The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

68 3. The derivation of nuc1ear forces from chiral Lagrangians
The corrections O(M) in eq. (3.134) correspond to higher order symmetry breaking terms that
include two and more insert ions of the quark mass matrix M. Eq. (3.134) allows one to express
the pion mass in terms of the quark condensate as follows
This expression is known as the Gell-Mann-Oakes-Renner relation [163].
(3.135)
Let us finally make aremark concerning the constant in: . For that we can calculate the axial-vector
current corresponding to the Lagrangian (3.124), that can be equivalently rewritten as
(3.136)
To calculate the symmetry current we use eq. (3.13). Note that in the case of a SU(2)A transformation
one has (Jv = 0, (JR = (JA and (JL = -(JA, see eqs. (3.92), (3.93). Consequently, an
infinitesimal axial-vector transformation 90 E SU(2)A of the matrices U, ut is given by
The related axial-vector current iS21
U � U' = hRUhR = U - �(J� (TiU + UTi) ,
ut' - h-1Uth-1 R R - ut + �(Ji A (TiUt + UtTi) ,
2
(3.137)
(3.138)
where the ellipses stay for terms with more pion fields. Clearly, the axial current connects the
vacuum state with the one-pion state:22
(3.139)
Here, n hys 23 denotes the experimentally observed value of the pion decay constant, which can be
calculated from the rate for pion decay. The dots correspond to corrections, which are proportional
to powers of p2 / i; . 24 Consequently, these corrections are proportional to the pion mass squared
M; , which is zero in chiral limit. Thus, in the chiral limit in: is not renormalized and in: = n hys.
To perform later the expansion in powers of small momenta Q we need an ordering scheme for
various interactions in the effective Lagrangian, which can be viewed as the derivative expansion
and the expansion in powers of quark masses corresponding to the symmetry breaking terms.
One commonly counts the squared pion mass M; rv (140MeV)2 or, equivalently, the quark mass
insertion as two powers of small external momenta (two derivatives). Therefore, the complete
Lagrangian to leading order is given by
(3.140)
21
Note that BA corresponds to _ca in the notation of eq. (3.12).
22This is a general consequence of Goldstone theorem.
23
0ne commonly uses a different notation, in which OUf Irr (I!:hys) is denoted by F or I (Irr).
24This can be demonstrated from Lorentz invariance and the power counting scheme, that will be introduced in
the following section.