The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

86 3. The derivation of nuclear forces from chiral Lagrangians
eqs. (B.2), (B.3) that l ::; r. For 4k + i = 4k + i, eq. (B.4) leads to l ::; r - 2. Finally, for
4k + i > 4k + i eqs. (B.5)-(B.8) require l ::; r - 2.
Eq. (3.218) is also valid for i = 3 apart from the single term _,\4k +3 Hl,\4( k+l) Al=r'l]. This is due
to eq. (B.7). In the ease i = 0, only the operators ,\4 k + i Al'l], restrieted by the eonditions
i = 0, k < k for l = r , or l < r (3.219)
ean enter the right hand side of eq. (3.217). Furthermore, the value of k is bounded from above
in all eases by the inequality
- 1
k ::; k + 6 (r - l + 8) ,
(3.220)
as ean be deferred from the seeond inequality in (B.8) and the equality in (B.16).
Now it is clear how to deal with the system of equations (3.205). The equations have to be solved
order by order, starting from r = O. At eaeh fixed order r one solves the equations with inereasing
number 4k + i starting from 4k + i = 1 to obtain all operators ,\4k + iAl=r'l]. This requires the
knowledge of operators ,\ 4k + i Al'l], 4k + i < 4k + i, of the same order l = r and a finite number
of operators ,\(4k + i ) Al 'I] at lower orders l < r. The only exeeptions of this proeedure are the
equations with arbitrary k's and i = 3, that require the knowledge of the operators ,\ 4( k +l) Al=r'l].
Such equations have, therefore, to be solved after those ones with 4( k + 1) external pions. The
number of equations to be solved at each order can be estimated by use of eq. (3.220) and the
inequalities of appendix B. After solving the required number of equations at order r one ean go
to the next order r + 1. These rules for the reeursive solution of eqs. (3.205) are summarized onee
more graphically in fig. 3.5.
To justify our ansatz about the strueture of the operator A, eonsider the starting equations at
order r = 0, given by
E(,\4),\4Ao'l] = ,\4H 2 '1]
E(,\l),\l Ao'l] = ,\ 1 Hl'l]
(3.221)
(3.222)
From eq. (3.217) one can see that the unknown operator ,\4k + i Al=r'l] has the structure which we
assumed at the beginning of this seetion, if and only if the already known operators '\A'I] entering
the right hand side of this equation have precisely this form. Therefore, to proof our ansatz
recursively for all operators '\A'I] it is sufficient to see that it holds for the eorresponding starting
operators in eqs. (3.221) and (3.222), whieh is obviously the ease.
Having ealculated the operators ,\ 4k + i A'I] it is straight forward to obtain the expansion in powers
of Q for the transformed Hamiltonian eq. (3.197). To do that we have to estimate the order of all
terms on its right hand side. Introducing the operators
and their ehiral power (the power of Q)
with
L = 'I1At \ 4kt '11
t - ., lt /\ +it A l't "
Vt = 3 - 3N + Vt
Vt = 4kt + 2it + lt + l' t
(3.223)
(3.224)
(3.225)