View source for Gluon field strength tensor

{{Short description|Second-rank tensor in quantum chromodynamics}}
{{Further|Ricci calculus|Special unitary group|Quantum chromodynamics}}
{{quantum field theory}}
In [[theoretical physics|theoretical]] [[particle physics]], the '''gluon field strength tensor''' is a second order [[tensor field]] characterizing the [[gluon]] interaction between [[quark]]s.

The [[strong interaction]] is one of the [[fundamental interaction]]s of nature, and the [[quantum field theory]] (QFT) to describe it is called ''[[quantum chromodynamics]]'' (QCD). [[Quark]]s 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.

The gluon field strength tensor is a [[tensor rank|rank]] 2 tensor field on the [[spacetime]] with values in the [[adjoint bundle]] of the chromodynamical SU(3) [[gauge group]] (see [[vector bundle]] for necessary definitions).

==Convention==
Throughout this article, Latin indices (typically {{math|''a'', ''b'', ''c'', ''n''}}) take values 1, 2, ..., 8 for the eight gluon [[color charge]]s, while Greek indices (typically {{math|''α'', ''β'', ''μ'', ''ν''}}) take values 0 for timelike components and 1, 2, 3 for spacelike components of [[four-vector]]s and four-dimensional spacetime tensors. In all equations, the [[Einstein notation|summation convention]] is used on all color and tensor indices, unless the text explicitly states that there is no sum to be taken (e.g. “no sum”).

==Definition==
Below the definitions (and most of the notation) follow K. Yagi, T. Hatsuda, Y. Miake<ref name="Yagi, Hatsuda, Miake">{{cite book
 |author1=Yagi, K. |author2=Hatsuda, T. |author3=Miake, Y. |year=2005
 |title=Quark-Gluon Plasma: From Big Bang to Little Bang
 |pages=17–18
 |series=Cambridge monographs on particle physics, nuclear physics, and cosmology
 |volume=23
 |publisher=Cambridge University Press
 |isbn=978-0-521-561-082
 |url=https://books.google.com/books?id=C2bpxwUXJngC&q=equation+of+motion+gluon+field+strength&pg=PA18
}}</ref> and Greiner, Schäfer.<ref name="Greiner, Schäfer">{{cite book
 |author1=Greiner, W. |author2=Schäfer, G. |year=1994
 |title=Quantum Chromodynamics
 |chapter=4
 |publisher=Springer
 |isbn=978-3-540-57103-2
 |chapter-url=https://books.google.com/books?id=gzYHNhp9KTIC
}}</ref>

===Tensor components===

The tensor is denoted {{math|''G''}}, (or {{math|''F''}}, {{math|{{overline|''F''}}}}, or some variant), and has components defined [[Proportionality (mathematics)|proportional]] to the [[commutator]] of the quark [[gauge covariant derivative|covariant derivative]] {{math|''D<sub>μ</sub>''}}:<ref name="Greiner, Schäfer"/><ref>{{cite journal
 |author1=Bilson-Thompson, S.O. 
 |author2=Leinweber, D.B. 
 |author3=Williams, A.G. 
 |year=2003
 |title=Highly improved lattice field-strength tensor
 |journal=Annals of Physics
 |volume=304 |issue=1 |pages=1–21
 |arxiv =hep-lat/0203008 
 |doi=10.1016/s0003-4916(03)00009-5
 |bibcode=2003AnPhy.304....1B
|s2cid=119385087 
 }}</ref>

:<math> G_{\alpha\beta} = \pm \frac{1}{i g_\text{s}} [D_\alpha, D_\beta]\,,</math>

where:

:<math>D_\mu =\partial_\mu \pm ig_\text{s} t_a \mathcal{A}^a_\mu\,,</math>

in which

*{{math|''i''}} is the [[imaginary unit]]; 
*{{math|''g''<sub>s</sub>}} is the [[coupling constant#QCD and asymptotic freedom|coupling constant]] of the strong force; 
*{{math|''t<sub>a</sub>'' {{=}} ''λ<sub>a</sub>''/2}} are the [[Gell-Mann matrices]] {{math|''λ<sub>a</sub>''}} divided by 2; 
*{{math|''a''}} is a color index in the [[adjoint representation]] of [[special unitary group|SU(3)]] which take values 1, 2, ..., 8 for the eight generators of the group, namely the [[Gell-Mann matrices]]; 
*{{math|''μ''}} is a spacetime index, 0 for timelike components and 1, 2, 3 for spacelike components; 
*<math>\mathcal{A}_\mu = t_a \mathcal{A}^a_\mu </math> expresses the [[gluon field]], a [[Spin (physics)|spin]]-1 gauge field or, in differential-geometric parlance, a [[connection (principal bundle)|connection]] in the SU(3) [[principal bundle]];
* <math>\mathcal{A}_\mu</math> are its four (coordinate-system dependent) components, that in a fixed gauge are {{gaps|3|×|3}} traceless [[Hermitian matrix]]<!-- mathematicians use skew-Hermitian as generators, but a physical convention is to use Hermitian -->-valued functions, while <math>\mathcal{A}^a_\mu</math> are 32 [[real-valued function]]s, the four components for each of the eight four-vector fields.

Note that different authors choose different signs.

Expanding the commutator gives;

:<math>G_{\alpha\beta} =\partial_{\alpha}\mathcal{A}_\beta-\partial_\beta\mathcal{A}_\alpha \pm ig_\text{s}[\mathcal{A}_{\alpha}, \mathcal{A}_{\beta}]</math>

Substituting <math>t_a \mathcal{A}^a_\alpha = \mathcal{A}_{\alpha} </math> and using the [[commutation relation]] <math>[t_a, t_b ] = i f_{ab}{}^{c} t_c </math> for the Gell-Mann matrices (with a relabeling of indices), in which {{math|''f <sup>abc</sup>''}} are the [[structure constants]] of SU(3), each of the gluon field strength components can be expressed as a [[linear combination]] of the Gell-Mann matrices as follows:

:<math>\begin{align}
G_{\alpha \beta} & = \partial_\alpha t_a \mathcal{A}^a_{\beta} - \partial_\beta t_a \mathcal{A}^a_\alpha \pm i g_\text{s} \left[t_b ,t_c \right ] \mathcal{A}^b_\alpha \mathcal{A}^c_\beta \\
 & = t_a \left( \partial_\alpha \mathcal{A}^a_{\beta} - \partial_\beta \mathcal{A}^a_\alpha \pm i^2 f_{bc}{}^ag_\text{s} \mathcal{A}^b_\alpha \mathcal{A}^c_\beta \right) \\
 & = t_a G^a_{\alpha \beta} \\
\end{align}\,,</math>

so that:<ref>{{cite journal|title=The field strength correlator from QCD sum rules
|author1=M. Eidemüller |author2=H.G. Dosch |author3=M. Jamin |orig-year=1999|year=2000
|location=Heidelberg, Germany
|journal=Nucl. Phys. B Proc. Suppl. |volume=86 |issue=1–3 |pages=421–425 
|arxiv=hep-ph/9908318|bibcode=2000NuPhS..86..421E|doi=10.1016/S0920-5632(00)00598-3}}</ref><ref>{{cite book|title = Advanced Topics in Quantum Field Theory: A Lecture Course |author= M. Shifman|publisher = Cambridge University Press|year=2012 |isbn = 978-0521190848|url= https://books.google.com/books?id=zeQuWycXV3oC&q=gluon+field+strength+tensor&pg=PA473}}</ref>

:<math>G^a_{\alpha \beta} = \partial_\alpha \mathcal{A}^a_{\beta} - \partial_\beta \mathcal{A}^a_\alpha \mp g_\text{s} f^{a}{}_{bc} \mathcal{A}^b_\alpha \mathcal{A}^c_\beta \,,</math>

where again {{math|''a, b, c'' {{=}} 1, 2, ..., 8}} are color indices. As with the gluon field, in a specific coordinate system and fixed gauge {{math|''G<sub>αβ</sub>''}} are {{gaps|3|×|3}} traceless Hermitian matrix-valued functions, while {{math|''G<sup>a</sup><sub>αβ</sub>''}} are real-valued functions, the components of eight four-dimensional second order tensor fields.

===Differential forms===

The gluon color field can be described using the language of [[differential form]]s, specifically as an adjoint bundle-valued [[curvature form|curvature 2-form]] (note that fibers of the adjoint bundle are the '''su'''(3) [[Lie algebra]]);

:<math>\mathbf{G} =\mathrm{d}\boldsymbol{\mathcal{A}} \mp g_\text{s}\,\boldsymbol{\mathcal{A}}\wedge \boldsymbol{\mathcal{A}}\,,</math>

where <math>\boldsymbol{\mathcal{A}}</math> is the gluon field, a [[vector potential]] 1-form corresponding to {{math|'''G'''}} and {{math|&and;}} is the (antisymmetric) [[Exterior algebra|wedge product]] of this algebra, producing the structure constants {{math|''f <sup>abc</sup>''}}. The [[Élie Cartan|Cartan]]-derivative of the field form (i.e. essentially the divergence of the field) would be zero in the absence of the "gluon terms", i.e. those <math>\boldsymbol{\mathcal{A}}</math> which represent the non-abelian character of the SU(3).

A more mathematically formal derivation of these same ideas (but a slightly altered setting) can be found in the article on [[metric connection]]s.

===Comparison with the electromagnetic tensor===

This almost parallels the [[electromagnetic field tensor]] (also denoted {{math|''F'' }}) in [[quantum electrodynamics]], given by the [[electromagnetic four-potential]] {{math|''A''}} describing a spin-1 [[photon]];

:<math>F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\,,</math>

or in the language of differential forms:

:<math>\mathbf{F} = \mathrm{d}\mathbf{A}\,.</math>

The key difference between quantum electrodynamics and quantum chromodynamics is that the gluon field strength has extra terms which lead to [[self-interaction]]s between the gluons and [[asymptotic freedom]]. This is a complication of the strong force making it inherently [[Nonlinear system|non-linear]], contrary to the linear theory of the electromagnetic force. QCD is a [[non-abelian gauge theory]]. The word ''non-abelian'' in [[group theory|group-theoretical]] language means that the group operation is not [[commutative property|commutative]], making the corresponding Lie algebra non-trivial.

==QCD Lagrangian density==

{{see also|Classical field theory}}

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#Analogues for waves and fields|equation of motion for the field]]. The Lagrangian density for massless quarks, bound by gluons, is:<ref name="Greiner, Schäfer"/>

:<math>\mathcal{L}=-\frac{1}{2}\mathrm{tr}\left(G_{\alpha\beta}G^{\alpha\beta}\right)+ \bar{\psi}\left(iD_\mu \right)\gamma^\mu\psi </math>

where "tr" denotes [[Trace (linear algebra)|trace]] of the {{gaps|3|×|3}} matrix {{math|''G<sub>αβ</sub>G<sup>αβ</sup>''}}, and {{math|''γ<sup>μ</sup>''}} are the {{gaps|4|×|4}} [[gamma matrices]]. In the fermionic term <math>i\bar{\psi}\left(iD_\mu\right)\gamma^{\mu}\psi</math>, both color and spinor indices are suppressed. With indices explicit, <math>\psi_{i,\alpha}</math> where <math>i=1,\ldots ,3</math> are color indices and <math>\alpha=1,\ldots,4</math> are Dirac spinor indices.

==Gauge transformations==

{{main|Gauge theory}}

In contrast to QED, the gluon field strength tensor is not gauge invariant by itself. Only the product of two contracted over all indices is gauge invariant.<!--- to be filled in later, specifically for the gluon field strength tensor --->

==Equations of motion==

Treated as a classical field theory, the equations of motion for the<ref name="Yagi, Hatsuda, Miake"/> quark fields are:

:<math>( i\hbar \gamma^\mu D_\mu - mc ) \psi = 0 </math>

which is like the [[Dirac equation]], and the equations of motion for the gluon (gauge) fields are:

:<math>\left[D_\mu , G^{\mu\nu} \right] = g_\text{s} j^\nu </math>

which are similar to the [[Maxwell equations]] (when written in tensor notation). More specifically, these are the [[Yang–Mills theory|Yang–Mills equations]] for quark and gluon fields. The [[color charge four-current]] is the source of the gluon field strength tensor, analogous to the electromagnetic [[four-current]] as the source of the electromagnetic tensor. It is given by

:<math>j^\nu = t^b  j_b^\nu \,, \quad j_b^\nu = \bar{\psi}\gamma^\nu t^b  \psi,</math>

which is a conserved current since color charge is conserved. In other words, the color four-current must satisfy the [[continuity equation]]:

:<math>D_\nu j^\nu = 0 \,.</math>

==See also==
*[[Quark confinement]]
*[[Gell-Mann matrices]]
*[[Field (physics)]]
*[[Yang–Mills field]]
*[[Eightfold Way (physics)]]
*[[Einstein tensor]]
*[[Wilson loop]]
*[[Wess–Zumino gauge]]
*[[Quantum chromodynamics binding energy]]
*[[Ricci calculus]]
*[[Special unitary group]]

==References==
===Notes===
{{reflist}}

===Further reading===
====Books====
*{{cite book|title = Quarks: the stuff of matter|author = H. Fritzsch|publisher = Allen lane|year=1982|isbn = 978-0-7139-15334}}
*{{cite book|title = Particle Physics |edition = 3rd |author1=B.R. Martin |author2=G. Shaw | series = Manchester Physics Series|publisher = John Wiley & Sons|year=2009|isbn = 978-0-470-03294-7}}
*{{cite book|title = The Physics of the Quark-Gluon Plasma: Introductory Lectures|author1=S. Sarkar |author2=H. Satz |author3=B. Sinha |publisher = Springer|year=2009|isbn = 978-3642022852|url=https://books.google.com/books?id=1WPV0NSenZ4C&q=gluon+field+strength+tensor&pg=PA344}}
*{{cite book|title = Hadrons, Quarks and Gluons: Proceedings of the Hadronic Session of the Twenty-Second Rencontre de Moriond, Les Arcs-Savoie-France|editor= J. Thanh Van Tran |publisher = Atlantica Séguier Frontières|year=1987|isbn = 978-2863320488|url=https://books.google.com/books?id=Vo48UfeBaG4C&q=gluon+field+strength+tensor&pg=PA615}}
*{{cite book|title = Chiral Quark Dynamics|author1=R. Alkofer |author2=H. Reinhart |publisher = Springer|year=1995 |isbn = 978-3540601371|url=https://books.google.com/books?id=fDENwIMAc50C&q=gluon+field+strength+tensor&pg=PA7}}
*{{cite book|title = Hadronic Production of ''ψ''(2S) Cross Section and Polarization|author= K. Chung|year=2008 |isbn = 978-0549597742|url=https://books.google.com/books?id=ETrMNvqLrXUC&q=gluon+field+strength+tensor&pg=PA2}}
*{{cite book|title = Foundations of Perturbative QCD|author= J. Collins|publisher = Cambridge University Press|year=2011 |isbn = 978-0521855334|url= https://books.google.com/books?id=0xGi1KW9vykC&q=gluon+field+strength+tensor&pg=PA233}}
*{{cite book|title = Standard Model of Particle Physics|author1=W.N.A. Cottingham |author2=D.A.A. Greenwood |publisher = Cambridge University Press|year=1998 |isbn = 978-0521588324|url= https://books.google.com/books?id=XSseBp7y19MC&q=gluon+field+strength+tensor&pg=PR16}}

====Selected papers====
*{{cite journal
|title=QCD evolutions of twist-3 chirality-odd operators
|author1=J.P. Maa |author2=Q. Wang |author3=G.P. Zhang |year=2012
|journal=Physics Letters B
|doi=10.1016/j.physletb.2012.12.007 
|arxiv = 1210.1006 |bibcode = 2013PhLB..718.1358M |volume=718 |issue=4–5 |pages=1358–1363|s2cid=118575585 }}
*{{cite journal
|title=Field strength correlators in full QCD
|author=M. D’Elia, A. Di Giacomo, E. Meggiolaro
|year=1997
|journal=Physics Letters B
|arxiv=hep-lat/9705032
|bibcode = 1997PhLB..408..315D |doi = 10.1016/S0370-2693(97)00814-9
 |volume=408
 |issue=1–4
|pages=315–319|s2cid=119533874
}}
*{{cite arXiv
|title=Gauge Invariant Field Strength Correlators In QCD
|author1=A. Di Giacomo |author2=M. D’elia |author3=H. Panagopoulos |author4=E. Meggiolaro |year=1998
|eprint = hep-lat/9808056
}}
*{{cite journal
|title=A Virial Theorem for the Kinetic Energy of a Heavy Quark inside Hadrons
|author=M. Neubert
|year=1993
|journal=Physics Letters B
|arxiv=hep-ph/9311232
|bibcode = 1994PhLB..322..419N |doi = 10.1016/0370-2693(94)91174-6
 |volume=322
 |issue=4
|pages=419–424|s2cid=14214029
}}
*{{cite journal
|title=Field strength correlators and dual effective dynamics in QCD
|author1=M. Neubert |author2=N. Brambilla |author2-link=Nora Brambilla|author3=H.G. Dosch |author4=A. Vairo |year=1998
|journal=Physical Review D
|arxiv = hep-ph/9802273
|doi=10.1103/PhysRevD.58.034010
|bibcode = 1998PhRvD..58c4010B |volume=58|issue=3 |pages=034010 |s2cid=1824834 }}
*{{cite news
|title=QCD sum-rule calculation of the kinetic energy and chromo-interaction of heavy quarks inside mesons
|author=M. Neubert
|year=1996
|journal=Physics Letters B
|url=ftp://79.110.128.93/books/physics,%20math/%D0%96%D1%83%D1%80%D0%BD%D0%B0%D0%BB%D1%8B/Phys%20Letters%20B/Volume%20389,%20Issue%204,%20pp.%20631-830%20(26%20December%201996)/Matthias%20Neubert%20-%20QCD%20sum-rule%20calculation%20of%20the%20kinetic%20energy%20and%20chromo-interaction%20of%20heavy%20quarks%20inside%20mesons.pdf
}}

==External links==
*{{cite news|title=QCD |author=K. Ellis |year=2005 |publisher=[[Fermilab]] |url=http://theory.fnal.gov/people/ellis/Talks/FNAL/l1_s.pdf |url-status=unfit |archive-url=https://web.archive.org/web/20060926033047/http://theory.fnal.gov/people/ellis/Talks/FNAL/l1_s.pdf |archive-date=September 26, 2006 }}
*{{cite web|access-date=2013-10-17
|title=Chapter 2: The QCD Lagrangian
|location=Technische Universität München
|url=http://www.t39.ph.tum.de/T39_files/Lectures_files/StrongInteraction2011/QCDkap2.pdf}}

{{tensors}}

[[Category:Gauge theories]]
[[Category:Quantum chromodynamics]]
[[Category:Gluons]]
[[Category:Tensors]]