Super QCD
Encyclopedia
In theoretical physics
, super QCD is a supersymmetric gauge theory which resembles quantum chromodynamics
(QCD) but contains additional particles and interactions which render it supersymmetric
.
The most commonly used version of super QCD is in 4 dimensions and contains one Majorana spinor supercharge. The particle content consists of vector supermultiplets, which include gluons
and gluinos
and also chiral supermultiplets
which contain quarks
and squarks transforming in the fundamental representation
of the gauge group. This theory has many features in common with real world QCD, for example in some phases it manifests confinement and chiral symmetry breaking
. The supersymmetry of this theory means that, unlike QCD, one may use nonrenormalization theorems to analytically demonstrate the existence of these phenomena and even calculate the condensate
which breaks the chiral symmetry.
and baryons
, and the moduli space
of vacua
of the theory may be parametrized by their vacuum expectation values
. On most of the moduli space the Higgs mechanism
makes all of the fields massive, and so they may be integrated out
. Classically, the resulting moduli space is singular
. The singularities correspond to points where some gluons are massless, and so could not be integrated out. In the full quantum
moduli space is nonsingular, and its structure depends on the relative values of M and N. For example, when M is less than or equal to N+1, the theory exhibits confinement.
When M is less than N, the effective action
differs from the classical action
. More precisely, while the perturbative
nonrenormalization theory forbids any perturbative correction to the superpotential
, the superpotential receives nonperturbative corrections. When N=M+1, these corrections result from a single instanton
. For larger values of N the instanton calculation suffers from infrared divergences, however the correction may nonetheless be determined precisely from the gaugino condensation
. The quantum correction to the superpotential was calculated in The Massless Limit Of Supersymmetric Qcd. If the chiral multiplets are massless, the resulting potential energy
has no minimum and so the full quantum theory has no vacuum. Instead the fields role forever to larger values.
When M is equal to or greater than N, the classical superpotential is exact. When M is equal to N, however, the moduli space receives quantum corrections from a single instanton. This correction renders the moduli space nonsingular, and also leads to chiral symmetry breaking. Then M is equal to N+1 the moduli space is not modified and so there is no chiral symmetry breaking, however there is still confinement.
When M is greater than N+1 but less than 3N/2, the theory is asymptotically free
. However at low energies the theory becomes strongly coupled, and is better described by a Seiberg dual
description in terms of magnetic variables
with the same global flavor symmetry group but a new gauge symmetry SU(M-N). Notice that the gauge group is not an observable
, but simply reflects the redundancy or a description and so may well differ in various dual theories, as it does in this case. On the other hand the global symmetry
group is an observable so it is essential that it is the same, SU(M), in both descriptions. The dual magnetic theory is free in the infrared
, the coupling constant
shrinks logarithmically, and so by the Dirac quantization condition the electric coupling constant grows logarithmically in the infrared. This implies that the potential between two electric charges, at long distances, scales as the logarithm of their distance divided by the distance.
When M is between 3N/2 and 3N, in the theory has an infrared fixed point
where it becomes a nontrivial conformal field theory
. The potential between electric charges obeys the usual Colomb law, it is inversely proportional to the distance between the charges.
When M is greater than 3N, the theory is free in the infrared, and so the force between two charges is inversely proportional to the product of the distance times the logarithm of the distance between the charges. However the theory is ill-defined in the ultraviolet, unless one includes additional heavy degrees of freedom which lead, for example, to a Seiberg dual theory of the type described above at N+1The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, super QCD is a supersymmetric gauge theory which resembles quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(QCD) but contains additional particles and interactions which render it supersymmetric
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
.
The most commonly used version of super QCD is in 4 dimensions and contains one Majorana spinor supercharge. The particle content consists of vector supermultiplets, which include gluons
Gluon
Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....
and gluinos
Gluino
A gluino is the hypothetical supersymmetric partner of a gluon. Gluinos are expected by supersymmetry theorists to be pair produced in particle accelerators such as the Large Hadron Collider if they exist....
and also chiral supermultiplets
Chiral superfield
In theoretical physics, one often analyzes theories with supersymmetry in which chiral superfields play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using the notion of superspace...
which contain quarks
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
and squarks transforming in the fundamental representation
Fundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group...
of the gauge group. This theory has many features in common with real world QCD, for example in some phases it manifests confinement and chiral symmetry breaking
Chiral symmetry breaking
In particle physics, chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of gauge theory such as Quantum Chromodynamics. The origin may be described as a fermion condensate...
. The supersymmetry of this theory means that, unlike QCD, one may use nonrenormalization theorems to analytically demonstrate the existence of these phenomena and even calculate the condensate
Condensate
Condensate may refer to:* The liquid phase produced by the condensation of steam or any other gas* The product of a chemical condensation reaction, other than water* Natural gas condensate, in the natural gas industry- Quantum physics :...
which breaks the chiral symmetry.
Phases of super QCD
Consider 4-dimensional SQCD with gauge group SU(N) and M flavors of chiral multiplets. The vacuum structure depends on M and N. The squarks may be reorganized into mesonsMeson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
and baryons
Baryon
A baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...
, and the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of vacua
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
of the theory may be parametrized by their vacuum expectation values
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
. On most of the moduli space the Higgs mechanism
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
makes all of the fields massive, and so they may be integrated out
Path integral
Path integral may refer to:* Line integral, the integral of a function along a curve* Functional integration, the integral of a functional over a space of curves...
. Classically, the resulting moduli space is singular
Singularity
Singularity or Singular points may refer to:-Mathematics:* Mathematical singularity, a point at which a given mathematical object is not defined or not well-behaved, for example infinite or not differentiable-Geometry:...
. The singularities correspond to points where some gluons are massless, and so could not be integrated out. In the full quantum
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
moduli space is nonsingular, and its structure depends on the relative values of M and N. For example, when M is less than or equal to N+1, the theory exhibits confinement.
When M is less than N, the effective action
Effective action
In quantum field theory, the effective action is a modified expression for the action, which takes into account quantum-mechanical corrections, in the following sense:...
differs from the classical action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
. More precisely, while the perturbative
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
nonrenormalization theory forbids any perturbative correction to the superpotential
Superpotential
Superpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
, the superpotential receives nonperturbative corrections. When N=M+1, these corrections result from a single instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
. For larger values of N the instanton calculation suffers from infrared divergences, however the correction may nonetheless be determined precisely from the gaugino condensation
Gaugino condensation
In particle physics, gaugino condensation is the nonzero vacuum expectation value in some models of a bilinear expression constructed in theories with supersymmetry from the superpartner of a gauge boson called the gaugino. The gaugino and the bosonic gauge field and the D-term are all components...
. The quantum correction to the superpotential was calculated in The Massless Limit Of Supersymmetric Qcd. If the chiral multiplets are massless, the resulting potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
has no minimum and so the full quantum theory has no vacuum. Instead the fields role forever to larger values.
When M is equal to or greater than N, the classical superpotential is exact. When M is equal to N, however, the moduli space receives quantum corrections from a single instanton. This correction renders the moduli space nonsingular, and also leads to chiral symmetry breaking. Then M is equal to N+1 the moduli space is not modified and so there is no chiral symmetry breaking, however there is still confinement.
When M is greater than N+1 but less than 3N/2, the theory is asymptotically free
Asymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
. However at low energies the theory becomes strongly coupled, and is better described by a Seiberg dual
Seiberg duality
In quantum field theory, Seiberg duality, conjectured by Nathan Seiberg, is an S-duality relating two different supersymmetric QCDs. The two theories are not identical, but they agree at low energies. More precisely under a renormalization group flow they flow to the same IR fixed point, and so...
description in terms of magnetic variables
Montonen-Olive duality
In theoretical physics, Montonen–Olive duality is the oldest known example of S-duality or a strong-weak duality. It generalizes the electro-magnetic symmetry of Maxwell's equations...
with the same global flavor symmetry group but a new gauge symmetry SU(M-N). Notice that the gauge group is not an observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
, but simply reflects the redundancy or a description and so may well differ in various dual theories, as it does in this case. On the other hand the global symmetry
Global symmetry
A global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.Global symmetries require conservation laws, but not forces, in physics.-See also:...
group is an observable so it is essential that it is the same, SU(M), in both descriptions. The dual magnetic theory is free in the infrared
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...
, the coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
shrinks logarithmically, and so by the Dirac quantization condition the electric coupling constant grows logarithmically in the infrared. This implies that the potential between two electric charges, at long distances, scales as the logarithm of their distance divided by the distance.
When M is between 3N/2 and 3N, in the theory has an infrared fixed point
Infrared fixed point
In physics, an infrared fixed point is a set of coupling constants, or other parameters that evolve from initial values at very high energies , to fixed stable values, usually predictable, at low energies...
where it becomes a nontrivial conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
. The potential between electric charges obeys the usual Colomb law, it is inversely proportional to the distance between the charges.
When M is greater than 3N, the theory is free in the infrared, and so the force between two charges is inversely proportional to the product of the distance times the logarithm of the distance between the charges. However the theory is ill-defined in the ultraviolet, unless one includes additional heavy degrees of freedom which lead, for example, to a Seiberg dual theory of the type described above at N+1