In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics.
It is a special case of 4D N = 1 global supersymmetry.
Super Yang–Mills was studied by Julius Wess and Bruno Zumino in which they demonstrated the supergauge-invariance of the theory and wrote down its action,[1] alongside the action of the Wess–Zumino model, another early supersymmetric field theory.
The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry[2] and of Tong.
A first treatment can be done without defining superspace, instead defining the theory in terms of familiar fields in non-supersymmetric quantum field theory.
can be written in terms of two Weyl spinors which are conjugate to one another:
, and the theory can be formulated in terms of the Weyl spinor field
, the conceptual difficulties simplify somewhat, and this is in some sense the simplest gauge theory.
and a auxiliary real scalar field
The field strength tensor is defined as usual as
The Lagrangian written down by Wess and Zumino[1] is then This can be generalized[3] to include a coupling constant
is the dual field strength tensor and
μ ν ρ σ
In full generality, we must define the gluon field strength tensor, and the covariant derivative of the adjoint Weyl spinor,
To write down the action, an invariant inner product on
is such an inner product, and in a typical abuse of notation we write
, suggestive of the fact that the invariant inner product arises as the trace in some representation of
The theory is defined in terms of a single adjoint-valued real superfield
The theory is defined in terms of a superfield arising from taking covariant derivatives of
: The supersymmetric action is then written down, with a complex coupling constant
For the simplified Yang–Mills action on Minkowski space (not on superspace), the supersymmetry transformations are where
Then, after a supersymmetry transformation and the compensating gauge transformation, the superfields transform as The preliminary theory defined on spacetime is manifestly gauge invariant as it is built from terms studied in non-supersymmetric gauge theory which are gauge invariant.
The superfield formulation requires a theory of generalized gauge transformations.
, such that The chiral superfield used to define the action, is gauge invariant.
As a classical theory, supersymmetric Yang–Mills theory admits a larger set of symmetries, described at the algebra level by the superconformal algebra.
Matter can be added in the form of Wess–Zumino model type superfields
as the Lagrangian as in the Wess–Zumino model, for gauge invariance it must be replaced with
(distinct from the idea of superpartners, and from conjugate superfields), which has opposite charge.
This gives the action For non-abelian gauge, matter chiral superfields
to be the fundamental representation, and add the Wess–Zumino term: More general and detailed forms of the super QCD action are given in that article.