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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156 5. Isosp<strong>in</strong> violation <strong>in</strong> the two-nuc1eon system<br />
Born amplitudes for the lowest order eIB and eSB operators between the various two-nucleon<br />
states takes the form<br />
_ (�:)<br />
_ (�:)<br />
a (E�l) + E�2))<br />
a (E�l) _<br />
E�2))<br />
(5.13)<br />
where we will determ<strong>in</strong>e the coupl<strong>in</strong>g constant E�1,2) later on and also derive the scal<strong>in</strong>g properties<br />
of the E�;,2). <strong>The</strong> terms with the superscript '(1)' refer to eIB whereas the second ones relevant<br />
for eSB are denoted by the superscript '(2)'. Higher order operators are denoted accord<strong>in</strong>gly.<br />
<strong>The</strong>re is, of course, also a eIB contribution to the np matrix element. To be consistent with the<br />
charge symmetrie calculation of ref. [91J, we absorb its effect <strong>in</strong> the constant D2, i.e. it amounts<br />
to a f<strong>in</strong>ite renormalization of D2 and is thus not observable. In eq. (5.13), p = JmEcms is the<br />
nucleon cms moment um.<br />
5.2 Brief <strong>in</strong>troduction <strong>in</strong>to the KSW approach<br />
We will now give a very brief <strong>in</strong>troduction <strong>in</strong>to the KSW formalism and repeat so me basic po<strong>in</strong>ts of<br />
ref. [91J. It is weIl known that the scatter<strong>in</strong>g lenghts a <strong>in</strong> both np S-channels take unnatural large<br />
values. Let us first consider a pionless theory. In such a case one can calculate the two-nucleon<br />
T-matrix analytically, see sec. 2.2, s<strong>in</strong>ce all <strong>in</strong>teractions <strong>in</strong> the effective Lagrangian are of contact<br />
type. As discussed <strong>in</strong> section 2.2, the range of applicability of such an effective theory is zero<br />
<strong>in</strong> the case of <strong>in</strong>f<strong>in</strong>itly large scatter<strong>in</strong>g length, if ord<strong>in</strong>ary dimensional regularization is used. In<br />
what follows we will adopt the notation of ref. [91J and work with the amplitude A <strong>in</strong>stead of the<br />
T-matrix def<strong>in</strong>ed <strong>in</strong> eq. (2.6). <strong>The</strong> connection between the amplitude and the on-shell T-matrix<br />
Ton (p) is gi yen by<br />
= A( ) _ Ton(p) p 27r2 '<br />
<strong>The</strong> tree level amplitude <strong>in</strong> the S-channels can be expressed as<br />
Atree =<br />
00<br />
- L C2np2n ,<br />
n=O<br />
(5.14)<br />
(5.15)<br />
where C2n are some constants. Us<strong>in</strong>g dimensional regularization with the m<strong>in</strong>imal subtraction<br />
= scheme (MS), which amounts to subtract<strong>in</strong>g poles <strong>in</strong> the physical dimension (D 4), one f<strong>in</strong>ds<br />
the amplitude A <strong>in</strong> the form:<br />
(5.16)<br />
Here we have used the fact that the power law divergent <strong>in</strong>tegrals (like those one <strong>in</strong> eq. (2.49))<br />
vanish after perform<strong>in</strong>g dimensional regularization. If the scatter<strong>in</strong>g length would be of a natural<br />
size, i.e. of the order a rv 1/ A, where A is ascale enter<strong>in</strong>g the values of the rema<strong>in</strong><strong>in</strong>g effective range<br />
and shape parameters, which is comparable with the pion mass, one could rewrite the effective<br />
range expansion (2.21) for the <strong>in</strong>verse T-matrix <strong>in</strong>to the expansion for the amplitude A:<br />
A =<br />
�<br />
An =<br />
00<br />
- ---;<br />
7ra . ar 2 2 P<br />
4 [ ( )<br />
;: 1 -<br />
zap<br />
( 3)]<br />
+ 2 - a p + 0<br />
A 3 . (5.17)