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
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
62 3. <strong>The</strong> derivation of nuc1ear forces from chiral Lagrangians<br />
where the quantity EJ.t is def<strong>in</strong>ed as<br />
(3.88)<br />
A more detailed discussion concern<strong>in</strong>g the stereographical parametrization can be found <strong>in</strong> [161].<br />
<strong>The</strong> use of the stereographical coord<strong>in</strong>ates requires sometimes rat her tedious calculations. We will<br />
now briefly discuss another more elegant way, which is commonly used <strong>in</strong> the analysis of processes<br />
with one nucleon. Let us first <strong>in</strong>troduce the standard notation and def<strong>in</strong>e the matrix u as<br />
(3.89)<br />
For the generators V and A we will aga<strong>in</strong> use the representation (3.78) with l = 1'5 , To establish<br />
the transformation properties of the u we will apply eq. (3.55), which def<strong>in</strong>es how the Cs transform<br />
under go E SU(2)v X SU(2)A. An arbitrary group element go needs not necessarily to be<br />
parametrized <strong>in</strong> terms of s and � via eq. (3.51). Equally weIl we can write<br />
(3.90)<br />
Here, eV,A are transformation parameters of the SU(2)V,A groups. We can now rewrite eq. (3.55)<br />
us<strong>in</strong>g the parametrization (3.90) as<br />
(3.91)<br />
where s' is, as usual, a function of � and eV,A. Us<strong>in</strong>g the projectors PR and PL onto the states<br />
with a def<strong>in</strong>ite chirality def<strong>in</strong>ed <strong>in</strong> eq. (3.5) we can express (ev . V + eA . A) as<br />
<strong>The</strong> parameters e R,L are def<strong>in</strong>ed by<br />
(3.92)<br />
(3.93)<br />
and VR,L == VPR,L are the generators of SU(2)R,L. Project<strong>in</strong>g eq. (3.91) onto states with adef<strong>in</strong>ite<br />
chirality we obta<strong>in</strong> the follow<strong>in</strong>g equations:<br />
PRi'R" V e�' v<br />
PLih' V e-�' v<br />
Here we have used the properties of the projectors PR,L:<br />
and the follow<strong>in</strong>g equalities:<br />
P e·v Re e s'·V ,<br />
P -e·v<br />
Le e s'·V .<br />
(3.94)<br />
(3.95)<br />
(3.96)<br />
(3.97)<br />
S<strong>in</strong>ce PR,L commute with the V, one can completely drop these projectors <strong>in</strong> eqs. (3.94), (3.95).<br />
We can now easily read off the transformation properties of the u from eqs. (3.94), (3.95):<br />
90 , h h- 1 h h- 1<br />
U -=---7 U = RU = U L . (3.98)