•Reduction

nonCDterms = ampfinalren /. LeutwylerLambda[] -> 0 /. (C0 | D0)[__] -> 0 ;

nonCDterms // Length

394

nonCDterms1 = Collect[nonCDterms, {_LeutwylerJBar, _Log, _CouplingConstant}] ;

LeafCount /@ List @@ nonCDterms1

{17, 16, 17, 9, 22, 22, 21, 53, 92, 22, 16, 28, 21, 22, 90, 60, 74, 864, 906, 127, 3587, 3686, 1629, 1352, 7091}

nonCDterms1 // Length

25

nonCDterms2 = (WriteString["stdout", "."] ; Simplify[#]) & /@ nonCDterms1 ;

.........................

nonCDterms2 // LeafCount

1841

nonCDterms2 // Length

25

LeafCount /@ List @@ nonCDterms2

{17, 16, 17, 9, 22, 22, 21, 28, 50, 22, 16, 28, 21, 22, 33, 53, 43, 52, 58, 49, 86, 194, 538, 241, 182}

nonCDterms4CTs = nonCDterms2 - (nonCDterms2 /. {CouplingConstant[_[4], ___] -> 0}) // Expand // manred[MandelstamS] // Together // Simplify

1/(9 (C^( ))^2 (f _ π^(ó  ))^4) (18 k _ 14^(r ) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^8 - C^( ) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) (10 k _ 1^(r ) (t - (m _ π^0^(ó  r ))^2) - 2 k _ 2^(r ) (-36 (m _ π^+^(ó  r ))^2 + 5 (m _ π^0^(ó  r ))^2 + 13 t) - 9 (8 (k _ 6^(r ) + k _ 8^(r )) (m _ π^0^(ó  r ))^2 + k _ 4^(r ) (-4 (m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2 + 3 t) + 2 k _ 3^(r ) (-4 (m _ π^+^(ó  r ))^2 + (m _ π^0^(ó  r ))^2 + 3 t))) (f _ π^(ó  ))^4 + 9 (C^( ))^2 (2 l _ 3^(r ) (m _ π^0^(ó  r ))^4 + 2 l _ 1^(r ) (t - 2 (m _ π^+^(ó  r ))^2)^2 + l _ 2^(r ) (8 (m _ π^+^(ó  r ))^4 - 4 (t + 2 u) (m _ π^+^(ó  r ))^2 + t^2 + 2 u^2 + 2 t u)))

nonCDterms4Polys = nonCDterms2 - (lowestamp /. ren4Rule // ChargeEliminate) /. {CouplingConstant[_[4], ___] -> 0, _Log -> 0, _LeutwylerJBar -> 0} // Expand // manred[MandelstamU] // Together // Simplify

-(45 (s + t) (s - 4 (m _ π^+^(ó  r ))^2) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^8 - 9 C^( ) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) ((48 t - 80 s) (m _ π^+^(ó  r ))^4 + 4 (5 s^2 - 14 t s - 11 t^2 + 10 (s + t) (m _ π^0^(ó  r ))^2) (m _ π^+^(ó  r ))^2 + s (s + t) (19 t - 10 (m _ π^0^(ó  r ))^2)) (f _ π^(ó  ))^4 + (C^( ))^2 (s - 4 (m _ π^+^(ó  r ))^2) (72 (s + t) (m _ π^+^(ó  r ))^4 - 2 (20 s^2 - 17 t s - 37 t^2 + 72 (s + t) (m _ π^0^(ó  r ))^2) (m _ π^+^(ó  r ))^2 + 99 (s + t) (m _ π^0^(ó  r ))^4 + 10 s^3 + 13 t^3 + 23 s t^2 - 90 t (s + t) (m _ π^0^(ó  r ))^2 + 20 s^2 t))/(288 (s + t) π^2 (C^( ))^2 (f _ π^(ó  ))^4 (s - 4 (m _ π^+^(ó  r ))^2))

DiscardOrders[nonCDterms4Polys, PerturbationOrder -> 2] // Simplify

0

nonCDtermsLogs = FullSimplify /@ Collect[manred[MandelstamU] /@ (nonCDterms2 - (nonCDterms2 /. {_Log -> 0}) // Expand), {_Log}]

-(log((m _ γ^(ó  r ))^2/μ^2) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4)/(16 π^2 (C^( ))^2) + (log((m _ γ^(ó  r ))^2/(m _ π^+^(ó  r ))^2) (t - (m _ π^0^(ó  r ))^2) ((m _ π^0^(ó  r ))^2 - (m _ π^+^(ó  r ))^2))/(8 π^2 C^( )) - (log((m _ π^0^(ó  r ))^2/μ^2) (5 (m _ π^0^(ó  r ))^4 - 4 t (m _ π^0^(ó  r ))^2 + t^2))/(32 π^2 (f _ π^(ó  ))^4) - 1/(96 π^2 (C^( ))^2 (f _ π^(ó  ))^4) (log((m _ π^+^(ó  r ))^2/μ^2) (3 ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^8 - 9 C^( ) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) (-4 (m _ π^+^(ó  r ))^2 + 2 (m _ π^0^(ó  r ))^2 + t) (f _ π^(ó  ))^4 + (C^( ))^2 (30 (m _ π^0^(ó  r ))^4 - 24 (2 (m _ π^+^(ó  r ))^2 + t) (m _ π^0^(ó  r ))^2 + (2 s + t)^2 + 4 (m _ π^+^(ó  r ))^2 (6 (m _ π^+^(ó  r ))^2 - 4 s + 5 t))))

nonCDtermsJBars = FullSimplify /@ Collect[(nonCDterms2 - (nonCDterms2 /. {_LeutwylerJBar -> 0}) // Expand // manred[MandelstamU]) // manred[None], {_LeutwylerJBar}]

(Overscript[J, _] _ (m _ γ^(ó  r ))^2(t) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4)/(C^( ))^2 + (Overscript[J, _] _ (m _ π^0^(ó  r ))^2(t) (t - (m _ π^0^(ó  r ))^2)^2)/(2 (f _ π^(ó  ))^4) + (Overscript[J, _] _ (m _ π^+^(ó  r ))^2(t) (4 (m _ π^+^(ó  r ))^2 - 3 (m _ π^0^(ó  r ))^2) (((m _ π^0^(ó  r ))^2 - (m _ π^+^(ó  r ))^2) (f _ π^(ó  ))^4 + C^( ) (t - (m _ π^0^(ó  r ))^2)))/(C^( ) (f _ π^(ó  ))^4) - 1/(12 (C^( ))^2 (u - 4 (m _ π^+^(ó  r ))^2) (f _ π^(ó  ))^4) (Overscript[J, _] _ (m _ π^+^(ó  r ))^2(u) (3 (s + t) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^8 + 6 C^( ) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) ((s + t) (-2 (m _ π^0^(ó  r ))^2 + 5 s - 3 t) - 8 (s - t) (m _ π^+^(ó  r ))^2) (f _ π^(ó  ))^4 + 2 (s + t) (C^( ))^2 (6 (m _ π^0^(ó  r ))^4 - 6 (s + t) (m _ π^0^(ó  r ))^2 + (s + t) (2 s + t)))) + 1/(12 (C^( ))^2 (s - 4 (m _ π^+^(ó  r ))^2) (f _ π^(ó  ))^4) (Overscript[J, _] _ (m _ π^+^(ó  r ))^2(s) (3 (s - 4 (m _ π^+^(ó  r ))^2) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^8 - 6 C^( ) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) (48 (m _ π^+^(ó  r ))^4 - 8 ((m _ π^0^(ó  r ))^2 + 4 s + 2 t) (m _ π^+^(ó  r ))^2 + s (2 (m _ π^0^(ó  r ))^2 + 5 s + 8 t)) (f _ π^(ó  ))^4 + 2 (C^( ))^2 (s - 4 (m _ π^+^(ó  r ))^2) (6 (m _ π^0^(ó  r ))^4 + 6 (s - 4 (m _ π^+^(ó  r ))^2) (m _ π^0^(ó  r ))^2 + (-8 (m _ π^+^(ó  r ))^2 + 2 s + t) (s - 4 (m _ π^+^(ó  r ))^2))))

CDterms = Simplify /@ Collect[manred[MandelstamU] /@ (ampfinalren - (ampfinalren /. LeutwylerLambda[___] -> 0 /. (C0 | D0)[__] -> 0) /. LeutwylerLambda[___] -> 0 // Expand), {_C0, _D0}] // Simplify

1/(64 π^2 (C^( ))^2) (4 D _ 0 ( (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , s , t , (m _ π^+^(ó  r ))^2 , 0 , (m _ π^+^(ó  r ))^2 , 0 ) (s - 2 (m _ π^+^(ó  r ))^2)^2 ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4 + 4 D _ 0 ( (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , 4 (m _ π^+^(ó  r ))^2 - s - t , t , (m _ π^+^(ó  r ))^2 , 0 , (m _ π^+^(ó  r ))^2 , 0 ) (-2 (m _ π^+^(ó  r ))^2 + s + t)^2 ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4 + 3 t C _ 0 ( (m _ π^+^(ó  r ))^2 , t , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , 0 , 0 ) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4 + C _ 0 ( (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , t , 0 , (m _ π^+^(ó  r ))^2 , 0 ) (5 t - 16 (m _ π^+^(ó  r ))^2) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4 + 2 C _ 0 ( s , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , 0 ) (s - 2 (m _ π^+^(ó  r ))^2) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4 - 2 C _ 0 ( (m _ π^+^(ó  r ))^2 , 4 (m _ π^+^(ó  r ))^2 - s - t , (m _ π^+^(ó  r ))^2 , 0 , (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 ) (-2 (m _ π^+^(ó  r ))^2 + s + t) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2)^2 (f _ π^(ó  ))^4 - 8 C _ 0 ( (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , t , (m _ π^+^(ó  r ))^2 , 0 , (m _ π^+^(ó  r ))^2 ) C^( ) (t - 2 (m _ π^+^(ó  r ))^2) (4 (m _ π^+^(ó  r ))^4 - 7 (m _ π^0^(ó  r ))^2 (m _ π^+^(ó  r ))^2 + 3 (m _ π^0^(ó  r ))^4) - 2 C _ 0 ( (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , 4 (m _ π^+^(ó  r ))^2 - s - t , (m _ π^+^(ó  r ))^2 , 0 , (m _ π^+^(ó  r ))^2 ) (-2 (m _ π^+^(ó  r ))^2 + s + t) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) (((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) (f _ π^(ó  ))^4 + 4 C^( ) (s - (m _ π^0^(ó  r ))^2)) - 2 C _ 0 ( (m _ π^+^(ó  r ))^2 , (m _ π^+^(ó  r ))^2 , s , (m _ π^+^(ó  r ))^2 , 0 , (m _ π^+^(ó  r ))^2 ) (s - 2 (m _ π^+^(ó  r ))^2) ((m _ π^+^(ó  r ))^2 - (m _ π^0^(ó  r ))^2) (((m _ π^0^(ó  r ))^2 - (m _ π^+^(ó  r ))^2) (f _ π^(ó  ))^4 + 4 C^( ) (-4 (m _ π^+^(ó  r ))^2 + (m _ π^0^(ó  r ))^2 + s + t)))

CheckF[{lowestamp, CDterms, nonCDterms4CTs, nonCDterms4Polys, nonCDtermsLogs, nonCDtermsJBars}, ("PiPiEMResult" <> ToString[d1] <> ToString[d2] <> ToString[d3] <> ToString[d4])] ;

Check crossing symmetry

TimeConstrained[manred[MandelstamU][{CDterms, nonCDterms4CTs, nonCDterms4Polys, nonCDtermsLogs, nonCDtermsJBars}] - manred[MandelstamU][({CDterms, nonCDterms4CTs, nonCDterms4Polys, nonCDtermsLogs, nonCDtermsJBars} /. {MandelstamS -> MandelstamT, MandelstamT -> MandelstamS})] // Simplify // PaVeOrder // Simplify, 5, "Not symmetric"]

Not symmetric

TimeConstrained[manred[MandelstamT][{CDterms, nonCDterms4CTs, nonCDterms4Polys, nonCDtermsLogs, nonCDtermsJBars}] - manred[MandelstamT][({CDterms, nonCDterms4CTs, nonCDterms4Polys, nonCDtermsLogs, nonCDtermsJBars} /. {MandelstamS -> MandelstamU, MandelstamU -> MandelstamS})] // Simplify // PaVeOrder // Simplify, 5, "Not symmetric"]

{0, 0, 0, 0, 0}

TimeConstrained[manred[MandelstamS][{CDterms, nonCDterms4CTs, nonCDterms4Polys, nonCDtermsLogs, nonCDtermsJBars}] - manred[MandelstamS][({CDterms, nonCDterms4CTs, nonCDterms4Polys, nonCDtermsLogs, nonCDtermsJBars} /. {MandelstamU -> MandelstamT, MandelstamT -> MandelstamU})] // Simplify // PaVeOrder // Simplify, 5, "Not symmetric"]

Not symmetric

Converted by Mathematica  (July 10, 2003)