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where Δ mc 2 is the difference of the rest energies between a proton and a neutron. As most of neutrons that had existed before the BBN era were brought into helium nu- clei, a negative ξ$_{e}$ enhances helium synthesis. We treat ξ$_{e}$ as a free parameter restricted from observations [26] and assume β -equilibrium at the beginning of calculations. Allowed range with 2 σ CL is 

Note that the Affleck-Dine mechanism prefers negative ξ$_{e}$ with the absolute value of the order of η because η is supposed to be positive [27]. 

Because the horizon scale right after the BBN is much smaller than 10 $^{6}$M ⊙ in mass, the effect of diffusion is negligible. Thus, we calculate the BBN with high η in the same way as the homogeneous BBN ignoring inho- mogeneities inside the region and adopt the same method of numerical calculations as that of the ordinary homo- geneous BBN. We first calculated BBN with the reaction network including 61 nuclei shown in Table. I. The nu- cleosynthesis with a larger reaction network with nuclei in Table. II is also calculated to see how much heavy el- ements are produced. Heavy element production during BBN with high η is discussed in [22–24]. To calculate BBN, we used the Kawano code introduced in [28], with revised reaction rates taking from [29] whenever they are available. Reaction rates of heavy elements are taken from [30]. 

We calculate the BBN with various parameters. The calculations are started from the time when the tempera- ture is 10 11 K and we obtained the abundances at 10 7 K. The nuclear statistical equilibrium and the β -equilibrium are assumed at the initial state. 

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## B. Numerical Calculations 

## III. RESULTS 

To locate a star on the bMS in ω Centauri by enhancing the helium content, the helium fraction should be as high as 0.35 [3]. Therefore we require η to be greater than η$_{b}$ that reproduces Y$_{p}$ = 0 . 35. To estimate the amount of helium produced during BBN, only the reactions of light elements are essential. For this reason, we use the small reaction network to estimate Y$_{p}$ . 

The results of calculations are shown in Fig. 1. We performed the calculation with three different ξ$_{e}$ ; ξ$_{e}$ = 0, 0 . 01, and − 0 . 1. The horizontal line indicates Y$_{p}$ = 0 . 35. For ξ$_{e}$ = 0, η yielding Y$_{p}$ > 0 . 35 is in the region 

Thus η$_{b}$ = 3 × 10 − $^{5}$. For ξ$_{e}$ = − 0 . 01 and ξ$_{e}$ = 0 . 1, η$_{b}$ = 2 × 10 − 5 and η$_{b}$ = 2 × 10 − $^{4}$, respectively. η above these values are required to explain the bMS stars. 

As previously predicted by Matsuura et al. [22–24], heavy elements are produced during BBN with high η . The amount of heavy elements produced during BBN gives the upper limit of η . Too large η might overproduce heavy elements incompatible with the metallicity of bMS, which is at least a factor of a few smaller than the solar abundances. As nuclei with the mass number ( A ) of ∼ 80 or greater are important for this restriction, we use a large reaction network for this purpose. For calculations of the large reaction network, we only consider the case of ξ$_{e}$ = 0. 

In Fig. 2, we show the results of BBN calculations for some values of η . Between η $^{= 1}$×10 − 5 and η $^{= 2}$×10 − $^{5}$, 

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## A. Lower Limits 

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## B. Upper Limits 

