,0};

PauliVector={IdentityMatrix[2],PauliMatrix[1],PauliMatrix[2],PauliMatrix[3]};

uR[p′]={1,0};
uR[p]={0,1};
uL[p′]={0,1};

uR[p′].(ϵMinusk.PauliVector)
uR[p′].(ϵPlusk.PauliVector)

(Conjugate[ϵMinusk′].PauliVector).uR[p]
(Conjugate[ϵPlusk′].PauliVector).uR[p]

Table[(uR[p′].(ϵPlusk.PauliVector)).PauliVector[[i]].((Conjugate[ϵPlusk′].PauliVector).uR[p]),{i,1,4}]

Estrutura de Spin (termo com a massa)

uL[p′].(ϵMinusk.PauliVector)
uL[p′].(ϵPlusk.PauliVector)