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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190 F. <strong>The</strong> complete set of the N N contact <strong>in</strong>teractions with two derivatives<br />
where the matrix R is given by<br />
(F.26)<br />
We have used eq. (F.16) to obta<strong>in</strong> the seeond l<strong>in</strong>e of eq. (F.25). <strong>The</strong> term vp vQSrr <strong>in</strong> the<br />
squared braekets <strong>in</strong> the seeond l<strong>in</strong>e of this equation vanishes modulo higher order terms after<br />
apply<strong>in</strong>g the EOM for Hv . Sueh higher order terms are denoted <strong>in</strong> eq. (F.25) by the dots.<br />
To end up with the third l<strong>in</strong>e of eq. (F.25) we have used the last equality <strong>in</strong> eq. (F.2). <strong>The</strong><br />
seeond term <strong>in</strong> the parenthesis (SQ {Sp, Srr}) vanishes beeause of the third equality <strong>in</strong> (F.2)<br />
and the anti-symmetrie property of the f/lyprr . F<strong>in</strong>ally, the rema<strong>in</strong><strong>in</strong>g term <strong>in</strong> the last l<strong>in</strong>e<br />
of eq. (F.25) <strong>in</strong>volves the operator v . 1J or v . 71, as is obvious from eq. (F.26), and henee<br />
vanishes modulo higher order terms.<br />
• Terms with 8/l' 8y, 8p and Vrr.<br />
<strong>The</strong> terms <strong>in</strong>volv<strong>in</strong>g the Levy-Civita tensor f /lyprr violate parity <strong>in</strong>varianee, whereas the<br />
rema<strong>in</strong><strong>in</strong>g terms ean be elim<strong>in</strong>ated us<strong>in</strong>g the EOM for the nucleon field.<br />
• Terms with 8/l' 8y, 8p and Vrr , VQ or Vrr, SQ or Srr, SQ are not Lorentz <strong>in</strong>variant.<br />
• Terms with 8/l' 8y, 8p, Srr, SQ and vß .<br />
Sueh terms <strong>in</strong>volv<strong>in</strong>g f/l ' Y 'p'rr' are not parity <strong>in</strong>variant, whereas the rema<strong>in</strong><strong>in</strong>g terms eontribute<br />
at higher orders .<br />
• All rema<strong>in</strong><strong>in</strong>g terms with more than one <strong>in</strong>sertion of the velo city operator ean be elim<strong>in</strong>ated<br />
us<strong>in</strong>g the EOM for the nucleon field.<br />
Thus, there are no eontaet terms with three derivatives at order ßi = 3.<br />
Another possible k<strong>in</strong>d of eontributions at this order with<strong>in</strong> our power eount<strong>in</strong>g seheme is from the<br />
terms with two derivatives suppressed by one power ofthe <strong>in</strong>verse nucleon mass (ljm-eorrections).<br />
Let us now take a closer look at sueh terms, whieh ean be written as<br />
(F.27)<br />
where the eonstants Bi are expressed <strong>in</strong> terms of the eoupl<strong>in</strong>g eonstants of the lower order Lagrangian.<br />
Note that s<strong>in</strong>ee we do not eonsider the terms with six and more nucleon legs <strong>in</strong> the<br />
effeetive Lagrangian, whieh do not affeet the <strong>in</strong>teraction between two nucleons, the only eonstants<br />
enter<strong>in</strong>g the B's are from LN and L�/v