Nonperturbative Faddeev-Popov formula and the infrared limit of QCD

Daniel Zwanziger

    Research output: Contribution to journalArticle

    Abstract

    We show that an exact nonperturbative quantization of continuum gauge theory is provided by the Faddeev-Popov formula in the Landau gauge, δ(∂·)det[-∂·D(A)]exp[-SYM(A)], restricted to the region where the Faddeev-Popov operator is positive -∂·D(A)>0 (Gribov region). Although there are Gribov copies inside this region, they have no influence on expectation values. The starting point of the derivation is stochastic quantization which determines the Euclidean probability distribution P (A) by a method that is free of the Gribov critique. In the Landau-gauge limit the support of P(A) shrinks down to the Gribov region with Faddeev-Popov weight. The cutoff of the resulting functional integral on the boundary of the Gribov region does not change the form of the Dyson-Schwinger (DS) equations because det[-∂·D(A)] vanishes on the boundary, so there is no boundary contribution. However this cutoff does provide supplementary conditions that govern the choice of solution of the DS equations. In particular the "horizon condition," though consistent with the perturbative renormalization group, puts QCD into a nonperturbative phase. The infrared asymptotic limit of the DS equations of QCD is obtained by neglecting the Yang-Mills action SYM. We sketch the extension to a BRST-invariant formulation. In the infrared asymptotic limit, the BRST-invariant action becomes BRST exact, and defines a topological quantum field theory with an infinite mass gap. Confinement of quarks is discussed briefly.

    Original languageEnglish (US)
    Article number016002
    JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
    Volume69
    Issue number1
    DOIs
    StatePublished - 2004

    Fingerprint

    quantum chromodynamics
    cut-off
    horizon
    gauge theory
    derivation
    quarks
    continuums
    formulations
    operators

    ASJC Scopus subject areas

    • Physics and Astronomy (miscellaneous)

    Cite this

    Nonperturbative Faddeev-Popov formula and the infrared limit of QCD. / Zwanziger, Daniel.

    In: Physical Review D - Particles, Fields, Gravitation and Cosmology, Vol. 69, No. 1, 016002, 2004.

    Research output: Contribution to journalArticle

    @article{3b2f000d4fce438d9744a8041d35e478,
    title = "Nonperturbative Faddeev-Popov formula and the infrared limit of QCD",
    abstract = "We show that an exact nonperturbative quantization of continuum gauge theory is provided by the Faddeev-Popov formula in the Landau gauge, δ(∂·)det[-∂·D(A)]exp[-SYM(A)], restricted to the region where the Faddeev-Popov operator is positive -∂·D(A)>0 (Gribov region). Although there are Gribov copies inside this region, they have no influence on expectation values. The starting point of the derivation is stochastic quantization which determines the Euclidean probability distribution P (A) by a method that is free of the Gribov critique. In the Landau-gauge limit the support of P(A) shrinks down to the Gribov region with Faddeev-Popov weight. The cutoff of the resulting functional integral on the boundary of the Gribov region does not change the form of the Dyson-Schwinger (DS) equations because det[-∂·D(A)] vanishes on the boundary, so there is no boundary contribution. However this cutoff does provide supplementary conditions that govern the choice of solution of the DS equations. In particular the {"}horizon condition,{"} though consistent with the perturbative renormalization group, puts QCD into a nonperturbative phase. The infrared asymptotic limit of the DS equations of QCD is obtained by neglecting the Yang-Mills action SYM. We sketch the extension to a BRST-invariant formulation. In the infrared asymptotic limit, the BRST-invariant action becomes BRST exact, and defines a topological quantum field theory with an infinite mass gap. Confinement of quarks is discussed briefly.",
    author = "Daniel Zwanziger",
    year = "2004",
    doi = "10.1103/PhysRevD.69.016002",
    language = "English (US)",
    volume = "69",
    journal = "Physical review D: Particles and fields",
    issn = "1550-7998",
    publisher = "American Institute of Physics",
    number = "1",

    }

    TY - JOUR

    T1 - Nonperturbative Faddeev-Popov formula and the infrared limit of QCD

    AU - Zwanziger, Daniel

    PY - 2004

    Y1 - 2004

    N2 - We show that an exact nonperturbative quantization of continuum gauge theory is provided by the Faddeev-Popov formula in the Landau gauge, δ(∂·)det[-∂·D(A)]exp[-SYM(A)], restricted to the region where the Faddeev-Popov operator is positive -∂·D(A)>0 (Gribov region). Although there are Gribov copies inside this region, they have no influence on expectation values. The starting point of the derivation is stochastic quantization which determines the Euclidean probability distribution P (A) by a method that is free of the Gribov critique. In the Landau-gauge limit the support of P(A) shrinks down to the Gribov region with Faddeev-Popov weight. The cutoff of the resulting functional integral on the boundary of the Gribov region does not change the form of the Dyson-Schwinger (DS) equations because det[-∂·D(A)] vanishes on the boundary, so there is no boundary contribution. However this cutoff does provide supplementary conditions that govern the choice of solution of the DS equations. In particular the "horizon condition," though consistent with the perturbative renormalization group, puts QCD into a nonperturbative phase. The infrared asymptotic limit of the DS equations of QCD is obtained by neglecting the Yang-Mills action SYM. We sketch the extension to a BRST-invariant formulation. In the infrared asymptotic limit, the BRST-invariant action becomes BRST exact, and defines a topological quantum field theory with an infinite mass gap. Confinement of quarks is discussed briefly.

    AB - We show that an exact nonperturbative quantization of continuum gauge theory is provided by the Faddeev-Popov formula in the Landau gauge, δ(∂·)det[-∂·D(A)]exp[-SYM(A)], restricted to the region where the Faddeev-Popov operator is positive -∂·D(A)>0 (Gribov region). Although there are Gribov copies inside this region, they have no influence on expectation values. The starting point of the derivation is stochastic quantization which determines the Euclidean probability distribution P (A) by a method that is free of the Gribov critique. In the Landau-gauge limit the support of P(A) shrinks down to the Gribov region with Faddeev-Popov weight. The cutoff of the resulting functional integral on the boundary of the Gribov region does not change the form of the Dyson-Schwinger (DS) equations because det[-∂·D(A)] vanishes on the boundary, so there is no boundary contribution. However this cutoff does provide supplementary conditions that govern the choice of solution of the DS equations. In particular the "horizon condition," though consistent with the perturbative renormalization group, puts QCD into a nonperturbative phase. The infrared asymptotic limit of the DS equations of QCD is obtained by neglecting the Yang-Mills action SYM. We sketch the extension to a BRST-invariant formulation. In the infrared asymptotic limit, the BRST-invariant action becomes BRST exact, and defines a topological quantum field theory with an infinite mass gap. Confinement of quarks is discussed briefly.

    UR - http://www.scopus.com/inward/record.url?scp=1342302981&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=1342302981&partnerID=8YFLogxK

    U2 - 10.1103/PhysRevD.69.016002

    DO - 10.1103/PhysRevD.69.016002

    M3 - Article

    VL - 69

    JO - Physical review D: Particles and fields

    JF - Physical review D: Particles and fields

    SN - 1550-7998

    IS - 1

    M1 - 016002

    ER -