- Email: [email protected]
A review of supersymmetric quantum mechanics
Physica 15D (1985) 138-146 North-Holland, Amsterdam
A REVIEW O F S U P E R S Y M M E T R I C Q U A N T U M MECHANICS Barry F R E E D M A N Department of Physics, Unioersity of Illinois, Champaign-Urbana, IL 61801, USA
and Fred C O O P E R Theoretical Dioision, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
1. Introduction Recently, there has been renewed interest in supersymmetric field theory [1] as a possible vehicle for making realistic phenomenological models of particle physics which do not suffer from the need to fine-tune the bare masses in the boson sector of the theory. In ordinary theories the meson radiative correction are of the order of A 2 where A is the cutoff, usually assumed to be the grand unified scale ( - 10 is GeV). The fact that the generators of supersymmetry transformations convert bosons into fermions and that fermion masses can stay zero because of chiral symmetries allows a solution of the fine-tuning problems [2]. In a realistic model based on supersymmetry (SUSY), supersymmetry must be broken because of the lack of observed boson partners to the light leptons. One can prove that in weak coupling perturbation theory, if the classical (tree level) approximation to the theory is supersymmetric, then perturbative radiative corrections do not break supersymmetry. Thus, in making realistic model field theories, one must break SUSY at the tree level or nonperturbatively. It is thus important to have a nonperturbative method of determining if supersymmetry is broken in a given theory.
One reason for studying supersymmetric quantum mechanics is that there is a class of superpotentials W(x) which behave at large x as x ~, for integer a, and for which we know from general arguments whether SUSY is broken or unbroken. Thus, one can use these superpotentials to test various methods for determining if supersymmetry is broken in an arbitrary model. Recently [3], Witten proposed a topological invariant, the Witten index A, which counts the number of bosons minus the number of fermions having ground state energy zero. Since the ground state energy cannot be zero if supersymmetry is broken, one expects SUSY to be preserved and hence the theory to be a poor candidate for a realistic model if A is not zero. In this study we evaluate A for several examples, and show some unexpected peculiarities of the Witten index for certain choice of superpotentials W(x) [4]. In this survey we also discuss two other nonperturbative methods of studying supersymmetry breakdown. One involves relating supersymmetric quantum mechanics to a stochastic classical problem [5, 6] and the other involves considering a discrete (but not supersymmetric) version of the theory and studying its behavior as one removes the lattice cutoff [7, 8].
0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
In this survey we review both the Hamiltonian and the path-integral approaches to supersymmetric quantum mechanics [2, 5, 9]. We then discuss the related path integrals for the Witten index and for stochastic processes and show how they are indications for supersymmetry breakdown. We then discuss a system where the superpotential W(x) has asymmetrical values at x = ___o0 [4, 14]. In this case, we find that pairing is broken as there is a mismatch between the fermionic and bosonic continuum density of states. Also, the Witten index is dependent on the regulation parameter/3. We finally discuss nonperturbative strategies for studying supersymmetry breakdown based on introducing a lattice and studying the behavior of the ground state energy as the lattice cutoff is removed [7, 8].
One can also define a "fermion" number operator ( - 1)F: = 1 - 2~/'xP*,
(4)
which anticommutes with Q, and Q*. One can realize the above algebra in one dimension by the following matrix representation for ~/'* and g':
~*=a-=( 01 0), ff,=o+=(1
0).
(5)
In this representation
[m*,m] ( - 1)F = - 03'
2
= - ½°3'
(6)
and the Hamiltonian is diagonal: H= ½[pZ+ W2(x)]
2. S u p e r s y m m e t r i c quantum mechanics and spontaneous supersymmetry breaking in the Hamiltonian formalism [2, 5, 9]
139
+~W'(x),
(7)
or
= __z, . . . . ,,.+
+
(8) '
{Q _Q..
The supersymmetry algebra in D = 1 is generated by the charges
D±--
Q*=(p+iw(~))~*, { Q * , Q * ) O=(p-iW(Yc))¢I", { Q , Q } = 0 ,
The eigenstates of H can be written as the vector
= 0,
(a)
W(x)=i
(9)
where [p,;2] =-i,
{~*,x~} = 1.
where the _+ corresponds to ( - 1 ) F being + 1, i.e., the + corresponds to the bosonic states and the to the fermionic states. From eq. (8) and
One has that {Q*,Q} =2H,
(2)
so that
H =' ~t':2 -, wz(:~) <¢I'*, ~> W'(x) ' -'-~ 2 [H, QI=[H,Q*]=O.
H~P =
ESO
(10)
we obtain
(3)
O_~-)
= 2v~-E@(+) = - i Q * f f , ( - ) ,
D + ~ +) = 2v~-E-~ ( - ) = iQff,~+),
(11)
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
140
which shows how supersymmetry pairs the positive energy solutions. Next, we want to discuss supersymmetry breaking. For supersymmetry to be a good symmetry (12)
QI0> = Q*I0> =010>,
where 10) denotes the vacuum state. From (12) and (2) we find that for supersymmetry to be a good symmetry the ground state energy must be zero. From (11) we then see when supersymmetry is broken, Eg 4: 0, the ground state must be degenerate. One measure of supersymmetry breaking is the Witten index [3] A = Tr( - 1) F. Since the finite energy states are fermi-bose paired, this quantity measures N ÷ ( E = 0 ) - N _ ( E = 0), the difference of zero energy boson and fermion states. Thus, if A 4: 0, supersymmetry is unbroken. Usually Tr ( - 1) F is ill defined, so instead one considers
where p ± ( E ) are the corresponding densities of states. Thus, only if the density of states for bosons and fermions is different will A(fl) depend on ft. This will be important for our later discussion. When the superpotential W ( x ) - x a for large values of x, then it is easy to discuss supersymmetry breaking. For supersymmetry to be preserved one has
H
'o =
O,
=
%(+)(x)
)
'
(17)
or, from (11), o
_
(18)
O ( ~ x + W ( x ) ) ~o~+) = O.
A ( f l ) = Tr ( - 1) F e - a n = T r ( e - a " ÷ - e - a n - ) . (131 Following Akhoury and Comtet [4] it is useful to introduce the heat kernels K ± ( x , y , fl) which satisfy -+ WZ + W' dx 2 -
K
±=
O,
(14)
Thus, we obtain
ql(o-)= A1 efw(x)dx,
~o+)= A2 e-fwCx)dx.
If W ( x ) - bx ~ for large x then
vrtt(-)_ AlebX"+t/(a+l),
vtt(2+)_ A2e-bx"+t/(a+l). (20)
K ±(x, y , f l ) = (Yl e-an~lx) = ~ e-aE.~.± ( x ) ko~ ( y ) .
We see when a is even there is no normalizable ground state with zero energy. When a is odd there is one normalizable ground state consisting of a single boson state:
P/
Using states normalized to one we obtain
a(B)=f dx[K+(x,x,B)-K_(x,x,B)].
(151 ~l'to=
If there are also continuum states in the spectrum, then one has
a(B)=N+(E=O)-N_(E=O)
+
Eo
(19)
dEe-aE[o+(E)-0_(E)], (16)
0 )) , ~lto(+
ff,o~+)= A 2 e - f W d x ,
(21)
such that H g ' o = 0. Thus, we get the following picture of the spectrum when W(x) ~ x" for large x: for a even SUSY is broken, A = 0, Eg > 0 and all the eigenvalues have fermi-bose degeneracy; for a odd SUSY is a good symmetry, A = 1, the
B. Freedman and F. Cooper/Supersymmetric quantum mechanics
ground state has E = 0 and is a bose state, and all excited states have fermi-bose degeneracy.
141
The determinant has been evaluated by Gildener and Patrascioiu: [10] det[ O , - W'] = 1-Ix,.,
(26)
m
3. Path integral formalism [5,10]
where the X" satisfy
[ a, - w ' ( x ) ] % = am%,
From the Hamiltonian (3) we have that
[ - - O r --
L={:c2_½W2(x)+i,i,.a,ff " [,t,*,,/,] W'(x). 2
(27)
WZ(X)]~It~m~-X"~"¢ m.
Thus,
(22)
'l'"=C,.exp f[dr'[X,,+ W'(x(r'))].
(28)
Letting t --+ it, the euclidean path integral is Imposing antiperiodic boundary conditionsg%(T) = -~/'~(0) yields
z[j.,.,*] =f[dx]ftd*lf[d~'*] x e - s~ + f(Jx+n~'*+n*~')d",
(23)
Xm = i(2m T+ 1)~r _ 1" f 0 r T dr W,(x)
where ~/', ~/'* are now elements of a Grassmann algebra
Thus,
(~*,~}
HX"(g)//Xm(O)
= {~,~,} = (~*,~*}
= o,
ford'r W'(x) 2
(30)
m
and S E -~
= cosh
(29)
Thus,
roLEd'r,
(24)
Z = Z _ + Z+,
z~=ftdxlexp[-Sy].
where W 2
2
W 2
(31)
w'(x) )
LE= ~( O,x)E +--V(x)-- ~'*[ O, - W'(x)] 'Z'.
sY = ford'r --~-+ --~-( x ) +_ 2
For the path integral Z[j], the boundary conditions on the fermion fields are 9(0) = - '/'(r). We next want to integrate out the fermi field. We obtain
Thus, as expected, from (7)
"
Z = Tre - H r = T r e - t t + r + Tre - n - r
= Z + + Z_
= E e-E'.+'T+ E e-~'.-'T= E e-E. T,
Z[j,n,n*] = f [ d x ] d e t [ O , - W ' ( x ) ]
(32)
{e.}=(e~.+,,e.~.-~}. If SUSY is unbroken, then Eo(+) = 0, Eo(-) > 0 and to satisfy the degeneracy condition for n > 1,
×ox,[/oVt0,(25)
E(+)=
En-(-)l •
(33)
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
142
If SUSY is broken, then E (+) = E (-) > 0 for all n, (n = 0 , 1 , 2 . . . . ). Now, from this last discussion we expect that the regulated Witten index also has a simple path integral form. Since from (13)
A(fl)
related path integral connected with the classical stochastic processes defined by the Langevin equation [11, 12, 13, 5]
k(~') = W(x('r)) +f(T),
(41)
where f(~') is a random stirring force having Gaussian statistics:
= T r ( - 1)Fe-#n = Tr(e - a n + - e-On-), (34)
we expect
1fT
P[ f ]= Nexp
A(fl)= Z+- Z_,
-2Jto
dt f2(t ) ] Fo J'
(42)
(35) SO
where in S + we integrate from zero to fl instead of T. This is indeed so. To evaluate Tr ( - 1)Fe -oH we need to insert eigenstates of Ix) and Iq,) into the trace. Since ( - 1 ) F I b) -- Ib) and ( - 1 ) F I f ) = If), one obtains for A(fl)
(36) but now, because of the ( - 1) F, x(0) = x ( $ ) ,
~(0) = ~ ( f l ) ,
~*(0) = xo*(fl), (37)
To determine correlation functions in x resulting from the statistics of the forcing term one has
Ptf]Df=P[~-
• 2m
IT'/?
1 --
r
(38)
so that
,~
(44)
: [d_ -- G-I(~
= d
sinh/rd~ J0
Dx.
8x(,')
,
T f0 d~'W ( x ) ,
1-ihm(g)/X,.(O) =
W(x)]det]~
Now, we have
so we have a determinant with periodic boundary conditions. Imposing ~/%(T) = q%(0) in (17) yields ~km =
(43)
f DfP[flf(,)f(,') = Foa(~-,9.
ftdx]ftd ]ftd *]
a(a)
f Dfe[f] = 1, fDZP[ZIy(,)=O,
W'(x) 2
(39)
and we obtain
A ( f l ) = z + - z _ = Z e - E ~ * ' # - Z e -E~-'#.
(40)
When SUSY is unbroken then ~'(+)= E l - ) and ~t-,n + 1 Eo(+) = 0. In that case, A = 1, independent of ft. When SUSY is broken, then E l - ) = El +) > 0 for all n and za = 0, independent of/3. There is also a
x
- - ,r')
[ 8(~-
~') -o(~- ~') w'(x)]
dK(~,~'),
(45)
where we have used the fact that this is a causal system with forward propagation in time
Co(*-*') = e(*-*'),
dG° = 8 ( z - z ' ) . dr
(46)
Using this boundary condition, only the first term in the expansion of the Tr In survives, and, apart
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
from overall normalization,
I 8f det Ix-I#x
= e Tr(lnK)
= e 1/2fW'(x)dr.
143
even for certain x 0, but not when n is odd. Furthermore, one can define a classical probability (47)
Pd(Y, ,r) = (8(y - x(r)))/
Parisi and Sourlas [6] introduced Grassmann variables to represent the determinant
= f DfP(f)~(y - x(z)),
(53)
such that
I=f[dx][d~]exp(-½f[jc2+W z +~(t)[
0 -0--~- W'] g ' ( t ) ] ) .
(48)
f d yy"Pd(y) =-f DfP[f ](x( z))"= ( x " ( r ) ) . (54)
However, antiperiodic b.c. on ~/" give Z + - Z_ and periodic b.c. on ~/" give Z + + Z_. The forward propagation boundary condition leads to
W2(x) - W'(x)]dt), (49)
= z+ = E
(50)
where we have identified F 0 = h and have used f3cW(x)dr = fO,F(x(¢))dr = 0 for polynomial W(x) and periodic x(~'). In deriving (49) we also assumed that eq. (41) had only one solution for a given f. When supersymmetry is broken and T ~ m, Eo~+) > 0 and one cannot satisfy Z+= 1. Thus, the stochastic problem must become non-invertible [6]. In fact, the equation Jc = - g x " ,
X ( t o ) = Xo,
(51)
has solution [5] x = [(n - 1)(gt + c)] 1/0-"),
(52)
where x~-" c = (n - 1)
ap
Ot =½F°---sP+ff-f (W(y)P(y't))"
l=Zs =fe[flDf fdx × exp ( - ½/o[:t2+
Pc~ satisfies the Fokker-Planck equation (55)
For an equilibrium distribution to exist at long times t one has
P(y,t)~ i(y)
(56)
and
fi(y) dy = 1. Setting
OP/Ot = 0, we obtain
if(y) = Ne-2/hfW{y)dy = Ig'o(Y)l 2,
(57)
since ~o(y)=Ae -yw(y)dy from (21) and F0= h. Thus, only when SUSY is unbroken (E 0 = 0) does an equilibrium distribution exist for the classical stochastic system.
4. S U S Y without pairing [4, 14] When we allow W(x) to exhibit the nontrivial topological behaviour
gt°" O(x)--+~+
This solution can blow up for t > t o when n is
atx=
+~,
O(x)--+O_ at x = - ~ ,
(58)
144
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
then the potentials W 2 " ~ - ~t"7, have spectra with a continuum as well as a finite number of bound states. In these cases supersymmetry actually forces an asymmetry in the density of states p + ( E ) # p - ( E ) and therefore also the Witten index becomes fl dependent. If we choose W(x) = tanh x then the Hamiltonians H± are H÷
p+(E)-o-(E)=
-
~ E ( E 2 - 1) 1/2"
(63)
We have from (16)
A(fl) = 1 + fl ~ d E e - # e ( p + ( E ) - p - ( E ) )
d2
--+1 dx 2
- 2 sech2 x,
1
(59)
d2 H_= ---+1. dx 2
For H _ there are only scattering states ~(-), which are defined by their asymptotic behavior,
f_(x,k)=e
Jost functions and derive
-ikx,
g _ ( x , k ) - - e -ikx,
= 1 - ~-r(:,/3).
(64)
This explicitly shows the/3 dependence of A(/3). Thus, in this case the index is not an integer and is fl dependent.
x~o¢,
x ~ oo,
(60)
5. Nonperturbative strategies for studying SUSY breaking
E_(k)=(l+k2), whereas for H+ there is one bound state '/'o~+) = sechx,
E0¢+) = 0
(61)
as well as the scattering states obtained by the supersymmetry operation (11) ~2(1 + k2) ~ ( + ) _ ( - -~x + 0 tanh x) ko(-),
(62)
which implies ~/2(l+k2)f+=(-ikx+tanhx)e
ikx, x ~ +o¢,
~2(1 + k 2) g+-- ( +ikx + tanh x) e -ikx, X ----~ - - O 0 ,
e+(k)=k2+ l. Now, the boundary conditions require that ff'(+)(k) and ~ ( - ) ( k ) have different phase shifts, making it impossible for the density of states p+(E) to be equal to p(-)(E). In fact, using the boundary condition that W ( x ) ~ +1 at 5: ~ and the supersymmetry relation between ~(+) and '/'(-), one can obtain a relationship between the
Two methods for exploring field theory in the strong coupling (nonperturbative) domain exist. One is strong coupling expansions of lattice versions of the field theory [15] (on the lattice a kinetic energy or strong coupling expansion is a non-singular expansion). The other is Monte Carlo evaluation of the latticized path integral [16]. Here, we ignore the problem of trying to make a supersymmetric lattice theory [17]. Instead, we introduce a time lattice with lattice spacing a. The lattice breaks all or part of the SUSY algebra. Clearly, the g r o u n d state energy of the lattice Hamiltonian will not be zero. However, we can ask what happens to Eo(a) as the lattice spacing a goes to zero. Although it might be difficult in an approximation scheme to see if E 0 = 0 because we have an extra parameter (the lattice spacing), one can ask whether
Eo(a ) = a ~
(65)
as a ~ 0. Here "y is a critical exponent. If we find that y > 0 within the accuracy of our calculational scheme, then one could say with confidence that the continuum theory is supersymmetric. If ~, --, 0
B. Freedman and F. Cooper/Supersymmetric quantum mechanics
145
and Eo(a ) ---, finite constant as a ~ 0, then we expect the continuum theory to break supersymmetry. So
where
y>0oE
o=0
(SUSY),
=0oE
0>0
(BokenSUSY).
is a dimensionless correlation length. One needs to extrapolate this finite series from small z to infinite z. To do this one uses Pad6 approximants [18]. Assuming
(66)
If we perform a Monte Carlo calculation at two decreasing small lattice spacings
E o - z ~, y = In
0( )
/In --
a=
E~(z)
(73)
z~limZEo(z ) ,
(67) we obtain a sequence of approximants
in the limit a' and a ~ 0. An analytic procedure is the strong coupling expansion of the Langevin equation [7]. For example, for W(x) = gx 3, eq. (41) becomes
e( x . - Xn_x) + gx 3 = fn
(68)
on the lattice, where e = 1/a. One then calculates
x =E mx(.m and finds #/3 E-----ff-~i~cl/3 f - 2 / 3 ~(e2) " Xn = gl/3 + 3g2/3 k J n - l J n - - f n 1/3) +
a = + 1, + 0.4766, - 3.507: - 3.4903, - 3.4997, - 3.4763.
(74)
Thus, since y = -or~3, we estimate (75)
g 0 -- a 1'16.
This agrees with our knowledge that SUSY is good when W = gx 3. It also gives credence to the idea that one can naively put SUSY on a lattice and have it restored in the continuum. We can also check this method when SUSY is broken. If we choose W(x) = gx2/2, • we can write [8] a strong coupling series for E o using path integral techniques:
(69) The ground state energy for a quantum mechanical system with V(x) = g2x6/2 - ~gx 2 is given by the virial theorem to be
X 8J(t'-----) Z ° [ J ] ' E 0 = 2 g 2 ( x 6) - 3 g ( x 2 ) .
(70)
Zo[Jl=f Dxexp(- f dt [½W~
To perform the average over the noise we use
P [ f ] = I-'[ e-a/2(i)/2"/ a-a-i V2~r "
(71)
The integrals over the noise are just g a m m a functions. We find at the m ' t h order in e eo =
c.z°, n~0
(72)
W'(x) 2
(76)
J(t)x(t)])
On the lattice, t = na and W(x) = W(Xn), SO
.--,F( Jn) Z o [ J l = Jl F - - ~ = Uexp ~ In [V(Jn)/F(O)], n
(77)
B. Freedmanand F. Cooper/Supersymmetric quantum mechanics
146
F(y) =
dtexp(-½aW2
+ ½aW' + ayt)
References
= EA,y"/n],
A. = f
d t t" exp
[ - a g t a / 8 + agt/2],
A___~ n = ( 3~e)nEanrne 2m, Ao e = g - 1 / 3 ( 3 a ) -1/2. The inverse propagator
0 2 8 ( t - t') b e c o m e s well
b e h a v e d o n t h e lattice:
1 ~, , , - + 1 + 8m, .+1 - 2 8 , m ) . G o l ( ? / , m ) = a-5(
(78)
T h e e x p a n s i o n i n p o w e r s of G o x gives a series i n ~. W e o b t a i n
Eo= ~ < x 4 > -- 3g
C.e 2".
A n a l y z i n g this series we find E0 4 : 0 a n d , / < 0 , s h o w i n g n o e v i d e n c e for S U S Y , as expected.
(79)
[1] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. P. Fayet, Phys. Lett. 70B (1977) 461. [2] E. Witten, Nucl. Phys. B185 (1981) 513. [3] E. Witten, Nucl. Phys. B202 (1982) 253. [4] R. Akhoury and A. Comtet, Anomalous Behavior of the Witten Index, UCLA Preprint UCLA/84/Tep 11 (February 1984). [5] F. Cooper and B. Freedman, Ann. Phys. 146 (1983) 262. [6] G. Parisi and N. Sourlas, Nucl. Phys. B206 (1982) 321. See also N. Sourlas, Physica 15D (1985) 115 (these proceedings). [7] C.M. Bender, F. Cooper and B. Freedman, Nucl. Phys. B219 (1983) 61. [8] C.M. Bender, F. Cooper and A. Das, Phys. Rev. D15 (1983) 28, 1473. [9] P. Salomonson and J.W. van Holten, Nucl. Phys. B190 (1982) 509. [10] E. Gildner and A. Patrascioiu, Phys. Rev. D16 (1977) 1802. [11] P.C. Martin, E. Siggia and H. Rose, Phys. Rev. A8 (1973) 423. [12] R. Phythian, J. Phys. A10 (1977) 777. [13] C.M. Bender et al., J. Stat. Phys. 22 (1980) 647. [14] M. Stone and W. Su, Univ. of Illinois Preprint (1983). [15] C.M. Bender, F. Cooper, G. Guralnik and D. Sharp, Phys. Rev. D19 (1979) 1865. [16] N. Metropolis et al., J. Chem. Phys. 21 (1953) 1087. [17] P. Dondi and H. Nicolai, Nuovo Cimento 41A (1977) 1. See also V.A. Kosteleck~ and J.M. Rabin, Physica 15D (1985) 213 (these proceedings). [18] C. Bender, F. Cooper, G. Guralnik, R. Roskies and D. Sharp, Phys. Rev. Lett. 43 (1979) 537.
A REVIEW O F S U P E R S Y M M E T R I C Q U A N T U M MECHANICS Barry F R E E D M A N Department of Physics, Unioersity of Illinois, Champaign-Urbana, IL 61801, USA
and Fred C O O P E R Theoretical Dioision, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
1. Introduction Recently, there has been renewed interest in supersymmetric field theory [1] as a possible vehicle for making realistic phenomenological models of particle physics which do not suffer from the need to fine-tune the bare masses in the boson sector of the theory. In ordinary theories the meson radiative correction are of the order of A 2 where A is the cutoff, usually assumed to be the grand unified scale ( - 10 is GeV). The fact that the generators of supersymmetry transformations convert bosons into fermions and that fermion masses can stay zero because of chiral symmetries allows a solution of the fine-tuning problems [2]. In a realistic model based on supersymmetry (SUSY), supersymmetry must be broken because of the lack of observed boson partners to the light leptons. One can prove that in weak coupling perturbation theory, if the classical (tree level) approximation to the theory is supersymmetric, then perturbative radiative corrections do not break supersymmetry. Thus, in making realistic model field theories, one must break SUSY at the tree level or nonperturbatively. It is thus important to have a nonperturbative method of determining if supersymmetry is broken in a given theory.
One reason for studying supersymmetric quantum mechanics is that there is a class of superpotentials W(x) which behave at large x as x ~, for integer a, and for which we know from general arguments whether SUSY is broken or unbroken. Thus, one can use these superpotentials to test various methods for determining if supersymmetry is broken in an arbitrary model. Recently [3], Witten proposed a topological invariant, the Witten index A, which counts the number of bosons minus the number of fermions having ground state energy zero. Since the ground state energy cannot be zero if supersymmetry is broken, one expects SUSY to be preserved and hence the theory to be a poor candidate for a realistic model if A is not zero. In this study we evaluate A for several examples, and show some unexpected peculiarities of the Witten index for certain choice of superpotentials W(x) [4]. In this survey we also discuss two other nonperturbative methods of studying supersymmetry breakdown. One involves relating supersymmetric quantum mechanics to a stochastic classical problem [5, 6] and the other involves considering a discrete (but not supersymmetric) version of the theory and studying its behavior as one removes the lattice cutoff [7, 8].
0167-2789/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
In this survey we review both the Hamiltonian and the path-integral approaches to supersymmetric quantum mechanics [2, 5, 9]. We then discuss the related path integrals for the Witten index and for stochastic processes and show how they are indications for supersymmetry breakdown. We then discuss a system where the superpotential W(x) has asymmetrical values at x = ___o0 [4, 14]. In this case, we find that pairing is broken as there is a mismatch between the fermionic and bosonic continuum density of states. Also, the Witten index is dependent on the regulation parameter/3. We finally discuss nonperturbative strategies for studying supersymmetry breakdown based on introducing a lattice and studying the behavior of the ground state energy as the lattice cutoff is removed [7, 8].
One can also define a "fermion" number operator ( - 1)F: = 1 - 2~/'xP*,
(4)
which anticommutes with Q, and Q*. One can realize the above algebra in one dimension by the following matrix representation for ~/'* and g':
~*=a-=( 01 0), ff,=o+=(1
0).
(5)
In this representation
[m*,m] ( - 1)F = - 03'
2
= - ½°3'
(6)
and the Hamiltonian is diagonal: H= ½[pZ+ W2(x)]
2. S u p e r s y m m e t r i c quantum mechanics and spontaneous supersymmetry breaking in the Hamiltonian formalism [2, 5, 9]
139
+~W'(x),
(7)
or
= __z, . . . . ,,.+
+
(8) '
{Q _Q..
The supersymmetry algebra in D = 1 is generated by the charges
D±--
Q*=(p+iw(~))~*, { Q * , Q * ) O=(p-iW(Yc))¢I", { Q , Q } = 0 ,
The eigenstates of H can be written as the vector
= 0,
(a)
W(x)=i
(9)
where [p,;2] =-i,
{~*,x~} = 1.
where the _+ corresponds to ( - 1 ) F being + 1, i.e., the + corresponds to the bosonic states and the to the fermionic states. From eq. (8) and
One has that {Q*,Q} =2H,
(2)
so that
H =' ~t':2 -, wz(:~) <¢I'*, ~> W'(x) ' -'-~ 2 [H, QI=[H,Q*]=O.
H~P =
ESO
(10)
we obtain
(3)
O_~-)
= 2v~-E@(+) = - i Q * f f , ( - ) ,
D + ~ +) = 2v~-E-~ ( - ) = iQff,~+),
(11)
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
140
which shows how supersymmetry pairs the positive energy solutions. Next, we want to discuss supersymmetry breaking. For supersymmetry to be a good symmetry (12)
QI0> = Q*I0> =010>,
where 10) denotes the vacuum state. From (12) and (2) we find that for supersymmetry to be a good symmetry the ground state energy must be zero. From (11) we then see when supersymmetry is broken, Eg 4: 0, the ground state must be degenerate. One measure of supersymmetry breaking is the Witten index [3] A = Tr( - 1) F. Since the finite energy states are fermi-bose paired, this quantity measures N ÷ ( E = 0 ) - N _ ( E = 0), the difference of zero energy boson and fermion states. Thus, if A 4: 0, supersymmetry is unbroken. Usually Tr ( - 1) F is ill defined, so instead one considers
where p ± ( E ) are the corresponding densities of states. Thus, only if the density of states for bosons and fermions is different will A(fl) depend on ft. This will be important for our later discussion. When the superpotential W ( x ) - x a for large values of x, then it is easy to discuss supersymmetry breaking. For supersymmetry to be preserved one has
H
'o =
O,
=
%(+)(x)
)
'
(17)
or, from (11), o
_
(18)
O ( ~ x + W ( x ) ) ~o~+) = O.
A ( f l ) = Tr ( - 1) F e - a n = T r ( e - a " ÷ - e - a n - ) . (131 Following Akhoury and Comtet [4] it is useful to introduce the heat kernels K ± ( x , y , fl) which satisfy -+ WZ + W' dx 2 -
K
±=
O,
(14)
Thus, we obtain
ql(o-)= A1 efw(x)dx,
~o+)= A2 e-fwCx)dx.
If W ( x ) - bx ~ for large x then
vrtt(-)_ AlebX"+t/(a+l),
vtt(2+)_ A2e-bx"+t/(a+l). (20)
K ±(x, y , f l ) = (Yl e-an~lx) = ~ e-aE.~.± ( x ) ko~ ( y ) .
We see when a is even there is no normalizable ground state with zero energy. When a is odd there is one normalizable ground state consisting of a single boson state:
P/
Using states normalized to one we obtain
a(B)=f dx[K+(x,x,B)-K_(x,x,B)].
(151 ~l'to=
If there are also continuum states in the spectrum, then one has
a(B)=N+(E=O)-N_(E=O)
+
Eo
(19)
dEe-aE[o+(E)-0_(E)], (16)
0 )) , ~lto(+
ff,o~+)= A 2 e - f W d x ,
(21)
such that H g ' o = 0. Thus, we get the following picture of the spectrum when W(x) ~ x" for large x: for a even SUSY is broken, A = 0, Eg > 0 and all the eigenvalues have fermi-bose degeneracy; for a odd SUSY is a good symmetry, A = 1, the
B. Freedman and F. Cooper/Supersymmetric quantum mechanics
ground state has E = 0 and is a bose state, and all excited states have fermi-bose degeneracy.
141
The determinant has been evaluated by Gildener and Patrascioiu: [10] det[ O , - W'] = 1-Ix,.,
(26)
m
3. Path integral formalism [5,10]
where the X" satisfy
[ a, - w ' ( x ) ] % = am%,
From the Hamiltonian (3) we have that
[ - - O r --
L={:c2_½W2(x)+i,i,.a,ff " [,t,*,,/,] W'(x). 2
(27)
WZ(X)]~It~m~-X"~"¢ m.
Thus,
(22)
'l'"=C,.exp f[dr'[X,,+ W'(x(r'))].
(28)
Letting t --+ it, the euclidean path integral is Imposing antiperiodic boundary conditionsg%(T) = -~/'~(0) yields
z[j.,.,*] =f[dx]ftd*lf[d~'*] x e - s~ + f(Jx+n~'*+n*~')d",
(23)
Xm = i(2m T+ 1)~r _ 1" f 0 r T dr W,(x)
where ~/', ~/'* are now elements of a Grassmann algebra
Thus,
(~*,~}
HX"(g)//Xm(O)
= {~,~,} = (~*,~*}
= o,
ford'r W'(x) 2
(30)
m
and S E -~
= cosh
(29)
Thus,
roLEd'r,
(24)
Z = Z _ + Z+,
z~=ftdxlexp[-Sy].
where W 2
2
W 2
(31)
w'(x) )
LE= ~( O,x)E +--V(x)-- ~'*[ O, - W'(x)] 'Z'.
sY = ford'r --~-+ --~-( x ) +_ 2
For the path integral Z[j], the boundary conditions on the fermion fields are 9(0) = - '/'(r). We next want to integrate out the fermi field. We obtain
Thus, as expected, from (7)
"
Z = Tre - H r = T r e - t t + r + Tre - n - r
= Z + + Z_
= E e-E'.+'T+ E e-~'.-'T= E e-E. T,
Z[j,n,n*] = f [ d x ] d e t [ O , - W ' ( x ) ]
(32)
{e.}=(e~.+,,e.~.-~}. If SUSY is unbroken, then Eo(+) = 0, Eo(-) > 0 and to satisfy the degeneracy condition for n > 1,
×ox,[/oVt0,(25)
E(+)=
En-(-)l •
(33)
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
142
If SUSY is broken, then E (+) = E (-) > 0 for all n, (n = 0 , 1 , 2 . . . . ). Now, from this last discussion we expect that the regulated Witten index also has a simple path integral form. Since from (13)
A(fl)
related path integral connected with the classical stochastic processes defined by the Langevin equation [11, 12, 13, 5]
k(~') = W(x('r)) +f(T),
(41)
where f(~') is a random stirring force having Gaussian statistics:
= T r ( - 1)Fe-#n = Tr(e - a n + - e-On-), (34)
we expect
1fT
P[ f ]= Nexp
A(fl)= Z+- Z_,
-2Jto
dt f2(t ) ] Fo J'
(42)
(35) SO
where in S + we integrate from zero to fl instead of T. This is indeed so. To evaluate Tr ( - 1)Fe -oH we need to insert eigenstates of Ix) and Iq,) into the trace. Since ( - 1 ) F I b) -- Ib) and ( - 1 ) F I f ) = If), one obtains for A(fl)
(36) but now, because of the ( - 1) F, x(0) = x ( $ ) ,
~(0) = ~ ( f l ) ,
~*(0) = xo*(fl), (37)
To determine correlation functions in x resulting from the statistics of the forcing term one has
Ptf]Df=P[~-
• 2m
IT'/?
1 --
r
(38)
so that
,~
(44)
: [d_ -- G-I(~
= d
sinh/rd~ J0
Dx.
8x(,')
,
T f0 d~'W ( x ) ,
1-ihm(g)/X,.(O) =
W(x)]det]~
Now, we have
so we have a determinant with periodic boundary conditions. Imposing ~/%(T) = q%(0) in (17) yields ~km =
(43)
f DfP[flf(,)f(,') = Foa(~-,9.
ftdx]ftd ]ftd *]
a(a)
f Dfe[f] = 1, fDZP[ZIy(,)=O,
W'(x) 2
(39)
and we obtain
A ( f l ) = z + - z _ = Z e - E ~ * ' # - Z e -E~-'#.
(40)
When SUSY is unbroken then ~'(+)= E l - ) and ~t-,n + 1 Eo(+) = 0. In that case, A = 1, independent of ft. When SUSY is broken, then E l - ) = El +) > 0 for all n and za = 0, independent of/3. There is also a
x
- - ,r')
[ 8(~-
~') -o(~- ~') w'(x)]
dK(~,~'),
(45)
where we have used the fact that this is a causal system with forward propagation in time
Co(*-*') = e(*-*'),
dG° = 8 ( z - z ' ) . dr
(46)
Using this boundary condition, only the first term in the expansion of the Tr In survives, and, apart
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
from overall normalization,
I 8f det Ix-I#x
= e Tr(lnK)
= e 1/2fW'(x)dr.
143
even for certain x 0, but not when n is odd. Furthermore, one can define a classical probability (47)
Pd(Y, ,r) = (8(y - x(r)))/
Parisi and Sourlas [6] introduced Grassmann variables to represent the determinant
= f DfP(f)~(y - x(z)),
(53)
such that
I=f[dx][d~]exp(-½f[jc2+W z +~(t)[
0 -0--~- W'] g ' ( t ) ] ) .
(48)
f d yy"Pd(y) =-f DfP[f ](x( z))"= ( x " ( r ) ) . (54)
However, antiperiodic b.c. on ~/" give Z + - Z_ and periodic b.c. on ~/" give Z + + Z_. The forward propagation boundary condition leads to
W2(x) - W'(x)]dt), (49)
= z+ = E
(50)
where we have identified F 0 = h and have used f3cW(x)dr = fO,F(x(¢))dr = 0 for polynomial W(x) and periodic x(~'). In deriving (49) we also assumed that eq. (41) had only one solution for a given f. When supersymmetry is broken and T ~ m, Eo~+) > 0 and one cannot satisfy Z+= 1. Thus, the stochastic problem must become non-invertible [6]. In fact, the equation Jc = - g x " ,
X ( t o ) = Xo,
(51)
has solution [5] x = [(n - 1)(gt + c)] 1/0-"),
(52)
where x~-" c = (n - 1)
ap
Ot =½F°---sP+ff-f (W(y)P(y't))"
l=Zs =fe[flDf fdx × exp ( - ½/o[:t2+
Pc~ satisfies the Fokker-Planck equation (55)
For an equilibrium distribution to exist at long times t one has
P(y,t)~ i(y)
(56)
and
fi(y) dy = 1. Setting
OP/Ot = 0, we obtain
if(y) = Ne-2/hfW{y)dy = Ig'o(Y)l 2,
(57)
since ~o(y)=Ae -yw(y)dy from (21) and F0= h. Thus, only when SUSY is unbroken (E 0 = 0) does an equilibrium distribution exist for the classical stochastic system.
4. S U S Y without pairing [4, 14] When we allow W(x) to exhibit the nontrivial topological behaviour
gt°" O(x)--+~+
This solution can blow up for t > t o when n is
atx=
+~,
O(x)--+O_ at x = - ~ ,
(58)
144
B. Freedman and F. Cooper/ Supersymmetric quantum mechanics
then the potentials W 2 " ~ - ~t"7, have spectra with a continuum as well as a finite number of bound states. In these cases supersymmetry actually forces an asymmetry in the density of states p + ( E ) # p - ( E ) and therefore also the Witten index becomes fl dependent. If we choose W(x) = tanh x then the Hamiltonians H± are H÷
p+(E)-o-(E)=
-
~ E ( E 2 - 1) 1/2"
(63)
We have from (16)
A(fl) = 1 + fl ~ d E e - # e ( p + ( E ) - p - ( E ) )
d2
--+1 dx 2
- 2 sech2 x,
1
(59)
d2 H_= ---+1. dx 2
For H _ there are only scattering states ~(-), which are defined by their asymptotic behavior,
f_(x,k)=e
Jost functions and derive
-ikx,
g _ ( x , k ) - - e -ikx,
= 1 - ~-r(:,/3).
(64)
This explicitly shows the/3 dependence of A(/3). Thus, in this case the index is not an integer and is fl dependent.
x~o¢,
x ~ oo,
(60)
5. Nonperturbative strategies for studying SUSY breaking
E_(k)=(l+k2), whereas for H+ there is one bound state '/'o~+) = sechx,
E0¢+) = 0
(61)
as well as the scattering states obtained by the supersymmetry operation (11) ~2(1 + k2) ~ ( + ) _ ( - -~x + 0 tanh x) ko(-),
(62)
which implies ~/2(l+k2)f+=(-ikx+tanhx)e
ikx, x ~ +o¢,
~2(1 + k 2) g+-- ( +ikx + tanh x) e -ikx, X ----~ - - O 0 ,
e+(k)=k2+ l. Now, the boundary conditions require that ff'(+)(k) and ~ ( - ) ( k ) have different phase shifts, making it impossible for the density of states p+(E) to be equal to p(-)(E). In fact, using the boundary condition that W ( x ) ~ +1 at 5: ~ and the supersymmetry relation between ~(+) and '/'(-), one can obtain a relationship between the
Two methods for exploring field theory in the strong coupling (nonperturbative) domain exist. One is strong coupling expansions of lattice versions of the field theory [15] (on the lattice a kinetic energy or strong coupling expansion is a non-singular expansion). The other is Monte Carlo evaluation of the latticized path integral [16]. Here, we ignore the problem of trying to make a supersymmetric lattice theory [17]. Instead, we introduce a time lattice with lattice spacing a. The lattice breaks all or part of the SUSY algebra. Clearly, the g r o u n d state energy of the lattice Hamiltonian will not be zero. However, we can ask what happens to Eo(a) as the lattice spacing a goes to zero. Although it might be difficult in an approximation scheme to see if E 0 = 0 because we have an extra parameter (the lattice spacing), one can ask whether
Eo(a ) = a ~
(65)
as a ~ 0. Here "y is a critical exponent. If we find that y > 0 within the accuracy of our calculational scheme, then one could say with confidence that the continuum theory is supersymmetric. If ~, --, 0
B. Freedman and F. Cooper/Supersymmetric quantum mechanics
145
and Eo(a ) ---, finite constant as a ~ 0, then we expect the continuum theory to break supersymmetry. So
where
y>0oE
o=0
(SUSY),
=0oE
0>0
(BokenSUSY).
is a dimensionless correlation length. One needs to extrapolate this finite series from small z to infinite z. To do this one uses Pad6 approximants [18]. Assuming
(66)
If we perform a Monte Carlo calculation at two decreasing small lattice spacings
E o - z ~, y = In
0( )
/In --
a=
E~(z)
(73)
z~limZEo(z ) ,
(67) we obtain a sequence of approximants
in the limit a' and a ~ 0. An analytic procedure is the strong coupling expansion of the Langevin equation [7]. For example, for W(x) = gx 3, eq. (41) becomes
e( x . - Xn_x) + gx 3 = fn
(68)
on the lattice, where e = 1/a. One then calculates
x =E mx(.m and finds #/3 E-----ff-~i~cl/3 f - 2 / 3 ~(e2) " Xn = gl/3 + 3g2/3 k J n - l J n - - f n 1/3) +
a = + 1, + 0.4766, - 3.507: - 3.4903, - 3.4997, - 3.4763.
(74)
Thus, since y = -or~3, we estimate (75)
g 0 -- a 1'16.
This agrees with our knowledge that SUSY is good when W = gx 3. It also gives credence to the idea that one can naively put SUSY on a lattice and have it restored in the continuum. We can also check this method when SUSY is broken. If we choose W(x) = gx2/2, • we can write [8] a strong coupling series for E o using path integral techniques:
(69) The ground state energy for a quantum mechanical system with V(x) = g2x6/2 - ~gx 2 is given by the virial theorem to be
X 8J(t'-----) Z ° [ J ] ' E 0 = 2 g 2 ( x 6) - 3 g ( x 2 ) .
(70)
Zo[Jl=f Dxexp(- f dt [½W~
To perform the average over the noise we use
P [ f ] = I-'[ e-a/2(i)/2"/ a-a-i V2~r "
(71)
The integrals over the noise are just g a m m a functions. We find at the m ' t h order in e eo =
c.z°, n~0
(72)
W'(x) 2
(76)
J(t)x(t)])
On the lattice, t = na and W(x) = W(Xn), SO
.--,F( Jn) Z o [ J l = Jl F - - ~ = Uexp ~ In [V(Jn)/F(O)], n
(77)
B. Freedmanand F. Cooper/Supersymmetric quantum mechanics
146
F(y) =
dtexp(-½aW2
+ ½aW' + ayt)
References
= EA,y"/n],
A. = f
d t t" exp
[ - a g t a / 8 + agt/2],
A___~ n = ( 3~e)nEanrne 2m, Ao e = g - 1 / 3 ( 3 a ) -1/2. The inverse propagator
0 2 8 ( t - t') b e c o m e s well
b e h a v e d o n t h e lattice:
1 ~, , , - + 1 + 8m, .+1 - 2 8 , m ) . G o l ( ? / , m ) = a-5(
(78)
T h e e x p a n s i o n i n p o w e r s of G o x gives a series i n ~. W e o b t a i n
Eo= ~ < x 4 > -- 3g
C.e 2".
A n a l y z i n g this series we find E0 4 : 0 a n d , / < 0 , s h o w i n g n o e v i d e n c e for S U S Y , as expected.
(79)
[1] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. P. Fayet, Phys. Lett. 70B (1977) 461. [2] E. Witten, Nucl. Phys. B185 (1981) 513. [3] E. Witten, Nucl. Phys. B202 (1982) 253. [4] R. Akhoury and A. Comtet, Anomalous Behavior of the Witten Index, UCLA Preprint UCLA/84/Tep 11 (February 1984). [5] F. Cooper and B. Freedman, Ann. Phys. 146 (1983) 262. [6] G. Parisi and N. Sourlas, Nucl. Phys. B206 (1982) 321. See also N. Sourlas, Physica 15D (1985) 115 (these proceedings). [7] C.M. Bender, F. Cooper and B. Freedman, Nucl. Phys. B219 (1983) 61. [8] C.M. Bender, F. Cooper and A. Das, Phys. Rev. D15 (1983) 28, 1473. [9] P. Salomonson and J.W. van Holten, Nucl. Phys. B190 (1982) 509. [10] E. Gildner and A. Patrascioiu, Phys. Rev. D16 (1977) 1802. [11] P.C. Martin, E. Siggia and H. Rose, Phys. Rev. A8 (1973) 423. [12] R. Phythian, J. Phys. A10 (1977) 777. [13] C.M. Bender et al., J. Stat. Phys. 22 (1980) 647. [14] M. Stone and W. Su, Univ. of Illinois Preprint (1983). [15] C.M. Bender, F. Cooper, G. Guralnik and D. Sharp, Phys. Rev. D19 (1979) 1865. [16] N. Metropolis et al., J. Chem. Phys. 21 (1953) 1087. [17] P. Dondi and H. Nicolai, Nuovo Cimento 41A (1977) 1. See also V.A. Kosteleck~ and J.M. Rabin, Physica 15D (1985) 213 (these proceedings). [18] C. Bender, F. Cooper, G. Guralnik, R. Roskies and D. Sharp, Phys. Rev. Lett. 43 (1979) 537.