Fermion Propagators in XD

Flip Tanedo
30 July 2010; updated 1 Aug 2010 (sign error on RH massive case)
3 Aug 2010 (Changed notation on +/- in flat case, included warped interval)
4-5 Aug 2010: corrected boundary conditions for all propagators.

18 Aug 2010: I've updated my DoublyCompositeHiggs.tex document to include an overall sign on the F functions and the D-bar operator such that the overall propagators are the same, but the functions now have a more natural sign. I have not yet updated this document! (The results should be trivial though: F gets an overall sign difference which is countered by the overall sign on the Dirac-bar operator.)

29 Aug 2010: I'm redoing these again using proper sign conventions. It appears that my i conventions are a bit off, but these are trivial prefactors. I also derive the Wick-rotated propagators. I've not updated the flat interval to the correct sign conventions because I'm a selfish jerk and just don't care about them.

These are calculations to derive the fermion propagator in a flat and warped interval extra dimension. The relevant calculations leading up to the Mathematica step are written out explicitly elsewhere. See DoubleTop.pdf for documentation.

16 April 2011: Included Mixed BC

Notation: FpgL = ((Subscript[F, L+])^(>))

Flat Interval

These use the old minus sign conventions... resulting F functions are off by a minus sign which is compensated by the definition of curly-bar-D in my old notes.

Left Massless

ClearAll["Global`*"]
FmgL[z_] := AmgL Cos[x z] + BmgL Sin[x z]
FmlL[z_] := AmlL Cos[x z] + BmlL Sin[x z]
FpgL[z_] := ApgL Cos[x z] + BpgL Sin[x z]
FplL[z_] := AplL Cos[x z] + BplL Sin[x z]
dFmgL[z_] := (∂_y) FmgL[y] /. {y -> z}
dFmlL[z_] := (∂_y) FmlL[y] /. {y -> z}
dFpgL[z_] := (∂_y) FpgL[y] /. {y -> z}
dFplL[z_] := (∂_y) FplL[y] /. {y -> z}

(* Left Chiral Zero Mode *)
BpgLSol=Solve[dFpgL[L]==0,BpgL][[1]];
AmgLSol=Solve[FmgL[L]==0,AmgL][[1]];
BplLSol=Solve[dFplL[0]==0,BplL][[1]];
AmlLSol=Solve[FmlL[0]==0,AmlL][[1]];

LeftChiralSol=Join[BpgLSol,BplLSol,AmgLSol,AmlLSol];

(*Matching Conditions*)
BmlLSol=FullSimplify[Solve[FmgL[zp]-FmlL[zp]==0/.LeftChiralSol,BmlL]][[1]];
AplLSol=FullSimplify[Solve[FpgL[zp]-FplL[zp]==0/.LeftChiralSol,AplL]][[1]];

BmgLSol=FullSimplify[Solve[-dFmgL[zp]+dFmlL[zp]==I/.LeftChiralSol/.BmlLSol/.AplLSol,BmgL][[1]]];
ApgLSol=FullSimplify[Solve[-dFpgL[zp]+dFplL[zp]==I/.LeftChiralSol/.BmlLSol/.AplLSol,ApgL][[1]]];

(*BmgL and ApgL are fully determined,plug into the other values*)
LeftChiralSol=LeftChiralSol/.BmgLSol/.ApgLSol;
BmlLSol=BmlLSol/.BmgLSol/.ApgLSol;
AplLSol=AplLSol/.BmgLSol/.ApgLSol;
FullLeftSol=FullSimplify[Join[LeftChiralSol,BmlLSol,AplLSol,BmgLSol,ApgLSol]];

(*Write out functions*)
FullSimplify[FmgL[z]/.FullLeftSol]
FullSimplify[FmlL[z]/.FullLeftSol]
FullSimplify[FpgL[z]/.FullLeftSol]
FullSimplify[FplL[z]/.FullLeftSol]

Right Massless

ClearAll["Global`*"]
FmgR[z_] := AmgR Cos[x z] + BmgR Sin[x z]
FmlR[z_] := AmlR Cos[x z] + BmlR Sin[x z]
FpgR[z_] := ApgR Cos[x z] + BpgR Sin[x z]
FplR[z_] := AplR Cos[x z] + BplR Sin[x z]

dFmgR[z_] := (∂_y) FmgR[y] /. {y -> z}
dFmlR[z_] := (∂_y) FmlR[y] /. {y -> z}
dFpgR[z_] := (∂_y) FpgR[y] /. {y -> z}
dFplR[z_] := (∂_y) FplR[y] /. {y -> z}

(*Left Chiral Zero Mode*)
BmgRSol=Solve[dFmgR[L]==0,BmgR][[1]];
ApgRSol=Solve[FpgR[L]==0,ApgR][[1]];
BmlRSol=Solve[dFmlR[0]==0,BmlR][[1]];
AplRSol=Solve[FplR[0]==0,AplR][[1]];
RightChiralSol=Join[BmgRSol,BmlRSol,ApgRSol,AplRSol];

(*Matching Conditions*)
AmlRSol=FullSimplify[Solve[FmgR[zp]-FmlR[zp]==0/.RightChiralSol,AmlR]][[1]];
BplRSol=FullSimplify[Solve[FpgR[zp]-FplR[zp]==0/.RightChiralSol,BplR]][[1]];

AmgRSol=FullSimplify[Solve[-dFmgR[zp]+dFmlR[zp]==I/.RightChiralSol/.AmlRSol/.BplRSol,AmgR][[1]]];
BpgRSol=FullSimplify[Solve[-dFpgR[zp]+dFplR[zp]==I/.RightChiralSol/.AmlRSol/.BplRSol,BpgR][[1]]];

(*AmgR and BpgR are fully determined,plug into the other values*)
RightChiralSol=RightChiralSol/.AmgRSol/.BpgRSol;
AmlRSol=AmlRSol/.AmgRSol/.BpgRSol;
BplRSol=BplRSol/.AmgRSol/.BpgRSol;
FullRightSol=FullSimplify[Join[RightChiralSol,AmlRSol,BplRSol,AmgRSol,BpgRSol]];

(*Write out functions*)
FullSimplify[FmgR[z]/.FullRightSol]
FullSimplify[FmlR[z]/.FullRightSol]
FullSimplify[FpgR[z]/.FullRightSol]
FullSimplify[FplR[z]/.FullRightSol]

Left Massive

ClearAll["Global`*"]
FmgL[z_] := AmgL Cos[x z] + BmgL Sin[x z]
FmlL[z_] := AmlL Cos[x z] + BmlL Sin[x z]
FpgL[z_] := ApgL Cos[x z] + BpgL Sin[x z]
FplL[z_] := AplL Cos[x z] + BplL Sin[x z]

dFmgL[z_] := (∂_y) FmgL[y] /. {y -> z}
dFmlL[z_] := (∂_y) FmlL[y] /. {y -> z}
dFpgL[z_] := (∂_y) FpgL[y] /. {y -> z}
dFplL[z_] := (∂_y) FplL[y] /. {y -> z}

(*Left Chiral Zero Mode*)
BpgLSol=Solve[dFpgL[L]+m FpgL[L]==0,BpgL][[1]];
AmgLSol=Solve[FmgL[L]==0,AmgL][[1]];
BplLSol=Solve[dFplL[0]+m FplL[0]==0,BplL][[1]];
AmlLSol=Solve[FmlL[0]==0,AmlL][[1]];

LeftChiralSol=Join[BpgLSol,BplLSol,AmgLSol,AmlLSol];

(*Matching Conditions*)
BmlLSol=FullSimplify[Solve[FmgL[zp]-FmlL[zp]==0/.LeftChiralSol,BmlL]][[1]];
AplLSol=FullSimplify[Solve[FpgL[zp]-FplL[zp]==0/.LeftChiralSol,AplL]][[1]];

(*New for m≠0:re-insert AmlL solution into LeftChiralSol*)
LeftChiralSol=FullSimplify[LeftChiralSol/.AplLSol];

BmgLSol=FullSimplify[Solve[-dFmgL[zp]+dFmlL[zp]==I/.LeftChiralSol/.BmlLSol/.AplLSol,BmgL][[1]]];
ApgLSol=FullSimplify[Solve[-dFpgL[zp]+dFplL[zp]==I/.LeftChiralSol/.BmlLSol/.AplLSol,ApgL][[1]]];

(*BmgL and ApgL are fully determined,plug into the other values*)
LeftChiralSol=LeftChiralSol/.BmgLSol/.ApgLSol;
BmlLSol=BmlLSol/.BmgLSol/.ApgLSol;
AplLSol=AplLSol/.BmgLSol/.ApgLSol;
FullLeftSol=FullSimplify[Join[LeftChiralSol,BmlLSol,AplLSol,BmgLSol,ApgLSol]];

(*Write out functions*)
FullSimplify[FmgL[z]/.FullLeftSol]
FullSimplify[FmlL[z]/.FullLeftSol]
FullSimplify[FpgL[z]/.FullLeftSol]
FullSimplify[FplL[z]/.FullLeftSol]

Right Massive

ClearAll["Global`*"]
FmgR[z_] := AmgR Cos[x z] + BmgR Sin[x z]
FmlR[z_] := AmlR Cos[x z] + BmlR Sin[x z]
FpgR[z_] := ApgR Cos[x z] + BpgR Sin[x z]
FplR[z_] := AplR Cos[x z] + BplR Sin[x z]

dFmgR[z_] := (∂_y) FmgR[y] /. {y -> z}
dFmlR[z_] := (∂_y) FmlR[y] /. {y -> z}
dFpgR[z_] := (∂_y) FpgR[y] /. {y -> z}
dFplR[z_] := (∂_y) FplR[y] /. {y -> z}

(*Left Chiral Zero Mode*)
BmgRSol=Solve[dFmgR[L]-m FmgR[L]==0,BmgR][[1]];
ApgRSol=Solve[FpgR[L]==0,ApgR][[1]];
BmlRSol=Solve[dFmlR[0]-m FmlR[0]==0,BmlR][[1]];
AplRSol=Solve[FplR[0]==0,AplR][[1]];

RightChiralSol=Join[BmgRSol,BmlRSol,ApgRSol,AplRSol];

(*Matching Conditions*)
AmlRSol=FullSimplify[Solve[FmgR[zp]-FmlR[zp]==0/.RightChiralSol,AmlR]][[1]];
BplRSol=FullSimplify[Solve[FpgR[zp]-FplR[zp]==0/.RightChiralSol,BplR]][[1]];

(*New for m≠0:re-insert AmlR solution into RightChiralSol*)
RightChiralSol=FullSimplify[RightChiralSol/.AmlRSol];

AmgRSol=FullSimplify[Solve[-dFmgR[zp]+dFmlR[zp]==I/.RightChiralSol/.AmlRSol/.BplRSol,AmgR][[1]]];
BpgRSol=FullSimplify[Solve[-dFpgR[zp]+dFplR[zp]==I/.RightChiralSol/.AmlRSol/.BplRSol,BpgR][[1]]];

(*AmgR and BpgR are fully determined,plug into the other values*)
RightChiralSol=RightChiralSol/.AmgRSol/.BpgRSol;
AmlRSol=AmlRSol/.AmgRSol/.BpgRSol;
BplRSol=BplRSol/.AmgRSol/.BpgRSol;
FullRightSol=FullSimplify[Join[RightChiralSol,AmlRSol,BplRSol,AmgRSol,BpgRSol]]

(*Write out functions*)
FullSimplify[FmgR[z]/.FullRightSol]
FullSimplify[FmlR[z]/.FullRightSol]
FullSimplify[FpgR[z]/.FullRightSol]
FullSimplify[FplR[z]/.FullRightSol]

Warped Interval

Solution to the Homogeneous Equation

ClearAll["Global`*"]
DSolve[-(p^2)F[z]-D[F[z],{z,2}]+(4/z)D[F[z],z]+(((c^2)+c-6)/(z^2))F[z]==0,F[z],z][[1]]
DSolve[-(p^2)F[z]-D[F[z],{z,2}]+(4/z)D[F[z],z]+(((c^2)-c-6)/(z^2))F[z]==0,F[z],z][[1]]

Left-handed Warped Fermion

ClearAll["Global`*"]
FmgL[z_] := AmgL (z^(5/2))BesselJ[c-1/2,x z] + BmgL (z^(5/2))BesselY[c-1/2,x z]
FmlL[z_] := AmlL (z^(5/2))BesselJ[c-1/2,x z] + BmlL (z^(5/2))BesselY[c-1/2,x z]
FpgL[z_] := ApgL (z^(5/2))BesselJ[c+1/2,x z] + BpgL (z^(5/2))BesselY[c+1/2,x z]
FplL[z_] := AplL (z^(5/2))BesselJ[c+1/2,x z] + BplL (z^(5/2))BesselY[c+1/2,x z]
dFmgL[z_] := (∂_y) FmgL[y] /. {y -> z}
dFmlL[z_] := (∂_y) FmlL[y] /. {y -> z}
dFpgL[z_] := (∂_y) FpgL[y] /. {y -> z}
dFplL[z_] := (∂_y) FplL[y] /. {y -> z}

(*Left Chiral Zero Mode*)
BpgLSol=Solve[dFpgL[Rp]+((-2+c)/Rp) FpgL[Rp]==0,BpgL][[1]];
AmgLSol=Solve[FmgL[Rp]==0,AmgL][[1]];
BplLSol=Solve[dFplL[R]+((-2+c)/R) FplL[R]==0,BplL][[1]];
AmlLSol=Solve[FmlL[R]==0,AmlL][[1]];

LeftChiralSol=FullSimplify[Join[BpgLSol,BplLSol,AmgLSol,AmlLSol]];

(*Matching Conditions*)
BmlLSol=FullSimplify[Solve[FmgL[zp]-FmlL[zp]==0/.LeftChiralSol,BmlL]][[1]];
AplLSol=FullSimplify[Solve[FpgL[zp]-FplL[zp]==0/.LeftChiralSol,AplL]][[1]];

LeftChiralSol=FullSimplify[LeftChiralSol/.BmlLSol/.AplLSol];

BmgLSol=FullSimplify[Solve[dFmgL[zp]-dFmlL[zp]==I (zp/R)^4/.LeftChiralSol/.BmlLSol/.AplLSol,BmgL][[1]]];
ApgLSol=FullSimplify[Solve[dFpgL[zp]-dFplL[zp]==I (zp/R)^4/.LeftChiralSol/.BmlLSol/.AplLSol,ApgL][[1]]];

(*BmgL and ApgL are fully determined,plug into the other values*)
LeftChiralSol=LeftChiralSol/.BmgLSol/.ApgLSol;
BmlLSol=BmlLSol/.BmgLSol/.ApgLSol;
AplLSol=AplLSol/.BmgLSol/.ApgLSol;
FullLeftSol=FullSimplify[Join[LeftChiralSol,BmlLSol,AplLSol,BmgLSol,ApgLSol]];

(*Write out functions*)
FullSimplify[FmgL[z]/.FullLeftSol]
FullSimplify[FmlL[z]/.FullLeftSol]
FullSimplify[FpgL[z]/.FullLeftSol]
FullSimplify[FplL[z]/.FullLeftSol]

(*Here are the old propagators for reference; they should differ by an overall sign*)

Right-handed Warped Fermion

ClearAll["Global`*"]
FmgR[z_] := AmgR (z^(5/2))BesselJ[c-1/2,x z] + BmgR (z^(5/2))BesselY[c-1/2,x z]
FmlR[z_] := AmlR (z^(5/2))BesselJ[c-1/2,x z] + BmlR (z^(5/2))BesselY[c-1/2,x z]
FpgR[z_] := ApgR (z^(5/2))BesselJ[c+1/2,x z] + BpgR (z^(5/2))BesselY[c+1/2,x z]
FplR[z_] := AplR (z^(5/2))BesselJ[c+1/2,x z] + BplR (z^(5/2))BesselY[c+1/2,x z]
dFmgR[z_] := (∂_y) FmgR[y] /. {y -> z}
dFmlR[z_] := (∂_y) FmlR[y] /. {y -> z}
dFpgR[z_] := (∂_y) FpgR[y] /. {y -> z}
dFplR[z_] := (∂_y) FplR[y] /. {y -> z}

(*Right Chiral Zero Mode*)
BmgRSol=Solve[-dFmgR[Rp]+((2+c)/Rp) FmgR[Rp]==0,BmgR][[1]];
ApgRSol=Solve[FpgR[Rp]==0,ApgR][[1]];
BmlRSol=Solve[-dFmlR[R]+((2+c)/R) FmlR[R]==0,BmlR][[1]];
AplRSol=Solve[FplR[R]==0,AplR][[1]];
RightChiralSol=FullSimplify[Join[BmgRSol,BmlRSol,ApgRSol,AplRSol]];


(*Matching Conditions*)
AmlRSol=FullSimplify[Solve[FmgR[zp]-FmlR[zp]==0/.RightChiralSol,AmlR]][[1]];
BplRSol=FullSimplify[Solve[FpgR[zp]-FplR[zp]==0/.RightChiralSol,BplR]][[1]];

RightChiralSol=FullSimplify[RightChiralSol/.AmlRSol/.BplRSol];

AmgRSol=FullSimplify[Solve[dFmgR[zp]-dFmlR[zp]==I (zp/R)^4/.RightChiralSol/.AmlRSol/.BplRSol,AmgR][[1]]];
BpgRSol=FullSimplify[Solve[dFpgR[zp]-dFplR[zp]==I (zp/R)^4/.RightChiralSol/.AmlRSol/.BplRSol,BpgR][[1]]];

(*AmgR and BpgR are fully determined,plug into the other values*)
RightChiralSol=RightChiralSol/.AmgRSol/.BpgRSol;
AmlRSol=AmlRSol/.AmgRSol/.BpgRSol;
BplRSol=BplRSol/.AmgRSol/.BpgRSol;
FullRightSol=FullSimplify[Join[RightChiralSol,AmlRSol,BplRSol,AmgRSol,BpgRSol]];

(*Write out functions*)
FullSimplify[FmgR[z]/.FullRightSol]
FullSimplify[FmlR[z]/.FullRightSol]
FullSimplify[FpgR[z]/.FullRightSol]
FullSimplify[FplR[z]/.FullRightSol]

(* Here are the old wrong-sign functions for comparison *)

Wick-Rotated Warped Interval

Solution to the Homogeneous Equation

ClearAll["Global`*"]
DSolve[(pe^2)F[z]-D[F[z],{z,2}]+(4/z)D[F[z],z]+(((c^2)+c-6)/(z^2))F[z]==0,F[z],z][[1]]
DSolve[(pe^2)F[z]-D[F[z],{z,2}]+(4/z)D[F[z],z]+(((c^2)-c-6)/(z^2))F[z]==0,F[z],z][[1]]

These imaginary Bessels are really just I's and K's.

F[z_]:=(z^(5/2)) BesselI[(1/2) (-1+2 c), pe z]
F[z_]:=a (z^(5/2)) BesselI[(1/2) (-1+2 c), pe z]+ b (z^(5/2)) BesselK[(1/2) (-1+2 c), pe z]
FullSimplify[(pe^2)F[z]-D[F[z],{z,2}]+(4/z)D[F[z],z]+(((c^2)-c-6)/(z^2))F[z]]

Left-handed Warped Fermion

ClearAll["Global`*"]
FmgL[z_] := AmgL (z^(5/2))BesselI[c-1/2,xe z] + BmgL (z^(5/2))BesselK[c-1/2,xe z]
FmlL[z_] := AmlL (z^(5/2))BesselI[c-1/2,xe z] + BmlL (z^(5/2))BesselK[c-1/2,xe z]
FpgL[z_] := ApgL (z^(5/2))BesselI[c+1/2,xe z] + BpgL (z^(5/2))BesselK[c+1/2,xe z]
FplL[z_] := AplL (z^(5/2))BesselI[c+1/2,xe z] + BplL (z^(5/2))BesselK[c+1/2,xe z]
dFmgL[z_] := (∂_y) FmgL[y] /. {y -> z}
dFmlL[z_] := (∂_y) FmlL[y] /. {y -> z}
dFpgL[z_] := (∂_y) FpgL[y] /. {y -> z}
dFplL[z_] := (∂_y) FplL[y] /. {y -> z}

(*Left Chiral Zero Mode*)
BpgLSol=Solve[dFpgL[Rp]+((-2+c)/Rp) FpgL[Rp]==0,BpgL][[1]];
AmgLSol=Solve[FmgL[Rp]==0,AmgL][[1]];
BplLSol=Solve[dFplL[R]+((-2+c)/R) FplL[R]==0,BplL][[1]];
AmlLSol=Solve[FmlL[R]==0,AmlL][[1]];

LeftChiralSol=FullSimplify[Join[BpgLSol,BplLSol,AmgLSol,AmlLSol]];

(*Matching Conditions*)
BmlLSol=FullSimplify[Solve[FmgL[zp]-FmlL[zp]==0/.LeftChiralSol,BmlL]][[1]];
AplLSol=FullSimplify[Solve[FpgL[zp]-FplL[zp]==0/.LeftChiralSol,AplL]][[1]];

LeftChiralSol=FullSimplify[LeftChiralSol/.BmlLSol/.AplLSol];

BmgLSol=FullSimplify[Solve[dFmgL[zp]-dFmlL[zp]==I (zp/R)^4/.LeftChiralSol/.BmlLSol/.AplLSol,BmgL][[1]]];
ApgLSol=FullSimplify[Solve[dFpgL[zp]-dFplL[zp]==I (zp/R)^4/.LeftChiralSol/.BmlLSol/.AplLSol,ApgL][[1]]];

(*BmgL and ApgL are fully determined,plug into the other values*)
LeftChiralSol=LeftChiralSol/.BmgLSol/.ApgLSol;
BmlLSol=BmlLSol/.BmgLSol/.ApgLSol;
AplLSol=AplLSol/.BmgLSol/.ApgLSol;
FullLeftSol=FullSimplify[Join[LeftChiralSol,BmlLSol,AplLSol,BmgLSol,ApgLSol]];

(*Write out functions*)
FullSimplify[FmgL[z]/.FullLeftSol]
FullSimplify[FmlL[z]/.FullLeftSol]
FullSimplify[FpgL[z]/.FullLeftSol]
FullSimplify[FplL[z]/.FullLeftSol]

Right-handed Warped Fermion

ClearAll["Global`*"]
FmgR[z_] := AmgR (z^(5/2))BesselI[c-1/2,xe z] + BmgR (z^(5/2))BesselK[c-1/2,xe z]
FmlR[z_] := AmlR (z^(5/2))BesselI[c-1/2,xe z] + BmlR (z^(5/2))BesselK[c-1/2,xe z]
FpgR[z_] := ApgR (z^(5/2))BesselI[c+1/2,xe z] + BpgR (z^(5/2))BesselK[c+1/2,xe z]
FplR[z_] := AplR (z^(5/2))BesselI[c+1/2,xe z] + BplR (z^(5/2))BesselK[c+1/2,xe z]
dFmgR[z_] := (∂_y) FmgR[y] /. {y -> z}
dFmlR[z_] := (∂_y) FmlR[y] /. {y -> z}
dFpgR[z_] := (∂_y) FpgR[y] /. {y -> z}
dFplR[z_] := (∂_y) FplR[y] /. {y -> z}

(*Right Chiral Zero Mode*)
BmgRSol=Solve[-dFmgR[Rp]+((2+c)/Rp) FmgR[Rp]==0,BmgR][[1]];
ApgRSol=Solve[FpgR[Rp]==0,ApgR][[1]];
BmlRSol=Solve[-dFmlR[R]+((2+c)/R) FmlR[R]==0,BmlR][[1]];
AplRSol=Solve[FplR[R]==0,AplR][[1]];
RightChiralSol=FullSimplify[Join[BmgRSol,BmlRSol,ApgRSol,AplRSol]];


(*Matching Conditions*)
AmlRSol=FullSimplify[Solve[FmgR[zp]-FmlR[zp]==0/.RightChiralSol,AmlR]][[1]];
BplRSol=FullSimplify[Solve[FpgR[zp]-FplR[zp]==0/.RightChiralSol,BplR]][[1]];

RightChiralSol=FullSimplify[RightChiralSol/.AmlRSol/.BplRSol];

AmgRSol=FullSimplify[Solve[dFmgR[zp]-dFmlR[zp]==I (zp/R)^4/.RightChiralSol/.AmlRSol/.BplRSol,AmgR][[1]]];
BpgRSol=FullSimplify[Solve[dFpgR[zp]-dFplR[zp]==I (zp/R)^4/.RightChiralSol/.AmlRSol/.BplRSol,BpgR][[1]]];

(*AmgR and BpgR are fully determined,plug into the other values*)
RightChiralSol=RightChiralSol/.AmgRSol/.BpgRSol;
AmlRSol=AmlRSol/.AmgRSol/.BpgRSol;
BplRSol=BplRSol/.AmgRSol/.BpgRSol;
FullRightSol=FullSimplify[Join[RightChiralSol,AmlRSol,BplRSol,AmgRSol,BpgRSol]];

(*Write out functions*)
FullSimplify[FmgR[z]/.FullRightSol]
FullSimplify[FmlR[z]/.FullRightSol]
FullSimplify[FpgR[z]/.FullRightSol]
FullSimplify[FplR[z]/.FullRightSol]

Mixed BC: (+ -)

(X Y) mixed boundary conditions means X on the UV brane (z=R) and Y on the IR brane (z=R'). (X=+) means Neumann BC for the left-chiral spinor (χ) whereas (X=-) means Dirichlet BC for the left-chiral spinor. For example, the LH chiral BC are (++) while RH chiral BC are (--).

(*
(*Left Chiral Zero Mode*)
BpgLSol=Solve[dFpgL[Rp]+((-2+c)/Rp) FpgL[Rp]==0,BpgL][[1]];
AmgLSol=Solve[FmgL[Rp]==0,AmgL][[1]];
BplLSol=Solve[dFplL[R]+((-2+c)/R) FplL[R]==0,BplL][[1]];
AmlLSol=Solve[FmlL[R]==0,AmlL][[1]];

(*Right Chiral Zero Mode*)
BmgRSol=Solve[-dFmgR[Rp]+((2+c)/Rp) FmgR[Rp]==0,BmgR][[1]];
ApgRSol=Solve[FpgR[Rp]==0,ApgR][[1]];
BmlRSol=Solve[-dFmlR[R]+((2+c)/R) FmlR[R]==0,BmlR][[1]];
AplRSol=Solve[FplR[R]==0,AplR][[1]];
*)

ClearAll["Global`*"]
Fmg[z_] := Amg (z^(5/2))BesselI[c-1/2,xe z] + Bmg (z^(5/2))BesselK[c-1/2,xe z]
Fml[z_] := Aml (z^(5/2))BesselI[c-1/2,xe z] + Bml (z^(5/2))BesselK[c-1/2,xe z]
Fpg[z_] := Apg (z^(5/2))BesselI[c+1/2,xe z] + Bpg (z^(5/2))BesselK[c+1/2,xe z]
Fpl[z_] := Apl (z^(5/2))BesselI[c+1/2,xe z] + Bpl (z^(5/2))BesselK[c+1/2,xe z]
dFmg[z_] := (∂_y) Fmg[y] /. {y -> z}
dFml[z_] := (∂_y) Fml[y] /. {y -> z}
dFpg[z_] := (∂_y) Fpg[y] /. {y -> z}
dFpl[z_] := (∂_y) Fpl[y] /. {y -> z}

(* +- conditions on the boundary *)
BplSol=Solve[dFpl[R]+((-2+c)/R) Fpl[R]==0,Bpl][[1]];
AmlSol=Solve[Fml[R]==0,Aml][[1]];
(*BpgSol=Solve[dFpg[Rp]+((-2+c)/Rp) Fpg[Rp]==0,Bpg][[1]];
AmgSol=Solve[Fmg[Rp]==0,Amg][[1]];*)
BmgSol=Solve[-dFmg[Rp]+((2+c)/Rp) Fmg[Rp]==0,Bmg][[1]];
ApgSol=Solve[Fpg[Rp]==0,Apg][[1]];

PMSol=FullSimplify[Join[BmgSol,BplSol,ApgSol,AmlSol]];

(*Matching Conditions*)
BmlSol=FullSimplify[Solve[Fmg[zp]-Fml[zp]==0/.PMSol,Bml]][[1]];
AplSol=FullSimplify[Solve[Fpg[zp]-Fpl[zp]==0/.PMSol,Apl]][[1]];

PMSol=FullSimplify[PMSol/.BmlSol/.AplSol];

AmgSol=FullSimplify[Solve[dFmg[zp]-dFml[zp]==I (zp/R)^4/.PMSol/.BmlSol/.AplSol,Amg][[1]]];
BpgSol=FullSimplify[Solve[dFpg[zp]-dFpl[zp]==I (zp/R)^4/.PMSol/.BmlSol/.AplSol,Bpg][[1]]];

PMSol=PMSol/.AmgSol/.BpgSol;

(* Amg and Bpg are fully determined,plug into the other values*)
BmlSol=BmlSol/.AmgSol/.BpgSol;
AplSol=AplSol/.AmgSol/.BpgSol;

FullPMSol=FullSimplify[Join[PMSol,BmlSol,AplSol,AmgSol,BpgSol]];

(*Write out functions*)
FullSimplify[Fmg[z]/.FullPMSol]
FullSimplify[Fml[z]/.FullPMSol]
FullSimplify[Fpg[z]/.FullPMSol]
FullSimplify[Fpl[z]/.FullPMSol]

Mixed BC: (-+), Bibhushan's homework

I went ahead and did this because Bibhushan was too busy. -F

(*
(*Left Chiral Zero Mode*)
BpgLSol=Solve[dFpgL[Rp]+((-2+c)/Rp) FpgL[Rp]==0,BpgL][[1]];
AmgLSol=Solve[FmgL[Rp]==0,AmgL][[1]];
BplLSol=Solve[dFplL[R]+((-2+c)/R) FplL[R]==0,BplL][[1]];
AmlLSol=Solve[FmlL[R]==0,AmlL][[1]];

(*Right Chiral Zero Mode*)
BmgRSol=Solve[-dFmgR[Rp]+((2+c)/Rp) FmgR[Rp]==0,BmgR][[1]];
ApgRSol=Solve[FpgR[Rp]==0,ApgR][[1]];
BmlRSol=Solve[-dFmlR[R]+((2+c)/R) FmlR[R]==0,BmlR][[1]];
AplRSol=Solve[FplR[R]==0,AplR][[1]];
*)

ClearAll["Global`*"]
Fmg[z_] := Amg (z^(5/2))BesselI[c-1/2,xe z] + Bmg (z^(5/2))BesselK[c-1/2,xe z]
Fml[z_] := Aml (z^(5/2))BesselI[c-1/2,xe z] + Bml (z^(5/2))BesselK[c-1/2,xe z]
Fpg[z_] := Apg (z^(5/2))BesselI[c+1/2,xe z] + Bpg (z^(5/2))BesselK[c+1/2,xe z]
Fpl[z_] := Apl (z^(5/2))BesselI[c+1/2,xe z] + Bpl (z^(5/2))BesselK[c+1/2,xe z]
dFmg[z_] := (∂_y) Fmg[y] /. {y -> z}
dFml[z_] := (∂_y) Fml[y] /. {y -> z}
dFpg[z_] := (∂_y) Fpg[y] /. {y -> z}
dFpl[z_] := (∂_y) Fpl[y] /. {y -> z}

(* -+ conditions on the boundary *)
BmlSol=Solve[-dFml[R]+((2+c)/R) Fml[R]==0,Bml][[1]];
AplSol=Solve[Fpl[R]==0,Apl][[1]];
BpgSol=Solve[dFpg[Rp]+((-2+c)/Rp) Fpg[Rp]==0,Bpg][[1]];
AmgSol=Solve[Fmg[Rp]==0,Amg][[1]];

PMSol=FullSimplify[Join[AmgSol,AplSol,BpgSol,BmlSol]];

(*Matching Conditions*)
AmlSol=FullSimplify[Solve[Fmg[zp]-Fml[zp]==0/.PMSol,Aml]][[1]];
BplSol=FullSimplify[Solve[Fpg[zp]-Fpl[zp]==0/.PMSol,Bpl]][[1]];

PMSol=FullSimplify[PMSol/.AmlSol/.BplSol];

BmgSol=FullSimplify[Solve[dFmg[zp]-dFml[zp]==I (zp/R)^4/.PMSol/.AmlSol/.BplSol,Bmg][[1]]];
ApgSol=FullSimplify[Solve[dFpg[zp]-dFpl[zp]==I (zp/R)^4/.PMSol/.AmlSol/.BplSol,Apg][[1]]];

PMSol=PMSol/.BmgSol/.ApgSol;

(* Bmg and Apg are fully determined,plug into the other values*)
AmlSol=AmlSol/.BmgSol/.ApgSol;
BplSol=BplSol/.BmgSol/.ApgSol;

FullPMSol=FullSimplify[Join[PMSol,AmlSol,BplSol,BmgSol,ApgSol]];

(*Write out functions*)
FullSimplify[Fmg[z]/.FullPMSol]
FullSimplify[Fml[z]/.FullPMSol]
FullSimplify[Fpg[z]/.FullPMSol]
FullSimplify[Fpl[z]/.FullPMSol]