The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

190 F. The complete set of the N N contact interactions with two derivatives
where the matrix R is given by
(F.26)
We have used eq. (F.16) to obtain the seeond line of eq. (F.25). The term vp vQSrr in the
squared braekets in the seeond line of this equation vanishes modulo higher order terms after
applying the EOM for Hv . Sueh higher order terms are denoted in eq. (F.25) by the dots.
To end up with the third line of eq. (F.25) we have used the last equality in eq. (F.2). The
seeond term in the parenthesis (SQ {Sp, Srr}) vanishes beeause of the third equality in (F.2)
and the anti-symmetrie property of the f/lyprr . Finally, the remaining term in the last line
of eq. (F.25) involves the operator v . 1J or v . 71, as is obvious from eq. (F.26), and henee
vanishes modulo higher order terms.
• Terms with 8/l' 8y, 8p and Vrr.
The terms involving the Levy-Civita tensor f /lyprr violate parity invarianee, whereas the
remaining terms ean be eliminated using the EOM for the nucleon field.
• Terms with 8/l' 8y, 8p and Vrr , VQ or Vrr, SQ or Srr, SQ are not Lorentz invariant.
• Terms with 8/l' 8y, 8p, Srr, SQ and vß .
Sueh terms involving f/l ' Y 'p'rr' are not parity invariant, whereas the remaining terms eontribute
at higher orders .
• All remaining terms with more than one insertion of the velo city operator ean be eliminated
using the EOM for the nucleon field.
Thus, there are no eontaet terms with three derivatives at order ßi = 3.
Another possible kind of eontributions at this order within our power eounting seheme is from the
terms with two derivatives suppressed by one power ofthe inverse nucleon mass (ljm-eorrections).
Let us now take a closer look at sueh terms, whieh ean be written as
(F.27)
where the eonstants Bi are expressed in terms of the eoupling eonstants of the lower order Lagrangian.
Note that sinee we do not eonsider the terms with six and more nucleon legs in the
effeetive Lagrangian, whieh do not affeet the interaction between two nucleons, the only eonstants
entering the B's are from LN and L�/v