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