The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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164 5. Isosp<strong>in</strong> violation <strong>in</strong> the two�nuc1eon system<br />
potential models (if one assurnes that there is <strong>in</strong>deed some charge dependence <strong>in</strong> the pion�nucleon<br />
coupl<strong>in</strong>g constants, see e.g. the discussion <strong>in</strong> ref. [216].) We remark that the TPE contribution is<br />
nom<strong>in</strong>ally suppressed by one order <strong>in</strong> the small parameter, Le. one would expect its contribution<br />
to be one third of the lead<strong>in</strong>g OPE with different pion masses.<br />
It is also <strong>in</strong>structive to see how the corrections due to the np mass difference come about. For<br />
that, let us write the nucleon mass as m + 6m, where 6m subsurnes the em and strong contribution<br />
to the np mass splitt<strong>in</strong>g, Le. these terms are 0(0:') and O( E) . Differentiat<strong>in</strong>g the lead<strong>in</strong>g isosp<strong>in</strong>�<br />
symmertic amplitude eq. (5.31) with respect to the nucleon mass shift gives<br />
from which it follows that<br />
8A�1 A�1 .<br />
-- = --(/1+Zp) ,<br />
8m 47f<br />
We now turn to CSB. <strong>The</strong> pattern of the various contributions looks different:<br />
LO<br />
Q�2 0:' ,E Q�2<br />
NLO<br />
Q�1 Q�1 0:' ,E<br />
Electromagnetic four�nucleon contact <strong>in</strong>teractions<br />
with no derivatives.<br />
Ern four-nucleon contact terms with two derivatives,<br />
strong isosp<strong>in</strong>�break<strong>in</strong>g contact terms with two derivatives,<br />
<strong>in</strong>sert ions proportional to the np mass difference.<br />
(5.42)<br />
(5.43)<br />
We remark that the lead<strong>in</strong>g order CSB effects do not modify the effective range but the correspond<strong>in</strong>g<br />
scatter<strong>in</strong>g length. Note furt her that we have not considered effects due to virtual<br />
photons (like, for <strong>in</strong>stance, 7f"( exchanges), see ref. [228]. Such diagrams were calculated with<strong>in</strong><br />
the We<strong>in</strong>berg power count<strong>in</strong>g approach <strong>in</strong> ref. [218]. <strong>The</strong> ISO np low�energy parameters appear<br />
to be very little affected.