Feynman Path Integral Formulation

36 1 Continuum Formulation
{Q i α, ¯Q j β } = 2δ ij γ μ αβ P μ , (1.172)
with i, j = 1...N . The original case of Eq. (1.168) then corresponds to the simplest
choice, N = 1 supersymmetry, whereas N > 1 is referred to as extended supersymmetry.
So different supersymmetric theories can be labeled by the number N
of supersymmetric charges, but it turns out that this number is highly constrained, as
in any given spacetime dimensions only certain values of N are possible. As shown
above, in four dimensions N = 1 supersymmetry has a complex pair Q, ¯Q of supersymmetry
charges, which are each two-component Weyl spinors, thus giving a total
of four real supercharges. On the other hand, still in four dimensions, N = 4 supersymmetry
has four complex pairs Q, ¯Q of supersymmetry charges, again with each a
two-component Weyl spinors, thus giving now a total of sixteen real supercharges.
Accordingly the renormalization properties of supersymmetric field theories vary
dramatically, depending on which type of supersysmmetry is actually being implemented.
For example, for N = 2 supersymmetry the vanishing of the β-function
at leading order implies that it will vanish to all orders. For N = 4 supersymmetry
the situation is even more remarkable, since there one has β(g) =0 to all orders
in g without any need to fine-tune the interaction. The latter provides an example
of a theory with no ultraviolet divergences, and truly constant coupling constant.
Ultimately whether any of these theories are just ingenious elaborate mathematical
recreations, or appear instead as parts of physical theories realized in nature in some
form or another remains so far still an open question (for a recent survey of phenomenological
opportunities for supersymmetric theories see, for example Zumino
and Gaillard, 2008). After all QED or QCD are not finite theories, and still lead to
perfectly acceptable, non-trivial and experimentally verifiable predictions once the
problem of ultraviolet divergences is treated correctly via the renormalization procedure.
The danger in the case of supersymmetric theories is that after all the elaborate
work done to construct such theories one might be left with an empty shell: a trivial
theory and a complicated way of re-writing an essentially non-interacting, Gaussian
theory.
Of great phenomenological interests are supersymmetric Yang-Mills theories in
four dimensions. The simplest corresponds to an SU(N c ) pure gauge theory with
N = 1 supersymmetry. The theory contains gauge bosons A a μ (the ordinary gluons,
with a = 1...N 2 c − 1) and a single 4-component Majorana spinor λ a , the gluino,
satisfying the Majorana condition ¯λ a = λ aT C. The gluino is the supersymmetric
partner of the gluon, and, like the gluon itself, transforms under the adjoint representation
of the group (thus in the case of SU(3) both the gluon and the gluino are
in a color octet representation). The susy-Yang-Mills Lagrangian is
L = − 1 4 Fa μν F a μν + 1 2 ¯λγ μ D μ (A)λ , (1.173)
with F a μν the usual Yang-Mills field strength tensor, and D μ (A) the usual gauge covariant
derivative acting on λ a . The action is locally invariant under supersymmetry
transformations