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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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To keep the presentation self-conta<strong>in</strong>ed, we also show the contact <strong>in</strong>teractions written <strong>in</strong> ref. [78]:<br />

-�CS(NtN)(NtN) - �CT(NtaN) . (Nt aN) ,<br />

- C� [(Nt V N)2 + (V Nt N)2] - C�(NtV N) . (V Nt N)<br />

- C�(NtN) [NtV2N + V2NtN]<br />

- iC� [(NtVN) ' (VNt x aN) + (VNtN) . (Nta x VN)]<br />

- iC�(NtN)(VNt . a x VN) - iC�(NtaN) . (VNt x VN)<br />

- (C�OikOjl + C�OilOkj + C�OijOkz)<br />

x [(NtO"kOiN)(NtO"IOjN) + (OiNtO"kN)(OjNtO"IN)]<br />

- (C�OOikOjl + C�10ilOkj + C�<br />

2<br />

0ijOkl) (NtO"kOiN)(ojNtO"IN)<br />

( 1 I<br />

- 2"C13 (OikOjl + OilOkj) + C140ijOki<br />

, )<br />

x [(OiNtO"kOjN) + (OjNtO"kOiN)] (NtO"IN) .<br />

187<br />

(F.12)<br />

(F.13)<br />

To be able to compare the Lagrangian (F.11), which is expressed <strong>in</strong> terms ofthe velo city dependent<br />

field Hv , with eq. (F.13), one has to go <strong>in</strong> eq. (F.11) <strong>in</strong>to the rest-frame system of the nucleon by<br />

choos<strong>in</strong>g<br />

= (1,0,0,0) . (F.14)<br />

<strong>The</strong>n, the sp<strong>in</strong> operator takes the form SJ.t = (0, 8)7, where Si is given by8<br />

vJ.t<br />

(F.15)<br />

Note furt her that the four-derivative öJ.t (oJ.t) reads öJ.t = (öD, -V) (ö J.t = (ÖD, V)) . It is easy to<br />

demonstrate that the zeroth component of the derivative operator does not appear <strong>in</strong> such contact<br />

<strong>in</strong>teractions. For that we express öJ.t us<strong>in</strong>g the third equality <strong>in</strong> eq. (F.2) as:<br />

(F.16)<br />

Now, all terms that <strong>in</strong>volve the operator v . 0 act<strong>in</strong>g onto the nucleon field Hv are redundant and<br />

can be dropped <strong>in</strong> the effective Lagrangian. Further, contract<strong>in</strong>g Öv with the second term <strong>in</strong> the<br />

parenthesis <strong>in</strong> eq. (F.16) yields only the spatial derivative, s<strong>in</strong>ce SO = ° <strong>in</strong> the rest-frame system.<br />

F<strong>in</strong>ally, we po<strong>in</strong>t out that the large (small) component field Hv (hv) def<strong>in</strong>ed <strong>in</strong> eq. (3.160), which<br />

is, <strong>in</strong> general, a four component Dirac sp<strong>in</strong>or, turns <strong>in</strong>to the two component Pauli sp<strong>in</strong>or N (N):<br />

(F.17)<br />

With these rules it is easy to rewrite the Lagrangian (F.11) <strong>in</strong> the same notation as <strong>in</strong> eq. (F.13)<br />

and to establish the connection between the constants CL.,14 and CY1, ... ,18' Concretely, one obta<strong>in</strong>s:<br />

7 <strong>The</strong> zeroth component of SJl. vanishes because of its def<strong>in</strong>ition (F.l) and eq. (F.14).<br />

8 We use here Lat<strong>in</strong> letters to denote the components of various three-vectors. Clearly, we do not dist<strong>in</strong>guish<br />

between co- and contravariant quantities <strong>in</strong> such cases.

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