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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116 3. <strong>The</strong> derivation o{ nuc1ear {orces {rom chiral Lagrangians<br />
2 3<br />
Figure 3.25: Three-nucleon force: TPE, OPE and contact <strong>in</strong>teraction. In the case<br />
of diagrams 1 and 2, all possible time order<strong>in</strong>gs should be considered. For notations<br />
see figs. 3.6, 3.23 and fig. 3.14.<br />
an exact cancelation between the contributions from the "reducible" and irreducible OPE and<br />
TPE diagrams. This cancelation was recently po<strong>in</strong>ted out <strong>in</strong> [185] for the case of an expansion<br />
<strong>in</strong> the pion-nucleon coupl<strong>in</strong>g constant and adopt<strong>in</strong>g the static approximation for nucleons. To<br />
understand why the lead<strong>in</strong>g 3N force vanishes, let us take a closer look at the terms <strong>in</strong> the third<br />
l<strong>in</strong>e of eq. (3.254). <strong>The</strong> contributions from the irreducible TPE diagrams 1-8 <strong>in</strong> fig. 3.23 can be<br />
expressed schematically as:<br />
(3.336)<br />
where we pulled out the common factor M represent<strong>in</strong>g the sp<strong>in</strong>, isosp<strong>in</strong> and momentum structure,<br />
which is obviously the same for all graphs 1-8 <strong>in</strong> fig. 3.23. <strong>The</strong> contribution from the "reducible"<br />
diagrams 1-4 <strong>in</strong> fig. 3.24 can be expressed as<br />
[ 2 2 ] Wl + W2<br />
+ -2- w1 w2 WIW2<br />
--2 M=2 w1 22M. w2<br />
(3.337)<br />
<strong>The</strong> cancelation is now evident. <strong>The</strong> same sort of cancelation can be observed for the irreducible<br />
diagram 10 <strong>in</strong> fig. 3.23 and the "reducible" graphs 5 and 6 <strong>in</strong> fig. 3.24 <strong>in</strong>volv<strong>in</strong>g contact <strong>in</strong>teractions.<br />
We conclude that there is no three-nucleon force at the order v = -l.<br />
Let us now consider the 3N force at order v = 0 related to the effective potential (3.255). <strong>The</strong><br />
terms <strong>in</strong> the second l<strong>in</strong>e of eq. (3.255) refer to the two-pion exchange diagram 1 <strong>in</strong> fig. 3.25. <strong>The</strong><br />
first term <strong>in</strong> this equation corresponds to the contact force shown <strong>in</strong> fig. 3.25 (3). F<strong>in</strong>ally, the<br />
one pion exchange graph 2 <strong>in</strong>volv<strong>in</strong>g contact <strong>in</strong>teractions with four nucleon legs arises from the<br />
second and third terms <strong>in</strong> eq. (3.255). In all cases the method of unitary transformation does not<br />
<strong>in</strong>troduce any new aspects, s<strong>in</strong>ce it yields the same result as time-ordered perturbation theory.<br />
With<strong>in</strong> the last approach, this 3N force has been calculated and discussed by van Kolck [77].<br />
In that reference a complete expression for the lead<strong>in</strong>g chiral 3N force is given. Unfortunately,<br />
the <strong>in</strong>clusion of the lead<strong>in</strong>g 3N <strong>in</strong>teraction <strong>in</strong>tro duces several new parameters. While the 1f N N<br />
coupl<strong>in</strong>gs D1 and D2 <strong>in</strong> eq. (3.236) can, <strong>in</strong> pr<strong>in</strong>ciple, be determ<strong>in</strong>ed from the processes like 1fdeuteron<br />
scatter<strong>in</strong>g or 1f production and absorption on N N system, the rema<strong>in</strong><strong>in</strong>g three contact<br />
<strong>in</strong>teractions El,2,3 can only be fixed from data <strong>in</strong>volv<strong>in</strong>g systems with more than two nucleons.<br />
At present, this 3N force has not yet been applied <strong>in</strong> systematic calculations of the three- and