The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.6. Application to chiral invariant Hamiltonians 83
where Pi (ni) is the number ofpion (nucleon) fields at a vertex oftype i. Now we use the topological
identity (3.175) with I = In + Ip to cast eq. (3.208) into the form
vA = 3 -3N - Ep + L l'i /'\,i (3.210)
where
(3.211)
Again we point out that this result takes its form due to the ansatz that the projected operators
)..i A17 consist of an equal number of vertices and energy denominators. Also, it should be mentioned
that we use the number of external pion lines to express the counting index VA instead of the
number of loops and of separately connected pieces as it has been done in [73], [74], [76], [78].
This is more natural in the projection formalism employed here. Let us take a doser look at vertex
dimension /'\,i, which is related to the canonical field dimension. It is weIl known, see e.g. ref. [132],
that all interactions can be classified with respect to /'\,i. Those with /'\,i < 0 are called relevant
(superrenormalizable), with /'\,i = 0 marginal (renormalizable) and with /'\,i > 0 irrelevant (non­
renormalizable). The last ones are "harmless" and can be wen treated within low-energy effective
field theories. In contrast to these, the relevant and marginal interactions lead to complications
with the power counting. This becomes immediately clear if one takes a look at eq. (3.210):
an infinite number of diagrams contributes to a given process at a given order. Furthermore, for
relevant interactions the number VA is even not bounded from below. That is why the perturbative
treatment in powers of the moment um scale Q is not possible in this case. For more details about
the role of such interactions in effective field theories see ref. [192]. Chiral symmetry does not allow
any relevant or marginal interactions in the interaction Hamiltonian Hf. The minimal possible
value of /'\,i for vertices in Hf is one. Such a vertex with /'\,i = 1 contains two nucleons, one pion
and one derivative. In general, for vertices with two nucleons and Pi pions one has /'\,i 2 Pi. The
interactions with pions only have the vertex dimension /'\,i 2 2, where /'\,i = 2 corresponds to the
interaction with four pion fields and two derivatives. The purely pionic interactions have an even
number Pi of pions and /'\,i 2 max(2, -2 + Pi).
Now we would like to determine the minimal value of VA for )..a A17. For that it is convenient to
express vA in terms of L, C and ßi defined in eq. (3.178). Making use of the relations (3.208),
(3.209), (3.175) we obtain
(3.212)
where En is again the number of external nudeon lines. The value of VA is one less than the one
of v for an irreducible time-ordered graph in Weinberg's formula eq. (3.177). This difference is
because the number of energy denominators equals now the number of intermediate states plus
one, as follows from our ansatz about the structure of )..a A17. It is now easy to find the minimal
value of VA. Different to the case of few-nudeon scattering considered in the section 3.4, VA is
not bounded from below for a fixed number En of external nudeon lines. This is because each
additional disconnected purely pionic tree subdiagram with all vertices with two derivatives yields
the factor -2, as can be seen from eq. (3.212). Therefore, the minimal value of VA requires
min(VA) = 3 -3N -2k (3.213)
the maximal possible number of such disconnected pieces. Correspondingly, one finds for matrix
elements of the operator )..4k+i AT] with k a positive integer or zero and i = 0,1,2,3: