Perturbative renormalization of the axial charge in a relativistic approach.

A realistic NN interaction contains a large attractive scalar isoscalar component (due to σ-exchange in OBE model) and a repulsive vector isovector component (ω-exchange). Evaluating the selfenergy of a nucleon in a medium of nuclear matter with such an interaction using the mean field approximation, one finds that it contains a large attractive scalar component and a repulsive component, which under Lorentz transformation transforms like the timelike component of a vector [#2##1###]

with ρ the nuclear density and ρ₀ the saturation density of nuclear matter #math2#(ρ₀ = 0.17fm^-3). Taking into account the Fock-exchange terms in the Hartree-Fock approximation or accounting for correlation effects in the DBHF approximation one obtains a small spacelike vector component and finds that all the terms depend slightly on the momentum of the nucleon [#10##1###]. We now want to calculate matrix elements for the axial operator #math3#g_Aγ^μγ₅, concentrating on the axial charge #math4#g_Aγ⁰γ₅ for nucleons moving in the nuclear medium. We can immediately write the perturbative corrections to the axial charge due to the nucleon selfenergy, which are depicted diagramatically in fig. 1, where we have separated the contribution from positive and negative intermediate states in the nucleon propagator. Analytically this decomposition is given by

where #math6#M, E(#tex2html_wrap_inline1213#
 ) are the mass and on shell energy of the free nucleon and #math7#u_r, v_r the ordinary free spinors in Mandl-Shaw representation [#9##1###]. The axial charge matrix element is reduced to a bispinor representation assuming #math8#E(#tex2html_wrap_inline1216#
 ) #tex2html_wrap_inline1217#
M by means of

Now the a) and b) diagrams from fig. 1 with positive intermediate nucleon components are automatically absorbed into the calculation with dressed non relativistic wave functions but genuine corrections from the negative intermediate states c) and d) remain. One can easily see that the renormalization with the #math10#Σ^vγ⁰ term of (#self#97>) vanishes identically and only the renormalization with the #math11#Σ^s term remains. One immediately gets a renormalized axial charge matrix element corresponding to bare matrix element plus figs. 1c and 1d given by

or equivalently a renormalization of the axial charge by the amount #math13#(1 - #tex2html_wrap_inline1230#
#tex2html_wrap_inline1231#
). This is the result obtained in [#7##1###]. Note that since the relativistic potential of (1) implicitly accounts for direct and exchange terms no further corrections have to be done in contrast to [#7##1###] where, because one starts from a NN interaction, direct and exchange terms are explicitly evaluated. With standard values of #math14#Σ^s of the order of -400MeV and taking #math15#ρ #tex2html_wrap_inline1236#
ρ₀ one obtains a renormalization factor of the order of 1.4 in qualitative agreement with [#7##1###,#8##1###].

Another way to arrive at eq.(#pertur1#120>) is to realize that the solution of the Dirac equation for with a selfenergy of the kind displayed in eq.(#self#121>) yields Dirac spinors for the nucleons in the nuclear medium, which are identical to Dirac spinors of free nucleons, except that the mass of the nucleon M has to be replaced by an effective mass #math16#M^* = M + Σ^sρ/ρ₀. Calculating the matrix element for the axial charge operator with these dressed Dirac spinors and reducing it to a bispinor representation one finds as in eq.(#bare#124>)