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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6 1. Introduction<br />
pion-nucleon system as well as for the processes with electroweak probes [71]. For arecent review<br />
see ref. [72].<br />
Motivated by successful applications of CHPT <strong>in</strong> the 7f7f and 7f N sectors We<strong>in</strong>berg proposed <strong>in</strong><br />
1990 to extend the formalism to the N N <strong>in</strong>teraction [73]. <strong>The</strong> crucial difference is, however,<br />
that the nucleon-nucleon <strong>in</strong>teraction is non-perturbative at low energies: it is strong enough to<br />
b<strong>in</strong>d two nucleons <strong>in</strong> the deuteron. Thus, direct application of CHPT to the N N amplitude will<br />
necessarily fail. <strong>The</strong> way out of this problem, proposed by We<strong>in</strong>berg, is to apply CHPT not to<br />
the amplitude but to a kernel of the correspond<strong>in</strong>g <strong>in</strong>tegral Lippmann-Schw<strong>in</strong>ger (LS) equation.<br />
Such a kernel (or potential) can be def<strong>in</strong>ed with<strong>in</strong> time-ordered perturbation theory as a sum over<br />
all irreducible diagrams with two <strong>in</strong>com<strong>in</strong>g and outgo<strong>in</strong>g nucleon l<strong>in</strong>es. Irreducible means here<br />
that no pure 2N <strong>in</strong>termediate states are allowed. Reducible diagrams are then generated due to<br />
iterations of the kernel <strong>in</strong> the LS equation. We<strong>in</strong>berg has shown that a systematic power count<strong>in</strong>g<br />
for the potential can be derived <strong>in</strong> a similar manner as <strong>in</strong> the case of the 7f7f and 7f N systems.<br />
<strong>The</strong> correspond<strong>in</strong>g Lagrangian should be extended to allow apart from the 7f7f and 7f N also N N<br />
contact <strong>in</strong>teractions, which are not constra<strong>in</strong>ed by chiral but only by isosp<strong>in</strong> symmetry.<br />
<strong>The</strong>se ideas have been extensively studied by the Texas-Seattle group [74], [75], [76], [77], [78].<br />
<strong>The</strong>y have obta<strong>in</strong>ed an energy dependent two-nucleon potential at next-to-next-to-lead<strong>in</strong>g order.<br />
With altogether 26 free parameters6 they were able to achieve a qualitative agreement with the<br />
N N scatter<strong>in</strong>g data and some deuteron properties. Also three-body forces have been considered<br />
by this group. Apply<strong>in</strong>g this formalism to many-body problems allows to establish a beautiful<br />
hierarchy of the two- and many-body forces: the three-body forces should be weaker than the<br />
two-body ones, the four-body forces should be weaker than the three-body ones and so on.<br />
S<strong>in</strong>ce that time the derivation of the nucleon-nucleon <strong>in</strong>teraction us<strong>in</strong>g the technique of effective<br />
theories became a subject of quite remarkable <strong>in</strong>terest and <strong>in</strong>tense discussions. Cohen and collab<br />
orators [79], [80], [81], [83] considered <strong>in</strong> detail an effective theory with pions <strong>in</strong>tegrated out.<br />
In such a case simple analytic calculations for the two-nucleon S-matrix can be performed, s<strong>in</strong>ce<br />
all <strong>in</strong>teractions between nucleons are of the contact type. <strong>The</strong>y exam<strong>in</strong>ed different regularization<br />
schemes for divergent <strong>in</strong>tegrals <strong>in</strong> the LS equation like dimensional and cut-off regularizations and<br />
made some <strong>in</strong>terest<strong>in</strong>g observations. First, it turned out that <strong>in</strong> such a pionless effective theory<br />
the cut-off could be taken to <strong>in</strong>f<strong>in</strong>ity only if the effective range parameter of the <strong>in</strong>teraction is<br />
negative. This conclusion follows, <strong>in</strong> fact, from a theorem which had been proven by Wigner long<br />
time aga [84] and is based only on such general pr<strong>in</strong>ciples like causality and unitarity. <strong>The</strong> theorem<br />
says, that if a potential vanishes beyond some range R, then the rate d6(k)/dk at which the phase<br />
shift can change with energy is bounded from below by some function of R, k and 6(k). Secondly,<br />
an effective theory with a f<strong>in</strong>ite cut-off was found not to be "systematic" <strong>in</strong> the sense, that higherorder<br />
terms <strong>in</strong> the potential do not get systematically smaller as the order is <strong>in</strong>creased. Thus, the<br />
expansion does not seem to converge. We will comment more on that <strong>in</strong> the chapter 2. F<strong>in</strong>ally,<br />
the use of cut-off schemes and dimensional regularization was shown to lead to different results<br />
for the scatter<strong>in</strong>g amplitude. <strong>The</strong> scatter<strong>in</strong>g amplitude calculated with dimensional regularization<br />
only maps to the effective range expansion for on-shell momenta k « 1/ JaTe, where a and Te<br />
are the scatter<strong>in</strong>g length and the effective range. For such small momenta both schemes produce<br />
identical results <strong>in</strong> agreement with the effective range expansion. <strong>The</strong> experimental values for the<br />
S-wave np scatter<strong>in</strong>g lengths and effective ranges are<br />
6 Some of these parameters are redundant.<br />
as = (-23.758 ± 0.010) fm ,<br />
at = (5.424 ± 0.004) fm ,<br />
Ts = (2.75 ± 0.05) fm ,<br />
Tt = (1.759 ± 0.005) fm .<br />
(1.1 )<br />
(1.2)