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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108 3. <strong>The</strong> derivation of nuclear forces from chiral Lagrangians<br />
A and B are valid also <strong>in</strong> this case. Furthermore, one has to <strong>in</strong>clude states conta<strong>in</strong>ed <strong>in</strong> A ° with<br />
.6.'s and without pions. This is discussed <strong>in</strong> appendix C. Let us now take a closer look at these<br />
modifications.<br />
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Figure 3.18: First corrections to the NN potential with .6.-excitations: irreducible<br />
(1-4) and reducible (5-8) vertex and self-energy corrections to the one-pion exchange.<br />
For notations see figs. 3.6, 3.16.<br />
For the projected operator Aa A1] we make the same ansatz as <strong>in</strong> the case without .6., namely that<br />
it consists of an equal number of vertices and energy denom<strong>in</strong>ators. S<strong>in</strong>ce the <strong>in</strong>clusion of the .6.<br />
does not affect the power count<strong>in</strong>g scheme, equations (3.210) and (3.212) are not modified. Here<br />
and <strong>in</strong> what follows, we will denote by N the baryon number. Consequently, the m<strong>in</strong>imal value<br />
of VA is aga<strong>in</strong> given by eq. (3.213). Furthermore, m<strong>in</strong>(vA) = 2 for the operator AOA1], as follows<br />
from eq. (3.210). <strong>The</strong>refore, we will aga<strong>in</strong> def<strong>in</strong>e the order l of the operator Aa A1] via eq. (3.214).<br />
As discussed <strong>in</strong> appendix C, the <strong>in</strong>clusion of the ß's does not change the m<strong>in</strong>imal value of the<br />
chiral power v for the projected decoupl<strong>in</strong>g equation (3.206). <strong>The</strong>refore, we can apply eqs.<br />
(3.216)<br />
(3.215),<br />
without any changes. Thus, the system of equations (3.205) can be solved perturbatively<br />
precisely <strong>in</strong> the same way as <strong>in</strong> the case without ß. <strong>The</strong> only modification is that at each order<br />
r one has to start by solv<strong>in</strong>g the equation (3.217) projected with AO • It is shown <strong>in</strong> appendix C<br />
that all operators Aa A/1] of the right-hand side of this equation are of higher orders l 2: r + 2.<br />
<strong>The</strong> result<strong>in</strong>g operator A ° Ar 1] is then used to solve the rema<strong>in</strong><strong>in</strong>g equations at order r projected<br />
onto the states with pions.<br />
S<strong>in</strong>ce equation (3.217) with 4k + i = 0 vanishes at orders r = 0 and r = 1, the start<strong>in</strong>g equations<br />
are aga<strong>in</strong> (3.221), (3.222). <strong>The</strong>refore, the ansatz about the structure of the operator A can be<br />
justified <strong>in</strong> the same way as <strong>in</strong> sec. 3.6. F<strong>in</strong>ally, all expressions (3.225)-(3.229) can be applied<br />
without any changes to obta<strong>in</strong> the effective potential.<br />
We are now <strong>in</strong> the position to calculate the effective potential <strong>in</strong>clud<strong>in</strong>g .6.-excitations. We will<br />
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