REVISED BIG BANG NUCLEOSYNTHESIS WITH LONG-LIVED, NEGATIVELY CHARGED MASSIVE PARTICLES: UPDATED RECOMBINATION RATES, PRIMORDIAL ⁹Be NUCLEOSYNTHESIS, AND IMPACT OF NEW ⁶Li LIMITS

Table 7 shows the binding energies of ⁷Be_X atomic states with main quantum numbers n ranging from one to seven. Since the ⁷Be nuclear charge distribution has a finite size, the amplitude of the Coulomb potential at small r is less than that for two point-charges. Wave functions at small radii and binding energies of tightly bound states with small n values therefore deviate from those of the Bohr model. Binding energies in the Bohr model are given by $E_{\rm B}^{\rm Bohr}= Z^2 \alpha ^2 \mu /2n^2$, where α is the fine structure constant. On the other hand, the binding energies of loosely bound states with large n values are similar to those of the Bohr model.

The resonant rates of the reaction ⁷Be(X⁻, γ)⁷Be_X are calculated for m_X = 1, 10, 100, and 1000 GeV adopting the WS40 model for the nuclear charge distribution. The normalization of the total charge leads to a radius parameter, R = 2.63 fm. Radiative decay widths for E1 transitions are calculated taking into account the change of the E1 effective charge as a function of m_X.

In general, the recombination can efficiently proceed via resonant reactions through atomic states ${^7Z^\ast }_X^{\ast {\rm a}}$ composed of a nuclear excited state ⁷Z* and an X⁻ (Bird et al. 2008). In these reactions, the resonances radiatively decay to lower energy states of ${^7Z^\ast }_X^{\ast {\rm a}}$, ⁷Z*_X, ${^7Z}_X^{\ast {\rm a}}$, and ⁷Z_X that have larger binding energies. Once bound states are produced in the reaction, subsequent transitions via radiative decays to lower energy states occur quickly. Finally, the GS ⁷Z_X is produced after atomic states are converted to the atomic GS, and the nuclear excited state ⁷Z* inside the atomic states is converted to the nuclear GS (Bird et al. 2008).

Table 8 shows calculated parameters of important transitions related to the reaction ⁷Be(X⁻, γ)⁷Be_X for m_X = 1, 10, 100, and 1000 GeV. There are an infinite number of atomic states of ⁷Be$^\ast _X$, composed of the first nuclear excited state ⁷Be*[≡ ⁷Be*(0.429 MeV, 1/2⁻)] and an X⁻. Among them states that satisfy E_B ≲ 0.4291 MeV are important resonances in the recombination. We take into account atomic resonances with binding energies of 0.23 MeV ⩽E_B ⩽ 0.43 MeV. They are the 1S state for m_X = 1 GeV, the 2S and the 2P states for m_X = 10 GeV, and the 3S, 3P, and 3D states for m_X = 100 GeV and 1000 GeV. The transitions, matrix elements, radiative decay widths of the resonance, and resonance energies are listed in the second to fifth columns, respectively.

Table 8. Calculated Parameters for ⁷Be(X⁻, γ)⁷Be_X in the WS40 Model

mX	Transition	τif	Γγ	Er
(GeV)		(fm)	(eV)	(MeV)
1	7Be$^\ast _X$(1S)→ 7BeX(1S)	⋅⋅⋅	0.00343a	0.0881
10	7Be$^\ast _X$(2P)→ 7Be$^\ast _X$(1S)	4.29	2.80	0.0198
100	7Be$^\ast _X$(3D)→ 7Be$^\ast _X$(2P)	6.04	1.64	0.140
100	7Be$^\ast _X$(3P)→ 7Be$^\ast _X$(2S)	8.09	0.438	0.155
100	7Be$^\ast _X$(3P)→ 7Be$^\ast _X$(1S)	0.738	0.653	0.155
1000	7Be$^\ast _X$(3D)→ 7Be$^\ast _X$(2P)	5.80	1.84	0.123
1000	7Be$^\ast _X$(3P)→ 7Be$^\ast _X$(2S)	7.84	0.481	0.141
1000	7Be$^\ast _X$(3P)→ 7Be$^\ast _X$(1S)	0.693	0.662	0.141

Note. ^aGiven by $\Gamma _\gamma = \tau _\gamma ^{-1}$ with a lifetime of 192 fs taken from that of the first excited 1/2⁻ state in ⁷Be (Tilley et al. 2002).

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Binding energies of ⁷Be$^\ast _X$ are taken to be the same as those of ⁷Be_X. This approximation is justified since the quantum three-body model (Kamimura et al. 2009) for α+³He+X⁻ showed that the rms charge radii of ⁷Be and ⁷Be* differ by only 0.05 fm. For the case of m_X = 1 GeV, there is no important resonance of atomic excited states because of the relatively small binding energies of ⁷Be_X. The most important resonance is then the atomic GS of ⁷Be$^\ast _X$(1S), which can decay only into atomic states of the nuclear GS, i.e., ⁷Be$_X^{\ast {\rm a}}$ and ⁷Be_X. We take the measured rate for the radiative decay of ⁷Be* (Tilley et al. 2002) as that for the decay of ⁷Be$^\ast _X$(1S) into the GS ⁷Be_X(1S). This rate is listed although this transition is a magnetic dipole transition and therefore relatively weak (Tilley et al. 2002).

We note that if a final state of the resonance decay is a resonance above the energy threshold of the A + X⁻ separation channel, the final state instantaneously decays into the separation channel. The resonant reaction with the final state is therefore not an available path to the GS A_X. For example, in the case of m_X = 10 GeV, the state of ⁷Be$_{X}^{\ast {\rm a}}$(2P) can be produced via the resonance ⁷Be$_{X}^{\ast {\rm a}}$(2S) with a resonance energy of E_r = 0.103 MeV. However, the 2P state quickly decays into the separation channel before it can radiatively decay to the GS.

Pathways in the resonant reaction ⁷Be(X⁻, γ)⁷Be_X are divided into three types according to the final states in the transitions from atomic state resonances $^7Z_X^\ast$ or ${^7Z^\ast }_X^{\ast {\rm a}}$. Type 1 involves transitions to atomic states of the same nuclear state ($^7Z_X^\ast$ or ${^7Z^\ast }_X^{\ast {\rm a}}$). For Type 1, decay widths for the transitions can be approximately calculated by taking into account only the atomic wave functions. Type 2 involves transitions to the nuclear GS of the same atomic state (⁷Z_X or ${^7Z}_X^{\ast {\rm a}}$). For Type 2, the decay widths can be approximately calculated by taking into account only the nuclear wave functions. Type 3 denotes transitions to different atomic states of the nuclear GS (⁷Z_X or ${^7Z}_X^{\ast {\rm a}}$). This transition type simultaneously involves both atomic and nuclear transitions, and the number of possible final states can be very large. In addition, calculations of decay widths for the transitions need both nuclear and atomic wave functions. Although a precise calculation of decay widths is beyond the scope of this study, we show in the Appendix that the E1 widths for Type 3 transitions are significantly smaller than those of Type 1. In the Appendix, we suggest that the E1 width for Type 3 transitions can be interestingly large for exotic atomic systems involving a negatively charged particle with a mass equal to or larger than the nuclear mass. Most importantly, Type 3 transition widths can be much larger than those of normal atomic systems composed of nuclei and electrons.

We suppose that in Type 1 transitions the nuclear states do not significantly change and only their atomic states change. Then, one can simply take atomic wave functions expressed as $\Psi _{\rm i}({{\boldsymbol r}})=\psi _{\rm i}(r)Y_{l_{\rm i}m_{\rm i}}(\hat{r})$ and $\Psi _{\rm f}({{\boldsymbol r}})=\psi _{\rm f}(r)Y_{l_{\rm f}m_{\rm f}}(\hat{r})$, where ψ_i(r) and ψ_f(r) are radial wave functions of initial and final states, respectively, l_i and m_i are the azimuthal and magnetic quantum numbers, respectively, of the initial state, and l_f and m_f are those of the final state. The radiative decay width (Equation (15)) of the resonance ${^7Z^\ast _X}^{\ast {\rm a}}$ is then rewritten in the form

$$\begin{eqnarray} \Gamma _\gamma &=&C(l_{\rm i}, l_{\rm f}) e_1^2 E_\gamma ^3 \tau _{\rm if}^2 , \end{eqnarray} \tag{ 19 }$$

where C(l_i, l_f) is a constant that depends on angular momenta l_i and l_f. The values C(0, 1) = 4/3, C(1, 0) = 4/9, and C(2, 1) = 8/15 are used in deriving the following rates.

The thermal resonant rate is given by Equation (11), where in the ⁷Z+X⁻ recombination (for Z = Li or Be) the reduced mass in amu is A = A_AA_X/(A_A + A_X), and the statistical factor is

$$\begin{eqnarray} \omega &=&\frac{2J+1}{(2I_A+1)(2I_X+1)} \nonumber \\ &=&\frac{(2l_{\rm res}+1)[2I(A^\ast (1/2^-))+1]}{2I(A(3/2^-))+1}=\frac{(2l_{\rm res}+1)}{2}, \end{eqnarray} \tag{ 20 }$$

where l_res is the azimuthal quantum number of the resonance, and I(A(3/2⁻)) = 3/2 and I(A(1/2⁻)) = 1/2 are the spins of the GS and the first nuclear excited state of ⁷Z, respectively.

The resonant rates via Types 1 and 2 (for m_X = 1 GeV) transitions are derived as

$$\begin{eqnarray*} &&N_{\rm A} \langle \sigma v \rangle _{\rm R} =\left\{ \begin{array}{@{}l} 2.94 \times 10^2\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-1.02/T_9)\\ \quad\qquad\qquad ({\rm for}\; m_X=1\; \rm GeV)\qquad\qquad\ \ \ (21)\\ 7.40 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-0.230/T_9)\\ \quad\qquad\qquad ({\rm for}\; m_X=10\; \rm GeV)\qquad\qquad\ (22) \\ 3.73 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-1.62/T_9) \\ +1.49 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-1.80/T_9)\\ \quad\qquad\qquad ({\rm for}\; m_X=100\; \rm GeV) \qquad\qquad\! (23)\\ 3.86 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-1.43/T_9) \\ +1.44 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-1.64/T_9)\\ \quad\qquad\qquad ({\rm for}\; m_X=1000\; \rm GeV). \qquad\quad\ \ \ (24) \end{array}\right. \end{eqnarray*} \tag{ 21 }$$

The rate for m_X = 1 GeV corresponds to the pure nuclear transition from the resonance ⁷Be$^\ast _X$(1S) to the GS ⁷Be_X(1S). The rate for m_X = 10 GeV corresponds to the atomic transition from the resonance ⁷Be${^\ast _X}^{\ast {\rm a}}$(2P) to the GS ⁷Be$^\ast _X$(1S). The first terms in the rates for m_X = 100 and 1000 GeV correspond to the atomic transition from the resonance ⁷Be${^\ast _X}^{\ast {\rm a}}$(3D) to ⁷Be${^\ast _X}^{\ast {\rm a}}$(2P), while the second terms correspond to sums of the atomic transitions from the resonance ⁷Be${^\ast _X}^{\ast {\rm a}}$(2P) to ⁷Be${^\ast _X}^{\ast {\rm a}}$(2S) and ⁷Be$^\ast _X$(1S).

This calculated rate is compared to the previous rate derived in the limit of infinite m_X (Equation (2.9) of Bird et al. 2008).¹¹ We take the rate for m_X = 1000 GeV for this comparison. Our first term for the transition 3D → 2P is a factor of ∼2 higher than that of Bird et al. (2008). Our second term for the transition 3P → 2S and 1S is roughly the same as that of Bird et al. (2008).

We fitted the function, i.e., $N_{\rm A} \langle \sigma v \rangle =(a+b T_9)/T_9^{1/2}$, to nonresonant rates calculated for the recombination of nuclei and X⁻ particles in the temperature region of T₉ = [10⁻³, 1], and obtained approximate analytical expressions.

With higher CM energy, the frequencies for the oscillations of continuum-state wave functions increase. Thus, it takes more computational time to precisely calculate the radial matrix elements or cross sections at larger energy. In the present study, we derived the cross sections only in the energy range of 10⁻⁵ MeV <E < 1 MeV, and the recombination rates are calculated in the temperature range of T₉ ⩽ 1 using the derived cross sections and just setting cross sections for E > 1 MeV to be zero. Since the nucleosynthesis as well as recombinations of ⁴He and heavier nuclei with X⁻ proceed after the temperature of the universe decreases down to T₉ < 1, the reaction rates for higher temperatures T₉ > 1 are not necessary in BBN calculations. Considering that at the relevant temperatures, the contribution to the thermal rates from reactions at CM energies greater than the temperature is small, our reaction rates can be safely used in the desired temperature regime.

The nonresonant rate for the reaction ⁷Be(X⁻, γ)⁷Be_X is then derived to be

$$\begin{eqnarray*} &&N_{\rm A} \langle \sigma v \rangle _{\rm NR} =\left\{ \begin{array}{@{}l} 2.44 \times 10^5\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.344 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=1\; \rm GeV) \qquad\qquad\ \ \ \! (25)\\ 5.98 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.211 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=10\; \rm GeV) \qquad\qquad\ \!(26)\\ 4.07 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.196 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=100\; \rm GeV) \qquad\qquad\!\!\! (27)\\ 3.86 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.194 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=1000\; \rm GeV). \qquad\quad\ \ (28) \end{array} \right. \end{eqnarray*} \tag{ 25 }$$

Nonresonant cross sections are calculated with RADCAP taking into account the multiple components of partial waves for scattering states. We show continuum wave functions at the CM energy E = 0.07 MeV, which is the average energy corresponding to the temperature of the recombination of ⁷Be+X⁻ for the case of m_X = 1000 GeV, i.e., E = 3T/2 with T ∼ 0.4 × 10⁹ K.

The total cross section for the absorption of an unpolarized photon with frequency ν via an E1 transition from a bound state (n, l) to a continuum state (E) is given (Gaunt 1930; Karzas & Latter 1961) by

$$\begin{eqnarray} \sigma _{nl\rightarrow E}&=&\frac{16 \pi ^2}{3} e_1^2 \mu k \nu \nonumber \\ &&\times \left[\frac{l+1}{2l+1} \left(\tau _{nl}^{E, l+1}\right)^2 + \frac{l}{2l+1} \left(\tau _{nl}^{E, l-1}\right)^2 \right], \end{eqnarray} \tag{ 29 }$$

where $k=\sqrt{\vphantom{A^A}\smash{{{\strut 2\mu E}}}}$ is the wave number and

$$\begin{equation} \tau _{nl}^{E, l\pm 1}=\int r^2 dr \psi _{E, l\pm 1}(r) r \psi _{nl}(r) \end{equation} \tag{ 30 }$$

is the radial matrix element for the radius r, and wave functions are normalized as

$$\begin{equation} \int r^2 dr \left|\psi _{nl}(r)\right|^2=1, \end{equation} \tag{ 31 }$$

and asymptotically

$$\begin{equation} \psi _{E, l}(r)\sim \frac{\sin \left[kr -\eta \ln (2kr) -l\pi /2+ \sigma _l +\delta _l\right]}{kr} \end{equation} \tag{ 32 }$$

at large r, where η is defined by

$$\begin{equation} \eta =\frac{Z}{ka_{\rm B}}=\left(\frac{Z^2 \alpha ^2 \mu }{2E} \right)^{1/2} \end{equation} \tag{ 33 }$$

with a_B = 1/(μα) the Bohr radius, σ_l is the Coulomb phase shift, and δ_l is the phase shift due to the difference in Coulomb potential between cases of the point charge and finite size nuclei (Burke 2011). The parameter e₁ is the effective charge as defined in Equation (16). We note that the precise cross section (Equation (29)) includes $e_1^2$ instead of α which is usually adopted for hydrogen-like normal atoms.

We compare the calculated cross sections with those for the recombination of two point-charges. Wave functions of scattering and bound states and the bound-free absorption cross section in a pure Coulomb field have been derived analytically. The bound and continuum state wave functions are given (Karzas & Latter 1961) by

$$\begin{eqnarray} \psi _{nl}(r)&=& \left(\frac{2Z}{na_{\rm B}}\right)^{3/2} \left[\frac{\Gamma (n+l+1)}{\Gamma (n-l)2n}\right]^{1/2} \frac{(2Zr/na_{\rm B})^l}{(2l+1)!} \nonumber \\ &&\nonumber \\ &&\times \mathrm{e}^{-Zr/na_{\rm B}} {}_1F_1\left(l+1-n; 2l+2; \frac{2Zr}{na_{\rm B}}\right), \end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray} \psi _{E, l}(r)&=& \exp \left(\frac{\eta \pi }{2}\right) \frac{\left|\Gamma (l+1-i\eta)\right|}{(2l+1)!} \left(2kr \right)^l \mathrm{e}^{{\rm ikr}} \nonumber \\ &&\times {}_1F_1\left(l+1-i\eta ; 2l+2; -2ikr \right), \end{eqnarray} \tag{ 35 }$$

where ₁F₁ is the regular confluent hypergeometric function.

The cross section for absorption or ionization is analytically given (Equations (36) and (37) of Karzas & Latter 1961) by

$$\begin{eqnarray} && \sigma _{nl\rightarrow E, l-1}= \frac{\pi e_1^2}{\mu \nu } \frac{2^{4l}}{3} \nonumber\\ &&\quad \times \frac{l^2 (n+l)! \lbrace (1^2+\eta ^2 ) (2^2+\eta ^2 )... [(l-1)^2+\eta ^2 ]\rbrace }{(2l+1)! (2l-1)! (n-l-1)!} \nonumber \\ &&\quad \times \frac{\exp (-4\eta \cot ^{-1} \rho)}{1-\mathrm{e}^{-2\pi \eta }} \frac{\rho ^{2l+2}}{(1+\rho ^2)^{2n-2}} \nonumber\\ &&\quad \times [G_l(l+1-n; \eta ; \rho) -(1+\rho ^2)^{-2} G_l(l-1-n; \eta ; \rho)]^2, \nonumber\\ \end{eqnarray} \tag{ 36 }$$

where the quantity in the curly brackets is unity when l = 1, and

$$\begin{eqnarray} && \sigma _{nl\rightarrow E, l+1} = \frac{\pi e_1^2}{\mu \nu } \frac{2^{4l+6}}{3} \nonumber\\ &&\quad \times \frac{(l+1)^2 (n+l)! (1^2+\eta ^2 ) (2^2+\eta ^2 )... [(l+1)^2+\eta ^2 ]}{(2l+1) (2l+1)! (2l+2)! (n-l-1)! [(l+1)^2+\eta ^2 ]^2} \nonumber\\ &&\quad \times \frac{\exp (-4\eta \cot ^{-1} \rho)}{1-\mathrm{e}^{-2\pi \eta }} \frac{\rho ^{2l+4} \eta ^2}{(1+\rho ^2)^{2n}} \nonumber\\ &&\quad \times \left[\left(l+1-n \right)G_{l+1} (l+1-n; \eta ; \rho) \right. \nonumber\\ &&\quad \left. +\frac{l+1+n}{1+\rho ^2} G_{l+1}(l-n; \eta ; \rho)\right]^2. \end{eqnarray} \tag{ 37 }$$

Equations  (36) and (37) correspond to transitions to the continuum states with angular momenta l − 1 and l + 1, respectively. The parameter ρ is defined ρ ≡ η/n, and the real polynomial G_l is given by

$$\begin{equation} G_l(-m, \eta , \rho)=\sum _{s=0}^{2m} b_s \rho ^s, \end{equation} \tag{ 38 }$$

with coefficients

$$\begin{equation} b_0=1,\quad b_1=\frac{2m\eta }{l}, \end{equation} \tag{ 39 }$$

$$\begin{eqnarray} b_s&=&-\frac{1}{s(s+2l-1)} \left[ 4\eta (s-1-m) b_{s-1} \right. \nonumber \\ &&\left. +(2m+2-s) (2m+2l+1-s) b_{s-2} \right]. \end{eqnarray} \tag{ 40 }$$

The recombination cross section can be derived using the principle of detailed balance (Blatt & Weisskopf 1991; Rybicki & Lightman 1979)¹²:

$$\begin{equation} \frac{\sigma _{E, l\pm 1\rightarrow nl}}{\sigma _{nl\rightarrow E, l\pm 1}}=\frac{[2I(n,l)+1]}{(2I_1+1)(2I_2+1)} \left(\frac{E_\gamma ^2}{\mu E}\right), \end{equation} \tag{ 41 }$$

where I₁ and I₂ are spins of particles 1 and 2 constituting the bound state, I(n, l) is the spin of the bound state (n, l), and the radiation energy is related to the CM energy and the binding energy by E_γ = E + E_B.

The thermal recombination rate is derived as a function of temperature T by integrating the calculated cross section σ(E) over the Maxwellian energy distribution (Equation (10)). The analytical expression for the wave function in the case of a point-charge nucleus (Equations (31) and (32) of Karzas & Latter 1961) is derived using the confluent hypergeometric function calculated with algorithm 707 of Nardin et al. (1992).

Figure 6 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV for the ⁷Be+X⁻ system as a function of radius r for the case of m_X = 1 GeV. Solid lines correspond to calculated wave functions while the dotted lines correspond to the analytical formula for hydrogen-like atomic states composed of two point-charges (Equations (34) and (35)). In the upper panel, wave functions for the GS (1S state), 2S, 2P, 3P, 3D, and 4F states are plotted. Here, one can see that the wave functions for the GS and 2S state in the finite charge distribution case (solid lines) deviate from those of the point-charge case (dotted lines). The wave functions of other states agree with those for the point-charge case. The scattering wave functions for the s-, p-, d-, and f-waves are plotted in the middle panel. Note that the normalization for the amplitude of the wave function adopted in RADCAP is different from that in Karzas & Latter (1961). Hence, the latter wave functions are normalized to satisfy the former normalization. In addition, wave functions derived with RADCAP are multiplied by exp (iθ), where θ are arbitrary real constants and then transformed into real numbers. Only the wave function of the l = 0 state for the finite charge distribution case (solid lines) deviates from that of the point-charge case (dotted lines).

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The bottom panel shows the recombination cross section as a function of the energy E. The solid lines correspond to the calculated results, while the dotted lines correspond to the analytical solution for the two point-charges (Equations (36), (37), and (41)). Partial cross sections for the following transitions are drawn: scattering p-wave → bound 1S state (1 black lines); p-wave → 2S (2 red); s-wave → 2P (3 green); d-wave → 2P (4 blue); s-wave → 3P (5 gray); d-wave → 3P (6 sky blue); p-wave → 3D (7 orange); f-wave → 3D (8 cyan); d-wave → 4F (9 violet); and g-wave → 4F (10 magenta). Dotted lines for point-charge nuclei correspond to transitions 1, 4, 8, 2 and 6 overlapping, 10, 3, 5, 7, and 9 in descending order of cross sections at E = 10⁻⁵ MeV. This order is true in all figures of recombination cross sections shown in this paper. The cross section of transition 2 is higher than that of transition 6 at high energies although they overlap at low energies. The order of solid lines at E = 10⁻⁵ MeV is the same as that of dotted lines. Since the mass m_X is relatively small, the reduced mass is small and the spatial extent of the bound-state wave functions is large. The effect of a finite size charge distribution is only important for small r and is therefore small. Small differences in bound and scattering state wave functions lead to small changes in the cross sections through differences in the binding energies and wave function shapes. The largest differences in the cross sections are found for the two transitions starting from an initial s-wave, i.e., s-wave → 2P and s-wave → 3P. This is caused by differences in the scattering s-wave function.

Figure 7 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV of the ⁷Be+X⁻ system as a function of radius r for the case of m_X = 10 GeV. Line types indicate the same quantities as in Figure 6. In the upper panel, the wave functions for the GS and 2S state in the finite charge distribution case (solid lines) deviate significantly from those for the point-charge case (dotted lines). Also, the wave functions for the 2P and 3P states deviate slightly. In the middle panel, the difference in the wave function for the l = 0 state is very large. A difference in the l = 1 state exists although it is not large. The bottom panel shows the recombination cross section as a function of energy E. Line types indicate the same quantities as in Figure 6. The order of solid lines at E = 10⁻⁵ MeV is 4, 1, 8, 6, 10, 2, 7, 9, 3, 5. Because of the larger m_X value, the effect of a finite-size charge distribution is more important. Bound- and scattering-state wave functions and recombination cross sections are then significantly different from those for the point-charge case. Because of the large difference in the scattering s-wave function, the cross sections for transitions from an initial s-wave, i.e., s-wave → 2P and s-wave → 3P, are much smaller than those in the point-charge case. Partial cross sections for transitions from an initial p-wave to bound 1S, 2S and 3D states are also altered by the finite-size charge distribution. The cross sections for transitions to 1S and 2S states are also affected by differences in binding energies of the states between the finite- and point-charge cases.

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Figure 8 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV for the ⁷Be+X⁻ system as a function of radius r for the case of m_X = 100 GeV. Line types indicate the same quantities as in Figure 6. It is clear from a comparison of Figures 6–8 that deviations of the wave functions from those in the point-charge cases become larger as m_X increases. We can see that deviations of wave functions for bound GS, 2S, 2P, and 3P states and scattering wave functions of l = 0 and l = 1 states are very large, and that a deviation exists for the l = 1 state, but it is not large. The bottom panel shows the recombination cross section as a function of the energy E. Line types indicate the same quantities as in Figure 6. The order of solid lines at E = 10⁻⁵ MeV is 4, 8, 6, 1, 10, 2, 9, 7, 3, 5. Differences in the solid and dotted lines are even larger than in the case of m_X = 10 GeV (Figure 7).

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Figure 9 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV as a function of radius r for the ⁷Be+X⁻ system in the case of m_X = 1000 GeV. Also shown is the recombination cross section as a function of the energy E (bottom panel). Thick solid and dotted lines indicate the same quantities as in Figure 6. The order of solid lines at E = 10⁻⁵ MeV is 4, 8, 6, 10, 1, 2, 9, 7, 3, 5. Since the reduced mass is similar to that in the case of m_X = 100 GeV, this figure is rather similar to Figure 8. In order to check our calculations, we also calculate the wave functions and the cross sections for case of point-charge nuclei using the same code (a modified version of RADCAP) as used for the finite charge distribution case. Thin solid lines in the upper and middle panels show the calculated results which agree with analytical solutions (dotted lines) quite well.

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We found an important characteristic of the ⁷Be+X⁻ recombination based upon our precise calculation including many transition channels. In the limit of a heavy X⁻ particle, i.e., m_X ≳ 100 GeV, the most important transition in the recombination is the d-wave → 2P. This fact does not hold in the case of the point-charge model. In that case, the transition p-wave → 1S is predominant (see the dotted lines in Figures 6–9). In the case of a finite size charge distribution, in addition to the main pathway of d-wave → 2P, cross sections for the transitions f-wave → 3D and d-wave → 3P are also larger than that for the GS formation. It is thus found that estimations of recombination cross sections taking into account only the GS as the final state may not be correct.

We note that our rate for m_X = 1000 GeV is more than six times larger than the previous rate (Bird et al. 2008). We confirmed that the previous rate (Bird et al. 2008) is somewhat close to our rate when only taking into account the transition from the scattering p-wave to the bound 1S and 2S states. In Bird et al. (2008), it is described that the capture of ⁷Be directly to the GS of ⁷Be_X has the largest cross section, closely followed by the capture to the 2S level. This is true for hydrogen-like ions composed of point-charged particles. However, we found that the most important transition is from the scattering d-wave to the bound 2P state. The previous rate (Bird et al. 2008) was adopted in most previous studies on BBN involving the X⁻ particle, including studies by part of the present authors (Kusakabe et al. 2007, 2008, 2010). The nonresonant recombination rate is important for the ⁷Be destruction and also for constraining the parameter region for solving the Li problem. The significant improvement in the rate found in the present work therefore makes it possible to derive an improved constraint on the X⁻ particle as shown in Section 8.

Table 9 shows binding energies for the ⁷Li_X atomic states having main quantum numbers n from one to seven.

The resonant rates of the reaction ⁷Li(X⁻, γ)⁷Li_X were calculated for m_X = 1, 10, 100, and 1000 GeV in the WS40 model. The radius parameter for the WS40 model is R = 2.48 fm.

Table 10 shows calculated parameters of important transitions related to the reaction ⁷Li(X⁻, γ)⁷Li_X for m_X = 1, 100, and 1000 GeV. Similar to the recombination of ⁷Be+X⁻, the recombination can efficiently proceed via resonant reactions involving atomic states of ${^7{\rm Li}^\ast _X}^{\ast {\rm a}}$ composed of the first nuclear excited state ⁷Li*[≡ ⁷Li*(0.478 MeV, 1/2⁻)]. Important resonances for ${^7{\rm Li}^\ast _X}^{\ast {\rm a}}$ satisfy E_B ≲ 0.47761 MeV. We take into account atomic resonances with binding energies of 0.28 MeV ⩽E_B ⩽ 0.48 MeV except for the atomic GS for the case of m_X = 1 GeV. The transitions, matrix elements, radiative decay widths of the resonances, and resonance energies are listed in the second to fifth columns, respectively. For the case of m_X = 1 GeV, there are no important resonances of atomic excited states because of the relatively small binding energies of ⁷Li_X. The most important resonance in the recombination reaction is then the atomic GS of ⁷Li$^\ast _X$(1S), which can only decay into atomic states of the nuclear GS, i.e., ⁷Li$_X^{\ast {\rm a}}$ and ⁷Li_X. We take the measured rate for the radiative decay of ⁷Li* (Tilley et al. 2002) for the decay of ⁷Li$^\ast _X$(1S) into the GS ⁷Li_X(1S).

Table 10. Calculated Parameters for ⁷Li(X⁻, γ)⁷Li_X in the WS40 Model

mX	Transition	τif	Γγ	Er
(GeV)		(fm)	(eV)	(MeV)
1	7Li$^\ast _X$(1S)→ 7LiX(1S)	⋅⋅⋅	0.00627a	0.280
100	7Li$^\ast _X$(2P)→ 7Li$^\ast _X$(1S)	3.91	1.26	0.124
1000	7Li$^\ast _X$(2P)→ 7Li$^\ast _X$(1S)	3.81	1.34	0.105

Note. ^aGiven by $\Gamma _\gamma = \tau _\gamma ^{-1}$ with the lifetime 105 fs taken from that of the first excited 1/2⁻ state of ⁷Li (Tilley et al. 2002).

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The resonant rates for the reaction ⁷Li(X⁻, γ)⁷Li_X via Types 1 and 2 (only for m_X = 1 GeV) transitions are derived to be

$$\begin{eqnarray*} &&N_{\rm A} \langle \sigma v \rangle _{\rm R} = \left\{\begin{array}{@{}l} 5.37 \times 10^2\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-3.24/T_9)\\ \quad\qquad\qquad ({\rm for}\; m_X=1\; \rm GeV) \qquad\qquad\ \ \ \,(42)\\ 1.72 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-1.44/T_9)\\ \quad\qquad\qquad ({\rm for}\; m_X=100\; \rm GeV) \qquad\qquad\!\, (43)\\ 1.68 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} T_9^{-3/2} \exp (-1.22/T_9)\\ \quad\qquad\qquad ({\rm for}\; m_X=1000\; \rm GeV). \qquad\qquad\!\!\!\!\! (44) \end{array} \right. \end{eqnarray*} \tag{ 42 }$$

The rate for m_X = 1 GeV corresponds to a pure nuclear transition from the resonance ⁷Li$^\ast _X$(1S) to the GS ⁷Li_X(1S) (magnetic dipole transition Tilley et al. 2002). The rate for m_X = 10 GeV is zero since there are no important resonances operating as a path for the recombination reaction. The rates for m_X = 100 and 1000 GeV correspond to the atomic transition from the resonance ⁷Li${^\ast _X}^{\ast {\rm a}}$(2P) to ⁷Li$^\ast _X$(1S).

The thermal nonresonant rate for the reaction ⁷Li(X⁻, γ)⁷Li_X was derived as a function of temperature T by integrating the calculated cross section σ(E) over energy (Equation (10)). The derived rates are

$$\begin{eqnarray*} &&N_{\rm A} \langle \sigma v \rangle _{\rm NR} =\left\{ \begin{array}{@{}l} 1.15 \times 10^5\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.453 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=1\; \rm GeV) \qquad\qquad\ \, (45) \\ 2.50 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.271 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=10\; \rm GeV) \qquad\qquad (46)\\ 1.70 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.247 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=100\; \rm GeV) \qquad\qquad\!\!\! (47)\\ 1.62 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.245 T_9 \right) T_9^{-1/2}\\ \quad \qquad\qquad ({\rm for}\; m_X=1000\; \rm GeV). \qquad\quad\ \,\, (48) \end{array} \right. \end{eqnarray*} \tag{ 45 }$$

Figure 10 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV as a function of radius r for the ⁷Li+X⁻ system with m_X = 1 GeV. Also shown is the recombination cross section as a function of energy E (bottom panel). Line types indicate the same quantities as in Figure 6. The order of solid lines at E = 10⁻⁵ MeV is the same as that of dotted lines. In general, the trends of calculated results are similar to those of the ⁷Be+X⁻ system (Figure 6). However, the Coulomb potential in the ⁷Li+X⁻ system is smaller than that in the ⁷Be+X⁻ system. Therefore, the spatial widths of wave functions in the former system are larger. The effect of the finite size of the charge distribution as shown by differences between the solid and dotted lines is then somewhat smaller in the ⁷Li+X⁻ system than in the ⁷Be+X⁻ system.

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Figures 11–13 show bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV as a function of radius r for the ⁷Li+X⁻ system in the case of m_X = 10 GeV, 100 GeV, and 1000 GeV, respectively. Also shown is the recombination cross section as a function of the energy E (bottom panel). Line types indicate the same quantities as in Figure 6. The orders of solid lines at E = 10⁻⁵ MeV are 1, 4, 8, 6, 2, 10, 7, 9, 5, 3 in Figure 11, and 4, 1, 8, 6, 10, 2, 7, 9, 3, 5 in Figures 12 and 13, respectively. Similar to the case of the ⁷Be+X⁻ system, larger m_X values lead to larger differences in both the wave functions and the recombination cross sections between the finite-size charge and point-charge cases. We also find that this ⁷Li+X⁻ system has the important characteristic that the transition d-wave → 2P is the most important for m_X ≳ 100 GeV.

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Table 11 shows the binding energies of ⁹Be_X atomic states that have main quantum numbers n from one to seven.

The nonresonant reaction rates for ⁹Be(X⁻, γ)⁹Be_X were also calculated for m_X = 1, 10, 100, and 1000 GeV in the WS40 model. The radius parameter for the WS40 model is R = 2.59 fm.

In the estimation of recombination rate for ⁹Be, the resonant reactions involving atomic states and nuclear excited states for ⁹Be* were neglected since even the first nuclear excited state has a large excitation energy of 1.684 MeV. We therefore only calculated the nonresonant rate.

The thermal nonresonant rate was derived as a function of temperature T by integrating the calculated cross section σ(E) over energy (Equation (10)). It is then

$$\begin{eqnarray*} &&N_{\rm A} \langle \sigma v \rangle _{\rm NR} =\left\{ \begin{array}{@{}l} 2.07 \times 10^5\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.339 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=1\; \rm GeV) \qquad\qquad\ \,(49)\\ 3.89 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.199 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=10\; \rm GeV) \qquad\qquad (50)\\ 2.32 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.180 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=100\; \rm GeV) \qquad\qquad\!\!\! (51)\\ 2.14 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.179 T_9 \right) T_9^{-1/2}\\ \quad\qquad\qquad ({\rm for}\; m_X=1000\; {\rm GeV}). \qquad\quad\,\,(52) \end{array}\right. \end{eqnarray*} \tag{ 49 }$$

Figure 14 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV as a function of radius r for the ⁹Be+X⁻ system for the case of m_X = 1 GeV. We also show recombination cross section as a function of the energy E (bottom panel). Line types indicate the same quantities as in Figure 6. The order of solid lines at E = 10⁻⁵ MeV is the same as that of the dotted lines. Trends of calculated results are similar to those of the ⁷Be+X⁻ system (Figure 6).

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Figures 15–17 show bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV as a function of radius r, and recombination cross sections as a function of the energy E (bottom panel) of the ⁹Be+X⁻ system for the cases of m_X = 10 GeV, 100 GeV, and 1000 GeV, respectively. Line types indicate the same quantities as in Figure 6. The orders of solid lines at E = 10⁻⁵ MeV are 4, 1, 8, 6, 10, 2, 7, 9, 3, 5 in Figure 15, and 4, 8, 6, 10, 1, 2, 9, 7, 3, 5 in Figures 16 and 17, respectively. Similar to the case of the ⁷Be+X⁻ system, larger m_X values lead to larger differences in wave functions and recombination cross sections between the finite-size charge and point-charge cases. We also find that the transition, d-wave → 2P, is most important for m_X ≳ 100 GeV in this ⁹Be+X⁻ system.

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Table 12 shows the binding energies of ⁴He_X atomic states that have main quantum numbers n from one to seven.

The nonresonant rates of the reaction ⁴He(X⁻, γ)⁴He_X were calculated for m_X = 1, 10, 100, and 1000 GeV using the WS40 model. For this case, the radius parameter is R = 1.31 fm. Since all excited states of ⁴He* have excitation energies larger than 20 MeV, atomic states of nuclear excited states ⁴He$_X^\ast$ are never important resonances in the recombination process. We then calculate only the nonresonant rate.

The thermal nonresonant rates were derived as a function of temperature T by integrating the calculated cross section σ(E) over energy (Equation (10)). The resultant rates are

$$\begin{eqnarray*} &&N_{\rm A} \langle \sigma v \rangle _{\rm NR} =\left\{ \begin{array}{@{}l} 5.38 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.648 T_9 \right) T_9^{-1/2} \\ \quad\qquad ({\rm for}\; m_X=1\; \rm GeV) \qquad\qquad\quad\ (53) \\ 1.63 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.404 T_9 \right) T_9^{-1/2} \\ \quad\qquad ({\rm for}\; m_X=10\; \rm GeV)\qquad\qquad\ \ \ (54) \\ 1.32 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.367 T_9 \right) T_9^{-1/2} \\ \quad\qquad ({\rm for}\; m_X=100\; \rm GeV) \qquad\qquad\ (55)\\ 1.29 \times 10^4\ {\rm cm}^3 \ {\rm mol}^{-1} \ {\rm s}^{-1} \left(1-0.363 T_9 \right) T_9^{-1/2} \\ \quad\qquad ({\rm for}\; m_X=1000\; \rm GeV). \qquad\quad\ \ \ \, (56) \end{array} \right. \end{eqnarray*} \tag{ 53 }$$

Figure 18 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV as a function of radius r for the ⁴He+X⁻ system in the case of m_X = 1 GeV. The recombination cross section is also given as a function of the energy E (bottom panel). Line types indicate the same quantities as in Figure 6. The order of solid lines at E = 10⁻⁵ MeV is the same as that of the dotted lines. Since the Coulomb potential in the ⁴He+X⁻ system is small, the wave functions are more extended spatially. Therefore, the effect of the finite-size charge distribution is small as evidenced by the fact that the solid and dotted lines almost overlap in this figure.

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Figures 19–21 show bound-state wave functions (upper panel) and continuum wave functions (middle panel) at E = 0.07 MeV as a function of radius r for the ⁴He+X⁻ system in the case of m_X = 10 GeV, 100 GeV, and 1000 GeV, respectively. The recombination cross section is also shown as a function of the energy E (bottom panel). Line types indicate the same quantities as in Figure 6. In Figures 19–21, the orders of solid lines at E = 10⁻⁵ MeV are the same as that of the dotted lines. It is apparent that larger m_X values lead to larger differences in the wave functions and recombination cross sections due to a finite-size versus a point-charge distribution. However, because of the small amplitude of the Coulomb potential, the effect of the finite-size nuclear charge does not significantly affect the wave functions and cross sections. As a result, even in the case of heavy X⁻ particles (m_X ≳ 100 GeV), the dominant transition contributing to the recombination is the p-wave → 1S, similarly to the case of the Coulomb potential for point charges.

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Figure 22 shows the calculated abundances of normal nuclei (panel (a)) and X nuclei (panel (b)) as a function of T₉ for m_X = 1 GeV. Curves for ¹H and ⁴He correspond to the mass fractions, X_p (¹H) and Y_p (⁴He) in total baryonic matter, while the other curves correspond to number abundances with respect to that of hydrogen. The dotted lines show the result of the SBBN model. The abundance and the lifetime of the X⁻ particle are assumed to be Y_X = 0.05 and τ_X = ∞, respectively, for this example.

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Early in the BBN epoch (T₉ ≳ 1), p_X is the only X-nuclide with an abundance larger than A_X/H >10⁻¹⁷. Its abundance is the equilibrium value determined by the balance between the recombination of p and X⁻ and the photoionization of p_X. When the temperature decreases to T₉ ≲ 1, ⁴He is produced as in SBBN (panel (a)). Simultaneously, the abundance of ⁴He_X increases through the recombination of ⁴He and X⁻ (panel (b)). As the temperature decreases further, the recombination of nuclei with X⁻ gradually proceeds in order of decreasing binding energies of A_X, similar to the recombination of nuclei with electrons.

⁷Be first recombines with X⁻ via the ⁷Be(X⁻, γ)⁷Be_X reaction at T₉ ≲ 0.1. The produced ⁷Be_X nuclei are then slightly destroyed via the ⁷Be_X(p, γ)⁸B_X reaction. In the late epoch, the ⁷Be abundance increases through the reaction ⁴He_X(³He, X⁻)⁷Be at T₉ ∼ 0.03–0.02.

The ⁶Li abundance decreases through the recombination reaction ⁶Li(X⁻, γ)⁶Li_X operating at T₉ ≲ 0.05. However, soon after the start of recombination, it is produced through the reaction ⁴He_X(d, X⁻)⁶Li at T₉ ∼ 0.03–0.02. After this production, the ⁶Li_X abundance also increases through recombination. In the late epoch, the ⁶Li abundance increases through the reaction ²H_X(α, X⁻)⁶Li at T₉ ∼ 4 × 10⁻³.

At T₉ ≲ 0.05, the ⁷Li abundance decreases through the recombination reaction ⁷Li(X⁻, γ)⁷Li_X. A small amount of ⁷Li is later produced through the reactions ⁴He_X(t, X⁻)⁷Li (T₉ ∼ 0.02–0.01), and ³H_X(α, X⁻)⁷Li [T₉ ∼ (6–5) × 10⁻³].

⁹Be is predominantly produced through the reaction ⁷Li_X(d, X⁻)⁹Be at T₉ ∼ 0.06–0.05. At T₉ ≲ 0.05, the recombination ⁹Be(X⁻, γ)⁹Be_X enhances the abundance of ⁹Be_X. The abundance of ⁹Be is small and not seen in this figure since it is converted to ⁹Be_X via the recombination. We note that the proton capture reaction ⁹Be_X(p, ⁶Li)⁴He_X does not work efficiently at this temperature of ⁹Be_X production.

The recombination of ⁷Be with an X⁻ particle and the subsequent radiative proton capture of ⁷Be_X occurs at T₉ ∼ 0.1 although the effect of the latter reaction cannot be seen well in this figure. In this case, the abundances of ⁷Be and ⁷Be_X only change through the recombination at T₉ ∼ 0.1. The abundance ratio of ⁷Be to ⁷Be_X is then simply described with chemical equilibrium (Rybicki & Lightman 1979) as

$$\begin{eqnarray} \frac{n_A n_X}{n_{A_X}} &=&\frac{g_A g_X}{g_{A_X}} \left(\frac{m_A m_X}{m_{A_X}} \frac{T}{2\pi }\right)^{3/2} \mathrm{e}^{(m_{A_X}-m_A-m_X)/T} \nonumber \\ &\approx & \left(\frac{\mu T}{2\pi }\right)^{3/2} \mathrm{e}^{-E_{\rm B}(A_X)/T}, \end{eqnarray} \tag{ 61 }$$

where the spin factor of X⁻ is g_X = 1 and only the GS of ⁷Be_X is considered so that $g_{A_X}=g_A$.

The baryon number density determined from the CMB WMAP measurement is

$$\begin{eqnarray} n_b&\approx & \frac{\rho _b}{m_p}=\frac{\rho _c\Omega _b}{m_p}\left(1+z\right)^3\nonumber \\ &=&1.26 \times 10^{19}\ {\rm cm}^{-3}\left(\frac{h}{0.700}\right)^2 \left(\frac{\Omega _b}{0.0463}\right)T_9^3, \end{eqnarray} \tag{ 62 }$$

where ρ_b and ρ_c are the baryon density and the present critical density, respectively. Ω_b = 0.0463 ± 0.0024 is the baryon density parameter, z is the redshift of the universe, which is related to temperature as (1 + z) = T/T₀ where T₀ = 2.7255 K is the present radiation temperature of the universe (Fixsen 2009), and h = H₀/(100 km s⁻¹ Mpc⁻¹) =0.700 ± 0.022 is the reduced Hubble constant with Hubble constant H₀. The cosmological parameters have been taken from values determined from the WMAP (Spergel et al. 2003, 2007; Larson et al. 2011; Hinshaw et al. 2013) (ΛCDM model; WMAP9 data only).

We define the recombination temperature T_rec(A) at which abundances of the ionized nuclei A and the bound state A_X are equal. The T_rec(A) value is determined as a function of the abundance of X⁻, Y_X, using Equations (61) and (62). For example, the recombination temperature of ⁷Be for the case of m_X = 1 GeV and Y_X = 0.05 is T_rec(⁷Be) = 8.49 keV (corresponding to T₉ = 0.0985). Since the recombination proceeds at temperatures lower than in the case of larger m_X, the number density of protons at recombination is smaller. As a result, the rate for ⁷Be_X to experience radiative proton capture in the temperature range of T₉ ≲ 0.1 is small. The reduction of the ⁷Be_X abundance through the proton capture is therefore less efficient than in the cases with m_X = 100 GeV and 1000 GeV. However, it is still much more efficient than the case with m_X = 10 GeV because of the smaller resonant energy in the resonant reaction ⁷Be_X(p, γ)⁸B_X (see Section 4).

Figure 23 shows contours of calculated final lithium abundances for the case of m_X = 1 GeV. These are normalized to the values observed in MPSs, i.e., d(⁶Li) = ⁶Li^Cal/⁶Li^Obs (blue lines) and d(⁷Li) = ⁷Li^Cal/⁷Li^Obs (red lines). The final ⁷Li abundance is a sum of the abundances of ⁷Li and ⁷Be produced in BBN. This is because ⁷Be is converted to ⁷Li via the electron capture at a later epoch. The dashed lines around the line of d(⁷Li) = 1 correspond to the 2σ uncertainty in the observational constraint. The gray region located to the right of the contours for d(⁶Li) = 10 and/or the 2σ lower limit, d(⁷Li) = 0.67, are excluded by the overproduction of ⁶Li and underproduction of ⁷Li, respectively. The orange region is the interesting parameter region in which a significant ⁷Li reduction occurs without inducing an overproduction of ⁶Li. Dotted lines are contours of the calculated abundance ratio of ⁹Be/H assuming a rate for the reaction ⁷Li_X(d, X⁻)⁹Be as described above in Section 6.

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If the X⁻ lifetime is long enough, larger X⁻ abundances lead to more production of ⁶Li. This is because the production rate of ⁶Li through the reaction ⁴He_X(d, X⁻)⁶Li is proportional to the abundance of ⁴He_X which is proportional to the X⁻ abundance as long as n_X ⩽ n_α. On the other hand, for large Y_X values the amount of ⁷Be destruction is not proportional to the X⁻ abundance. The reason is that the amount of ⁷Be destruction roughly depends upon the recombination temperature and the conversion rate of ⁷Be_X through the reaction ⁷Be_X(p, γ)⁸B_X. However, the recombination temperature is almost independent of Y_X. Therefore, the conversion rate is independent of Y_X although it is dependent on n_p. The destruction is therefore not efficient even if the Y_X values were very high.

The excluded gray region corresponds to Y_X ≳ 10⁻³ in the limit of a long X⁻-particle lifetime τ_X ≳ 10⁸ s. This region is determined from the overproduction of ⁶Li. The orange region corresponding to a solution to the ⁷Li problem is located at Y_X ≳ 0.1 and τ_X ∼ 5 × 10³–10⁵ s. Within this region, the primordial ⁹Be abundance is predicted to be ⁹Be/H ≲ 3 × 10⁻¹⁶. This is much larger than the SBBN value of 9.60 × 10⁻¹⁹ (Coc et al. 2012). Since the abundances of D, ³He, and ⁴He are not significantly altered, the adopted constraints on their primordial abundances are satisfied in this region.

Figure 24 shows the same contours for calculated abundances of ^{6, 7}Li and ⁹Be as in Figure 23. In this case, the instantaneous charged-current decay of ⁷Be_X → ⁷Li+X⁰ (Jittoh et al. 2007, 2008, 2010; Bird et al. 2008) is also taken into account. In this case, the X⁻ particle interacts not only via its charge but also a weak interaction (Jittoh et al. 2007, 2008, 2010). ⁷Be_X can then be converted to ⁷Li plus a neutral particle X⁰. Other X-nuclei may also decay depending upon the mass of the X⁰. The prohibition of ⁶Li overproduction, however, limits the length of the lifetime of the X⁻ as seen in Figure 23. Effects of the weak decay catalyzed by the X⁻ then appear through the conversion of X-nuclei produced just before the epoch of ⁶Li production.

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Above the recombination temperature for ⁴He_X at which ⁶Li production also proceeds, ⁶Li_X, ⁷Li_X, and ⁷Be_X can all be produced with large fractions of bound states (see Figure 22). Among these three X-nuclei, ⁷Be_X is the most abundant and its abundance evolution affects the parameter region for a solution to the Li problem. Therefore, for simplicity, we only consider the decay of ⁷Be_X here.

The contours for the ⁶Li abundance are similar to those in Figure 23. On the other hand, the ⁷Li abundance is much different from that in Figure 23 because of the different processes for ⁷Be destruction. Including the charged-current decay of ⁷Be_X, the destruction rate of ⁷Be in this model is the same as the recombination rate of ⁷Be itself. In the model without the decay, the destruction rate requires that ⁷Be_X nuclei produced via the recombination then experience a proton capture reaction without being re-ionized to a ⁷Be+X⁻ free state. The different processes of ⁷Be_X destruction therefore cause a difference in the efficiency for the final ⁷Li reduction. In this model with the decay, the amount of ⁷Be destruction roughly scales as Y_X unlike the model without the decay.

The excluded region is wider than in Figure 23. This region is determined from the ⁷Li underproduction. This region also involves lower values of τ_X than in Figure 23. The solution to the ⁷Li problem is at Y_X ≳ 8 × 10⁻⁴ and τ_X ≳ 10² s (orange region). In this region, the ⁹Be abundance is calculated to be ⁹Be/H  ≲3 × 10⁻¹⁷.

Figure 25 shows the same abundances as a function of T₉ as in Figure 22, but for the case of m_X = 10 GeV and without the decay of ⁷Be_X.