WaveFunctionFactor.nb

•Renormalization

The beta functions:

${e^( ), L _ 4^(r ) + λ/8, L _ 5^(r ) + λ/4, L _ 6^(r ) + (3 λ)/32, L _ 8^(r ), k _ 1^(r ) + ((12 (C^( ))^2)/(f _ π^(ó ))^8 + (3 C^( ))/(f _ π^(ó ))^4 + 3/2) λ, k _ 2^(r ) + (2 C^( ) λ)/(f _ π^(ó ))^4, k _ 3^(r ) - (3 λ)/4, k _ 4^(r ) - (2 C^( ) λ)/(f _ π^(ó ))^4, k _ 7^(r ) + ((2 C^( ))/(f _ π^(ó ))^4 + 1/4) λ, k _ 8^(r ) + λ/8, k _ 9^(r ) + (-(12 (C^( ))^2)/(5 (f _ π^(ó ))^8) - (3 C^( ))/(5 (f _ π^(ó ))^4) - 3) λ, k _ 10^(r ) + (-C^( )/(5 (f _ π^(ó ))^4) - 27/20) λ, k _ 11^(r ) + (-C^( )/(5 (f _ π^(ó ))^4) - 1/4) λ, k _ 14^(r )}$

The contributions:

The full amplitude (to fourth order) ff4 differs from the lowest order amplitude ff2 by a factor Z, ff4 = ff2. This is equivalent to a redefinition of the pion field, = π.

We demand that ff4 be zero on the mass shell with =, where = +Cm is the renormalized mass. Since we are only working to O(), we only need Cm to first order in .

Here follow then the mass renormalization of the charged and neutral pions in terms of scale independent coupling constants:

Change variables to compare the neutral mass renormalization with Meissner, Müller and Steininger:

1/(288 π^2 Z (f _ π^(ó ))^4) ((m _ π^0^(ó ))^2 (C^( ) (e^( ))^2 (-40 Z k _ 2^(r ) - 27 k _ 3^(r ) - 36 Z k _ 4^(r ) + 40 Z k _ 7^(r ) + 5 k _ 7^(r ) + 4 Z k _ 10^(r ) + 27 k _ 10^(r ) - 4 Z k _ 11^(r ) - 5 k _ 11^(r ) + 8 k _ 14^(r )) - 9 Z (f _ π^(ó ))^2 ((2 L _ 4^(r ) + 2 L _ 5^(r ) - 3 L _ 6^(r ) - 32 L _ 8^(r )) (m _ π^0^(ó ))^2 - 2 log((m _ π^+^(ó ))^2/(m _ π^0^(ó ))^2) (m _ π^+^(ó ))^2)))

The factor Z.

Adding the mass shift to the full amputated two-point function, times the two-point function (compare Urech's thesis formula 1.55) gives (1-Z). Thus Z is found by dividing off .

Check that the terms go:

The neutral factor:

1/(72 π^2 C^( ) (f _ π^(ó ))^2) (8 π^2 (10 k _ 2^(r ) - 18 k _ 3^(r ) - 9 k _ 4^(r ) + 10 k _ 10^(r )) ((m _ π^+^(ó ))^2 - (m _ π^0^(ó ))^2) (f _ π^(ó ))^4 + 3 C^( ) (24 π^2 (f _ π^(ó ))^2 + (64 π^2 λ - log((m _ π^+^(ó ))^2/μ^2)) (m _ π^+^(ó ))^2 + 192 π^2 (2 L _ 4^(r ) + L _ 5^(r )) (m _ π^0^(ó ))^2))

The charged amputated two-point function with the mass shift subtracted off:

The charged factor:

Check that it has a finite limit:

1/(144 π^2 C^( ) (f _ π^(ó ))^2) ((160 π^2 k _ 2^(r ) + 160 π^2 k _ 10^(r ) + 9 (8 π^2 λ + log((m _ π^+^(ó ))^2/μ^2) - 1)) ((m _ π^+^(ó ))^2 - (m _ π^0^(ó ))^2) (f _ π^(ó ))^4 + 3 C^( ) (48 π^2 (f _ π^(ó ))^2 + (64 π^2 λ - log((m _ π^+^(ó ))^2/μ^2)) (m _ π^+^(ó ))^2 + (768 π^2 L _ 4^(r ) + 384 π^2 L _ 5^(r ) + 64 π^2 λ - log((m _ π^0^(ó ))^2/μ^2)) (m _ π^0^(ó ))^2))