Quantization of a Six-Dimensional Wess-Zumino Model

-We examine a six-dimensional Wess-Zumino model. The equations of motion are of the fourth order, implying two modes of propagation; a normal bradyonic mode and a tachyonic mode. The conserved fermion current is constructed. The component fields are quantized in such a way that the operator representing the supercharge is the generator of supersymmetry transformations. The quantization is complemented by the definition of the vacuum for both modes. The evaluation of vacuum expectation values leads to a Feynman propagator for the normal mode and a half-advanced and half-retarded propagator for the tachyon mode. Convolutions between these propagators show consistency with unitarity.


-Introduction.
It is well known that the consideration of higher-order theories gives rise to difficulties and problems which are hard to overcome. However, it also leads to a better ultraviolet behaviour of the propagators. When the model is also supersymmetric, we have in addition the compensating effects produced by boson and fermion loops. The Lagrangian procedures are known[l] and the classical retarded solution was given in ref. [2].
In this paper we are going to study the simple Wess-Zumino model [3], extended to higher dimensions in ref. [4]. This kind of extension was also examined in ref. [5] where a different alternative was adopted. In ref. [4] the fields obey higher-order equations of motion. The order increases with the dimensionality of space-time. In our case (six dimensions) the equations are of fourth order. They have normal free-wave solutions as well as real exponential solutions. The latter are outside the (*) Also at Comisi5n de Investigaciones Cientificas de la Provincia de Buenos Aires, Argentina. usual framework of field theory, as, for example, in ref. [6], where the function space is of type S (tempered distributions [7]).
There is no doubt that the field theory of higher-order equations requires the consideration of another framework for its correct mathematical development. The extension appropriate for our purpose seems to be that of the ,<Rigged Hilbert Spaces,~ [8]. Those spaces have been used by several authors in a variety of cases, such as the description of resonances, virtual states, scattering theory and Gamow states [9][10][11][12][13]. The Fourier transforms of our wave functions are analytic distributions defined over the set of entire analytic functions [14], rapidly decreasing on the real axis. A more detailed description can be found in ref. [15].

-The model.
For the properties of spinors in higher dimensions one can consult the classical book on the subject by Cartan [16].
We have two types of Weyl spinors, which in six dimensions have four components each. In contradistinction to the four-dimensional case, the conjugate of a Weyl spinor is another Weyl spinor of the same type and a scalar product can be formed by the product of spinors of different types.
c) Vector  It is easy to eliminate F or ¢ from the scalar equation (10). If we multiply the first spinor equation (11) by 8~ and take the adjoint of the second equation we see that we can eliminate ~ (or ¢~). Analogously, we can eliminate ~1~ from the vector equations (12). In this way we get, for any component Z(x), the equation Equation (15) shows that any component of the chiral superfield has two modes of propagation. A normal bradyon mode with mass m 2 and a tachyon mode with ,mass, -ms. Therefore, the Fourier development of any component has the form (0 is the Heaviside step function). From now on, for the sake of simplicity, we are not going to specify the mode of prolaagation, except of course when we evaluate the respective propagators.
Following procedures for higher-order Lagrangians [17] it is possible to write the energy-momentum vector. Also, due to supersymmetry there is a conserved fermion charge which can be obtained from a spinor NSether current j2. From the law of change of the chiral superflied under infmitesimal supersymmetry transformation with parameter ~", we get In the usual quantization method, the commutation relations imply that the energy-momentum vector is the infinitesimal generator of displacements: However, when supersymmetry is present the supersymmetric charge is more fundamental in the sense that P, can be deduced from Q~ through the relation: For this reason we can choose the commutation rules in such a way that Q~ is the infinitesimal generator of supersymmetry transformations. Therefore (cf. (21) and (22) With these relations one can check that (29), with (25) and (26), reproduce the symmetrized form of expression (27) for the energy-momentum vector. Note also that in (25) and (26) the order of factors is immaterial.

-Propagators.
For the complete specification of the quantum states of the fields, we have to add to the quantum rules (33) the definition of the vacuum.
As was pointed out before, any component of the chiral superfield has two modes of propagation. The normal or bradyon mode obeys the normal Klein-Gordon equation, so that one can def'me the usual creation and annihilation operators. The vacuum of the bradyon mode is then defined as the state that is annihilated by the destruction operators. It follows that the normal mode propagates according to Feynman's Green function.
The tachyon mode cannot be treated in the same way. For example the elementary exponential exp [ikx] is a solution of the free-wave equation if ko = o~ = = (k 2 -m2) 1/2. But, when k 2 < m 2 the exponential blows up either for t--* ~ or for t--* -~. Further, while for the bradyon mode, the set of k~ obeying k 2 = k2+ m 2 form a two-sheeted hyperboloid (each sheet characterized by the sign of k0), for the tachyon mode the set of all k~ obeying k~ = k 2 -m 2 is one-sheeted and there is no Lorentz-invariant way of separating positive from negative energies. These and other considerations have been given in ref. [19] and [20] to find the propagators of tachyon modes. The essential point is that the vacuum state is not annihilated by any of the Fourier components of the mode, but rather by the symmetrical product defining the energy density.
The right-hand side of (46) defines the Wheeler function: (47) f kd It is not difficult to show that for the tachyon mode, (47) is the half-advanced and halfretarded Fourier transform of (k 2 + m2) -1 = (k~ -k 2 + m2) -1 = (ko 2 -o~2) -1. This kind of Green's function was adopted in ref. [21] to describe the interaction between charge particles in a complete absorber (see also ref. [22]). The condition on the absorber is easily understandable, as the W-function is equivalent, for o) real, to the principal-value Green function which lacks the on-shell pole that represents the free propagation of any excitation. The propagator in momentum space can be found directly from eq. (14). The Green function A(k) is a solution of (k a-m4)A(k) = 1, When the perturbative S-matrix is constructed following standard procedures, the propagator d is represented by the internal lines of the corresponding Feynman diagrams. The external legs corresponds only to normal modes of propagation, as the tachyon excitation cannot occupy free asymptotic states. The W-propagator is compatible with the situation as the half-advanced and half-retarded Green functions are equivalent to Cauchy's principal value on the real ko-axes. In other words, when k0 and k are such that ko 2 -k 2 = -m 2, the W-propagator is zero. This property also means that the S-matrix is automatically unitary at tree level.
For the simplest one-loop diagram we must take the convolution of 3-propagators. Both, the Feynman and the Wheeler Green-functions have the form G(x): dq dqo q~_¢o s , F where F is either Feynman path or the half-advanced and half-retarded path. ~o is defined, respectively, by (18) and (20).
It is well known that the Feynman path can be made to run along the real k0-axes (F = R), by adding a small negative imaginary part to the mass: oJ 2 = q2 + m 2 _ i~. For the Wheeler function we will follow a similar procedure. First, we take a complex mass parameter, -m2---)~ s. Then, the path of integration is deformed to make it coincide with the real axis. When a pole is crossed, a loop must be added so as to compensate for that fact. Finally, for the Wheeler propagator we get (51) o2 : q2 + t~2, ~o = (q2 + ~2)1/2 (Imo) > 0), where Lz is a loop in the positive sense around z. The loop L,o comes from the retarded path, and the loop L_~ from the advanced path.
It is then clear that any propagator can be expressed in the form (50) (or its derivatives when the field is not scalar).
When the multiply together two Green functions, the exponentials join in a single one. We can change variables in the space part (ql + q2 = q), but the time component needs some care as the q0-integration is not necessarily real. where F12 is a path surrounding q~ + q2 with ql e F1 and q2 e F2.
In this way we can define the convolution product of zll and ~12 as where ~Ol 2 = (k -q)2 + t~12; ~o~ = q2 + t~ and ko is to be integrated along a path (F12) which encloses rl + F2. Now we use With (55), the integrand of (54) can be written as a sum of four terms of the form The dql and dq2 integration can then be performed by using Cauchy's theorem, or by using (ref. [23]): where 1 (58) V, = ~-sg Ira(z). Z With these formulae we can find the main properties of convolutions. We are not going to deduce those properties here. They can be found in ref. [24].
The result of the convolution of two Feynman propagators is well known. For real energy it has an absortive part on the real energy axis. These facts are related to the resultant F12 path. For two causal functions F12 is equivalent to the real axis (R), while for two Wheeler functions r12 is a closed loop enclosing all singularities, which are scattered over a region of the complex k0-plane (see [24]).

-Discussion.
A consistent quantization of the Wess-Zumino model in six dimensions shows some of the main characteristics of the general n-dimensional case. The equations of motion are of higher order. It is clear that a Lorentz-invariant higher-order equation implies that there are several modes of propagation for the virtual excitations of the field. There is a normal bradyon and other complementary modes. In six dimensions we have the simplest higher-order case, where besides the normal mode there is a tachyonic mode. When we expand the fields in terms of elementary ,,plane wave~) solutions exp [ikx], the time component (k0 = ~o) is not real for the abnormal modes. For a correct mathematical treatment it is necessary to extend the usual framework provided by the set of tempered distributions. The appropriate scheme seems to be that of the Rigged Hilbert Spaces. Particularly those in which the Hi]bert triplet (W, L 2, l~z) includes in W, all the exponentials (real or complex exp[ilx]), and in I~ z the corresponding dual functionals. The exposition of this framework is not within the scope of the present paper. We take for granted that the operations which we perform are allowed in an appropriate functional space, The normal mode is the only one that can show up asymptotically as a free wave. Further, it has the causal Feynman function as propagator. The rest of the modes (including the tachyon mode) can only exist as virtual excitations that propagate according to the Wheeler function (half-advanced and half-retarded). This Green function coincides with Cauchy's principal value on the real energy axis. For this reason it lacks the on-shell free-particle ~-function and it is compatible with the absence of asymptotically free propagation of the mode. These facts imply unitarity at tree level. When we examine the convolutions between those Green functions it is possible to see that only the absortive parts corresponding to bradyons appear. Thus showing that the simplest one-loop diagrams do not violate unitarity.
In ref. [25] it is shown that the advanced and the retarded Green functions are Lorentz invariant. It follows that the Wheeler propagator is also Lorentz invariant. As the S-matrix is constructed only with bradyon wave functions and Feynman and Wheeler propagators, it follows ,,a posteriori~ that it also Lorentz invariant.
We have shown how to construct the conserved fermion current. The fields are quantized so as to obtain a fermion charge operator that generates the correct supersymmetry transformations. This procedure is equivalent to the usual canonical quantization method, except for the fact that the supersymmetry relations gives automatically a symmetric energy-momentum operator.
Furthermore, supersymmetry assures that there is no mass or coupling constant renormalization. $$$ This work was partially supported by the Comision de Investigaciones Cientificas, Provincia Buenos Aires, Argentina.