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.
3.4. Includ<strong>in</strong>g nucleons 77<br />
Next, we write the number of loops <strong>in</strong> the form<br />
L=I- :L Vi+C. (3.175)<br />
Here, we modified the earlier expression (3.145) to be valid also for C disconnected diagrams.<br />
F<strong>in</strong>ally, the numbers of <strong>in</strong>ternal and external particle l<strong>in</strong>es l and E = En + Ep , where En = 2N,<br />
can be collected via the follow<strong>in</strong>g identity:<br />
(3.176)<br />
where ni is the number of nucleon fields at the vertex of type i. Us<strong>in</strong>g eqs. (3.174)-(3.176) and<br />
eq. (3.173) we end up with the result<br />
where<br />
v = 4 - N + 2(L - C) + :L Vi�i , (3.177)<br />
(3.178)<br />
As already discussed above, the correspond<strong>in</strong>g <strong>in</strong>dices �i are not negative for the purely pionic<br />
Lagrangian.36 Analogously, �i are not negative for pion-nucleon as weIl as for nucleon-nucleon<br />
<strong>in</strong>teractions. In particular, the m<strong>in</strong>imal possible value �i = 0 is achieved for the vertices with two<br />
nucleons and one derivative (and any number of pion fields) or for the four nucleon <strong>in</strong>teractions<br />
without derivatives and pion mass <strong>in</strong>sert ions (and, therefore, also without pion fields). Thus, v<br />
<strong>in</strong> eq. (3.177) is aga<strong>in</strong> bounded from below and, therefore, it seems that the scatter<strong>in</strong>g amplitude<br />
für the process N N ----7 N N can be calculated perturbatively <strong>in</strong> powers of low external threemomenta<br />
of nucleons <strong>in</strong> a similar way as <strong>in</strong> the case of pion-pion scatter<strong>in</strong>g discussed <strong>in</strong> the last<br />
section. This already sounds suspicious, s<strong>in</strong>ce the presence of the low-energy bound state <strong>in</strong> the<br />
np 3S 1 _3 D 1 channel clearly signals the failure of perturbation theory .<br />
.. J .. .. J 11 J ..<br />
p ! I p' P I l I p'<br />
't<br />
I!<br />
I !<br />
I<br />
.. .. .. .. i ..<br />
-p _p' -p -f -I<br />
-p<br />
a) b)<br />
Figure 3.3: Irreducible and reducible time-ordered diagrams. <strong>The</strong> vertical dotdashed<br />
l<strong>in</strong>es are the energy cuts. <strong>The</strong> solid (dashed) l<strong>in</strong>es correspond to nucleons<br />
(pions).<br />
To solve this paradox let us take a closer look at old-fashioned perturbation theory. In the above<br />
derivation of the power count<strong>in</strong>g rule (3.177) we made an assumption that all energy denom<strong>in</strong>ators<br />
36 More precisely, it is zero for the Lagrangian for free pions and positive für <strong>in</strong>teractions (<strong>in</strong> the absence of external<br />
sources).