1 ν

bb+ +

µ < 0 : h’ =

2 2

1 ν

cc+ +

h0 =

2 2

1+ ν

h+ = c c’ (4.112)

2 2

along with

p ν

U± = ± + 2p “ (4.113)

2p 8p

(4.112) indicates that there exists an intermediate Hamiltonian h0

which is superpartner to both h’ and h+ . However, if µ > 0 then ν

turns imaginary and there can be no hermitean intermediate Hamil-

tonian.

4.10 Explicit SUSY Breaking and Singular

Superpotentials

So far we have considered unbroken and spontaneously broken cases

of SUSY. Let us now make a few remarks on the possibility of explicit

breaking of SUSY. This is to be distinguished from the spontaneous

breaking in that for the explicit breaking the SUSY algebra does not

work in the conventional sense in the Hilbert space but rather as

an algebra of formal di¬erential operators [81,82]. Explicit breaking

of SUSY can be accompained by negative ground state energy with

© 2001 by Chapman & Hall/CRC

unpairing states in contrast to spontaneously broken SUSY when all

levels are paired. Explicit breaking of SUSY can be caused by the

presence of singular superpotentials. Note that so far in our discus-

sions we were concerned with continuously di¬erentiable superpo-

tentials which in turn led to nonsingular supercharges and partner

supersymmetric Hamiltonians in (’∞, ∞).

Let us consider now a singular superpotential of the type

ν

W (x) = ’x (4.114)

x

where ν ∈ R. With W (x) given above one can easily work out the

partner Hamiltonians

1

W2 “ W

H± =

2

d2

1 ν(ν “ 1)

’ 2 + x2 +

= ’ (2ν “ 1) (4.115)

x2

2 dx

At the point ν = 1, the component H+ is found to shed o¬ the

singular term and to acquire the form of the oscillator Hamiltonian

except for a constant term and to acquire the form of the oscillator

Hamiltonian except for a constant term

d2

1

’ 2 + x2 ’ 3

H+ = (4.116)

2 dx

However, H’ is singular.

The interesting point is that if we focus on H+ we ¬nd that it

possesses the spectrum

1 3

n

E+ = n + ’ (4.117)

2 2

for n = 0, 1, 2, . . .. For n = 0 one is naturally led to a negative

0

ground state energy E+ = ’1 and in consequence SUSY breaking.

Actually SUSY remains broken in the entire interval 1 < ν < 3 with

2 2

0 = ’2ν + 1 < 0. Note that no

the ground state energy given by E+

negative norm state is associated with ν ∈ ( 1 , 3 ). However, Q|0 >

22

turns out to be nonnormalizable and not belonging to the Hilbert

space. Indeed it is the nonnormalizability of Q|0 > which causes

© 2001 by Chapman & Hall/CRC

the ground state to lose its semi-positive de¬niteness character. Je-

vicki and Rodrigues [83] have made a detailed analysis of the model

proposed by (4.114) and have found several ranges of the coupling

ν < ’ 3 and ’ 3 < ν < ’ 1 apart from the one we have just now

2 2 2

mentioned. In both these cases, however, the ground state energy

remains positive.

Casahorran and Nam [81,82] have made further studies on the

explicit nature of SUSY breaking. They have obtained a new family

of singular superpotentials which include the class of P¨schl-Teller

o

potentials. In particular for the system (l < 0)

1 d2 1 1

2

1 + |l|2 ’ |l| (|l| ’ 1) sech2 x

H+ =’ +

2 dx2 2 2

1

(1 + |l|)2 ’ (|l| ’ 1 ’ n)2 ,

n

E+ =

2

n = 0, 1, . . . < |l| ’ 1 (4.118)

and its partner

1 d2 1 1

+ (1 + |l|)2 ’ |l| (|l| + 1) sech2 x + cosech2 x

H’ =’

2 dx2 2 2

1

(1 + |l|)2 ’ (|l| ’ 2 ’ 2n)2 ,

n

E’ =

2

|l|

n = 0, 1, . . . , < ’1 (4.119)

2

SUSY can be seen to be explicitly broken for l < ’1 with unpaired

states. Indeed one can see that with l < ’1, H+ possesses positive

energy eigenstates. The condition for H’ to possess positive energy

eigenstates is, however, l < ’2.

There have also been other proposals in the literature with sin-

gular superpotentials. The one suggested by Roy and Roychoudhury,

namely [84]

n

2νx µ 2»i x

W (x) = ’x + ++ (4.120)

1 + νx2 x i=1 1 + »1 x2

exhibits two negative eigenstates. Occurrence of negative energy

states in SUSY models has also been discussed in [85].

© 2001 by Chapman & Hall/CRC

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