aa r X i v : . [ nu c l - t h ] J un Pseudospin Conserving Shell Model Interactions
Joseph N. Ginocchio
MS B283, Theoretical Division, Los Alamos National Laboratory,Los Alamos, New Mexico 87545, USA (Dated: April 11, 2018)Pseudospin symmetry is approximately conserved in nuclei. Normally shell modelinteractions are written in terms of spin an orbital angular momentum operators,not in terms of pseudospin and pseudo-orbital angular momentum operators. Wedetermine the shell model interactions which conserve pseudospin symmetry andpseudo-orbital angular momentum symmetry and write them in terms of spin andorbital angular momentum operators including the tensor interaction. We show that,although the tensor interaction by itself does not conserve pseudo-orbital angularmomentum, certain combinations of the tensor interaction with the two body angularmomentum squared interaction and the two body spin orbit interaction do conservepseudo-orbital angular momentum.
PACS numbers: 21.60.-n, 21.10.-k, 02.20.-aKeywords: Symmetry, Pseudospin, Effective field theory, Chiral perturbation theory, Rel-ativistic mean field theory, Spin, Shell model interactions
I. INTRODUCTION
Pseudospin symmetry is a relativistic symmetry of the Dirac Hamiltonian that occurs whenthe scalar potential plus a constant is equal in magnitude to the vector potential but oppositein sign [1]. This condition approximately holds for the relativistic mean fields of nuclei [2].Indeed, nuclear energy levels and transition rates in both spherical and deformed nucleiare consistent with approximate pseudospin symmetry [3]. Beyond the mean field the non-relativistic shell model with effective interactions has been very successful in describingnuclei. However, the bare nuclear interaction and the effective shell model interactionsbetween nucleons are expressed in terms of spin operators and not pseudospin operators. Inthis paper we shall determine the interactions which conserve pseudospin symmetry. Theseinteractions are written in momentum space rather than coordinate space because pseudospininvolves the intertwining of spin and momentum. However, this may be an advantage sincerecent effective interactions, including effective field theories with and without pions [4] andthe low-k effective interactions [5], are written terms of momentum.
II. PSEUDOSPIN SYMMETRY
Pseudospin symmetry is an SU(2) symmetry as is spin symmetry. The relativistic generatorsfor the pseudospin algebra, ˜ S i,k ( i = x, y, z ), where k is the nucleon number, are [6]˜ S i,k = ˜ s i,k s i,k = U p s i,k U p s i,k , (1)where s i,k = σ i,k / σ i the Pauli matrices, and U p = σ k · ˆ p isthe momentum-helicity unitary operator [7], and ˆ p i,k = p i,k p is the unit three momentum ofa single nucleon. The four by four nature of the generators results from the fact that theyare relativistic generators. The generators for the non-relativistic pseudospin algebra are˜ s i,k = U p s i,k U p = 2 s k · ˆ p ˆ p i,k − s i,k . (2)We note that, although the pseudospin generators depend on momentum, they depend onthe unit vector of momentum and therefore are equivalent to spin as far as momentum powercounting in effective field theory.For the two nucleons, the total non-relativistic pseudospin is˜ s i = ˜ s i, + ˜ s i, = U ,p s i, U ,p + U ,p s i, U ,p = U ,p U ,p s i U ,p U ,p (3)where the total spin is s i = s i, + s i, . In the rest frame p i, = p i , p i, = − p i , where p i is therelative momentum. Hence the generators are˜ s i = U ,p U ,p s i U ,p U ,p = 2 s · ˆ p ˆ p i − s i (4)Likewise the non-relativistic pseudo-orbital angular momentum is [3]˜ ℓ i = σ · ˆ p σ · ˆ p ℓ i σ · ˆ p σ · ˆ p (5)where the orbital angular momentum is ℓ i = ( r × p ) i ~ where r is the relative coordinate. III. PSEUDOSPIN SYMMETRY CONSERVING INTERACTION
The general interaction which conserves pseudospin, pseudo-orbital angular momentum, andthe total angular momentum, j i = ˜ ℓ i + ˜ s i , is V ( p ) =( ˜ V (0) c ( p ) + ˜ V (0) ps ( p )˜ s · ˜ s + ˜ V (0) po ( p )˜ ℓ · ˜ ℓ + ˜ V (0) pso ( p )˜ s · ˜ ℓ ) (1 − τ · τ )4 +( ˜ V (1) c ( p ) + ˜ V (1) ps ( p )˜ s · ˜ s + ˜ V (1) po ( p )˜ ℓ · ˜ ℓ + ˜ V (1) pso ( p )˜ s · ˜ ℓ ) (3+ τ · τ )4 , (6)where τ are isospin Pauli matrices and we include the possibility that the coefficients V ( T ) A ( p ), A = c, ps, po, pso , could depend on isospin, T = 0 , V ( T ) pso ( p ) = 0 then pseudospin and pseudo-orbital angular momentum are invariant sym-metries; that is, the eigenfunctions have conserved pseudospin and pseudo-orbital angularmomentum quantum numbers and the energies do not depend on the orientation of thepseudospin. In this case the generators in Eq (4) and Eq (5) commute with the interaction.If V ( T ) pso ( p ) = 0 then pseudospin and pseudo-orbital angular momentum are dynamical sym-metries; that is, the eigenfunctions have conserved pseudospin and pseudo-orbital angularmomentum quantum numbers but the energies are not degenerate. In this case the gener-ators in Eq (4) and Eq (5) do not commute with the interaction. This is the most realisticpossibility. IV. PSEUDO-OPERATORS IN TERMS OF NORMAL OPERATORS
Since nuclear interactions are usually written in terms of spin and orbital angular momen-tum (normal operators), we rewrite the pseudospin and pseudo-orbital angular momentumoperators (pseudo-operators) in the interaction in Eq (6) in terms of these normal operators.First we consider the pseudospin-pseudospin interaction˜ s · ˜ s = ( σ · ˆ p ˆ p − s ) · ( σ · ˆ p ˆ p − s ) = σ · ˆ p σ · ˆ p − σ · ˆ p s · ˆ p − σ · ˆ p s · ˆ p + s · s , (7)which leads to ˜ s · ˜ s = s · s ; (8)that is, the two are equivalent. This is consistent with the study of the nucleon-nucleoninteraction [8] in which it was shown that the pseudospin transformation on two nucleons doesnot change the spin. However, the mixing angle between states with the same pseudospin butdifferent pseudo-orbtial angular momentum can be different than the mixing angle betweenstates of with the same spin but different orbital angular momentum, which comes aboutthrough other terms involving the pseudo-orbital angular momentum operator and the orbitalangular momentum operator.Furthermore, from Eq.(3), the tensor interaction becomes σ · p σ · p = p [˜ s · s + s · ˜ s + ˜ s · ˜ s + s · s ] . (9)That is, the tensor interaction is symmetrical in pseudospin and spin and it is an interactionbetween the spin and pseudospin, which is an interesting insight.We consider now interactions involving the pseudo-orbital angular momentum. The twobody pseudospin-pseudo-orbit interaction becomes˜ s · ˜ ℓ = − s · ℓ + σ · ˆ p σ · ˆ p + 1 − s · s, (10)This means that this interaction can be written in terms of the two body spin-orbit interac-tion and the tensor interaction. On the other hand the pseudo-orbital angular momentumsquared is ˜ ℓ · ˜ ℓ = ℓ · ℓ + 4 s · ℓ − s · s − σ · ˆ p σ · ˆ p (11)which means that the pseudo-orbital angular momentum squared can also be written interms of the the orbital angular momentum squared, two body spin-orbit interaction andthe tensor interaction. So the pseudospin and pseudo-orbital angular momentum do notintroduce any new terms that are not already present with the spin and orbital angularmomentum. Hence the interaction in Eq.(6) can be written in terms of operators involvingspin and orbital angular momentum. V. THE INTERACTION IN TERMS OF NORMAL OPERATORS
Using the relations between peudo-operators and normal operators given in Eq.(8)-Eq.(11)we can rewrite Eq.(6) in terms of normal operators. This interaction becomes V ( p ) =( V (0) c ( p ) + V (0) s ( p ) s · s + V (0) o ( p ) ℓ · ℓ + V (0) so ( p ) s · ℓ + V (0) t ( p ) σ · ˆ p σ · ˆ p ) (1 − τ · τ )4 +( V (1) c ( p ) + V (1) s ( p ) s · s + V (1) o ( p ) ℓ · ℓ + V (1) so ( p ) s · ℓ + V (0) t ( p ) σ · ˆ p σ · ˆ p ) (3+ τ · τ )4 , (12)with V ( T ) c ( p ) = ˜ V ( T ) c ( p ) + ˜ V ( T ) pso ( p ) − V ( T ) po ( p ) (13) V ( T ) s ( p ) = ˜ V ( T ) ps ( p ) + 4 ˜ V ( T ) po ( p ) − V ( T ) pso ( p ) (14) V ( T ) o ( p ) = ˜ V ( T ) po ( p ) (15) V ( T ) so ( p ) = 4 ˜ V ( T ) po ( p ) − ˜ V ( T ) pso ( p ) (16) V ( T ) t ( p ) = ˜ V ( T ) pso ( p ) − V ( T ) po ( p ) (17)In particular the tensor interaction is part of the interaction which conserves pseudo-orbitalangular momentum, although by itself the tensor interaction breaks pseudo-orbital angularmomentum. The conservation comes with a certain combination of tensor interaction, two-body angular momentum squared, and two-body spin-orbit. In fact from the relations abovewe derive, V ( T ) t ( p ) + V ( T ) so ( p ) − V ( T ) o ( p ) = 0 . (18)This also implies that for V ( T ) so ( p ) = 2 V ( T ) o ( p ) , (19)the tensor interaction vanishes and both spin and pseudospin are conserved. VI. SUMMARY
We have shown that there exist shell model interactions with tensor interactions which con-serve pseudospin and pseudo-orbital angular momentum. These interactions have a relationbetween the strength of the tensor interaction, the strength of the two-body orbital angularmomentum squared, and the strength of two-body spin orbit which is given in Eq.(18). Theseinteractions include those for which pseudospin and pseudo-orbital angular momentum aredynamical symmetries, which are the most realistic interactions.The tensor interaction has been shown to be important for shell evolution in exotic nuclei[9]. At the same time pseudospin doublets are also seen in these nuclei [10]. Perhaps theinteractions discussed in this paper will be able to explain both effects in a unified way.This research was supported by the United States Department of Energy under contractW-7405-ENG-36. [1] J.N. Ginocchio, Phys. Rev. Lett. , 436 (1997).[2] B. A. Nikolaaus, T. Hoch, and D. G. Madland, Phys. Rev. C 46 , 1757 (1992).[3] J. N. Ginocchio, Physics Reports , 165 (2005).[4] P. F. Bedaque and U. van Klock, Annu. Rev. Nucl. Part. Sci. , 339 (2002).[5] S.K. Bogner, R.J. Furnstahl, and A. Schwenk, Prog. in Part. and Nucl. Phy. , 94 (2010).[6] J. N. Ginocchio and A. Leviatan, Phys. Lett. B 425 , 1 (1998).[7] A. L. Blokhin, C. Bahri, and J. P. Draayer, Phys. Rev. Lett. , 4149 (1995).[8] J.N. Ginocchio, Phys. Rev. C 65 , 054002 (2002).[9] T. Otsuka and D. Abe, Prog. Part. Nucl. Phys. , 425 (2007).[10] O. Sorlin and M. G. Porquet, Prog. Part. Nucl. Phys.61