•Diagonalization of mass terms

The mass term is selected, the placeholder variables U and χ are substituted with the PHI variables representing these quantitites and all is expanded to order one  in the meson fields:

eqsPhi = (I eqsU /. {UMatrix[U][x] -> MM[x, Explicit -> True, ExpansionOrder -> 1, DropOrder -> 1], UMatrix[χ][x] -> 2 QuarkCondensate[] UQuarkMassMatrix[DiagonalToU -> False, QuarkToMesonMasses -> False]} // NMExpand // ExpandAll // CycleUTraces // Simplify) /. {_DropFactor -> 1, _[QuantumField[PartialD[LorentzIndex[μ]], PartialD[LorentzIndex[μ]], Particle[PhiMeson]], ___][_] -> 0}

(2 !, _ 0^( ) (3 (Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 m) + 3 (m '6 Overscript[ϕ^( ), ->] · Overscript[σ, ->]) - 2 "÷¬öé"<br /> < Overscript[ϕ^( ), ->] · Overscript[σ, ->] '6 m >))/(3 f _ ϕ^(ó  ))

This is written out in components:

lh = eqsPhi // WriteOutIsoVectors // WriteOutUMatrices // Simplify

( ó  3 8 ó  8 ó  3 8 ó  ó  1 2 ó  ó  4 5 ) 4 !, (m (3 (ϕ ) - Sqrt[3] (ϕ ) ) + 2 (Sqrt[3] m (ϕ ) + m (3 (ϕ ) + Sqrt[3] (ϕ ) ))) 2 (m + m ) !, ((ϕ ) - i (ϕ ) ) 2 (m + m ) !, ((ϕ ) - i (ϕ ) ) 0 d s u d u 0 u s 0 ---------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------- ------------------------------------------------- ó  ó  ó  9 f f f ϕ ϕ ϕ ó  ó  1 2 ó  8 ó  3 8 ó  3 8 ó  ó  6 7 2 (m + m ) !, ((ϕ ) + i (ϕ ) ) 4 !, (-2 Sqrt[3] m (ϕ ) + m (6 (ϕ ) - 2 Sqrt[3] (ϕ ) ) + m (3 (ϕ ) + Sqrt[3] (ϕ ) )) 2 (m + m ) !, ((ϕ ) - i (ϕ ) ) d u 0 0 s d u d s 0 ------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------- ó  ó  ó  f 9 f f ϕ ϕ ϕ ó  ó  4 5 ó  ó  6 7 ó  8 ó  8 3 ó  3 8 2 (m + m ) !, ((ϕ ) + i (ϕ ) ) 2 (m + m ) !, ((ϕ ) + i (ϕ ) ) 4 !, (4 Sqrt[3] m (ϕ ) + m (Sqrt[3] (ϕ ) - 3 (ϕ ) ) + m (3 (ϕ ) + Sqrt[3] (ϕ ) )) u s 0 d s 0 0 s d u ------------------------------------------------- ------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------- ó  ó  ó  f f 9 f ϕ ϕ ϕ

m Overscript[ϕ^( ), ->] · Overscript[ϕ^( ), ->] is also written out in components:

rh = m * (IsoDot[IsoVector[QuantumField[Particle[PhiMeson]]][x], IsoVector[UMatrix[UGenerator[]]]] // WriteOutIsoVectors // WriteOutUMatrices)

( 8 ) 3 (ϕ ) m ((ϕ ) + ----------) 1 2 4 5 Sqrt[3] m ((ϕ ) - i (ϕ ) ) m ((ϕ ) - i (ϕ ) ) 8 (ϕ ) 3 1 2 m (---------- - (ϕ ) ) 6 7 m ((ϕ ) + i (ϕ ) ) Sqrt[3] m ((ϕ ) - i (ϕ ) ) 8 2 m (ϕ ) 4 5 6 7 --------------- m ((ϕ ) + i (ϕ ) ) m ((ϕ ) + i (ϕ ) ) Sqrt[3]


We equal these two matrices and find the the solutions that have diagonal
mass matrices. If we further enforce orthogonality, we get the standard
charged fields.

sol = Solve[lh == rh, {List @@ WriteOutIsoVectors[IsoVector[QuantumField[Particle[PhiMeson]]][x]], m} // Flatten] // Simplify

Solve :: svars : Equations may not give solutions for all \"solve\" variables.

{{ϕ^( )^1 -> 0, ϕ^( )^4 -> 0, ϕ^( )^2 -> 0, ϕ^( )^5 -> 0, m -> (2 (m _ d^(ó  ) + m _ s^(ó  )) !, _ 0^( ))/f _ ϕ^(ó  ), ϕ^( )^3 -> 0, ϕ^( )^8 -> 0}, {ϕ^( )^1 -> 0, ϕ^( )^6 -> 0, ϕ^( )^2 -> 0, ϕ^( )^7 -> 0, m -> (2 (m _ u^(ó  ) + m _ s^(ó  )) !, _ 0^( ))/f _ ϕ^(ó  ), ϕ^( )^3 -> 0, ϕ^( )^8 -> 0}, {ϕ^( )^4 -> 0, ϕ^( )^6 -> 0, ϕ^( )^5 -> 0, ϕ^( )^7 -> 0, m -> (2 (m _ d^(ó  ) + m _ u^(ó  )) !, _ 0^( ))/f _ ϕ^(ó  ), ϕ^( )^3 -> 0, ϕ^( )^8 -> 0}, {ϕ^( )^1 -> 0, ϕ^( )^4 -> 0, ϕ^( )^6 -> 0, ϕ^( )^2 -> 0, ϕ^( )^5 -> 0, ϕ^( )^7 -> 0, m -> (4 m _ u^(ó  ) !, _ 0^( ))/f _ ϕ^(ó  ), ϕ^( )^8 -> 0}, {ϕ^( )^1 -> 0, ϕ^( )^4 -> 0, ϕ^( )^6 -> 0, ϕ^( )^2 -> 0, ϕ^( )^5 -> 0, ϕ^( )^7 -> 0, ϕ^( )^3 -> 0, ϕ^( )^8 -> 0}, {ϕ^( )^1 -> 0, ϕ^( )^4 -> 0, ϕ^( )^6 -> 0, ϕ^( )^2 -> 0, ϕ^( )^5 -> 0, ϕ^( )^7 -> 0, m -> (4 (m _ d^(ó  ) + m _ u^(ó  ) + m _ s^(ó  ) + ((m _ d^(ó  ))^2 - (m _ u^(ó  ) + m _ s^(ó  )) m _ d^(ó  ) + (m _ u^(ó  ))^2 + (m _ s^(ó  ))^2 - m _ u^(ó  ) m _ s^(ó  ))^(1/2)) !, _ 0^( ))/(3 f _ ϕ^(ó  )), ϕ^( )^3 -> -((m _ d^(ó  ) + m _ u^(ó  ) - 2 m _ s^(ó  ) + 2 ((m _ d^(ó  ))^2 - (m _ u^(ó  ) + m _ s^(ó  )) m _ d^(ó  ) + (m _ u^(ó  ))^2 + (m _ s^(ó  ))^2 - m _ u^(ó  ) m _ s^(ó  ))^(1/2)) ϕ^( )^8)/(3^(1/2) (m _ d^(ó  ) - m _ u^(ó  )))}, {ϕ^( )^1 -> 0, ϕ^( )^4 -> 0, ϕ^( )^6 -> 0, ϕ^( )^2 -> 0, ϕ^( )^5 -> 0, ϕ^( )^7 -> 0, m -> (4 (m _ d^(ó  ) + m _ u^(ó  ) + m _ s^(ó  ) - ((m _ d^(ó  ))^2 - (m _ u^(ó  ) + m _ s^(ó  )) m _ d^(ó  ) + (m _ u^(ó  ))^2 + (m _ s^(ó  ))^2 - m _ u^(ó  ) m _ s^(ó  ))^(1/2)) !, _ 0^( ))/(3 f _ ϕ^(ó  )), ϕ^( )^3 -> -((m _ d^(ó  ) + m _ u^(ó  ) - 2 (m _ s^(ó  ) + ((m _ d^(ó  ))^2 - (m _ u^(ó  ) + m _ s^(ó  )) m _ d^(ó  ) + (m _ u^(ó  ))^2 + (m _ s^(ó  ))^2 - m _ u^(ó  ) m _ s^(ó  ))^(1/2))) ϕ^( )^8)/(3^(1/2) (m _ d^(ó  ) - m _ u^(ó  )))}}

Converted by Mathematica  (July 10, 2003)