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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 List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a supermultiplet is a representation of a supersymmetry algebra, possibly with extended supersymmetry. Then a superfield is a field on
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 num ...
which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply be viewed as a function on superspace. Formally, it is a
section Section, Sectioning, or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of an associated supermultiplet bundle. Phenomenologically, superfields are used to describe
particles In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
. It is a feature of supersymmetric field theories that particles form pairs, called
superpartner In particle physics, a superpartner (also sparticle) is a class of hypothetical elementary particles predicted by supersymmetry, which, among other applications, is one of the well-studied ways to extend the Standard Model of high-energy physics. ...
s where
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-integer ...
are paired with
fermions In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin ( spin , spin , etc.) and obey the Pauli exclusion principle. These particles include all quarks and leptons and ...
. These supersymmetric fields are used to build supersymmetric
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatom ...
, where the fields are promoted to operators.


History

Superfields were introduced by
Abdus Salam Mohammad Abdus Salam Salam adopted the forename "Mohammad" in 1974 in response to the anti-Ahmadiyya decrees in Pakistan, similarly he grew his beard. (; ; 29 January 192621 November 1996) was a Pakistani theoretical physicist. He shared the 1 ...
and J. A. Strathdee in a 1974 article. Operations on superfields and a partial classification were presented a few months later by
Sergio Ferrara Sergio Ferrara (born 2 May 1945) is an Italian physicist working on theoretical physics of elementary particles and mathematical physics. He is renowned for the discovery of theories introducing supersymmetry as a symmetry of elementary particles ( ...
, Julius Wess and Bruno Zumino.


Naming and classification

The most commonly used supermultiplets are vector multiplets, chiral multiplets (in d = 4,\mathcal = 1 supersymmetry for example), hypermultiplets (in d = 4,\mathcal = 2 supersymmetry for example), tensor multiplets and gravity multiplets. The highest component of a vector multiplet is a
gauge boson In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles whose interactions are described by a gauge theory interact with each other by the exchange of gauge ...
, the highest component of a chiral or hypermultiplet is a
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
, the highest component of a gravity multiplet is a
graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...
. The names are defined so as to be invariant under dimensional reduction, although the organization of the fields as representations of the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physi ...
changes. The use of these names for the different multiplets can vary in literature. A chiral multiplet (whose highest component is a spinor) may sometimes be referred to as a ''scalar multiplet'', and in d = 4,\mathcal = 2 SUSY, a vector multiplet (whose highest component is a vector) can sometimes be referred to as a chiral multiplet.


Superfields in d = 4, N = 1 supersymmetry

Conventions in this section follow the notes by . A general complex superfield \Phi(x, \theta, \bar \theta) in d = 4, \mathcal = 1 supersymmetry can be expanded as :\Phi(x, \theta, \bar\theta) = \phi(x) + \theta\chi(x) + \bar\theta \bar\chi'(x) + \bar \theta \sigma^\mu \theta V_\mu(x) + \theta^2 F(x) + \bar \theta^2 \bar F'(x) + \bar\theta^2 \theta\xi(x) + \theta^2 \bar\theta \bar \xi' (x) + \theta^2 \bar\theta^2 D(x), where \phi, \chi, \bar \chi' , V_\mu, F, \bar F', \xi, \bar \xi', D are different complex fields. This is not an irreducible supermultiplet, and so different constraints are needed to isolate irreducible representations.


Chiral superfield

A (anti-)chiral superfield is a supermultiplet of d=4, \mathcal = 1 supersymmetry. In four dimensions, the minimal \mathcal=1 supersymmetry may be written using the notion 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 num ...
. Superspace contains the usual space-time coordinates x^, \mu=0,\ldots,3, and four extra fermionic coordinates \theta_\alpha,\bar\theta^\dot\alpha with \alpha, \dot\alpha = 1,2, transforming as a two-component (Weyl)
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
and its conjugate. In d = 4,\mathcal = 1
supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
, a chiral superfield is a function over chiral superspace. There exists a projection from the (full) superspace to chiral superspace. So, a function over chiral superspace can be pulled back to the full superspace. Such a function \Phi(x, \theta, \bar\theta) satisfies the covariant constraint \overline\Phi=0, where \bar D is the covariant derivative, given in index notation as :\bar D_\dot\alpha = -\bar\partial_\dot\alpha - i\theta^\alpha \sigma^\mu_\partial_\mu. A chiral superfield \Phi(x, \theta, \bar\theta) can then be expanded as : \Phi (y , \theta ) = \phi(y) + \sqrt \theta \psi (y) + \theta^2 F(y), where y^\mu = x^\mu + i \theta \sigma^\mu \bar . The superfield is independent of the 'conjugate spin coordinates' \bar\theta in the sense that it depends on \bar\theta only through y^\mu. It can be checked that \bar D_\dot\alpha y^\mu = 0. The expansion has the interpretation that \phi is a complex scalar field, \psi is a Weyl spinor. There is also the auxiliary complex scalar field F, named F by convention: this is the F-term which plays an important role in some theories. The field can then be expressed in terms of the original coordinates (x,\theta, \bar \theta) by substituting the expression for y: :\Phi(x, \theta, \bar\theta) = \phi(x) + \sqrt \theta \psi (x) + \theta^2 F(x) + i\theta\sigma^\mu\bar\theta\partial_\mu\phi(x) - \frac\theta^2\partial_\mu\psi(x)\sigma^\mu\bar\theta - \frac\theta^2\bar\theta^2\square\phi(x).


Antichiral superfields

Similarly, there is also antichiral superspace, which is the complex conjugate of chiral superspace, and antichiral superfields. An antichiral superfield \Phi^\dagger satisfies D \Phi^\dagger = 0, where :D_\alpha = \partial_\alpha + i\sigma^\mu_\bar\theta^\dot\alpha\partial_\mu. An antichiral superfield can be constructed as the complex conjugate of a chiral superfield.


Actions from chiral superfields

For an action which can be defined from a single chiral superfield, see Wess–Zumino model.


Vector superfield

The vector superfield is a supermultiplet of \mathcal = 1 supersymmetry. A vector superfield (also known as a real superfield) is a function V(x,\theta,\bar\theta) which satisfies the reality condition V = V^\dagger. Such a field admits the expansion :V = C + i\theta\chi - i \overline\overline + \tfrac\theta^2(M+iN)-\tfrac\overline(M-iN) - \theta \sigma^\mu \overline A_\mu +i\theta^2 \overline \left( \overline + \tfrac\overline^\mu \partial_\mu \chi \right) -i\overline^2 \theta \left(\lambda + \tfrac\sigma^\mu \partial_\mu \overline \right) + \tfrac\theta^2 \overline^2 \left(D + \tfrac\Box C\right). The constituent fields are * Two real scalar fields C and D * A complex scalar field M + iN * Two Weyl spinor fields \chi_\alpha and \lambda^\alpha * A real vector field (
gauge field In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
) A_\mu Their transformation properties and uses are further discussed in
supersymmetric gauge theory In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion. Gauge theory A gauge theory is a field theory with gauge symmetry. Rough ...
. Using gauge transformations, the fields C, \chi and M + iN can be set to zero. This is known as
Wess–Zumino gauge In particle physics, the Wess–Zumino gauge is a particular choice of a gauge transformation in a gauge theory with supersymmetry. In this gauge, the supersymmetrized gauge transformation is chosen in such a way that most components of the vect ...
. In this gauge, the expansion takes on the much simpler form : V_ = \theta\sigma^\mu\bar\theta A_\mu + \theta^2 \bar\theta \bar\lambda + \bar\theta^2 \theta \lambda + \frac\theta^2\bar\theta^2 D. Then \lambda is the
superpartner In particle physics, a superpartner (also sparticle) is a class of hypothetical elementary particles predicted by supersymmetry, which, among other applications, is one of the well-studied ways to extend the Standard Model of high-energy physics. ...
of A_\mu, while D is an auxiliary scalar field. It is conventionally called D, and is known as the D-term.


Scalars

A scalar is never the highest component of a superfield; whether it appears in a superfield at all depends on the dimension of the spacetime. For example, in a 10-dimensional N=1 theory the vector multiplet contains only a vector and a
Majorana–Weyl spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotati ...
, while its dimensional reduction on a d-dimensional
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
is a vector multiplet containing d real scalars. Similarly, in an 11-dimensional theory there is only one supermultiplet with a finite number of fields, the gravity multiplet, and it contains no scalars. However again its dimensional reduction on a d-torus to a maximal gravity multiplet does contain scalars.


Hypermultiplet

A hypermultiplet is a type of representation of an extended supersymmetry algebra, in particular the matter multiplet of \mathcal = 2 supersymmetry in 4 dimensions, containing two complex
scalars 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 a ...
''A''''i'', a Dirac
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
ψ, and two further
auxiliary Auxiliary may refer to: In language * Auxiliary language (disambiguation) * Auxiliary verb In military and law enforcement * Auxiliary police * Auxiliaries, civilians or quasi-military personnel who provide support of some kind to a military se ...
complex scalars ''F''''i''. The name "hypermultiplet" comes from old term "hypersymmetry" for ''N''=2 supersymmetry used by ; this term has been abandoned, but the name "hypermultiplet" for some of its representations is still used.


Extended supersymmetry (N > 1)

This section records some commonly used irreducible supermultiplets in extended supersymmetry in the d = 4 case. These are constructed by a
highest-weight representation In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multipl ...
construction in the sense that there is a vacuum vector annihilated by the supercharges Q^A, A = 1, \cdots, \mathcal. The irreps have dimension 2^\mathcal. For supermultiplets representing massless particles, on physical grounds the maximum allowed \mathcal is \mathcal = 8, while for
renormalizability Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
, the maximum allowed \mathcal is \mathcal = 4.


N = 2

The \mathcal = 2 vector or chiral multiplet \Psi contains a
gauge field In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
A_\mu, two Weyl fermions \lambda, \psi, and a scalar \phi (which also transform in the adjoint representation of a
gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...
). These can also be organised into a pair of \mathcal = 1 multiplets, an \mathcal = 1 vector multiplet W = (A_\mu, \lambda) and chiral multiplet \Phi = (\phi, \psi). Such a multiplet can be used to define Seiberg–Witten theory concisely. The \mathcal = 2 hypermultiplet or scalar multiplet consists of two Weyl fermions and two complex scalars, or two \mathcal = 1 chiral multiplets.


N = 4

The \mathcal = 4 vector multiplet contains one gauge field, four Weyl fermions, six scalars, and CPT conjugates. This appears in N = 4 supersymmetric Yang–Mills theory.


See also

*
Supersymmetric gauge theory In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion. Gauge theory A gauge theory is a field theory with gauge symmetry. Rough ...
* D-term * F-term


References

* * Stephen P. Martin. ''A Supersymmetry Primer'', arXiv:hep-ph/9709356 . * Yuji Tachikawa. ''N=2 supersymmetric dynamics for pedestrians'', arXiv:1312.2684. * {{Supersymmetry topics Supersymmetry