4.3. Quark Masses and the Range of Nuclear Forces: Diproton Stability

We have explored two of the three dimensions in m_u, m_d, m_e space: m_d - m_u and m_e. In addition there is a third dimension to explore, m_d + m_u. This quantity affects the pion mass and therefore the range of the nuclear interactions; this does not affect the np stability arguments but does affect the D stability.

The dependence on this third dimension of fermion mass variation can be estimated through the effect of changes in nucleon potential through the pion mass, m² (m_u + m_d) _QCD. In this framework Agrawal et al. (1998) investigated the effect of varying the Higgs expectation value v, which changes all the fermion masses in proportion. Using a simple model of the deuteron potential (range 2 fm, depth 35MeV) they found no bound states anymore if the range is reduced by 20%, or the quark mass sum is increased by 40%. This corresponds to a change v / v₀ = 1.4 or m_i = 0.4 m_i, or approximately m_d + m_u 0.4 (m_d + m_u) 7 MeV. (See also the earlier discussion of light nuclei stability by Pochet et al. 1991). On the side of decreasing quark masses or increasing range (i.e. m_d + m_u < 0), the effects are opposite; at about m_d + m_u -0.25 (m_d + m_u) -4 MeV, the diproton ²He or the dineutron become bound (Dyson 1971). (Which one is stable depends on the mass difference m_d-u.) However, a tighter constraint in this dimension is likely to arise from the behavior of heavier nuclei.