$\bar {\sf {H}}^{+}$ ion production from collisions between antiprotons and excited positronium: cross sections calculations in the framework of the GBAR experiment

The capture of an electron with Ps formation in e⁺–H collisions is considered (reaction (1)). Figure 1 describes the coordinates used in this section; for further discussion and details about the CDW-FS model of this reaction and on the choice of the coordinates, see [23].

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The wave function corresponding to the entrance channel is

$$\begin{equation} \xi _{\alpha}^{(+)}=\varphi_{n_{\rm h} l_{\rm h} m_{\rm h}}({\boldsymbol{r}}) {\cal F}_{{\boldsymbol{k}}_{\alpha}}^{(+)}({\boldsymbol{R}}), \end{equation} \tag{ 5 }$$

where $\varphi _{n_{\mathrm { h}}l_{\mathrm { h}}m_{\mathrm { h}}}({\boldsymbol {r}}) = R_{n_{\mathrm { h}}l_{\mathrm { h}}}(r)Y_{l_{\mathrm { h}}m_{\mathrm { h}}}(\hat {{\boldsymbol {r}}})$ is the wave function of the electron in the hydrogen atom in the (n_hl_hm_h) state and ${\cal F}_{{\boldsymbol {k}}_{\alpha }}^{(+)}({\boldsymbol {R}})$ is the Coulomb wave function of the positron in the continuum of the hydrogen target. The final state is described by the wave function

$$\begin{equation} \xi _{\beta}^{(-)}=\psi _{n_{\rm p}l_{\rm p}m_{\rm p}}({\boldsymbol{\rho}}) {\cal F}_{{\boldsymbol{k}}_{-}}^{(-)}({\boldsymbol{r}}) {\cal F}_{{\boldsymbol{k}}_{+}}^{(-)}({\boldsymbol{R}}), \end{equation} \tag{ 6 }$$

where $\psi _{n_{\mathrm { p}}l_{\mathrm { p}}m_{\mathrm { p}}}({\boldsymbol {\rho }}) = \tilde {R}_{n_{\mathrm { p}}l_{\mathrm { p}}}(\rho )Y_{l_{\mathrm { p}}m_{\mathrm { p}}}(\hat {\boldsymbol {\rho }})$ is the wave function of the electron bound in the Ps atom, formed in the (n_pl_pm_p) state, ${\cal F}_{{\boldsymbol {k}}_{-}}^{(-)}({\boldsymbol {r}})$ and ${\cal F}_{{\boldsymbol {k}}_{+}}^{(-)}({\boldsymbol {R}})$ are, respectively, the Coulomb wave functions of the outgoing electron and the positron in the continuum of the residual proton of charge Z_T = 1. The Coulomb wave functions write as

$$\begin{equation} {\cal F}_{{\boldsymbol{k}}_{\alpha}}^{(+)}({\boldsymbol{R}})=N_{\alpha +}\exp \left[ {\rm i}\frac{{\boldsymbol{k}}_{\alpha}}{\mu_{\alpha}}.{\boldsymbol{R}}\right] {}_{1}F_{1}\left(-{\rm i}\alpha_{+}; 1; -{\rm i}\frac{{\boldsymbol{k}}_{\alpha}}{\mu_{\alpha}}.{\boldsymbol{R}} + {\rm i}\frac{k_{\alpha}}{\mu_{\alpha}}R\right) \end{equation} \tag{ 7 }$$

with $\alpha _{+}=(Z_{\mathrm {T}}-1)\frac {\mu _{\alpha }}{k_{\alpha }}=0$ since the charge Z_T − 1 of the hydrogen target is 0 (in which case, the Coulomb wave function actually reduces to a plane wave) and $N_{\alpha +}= \Gamma (1+{\mathrm { i}}\alpha _{+})\exp \left (-\frac {\pi }{2}\alpha _{+}\right )$. k_α is the wavevector of the reduced positron in the entrance channel and μ_α is the reduced mass of the positron–target system ($\mu _{\alpha }=\frac {m(M+m)}{M+2m}\simeq m$);

$$\begin{equation} {\cal F}_{{\boldsymbol{k}}_{+}}^{(-)}({\boldsymbol{R}})=N_{\beta +}\exp \left[ {\rm i}\frac{{\boldsymbol{k}} _{+}} {\mu_{\beta}} .{\boldsymbol{R}}\right] {}_{1}F_{1}\left({\rm i}\beta_{+}; 1; -{\rm i}\frac{{\boldsymbol{k}}_{+}}{\mu_{\beta}}.{\boldsymbol{R}} - {\rm i}\frac{k_{+}}{\mu_{\beta}}R\right) \end{equation} \tag{ 8 }$$

with $\beta _{+}=Z_{\mathrm {T}}\frac {\mu _{\beta }}{k_{+}}$ and $N_{\beta +}= \Gamma (1-{\mathrm { i}}\beta _{+})\exp \left (-\frac {\pi }{2}\beta _{+}\right )$. k₊ is the wavevector of the reduced positron in the exit channel and μ_β is the reduced mass of the Ps–residual target system ($\mu _{\beta }=\frac {2m\times M}{M+2m}\simeq 2m$);

$$\begin{equation} {\cal F}_{{\boldsymbol{k}}_{-}}^{(-)}({\boldsymbol{r}})=N_{\beta -}\exp \left[ {\rm i}\frac{{\boldsymbol{k}}_{-}}{\mu_{\beta}}\cdot{\boldsymbol{r}}\right] {}_{1}F_{1}\left(-{\rm i}\beta_{-}; 1; -{\rm i}\frac{{\boldsymbol{k}}_{-}}{\mu_{\beta}}\cdot{\boldsymbol{r}} - {\rm i}\frac{k_{-}}{\mu_{\beta}}r\right) \end{equation} \tag{ 9 }$$

with $\beta _{-}=Z_{\mathrm {T}}\frac {\mu _{\beta }}{k_{-}}$ and $N_{\beta -}= \Gamma (1+{\mathrm { i}}\beta _{-})\exp \left (+\frac {\pi }{2}\beta _{-}\right )$. k₋ is the wavevector of the reduced electron in the exit channel. ${\boldsymbol {k}}_{+} \simeq {\boldsymbol {k}}_{-} \simeq \frac {{\boldsymbol {k}}_{\beta }}{\mu _{\beta }}$, where k_β is the wavevector of the reduced Ps in the exit channel. Even if the exit channel of the reaction (1) is not Coulombic as it contains two species with only one having the overall charge, the CDW-FS model includes distortions in the final channel which are related to the Coulomb continuum states of the positron and the electron (of the Ps atom) in the field of the residual target (the proton). Therefore, the continuum wave functions (8) and (9) do not reduce to plane waves. If no distortions are included in the final channel, the CBA is obtained. This situation is similar to the one described in [1, 26]. By artificially fixing the charges β₋ and β₊ to zero, we have got back almost the same results, which are quoted in [1]. The latter will be discussed in section 3.1.2.

In order to compute the transition matrix element, the following partial wave expansion of the Coulomb wave functions has been used:

$$\begin{eqnarray} &&{\cal F}_{{\boldsymbol{k}}_{}}^{(\pm)}({\boldsymbol{r}})=4\pi \; \sum_{lm}(\mathrm{i})^l\,{\rm e}^{\pm {\rm i} \delta _l}\; \frac{1}{kr}\;F_l(kr)\;Y_{lm}^{*}(\hat{{\boldsymbol{k}}})\; Y_{lm}^{}(\hat {{\boldsymbol{r}}}), \end{eqnarray} \tag{ 10 }$$

where the functions F_l(kr) are the Coulomb radial functions with the Sommerfeld parameters η = β_± or α₊ and δ_l are the usual Coulomb phase shifts $\delta _l=\arg \Gamma (l+1+\mathrm {i}\eta )$. The spherical harmonic function $Y_{l_{\mathrm { p}}m_{\mathrm { p}}}(\hat {{\boldsymbol {\rho }}})$ has been treated using the development [27]

$$\begin{eqnarray}\fl Y_{LM}(\hat{{\boldsymbol{\rho}}})\kern-1pt =\kern-1pt \rho ^{-L}(-1)^{L+M} \sum_{\lambda \mu} \left(\frac{\hat{L}! \hat{L} 4\pi}{\hat{\lambda}! \widehat{(L - \lambda)}!} \right)^{\frac{1}{2}} R^{L-\lambda}r^{\lambda}\kern-1pt\left(\kern-1pt\begin{array}{@{}ccc@{}} L-\lambda & \lambda & L\\ M-\mu & \mu & -M \end{array}\kern-1pt\right)\kern-1pt Y_{L-\lambda M-\mu}(\hat{{\boldsymbol{R}}}) Y_{\lambda\mu}(-\hat{{\boldsymbol{r}}}),\nonumber\\ \end{eqnarray} \tag{ 11 }$$

where 0 ⩽ λ ⩽ L and −λ ⩽ μ ⩽ λ.

The perturbative potential in the entrance channel (initial state) is

$$\begin{equation} V_{\alpha} = \left(\frac{1}{R}-\frac{1}{\rho}\right). \end{equation} \tag{ 12 }$$

The transition matrix element is then

$$\begin{eqnarray} T_{\alpha\beta}^{(-)} & = & \left< \xi _{\beta}^{(-)} \right| V_{\alpha} \left| \xi _{\alpha}^{(+)} \right> \nonumber \\ & = & (-1)^{l_{\rm p}+m_{\rm h}} \frac{(4\pi)^{\frac{3}{2}}}{k_{+}k_{\alpha}k_{-}} \hat{l}_{\rm h}^{\frac{1}{2}}\hat{l}_{\rm p}^{\frac{1}{2}}(\hat{l}_{\rm p}!)^{\frac{1}{2}} \sum_{l_{\mathrm{i}} \mathcal{L}} \mathrm{i}^{l_{\mathrm{i}}} {\rm e}^{{\rm i}\delta _{l_{\mathrm{i}}}} \hat{\cal{L}}^{\frac{1}{2}}\hat{l}_{\mathrm{i}} {\cal{S}}_{l_{\mathrm{i}} \cal{L}} Y_{{\cal{L}} m_{\rm h}-m_{\rm p}}(\hat{{\boldsymbol{k}}}_{\beta})\qquad \end{eqnarray} \tag{ 13 }$$

with

$$\begin{eqnarray} &&\fl{\cal{S}}_{l_{\mathrm{i}} \cal{L}} = \sum_{l_{\mathrm{f}} l l' L L'} \sum_{0 \leqslant \lambda \leqslant l_{\rm p}} \mathrm{i}^{-l-l_{\mathrm{f}}} {\rm e}^{{\rm i}(\delta_{l}+ \delta_{l_{\mathrm{f}}})} {\cal{A}} ^{\lambda l_{\mathrm{f}} l l' L L'}_{l_{\mathrm{i}}{\cal{L}}} {\cal{R}} ^{\lambda l l'}_{l_{\mathrm{i}}l_{\mathrm{f}}} , \nonumber\\ &&\fl {\cal{A}} ^{\lambda l_{\mathrm{f}} l l' L L'}_{l_{\mathrm{i}}{\cal{L}}} =(-1)^{\lambda} \left(\frac{\hat{l}_{\mathrm{f}} \hat{l} \hat{l}' \hat{L} \hat{L}'}{((2\lambda)!(2(l_{\rm p}-\lambda))!)^{\frac{1}{2}}}\right) ^{\frac{1}{2}} \nonumber\\ \times \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l & {\cal{L}}\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l' & l_{\rm h} & L\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\rm p}-\lambda & l' & L'\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} \lambda & l & L\\ 0 & 0 & 0 \end{array} \right)\nonumber\\ \times \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & L' & l_{\mathrm{i}}\\ 0 & 0 & 0 \end{array} \right) \sum_{\mu m_{\mathrm{f}}} (-1)^{m_{\mathrm{f}}} \left( \begin{array}{@{}c c c@{}} \lambda & l & L\\ \mu & m_{\mathrm{f}}+m_{\rm p}-m_{\rm h} & m_{\rm h}-m_{\rm p}-\mu-m_{\mathrm{f}} \end{array} \right)\nonumber\\ \times \left( \begin{array}{@{}c c c@{}} l_{\rm p}-\lambda & \lambda & l_{\rm p}\\ -m_{\rm p}-\mu & \mu & m_{\rm p} \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\rm p}-\lambda & l' & L'\\ -m_{\rm p}-\mu & \mu+m_{\rm p}+m_{\mathrm{f}} & -m_{\mathrm{f}} \end{array} \right)\nonumber\\ \times \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & L' & l_{\mathrm{i}}\\ -m_{\mathrm{f}} & m_{\mathrm{f}} & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l & {\cal{L}}\\ m_{\mathrm{f}} & m_{\rm h}-m_{\rm p}-m_{\mathrm{f}} & m_{\rm p}-m_{\rm h} \end{array} \right)\nonumber\\ \times \left( \begin{array}{@{}c c c@{}} l' & l_{\rm h} & L\\ -m_{\mathrm{f}}-m_{\rm p}-\mu & m_{\rm h} & m_{\mathrm{f}}+m_{\rm p}+\mu-m_{\rm h} \end{array} \right),\nonumber\\ &&\fl {\cal{R}} ^{\lambda l l'}_{l_{\mathrm{i}}l_{\mathrm{f}}} =\int\nolimits_{0}^{\infty} {\rm d}R\ R^{l_{\rm p}-\lambda} F_{l_{\mathrm{f}}}(k_{+}R){\cal V}_{\lambda l l'}(R) F_{l_{\mathrm{i}}}(k_{\alpha}R) , \nonumber\\ &&\fl {\cal V}_{\lambda l l'}(R) = \int\nolimits_{0}^{\infty} {\rm d}r\ r^{\lambda + 1} {\cal R}_{n_{\rm h}l_{\rm h}}(r){\cal J}^{l_{\rm p}}_{l'}(r, R) F_{l}(k_{-}r) , \nonumber\\ &&\fl {\cal J}^{l_{\rm p}}_{l'}(r, R) = \frac{1}{2} \int\nolimits_{-1}^{1} {\rm d}u\,\rho ^{-l_{\rm p}} \tilde{\cal {R}}_{n_{\rm p}l_{\rm p}}(\rho) \left( \frac{1}{R} - \frac{1}{\rho} \right) P_{l'}(u) . \end{eqnarray} \tag{ 14 }$$

In the above expression, the variable u is defined as $\rho = \sqrt {r^{2} + R^{2} + 2rRu}$ and $\hat {l}$ indicates $\hat {l}=2l + 1$. P_l' is the Legendre polynomial of degree l', coming from the multipole expansion

$$\begin{equation} \rho ^{-l_{\rm p}} \tilde{\cal {R}}_{n_{\rm p}l_{\rm p}}(\rho) \left(\frac{1}{R} - \frac{1}{\rho} \right) = 4\pi \sum_{l'm'} {\cal J}^{l_{\rm p}}_{l'}(r, R) Y^{*}_{l'm'}(\hat{\boldsymbol{R}}) Y_{l'm'}(\hat {{\boldsymbol{r}}}). \end{equation} \tag{ 15 }$$

The CDW-FS total cross section for Ps(n_p,l_p) formation from H(n_h,l_h) (reaction (1)) is given by

$$\begin{eqnarray} \sigma^{3\mathrm{B}, 1} _{n_{\rm h}l_{\rm h}; n_{\rm p}l_{\rm p}} & = & \frac{1}{\hat{l}_{\rm h}} \sum_{m_{\rm h} m_{\rm p}} \frac{1}{4\pi^{2}} \frac{k_{\beta}}{k_{\alpha}} \mu _{\alpha} \mu _{\beta} \int {\rm d}\hat{{\boldsymbol{k}}}_{\beta} \left| T^{(-)}_{\alpha\beta}\right|^{2} \nonumber \\ & = & 16\pi \frac{ k_{\beta}}{k_{-}^{2}k_{+}^{2}k_{\alpha}^{3}} \mu _{\alpha} \mu _{\beta} \hat{l}_{\rm p}! \hat{l}_{\rm p} \sum_{m_{\rm h} m_{\rm p}} \sum_{l_{\mathrm{i}} l'_{\mathrm{i}} {\cal L}} \mathrm{i}^{l_{\mathrm{i}}-l'_{\mathrm{i}}} \,{\rm e}^{{\rm i}(\delta_{l_{\mathrm{i}}} - \delta_{l'_{\mathrm{i}}})} \hat{l}_{i} \hat{l}'_{i} \hat{{\cal L}} \left({\cal S}^{*}_{l'_{\mathrm{i}}{\cal L}}\times{\cal S}_{l_{\mathrm{i}}{\cal L}}\right).\qquad \end{eqnarray} \tag{ 16 }$$

These expressions can be simplified in the cases when either l_h or l_p, or both, are taken to be 0; details can be found in appendix A.

We develop here the very general case where both l_h and l_p are different from 0 (the particular cases when l_h = 0, l_p = 0 and l_h = l_p = 0 are described in appendix B) and the wave function for H⁻ is of the form

$$\begin{eqnarray} \Phi _{\alpha}({\boldsymbol{r}}_{1},{\boldsymbol{r}}_{2}) & = & f (r_{1},r_{2})\left( 1 + g(r_{12})\right), \end{eqnarray} \tag{ 17 }$$

where r₁₂ is defined as $r_{12}= \left | {\boldsymbol {r}}_{1}-{\boldsymbol {r}}_{2} \right |$. This wave function has been treated using a partial wave expansion, which can be written as follows:

$$\begin{eqnarray} &&\Phi _{\alpha}({\boldsymbol{r}}_{1},{\boldsymbol{r}}_{2}) = 4\pi \sum_{l_{\mathrm{t}} m_{\mathrm{t}}} \tilde{h}_{l_{\mathrm{t}}}(r_{1},r_{2}) Y^{*}_{l_{\mathrm{t}} m_{\mathrm{t}}}(\hat{{\boldsymbol{r}}}_{1}) Y_{l_{\mathrm{t}} m_{\mathrm{t}}}(\hat{{\boldsymbol{r}}}_{2}) \end{eqnarray} \tag{ 18 }$$

with

$$\begin{equation} \left\lbrace \begin{array}{@{}l@{\quad}l@{\quad}l@{}} \tilde{h}_{0}=f + \tilde{g}_{0} & \mathrm{if}& l_{\mathrm{t}} = 0, \\ \tilde{h}_{l_{\mathrm{t}}}=\tilde{g}_{l_{\mathrm{t}}}& & \mathrm{else} \end{array} \right. \end{equation} \tag{ 19 }$$

and

$$\begin{eqnarray} &&\tilde{g}_{l_{\mathrm{t}}}(r_{1},r_{2}) = \frac{f(r_{1},r_{2})}{2} \int\nolimits_{-1}^{1} {\rm d}u\ g(r_{12}) P_{l_{\mathrm{t}}}(u), \end{eqnarray} \tag{ 20 }$$

where $r_{12}= \sqrt {r_{1}^{2}+r_{2}^{2}+2r_{1}r_{2}u}$. Three different wave functions have been used for modelling H⁻. The first one is the 'uncorrelated' Chandrasekhar wave function (labelled UC in the following) [28]. This is a simple and convenient wave function which takes into account the radial correlations between the two electrons of H⁻ but no angular correlations. It has been used in several papers already cited and is usually considered as a sufficient level of description for H⁻. Φ^UC_α(r₁,r₂) is given by

$$\begin{eqnarray*} &&\left\lbrace \begin{array}{@{}l@{}} f (r_{1},r_{2})=\displaystyle\frac{N_{1}}{4\pi} \left( {\rm e}^{-a_{\mathrm{uc}}r_{1}-b_{\mathrm{uc}}r_{2}} + {\rm e}^{-a_{\mathrm{uc}}r_{2}-b_{\mathrm{uc}}r_{1}}\right), \\ g(r_{12})=0 , \end{array} \right. \end{eqnarray*}$$

where a_uc = 1.0392,b_uc = 0.2831 and the normalization constant N₁ = 0.3948. The binding energy of H⁻ computed with this function is E_H⁻ = −0.5133, which has to be compared with the exact theoretical value, −0.5277 [29]. This wave function can be modified by introducing g ≠ 0, so that now it takes into account angular correlations. This 'correlated' Chandrasekhar wave function Φ^CC_α(r₁,r₂) is defined by [28]

$$\begin{eqnarray*} &&\left\lbrace \begin{array}{@{}l@{}} f (r_{1},r_{2})=\displaystyle\frac{N_{2}}{4\pi} \left( {\rm e}^{-a_{\rm cc}r_{1}-b_{\rm cc}r_{2}} + {\rm e}^{-a_{\rm cc}r_{2}-b_{\rm cc}r_{1}}\right), \\ g(r_{12})=D r_{12} , \end{array} \right. \end{eqnarray*}$$

where a_cc = 1.0748, b_cc = 0.4776, D = 0.3121 and the normalization constant N₂ = 0.3942. The binding energy is then E_H⁻ = −0.5259. Note that a misprint turned the sign of D into negative in several papers (in [17], for instance) whereas it is positive in the original Chandrasekhar paper, as it should be. The last wave function chosen to describe H⁻ is the Le Sech wave function (labelled LS in the following) [30]. Φ^LS_α(r₁,r₂) decomposes as follows:

$$\begin{eqnarray*} &&\left\lbrace \begin{array}{@{}l@{}} f (r_{1},r_{2})=N_{3} \,{\rm e}^{-r_{1}-r_{2}}\left(\cosh (\lambda r_{1}) + \cosh (\lambda r_{2}) +\beta \left( r_{1}-r_{2}\right)^{2}\right), \\ g(r_{12})=\gamma r_{12}\,{\rm e}^{-\alpha r_{12}}, \end{array} \right. \end{eqnarray*}$$

where α = 0.05, β = 0.04, γ = 0.50, λ = 0.57 and the normalization constant N₃ = 0.03615 [31]. The binding energy calculated with the LS wave function is E_H⁻ = −0.5270. This wave function has been emphasized as a very accurate description of H⁻ [32]. Table 1 shows the contribution of the different angular components in the H⁻ wave function and the convergence of the normalization when using the partial wave expansion (18); in this table, a_lt is defined by $a_{l_{\mathrm {t}}} = (4\pi )^2 \int \int {\mathrm { d}}r_1\, {\mathrm { d}}r_2\ r_1 ^2 r_2 ^2 | \tilde {h}_{l_{\mathrm {t}}} (r_1 , r_2 ) | ^2$.

Figure 2 describes the coordinates used for the four-body reaction, similarly to [24] and derived from [33]. The wave function of the initial state is

$$\begin{equation} \xi_{\alpha}^{(+)}=\Phi _{\alpha}({\boldsymbol{r}}_{1},{\boldsymbol{r}}_2) {\cal F}_{{\boldsymbol{k}}_{\alpha}}^{(+)}({\boldsymbol{R}}) \end{equation} \tag{ 21 }$$

with the Coulomb wave function ${\cal F}_{{\boldsymbol {k}}_{\alpha }}^{(+)}({\boldsymbol {R}})$ describing the positron in the continuum of H⁻ (see equation (7)). k_α is the wavevector of the reduced positron in the entrance channel with $\mu _{\alpha }=\frac {m(M+2m)}{M+3m}\simeq m$ the reduced mass of the positron–H⁻ target system. The wave function for the final state is

$$\begin{equation}\fl \xi _{\beta}^{(-)}=\frac{1}{\sqrt{2}}\left\lbrace \psi _{\beta}({\boldsymbol{\rho}}_{1}) {\cal F}_{{\boldsymbol{k}}_{-}} ^{(-)}({\boldsymbol{r}}_{1}) \varphi _{n_{\rm h}l_{\rm h}m_{\rm h}}({\boldsymbol{r}}_{2}) + \psi _{\beta}({\boldsymbol{\rho}}_{2}) {\cal F}_{{\boldsymbol{k}}_{-}}^{(-)}({\boldsymbol{r}}_{2}) \varphi _{n_{\rm h}l_{\rm h}m_{\rm h}}({\boldsymbol{r}}_{1}) \right\rbrace {\cal F}_{{\boldsymbol{k}}_{+}}^{(-)}({\boldsymbol{R}}), \end{equation} \tag{ 22 }$$

where $\psi _{\beta }({\boldsymbol {\rho }}_{i}) {\cal F}_{{\boldsymbol {k}}_{-}}^{(-)}({\boldsymbol {r}}_{i})$ describes electron i captured in the outgoing Ps while electron j remains in the residual hydrogen atom (φ_nhlhmh(r_j)); ${\cal F}_{{\boldsymbol {k}}_{+}}^{(-)}({\boldsymbol {R}})$ is the Coulomb wave function of the outgoing positron (see equation (8)). k_β is the wavevector of the reduced Ps in the exit channel; we also define $\mu _{\beta } = \frac {2m(M+m)}{M+3m}\simeq 2m$, the reduced mass of the Ps–residual target system. As defined in the previous section, ${\boldsymbol {k}}_{+} \simeq {\boldsymbol {k}}_{-} \simeq \frac {{\boldsymbol {k}}_{\beta }}{\mu _{\beta }}$. Again, a partial wave expansion of the Coulomb wave functions has been used (see equation (10)). The Sommerfeld parameters are $\alpha _{+} = (Z_{\mathrm {T}} - 2)\frac {\mu _{\alpha }}{k_{\alpha }}$, $\beta _+ = (Z_{\mathrm {T}} - 1)\frac {\mu _{\beta }}{k_{+}}$ and $\beta _- = (Z_{\mathrm {T}} - 1)\frac {\mu _{\beta }}{k_{-}}$; since the proton charge Z_T = 1, we have β_± = 0: therefore, it should be noted that the Coulomb wave functions for the electron and the positron in the exit channel are actually plane waves. The Coulomb phase shifts δ_l are defined as in the previous section. The ¹S symmetry of the initial state of H⁻ imposes the symmetry in the final state, hence the choice of expression (22) for ξ⁽⁻⁾_β.

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The chosen short-range perturbative potential is

$$\begin{equation} V_{\alpha}=\left( \frac{2}{R} - \frac{1}{\rho _{1}} - \frac{1}{\rho _{2}}\right) \end{equation} \tag{ 23 }$$

and the four-body CDW-FS matrix element T⁽⁻⁾_αβ reads as

$$\begin{eqnarray} T^{(-)}_{\alpha \beta} & = & \left\langle \xi _{\beta}^{(-)} \right| V_{\alpha} \left| \xi _{\alpha}^{(+)} \right\rangle \nonumber \\ &=&\int {\rm d}{\boldsymbol{R}} {\cal F}_{{\boldsymbol{k}}_{+}}^{(-)*}({\boldsymbol{R}}) V_{T}({\boldsymbol{R}}){\cal F}_{{\boldsymbol{k}}_\alpha}^{(+)}({\boldsymbol{R}}) \end{eqnarray} \tag{ 24 }$$

with

$$\begin{eqnarray} &&\fl V_{T}({\boldsymbol{R}})= \sqrt{2}\int \int {\rm d}{\boldsymbol{r}}_{1} \,{\rm d}{\boldsymbol{r}}_{2} \tilde{R}_{n_{\rm p}l_{\rm p}}(\rho _{1}) Y_{l_{\rm p}m_{\rm p}}^{*}(\hat{{\boldsymbol{\rho}}}_{1}){\cal F}_{{\boldsymbol{k}}_{-}}^{(-)*}({\boldsymbol{r}}_{1})R_{n_{\rm h}l_{\rm h}}({\boldsymbol{r}}_{2})Y_{l_{\rm h}m_{\rm h}}^{*}(\hat{{\boldsymbol{r}}}_{2}) \left( \frac{1}{R}-\frac{1}{\rho _{1}} \right)\Phi _{\alpha}({\boldsymbol{r}}_{1},{\boldsymbol{r}}_{2}) \nonumber \\ &&\fl \hphantom{V_{T}({\boldsymbol{R}})=} + \sqrt{2}\int\kern-4pt \int\kern-1pt {\rm d}{\boldsymbol{r}}_{1} \,{\rm d}{\boldsymbol{r}}_{2} \tilde{R}_{n_{\rm p}l_{\rm p}}(\rho _{1}) Y_{l_{\rm p}m_{\rm p}}^{*}(\hat{{\boldsymbol{\rho}}}_{1}){\cal F}_{{\boldsymbol{k}}_{-}}^{(-)*}({\boldsymbol{r}}_{1})R_{n_{\rm h}l_{\rm h}}({\boldsymbol{r}}_{2})Y_{l_{\rm h}m_{\rm h}}^{*}(\hat{{\boldsymbol{r}}}_{2}) \left( \frac{1}{R}-\frac{1}{\rho _{2}} \right)\Phi _{\alpha}({\boldsymbol{r}}_{1},{\boldsymbol{r}}_{2}) \nonumber \\ &&\fl \hphantom{V_{T}({\boldsymbol{R}})}\equiv V_{\rm cap}({\boldsymbol{R}}) + V_{\rm exc}({\boldsymbol{R}}), \end{eqnarray} \tag{ 25 }$$

where 'cap' stands for capture and 'exc' for excitation, notations that are justified in the case of the uncorrelated Chandrasekhar wave function (see section 2.2.2). The transition matrix element T⁽⁻⁾_αβ can be thus written as a sum of two terms, t_cap and t_exc. After lengthy calculations, the former term may be expressed as

$$\begin{eqnarray} t_{\rm cap}&=& \int {\rm d}{\boldsymbol{R}}\ {\cal F}_{{\boldsymbol{k}}_{+}}^{(-)*}({\boldsymbol{R}})\ V_{\rm cap}({\boldsymbol{R}})\ {\cal F}_{{\boldsymbol{k}}_\alpha}^{(+)}({\boldsymbol{R}}) \nonumber \\ &=&(-1)^{l_{\rm p}} \sqrt{2} \frac{(4\pi)^{\frac{5}{2}}}{k_{-}k_{\alpha}k_{+}} \left( \hat{l}_{\rm p}! \hat{l}_{\rm p}\hat{l}_{\rm h}\right)^{\frac{1}{2}} \sum_{l_{\mathrm{i}} {\cal L}} \mathrm{i}^{l_{\mathrm{i}}}\, {\rm e}^{{\rm i} \delta _{l_{\mathrm{i}}}} \hat{l}_{\mathrm{i}}\hat{{\cal L}}^{\frac{1}{2}} {\cal S}_{l_{\mathrm{i}}{\cal L}} Y_{{\cal L} -m_{\rm p}-m_{\rm h}}(\hat{{\boldsymbol{k}}}_{\beta}),\qquad \end{eqnarray} \tag{ 26 }$$

where

$$\begin{eqnarray} &&\fl {\cal S}_{l_{\mathrm{i}}{\cal L}}=\sum_{l_{\mathrm{f}}ll'LL'\lambda} \mathrm{i}^{-l-l_{\mathrm{f}}}\, {\rm e}^{{\rm i}(\delta _{l} +\delta _{l_{\mathrm{f}}})} {\cal A}_{l_{\mathrm{i}}{\cal L}}^{l_{\mathrm{f}}ll'LL'\lambda} {\cal R}_{l_{\mathrm{f}}ll'\lambda l_{\mathrm{i}}}, \nonumber \\ \fl {\cal A}_{l_{\mathrm{i}}{\cal L}}^{l_{\mathrm{f}}ll'LL'\lambda}= (-1)^{\lambda} \frac{\hat{l}_{f}\hat{l}\hat{l}'\hat{L}\hat{L}'}{((2\lambda)!(2(l_{\rm p}-\lambda))!)^{\frac{1}{2}}}\left( \begin{array}{@{}c c c@{}} l_{\rm h} & l' & L\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l & \lambda & L\\ 0 & 0 & 0 \end{array} \right) \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} l' & l_{\mathrm{i}} & L'\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l_{\rm p}-\lambda & L'\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l & {\cal L}\\ 0 & 0 & 0 \end{array} \right)\nonumber \\ \times \sum_{\mu m'}(-1)^{m'} \left( \begin{array}{@{}c c c@{}} l_{\rm p}-\lambda & \lambda & l_{\rm p}\\ -m_{\rm p}-\mu & \mu & m_{\rm p} \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\rm h} & l' & L\\ -m_{\rm h} & m' & m_{\rm h}-m' \end{array} \right) \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} l' & l_{\mathrm{i}} & L'\\ -m' & 0 & m' \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l & {\cal L}\\ -m'-m_{\rm p}-\mu & m'+\mu-m_{\rm h} & m_{\rm p}+m_{\rm h} \end{array} \right)\nonumber \\ \times \left( \begin{array}{@{}c c c@{}} l & \lambda & L\\ m_{\rm h}-m'-\mu & \mu & m'-m_{\rm h} \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l_{\rm p}-\lambda & L'\\ m'+m_{\rm p}+\mu & -m_{\rm p}-\mu & -m' \end{array} \right), \nonumber \\ &&\fl {\cal R}_{l_{\mathrm{f}}ll'\lambda l_{\mathrm{i}}}= \int\nolimits_{0}^{\infty} {\rm d}R\ R^{l_{\rm p}-\lambda}\ F_{l_{\mathrm{f}}}(k_{+}R)\ {\cal V}_{\lambda l l'}(R)\ F_{l_{\mathrm{i}}}(k_{\alpha}R), \nonumber \\ &&\fl {\cal V}_{\lambda l l'}(R) = \int\nolimits_{0}^{\infty} {\rm d}r\ r^{\lambda + 1}\ {\cal L}_{n_{\rm h}l_{\rm h}}(r)\ {\cal J}^{l_{\rm p}}_{l'}(r, R)\ F_{l}(k_{-}R), \nonumber \\ &&\fl {\cal J}^{l_{\rm p}}_{l'}(r, R) = \frac{1}{2} \int\nolimits_{-1}^{1} {\rm d}u\ \rho ^{-l_{\rm p}}\ R_{n_{\rm p}l_{\rm p}}(\rho) \left( \frac{1}{R} - \frac{1}{\rho} \right) P_{l'}(u), \nonumber \\ &&\fl {\cal L}_{n_{\rm h}l_{\rm h}}(r_{1}) = \int\nolimits_{0}^{\infty} {\rm d}\ r_{2} r_{2}^{2}\ R_{n_{\rm h}l_{\rm h}}(r_{2}) \tilde{h}_{l_{\rm h}}(r_{1},r_{2}). \end{eqnarray} \tag{ 27 }$$

Similarly the excitation transition matrix element is given by

$$\begin{eqnarray} t_{\rm exc} & = & \int {\rm d}{\boldsymbol{R}}\ {\cal F}_{{\boldsymbol{k}}_{+}}^{(-)*}({\boldsymbol{R}}) V_{\rm exc}({\boldsymbol{R}}){\cal F}_{{\boldsymbol{k}}_\alpha}^{(+)}({\boldsymbol{R}}) \nonumber \\ &=&(-1)^{l_{\rm p}} \sqrt{2} \frac{(4\pi)^{\frac{5}{2}}}{k_{-}k_{\alpha}k_{+}} \left( \hat{l}_{\rm p}! \hat{l}_{\rm p}\hat{l}_{\rm h}\right)^{\frac{1}{2}} \sum_{l_{\mathrm{i}} \tilde{l}} \mathrm{i}^{l_{\mathrm{i}}}\, {\rm e}^{{\rm i} \delta _{l_{\mathrm{i}}}} \hat{l}_{\mathrm{i}}\hat{\tilde{l}}^{\frac{1}{2}} \tilde{{\cal S}}_{l_{\mathrm{i}}\tilde{l}} Y_{\tilde{l}-m_{\rm p}-m_{\rm h}}(\hat{{\boldsymbol{k}}}_{\beta})\qquad \end{eqnarray} \tag{ 28 }$$

with

$$\begin{eqnarray} &&\fl \tilde{{\cal S}}_{l_{\mathrm{i}}\tilde{l}}= \sum_{l_{\mathrm{f}}l_{\mathrm{t}}\Lambda l l' \lambda L L' \tilde{L}} \mathrm{i}^{-l-l_{\mathrm{f}}} {\rm e}^{{\rm i}(\delta_{l} +\delta_{l_{\mathrm{f}}})}\ \tilde{{\cal A}}^{l_{\mathrm{f}}l_{\mathrm{t}}\Lambda l l' \lambda L L' \tilde{L}}_{l_{\mathrm{i}}\tilde{l}}\ \tilde{{\cal R}}_{l_{\mathrm{f}}l_{\mathrm{t}}\Lambda l l' \lambda l_{\mathrm{i}}}, \nonumber \\ \fl \tilde{{\cal A}}^{l_{\mathrm{f}}l_{\mathrm{t}}\Lambda l l' \lambda L L' \tilde{L}}_{l_{\mathrm{i}}\tilde{l}}=(-1)^{\lambda} \frac{\hat{l_{\mathrm{f}}}\hat{l_{\mathrm{t}}}\hat{\Lambda}\hat{l}\hat{l'}\hat{L}\hat{L'}\hat{\tilde{L}}}{\left( (2\lambda)!(2(l_{\rm p}-\lambda))! \right)^{\frac{1}{2}}} \left( \begin{array}{@{}c c c@{}} l_{\rm h} & l_{\mathrm{t}} & \Lambda\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l' & l_{\mathrm{t}} & \tilde{L}\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l & \lambda & \tilde{L}\\ 0 & 0 & 0 \end{array} \right) \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} \Lambda & l_{\mathrm{i}} & L\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l' & L & L'\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l_{\rm p}-\lambda & L'\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l & \tilde{l}\\ 0 & 0 & 0 \end{array} \right) \nonumber \\ \times \sum_{\mu m m_{\mathrm{t}} } (-1)^{m+\mu} \left( \begin{array}{@{}c c c@{}} \Lambda & l_{\mathrm{i}} & L\\ m_{\mathrm{t}}-m_{\rm h} & 0 & m_{\rm h}-m_{\mathrm{t}} \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\rm h} & l_{\mathrm{t}} & \Lambda\\ -m_{\rm h} & m_{\mathrm{t}} & m_{\rm h}-m_{\mathrm{t}} \end{array} \right) \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} l' & l_{\mathrm{t}} & \tilde{L}\\ m+m_{\mathrm{t}}-\mu & -m_{\mathrm{t}} & \mu-m \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l & \tilde{l}\\ -m_{\rm p}-m_{\rm h}-m & m & m_{\rm p}+m_{\rm h} \end{array} \right) \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} l_{\rm p}-\lambda & \lambda & l_{\rm p}\\ -m_{\rm p}-\mu & \mu & m_{\rm p} \end{array} \right) \left( \begin{array}{@{}c c c@{}} l' & L & L'\\ \mu-m-m_{\mathrm{t}} & m_{\mathrm{t}}-m_{\rm h} & m+m_{\rm h}-\mu \end{array} \right) \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l_{\rm p}-\lambda & L'\\ m_{\rm p}+m_{\rm h}+m & -m_{\rm p}-\mu & \mu-m-m_{\rm h} \end{array} \right) \left( \begin{array}{@{}c c c@{}} l & \lambda & \tilde{L}\\ -m & \mu & m-\mu \end{array} \right), \nonumber \\ &&\fl \tilde{{\cal R}}_{l_{\mathrm{f}}l_{\mathrm{t}}\Lambda l l' \lambda l_{\mathrm{i}}} = \int\nolimits_{0}^{\infty} {\rm d}R\ R^{l_{\rm p}-\lambda}\ F_{l_{\mathrm{f}}}(k_{+}R)\ \tilde{{\cal V}}^{\Lambda l_{\mathrm{t}}}_{\lambda l l'}(R)\ F_{l_{\mathrm{i}}}(k_{\alpha}R), \nonumber \\ &&\fl \tilde{{\cal V}}^{\Lambda l_{\mathrm{t}}}_{\lambda l l'}(R) =\int\nolimits_{0}^{\infty} {\rm d}r_{1}\ r_{1}^{\lambda+1}\ \tilde{{\cal J}}^{l_{\rm p}}_{l'}(r_{1},R)\ F_{l}(k_{-}r_{1})\ \tilde{{\cal L}}^{\Lambda l_{\mathrm{t}}}_{n_{\rm h}l_{\rm h}}(r_{1},R), \nonumber \\ &&\fl \tilde{{\cal L}}^{\Lambda l_{\mathrm{t}}}_{n_{\rm h}l_{\rm h}}(r_{1},R) = \frac{1}{\hat{\Lambda}} \int\nolimits_{0}^{\infty} {\rm d}r_{2}\ r_{2}^{2}\ R_{n_{\rm h}l_{\rm h}}(r_{2}) \left( \frac{r_{<}^{\Lambda}}{r_{>}^{\Lambda+1}}-\frac{\delta_{\Lambda 0}}{R}\right) \tilde{h}_{l_{\mathrm{t}}}(r_{1},r_{2}), \nonumber \\ &&\fl \tilde{{\cal J}}^{l_{\rm p}}_{l'}(r_{1},R) =\frac{1}{2} \int\nolimits_{-1}^{1} {\rm d}u\ R_{n_{\rm p}l_{\rm p}}\ (\rho_{1}) \rho_{1}^{l_{\rm p}}\ P_{l'}(u). \end{eqnarray} \tag{ 29 }$$

r_< (respectively r_>) is defined as min(r₂,R) (respectively max(r₂,R)). The total cross section for a given state (n_h,l_h) of H and a given state (n_p,l_p) of Ps (reaction (3)) is given by

$$\begin{eqnarray} \sigma^{4\mathrm{B}, 3} _{n_{\rm h}l_{\rm h}; n_{\rm p}l_{\rm p}} & = & \frac{1}{4\pi^{2}} \frac{k_{\beta}}{k_{\alpha}} \mu _{\alpha} \mu _{\beta} \sum_{m_{\rm p}m_{\rm h}} \int {\rm d}\hat{{\boldsymbol{k}}}_{\beta} \left| T_{\alpha\beta}\right|^{2} \nonumber \\ & = & \frac{1}{4\pi^{2}} \frac{k_{\beta}}{k_{\alpha}} \mu _{\alpha} \mu _{\beta} \frac{2(4\pi)^{5}}{(k_{-}k_{\alpha}k_{+})^{2}} \hat{l}_{\rm p}!\hat{l}_{\rm p}\hat{l}_{\rm h} \sum_{m_{\rm p}m_{\rm h}} \sum_{l_{\mathrm{i}}l'_{\mathrm{i}}\tilde{l}} \mathrm{i}^{l_{\mathrm{i}}-l'_{\mathrm{i}}} {\rm e}^{{\rm i}(\delta_{l_{\mathrm{i}}}-\delta_{l'_{\mathrm{i}}})} \hat{l}_{\mathrm{i}} \hat{l}'_{\mathrm{i}} \hat{\tilde{l}} \nonumber \\ & & \times \left\lbrace ({\cal S}^{*}_{l'_{\mathrm{i}}\tilde{l}}\times {\cal S}_{l_{\mathrm{i}}\tilde{l}} + \tilde{{\cal S}}^{*}_{l'_{\mathrm{i}}\tilde{l}}\times \tilde{{\cal S}}_{l_{\mathrm{i}}\tilde{l}} + \left[ {\cal S}^{*}_{l'_{\mathrm{i}}\tilde{l}}\times \tilde{{\cal S}}_{l_{\mathrm{i}}\tilde{l}} + \mathrm{c.c.} \right] \right\rbrace \end{eqnarray} \tag{ 30 }$$

with

$$\begin{equation} \left| T_{\alpha \beta}\right|^{2} = t_{\rm exc}^{*} \times t_{\rm exc} + t_{\rm exc}^{*} \times t_{\rm cap} + t_{\rm cap}^{*} \times t_{\rm exc} + t_{\rm cap}^{*} \times t_{\rm cap}. \end{equation} \tag{ 31 }$$

Because of the absence of angular correlations in the uncorrelated Chandrasekhar wave function, further simplifications can be applied to the transition matrix element. The H⁻ wave function can be treated slightly differently. Indeed, one notices that the UC wave function can also be written as

$$\begin{equation} \Phi _{\alpha}({\boldsymbol{r}}_{1},{\boldsymbol{r}}_{2})=\frac{1}{\sqrt{2}} \left\lbrace \varphi_{a_{\rm uc}}({\boldsymbol{r}}_{1})\varphi_{b_{\rm uc}}({\boldsymbol{r}}_{2}) + \varphi_{a_{\rm uc}}({\boldsymbol{r}}_{2})\varphi_{b_{\rm uc}}({\boldsymbol{r}}_{1}) \right\rbrace. \end{equation} \tag{ 32 }$$

Now, in T⁻_αβ, terms depending on r₁ and r₂ are fully separable and the transition matrix element can be written as

$$\begin{equation} T^{-}_{\alpha\beta} = t^{ab} + t^{ba}, \end{equation} \tag{ 33 }$$

where a_uc is shortened into a and b_uc in b. t^ij describes the capture into the Ps of the electron initially in state i in H⁻ while, simultaneously, the electron initially in state j is excited to a final state H(n_h,l_h,m_h). Since both electrons in H⁻ are indistinguishable, T⁻_αβ is the coherent sum of two terms. We have

$$\begin{eqnarray} &&t^{ij}=\int{{\rm d} {\boldsymbol{R}} \; {{\cal F}_{{\boldsymbol{k}} _{\beta}}^{(-)}}^*({\boldsymbol{R}}) \;^{}V^{ij}({\boldsymbol{R}} )\; {\cal F}_{{\boldsymbol{k}}_{\alpha}}^{(+)}({\boldsymbol{R}})}, \end{eqnarray} \tag{ 34 }$$

where

$$\begin{eqnarray} \fl ^{}V^{ij}({\boldsymbol{R}})&=&\int{{\rm d}{\boldsymbol{r}}_1\; \Psi_{f}^*({\boldsymbol{\rho}}_1)\; {{\cal F}_{{\boldsymbol{k}}_{\beta}}^{(-)}}^*({\boldsymbol{r}}_1)\; \left(\frac{1}{R}-\frac{1}{\rho_1}\right)\; ^{}\varphi_{i}({\boldsymbol{r}}_1) } \times \int{{\rm d}{\boldsymbol{r}}_2\; {\varphi}_{n_{\rm h} l_{\rm h} m_{\rm h}}^*({\boldsymbol{r}}_2) \;^{}\varphi_{j}({\boldsymbol{r}}_2)} \nonumber \\ \fl & &+\int{{\rm d}{\boldsymbol{r}}_1 \;\Psi_{f}^*({\boldsymbol{\rho}}_1)\; {{\cal F}_{{\boldsymbol{k}}_{\beta}}^{(-)}}^*({\boldsymbol{r}}_1)\; ^{}\varphi_{i}({\boldsymbol{r}}_1) } \times \int{{\rm d}{\boldsymbol{r}}_2\; {\varphi}_{n_{\rm h} l_{\rm h} m_{\rm h}}^*({\boldsymbol{r}}_2)\; \left(\frac{1}{R}-\frac{1}{\rho_2}\right)\; ^{}\varphi_{j}({\boldsymbol{r}}_2) }\nonumber \\ \fl &=& \left\{ \int{{\rm d}{\boldsymbol{r}}_1\;\Psi_{f}^*({\boldsymbol{\rho}}_1) \;{{\cal F}_{{\boldsymbol{k}}_{\beta}}^{(-)}}^*({\boldsymbol{r}}_1)\; \left(\frac{1}{R}-\frac{1}{\rho_1}\right)\;^{}\varphi_{i}({\boldsymbol{r}}_1) } \right\} \times \left< n_{\rm h} l_{\rm h} m_{\rm h} |j\right>\nonumber \\ \fl & &+ \left\{ \int{{\rm d}{\boldsymbol{r}}_1\;\Psi_{f}^*({\boldsymbol{\rho}}_1)\; {{\cal F}_{{\boldsymbol{k}}_{\beta}}^{(-)}}^*({\boldsymbol{r}}_1)\; ^{}\varphi_{i}({\boldsymbol{r}}_1) }\right\} \times \, ^{}{\cal K}_{n_{\rm h} l_{\rm h} m_{\rm h}}^{j}({\boldsymbol{R}}) \nonumber \\ \fl &\equiv & ^{}V_{\rm cap}^{ij}({\boldsymbol{R}})+\, ^{}{V_{\rm exc}^{ij}}({\boldsymbol{R}}) \;. \end{eqnarray} \tag{ 35 }$$

According to this, the matrix element t^ij may be expressed as

$$\begin{eqnarray} &&^{}t^{ij}=\;^{}t^{ij}_{\rm cap}+\;^{}t^{ij}_{\rm exc}\;. \end{eqnarray} \tag{ 36 }$$

The matrix elements t^ij_cap and t^ij_exc may be obtained by replacing V^ij in equation (34) with V^ij_cap and V^ij_exc, respectively. The term t^ij_cap mostly describes the capture of the electron in state i in H⁻ into the Ps; the process is pondered by the overlap between the wave function of the electron in the hydrogen atom and the wave function of the H⁻ electron initially in state j. Due to the scalar product $\left < n_{\mathrm { h}} l_{\mathrm { h}} m_{\mathrm { h}} |j \right >$, it is worthwhile noticing that the term of 'capture' is equal to zero for l_h ≠ 0. The term t^ij_exc encloses the excitation of the electron that remains bound to the proton, from state j in H⁻ to state (n_h,l_h,m_h) in the hydrogen atom. Indeed, ${\cal K}_{n_{\mathrm { h}} l_{\mathrm { h}} m_{\mathrm { h}}}^{j}({\boldsymbol {R}})$ can be written as

$$\begin{equation} ^{}{\cal K}_{n_{\rm h} l_{\rm h} m_{\rm h}}^{j}({\boldsymbol{R}}) = {\cal L}_{n_{\rm h} l_{\rm h}}^{j}(R) \;Y_{l_{\rm h} m_{\rm h}}^*(\hat{\boldsymbol{R}}) \end{equation} \tag{ 37 }$$

with

$$\begin{equation} {\cal L}_{n_{\mathrm{h}} l_{\mathrm{h}}}^{j}(R) = \frac{\sqrt{4\pi}}{\hat{l_{\mathrm{h}}}}\;\int_0^{\infty}{{\rm d}r\,r^2\, {R}_{n_{\mathrm{h}} l_{\mathrm{h}}}(r)\;\left(\frac{r_{<}^{l_{\mathrm{h}}}}{r_{>}^{l_{\mathrm{h}}+1}}-\frac{\delta_{l_{\mathrm{h}} 0}}{R} \right) \; R_{j}(r)}. \end{equation} \tag{ 38 }$$

The total cross section with the uncorrelated Chandrasekhar wave function can be written as

$$\begin{eqnarray} &&\fl \sigma^{4\mathrm{B},3\mbox{-}\mathrm{UC}} _{n_{\rm h}l_{\rm h}; n_{\rm p}l_{\rm p}} =\frac{1}{4\pi^{2}} \frac{k_{\beta}}{k_{\alpha}} \mu _{\alpha} \mu _{\beta} \sum_{m_{\rm p}m_{\rm h}} \int {\rm d}\hat{{\boldsymbol{k}}}_{\beta} \left| T_{\alpha\beta}\right|^{2}, \nonumber \\ &&\fl \left| T_{\alpha\beta}\right|^{2}= t^{ab *}_{\rm cap}\times t^{ab}_{\rm cap} + t^{ab *}_{\rm exc}\times t^{ab}_{\rm exc} + \left[ t^{ab *}_{\rm cap}\times t^{ab}_{\rm exc} + {\rm c.c.} \right]\\ &&+ t^{ab *}_{\rm cap}\times t^{ba}_{\rm cap} + t^{ab *}_{\rm exc}\times t^{ba}_{\rm exc} + \left[ t^{ab *}_{\rm cap}\times t^{ba}_{\rm exc} + {\rm c.c.} \right]\nonumber\\ &&+ \left\lbrace a \leftrightarrow b \right\rbrace. \nonumber \end{eqnarray} \tag{ 39 }$$

If m_h or l_h are different from zero, most of these terms are null. The expression of the pure capture terms is

$$\begin{eqnarray} \sum_{m_{\rm p}m_{\rm h}} \int {\rm d}\hat{{\boldsymbol{k}}}_{\beta}\ t^{ij *}_{\rm cap}\times t^{kq}_{\rm cap} & = & \frac{(4\pi)^{3}}{(k_{+}k_{-}k_{\alpha})^{2}} \hat{l}_{\rm p}!\hat{l}_{\rm p} \sum_{l_{\mathrm{i}}\tilde{l}} \hat{l}_{\mathrm{i}}\hat{\tilde{l}} \left({\cal W}^{ij *}_{l_{\mathrm{i}}\tilde{l}}\times {\cal W}^{kq}_{l_{\mathrm{i}}\tilde{l}}\right) \end{eqnarray} \tag{ 40 }$$

with

$$\begin{eqnarray} &&\fl {\cal W}^{ij}_{l_{\mathrm{i}}\tilde{l}} = \sum_{l_{\mathrm{f}} ll'\lambda L} \mathrm{i}^{-l-l_{\mathrm{f}}}\, {\rm e}^{{\rm i}(\delta_l + \delta_{l_{\mathrm{f}}})}\ {\cal B}_{l_{\mathrm{i}} \tilde{l}}^{l_{\mathrm{f}} ll'\lambda L}\ {\cal R}^{ij}_{l_{\mathrm{f}} ll'\lambda l_{\mathrm{i}}} ,\nonumber \\ \fl {\cal B}_{l_{\mathrm{i}} \tilde{l}}^{l_{\mathrm{f}} ll'\lambda L} = (-1)^{\lambda} \frac{\hat{l}_{\mathrm{f}} \hat{l} \hat{l}' \hat{L}}{\left( (2\lambda)!(2(l_{\rm p} -\lambda))! \right)^{\frac{1}{2}}} \left\lbrace \begin{array}{@{}c c c@{}} l & \lambda & l'\\ l_{\mathrm{f}} & l_{\rm p}-\lambda & L \\ \tilde{l} & l_{\rm p} & l_{\mathrm{i}} \end{array} \right\rbrace \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} \lambda & l' & l\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l' & l_{\mathrm{i}} & L \\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l_{\rm p}-\lambda & L\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l & l_{\mathrm{f}} & \tilde{l}\\ 0 & 0 & 0 \end{array} \right), \\ &&\fl {\cal R}^{ij}_{l_{\mathrm{f}} ll'\lambda l_{\mathrm{i}}} = \int\nolimits_{0}^{\infty} {\rm d}R\ R^{l_{\rm p} -\lambda}\ F_{l_{\mathrm{f}} }(k_+ R)\ {\cal V}^{ij}_{ll'\lambda}(R)\ F_{l_{\mathrm{i}} }(k_{\alpha}R),\nonumber \\ &&\fl {\cal V}^{ij}_{ll'\lambda}(R) = \left\langle n_{\rm h} l_{\rm h} m_{\rm h} | j \right\rangle \times \int\nolimits_{0}^{\infty} {\rm d}r\ r^{\lambda + 1}\ F_l (k_- r)\ {\cal J}^{l_{\rm p} }_{l'}(r,R)\ R_i (r). \nonumber \end{eqnarray} \tag{ 41 }$$

Pure excitation terms can be written as follows:

$$\begin{eqnarray} \sum_{m_{\rm p}m_{\rm h}} \int {\rm d}\hat{{\boldsymbol{k}}}_{\beta}\ t^{ij *}_{\rm exc}\times t^{kq}_{\rm exc} & = & \frac{(4\pi)^{3}}{(k_{+}k_{-}k_{\alpha})^{2}} \hat{l}_{\rm p}!\hat{l}_{\rm p}\hat{l}_{\rm h} \sum_{l_{\mathrm{i}}\tilde{l} L} \hat{l}_{\mathrm{i}}\hat{\tilde{l}}\hat{L} \left(\tilde{{\cal W}}^{ij *}_{l_{\mathrm{i}}\tilde{l}L}\times \tilde{{\cal W}}^{kq}_{l_{\mathrm{i}}\tilde{l}L}\right) \end{eqnarray} \tag{ 42 }$$

with

$$\begin{eqnarray} &&\fl \tilde{{\cal W}}^{ij}_{l_{\mathrm{i}}\tilde{l}L} =\sum_{l_{\mathrm{f}} ll' \lambda L'} \mathrm{i}^{-l-l_{\mathrm{f}}}\, {\rm e}^{{\rm i}(\delta_l + \delta_{l_{\mathrm{f}}})}\ \tilde{{\cal B}}_{l_{\mathrm{i}} \tilde{l}L}^{l_{\mathrm{f}} ll'\lambda L'}\ \tilde{{\cal R}}^{ij}_{l_{\mathrm{f}} ll'\lambda l_{\mathrm{i}}} , \nonumber \\ \fl \tilde{{\cal B}}_{l_{\mathrm{i}} \tilde{l}L}^{l_{\mathrm{f}} ll'\lambda L'} = (-1)^{\lambda} \frac{\hat{l}_{\mathrm{f}} \hat{l} \hat{l}' \hat{L}'}{((2\lambda)!(2(l_{\rm p} -\lambda))!)^{\frac{1}{2}}} \left\lbrace \begin{array}{@{}c c c@{}} l & \lambda & L'\\ l_{\mathrm{f}} & l_{\rm p}-\lambda & L \\ \tilde{l} & l_{\rm p} & L \end{array} \right\rbrace \left( \begin{array}{@{}c c c@{}} l_{\rm h} & l_{\mathrm{i}} & L\\ 0 & 0 & 0 \end{array} \right) \nonumber \\ \times \left( \begin{array}{@{}c c c@{}} \lambda & l' & l\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l' & L & L' \\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l_{\mathrm{f}} & l_{\rm p}-\lambda & L'\\ 0 & 0 & 0 \end{array} \right) \left( \begin{array}{@{}c c c@{}} l & l_{\mathrm{f}} & \tilde{l}\\ 0 & 0 & 0 \end{array} \right), \nonumber \\ &&\fl \tilde{{\cal R}}^{ij}_{l_{\mathrm{f}} ll'\lambda l_{\mathrm{i}}} = \int\nolimits_{0}^{\infty} {\rm d}R\ R^{l_{\rm p} -\lambda}\ F_{l_{\mathrm{f}} }(k_+ R)\ \tilde{{\cal V}}^{i}_{ll'\lambda}(R)\ \tilde{{\cal L}}^{j}_{n_{\rm h} l_{\rm h} }(R)\ F_{l_{\mathrm{i}} }(k_{\alpha}R), \nonumber \\ &&\fl \tilde{{\cal V}}^{i}_{ll'\lambda}(R) =\int\nolimits_{0}^{\infty} {\rm d}r\ r^{\lambda + 1} F_l (k_- r)\ \tilde{{\cal J}}^{l_{\rm p} }_{l'}(r,R)\ R_i (r), \nonumber \\ &&\fl \tilde{{\cal L}}^{j}_{n_{\rm h} l_{\rm h} }(R) = \frac{{\cal L}^{j}_{n_{\rm h} l_{\rm h} }(R)}{\sqrt{4\pi}}. \end{eqnarray} \tag{ 43 }$$

Finally, the cross terms can be expressed with the previously introduced quantities. For instance

$$\begin{eqnarray} &&\sum_{m_{\rm p}m_{\rm h}} \int {\rm d}\hat{{\boldsymbol{k}}}_{\beta} t^{ij *}_{\rm cap}\times t^{kq}_{\rm exc} = \frac{(4\pi)^{3}}{(k_{+}k_{-}k_{\alpha})^{2}} \hat{l}_{\rm p}!\hat{l}_{\rm p} \sum_{l_{\mathrm{i}}\tilde{l}} \hat{l}_{\mathrm{i}}\hat{\tilde{l}} \left({\cal W}^{ij *}_{l_{\mathrm{i}}\tilde{l}}\times \tilde{{\cal X}}^{kq}_{l_{\mathrm{i}}\tilde{l}}\right), \nonumber \\ &&\tilde{{\cal X}}^{kq}_{l_{\mathrm{i}}\tilde{l}} = (-1)^{l_{\mathrm{i}}} \ \hat{l}_{\mathrm{i}}^{\frac{1}{2}}\ \tilde{{\cal W}}^{kq}_{l_{\mathrm{i}}\tilde{l} l_{\mathrm{i}} , l_{\rm h} = 0}. \end{eqnarray} \tag{ 44 }$$

Cross sections for reaction (1) have been computed for 6 states of the Ps atom, from Ps(1s) to Ps(3d) and for 13 states of hydrogen, from H(1s) to H(5d) (and up to H(5g) for ground-state Ps). Higher states have not been investigated yet since the calculation for each (n_p, l_p, m_p)–(n_h, l_h, m_h) pair is very time-consuming. To obtain the cross sections for the reverse reaction (2), the following kinetic transformation is applied, assuming microreversibility and invariance by charge conjugation [1]:

$$\begin{equation} \sigma ^{3\mathrm{B},2} _{n_{\rm h} l_{\rm h} ; n_{\rm p} l_{\rm p}} = \frac{\hat{l}_{\rm h}}{\hat{l}_{\rm p}} \frac{k_{\mathrm{f}} ^2}{k_{\mathrm{i}} ^2} \sigma ^{3B,1} _{n_{\rm h} l_{\rm h} ; n_{\rm p} l_{\rm p}}, \end{equation} \tag{ 45 }$$

where $\frac {1}{2}\frac {k_{\mathrm {f}} ^2}{2m}$ is almost the kinetic energy of the Ps atom in the centre of mass and $\frac {1}{2}\frac {k_{\mathrm {i}} ^2}{m}$ is almost the kinetic energy of the electron/positron in the centre of mass. The energy considered for reaction (1) ranges from 0 to 50 eV positron energy and the results presented here for reaction (2) focus on the 0 and 30 keV antiproton energy region.

Figures 3–5 present, for Ps(1s)–Ps(3d), the cross sections of antihydrogen formation σ^3B,2_nhlh;nplp for n_h = 1–4. They demonstrate the large production of the higher excited states of $\bar {\mathrm {H}}$ (this general behaviour is in agreement with other computations [3, 4]), $\bar {\mathrm {H}}$(1s) production being almost negligible at low energies in all cases. Figure 6 details, in the case of Ps(2p) and n_h = 4, the different contributions of the l_h states. The relative behaviour of the cross sections in this example are similar in any other (n_h,l_h)–(n_p,l_p) case. It shows that, for a given n_h, the production of s-state antihydrogen is always the lowest contribution, whereas the formation of states with non-zero angular momentum quantum numbers gets more and more favoured as l_h increases. For the highest l_h state, this hierarchy can be slightly perturbed towards the threshold and towards the intermediate energy region, as can be seen in figure 6 for the $\bar {\mathrm {H}}$(4f).

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In the prospect of the GBAR experiment for which the highest production rate of antihydrogen atoms is sought, figure 7 compares the different states of Ps, when all the contributions of $\bar {\mathrm {H}}$ states are summed from (1s) to (5d). The maximum of $\bar {\mathrm {H}}$ production occurs around 6 keV antiproton energy when the Ps is in its ground state and, because of the threshold constraints for n_h > 1, $\bar {\mathrm {H}}$ formation from Ps(1s) appears to be completely negligible below 5 keV. For excited states of Ps, the production of antihydrogen atoms peaks in the region between 2 and 3 keV. The highest cross section is obtained for Ps(2p) at 2 keV: this maximum dominates the others by at least a factor of 2.

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As already explained in the introduction, cross sections for reaction (2) can only be compared to one experiment performed by Merrison et al [10] for ground state Ps in the energy range 11–15 keV. In these inclusive measurements, no distinction between the hydrogen states produced could be done. The comparison, shown in figure 8 will thus take place between these experimental data and the sum of our cross sections over the (anti)hydrogen states. Despite a slight over-estimation from the summed cross sections, the theoretical calculations and the experimental data are in rather good agreement. To give another perspective, the CC($\bar {13}$, $\bar {8}$) and CC($\bar {28}$, 3) calculations by Mitroy and Ryzhikh [4] are also included in figure 8. As mentioned in the introduction, these calculations and the experimental data are in good agreement, except for the value at 11.3 keV. In that region, the CDW-FS results seem to better reproduce the behaviour of the total hydrogen production. On the other hand, CDW-FS largely underestimates the production of ground state hydrogen, which nonetheless remains small compared to the production of excited states. In the absence of other experimental results for reaction (2) (or its matter counterpart), in particular in the case of excited Ps, this positive comparison validates the use of CDW-FS to address the needs of the GBAR experiment.

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The predictions of the CBA model (obtained by setting β₋ = β₊ = 0) are also depicted in figure 8, along with the cross section given in [1]. As expected, the two models exhibit very similar results.

Uncorrelated Chandrasekhar wave function

The cross sections for reaction (3) were first computed using the uncorrelated Chandrasekhar wave function. Ps in states (1s)–(3d) were investigated and excited states of hydrogen up to (4f) were considered. The results translated for reaction (4) (see equation (46) below): relations between the cross sections of reactions (3) and (4), assuming microreversibility and invariance by charge conjugation, are presented in figures 9–11, for energies between 0 and 30 keV antihydrogen energy and the contribution of the antihydrogen excited states have been summed over l_h. As an example of the cross sections dependence in the orbital quantum number l_h, more detailed results are shown in figure 13 for Ps(2p) and in a state n_h = 4:

$$\begin{equation} \sigma ^{4\mathrm{B},4} _{n_{\rm h} l_{\rm h} ; n_{\rm p} l_{\rm p}} = \frac{1}{\hat{l}_{\rm p} \hat{l}_{\rm h}} \frac{k_{\mathrm{f}} ^2}{k_{\mathrm{i}} ^2} \sigma ^{4\mathrm{B},3} _{n_{\rm h} l_{\rm h} ; n_{\rm p} l_{\rm p}}. \end{equation} \tag{ 46 }$$

The general behaviour of the four-body reaction cross sections is a dramatic increase of the $\bar {\mathrm {H}}^{+}$ production towards the thresholds (see table 2) when the antihydrogen is in its ground state. This had already been noted by McAlinden et al (who gave a $\frac {1}{E}$ law to estimate the cross section of reaction (3) at very low energies) and Roy et al for Ps(1s) to (2p) [19, 20]; it is now also demonstrated for n_p = 3. The other tendency, over all the energy range considered, is a shift of the cross sections towards lower values when n_h increases. This decrease of the cross sections with n_h is amplified as l_p goes up. Two cases can be distinguished: l_p = 0 and l_p ≠ 0. Indeed, for s-states of Ps, values of the cross sections for $\bar {\mathrm {H}}^{+}$ formation from $\bar {\mathrm {H}}$(1s) are very similar: they are all of the order of 10 πa²₀ at the threshold and decrease with the impact energy, until, around 10–15 keV, they become comparable to the cross sections when $\bar {\mathrm {H}}$ is in the n_h = 2 states in the entrance channel. For the l_p ≠ 0 states of Ps, the channel where the antihydrogen is in its ground state is always dominant; below 5 keV, these cross sections are one or two orders of magnitude higher than the ones with s-state Ps, as can be seen in figure 12.

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Table 2. Thresholds (keV) for reaction (4) when $\bar {\mathrm {H}}$ is in its ground state, given for each H⁻ ($\bar {\mathrm {H}}^{+}$) wave function used in this work.

np	UC	CC	LS	exact
1	5.887	5.590	5.562	5.545
2	1.205	0.908	0.880	0.863
3	0.346	48.6×10−3	20.6×10−3	3.67×10−3

The cross sections presented in figure 13 are representative of the behaviour for n_h ⩾ 3 and l_p > 0. The $\bar {\mathrm {H}}^{+}$ ions are preferentially produced from s-state and p-state antihydrogen atoms in close competition and then, to a lower extent, from d-states. For s-state antihydrogen atoms, the cross sections always exhibit the same structure with a maximum in the region between 10 and 15 keV. In the case of s-state Ps (l_p = 0), not presented here, $\bar {\mathrm {H}}^{+}$ production will be favoured for l_h = 0, but when n_h = 4, it is now the p-state antihydrogen channel which dominates the one with s-state antihydrogen. For any state of Ps, when $\bar {\mathrm {H}}$ is in a state n_h = 2 in the entrance channel, it is $\bar {\mathrm {H}}$(2s) that leads preferentially to $\bar {\mathrm {H}}^{+}$.

Using equation (39) given for the UC wave function, it is possible to investigate the respective contributions of the capture and the excitation terms to the total cross section by taking either the t^ij_exc or t^ij_cap terms equal to zero. This is shown in figure 14(a) in the case of Ps(2p) and $\bar {\mathrm {H}}$(4s). The maximum observed in the cross section is explained by the preponderance on the capture process at these energies. The behaviour of the capture cross section is typical and the energy value at the maximum (11 keV) corresponds, as expected, to a projectile velocity twice larger than the velocity of the positron in the n_p = 2 level of the Ps atom. The excitation is largely dominant below 5 keV. The cross terms, not presented here, are completely negligible above 10 keV. Similarly, the contributions of the processes 'ab' and 'ba' can be investigated by taking respectively t^ba or t^ab equal to zero. The results are presented in figure 14(b) and show that the major contribution to the total cross section of $\bar {\mathrm {H}}^{+}$ is the 'ba' process, which corresponds to the de-excitation of the positron initially in the antihydrogen atom towards the lower level a of $\bar {\mathrm {H}}^{+}$ while, simultaneously, the positron in the Ps is captured in the outer level b of $\bar {\mathrm {H}}^{+}$.

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Correlated Chandrasekhar and Le Sech wave functions.

From the previous results with the uncorrelated Chandrasekhar wave function, cases of interest for GBAR have been selected to be investigated with the correlated Chandrasekhar wave function and the LS wave function. Since the formation of $\bar {\mathrm {H}}^{+}$ ions from excited states of antihydrogen atoms is negligible, only $\bar {\mathrm {H}}$(1s) and (2s) have been considered. Other cases can of course be calculated using the formulas given in section 2.2 and in appendix B but at the cost of a long computational time. The results are presented in figures 15–17, where they are also compared to the ones obtained with the uncorrelated Chandrasekhar wave function. In the case of the LS wave function, figure 18 compares, for each state of Ps investigated, the cross sections of $\bar {\mathrm {H}}^{+}$ production when $\bar {\mathrm {H}}$ is in its ground state in the entrance channel.

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The first observation is that the angular correlations taken into account in the CC and LS wave functions have an important effect on the total cross sections in all the energy ranges investigated and thus cannot be overlooked as a small correction. However, there is little difference between the CC and the LS wave functions: for $\bar {\mathrm {H}}$(1s) in the entrance channel, the CC and LS wave functions give the same results within 1% below 10 keV. The most notable difference is observed for $\bar {\mathrm {H}}$(2s) above, roughly, 5 keV; below, the cross sections obtained with the CC and the LS wave functions converge towards the threshold. This means that the collisional model is not sensitive to the level of description of these angular correlations at low energy, close to the reaction thresholds. The observed effect of the angular correlations is, when $\bar {\mathrm {H}}$ is in its ground state, to increase the cross section for ground state Ps by almost a factor of 2, to slightly increase the cross section above 3 keV for Ps(2s) and (3s) (and slightly decrease them below 3 keV), to decrease it for Ps(2p) and (3p) below, roughly, 10 keV and finally to largely decrease that cross section for Ps(3d), losing up to one order of magnitude close to the threshold. For $\bar {\mathrm {H}}$(2s) in the entrance channel, the use of the correlated wave functions leads to a decrease of the total cross sections compared to the UC results. The predominance of the $\bar {\mathrm {H}}$(1s) channel below 20 keV is confirmed. The expected nearly resonant behaviour for n_p = 3 is indeed observed, but, in the prospect of the GBAR experiment, does not lead to a very sharp increase of the $\bar {\mathrm {H}}^{+}$ production, unless using Ps(3p) or (3d) well below 2 keV antiproton energy. However, as can be remarked in figure 18, $\bar {\mathrm {H}}$(1s) with the (3p) or, above all, the (2p) states of Ps dominate all the other processes below 6 keV. Above 10 keV, Ps(1s) is the dominant channel.

Since no experimental results are yet available for the four-body reaction, only a comparison with other theoretical models can be undertaken. The coupled pseudo-states computations of McAlinden et al, using an approached wave function for H⁻, and the CMEA calculations of Roy and Sinha, who chose the uncorrelated Chandrasekhar wave function, are thus compared to our four-body CDW-FS results with both uncorrelated Chandrasekhar and LS wave functions. So far, others authors kept the hydrogen atom in its ground state. The CC calculation at the threshold for the Ps(1s) + H(1s) channel performed by Blackwood et al [34] is also included. Figure 19 details the cases of Ps(1s), Ps(2s) and Ps(2p).

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In most of the cases, although the general behaviour of the cross sections is similar, the CDW-FS results are in disagreement with the other theories' calculations, giving a much higher cross section. The notable exception is for Ps(2s) below 5 keV, where all the models seem to agree towards the threshold. Otherwise, the discrepancy can be higher than one order of magnitude. It could be expected, since CDW-FS, like the CMEA model used by Roy and Sinha, is not a low energy theory and is subdued to the post/prior discrepancy (which is emphasized towards the low energy region). In the medium energy region above 20 keV, both CDW-FS and CMEA should be valid but still disagree. The main difference between the two models is the treatment of the asymptotic states, which is exact in the case of CDW-FS and could be a reasonable explanation. However, the latter argument cannot be used when comparing to the CC calculation, which is intended to be accurate in the low energy region. Nonetheless, the model developed by McAlinden et al is itself not free from approximations: in particular, the wave function they used for H⁻ (a split shell function involving one electron in the H(1s) state and the other in a linear combination of s-orbitals) gives a binding energy of −0.513, similar to the uncorrelated Chandrasekhar wave function. These observations stress that our results should be handled with particular care when used to predict experimental behaviours, either for GBAR or for other future projects.