**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,
*m*_{}^{2}
(*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* / *v*_{0} = 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 ^{2}He 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.