The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

164 5. Isospin violation in the two�nuc1eon system
potential models (if one assurnes that there is indeed some charge dependence in the pion�nucleon
coupling constants, see e.g. the discussion in ref. [216].) We remark that the TPE contribution is
nominally suppressed by one order in the small parameter, Le. one would expect its contribution
to be one third of the leading OPE with different pion masses.
It is also instructive to see how the corrections due to the np mass difference come about. For
that, let us write the nucleon mass as m + 6m, where 6m subsurnes the em and strong contribution
to the np mass splitting, Le. these terms are 0(0:') and O( E) . Differentiating the leading isospin�
symmertic amplitude eq. (5.31) with respect to the nucleon mass shift gives
from which it follows that
8A�1 A�1 .
-- = --(/1+Zp) ,
8m 47f
We now turn to CSB. The pattern of the various contributions looks different:
LO
Q�2 0:' ,E Q�2
NLO
Q�1 Q�1 0:' ,E
Electromagnetic four�nucleon contact interactions
with no derivatives.
Ern four-nucleon contact terms with two derivatives,
strong isospin�breaking contact terms with two derivatives,
insert ions proportional to the np mass difference.
(5.42)
(5.43)
We remark that the leading order CSB effects do not modify the effective range but the corresponding
scattering length. Note furt her that we have not considered effects due to virtual
photons (like, for instance, 7f"( exchanges), see ref. [228]. Such diagrams were calculated within
the Weinberg power counting approach in ref. [218]. The ISO np low�energy parameters appear
to be very little affected.