The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

76 3. The derivation of nuclear forces from chiral Lagrangians
The advantage of this formulation is that the new object Hv transforms in a covariant way under
the reparametrization (3.169):
H- w - eiq'xHv
· (3.172)
The requirements on the l/m-coefficients following from the reparametrization invariance of Leff
can now easily be worked out. A more detailed discussion on the construction of the effective
Lagrangian for pions and nucleons can be found in references [73], [75], [78], [161]. It goes without
saying that as soon as one has Leff, the corresponding Hamiltonian can be obtained applying the
usual rules of the canonical formalism [75].34 We will not discuss the construction of the effective
Lagrangian (Hamiltonian) for pions and nucleons any furt her and simply adopt the corresponding
expressions from refs. [75], [77], [78]. Below, we will show explicitly all terms in the Hamiltonian
we need to calculate the N N potential to a required order and consider some of these structures
in more detail.
Let us now repeat the arguments of Weinberg [73] and derive the power counting rules for a
general process with N nucleons and an arbitrary number of extern al pions. According to the
nonrelativistic treatment and to the fact that the low component nucleon fields are integrated
out, no nucleon-antinucleon pairs can be created or destroyed. In his original work [73], Weinberg
used the covariant perturbation theory (Feynman diagram technique) to estimate the power v
of small external momenta Q for an arbitrary N-nucleon scattering process. We will now use
time-ordered35 perturbation theory to calculate v, since it makes more transparent the problems
associated with the perturbative treatment of the few-nucleon problem. We will proceed similarly
to the case of pion-pion scattering considered in the last section. Instead of covariant propagators
one has in old-fashioned perturbation theory energy denominators and phase-space factors. One
also has to remember that all integrations are now three dimensional and the particles are always
on their mass shell but can be off the energy shell in the intermediate states. The power of
momenta v can be found via
L L ( Pi) Ep
v = 31 - 3 11,. - D + v: d· - - + - . z . z Z
+ 3
(3.173)
Z Z
2 2
where Dis the number of energy denominators (or, equivalently, the number of intermediate states
between time-ordered vertices), 1 is the number of internal lines and Ep denotes the number of
external pion lines. Each of the Vi vertices of type i contains di derivatives or pion mass insertions
and Pi pions. For the moment, we assurne that all energy denominators scale as l/Q. The first
term in eq. (3.173) gives the number of moment um integrations. Not all of these momentum
integrations are independent of each other because of the 6-functions accompanying vertices. We
take into account the reduction of the v due to such 6-functions subtracting the second term in
eq. (3.173). Further , we do not count the phase-space factors of external pions and, therefore,
add Ep/2 to the right-hand side of eq. (3.173). Finally, we also do not count the overall fourdimensional
6-function in the S-matrix elements. However, we should add the factor 3 and not 4
to the right-hand side of eq. (3.173), since this expression for the power v of momenta corresponds
to aT-matrix element, which does not include the overall energy conserving 6-function present in
the S-matrix. The meaning of the last term in eq. (3.173) is now clarified. To make the formula
(3.173) suitable for practical calculations we need few topological identities. First, note that the
number of intermediate states D can be expressed as
D=L Vi-l. (3.174)
34 It was shown by Ostrogradski [180] how to derive a Hamiltonian from a Lagrangian that includes higher
derivatives. See also [178], [181].
35 Sometimes it is also called "old-fashioned" perturbation theory.
'