Charge independence breaking and charge symmetry breaking in the nucleon–nucleon interaction from effective field theory

We discuss charge symmetry and charge independence breaking in an effective field theory approach for few–nucleon systems. We systematically introduce strong isospin–violating and electromagnetic operators in the theory. The charge dependence observed in the nucleon–nucleon scattering lengths is due to one–pion exchange and one electromagnetic four–nucleon contact term. This gives a parameter free expression for the charge dependence of the corresponding effective ranges, which is in agreement with the rather small and uncertain empirical determinations. We also compare the low energy phase shifts of the nn and the np system. DEDICATED TO WALTER GLÖCKLE ON THE OCCASION OF HIS 60th BIRTHDAY email: [email protected] email: [email protected] 1. It is well established that the nucleon–nucleon interactions are charge dependent (for a review, see e.g.[1]). For example, in the S0 channel one has for the scattering lengths a and the effective ranges r (n and p refers to the neutron and the proton, in order) ∆aCIB = 1 2 (ann + app)− anp = 5.7± 0.3 fm , ∆rCIB = 1 2 (rnn + rpp)− rnp = 0.05± 0.08 fm . (1) These numbers for charge independence breaking (CIB) are based on the Nijmegen potential and the Coulomb effect for pp scattering is subtracted based on standard methods. The charge independence breaking in the scattering lengths is large, of the order of 25%, since anp = (−23.714±0.013) fm. In addition, there are charge symmetry breaking (CSB) effects leading to different values for the pp and nn threshold parameters, ∆aCSB = app − ann = 1.5± 0.5 fm , ∆rCSB = rpp − rnn = 0.10± 0.12 fm . (2) Both the CIB and CSB effects have been studied intensively within potential models of the nucleon–nucleon (NN) interactions. In such approaches, the dominant CIB comes from the charged to neutral pion mass difference in the one–pion exchange (OPE), ∆a CIB ∼ 3.6± 0.2 fm. Additional contributions come from γπ and 2π (TPE) exchanges. Note also that the charge dependence in the pion–nucleon coupling constants in OPE and TPE almost entirely cancel. In the meson–exchange picture, CSB originates mostly from ρ−ω mixing, ∆a CSB ∼ 1.2 ± 0.4 fm. Other contributions due to π − η, π − η mixing or the proton–neutron mass difference are known to be much smaller. Within QCD, CSB and CIB are of course due to the different masses and charges of the up and down quarks. Such isospin violating effects can be systematically analyzed within the framework of chiral effective field theories. In the two–nucleon sector, a complication arises due to the unnaturally large S–wave scattering lengths. This can be dealt with in two ways. One is the “hybrid” approach due to Weinberg [2], in which chiral perturbation theory is applied to the interaction kernel sewed to realistic nuclear wave functions obtained by conventional means. This approach has been successfully applied to a variety of processes which are dominated by OPE, a particularly striking one being the prediction of the electric dipole amplitude for neutral pion production off deuterons [3] recently measured at SAL [4]. A different and more systematic fashion to deal with the unnaturally large scattering lengths is the recently proposed power divergence subtraction scheme (PDS) proposed by Kaplan, Savage and Wise (KSW) [5]. Essentially, one resums the lowest order local four–nucleon contact terms ∼ C0(N N) (in the S–waves) to generate the large scattering lengths and treats the remaining effects perturbatively, in particular also pion exchange. This means that most low–energy observables are dominated by contact interactions. The chiral expansion for NN scattering entails a new scale ΛNN of the order of 300 MeV, so that one can systematically treat external momenta up to the size of the pion mass. There have been suggestions that the radius of convergence can be somewhat enlarged [6], but in any case ΛNN is considerably smaller than There exist by now modifications of this approach and it as been argued that it is equivalent to cut–off schemes. We do not want to enter this discussion here but rather stick to its original version.