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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82 3. <strong>The</strong> derivation of nuclear forces from chiral Lagrangians<br />
Because of the property eq. (3.191) of the operator A each s<strong>in</strong>gle equation <strong>in</strong> eqs. (3.205) can be<br />
expressed as<br />
00 00<br />
). iHrfJ + L ). iHj). j AT] + ). iHo). iAT] - ). iAT]HjT] - ). iAT]HoT] - L ). iAT]Hj). j AT] = 0<br />
j=l j=l<br />
(3.206)<br />
<strong>The</strong> system of eqs. (3.205) is, clearly, too complicated to be solved exactly. In what follows, we will<br />
apply the usual philosophy of effective theories. We are <strong>in</strong>terested only <strong>in</strong> low-energy processes.<br />
<strong>The</strong>refore, we will expand the matrix elements of an effective potential <strong>in</strong> powers of the small<br />
moment um scale Q, as it was proposed by We<strong>in</strong>berg [73]. To calculate the operator A from the<br />
system of coupled equations (3.205), we will aga<strong>in</strong> make use of the expansion <strong>in</strong> powers of Q. In<br />
particular, the matrix elements of A will be classified by powers of Q by use of simple dimensional<br />
analysis:<br />
(3.207)<br />
Here Q is aga<strong>in</strong> the mass scale correspond<strong>in</strong>g to three-momenta of nucleons and four-momenta<br />
of pions. <strong>The</strong> effective Hamiltonian act<strong>in</strong>g on the purely nucleonic Fock space is def<strong>in</strong>ed <strong>in</strong><br />
eq. (3.197) and can be found from the orig<strong>in</strong>al one if the operator A is known. It will be clear<br />
from the power count<strong>in</strong>g arguments that the matrix elements (3.207) with larger 1I A lead to an<br />
effective potential suppressed by additional powers of Q. At first sight, this appears to be a<br />
miracle, s<strong>in</strong>ce (�I and 1