The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

5.1 Isospin violating effective Lagrangian 155
with Q± = 1/2( uQut ± ut Qu) and A = A - (A) /2 projects onto the off-diagonal elements of the
operator A. It can be checked [226] that the Q± transform covariantly under chiral SU(2) v x
SU(2)A rotation, Le.:
(5.9)
Further, under parity and charge conjugation transformations one finds:
Q± ---+ Q� =
±Q�
. (5.10)
Evidently, the charge matrices always have to appear quadratic since a virtual photon can never
leave a diagram. The last two terms in eq. (5.8) are not observable since they lead to an equal em
mass shift for the proton and the neutron, whereas the operator rv h to this order gives the em
proton-neutron mass difference. In what follows, we will refrain from writing down such types of
operators which only lead to an overall shift of masses or coupling constants. We note that in the
pion and pion-nucleon sector, one can effectively count the electric charge as a small moment um
or meson mass. This is based on the observation that Mn / Ax rv e/...(47r = fo rv 1/10 since
Ax :::: 41ffn = 1.2 GeV. It is thus possible to turn the dual expansion in small momenta/meson
masses on one side and in the electric coupling e on the other side into an expansion with one
generic small parameter. We also remark that from here on we use the fine structure constant
a = e2 /41f as the em expansion parameter.
We now turn to the two-nucleon sector, i.e. the four-fermion contact interactions without pion
fields. Consider first the strong terms. The effective Lagrangian takes the form
Cf:tN
h(Nt N)2 + l 2 (NtiJN)2 + l 3 (Nt (X+)N)(Nt N) + l4(Nt X+N)(Nt N)
+ l5(Nt iJ(X+)N)(Nt iJN) + l6(Nt iJx+N) (Nt iJN) + . .. , (5.11)
w here the ellipsis denotes terms with two (or more) derivatives acting on the nucleon fields.
Similarly, one can construct the em terms. The ones without derivatives on the nucleon fields
read
CNN
Nt {r1 (Q� - Q�) + r2 Q + (Q +) } N (Nt N)
+ NtiJ {r3(Q� - Q�) + r4Q+(Q+)} N(NtiJN)
+ Nt {r5Q+ + r6(Q+)} N(Nt Q+N) + NtiJ {r7Q+ + r8(Q+)} N(Nt iJQ+N)
+ rg(Nt Q+N)2 + rlO(Nt iJQ+N)2 . (5.12)
There are also various terms resulting from the insertion of the Pauli isospin matrices i in different
Nt N binomials. Some of these can be eliminated by Fierz reordering (or anti-symmetrization, see
appendix E), while the others are of no importance for our considerations. Note that from now on
we consider em effects. The leading CSB rv mu - md has the same structure as the corresponding
em term and thus its contribution can be effectively absorbed in the value of E62) , as defined below.
We re mark since ANN is significantly smaller than Ax' it does not pay to treat the expansion in
the generic KSW moment um Q simultaneously with the one in the fine structure constant (as it is
done e.g. in the pion-nucleon sector ). Instead, one has to assign to each term a double expansion
parameter Qnam, with n and m integers. Lowest order charge independence breaking is due to
a term rv
(Nt
T3 N)2 whereas charge symmetry breaking at that order is given by a structure
rv (NtT3N)(NtN). In the KSW approach, it is customary to project the Lagrangian terms on the
pertinent NN partial waves. Denoting by ß the 1 So partial wave for a given cms energy Ecms, the