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.

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.

The supersymmetric Yang–Mills action

Preliminary treatment

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

Spacetime and matter content

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 ${\displaystyle \Psi }$.

Supersymmetric pure electromagnetic theory

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.

Supersymmetric Yang–Mills theory (preliminary treatment)

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

Superspace treatment

Superspace and superfield content

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.

Supersymmetric Maxwell theory on superspace

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.

Supersymmetric Yang–Mills on superspace

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

Symmetries of the action

Supersymmetry

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

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

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

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

Gauge symmetry

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).

Generalized abelian gauge transformations

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.

Generalized non-abelian gauge transformations

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.

Extra classical symmetries

Superconformal symmetry

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.

R-symmetry

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.

Adding matter

Abelian gauge

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

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

Non-Abelian gauge

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.

Fayet–Iliopoulos term

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.

References