Color charge

Concept colors for the color charge: red, green, blue / cyan, magenta, yellow

The color charge , also briefly color , a particle is in the physics of a size that in the quantum describes how the particles under the strong interaction behaves. All strongly interacting particles are colored; in the standard model of particle physics these are the quarks and the gluons . All other elementary particles are colorless. Physically speaking, the quarks and gluons are in a nontrivial representation of the symmetry group of quantum chromodynamics, the other elementary particles in the trivial one. The concept was proposed in 1964 by Oscar Wallace Greenberg and independently in 1965 by Moo-Young Han and Yoichiro Nambu .

The designation as color is just as misleading as the designation as charge: the color of quantum chromodynamics does not correspond to the optical color of a macroscopic object, nor is the color charge the charge of the strong interaction. Instead, color can best be understood in analogy to the spin of a particle, in which a classical particle in quantum mechanics is represented as a two-component wave function : In the description by quantum chromodynamics, the wave function of a quark has three components, with the three primary colors red and green and blue, the colors of an antiquark correspond to the three anti- colors (secondary colors) anti-red (cyan), anti-green (magenta) and anti-blue (yellow). Gluons consist of a combination of colors and anti-colors and are described by matrices in the color space.

Nobel laureate in physics Richard P. Feynman writes about the designation of this property as "color" :

"The idiot physicists, unable to come up with any wonderful Greek words anymore, call this type of polarization by the unfortunate name of 'color,' [sic!] Which has nothing to do with the color in the normal sense."

"Those physicist idiots, incapable of coming up with any wonderful Greek words, refer to this kind of polarization with the unfortunate term 'color,' [sic!] Which has nothing to do with color in the usual sense."

- Richard P. Feynman : QED: The Strange Theory of Light and Matter

The analogy between optical color and quantum chromodynamic color is as follows: Just as the three optical basic colors add up to white , an object that is composed of color and the associated anti-color, of three colors or of three anti-colors, has no strong charge.

Historical background

The -Baryon was discovered in 1951 . It consists of three identical u-quarks that have no orbital angular momentum , so that the baryon is spatially symmetrical and has a symmetrical spin wave function with a total spin of  3/2. As a baryon, however, the particle is a fermion and must therefore have an antisymmetric overall wave function. It was therefore natural to introduce an additional discrete degree of freedom . In order for this additional degree of freedom to lead to an antisymmetric state with three quarks, it must be able to assume at least three different values. ${\displaystyle \Delta ^{++}}$

Another indication of hidden degrees of freedom of quarks came in from the measurement of scattering cross sections of electron- positron - colliders . When comparing the reactions of electron-positron pairs to hadrons and to muon- antimyon pairs, one naively expects that the scattering cross-sections relate to one another like the square of the electric charges of the particles involved, summed up over the different particle types or flavors : ${\displaystyle {\mathcal {BR}}}$ ${\displaystyle Q}$

${\displaystyle \left.{\frac {{\mathcal {BR}}(e^{+}e^{-}\to {\text{had}})}{{\mathcal {BR}}(e^{+}e^{-}\to \mu ^{+}\mu ^{-})}}\right|_{\text{naiv}}=\left({\frac {\sum _{\text{flavours}}Q_{i}}{e}}\right)^{2}}$

with the elementary charge . ${\displaystyle e}$

Experimentally, however, a value was found that was increased by a factor of 3:

${\displaystyle \left.{\frac {{\mathcal {BR}}(e^{+}e^{-}\to {\text{had}})}{{\mathcal {BR}}(e^{+}e^{-}\to \mu ^{+}\mu ^{-})}}\right|_{\text{exp}}=3\left({\frac {\sum _{\text{flavours}}Q_{i}}{e}}\right)^{2}}$

This indicates that quarks have an additional degree of freedom with three manifestations, that in other words there are three different quarks of each flavor. On the other hand, spectroscopic measurements made it possible to rule out that the masses of the different quarks of the same flavor differ.

Mathematical description

The symmetry group of quantum chromodynamics is . The fundamental representation consists of complex matrices. As a Lie group of the dimension , the representation matrices can be written as exponentials of elements of a Lie algebra with eight generators. These eight generators are the Gell-Mann matrices . ${\displaystyle SU(3)}$ ${\displaystyle 3\times 3}$ ${\displaystyle 8}$ ${\displaystyle \lambda ^{a}}$

A gauge transformation acts on a fermion by means of ${\displaystyle \psi }$

${\displaystyle \psi \to \psi '=\exp \left(-\mathrm {i} \sum _{a=1}^{8}\theta ^{a}\lambda ^{a}\right)\psi }$

with eight real parameters . For dimensional reasons it follows that either forms a triplet under this transformation, that is, is a three-component vector, or a singlet , is a scalar. The color singlet thus consists of particles on which quantum chromodynamics has no effect; they are in a trivial representation. ${\displaystyle \theta ^{a}}$ ${\displaystyle \psi }$ ${\displaystyle \psi }$ ${\displaystyle \psi }$

The eight strong charges of the fermions are defined by

${\displaystyle \rho ^{a}=\psi _{i}^{\dagger }{\tfrac {1}{2}}\lambda _{ij}^{a}\psi _{j}}$

with the charge densities . Since the Gell-Mann matrices follow a commutator relation with the structural constants of Lie algebra, the charges cannot be measured together . So one has to look for a maximal subset of commuting observables . In the case of strong charges, these are by convention the components and ; the matrices and are diagonal . The common set of eigenvectors to and are the three colors ${\displaystyle \rho ^{a}}$ ${\displaystyle [\lambda ^{a},\lambda ^{b}]=2\mathrm {i} f^{abc}\lambda ^{c}}$ ${\displaystyle f^{abc}}$ ${\displaystyle q^{3}}$ ${\displaystyle q^{8}}$ ${\displaystyle \lambda ^{3}}$ ${\displaystyle \lambda ^{8}}$ ${\displaystyle \lambda ^{3}}$ ${\displaystyle \lambda ^{8}}$

${\displaystyle {\begin{aligned}\psi _{r}={\vec {r}}\otimes \psi ={\begin{pmatrix}1\\0\\0\end{pmatrix}}\otimes \psi \\\psi _{g}={\vec {g}}\otimes \psi ={\begin{pmatrix}0\\1\\0\end{pmatrix}}\otimes \psi \\\psi _{b}={\vec {b}}\otimes \psi ={\begin{pmatrix}0\\0\\1\end{pmatrix}}\otimes \psi \end{aligned}}}$

with the designations for red, green and blue. ${\displaystyle r,g,b}$

The strong charge of a red, green and blue particle is corresponding

${\displaystyle {\begin{alignedat}{2}{\vec {q}}_{r}&=+{\frac {1}{2}}{\vec {e}}_{3}&&+{\frac {1}{2{\sqrt {3}}}}{\vec {e}}_{8}\\{\vec {q}}_{g}&=-{\frac {1}{2}}{\vec {e}}_{3}&&+{\frac {1}{2{\sqrt {3}}}}{\vec {e}}_{8}\\{\vec {q}}_{b}&=&&-{\frac {1}{\sqrt {3}}}{\vec {e}}_{8}\end{alignedat}}}$

and addition of red, green and blue particles gives a total of an object neutral under the strong charge.

The antiparticles transform under the conjugated representation ; which are color vectors

${\displaystyle {\begin{aligned}{\vec {c}}={\begin{pmatrix}1&0&0\end{pmatrix}}\\{\vec {m}}={\begin{pmatrix}0&1&0\end{pmatrix}}\\{\vec {y}}={\begin{pmatrix}0&0&1\end{pmatrix}}\end{aligned}}}$

corresponding to the secondary colors for cyan , magenta and yellow. ${\displaystyle c,m,y}$

Description of the gluons

The gluons transform under gauge transformations in the adjoint representation of the symmetry group. The representation matrices are the structural constants, that is , and the gauge transformation is ${\displaystyle {f^{a}}_{bc}}$

${\displaystyle A^{\mu a}\to A'^{\mu a}=A^{\mu a}-{\frac {1}{g}}\partial ^{\mu }\theta ^{a}+f^{abc}\theta ^{b}A^{\mu c}}$,

from which it is evident that there are eight gluons participating in the strong interaction; they form an octet.

The gluons in the color space are represented as linearly independent, trace- free matrices, formally as a tensor product of a color and an anti-color. They can be chosen so that they (except for normalization) correspond to the Gell-Mann matrices. For example is ${\displaystyle 3\times 3}$

${\displaystyle A_{ij}^{\mu 1}=(r_{i}m_{j}+g_{i}c_{j})/{\sqrt {2}}\otimes A^{\mu }=\lambda _{ij}^{1}/{\sqrt {2}}\otimes A^{\mu }}$

a superposition of red-magenta and green-cyan.

The “ninth” gluon would be

${\displaystyle A_{ij}^{\mu 9}=(r_{i}c_{j}+g_{i}m_{j}+b_{i}y_{j})/{\sqrt {3}}\otimes A^{\mu }=I_{3}/{\sqrt {3}}\otimes A^{\mu }}$

with the three-dimensional identity matrix and thus a singlet under the . It would not take part in the strong interaction and would therefore be a sterile particle . Attempts to interpret the ninth gluon as the photon , i.e. as the gauge boson of the electromagnetic interaction , failed. ${\displaystyle I_{3}}$ ${\displaystyle SU(3)}$

Singlets and confinement

→ Main article : Confinement

No free gluons or quarks can be observed experimentally ; they are confined (ger .: imprisoned ). The force responsible for the confinement is the strong nuclear force , which grows with increasing distance : If you try to free a quark from a hadron, a quark-antiquark pair is formed, so that two new hadrons are created. The observable physical objects, which are made up of gluons or quarks, must therefore be singlets below the . ${\displaystyle SU(3)}$

The three allowed combinations that lead to singlets are:

${\displaystyle {\begin{aligned}\Psi _{\mathrm {B} }&={\frac {1}{\sqrt {6}}}\varepsilon _{abc}\psi _{a}\psi _{b}\psi _{c}\\\Psi _{\bar {\mathrm {B} }}&={\frac {1}{\sqrt {6}}}\varepsilon _{abc}{\bar {\psi }}_{a}{\bar {\psi }}_{b}{\bar {\psi }}_{c}\\\Psi _{\mathrm {M} }&={\frac {1}{\sqrt {3}}}{\bar {\psi }}_{a}\psi _{a}\end{aligned}}}$

With

• the Levi Civita symbol ${\displaystyle \varepsilon }$
• a baryonic wave function made up of three quarks ${\displaystyle \Psi _{\mathrm {B} }}$
• an antibaryonic wave function made up of three antiquarks ${\displaystyle \Psi _{\bar {\mathrm {B} }}}$
• ${\displaystyle \Psi _{\mathrm {M} }}$a mesonic wave function made up of quark-antiquark pairs.

In addition, theoretically exotic structures can occur such as tetraquarks , pentaquarks or particles with a higher quark content, which can be composed of the above states, as well as glueballs as purely gluonic structures. There is experimental evidence of the existence of tetraquarks at COZY , but not of the existence of glueballs.

The structure of the gauge theory is also responsible for the fact that the strong interaction is so short-range, although the gluons are massless like the photons of quantum electrodynamics : Since the gluons, as an adjoint representation of the gauge group, carry color themselves, they interact with themselves in electrodynamics, on the other hand, trivial and adjoint representation coincide so that photons do not influence each other. ${\displaystyle SU(3)}$ ${\displaystyle U(1)}$

Web links

• OW Greenberg: Color charge , Scholarpedia

literature

• Ian JR Aitchison and Anthony JG Hey: Gauge Theories in Particle Physics . 2nd Edition. Institute of Physics Publishing, Bristol 1989, ISBN 0-85274-329-7 , pp. 281-288 (English).
• Peter Becher, Manfred Böhm and Hans Joos: gauge theories of strong and electroweak interaction . 2nd Edition. Vieweg + Teubner, 1983, ISBN 978-3-519-13045-1 .
• David Griffiths: Introduction to Elementary Particle Physics . John Wiley & Sons, New York 1987, ISBN 0-471-60386-4 (English).
• Jarrett L. Lancaster: Introduction to Classical Field Theory: A Tour of the fundamental interactions . Morgan & Claypool, San Rafael, ISBN 978-1-64327-081-4 , pp. 4.8-4.12 (English).

Individual evidence

1. ↑ Oscar W. Greenberg: Spin and Unitary-Spin Independence in a Paraquark Model of Baryons and Mesons . In: Physical Review Letters . tape 13 , no. 20 , 1964, pp. 598-602 (English).
2. ^ Moo-Young Han and Yoichiro Nambu: Three-Triplet Model with Double SU (3) Symmetry . In: Physical Review . tape 139 , 4B, 1965, pp. B 1006-B 1010 (English).
3. ^ Richard P. Feynman: QED: The Strange Theory of Light and Matter . Princeton University Press, Princeton Oxford 2006, ISBN 978-0-691-12575-6 .
4. ^ WASA-at-COZY Collaboration: Evidence for a New Resonance from Polarized Neutron-Proton Scattering . In: Physical Review Letters . tape 112 , 2014, p. 202401 ff .