•EM counter-terms

OrellanaCTs0 = FullSimplify /@ (Collect[(Select[Expand[cts0], FreeQ[Denominator[#], MandelstamS | MandelstamT | MandelstamU] &] /. CouplingConstant[ChPTVirtualPhotons2[4], _ ? ((# < 11) &), ___] -> 0 // ChargedMassesEliminate // Expand) /. CouplingConstant[QED[_], ___]^(_ ? ((# > 2) &)) -> 0, {MandelstamU, ParticleMass[__]}])

(4 (e^( ))^2 (10 k _ 1^(r ) + 10 k _ 2^(r ) + 54 k _ 3^(r ) - 27 k _ 4^(r ) - 2 (5 k _ 5^(r ) + 77 k _ 6^(r ) + k _ 7^(r ) + 72 k _ 8^(r ))) (m _ π^0^(ó  r ))^2)/(27 (f _ π^(ó  ))^2)

OrellanaCTs1 = FullSimplify /@ (Collect[(cts1 /. CouplingConstant[ChPTVirtualPhotons2[4], _ ? ((# < 11) &), ___] -> 0 // ChargedMassesEliminate // Expand) /. CouplingConstant[QED[_], ___]^(_ ? ((# > 2) &)) -> 0, {MandelstamU, ParticleMass[__]}])

(2 u (e^( ))^2 (10 k _ 1^(r ) - 26 k _ 2^(r ) - 27 (2 k _ 3^(r ) + k _ 4^(r ))))/(9 (f _ π^(ó  ))^2) - (8 (e^( ))^2 (20 k _ 1^(r ) + 20 k _ 2^(r ) - 54 k _ 3^(r ) - 27 k _ 4^(r ) - 5 k _ 5^(r ) - 185 k _ 6^(r ) - k _ 7^(r ) - 180 k _ 8^(r )) (m _ π^0^(ó  r ))^2)/(27 (f _ π^(ó  ))^2)

test1 = FullSimplify /@ Collect[OrellanaCTs0 + OrellanaCTs1 // Expand, {MandelstamU, ParticleMass[__]}]

(2 u (e^( ))^2 (10 k _ 1^(r ) - 26 k _ 2^(r ) - 27 (2 k _ 3^(r ) + k _ 4^(r ))))/(9 (f _ π^(ó  ))^2) - (4 (e^( ))^2 (10 k _ 1^(r ) + 10 k _ 2^(r ) - 9 (6 k _ 3^(r ) + k _ 4^(r ) + 8 (k _ 6^(r ) + k _ 8^(r )))) (m _ π^0^(ó  r ))^2)/(9 (f _ π^(ó  ))^2)

zz = CouplingConstant[ChPTVirtualPhotons2[2], RenormalizationState[1]]/DecayConstant[PseudoScalar[2], RenormalizationState[0]]^4 ;

KnechtNehmesKappaPM = (3 + 4 zz/9) K1[1] - 40 zz/9 K2[1] - 9 K3[1] + 4 zz K4[1] + 4 (1 + 8 zz) K6[1] + 2 (1 - 8 zz) K8[1] /. scaleRuleinv /. _Log -> 0 // Simplify

-64/9 π^2 (10 k _ 1^(r ) + 10 k _ 2^(r ) - 9 (6 k _ 3^(r ) + k _ 4^(r ) + 8 (k _ 6^(r ) + k _ 8^(r ))))

KnechtNehmesKappaPP = -(3 + 4 zz/9) K1[1] - 248 zz/9 K2[1] + 9 K3[1] - 20 zz K4[1] + 4 (1 + 8 zz) K6[1] + 2 (1 - 8 zz) K8[1] /. scaleRuleinv /. _Log -> 0 // Simplify

64/9 π^2 (10 k _ 1^(r ) - 62 k _ 2^(r ) - 9 (6 k _ 3^(r ) + 5 k _ 4^(r ) - 8 k _ 6^(r ) - 8 k _ 8^(r )))

KnechtNehmesKappaCTs = (MandelstamS - ParticleMass[PionZero, RenormalizationState[1]]^2)/DecayConstant[PseudoScalar[2], RenormalizationState[0]]^2 CouplingConstant[QED[1], RenormalizationState[0]]^2/(32 π^2) (1/2 (KnechtNehmesKappaPM0 - KnechtNehmesKappaPP0)) + 1/(32 π^2 DecayConstant[PseudoScalar[2], RenormalizationState[0]]^2) CouplingConstant[QED[1], RenormalizationState[0]]^2 ParticleMass[PionZero, RenormalizationState[1]]^2 (1/2 (KnechtNehmesKappaPM0 + KnechtNehmesKappaPP0)) // Expand // Simplify

1/(64 π^2 (f _ π^(ó  ))^2) ((e^( ))^2 (2 KnechtNehmesKappaPP0 (m _ π^0^(ó  r ))^2 + (KnechtNehmesKappaPM0 - KnechtNehmesKappaPP0) s))

test20 = FullSimplify /@ ((KnechtNehmesKappaCTs + (KnechtNehmesKappaCTs /. {MandelstamS -> MandelstamT, MandelstamT -> MandelstamS}) // manred[MandelstamS] // Expand // Simplify // ChargedMassesEliminate // Expand) /. CouplingConstant[QED[_], ___]^(_ ? ((# > 2) &)) -> 0 // Collect[#, {MandelstamU, ParticleMass[__]}] &)

(KnechtNehmesKappaPM0 (m _ π^0^(ó  r ))^2 (e^( ))^2)/(16 π^2 (f _ π^(ó  ))^2) + ((KnechtNehmesKappaPP0 - KnechtNehmesKappaPM0) u (e^( ))^2)/(64 π^2 (f _ π^(ó  ))^2)

test2 = FullSimplify /@ Collect[test20 /. {KnechtNehmesKappaPM0 -> KnechtNehmesKappaPM, KnechtNehmesKappaPP0 -> KnechtNehmesKappaPP} // Expand, {MandelstamU, ParticleMass[__]}]

(2 u (e^( ))^2 (10 k _ 1^(r ) - 26 k _ 2^(r ) - 27 (2 k _ 3^(r ) + k _ 4^(r ))))/(9 (f _ π^(ó  ))^2) - (4 (e^( ))^2 (10 k _ 1^(r ) + 10 k _ 2^(r ) - 9 (6 k _ 3^(r ) + k _ 4^(r ) + 8 (k _ 6^(r ) + k _ 8^(r )))) (m _ π^0^(ó  r ))^2)/(9 (f _ π^(ó  ))^2)

Collect[Simplify[test1 - test2] // Expand, {CouplingConstant[QED[1], RenormalizationState[0]], MandelstamU, ParticleMass[__]}] // Simplify

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Converted by Mathematica  (July 10, 2003)