Abstract
The two-dimensional QCD in the large N limit, generally referred to as the ’t Hooft model, is numerically investigated in the axial gauge in a comprehensive manner. The corresponding Bethe-Salpeter equation for a bound \( q\overline{q} \) pair, originally derived by Bars and Green in 1978, was first numerically tackled by Li and collaborators in late 1980s, yet only for the stationary mesons. In this paper, we make further progress by numerically solving the Bars-Green equation for moving mesons, ranging from the chiral pion to charmonium. By choosing several different quark masses, we computed the corresponding quark condensates, meson spectra and their decay constants for a variety of meson momenta, and found satisfactory agreement with their counterparts obtained using light-cone gauge, thus numerically verifed the gauge and Poincaré invariance of the ’t Hooft model. Moreover, we have explicitly confirmed that, as the meson gets more and more boosted, the large component of the Bars-Green wave function indeed approaches the corresponding ’t Hooft light-cone wave function, while the small component of the wave function rapidly fades away.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.J. Hamer, SU(2) Yang-Mills Theory in (1+1)-dimensions: A Finite Lattice Approach, Nucl. Phys. B 195 (1982) 503 [INSPIRE].
F. Berruto, L. Giusti, C. Hölbling and C. Rebbi, A study of the ’t Hooft model with the overlap Dirac operator, Phys. Rev. D 65 (2002) 094516 [hep-lat/0201010] [INSPIRE].
K. Hornbostel, S.J. Brodsky and H.C. Pauli, Light Cone Quantized QCD in (1+1)-Dimensions, Phys. Rev. D 41 (1990) 3814 [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
E. Witten, Baryons in the 1/n Expansion, Nucl. Phys. B 160 (1979) 57 [INSPIRE].
S.R. Coleman, Aspects of Symmetry, Chapter 8, 1/N, Cambridge University Press, (1985).
G. ’t Hooft, A Two-Dimensional Model for Mesons, Nucl. Phys. B 75 (1974) 461 [INSPIRE].
C.G. Callan Jr., N. Coote and D.J. Gross, Two-Dimensional Yang-Mills Theory: A Model of Quark Confinement, Phys. Rev. D 13 (1976) 1649 [INSPIRE].
M.B. Einhorn, Form-Factors and Deep Inelastic Scattering in Two-Dimensional Quantum Chromodynamics, Phys. Rev. D 14 (1976) 3451 [INSPIRE].
R.C. Brower, W.L. Spence and J.H. Weis, Bound States and Asymptotic Limits for QCD in Two-dimensions, Phys. Rev. D 19 (1979) 3024 [INSPIRE].
F. Lenz, M. Thies, K. Yazaki and S. Levit, Hamiltonian formulation of two-dimensional gauge theories on the light cone, Annals Phys. 208 (1991) 1 [INSPIRE].
M. Burkardt, Off forward parton distributions in (1+1)-dimensional QCD, Phys. Rev. D 62 (2000) 094003 [hep-ph/0005209] [INSPIRE].
L. Ya. Glozman, V.K. Sazonov, M. Shifman and R.F. Wagenbrunn, How Chiral Symmetry Breaking Affects the Spectrum of the Light-Heavy Mesons in the ’t Hooft Model, Phys. Rev. D 85 (2012) 094030 [arXiv:1201.5814] [INSPIRE].
E. Witten, Chiral Symmetry, the 1/n Expansion and the SU(N) Thirring Model, Nucl. Phys. B 145 (1978) 110 [INSPIRE].
A.R. Zhitnitsky, On Chiral Symmetry Breaking in QCD in Two-dimensions (N c → Infinity), Phys. Lett. B 165 (1985) 405 [Sov. J. Nucl. Phys. 43 (1986) 999] [Yad. Fiz. 43 (1986) 1553] [INSPIRE].
M. Li, Large-N Two-dimensional QCD and Chiral Symmetry, Phys. Rev. D 34 (1986) 3888 [INSPIRE].
I. Bars and M.B. Green, Poincaré and Gauge Invariant Two-Dimensional QCD, Phys. Rev. D 17 (1978) 537 [INSPIRE].
S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rept. 301 (1998) 299 [hep-ph/9705477] [INSPIRE].
M. Li, L. Wilets and M.C. Birse, QCD in Two-dimensions in the Axial Gauge, J. Phys. G 13 (1987) 915 [INSPIRE].
X. Ji, Parton Physics on a Euclidean Lattice, Phys. Rev. Lett. 110 (2013) 262002 [arXiv:1305.1539] [INSPIRE].
Yu. S. Kalashnikova and A.V. Nefediev, Two-dimensional QCD in the Coulomb gauge, Phys. Usp. 45 (2002) 347 [Usp. Fiz. Nauk 172 (2002) 378] [hep-ph/0111225] [INSPIRE].
V. Baluni, The Bose Form of Two-dimensional Quantum Chromodynamics, Phys. Lett. B 90 (1980) 407 [INSPIRE].
P.J. Steinhardt, Baryons and Baryonium in QCD in Two-dimensions, Nucl. Phys. B 176 (1980) 100 [INSPIRE].
D. Amati and E. Rabinovici, On Chiral Realizations of Confining Theories, Phys. Lett. B 101 (1981) 407 [INSPIRE].
G. Bhattacharya, Equivalence Between Two-dimensional SU(n) QCD and Schwinger Model: An Analysis of Strong Coupling Phase, Nucl. Phys. B 205 (1982) 461 [INSPIRE].
F. Berruto, G. Grignani and P. Sodano, The strongly coupled ’t Hooft model on the lattice, Phys. Rev. D 62 (2000) 054510 [hep-lat/9912038] [INSPIRE].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].
V.L. Berezinsky, Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. 1. Classical systems, Zh. Eksp. Teor. Fiz. 59 (1970) 907 [Sov. Phys. JETP 32 (1971) 493] [INSPIRE].
J.M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181 [INSPIRE].
M. Burkardt, Trivial vacua, high orders in perturbation theory and nontrivial condensates, Phys. Rev. D 53 (1996) 933 [hep-ph/9509226] [INSPIRE].
B. Grinstein and R.F. Lebed, Explicit quark - hadron duality in heavy-light meson weak decays in the ’t Hooft model, Phys. Rev. D 57 (1998) 1366 [hep-ph/9708396] [INSPIRE].
T. Kuramoto and K. Yazaki, Interaction between two heavy mesons in QCD in two-dimensions, Nucl. Phys. A 572 (1994) 621 [INSPIRE].
J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Dover, New York, U.S.A. (1923).
Y. Jia, S.-R. Liang, X. Xiong and R. Yu, in preparation.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1708.09379
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Jia, Y., Liang, S., Li, L. et al. Solving the Bars-Green equation for moving mesons in two-dimensional QCD. J. High Energ. Phys. 2017, 151 (2017). https://doi.org/10.1007/JHEP11(2017)151
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2017)151