N = 1 supersymmetric Yang–Mills theory

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.^[3]

While N = 4 supersymmetric Yang–Mills theory is also a supersymmetric Yang–Mills theory, it has very different properties to ${\displaystyle {\mathcal {N}}=1}$ supersymmetric Yang–Mills theory, which is the theory discussed in this article. The ${\displaystyle {\mathcal {N}}=2}$ supersymmetric Yang–Mills theory was studied by Seiberg and Witten in Seiberg–Witten theory. All three theories are based in ${\displaystyle d=4}$ super Minkowski spaces.

A first treatment can be done without defining superspace, instead defining the theory in terms of familiar fields in non-supersymmetric quantum field theory.

The base spacetime is flat spacetime (Minkowski space).

SYM is a gauge theory, and there is an associated gauge group ${\displaystyle G}$ to the theory. The gauge group has associated Lie algebra ${\displaystyle {\mathfrak {g}}}$.

The field content then consists of

• a ${\displaystyle {\mathfrak {g}}}$-valued gauge field ${\displaystyle A_{\mu }}$
• a ${\displaystyle {\mathfrak {g}}}$-valued Majorana spinor field ${\displaystyle \Psi }$ (an adjoint-valued spinor), known as the 'gaugino'
• a ${\displaystyle {\mathfrak {g}}}$-valued auxiliary scalar field ${\displaystyle D}$.

For gauge-invariance, the gauge field ${\displaystyle A_{\mu }}$ is necessarily massless. This means its superpartner ${\displaystyle \Psi }$ is also massless if supersymmetry is to hold. Therefore ${\displaystyle \Psi }$ can be written in terms of two Weyl spinors which are conjugate to one another: ${\displaystyle \Psi =(\lambda ,{\bar {\lambda }})}$, and the theory can be formulated in terms of the Weyl spinor field ${\displaystyle \lambda }$ instead of ${\displaystyle \Psi }$.

When ${\displaystyle G=U(1)}$, the conceptual difficulties simplify somewhat, and this is in some sense the simplest gauge theory. The field content is simply a (co-)vector field ${\displaystyle A_{\mu }}$, a Majorana spinor ${\displaystyle \Phi }$ and a auxiliary real scalar field ${\displaystyle D}$.

The field strength tensor is defined as usual as ${\displaystyle F_{\mu \nu }:=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}$.

The Lagrangian written down by Wess and Zumino^[1] is then

${\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {i}{2}}{\bar {\Psi }}\gamma ^{\mu }\partial _{\mu }\Psi +{\frac {1}{2}}D^{2}.}$

This can be generalized^[3] to include a coupling constant ${\displaystyle e}$, and theta term ${\displaystyle \propto \vartheta F_{\mu \nu }*F^{\mu \nu }}$, where ${\displaystyle *F^{\mu \nu }}$ is the dual field strength tensor

${\displaystyle *F^{\mu \nu }={\frac {1}{2}}\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }.}$

and ${\displaystyle \epsilon ^{\mu \nu \rho \sigma }}$ is the alternating tensor or totally antisymmetric tensor. If we also replace the field ${\displaystyle \Psi }$ with the Weyl spinor ${\displaystyle \lambda }$, then a supersymmetric action can be written as

This can be viewed as a supersymmetric generalization of a pure ${\displaystyle U(1)}$ gauge theory, also known as Maxwell theory or pure electromagnetic theory.

In full generality, we must define the gluon field strength tensor,

${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }-i[A_{\mu },A_{\nu }]}$

and the covariant derivative of the adjoint Weyl spinor, ${\displaystyle D_{\mu }\lambda =\partial _{\mu }\lambda -i[A_{\mu },\lambda ].}$

To write down the action, an invariant inner product on ${\displaystyle {\mathfrak {g}}}$ is needed: the Killing form ${\displaystyle B(\cdot ,\cdot )}$ is such an inner product, and in a typical abuse of notation we write ${\displaystyle B}$ simply as ${\displaystyle {\text{Tr}}}$, suggestive of the fact that the invariant inner product arises as the trace in some representation of ${\displaystyle {\mathfrak {g}}}$.

Supersymmetric Yang–Mills then readily generalizes from supersymmetric Maxwell theory. A simple version is

${\displaystyle S_{\text{SYM}}=\int d^{4}x{\text{Tr}}\left[-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {1}{2}}{\bar {\Psi }}\gamma ^{\mu }D_{\mu }\Psi \right]}$

while a more general version is given by

The base superspace is ${\displaystyle {\mathcal {N}}=1}$ super Minkowski space.

The theory is defined in terms of a single adjoint-valued real superfield ${\displaystyle V}$, fixed to be in Wess–Zumino gauge.

The theory is defined in terms of a superfield arising from taking covariant derivatives of ${\displaystyle V}$:

${\displaystyle W_{\alpha }=-{\frac {1}{4}}{\mathcal {{\bar {D}}^{2}}}{\mathcal {D}}_{\alpha }V}$.

The supersymmetric action is then written down, with a complex coupling constant ${\displaystyle \tau ={\frac {\vartheta }{2\pi }}+{\frac {4\pi i}{e}}}$, as

where h.c. indicates the Hermitian conjugate of the preceding term.

For non-abelian gauge theory, instead define

${\displaystyle W_{\alpha }=-{\frac {1}{8}}{\bar {\mathcal {D}}}^{2}(e^{-2V}{\mathcal {D}}_{\alpha }e^{2V})}$

and ${\displaystyle \tau ={\frac {\vartheta }{2\pi }}+{\frac {4\pi i}{g}}}$. Then the action is

For the simplified Yang–Mills action on Minkowski space (not on superspace), the supersymmetry transformations are

${\displaystyle \delta _{\epsilon }A_{\mu }={\bar {\epsilon }}\gamma _{\mu }\Psi }$

${\displaystyle \delta _{\epsilon }\Psi =-{\frac {1}{2}}F_{\mu \nu }\gamma ^{\mu \nu }\epsilon }$

where ${\displaystyle \gamma ^{\mu \nu }={\frac {1}{2}}(\gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu })}$.

For the Yang–Mills action on superspace, since ${\displaystyle W_{\alpha }}$ is chiral, then so are fields built from ${\displaystyle W_{\alpha }}$. Then integrating over half of superspace, ${\displaystyle \int d^{2}\theta }$, gives a supersymmetric action.

An important observation is that the Wess–Zumino gauge is not a supersymmetric gauge, that is, it is not preserved by supersymmetry. However, it is possible to do a compensating gauge transformation to return to Wess–Zumino gauge. Then, after a supersymmetry transformation and the compensating gauge transformation, the superfields transform as

${\displaystyle \delta A_{\mu }=\epsilon \sigma _{\mu }{\bar {\lambda }}+\lambda \sigma _{\mu }{\bar {\epsilon }},}$

${\displaystyle \delta \lambda =\epsilon D+(\sigma ^{\mu \nu }\epsilon )F_{\mu \nu }}$

${\displaystyle \delta D=i\epsilon \sigma ^{\mu }\partial _{\mu }{\bar {\lambda }}-i\partial _{\mu }\lambda {\bar {\sigma }}^{\mu }{\bar {\epsilon }}.}$

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. (Not supergauge transformations, which would be transformations in a theory with local supersymmetry).

Such a transformation is parametrized by a chiral superfield ${\displaystyle \Omega }$, under which the real superfield transforms as

${\displaystyle V\mapsto V+i(\Omega -\Omega ^{\dagger }).}$

In particular, upon expanding ${\displaystyle V}$ and ${\displaystyle \Omega }$ appropriately into constituent superfields, then ${\displaystyle V}$ contains a vector superfield ${\displaystyle A_{\mu }}$ while ${\displaystyle \Omega }$ contains a scalar superfield ${\displaystyle \omega }$, such that

${\displaystyle A_{\mu }\mapsto A_{\mu }-2\partial _{\mu }({\text{Re}}\,\omega )=:A_{\mu }+\partial _{\mu }\alpha .}$

The chiral superfield used to define the action,

${\displaystyle W_{\alpha }=-{\frac {1}{4}}{\bar {\mathcal {D}}}^{2}{\mathcal {D}}_{\alpha }V,}$

is gauge invariant.

The chiral superfield is adjoint valued. The transformation of ${\displaystyle V}$ is prescribed by

${\displaystyle e^{2V}\mapsto e^{-2i\Omega ^{\dagger }}e^{2V}e^{2i\Omega }}$,

from which the transformation for ${\displaystyle V}$ can be derived using the Baker–Campbell–Hausdorff formula.

The chiral superfield ${\displaystyle W_{\alpha }=-{\frac {1}{8}}{\bar {\mathcal {D}}}^{2}(e^{-2V}{\mathcal {D}}_{\alpha }e^{2V})}$ is not invariant but transforms by conjugation:

${\displaystyle W_{\alpha }\mapsto e^{2i\Omega }W_{\alpha }e^{-2i\Omega }}$,

so that upon tracing in the action, 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. Just as the super Poincaré algebra is a supersymmetric extension of the Poincaré algebra, the superconformal algebra is a supersymmetric extension of the conformal algebra which also contains a spinorial generator of conformal supersymmetry ${\displaystyle S_{\alpha }}$.

Conformal invariance is broken in the quantum theory by trace and conformal anomalies.

While the quantum ${\displaystyle {\mathcal {N}}=1}$ supersymmetric Yang–Mills theory does not have superconformal symmetry, quantum N = 4 supersymmetric Yang–Mills theory does.

The ${\displaystyle {\text{U}}(1)}$ R-symmetry for ${\displaystyle {\mathcal {N}}=1}$ supersymmetry is a symmetry of the classical theory, but not of the quantum theory due to an anomaly.

Matter can be added in the form of Wess–Zumino model type superfields ${\displaystyle \Phi }$. Under a gauge transformation,

${\displaystyle \Phi \mapsto \exp(-2iq\Omega )\Phi }$,

and instead of using just ${\displaystyle \Phi ^{\dagger }\Phi }$ as the Lagrangian as in the Wess–Zumino model, for gauge invariance it must be replaced with ${\displaystyle \Phi ^{\dagger }e^{2qV}\Phi .}$

This gives a supersymmetric analogue to QED. The action can be written

${\displaystyle S_{\text{SMaxwell}}+\int d^{4}x\,\int d^{4}\theta \,\Phi ^{\dagger }e^{2qV}\Phi .}$

For ${\displaystyle N_{f}}$ flavours, we instead have ${\displaystyle N_{f}}$ superfields ${\displaystyle \Phi _{i}}$, and the action can be written

${\displaystyle S_{\text{SMaxwell}}+\int d^{4}x\,\int d^{4}\theta \,\Phi _{i}^{\dagger }e^{2q_{i}V}\Phi _{i}.}$

with implicit summation.

However, for a well-defined quantum theory, a theory such as that defined above suffers a gauge anomaly. We are obliged to add a partner ${\displaystyle {\tilde {\Phi }}}$ to each chiral superfield ${\displaystyle \Phi }$ (distinct from the idea of superpartners, and from conjugate superfields), which has opposite charge. This gives the action

${\displaystyle S_{\text{SQED}}=S_{\text{SMaxwell}}+\int d^{4}x\,\int d^{4}\theta \,\Phi _{i}^{\dagger }e^{2q_{i}V}\Phi _{i}+{\tilde {\Phi }}_{i}^{\dagger }e^{-2q_{i}V}{\tilde {\Phi }}_{i}.}$

For non-abelian gauge, matter chiral superfields ${\displaystyle \Phi }$ are now valued in a representation ${\displaystyle R}$ of the gauge group: ${\displaystyle \Phi \mapsto \exp(-2i\Omega )\Phi }$.

The Wess–Zumino kinetic term must be adjusted to ${\displaystyle \Phi ^{\dagger }e^{2V}\Phi }$.

Then a simple SQCD action would be to take ${\displaystyle R}$ to be the fundamental representation, and add the Wess–Zumino term:

${\displaystyle S_{\text{SYM}}+\int d^{4}x\,d^{4}\theta \,\Phi ^{\dagger }e^{2V}\Phi }$.

More general and detailed forms of the super QCD action are given in that article.

When the center of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ is non-trivial, there is an extra term which can be added to the action known as the Fayet–Iliopoulos term.