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

We have explored two of the three dimensions in deltamu, deltamd, deltame space: deltamd - deltamu and me. In addition there is a third dimension to explore, deltamd + deltamu. 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, mpi2 propto (mu + md) LambdaQCD. 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 deltami = 0.4 mi, or approximately deltamd + deltamu approx 0.4 (md + mu) approx 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. deltamd + deltamu < 0), the effects are opposite; at about deltamd + deltamu approx -0.25 (md + mu) approx -4 MeV, the diproton 2He or the dineutron become bound (Dyson 1971). (Which one is stable depends on the mass difference delta md-u.) However, a tighter constraint in this dimension is likely to arise from the behavior of heavier nuclei.