•A few checks:

melamp[i1_, i2_, i3_, i4_] := melsimplified /. {I1 -> i1, I2 -> i2, I3 -> i3, I4 -> i4} ;

melamp1[i1_, i2_, i3_, i4_] := res1 /. {I1 -> i1, I2 -> i2, I3 -> i3, I4 -> i4} ;

MesonTab = Table[If[i == j || i == 3 && j == 8 || i == 8 && j == 3, melamp[i, j, 3, 3] /. p2 -> p - p1 /. p1 -> q // SUNReduce[#, FullReduce -> True] & // Simplify, 0], {i, 8}, {j, 8}]

( ) ó  2 i ((m ) + 2 p · p - p · ( p - q ) - p · q - p · ( p - q ) - p · q + 2 ( p - q ) · q) π 3 4 3 3 4 4 ------------------------------------------------------------------------------------------------------------------------------------------------------ ó  2 3 (f ) ϕ 0 0 0 0 0 0 0 ó  2 i ((m ) + 2 p · p - p · ( p - q ) - p · q - p · ( p - q ) - p · q + 2 ( p - q ) · q) π 3 4 3 3 4 4 ------------------------------------------------------------------------------------------------------------------------------------------------------ ó  2 3 (f ) 0 ϕ 0 0 0 0 0 0 ó  2 ó  2 ó  2 i ((m ) - (m ) ) i (m ) + 0 π K K ------------ ----------------------- ó  2 ó  2 (f ) Sqrt[3] (f ) 0 0 ϕ 0 0 0 0 ϕ ó  2 ó  2 ó  2 i (2 (m ) + 4 (m ) - 2 (m ) + 2 p · p - p · ( p - q ) - p · q - p · ( p - q ) - p · q + 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ó  2 12 (f ) 0 0 0 ϕ 0 0 0 0 ó  2 ó  2 ó  2 i (2 (m ) + 4 (m ) - 2 (m ) + 2 p · p - p · ( p - q ) - p · q - p · ( p - q ) - p · q + 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ó  2 12 (f ) 0 0 0 0 ϕ 0 0 0 ó  2 ó  2 ó  2 i (2 (m ) - 2 (m ) + 4 (m ) + 2 p · p - p · ( p - q ) - p · q - p · ( p - q ) - p · q + 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ó  2 12 (f ) 0 0 0 0 0 ϕ 0 0 ó  2 ó  2 ó  2 i (2 (m ) - 2 (m ) + 4 (m ) + 2 p · p - p · ( p - q ) - p · q - p · ( p - q ) - p · q + 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ó  2 12 (f ) 0 0 0 0 0 0 ϕ 0 ó  2 ó  2 i ((m ) - (m ) ) ó  2 + 0 i (m ) K K π ----------------------- ------------ ó  2 ó  2 Sqrt[3] (f ) 3 (f ) 0 0 ϕ 0 0 0 0 ϕ

MesonTab1 = I * Table[If[i == j || i == 3 && j == 8 || i == 8 && j == 3, melamp1[i, j, 3, 3] /. p2 -> p - p1 /. p1 -> q // WriteOutUMatrices // SUNReduce // Simplify, 0], {i, 8}, {j, 8}]

( ) ó  2 i (-(m ) - 2 p · p + p · ( p - q ) + p · q + p · ( p - q ) + p · q - 2 ( p - q ) · q) π 3 4 3 3 4 4 -------------------------------------------------------------------------------------------------------------------------------------------------------- ó  2 3 (f ) ϕ 0 0 0 0 0 0 0 ó  2 i (-(m ) - 2 p · p + p · ( p - q ) + p · q + p · ( p - q ) + p · q - 2 ( p - q ) · q) π 3 4 3 3 4 4 -------------------------------------------------------------------------------------------------------------------------------------------------------- ó  2 3 (f ) 0 ϕ 0 0 0 0 0 0 ó  2 ó  2 ó  2 i ((m ) - (m ) ) i (m ) + 0 π K K ------------ ----------------------- ó  2 ó  2 (f ) Sqrt[3] (f ) 0 0 ϕ 0 0 0 0 ϕ ó  2 ó  2 ó  2 i (-2 (m ) - 4 (m ) + 2 (m ) - 2 p · p + p · ( p - q ) + p · q + p · ( p - q ) + p · q - 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ó  2 12 (f ) 0 0 0 ϕ 0 0 0 0 ó  2 ó  2 ó  2 i (-2 (m ) - 4 (m ) + 2 (m ) - 2 p · p + p · ( p - q ) + p · q + p · ( p - q ) + p · q - 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ó  2 12 (f ) 0 0 0 0 ϕ 0 0 0 ó  2 ó  2 ó  2 i (-2 (m ) + 2 (m ) - 4 (m ) - 2 p · p + p · ( p - q ) + p · q + p · ( p - q ) + p · q - 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ó  2 12 (f ) 0 0 0 0 0 ϕ 0 0 ó  2 ó  2 ó  2 i (-2 (m ) + 2 (m ) - 4 (m ) - 2 p · p + p · ( p - q ) + p · q + p · ( p - q ) + p · q - 2 ( p - q ) · q) π + 0 3 4 3 3 4 4 K K ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ó  2 12 (f ) 0 0 0 0 0 0 ϕ 0 ó  2 ó  2 i ((m ) - (m ) ) ó  2 + 0 i (m ) K K π ----------------------- ------------ ó  2 ó  2 Sqrt[3] (f ) 3 (f ) 0 0 ϕ 0 0 0 0 ϕ

MesonTab1 - MesonTab // Simplify

( 0 0 0 0 0 0 0 0 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Table[melamp[1, 1, i, i] /. p4 -> -p3 // SUNReduce[#, FullReduce -> True] & // Simplify, {i, 8}]

{(i (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^2, (i ((m _ π^(ó  ))^2 + 2 p _ 1 · p _ 2 - 2 p _ 3^2))/(3 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + 2 p _ 1 · p _ 2 - 2 p _ 3^2))/(3 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i (m _ π^(ó  ))^2)/(3 (f _ ϕ^(ó  ))^2)}

Table[melamp[2, 2, i, i] /. p4 -> -p3 // SUNReduce[#, FullReduce -> True] & // Simplify, {i, 8}]

{(i ((m _ π^(ó  ))^2 + 2 p _ 1 · p _ 2 - 2 p _ 3^2))/(3 (f _ ϕ^(ó  ))^2), (i (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^2, (i ((m _ π^(ó  ))^2 + 2 p _ 1 · p _ 2 - 2 p _ 3^2))/(3 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i ((m _ π^(ó  ))^2 + (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 - p _ 3^2))/(6 (f _ ϕ^(ó  ))^2), (i (m _ π^(ó  ))^2)/(3 (f _ ϕ^(ó  ))^2)}

melamp[1, 1, 1, 1] // SUNReduce[#, FullReduce -> True] & // Simplify

(i (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^2

melamp[7, 7, 3, 3] // SUNReduce[#, FullReduce -> True] & // Simplify

(i (2 (m _ π^(ó  ))^2 - 2 (m _ K^+^(ó  ))^2 + 4 (m _ K^0^(ó  ))^2 + 2 p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 - p _ 2 · p _ 3 - p _ 2 · p _ 4 + 2 p _ 3 · p _ 4))/(12 (f _ ϕ^(ó  ))^2)

Select[Flatten[Outer[List, {3, 7}, {3, 7}, {3, 7}, {3, 7}], 3], (Count[#, 3, Infinity] == 2 && Count[#, 7, Infinity] == 2) &] // StandardForm

{{3, 3, 7, 7}, {3, 7, 3, 7}, {3, 7, 7, 3}, {7, 3, 3, 7}, {7, 3, 7, 3}, {7, 7, 3, 3}}

((melamp @@ #) & /@ {{3, 3, 7, 7}, {3, 7, 3, 7}, {3, 7, 7, 3}, {7, 3, 3, 7}, {7, 3, 7, 3}, {7, 7, 3, 3}} // SUNReduce[#, FullReduce -> True] & // Simplify) // Simplify

{(i (2 (m _ π^(ó  ))^2 - 2 (m _ K^+^(ó  ))^2 + 4 (m _ K^0^(ó  ))^2 + 2 p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 - p _ 2 · p _ 3 - p _ 2 · p _ 4 + 2 p _ 3 · p _ 4))/(12 (f _ ϕ^(ó  ))^2), -(i (-2 (m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 - 4 (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 - 2 p _ 1 · p _ 3 + p _ 1 · p _ 4 + p _ 2 · p _ 3 - 2 p _ 2 · p _ 4 + p _ 3 · p _ 4))/(12 (f _ ϕ^(ó  ))^2), -(i (-2 (m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 - 4 (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 + p _ 1 · p _ 3 - 2 p _ 1 · p _ 4 - 2 p _ 2 · p _ 3 + p _ 2 · p _ 4 + p _ 3 · p _ 4))/(12 (f _ ϕ^(ó  ))^2), -(i (-2 (m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 - 4 (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 + p _ 1 · p _ 3 - 2 p _ 1 · p _ 4 - 2 p _ 2 · p _ 3 + p _ 2 · p _ 4 + p _ 3 · p _ 4))/(12 (f _ ϕ^(ó  ))^2), -(i (-2 (m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 - 4 (m _ K^0^(ó  ))^2 + p _ 1 · p _ 2 - 2 p _ 1 · p _ 3 + p _ 1 · p _ 4 + p _ 2 · p _ 3 - 2 p _ 2 · p _ 4 + p _ 3 · p _ 4))/(12 (f _ ϕ^(ó  ))^2), (i (2 (m _ π^(ó  ))^2 - 2 (m _ K^+^(ó  ))^2 + 4 (m _ K^0^(ó  ))^2 + 2 p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 - p _ 2 · p _ 3 - p _ 2 · p _ 4 + 2 p _ 3 · p _ 4))/(12 (f _ ϕ^(ó  ))^2)}

Plus @@ % // Simplify

(i ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2 + 2 (m _ K^0^(ó  ))^2))/(f _ ϕ^(ó  ))^2

AmplitudeProjection[f, Channel -> {{PionPlus, KaonMinus} -> {PionPlus, KaonMinus}}, OnMassShell -> True]

1/4 f(4, 1, 1, 4) + 1/4 i f(4, 1, 1, 5) - 1/4 i f(4, 1, 2, 4) + 1/4 f(4, 1, 2, 5) + 1/4 i f(4, 2, 1, 4) - 1/4 f(4, 2, 1, 5) + 1/4 f(4, 2, 2, 4) + 1/4 i f(4, 2, 2, 5) - 1/4 i f(5, 1, 1, 4) + 1/4 f(5, 1, 1, 5) - 1/4 f(5, 1, 2, 4) - 1/4 i f(5, 1, 2, 5) + 1/4 f(5, 2, 1, 4) + 1/4 i f(5, 2, 1, 5) - 1/4 i f(5, 2, 2, 4) + 1/4 f(5, 2, 2, 5)

AmplitudeProjection[melamp, Channel -> {{PionPlus, KaonMinus} -> {PionPlus, KaonMinus}}, OnMassShell -> True] /. p3 -> -p2 /. p4 -> -p1 // SUNReduce[#, FullReduce -> True] & // Simplify

-(i (-(m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2 + p _ 1^2 - 6 p _ 1 · p _ 2 + p _ 2^2))/(6 (f _ ϕ^(ó  ))^2)

AmplitudeProjection[melamp, Channel -> {{PionPlus, KaonPlus} -> {PionPlus, KaonPlus}}, OnMassShell -> True] /. p3 -> -p2 /. p4 -> -p1 // SUNReduce[#, FullReduce -> True] & // Simplify

-(i (-(m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2 + p _ 1^2 + 6 p _ 1 · p _ 2 + p _ 2^2))/(6 (f _ ϕ^(ó  ))^2)

% + %% // Simplify

-(i (-(m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2 + p _ 1^2 + p _ 2^2))/(3 (f _ ϕ^(ó  ))^2)

AmplitudeProjection[melamp, Channel -> {{KaonZeroBar, EtaMeson} -> {KaonMinus, PionPlus}}, OnMassShell -> True] // SUNReduce[#, FullReduce -> True] & // MandelstamReduce[#, Masses -> {ParticleMass[KaonZero, RenormalizationState[0]], ParticleMass[EtaMeson, RenormalizationState[0]], ParticleMass[KaonPlus, RenormalizationState[0]], ParticleMass[Pion, RenormalizationState[0]]}] & // Simplify

(i (-(m _ π^(ó  ))^2 - 4 (m _ K^+^(ó  ))^2 - 4 (m _ K^0^(ó  ))^2 - 3 (m _ η^(ó  ))^2 + 9 t))/(6 6^(1/2) (f _ ϕ^(ó  ))^2)

Flatten[Outer[List, {1, 2}, {1, 2}, {1, 2}, {1, 2}], 3] // StandardForm

{{1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 2, 1}, {1, 1, 2, 2}, {1, 2, 1, 1}, {1, 2, 1, 2}, {1, 2, 2, 1}, {1, 2, 2, 2}, {2, 1, 1, 1}, {2, 1, 1, 2}, {2, 1, 2, 1}, {2, 1, 2, 2}, {2, 2, 1, 1}, {2, 2, 1, 2}, {2, 2, 2, 1}, {2, 2, 2, 2}}

((melamp @@ #) & /@ {{1, 2, 2, 1}, {1, 2, 1, 2}, {2, 1, 1, 2}, {2, 1, 2, 1}}) // SUNReduce[#, FullReduce -> True] & // Simplify

{-(i (-(m _ π^(ó  ))^2 + p _ 1 · p _ 2 + p _ 1 · p _ 3 - 2 p _ 1 · p _ 4 - 2 p _ 2 · p _ 3 + p _ 2 · p _ 4 + p _ 3 · p _ 4))/(3 (f _ ϕ^(ó  ))^2), -(i (-(m _ π^(ó  ))^2 + p _ 1 · p _ 2 - 2 p _ 1 · p _ 3 + p _ 1 · p _ 4 + p _ 2 · p _ 3 - 2 p _ 2 · p _ 4 + p _ 3 · p _ 4))/(3 (f _ ϕ^(ó  ))^2), -(i (-(m _ π^(ó  ))^2 + p _ 1 · p _ 2 + p _ 1 · p _ 3 - 2 p _ 1 · p _ 4 - 2 p _ 2 · p _ 3 + p _ 2 · p _ 4 + p _ 3 · p _ 4))/(3 (f _ ϕ^(ó  ))^2), -(i (-(m _ π^(ó  ))^2 + p _ 1 · p _ 2 - 2 p _ 1 · p _ 3 + p _ 1 · p _ 4 + p _ 2 · p _ 3 - 2 p _ 2 · p _ 4 + p _ 3 · p _ 4))/(3 (f _ ϕ^(ó  ))^2)}

MandelstamReduce /@ % // Simplify

{(i (t - (m _ π^(ó  ))^2))/(f _ ϕ^(ó  ))^2, -(i (-3 (m _ π^(ó  ))^2 + s + t))/(f _ ϕ^(ó  ))^2, (i (t - (m _ π^(ó  ))^2))/(f _ ϕ^(ó  ))^2, -(i (-3 (m _ π^(ó  ))^2 + s + t))/(f _ ϕ^(ó  ))^2}

Plus @@ % // MandelstamReduce // Simplify

-(2 i (s - 2 (m _ π^(ó  ))^2))/(f _ ϕ^(ó  ))^2

Converted by Mathematica  (July 10, 2003)