The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

74 3. The derivation o{ nuc1ear {orces {rom chiral Lagrangians
The exponential faetor in (3.160) eliminates the trivial kinematic dependenee of the nucleon field
on the moment um mvw The quantities H and h are usually ealled the large and small eomponents,
respeetively. The free field Lagrangian (3.159) ean be expressed in terms of H, h as
.c = fI(iv . ö)H - h(iv . ö + 2m)h + fIi�h + hi�H
This leads to the following equations of motions for H and h:
(v · ö)H
(v . ö)h
From the second equation one sees that
-Pv+�h ,
2imh + Pv - �H .
Therefore, the large eomponent field H obeys the free equation of motion
(v · ö)H = 0 ,
(3.162)
(3.163)
(3.164)
(3.165)
up to l/m eorreetions. The nucleon mass does not enter this equation of motion to leading order.
Note that in the nucleon rest-frame with vJ.l = (1,0, 0,0), H and h ean be identified with the usual
upper and lower eomponents of a Dirae spinor modulo the faetor eimt. The small eomponent field
h ean now be eompletely eliminated from the Lagrangian (3.162) using the equations of motion
(3.163). A more elegant path integral formulation of this nonrelativistie reduetion ean be found
in the referenee [173].
For the ease of nucleons interaeting with pions one ean proeeed in a similar way as for free
nucleons to eliminate h. The equations of motion (3.163) are then modified by terms including
pion and nucleon fields. Sinee the ehiral symmetry only allows for derivative pion-nucleon and
pion-pion eouplings, these additional terms ean be regarded as small eorrections of order 1/ Ax to
the leading order equations ofmotions (3.163).31 There is one important problem in this approach:
time derivatives of the nucleon fields eontribute large factors of order m. In fact, it ean be proven
[68], [174], [175] that the leading order Lagrangian .cS 1 Jv for nucleons 'l1 interacting with pions 7r,
Le. the Lagrangian of the minimal low-energy dimension, is given by
r(l) _ ,1" ( ' J.l D
9A J.I ) 'T'
'-'7rN - 'i' n J.I - m + TI' I' 5 UJ.l 'i'. (3.166)
Here, the low-energy dimension of an operator has to be understood in the following sense: let Q be
a generic four-momentum of (external) pions or a generie three-momentum of (external) nucleons.
Then, U, 'l1, DJ.I'l1, m count as order one, whereas, for example, uJ.l ' öJ.lU, M7r , (i--y J.l DJ.I - m)'l1 as
order Q. Consequently, one ean apply the equation of motion
(i�- m)'l1 = ... , (3.167)
31 Clearly, one should not understand this statement ab out the smallness of the higher derivative terms too literal:
the operators with more field derivatives are, of course, by no means "smaller" than those with less derivatives.
Rather, one should understand this jargon in the way it was explained in the section 3.3. For a purely pionic
effective theory, the interactions in the Lagrangian with more field derivatives lead to low-energy observables (8matrix
elements) that are suppressed by additional powers of small external momenta. This can be seen from the
power counting (3.146). In this sense, the operators with larger index d - 2 that was defined in eq. (3.146) are of
higher order and called small corrections.