The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.9. Three-nucleon potential
Single ß-excitation in the box graphs:
VTPEP A,box-1 =
3g�
(2M2 + q2)2 A(q)
327f f: 7f ß
19::2 f: (Ti · T2) {(12ß2 - 20M; - Ul)L(q) + 6E2 D(q)} (3.333)
12��� f : ((51 . if)(52 . if) - q2(51 . (2)) { -2L(q) + (s2 -4ß2)D(q)}
128��: (Tl · T2) ((51 . if)(52 · if) -q2(51 · 52)) s2 A(q) ,
ß
with A(q) given in eq. (3.311).
Double ß-excitation in the box graphs:
VTPEP A,box-2 = -6 :::f: { -4ß2 L(q) + E[H(q) + (E + 8ß2)D(q)]}
with
38 ::2 f: (Tl · T2) { (12E - s2)L(q) + 3E[H(q) + (8ß2 -E)D(q)]}
51�� f: ((51 . if)(52 . if) - q2(51 . (2)) {6L(q) + (12ß 2 - s2)D(q)} (3.334)
102��2 f: (Tl · T2) ((51 . if)(52 · if) - q2(51 · 52)) {2L(q) + (4ß2 + s2)D(q)}
113
(3.335)
Note that we have not shown the polynomial part of the potential since it has the structure given
by eq. (3.270). Further note that some of the contributions, in particular, those ones corresponding
to the first and the last lines in eq. (3.333) have the same structure as the corresponding NNLO
terms in eq. (3.310) with -C3 = 2C4 = g�/(2ß). This is discussed in detail in ref. [109]. However,
the remaining contributions show a non-trivial dependence of the N N potential on the N ß mass
splitting. In the next chapter we will furt her discuss effects of the explicit inclusion of the ß-isobar
excitations for the low-energy N N observables. It would also be interesting to perform NNLO
analysis of the TPEP with explicit ß's. In that case one should refit the LEC's Cl , C3 and C4 from
pion-nucleon scattering using the small scale expansion. This work is under way [190].
3.9 Three-nucleon potential
Let us consider the three-nucleon force. Obviously, the leading part of the effective potential,
Ve�-3N in eq. (3.253), does not involve three-nucleon operators. The first contributions to the 3N
force start at order 8-3N = -1. Let us now take a closer look at eq. (3.254). The first operator in
this equation corresponds to contact interactions with four nucleon legs and without derivatives.
It does not lead to a three-body force. The next three terms in the first line of eq. (3.254) yield
the two-pion exchange three-nucleon interaction represented by the graph 9 in fig. 3.23. Here, one
has to sum over all possible time orderings. It is interesting that after performing such summation
this three-nucleon force vanishes completely, as has been pointed out by Weinberg [73]. This can
be shown in an elegant way [114] without performing the explicit calculation. Indeed, one obtains