3 Z A Z A Z A n 1 K 29-70 Sr 68-131 H 1-3 Ca 30-73 Y 70-134 He 3,4,6 Sc 32-76 Zr 72-137 Li 6-9 Ti 34-80 Nb 74-140 Be 7-12 V 36-83 Mo 77-144 B 8,10-14 Cr 38-86 Tc 79-147 C 9-18 Mn 40-89 Ru 81-150 N 11-21 Fe 42-92 Rh 83-153 O 13-22 Co 44-96 Pd 86-156 F 14,16-26 Ni 46-99 Ag 88-160 Ne 15-41 Cu 48-102 Cd 90-163 Na 17-44 Zn 51-105 In 92-166 Mg 19-47 Ga 53-108 Sn 94-169 Al 21-51 Ge 55-112 Sb 97-172 Si 22-54 As 57-115 Te 99-176 P 23-57 Se 59-118 I 101-179 S 24-60 Br 61-121 Xe 103-182 Cl 26-63 Kr 63-124 Ar 27-67 Rb 66-128 Z A Z A Z A n 1 C 11-15 Mg 24-27 H 1-3 N 12-16 Al 25-28 He 3,4 O 14-19 Si 28-31 Li 6-8 F 17-20 P 29-32 Be 7,8 Ne 19-23 S 32 B 8,10-12 Na 21-24 Table. I: Nuclei included in the small reaction network. Z is the name of nuclei and A is the mass number of Z included in the network. ’n’ refers to neutron. Table. II: Nuclei included in the large reaction network. See the caption of Table. I for details. 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]. 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$_{p}$ . A. Lower Limits B. Numerical Calculations − 0 . 01 ≤ ξ$_{e}$ ≤ 0 . 1 . (3) η > $^{3}$× 10 − $^{5}$. (4) 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 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]. 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 − $^{5}$, 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. B. Upper Limits III. RESULTS