User:Maschen/Gluon field strength tensor
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In theoretical particle physics, the gluon field strength tensor is a tensor field characterizing the gluon interaction between quarks.
Background
[edit]The quantum field theory (QFT) of the strong interaction (one of the fundamental interactions of nature) is called quantum chromodynamics (QCD). Quarks interact with each other by the strong force due to their color charge, mediated by gluons. Gluons themselves possess color charge and can mutually interact.
Definition
[edit]The gauge covariant derivative
[edit]The gauge covariant derivative is required to transform quark fields in manifest covariance, the four-gradient ∂μ alone is not enough. The components are given by:[1][2]
wherein i is the imaginary unit and is the spin-1 gluon field. Different authors choose different signs.
The field strength
[edit]The tensor is denoted G, (or F, F, or some variant), and has components defined proportional to the commutator of gauge covariant derivative above;[2][3][4]
The proportionality constant is ± 1/igs, in which
is the coupling constant for QCD. Again, different authors choose different signs.
Expanding the commutator gives;
This almost parallels the electromagnetic field tensor (also denoted F) in quantum electrodynamics, given by the electromagnetic four-potential A describing a spin-1 photon;
The gluon field strength has extra terms which lead to self-interactions between the gluons and asymptotic freedom. This is a complication of the strong force making it inherently non-linear, contrary to the linear theory of the electromagnetic force.
QCD Lagrangian densities
[edit]Characteristic of field theories, the dynamics of the field strength are summarized by a suitable Lagrangian density and substitution into the Euler-Lagrange equation (for fields) obtains the equation of motion for the field. The Lagrangian for quarks, bound by gluons, is[2]
where "tr" denotes trace, and the quark fields are in triplet representation,
The quark field ψ belongs to the fundamental representation (3) and the antiquark field ψ belongs to the complex conjugate representation (3*), complex conjugate is denoted by * (not overbar).
Gauge transformations
[edit]The quark fields are invariant under the gauge transformation;[2]
where
in turn n = 1, 2,... 8, λn are the eight matrix representations of the SU(3) group, and θn(x) are eight global phase factors as a function of spatial position x. The matrix is exponentiated. The gauge covariant derivative transforms similarly.
The gluon field strength tensor is not gauge invariant, due to the extra coupled terms . The transformation of each alone is;
See also
[edit]References
[edit]Notes
[edit]- ^ M. Neubert (1996). "QCD sum-rule calculation of the kinetic energy and chromo-interaction of heavy quarks inside mesons" (PDF). Physics Letters B. CERN, Geneva, Switzerland: Elsevier.
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(help) - ^ a b c d W. Greiner, G. Schäfer (1994). "4". Quantum Chromodynamics. Springer. ISBN 3-540-57103-5.
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(help) - ^ S.O. Bilson-Thompson, D.B. Leinweber, A.G. Williams (2003). "Highly improved lattice field-strength tensor". Annals of Physics. Vol. 304(1). Adelaide, Australia: Elsevier. p. 1-21.
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(help)CS1 maint: multiple names: authors list (link) - ^ M. Eidemüller, H.G. Dosch, M. Jamin (1998). "The field strength correlator from QCD sum rules" (PDF). Physics Letters B. Heidelberg, Germany. arXiv:hep-ph/9812417.
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Further reading
[edit]Books
[edit]- H. Fritzsch (1982). Quarks: the stuff of matter. Allen lane. ISBN 0-7139-15331.
- B.R. Martin, G. Shaw (2009). Particle Physics. Manchester Physics Series (3rd ed.). John Wiley & Sons. ISBN 978-0-470-03294-7.
- S. Sarkar, H. Satz, B. Sinha (2009). The Physics of the Quark-Gluon Plasma: Introductory Lectures. Springer. ISBN 3642022855.
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: CS1 maint: multiple names: authors list (link) - J. Thanh Van Tran (editor) (1987). Hadrons, Quarks and Gluons: Proceedings of the Hadronic Session of the Twenty-Second Rencontre de Moriond, Les Arcs-Savoie-France. Atlantica Séguier Frontières. ISBN 2863320483.
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has generic name (help) - R. Alkofer, H. Reinhart (1995). Chiral Quark Dynamics. Springer. ISBN 3540601376.
- K. Chung (2008). Hadronic Production of ψ(2S) Cross Section and Polarization. ProQuest. ISBN 0549597743.
- M. Shifman (2012). Advanced Topics in Quantum Field Theory: A Lecture Course. Cambridge University Press. ISBN 0521190843.
- J. Collins (2011). Foundations of Perturbative QCD. Cambridge University Press. ISBN 0521855330.
- W.N.A. Cottingham, D.A.A. Greenwood (1998). Standard Model of Particle Physics. Cambridge University Press. ISBN 0521588324.
Selected papers
[edit]- J.P. Maa, Q. Wang, G.P. Zhang (2012). "QCD evolutions of twist-3 chirality-odd operators". Physics Letters B. Beijing, China: Elsevier.
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(help)CS1 maint: multiple names: authors list (link) - M. D’Elia, A. Di Giacomo, E. Meggiolaro (1997). "Field strength correlators in full QCD" (PDF). Physics Letters B. Pisa, Italy.
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suggested) (help)CS1 maint: multiple names: authors list (link) - A. Di Giacomo, M. D’elia, H. Panagopoulos, E. Meggiolaro (1998). "Gauge Invariant Field Strength Correlators In QCD" (PDF). Pisa (Italy), Nicosia (Cyprus), Heidelberg (Germany).
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suggested) (help)CS1 maint: multiple names: authors list (link) - M. Neubert (1993). "A Virial Theorem for the Kinetic Energy of a Heavy Quark inside Hadrons" (PDF). Physics Letters B. Geneva (CERN), Switzerland.
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suggested) (help) - M. Neubert, N. Brambilla, H.G. Dosch, A. Vairo (1998). "Field strength correlators and dual effective dynamics in QCD". Physical Review D. Washington, USA. doi:10.1103/PhysRevD.58.034010.
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External links
[edit]- K. Ellis (2005). "QCD" (PDF). Fermilab, USA.
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Categories: QFT, QCD