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Submitted on 1 Jan 1982
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DYNAMICAL SUPERSYMMETRY BREAKING
J. van Holten
To cite this version:
J. van Holten. DYNAMICAL SUPERSYMMETRY BREAKING. Journal de Physique Colloques, 1982, 43 (C3), pp.C3-263-C3-265. �10.1051/jphyscol:1982352�. �jpa-00221907�
JOURNAL DE PHYSIQUE
Colloque C3, supplgment au n o 12, Tome 43, d6cembre 1982 page C3-263
D Y N A M I C A L SUPERSYMMETRY B R E A K I N G
J.W. Van Holten *
CERN, Geneva, Switzer Zand
Supersymmetry is at present of interest in two related areas of physics: N = 1 globally supersymmetric gauge theories may have important applications for Grand Unified Theories, while local supersymmetry leads naturally to quantum theories of gravitation. On the other hand, it is clear that at low energies the world is not manifestly supersymmetric. Therefore, the question of supersymmetry breaking is an important issue in these theories. The construction of phenomenologically interes- ting models is greatly hampered by the existence of a number of non-renormalization theoremsl), which forbid perturbative spontaneous breaking of rigid supersymmetry beyond the tree-level. As a result the only viable mechanisms for supersymmetry breaking through quantum effects are non-perturbative ones, unless one considers local supersymmetry. Supersymmetry breaking induced by supergravity effects has recently been studied by a number of authors2). In the following I discuss the pos- sibility of dynamical supersymmetry breaking.
There are several related criteria for spontaneous supersymmetry breaking to occur in a theory. These include the following3) ,4) :
1. The vacuum energy is positive: E,, . =
<oI {Q,Q+} Io>
> 0;2. There is an auxiliary field F acquiring an expectation value: <F> :
=
<oI{Q,$}Io>
f 0;3. There is a Goldstone fermion: {Q,$} = <F> +
...
;4. The index of a certain operator, denoted by A z ~ r ( - ) ~ , must vanish 4 ) , 5 ) To derive the above results one uses the algebra
[Q,
.Qj).H s9
+ (off diagonal terms) ,where H is the Hamiltonian. From this it follows that one cannot break one super- symmetry without breaking all of them (no hierarchy in supersymmetry breaking).
The fourth criterion can be derived by considering the theory in a finite volume, whence the low-lying energy levels become discrete. The supersymmetry al- gebra then implies a pairing between bosonic and fermionic states of non-zero energy:
Q + ~ b > = E l f > , Q =Klb>
.
Therefore, adiabatic changes in the parameters of the theory (including the infinite volume limit) leave invariant the quantity
where ng are the numbers of bosonic and fermionic zero-energy states. It follows that A f 0 implies the existence of an unpaired zero-energy state, which must remain in the infinite volume limit, proving that supersymmetry is unbroken. Note that A = 0 shows the possibility of creating a zero-energy fermion in any bosonic state (including the ground state) of the model. Whether this massless fermion is a Goldstone fermion for broken supersymmetry cannot be established from this criterion.
"Institut fiir Theor. Physik, ~niversitzt Wuppertal, Gaussstrasse 20, Wuppertal, F.R.G.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982352
C3-264 JOURNAL DE PHYSIQUE
The most interesting applications of Witten's index include supersymmetric gauge theories and non-linear U models. For gauge theories it leads to the conclusion that supersymmetry breaking requires an Abelian factor in the gauge group, with an asso- ciated D-term, or in the case of a simple non-Abelian group, the existence of charged fermions in a complex representation. For non-linear u models supersymmetry breaking is related to the topology of the field space. Notably the index A equals the Euler characteristic of the manifold. There are, however, other topological quantities playing a role as well.
Two possible mechanisms for non-perturbative supersymmetry breaking are instan- tons6)-9) and the formation of bound stateslo) ,I1). An explicit example of super- symmetry breaking by instanton effects is found in supersymmetric quantum mechanics 7)
.
To implement the same idea in field theories gives rise to a problem with the number of fermionic zero modes. In general, there are too many to make
However, it has been claimed that,instantons may give rise to explicit super- symmetry breaking terms of order e-l/g in the effective action61 78) 99). The authors of Ref. 9) show that an analysis of the Ward identities gives rise to inconsistencies which they ascribe to the lack of superconformal symmetry in the set of classical instanton solutions.
The possibility of supersymmetry breaking by generation of composite states in the particle spectrum has been demonstrated in some two- and three-dimensional modelslo). The idea there was to show that even if there is no supersymmetry breaking on the classical level, the effective action for the bound states may have spontaneous supersymmetry breaking at the tree level, thereby circumventing the no-go theorems of Ref, 1). The A parameter of these models has been shown to vanish12), proving the consistency of the analysis. A noteworthy point is that the breaking of supersymmetry is not signalled by the existence of a fermionic condensatell), but by the non-vanishing of a scalar-scalar two-point function, related to the auxiliary component of a supermultiplet of bound states which are generated.
In conclusion it has been shown that both instanton effects and bound states can give rise to supersymmetry breaking, but no realistic models exhibiting these phenomena have as yet been constructed.
REFERENCES
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3) See for a review: FAYET P. and FERRARA S., Phys. Rep. 32C (1977).
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6 ) ABBOTT L., GRISARU M.T. and SCHNITZER H.J., Phys. Rev. Q (1977) 3002.
7 ) WITTEN E., Nucl. Phys. B185 (1981) 513;
SALOMONSON P. and VAN HOLTEN J.W., Nucl. Phys. (1982) 509.
J.W. Van Holten
8) CASHER A., Brussels preprint (1982) ;
VAINSHTEIN A.I..and ZAKHAROV V.I., ITEP preprint 6 (1982).
9) NOVIKOV V.A., SHIFMAN M.A., VAINSHTEIN A.I. and ZAKHAROV V.I., ITEP preprint 93 (1982).
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VENEZIANO G. and Y A N K I E L O ~ S., Phys. Lett. B (to be published).
12) NOLAN D.M.E., Durham preprint DTP-82/5 (1982).
