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 simplestchoice, N = 1 supersymmetry, whereas N > 1 is referred to as extended supersymmetry.So different supersymmetric theories can be labeled by the number Nof supersymmetric charges, but it turns out that this number is highly constrained, asin any given spacetime dimensions only certain values of N are possible. As shownabove, in four dimensions N = 1 supersymmetry has a complex pair Q, ¯Q of supersymmetrycharges, which are each two-component Weyl spinors, thus giving a totalof four real supercharges. On the other hand, still in four dimensions, N = 4 supersymmetryhas four complex pairs Q, ¯Q of supersymmetry charges, again with each atwo-component Weyl spinors, thus giving now a total of sixteen real supercharges.Accordingly the renormalization properties of supersymmetric field theories varydramatically, depending on which type of supersysmmetry is actually being implemented.For example, for N = 2 supersymmetry the vanishing of the β-functionat leading order implies that it will vanish to all orders. For N = 4 supersymmetrythe situation is even more remarkable, since there one has β(g) =0 to all ordersin g without any need to fine-tune the interaction. The latter provides an exampleof a theory with no ultraviolet divergences, and truly constant coupling constant.Ultimately whether any of these theories are just ingenious elaborate mathematicalrecreations, or appear instead as parts of physical theories realized in nature in someform or another remains so far still an open question (for a recent survey of phenomenologicalopportunities for supersymmetric theories see, for example Zuminoand Gaillard, 2008). After all QED or QCD are not finite theories, and still lead toperfectly acceptable, non-trivial and experimentally verifiable predictions once theproblem of ultraviolet divergences is treated correctly via the renormalization procedure.The danger in the case of supersymmetric theories is that after all the elaboratework done to construct such theories one might be left with an empty shell: a trivialtheory and a complicated way of re-writing an essentially non-interacting, Gaussiantheory.Of great phenomenological interests are supersymmetric Yang-Mills theories infour dimensions. The simplest corresponds to an SU(N c ) pure gauge theory withN = 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 supersymmetricpartner of the gluon, and, like the gluon itself, transforms under the adjoint representationof the group (thus in the case of SU(3) both the gluon and the gluino arein a color octet representation). The susy-Yang-Mills Lagrangian isL = − 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 covariantderivative acting on λ a . The action is locally invariant under supersymmetrytransformations