VOLUME 74, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 1995

Origin of Pseudospin Symmetry

A. L. Blokhin,* C. Bahri,† and J. P. DraayerDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803-4001

(Received 15 August 1994)

A many-particle operator that affects a transformation to the pseudospin basis in heavy nuclei isidentified. Both mean-field and many-particle estimates demonstrate that in the helicity-transformedrepresentation the nucleons move in a finite-depth nonlocal potential with a reduced spin-orbit strength.Because of the close relation between the helicity and chirality operations, the results suggest that thepseudospin symmetry in heavy nuclei yields to the chiral symmetry of massless hadrons in the highenergy region.

PACS numbers: 24.80.Dc, 11.30.Rd, 21.30.+y, 21.60.Cs

(1) Introduction.—The pseudo spin-space concept innuclear theory [1,2] refers to a division of the single-particle total angular momentum into pseudo �j � l 1 s�rather than normal �j � l 1 s� orbital and spin parts.Such a division is favored by the observed approxi-mate degeneracy of pairs of single-particle states, ��l 2

1�j , �l 1 1�j11� with j � l 212 , within the major shells of

heavy �A $ 100� nuclei, where the spin-orbit interactionis known to be strong [3,4]. [The latter stands in con-trast to the relatively weak splitting of normal spin-orbitdoublets, �lj , lj11� with j � l 2

12 , in light �A # 28� nu-

clei.] By assigning new lj and lj11 labels to the �l 2 1�j

and �l 1 1�j11 states (for example, a �d5�2, g7�2� pair is as-signed new � f5�2, f7�2� “pseudo” labels), an advantage isgained because these paired states can then be interpretedas members of a weakly split pseudospin doublet. Indeed,the pseudospin and pseudospace quantum numbers appearto be reasonably well conserved, supporting a picture ofheavy nuclei as systems with weakly broken (dynamical)pseudospin symmetry. The oscillator shell model of allpseudospin doublets within a major shell, augmented witha quadrupole-quadrupole residual interaction, leads to themany-particle pseudo-SU(3) theory for heavy deformednuclei [1,4,5]. The decoupling of the pseudospace andpseudospin degrees of freedom has also been suggested asa possible explanation for the existence of identical bandsin superdeformed nuclei [6].

Good pseudospin symmetry in heavy nuclei, while ex-perimentally well corroborated and successfully used innumerous theoretical applications (see [5] for references),still lacks a sound microscopic justification. The usual un-derstanding is based on the single-particle Hamiltonian ofthe oscillator shell model, namely, on the “accidental” re-sult that deviations from the oscillator spectrum approxi-mately follow a 2j�j 1 1� 2 l�l 1 1� dependence, whichtransforms into l�l 1 1� under the normal ! pseudo re-labeling. Relativistic mean-field estimates were presented[7] in support of such a dependence in the limit of largenucleon numbers. Also, a unitary operator was proposed[8] which acts on the spin and angle variables and accom-plishes the normal ! pseudo relabeling within a given

shell. Later this approach was revisited [9] and resulted inthe introduction of another operator which is specificallydesigned for shell-model applications, being unitary onlywithin the normal-parity subspace of the oscillator [4,8].

This paper shows that the pseudospin symmetry, whichreveals itself on the single-particle (mean-field) level, hasa microscopic origin which is related to the nature ofthe internucleonic forces, perhaps with roots in chiralsymmetry. A microscopic transformation that is differentfrom those mentioned above [8,9] while reducing to themwhen restricted to a single major shell, is shown to fulfillkey requirements for the pseudospin transformation whenapplied to the nucleus as a whole.

(2) Microscopic pseudospin transformation.—To in-corporate both the single-particle and many-particle as-pects of the pseudospin picture, a microscopic operatorthat accomplishes the normal ! pseudo transformationshould be of the form

Utotal �AY

i�1U�ri , pi ,si� , (1)

where ri stand for the position, pi for the momentum,and s i for the Pauli spin matrices of the individ-ual nucleons. The structure of U�r, p,s � is fixedby the following constraints: (a) l2 � Ul2U21 �l2 1 2l ? s 1 2 � 2j2 2 l2 1

12 (this sets the transforma-

tion rule [8]); (b) �U, j� � 0(rotational invariance); (c)�U,P � � �U,T � � 0(parity and time-reversal symme-try); (d) UUy � UyU � 1(unitarity and conservation ofobservables); and (e) �U, p� � 0(translational invariance).

Once constraints (a), (b), (c), and (d) are applied, onlythree distinguishable choices for U remain:

U � �d ? dy�21�2d,

d � �cosu r0p 1 i sinu r�r0� ?s , (2)

where r0 is a characteristic length, and due to the optionof rescaling r0, the value of u can be fixed at 6

p

4 , 0, orp

2 . The first choice yields the boson annihilation (1 p

4 )�creation (2 p

4 ) operator form that is specifically designedfor shell-model applications [9]. However, these twooperators are unitary only within a subspace of normal

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VOLUME 74, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 22 MAY 1995

parity states while in the unique parity subspace theiraction is undefined. When global unitarity is required, onlytwo possibilities remain. The case u �

p

2 correspondsto the Ur � i s ? r�r operator (henceforth called the rhelicity) that was proposed in [8]. The u � 0 choiceis the p helicity Up � s ? p�p. This is the only formthat is compatible with the constraint of translationalinvariance and thus consistent with a realistic many-particle theory that is not confined to the shell-modelapproach. Additional arguments in favor of this choiceare given below.

(3) Single-particle Hamiltonian and wave functions.—If �1� accomplishes the pseudospin transformation, thenin addition to satisfying general constraints, it shoulddecouple the spin and orbital degrees of freedom inthe heavy nuclei. The applicability of the mean-fieldapproach allows for a reasonable direct check on this byconsidering transformed single-particle Hamiltonians andwave functions. Corrections for center-of-mass motionare relatively small in heavy nuclei and therefore notexpected to worsen such an argument.

For simplicity a spherically symmetric field is consid-ered. In this case the Hamiltonian and its wave functionsare given by

H �p2

2M1 V �r� 1 W�r�l ? s , (3)

cnjlm�r� � ilRnjl�r��Yl ≠ x�jm , (4)

where n is the radial quantum number (number of nodes),Yl is a spherical harmonic, and x is a Pauli spinor�s �

12 �.

In a coordinate representation the p helicity has thefollowing operator form:

Up � 2iK�l 2 L 2 1�21r�s ? =� , (5)

where

K � G

√l 1 L 1 2

2

!G

√l 2 L

2

!

3

"G

√l 1 L 1 3

2

!G

√l 2 L 2 1

2

!#21

is unitary, l �12 ��1 1 4l2�21�2 2 1� has the orbital mo-

mentum as its eigenvalues, L � r ? = � r≠�≠r generatesshear, and G�x� denotes the gamma function. The unitarityof K follows from the conjugation rules Ly � 2�L 1 3�and ly � l. Then (3) and (4) transform into

Hp � UpHUyp �

p2

2M1 V �r� 1 W �r�l ? s , (6)

c�p�njlm�r� � Upcnjlm�r� � il Rnjl�r��Yl ≠ x�jm , (7)

where V �r� and W �r� are now strongly nonlocal functionsgiven by

V �r� � K�V �r� 2 2W�r� 2 �l 1 1�y�r��Ky , (8)

W�r� � K�y�r� 2 W�r��Kyl ? s , (9)

with

y�r� � �l 2 L 2 1�21�rV 0�r� 2 �l 1 2�rW 0�r��

3 �l 1 L 1 2�21

(primes denoting derivatives) and

Rnjl�r� � 2 G

µl 1 L 1 3

2

∂G

√l 2 L

2

!

3

"G

√l 1 L 1 3

2

!G

µl 2 L

2

∂#21

Rnjl�r� .

Although (6) in its general form does not provide incon-trovertible evidence for a reduction in the magnitude ofthe spin-orbit splitting (see below), the latter is likely tohold at low l within the nuclear surface region so long asthe effective value of the L operator exceeds l 2 1.

To qualitatively understand the behavior of Rnjl�r�, ob-serve that the Up transformation involves three consecu-tive operations: a Fourier transform, a switch from l tol, and an inverse Fourier transform, which together deter-mine the mapping for the radial function. This mappinggenerates the following universal behavior:

Rnjl�r� ~

(rl , r ! 0,r2�l13�, r ! `.

(10)

The standard rl dependence in the interior region followsbecause deep in the bulk of a heavy nucleus V �r� is notexpected to deviate significantly from the flat behaviorof V �r�. The r2�l13� asymptotic behavior means a morediffuse surface. This and strong nonlocalities in thesurface region come with the p-helicity transformation.

(4) Dirac-Brueckner approach and the helicity trans-formation.—A relativistic extension of the Bruecknertheory (see [10] for references) provides parameter-freemicroscopic predictions for both infinite and finite nu-cleon systems. While this approach gives a good de-scription of nuclear matter, the gross features of finitenuclei (especially light species) are reproduced less well,but nonetheless much better than in nonrelativistic theo-ries [11]. For this reason, results of Dirac-Brueckner nu-clear matter calculations are used below for examining thep-helicity transformed two-body nuclear interaction, aswell as the mean field, in heavy nuclei.

For a wide range of nuclear densities, including thesaturation point, the nucleon-nucleon interaction in theinfinite medium is approximated perfectly by a one-bosonexchange potential (OBEP) with the boson parametersfitted to the Bonn model and the density-dependenteffective nucleon mass M� calculated in a self-consistentmanner [10]. And to a very good approximation, thedensity-dependent self-consistent field has the sameLorentz structure as the free Dirac Hamiltonian. Con-sequently, a single-particle Hamiltonian in the mediumcommutes with the p helicity, and the helicity transfor-mation does not affect the single-particle energies.

However, the two-body interaction changes dramati-cally. In the representation of plane-wave Dirac spinors

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for nucleon states, normalized to unity, the p-helicity op-eration is equivalent to ig5S when acting on right (ket)states. Here g5 is the usual product of Dirac matri-ces (g5 � ig0g1g2g3), and S is a formal operation forswitching the sign of the effective mass [Sf�M�, p� �f�2M�, p�]. (Thus there is no difference between the chi-ral and the helicity transformations in the M� ! 0 limit.)Since the g matrices change sign under the g5 gener-ated chiral transformation, and the OBEP is bilinear inthose matrices, the helicity transformation of the OBEPis reduced to changing the sign of M� in the momen-tum representation. This is easily accomplished in thetwo-nucleon center-of-mass frame and produces stronglyincident-energy dependent and therefore nonlocal inter-actions. Because only a rough estimate for the poten-tials is required for this analysis, here these potentials areconverted into local approximations by averaging over al-lowed values of the relative momentum q with an appro-priate distribution of q at a fixed momentum transfer k.The localized helicity-transformed OBEP in the momen-tum space is found to converge rapidly in the shortwaveregion �k . 2kF� to the initial potential, averaged with thesame distribution. The values of the localized central partof the internucleon potential also coincide at k � 0 be-fore and after the transformation in accordance with thehelicity-invariance of the single-nucleon energy in the in-finite medium.

The localized estimates for transformed single-particlepotentials in coordinate space, shown in Figs. 1 and 2,were calculated in first order perturbation theory with re-spect to dV�k�, the localized difference between the trans-formed and initial OBEP. Unperturbed potentials weretaken in the standard Woods-Saxon parametrization [12]

FIG. 1. Localized estimates for the neutron central and spin-orbit potentials of 208Pb before and after the helicity transfor-mation (continuous lines and shaded areas, respectively). Thetwo curves that define the borders of the shaded areas were de-termined by using different reasonable approximations for therelative momentum distribution in a finite nucleus.

with a slight adjustment for the radial dependence whichallows for a simple analytic Fourier transform along witha quantitative fit. The estimate was done analytically us-ing a zero-order nuclear density distribution of the sameparametrization but with a lesser diffuseness [3] and aSkyrme-type low momentum expansion for dV�k�. Be-cause of the strong nonlocality of transformed OBEP, theanalytic formulas for single-particle potentials are morecomplicated than in a conventional scheme with Skyrmeforces [13]. Basic complications and approximations ofthis analysis are the following: (a) d and f waves of therelative motion make an impact that is on the same or-der of magnitude as the normally included s and p waves;(b) dG�k�, a difference between the localized G matrixin the transformed space and the physical G matrix, co-incides with dV�k� in the first order because changes inthe short wavelength region are small (see previous para-graph); (c) the ratio of proton and neutron densities istaken equal for all r, and Coulomb corrections are notconsidered; (d) M� and kF are fixed at their saturationpoint values [10].

Although the single-nucleon potentials shown in thefigures are only rough local estimates for strongly non-local fields, they display several features that are char-acteristic of pseudospin symmetry. First, in accordancewith Sec. 3, the transformation preserves the finite depthof the central potential and increases the surface diffuse-ness. And because the kinetic energy is conserved by thetransformation, this in turn implies that the transformedradial wave functions associated with higher energy or-bitals (which are most important for heavier nuclei) mustbe localized at a larger radial distance than for the corre-sponding conventional functions. Second, a minimum of

FIG. 2. The same as Fig. 1 but in this case for protons.

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the spin-orbit potential, which is located in the surface re-gion in the normal representation, gets shifted deeper intothe bulk as a result of the helicity transformation. Andfrom this it follows that the magnitude of the spin-orbitpotential in the region where the wave functions are local-ized and which is primarily responsible for the interactionstrength, exhibits a dramatic decrease. Also note that theeffective pseudo spin-orbit interaction of the neutrons ismore repulsive than one of the protons—in consonancewith experiment [8].

(5) Conclusions.—The microscopic origin of thepseudospin symmetry is considered. The many-particlep-helicity operator is found to be the only one that gen-erates the normal ! pseudo relabeling of the spin andorbital momenta while satisfying all other global symme-try requirements. In addition, it is shown to transformwave functions in a physically reasonable manner and toeffectively compensate for the single-particle spin-orbitinteraction strength that is observed in the normal (notpseudo) picture.

The approximate independence of the single-nucleonspectrum in an infinite medium on the helicity trans-formation and the consistency of the microscopic es-timates for the single-particle nuclear potentials withthe Dirac-Brueckner calculations, is used to connect thepseudospin symmetry to the boson-exchange nature ofnucleon-nucleon interaction. Based on the results of thatanalysis and because of the close relation (coincidence inthe chiral symmetry limit) of the helicity and chirality op-erations, the goodness of pseudospin symmetry may beexpected to increase with rising densities (or energy perparticle) in hadronic systems, and actually yield to chiralsymmetry in the region of asymptotic freedom.

The authors acknowledge useful discussions with W.Nazarewicz and W. Greiner. This work was supported

by Grants (PHY-9312628 and PHY-9317377) from theU.S. National Science Foundation and the DeutscheForschungsgemeinschaft (DFG).

*On leave from Bogolyubov Institute for TheoreticalPhysics, Academy of Sciences of Ukraine, 252143 Kiev,Ukraine.

†Current address: Department of Physics, University ofToronto, Toronto, Ontario, Canada M5S 1A7.

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