The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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