The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.2. Two nuc1eons at very low energies
2.2 Two nucleons at very low energies
In this section we would like to apply the ideas formulated above to the scattering problem of two
nucleons at very low energies. We will consider proton-neutron scattering and concentrate only
on the strong interaction between these two particles. Possible electromagnetic corrections due
to the magnetic moments of neutron and proton are very small and will be neglected. Here we
will, basically, follow the considerations of refs. [81], [83]. A detailed analysis of the two-nucleon
system within a pionless effective theory at next-to-next-to-Ieading order in the low-momentum
expansion can be found in ref. [98].
The longest range part of the nuclear force is due to the exchange of pions. Since we are interested
in very low energies, even pionic degrees of freedom can be integrated out. Thus, the effective
Hamiltonian entirely consists of contact interactions with four nucleon legs. These terms may
contain any even number of derivatives (terms with an ocid number of derivatives would break
parity invariance) and insertions of spin and isospin matrices. Apart from parity invariance, one
should also provide rotational invariance as weIl as symmetry under time reversal operation. In
many cases it appears to be very useful to require also exact isospin invariance of the effective
Hamiltonian. Because of the low-energy limit, a nonrelativistic treatment of nucleons is weIl
justified. Relativistic corrections may be systematically incorporated in the same way as the
corrections to interaction between nucleons, i.e. by putting local one-nucleon terms with more
than two derivatives in the Hamiltonian.
In what follows, we will only consider S-waves. For nonrelativistic nucleons, one can express the
effective Hamiltonian in the two-nucleon center of mass system in the form
H(p',p) == (S p'lHIS p) = J(p' -p) + V(p',p) ,
m
where m is the nucleon mass. Here, we have cancelled in the first term in eq. (2.4) the factors
p2 resulting from the kinetic energy operator and from the three-dimensional J-function. The
potential V(p',p) is given by
p ,p = 0 + 2 P + p + 4 P + p + 4P P + . .. , (2.5)
V( ' ) C C ( ,2 2 ) C1( ,4 4 ) C 2 ,2 2
where the C's are (unknown) coefficients.8 Note that terms like p'p do not arise in the S-channel
after performing a partial wave decomposition. This can be seen from the formulae of appendix
G. For instance, the term p' . p, which could lead to such a structure, vanishes after projecting
onto the S-state because of its angular dependence. The off-shell g two-body T-matrix can be
found from the Lippmann-Schwinger integral equation
T(p',p;k) = V(p',p) + m (OO q2dqV(p',q) k2 \ .
Jo -
q T(q,p;k) ,
+ ZE
where k is the off-shell momentum and E ---+ 0+. An iterative solution to this equation is represented
diagrammatically in fig. 2.2. Clearly, this equation is not well-defined if one uses the potential
(2.5): each iteration would produce ultraviolet divergences. As already stressed in the last section,
one should cut off the integral entering the LS equation (2.6). Alternatively, one can introduce a
regularized potential, for instance, like
8 This definition of C's differs by the factor 21r2 from the one given in [81], [83).
9 The matrix TUb, eh , q) is called on-shell if Iql I = 1il2 I = q, half-shell if Iq 2 1 = q i=- ql and off-shell otherwise.
15
(2.4)
(2.6)
(2.7)