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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3.8. Two-nucleon potential<br />
Here,<br />
g� (1 - g�) ( ) .<br />
641fmf� Tl ' T2 Z (Tl + (T2 . P x P 1f + q q .<br />
( � � ) ( � I �) (4M 2 2)A( )<br />
A(q) =<br />
1 q<br />
2q 2M1f<br />
- arctan -- .<br />
Furt her , P(k, if) <strong>in</strong> eq. (3.309) is a polynomial <strong>in</strong> momenta of at most second degree and has the<br />
same structure as the expressions <strong>in</strong> eq. (3.270).44 Thus, its explicit form is of no relevance here,<br />
s<strong>in</strong>ce it only contributes to the renormalization of the coupl<strong>in</strong>gs Cs , CT , Ci . Aga<strong>in</strong>, no explicit<br />
dependence on the renormalization scale appears <strong>in</strong> the expression (3.310) für the non-polynomial<br />
terms. In ref. [108] it is also po<strong>in</strong>ted out that all NNLO one-pion exchange diagrams conta<strong>in</strong><strong>in</strong>g<br />
vertex corrections do not yield any pion-nucleon form factor and only lead to renormalization of<br />
the 1fN coupl<strong>in</strong>g gA (i.e. they have the same form as the proper one-pion exchange). Clearly, the<br />
same holds true for all NNLO graphs represent<strong>in</strong>g vertex corrections to short range <strong>in</strong>teractions.<br />
For the LECs Cl,3,4 enter<strong>in</strong>g eq. (3.310) we should take the values obta<strong>in</strong>ed from fitt<strong>in</strong>g 1fN phases<br />
<strong>in</strong> the threshold region, see e.g. ref. [71] or, alternatively, from fitt<strong>in</strong>g the <strong>in</strong>variant amplitudes<br />
<strong>in</strong>side the Mandelstarn triangle, i.e. <strong>in</strong> the unphysical region [195]. <strong>The</strong> so determ<strong>in</strong>ed parameters<br />
are only slightly different, but these small differences will play a role later on. For example, the<br />
LECs CI,3,4 from fit 1 of ref. [71] are Cl = -1.23 GeV-l, C3 = -5.94 GeV-l and C4 = 3.47 GeV-l.<br />
A recent <strong>in</strong>vestigation of the subthreshold amplitudes [195] leads to slightly different values, Cl =<br />
-0.81 GeV-l, C3 = -4.70 GeV-l and C4 = 3.40 GeV-l. It is this latter set we will use <strong>in</strong> the<br />
follow<strong>in</strong>g. <strong>The</strong>se values are also consistent with the re cent determ<strong>in</strong>ation from the proton-proton<br />
<strong>in</strong>teraction based on the chiral two-pion exchange potential [198].<br />
Kaiser et al. also <strong>in</strong>cluded l/m corrections <strong>in</strong>to the expression (3.310) for the two-pion exchange<br />
potential an NNLO. This is consistent with the power count<strong>in</strong>g scheme <strong>in</strong> the one-nucleon sector.<br />
In our power count<strong>in</strong>g scheme with Q/m rv Q2 / A�, where Q corresponds to the low moment um<br />
scale, such l/m terms appear first one order higher. Nevertheless, we have decided to keep these<br />
corrections, s<strong>in</strong>ce otherwise one cannot directly use the values of the LECs Cl,3,4 as determ<strong>in</strong>ed<br />
from the 1fN sector <strong>in</strong> the presence of the l/m terms. By do<strong>in</strong>g so we only <strong>in</strong>clude some sm aller<br />
terms of higher order <strong>in</strong> the potential. It can easily be checked, that those terms are <strong>in</strong>deed small<br />
as compared to the Cl,3,ccontributions. For example, the constant C4 enters eq. (3.310) only <strong>in</strong><br />
the comb<strong>in</strong>ation C4 + 1/ ( 4m). Correspond<strong>in</strong>gly, the pert<strong>in</strong>ent C4 contribution is numerically more<br />
than 10 times larger than the l/m correction.<br />
To end this section we would like to po<strong>in</strong>t out once more that eqs. (3.269), (3.300), (3.270) and<br />
(3.310) def<strong>in</strong>e the unique expressions for the effective potential up to NNLO, i.e. the results do<br />
not depend on the regularization scheme, as discussed above.<br />
105<br />
(3.310)<br />
(3.311)<br />
3.8.3 Phenomenological <strong>in</strong>terpretation of some of the 1f N LEes and the role<br />
of the ß(1232)<br />
It is well known that the values of some of the LECs can be understood phenomenologically <strong>in</strong><br />
terms of the resonance saturation hypotheses. This is discussed <strong>in</strong> detail <strong>in</strong> the reference [199]<br />
for the purely meson sector and <strong>in</strong> ref. [200] for pion-nucleon <strong>in</strong>teractions. <strong>The</strong> idea of such<br />
a phenomenological <strong>in</strong>terpretation is very simple. Consider the most general chiral <strong>in</strong>variant<br />
Lagrangian for pions, nucleons and their excitations. <strong>The</strong> resonances can be <strong>in</strong>tegrated out from<br />
44 More precisely, after perform<strong>in</strong>g the partial wave decomposition, fCk, if) leads <strong>in</strong> each partial wave to polynomials<br />
<strong>in</strong> p and pi of at most second degree.