The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

46 2. Low-momentum effective theories for two nucleons
LO 2.40 5.514 2.084 0.345 0.210 0.063
NLO 2.23 5.514 1.893 0.105 0.005 -0.055
NNLO 2.23 5.514 1.896 0.130 0.059 -0.018
true values 2.23 5.514 1.893 0.130 0.059 -0.026
Table 2.5: The values of the binding energy E and effective range parameters from the unitarily
transformed potential V' and the truncated expansion (2.116) for A =300 MeV. The couplings
are fixed by the requirement that the first terms in the effective range expansion are exactly
reproduced.
parameters calculated with the truncated expansion (2.116) using the values of the couplings given
in table 2.4, which are determined requiring that the first terms in the effective range expansion
are exactly reproduced. For example, at NLO the two constants Co and C2 are fixed requiring
that a and r are exactly reproduced. This allows to make a prediction for V2.
2.3.6 Coordinate space representation
Up to now we have only considered the potentials in momentum space. It is also interesting to
see how the unitarily transformed potential looks like in coordinate space, which usually provides
more physical insights. Denote by Vi(p' ,p) the effective moment um space potential for angular
momentum l. The corresponding r-space expression is obtained from
Vi(x', x) = / p 2 dpp, 2 dp' jl(pX) Vi(P' ,p) jl(P'X') ,
(2.125)
with jl(Y) the conventional l t h spherical Bessel function. The S-wave potential (l = 0) for A =
400 MeV is shown in the left panel of fig. 2.12. It is, of course, symmetrie under the interchange of
x and x', but looks very different from the original local potential. In particular, for A = 400 MeV
the unitarily transformed potential is strongly non-local and purely attractive for short distances.
Of course, if one increases the value of the the cut-off A, a peak along the diagonal x = x'
resembling the delta function should develop and the superposition of the two Yukawa potentials
related to the light and heavy meson exchanges, respectively, should appear along the diagonal.
This is indeed the case as demonstrated in the right panel of fig. 2.11 for A = 5.5 GeV. Note also
that in this case, where U rv 1, the range of the potential is essentially given by the inverse of the
light meson mass. One can also construct the momentum and coordinate representations of the
deuteron wave function. For a momenturn space picture, we refer to ref. [123].
The unitarily transformed potential shows an oscillating behavior in coordinate space. This is
because it is defined only in a certain range of momenta. Due to the cut-off we unavoidably
introduce discontinuities in higher derivatives of the potential over momenta. For instance, the
second derivative of the smooth regulating function f(k) given in eq. (2.98) has discontinuity
points at k = A - a, k
= A - a + b, k = A + a - b and k = A + a. This problem somewhat
rest riets the applicability of the described projection method in moment um space. In particular,
after performing the unitary transformation one could not calculate such quantity for the deuteron
like the quadrupole momentum, which requires the knowledge of second derivatives of the wave
function. However, one can always choose the regulating function f (k) such that the first n
derivatives are continuous.