Download Thesis in Pdf Format - Theoretical Nuclear Physics and ...

34 3.2. The unbound neutron as kaon-production target3.2.2 Helicity amplitudesIn contrast to the hadronic parameters, the relations between electromagnetic couplings have tobe distilled from experimental information. The partial decay width for the radiative decay of aresonance of spin S to the ground-state nucleon is given by [1]Γ(N ∗ → Nγ) = p∗2 γπ2m N(2S + 1)m N ∗(|A N 1/2 |2 + |A N 3/2 |2) , (3.9)in terms of photocoupling helicity amplitudes A N J. These helicity amplitudes can be directly linkedwith current matrix elements. Using the conventions of Ref. [129], we have√A N 1/2 = πα〈p2m N (m 2 N− m 2 ∗ N ) N ∗, λ N ∗ = 1 2∣ j x(0) + i j y (0)∣ p N, λ N = 1 〉,2√A N 3/2 = πα〈p2m N (m 2 N− m 2 ∗ N ) N ∗, λ N ∗ = 3 2∣ j x(0) + i j y (0)∣ p N, λ N = 1 〉 (3.10).2Here, j µ (x) is the current operator. It speaks for itself that A N 3/2is zero when S = 1/2. The currentmatrix elements can be calculated within a quark model (see Ref. [129] for example), or using thephenomenological interaction Lagrangians defined in Section D.3.2. In this way, the N ∗ and ∆ ∗transition moments can be related to the photocoupling helicity amplitudes A N J. One has [147]√A N 1/2 = ∓ e m 2 N ∗ − m2 Nκ N2m N 2m ∗ N , (3.11)Nfor spin-1/2 resonances andA N 1/2 = e4m N ∗A N 3/2 =√m 2 N ∗ − (m2 N±κ (1)3m NN ∗ N − m N ∗(m N ∗ ∓ m N)4m 2 N√e m 2 N ∗ − (m2 N±κ (1)N4m N m ∗ N ∓ m N ∗ ∓ m Nκ (2)N N 4m ∗ NN),)κ (2)N ∗ N,(3.12)for spin-3/2 resonances. In Eqs. (3.11) and (3.12), the upper (lower) sign corresponds to positive-(negative-) parity resonances. Inverting these relations and neglecting the small proton-neutron massdifference, we findfor spin-1/2 resonances andκ (1)N ∗ nκ (1)N ∗ pκ (2)N ∗ nκ (2)N ∗ pκ N ∗ nκ N ∗ p= An 1/2A p , (3.13)1/2√3An1/2± A n 3/2= √3Ap,1/2 ± Ap 3/2√3An1/2− mpm=N ∗ An 3/2√3Ap1/2 − .mpm N ∗ Ap 3/2(3.14)for spin-3/2 resonances. Note that these conversion rules are only meaningful for N ∗ ’s, since the∆-nucleon magnetic transition moments κ (1,2)∆ ∗ Nare isospin independent.Values for the published helicity amplitudes of the S 11 (1650), P 11 (1710), P 13 (1720), P 13 (1900) andD 13 (1900) resonance are presented in Table 3.1. The listed numbers are from the RPP [1] and two