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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5.3 <strong>The</strong> lead<strong>in</strong>g CIB and CSB effects <strong>in</strong> the np 1 So channel 161<br />
a counterterm of the structure E61) (J.l)a(Nt T3 N)(Nt T3 N) (cf. sec. 5.2) s<strong>in</strong>ce it is needed to make<br />
the amplitude scale-<strong>in</strong>dependent. Note that for the graph IV we have used the same subtraction<br />
as performed <strong>in</strong> [91] . Consequently, for operators of this type with 2n derivatives we can establish<br />
the scal<strong>in</strong>g property E�� rv Q-2+n. This does not contradict the KSW power count<strong>in</strong>g for the<br />
isosp<strong>in</strong> symmetrie theory s<strong>in</strong>ce a « 1. Stated differently, the lead<strong>in</strong>g CIB term of order aQ-2 is<br />
numerically much smaller than the strong lead<strong>in</strong>g order contribution rv Q-l. <strong>The</strong> <strong>in</strong>sertion from<br />
this contact term is shown <strong>in</strong> the last diagram of fig. 5.3 and leads to an additional contribution to<br />
ßA. In complete analogy, we can treat the lead<strong>in</strong>g order CSB effect which is due to an operator<br />
of the form aE6 2) (J.l) (Nt T3N) (Nt N). This term is, however, f<strong>in</strong>ite. Putt<strong>in</strong>g pieces together, we<br />
get<br />
ßA V 1 = -a<br />
(E(l) + E(2)) [A_l] 2<br />
1,-2,pp 0 0 Co ' ßAV 1 = _ (E(l<br />
_ ) E(2)) [A-l] 2<br />
1,-2,nn a 0 0 Co '<br />
where the coupl<strong>in</strong>g constants E61,2) (J.l) obey the renormalization group equations,<br />
(5.36)<br />
(5.37)<br />
Note that from here on we do no longer exhibit the scale dependence of the various coupl<strong>in</strong>gs<br />
constants E61,2),CO,2,D2. We can now relate the pp and nn scatter<strong>in</strong>g lengths to the np one (of<br />
course, <strong>in</strong> the pp system Coulomb subtraction is assumed),<br />
1<br />
1<br />
For the effective ranges, we have only CIB<br />
(5.38)<br />
(5.39)<br />
Note that this last relation is scale-<strong>in</strong>dependent and that it does not conta<strong>in</strong> any unknown parameter.<br />
We re mark that for the CIB scatter<strong>in</strong>g lengths difference the pion contribution alone is<br />
not scale-<strong>in</strong>dependent and can thus never be uniquely disentangled from the contact term contribut<br />
ion rv While the lead<strong>in</strong>g OPE contribution resembles the result obta<strong>in</strong>ed <strong>in</strong> meson<br />
E61).<br />
exchange models, the mandatory appearance of this contact term is a dist<strong>in</strong>ctively new feature<br />
of the effective field theory approach. It is easy to classify the lead<strong>in</strong>g and next-to-lead<strong>in</strong>g em<br />
corrections to these results. At order aQ-l, one has the contribution from two potential pions<br />
with the pion mass difference and also contact <strong>in</strong>teractions with two derivatives. Effects due to the<br />
charge dependence of the pion-nucleon coupl<strong>in</strong>g constants, i.e. isosp<strong>in</strong> break<strong>in</strong>g terms from .c�N,<br />
only start to contribute at order aQo . Such effects are therefore suppressed by two orders of Q<br />
compared to the lead<strong>in</strong>g terms. This f<strong>in</strong>d<strong>in</strong>g is <strong>in</strong> agreement with the various numerical analyses