The nucleon-nucleon interaction with the leading order relativistic effects Petr Vesel´y Institute of Particle and Nuclear Physics Faculty of Mathematics and Physics Charles University, Prague Preface This text was written as a part of the lecture course1 The Relativistic Description of Nuclear Systems. We briefly summarize the methods employed in a quantum-mechanical (QM) description of the nucleon-nucleon problem. The QM approach means the Hamil- tonian description in a Hilbert space with a fixed number of nucleons (from which all other degrees of freedom are “integrated out”). Here we consider only its version in which the relativistic effects are included in a Hamiltonian (both in its kinetic and interaction parts) perturbatively in their leading order. After a short historical introduction given in the first chapter we consider the one- scalar-meson-exchange (OSE) interaction between two nucleons. The relativistic OSE amplitude is derived from the corresponding Feynman diagram of the covariant field theory. We briefly discuss the relation of this amplitude to the quantum-mechanical potential, in particular the techniques of the off-energy-shell continuation. Separation of the center-of-mass motion in Hamiltonian with relativistic effects is also considered. In the third chapter we deal with the subsequent transformation of the OSE potential from the momentum to the coordinate representation. Dubna, March 2, 2006 Petr Vesel´y 1This course was read by Jiˇr´ıAdam in 2005/2006 at the Institute of Particle and Nuclear Physics i Contents 1 Brief review of the NN potentials 1 2 One scalar particle exchange - OSE 5 2.1 Amplitudeoftheexchangeprocess . 5 2.2 Off-energy-shellcontinuation. .... 9 2.3 Separation of the center-of-mass motion . ...... 10 2.3.1 ComparisontotheBonnpotential. 17 3 The OSE potential, x-representation 19 3.1 Central ~q-dependentpart ........................... 22 3.2 Spin-orbitpart ................................. 24 3.3 Centralanticommutatorpart. 25 3.4 The OSE potential in the Schr¨odinger equation . ....... 25 4 Summary 27 A Poincare and Galilean algebras 28 B Commutators needed for the c.m. separation 32 C Anticommutator term in the Schr¨odinger equation 34 ii Chapter 1 Brief review of the NN potentials Since the Rutherford’s discovery of nucleus and the subsequently confirmed nucleonic structure of nuclei the crucial task - the explanation of the origin of attractive nuclear forces - has been haunting nuclear physicists. In 1930’s the quantum field theory was emerging and in its spirit Yukawa suggested the first microscopic derivation. He suggested the massive elementary1 particle – meson – which would intermediate the finite range nucleon-nucleon (NN) interaction. In 1947 a pion - the particle identified later with the Yukawa meson - was found in cosmic rays. Subsequent effort of theoretical physicists led to introduction of the one-pion- exchange (OPE) and later the two-pion-exchange (TPE) models of the NN interaction. Comparison to experimental results (phase-shift analysis) established the OPE potential as the long range part (“tail”) of the nuclear force. The full OPE+TPE program failed, however: the historical circumstances are in more detail explained in the introduction of the survey [1]. From the theoretical point of view it is important that the way the TPE processes were included in 1950’s did not respect the chiral symmetry2. In 1960’s heavy mesons (in particular, the vector ρ- a ω-mesons) were discovered and the attention of physicists turned to newly established OBE (one-boson-exchange) models. Recall that although these days we consider (most of the) heavy mesons to be quark-antiquark bound states, they are experimentally seen as the two- or three-pion resonances either in a pion- pion scattering or in processes with several pions in a final state. One says that heavy meson exchanges between nucleons represent correlated multipion exchanges. The OBE potentials are based on the idea that the single one-meson exchanges (of various meson types) can realistically represent most features of the full NN interaction, to which also exchanges of multiple mesons contribute. The meson-nucleon interaction is strong, i.e., a perturbative expansion in powers of coupling constants cannot be expected to work and the potential has to be iterated to all orders with the help of some dynamical equation – either non-relativistic Lippmann-Schwinger equation or some of analogous relativistic ones. There is, nevertheless, an important organizing principle for various contributions to meson-mediated NN potential: their importance decreases with their decreasing range, which in turn is inversely proportional to the total exchanged mass 1Elementary until the discovery of the hadron substructure and of the quark model. 2Let us note that the OPE+TPE approach was not rejected forever. Recently, it was revived in the framework of the effective (chiral) field theory. 1 CHAPTER 1. BRIEF REVIEW OF THE NN POTENTIALS 2 (i.e., sum of the masses of all exchanged mesons). In the framework of OBE potentials, the scalar, pseudoscalar and vector mesons with masses 1 GeV are usually taken into account. All mesons, or rather all their parameters, need≤ not necessarily be identified with physical mesons seen in experiments and listed in the Particle Data tables: some of them can actually effectively represent a sum of all (also multiple) exchanges in a particular spin/isospin channel. Thus, the scalar (isoscalar) field plays rather important role in the OPE models: it delivers a necessary intermediate range attraction. Still, the existence of the corresponding physical particle was not confirmed, at best it is a very broad and unstable resonance. The Bonn group (mentioned below) demonstrated (among other similar facts) that one should consider the scalar field as an effective field describing the correlated two pion exchange (with spin and isospin equal to zero). Although it was possible to fit the NN scattering data well by using OBE potentials, it was not actually clear that the uncorrelated multiparticle exchanges can be completely neglected. Also the role of nucleon excitation was not very clear. In the mid 1970’s the Bonn group [2, 1] started to evaluate the multipion exchange diagrams and in the following decade they computed all 2π diagrams (including those with virtual ∆ - resonances), relevant 3π and 4π diagrams and also combined crossed meson exchanges - ωπ, ρπ. This calculations demonstrated – though admittedly in a model-dependent way – that with rather good precision the NN potential can be indeed represented by OBE potential, with some of the exchanges being effective in a sense described above. This developments and other successful descriptions of the NN interaction in terms of meson exchanges (e.g., [3]) encouraged a birth of a new generation of realistic semi- phenomenological NN models called high-precision potentials. In chronological order they are Nijmegen Nijm-I, Nijm-II and Reid93 potentials [4]; • Argonne V potential [5]; • 18 CD-Bonn potential [6]. • All these potentials are inspired by the meson-exchange picture, in particular, all of them include the OPE part. They are also all charge-dependent, as required by deviations of the pp and np data. The main differences among them are in meson exchanges explicitly included and in parts of the force which are described completely phenomenologically. Also in their OBE components various approximations are made, in particular, to en- able the Fourier transform of the corresponding Feynman amplitudes the non-relativistic reduction (resulting in the local approximation) is often employed. The models Nijm-I and Nijm-II are based on the Nijmegen78 OBE potential [3]. Nijm- II uses the local approximation for all OBE amplitudes, while Nijm-I keeps some terms non-local. The Reid93 and V18 do not use meson exchanges for the intermediate and short-range parts of the potentials and describe them purely phenomenologically. The V18 potential employs the functions of the Wood-Saxon type, while Reid93 uses local Yukawa functions of multiples of the pion mass. Unlike other models, the CD-Bonn employs the full nonlocal Feynman amplitudes for the OBE potentials. Apart from the pion, the physical vector mesons ρ(769) and ω(783) and less important δ and η exchanges, two fictitious scalar-isoscalar σ-mesons are used. CHAPTER 1. BRIEF REVIEW OF THE NN POTENTIALS 3 To sum it up, the current model descriptions of the NN interaction, based on the meson-exchange picture with semi-phenomenological adjustments at the short-range can fit very successfully the NN data below pion production threshold. Extension of these ideas to the 3N force and e.m. and weak nuclear currents is, in principle, also phenomenologically successful, but not yet as conclusive. More important, it is rather difficult to deduce from these models some unambiguous information on related aspects of hadronic dynamics, e.g., on the importance of various physical (heavy) mesons and/or nucleon resonances, on their coupling constants, on whether it is adequate to include them as stable particles etc. It is also difficult to relate assumptions and the phenomenological input of these models to the underlying fundamental theory, the QCD. Even the important symmetries of the QCD – the approximate isospin symmetry and the approximate spontaneously-broken chiral symmetry – are reflected only purely phenomenologically by adopting the experimental values of the meson masses and coupling constants, considering the phenomenological mixing of heavier mesons and recognizing the leading role of the pion as the lightest meson. Recently, the Chiral Perturbation Theory (ChPT) has been established as an attempt to describe the hadron dynamics in the low-energy regime in more systematic and fun- damental way. The ChPT in relation to the fundamental theory of strong interaction - QCD - is an effective field theory satisfying all symmetries included in QCD with nucleons and pions (and in some formulations also ∆-isobars) used as the degrees of freedom. The applicability of this effective theory is then limited by the momentum scale of Λ 1 GeV.