21.12.2012 Views

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

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

3.8. Two-nucleon potential 99<br />

where E is an <strong>in</strong>itial energy ofthe two nucleons (full energy). One should stress that on-shell, this<br />

contribution vanishes. This on-shell center-of-mass k<strong>in</strong>ematics is often used as an approximation<br />

for calculations of N N scatter<strong>in</strong>g or few-nucleon forces. However, if one iterates this potential <strong>in</strong><br />

the Lippmann-Schw<strong>in</strong>ger equation, the energy E can no longer be simply related to the momenta<br />

k and ij. A similar comment applies to us<strong>in</strong>g this recoil correction <strong>in</strong> the calculation of few-nucleon<br />

forces (for N � 3).<br />

Let us now look at the corrections aris<strong>in</strong>g <strong>in</strong> the framework of the projection formalism. <strong>The</strong> ones<br />

from the tree diagrams with contact <strong>in</strong>teractions with two derivatives do not change and are aga<strong>in</strong><br />

given by eq. (3.270). Apart from the corrections from irreducible one-Ioop graphs 1-8 <strong>in</strong> fig. 3.9<br />

given by eq. (3.272) one obta<strong>in</strong>s contributions from reducible diagrams 9 and 10, which can be<br />

expressed by<br />

(3.278)<br />

V(2 )<br />

211",1-1oop,red.<br />

Summ<strong>in</strong>g up eqs. (3.272) and (3.278) we obta<strong>in</strong> the two-pion exchange contributions to the potential<br />

with<strong>in</strong> the projection formalism:<br />

(2 )<br />

V211", 1-1oop<br />

(3.279)<br />

As was first noted <strong>in</strong> ref. [108] us<strong>in</strong>g a different formalism, the isoscalar sp<strong>in</strong> <strong>in</strong>dependent central<br />

and the isovector sp<strong>in</strong> dependent parts of the two-nucleon potential correspond<strong>in</strong>g to the two-pion<br />

exchange adds up to zero. It is comfort<strong>in</strong>g that we f<strong>in</strong>d the same result. Note that this is different<br />

from the energy-dependent potential derived with<strong>in</strong> time-ordered perturbation theory.<br />

<strong>The</strong> contributions from one-Ioop "reducible" diagrams 1-4 <strong>in</strong> fig. 3.11 and 3-6 <strong>in</strong> fig. 3.13, which<br />

<strong>in</strong>volve the nucleon self-energy and loop corrections to the four-fermion <strong>in</strong>teractions, are given by<br />

(3.280)<br />

and by<br />

(3.281)<br />

V(2 )<br />

NN, 1-1oop,red.

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!