The Nucleon-Nucleon Interaction with the Leading Order Relativistic Effects

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 fundamental 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. The Lagrangian of ChPT involves all possible terms (containing the nucleon fields,≈ pionic fields and its derivatives) that are allowed by the symmetries. This means that the Lagrangian actually consists of an infinite number of terms. However, in practical calculations we take into account only terms up to some given order of p/Λ depending on the required accuracy. The main goal of the ChPT is, in principle, to explain the spectrum of (multi)hadron states and the low-energy hadron dynamics (including, e.g. the NN scattering) relating it to the fundamental theory of strong interaction. This should be achieved by calculating the coefficients (coupling constants) of the ChPT Lagrangian directly from QCD. In prac- tice, however, we still cannot solve this task, so the parameters are obtained by fitting experimental data. As for the NN interaction, the ChPT these days provides (at its next-to-next-to-next- to-leading-order; N 3LO !) the potentials [7, 8] which describe the data as accurately as the phenomenological high-precision ones mentioned above (with roughly the same number of fitted parameter). But at least some of these parameters are connected to other hadronic processes and all of them are in principle calculable from the lattice QCD. Derivations of the NN potentials in the framework of the ChPT also demonstrated the convergence: the contributions of higher orders are indeed in the energy region considered yielding increasingly smaller corrections. The ChPT also explains why are the forces between more than two nucleons less important (and the leading order contributions to the three- nucleon force were derived and tested). At the end of this chapter let us note that very detailed historical overview can be found at the beginning of the text [1]. For more recent review articles and summary talks see e.g. [9]. CHAPTER 1. BRIEF REVIEW OF THE NN POTENTIALS 4

In our text we brieﬂy review the main steps in deriving the one-meson-exchange potential (for simplicity for the scalar meson) in a framework very close to the Bonn OBEP-R potentials. Its purpose is to give clear account of the necessary technical steps as well as of the underlying assumptions/physical principles. Chapter 2

One scalar particle exchange - OSE

In this chapter we demonstrate the relativistic techniques used for the derivation of the OBEP on the simplest example of the isoscalar scalar σ particle1 exchange in the framework of the extended S-matrix technique [10]. We follow the derivation from the corresponding Feynman diagram of the OSE exchange to the quantum-mechanical NN potential.

2.1 Amplitude of the exchange process

Let us construct matrix element of the process NN NN mediated by a single scalar meson exchange using the relevant Feynman diagram,→ according to the standard QFT rules. We assume the simplest interaction Lagrangian between the Dirac bispinor ﬁelds

Figure 2.1: The Feynman diagram of the OSE exchange.

1Although, as mentioned above, this particle is only hypothetical.

5 CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 6

Ψ representing the nucleons and a scalar ﬁeld σ representing the intermediate meson

= g ΨΨ¯ σ , (2.1) Lint s where the constant gs determines the strength of this interaction. As an useful building block let us ﬁrst introduce the amplitude of the nucleon-meson vertex (the “vertex function”) described by ﬁg. 2.2. This amplitude reads: g g (s; p′,p)= s u¯(p′)Γ(ˆ s)u(p)= s Γ(s; p′,p) , Γ(ˆ s)= i . (2.2) M (2π)3/2 (2π)3/2

From now on we use the following notation for the three- and four-momenta:

q (p′ p) , Q (p′ + p) , (2.3) ≡ − ≡ and analogously for the momenta at 1st and 2nd vertices of Fig. 2.1:

q (p′ p) , Q (p′ + p) , i = 1, 2 . (2.4) i ≡ − i i ≡ i

On-mass-shell energies will be denoted by E, oﬀ-shell ones by p0 or q0. From the Feynman

Figure 2.2: The meson-nucleon vertex. rules we construct the S-matrix of the OSE (see ﬁg. 1), using the one-vertex amplitudes (2.2) and the scalar propagator

S = = 2πiδ(E′ + E′ E E ) δ(~q + ~q ) (p′ ,p′ ,p ,p ) , fi 1 2| | 1 2 ∼ − 1 2 − 1 − 2 1 2 V 1 2 1 2 g2 Γ(s; p′ ,p )Γ(s; p′ ,p ) (p′ ,p′ ,p ,p ) = s 1 1 2 2 , (2.5) V 1 2 1 2 (2π)3 µ2 q2 − CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 7

2 2 2 where q = q1 = q2 and q = q0 ~q . The δ-functions express the conservation of the energy and 3-momentum− in the S-matrix− element. The presence of the energy δ-function ′ ′ mean that the S-matrix approach deﬁnes the amplitude only for E1 + E2 = E1 + E2, i.e., on the energy-shell. In the next section we discuss this limitation. The 3-momentum conservation is here written in a form suggesting the introduction of q = q1 = q2, but we should keep in mind that: −

δ(~q + ~q )= δ(~p ′ + ~p ′ ~p ~p )= δ(P~ ′ P~ ) , (2.6) 1 2 1 2 − 1 − 2 − ′ ′ ′ where P~ = ~p1 + ~p2 and P~ = ~p1 + ~p2 are the total momenta in the final and initial state, respectively. The non-relativistic normalization of nucleon spinors is assumed u+(p)u(p)= 1, i.e., E + m 1 u(p)= . (2.7) 2E ~σ·~p r E+M In this normalization the S-matrix element (2.5) does not contain any factors of m/E. ′ ′ By eq. (2.5) we introduced the amplitude (p1,p2,p1,p2), which physically plays the role of the NN potential. The last sentence acquiresV the exact meaning only after the dynamical equation for the NN T-matrix is specified, that is, after we define how to sum up the infinite series of terms as required by the non-perturbative nature of the strong NN interaction. Rather ambitious possibility would be to treat the problem in a manifestly Lorentz covariant way, making use of the Bethe-Salpeter equation (or some of its quasi- potential reductions). In this approach the amplitude serves as (an OSE approximation to) the irreducible kernel of the relativistic dynamicalV equation, which can be in symbolic notation written as = + rel , (2.8) T V VG2 T rel where 2 is the relativistic two-nucleon propagator. The advantage of the covariant treatmentG is unambiguous connection to the field theory: the Feynman rules allow to calculate the kernel also for the nucleons off-mass-shell (when the diagram fig. 2.1 is embedded into moreV complicated one, e.g. into a term of the ladder series): intermediate nucleons are off-mass-shell and four-momenta are conserved at each vertex, exactly as in the standard Feynman diagrams. This approach is, however, rather technically involved and it does not seem that the fully relativistic treatment is really required, at least for energies below the pion production threshold. Here, we instead adopt more common strategy: we accept the non-relativistic reduction in which we take into account only the main relativistic corrections up to the second order in powers of momenta. Thus, the normalization factor is expanded as:

2 1/2 ~p 2 E + m 2m + 2m ~p = 2 = 1 , 2E ∼ ~p ∼ − 8m2 r 2m + m ! the matrix element as ~σ ~p ′ 1 σ ~p ′ σ ~p ~p ′ ~p + i~σ ~p ′ ~p 1, · = 1 · · = 1 · · × , −E′ + M ~σ·~p ∼ − 4m2 − 4m2 E+M CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 8 and by simple algebraic steps we get the non-relativistically reduced vertex function:

Q~ 2 i Γ(s; p′,p) i 1 ~σ ~q Q~ . (2.9) ∼= 2 2 " − 8m − 8m · × # Substituting this into (2.5) yields

2 ~ 2 ~ 2 ~ ~ gs 1 Q1 + Q2 + i~σ1 ~q Q1 i~σ2 ~q Q2 − = 1 × · − × · . (2.10) Vred prop ∼ −(2π)3 µ2 + ~q 2 q2 − 8m2 − 0 " # Thus, after the reduction the vertex function Γ is the function of only 3-momenta (this holds only for this simple scalar exchange, e.g., the πNN coupling is more complicated even at this order [10]). This indicates the possibility to identify the amplitude with V the QM potential in the momentum space, acting in the Hilbert space of two nucleons and therefore depending only on their coordinates (momenta) and spins:

′ ′ ′ ′ V (~p , ~p , ~p , ~p )=< ~p , ~p Vˆ ~p , ~p >= − . (2.11) 1 2 1 2 1 2| | 1 2 Vred prop

However, the energy of the exchanged meson q0 is still not clearly expressed in terms of the nucleon momenta. We started from the Feynman amplitude (2.5) in which the energy is conserved at each vertex and hence:

q =∆ = ∆ , ∆ = E′ E , i = 1, 2 , (2.12) 0 1 − 2 i i − i ′ where Ei(Ei) is the on-mass-shell energy of the i-th nucleon in the initial (ﬁnal) state, i.e.,: ~p 2 E = ~p 2 + m2 = m + i + ... , (2.13) i i ∼ 2m ′ q 2 and similarly for Ei. Notice that the energy-transfer ∆i is of the order of p /m:

~p ′2 ~p 2 ~q Q~ ∆ = E′ E = i − i + = i · i + ... . (2.14) i i − i ∼ 2m ··· 2m The potential Vˆ has to commute with the operator interchanging coordinates and spins of the 1st and 2nd nucleons, i.e., it has to be symmetric in respect to interchange of the subscripts 1 and 2. Thus, we can write from (2.12)

∆ ∆ (∆ ∆ )2 q = 1 − 2 q2 = 1 − 2 . (2.15) 0 2 → 0 4 2 This expresses q0 in terms of the squares of the nucleon momenta in a symmetric way and, as we will demonstrate below, yields a consistent deﬁnition of the NN potential. Since ∆ p2/m, also q p2/m. The average nucleon momentum is of the order 100MeV, i ∼ 0 ∼ ∼ hence q0 is small compared to the meson mass (several hundreds MeVs). Thus, we can expand the meson propagator into the Taylor series: 1 1 q2 = + 0 . (2.16) µ2 + ~q 2 q2 ∼ µ2 + ~q 2 (µ2 + ~q 2)2 − 0 CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 9

′ ′ It is worth noting that in the center-of-mass (c.m.) frame E1 = E2 and E1 = E2, hence ∆1 = ∆2. That is, the expression (2.15) leads in the c.m. frame to the NN potential 2 without meson retardation, i.e., q0 = 0 and the meson propagator in this frame reduces to the purely static one 1/(µ2 + ~q 2). Let us emphasize that this is valid only in the c.m. frame and does not hold in an arbitrary one, in accordance with an approximate Lorentz invariance.

2.2 Oﬀ-energy-shell continuation

The definition of the potential described at the end of the previous section is consistent, but not general enough. The point is that there exist alternative methods of derivations (for a detailed discussion see e.g. last two papers of Ref. [10]), which yield different (though approximately unitarily equivalent) results. The S-matrix method in its extended version generalizes the prescription above in a way which covers all other approaches. It this section we describe this reasoning on the example of the meson retardation, more general discussion which covers also the possible energy-dependence of vertices is given in Ref. [10]. What do we mean by the extended S-matrix method? The point is that our starting point – the diagram fig. 2.1 – defines the amplitude (2.5) on-the-energy shell, i.e., for ′ ′ E1 + E2 = E1 + E2, and this would be true even if this amplitude is just a sub-block of more complicated Feynman diagrams, since in the covariant Feynman diagrams the energy is necessarily conserved at each vertex. On the other hand, the QM potential generates the corresponding NN-scattering S-matrix element through the T-matrix, which results from the summation of the infinite Born series via the Lippmann-Schwinger equation:

T = V + VG0T , (2.17) where G = 1/(E T ) is the QM free two-nucleon propagator with E being the energy 0 − 0 of the process and T0 is the kinetic energy operator of two-nucleons (non-relativistic or relativistic). Unlike in the relativistic equation (2.8), in eq. (2.17) the nucleons in the intermediate state are on their mass shell, but the energy of the intermediate state is not equal to E (since [T0, V ] = 0 = [H, V ]). Thus, to calculate T we have to specify V also off the energy shell. It can6 be done6 as in the previous section, just by demanding that the formula for V derived there is valid not just on energy shell, but for any initial and final energies. Since the vertex functions depend only on three-momenta which are in both frameworks fixed by the three-momentum conservation in a same way, this discussion in fact concerns only the treatment of the meson energy (2.15). More general off-energy- shell continuation of the potential follows from more careful analysis of prescription for 2 the meson energy (in our case of the scalar exchange, it is enough to consider q0 in the meson propagator, in general one also has to deal with dependence of the vertices on the 2 energy transfer). Obviously, q0 has to be expressed in terms of nucleon energy transfers 2 ∆i. Considering the orders of the relativistic effects taken into account, q0 has to be bilinear function of ∆1 and ∆2, which is symmetric in respect to 1 2, • ↔ properly normalized at the on-energy-shell point ∆ = ∆ . • 1 − 2 CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 10

2 The most general expression for q0 satisfying these criteria is: 1 q2 = (∆ ∆ )2 + (1 2ν)(∆ +∆ )2 , (2.18) 0 4 1 − 2 − 1 2 where ν is an arbitrary real parameter. For ν = 1/2 this expression reduces to (2.15); notice that only for this value of ν we get the propagator without the retardation in the c.m. frame. Other values of ν reproduce the results of alternative techniques of derivation, e.g., the “old time-ordered” perturbation theory (typically for ν = 0). On-energy-shell ∆1 +∆2 = 0 and we get (2.15), as required. 2 Writing q0 in terms of the nucleon momenta (up to the order considered):

[~q (Q~ + Q~ )]2 + (1 2ν)[~q (Q~ Q~ )]2 q2 = · 1 2 − · 1 − 2 (2.19) 0 16m2 and substituting (2.16) into (2.10) we obtain the ﬁnal expression for the OSE potential with the leading order relativistic corrections in an arbitrary reference frame:

V (~p ′, ~p ′ , ~p , ~p )δ(~q + ~q ) = < ~p ′, ~p ′ Vˆ ~p , ~p > δ(~q + ~q )= 1 2 1 2 1 2 1 2| | 1 2 ≡ Vred 1 2 ~ 2 ~ 2 ~ ~ (1) 2 Q1 + Q2 + i~σ1 ~q Q1 i~σ2 ~q Q2 = δ(~q1 + ~q2) V (~q ) 1 × · 2 − × · " − 8m 2 2 [~q (Q~ 1 + Q~ 2)] + (1 2ν)[~q (Q~ 1 Q~ 2)] + · 2 −2 2 · − , (2.20) 16m (µ + ~q ) # g2 1 V (1)(~q 2)= s . (2.21) −(2π)3 µ2 + ~q 2 In what follows we show: how to simplify this expression by separating the c.m. motion, • that our result respects – up to the order considered– the Lorentz invariance, • how to transform the potential into the coordinate representation. • 2.3 Separation of the center-of-mass motion

In this section we discuss the separation of the center of mass motion (c.m.) of the NN system. Let us ﬁrst explain how does it work in the QM framework. Having the c.m. motion separated means that the system as a whole moves as a single (composite) particle. That is, any eigen-state ψ > of the Hamiltonian is also eigen-state ˆ | of the total momentum P~ and it factorizes into

ψ >= P~ > ǫ > , (2.22) | | | where P~ > is a plane wave describing the motion of the system as a whole and ǫ > describes| its internal state with an (intrinsic) energy ǫ. The latter part is not affected| CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 11 by the velocity or position of the whole system, i.e., it is reference frame-independent. The former part depends only on the velocity of the center-of-mass and on the (frame- independent) invariant mass ǫ, which is determined by the intrinsic state of the system. The obvious advantage of such factorization is that the state is known in any reference frame: the frame-dependence is contained in a rather trivial way in the plane wave P~ >, while the intrinsic wave functions ǫ> can be conveniently calculated in the c.m. frame.| Let us emphasize that all our discussion| below is limited to the loosely bound systems for which the intrinsic energy ǫ, which plays the role of the invariant mass, differs only slightly from the sum of the constituent masses (i.e., for two particles with equal masses ǫ 2m). For such systems the kinetic energy is of the order of the potential energy and the∼ intrinsic energy with the sum of rest masses subtracted (i.e., the binding energy or the kinetic energy of the relative motion) is a small number (resulting from the cancelation of < Tˆ > + < Vˆ >, < Tˆ > being positive and < Vˆ > negative). Then, we can assume that the kinetic and potential energies are of the same order, which yields the convenient consistent counting scheme, in which the non-relativistic kinetic and potential energies are roughly the same Tˆ(1) Vˆ (1). The superscripts will from now on indicate the order of various terms in the Hamiltonian,∼ their values refer to the power of 1/m for the corresponding kinetic energy. The relativistic effects appear first at the order 1/m3 and can be also arranged according to the respective orders of 1/m. In particular,∼ the leading order relativistic correction to the Hamiltonian are Tˆ(3) Vˆ (3) (by assumption ∼ they are smaller than the non-relativistic terms). Let us briefly remind how is the separation of the c.m. motion realized in the non- relativistic QM. The non-relativistic two-particle Hamiltonian reads (for the sake of briefness we restrict ourselves to the case of two particles with the same masses, but the non-equal-mass case can be treated analogously):

~pˆ2 ~pˆ2 Hˆ = 2m + Hˆ (1) = 2m + 1 + 2 + Vˆ (1) , (2.23) 2m 2m where, compared to the usual form Hˆ (1), we added to the Hamiltonian the rest masses of the particles. To separate the c.m. motion one expresses the individual particle coordinates and momenta in terms of the relative and c.m. coordinates and momenta: 1 1 1 ~r = ~r ~r , R~ = (~r + ~r ) ~r = R~ + ~r, ~r = R~ ~r (2.24) 1 − 2 2 1 2 1 2 2 − 2 1 1 1 ~p = (~p ~p ) , P~ = ~p + ~p ~p = P~ + ~p, ~p = P~ ~p. (2.25) 2 1 − 2 1 2 1 2 2 2 − In terms of these variables the non-relativistic Hamiltonian (2.23) becomes:

ˆ ˆ ˆ P~ 2 ~pˆ2 P~ 2 P~ 2 Hˆ = M + + + Vˆ (1) M + + hˆ(1) = Mˆ + , (2.26) 2M 2µ ≡ 2M 2M where M = 2m is the total mass and µ = m/2 is the reduced mass. If now the potential Vˆ (1) does not depend on R~ and P~ (i.e., on the position of the center-of-mass and on the total momentum of the system), we can write eigen-state of the Hamiltonian as (2.22) CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 12 with

The invariant mass operator is deﬁned as the Hamiltonian in the c.m. frame, i.e.:

Mˆ = Hˆ (P~ =0)= M + hˆ(1) . (2.29)

However, notice that in the non-relativistic description the energy of the c.m. motion is just P~ 2/2M, as if the invariant mass is just the sum of the constituent masses and does not contain the intrinsic energy of the relative motion (e.g., the binding effects). In this framework the deuteron would move with the mass M = mp + mn, instead of Md = mp +mn Ed (where Ed > 0 is the deuteron binding energy). Thus, in the non-relativistic description− we have to include the binding effects (e.g. in the kinematical relations) by hands, they cannot be obtained formally from the non-relativistic Hamiltonian. To include the intrinsic energy into the c.m. kinetic energy we would have to replace ˆ ˆ P~ 2 P~ 2 . (2.30) 2M → 2Mˆ But 1 1 1 hˆ(1) Mˆ = M + hˆ(1) = 1 , (2.31) → Mˆ M 1+ hˆ(1)/M ≃ M − M ! and the second term hˆ(1)/M is of the order of the relativistic correction, hence it cannot emerge from the purely∼ non-relativistic Hamiltonian. We will see below that exactly such term is recovered when the leading order relativistic effects are taken onto account. Now we can formulate the problem of the separation of the c.m. motion more precisely: starting from the Hamiltonian expressed in terms of coordinates, momenta and spins of individual particles one looks for a set of a new canonical variables in terms of which the Hamiltonian takes a one particle form, i.e., the form of the the one-particle kinetic energy operator with the total momentum and with a constant mass replaced by the invariant mass operator. (In fact, as we briefly remind in Appendix A, also other generators of the Poincare algebra – the angular momentum and the Lorentz boost – take on in terms of the new variables their respective one-particle forms.) Let us now consider the Hamiltonian with the relativistic kinetic energy:

ˆ ˆ2 2 ˆ2 2 ˆ H = ~p1 + m + ~p2 + m + V . (2.32) q q After the c.m. motion is separated it should take on the form:

ˆ Hˆ = P~ 2 + Mˆ 2 . (2.33) q CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 13

However, it can be shown that even for free particles (Vˆ = 0) there does not exist any c.m. and relative momenta (as functions of the individual ones) which would achieve this transformation. On the other hand, R.A. Krajcik and L.L. Foldy [11] derived an explicit construction of the c.m. and relative variables (which bring all Poincare generators into their one- particle form) for any fixed number N of particles with arbitrary (non-zero) masses and arbitrary spins up to any finite order in 1/m. The total momentum is still the sum of the individual momenta, but all other c.m. and relative variables are complicated functions of individual coordinates, momenta and spins. The explicit construction of these variables is rather tedious complicated procedure, but the results can be re-phrased in a following way (see also a paper by J.L. Friar [12]). Assume that we have a set of operators expressed in terms of individual coordinates, momenta and spins (it can be Hamiltonian, other Poincare generators, but also e.g. electromagnetic currents or other transition operators, derived typically from the field theory). Let as denote such generic operator by AFW (ri,pi,si). The subscript FW refers to Foldy-Wouthuysen and indicates that these operators already act in the space with a fixed number of particles (i.e., they are diagonalized in particle-antiparticle space). The corresponding operator in terms of the c.m. and relative coordinates is obtained by substituting ri,pi by the non-relativistic c.m. and relative variables (e.g. for two equal-mass particles with the help of (2.24,2.25)) and then by applying the unitary transformation (up to the order in 1/m considered) with the operator Uˆχ:

Aˆ = Uˆ Aˆ (r ,p ,s non-rel c.m.) Uˆ † , (2.34) KF χ FW i i i → χ † ˆ (2) (4) Uˆχ = exp(iχˆ) , χˆ =χ ˆ , P,~ χˆ = 0 , χˆ =χ ˆ +χ ˆ + ... , (2.35) h i where the operators Uˆχ and AˆKF are already expressed in terms of the c.m. and relative coordinates. Accepting this result, all we need is the operator χˆ. The procedure will become more transparent when we illustrated it below on our example of the Hamiltonian for a nucleon-nucleon system. Considering now a two-nucleon system and taking (2.33) up to the order 1/m3 we ﬁnd:

~ˆ2 ~ˆ4 ˆ2 2 P P HˆKF = P~ + Mˆ Mˆ + (2.36) ˆ ˆ 3 q ≃ 2M − 8M ~ˆ2 ˆ(1) ~ˆ4 ˆ P h P M + 1 3 (2.37) ≃ 2M − M ! − 8M ˆ ˆ ˆ ~pˆ2 ~pˆ4 ~pˆ2P~ 2 P~ 2 P~ 4 = 2m + + m − 4m3 − 8m3 4m − 64m3 ˆ P~ 2vˆ(1) +ˆv(1) +v ˆ(3) , (2.38) − 8m2 where in the last two lines we separated the kinetic and interaction-dependent parts of CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 14 the Hamiltonian up to the order considered, making use of the invariant mass operator:

ˆ2 2 ˆ ˆ Mˆ = 2 ~p + m +v ˆ(~r, ~p,~s1,~s2) (2.39) q ~pˆ2 ~pˆ4 2m + +v ˆ(1) +v ˆ(3) = 2m + hˆ(1) + hˆ(3) , (2.40) ≃ m − 4m3 and (2.31) was used for 1/Mˆ , while 1/Mˆ 3 1/M 3. The Hamiltonian expressed in terms of the c.m. and relative coordinates has to≃ be exactly of the form (2.38) to comply with the (approximate) Lorentz invariance, in particular, the dependence of the interaction on the total momentum has to be of the form of the last term in (2.38) (recall, thatv ˆ does not depend on the total momentum). We will now verify this both for the kinetic energy and for the OSE interaction derived before. That is, we will show that

ˆ ˆ (1) ˆ (3) ˆ (1) ˆ (3) (2) ˆ (1) HKF = 2m + HKF + HKF = 2m + HFW + HFW + iχˆ , HFW , (2.41) h i taking into account that the functionχ ˆ for this case does not depend on the interaction and is (at the required order in 1/m) given by [11, 12]:

ˆ 1 ˆ 1 ˆ ˆ ˆ ˆ χˆ(2)(~r,ˆ ~p,ˆ P~ )= (~σ ~σ ) ~pˆ P~ (~rˆ P~ )(~pˆ P~ )+(~pˆ P~ )(~rˆ P~ ) . (2.42) 8m2 1 − 2 × · − 16m2 · · · · h i Starting with the kinetic part of the Hamiltonian, we can write:

~pˆ2 + ~pˆ2 ~pˆ4 + ~pˆ4 Tˆ = 2m + Tˆ(1) + Tˆ(3) = 2m + 1 2 1 2 (2.43) FW FW FW 2m − 8m3 ˆ ˆ ˆ ˆ ~pˆ2 P~ 2 ~pˆ4 P~ 4 ~pˆ2P~ 2 (~pˆ P~ )2 = 2m + + · , (2.44) m 4m − 4m3 − 64m3 − 8m3 − 4m3 where in the ﬁrst line TˆFW is expressed in terms of the individual momenta and in the second one in terms of the non-relativistic total and relative momenta (2.25). Clearly, the second line diﬀers by the last term from the kinetic part of (2.38), but it is easy to verify (see Appendix B) that – as it follows from the kinetic part of (2.41)– it holds

ˆ ˆ(1) ˆ(3) ˆ(1) ˆ(3) (2) ˆ(1) TKF = 2m + TKF + TKF = 2m + TFW + TFW + iχˆ , TFW , (2.45) h i since ˆ (~pˆ P~ )2 iχˆ(2), Tˆ(1) = · (2.46) FW 4m3 and it cancels the last term of (2.44).h Thei interaction-dependent part of (2.41) reads (taking into account (2.38):

(1) ˆ (1) ˆ (1) vˆ = VKF = VFW , (2.47) ˆ P~ 2vˆ(1) vˆ(3) = Vˆ (3) + iχˆ(2), vˆ(1) . (2.48) − 8m2 FW CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 15

The second line is called the Foldy constraint for the relativistic potential, it restricts the ~ ˆ (3) (3) P dependence of VFW and defines the frame-independentv ˆ , which together with the non-relativisticv ˆ(1) determines the interaction in the invariant mass operator. For a par- ˆ (1) ticular potential derived from the field theory we have to show that VFW depends only on ˆ (3) relative coordinates and the leading order relativistic correction VFW satisfies the Foldy constraint. It is a bit more complicated than for the kinetic part of the Hamiltonian, in particular, since we do not derive the potential VFW directly in the operator form. Recall that the extended S-matrix technique allows us to identify only the momentum-space representation of the potential (e.g., its matrix elements – for our OSE given by (2.20,2.21)). Besides, these momentum-space matrix elements are written (e.g. (2.20,2.21)) in terms of the individual momenta, to check the Foldy constraint it is first convenient to re-write them in terms of the total and relative momenta. Recall, that the Lorentz (or Galilei) invariance requires that the potential commutes with the total momentum (see Appendix A; the same also follows from the S-matrix definition (2.5)). Hence we can always (at any order in 1/m) write:

< ~p ′, ~p ′ Vˆ ~p , ~p >=< P~ ′ < ~p ′ Vˆ ~p> P~ >= δ(P~ ′ P~ ) V (~p ′, ~p, P~ ) . (2.49) 1 2| | 1 2 | | | | − The δ-function for the total momenta will be suppressed from now on (but we should keep ˆ in mind that in any kinematical relation P~ ′ = P~ and in all operators P~ P~ ), also the → corresponding plane waves P~ > will be skipped for the sake of briefness. Moreover, it is convenient to replace the relative| momenta in the initial and ﬁnal state by their diﬀerence and sum: ~q = ~p ′ ~p, Q~ = ~p ′ + ~p, (2.50) − so we will replace V (~p ′, ~p, P~ ) V (~q, Q,~ P~ ) . (2.51) → Notice, that for momenta used in (2.20,2.21) it holds

Q~ = Q~ + P~ , Q~ = Q~ + P~ Q~ + Q~ = 2P~ , Q~ Q~ = 2Q~ . (2.52) 1 2 − → 1 2 1 − 2 Using these notations we can re-write (2.20,2.21):

g2 1 V (1) (~q 2) = V (1) (~q 2)= v(1)(~q 2)= s , (2.53) FW KF −(2π)3 µ2 + ~q 2 (1) 2 ~ 2 (3) ~ ~ v (~q ) ~ 2 i ~ (1 2ν)(~q Q) VFW (~q, Q, P ) = 2 Q (~σ1 + ~σ2) ~q Q + − 2 2· 4m "− − 2 · × µ + ~q ~ 2 ~ 2 i ~ (~q P ) P (~σ1 ~σ2) ~q P + 2 · 2 (2.54) − − 2 − · × µ + ~q # (1) 2 ~ 2 (3) ~ v (~q ) ~ 2 i ~ (~q P ) = v (~q, Q,~s1,~s2)+ 2 P (~σ1 ~σ2) ~q P + 2 · 2 . (2.55) 4m "− − 2 − · × µ + ~q #

′ ˆ (1) ~ The fact that < ~p VFW ~p> does not depend on P allows us to identify it consistently with the non-relativistic| | contribution to the intrinsic potential v (it just happens for the CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 16

OSE that this potential depends only on ~q 2, in principle it can also depend on spins – as it indeed does for the one-pion-exchange potential– and/or on Q~ ). For the same reason we can identify the ﬁrst line of (2.54) with the leading order relativistic contribution (3) v (~q, Q,~s~ 1,~s2) to the intrinsic potential v (for the sake of briefness, we will suppress the dependence on spins in what follows). The P~ -dependent part of (2.55) has to be consistent with the Foldy condition (2.48). Taking the momentum space matrix element of (2.48) we can rearrange it into:

P~ 2v(1)(~q 2) < ~p ′ iχˆ(2), vˆ(1) ~p>= V (3) (~q, Q,~ P~ ) v(3)(~q, Q~ ) (2.56) − 8m2 − | | FW − (1) 2 ~ 2 v (~q ) ~ 2 i ~ (~q P ) = 2 P (~σ1 ~σ2) ~q P + 2 · 2 , (2.57) 4m "− − 2 − · × µ + ~q # where the first line holds in general and the second one was obtained with the help of (3) (2.55) just for the OSE potential. Let us point out once more that just identifying VFW with the relativistic potential in terms of the total and relative momenta would not work: the P~ -dependence of (2.55) is clearly not the same as in (2.38) and the commutator term in (2.56) is essential. In the Appendix B we show explicitly that eqs. (2.56,2.57) are satisfied with the operatorχ ˆ given by (2.42) and the OSE with the leading-order relativistic effects and with the separated c.m. motion is therefore consistent with the requirement of Lorentz invariance. On the other, if one is interested only in the two-nucleon problem and needs to find the intrinsic wave functions, it is enough to replace in eqs. (2.21,2.20) the momenta Q~ Q~ 1 → and Q~ Q~ to get 2 → − v(~q, Q~ ) = < ~p ′ vˆ(1) +v ˆ(3) ~p> | | 2 ~ 2 ~ 2 gs 1 Q i ~ (1 2ν)(~q Q) = 3 2 2 1 2 2 (~σ1 + ~σ2) ~q Q + −2 2 · 2 . (2.58) −(2π) µ + ~q " − 4m − 8m · × 4m (µ + ~q ) # Comparison of this potential with the OSE of the Bonn OBEP-R is given in next subsec- tion, the interpretation of its particular terms and the transformation into the coordinate representation are discussed in the next chapter. Notice that the dependence on the “off- shell” parameter ν enters only into this frame-independent part of the potential, the c.m. motion is completely ν-independent. Below, we will put ν = 1/2, so that the potential v is instantaneous (does not contain the retardation effects). This choice is common for most semi-phenomenological NN potentials. One should keep in mind, that this does not completely eliminate ν-dependence from all operators: in particular those which carry non-zero momentum transfer (e.g., em currents) still have to contain contribution due to a meson retardation even for this choice ν = 1/2 (see, e.g. [10] and references therein). Let us at the end of this section emphasize the following point: the short discussion given here should just remind how is the c.m. motion properly separated when the relativistic effects are included perturbatively. It shows how to construct a consistent two particle states in an arbitrary reference frame. It is clear that for systems of more than two particles the situation is much more complicated, since the pair-wise interaction cannot be taken for the zero pair momentum for all interacting pairs (the discussion of CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 17 perturbative relativistic effects for few nucleon systems is given e.g. in [17], where their importance was also assessed in numerical calculations). Finally, let us point out that the construction of this section is not supposed to be the proof of the approximate Poincare invariance of the relativistic Hamiltonian for the OSE. We started from the explicitly Poincare-invariant S-matrix element, so the question of the proper transformation properties of the potential arises at the following points:

does the deﬁnition of the amplitude in (2.5) contain a correct normalization • factors? (Here, it is technically achievedV by the non-relativistic normalization of nucleon spinors).

does the oﬀ-energy shell continuation violate the transformation properties of the • potential?

Both these questions can be answered (yes and no) already before the c.m. motion is separated: one can check the relevant commutator relations of the Poincare algebra already for the operators written in terms of the individual coordinates. This is in fact useful, since it determines the interaction-dependent part of the Lorentz boost operator K~ . This step is skipped in these notes, we just show that we can keepχ ˆ(2) to be purely kinematical to consistently separate the c.m. motion.

2.3.1 Comparison to the Bonn potential Our intrinsic OSE potential (2.58) for ν = 1/2 reads:

2 ~ 2 ~ gs 1 Q i ~ v(~q, Q)= 3 2 2 1 2 2 (~σ1 + ~σ2) ~q Q . (2.59) −(2π) µ + ~q " − 4m − 8m · × #

The OSE component of the Bonn OBEP-R (Appendix A of [1]) is given in a slightly diﬀerent form (using our notation):

2 ~ 2 2 ~ gs 1 Q ~q i ~ vBonn(~q, Q)= 3 2 2 1 2 + 2 2 (~σ1 + ~σ2) ~q Q . (2.60) −(2π) µ + ~q " − 8m 8m − 8m · × #

How to explain this difference of the potentials, following obviously from the same physical mechanism? The point is that all Bonn OBE potentials are derived from different dynamical equation (the Blankenbecler-Sugar equation) which is then with the help of the so-called “minimal-relativity” re-definition transformed into the form of the Lippmann-Schwinger equation with the non-relativistic kinetic energy (this works only for the case of two particles!). Hence, to be able to compare, we have to transform our equation into the form with the same kinetic operator. This can be done by a trick due to Coester. One starts with the two particle equation written in the c.m. frame (here for the scattering state):

2 ~pˆ2 + m2 +v ˆ ǫ>= ~p 2 + m2 ǫ > . (2.61) | | q p CHAPTER 2. ONE SCALAR PARTICLE EXCHANGE - OSE 18

Applying once more the Hamiltonian this equation can be squared:

4(~pˆ2 + m2) + 2 ~pˆ2 + m2, vˆ +v ˆ2 = 4(~p 2 + m2) ǫ > . (2.62) | q Decomposing now the square root in the anticommutator into the Taylor series and dividing by 4m one gets

ˆ2 2 2 ~p 1 ˆ2 vˆ ~p +v ˆ + 2 ~p , vˆ + = ǫ > , (2.63) " m 4m 4m# m | n o which is a Schrödinger equation with the non-relativistic kinetic operator (and a non- relativistic relation between the momentum and energy in the eigen-value) with an effective potential 1 vˆ2 vˆ =v ˆ + ~pˆ2, vˆ + . (2.64) eff 4m2 4m n o If one is allowed to neglect the last term ( g4) – which might not be a good approximation ∼ – and if one identifiesv ˆ with our result (2.59) andv êff with the Bonn one (2.60), it should hold in the matrix element up to the order 1/m3:

~p ′2 + ~p 2 Q~ 2 + ~q 2 v (~q, Q~ )= 1+ v(~q, Q~ )= 1+ v(~q, Q~ ) , (2.65) Bonn 4m2 8m2 ! which can be easily veriﬁed from the explicit form of (2.59,2.60). Once more, it should be clear that this correspondence holds only for two nucleons. Since the Bonn potentials are derived in the Blankenbecler-Sugar framework, it is not an easy task to use them (consistently) in more complex systems. Chapter 3

The OSE potential, x-representation

In the previous parts of our text we derived the OSE potential in the momentum representation. This potential can be inserted into the Lippmann-Schwinger equation for the T -matrix, although some care is needed to ensure the convergence of the integral over the intermediate momentum. That is, due to the presence of several terms which involve powers of momenta, the integration has to be regularized: either by a momentum cut-off or by introduction of form factors, modeling a finite size of the σNN vertices. For many practical reasons it is also worth to evaluate this potential in the coordinate representation. In this chapter we demonstrate such a transformation for the OSE potential in the c.m. frame, i.e., for the intrinsic potential to be used in the two-nucleon Schrödinger equation. Let us take, for example, the potential in the form (2.60) – corresponding to the OSE part of Bonn OBEP-R potential. The operators are transformed from the momentum to the coordinate representation in a standard way via the Fourier transformation, following from the completeness relation:

d3p d3p ~p ~p >< ~p ~p = 1ˆ , 1 2 | 1 2 1 2| Z with the help of the matrix elements ei~p·~x < ~x ~p>= . | (2π)3/2 Explicitly,

< ~x ′~x ′ vˆ ~x ~x > = d3p′ d3p′ d3p d3p < ~x ′~x ′ ~p ′ ~p ′ >< ~p ′ ~p ′ vˆ ~p ~p >< ~p ~p ~x ~x > 1 2| | 1 2 1 2 1 2 1 2| 1 2 1 2| | 1 2 1 2| 1 2 Z 3 ′ 3 ′ 3 3 d p d p d p d p ′ ′ ′ ′ = 1 2 1 2 ei(~p1·~x1+~p2·~x2−~p1·~x1−~p2·~x2) (2π)3/2 (2π)3/2 (2π)3/2 (2π)3/2 Z < ~p ′ ~p ′ vˆ ~p ~p > . (3.1) 1 2| | 1 2 Since we know that the potential always commutes with the total momentum (see Appen- ˆ dix A) and the intrinsic potential does not depend on P~ , it is convenient to use instead of the individual particle coordinates and momenta their linear combinations deﬁned by eqs. (2.24,2.25). In terms of these variables we can with the help of: ~p ~x + ~p ~x = ~p ~r + P~ R~ , (3.2) 1 · 1 2 · 2 · · 19 CHAPTER 3. THE OSE POTENTIAL, X-REPRESENTATION 20 write: ′ ′ ~ ′ ′ ~ < ~x1~x2 vˆ ~x1~x2 > < R ~r vˆ ~rR>= | | 3 ′ → 3 3| ′| 3 d P d P d p d p ′ ′ ′ ′ = ei(~p ·~r −~p·~r+P~ ·R~ −P~ ·R~) < P~ ′~p ′ vˆ ~pP~ > . (3.3) (2π)3/2 (2π)3/2 (2π)3/2 (2π)3/2 | | Z Let us use for the momentum matrix element of the potential slightly diﬀerent notation from that of the previous chapter (for the reasons which are explained below):

< P~ ′~p ′ vˆ ~p P~ >= δ(P~ ′ P~ ) v(~p ′, ~p ) . (3.4) | | − Then we can integrate over P~ ′ with the help of the δ-function and consequently over P~ (using the fact that v(~p ′, ~p ) does not depend on P~ ):

3 ′ 3 d p d p ′ ′ < R~ ′~r ′ vˆ ~rR>~ = δ(R~ ′ R~ ) ei(~p ·~r −~p·~r) v(~p ′, ~p ) . (3.5) | | − (2π)3/2 (2π)3/2 Z Obviously, the potential is always local in the coordinate R~. Now it appears natural to use instead of ~p ′ and ~p their linear combinations ~q = ~p ′ ~p and Q~ = ~p ′ + ~p to re-arrange: − 1 1 1 1 ~p ′ ~r ′ ~p ′ ~r = (~p ′ ~p ) (~r ′ +~r )+ (~p ′ +~p ) (~r ′ ~r )= ~q (~r ′ +~r )+ Q~ (~r ′ ~r ) . (3.6) · − · 2 − · 2 · − 2 · 2 · − Hence, skipping the c.m. coordinate R~ and the corresponding δ-function and introducing for convenience K~ = Q/~ 2, we can write:

1 ~ ′ i ′ < ~r ′ vˆ ~r>= d3K eiK·(~r −~r) d3q e 2 ~q·(~r +~r) v(~p ′, ~p ) . (3.7) | | (2π)3 Z Z For the potentials which depend only on ~q this reduces to a simple formula:

< ~r ′ vˆ ~r>= δ(~r ′ ~r ) v(~r )= δ(~r ′ ~r ) d3q ei~q·~r v(~q ) , (3.8) | | − − Z deﬁning in the coordinate space the potential local in ~r and hence easy to use in an usual coordinate-space Schr¨odinger equation. If the momentum-space intrinsic potential depends on Q~ = 2K~ it is, in general, nonlocal in the relative coordinate. However, for the particular form with only a polynomial dependence on Q~ it can be still transformed to some usable form. Here, we restrict ourselves to the potential (2.60):

2 ~ 2 2 ′ gs 1 Q ~q i ~ vBonn(~p , ~p )= 3 2 2 1 2 + 2 2 (~σ1 + ~σ2) ~q Q −(2π) µ + ~q " − 8m 8m − 8m · × # ~q 2 ~p ′2 + ~p 2 i = v(1)(~q 2 ) 1+ S~ ~q ~p , (3.9) 4m2 − 4m2 − 2m2 · × ~ 1 where the second line is obtained by introducing the two-nucleon spin S = 2 (~σ1 + ~σ2), using ~q Q~ = 2~q ~p, and substituting Q~ 2 = ~q 2 + 2(~p ′2 + ~p 2). We will argue below that this particular× form× is the most suitable for the− transformation into the coordinate space. CHAPTER 3. THE OSE POTENTIAL, X-REPRESENTATION 21

Let us also note that: although we do not consider here the retardation part of the potential (which van- • ishes in the c.m. frame for ν = 1/2), it can be also transformed into the local potential in the coordinate space (after lengthy and tedious algebra),

also the OBE potential for the vector mesons is treatable by exactly the same tricks • as the scalar-exchange one considered here. Further, as mentioned above, to ensure the proper behavior of the potential near the origin (it should not be more singular than 1/r2), the OBE potentials are usually modiﬁed by the meson-nucleon form factors. That is, each vertex is multiplied by a function of the square of the meson momentum f(~q 2), such that f(~q 2) 0 for large ~q 2 and should approach 1 for small ~q 2. In the potential one replaces → 1 f 2(~q 2) . (3.10) µ2 + ~q 2 → µ2 + ~q 2 This modiﬁes the form of v(r) for small r, but leaves its asymptotic behavior for r unchanged. The functional dependence of f(~q 2) is usually chosen such that the transfor-→ ∞ mation of the potential to the coordinate representation does not become more complicated. The simplest typical example of the form factor is:

2 2 Λ fsq(~q )= , (3.11) sΛ2 + ~q 2 where Λ is the cut-oﬀ parameter, assumed to be much larger than the mass of the exchanged meson. For this form factor eq. (3.10) becomes

1 1 Λ2 Λ2 1 1 = , (3.12) µ2 + ~q 2 → (µ2 + ~q 2) (Λ2 + ~q 2) Λ2 µ2 µ2 + ~q 2 − Λ2 + ~q 2 − so we can just transform our “bare” potential (without the form factors) into the coordinate space (without caring much about convergence of the integrals), and then subtract exactly the same result with µ Λ and multiply by a constant. Although this way of introducing→ the form factors serves its purpose, it might be bit too detached from the relativistic amplitude we started from and from the way the meson-nucleon coupling constants are actually determined – ideally from experiments with physical mesons for which q2 = µ2 = ~q 2. Therefore, in some potentials (in particular, in the Bonn one) the invariant form factors6 dependent on 4-vector squared are introduced, with a normalization f(q2) = 1. In our definition of the potential in the c.m. frame (in which q0 = 0), the only difference would be the normalization of the form factor, since now f(~q 2 ) = 1 (just keep in mind that also the discussion of the previous chapter has 6 2 to be slightly modified due to the dependence of f(q ) on q0). The Bonn group employs instead of (3.11) the following functional form:

Λ2 µ2 n f(q2)= − , (3.13) Λ2 q2 − CHAPTER 3. THE OSE POTENTIAL, X-REPRESENTATION 22 where (for various versions of the Bonn model and for diﬀerent mesons) n can be either 1/2 or some integer number (usually n = 1 or 2). In the c.m. frame, where we put q2 ~q 2, it holds (using the notation z = ~q 2): → − Λ2 µ2 1 1 1 − = , (3.14) Λ2 + z µ2 + z µ2 + z − Λ2 + z Λ2 µ2 2 1 1 1 d 1 − = + (Λ2 µ2) . (3.15) Λ2 + z µ2 + z µ2 + z − Λ2 + z − dΛ2 Λ2 + z These equations (and analogous ones for higher integer n) can be obtained from the identity:

Λ2 µ2 (m+1) 1 Λ2 µ2 m 1 (Λ2 µ2)m − = − − , (3.16) Λ2 + z µ2 + z Λ2 + z µ2 + z − (Λ2 + z)(m+1) for m = 0, 1,... . For what follows, it is still important that the regularized potential is obtained simply from the “bare” one (from which µ Λ term, and – in some cases – its derivative(s) in respect to Λ2 are subtracted). → Now we consider explicitly term after term of the potential (3.9) and derive their coordinate space representations.

3.1 Central ~q- dependent part

Let us remind the well-known Fourier transform for the leading order central potential g2 1 v(1)(~q 2 )= s . (3.17) −(2π)3 µ2 + ~q 2 For (3.8) we calculate the following integral (using for d3q a representation in terms of the spherical coordinates in a frame with the z-axis along ~r ):

2π ∞ 1 1 eiqr cos θ d3q ei~q·~r = dφ dqq2 d(cos θ) µ2 + ~q 2 µ2 + q2 Z Z0 Z0 −Z1 ∞ 1 ∞ q2 2π q (eiqr e−iqr) = 2π dq dz eiqrz = dq − µ2 + q2 ir µ2 + q2 Z0 −Z1 Z0 ∞ π dq q eiqr e−µr = = 2π2 = 2π2µY (µr) , (3.18) ir q iµ q + iµ r 0 −∞Z − where the last integral was calculated from the residuum at q = iµ and we have introduced −x (1) a function Y0(x) = e /x. This gives for the v in the coordinate space: g2 e−µr g2 v(1)(r)= s = s µY (µr) . (3.19) −4π r −4π 0 CHAPTER 3. THE OSE POTENTIAL, X-REPRESENTATION 23

Let us note that: the potential (3.19) is a local, central, spin-independent one. Its radial dependence • is that of the Yukawa potential: it behaves as 1/r near origin and decreases as e−µr for large r; it is proportional to a typical factor g2/(4π);

if Λ >µ, the form factors (3.11) or (3.13) make the potential ﬁnite for r = 0, while • its asymptotic for r remains the same; → ∞ the potential is attractive (unlike the potential due to the vector exchange between • two particles with the same “charge”, i.e., the sign of the coupling constant to the meson). This feature does not depend on the spin of the constituents, for the scalar exchange between two scalar heavy particles we would also get an attractive central Yukawa potential;

the operator vˆ(1) depends only on the operatorr ˆ (operator of the absolute value of • the relative coordinate) and it is given by (3.19) with r rˆ. → Next, let us take from (3.9) the relativistic term proportional to ~q 2. The Fourier transform of ~q 2/(µ2 + ~q 2) can be done in two equivalent ways. We can either write

~q 2 1 d3q ei~q·~r = d3q ei~q·~r µ2 + ~q 2 −△ µ2 + ~q 2 Z Z e−µr e−µr = 2π2 = 2π2 µ2 + (2π)3 δ(~r ) . (3.20) − △ r − r In the last step we used: 1 e−µr ( µ2) = δ(~r ) , (3.21) △ − −4π r i.e., 1/(4π)e−µr/r is the Green function of µ2 (see, e.g., textbook [13], Section 19.4, − △ − with k iµ). Alternatively,→ we can re-write still in the momentum representation: ~q 2 µ2 = 1 , (3.22) µ2 + ~q 2 − µ2 + ~q 2 from which we again get: ~q 2 e−µr d3q ei~q·~r = (2π)3 δ(~r ) 2π2 µ2 . (3.23) µ2 + ~q 2 − r Z Thus, for the central ~q-dependent part of (3.9) one gets:

g2 1 ~q 2 µ2 g2 s 1+ 1 v(1)(r) s δ(~r ) . (3.24) −(2π)3 µ2 + ~q 2 4m2 → − 4m2 − 4m2 Notice that the last term does not depend on µ, it therefore cancels when the potential is regularized according to (3.12,3.14) or (3.15). CHAPTER 3. THE OSE POTENTIAL, X-REPRESENTATION 24

3.2 Spin-orbit part

Now we consider the term: v(1)(~q 2 ) iS~ ~q ~p = w(~q 2 ) iS~ ~q ~p. (3.25) − 2m2 · × · × Substituting into (3.5,3.7):

1 ′ ′ d3p′d3pw(~q 2 ) iS~ ~q i~ (r)ei(~p ·~r −~p·~r) (2π)3 · × ∇ Z 1 ′ ′ = ǫ S (r) d3p′d3p q w(~q 2 ) ei(~p ·~r −~p·~r) −(2π)3 ijk i∇ k j Z 1 ~ ′ i ′ = ǫ S (r) d3KeiK·(~r −~r) d3q q w(~q 2 ) e 2 ~q·(~r +~r) −(2π)3 ijk i∇ k j Z Z = ǫ S (r) δ(~r ′ ~r ) i (r) d3q w(~q 2 ) ei~q·~r − ijk i∇ k − − ∇ j Z w′(r) = iǫ S (r) δ(~r ′ ~r ) [ (r) w(r)] = iǫ S (r) δ(~r ′ ~r ) r ijk i∇ k − ∇ j ijk i∇ k − j r w′(r) w′(r) = ǫ S r [i (r) δ(~r ′ ~r )] = S~ ~r i~ (r) δ(~r ′ ~r ) . (3.26) r ijk i j ∇ k − r · × ∇ − h i Taking into account that:

< ~r ′ S~ L~ ~r> = ǫ S r < ~r ′ p ~r> | · | ijk i j | k| 1 ′ = ǫ S r d3p p ei~p·(~r −~r) = S~ ~r i~ (r) δ(~r ′ ~r ) , (3.27) ijk i j (2π)3 k · × ∇ − Z h i we get for the operator form of (3.25):

w′(r) 1 dv(1)(r) L~ S~ = L~ S~ r · −2m2r dr · 1 µ g2 dY (x) g2 µ3 Y (x) = s µ µ 0 L~ S~ = s 1 L~ S~ , (3.28) −2m2 x −4π dx · −4π 2m2 x · −x where x = µr and for the derivative of Y0(x) = e /x it holds:

dY (x) 1 0 = 1+ Y (x) Y (x) . (3.29) dx − x 0 ≡ − 1 Notice, that for x 0 this potential 1/x3, therefore it has to be regularized, as described earlier in this→ chapter. ∼ CHAPTER 3. THE OSE POTENTIAL, X-REPRESENTATION 25

3.3 Central anticommutator part

The remaining term of (3.9) does not depend only on ~q and hence the formula (3.8) cannot be applied directly. One can however factorize any polynomial dependence on ~p or ~p ′ of the momentum matrix element of the potential by considering these factors to come from the explicit action of the operator ~pˆ multiplying the rest of the operator. In particular, one can write: Qnf(~q ) p,ˆ p,ˆ ... p,ˆ f (~r) ... , (3.30) → { { { F } }} where the function fF (~r ) is the Fourier image of the function f(~q )

i~q·~r fF (~r )= f(~q ) e d~q. (3.31) Z The multiple anticommutators with ~pˆ are not easy to handle in the usual coordinate space Schr¨odinger equation. Therefore, it is more convenient to use the second line of (3.9) and represent its third term as : ~p ′2 + ~p 2 1 v(1)(~q 2 ) ~pˆ2,v(1)(r) , (3.32) − 4m2 → −4m2 n o where v(1)(r) is given by (3.19). The anticommutator of ~pˆ2 with the central potential v(1)(r) can be treated as described in Appendix C.

3.4 The OSE potential in the Schr¨odinger equation

Summing up the results of the previous sections, we get for the potential (3.9):

µ2 1 vˆ (~r,ˆ ~pˆ) = 1 v(1)(r) ~pˆ2,v(1)(r) + v(3)(r) L~ S~ = (3.33) Bonn − 4m2 − 4m2 LS · g2 µ2 1 n o µ2 1 1 = s µ 1 Y (x) ~pˆ2,Y (x) + + Y (x) L~ S~ , −4π − 4m2 0 − 4m2 0 2m2 x x2 0 · g2 e−µr g2 n o v(1)(r) = s = s µY (x) , (3.34) −4π r −4π 0 g2 µ 1 1 g2 µ3 1 1 v(3)(r) = s 1+ e−µr = s + Y (x) , (3.35) LS −4π 2m2 r2 µr −4π 2m2 x x2 0 e−x x = µr , Y (x)= , (3.36) 0 x where we skipped the δ-function term from (3.24), since it is cancelled by the regulariza- tion. The second line of (3.33) coincides exactly with eq. (A.21) of [1]. After getting rid of the anticommutator with ~pˆ2 as described in the Appendix C, it is very easy to use this potential in the radial form of the Schr¨odinger equation: all terms are diagonal in the usual J(LS)M > basis, one would only need |

′ ′ ′ ′ 1 = δ ′ δ ′ δ ′ δ ′ [J(J + 1) L(L + 1) S(S + 1)] . | · | J J L L S S M M 2 − − CHAPTER 3. THE OSE POTENTIAL, X-REPRESENTATION 26

Of course, our one-scalar-meson-exchange potential cannot realistically describe the physics of the NN interaction, it has to be supplemented by parts following from exchanges of pseudoscalar and vector mesons: pions, ρ- and ω mesons, as well as of some heavier less important ones. − Chapter 4

Summary

In the previous chapters we got the first taste of a complicated journey from the field theory to the quantum mechanical potential which can be used in a familiar coordinate- space Schrödinger equation. We followed how the simplest one-meson-exchange diagram determines the corresponding momentum-space matrix elements of the QM potential, keeping an eye in particular on a freedom in fixing of its off-energy-shell continuation. Already the simplest scalar exchange demonstrates some general features: (i) the lowest order non-relativistic potential is a simple central Yukawa function, independent on the spin of the constituents, (ii) to get more general structures in the potential (spin-orbit, tensor, etc) one has to take into account at least the leading order relativistic effects, (iii) this in general makes the potentials velocity and spin dependent, which complicates a separation of the center-of-mass motion and transformation of the potential into the coordinate space. At least for OBE potentials at the leading relativistic order all these complications can be overcome. Even in this brief notes we could reproduce the full coordinate space potential for the one-scalar-exchange and for the two nucleon system (i.e., in its c.m. frame). The same can be achieved for one vector exchange, the expressions become somewhat more complicated, but there are no new obstacles emerging on the way. Ironically, the same cannot be said about the most important one-pion-exchange potential: due to the pseudoscalar nature of the pion the πNN vertex function is more complicated even at the lowest order and although the OPE potential can be derived up to the same order as the scalar and vector ones its seldom used in that form in the OBE models. On the other hand, the modern NN potentials stemming from the chiral perturbation theory are inherently more complicated, since they systematically extend beyond the simplest one-meson-exchange picture. The higher order terms, of which many are of relativistic nature, are included order by order and the potentials should comply with general symmetry principles, employed in our text. However, we are unaware of systematic study of these aspects of the ChPT potentials, although they are now increasingly often applied also for systems of more than two nucleons. Such a study would be a worthwhile extension of this compilation.

27 Appendix A

Poincare and Galilean algebras

In this appendix we brieﬂy introduce the algebras of the generators of the Poincare and Galilean groups. In the section of the main text on the c.m. motion many results collected in this Appendix are used. In both groups the operations of the space and time transla- tions, rotations and Lorentz (Galilean) boosts are included. The corresponding generators are the Hamiltonian H, the momentum P~ , the angular momentum J~ and the “boost” K~ . The generators satisfy the commutator relations of the corresponding algebra:

[Pˆi, Pˆj] = [Pˆi, Hˆ ] = 0 , (A.1)

[Jˆi, Hˆ ] = 0 , [Jˆi, Gˆj]= iǫijkGˆk , G~ = P~ , J,~ K~ , (A.2)

[Kî, Hˆ ]= iPî , (A.3) [Kˆ , Pˆ ]= iδ Hˆ , [Kˆ , Kˆ ]= iǫ Jˆ , for the Poincare group , (A.4) i j ij i j − ijk k [Kî, Pˆj]= iδijM , [Kî, Kˆj] = 0 , for the Galilean group , (A.5) where the first three lines are common for both groups, only the commutators in the last two lines differ. M stands for the total mass of the system. For a single particle we denote the generators by small characters (and Hamiltonian is replaced by a kinetic energy tˆ), in terms of the momentum ~p, a conjugated coordinate ~r, a spin ~s, and a mass m they are given as [14]:

~ˆ = ~rˆ ~pˆ + ~s,ˆ (A.6) × ~pˆ2 tˆ= ~pˆ2 + m2 tˆ= m + , (A.7) → 2m q ˆ 1 ~sˆ ~pˆ ˆ ~k = (~rˆtˆ+ tˆ~rˆ) × t~pˆ ~k = m~rˆ t~p,ˆ (A.8) 2 − tˆ+ m − → − where the arrow indicates Poincare Galilean transition. For a system of N non- interacting particles the generators equal→ just to the sums of the corresponding single particle generators. The instantaneous interaction can be introduced into the Poincare generators in several ways [16]. Here we are interested only in the so-called instant form of dynamics, in which only Hamiltonian and the boost operators depend on the interaction, while the momentum

28 APPENDIX A. POINCARE AND GALILEAN ALGEBRAS 29 and the angular momentum stay purely kinematical. That is,

N N ˆ ˆ P~ = ~pˆ(i) , J~ = ~ˆ(i) , (A.9) i=1 i=1 XN X Hˆ = tˆ(i)+ Vˆ , (A.10) i=1 XN ˆ ˆ ˆ ˆ 1 ˆ K~ = ~k(i)+ W~ , W~ = R,~ Vˆ + ~w,ˆ (A.11) 2 i=1 X n o where V is the potential, W~ is an interaction-dependent part of the boost. It follows from the commutators of the Poincare (Galilean) algebra that

[Pî, Vˆ ] = [Jî, Vˆ ] = 0 , (A.12) [Wˆ , Pˆ ]= iδ Vˆ [Wˆ , Pˆ ] = 0 , (A.13) i j ij → i j and (1) [Rî, vˆ ] = 0 , (A.14) wherev ˆ(1) is the non-relativistic potential. For the Galilean group W~ can be set to zero, for the Poincare group it is convenient to split it up into the anticommutator of the c.m. coordinate and the potential and an unknown vector ~w. This function ~w has to be determined/chosen so that the Poincare commutators are preserved; it then enters [11, 12, 15] the solution for the operatorχ ˆ defined below. As mentioned in the main text, Krajcik and Foldy [11, 12] showed that the generators of the Poincare algebra can be brought to their single-particle form. In symbolical notation: Gˆ = exp(iχˆ)Gˆ exp( iχˆ) , (A.15) KF FW − where GˆFW are the generators above, but with the coordinates ~r(i) and momenta ~p(i) expressed in terms of the non-relativistic c.m. and relative coordinates and momenta and the hermitean operatorχ ˆ has to be explicitly constructed by solving the Poincare commutators order by order. For more detail see refs. [11, 12, 15], here we only need the lowest orderχ ˆ(2) for two equal-mass particles given by (2.42). Let us just point out that it holds in general [Pî, χˆ] = [Jî, χˆ] = 0 , (A.16) and remind once more that the interaction-dependent part of the boost ~w (which is zero for our OSE at the order considered) enters in general the solution forχ ˆ(2). Finally, let us list for completeness the generators in the single-particle form GˆKF : ˆ ˆ ˆ ˆ J~ = R~ P~ + S~ , (A.17) × ˆ ˆ P~ 2 Hˆ = P~ 2 + Mˆ 2 Hˆ = Mˆ + , (A.18) q → 2M ˆ ˆ ˆ 1 ˆ ˆ S~ P~ ˆ ˆ ˆ ˆ K~ = (R~ Hˆ + Hˆ R~) × tP~ K~ = MR~ tP~ , (A.19) 2 − Hˆ + Mˆ − → − APPENDIX A. POINCARE AND GALILEAN ALGEBRAS 30 where P~ is the total momentum, R~ is a conjugate coordinate, S~ is an internal angular momentum (the spin of the composite system), and Mˆ is the invariant mass operator. The following commutator relations has to be satisfied: ˆ ˆ ˆ [P,~ Mˆ ] = [R,~ Mˆ ] = [S,~ Mˆ ] = 0 , (A.20) ˆ [S,~ Hˆ ] = 0 , [Sî, Sˆj]= iǫijkSˆk . (A.21)

Notice, that the Hamiltonian depends on the interaction only through Mˆ , and the boost only through Mˆ and Hˆ . Using the relations (A.20,A.21) it is instructive to verify the Poincare commutators (A.1-A.4) for the generators (A.17-A.19). In fact, the relations (A.1) are satisﬁed trivially, the relations (A.2) just express the fact that H is a scalar and P~ , J,~ K~ are vectors and are also very easy to check. Further, let us write: ˆ ˆ ˆ 1 ˆ ˆ S~ P~ ˆ ˆ ˆ ˆ K~ = (R~ Hˆ + Hˆ R~) × tP~ K~ R + K~ S + K~ P . (A.22) 2 − Hˆ + Mˆ − ≡ ˆ Then, the commutator (A.3) follows from the fact that only R~ in K~ R does not commute with the Hamiltonian and

~ˆ ˆ ˆ ∂f(P ) Pi [Rî, f(P~ )] = i [Rî, Hˆ ]= i . (A.23) ∂Pî ⇒ Hˆ

Similarly, for the commutator of K~ with P~ (A.4) only the canonical commutator relation ˆ of R~ (from K~ R) and P~ is needed. Thus, the only commutator, which takes some algebra to be proven, is that of two components of the boost (A.4). With the help of (A.22) it can be written as:

[Kˆi, Kˆj] = [KˆR,i, KˆR,j]

+ [KˆR,i, KˆP,j] + [KˆP,i, KˆR,j]

+ [KˆR,i, KˆS,j] + [KˆS,i, KˆR,j] + [KˆS,i, KˆS,j] , (A.24) ˆ where all other commutators between K~ X are equal to zero. It is also easy to show that the two commutators in the second line of (A.24) cancel. Then, the only remaining S~-independent contribution is the ﬁrst term on the r.h.s. Let us show that ˆ ˆ [Kˆ , Kˆ ]= iǫ (R~ P~ ) . (A.25) R,i R,j − ijk × k Using the identity [A,ˆ BˆCˆ]= Bˆ[A,ˆ Cˆ] + [A,ˆ Bˆ]Cˆ , (A.26) and (A.23) one ﬁnds:

[Rˆ H,ˆ Rˆ Hˆ ] = [Hˆ Rˆ , Hˆ Rˆ ]= i Rˆ Pˆ Rˆ Pˆ , (A.27) i j i j − i j − j i APPENDIX A. POINCARE AND GALILEAN ALGEBRAS 31

PîPˆj [RîH,ˆ Hˆ Rˆj]= i RîPˆj RˆjPî + δij , (A.28) − − − Hˆ 2 from which (A.25) follows (using RiPj RjPi = ǫijk(R~ P~ )k). Finally, consider the last line of (A.24).− First, using× (A.23) one gets: ˆ ˆ iǫijkSˆk i Pî (S~ P~ )j [Rî, KˆS,j]= + × , (A.29) −Hˆ + Mˆ (Hˆ + Mˆ )2 Hˆ which commutes with Hˆ . Therefore, ˆ ˆ Hˆ i Pî (S~ P~ )j [KˆR,i, KˆS,j]= iǫijkSˆk + × , − Hˆ + Mˆ (Hˆ + Mˆ )2 ˆ ˆ Hˆ i Pˆj (S~ P~ )i [KˆR,j, KˆS,i]=+iǫijkSˆk + × , Hˆ + Mˆ (Hˆ + Mˆ )2 and the first two terms of the last line of (A.24) give: ˆ ˆ ˆ ˆ 2Hˆ Pî (S~ P~ )j Pˆj (S~ P~ )i [KˆR,i, KˆS,j] [KˆR,j, KˆS,i]= iǫijkSˆk + i × − × − − Hˆ + Mˆ (Hˆ + Mˆ )2 ˆ ˆ ˆ 2Hˆ (P~ (P~ S~))k = iǫijkSˆk iǫijk × × − Hˆ + Mˆ − (Hˆ + Mˆ )2 ˆ ˆ ˆ 2Hˆ P~ 2 P~ S~ = iǫijkSˆk iǫijkPˆk · − Hˆ + Mˆ − (Hˆ + Mˆ )2 ! − (Hˆ + Mˆ )2 ˆ ˆ P~ S~ = iǫijkSˆk iǫijkPˆk · , (A.30) − − (Hˆ + Mˆ )2 ˆ where the last line follows from P~ 2 = Hˆ 2 Mˆ 2. The last term is easily calculated: − PˆbPˆd [KˆS,i, KˆS,j]= ǫiab ǫjcd[Sâ, Sˆc] (Hˆ + Mˆ )2

PˆbPˆd = i ǫiab ǫjcd ǫacf Sˆf (Hˆ + Mˆ )2 ˆ ˆ P~ S~ = i ǫijk Pˆk · , (A.31) (Hˆ + Mˆ )2 and it obviously cancels the last term of (A.30). Therefore, the last line of (A.24) equals: [Kˆ , Kˆ ] [Kˆ , Kˆ ] + [Kˆ , Kˆ ]= iǫ Sˆ , (A.32) R,i S,j − R,j S,i S,i S,j − ijk k which together with (A.24) and (A.25) completes the proof of: ˆ ˆ ˆ [Kˆi, Kˆj]= iǫijk R~ P~ + S~ = iǫijkJˆk , (A.33) − × k − Appendix B

Commutators needed for the c.m. separation

In this appendix we show that the separation of the c.m. motion with the help of the hermitean function (2.42) really works, i.e., we prove explicitly eqs. (2.46,2.57). To this end, we split the operatorχ ˆ(2) into two parts:

(2) ˆ ˆ ~ˆ (2) (2) χˆ (~r, ~p, P ) =χ ˆσ +χ ˆr , (B.1) 1 ˆ χˆ(2) = (~σ ~σ ) ~pˆ P~ , (B.2) σ 8m2 1 − 2 × · 1 ˆ ˆ ˆ ˆ χˆ(2) = (~rˆ P~ )(~pˆ P~ )+(~pˆ P~ )(~rˆ P~ ) r −16m2 · · · · ˆ h ˆ i iP~ 2 1 ˆ ˆ iP~ 2 1 ˆ ˆ = (~pˆ P~ )(~rˆ P~ )=+ (~rˆ P~ )(~pˆ P~ ) , (B.3) −16m2 − 8m2 · · 16m2 − 8m2 · · where the last line follows from the canonical commutation relation

[ˆri, pˆj]= iδij . (B.4)

(2) Obviously,χ ˆσ commutes with the kinetic energy and, using ∂f(ˆp) [ˆri, f(ˆp)] = i , (B.5) ∂pˆi one gets for the commutator of the non-relativistic kinetic energy withχ ˆ(2):

ˆ2 ~ˆ 2 2 (2) ~p P i ˆ ~ˆ ˆ ˆ2 1 ˆ ~ˆ iχˆr , + = 3 (~p P )Pi rˆi, ~p = 3 ~p P , (B.6) " m 4m# −8m · 4m · h i which proves eq. (2.46). As for the interaction-dependent part, eq. (2.57) is satisﬁed if: v(1)(~q 2) < ~p ′ iχˆ(2), vˆ(1) ~p> = i (~σ ~σ ) ~q P~ , (B.7) | σ | 4m2 1 − 2 · × v(1)(~q 2)P~ 2 v(1)(~q 2)(~q P~ )2 < ~p ′ iχˆ(2), vˆ(1) ~p> = · , (B.8) | r | 8m2 − 4m2(µ2 + ~q 2) 32 APPENDIX B. COMMUTATORS NEEDED FOR THE C.M. SEPARATION 33

ˆ where – as in the main text – we skip the eigen-states of the total momentum and P~ P~ . → The ﬁrst of these relations easily follows:

< ~p ′ iχˆ(2), vˆ(1) ~p> | σ | i = (~σ ~σ ) ~p ′ P~ < ~p ′ vˆ(1) ~p> < ~p ′ vˆ(1) ~p> (~σ ~σ ) ~p P~ 8m2 1 − 2 · × | | − | | 1 − 2 · × v(1)(~qh2) i = i (~σ ~σ ) ~q P~ , 4m2 1 − 2 · × where the last line results from ~q = ~p ′ ~p and from the fact that v(1) commutes with spins. − To get the relation (B.8) we ﬁrst write

(2) (1) iχˆr , vˆ ˆ ˆ P~2 vˆ(1) i ˆ ˆ P~ 2 vˆ(1) i ˆ ˆ = (~pˆ P~ )(~rˆ P~ )ˆv(1) + + vˆ(1)(~rˆ P~ )(~pˆ P~ ) , 16m2 − 8m2 · · 16m2 8m2 · · which in the matrix element yields (using also [~r,ˆ vˆ(1)] = 0):

(1) 2 ′ (1) ∂ (1) ′ 2 ∂ (1) 2 ∂v (~q ) < ~p rˆi vˆ ~p>= i ′ v ((~p ~p ) )= i v (~q ) = 2iqi 2 . | | ∂pi − ∂qi ∂~q Finally, since const ∂v(1)(x) v(1)(x) v(1)(x)= = , µ2 + x → ∂x −µ2 + x one gets v(1)(~q 2) < ~p ′ rˆ vˆ(1) ~p>= 2iq , | i | − i µ2 + ~q 2 which together with (B.9) yield the commutator (B.8). Obviously, the relations (B.7,B.8) hold for any central Yukawa-like potential. Appendix C

Anticommutator term in the Schr¨odinger equation

Consider the Schr¨odinger equation of the form:

ˆ2 ~p ˆ2 + a ~p , V1(r) + V2(r) ψ(~x )= Eψ(~x ) , (C.1) "2µ # n o 2 where µ is the reduced mass, a is a constant with a dimension MeV and V1,2(r) are two diﬀerent radial functions. Unlike in the main text, we keep here for the sake of comparison with literature the factors ~, the dimension of ~ is MeV fm. Using ~pˆ2 = ~2 we can re-write (C.1) as: · − △

+( g(r)+ g(r) ) U (r)+ ǫ ψ(~x )= △ △ △ − 2 h i (1+2g(r)) + 2(~ g(r)) ~ +( g(r)) U (r)+ ǫ ψ(~x ) = 0 , (C.2) △ ∇ · ∇ △ − 2 h 2µ i 2µ g(r)= a 2µ V (r)= ~2 a U (r) , U (r)= V (r) , ǫ = E . (C.3) 1 1 i ~2 i ~2 where g(r) is a dimensionless radial function, and we have used

(gψ)=( g)ψ + 2(~ g) (~ ψ)+ g ψ . △ △ ∇ · ∇ △ To simplify (C.2), one introduces [18] a “modified wave function” φ(~x ): φ(~x ) ψ(~x )= . (C.4) 1 + 2g(r) Notice that since g(r) is a scalar factor dependentp only on the magnitude r of the vector ~x, this re-definition affects only the radial parts of the wave functions ψ and φ. For the derivatives of the wave function we then get: ~ φ(~x ) (~ g) φ(~x ) ~ ψ(~x ) = ∇ ∇ , (C.5) ∇ (1+2g)1/2 − (1+2g)3/2 φ(~x ) 2(~ g) (~ φ) ( g) φ(~x ) 3(~ g)2 φ(~x ) ψ(~x ) = △ ∇ · ∇ △ + ∇ . (C.6) △ (1+2g)1/2 − (1+2g)3/2 − (1+2g)3/2 (1+2g)5/2

34 APPENDIX C. ANTICOMMUTATOR TERM IN THE SCHRODINGER¨ EQUATION35

This does not look like a big improvement, but evaluating the factors appearing in (C.2):

2(~ g) (~ φ) ( g) φ 3(~ g)2 φ (1+2g) ψ = 1 + 2g φ ∇ · ∇ △ + ∇ , (C.7) △ △ − (1+2g)1/2 − (1+2g)1/2 (1+2g)3/2 p2(~ g)(~ φ) 2(~ g)2 φ 2(~ g)(~ ψ) = ∇ ∇ ∇ , (C.8) ∇ ∇ (1+2g)1/2 − (1+2g)3/2 ( g) φ ( g) ψ = △ , (C.9) △ (1+2g)1/2 it is easy to see that their sum simpliﬁes considerably and, after substituting it into (C.2) and dividing by √1 + 2g, the Schr¨odinger equation becomes:

2 U2 (~ g) ǫ + ∇ 2 + φ(~x ) = 0 . (C.10) "△ − 1 + 2g (1+2g) 1 + 2g #

Writing now ǫ 2ǫg = + ǫ , 1 + 2g −1 + 2g using ~x dg(r) ~ g(r)= g′(r) (~ g(r))2 =(g′(r))2 , g′(r) , ∇ r → ∇ ≡ dr and introducing an eﬀective energy-dependent potential:

U (r) g′(r) 2 2ǫg(r) U (r, ǫ)= 2 + , (C.11) eff 1 + 2g(r) − 1 + 2g(r) 1 + 2g(r) we can write (C.10) as [ U (r, ǫ)+ ǫ ] φ(~x ) = 0 . (C.12) △ − eff The last equation has a familiar form of the usual Schrödinger equation (multiplied by 2µ/~2) and can be solved by standard methods. One only has to bear in mind that the − effective potential Ueff depends on energy. Hence, the modified φ(~x ), defined by (C.4), has to satisfy properly modified normalization and orthogonality relations (in particular, the solutions φ(~x ) for different energies are not orthogonal, unlike the original ψ(~x )). Since g(r) is proportional to V1(r), the radial dependence of Ueff differs from U2(r) only below the range of V1(r). We can also write:

~2 V (r) ~2 g′(r) 2 2Eg(r) V (r, ǫ)= U (r, ǫ)= 2 + , (C.13) eff 2µ eff 1 + 2g(r) − 2µ 1 + 2g(r) 1 + 2g(r) which is identical to eq. (39) of [3] (deﬁning the old Nijmegen potential). Bibliography

[1] R. Machleidt, Advances in Nuclear Physics 19 (1988) 1.

[2] R. Machleidt, K. Holinde, C. Elster, Phys. Rept. 149 (1987) 1.

[3] M.M. Nagels, T.A. Rijken and J.J. de Swart, Phys. Rev. D17 (1978) 768.

[4] V.G.J. Stoks, R.A.M. Klomps, C.P.F. Terhegen and J.J. de Swart, Phys. Rev. C49 (1994) 2950.

[5] R.B. Wiringa, V.G.J. Stoks and R. Schiavilla, Phys. Rev. C51 (1995) 38.

[6] R. Machleidt, F. Sammarruca and Y. Song, Phys. Rev. C53 (1996) 1483.

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[9] R. Machleidt and I. Slaus, J. Phys. G27 (2001) R69; [nucl-th/0101056]; J. Haidenbauer, Braz. J. Phys 34 (2004) 845-849, available from Braz. J. Phys. Server, link from the Spires: http://www.slac.stanford.edu/spires/ﬁnd/hep/www?j=BJPHE,34,845 U. van Kolck, Nucl. Phys. A752 (2005) 145-154; [nucl-th/0409064]; R. Machleidt,The Nuclear Force Problem: Is the Never-Ending Story Coming to an End?, [nucl-th/0609050]; E. Epelbaum, Prog. Part. Nucl. Phys. 57 (2006) 654-741; [nucl-th/0509032]

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[11] R.A. Krajcik, L.L. Foldy, Phys. Rev. D10 (1974) 1777.

36 BIBLIOGRAPHY 37

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[17] J.L. Forest, V.R. Pandharipande, J.L. Friar, nucl-th/9410011; J.L. Forest, V.R. Pandharipande, J.L. Friar, Phys. Rev. C52 (1995) 568-575; J.L. Forest, V.R. Pandharipande, A. Arriaga, Phys. Rev. C60 (1999) 014002; nucl-th/9805033

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