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 ...

, super QCD is a supersymmetric gauge theory which resembles

quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...

(QCD) but contains additional particles and interactions which render it

supersymmetric 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 ...

. The most commonly used version of super QCD is in 4 dimensions and contains one

Majorana spinor In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this e ...

supercharge. The particle content consists of vector supermultiplets, which include

gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...

s and gluinos and also chiral supermultiplets which contain

quark A quark () is a type of 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. All commonly o ...

s and

squark In supersymmetric extension to the Standard Model (SM) of physics, a sfermion is a hypothetical spin-0 superpartner particle (sparticle) of its associated fermion. Each particle has a superpartner with spin that differs by . Fermions in the SM h ...

s 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 Confinement may refer to * With respect to humans: ** An old-fashioned or archaic synonym for childbirth ** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods ** Civil confi ...

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.

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 (spin-zero) squarks may be reorganized into hadrons, and the

moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...

vacua A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...

of the theory may be parametrized by their vacuum expectation values. On most of the moduli space the

Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...

makes all of the fields massive, and so they may be

integrated out Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...

. Classically, the resulting moduli space is

singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, ...

. The singularities correspond to points where some gluons are massless, and so could not be integrated out. In the full

quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...

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 In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective act ...

differs from the classical action. More precisely, while the

perturbative In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for whi ...

nonrenormalization theory forbids any perturbative correction to the superpotential, the superpotential receives

nonperturbative In mathematics and physics, a non-perturbative function (mathematics), function or process is one that cannot be described by perturbation theory. An example is the function : f(x) = e^, which does not have a Taylor series at ''x'' = 0. Every c ...

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 i
The Massless Limit Of Supersymmetric Qcd
If the chiral multiplets are massless, the resulting

potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

has no minimum and so the full quantum theory has no vacuum. Instead the fields roll 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 In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases. Asymptotic fr ...

. 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 In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuo ...

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 (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...

, the

coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...

shrinks logarithmically, and so by the

Dirac quantization condition In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...

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 In physics, an infrared fixed point is a set of coupling constants, or other parameters, that evolve from initial values at very high energies (short distance) to fixed stable values, usually predictable, at low energies (large distance). This usu ...

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+1

References

Lectures on supersymmetric gauge theories and electric-magnetic duality
by

Nathan Seiberg Nathan "Nati" Seiberg (; born September 22, 1956) is an Israeli American theoretical physicist who works on quantum field theory and string theory. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, United ...

and Kenneth Intriligator. {{Quantum field theories Supersymmetric quantum field theory Quantum chromodynamics