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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3.9. Three-nuc1eon potential<br />
2 3<br />
Figure 3.26: Lead<strong>in</strong>g three-nuc1eon force with b,-isobar excitations. All possible<br />
time order<strong>in</strong>gs should be considered. For notations see figs. 3.6, 3.16 and fig. 3.23.<br />
more-nuc1eon system. First steps <strong>in</strong> this direction have been done <strong>in</strong> refs. [118], [196]. In the<br />
first work, Friar et al. consider the constra<strong>in</strong>ts on a possible form of the TPE force aris<strong>in</strong>g from<br />
requirement of its (approximate) chiral symmetry. <strong>The</strong>y po<strong>in</strong>ted out that any TPE 3N force can<br />
be expressed as<br />
where<br />
117<br />
(3.338)<br />
(3.339)<br />
Here ij and ij' are the moment um transfers between the nuc1eons 1, 3 and 3, 2. Note that these<br />
expressions are valid modulo l/m corrections. <strong>The</strong> authors of ref. [118] extracted the coefficients<br />
a, b, C and d from various exist<strong>in</strong>g 3N forces, which are based on different models, and compared<br />
them with the predictions obta<strong>in</strong>ed with<strong>in</strong> the framework of chiral perturbation theory without<br />
explicit b,'s. In the latter case, the lead<strong>in</strong>g order values of these coefficients can be calculated<br />
from the 3N force shown <strong>in</strong> fig. 3.25 (1). <strong>The</strong> values of the relevant 7r7rNN coupl<strong>in</strong>gs Cl , C3 and<br />
C4 are known from the pion-nuc1eon sector, as discussed <strong>in</strong> sec. 3.8.2 and thus the coefficients a,<br />
b, c and d can be predicted without free parameters. It turns out that although the signs of these<br />
coefficients are always the same, their values differ substantially (by factors of two) for different<br />
three-nuc1eon forces. Furthermore, to lead<strong>in</strong>g order <strong>in</strong> low-momentum expansion CHPT leads to<br />
the largest absolute values for the coefficients a, b and d48 and to c = O. <strong>The</strong> c-term does not<br />
vanish for the Tucson-Melbourne (TM) force [43] and thus violates chiral constra<strong>in</strong>ts. <strong>The</strong>refore<br />
<strong>in</strong> ref. [118] it is recommended to drop this c-term <strong>in</strong> the TM force.<br />
In the second work, Hüber et al. analyse the effects of the 3N force correspond<strong>in</strong>g to the second<br />
graph <strong>in</strong> fig. (3.25) <strong>in</strong> elastic neutron-deuteron scatter<strong>in</strong>g at low-energies. <strong>The</strong>y found that the<br />
<strong>in</strong>clusion of such a short-range-long-range force might considerably improve the agreement for<br />
the analyz<strong>in</strong>g power Ay• This study is, however, merely qualitative, s<strong>in</strong>ce only a part of the<br />
lead<strong>in</strong>g-order 3N force result<strong>in</strong>g from CHPT has been <strong>in</strong>cluded and no attempt was made to<br />
obta<strong>in</strong> a reasonable fit to data.<br />
An important observation has been done by van Kolck <strong>in</strong> ref. [77]. He po<strong>in</strong>ted out that several<br />
vertices enter<strong>in</strong>g the expressions for the lead<strong>in</strong>g 3N force are saturated by virtual b,-isobar ex-<br />
4 8 Note, however, that the authors of ref. [118) used the values for Cl,3,4 as given <strong>in</strong> [71) that are somewhat larger<br />
than the ones from the later publication [195], which we adopt <strong>in</strong> this work.