The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.8. Two-nuc1eon potential
with
L(q) = � 14M2 + q2 In J4M; + q2 + q .
q V 7r 2M7r
103
(3.301)
The polynomial terms, obviously, renormalize the coupling constants of the dimension zero and
two contact terms with four nucleon legs. Again, we perform anti-symmetrization to map the
terms appearing in eq. (3.296) via eq. (E.13) onto the basis used in the polynomial part of the
V(O ) and in the V�2h,tree' cf. eqs. (3.269), (3.270). Including the contribution from eq. (3.293),
the complete renormalization of the four-nucleon couplings takes the form
Cf
cr
cr
C� - 38 -81 , Cs = C� -38 -281 ,
Cp - 82 , C� = cg - 482 , Cf = cg -83 ,
482 , C� = cg , C6 = cg + 83 , C; = C$ .
c2 -
(3.302)
Note that C5 and C7 do not get renormalized to this order.
Now we would like to comment on the obtained results. The effective N N potential at NLO is
given by the non-polynomial and the polynomial parts shown in eqs. (3.269), (3.300) and (3.270).
The coupling constants Ci 's are renormalized via eq. (3.302). Note that each of these two parts, the
polynomial and the non-polynomial ones, separately does not depend on the renormalization scale
fl,. This fl,-independence is not accidental. Indeed, assume that the potential can be expressed as
(3.303)
where CT and gT correspond to sets of renormalized four-nucleon and remaining couplings, which
in our case do not depend on fl" Q denotes generic nucleon three-momenta and VI and V2 are
non-polynomial and polynomial parts of the potential. For simplicity, we consider here the case
m -+ 00. Since V2 is a polynomial, there exists some non-negative number nmin such that for all
n > nmin: 43
dnV2( Q, M7r, fl" gT, CT)
=
dQn 0 . (3.304)
We now require that the whole effective potential V does not depend on fl,:
dV
dfl, = 0
Therefore, we obtain for all n > nmin:
(3.306)
Here we assume the usual continuity properties of the function V2( Q, M7r, fl" gT, CT). Consequently,
one can express VI as
(3.307)
Further, the function V? (M7r , fl" gT, CT) should vanish since VI does not contain a polynomial part.
That is why both VI and V2 should be separately independent on fl,. In our case it turns out that
the explicit fl,-dependence even vanishes completely if both VI and V2 are expressed in terms of the
renormalized coupling constants. All these couplings should be fixed from fitting to data. This will
.
(3.305)
43We use here a symbolic notation for derivatives. It should be kept in mind that Q represents vectorial quantities.