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Collecton of cal formulae Lublana, 12 September 2016, p.: 7 In a compact notaton (see precse form of ε n the prevous table): K (k, k ) ϕ (k) ξ (k ) Below V N and V Δ stand for the pon nter., for sgma. ξ (k) V (k) ω k ε, ϕ (k) m (ω 0 ε ) V (k) ω k ε g, ω 0 ε ω 0 ε Furthermore, g NN g, g ΔN g NΔ g, g ΔΔ g Δ, g σ g σ g σσ 1. ϕ (k ) g m V (k ), ϕ σ(k ) m σ V N (k ), ϕ σ (k σ ) m (k σ ). From m 2 u m 2 [( ω ) 2 k 2 k 2 ] m 2 m 2 2 ω µ 2 2 ( ω (m 2 m 2 µ 2 )/2 ) : ε m2 m 2 µ 2 2, µ N,Δ m π, µ σ µ, N W ω 0, W ω 1, σ µ W ω µ0 U (k) U (k) U (k) U (k) K (k, k )U (k ) r (ω k ω 0 ) ξ (k )U (k ) r (ω k ω 0 ) ϕ (k) x ϕ (k) x b γ A,γ x γ A,γ b ξ (k)ϕ γ(k) r (ω k ω 0 ) U (k)ξ (k) r (ω k ω 0 ) Old notaton: g x (), g x ()

Collecton of cal formulae Lublana, 12 September 2016, p.: 8 D δ (k, k δ ) K δ (k, k δ ) K δ (k, k δ ) K δ (k, k δ ) K (k, k )D δ (k, k δ ) r (ω k ω 0 ) Dδ (k, k δ )ξ (k ) r (ω k ω 0 ) z δ ϕ (k) ϕ (k) z δ b δ γ A,γ z δ γ b δ K δ (k, k δ )ξ (k ) r (ω k ω 0 ) ξ(k δ δ ) ϕ δ (k )ξ (k ) r (ω k ω 0 ) Note that all values correspond to the nomnal Δ and σ masses when under the ntegral; whle values always to averaged W -dependent masses. Solvng the coupled system for c s A RR (W )c N R (W ) A RN (W )c N N(W ) b N R (W ) A NR (W )c N R (W ) A NN (W )c N N(W ) b N N(W ) (0.35) A RR W m 0 R V RN(k)V RN (k) A RN V NN(k)V RN (k) Vm Δ NΔ(k)V RΔ (k) b N R V RN (k 0 ), A NN W m N V NN(k)ṼNN(k) A NR V RN(k)ṼNN(k) Vm Δ RΔ (k)ṽnδ(k) Vm Δ RΔ (k)v RΔ (k) Vm Δ NΔ(k)ṼNΔ(k) Vmσ Nσ (k)v Rσ (k) Vmσ Rσ (k)ṽnσ(k) Vmσ Rσ (k)v Rσ (k) Vmσ Nσ (k)ṽnσ(k) b N N V NN (k 0 ) (0.36) A RR (W ) ĉ Δ R(W, m) A RN (W ) ĉ Δ N(W, m) b Δ R(W, m) VRΔ(k m 1 ) A NR (W ) ĉ Δ R(W, m) A NN (W ) ĉ Δ N(W, m) b Δ N(W, m) VNΔ(k m 1 ) (0.37) A RR (W ) ĉ σ R(W, µ) A RN (W ) ĉ σ N(W, µ) b σ R(W, µ) V Rσ(k µ µ0 ) A NR (W ) ĉ σ R(W, µ) A NN (W ) ĉ σ N(W, µ) b σ N(W, µ) VNσ(k m µ0 ) (0.38) Fnally, the resonance part of the K matrx acqures the form: (h s the channel eventually dependent on m and µ) K hh [ ] VRh V Nh [ ] c h R (W ) c h N(W ) V Rh V Rh Z R (W )(m R W ) V Nh V Nh Z N (W )(m N W ) (0.39)

Collecton of cal formulae Lublana, 12 September 2016, p.: 9 where V Xh u XX V Xh u XY V Y h, V Y h u Y X V Xh u Y Y V Y h (0.40) and U dgonalzes the A matrx UAU T dag[λ R, λ N ] dag[z R (W )(W m R ), Z N (W )(W m N )] (0.41) V R (k) V R (k) V N (k) V N (k) x ϕ (k) y ϕ (k) D δ (k, k δ ) K δ (k, k δ ) z ϕ (k)ξ δ (k δ ) A RR W m 0 R V R (k)vr (k) r (ω k ω 0 ) A RN V N (k)vr (k) r (ω k ω 0 ) y A NN W m N V N (k) V R (k) r (ω k ω 0 ) A NR V N (k) V R (k) r (ω k ω 0 ) x We have (note that n the old notes the factor m x V R (k)ϕ (k) r (ω k ω 0 ) y V R (k)ϕ (k) r (ω k ω 0 ) V N (k)ϕ (k) r (ω k ω 0 ) s mssng): V N (k)ϕ (k) r (ω k ω 0 ) V (k)ϕ (k) r (ω k ω 0 ) g g m (ω 0 ε ) m (ω 0 ε )b g V (k)v(k) r (ω k ω 0 )(ω k ε ) m (ω 0 ε )b From x γ a 1,γb γ x b g m (ω 0 ε ) γ γ a 1,γ b γ b g m (ω 0 ε ) a 1,γ b γ b g m (ω 0 ε )

Collecton of cal formulae Lublana, 12 September 2016, p.: 10 NNB NNB NΔB ΔNB ΔΔB NσN ΔσΔ σnn σδδ σσn A NNN, NΔB 0 0 A ΔNN, ΔNB A NΔN, ΔΔB 0 0 A ΔΔN, A NNΔ, 0 0 A NNσ 0 0 0 0 A ΔNΔ, 0 A ΔNσ 0 0 0 A NΔΔ, 0 0 A NΔσ 0 0 0 0 A ΔΔΔ, 0 A ΔΔσ 0 0 0 NσN 0 0 0 0 0 0 A σnn A σnδ A σnσ ΔσΔ 0 0 0 0 0 0 A σδn A σδδ A σδσ σnn A NσN σδδ 0 0 A ΔσN A NσΔ 0 0 A Nσσ 0 0 0 0 A ΔσΔ 0 A Δσσ 0 0 0 σσn 0 0 0 0 0 0 A σσn A σσδ A σσσ Below,, γ,, {N, Δ}, the bar values may depend on m for the πδ channel or on µ for σn. Note (ω γ0 ε γ )/ γ (ω 0 ε γ)/. A ᾱγ, Aᾱ,γ A σγ A σ,γ A ᾱσ Aᾱ,σ A σσ A σ,σ A σᾱγ Aᾱσ,σγγ A σᾱσ Aᾱσ,σσN A σσγ A σσn,σγγ A σσσ A σσn,σσn r (ω k ω 0 ) r (ω k ω 0 ) r (ω k ω 0 ) r (ω k ω 0 ) σ σ σ σ V ᾱ(k) (ω k ε ᾱ) V γ(k) (ω k ε γ) V N (k) Vγ(k) (ω k ε σ ) (ω k ε γ) V ᾱ(k) (ω k ε ᾱ) m (ω γ0 ε γ ) gγ γ m (ω γ0 ε γ ) gγ γ V N (k) (ω k ε σ ) m (ω 0 ε σ ) V N (k) VN (k) m (ω k ε σ ) (ω k ε σ ) (ω 0 ε σ ) (k σ ) (ω σk ε σ ᾱᾱ) (k σ ) (ω σk ε σ ᾱ) NN(k σ ) (ω σk ε σ N σ) NN(k σ ) (ω σk ε σ N σ) γγ(k σ ) (ω σk ε σ γγ) VNN(k σ σ ) (ω σk ε σ Nσ) Vγγ(k σ σ ) (ω σk ε σ γγ) NN(k σ ) (ω σk ε σ Nσ) m γ σ (ω σ0 ε σ γγ) m N σ (ω σ0 ε σ Nσ) m γ σ (ω σ0 ε σ γγ) m N σ (ω σ0 ε σ Nσ)

Collecton of cal formulae Lublana, 12 September 2016, p.: 11 Below,, δ,, {N, Δ} and U V R or V N : bᾱ b σ bᾱσ b σ σn U (k) ξ ᾱ(k) r (ω k ω 0 ) U (k) r (ω k ω 0 ) V ᾱ(k) (ω k ε ᾱ) U (k) ξ σ (k) U (k) V (k) r (ω k ω 0 ) r (ω k ω 0 ) (ω k ε N ) σ U σ (k σ ) ξᾱ(k) σ σ U σ (k σ ) V (k σ σ ) (ω σk ε σ ᾱ) σ U σ (k σ ) ξ σ σn(k) σ U σ (k σ ) V NN(k σ σ ) (ω σk ε σ N σ) Below ξ s always evaluated wth the averaged nvarant masses (m or µ): bᾱδ b σδ Aᾱ,δ ξ δ (k δ ) A σ,δ ξ δ (k δ ) bᾱσ Aᾱ,σ ξ(k σ δ ) A ᾱδ bᾱδ A ᾱδ, A σδ V δ (k δ ) (ω δ ε δ ) (k σ ) (ω σ ε σ ) V δ (k δ ) (ω δ ε δ ) σ Aᾱσ,σδδ ξσδ(k δ δ ) A σᾱδ VδN(k δ δ ) (ω δ ε δ δσ ) A σ,σ ξ(k σ σ ) A σσ V(k σ σ ) (ω σ ε σ ) b σσ b σδ σn A σσn,σδδ ξσδ(k δ δ ) A σ σδ VδN(k δ δ ) (ω δ ε δ δσ ) bᾱσ σ Aᾱσ,σσN ξσn(k σ σ ) A σᾱσ VNN(k σ σ ) (ω σ ε σ Nσ) b σσ σn A σσn,σσn ξσn(k σ σ ) A σ σσ VNN(k σ σ ) (ω σ ε σ Nσ) The equatons for z δ consst of three sets of equatons Az b δ wth the same A but wth three b δ for δ N, Δ, σ.

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