Structure of vacuum in chiral supersymmetric quantum electrodynamics A. V. Smilga Institute of Theoretical and Experimental Physics, Academy of Sciences of the USSR, Moscow (Submitted 21 January 1986) Zh. Eksp. Teor. Fiz. 91, 14-24 (July 1986) An effective Hamiltonian is found for supersymmetric quantum electrodynamics defined in a finite volume. When the right and left fields appear in the theory nonsymmetrically (but so that anomalies cancel out), this Hamiltonian turns out to be nontrivial and describes the motion of a particle in the "ionic crystal" consisting of magnetic charges of different sign with an additional scalar potential of the form U = K '/2, where aiK = z..Despite the nontrivial form of the effective Hamiltonian, supersymmetry remains unbroken. 1. INTRODUCTION analysis of nonchiral theories. We shall investigate the sim- One of the most acute problems in theoretical elemen- plest example of chiral supersymmetric QCD. Unfortunate- tary-particle physics is the question of supersymmetry ly, our original hopes have not been justified, and supersym- breaking, which is generally believed to occur in nature in metry remains unbroken in this theory. The effective one form or another at high enough energies. The most at- Hamiltonian has, however, turned out to be highly nontri- tractive mechanism of supersymmetry breaking is spontane- vial and does not reduce to free motion, as was the case for ous breaking by dynamic effects, not manifest at the "tree" nonchiral theories. The methods developed below can be level. A well-known example of this type of breaking occurs used in the analysis of more complicated chiral theories, in- in the Witten quantum mechanics with superpotential of the cluding those in which supersymmetry is definitely broken. form V(X)= A (X 3/3 - a2X) (Ref. 1) . However, for a long In Section 1, we examine a chiral supersymmetric QCD time, searches for the dynamic breaking of supersymmetry in the quantum-mechanical limit in which the fields are as- in four-dimensional field theories remained unsuccessful. sumed to be independent of spatial coordinates. We obtain Thus, this type of breaking does not occur in supersymme- the effective Hamiltonian that turns out to describe the mo- tric Yang-Mills theories without matter.2 The situation is no tion of a charged particle in the field of a Dirac monopole better in gauge theories with extended supersymmetry, or in with an additional scalar potential ofthe form - l/r '. States supersymmetric QCD with different numbers of right and of arbitrarily low energy are present in this problem, but left multiplets of matter in the fundamental representation. there are no states with zero energy. In Section 3, we turn to However, nontrivial field theories in which supersym- field theory in a finite volume. The effective Hamiltonian for metry is dynamically broken have recently been f~und.~.~the zeroth harmonic of the gauge field, sought by precisely The simplest example of this type of theory is the SU(5) the same method, then describes motion in a lattice of mono- supersymmetric Yang-Mills theory that includes chiral mat- poles. It is interesting that the final expression for H "can be ter fields in the 5, and 10, representations. The triangular written only for a theory free of anomalies (e.g., a theory anomaly cancels out in this theory, but right and left fields with eight left fermions of unit charge and one right fermion appear in different representations, so that matter remains with twice this charge). The lattice then contains monopoles massless. It is precisely these "theories with irremovable of different sign, and the average magnetic space charge is chirality" that were not fully examined by Witten in his well- zero. Analysis of the effective Hamiltonian for an anomaly- known paper, in which he analyzed vacuum states for a wide free theory in a finite volume shows that the Witten index is class of theories.' then nonzero and supersymmetry is not broken. A briefsum- The arguments advanced in Refs. 3 and 4 in favor of mary of all the results is given in Section 4. supersymmetry breaking in this theory are indirect. Instan- 2. QUANTUM MECHANICS OF SUPERSYMMETRIC QCD ton calculations of particular correlators support the forma- tion of a gluino condensate (A :A ""),, suggesting spontane- Consider a theory describing the interaction of a photon ous breaking of chiral symmetry. If supersymmetry is not A, and a photino A, with one chiral left field ( p,qa) in the broken, the pseudoscalar goldstone should be part of the limit where the fields do not depend on spatial coordinates chiral multiplet, which also contains a massless scalar parti- [in one sense, a four-dimensional theory with one left field cle. However, there is no place for this particle in the present does not exist because of the anomaly, but in (0 + 1)-dimen- case because the theory has no valleys, i.e., directions in the sional situations, this anomaly is not manifest]. The total space of scalar fields for which the classical potential vanish- hypercharges and the Hamiltonian for this theory are" es (a detailed discussion of this range of questions can be found in the review in Ref. 5). Our ultimate aim is to examine directly the question of supersymmetry breaking in different chiral theories by con- structing the effective Hamiltonian and then analyzing the Q~=2-%~d[ifj(oh ) dp-ecp(P66p]+~[-<6~.. spectrum of states of lowest energy by analogy with Witten's -ieqAk (or)21, 8 Sov. Phys. JETP 64 (I),July 1986 0038-5646/86/070008-06$04.00 @ 1987 American Institute of Physics 8 where $, and A, are two-component Weyl spinors and where B and q, are angles that parametrize the direction of A. The freedom in choosing the common phase will be ex- plained below. The factor A 'I2appears in (7) for conven- ience, in order to ensure that the normalization integral of 1 a, 1 over the charged field does not depend on A. The char- acteristic angles q, in the wave function (7) are given by The supersymmetry algebra has the form lpc,,, 1 - (eA ) -'I2. It is clear that the approximation A ) (q,I {Q,, @) +=baBfi-eAk (ah)uBB, (3) is self-consistent for y = l/eA < 1. The quantity y is, in fact, the true expansion parameter in the effective Hamiltonian. where The complete wave function of the system is is the coupling that commutes with the hypercharge and the Hamiltonian. The significance of (3) is that the superposi- h tion of the transformed supersymmetry yields both airansla- where alf, are the wave functions of excited states of H,. The effective Hamiltonian acting on fo(AJ, ) is given by tion and a gauge transformation. On physical states GY = 0, so that the second term on the right of (3) does not ~ntri- bute to the physical matrix elements. We note that the G that follows from (3) is automatically "correctly ordered'zn the sense of Ref. 6. It is also clear that all the remaining G that In our case, in which the problem is s~per~ym~metric,we can differ from (4) by a constant dozot yield the correct super- also define an effective hypercharge Q z" = (Q "'), + ..., so symmetry algebra on the space GV, = 0. that It is clear that the classical potential following from the Hamiltonian (2) vanishes when q, = @ = 0 for any A,, i.e., there is a valley along which the wave function tends to A [We note that ( 11) does not contain a component due to G "spread out." In other words, in our problem, A is a "slow" because, in the quantum-mechanical limit, the gauge trans- variable and the lowest-energy states of the system are asso- ciated precisely with the excitation of this degree of freedom. formation rotates only the charged fields.] At the end of the present paper, we shall produce arguments showing %at The charged fields q, and 6,on the other hand, are "fast" high%r-orderperturbation-theory terms are absent from Q variables and their excitation energies are high. The Born- "" Oppenheimer approximation, discussed in detail in Ref. 2 in and H ''. Direct calculation yields relation to field theories, is valid in this case. It is therefore assumed that (A(B (q,1. We shall classify the terms in the hypercharges and in the Hamiltonians in terms of the parameter (q,(/(A(. We have Q, = Q:) 1.. - 1 A- ge" = - (P-&)' + +; .Ad , + QY, where 2 8Az- 8A where a! is a vector function of the fields A, which is identi- cal with the vector potential of a Dirac monopole placed at the origin A = 0, with a filament running along the positive h A h direction of thez axis, chosen in accordance with (8).Other and, similarly, H = Ho + H, + H,,where choices of the phase of the spinor w, correspond to other gauges for the vector potential d!The last term in the Ham- iltonian will work for wave functions in the sector with fer- mion charge F = 1, and describes the interaction between the magnetic field GV= - A/2A and a spin 5 particle, the h gyromagnetic ratio being greater by a factor of two than the The Hamiltonian Ho (in which Ai appears as a parameter) is Dirac value. (We note that, in the F = 0 and F = 2 sectors, quadratic in the charged fields e, and $. It is not difficult to the Hamiltonian corresponds to the scattering of a scalar find the explicit form of the vacuum wave function of this particle.) The scalar potential 1/8A is necessary to ensure Hamiltonian (its energy is zero) : the supersymmetry of the Hamiltonian.