Open Access
Erratum
This article is an erratum for:
[https://doi.org/10.1051/0004-6361/201630169]


Issue
A&A
Volume 619, November 2018
Article Number C1
Number of page(s) 3
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/201630169e
Published online 16 November 2018

A sign factor was lacking in the expressions of the redistribution matrix in the case of incomplete Paschen-Back effect, Eqs. (A.1) and (A.6) of Bommier (2017). This sign factor is unity in the absence of incomplete Paschen-Back effect. A (2I + 1) denominator was also missing in Eqs. (A.6) and (40), and some typos occurred in Eq. (A.6). The correct formulæ are provided below.

The corrected Eq. (A.1) is:

R ij ( ν , ν 1 , Ω , Ω 1 ; B ) = J u J ¯ u J u M u J u J ¯ u J u M u J J ¯ J M J J ¯ J M K K Q f ( υ ) d 3 υ ( 1 ) Q T Q K ( j , Ω 1 ) T Q K ( i , Ω ) × 3 2 L u + 1 2 S + 1 ( 2 K + 1 ) ( 2 K + 1 ) ( 1 ) M M ( 1 ) J u + J ¯ u + J u + J ¯ u ( 1 ) J + J ¯ + J + J ¯ × ( 2 J u + 1 ) ( 2 J ¯ u + 1 ) ( 2 J u + 1 ) ( 2 J ¯ u + 1 ) ( 2 J + 1 ) ( 2 J ¯ + 1 ) ( 2 J + 1 ) ( 2 J ¯ + 1 ) × C J u M u J u ( B ) C J u M u J ¯ u ( B ) C J u M u J u ( B ) C J u M u J ¯ u ( B ) C J M J ( B ) C J M J ¯ ( B ) C J M J ( B ) C J M J ¯ ( B ) × { J u 1 J L S L u } { J u 1 J ¯ L S L u } { J ¯ u 1 J L S L u } { J ¯ u 1 J ¯ L S L u } × ( J u 1 J M u p M ) ( J u 1 J ¯ M u p M ) ( J ¯ u 1 J M u p M ) ( J ¯ u 1 J ¯ M u p M ) × ( 1 1 K p p Q ) ( 1 1 K p p Q ) × { Γ R Γ R + Γ I + Γ E + i Δ E M u M u ħ δ ( ν ~ ν ~ 1 ν M M ) [ 1 2 Φ ba ( ν M u M ν ~ 1 ) + 1 2 Φ ba ( ν M u M ν ~ 1 ) ] + [ Γ R Γ R + Γ I + i Δ E M u M u ħ Γ R Γ R + Γ I + Γ E + i Δ E M u M u ħ ] × [ 1 2 Φ ba ( ν M u M ν ~ 1 ) + 1 2 Φ ba ( ν M u M ν ~ 1 ) ] [ 1 2 Φ ba ( ν M u M ν ~ ) + 1 2 Φ ba ( ν M u M ν ~ ) ] } . $$ \begin{aligned} \begin{array}{ll} \mathcal{R} _{ij}\left( \nu ,\nu _{1},\boldsymbol{\Omega },\boldsymbol{\Omega }_{1}; \boldsymbol{B}\right)&=\mathop {\sum }\limits_{J_{u}\bar{J}_{u}J_{u}^{*}M_{u}J_{u}^{\prime }\bar{J}_{u}^{\prime }J_{u}^{\prime *}M_{u}^{\prime }J_{\ell }\bar{J}_{\ell }J_{\ell }^{*}M_{\ell }J_{\ell }^{\prime }\bar{J} _{\ell }^{\prime }J_{\ell }^{\prime *}M_{\ell }^{\prime }KK^{\prime }Q}\int f(\boldsymbol{\upsilon}){\mathrm{d} }^{3}\boldsymbol{\upsilon}\ (-1)^{Q}\mathcal{T} _{-Q}^{K^{\prime }}(j,\boldsymbol{\Omega }_{1})\mathcal{T} _{Q}^{K}(i,\boldsymbol{\Omega }) \\&\quad \times 3\dfrac{2L_{u}+1}{2S+1}\sqrt{(2K+1)(2K^{\prime }+1)}\ (-1)^{M_{\ell }-M_{\ell }^{\prime }} \left( -1\right) ^{J_{u}+\bar{J}_{u}+J_{u}^{\prime }+\bar{J}_{u}^{\prime }} \left( -1\right) ^{J_{\ell }+\bar{J}_{\ell }+J_{\ell }^{\prime }+\bar{J}_{\ell }^{\prime }} \\&\quad \times \sqrt{(2J_{u}+1)(2\bar{J}_{u}+1)(2J_{u}^{\prime }+1)(2\bar{J} _{u}^{\prime }+1)(2J_{\ell }+1)(2\bar{J}_{\ell }+1)(2J_{\ell }^{\prime }+1)(2 \bar{J}_{\ell }^{\prime }+1)} \\&\quad \times C_{J_{u}^{*}M_{u}}^{J_{u}}\left( B\right) C_{J_{u}^{*}M_{u}}^{\bar{J}_{u}}\left( B\right) C_{J_{u}^{\prime *}M_{u}^{\prime }}^{J_{u}^{\prime }}\left( B\right) C_{J_{u}^{\prime *}M_{u}^{\prime }}^{ \bar{J}_{u}^{\prime }}\left( B\right) C_{J_{\ell }^{*}M_{\ell }}^{J_{\ell }}\left( B\right) C_{J_{\ell }^{*}M_{\ell }}^{\bar{J}_{\ell }}\left( B\right) C_{J_{\ell }^{\prime *}M_{\ell }^{\prime }}^{J_{\ell }^{\prime }}\left( B\right) C_{J_{\ell }^{\prime *}M_{\ell }^{\prime }}^{ \bar{J}_{\ell }^{\prime }}\left( B\right) \\&\quad \times \left\{ \begin{array}{ccc} J_{u}&1&J_{\ell } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} J_{u}^{\prime }&1&\bar{J}_{\ell } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} \bar{J}_{u}&1&J_{\ell }^{\prime } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} \bar{J}_{u}^{\prime }&1&\bar{J}_{\ell }^{\prime } \\ L_{\ell }&S&L_{u} \end{array} \right\} \\&\quad \times \left( \begin{array}{ccc} J_{u}&1&J_{\ell } \\ -M_{u}&p&M_{\ell } \end{array} \right) \left( \begin{array}{ccc} J_{u}^{\prime }&1&\bar{J}_{\ell } \\ -M_{u}^{\prime }&p^{\prime }&M_{\ell } \end{array} \right) \left( \begin{array}{ccc} \bar{J}_{u}&1&J_{\ell }^{\prime } \\ -M_{u}&p^{\prime \prime \prime }&M_{\ell }^{\prime } \end{array} \right) \left( \begin{array}{ccc} \bar{J}_{u}^{\prime }&1&\bar{J}_{\ell }^{\prime } \\ -M_{u}^{\prime }&p^{\prime \prime }&M_{\ell }^{\prime } \end{array} \right) \\&\quad \times \left( \begin{array}{ccc} 1&1&K^{\prime } \\ -p&p^{\prime }&Q \end{array} \right) \left( \begin{array}{ccc} 1&1&K \\ -p^{\prime \prime \prime }&p^{\prime \prime }&Q \end{array} \right) \\&\quad \times \left\{ \dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\Gamma _{E}+\dfrac{{\mathrm{i} }\Delta E_{M_{u}M_{u}^{\prime }}}{\hbar }}\ \delta \left( \tilde{\nu }-\tilde{\nu }_{1}-\nu _{M_{\ell }M_{\ell }^{\prime }}\right) \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{M_{u}^{\prime }M_{\ell }}-\tilde{\nu }_{1}\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{M_{u}M_{\ell }}-\tilde{\nu }_{1}\right) \right] \right. \\&\quad +\left[ \dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\dfrac{{\mathrm{i } }\Delta E_{M_{u}M_{u}^{\prime }}}{\hbar }}-\dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\Gamma _{E}+\dfrac{{\mathrm{i} }\Delta E_{M_{u}M_{u}^{\prime }} }{\hbar }}\right] \\&\quad \times \left. \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{M_{u}^{\prime }M_{\ell }}-\tilde{\nu }_{1}\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{M_{u}M_{\ell }}-\tilde{\nu }_{1}\right) \right] \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{M_{u}^{\prime }M_{\ell }^{\prime }}-\tilde{\nu }\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{M_{u}M_{\ell }^{\prime }}-\tilde{\nu }\right) \right] \right\} \end{array} \ . \end{aligned} $$(A.1)

The corrected Eq. (A.6) is:

R ij ( ν , ν 1 , Ω , Ω 1 ; B ) = J u J ¯ u J u F u F ¯ u F u M u J u J ¯ u J u F u F ¯ u F u M u J J ¯ J F F ¯ F M J J ¯ J F F ¯ F M K K Q × f ( υ ) d 3 υ ( 1 ) Q T Q K ( j , Ω 1 ) T Q K ( i , Ω ) × 3 2 L u + 1 ( 2 I + 1 ) ( 2 S + 1 ) ( 2 K + 1 ) ( 2 K + 1 ) ( 1 ) M M × ( 1 ) J u + J ¯ u + J u + J ¯ u ( 1 ) J + J ¯ + J + J ¯ ( 1 ) F u + F ¯ u + F u + F ¯ u ( 1 ) F + F ¯ + F + F ¯ × ( 2 J u + 1 ) ( 2 J ¯ u + 1 ) ( 2 J u + 1 ) ( 2 J ¯ u + 1 ) ( 2 J + 1 ) ( 2 J ¯ + 1 ) ( 2 J + 1 ) ( 2 J ¯ + 1 ) × ( 2 F u + 1 ) ( 2 F ¯ u + 1 ) ( 2 F u + 1 ) ( 2 F ¯ u + 1 ) ( 2 F + 1 ) ( 2 F ¯ + 1 ) ( 2 F + 1 ) ( 2 F ¯ + 1 ) × C J u M J u ( F u M u ) J u ( B ) C J u M J u ( F u M u ) J ¯ u ( B ) C J u M J u ( F u M u ) J u ( B ) C J u M J u ( F u M u ) J ¯ u ( B ) × C J M J ( F M ) J ( B ) C J M J ( F M ) J ¯ ( B ) C J M J ( F M ) J ( B ) C J M J ( F M ) J ¯ ( B ) × C F u ( J u M J u ) M u F u ( B ) C F ¯ u ( J ¯ u M J u ) M u F ¯ u ( B ) C F u ( J u M J u ) M u F u ( B ) C F ¯ u ( J ¯ u M J u ) M u F ¯ u ( B ) × C F ( J M J ) M F ( B ) C F ¯ ( J ¯ M J ) M F ¯ ( B ) C F ( J M J ) M F ( B ) C F ¯ ( J ¯ M J ) M F ¯ ( B ) × { J u 1 J L S L u } { J u 1 J ¯ L S L u } { J ¯ u 1 J L S L u } { J ¯ u 1 J ¯ L S L u } × { F u 1 F J I J u } { F u 1 F ¯ J ¯ I J u } { F ¯ u 1 F J I J ¯ u } { F ¯ u 1 F ¯ J ¯ I J ¯ u } × ( F u 1 F M u p M ) ( F u 1 F ¯ M u p M ) ( F ¯ u 1 F M u p M ) ( F ¯ u 1 F ¯ M u p M ) × ( 1 1 K p p Q ) ( 1 1 K p p Q ) × { Γ R Γ R + Γ I + Γ E + i Δ E M u M u ħ δ ( ν ~ ν ~ 1 ν M M ) [ 1 2 Φ ba ( ν M u M ν ~ 1 ) + 1 2 Φ ba ( ν M u M ν ~ 1 ) ] + [ Γ R Γ R + Γ I + i Δ E M u M u ħ Γ R Γ R + Γ I + Γ E + i Δ E M u M u ħ ] × [ 1 2 Φ ba ( ν M u M ν ~ 1 ) + 1 2 Φ ba ( ν M u M ν ~ 1 ) ] [ 1 2 Φ ba ( ν M u M ν ~ ) + 1 2 Φ ba ( ν M u M ν ~ ) ] } . $$ \begin{aligned} \begin{array}{ll} \mathcal{R} _{ij}\left( \nu ,\nu _{1},\boldsymbol{\Omega },\boldsymbol{\Omega }_{1}; \boldsymbol{B}\right)&= \mathop {\displaystyle \sum }\limits_{J_{u}\bar{J}_{u}J_{u}^{*}F_{u}\bar{F} _{u}F_{u}^{*}M_{u}J_{u}^{\prime }\bar{J}_{u}^{\prime }J_{u}^{\prime *}F_{u}^{\prime }\bar{F}_{u}^{\prime }F_{u}^{\prime *}M_{u}^{\prime }J_{\ell }\bar{J}_{\ell }J_{\ell }^{*}F_{\ell }\bar{F}_{\ell }F_{\ell }^{*}M_{\ell }J_{\ell }^{\prime }\bar{J}_{\ell }^{\prime }J_{\ell }^{\prime *}F_{\ell }^{\prime }\bar{F}_{\ell }^{\prime }F_{\ell }^{\prime *}M_{\ell }^{\prime }KK^{\prime }Q} \\&\quad \times \int f(\boldsymbol{\upsilon}){\mathrm{d} }^{3}\boldsymbol{\upsilon}\ (-1)^{Q}\mathcal{T} _{-Q}^{K^{\prime }}(j,\boldsymbol{\Omega }_{1})\mathcal{T} _{Q}^{K}(i,\boldsymbol{\Omega }) \\&\quad \times 3\dfrac{2L_{u}+1}{(2I+1)(2S+1)}\sqrt{(2K+1)(2K^{\prime }+1)}\ (-1)^{M_{\ell }-M_{\ell }^{\prime }} \\&\quad \times \left( -1\right) ^{J_{u}+\bar{J}_{u}+J_{u}^{\prime }+\bar{J}_{u}^{\prime }} \left( -1\right) ^{J_{\ell }+\bar{J}_{\ell }+J_{\ell }^{\prime }+\bar{J}_{\ell }^{\prime }} \left( -1\right) ^{F_{u}+\bar{F}_{u}+F_{u}^{\prime }+\bar{F}_{u}^{\prime }} \left( -1\right) ^{F_{\ell }+\bar{F}_{\ell }+F_{\ell }^{\prime }+\bar{F}_{\ell }^{\prime }} \\&\quad \times \sqrt{(2J_{u}+1)(2\bar{J}_{u}+1)(2J_{u}^{\prime }+1)(2\bar{J} _{u}^{\prime }+1)(2J_{\ell }+1)(2\bar{J}_{\ell }+1)(2J_{\ell }^{\prime }+1)(2 \bar{J}_{\ell }^{\prime }+1)} \\&\quad \times \sqrt{(2F_{u}+1)(2\bar{F}_{u}+1)(2F_{u}^{\prime }+1)(2\bar{F} _{u}^{\prime }+1)(2F_{\ell }+1)(2\bar{F}_{\ell }+1)(2F_{\ell }^{\prime }+1)(2 \bar{F}_{\ell }^{\prime }+1)} \\&\quad \times C_{J_{u}^{*}M_{J_{u}}\left( F_{u}^{*}M_{u}\right) }^{J_{u}}\left( B\right) C_{J_{u}^{*}M_{J_{u}}\left( F_{u}^{*}M_{u}\right) }^{\bar{J}_{u}}\left( B\right) C_{J_{u}^{\prime *}M_{J_{u}}^{\prime }\left( F_{u}^{\prime *}M_{u}^{\prime }\right) }^{J_{u}^{\prime }}\left( B\right) C_{J_{u}^{\prime *}M_{J_{u}}^{\prime }\left( F_{u}^{\prime *}M_{u}^{\prime }\right) }^{\bar{J}_{u}^{\prime }}\left( B\right) \\&\quad \times C_{J_{\ell }^{*}M_{J_{\ell }}\left( F_{\ell }^{*}M_{\ell }\right) }^{J_{\ell }}\left( B\right) C_{J_{\ell }^{*}M_{J_{\ell }}\left( F_{\ell }^{*}M_{\ell }\right) }^{\bar{J}_{\ell }}\left( B\right) C_{J_{\ell }^{\prime *}M_{J_{\ell }}^{\prime }\left( F_{\ell }^{\prime *}M_{\ell }^{\prime }\right) }^{J_{\ell }^{\prime }}\left( B\right) C_{J_{\ell }^{\prime *}M_{J_{\ell }}^{\prime }\left( F_{\ell }^{\prime *}M_{\ell }^{\prime }\right) }^{\bar{J}_{\ell }^{\prime }}\left( B\right) \\&\quad \times C_{F_{u}^{**}\left( J_{u}M_{J_{u}}\right) M_{u}}^{F_{u}}\left( B\right) C_{\bar{F}_{u}^{**}\left( \bar{J} _{u}M_{J_{u}}\right) M_{u}}^{\bar{F}_{u}}\left( B\right) C_{F_{u}^{\prime **}\left( J_{u}^{\prime }M_{J_{u}}^{\prime }\right) M_{u}^{\prime }}^{F_{u}^{\prime }}\left( B\right) C_{\bar{F}_{u}^{\prime **}\left( \bar{J}_{u}^{\prime }M_{J_{u}}^{\prime }\right) M_{u}^{\prime }}^{\bar{F} _{u}^{\prime }}\left( B\right) \\&\quad \times C_{F_{\ell }^{**}\left( J_{\ell }M_{J_{\ell }}\right) M_{\ell }}^{F_{\ell }}\left( B\right) C_{\bar{F}_{\ell }^{**}\left( \bar{J}_{\ell }M_{J_{\ell }}\right) M_{\ell }}^{\bar{F}_{\ell }}\left( B\right) C_{F_{\ell }^{\prime **}\left( J_{\ell }^{\prime }M_{J_{\ell }}^{\prime }\right) M_{\ell }^{\prime }}^{F_{\ell }^{\prime }}\left( B\right) C_{\bar{F}_{\ell }^{\prime **}\left( \bar{J}_{\ell }^{\prime }M_{J_{\ell }}^{\prime }\right) M_{\ell }^{\prime }}^{\bar{F} _{\ell }^{\prime }}\left( B\right) \\&\quad \times \left\{ \begin{array}{ccc} J_{u}&1&J_{\ell } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} J_{u}^{\prime }&1&\bar{J}_{\ell } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} \bar{J}_{u}&1&J_{\ell }^{\prime } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} \bar{J}_{u}^{\prime }&1&\bar{J}_{\ell }^{\prime } \\ L_{\ell }&S&L_{u} \end{array} \right\} \\&\quad \times \left\{ \begin{array}{ccc} F_{u}&1&F_{\ell } \\ J_{\ell }&I&J_{u} \end{array} \right\} \left\{ \begin{array}{ccc} F_{u}^{\prime }&1&\bar{F}_{\ell } \\ \bar{J}_{\ell }&I&J_{u} \end{array} \right\} \left\{ \begin{array}{ccc} \bar{F}_{u}&1&F_{\ell }^{\prime } \\ J_{\ell }&I&\bar{J}_{u} \end{array} \right\} \left\{ \begin{array}{ccc} \bar{F}_{u}^{\prime }&1&\bar{F}_{\ell }^{\prime } \\ \bar{J}_{\ell }&I&\bar{J}_{u} \end{array} \right\} \\&\quad \times \left( \begin{array}{ccc} F_{u}&1&F_{\ell } \\ -M_{u}&p&M_{\ell } \end{array} \right) \left( \begin{array}{ccc} F_{u}^{\prime }&1&\bar{F}_{\ell } \\ -M_{u}^{\prime }&p^{\prime }&M_{\ell } \end{array} \right) \left( \begin{array}{ccc} \bar{F}_{u}&1&F_{\ell }^{\prime } \\ -M_{u}&p^{\prime \prime \prime }&M_{\ell }^{\prime } \end{array} \right) \left( \begin{array}{ccc} \bar{F}_{u}^{\prime }&1&\bar{F}_{\ell }^{\prime } \\ -M_{u}^{\prime }&p^{\prime \prime }&M_{\ell }^{\prime } \end{array} \right) \\&\quad \times \left( \begin{array}{ccc} 1&1&K^{\prime } \\ -p&p^{\prime }&Q \end{array} \right) \left( \begin{array}{ccc} 1&1&K \\ -p^{\prime \prime \prime }&p^{\prime \prime }&Q \end{array} \right) \\&\quad \times \left\{ \dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\Gamma _{E}+\dfrac{{\mathrm{i} }\Delta E_{M_{u}M_{u}^{\prime }}}{\hbar }}\ \delta \left( \tilde{\nu }-\tilde{\nu }_{1}-\nu _{M_{\ell }M_{\ell }^{\prime }}\right) \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{M_{u}^{\prime }M_{\ell }}-\tilde{\nu }_{1}\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{M_{u}M_{\ell }}-\tilde{\nu }_{1}\right) \right] \right. \\&+\left[ \dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\dfrac{{\mathrm{i } }\Delta E_{M_{u}M_{u}^{\prime }}}{\hbar }}-\dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\Gamma _{E}+\dfrac{{\mathrm{i} }\Delta E_{M_{u}M_{u}^{\prime }} }{\hbar }}\right] \\&\times \left. \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{M_{u}^{\prime }M_{\ell }}-\tilde{\nu }_{1}\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{M_{u}M_{\ell }}-\tilde{\nu }_{1}\right) \right] \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{M_{u}^{\prime }M_{\ell }^{\prime }}-\tilde{\nu }\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{M_{u}M_{\ell }^{\prime }}-\tilde{\nu }\right) \right] \right\} \end{array} \ . \end{aligned} $$(A.6)

This equation is in excellent agreement as for the Racah algebra with Eq. (30) of Casini et al. (2014). The product of two coefficients C J * M J ( F * M ) J ( B ) C F * * ( J M J ) M F ( B ) $ C_{J^{\ast }M_{J}\left( F^{\ast }M\right)}^{J}\left( B\right) C_{F^{\ast \ast }\left( JM_{J}\right)M}^{F}\left( B\right) $ is equal to the coefficient C μ JF ( M ) $ C_{\mu}^{JF}\left( M\right) $ of Casini et al. (2014), because these coefficients all result from matrix diagonalization, performed in one step (FS + HFS) in Casini et al. (2014) and in two steps (FS and HFS) in our case. A similar coefficient is visible in Eq. (3.58) of Landi Degl’Innocenti & Landolfi (2004).

The following equation replaces Eq. (40) of Bommier (2017), by introducing the (2I + 1) denominator

R ij ( ν , ν 1 , Ω , Ω 1 ; B = 0 ) = J u F u J u F u J F J F K Q f ( υ ) d 3 υ ( 1 ) Q T Q K ( j , Ω 1 ) T Q K ( i , Ω ) × 3 2 L u + 1 ( 2 I + 1 ) ( 2 S + 1 ) ( 2 J u + 1 ) ( 2 J u + 1 ) ( 2 J + 1 ) ( 2 J + 1 ) ( 2 F u + 1 ) ( 2 F u + 1 ) ( 2 F + 1 ) ( 2 F + 1 ) ( 1 ) F F × { J u 1 J L S L u } { J u 1 J L S L u } { J u 1 J L S L u } { J u 1 J L S L u } × { F u 1 F J I J u } { F u 1 F J I J u } { F u 1 F J I J u } { F u 1 F J I J u } × { K F u F u F 1 1 } { K F u F u F 1 1 } × { Γ R Γ R + Γ I + Γ E + i Δ E F u F u ħ δ ( ν ~ ν ~ 1 ν F F ) [ 1 2 Φ ba ( ν F u F ν ~ 1 ) + 1 2 Φ ba ( ν F u F ν ~ 1 ) ] + [ Γ R Γ R + Γ I + 1 2 [ D ( K ) ( α u F u ) + D ( K ) ( α u F u ) ] + i Δ E F u F u ħ Γ R Γ R + Γ I + Γ E + i Δ E F u F u ħ ] × [ 1 2 Φ ba ( ν F u F ν ~ 1 ) + 1 2 Φ ba ( ν F u F ν ~ 1 ) ] [ 1 2 Φ ba ( ν F u F ν ~ ) + 1 2 Φ ba ( ν F u F ν ~ ) ] } , $$ \begin{aligned} \begin{array}{ll} \mathcal{R} _{ij}( \nu ,\nu _{1},\boldsymbol{\Omega },\boldsymbol{\Omega }_{1}; \boldsymbol{B} = \boldsymbol{0})&=\mathop {\displaystyle \sum }\limits_{J_{u}F_{u}J_{u}^{\prime }F_{u}^{\prime }J_{\ell }F_{\ell }J_{\ell }^{\prime }F_{\ell }^{\prime }KQ}\int f(\boldsymbol{\upsilon}) {\mathrm{d} }^{3}\boldsymbol{\upsilon}\ (-1)^{Q}\mathcal{T} _{-Q}^{K}(j,\boldsymbol{\Omega }_{1}) \mathcal{T} _{Q}^{K}(i,\boldsymbol{\Omega }) \\&\quad \times 3\dfrac{2L_{u}+1}{(2I+1)(2S+1)}(2J_{u}+1)(2J_{u}^{\prime }+1)(2J_{\ell }+1)(2J_{\ell }^{\prime }+1)(2F_{u}+1)(2F_{u}^{\prime }+1)(2F_{\ell }+1)(2F_{\ell }^{\prime }+1)\ (-1)^{F_{\ell }-F_{\ell }^{\prime }} \\&\quad \times \left\{ \begin{array}{ccc} J_{u}&1&J_{\ell } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} J_{u}^{\prime }&1&J_{\ell } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} J_{u}&1&J_{\ell }^{\prime } \\ L_{\ell }&S&L_{u} \end{array} \right\} \left\{ \begin{array}{ccc} J_{u}^{\prime }&1&J_{\ell }^{\prime } \\ L_{\ell }&S&L_{u} \end{array} \right\} \\&\quad \times \left\{ \begin{array}{ccc} F_{u}&1&F_{\ell } \\ J_{\ell }&I&J_{u} \end{array} \right\} \left\{ \begin{array}{ccc} F_{u}^{\prime }&1&F_{\ell } \\ J_{\ell }&I&J_{u}^{\prime } \end{array} \right\} \left\{ \begin{array}{ccc} F_{u}&1&F_{\ell }^{\prime } \\ J_{\ell }&I&J_{u} \end{array} \right\} \left\{ \begin{array}{ccc} F_{u}^{\prime }&1&F_{\ell }^{\prime } \\ J_{\ell }&I&J_{u}^{\prime } \end{array} \right\} \\&\quad \times \left\{ \begin{array}{ccc} K&F_{u}&F_{u}^{\prime } \\ F_{\ell }&1&1 \end{array} \right\} \left\{ \begin{array}{ccc} K&F_{u}&F_{u}^{\prime } \\ F_{\ell }^{\prime }&1&1 \end{array} \right\} \\&\quad \times \left\{ \dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\Gamma _{E}+\dfrac{{\mathrm{i} }\Delta E_{F_{u}F_{u}^{\prime }}}{\hbar }}\ \delta \left( \tilde{\nu }-\tilde{\nu }_{1}-\nu _{F_{\ell }F_{\ell }^{\prime }}\right) \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{F_{u}^{\prime }F_{\ell }}-\tilde{\nu }_{1}\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{F_{u}F_{\ell }}-\tilde{\nu }_{1}\right) \right] \right. \\&\quad +\left[ \dfrac{\Gamma _{R}}{\Gamma _{R}+\Gamma _{I}+\dfrac{1}{2} \left[ D^{(K)}(\alpha _{u}F_{u})+D^{(K)}(\alpha _{u}F_{u}^{\prime })\right] + \dfrac{{\mathrm{i} }\Delta E_{F_{u}F_{u}^{\prime }}}{\hbar }}-\dfrac{\Gamma _{R} }{\Gamma _{R}+\Gamma _{I}+\Gamma _{E}+\dfrac{{\mathrm{i} }\Delta E_{F_{u}F_{u}^{\prime }}}{\hbar }}\right] \\&\quad \times \left. \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{F_{u}^{\prime }F_{\ell }}-\tilde{\nu }_{1}\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{F_{u}F_{\ell }}-\tilde{\nu }_{1}\right) \right] \left[ \dfrac{1}{2}\Phi _{ba}\left( \nu _{F_{u}^{\prime }F_{\ell }^{\prime }}-\tilde{\nu }\right) +\dfrac{1}{2}\Phi _{ba}^{*}\left( \nu _{F_{u}F_{\ell }^{\prime }}-\tilde{\nu }\right) \right] \right\} \end{array}\ , \end{aligned} $$(40)

Acknowledgments

The author is very grateful to Ernest Alsina Ballester for having pointed out the errors. Ernest Alsina Ballester redid the calculations in the metalevels formalism (Landi Degl’Innocenti et al. 1997).

References

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