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