Roles of $^7$Be$(n,p)^7$Li resonances in big bang nucleosynthesis with time-dependent quark mass and Li reduction by a heavy quark mass

TL;DR

This study examines how variations in the quark mass during the early Universe influence nuclear reaction rates, especially 7Be(n,p)7Li, affecting primordial element abundances and potentially resolving the lithium problem.

Contribution

It introduces a detailed analysis of 8Be resonances' effects on BBN with a time-dependent quark mass, highlighting the impact on lithium abundance.

Findings

Resonance at 0.33 MeV significantly enhances reaction rates.

Negative quark mass variation reduces 7Li abundance.

Quark mass variation of (4--8)×10^{-3} aligns lithium predictions with observations.

Abstract

Big bang nucleosynthesis (BBN) has been used as a probe of beyond-standard physics in the early Universe, which includes a time-dependent quark mass m_q. We investigate effects of a quark mass variation delta m_q on the cross sections of the 7Be(n, p)7Li reaction and primordial light element abundances taking into account roles of 8Be resonances in the reaction during BBN. This resonant reaction has not been investigated although behaviors of low-lying resonances are not trivial. It is found that a resonance at the resonance energy E_r=0.33 MeV enhances the reaction rate and lowers the 7Li abundance significantly when the quark mass variation is negative. Based upon up-to-date observational limits on primordial abundances of D, 4He and Li, the quark mass variation in the BBN epoch are derived. In a model in which the resonance energies of the reactions 3He(d, p)4He and 3H(d, n)4He are insensitive to the quark mass, we find that the Li abundance can be consistent with observations for delta m_q/m_q = (4--8) times 10^{-3}.

Tables

Table 1. Table 1: The spins, resonance energies, corresponding excitation energies of 8 Be ∗ , and partial widths for the resonances in the 7 Be ( n , p ) 7 superscript 𝑛 𝑝 7 (n,p)^{7} Li reaction descouvemont .

$J$	$E_r$ [MeV]	$E_X$ [MeV]	${\Gamma }_i\left(E_r\right)$ [MeV]	${\Gamma }_f\left(E_r\right)$ [MeV]
$2^{-}$	0.00267	18.91	0.225	1.41
$3^{+}$	0.330	19.07	0.0767	0.088
		19.24
$3^{+}$	2.66	21.5	0.490	0.610

Table 2. Table 2: The cases considered in this BBN calculation. A, B, and C are explained in text. The 7 Be ( n , p ) 𝑛 𝑝 (n,p) reaction is treated independently from the other reactions, because the structure of its compound nucleus is different from the others.

	A	B	C
A	I	III	V
B/C	II	IV	VI

Table 3. Table 3: The 95 % C.L. of δ ​ m q / m q 𝛿 subscript 𝑚 𝑞 subscript 𝑚 𝑞 \delta m_{q}/m_{q} in units of 10 − 3 superscript 10 3 10^{-3} derived from observational constraints with (second column) and without (third column) Li abundances.

	D, He and Li	D and He
Case A	0.43–3.27	0.90–4.02
Case B	0.68–3.70	0.94–4.19
Case C	3.7–8.0	1.5–6.2

Equations

$$\sigma (E)\propto \sqrt{Q+E}\exp \left(-\sqrt{\frac{E_{\mathrm{G}}}{Q+E}}\right),$$

$$\frac{\sigma (E)}{{\sigma }_0}$$

$$N_{\mathrm{A}}⟨\sigma v⟩$$

$$\sigma (E)=\pi \text{lambdabar}(E{)}^2\frac{\omega {\Gamma }_{\mathrm{i}}(E){\Gamma }_{\mathrm{f}}(E)}{(E-E_{\mathrm{r}}{)}^2+[{\Gamma }_{\mathrm{r}}(E)/2{]}^2},$$

$$N_{\mathrm{A}}⟨\sigma v⟩=\sqrt{\frac{8}{\pi \mu }}\frac{N_{\mathrm{A}}}{(k_{\mathrm{B}}T{)}^{\frac{3}{2}}}{\int }_0^{\infty }EdE\sigma (E){\mathrm{e}}^{-E/k_{\mathrm{B}}T},$$

$$E_{\mathrm{r}}^{(\mathrm{d},\mathrm{p})}$$

$$E_{\mathrm{r}}^{(\mathrm{d},\mathrm{n})}$$

$$E_{\mathrm{r}}^{(\mathrm{n},\mathrm{p})}$$

$$\delta E_{\mathrm{r}}^{(\mathrm{n},\mathrm{p})}$$

Full text

The Standard Model of particle physics assumes that fundamental constants are time-independent over the cosmic history. However, possibility of time-dependent constants including bare coupling constants have been pursued for a long time dirac ; uzan .

The standard big bang nucleosynthesis (BBN) model kolb ; cyburt ; pitrou is characterized by a single cosmological parameter, the baryon-to-photon ratio. This parameter is precisely measured by cosmic microwave background observations Hinshaw:2012aka ; planck , and now BBN is a useful probe of beyond-standard physics in the early Universe. In addition, it has been pointed out that the observed 7Li abundance of metal-poor stars spite is significantly lower than the standard BBN (SBBN) prediction Ryan:1999vr ; Melendez:2004ni ; Asplund:2005yt ; Bonifacio:2006au ; shi2007 ; Aoki:2009ce ; Sbordone:2010zi . The solution to this “lithium problem” may come from systematic errors in inferring the primordial abundance from astronomical observations or lack of nuclear cross section data Cyburt:2009cf ; Chakraborty:2010zj . However, a reasonable solution has not been verified neither from astronomy nor from nuclear physics. Hence it is necessary to explore non-standard BBN to solve this problem.

BBN with a time-dependent quark mass has been studied by several authors berengut ; cheoun ; bedaque . A quark mass variation affects nuclear binding energies and hence resonance energies. Among main reactions in the BBN reaction network, 3He$(d,p)^{4}$He, 3H$(d,n)^{4}$He, and 7Be$(n,p)^{7}$Li are the only reactions whose cross sections are governed by resonances descouvemont . The resonances in the reactions 3He$(d,p)^{4}$He and 3H$(d,n)^{4}$He have been treated berengut ; cheoun ; bedaque . The 7Be$(n,p)^{7}$Li reaction is by far the strongest destruction reaction for 7Be nuclei in SBBN kawano . Therefore, the reaction cross section has been measured by many nuclear experiments exp1 ; exp2 ; exp3 ; exp4 ; exp5 ; exp6 . However, resonances in the reaction 7Be$(n,p)^{7}$Li have been ignored in the model of varying quark mass since the reaction is dominated by a broad resonance located around the separation threshold energy of the entrance channel. Nevertheless, a behavior of the near-threshold broad resonance at $E_{\mathrm{r}}=2.67$ keV should be treated carefully and another resonance at $E_{\mathrm{r}}=0.33$ MeV can play a role when the quark mass is changed, as shown in this letter.

Recently, new observations and reanalysis of astronomical data largely reduced uncertainties in primordial abundances sbordone ; cooke ; izotov ; aver . In this letter, we show the most stringent constraint on quark mass in the BBN epoch using these updated observational results.

The sensitivity of the binding energy of nucleus $A$, $E_{A}$, to the quark mass variation from the present value $\delta m_{q}$ is parameterized in terms of a sensitivity coefficient $K_{A}=\frac{\delta E_{A}/E_{A}}{\delta m_{q}/m_{q}}$, where $m_{q}$ is the quark mass and $\delta E_{A}$ is the variation of the binding energy due to $\delta m_{q}$. The $K_{A}$ values are adopted from Ref. flambaum for the case with the Argonne $v_{18}$ potential used as a two-nucleon potential and the Urbana model IX used as a three-nucleon potential.

For radiative capture reactions, usually the strongest electric dipole transition dominates, and cross sections depend on the kinetic energy $E$ and the $Q$-value berengut ; cheoun as $\sigma(E)\propto E_{\gamma}^{3}\sim(Q+E)^{3}$, where $E_{\gamma}$ is the energy of the emitted photon.

On the other hand, if reactions of two charged nuclei produce two charged nuclei in the final states, cross sections are proportional to the final state velocity $v\propto\sqrt{Q+E}$ and the penetration factor:

[TABLE]

where $E_{\mathrm{G}}=2\pi^{2}\mu_{34}c^{2}(\alpha Z_{3}Z_{4})^{2}$ is the Gamow energy for the exit channel with $\mu_{34}$ the reduced mass and $Z_{3}$ and $Z_{4}$ the exit-channel charges.

The $Q$ value is changed to $Q+\delta Q$ with the quark mass variation. When $E\ll Q$, which is usually realised in the BBN temperature, Eq. (1) can be expanded as

[TABLE]

where $\sigma_{0}$ is the cross sections for the case without quark mass variation, i.e., the SBBN values.

For narrow resonances, the reaction rates with a quark mass variation are given cheoun by

[TABLE]

where $N_{\mathrm{A}}$ is the Avogadro number, $\langle\sigma v\rangle$ is the reaction rates per particle pair, $[N_{\mathrm{A}}\langle\sigma v\rangle]_{0}$ is the reaction rate for the case without quark mass variation, $E_{0}=E_{\mathrm{G}}^{1/3}(k_{{\rm B}}T/2)^{2/3}$ is the Gamow peak with $k_{\mathrm{B}}$ the Boltzmann constant and $T$ the temperature, $\Gamma_{\mathrm{r}}$ is the total resonance width, and $E_{\mathrm{r}}$ and $E^{0}_{\mathrm{r}}$ are the resonance energy for the cases with and without a quark mass variation, respectively.

For broad resonances, Eq. (3) cannot be applied. The resonant cross sections are written by the Breit-Wigner formula iliadis

[TABLE]

where $\lambdabar(E)$ is the de Broglie wave length, $\omega=(2J+1)/(2j_{1}+1)(2j_{2}+1)$ is the spin factor with $J$ the spin of the resonant state, and $j_{1}$ and $j_{2}$ the spins of the two nuclei in the entrance channel, and $\Gamma_{\mathrm{i}}(E)$ and $\Gamma_{\mathrm{f}}(E)$ are the partial widths for the entrance and exit channels, respectively. The energy dependence of the partial widths is given as $\Gamma_{\mathrm{i,\;f}}(E)=2P_{\mathrm{i,\;f}}(E)\gamma_{\mathrm{i,\;f}}^{2}$, where $P_{\mathrm{i,\;f}}(E)$ is the penetration factor and $\gamma_{\mathrm{i,\;f}}^{2}$ is the reduced width. The reaction rates are written as

[TABLE]

where $\mu$ is the reduced mass of the two nuclei in the entrance channel. In the case of broad resonances, this integral should be performed numerically.

For the three reactions 3He$(d,p)^{4}$He, 3H$(d,n)^{4}$He, and 7Be$(n,p)^{7}$Li, the resonance energies are defined respectively as

[TABLE]

The shift of the resonance energy for the 7Be$(n,p)^{7}$Li reaction can then be written as

[TABLE]

For other two reactions, the shifts are written in the same manner cheoun . The sensitivity coefficient $K_{A}$ for the excited states is not studied flambaum , and we make assumptions of the following three cases separately for the three resonant reactions.

Case A

Variations in binding energies of excited states are the same as that of the ground state. In this case, the variation of the resonance energy is

[TABLE]

Case B

The resonance height does not change in the reverse reaction cheoun . In this case, the variation of the resonance energy in the forward reaction is

[TABLE]

Case C

The resonance energies do not change, i.e. $\delta E_{\mathrm{r}}=0$.

Our BBN calculation is based on Ref. kawano . Reaction rates have been updated for 4He($\alpha$, $\gamma$)7Be cyburt08 and 2H$(p,\gamma)^{3}$He ,2H$(d,n)^{3}$He, and 2H$(d,p)^{3}$H coc15 . Other reaction rates are taken from the JINA REACLIB Database reaclib . We adopt the neutron lifetime of 880.2 s pdg . The baryon-to-photon ratio $\eta=(6.108\pm 0.060)\times 10^{-10}$ is taken from Monte Carlo simulations cyburt based on the Planck 2015 observational results planck .

Table 1 shows adopted resonance energies and widths for the 7Be$(n,p)^{7}$Li reaction at the present time descouvemont .

Figure 1 shows calculated cross sections of the 7Be$(n,p)^{7}$Li reaction versus energy in Case A for $\delta m_{q}/m_{q}=$0.01 (dotted line), 0 (solid line), $-0.01$ (dashed-dotted line), and $-0.02$ (dashed line). The vertical axis $\sigma E^{1/2}$ is proportional to the reaction rate $\sigma v$ at the energy $E$. Shaded regions show partial cross sections by the second resonant component (Table 1) scaled by 1/2. In the energy range relevant to BBN, $E\lesssim{\mathcal{O}}(0.1)$ MeV, the first and second resonances predominantly contribute to the total cross section in SBBN.

In Case A, the resonance energy variation is $\delta E^{\mathrm{(n,p)}}_{\mathrm{r}}=17.2({\delta m_{q}}/{m_{q}})$ MeV. In the SBBN, the 7Be$(n,p)^{7}$Li cross section is predominantly contributed by a near-threshold resonance at $E_{\mathrm{r}}=2.67$ keV. However, the resonance energy of the next resonance at $E_{\mathrm{r}}=0.33$ MeV decreases to the kinetic energy at the 7Be synthesis, i.e. $E=3k_{\mathrm{B}}T/2\sim 0.1$ MeV, if $\delta m_{q}/m_{q}$ is negative. As seen in Fig. 1, all three resonances move to higher energies in the case of $\delta m_{q}/m_{q}=0.01$. The cross section is, therefore, smaller than in SBBN because of the hindered Boltzmann factor [Eq. (5)] for $E\lesssim 0.3$ MeV above which the second resonance contributes to the total cross section. Cross sections at low energies are higher than in SBBN for $\delta m_{q}/m_{q}=-0.02$ because of significant contributions of the second excited state. We note that the lowest resonance at $E_{\mathrm{r}}=2.67$ keV in the present universe is always important for the total cross section in the all four cases of $\delta m_{q}/m_{q}$. Even when the resonant state becomes a subthreshold bound state, it contributes to the total cross section because of its large width ($\Gamma=1.64$ MeV) and no hindrance of the neutron decay width $\Gamma_{n}$ from Coulomb potential.

Figure 2 shows the rate of the 7Be$(n,p)^{7}$Li reaction [Eq. (5)] as a function of $T_{9}=T/(10^{9}~{}{\rm K})$ for the same four cases as in Fig. 1. In the temperature range most relevant to the 7Be destruction during BBN, i.e., $T_{9}\lesssim 1$, the higher the quark mass is, the lower the reaction rate is. This results from the $\sigma v$ values at $E\lesssim 0.1$ MeV (see Fig. 1). Thus, the reaction rates are increased for negative values of $\delta m_{q}/m_{q}$.

In Case B, the resonance energy variation is $\delta E^{\mathrm{(n,p)}}_{\mathrm{r}}=-0.172({\delta m_{q}}/{m_{q}})$ MeV. This shift is much smaller than in Case A, and the effect on the reaction rate is negligible.

In the 3He$(d,p)$ and 3H$(d,n)$ reactions, compound nuclei are expected to have similar energy levels from the mirror conjugate states. We then assume that the same Case is applied to these two resonant reactions. This assumption is not applicable to the 7Be$(n,p)^{7}$Li reaction. Because the resonant energy shift of the 7Be$(n,p)^{7}$Li reaction is negligible in Case B, Case B and C are almost the same for the 7Be$(n,p)^{7}$Li reaction. Hence we consider only six cases (I to VI) shown in Table 2.

Figure 3 (upper panel) shows calculated Li abundance as a function of $\delta m_{q}/m_{q}$ for Cases I to VI. Cases II, IV, and VI are almost the same as cases investigated in Ref. cheoun for which behaviors of nuclear abundances have been analyzed. 7Li abundances are determined by changes in reaction rates of (i) 1H($n$,$\gamma$)2H, (ii) 3He($d$,$p$)4He and 3H($d$,$n$)4He, and (iii) 7Be($n$,$p$)7Li. For larger $\delta m_{q}/m_{q}$ values, the $Q$-value of the reaction (i) becomes smaller. Due to a delayed deuteron production in BBN, the neutron abundance is higher. Since the neutron is the dominant 7Be destroyer via 7Be($n$,$p$)7Li, the 7Be destruction rate is larger for larger $\delta m_{q}/m_{q}$. In addition, for larger $\delta m_{q}/m_{q}$ values, rates of reactions (ii) become smaller and the abundances of 3H and 3He become larger. As a result, abundances of 7Li and 7Be that are produced via 3H($\alpha$,$\gamma$) and 3He($\alpha$,$\gamma$) are larger. In previous studies, the rate of reaction (iii) has not been calculated accurately. However, it directly affects the 7Be and 7Li abundances.

The 7Li abundance decreases compared with Ref. cheoun in Cases I, III, and V if $\delta m_{q}/m_{q}$ is negative. This is because the resonance of the 7Be$(n,p)^{7}$Li reaction at $E_{\mathrm{r}}=0.33$ MeV experiences an energy shift and the reaction rates at the BBN temperature are enhanced. The reaction 7Be$(n,p)^{7}$Li converts 7Be to 7Li, and 7Li nuclei are easily burnt via the reaction 7Li$(p,\alpha)^{4}$He. Thus the primordial 7Li abundance, which is the sum of abundances of 7Li and 7Be during the BBN, decreases. In Cases II, IV, and VI, the 7Li abundance agrees with the previous work cheoun , because the 7Be$(n,p)^{7}$Li resonant reaction rates are not significantly affected by the quark mass variation.

The gray band in the upper panel of Fig. 3 shows the Spite plateau spite of 7Li abundance observed in metal-poor stars at $\log(\mathrm{Li}/\mathrm{H})+12=2.199\pm{0.086}$ sbordone . Excepting the region of $\delta m_{q}/m_{q}\gtrsim 0.01$ in Cases V and VI, the observed level is lower than the calculated results of the BBN model.

Figure 3 (lower panel) shows the deuterium abundance as a function of $\delta m_{q}/m_{q}$. The gray band corresponds to the observational 2$\sigma$ range of primordial D abundance cooke . It has been estimated from absorption lines in spectra of high-redshift quasars as $\mathrm{D/H}=(2.527\pm 0.030)\times 10^{-5}$. The error bar around the solid square shows an uncertainty in the SBBN model originating from the $2\sigma$ uncertainty in $\eta$ cyburt . The D abundance is larger for larger quark masses because of the delayed D production by the smaller $Q$ value of the 1H($n$,$\gamma$)2H rate resulting in a larger freezeout abundance of D cheoun .

Figure 4 shows the likelihood functions of the quark mass variation. Likelihood functions for respective nuclei $L_{i}$ ($i=$D, He, and Li) are assumed to be Gaussian, and the plotted ones are defined by products of $L_{\rm D}L_{\rm He}L_{\rm Li}$ (thick curves) and $L_{\rm D}L_{\rm He}$ (thin curves). We adopt the observational limit on the 4He abundance $Y_{p}=0.2449\pm 0.0040$ aver . When the Li abundance is used as a constraint, best fit values of $\delta m_{q}/m_{q}$ move to lower $\delta m_{q}/m_{q}$ regions (Cases A and B) and a higher region (Case C) in directions of reducing the Li abundances. In Case C, the likelihood function is wider since the constraint from the D abundance is weaker (Fig. 3).

Table 3 shows derived 95 % C.L. for Case A to C. We find a solution to the Li problem with a finite value of $\delta m_{q}/m_{q}=($4–8$)\times 10^{-3}$ in Case C. This result indicates that if the resonance energies of the reactions 3He$(d,p)^{4}$He and 3H$(d,n)^{4}$He are rather independent of the quark mass, the Li abundance becomes close to the observed value of metal-poor stars for $\delta m_{q}/m_{q}\lesssim 10^{-2}$.

In summary, the sensitivity of the 7Be$(n,p)^{7}$Li reaction rate to the change of a quark mass was investigated for the first time. We find that if the variation of the excitation energies of the compound nucleus 8Be∗ is the same as that of the ground state (i.e. Cases I, III, and V in Table 2), the shift of the resonance at $E_{\rm r}=0.33$ MeV can decrease the 7Be abundance significantly for $\delta m_{q}/m_{q}\lesssim-5\times 10^{-3}$.

The latest limit on the quark mass in the BBN epoch was derived for three models using updated observational constraints on primordial abundances. It is found that the Li abundance can be reduced toward the observed abundance level if the resonance energies of the reactions 3He$(d,p)^{4}$He and 3H$(d,n)^{4}$He does not depend on the quark mass.

This work is supported by NSFC Research Fund for International Young Scientists (11850410441).