Thrust at NLO

Plot conditions

Plot[{y2/.Solve[y1==1-y1-y2,y2][[1]],y2/.Solve[y2==1-y1-y2,y2][[1]],y1,y2/.Solve[0==1-y1-y2,y2][[1]]},{y1,0,1},AspectRatio->1,Frame->True,PlotRange->{{0,1},{0,1}}]

Computation

To do the computation, replace conditions by theta-functions

prefact =((CF (((4 π))^(-1+ϵ)) αs (((((μ2)/Q2)))^ϵ))/(Gamma[1-ϵ]));

rest=(y1^(-1-ϵ)) (y2^(-1-ϵ)) (y3^(-ϵ)) (2 ((y1^2)+(y2^2)) (1-ϵ)+4 (y3-y1 y2 ϵ));

rest0=rest/.ϵ->0

Replace conditions by θ functions, perform δ-function integral:

y1smallest=rest0 UnitStep[y2-y1]UnitStep[y3-y1]/.y3->1-y1-y2/.y1->τ

y2smallest=rest0 UnitStep[y1-y2]UnitStep[y3-y2]/.y3->1-y1-y2/.y2->τ

y3smallest=rest0 UnitStep[y1-y3]UnitStep[y2-y3]/.y2->1-y1-y3/.y3->τ

y1smallest2=Integrate[y1smallest,{y2,0,1},Assumptions->{τ>0,τ<1/3}]

y2smallest2=Integrate[y2smallest,{y1,0,1},Assumptions->{τ>0,τ<1/3}]

y3smallest2=Integrate[y3smallest,{y1,0,1},Assumptions->{τ>0,τ<1/3}]

Now add up...

dSigmadτ=FullSimplify[y1smallest2+y2smallest2+y3smallest2,{τ>0,τ<1/3}]

Note that τ < 1/3 at NLO, otherwise the cross section is zero.

Series[dSigmadτ,{τ,0,-1}]