Abstract
The main aspects of a gauge-invariant approach to the description of quark dynamics in the nonperturbative regime of quantum chromodynamics (QCD) are first reviewed. The role of the parallel transport operation in constructing gauge-invariant Green’s functions is then presented, and the relevance of Wilson loops for the representation of the interaction is emphasized. Recent developments, based on the use of polygonal lines for the parallel transport operation, are presented. An integro-differential equation, obtained for the quark Green’s function defined with a phase factor along a single, straight line segment, is solved exactly and analytically in the case of two-dimensional QCD in the large-N c limit. The solution displays the dynamical mass generation phenomenon for quarks, with an infinite number of branch-cut singularities that are stronger than simple poles.
Article PDF
Similar content being viewed by others
References
D. J. Gross and F. Wilczek, Ultraviolet behavior of non- Abelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973)
H. D. Politzer, Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30, 1346 (1973)
K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974)
J. B. Kogut, A review of the lattice gauge theory approach to quantum chromodynamics, Rev. Mod. Phys. 55, 775 (1983)
M. Kaku, Quantum field theory: A Modern Introduction, New York: Oxford University Press, 1993
F. J. Dyson, The S matrix in quantum electrodynamics, Phys. Rev. 75, 1736 (1949)
J. S. Schwinger, On the Green’s functions of quantized fields (1), Proc. Nat. Acad. Sci. USA 37, 452 (1951)
R. Alkofer and L. von Smekal, The infrared behavior of QCD Green’s functions, Phys. Rep. 353, 281 (2001)
C. S. Fischer, Infrared properties of QCD from Dyson–Schwinger equations, J. Phys. G 32, R253 (2006)
P. Maris, C. D. Roberts, and P. C. Tandy, Pion mass and decay constant, Phys. Lett. B 420, 267 (1998)
S. Mandelstam, Quantum electrodynamics without potentials, Ann. Phys. 19, 1 (1962)
S. Mandelstam, Quantization of the gravitational field, Ann. Phys. 19, 25 (1962)
I. Bialinicki-Birula, Gauge invariant variables in the Yang–Mills theory, Bull. Acad. Polon. Sci 11, 135 (1963)
S. Mandelstam, Feynman rules for electromagnetic and Yang–Mills fields from the gauge independent field theoretic formalism, Phys. Rev. 175, 1580 (1968)
Y. Nambu, QCD and the string model, Phys. Lett. B 80, 372 (1979)
L. S. Brown and W. I. Weisberger, Remarks on the static potential in quantum chromodynamics, Phys. Rev. D 20, 3239 (1979)
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72, 461 (1974)
J. C. Collins, Renormalization, UK: Cambridge University Press, 1984, p. 29
L. D. Faddeev and V. N. Popov, Feynman diagrams for the Yang–Mills fields, Phys. Lett. B 25, 29 (1967)
C. Becchi, A Rouet, and R. Stora, Renormalization of gauge theories, Ann. Phys. 98, 287 (1976)
I. V. Tyutin, Gauge invariance in field theory and statistical physics in operator formalism, preprint Lebedev-75-39 (1975), arXiv: 0812.0580
E. Corrigan and B. Hasslacher, A functional equation for exponential loop integrals in gauge theories, Phys. Lett. B 81, 181 (1979)
G. S. Bali, QCD forces and heavy quark bound states, Phys. Rep. 343, 1 (2001)
E. Eichten and F. Feinberg, Spin dependent forces in QCD, Phys. Rev. D 23, 2724 (1981)
N. Brambilla, P. Consoli, and G. M. Prosperi, A consistent derivation of the quark–antiquark and three quark potentials in a Wilson loop context, Phys. Rev. D 50, 5878 (1994)
N. Brambilla, A. Pineda, J. Soto, and A. Vairo, The QCD potential at O(1/m), Phys. Rev. D 63, 014023 (2001)
A. Yu. Dubin, A. B. Kaidalov, and Yu. A. Simonov, Dynamical regimes of the QCD string with quarks, Phys. Lett. B 323, 41 (1994)
Yu. A. Simonov, Vacuum background fields in QCD as a source of confinement, 1988, Nucl. Phys. B 307, 512 (1988)
A. M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B 164, 171 (1980)
Yu. M. Makeenko and A. A. Migdal, Exact equation for the loop average in multicolor QCD, Phys. Lett. B 88, 135 (1979)
Yu. M. Makeenko and A. A. Migdal, Self-consistent area law in QCD, Phys. Lett. B 97, 253 (1980)
Yu. M. Makeenko and A. A. Migdal, Quantum Chromodynamics as dynamics of loops, Nucl. Phys. B 188, 269 (1981)
A. A. Migdal, Loop equations and 1/N expansion, Phys. Rep. 102, 199 (1983)
V. S. Dotsenko and S. N. Vergeles, Renormalizability of phase factors in non-Abelian gauge theory, Nucl. Phys. B 169, 527 (1980)
R. A. Brandt, F. Neri, and Masa-aki Sato, Renormalization of loop functions for all loops, Phys. Rev. D 24, 879 (1981)
V. A. Kazakov and I. K. Kostov, Non-linear strings in twodimensional U(∞) gauge theory, Nucl. Phys. B 176, 199 (1980)
V. A. Kazakov, Wilson loop average for an arbitrary contour in two-dimensional U(N) gauge theory, Nucl. Phys. B 179, 283 (1981)
N. E. Bralić, Exact computation of loop averages in twodimensional Yang–Mills theory, Phys. Rev. D 22, 3090 (1980)
Yu. Makeenko, Large–N gauge theories, NATO Sci. Ser. C 556, 285 (2000), arXiv: hep-th/0001047
F. Jugeau and H. Sazdjian, Bound state equation in the Wilson loop approach with minimal surfaces, Nucl. Phys. B 670, 221 (2003)
H. Sazdjian, Integral equation for gauge invariant quark twopoint Green’s function in QCD, Phys. Rev. D 77, 045028 (2008)
L. Durand and E. Mendel, Functional equations for path dependent phase factors in Yang–Mills theories, Phys. Lett. B 85, 241 (1979)
G.’t Hooft, A two-dimensional model for mesons, Nucl. Phys. B 75, 461 (1974)
H. Sazdjian, Spectral properties of the gauge invariant quark Green’s function in two-dimensional QCD, Phys. Rev. D 81, 114008 (2010)
H. D. Politzer, Effective quark masses in the chiral limit, Nucl. Phys. B 117, 397 (1976)
A. S. Wightman, Quantum field theory in terms of vacuum expectation values, Phys. Rev. 101, 860 (1956)
S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Evanston: Row, Peterson and Co., 1961, pp. 721–742
G.’t Hooft and M. Veltman, Diagrammar, NATO Adv. Study Inst. Serv. Phys. 4, 177 (1974)
G. Källén, On the definition of the renormalization constants in quantum electrodynamics, Helv. Phys. Acta 25, 417 (1952)
H. Lehmann, On the properties of propagation functions and renormalization constants of quantized fields, Nuovo Cim. 11, 342 (1954)
Author information
Authors and Affiliations
Corresponding author
Additional information
This article is published with open access at www.springer.com/11467 and journal.hep.com.cn/fop
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Sazdjian, H. Gauge-invariant approach to quark dynamics. Front. Phys. 11, 111101 (2016). https://doi.org/10.1007/s11467-015-0515-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11467-015-0515-8