Non-perturbative chiral approach to Kp --gamma Y reactions(14)

A recently developed nonperturbative chiral approach to describe the S=-1 meson-baryon reactions has been extended to investigate the near threshold $K^- p \to \gamma \Lambda, \gamma \Sigma^0$ reactions. With the parameters governed by chiral SU(3) symmetr

APPENDIXA:RENORMALIZABILITYCHARACTEROFTHEOFFSHELL

PARTOFTHESTRONGAMPLITUDE.

WestartfromEq.(11)andtakejustoneloopintheBetheSalpeterequation.ThismeansthatwesubstitutetMB,K p(k,k′,P)bytheIij(k,k′,P)functionofEq.(5).LetustakethecontacttermfortheelectromagneticamplitudeAγY,MB(theprocedureandtheconclusionfortheothertermsarethesame).Bytakingthenonrelativisticreductionofthecontacttermandthepositiveenergypartofthenucleonpropagator,Eq.(6),asdoneintheevaluationofthemesonnucleonstrongamplitudein[19],weobtainfromtheloopacontributionproportionalto

i d4k

EB(k)1k2 µ2M+i e σ (A1)

Wenowseparatethestrongamplitude,representedbythefactork′0+k0intoitsonshellando shellpartsas

k′0+k0=2k′0+(k0 k′0)(A2)

Thepartoftheintegralcomingfromtheterm2k′0intheformerequationcorrespondstofactorizingthestrongamplitudewithitsonshellvalue,whichistheprocedurethatwehavefollowed.Thecontributionfromtheo shellpartwillbegivenby

i d4k

EB(k)1k2 µ2M+i e σ (A3)

Herewefollowthesamestepsthatledtoincludetheo shellcontributioninthestrongamplitudeintoarenormalizationofthefcouplingin[19].WeusetypicalapproximationsoftheheavybaryonformalismandsetM/EB( k)=1,p0 EB( k)=0.Inthiscasetheo shellterm(k0 k′0)andthenucleonpropagator(k′0 k0) 1cancelandwegetacontribution

i

= d4kd3kk2 µ2M+i

2w( k)e σ (A4) 2e σ ～qmaxe σ