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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


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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


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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].


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