Skip to main content
SpringerLink
Log in

The Light-Front Schrödinger Equation and the Determination of the Perturbative QCD Scale from Color Confinement: A First Approximation to QCD

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

The valence Fock-state wavefunctions of the light-front (LF) QCD Hamiltonian satisfy a relativistic equation of motion, analogous to the nonrelativistic radial Schrödinger equation, with an effective confining potential U which systematically incorporates the effects of higher quark and gluon Fock states. If one requires that the effective action which underlies the QCD Lagrangian remains conformally invariant and extends the formalism of de Alfaro, Fubini and Furlan to LF Hamiltonian theory, the potential U has a unique form of a harmonic oscillator potential, and a mass gap arises. The result is a nonperturbative relativistic LF quantum mechanical wave equation which incorporates color confinement and other essential spectroscopic and dynamical features of hadron physics, including a massless pion for zero quark mass and linear Regge trajectories with the same slope in the radial quantum number n and orbital angular momentum L. Only one mass parameter κ appears. The corresponding LF Dirac equation provides a dynamical and spectroscopic model of nucleons. The same LF equations arise from the holographic mapping of the soft-wall model modification of AdS5 space with a unique dilaton profile to QCD (3+1) at fixed LF time. LF holography thus provides a precise relation between the bound-state amplitudes in the fifth dimension of Anti-de Sitter (AdS) space and the boost-invariant LFWFs describing the internal structure of hadrons in physical space-time. We also show how the mass scale \({\kappa= m_\rho/\sqrt{2}}\) underlying confinement and the masses of light-quark hadrons determines the scale \({\Lambda_{\overline{MS}}^{(N_F=3)}}\) controlling the evolution of the perturbative QCD coupling. The relation between scales is obtained by matching the nonperturbative dynamics, as described by an effective conformal theory mapped to the LF and its embedding in AdS space, to the perturbative QCD regime computed to four-loop order. The data for the effective coupling defined from the Bjorken sum rule \({\alpha_{g_1}(Q^2)}\) are remarkably consistent with the Gaussian form predicted by LF holographic QCD. The result is an effective coupling defined at all momenta. The predicted value \({\Lambda^{(N_F=3)}_{\overline{MS}} = 0.423 m_\rho = 0.328 \pm 0.034}\) GeV is in agreement with the world average \({0.339 \pm 0.010}\) GeV. We thus can connect \({\Lambda_{\overline{MS}}^{(N_F=3)}}\) to hadron masses. The analysis applies to any renormalization scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dirac P.A.M.: Forms of relativistic dynamics. Rev. Mod. Phys. 21, 392 (1949)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Brodsky, S.J., Pauli, H.C., Pinsky, S.S.: Quantum chromodynamics and other field theories on the light cone. Phys. Rep. 301, 299 (1998). [hep-ph/9705477]

  3. Drell S.D., Yan T.M.: Connection of elastic electromagnetic nucleon form-factors at large Q**2 and deep inelastic structure functions near threshold. Phys. Rev. Lett. 24, 181 (1970)

    Article  ADS  Google Scholar 

  4. West G.B.: Phenomenological model for the electromagnetic structure of the proton. Phys. Rev. Lett. 24, 1206 (1970)

    Article  ADS  Google Scholar 

  5. Brodsky S.J., Drell S.D.: The Anomalous magnetic moment and limits on fermion substructure. Phys. Rev. D 22, 2236 (1980)

    Article  ADS  Google Scholar 

  6. Brodsky S.J., Ji C.R.: Factorization property of the deuteron. Phys. Rev. D 33, 2653 (1986)

    Article  ADS  Google Scholar 

  7. Brodsky, S.J., Hwang, D.S., Ma, B.Q., Schmidt, I.: Light cone representation of the spin and orbital angular momentum of relativistic composite systems. Nucl. Phys. B 593, 311 (2001). [hep-th/0003082]

  8. Lepage G.P., Brodsky S.J.: Exclusive processes in perturbative quantum chromodynamics. Phys. Rev. D 22, 2157 (1980)

    Article  ADS  Google Scholar 

  9. Efremov A.V., Radyushkin A.V.: Factorization and asymptotical behavior of pion form-factor in QCD. Phys. Lett. B 94, 245 (1980)

    Article  ADS  Google Scholar 

  10. Brodsky S.J., Primack J.R.: The electromagnetic interactions of composite systems. Ann. Phys. 52, 315 (1969)

    Article  ADS  Google Scholar 

  11. Cruz-Santiago, C.A., Stasto, A.M.: Recursion relations and scattering amplitudes in the light-front formalism. Nucl. Phys. B 875, 368 (2013). [arXiv:1308.1062[hep-ph]]

  12. Antonuccio, F., Brodsky, S.J., Dalley, S.: Light cone wave functions at small x. Phys. Lett. B 412, 104 (1997). [hep-ph/9705413]

  13. Bardeen W.A., Pearson R.B.: Local gauge invariance and the bound state nature of hadrons. Phys. Rev. D 14, 547 (1976)

    Article  ADS  Google Scholar 

  14. Burkardt, M., Seal, S.K.: A Study of light mesons on the transverse lattice. Phys. Rev. D 65, 034501 (2002). [hep-ph/0102245]

  15. Bratt, J., Dalley, S., van de Sande, B., Watson, E.M.: Small-x behaviour of lightcone wavefunctions in transverse lattice gauge theory. Phys. Rev. D 70, 114502 (2004). [hep-ph/0410188]

  16. Pauli H.C., Brodsky S.J.: Solving field theory in one space one time dimension. Phys. Rev. D 32, 1993 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  17. Hornbostel K., Brodsky S.J., Pauli H.C.: Light cone quantized QCD in (1+1)-dimensions. Phys. Rev. D 41, 3814 (1990)

    Article  ADS  Google Scholar 

  18. Hellerman, S., Polchinski, J.: Supersymmetric quantum mechanics from light cone quantization. In: Shifman, M.A. (ed.) The Many Faces of the Superworld*, pp. 142–155 [hep-th/9908202]

  19. Polchinski, J.: M theory and the light cone. Prog. Theor. Phys. Suppl. 134, 158 (1999). [hep-th/9903165]

  20. Vary, J.P., Honkanen, H., Li, J., Maris, P., Brodsky, S.J., Harindranath, A., de Teramond, G.F., Sternberg, P., et al.: Hamiltonian light-front field theory in a basis function approach. Phys. Rev. C 81, 035205 (2010). [arXiv:0905.1411[nucl-th]]

  21. Srivastava, P.P., Brodsky, S.J.: A unitary and renormalizable theory of the standard model in ghost free light cone gauge. Phys. Rev. D 66, 045019 (2002). [hep-ph/0202141]

  22. Brodsky, S.J., Shrock, R.: Condensates in quantum chromodynamics and the cosmological constant. Proc. Natl. Acad. Sci. 108, 45 (2011). [arXiv:0905.1151[hep-th]]

  23. Brodsky, S.J., Roberts, C.D., Shrock, R., Tandy, P.C.: Confinement contains condensates. Phys. Rev. C 85, 065202 (2012). [arXiv:1202.2376[nucl-th]]

  24. Brodsky, S.J., Roberts, C.D., Shrock, R., Tandy, P.C.: Essence of the vacuum quark condensate. Phys. Rev. C 82, 022201 (2010). [arXiv:1005.4610[nucl-th]]

  25. Brodsky, S.J., de Teramond, G.F.: Hadronic spectra and light-front wavefunctions in holographic QCD. Phys. Rev. Lett. 96, 201601 (2006). [hep-ph/0602252]

  26. de Teramond, G.F., Brodsky, S.J.: Light-front holography: a first approximation to QCD. Phys. Rev. Lett. 102, 081601 (2009). [arXiv:0809.4899[hep-ph]]

  27. Brodsky, S.J., De Tramond, G.F., Dosch, H.G.: Threefold complementary approach to holographic QCD. Phys. Lett. B 729, 3 (2014). [arXiv:1302.4105[hep-th]]

  28. de Alfaro V., Fubini S., Furlan G.: Conformal invariance in quantum mechanics. Nuovoa Cimento A 34, 569 (1976)

    Article  ADS  Google Scholar 

  29. Deur, A., Brodsky, S.J., de Teramond, G.F.: Scheme-independent determination of the perturbative QCD scale Λ s from confinement dynamics in holographic QCD. [arXiv:1409.5488[hep-ph]]

  30. Brodsky, S.J., de Teramond, G.F., Dosch, H.G.: Light-front holographic QCD and color confinement. Int. J. Mod. Phys. A 29, 1444013 (2014). [arXiv:1404.1529[hep-th]]

  31. Brodsky, S.J., de Téramond G.F., Dosch, H.G.: QCD on the light-front–a systematic approach to hadron physics. Few Body Syst. 55, 407 (2014). [arXiv:1310.8648[hep-ph]]

  32. Brodsky, S.J., de Téramond, G.F., Dosch, H.G., Erlich, J.: Light-front holographic QCD and emerging confinement. [arXiv:1407.8131[hep-ph]]

  33. Pauli, H.C.: On confinement in a light cone Hamiltonian for QCD. Eur. Phys. J. C 7, 289 (1999). [hep-th/9809005]

  34. Glazek, S.D., Trawinski, A.P.: Model of the AdS/QFT duality. Phys. Rev. D 88, no. 10, 105025 (2013). [arXiv:1307.2059[hep-ph]]

  35. Appelquist T., Dine M., Muzinich I.J.: The static potential in quantum chromodynamics. Phys. Lett. B 69, 231 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  36. Isgur N., Paton J.E.: A flux tube model for hadrons in QCD. Phys. Rev. D 31, 2910 (1985)

    Article  ADS  Google Scholar 

  37. Bjorken, J.D., Brodsky, S.J., Scharff Goldhaber, A.: Possible multiparticle ridge-like correlations in very high multiplicity proton-proton collisions. Phys. Lett. B 726, 344 (2013). [arXiv:1308.1435[hep-ph]]

  38. Chabysheva, S.S., Hiller, J.R.: A light-front coupled-cluster method for the nonperturbative solution of quantum field theories. Phys. Lett. B 711, 417 (2012). [arXiv:1103.0037[hep-ph]]

  39. Trawiński, A.P., Głazek, S.D., Brodsky, S.J., de Téramond, G.F., Dosch, H.G.: Effective confining potentials for QCD. Phys. Rev. D 90(7), 074017 (2014). [arXiv:1403.5651[hep-ph]]

  40. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113 (1999). [Adv. Theor. Math. Phys. 2, 231 (1998)] [hep-th/9711200]

  41. Polchinski, J., Strassler, M.J.: Deep inelastic scattering and gauge string duality. JHEP 0305, 012 (2003). [hep-th/0209211]

  42. Abidin, Z., Carlson, C.E.: Gravitational form factors of vector mesons in an AdS/QCD model. Phys. Rev. D 77, 095007 (2008). [arXiv:0801.3839[hep-ph]]

  43. Brodsky, S.J., de Teramond, G.F.: Light-front dynamics and AdS/QCD correspondence: the pion form factor in the space- and time-like regions. Phys. Rev. D 77, 056007 (2008). [arXiv:0707.3859[hep-ph]]

  44. Brodsky, S.J., de Teramond, G.F.: Light-front dynamics and AdS/QCD correspondence: gravitational form factors of composite hadrons. Phys. Rev. D 78, 025032 (2008). [arXiv:0804.0452[hep-ph]]

  45. de Teramond G.F., Dosch, H.G., Brodsky, S.J.: Kinematical and dynamical aspects of higher-spin bound-state equations in holographic QCD. Phys. Rev. D 87, no. 7, 075005 (2013). [arXiv:1301.1651[hep-ph]]

  46. de Teramond, G.F., Brodsky, S.J.: Gauge/gravity duality and hadron physics at the light-front. AIP Conf. Proc. 1296, 128 (2010). [arXiv:1006.2431[hep-ph]]

  47. Breitenlohner P., Freedman D.Z.: Stability in gauged extended supergravity. Ann. Phys. 144, 249 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Karch, A., Katz, E., Son, D.T., Stephanov, M.A.: Linear confinement and AdS/QCD. Phys. Rev. D 74, 015005 (2006). [hep-ph/0602229]

  49. Parisi G.: Conformal invariance in perturbation theory. Phys. Lett. B 39, 643 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  50. Leutwyler H., Stern J.: Relativistic dynamics on a null plane. Ann. Phys. 112, 94 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  51. de Téramond, G.F, Brodsky S.J., Dosch, H.G.: Hadron spectroscopy and dynamics from light-front holography and conformal symmetry. EPJ Web Conf. 73, 01014 (2014). [arXiv:1401.5531[hep-ph]]

  52. de Téramond, G.F., Brodsky, S.J., Dosch, H.G.: Light-front holography in QCD and hadronic physics. In: Proceedings of 49th Rencontres de Moriond on QCD and High Energy Interactions, La Thuile, Italy, 22–29 Mar 2014

  53. de Teramond, G.F., Brodsky, S.J.: Hadronic form factor models and spectroscopy within the gauge/gravity correspondence. In: Proceedings of the Ferrara International School Niccolò Cabeo : Hadronic Physics, Ferrara, Italy, 23–28 May 2011

  54. Brodsky, S.J., Cao, F.G., de Teramond, G.F.: Meson transition form factors in light-front holographic QCD. Phys. Rev. D 84, 075012 (2011). [arXiv:1105.3999[hep-ph]]

  55. de Teramond, G.F., Brodsky, S.J.: Excited baryons in holographic QCD. AIP Conf. Proc. 1432, 168 (2012). [arXiv:1108.0965[hep-ph]]

  56. Forshaw, J.R., Sandapen, R.: An AdS/QCD holographic wavefunction for the rho meson and diffractive rho meson electroproduction. Phys. Rev. Lett. 109, 081601 (2012). [arXiv:1203.6088[hep-ph]]

  57. Vega, A., Schmidt, I., Gutsche, T., Lyubovitskij, V.E.: Generalized parton distributions in AdS/QCD. Phys. Rev. D 83, 036001 (2011). [arXiv:1010.2815[hep-ph]]

  58. Chakrabarti, D., Mondal, C.: Nucleon and flavor form factors in a light front quark model in AdS/QCD. Eur. Phys. J. C 73, 2671 (2013). [arXiv:1307.7995[hep-ph]]

  59. Weinberg S.: Precise relations between the spectra of vector and axial vector mesons. Phys. Rev. Lett. 18, 507 (1967)

    Article  ADS  Google Scholar 

  60. Shifman M.A., Vainshtein A.I., Zakharov V.I.: QCD and resonance physics sum rules. Nucl. Phys. B 147, 385 (1979)

    Article  ADS  Google Scholar 

  61. Grunberg, G.: Renormalization group improved perturbative QCD. Phys. Lett. B 95, 70 (1980). [Erratum-ibid. B 110, 501 (1982)].

  62. Cvetic, G.: Techniques of evaluation of QCD low-energy physical quantities with running coupling with infrared fixed point. Phys. Rev. D 89, 036003 (2014). [arXiv:1309.1696[hep-ph]]

  63. Cornwall J.M.: Dynamical mass generation in continuum QCD. Phys. Rev. D 26, 1453 (1982)

    Article  ADS  Google Scholar 

  64. Shirkov, D.V., Solovtsov, I.L.: Analytic model for the QCD running coupling with universal alpha-s (0) value. Phys. Rev. Lett. 79, 1209 (1997). [hep-ph/9704333]

  65. Fischer, C.S., Alkofer, R.: Nonperturbative propagators, running coupling and dynamical quark mass of Landau gauge QCD. Phys. Rev. D 67, 094020 (2003). [hep-ph/0301094]

  66. Aguilar, A.C., Binosi, D., Papavassiliou, J., Rodriguez-Quintero, J.: Non-perturbative comparison of QCD effective charges. Phys. Rev. D 80, 085018 (2009). [arXiv:0906.2633[hep-ph]]

  67. Courtoy, A., Liuti, S.: Analysis of α s from the realization of quark-hadron duality. Int. J. Mod. Phys. Conf. Ser. 25, 1460046 (2014). [arXiv:1307.4211]

  68. Bjorken J.D.: Applications of the chiral U(6) x (6) algebra of current densities. Phys. Rev. 148, 1467 (1966)

    Article  ADS  Google Scholar 

  69. Deur, A., Burkert, V., Chen, J.P., Korsch, W.: Experimental determination of the effective strong coupling constant. Phys. Lett. B 650, 244 (2007). [hep-ph/0509113]

  70. Deur, A., Burkert, V., Chen, J.P., Korsch, W.: Determination of the effective strong coupling constant alpha(s,g(1))(Q**2) from CLAS spin structure function data. Phys. Lett. B 665, 349 (2008). [arXiv:0803.4119[hep-ph]]

  71. Drell S.D., Hearn A.C.: Exact sum rule for nucleon magnetic moments. Phys. Rev. Lett. 16, 908 (1966)

    Article  ADS  Google Scholar 

  72. Gerasimov, S.B.: A sum rule for magnetic moments and the damping of the nucleon magnetic moment in nuclei. Sov. J. Nucl. Phys. 2, 430 (1966). [Yad. Fiz. 2, 598 (1965)]

  73. Brodsky, S.J., de Teramond, G.F., Deur, A.: Nonperturbative QCD coupling and its β-function from light-front holography. Phys. Rev. D 81, 096010 (2010). [arXiv:1002.3948[hep-ph]]

  74. Brodsky, S.J., Lu, H.J.: Commensurate scale relations in quantum chromodynamics. Phys. Rev. D 51, 3652 (1995). [hep-ph/9405218]

  75. Brodsky, S.J., Gabadadze, G.T., Kataev, A.L., Lu, H.J.: The Generalized Crewther relation in QCD and its experimental consequences. Phys. Lett. B 372, 133 (1996). [hep-ph/9512367]

  76. Wu, X.G., Brodsky, S.J., Mojaza, M.: The renormalization scale-setting problem in QCD. Prog. Part. Nucl. Phys. 72, 44 (2013). [arXiv:1302.0599[hep-ph]]

  77. Deur, A., et al.: Experimental determination of the evolution of the Bjorken integral at low Q**2. Phys. Rev. Lett. 93, 212001 (2004). [hep-ex/0407007]

  78. Deur, A., Prok, Y., Burkert, V., Crabb, D., Girod, F.-X., Griffioen, K.A., Guler, N., Kuhn, S.E., et al.: High precision determination of the Q 2-evolution of the Bjorken Sum. Phys. Rev. D 90, 012009 (2014). [arXiv:1405.7854[nucl-ex]]

  79. Bloom E.D., Gilman F.J.: Scaling, duality, and the behavior of resonances in inelastic electron-proton scattering. Phys. Rev. Lett. 25, 1140 (1970)

    Article  ADS  Google Scholar 

  80. Kataev, A.L.: The Ellis-Jaffe sum rule: the estimates of the next to next-to-leading order QCD corrections. Phys. Rev. D 50, 5469 (1994). [hep-ph/9408248]

  81. Lepage G.P., Mackenzie P.B.: Renormalized lattice perturbation theory. Nucl. Phys. Proc. Suppl. 20, 173 (1991)

    Article  ADS  Google Scholar 

  82. Davies, C.T.H., et al.: High precision lattice QCD confronts experiment. Phys. Rev. Lett. 92, 022001 (2004). [hep-lat/0304004]

Download references

Author information

Authors and Affiliations

  1. SLAC National Accelerator Laboratory, Stanford University, Stanford, CA, 94309, USA

    Stanley J. Brodsky

  2. Universidad de Costa Rica, San José, Costa Rica

    Guy F. de Téramond

  3. Thomas Jefferson National Accelerator Facility, Newport News, VA, 23606, USA

    Alexandre Deur

  4. Institut für Theoretische Physik, Philosophenweg 16, 6900, Heidelberg, Germany

    Hans Günter Dosch

Authors
  1. Stanley J. Brodsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guy F. de Téramond
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandre Deur
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hans Günter Dosch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stanley J. Brodsky.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brodsky, S.J., de Téramond, G.F., Deur, A. et al. The Light-Front Schrödinger Equation and the Determination of the Perturbative QCD Scale from Color Confinement: A First Approximation to QCD. Few-Body Syst 56, 621–632 (2015). https://doi.org/10.1007/s00601-015-0964-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00601-015-0964-1

Keywords

Search

Navigation