Roles of ${}^7\mathrm{Be}(n,p){}^7\mathrm{Li}$ resonances in big bang nucleosynthesis with time-dependent quark mass and a possible Li reduction

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 the effects of a quark mass variation $\delta m_q$ on the cross sections of the ${}^7\mathrm{Be}(n,p){}^7\mathrm{Li}$ reaction and primordial light element abundances taking into account roles of ${}^8\mathrm{Be}$ 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 $E_r=0.33\mathrm{MeV}$ enhances the reaction rate and lowers the ${}^7\mathrm{Li}$ abundance significantly when the quark mass variation is negative. Based upon up-to-date observational limits on primordial abundances of D, ${}^4\mathrm{He}$, and Li, an updated constraint on the quark mass variation in the BBN epoch are derived. In a model in which the resonance energies of the reactions ${}^3\mathrm{He}(d,p){}^4\mathrm{He}$ and ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ are insensitive to the quark mass, we find that the Li abundance can be consistent with observations for $\delta m_q/m_q=(4–8)\times {10}^{-3}$.

Figure 1

The ${}^7\mathrm{Be}(n,p){}^7\mathrm{Li}$ cross sections multiplied by $E^{1/2}$ as a function of $E$ in Case A with quark mass variations as labeled. Shaded regions are contributions of the second resonant component scaled by $1/2$.

Figure 2

The ${}^7\mathrm{Be}(n,p){}^7\mathrm{Li}$ reaction rates versus $T_9=T/({10}^9\mathrm{K})$ in Case A with the same quark mass variations as in Fig. 1.

Figure 3

(Upper panel) The ${}^7\mathrm{Li}$ abundance as a function of $\delta m_q/m_q$ in Cases I to VI (see Table 2). The horizontal gray band shows the $2\sigma$ range of abundances observed in metal-poor stars [29]. (Middle panel) The deuterium abundance as a function of $\delta m_q/m_q$ for Cases A, B, and C for the ${}^3\mathrm{He}(d,p){}^4\mathrm{He}$ and ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$ reactions. The gray band shows the $2\sigma$ observational abundance [30]. The solid square with an error bar shows the central value and the uncertainty in the SBBN. (Lower panel) The ${}^4\mathrm{He}$ abundance as a function of $\delta m_q/m_q$ for Cases A, B, and C. The whole region is allowed by the observational $2\sigma$ limit on the ${}^4\mathrm{He}$ abundance [32].

Figure 4

(Upper panel) Likelihood functions of the quark mass variation $\delta m_q/m_q$ for Cases A to C of ${}^3\mathrm{He}(d,p){}^4\mathrm{He}$ and ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$. Thick and thin curves correspond to the functions derived with and without constraints from the observation of $\mathrm{Li}/\mathrm{H}$ abundance, respectively. The vertical solid line on $\delta m_q/m_q=0$ corresponds to the SBBN model. (Lower panel) Likelihood functions of the quark mass variation $\delta m_q/m_q$ for Case C of the reactions ${}^3\mathrm{He}(d,p){}^4\mathrm{He}$ and ${}^3\mathrm{H}(d,n){}^4\mathrm{He}$. Solid lines show arbitrarily normalized likelihood functions for respective nuclei $L_i$. Thick and thin dashed-dotted lines are products of the functions $L_{\mathrm{D}}{L}_{\mathrm{He}}{L}_{\mathrm{Li}}$ and $L_{\mathrm{D}}{L}_{\mathrm{He}}$, respectively.

Figure 5

Evolution of nuclear abundances as a function of $T_9=T/({10}^9\mathrm{K})$. $X$ and $Y$ are mass fractions of ${}^1\mathrm{H}$ and ${}^4\mathrm{He}$ in baryon in the universe, respectively. Other nuclear abundances are shown by the number ratios to ${}^1\mathrm{H}$, i.e., $A/\mathrm{H}$. The solid lines correspond to the heavy quark mass of $\delta m_q/m_q={10}^{-2}$ for Case C, while the dashed lines correspond to $\delta m_q=0$, i.e., the SBBN model.