Nucleon-nucleon potential from skyrmion dipole interactions
NNucleon-nucleon potential from skyrmion dipoleinteractions
Derek Harland ∗ and Chris Halcrow † January 8, 2021
Abstract
We derive the nucleon-nucleon interaction from the Skyrme model using secondorder perturbation theory and the dipole approximation to skyrmion dynamics. Unlikeprevious derivations, our derivation accounts for the non-trivial kinetic and potentialparts of the skyrmion-skyrmion interaction lagrangian and how they couple in thequantum calculation. We derive the eight low energy interaction potentials and comparethem with the phenomenological Paris model, finding qualitative agreement in sevencases. This is a substantial improvement on previous calculations and serves as anexcellent starting point for describing the nucleon-nucleon interaction from the Skyrmemodel.
A nucleon can be modelled as a point particle with spin and isospin degrees of freedom.The standard way to model the nucleon-nucleon interaction uses a hamiltonian that wasfirst written down in [1]. The form of this hamiltonian is prescribed by symmetries and itis specified by eight potentials that (in its simplest form) depend only on the separationof the nucleons. Much effort has been devoted to deriving these potentials from a morefundamental theory. It is well-established that the behaviour of the potentials at largeseparations is governed by Yukawa’s theory of pion exchange [2]. In contrast, at smallseparations, QCD effects are important and theorists frequently rely on phenomology, fittingthe potentials to experimental data. Several of these semi-phenomenological models havebeen proposed, such as the Paris and Argonne models [3, 4]. While parts of these models arefixed by theory, many parameters are not constrained by theory and must be fitted to data:for example, the Paris model has around sixty unconstrained parameters. More modernmodels based on effective field theory have a firmer theoretical foundation, but still involvemany unconstrained parameters [5, 6]. It seems to be very difficult to derive the nucleon-nucleon potentials from fundamental theory without introducing experimentally-determinedparameters.The Skyrme model is a model of nuclei with roots in QCD that, in its simplest form, hasonly three unconstrained parameters. It models nucleons using topologically nontrivial field ∗ email address: [email protected] † email address: [email protected] a r X i v : . [ h e p - t h ] J a n onfigurations called skyrmions [7]. A skyrmion is a spatially localised soliton that can bedescribed using six degrees of freedom: three for its position, and three for its orientation.In order to understand the nucleon-nucleon interaction from the Skyrme model oneshould start by understanding the classical dynamics of two skyrmions. The two-skyrmionsystem can be described using a configuration space parametrized by two positions and twoorientations, at least when the skyrmions are widely-separated. To extract the nucleon-nucleon interaction one needs to semiclassically quantise the two-skyrmion dynamics. Sotwo approximations are needed to derive the nucleon-nucleon interaction: an approximationto the classical dynamics of skyrmions, and a quantisation method.The problem of deriving the nucleon-nucleon interaction from the Skyrme model has along history and is not yet resolved. Early in his development of the model, Skyrme used theproduct approximation to understand the long range interaction of two skyrmions [8]. Here,the two-skyrmion field is given by the product of the fields of two one-skyrmions. While thisis a simple way of generating a two-skyrmion field, there is no reason to trust its validity whenthe skyrmions are close together. This approximation was used by Vinh Mau et al., whoshowed that the long-range interactions between skyrmions reproduce the one pion exchangepotential of Yukawa [9]. Subsequent papers attempted to extract shorter-range parts of thenucleon-nucleon potential, again using the product approximation [10, 11]. These papersall used what we will call first order perturbation theory to pass from classical skyrmionsto quantised nucleons. Disappointingly, this approach did not result in any medium-rangecentral attraction. This was a major failure: without central attraction there can be nonuclear binding.The resolution of this problem was found by Walet, Amado and Hosaka [12, 13], andcame in two parts. First, they replaced the product approximation, expanding the classicalinteraction potential as a Fourier series in relative orientation. The Fourier coefficientswere fixed by the Atiyah-Manton approximation, where Skyrme fields are generated usinginstantons [14]. Secondly, they improved the quantisation technique. The potential energyhas a low energy region and the wavefunction should be concentrated there. To accountfor this fact the authors used second order perturbation theory, and an attractive centralpotential was found. They focused entirely on the potential energy, assuming that the kineticparts of the interaction were subleading. A year later, Schroers and Gisiger–Paranjapecarefully studied the skyrmion-skyrmion interaction, and found that the kinetic energy isnot subleading and can dominate the classical dynamics [15, 16, 17]. The consequences ofthis fact for the nucleon-nucleon interaction are explored for the first time in this paper.Another problem with the Skyrme-derived nucleon-nucleon interaction remained un-solved until recently. The isoscalar spin-orbit potential is essential for describing experimen-tal data from nucleon-nucleon scattering [18, 19], and plays a vital role in the predictionof magic numbers for larger nuclei [20]. Riska and Nyman obtained a satisfactory resultfor the isovector spin-orbit potential, but their approach resulted in an isoscalar potentialwith the wrong sign [21, 22, 23]. Various modifications of the Skyrme model were proposedto correct this result, such as coupling the theory to a dilaton [24] and only including the L term [25] (now known as the BPS model [26]). These did not improve the situation,and the most promising approach was shown by Abada to ignore the dominant contribution[27], thereby nullifying the earlier positive conclusion. Although these attempts to fix thespin-orbit problem seem very different, they all share a common feature: they combine theproduct approximation with first order perturbation theory.2 solution to the spin-orbit problem was found by the authors of this paper [28]. Theresolution combines second order perturbation theory with the skyrmion-skyrmion interac-tion first found by Schroers [15]. This interaction includes the potential terms as well as thekinetic terms, whose significance was demonstrated by Gisiger and Paranjape [16, 17]. Infact, it is a coupling between potential and kinetic terms which provides the most importantcontribution to the spin-orbit potential. The results of this new method were previouslycalculated only for the isoscalar spin-orbit potential [28]. In this paper, we present the fullnucleon-nucleon interaction arising from this method. We find a significant improvementover previous attempts to derive the nucleon-nucleon interaction from the Skyrme model.Our approach is in some ways similar to that of Sugawara and von Hippel [29]. In theirmodel, pions can excite nucleons to delta resonances, and this results in a pion-mediatedcentral attraction between nucleons. Our model is similar but, unlike in [29], the nucleon-delta amplitude is determined by theory rather than experiment. Also, the model of [29]included an omega meson to account for the short-range parts of the nucleon-nucleon po-tential, whereas our model only captures long-range parts of the nucleon-nucleon potential.The Skyrme model does provide a framework to study short-range interactions, and we planto study these in the future.Although the results in this paper are for the standard Skyrme model, the methods pre-sented are also valid (perhaps with small modifications) for many modified Skyrme models.These are plentiful [26, 30, 31, 32]. The topic of this paper has a history of mistakes andsign errors in the literature [27, 33]. For both these reasons, we present our calculation inpainstaking detail.To understand the calculation we first must understand nuclei as quantised skyrmions.This is done in section 2. The dipole approximation, first studied by Schroers and Gisiger–Paranjape for massless pions, is derived in section 3 for massive pions. Section 4 derives aquantum hamiltonian from this classical dipole-dipole lagrangian. We present the calculationof the nucleon-nucleon potential from this hamiltonian in section 5, and draw our conclusionsin section 6. We include four appendices. These provide further details for our calculations,and present our explicit formulae for the nucleon-nucleon potential (which are too long toinclude in the main body of the article). In this section we review how a quantised skyrmion can be viewed as a nucleon, as was firstshown in [34]. The Skyrme model is a field theory described by the lagrangian (cid:90) R (cid:18) − F π (cid:126) Tr( L µ L µ ) + (cid:126) e Tr([ L µ , L ν ][ L µ , L ν ]) − F π m π (cid:126) Tr(1 − U ) (cid:19) d x. (2.1)Here U : R , → SU(2), L µ = U − ∂ µ U , 1 is the identity matrix, F π is the pion de-cay constant, m π is the pion mass and e is a dimensionless coupling constant. Boundaryconditions U ( t, x ) → as | x | → ∞ are imposed to allow for finite energy, and as aresult the model has a topologically conserved quantity, the winding number B ∈ Z of U : S ∼ = R ∪ {∞} → SU(2) ∼ = S . This winding number B has the physical interpretationof baryon number. 3tatic solutions of the equations of motion with B = 1 can be obtained using the hedgehogansatz: U H ( x ) = exp( − i σ j ˆ x j f ( r )) , (2.2)in which r = | x | , ˆ x j = x j /r , σ j are the Pauli matrices and f : R ≥ → R is chosen to minimiseenergy subject to the boundary conditions f (0) = π and f ( ∞ ) = 0. This hedgehog skyrmionis a soliton whose energy is concentrated at the origin. Further static B = 1 solutions canbe obtained by acting on the hedgehog with symmetries of the theory, namely translations,rotations, and isorotations (which take the form U (cid:55)→ QU Q − for Q ∈ SU(2)). In fact itsuffices to act with translations and isorotations only, as the hedgehog is invariant under acombination of rotations and isorotations. Thus we obtain solutions of the form U ( x ) = QU H ( x − X ) Q − (2.3)parametrised by X ∈ R and Q ∈ SU(2). The parameters X and Q respectively describethe position of the soliton and its orientation.The family (2.3) describes the lowest-energy static configurations in the B = 1 sec-tor. To a good approximation, low-energy dynamics in the B = 1 sector can be de-scribed by promoting the parameters X , Q to time-dependent functions, i.e. by writing U ( t, x ) = Q ( t ) U H ( x − X ( t )) Q ( t ) − . The lagrangian that governs this simplified dynamicsis L = M | ˙ X | + Λ2 | ω | , (2.4)in which M and Λ are constants which represent the classical mass and moment of interiaof the B = 1 skyrmion, and − i ω · σ = 2 Q − ˙ Q , (2.5)where ω is interpreted as the angular velocity of the skyrmion. The equations of motion arethat of a free spinning top. Hence, for small kinetic energies, the skyrmion simply moveswith constant linear and angular velocities.In order to make contact with nuclear physics we must quantise the low-energy dynamicsof a skyrmion. The quantum mechanical hamiltonian is H = 12 M | P | + (cid:126) | S | , (2.6)in which P j = − i (cid:126) ∂/∂X j and − i S j ψ ( X , Q ) = dd (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) =0 ψ ( X , Qe − i (cid:15)σ j / ) . (2.7)The operators S j satisfy [ S i , S j ] = i (cid:15) ijk S k and are interpreted physically as spin operators.In order to understand this hamiltonian we first diagonalise the operator | S | = S i S i . Theeigenvalues of this operator are known to be of the form ( n − / n ≥ ψ ( Q ) = ρ nαβ ( Q ), where ρ :SU(2) → GL( n, C ) is the n -dimensional irreducible representation of SU(2) and ρ nαβ are itsmatrix entries (with 1 ≤ α, β ≤ n ). Thus the eigenspace H n with eigenvalue ( n − / C n ⊗ C n . The wavefunction is required to satisfy the Finkelstein-Rubinstein constraint ψ ( X , − Q ) = − ψ ( X , Q ), and as a result only the eigenspaces with n ψ : R , → H ∼ = C ⊗ C . (2.8)These describe nucleons: the first factor of C corresponds to a spin doublet, and the secondto an isospin doublet. The next-lowest eigenspace corresponds to functions ψ : R , → H ∼ = C ⊗ C (2.9)and describes delta resonances.Before moving on to investigate the dynamics of two skyrmions we pause to describe someoperators acting on the 1-skyrmion Hilbert space that will be relevant to later calculations.Let R ij ( Q ) be defined by R ij ( Q ) = 12 Tr( σ i Qσ j Q − ) , ≤ i, j ≤ . (2.10)In other words, R ij are the matrix entries of the adjoint representation of SU(2). These acton skyrmion wavefunctions ψ by multiplication. If ψ ∈ H n then R ij ψ can be written R ij ψ = (cid:88) m κ mnj ⊗ λ mni ψ, (2.11)where κ mnj and λ mni are m × n matrices of Clebsch-Gordon coefficients and the sum is over m = 2 , n = 2 and over m = n − , n, n + 2 in the cases n ≥
4. Our calculationslater will involve the matrices κ mnj for m, n = 2 ,
4, and these are given explicitly in appendixA. The matrices λ mnj are identical to κ mnj , but denoted by a different symbol for clarity(the κ ’s correspond to spin and the λ ’s to isospin). We also make note of some identitiesinvolving these matrices; these can be derived using the matrices given in the appendix.These identities describe: commutators with spin operators,[ S i , κ mnj ⊗ λ mnl ] = i ε ijk κ mnk ⊗ λ mnl ; (2.12)contractions with spin operators and epsilon tensors, ε ijk S j κ k ⊗ λ l = i κ i ⊗ λ l ε ijk κ j ⊗ λ l S k = i κ i ⊗ λ l ε ijk S j κ k ⊗ λ l = − i2 κ i ⊗ λ l ε ijk κ j ⊗ λ l S k = 5i2 κ i ⊗ λ l ε ijk S j κ k ⊗ λ l = 5i2 κ i ⊗ λ l ε ijk κ j ⊗ λ l S k = − i2 κ i ⊗ λ l ; (2.13)and substitutions in terms of Pauli matrices, κ i = − √ σ i κ i κ j = 13 (cid:18) δ ij + i ε ijk σ k (cid:19) κ i κ j = − √ (cid:18) δ ij − i2 ε ijk σ k (cid:19) . (2.14)5 Dipole-dipole lagrangian
Having understood the dynamics of a single skyrmion, we now consider the dynamics oftwo well-separated skyrmions, following Schroers [15]. To do so, we first investigate theasymptotic tail of a single skyrmion. Far from the centre of a hedgehog skyrmion, the Skyrmefield U is close to the vacuum. In terms of pion fields, we may write U ( x ) = exp(i π ( x ) · σ )and the lagrangian (2.1) takes the form L = F π (cid:126) (cid:90) (cid:18) ∂ µ π · ∂ µ π − m π (cid:126) π · π + O ( π ) (cid:19) d x. (3.1)Thus, far from the centre of the skyrmion the Skyrme lagrangian reduces to the Klein-Gordonlagrangian. The asymptotic field of the skyrmion with position X takes the form π ( x ) ∼ − C (cid:18) (cid:126) F π e (cid:19) (cid:18) | x − X | + m π (cid:126) | x − X | (cid:19) e − mπ (cid:126) | x − X | R ( Q )( x − X ) , (3.2)where R ( Q ) is the orientation matrix defined in (2.10) and C is a dimensionless constantthat can be computed numerically. We can compare this with the field induced by a dipolewith dipole moment c : c · ∇ (cid:18) πr e − mπr (cid:126) (cid:19) = − (cid:18) r + m π (cid:126) r (cid:19) e − mπr (cid:126) c · x π . (3.3)We see that the j th pion field is a dipole with dipole moment c i = 4 π ( (cid:126) F π e ) C R ji .We have learned that, far from its center, a skyrmion resembles a triplet of orthogonaldipoles. As such, we can use the theory of relativistic dipoles to describe the interactionof well separated skyrmions. A single dipole with moment c and position X ( t ) which ismoving slowly with velocity ˙ X and rotating with angular velocity ω has charge distribution[15] ρ d ( x , X ( t )) = − T δ (3) ( x − X ( t )) (3.4) T := (cid:16)(cid:16) c − | ˙ X | c − ( ˙ X · c ) ˙ X (cid:17) · ∇ + ˙ X · c × ω (cid:17) . (3.5)Here and throughout the calculation we have neglected all terms with more than two timederivatives. The potential due to this dipole satisfies¨ φ d − (∆ − m ) φ d = − T δ (3) ( x − X ( t )) , (3.6)where m = m π / (cid:126) is the reduced pion mass. We formally invert this equation and expand inthe slow moving approximation φ d = (cid:0) (∆ − m ) − ∂ t (cid:1) − T δ (3) ( x − X ( t ))= T G m + d dt T (∆ − m ) − G m + ... (3.7)where G m is the Greens function for the Yukawa interaction, given by G m = − πr e − mr . (3.8)6o find the potential we are left to evaluate F m := (∆ − m ) − G m . We do this by solving G m = (∆ − m ) F m (3.9)whose unique decaying solution is F m = 18 πm e − mr . (3.10)It is worth considering the massless limit, which was studied earlier in [15]. In that case,(3.7) was solved using special properties of the Laplacian, resulting in F := ∆ − G = − r π . (3.11)The expansion of our solution about m = 0 islim m → F m = 18 πm − r π + O ( m ) . (3.12)Although this diverges as m →
0, the lagrangian only depends on derivatives of F m and thedivergent term is constant. Hence our result agrees with [15] in the limit m → T , the potential due to (3.4) is φ d ≈ − π (cid:16)(cid:16) c − ˙ X c − ˙ X · c ˙ X (cid:17) · ∇ + ˙ X · c × ω (cid:17) (cid:18) e − m | x − X | | x − X | (cid:19) + 18 πm d dt c · ∇ (cid:16) e − m | x − X | (cid:17) . (3.13)Once again, we have neglected terms with more than two time derivatives. Note that thisis not the result one finds by simply replacing the massless Greens function for the massiveGreens function in the expression for φ d from Schroers.A skyrmion is described by three orthogonal dipoles. So the charge density of, andthe potential due to, the skyrmion is simply the sum of those of the dipoles. Let the twoskyrmions be labeled by 1 and 2, each having their own positions X , X and angularvelocities ω , ω . The interaction lagrangian is given by L = 12 (cid:18)(cid:90) φ ρ + φ ρ (cid:19) d x . (3.14)This can be evaluated using (3.4) and (3.13). Once again we neglect terms with more thantwo derivatives, and the resulting expression is similar to equation (6.19) in [15]. The resultcan be expressed in terms of x := X − X , r = | x | , q := Q − Q , and ρ := 8 π (cid:126) C /e F π : L = M x i ˙ x i + Λ2 ω i ω i + Λ2 ω i ω i + ρC ij ˙ x i ˙ x j + ρA ij ˙ x i ω j + ρA ij ˙ x i ω j + ρB ij ω i ω j − ρD. (3.15)7ere A ij = A ab ; ij R ab , A ij = A ba ; ij R ab , B ij = B ab ; ij R ab , C ij = C ab ; ij R ab and D = D ab R ab ,with A ab ; ij = ε ajc ( δ ic ∇ b e − mr /r − ∇ ibc e − mr /m ) (3.16) B ab ; ij = − ε aic ε bjd ∇ cd e − mr /m (3.17) C ab ; ij = δ ij ∇ ab e − mr /r − ∇ abij e − mr /m (3.18)+ ( δ jb ∇ ia + δ ja ∇ ib + δ ib ∇ ja + δ ia ∇ jb ) e − mr /rD ab = ∇ ab e − mr /r. (3.19)Later, we will consider the massless ( m = 0) limit. Hence it is helpful to record the lagrangianin this limit, originally derived in [15]. It has the same structure as (3.15) but with A ab ; ij = ε ajc ( δ ic ∇ b /r + ∇ ibc r ) (3.20) B ab ; ij = ε aic ε bjd ∇ cd r (3.21) C ab ; ij = δ ij ∇ ab /r + ∇ abij r + ( δ jb ∇ ia + δ ja ∇ ib + δ ib ∇ ja + δ ia ∇ jb )1 /r (3.22) D ab = ∇ ab /r. (3.23)Although the lagrangian (3.15) is ungainly, its physical origin is simple: it describes the in-teraction of two slowly moving dipoles. It is therefore a good approximation to the dynamicsof two well-separated, slowly-moving skyrmions. Having obtained an approximate lagrangian for two skyrmions, we now calculate the corre-sponding hamiltonian.In general, the hamiltonian associated to a lagrangian for a particle moving on a Rie-mannian manifold with metric g under the influence of a potential V is (cid:126) (cid:52) g + V , with (cid:52) g being the Laplace-Beltrami operator for the metric g . If the metric is given in the form g = g µν e µ e ν , with e µ being a frame for the cotangent bundle, the Laplace-Beltrami operatoris (cid:52) g = − (det g ) − / E µ (det g ) / g µν E ν + f λµλ g µν E ν , (4.1)with E µ being the dual frame for the tangent bundle (such that e µ ( E ν ) = δ µν ) and f νλµ structure constants defined by [ E λ , E µ ] = f νλµ E ν . A derivation of this formula is given inappendix B. The operator (cid:52) g is manifestly self-adjoint with respect to the inner product (cid:104) ψ | ψ (cid:105) g = (cid:82) ψψ (det g ) / e ∧ . . . ∧ e n .If the metric is perturbed to g + δg then the correct hamiltonian is (cid:126) (cid:52) g + δg + V . This isself-adjoint with respect to the inner product (cid:104) ψ | ψ (cid:105) g + δg but not the inner product (cid:104) ψ | ψ (cid:105) g .If we want our deformed hamiltonian to still be self-adjoint with respect to (cid:104) ψ | ψ (cid:105) g we shouldinstead choose H = det(1 + g − δg ) / (cid:18) (cid:126) (cid:52) g + δg + V (cid:19) det(1 + g − δg ) − / . (4.2)8his can be expanded as a power series in δg . Assuming that f λµλ = 0, the terms up toquadratic order are H = (cid:126) (cid:52) g + V + (cid:126) E κ g κλ δg λµ g µν E ν − (cid:126) E κ g κλ δg λµ g µν δg νρ g ρσ E σ + (cid:126) g µν [ E µ , g κλ δg λκ ][ E ν , g ρσ δg σρ ] + O ( δg ) . (4.3)We will use equation (4.3) to calculate the hamiltonian for the lagrangian (3.15) asa power series in ρ . The ρ -independent kinetic terms determine a metric g , and the ρ -dependent kinetic terms determine a perturbation δg . For the frame e µ we choose e j = d x j , e j +3 = Ω j , e j +6 = Ω j ; j = 1 , , , (4.4)where − Ω jα i σ j = Ω α = 2 q − α d q α (4.5)matching (2.5). The dual frame is E j = ∂∂x j = i (cid:126) P j , E j +3 = − i S j , E j +6 = − i S j ; j = 1 , , . (4.6)It is important that the plus and minus signs in these equations are chosen correctly. Thesign of the spin terms is correct because − i σ k e k +3 ( E j +3 ) = − i σ k Ω k ( − i S j ) = Ω ( − i S j ) = 2 Q − dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 Q e − i tσ j / = − i σ j (4.7)and, more generally, e µ ( E ν ) = δ µν . From equations (4.3) and (3.15) we find H = (cid:126) | S | + (cid:126) | S | + 1 M | P | + H I , (4.8)where H I = 2 ρD − ρ (cid:126) B + ρ (cid:126) M Λ ( P i A i + A † i P i )+ ρ (cid:126) F + ρ (cid:126) M Λ A † i A i − ρ (cid:126) M Λ ( P i ˜ A i + ˜ A † i P i ) + O ( ρ ) + O ( M − ) (4.9)and B = S i B ij S j + S j B ij S i (4.10) A i = A ij S j + A ij S j (4.11) F = S i B ij B kj S k + S i B ji B jk S k (4.12)˜ A i = A ij B kj S k + A ij B jk S k . (4.13)9 Nucleon-nucleon potential
In the previous section we determined a hamiltonian that describes two interacting skyrmions.In this section we will apply perturbation theory to calculate a low-energy effective hamil-tonian acting on the nucleon-nucleon sector of the skyrmion-skyrmion Hilbert space.We begin by recalling some essential ideas from degenerate perturbation theory. Let H be a hamiltonian acting on a Hilbert space with energy eigenvalues E < E < E < . . . and eigenspaces H N . Consider a perturbed hamiltonian of the form H ( (cid:15) ) = H + (cid:15)H , (5.1)with (cid:15) small. When (cid:15) = 0 the E -eigenspace H is invariant under the action of H (0). As (cid:15) moves away from zero this eigenspace is deformed to a subspace H ( (cid:15) ) which is invariantunder H ( (cid:15) ) and which is canonically identified with H . Using this identification H ( (cid:15) ) ∼ = H one obtains an operator H E : H → H which describes the action of H ( (cid:15) ) on H ( (cid:15) ) andcan be thought of as a low-energy effective hamiltonian for H ( (cid:15) ). In appendix C we derivethe following perturbative formula for this effective hamiltonian: H E = E + (cid:15)H − (cid:15) (cid:88) N> E N − E H N H N + (cid:15) (cid:88) M,N (cid:54) =0 E N − E )( E M − E ) H N H NM H M − (cid:15) (cid:88) N> E N − E ) ( H N H N H + H H N H N ) + O ( (cid:15) ) . (5.2)Here H NM : H M → H N are the projections of H such that H = (cid:80) M,N H NM . Usuallyin degenerate perturbation theory one works in a basis in which the first (or second) orderterm is diagonal, and in that case the formula (5.2) reduces to well-known formulae for theperturbed eigenvalues. The advantage of using equation (5.2) is that it does not require oneto choose any particular basis.We will apply the formula (5.2) to the skyrmion-skyrmion hamiltonian (4.8), choosing H = (cid:126) ( | S | + | S | ) and (cid:15)H = M | P | + H I . This means that we have two deformationparameters: ρ and M − (recall that H I = O ( ρ )). The eigenspaces of H are the spaces H N = H m ⊗ H n labelled by N = ( m, n ), which describe particles of spin m − and n − . Theassociated eigenvalues of H are E N = ( m + n − (cid:126) / E corresponds to the nucleon-nucleon sector labelled by ( m, n ) = (2 , H E = E + | P | M + H I − (cid:88) N> H NI E N − E H N I + 12 M (cid:88) N> E N − E ) ( H NI [ | P | , H N I ] − [ | P | , H NI ] H N I )+ O ( ρ ) + O ( M − ) . (5.3)10he three terms involving H I will be referred to as first, second and third order. Noticethat the second order term does not involve | P | because | P | commutes with H and thus( | P | ) N = 0 if N >
0. The third order term is simpler than in equation (5.2) because weare only working up to order 2 in ρ and order 1 in M − .A key feature of the formula (5.3) is that the sums over N are finite. This is a special fea-ture of the dipole approximation, and happens because H I depends on Q , Q only throughthe combination R ab ( q ) = R ca ( Q ) R cb ( Q ). By the Clebsch-Gordon rules, multiplying astate ψ ∈ H ⊗ H with R ab ( q ) results in a state in (cid:76) m,n =2 , H m ⊗ H n . More explicitly,from equation (2.11), R ab ψ = R ca ( Q ) R cb ( Q ) ψ = (cid:88) m,n =2 , κ m a ⊗ λ m c ⊗ κ n b ⊗ λ n c ψ (5.4)(where the first subscript on κ and λ labels the particle). Thus in equation (5.2), and inwhat follows, the notation (cid:80) N> means a sum over ( m, n ) = (2 , , (4 , , (4 , (cid:80) N will mean a sum over ( m, n ) = (2 , , (2 , , (4 , , (4 , E N − E . The smallest energy difference is 3 (cid:126) / r > (cid:126) /m π , the approximation will be reliable provided that | P | M , ρr , ρ (cid:126) r Λ , ρ (cid:126) | P | M Λ r , ρ | P | M r < (cid:126) Λ . (5.5)The following conditions on | P | , r, M are sufficient to ensure that this is the case: | P | < M (cid:126) Λ , r > max (cid:40) (cid:114) ρ Λ (cid:126) , ρ Λ , (cid:126) m π (cid:41) , M > (cid:126) Λ . (5.6)The first inequality simply means that the skyrmions are moving slowly. The third inequalityis equivalent to the statement that the dominant contribution to the nucleon energy is therest mass of a skyrmion, rather than its spin energy. This is true in all proposed calibrationsof the Skyrme model. The condition that r > (cid:126) /m π is not very restrictive, because the pionCompton wavelength (cid:126) /m π is only slightly larger than the proton charge radius. We willexamine the remaining constraints in the next section, when we discuss calibrations.We wish to compare the hamiltonian (5.3) with the nucleon-nucleon potential. The latteris constrained by symmetry to be of the form [1] V NN = V ISC + V ISσσ σ σ + V IS S + (cid:126) V ISLS L · σ + (cid:0) V IVC + V IVσσ σ σ + V IV S + (cid:126) V IVLS L · σ (cid:1) τ τ . (5.7)The potentials are known as the isoscalar or isovector central, sigma, tensor and spin-orbitpotentials. Here σ i , σ i are the spin Pauli matrices and τ i , τ i are the isospin Pauli matrices,and we have used the shorthand τ τ = (cid:80) i τ i τ i and σ σ = (cid:80) i σ i σ i . The operator S is S = 3( σ · ˆ x )( σ · ˆ x ) − σ · σ . The total spin is σ = σ + σ , and L = x × P is the totalangular momentum. The coefficient functions V ∗∗∗∗ are functions of r = | x | , | P | and | L | only. Since the skyrmion-skyrmion system shares the symmetries of the nucleon-nucleonsystem, the hamiltonian (5.3) is guaranteed to be of this form.11e now proceed to describe the calculation of the terms in (5.7) from equation (5.3).This is a lengthy calculation, and in order to avoid errors we used two independent methods.The first method is algebraic in character and exploits the identities (2.13) and (2.14). Thesecond method is a direct calculation in Mathematica that uses the explicit formulae for thematrices κ mni given in appendix A. Both methods gave the same result in the case m π = 0.The result for m π > m π = 0 throughout this calculation. We work to order 2 in ρ and order 1in M : thus all equations are understood to be true only up to terms of O ( ρ ) and O ( M − ).Further details of the calculation are given in appendix D. At the end of this section we givethe final result for massless pions, and briefly explain the Mathematica-based calculation.The final result for massive pions is more complicated, and is given in appendix E. In this subsection we evaluate the first order term in equation (5.3), i.e. H I = 2 ρD − ρ (cid:126) B + ρ (cid:126) M Λ ( P i A i + A † i P i ) + ρ (cid:126) F + ρ (cid:126) M Λ ( A † i A i ) − ρ (cid:126) M Λ ( P i ˜ A i + ˜ A † i P i ) . (5.8)The first few terms can be evaluated using the identity R ab = κ a λ c κ b λ c = 19 σ a σ b τ τ , (5.9)which follows from (5.4) and the first equation in (2.14). Using identity (5.9) we find D = D ab R ab = 19 ∇ ab (1 /r ) R ab = 19 r S τ τ . (5.10)This reproduces the well-known result that the dipole potential for skyrmions induces theone-pion exchange potential between nucleons [9]. Using identities (5.9) and (2.13) we find B = B ab ; ij ( S i R ab S j + S j R ab S i ) (5.11)= 2 ∇ ab rR ab (5.12)= 427 r σ σ τ τ − r S τ τ . (5.13)12sing identity (5.9) and noting that ( S j ) = σ j and ( S j ) = σ j we obtain( P i A i + A † i P i ) = 12 { P i , A i + A † i } + 12 [ P i , A i − A † i ] (5.14)= 12 { P i , A ab ; ij } (cid:0) { R ab , S j } + { R ba , S j } (cid:1) − i (cid:126) ∇ i A ab ; ij (cid:0) [ R ab , S j ] + [ R ba , S j ] (cid:1) (5.15)= 118 { P i , A ab ; ij } δ aj ( σ b + σ b ) τ τ + (cid:126) ∇ i A ab ; ij ε ajk ( σ b σ k + σ b σ k ) τ τ (5.16)= 4 (cid:126) r S τ τ . (5.17)Note that the first term in equation (5.16) evaluates to 0, because A ab ; ij is skew-symmetricin a and j . This particular property of the dipole lagrangian means that the isovectorspin-orbit potential vanishes at order 1 in ρ .To evaluate the next few terms, we need an identity for ( R ab R cd ) = (cid:80) N R Nab R N cd .From the identities (5.4) and (2.14) we obtain R Nab R N cd = ( δ ac + i (cid:15) ace σ e )( δ bd + i (cid:15) bdf σ f )(1 − τ τ ) N = (2 , ( δ ac − i2 (cid:15) ace σ e )( δ bd + i (cid:15) bdf σ f )(1 + τ τ ) N = (4 , ( δ ac + i (cid:15) ace σ e )( δ bd − i2 (cid:15) bdf σ f )(1 + τ τ ) N = (2 , ( δ ac − i2 (cid:15) ace σ e )( δ bd − i2 (cid:15) bdf σ f )(1 − τ τ ) N = (4 , . (5.18)Therefore (cid:88) N R Nab R N cd = 13 δ ac δ bd + 118 (cid:15) ace (cid:15) bdf σ e σ f τ τ . (5.19)Using equation (5.19) and the fact that S αj = σ αj , we obtain F = B ij ; ab B kj ; cd (cid:88) N S i R Nab R N cd S k + B ji ; ab B jk ; cd (cid:88) N S i R Nab R N cd S k (5.20)= 43 r + 118 r ( S + σ σ ) τ τ . (5.21)Similarly, we obtain( P i ˜ A i + ˜ A † i P i ) = 12 { P i , ˜ A i + ˜ A † i } + 12 [ P i , ˜ A i − ˜ A † i ] (5.22)= 1 r (cid:18) − τ τ (cid:19) L · σ + (cid:126) r ( σ σ − S ) τ τ . (5.23)and ( A † i A i ) = 136 r − r S + 536 r σ σ − r τ τ + 5108 r S τ τ − r σ σ τ τ . (5.24)13ollecting everything together, we find H I = (cid:20) ρ (cid:126) r + ρ M (cid:126) r (cid:21) − ρ M (cid:126) r τ τ − ρ M (cid:126) r S + (cid:20) ρ (cid:18) (cid:126) r + 29 r (cid:19) + ρM (cid:126) r + ρ (cid:126) r + ρ M (cid:126) r (cid:21) S τ τ + ρ M (cid:126) r σ σ + (cid:20) − ρ (cid:126) r + ρ (cid:126) r − ρ M (cid:126) r (cid:21) σ σ τ τ − ρ M (cid:126) Λ r L · σ + ρ M (cid:126) r L · στ τ . (5.25) In this subsection we evaluate the second order term in equation (5.3), i.e. − (cid:88) N> E N − E H NI H N I = − ρ (cid:88) N> E N − E D N D N + ρ (cid:126) Λ (cid:88) N> E N − E (cid:0) D N B N + B N D N (cid:1) − ρ (cid:126) (cid:88) N> E N − E B N B N − ρ (cid:126) M Λ (cid:88) N> E N − E (cid:16) D N ( P i A i + A † i P i ) N + ( P i A i + A † i P i ) N D N (cid:17) + ρ (cid:126) M Λ (cid:88) N> E N − E (cid:16) B N ( P i A i + A † i P i ) N + ( P i A i + A † i P i ) N B N (cid:17) . (5.26)We will describe how to evaluate just a couple of the terms in this expression. The resultsfor all other terms can be found in appendix D, and their total appears at the end of thissection.The first term that we will evaluate is ρ (cid:126) Λ (cid:88) N> E N − E (cid:0) D N B N + B N D N (cid:1) = −∇ ab r ∇ cd r ε cij ε dkl (cid:88) N> E N − E (cid:104) R Nab S i R N jk S l + R Nab S k R N il S j + S i R Njk S l R N ab + S k R Nil S j R N ab (cid:105) . (5.27)All of the terms in the sum can be expressed using the identities (2.13), (2.14) and (5.4).For example,1 E N − E R Nab ε cij ε dkl S i R N jk S l = (cid:126)
3Λ 5i2 i ( δ ac − i2 (cid:15) ace σ e )( δ bd + i (cid:15) bdf σ f )(1 + τ τ ) N = (4 , (cid:126) i( − i2 ) ( δ ac + i (cid:15) ace σ e )( δ bd − i2 (cid:15) bdf σ f )(1 + τ τ ) N = (2 , (cid:126)
3Λ 5i2 ( − i2 ) ( δ ac − i2 (cid:15) ace σ e )( δ bd − i2 (cid:15) bdf σ f )(1 − τ τ ) N = (4 , . (5.28)14valuating the remaining terms in a similar manner results in an identity ε cij ε dkl (cid:88) N> E N − E (cid:104) R Nab S i R N jk S l + R Nab S k R N il S j + S i R Njk S l R N ab + S k R Nil S j R N ab (cid:105) = Λ (cid:126) δ ac δ bd (cid:18) − − τ τ (cid:19) + Λ (cid:126) ε ace ε bdf σ e σ f (cid:18) − − τ τ (cid:19) . (5.29)Substituting this into equation (5.27) leads to ρ (cid:126) Λ (cid:88) N> E N − E (cid:0) D N B N + B N D N (cid:1) = ρ Λ r (cid:20) − − τ τ + 181 (14 + 3 τ τ )( σ σ − S ) (cid:21) . (5.30)The other terms in equation (5.26) that do not involve P i can be evaluated by a similarmethod.Now we evaluate the term involving P i , A i and D . This requires some algebraic rear-rangement: − ρ (cid:126) M Λ (cid:88) N> E N − E (cid:16) D N ( P i A i + A † i P i ) N + ( P i A i + A † i P i ) N D N (cid:17) = − ρ (cid:126) M Λ (cid:88) N> E N − E (cid:110) P i , ( A i + A † i ) N D N + D N ( A i + A † i ) N (cid:111) + i ρ (cid:126) M Λ (cid:88) N> E N − E (cid:16) D N ∇ i ( A i − A † i ) N + ∇ i ( A i − A † i ) N D N (cid:17) + i ρ (cid:126) M Λ (cid:88) N> E N − E (cid:16) ( A i + A † i ) N ∇ i D N − ∇ i D N ( A i + A † i ) N (cid:17) . (5.31)Here we have used the identity [ P i , · ] = − i (cid:126) ∇ i . Each of these three terms can be evaluatedusing methods similar to those described above. For example, for the first term we use thefollowing identities, whose derivation is similar to that of (5.29): ε cij (cid:88) N> E N − E (cid:2) R Nab ( S i R N jd − R N id S j ) + ( S i R Njd − R Nid S j ) R N ab (cid:3) = Λ (cid:126) δ bd ε ace σ e (cid:18)
827 + 281 τ τ (cid:19) + Λ (cid:126) δ ac ε bdf σ f (cid:18) − − τ τ (cid:19) (5.32) ε dij (cid:88) N> E N − E (cid:2) R Nab ( S i R N cj − R N ci S j ) + ( S i R Ncj − R Nci S j ) R N ab (cid:3) = Λ (cid:126) δ bd ε ace σ e (cid:18) − − τ τ (cid:19) + δ ac ε bdf σ f (cid:18)
827 + 281 τ τ (cid:19) . (5.33)15he result is − ρ (cid:126) M Λ (cid:88) N> E N − E (cid:110) P i , ( A i + A † i ) N D N + D N ( A i + A † i ) N (cid:111) = ρ M (cid:126) r (cid:18) −
43 + 49 τ τ (cid:19) L · σ . (5.34)The full result for equation (5.31) is in appendix D. The remaining term in (5.26), whichinvolves P i , A and B , can be calculated by a similar method and is also given in the appendix.The complete result for equation (5.26) is − (cid:88) N> E N − E H NI H N I = (cid:20) − ρ (cid:18) (cid:126) r + 827Λ r + 32Λ9 (cid:126) r (cid:19) − ρ M (cid:18) (cid:126) r + 329 r (cid:19)(cid:21) + (cid:20) − ρ (cid:18) (cid:126) r + 2881Λ r + 16Λ27 (cid:126) r (cid:19) + ρ M (cid:18) (cid:126) r − r (cid:19)(cid:21) τ τ + (cid:20) − ρ (cid:18) (cid:126) r + 2881Λ r + 8Λ27 (cid:126) r (cid:19) − ρ M (cid:18) (cid:126) r + 2027 r (cid:19)(cid:21) S + (cid:20) − ρ (cid:18) (cid:126) r + 681Λ r + 4Λ27 (cid:126) r (cid:19) − ρ M (cid:18) (cid:126) r + 29 r (cid:19)(cid:21) S τ τ + (cid:20) ρ (cid:18) − (cid:126) r + 1481Λ r + 8Λ27 (cid:126) r (cid:19) + ρ M (cid:18) (cid:126) r + 2027 r (cid:19)(cid:21) σ σ + (cid:20) ρ (cid:18) − (cid:126) r + 127Λ r + 4Λ27 (cid:126) r (cid:19) + ρ M (cid:18) (cid:126) r + 29 r (cid:19)(cid:21) σ σ τ τ + ρ M (cid:20) (cid:126) r − (cid:126) r (cid:21) L · σ + ρ M (cid:20) (cid:126) r + 49 (cid:126) r (cid:21) L · στ τ . (5.35) In this subsection we evaluate the third order term in equation (5.3). We rearrange this asfollows:12 M (cid:88) N> E N − E ) ( H NI [ | P | , H N I ] − [ | P | , H NI ] H N I )= i (cid:126) M (cid:88) N> E N − E ) (cid:8) P i , ∇ i H NI H N I − H NI ∇ i H N I (cid:9) + (cid:126) M (cid:88) N> E N − E ) ∇ i H NI ∇ i H N I . (5.36)16ubstituting for H I leads to12 M (cid:88) N> E N − E ) ( H NI [ | P | , H N I ] − [ | P | , H NI ] H N I )= 4i ρ (cid:126) M (cid:88) N> E N − E ) (cid:8) P i , ∇ i D N D N − D N ∇ i D N (cid:9) − i ρ (cid:126) M Λ (cid:88) N> E N − E ) (cid:8) P i , ∇ i D NI B N I − D NI ∇ i B N I + ∇ i B NI D N I − B NI ∇ i D N I (cid:9) + i ρ (cid:126) M Λ (cid:88) N> E N − E ) (cid:8) P i , ∇ i B NI B N I − B NI ∇ i B N I (cid:9) + 4 ρ (cid:126) M (cid:88) N> E N − E ) ∇ i D NI ∇ i D N I − ρ (cid:126) M Λ (cid:88) N> E N − E ) (cid:0) ∇ i D NI ∇ i B N I + ∇ i B NI ∇ i D N I (cid:1) + ρ (cid:126) M Λ (cid:88) N> E N − E ) ∇ i B NI ∇ i B N I . (5.37)Each of these six terms can be evaluated by similar methods to those described above, andthe resulting expressions are given in appendix D. The end result is12 M (cid:88) N> E N − E ) ( H NI [ | P | , H N I ] − [ | P | , H NI ] H N I )= ρ M (cid:20) (cid:18) (cid:126) r + 8827 r + 800Λ (cid:126) r (cid:19) + (cid:18) (cid:126) r + 14881 r + 560Λ (cid:126) r (cid:19) τ τ + (cid:18) (cid:126) r + 7481 r + 224Λ (cid:126) r (cid:19) S + (cid:18) (cid:126) r + 59243 r + 272Λ (cid:126) r (cid:19) S τ τ + (cid:18) − (cid:126) r − r − (cid:126) r (cid:19) σ σ + (cid:18) − (cid:126) r − r − (cid:126) r (cid:19) σ σ τ τ + (cid:18) (cid:126) r + 5281 r + 16Λ (cid:126) r (cid:19) L · σ (cid:126) + (cid:18) (cid:126) r + 22243 r + 40Λ (cid:126) r (cid:19) L · σ (cid:126) τ τ (cid:21) . (5.38)17 .4 Results Adding the first, second and third order results together, we find the final expression of ourcalculation. The isoscalar potentials are V ISC = ρ (cid:0) − − (cid:126) r + 17 (cid:126) r (cid:1) (cid:126) r + ρ M (cid:0) − (cid:126) r + 71 (cid:126) r (cid:1) (cid:126) r (5.39) V IS = ρ (cid:0) − − (cid:126) r − (cid:126) r (cid:1) (cid:126) r + ρ M (cid:0) + 504Λ (cid:126) r − (cid:126) r (cid:1) (cid:126) r (5.40) V ISσσ = ρ (cid:0) + 112Λ (cid:126) r − (cid:126) r (cid:1) (cid:126) r + ρ M (cid:0) − − (cid:126) r + 2255 (cid:126) r (cid:1) (cid:126) r (5.41) V ISSO = ρ M (cid:0) − (cid:126) r (cid:1) (cid:0) + 7 (cid:126) r (cid:1) (cid:126) r , (5.42)while the isovector potentials are V IVC = ρ (cid:0) − − (cid:126) r − (cid:126) r (cid:1) (cid:126) r + ρ M (cid:0) + 336Λ (cid:126) r − (cid:126) r (cid:1) (cid:126) r (5.43) V IV = ρ + (cid:126) r r + ρ (cid:0) − − (cid:126) r + 19 (cid:126) r (cid:1) (cid:126) r (5.44)+ ρM (cid:126) r + ρ M (cid:0) + 360Λ r (cid:126) + 839 (cid:126) r (cid:1) (cid:126) r (5.45) V IVσσ = − ρ (cid:126) r + ρ (cid:0) + 48Λ (cid:126) r + 19 (cid:126) r (cid:1) (cid:126) r + ρ M (cid:0) − − (cid:126) r − (cid:126) r (cid:1) (cid:126) r (5.46) V IVSO = ρ M (cid:0) + 3120Λ (cid:126) r + 773 (cid:126) r (cid:1) (cid:126) r . (5.47)The equivalent expressions with a non-zero pion mass can be found in Appendix E.Finally, we explain our second method for evaluating this potential. This method startedwith the same expressions (5.8), (5.26) and (5.37) but differed in the way the terms in theseexpressions were evaluated. Using the identity (5.4), the operators R MNab and S αi werereplaced with the matrices given in appendix A. The identities listed in appendix E werethen obtained by computing the resulting matrix products in Mathematica, and the resultsadded together. We now compare the results of our calculation with the successful semi-phenomenologicalmodel proposed by the Paris group [3]. To compare models we must choose a calibration byfixing F π , e and m π . This is equivalent to fixing the energy scale, length scale and pion mass.Once these are chosen, all other constants are fixed by the Skyrme model. One calibration18e consider was proposed by Lau and Manton (LM), optimised to reproduce the Carbon-12energy spectrum [36]. In this case F π = 108 MeV , e = 3 .
93 and m π = 149 MeV (5.48)which fixes the constants M = 1096 MeV , Λ = 332 MeV fm and ρ = 229 MeV fm . (5.49)We will also consider a new calibration, optimised to reproduce the Paris potentials. Tofind this, we consider the sum of the L norms of the differences between ours and the Parispotentials for r = 1 . − . F π , e and m π and we minimise thefunction using a numerical gradient flow. We find that the optimal calibration is F π = 165 MeV , e = 3 .
75 and m π = 216 MeV (5.50)which fixes the constants M = 1752 MeV , Λ = 252 MeV fm and ρ = 124 MeV fm . (5.51)We call this the HH calibration. This calibration gives values of F π and m π reasonablyclose to their physical values. This is expected, since we are dealing with pionic physics.Unfortunately the skyrmion mass M is much too large. This is a common problem when onetries to describe the physics of the nucleon sector. Meier and Walliser proposed that one-loop corrections can significantly reduce the mass [37], although including this correction isdifficult and will affect the interaction potentials we have derived.Our calculation depended on two approximations: perturbation theory and the dipoleapproximation to skyrmion dynamics. Our use of perturbation theory is justified only whenthe inequalities (5.6) hold. In the Lau-Manton calibration the tightest constraint is r > (cid:126) m π = 1 .
33 fm , (5.52)while in the new calibration it is r > (cid:18) ρ Λ (cid:126) (cid:19) = 0 .
93 fm . (5.53)It is harder to quantify when the dipole approximation is valid. An initial test of its validitywas performed by Foster and Krusch, who compared numerically generated skyrmion dy-namics to the dipole approximation when the skyrmions are not spinning and are fixed inthe attractive channel [38]. Their results indicate that the dipole approximation is reliable atlarge separations, but unreliable at small separations of the order 1fm. They don’t estimatethe separation at which the dipole approximation ceases to be reliable.We plot the eight potentials from (5.7) for the LM calibration, the HH calibration andfrom the Paris model in Figure 1. For the long-range part of the interaction ( r (cid:38) r (fm). 20ign, though is too small in the HH calibration. The only major disagreement is withthe isovector spin-orbit potential. This was successfully described in early works from theSkyrme model [39], so perhaps nonlinear effects will resolve the disagreement.Let us compare our results with earlier calculations of the nucleon-nucleon potential fromthe Skyrme model. In our calculation the potentials are expressed in terms of the expansionparameters ρ and M − . At first order in ρ only the isovector tensor and sigma potentials arenon-zero. Hence second order perturbation theory was needed to generate non-trivial results.In calculations done using first order perturbation theory and the product approximation[11, 22, 23], all of the potentials are non-zero at first order in ρ . In these calculations, allfour isoscalar potentials had an incorrect sign – we believe this is a failing of the productapproximation. Our results, where only one isovector potential has an incorrect sign, are asubstantial improvement on those calculations.A direct comparison with the results of [12, 13] is difficult, as those papers only computepotentials in particular channels and do not compute V ISC etc. The method of [12, 13] couldin principle be used to compute six of the eight potentials in (5.7), but not the two spin-orbitpotentials. In this sense our method is more powerful. Another advantage of our approachover earlier methods is that it gives an explicit formula for the nucleon-nucleon interaction,but the price paid for this is that the formula doesn’t capture short-range effects.We also attempted to find a calibration of our model with m π = 0, by varying only F π and e only. Here, it was much more difficult to find agreement between our model and theParis potential. In particular, the isovector sigma potential V IVσσ had the wrong sign for allparameter choices that we tried. This suggests that a non-zero pion mass is an essentialingredient for producing realistic nucleon-nucleon interactions.
In summary, we have derived a nucleon-nucleon interaction from the Skyrme model us-ing a method recently introduced in [28]. Compared with earlier attempts based on theSkyrme model, we obtain a very good match with the long-range parts of the Paris potential.Overall, these results provide an excellent starting point for describing the nucleon-nucleoninteraction from the Skyrme model. Importantly, we can describe many features of thenucleon-nucleon interaction using a purely pionic theory.Our calculation is based on the dipole approximation to skyrmion dynamics, so doesnot capture the short-range part of the nucleon-nucleon interaction. Hence to reach smallervalues of r , we must first improve this approximation. The most complete description ofthe two-skyrmion configuration space is by Atiyah and Manton [14], who used instantons togenerate Skyrme fields. This gives a two-skyrmion configuration space which includes theminimal energy toroidal skyrmion as well as two widely separated skyrmions. Repeating thecalculation presented here using the Atiyah-Manton approximation would allow us to studythe intermediate- and short-range parts of the interaction.The results presented in Figure 1 are promising, but to seriously judge the success ofour calculation, we should compare directly with experimental data. This requires thecalculation of phase shifts from our model, found by solving a Schr¨odinger equation basedon the potentials. However, as explained above, we do not understand the potentials forsmall r and these are needed for the calculation. Walet calculated phase shifts by imposinghard-core boundary conditions at r = 1 [40]. An advantage of the Skyrme model is that we21hould not require a hard-core: the geometry of the configuration space does not allow theskyrmions to get too close. It would be preferable to incorporate this fact into any futurecalculation.The results suggest that the Skyrme model may provide an understanding of the nucleon-nucleon interaction using only pions. This is in contrast to the successful one-boson-exchangemodels which suggest that (cid:15) -, Ω- and ρ -mesons must be included. These can also be includedin the Skyrme model [41, 42, 32], and it would be interesting to see their effect on the resultspresented here. To proceed, one must first understand the classical asymptotic interactionof skyrmions in models coupled to mesons, generalising the results of [15, 16]. In fact, ourapproach could be adapted to any model which treats nuclei as quantised solitons. Thisincludes holographic QCD, where nuclei are described as instantons on a curved spacetime[43].Finally, there are many modified Skyrme models. Authors have included different pionicterms [26] and used modified potentials [30, 44] in the Skyrme lagrangian. Each modifi-cation will alter the results in Figure 1. Since our calculation yields an explicit formulafor the nucleon-nucleon interaction, it would be very easy to test these modified models bycomparing their predictions for the nucleon-nucleon interaction. We believe this new testwill provide valuable insights for Skyrme phenomenology and help find the Skyrme modelwhich best describes the physics of atomic nuclei. Acknowledgments
CJH is supported by The Leverhulme Trust as an Early Career Fellow.
A The Clebsch-Gordon matrices κ mnj In this appendix we present the matrices κ mnj that were used in our calculations. We chooseconventions such that the action of the spin operators S j on H n ∼ = C n ⊗ C n is given by S j,nn ⊗ n , where S = 12 σ = 12 (cid:18) (cid:19) , S = 12 σ = 12 (cid:18) − ii 0 (cid:19) , S = 12 σ = 12 (cid:18) − (cid:19) S = 12 √ √ √
30 0 √ S = 12 − i √ √ −
2i 00 2i 0 − i √
30 0 i √ S = 12 − − . Then κ = 1 √ (cid:18) − − (cid:19) κ = 1 √ (cid:18) − i 0 (cid:19) κ = 1 √ (cid:18) − (cid:19) κ = 1 √ −√ −
11 00 √ κ = 1 √ i √ √ κ = 1 √ κ = 1 √ (cid:18) √ − −√ (cid:19) κ = 1 √ (cid:18) i √ √ (cid:19) κ = 1 √ (cid:18) − − (cid:19) κ = 1 √ −√ −√ − − −√
30 0 −√ κ = 1 √ √ − i √ −
2i 0 i √
30 0 − i √ κ = 1 √ − − . The Laplace-Beltrami operator
In this appendix we derive the equation (4.1) for the Laplace Beltrami operator. Let e µ bea frame for the cotangent bundle of a manifold and let E µ be the dual frame for the tangentbundle. Let f λµν be (locally defined) functions such that [ E µ , E ν ] = f λµν E λ . Suppose thatthe metric is given by g = g µν e µ e ν . (B.1)The standard definition for the Laplace-Beltrami operator acting on a function ψ is (cid:52) g ψ = − ∗ d ∗ d ψ, (B.2)in which ∗ denotes the Hodge star operator, defined by u ∧ ∗ v = g ( u, v ) √ ge ∧ . . . ∧ e n . Onefinds that ∗ e ν = √ gg µν ι E µ (cid:121) ( e ∧ . . . ∧ e n ) , (B.3)in which ι denotes the interior product (so that ι E e ∧ e ∧ . . . ∧ e n = e ∧ . . . ∧ e n , ι E e ∧ e ∧ . . . ∧ e n = − e ∧ e . . . ∧ e n etc.). Therefore − ∗ d ψ = −√ gg µν ( E ν ψ ) ι E µ ( e ∧ . . . ∧ e n ) . (B.4)In order to evaluate d ∗ d ψ we need to to evaluate d ι E µ ( e ∧ . . . ∧ e n ). By the Cartanstructure equations,d ι E µ ( e ∧ . . . ∧ e n ) = − ι E µ d( e ∧ . . . ∧ e n ) + L E µ ( e ∧ . . . ∧ e n ) (B.5)= 0 + ( L E µ e ) ∧ e ∧ . . . ∧ e n + e ∧ ( L E µ e ) ∧ . . . ∧ e n + . . . , (B.6)with L denoting Lie derivative. Now ι E λ ( L E µ e ν ) = L E µ ( ι E λ e ν ) − ι [ E µ ,E λ ] e ν = 0 − f νµλ , (B.7)so L E µ e ν = − f νµλ E ν . (B.8)Inserting this into equation (B.6) givesd ι E µ ( e ∧ . . . ∧ e n ) = − ( f µλ e λ ) ∧ e ∧ . . . ∧ e n − e ∧ ( − f µλ e λ ) ∧ e ∧ . . . ∧ e n − . . . (B.9)= − f λµλ e ∧ . . . ∧ e n . (B.10)Combining equations (B.4) and (B.10) gives − d ∗ d ψ = √ gg µν ( E ν ψ ) f λµλ e ∧ . . . ∧ e n − E λ ( √ gg µν ( E ν ψ )) e λ ∧ ι E µ ( e ∧ . . . ∧ e n ) (B.11)and thus − ∗ d ∗ d ψ = − E µ ( √ gg µν ( E ν ψ )) + g µν f λµλ ( E ν ψ ) , (B.12)as claimed. 24 Degenerate perturbation theory
In this appendix we derive equation (5.2) for an effective hamiltonian using perturbationtheory. Let H F be a hamiltonian with eigenvalues E < E < E < . . . . Let | ψ α (cid:105) be anorthonormal basis for the E -eigenspace. Consider a deformation of H F of the form H F + (cid:15)H I . (C.1)We seek deformed basis vectors | ψ α ( (cid:15) ) (cid:105) whose span is invariant under H F + (cid:15)H I , and suchthat | ψ α (0) (cid:105) = | ψ α (cid:105) . (C.2)In other words, we require that( H F + (cid:15)H I ) | ψ β ( (cid:15) ) (cid:105) = | ψ α ( (cid:15) ) (cid:105) H αβ ( (cid:15) ) (C.3)for some (cid:15) -dependent matrix H αβ . We also require these vectors, like | ψ α (cid:105) , to be orthonor-mal: (cid:104) ψ α ( (cid:15) ) | ψ β ( (cid:15) ) (cid:105) = δ αβ . (C.4)In this situation H αβ ( (cid:15) ) is the hermitian matrix of H F + (cid:15)H I acting on the subspace spannedby | ψ α ( (cid:15) ) (cid:105) . We can regard H αβ ( (cid:15) ) as an effective hamiltonian describing the lowest eigen-values of H F + (cid:15)H I .To calculate this effective hamiltonian one must solve the system (C.2), (C.3), (C.4).This system does not have a unique solution, as one can make the replacement | ψ β ( (cid:15) ) (cid:105) →| ψ α ( (cid:15) ) (cid:105) U αβ ( (cid:15) ), for any unitary matrix U αβ ( (cid:15) ) and still have a solution. In order to fix thisdegeneracy we impose the constraint (cid:104) ψ α | ψ β ( (cid:15) ) (cid:105) = (cid:104) ψ α ( (cid:15) ) | ψ β (cid:105) . (C.5)We seek to solve the system (C.2), (C.3), (C.4), (C.5) within the framework of pertur-bation theory. That is, we seek a solution in the form | ψ α ( (cid:15) ) (cid:105) = | ψ α (cid:105) + (cid:15) | ψ α (cid:105) + (cid:15) | ψ α (cid:105) + . . . (C.6) H αβ ( (cid:15) ) = H αβ + (cid:15)H αβ + (cid:15) H αβ + . . . (C.7)which formally solves the system to all orders in (cid:15) . In order to construct the solution werewrite the equations in an iterative form. Let Π N denote the projection onto the E N -eigenspace of H F . Equations (C.4) and (C.5) imply thatΠ | ψ α ( (cid:15) ) (cid:105) − | ψ α (cid:105) = − | ψ β (cid:105) (cid:0) (cid:104) ψ β ( (cid:15) ) | − (cid:104) ψ β | (cid:1)(cid:0) | ψ α ( (cid:15) ) (cid:105) − | ψ α (cid:105) (cid:1) . (C.8)Equations (C.3) and (C.4) imply that H αβ ( (cid:15) ) = (cid:104) ψ α ( (cid:15) ) | ( H F + (cid:15)H I ) | ψ β ( (cid:15) ) (cid:105) . (C.9)Finally, equation (C.3) implies thatΠ N | ψ β ( (cid:15) ) (cid:105) = 1 E N − E (cid:0) Π N | ψ α ( (cid:15) ) (cid:105) (cid:0) H αβ ( (cid:15) ) − E δ αβ (cid:1) − (cid:15) Π N H I | ψ β ( (cid:15) ) (cid:105) (cid:1) . (C.10)25or N (cid:54) = 0.Now we generate the perturbative solution using equations (C.8), (C.9) and (C.10). First,equation (C.9) implies that H αβ = (cid:104) ψ α | H F | ψ α (cid:105) = E δ αβ . (C.11)Equation (C.8) implies that Π | ψ α (cid:105) = 0 (C.12)since the right hand side is O ( (cid:15) ). Equation (C.9) implies that H αβ = (cid:104) ψ α | H I | ψ β (cid:105) . (C.13)Equation (C.10) implies that Π N | ψ β (cid:105) = − E N − E H N I | ψ β (cid:105) , (C.14)with H MNI := Π M H I Π N . Thus altogether we have | ψ α (cid:105) = − (cid:88) N (cid:54) =0 E N − E H N I | ψ α (cid:105) . (C.15)This completes the solution to first order. Now we compute the second order terms. Equation(C.8) implies that Π | ψ α (cid:105) = − (cid:88) N (cid:54) =0 E N − E ) H NI H N I | ψ α (cid:105) . (C.16)Equation (C.9) then implies that H αβ = − (cid:88) N (cid:54) =0 E N − E (cid:104) ψ α | H NI H N I | ψ β (cid:105) . (C.17)Finally, equation (C.10) implies thatΠ N | ψ α (cid:105) = − E N − E ) H N I H I | ψ α (cid:105) + (cid:88) M (cid:54) =0 E N − E )( E M − E ) H NMI H M I | ψ α (cid:105) , (C.18)so that in total | ψ α (cid:105) = − (cid:88) N (cid:54) =0 E N − E ) H NI H N I | ψ α (cid:105) − (cid:88) N (cid:54) =0 E N − E ) H N I H I | ψ α (cid:105) + (cid:88) M,N (cid:54) =0 E N − E )( E M − E ) H NMI H M I | ψ α (cid:105) . (C.19)26his completes the solution to second order. Finally, to third order equation (C.8) impliesthatΠ | ψ α (cid:105) = − (cid:88) N (cid:54) =0 E N − E ) ( H NI H N I H I + H I H NI H N I ) | ψ α (cid:105) + 12 (cid:88) M,N (cid:54) =0 (cid:18) E M − E )( E N − E ) + 1( E M − E ) ( E N − E ) (cid:19) H NI H NMI H M I | ψ α (cid:105) (C.20)and equation (C.9) implies that H αβ = (cid:88) M,N (cid:54) =0 E N − E )( E M − E ) (cid:104) ψ α | H NI H NMI H M I | ψ β (cid:105)− (cid:88) N (cid:54) =0 E N − E ) (cid:104) ψ α | ( H NI H N I H I + H I H NI H N I ) | ψ β (cid:105) . (C.21)Thus, our solution for H αβ is H αβ = (cid:28) ψ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) H F + (cid:15)H I − (cid:15) (cid:88) N (cid:54) =0 E N − E H NI H N I + (cid:15) (cid:88) M,N (cid:54) =0 E N − E )( E M − E ) H NI H NMI H M I − (cid:15) (cid:88) N (cid:54) =0 E N − E ) ( H NI H N I H I + H I H NI H N I ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ψ β (cid:29) + O ( (cid:15) ) . (C.22) D Detailed calculation of potential with massless pions
In this section we presents the results of evaluating each of the terms in equations (5.26)and (5.37). We include these so that the reader can cross-check their calculations, shouldthey wish to re-derive our result. − ρ (cid:88) N> E N − E D N D N = ρ Λ (cid:126) r (cid:20) − − τ τ + 427 (2 + τ τ )( σ σ − S ) (cid:21) (D.1) ρ (cid:126) Λ (cid:88) N> E N − E (cid:0) D N B N + B N D N (cid:1) = ρ Λ r (cid:20) − − τ τ + 181 (14 + 3 τ τ )( σ σ − S ) (cid:21) . (D.2)27 ρ (cid:126) (cid:88) N> E N − E B N B N = ρ (cid:126) Λ r (cid:20) − − τ τ − τ τ )( S + σ σ ) (cid:21) (D.3) − ρ (cid:126) M Λ (cid:88) N> E N − E (cid:16) D N ( P i A i + A † i P i ) N + ( P i A i + A † i P i ) N D N (cid:17) = ρ M r (cid:20) − − τ τ + ( σ σ − S ) (cid:18) τ τ (cid:19)(cid:21) + ρ M (cid:126) r (cid:20) −
43 + 49 τ τ (cid:21) L · σ (D.4) ρ (cid:126) M Λ (cid:88) N> E N − E (cid:16) B N ( P i A i + A † i P i ) N + ( P i A i + A † i P i ) N B N (cid:17) = ρ (cid:126) M Λ r (cid:20) − τ τ + ( σ σ − S ) (cid:18) τ τ (cid:19)(cid:21) + ρ (cid:126) M Λ r (cid:20) τ τ (cid:21) L · σ . (D.5)4i ρ (cid:126) M (cid:88) N> E N − E ) (cid:8) P i , ∇ i D N D N − D N ∇ i D N (cid:9) = ρ Λ M (cid:126) r (cid:18) τ τ (cid:19) L · σ (D.6) − i ρ (cid:126) M Λ (cid:88) N> E N − E ) (cid:8) P i , ∇ i D NI B N I − D NI ∇ i B N I + ∇ i B NI D N I − B NI ∇ i D N I (cid:9) = ρ M (cid:126) r (cid:18) τ τ (cid:19) L · σ (D.7)i ρ (cid:126) M Λ (cid:88) N> E N − E ) (cid:8) P i , ∇ i B NI B N I − B NI ∇ i B N I (cid:9) = ρ (cid:126) M Λ r (cid:18) τ τ (cid:19) L · σ (D.8)4 ρ (cid:126) M (cid:88) N> E N − E ) ∇ i D NI ∇ i D N I = ρ Λ M (cid:126) r (cid:20) τ τ + (4 S − σ σ ) (cid:18) τ τ (cid:19)(cid:21) (D.9)28 ρ (cid:126) M Λ (cid:88) N> E N − E ) (cid:0) ∇ i D NI ∇ i B N I + ∇ i B NI ∇ i D N I (cid:1) = ρ M r (cid:20) τ τ + ( S − σ σ ) (cid:18) τ τ (cid:19)(cid:21) (D.10) ρ (cid:126) M Λ (cid:88) N> E N − E ) ∇ i B NI ∇ i B N I = ρ (cid:126) M Λ r (cid:20) τ τ + (2 S − σ σ ) (cid:18) τ τ (cid:19)(cid:21) (D.11) E Potentials with non-zero pion mass
Setting s = m π r/ (cid:126) , the isoscalar nucleon-nucleon potentials are given by V ISC = − ρ e − s r (cid:126) (cid:0) (cid:0) s + 4 s + 10 s + 12 s + 6 (cid:1) + 16Λ r (cid:126) (cid:0) s + 2 s + 4 s + 2 (cid:1) − r (cid:126) (cid:0) s + 2 (cid:1) (cid:1) + ρ M e − s r (cid:126) (cid:0) (cid:0) s + 6 s + 27 s + 84 s + 162 s + 180 s + 90 (cid:1) + 8Λ r (cid:126) (cid:0) s − s − s − s − s − (cid:1) + r (cid:126) (cid:0) s − s + 798 s + 1704 s + 852 (cid:1) (cid:1) (E.1) V IS = − ρ e − s r (cid:126) (cid:0) (cid:0) s + 4 s + 6 s + 3 (cid:1) + 56Λ r (cid:126) (cid:0) s + 5 s + 4 (cid:1) + 13 r (cid:126) ( s + 1) (cid:1) + ρ M e − s r (cid:126) (cid:0) (cid:0) s + 8 s + 30 s + 63 s + 72 s + 36 (cid:1) + 56Λ r (cid:126) (cid:0) s + 71 s + 76 s + 36 s + 18 (cid:1) + r (cid:126) (cid:0) s − s − s − (cid:1) (cid:1) (E.2) V ISσσ = ρ e − s r (cid:126) (cid:0) (cid:0) s + 5 s + 6 s + 3 (cid:1) + 112Λ r (cid:126) (cid:0) s + 2 s + 1 (cid:1) + 13 r (cid:126) (2 s − (cid:1) − ρ M e − s r (cid:126) (cid:0) (cid:0) s + 13 s + 42 s + 81 s + 90 s + 45 (cid:1) + 112Λ r (cid:126) (cid:0) s + 32 s + 28 s + 18 s + 9 (cid:1) + r (cid:126) (cid:0) s − s − s − (cid:1) (cid:1) (E.3)29 ISSO = ρ M e − s r (cid:126) (cid:0) (cid:0) s + 3 s + 3 (cid:1) − r (cid:126) (cid:0) s + 37 s + 42 s + 21 (cid:1) − r (cid:126) (cid:0) s + 274 s + 245 (cid:1) (cid:1) . (E.4)The isovector potentials are given by V IVC = − ρ e − s r (cid:126) (cid:0) (cid:0) s + 4 s + 10 s + 12 s + 6 (cid:1) + 112Λ r (cid:126) (cid:0) s + 2 s + 4 s + 2 (cid:1) + 13 r (cid:126) (cid:0) s + 2 (cid:1) (cid:1) + ρ M e − s r (cid:126) (cid:0) (cid:0) s + 6 s + 27 s + 84 s + 162 s + 180 s + 90 (cid:1) + 16Λ r (cid:126) (cid:0) s + 57 s + 156 s + 246 s + 252 s + 126 (cid:1) + r (cid:126) (cid:0) s − s − s − s − (cid:1) (cid:1) (E.5) V IV = ρ e − s r (cid:0) (cid:0) s + 3 s + 3 (cid:1) + r (cid:126) ( s + 1) (cid:1) − ρM (cid:126) e − s r (cid:0) s − s − s − (cid:1) + ρ e − s r (cid:126) (cid:0) − (cid:0) s + 4 s + 6 s + 3 (cid:1) − r (cid:126) (cid:0) s + 5 s + 4 (cid:1) + 19 r (cid:126) ( s + 1) (cid:1) + ρ M e − s r (cid:126) (cid:0) (cid:0) s + 8 s + 30 s + 63 s + 72 s + 36 (cid:1) + 8Λ r (cid:126) (cid:0) s + 1315 s + 1148 s + 180 s + 90 (cid:1) + r (cid:126) (cid:0) s + 2355 s + 3356 s + 1678 (cid:1) (cid:1) (E.6) V IVσσ = ρ e − s r (cid:0) s + r (cid:126) ( s − (cid:1) − ρM (cid:126) e − s r s ( s − ρ e − s r (cid:126) (cid:0) (cid:0) s + 5 s + 6 s + 3 (cid:1) + 48Λ r (cid:126) (cid:0) s + 2 s + 1 (cid:1) + 19 r (cid:126) (1 − s ) (cid:1) − ρ M e − s r (cid:126) (cid:0) (cid:0) s + 13 s + 42 s + 81 s + 90 s + 45 (cid:1) + 16Λ r (cid:126) (cid:0) s + 544 s + 332 s + 90 s + 45 (cid:1) + r (cid:126) (cid:0) s + 2679 s + 2650 s + 1325 (cid:1) (cid:1) (E.7) V IVSO = ρ M e − s r (cid:126) (cid:0) (cid:0) s + 3 s + 3 (cid:1) + 16Λ r (cid:126) (cid:0) s + 224 s + 390 s + 195 (cid:1) + r (cid:126) (cid:0) s + 1330 s + 773 (cid:1) (cid:1) . 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