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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158 5. Isosp<strong>in</strong> violation <strong>in</strong> the two-nucleon system<br />
lead<strong>in</strong>g to<br />
.. ..<br />
.. ...<br />
Figure 5.1: Resummation of the lowest order (NtN)2 contact terms ", Co.<br />
1 ( . mp )<br />
6 = 2i In 1 + 2 21f A .<br />
+<br />
• • •<br />
(5.25)<br />
Expand<strong>in</strong>g both sides <strong>in</strong> the small moment um scale Q '" p (for the amplitude A one uses the<br />
expansion (5.20)) one obta<strong>in</strong>s:<br />
(5.26)<br />
where<br />
1 ( mp )<br />
60 = ---; In 1 + i-A-1 ,<br />
22<br />
21f<br />
6 = 60 + 61 + ... ,<br />
(5.27)<br />
Up to now we have not yet <strong>in</strong>corporated pions <strong>in</strong>to the effective theory. <strong>The</strong> tree contribution to<br />
the amplitude due to the one-pion exchange is given by<br />
2 -+<br />
4f2 �2 M2<br />
-+ -+ -+<br />
gA 0"1 . q 0"2 . q<br />
n q + n<br />
Tl . T2 ,<br />
(5.28)<br />
which sc ales as 0(1) and thus occurs at the same order as an <strong>in</strong>sertion of C2p2. Consequently,<br />
pions can be treated perturbatively and the lead<strong>in</strong>g contributions to the amplitude due to the OPE<br />
are given by graphs 2, 3 and 4 <strong>in</strong> fig. 5.2. <strong>The</strong> last graph <strong>in</strong> this figure depicts the contribution from<br />
the contact <strong>in</strong>teraction D2M;, which can be, <strong>in</strong> pr<strong>in</strong>ciple, absorbed by the Co vertex.u To study<br />
at which energies such a perturbative treatment of pions would be no more adequate, the authors<br />
of ref. [91] used the renormalization group technique. We will not repeat their arguments here<br />
and refer the <strong>in</strong>terested reader to the orig<strong>in</strong>al publication. We only note that the scale enter<strong>in</strong>g<br />
the low-momentum expansion of the amplitude turns out to be<br />
161f l;<br />
ANN = - 2 - '" 300 MeV .<br />
gAm<br />
This leads to the expansion parameter Q/ ANN '" Mn/ ANN '" 0.46.<br />
(5.29)<br />
To end this section we would like to write down the expressions for A�I)_A�V) obta<strong>in</strong>ed <strong>in</strong> ref. [91]:<br />
11 However, one needs to keep this term explicitly <strong>in</strong> order to be able to perform consistent renormalization of the<br />
dressed OPE <strong>in</strong> the fourth graph <strong>in</strong> fig. 5.2.