Dynamical supersymmetry and collective nuclear structure physics

Nuclear Physics A421 (1984) 1679188~ North-Holland, Amsterdam

DYNAMICAL

SUPERSYMMETRY

Da Hsuan FENGt, Physics Division,

AND CDLLECTIVE

Naticnal

Science

NUCLEAR STRUCTURE

Foundation,

PHYSICS*

Washington

D.C.

20550

Hong-Zhou SUN Department of Physics, Eeijing University Beijing, People's Pepublic of China Michel VALLJEPES and Rcbert GILMDRE Department of Physics and Atmospheric Science Drexel University, Philadelphia, Pennsylvania A. FRANK and P. van ISACKER Centro de Estudios Nucleaires,

UNAM, Mexico

10104

D-F.

The symmetry principles of the Interacting Eoson Model (even-even nucleus) and the Interacting Bose-Fermi Model (even-odd nucleus) are reviewed. Dynamical supersyruhetry is introduced as the unifying symnetry for the Experimental tests of dynamical supersymmetry above two nuclear systems. are discussed in detail.

Although

not

tion

in one

part

fitting

of nature:

workshop,

which

Jiang-Nan

is held

(River South)

and

Fermi Model

(18FM).2

the

for the

In the present

symnetries

much

an

Oriental

of symnetry

discussed

is extensively city

of

concept.

and its manifesta-

Suzhou

in the

in this heart

of

Republic of China. about the Interacting

even-odd

nucleus,

the

Boson Model

Interacting

Bose-

talk, we wish to further this discussion

for the IEM and the IBFM.

can be achieved via supersynmetry briefly

very

exquisite

in this gathering,

a unifying s~metry

In Sec. 2, we will

nucleus,

in the People's

its counterpart

by seeking

is

that the beauty

in this

We have learned much, (IEW)l

symmetry

exclusively,

it is most

Therefore,

Such a unification

(SUSY).3

discuss

the

origins

of the

IBM and

the

IBFM

and indicate also a possible origin of supersyronetry in the nuclear

structure. experimental

The following

section,

tests of supers~etry.

Sec. 3, is devoted

to the discussion

Finally, a conclusion

is given

of the

in Sec. 4.

*Work partly supported by the National Science Foundation under grants PHYB304368 and INT83-05938. Based on the two invited talks of Da Hsuan Feng and Hong-Zhou Sun. Part of this material is also presented by DHF as an invited talk at the 1st Asia Pacific Physics Conference, Singapore, June, 1983. ~Pe~anent address: Drexel University, Philadelphia, leave-of-absence for the academic years 83185.

0375-9474/84/$03.00 @ Elsevier Science Publishers (Non-Homed Physics ~b~~g Division)

B.V,

Pennsylvania

19104.

On

D.H. Feng et al. / Llynamicat ~u~er~.vmmetry

168c

2. DYNAMICAL 2.1.

SYMMETRIES

FOR THE NUCLEUS

Symnetry of the even-even

The

symmetry,

or

group

(e-e) nucleus

structure,

which

arises

frcm the assunption

that the valence

couple

strongly

with

in pairs

regarded

(in the model)

bilinear

combination

R=O)

of the creation

this quantum

so-called

dynamical

particle

physics

formula

is

nuclear

an

mechanical

community

invariant

under

additional

invariant

symmetry,

as demanded

because

U(6)

Mathematically, so-called

problem,

(the

means

Arima

derivation of

by the system,

(now)

knows

be

that

operators,

introduced

the

Bell-Mann

Here,

broken

=

W(3)

U(6)

=

0(6)(c"SU(4))

U(6)

ZP

SU(5) =

rotational; of

states. 5 the

three

hamiltonian

=

the

O(6),

so-called

Thus,

collective is then

carry

out the

here.

Suffice

This

by

the

O(3),

nuclear there

the

system.

are

O(3)

three

(Ia) =

O(5) =

O(5) =

O(3)

(Ib)

O(3)

(Ic)

“classical

starting

means

that

below

described

state for this system

analysis

is regarded

from

of

the

all the

is well

via

interpretation

which the

nucleus.

(finite

documented

certain

by a finite

number

boson

The procedure

and will

not

energy,

of bosons.

be

bz . ..bAlO>

to

repeated

of the IBM, as

only to the e-e nucleus. an

e-e

"IBM"

nucleus

The wavefunction

is

of any

is schematically: E(constant)bi

for

invar-

in O(3).

it applicable

excitation

full

Casimir

here that the inherent assumptions render

coherent

symmetry

The

nunber)

is a conse-

U(6)

as the "unifying"

fran U(6) and ending

IBM calculations it to mention

vibrational)

limit"

manifestations

constructed

in the above discussions,

completely

SU(5),

Y unstable;

the U(6) symmetry

iants for each group

given

the

to be

until the lowest

It is now further known that each chain has its own geanetrical

quence

mass

to

group chains:

U(6)

W(3),

0kubo4

the

it is the O(3) group

for to

the

to

constructed

(or subgroups)

preserved

known

Applied

hamiltonian,

U(6)

are The

st and s (for

IachelloI

systematically

fran

pairs

of the Lie algebra U(6).

symmetry).

is reached.

must

It

and d(e=Z).

is well

of

is

These

which

dynamical

that the boson

is U(6).

the closed shell

2.

n(e=O)

and

a method

transformation,

invariance

one

or

and annihilation

terms with subsymmetries

rotational

outside

0

with signature

method,

example

case, this method

nucleons

form the 36 generators

symnetry

excellent

IBM possesses

momentum

as pure bosons

and dt and d (for l=2)

To solve

angular

the

(2)

D. H. Feng et al /Dynamical

NB the total number of bosons and

where bt can either be an st or dt operator, the vacuum

IO> the nearest

closed

to the left of Fig. 1 represent

shell

169~

Supersymmetry

for the nuclei.

The states

furthest

the states of such a nucleus.

-___-

Ns=N NF=O

-

A

NB=N-I

NB=N-1

Np=l

N,=O

A+1

A+2

FIGURE 1 Schematic

2.2.

Symmetry of the even-odd

On the other

is a mixed

unpaired

boson-fermion

Iachello

Bose-Fermi

(e-o) nucleus

the same highest

have a single

system,

for A, A+1 and A+2 nuclei

hand, the low lying states of the e-o nucleus

its e-e neighbor, must

spectrun

and

symmetry

nucleon

system.

Scholten

U(6).

which cannot

To account

extended

the

Model (IBF~J!).~ As one would

cannot have, as

By definition, be described

for the unpaired

fermion

IBM to the so-called

expect,

this system

as a boson.

It

in the

Interacting

the full IBFM hamiltonian

has

three parts:

"IBFM where

HIBM

represents

is self

q

of the IBFM hamiltonian signature

bilinear

products

the generators

"sp

+

"BF

H is the single particle haniltonian and VBF sP boson-fetmion interactions. Clearly, all three parts

must also satisfy

known that given

manentun

+

evident,

all possible

It is well

"IBM

j,,j,,....,jk

where

of the creation

of the U(m)

rotational

some single particle

algebra

k

is

invariance. level scheme with angular

the number

and annihilation

of such

operators

with m the total number

levels,

the

ai and a. form of degenerjacies.

17oc

D.il. Feng et al. / D~~mrnical

Therefore,

the

UB(6)xUF(m), bosons

highest

where

symmetry

B(F)

is

such

here

a

system

to denote

is

the

the

product

group

were carried

calculations

out by a brute

are inherent difficulties number

particle mately

of

force method

which

transition

and

IBFM are, single

has only in general,

particle

another

for the e-o nucleus, For example,

explore

An added

change

complication

here

highest

is no "universal" single particle

This

accordingly.

appropriate

to O(3) there

that the highest

the appropriate

then

single

particle

symnetry

for the

are many,

An

e-o nuclei. starting

with,

is that

implies

that

as in the IBM.

contribution

is UB(6)xUF(12).

On the

levels

for an odd proton

in the

The resulting

are many

Spin(3)B

(non-unique)

say,

the

table to

symmetry

There

is UB(6)xUF(20).

for the

levels for the odd parity states

sI,2, d3,2, d5,2 and g,,2.?

in order to reach the so-called

in

Unlike the

chains

This means

symmetries

physical

than the IBM be-

in this volume).

are

(energy

symmetry

in the system.

isotopes are p3,2, pI,2, and f5,2 (see Casten's

platinun mass region

of single in approxi-

fran one region of the periodic

of m will

the appropriate

the dynamical

for the platinun

hand,

interaction

meaningful

(1)). the possible

(see Eq.

greater.

there

result

here is more canplicated

level scheme changes

and thus the value

of

of data points

makes

degree of freedom

chains

much

This

is to

IBt!,

fermionic

three

the nunber

data).

the situation

cause of the additional

number

to the nmber would

there

one finds that

indeed.

way, as in the

However,

the problem.

level model

is roughly

transfer

very difficult

An alternate

IBM, which

which

the

is proportional

a three

However,

Empirically,

is approximately

For example,

?C or so parameters

interpretation

of diagonalization.

with such calculations.

parameters,

levels.

spectra,

group

for

of the IBFM for the e-o nuclei with HIBFM

terms in VBF, for the IBFM hamiltonian

other

algebra

appropriate

(fermions).

Some of the original

the

of

used

Supers_ptnmerry

example

UB(6)xUF(12)

ways

group,

symmetry

to break

such

the analogus

group

of such a chain,

as we have

and

just mentioned,

is: UB(6)xUF(12)

=

A

great deal of point

out

represented 2.3.

UB(6)xUF(6)xUF(2) 3

0B+F(6)xSUF(2)

decompositions. to

=

0B+F(5)xSUF(2)

mathematical We will

here

that

=

technology

return the

=

0B(6)xOF(6)xSUF(2) 0B+F(3)xSUF(2)

now exists

to this discussion states

given

in

-

Spin(3)

to carry out such chain

in Sec. 3.

the

(4)

middle

of

We would

like

Fig.

are

1

by the IBFM.

Supersymmetry

(SUSY):

unification

of the IBM and the IBFM symmetries

171c

D.II. Feng et ul. / Dvnumicai Supersymmetry

Clearly,

the bosonic

description

can only be an approximate Therefore,

fermions!). description

where

begins

UF(12) symnetry,

which

the

could

is the highest

can be raised.

conceivably

symmetry

begins

for the s-d mode?

physics space.

with the

space,

only as effective,

Iachello postulated

symmetry3,

This is quite

the mode?

model

its symmetry

a full

symmetry

in nuclear

within

on the N(3)

This is whether

In 1X0,

symnetry.

IBFM

are not

be the highest

symnetry

is recognized

(bosons

not admitting

symmetries

serve the role of a symmetry

higher symnetry. such a (higher)

highest

work of Elliott'

Once the IBM (IBFP) symmetry possibility

cannot

studies of dynamical with

For extmp'le, the seminal

IBM (IBFM),

IBV or the

system

symnetry,

it is only an "effective"

In a sense,

usually

(UB(6)xUF(m))

in the

from the previous

one

in either

one for the real physical

the UB(6)

of the nucleus

for the system. different

for the nucleus

an intriguing

(U@(6) or UB(6)xUF(m))

breaking

mechanism

of a still

the concept of supersymnetry

as

i.e.:

IBM symmetry IBFM s~metry

In a nut-she??, system.

This

discovered

SUSY

is the transformation

is a radically

in nuclear

different

physics

property

symmetry

(or physics,

fran

of a bose-fermi

all previous

for that matter).

mixed

symmetries

It is we?? known

that in a fermionic system with k orbits j,,j,,...,j,, the bilinear operator + are the generators of the UF(m = z 2j Clearly, the + 1) algebra. ajaj fermions transform among themselves only. All'the standard fetmionic models in physics operators the

have this bte

bosons)

bL

property.

for a bosonic

system,

the bilinear

(where 'I and L2 are the appropriate

Likewise,

angular

momenta

dre fh e generators

degeneracy).

Here the

sentation

the

for

for the

bosons

UF(m)

UB(n)

transform

algebra

is

algebra

(where

among themselves

antisymmetric

n is the

level

The repre-

only.

wtiile for

for

UB(6),

it

is

symmetric.

If, btaj,

in addition then

these

superalgebra.

to b'b and a).a., one adds on also the generators four

types

Technically,

0; &nerators one

refers

to

will the

form

the

generators

so-called btb

bosonic and afb and btaj as fermionic. The irreducible been

extensively

Young Supertableaux:

representation studied

by

(IR) of the U(n/m) superalgebra

Bars and 6alanteken.l'

They denoted

and

a!b and $(n/m) af.a J j

as

has recently the IR as a

172~

D.H. Feng et al. / Dynamical Supersymmetry

where

N is the total number

representation,

of bosons

one has the following

0 where each pair in the parenthesis

and fermions

in the system.

In such a

c~ponents:

and so on.

..I)

on the r.h.s. represents

a system of bosons

and fermions with fixed N. A special universal

property which the U(n/m)

to

all

commute

not only

(where

NB

and

the with

the

are

NF

superalgebra

superalgebras) nwnber

the

is

that

operator

number

has (and it is by no means

its

N but

operators

Casimir

with for

invariants

NB and bosons

C UNM NF separately

and

fermions

respectively):

t

cant

N 1 = C-’ NBI = 0 NFl = 0

3

c C”,,M r C”NM, 3

This

turns

s~metry As assuned

out

to be of crucial

in nuclear suggested

structure

by

Iachello,

the

Furthermore, but

multiplets).

this

higher

symmetry

This means,

UB(6~xUF(lZ)

the

dynamical

e-e

This

as

avoiding

synmetry

well

is because

that the wavefunctions thereby

in the study of dynamical

super-

beyond

for example,

IBM

and

IBFM

is

that Eq. (4) must

as follows:

U(6112) =

system

importance

physics.

to be the supersymnetry.

be altered

(64 WI (6~)

(and

2

is appropriate

other

of Eq.

the serious

consequence

not

components

(6a).

cannot be an admixture

I, the states below the dotted

the same

Also,

of

Equs.

(7) only the

for

of baryonic

(6~) imply

nonconservation. are

e-o

supersymmetric

(6b) and

of the states of various

line for A and A+1

the

two components

nuclei, In Fig. of the

173c

D.H. Feng ei al. / dynamical ~~persy~~e~r.v

supersymnetric

multiplet

(from now on we shall denote

only the states above the dotted the

states

belong

below

should

to a different

have

line belong

no

fermions

in the

therefore

it is through

and the IBFM dynamical

e-e

and

detected tion.

e-o

that dynamical

symmetry,

are SM of the dynamical

SUSY

SUSY that one is able to unify the

IBM

We wish insight earlier

this

to ask

opinion,

In our

Given a complete

The second

the Existence

to first

to

supersymmetry

needs

as the unifying

whether

there

such

a

chain with a superalgebra

address

question

which

one,

by

taking

it was not recognized,

three experimentally

verified

we

to

still

be

driven

is at a deeper

question.

a closer

probably

be

lJ(n/m) as its

level,

We can gain

look

They are given

in the beginning, chains

recognize

is not obviously

Casimir

for

(starting

the

positive,

invariants

are

is how one

of such a symmetry.

the second

its lowest, has three chains only.

question

can

are two levels to this ques-

at

the

only

scme wisdom IBM.

in this paper that the IBM, with U(6) as its highest

Suppose

symmetry

synmetry

can one find a set (or sets) of data which are well predic-

ted by the theory? might demonstrate

one

systems,

The first is:

(1)

must

SUPERSY~E~Y

experimentally.

highest

U(6)

therefore

General discussions

Having postulated the

and

symmetries.

3. TEST OF THE SYNDICAL 3.1.

system

For the A+Z, by definition,

SM.

Since the states of A and A+1 (and the others) hamiltonian,

it as SM).

to the SM because,

We

sy~etry

and

stated

and O(3)

in Equs. (la) through

(1~).

that there is a U(6) but only

at SU(3), O(6) and Sll(5)), will

higher

symnetry?

especially

since

constants.

The

answer

the eigenvalues

Nevertheless,

the

to this of the

answer

is

yes because of two reasons. The mathematical

structure

have U(6) as its "source".

of the three

In a sense,

chains

are now well

all three groups

known to

point "up-stream"

to

U(6).

(2) as

The fact that one can ex~rimentally

far as we know) strongly

suggest

that

verify each chain (and no others, there

exist a U(6)

as the unifying

symmetry. From

this

strongly, first

study

physically

discussion,

prove) all

the the

meaningful):

we

suggest

existence

of

mathematical

that sane

in order "glcbal"

chains

which

to demonstrate

symmetry Go

(or more

G o, say,

possesses

we must

(which

is

174

D.H. Feng et al. /Dynamical

Gil => G21

D

Supersymmetry

G12

3

622

3

.

.

GJ1 =

Gs2 3 . . .

Gal 2

Ga2 3 . . .

(8)

and so on.

Furthermore, This

each chain should be verified

is,

needless

experimentally,

say,

an

experimentally.

arduous

task

or in sane cases mathematically,

For this reason, out

to

the bulk of the tests

at the first

we

equipped

of dynamical

We will now discuss

level.

which

are

not

currently

to carry out fully.

supersymmetry

is carried

the tests which we have made

in

detail. 3.2.

Tests of the multi-j

In order

to

test

chain scenario.

the

The . . . . in the above fetmion

group

dynamical

The most obvious

UB(6)xUF(m)

lJ(6/m) 3

supersymmetry

(denoted

scenarios

supersymnetry,

one must

first

3

=

equation

imagine

a

ones are as follows:

means

that

somewhere

by F) should be coupled

down

Spin(3)

the

to the boson group

(9)

chain,

the

(denoted by

B) in order to reach Spin(3). Eq. (9) implies that the e-e canponent first

test

of

such a symmetry

is best to be found

of the SM has a good IBM dynamical

dynamical

supersymmetry

in

symnetry.

nuclear

et alll, involved a j = 3/2 fermion coupled

(U(‘5/41).

The chain they used is as follows: D

UB(4)xUF(4)a

As stated by Casten6, and Au nuclei, platinum. exist

0B(6)xSUF(4)

isolated than

more

realistic

j = 3/2

realistic. scheme

0B+F(6) =

this scheme was tested

whose masses

Unfortunately,

less

3

where

Therefore,

the

proposed

by

physics,

Balanteken

U(6/4)

in nuclei

to the 0B(6) “IBM” core

0B+F(3)s

Spin(3)

(with some success)

(10)

in the odd Ir

are in the region of the best "IBM" 0B(6) nuclei: as one

orbit

To test

learns

and thus the real

is called

for.

from this

shell

model,

supersymnetric

scope The

the

of dynamical

scheme

we

have

test

there

never

is somewhat

supersymmetry, in mind

a

is also

D.H. Feng et al. /Dynamical

suitable

for the platinun

states of the platinum

mass

region:

Supersymmetry

in particular,

175c

the

e-o

SM odd

parity

isotopes.

FIGURE 2 Supersymnetric

multiplets

The SM for the pllatinun isotopes sider

the SM with N = 6.

1°6Pt

is A, lg7Pt

since it corresponds

test lying

are given

isotopes

in Fig. 2.

For example,

con-

In the language of Fig. 1, below the dotted line,

is A+1 and the states above the dotted line for A+2 will 19gpt*. 198pt* to those in Of course, there are many states for

correspond

difficult

for the Platinum

to a large excitation

to make

correspondence

of d~amical

supers~metry

states

of

an

e-e

energy.

with theoretical is only

nucleus

and

Such data are, in general, predictions.

operationally

its

e-o

SM

Therefore,

possible

(A

and

the

for the low

A+1

in

Fig.

1

respectively).

The single particle level struccture for the valence "odd" nucleon in this mass

region

UF(12).

is P~,~,

U(6/12)

It should task.

~3~~

and f5/2.

Thus the chain appropriate =

U6(6)xUF(12)

be emphasized

An obvious

=

These

orbitals

transform

according

to

here is

whatever

that the "whatever"

until Spin(3)

in Eq. (II)

way to carry out such a decanposition

(11) is not

a trivial

is to generalize

the

D,H. Feng et al./ Dynamical Supersymmetry

176c

chain

U(6/4)

(Eq.

that there

l/2 structure

must

but

by Arima,

Shimizu

in 1969.

This

method orbital

explores

directly

exist

ingenious

part (k=0,2) and a pseudo-spin

Using Eq. (12), we opened

and simultaneously

that frcin the

proposed

first

method.

This

112, 312 and 512 are formed by a pseudcpart (s=1/2).

the "Pandora's

Thus

(12)

box" for this problem.

is summarized

in Fig. 3.

UF(12) Pseudo-orbital *eparetton

FIGURE Chains

into

values

by Hecht and Adler

decanposition

UF(6)xUF(Z)

-

group

that somehow a spin

of decanposition,

"pseudo-orbital"

the fact that orbits

for the UF(12) decomposition

SUF(4)

To this end, there exists a rather

method

Harvey and separately

is the so-called

UF(12) 2

an

on the allowable

This suggests

stroctures.6

in the dynamics.

nevertheless

embed

constraint

On the other hand, one observes

here.

are "couplet"

ancientl'

is to

due to the strong

of m, it is not possible data

That

(10)).

Unfortunately,

UF(12).

3

for the fermion group UF(12)

The result

I77c

D. H. Feng et al. j Dynamical Supersymmetry

FIGURE 4 Chains

The chains

for the supergroup

They &-e given UB(6)

=

dynamical

for the supergroup

UB(6) 2

symmetry

can now be written

The full line chains

in Fig. 4.

UB(5),

U(6/12)

U(6/12)

OB(5) and U'(6) 3

is preserved

SU*(3)

are chains where the boson

(Eq. (9) scenario).

The dotted

i.e., UB(6)xLJF(6)xUF(2) 3

UB+F(6)xSUF(2)

symmetry

There are some interesting,

tant,

is not preserved.

dynamics

inherent

in these

two

down iW?diatelY.

in Fig. 4, i.e.,

line chains,

are those where the boson dynamical

chains

which

and potentially

we will

briefly

impordiscuss

later on. For the platinum

mass region,

for the e-e SP, only two chains

Chain (I):

U(6112) =

where the 08(6) symmetry in Fig. 4 are important.

UB(6)xUF(12)

=> 0B+F(3)xSlJF(2) D

=

is the dominant They are:

UB(6)~UF(6)~UF(2)

Spin(3)

and U((6/12)

Chain (II): =

1

U~(6)~UF(lZ)

UB+F(6)xSUF(2)

a 0B+F(3)xSUF(2)

3

2

=

U~(6)~UF(6)~UF(2)

0B+F(6)xSUF(2)~

Spin(3)

0B+F(5)xSUF(2)

one

178c

D.H. Feng et al. /dynamical

S~~er~ymrn~try

h'e shall, fran now on, refer to these two chains as (I) and (II),

just as Ref.

6. Neglecting simplest

terms

=

A"+,(6)*

=

AC&

HI HII where

the

hamiltonians

contributing

for chains

+

+

HI is diagonal

BCD(5)'

in chain

The

(1 =

nucleus),

(I + QCD(~)~) Ha

symmetry

the

two

C”J(J+l)

(14a)

in chain

Casimir

referred

invariant a

II.

the

symbol

for the group

third hamiltonian

to as

in chain

G.

which

Ha for obvious reason:

A word on notation

II.

to the e-e nucleus

to a special

(independent

forrm of the IBM.

of the e-e nucleus), In either

hamiltonians

cases,

is in

of the e-o

Restricted

supers~metry

the restriction

for

chains

I and

For HII, there appears

way.

is NI(NI+5)

= (N+l,@]

(excited band). As discussed HI)

order

+ N2(N2+3)

to

reduces

to a

is called

spin

similar,

they

(the so-called

especially classified, successes,

when

the most

by Casten,

D(6)* are the UBiF(6) quantum

IR's for this group, one with

band)

and two with

the

data

arranged

are

still

important

of all the states

As theorists,

nicely there

term C

[NI,N2]

=

[N,l]

(See Figs. 6,7 for the details.)

is the prediction

below BOO KeV.

ground

look

II

an additional

where [NI,N*]

For the e-o SM, there are two possible

numbers.

via

the

(I4c)

reduces

IBFM.

in a nontrivial

whose eigenvalue

[NI,N2]

only,

(SPSY).

Although differ

energies

HII

(independent

form of the

+

we will also introduce

is also diagonal

supersymnetry

binding

I and HII is diagonal

When one is restricted

the e-o nucleus special

CL(L+l)

in the Casimir operators,

hamiltonian

order here.

+

a second

For the purpose of analysis,

H

the

HI

CG2 in Eq. (14) represents is nonlinear

to

I and III3 are:

we are very much cane

more

form serious

as

like in

predicted.

I has (i.e. (mass 195)

impressed Fig.

Fig.

difficulties

most serious one is that when this theory state is not correctly

feature that chain for the odd platinum

5

with this prediction,

rather

10 of

Ref.

than 6.

the

Despite

wellsuch

The H 's predictions. i to "Pt, even the ground

with

is applied

We shall discuss more about this later on.

17%

FIGURE 5 Experimental

A rather If.

interesting

For chain

physical

I, the

Og(6)

Geometrically,

preserved. For chain

II, however, SM

due to

the

do

support

such

plating follows:

If the core

the math~atics

for

addition a

of

a

One

3

UB(6)xUF(12)

=

The result (II).

reveals Chains

3

that the situation I'

and

II'

its y softness.

data this

for

for

the odd

discussion

as

instead of soft, vii?1

vibrator]

To test this case,

will,

UB(6)xUF(6)xUF(Z) 3

suB+F(5)xSUF(2)

0BtF(3)xSUF(2)

UB~6)xUF~lZ)

=r UB~~(6)xSUF~2)

and

3

SUB(5]xSUF~5)xSU~~Z)

U(6112)

extend

I and St4 is

the ana'logus chains for U(5):

=, 0B+F(5)xS!JF(2)=,

Chain (II'):

may

e-e

(or destroyed)

The

can be preserved?

investigated

U(6/12] 3

to chains

for the

is polarized

neutron.

[say a UB[5)

show that the s~etry

(I'):

IBF? core

that the core maintains

picture.

is rigid

can be given

the

the O'(6) core symnetry

several of usl' recently

Chain

interpretation

s~etry

this means

the

e-o

level scht3nes for 1g5Pt

3

3

Spin(3)

UB{6)xUF(6)xUF(2)

(... same as above)

is the opposite with

Wa)

a

suitable

(L5b)

to that of chains choice

of

(I)

parameters,

D.H. Feng et al. /Dynamical

18Oc essentially

produce

the U*(5) vibrator addition

identical

energy

(in geaetrical

of a neutron

Supersymmetry

spectrum.

language)

Physically

this

is rigid enough

implies

that

to withstand

the

to the system.

The two difficulties

that HI have, as far as the e-o SM data are concerned,

are:

(1) the position of the side bands; (2) a strong canpression

of the predicted

side bands.

As we have stated earlier, nuclei

(1) is more and more serious as the mass of the e-o 199 fact, for Pt, the excited band actually drops to become

increases.In

the "ground"

band.

The above-mentioned general

hamiltonian

reason chain

difficulties

H

and

((1)

This hamiltonian

.

II is priferred

is because

(2))

led us to consider

is still diagonal

in chain

which

(1 + "C"(6) 2, allows is characterized

The energy

where

E in Eq.

here

that

constant IR of

=

(1

(16)

for the UB+F(6)

previously

made

differentiate

between

inertia

according UBtF(6)

(WI

the two

which

Casimir must

(16)

x E for HII.

It deserves

in Eq.

on

crucial the

hamiltonians

point

e-e

implies

platinum

that

CU(6)2

this model should

have interaction

cannot

Only through

the e-o

of the variable

manent

HII and H . (I

in that

of the hamiltonian the

energies numbers

in the

spectrun

characterizing

the

are

Our choice

has

be

c"(6)2 by cO(6)2' for the experimental

As a rule

terms greater

question

is whether

than two. the data

of

A very

feature

a second

order

the hamiltonian

important

is telling

if one were

important

thumb,

Therefore,

It has

spectrum. suitable

There is another

a two body interaction.

Another

is based solely on

energy

that this choice may be more

overlooked.

scaled

IR's of the

invariant J2 of the O(3) symmetry).

to the other chains.

not

is essentially

a

all the

isotopes

Ha is not unique, at least for this set of nuclei.

fits

mention

(16) is in fact

show up.

is to replace

still unanswered

[NI,N2].

the 0B(6) symmetry all belong to the same

scales it with the Casimir

superior

for each band

quantum numbers

in brackets

rather

made

test15

been pointed out recentlyI

here

This

(WI)16

The hamiltonian

to connect

expression

factor

used to test

(1)) while the

scaling

for Ha is:

to the value of the quantum

simple choice the

energy

Ho is reminiscent

model

UB+F(6)

energy

a(NI(NI+5)+N2(N2+3)))

group.

SM will the difference

of

+

is the

IBM

The hamiltonian

expression

e-o SM, the

since the data

the

intraband

by the different

(eigenvalue)

E a

a separate

The

H II allows the control of the position

of the band head of the side band (thereby removing difficulty factor

a more

II.

Ho

and currently

us that our assump-

D.H. Feng et al. /Dynamical

2.0

r

1.8

-

(75) -I

jQ4i3

181c

Supersymmetry

-

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(53)

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--‘X,2’

,,‘--

-

f (521,” -

(73) -.

1.0 0.8

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(72)

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-___EXP

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912 q512,’

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6

Pt

i&Y Feng et al. j ~y~arn~~ai Supersymmetry

198

Pt

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-\

I'

0.2 0 0.0 t f50) --__-____ SPSY

EXP

SUSY

(MeV1

1.0 l/2

0.80.6 -

(3010 ,,.-'

fool0

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112

(2012

SPSY

-ii?!

EXP

SUSY

<300> (20)2 5’2

0.4-

\

/‘5/* “\,3LF;/2;f’

(312: ( lol2,.5~;.-~--

(11x4 5/2-(5/2?7/23 -______.

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(10)2-5f2-(5/23?f23

OJ’JL

(OO)O 112 3/2 -____SPSY

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<410> <600>

--J

~603 c411

Fig. 7 For details, see ref. 13

tion

of only

simplicity,

including

up to two body

is inadequate

The supersymnetric

The details

of

the

of the

test

shortage

are

discussed

of space,

extensively

will therefore

for some minor details,

test well.

also mention

We should

few electr~agnetic

transitions

are satisfactory

Other

U(F112) play

a vital

symmetry

role

(BE(Z)

is currently resutts

do

in the Tungstun e-e SM.

Also,

region,

and therefore

the O(6) limita

Other supersynmetry previously

emphasized such

important example

supergroup ally

denoted

U(6/m)

There

are

=

in

since

the results

A notable

so-called

indicate

that

which

SU(3)

one

is the

limit

of the

supers~metry

may

also

has a good SU(3) dynamical

is now underway

by some

states

of us

in the

in the Platinum mass

chain

scenarios

begin

are broken by a regular

principle

of baryon

Nevertheless,

which could

be of importance

symmetry

is

broken

by

literature

as

conservation

are

other

to nuclear another

is known as the orthosympletic

mathematical

the

the

super-

Lie algebra U(n)xU(m).

number there

with

is the

potentially

structure.

An The

supergroup. supergroup,

Usp(n/m)

gener-

superalgebra.

=

merits

0B(6)xSpF(m).=.

and

demerits

and so on until Spin(3)

for

this

chain

scenario.

(17) We

will

on them here. feature of the U(n/m~ superalgebra

conservation the Casimir

and NF separately. wavefunctions

the

to the

to this chain,

several

The salient number

the

SUSY

chosen.

in mind

Osp(6/m)

comment

the

U(n/m)'s

we have

Thus, according

briefly

are

SUSY chains when

and

chain scenarios

discussed

that

chains

is

17

Suffice

here.

and again

studied.

for the odd proton

algebra U(n/m) whose symnetries We have

not be given

for lo5Pt)

the

region, work

studies

reason

data

testing

supersymnetry

the

with Casten.

13 and

here that the test was also applied

U(6120)

3.3

Refs,

figure

Preliminary

for the

in

here.

who

model.

All

196pt - ?97pt (N=6)

such a SUSY model withstood

super-symmetric tests are also being WarnerI

on

The fit for the energy spectra are presented in 196pt _ 197 Pt S&l appears in Ref. 13 for the

(Ref. 17).

Figs. 6 and 7. P similar and is thus anitted

solely

(SPSY) tests, along the

HU for ?94Pt - ?9Spt (N=7),

with

it to say that except

of

is based

_ 199 Pt (N=S) were carried out by us, in collaboration

and 196Pt

work

which

(SUSY) as well as spin symnetric

lines we have discussed,

because

interations,

for the e-o system.

(see Eq* invariants

(6)).

Unfortunately,

of the Osp(n/m)

This means

that the chain

which will mix the various

is that it leads to baryonic

suck is not the case here

commute scenario

only with

N and

not

NB

of (17) will result

in

states of various nuclei:

LXX. Feng et al. /Dynamical

184~

IInteger angular mcmentun

S~persymmeir.y

state> = alA> + ElA+2*> + .,.

IOdd-half angular momentum

state> = SlAtl>

This is a serious,

fatal, consequence

the sacred principle accept nuclei,

such

of baryonic

we might

with a and

principle.

lying specctrlm

1A+2*, (or lA+3*>

chains

the

is violated.

6 and y could

thereby minimizing

) are, by definition,

the boron description

IA>(IA+l>)

because

theory

in fact

be

for the small

the violation could

in

of the

run as follows:

excited

states of the

therefore

states.

the low

In this way,

could be minimized.

superalgebra

We know, from

KS

is inadequate,

the

structure

Yet, if we were to

in the

for such smallness

should be predaninantely

nonconservation

Physically, structure.

arguement

(leb)

for nuclear

(as we do

ask whether

6 respectively,

A+2 (A+3) nuclei where

the baryonic

then

Cne plausible

since the states

conservation

to a degree

a violation,

say),

canparison

perhaps

Wa)

+ v(A+~*> + ,..

Osp(n/m)

may

be quite

important

the IBM, that one of the three

for nuclear

dynamical

s~metry

is:

LIB(S) 3

It is also well known

O*(6)

(since

tion for the fermion

unitary

.. ..

(lb) that a possible chain decanposi-

Racah's days)" algebra

UF(m)

is

the symplectic symmetry-breaking

mechanism:

UF(m) z

Hence, if a supergroup

U(6/m)

/

at the unitary level

UBW G@) UF(m)

the orthosymplectic

boson ortho-group,

(19)

could be "formed"

T

then perhaps

SpF(m)

supergroup

fermion symplectic-group

Osp(6/m)

J *\

Esp(6/m)

may also be formed at the

level:

O’(6) (21) SpF(m)

D.H

We may also

ask whether The answer

scenario?

Feng et 121./Dynamical

there

are any needs

is quite

possibly.

states of the e-o SM of the platinum state

in this case

is U(6/14) there

is the

(not the

U(b/lZ)

discussed

SIJSY and the microscopic

The highest

symnetry SUSY,

levels.

the resulting

superalgebra

case we discussed

previously),

down to Spin(3), by the

for this purpose

structure

a nucleus

iZ UF("m), where K is the total particle

Although

the positive

single particle

is via the Osp(b/14). 20 being studied by sane of us.

Work along such ideas are currently 3.4

The dominating

such a supergroup,

A route

methods.

for such a chain

consider

For example,

as the multi-j

is no known way to deccmpose

previously

experimentally

isotopes.

li13,B level.

185~

Supersymmetry

of the nuclei in the language

possesses,

nunber

of degeneracies

on the other

hand,

can only

of shell model,

of all possible exist

sanewhere

single further

dovm the chain:

UF(F) ___-____--___---__--> U(6/m) in order

Clearly,

to establish

clear understanding There

is a

of the microscopic

large

contributions

the link between

effort

in this

in this workshop.

these two symmetries,

structure

direction,

of the bosons

as witnessed

called

is

by

a very

the

for.

nlrmber of

Thus, such a link may be forthcoming

in the

near future.

4. CONCLUSION In this talk, we have presented symmetry

for the

IBM and the

the

recent experimental

evidences

in fact, be lurking

in the collective

out, there direct

still needed in nuclear

that indicated

of the

existence

tually most challenging

problems

sane may,

As we have pointed

Therefore,

the search

in our opinion,

in current

presented

supersymmetry

in particular,

a symnetry.

Nevertheless,

states remains,

as the unifying

also

of nuclei.

questions;

of such

in this direction.

collective

We have

that dynamical

structure

still exist some unanswered

proof

idea of supersymmetry

IBFM symmetries.

the lack of a much

work

for dynamical

is

SUSY

as one of the intellec-

low energy nuclear

physics.

ACKNOWLEDGEMENT Three of us, DHF, MV and Beijing

for this opportunity be the ancestrial gratitude

RG, would

for their warm hospitality. to visit

like to thank our hosts

One of us, DHF, is

the exquisite

hone of his mother

city of Suzhou,

and wife.

to his hosts at UNAM and Drexel

in Suzhou and

particularly which

grateful

happens

to

HZS would like to express his

University

for their hospitalities.

186~

LXH. Feng

Extensive

discussions

(who have collaborated A. Arima, 0 Schoften, N. Ginocchio,

et al. / ~ynurn~~a~ Supersymmetry

with many with

colleagues,

us on various

notably stages),

R. Casten,

D. D. Warner

J. Cizewski,

F. Iachello,

Y. S. Ling, G. M. Xu, M. Zhang, Q. 7. Han, C. L. Wu, J.

Lay Nam Chang, and M. ~shinsky

are much appreciated.

REFERENCES

1)

A. Arima and F. Iachello,

Phys. Rev. Lett. -35 (1074) 1069.

21

F. Xachello and 0. Scholten,

3)

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M. Gell-Mann

and Y. Ne'eman, Eight Fold

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R. Gilmore and D. H. Feng, Nucl. Phys. A301 (1078) 189; H. Feng, R. Gilmore and S. R. Deans,%'j%. Rev. E, (1981) 1254; :: N. Ginocchio, Phys. Rev. Lett. 44, (1980) 1970; Iachello, Phys. Rev. Lett. 44, A. E. L. Dieperink, 0, Scholten a&-F. (1980) 1974.

61

R. F. Casten, these proceedings.

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Y. S. Ling, M. Zhang, G. Ft. Xu, D. H. Feng, M. Vallieres Drexel University preprint (1984), to be published.

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R. Gilmore, Lie Groups, Lie Algebras Wiley and Sons, New York, lY14) .

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B. H. Flowers,

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101 A. 8. Balantekin and I. Bars, Invited talk at the Supersymmetry Conference, Los Alamos, New Mexico, 1983.

in Physics

III A. B. Balantekin, I. Bars and F. Iachello, Nucl. Phys. A370 (1981) 284; J. A. Cizewski, in Eoson Models in Nuclei, ed. 0. H. Feng, M. Vallieres and S. Pittel, (World Scientific Publication, Singapore, 1984) and references therein. 12) A. Arima, K. Shimizu and M. Harvey, Phys. Lett. 308, (1969) 517; K. T. Hecht and A. Adler, Nucl. Phys. A137, (196m29. 13) H. Z. Sun, M. Vallieres, Rev. C (in press).

D. H. Feng,

14) D. H. Feng, H. Z. Sun and M. Vallieres,

R. Gilmore

and

R.

Casten,

Phys.

to be published.

15) J. A. Cizewski,

R. F. Casten, G. %ith, M. Stelts, W. Kane, H. Bonner and W. F. Davidson, Phys. Rev. Lett. 40, (1978) 167; R. F. Casten and J. A. Cizewski, N&l. Phys. A309, (1978) 477.

16) G. Scharff-Goldhaber, C. Dover and A. L. Goodman, (1976) 239 and references therein.

Ann. Rev. Nucl. Sci, -26

D.H. Feng et al. /Dynamical

17)

H. Z. Sun, D. H. Feng, M. Vallieres, Isacker, to be published.

18) D. D. Warner,

187~

Supersymmetry

R. Gilmore,

A. Frank and P. van

Phys. Rev. Lett. (in press); and private communication.

19) I. Talmi and A. deshalit, 1963.

Nuclear Shell Theory,

(Academic

20) H. Z. Sun, Q. Z. Han, M. Zhang and D. H. Feng, unpublished.

Press, NY,