In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the Wess–Zumino model has become the first known example of an interacting four-dimensional
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
with linearly realised
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
. In 1974,
Julius Wess
Julius Erich Wess (5 December 19348 August 2007) was an Austrian theoretical physicist noted as the co-inventor of the Wess–Zumino model and Wess–Zumino–Witten model in the field of supersymmetry and conformal field theory. He was also a ...
and
Bruno Zumino
Bruno Zumino (28 April 1923 − 21 June 2014) was an Italian theoretical physicist and faculty member at the University of California, Berkeley. He obtained his DSc degree from the University of Rome in 1945.
He was renowned for his rigorous pro ...
studied, using modern terminology, dynamics of a single
chiral superfield
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra.
Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply ...
(composed of a complex
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
and a
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
) whose cubic
superpotential
In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
leads to a
renormalizable
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similarity, self-similar geometric structures, that are used to treat infinity, infinities arising in calculated ...
theory.
The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry,
and to some extent of Tong.
The model is an important model in supersymmetric quantum field theory. It is arguably the simplest supersymmetric field theory in four dimensions, and is ungauged.
The Wess–Zumino action
Preliminary treatment
Spacetime and matter content
In a preliminary treatment, the theory is defined on flat
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
(
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
). For this article, the metric has ''mostly plus'' signature. The matter content is a real
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
, a real
pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...
field
, and a real (
Majorana)
spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\colon ...
.
This is a preliminary treatment in the sense that the theory is written in terms of familiar scalar and spinor fields which are functions of spacetime, without developing a theory of
superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
or
superfield
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra.
Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply ...
s, which appear later in the article.
Free, massless theory
The Lagrangian of the free, massless Wess–Zumino model is
::
where
*
*
The corresponding action is
::
.
Massive theory
Supersymmetry is preserved when adding a mass term of the form
::
Interacting theory
Supersymmetry is preserved when adding an interaction term with coupling constant
:
::
The full Wess–Zumino action is then given by putting these Lagrangians together:
Alternative expression
There is an alternative way of organizing the fields. The real fields
and
are combined into a single ''complex'' scalar field
while the Majorana spinor is written in terms of two Weyl spinors:
. Defining the superpotential
:
the Wess–Zumino action can also be written (possibly after relabelling some constant factors)
Upon substituting in
, one finds that this is a theory with a massive complex scalar
and a massive Majorana spinor
of the ''same'' mass. The interactions are a cubic and quartic
interaction, and a
Yukawa interaction
In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki Yukawa, is an interaction between particles according to the Yukawa potential. Specifically, it is a scalar field (or pseudoscalar field) and a Dirac field of the t ...
between
and
, which are all familiar interactions from courses in non-supersymmetric quantum field theory.
Using superspace and superfields
Superspace and superfield content
Superspace consists of the direct sum of Minkowski space with 'spin space', a four dimensional space with coordinates
, where
are indices taking values in
More formally, superspace is constructed as the space of right cosets of the Lorentz group in the
super-Poincaré group.
The fact there is only 4 'spin coordinates' means that this is a theory with what is known as
supersymmetry, corresponding to an algebra with a single
supercharge
In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge in physics.
Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and v ...
. The
dimensional superspace is sometimes written
, and called
super Minkowski space
In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super Poincaré algeb ...
. The 'spin coordinates' are so called not due to any relation to angular momentum, but because they are treated as
anti-commuting numbers, a property typical of spinors in quantum field theory due to the
spin statistics theorem
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
.
A superfield
is then a function on superspace,
.
Defining the supercovariant derivative
:
a chiral superfield satisfies
The field content is then simply a single chiral superfield.
However, the chiral superfield ''contains'' fields, in the sense that it admits the expansion
:
with
Then
can be identified as a complex scalar,
is a
Weyl spinor
In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
and
is an auxiliary complex scalar.
These fields admit a further relabelling, with
and
This allows recovery of the preliminary forms, after eliminating the non-dynamical
using its equation of motion.
Free, massless action
When written in terms of the chiral superfield
, the action (for the free, massless Wess–Zumino model) takes on the simple form
:
where
are
integrals over spinor dimensions of
superspace
Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
.
Superpotential
Masses and interactions are added through a superpotential. The Wess–Zumino superpotential is
:
Since
is complex, to ensure the action is real its conjugate must also be added.
The full Wess–Zumino action is written
Supersymmetry of the action
Preliminary treatment
The action is invariant under the supersymmetry transformations, given in infinitesimal form by
::
::
::
where
is a Majorana spinor-valued transformation parameter and
is the
chirality operator.
The alternative form is invariant under the transformation
::
::
.
Without developing a theory of superspace transformations, these symmetries appear ad-hoc.
Superfield treatment
If the action can be written as
where
is a real superfield, that is,
, then the action is invariant under supersymmetry.
Then the reality of
means it is invariant under supersymmetry.
Extra classical symmetries
Superconformal symmetry
The ''massless'' Wess–Zumino model admits a larger set of symmetries, described at the algebra level by the
superconformal algebra
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superco ...
. As well as the Poincaré symmetry generators and the supersymmetry translation generators, this contains the conformal algebra as well as a conformal supersymmetry generator
.
The conformal symmetry is broken at the quantum level by trace and conformal anomalies, which break invariance under the conformal generators
for dilatations and
for
special conformal transformation
In projective geometry, a special conformal transformation is a linear fractional transformation that is ''not'' an affine transformation. Thus the generation of a special conformal transformation involves use of multiplicative inversion, which ...
s respectively.
R-symmetry
The
R-symmetry
In theoretical physics, the R-symmetry is the symmetry transforming different supercharges in a theory with supersymmetry into each other. In the simplest case of the ''N''=1 supersymmetry, such an R-symmetry is isomorphic to a global U(1) group o ...
of
supersymmetry holds when the superpotential
is a monomial. This means either
, so that the superfield
is massive but free (non-interacting), or
so the theory is massless but (possibly) interacting.
This is broken at the quantum level by anomalies.
Action for multiple chiral superfields
The action generalizes straightforwardly to multiple chiral superfields
with
. The most general
renormalizable
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similarity, self-similar geometric structures, that are used to treat infinity, infinities arising in calculated ...
theory is
: