The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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5.5 Classification scheme 163<br />
scatter<strong>in</strong>g length and effective range are exactly reproduced. <strong>The</strong> two new parameters E� I,2) are<br />
determ<strong>in</strong>ed from the nn and pp scatter<strong>in</strong>g lengths. <strong>The</strong> result<strong>in</strong>g parameters at f-l = Mn are of<br />
natural size,<br />
-3.46 fm2 , C2 = 2.75 fm4 , D 2 = 0.07 fm4 ,<br />
-6.47 fm2 , E� 2) = 1.10 fm2 . (5.40)<br />
To arrive at the curves shown <strong>in</strong> fig. 5.4, we have used the physical masses for the proton and the<br />
neutron. We stress aga<strong>in</strong> that the effect of the em terms rv E� I, 2) is small because of the explicit<br />
factor of a not shown <strong>in</strong> eq. (5.40). Hav<strong>in</strong>g fixed these parameters, we can now predict the nn ISO<br />
phase shift as depicted by the solid l<strong>in</strong>e <strong>in</strong> fig. 5.4. It agrees with nn phase shift extracted from<br />
the Argonne V18 potential (with the scatter<strong>in</strong>g length and effective range exactly reproduced) up<br />
to momenta of about 100 MeV. <strong>The</strong> analogous curve for the pp system (not shown <strong>in</strong> the figure) is<br />
elose to the solid l<strong>in</strong>e s<strong>in</strong>ce the CSB effects are very small. <strong>The</strong> lead<strong>in</strong>g CIB effect for the effective<br />
range, ßrclB = 1/2(rnn + rpp) - rnp, is given by the last term <strong>in</strong> eq. (5.39). With the values of<br />
the parameters shown <strong>in</strong> eq. (5.40) we obta<strong>in</strong><br />
ßrClB = 0.01 fm , (5.41)<br />
which agrees with the small value observed experimentally for this quantity, see eq. (5.1).<br />
5.5 Classification scheme<br />
In the framework presented here, it is straight forward to work out the various lead<strong>in</strong>g (LO), nextto-lead<strong>in</strong>g<br />
(NLO) and next-to-next-to-lead<strong>in</strong>g order (NNLO) contributions to CIB and CSB,<br />
with respect to the expansion <strong>in</strong> Q, to lead<strong>in</strong>g order <strong>in</strong> a and the light quark mass difference. Here,<br />
we will simply enumerate the pert<strong>in</strong>ent contributions which appear at a given order. This also<br />
naturally gives an estimate about their relative numerical importance based on simple dimensional<br />
analysis. Let us first consider CIB. In the elassification scheme given below, TPE/3PE stands for<br />
two / three-pion-exchange.<br />
LO<br />
aQ-2<br />
NLO<br />
a Q-l , c Q-l<br />
NNLO<br />
aQo<br />
Pion mass difference <strong>in</strong> OPE,<br />
four-nueleon contact <strong>in</strong>teraction with no derivatives.<br />
Pion mass difference <strong>in</strong> TPE and <strong>in</strong> the exchange of one radiation pion,<br />
four-nueleon contact <strong>in</strong>ter action with two derivatives,<br />
<strong>in</strong>sertions proportional to the np mass difference,<br />
Pion mass difference <strong>in</strong> 3PE and TPE,<br />
four-nueleon contact <strong>in</strong>teraction with four derivatives,<br />
CIB <strong>in</strong> the pion-nueleon coupl<strong>in</strong>g constants,<br />
Note that there are no CIB effects due to the light quark mass difference l<strong>in</strong>ear <strong>in</strong> c = mu - md<br />
from OPE and from the four-nucleon contact terms. <strong>The</strong>se type of terms can only appear to<br />
second order <strong>in</strong> c. Effects due to the charge dependence of the pion-nucleon coupl<strong>in</strong>g constants,<br />
i.e. isosp<strong>in</strong> break<strong>in</strong>g terms from .c��, only start to contribute at order aQo. Such effects are<br />
therefore suppressed by two orders of Q compared to the lead<strong>in</strong>g terms, Le. by a numerical factor<br />
of about (1/3)2. This f<strong>in</strong>d<strong>in</strong>g is <strong>in</strong> agreement with the various numerical analyses performed <strong>in</strong>