Non-Abelian gauge invariance and the infrared approximation

H. -T. Cho; H. M. Fried; T. Grandou

doi:10.1103/PhysRevD.37.960

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Non-Abelian gauge invariance and the infrared approximation

H. -T. Cho, H. M. Fried, and T. Grandou

Physique Théorique, Université de Nice, Physique Théorique, Parc Valrose, 06034 Nice Cedex, France

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Phys. Rev. D 37, 960 – Published 15 February, 1988

DOI: https://doi.org/10.1103/PhysRevD.37.960

Abstract

Two constructions are given of infrared approximations, defined by a nonlocal configuration-space restrictions, which preserve the local, non-Abelian gauge invariance of SU(N) two-dimensional QCD ( ${QCD}_{2}$ ). These continuum infrared methods are used to estimate the quenched order parameter 〈ψ¯ψ〉 in the strong-coupling, or chiral, limit and are compared to a previous calculation where gauge invariance was not manifest. Both constructions provide results which, in the chiral limit, differ from each other and from the previous estimation by an inessential, multiplicative scaling factor.

References (13)

T. Grandou, H.-T. Cho and H. M. Fried, preceding paper, Phys. Rev. D 37, 946 (1988).
F. Guérin and H. M. Fried, Phys. Rev. D 33, 3039 (1986).
H. M. Fried and T. Grandou, Phys. Rev. D 33, 1151 (1986).
E. S. Fradkin, Nucl. Phys. 76, 588 (1966); E. S. Fradkin, U. Esposito and S. Termini, Riv. Nuovo Cimento 2, 498 (1970).
L. Durand and E. Mendel, Phys. Rev. D 26, 1368 (1982).
J. Schwinger, Phys. Rev. 82, 664 (1951).
The original representation discussed in Ref. 4 has here been converted to a functional integral over a proper-time-dependent vector. When $φ_{μ}$ (s) is represented as ${dx}_{μ}$ (s)/ds, the Fradkin representations for Green's functions and fermion determinants reduce to corresponding path-integral representations for these quantities.
All of the conventional eikonal models for all particle processes can be obtained by suitable Bloch-Nordsieck approximations to the exact Fradkin representation. Applications analogous to those of this paper (and Refs. 1–3) for problems of viscous Navier-Stokes fluids can be found in H. M. Fried and J. Tessendorf, J. Math. Phys. 25, 1144 (1984); and H. M. Fried, Phys. Fluids 28, 3220 (1985).
Estimates by simpler infrared methods of the renormalization-group beta function in its strong-coupling regimes have been discussed in a modified $φ^{4}$ interaction, and for scalar QED by H. M. Fried, Nucl. Phys. B169, 329 (1980); Phys. Rev. D 27, 2956 (1983).
See, for example, S. Coleman, Ann. Phys. (N.Y.) 101, 239 (1976), and other references quoted therein.
The estimates of Ref. 3 suggest a diminution on the order of 25% in the magnitude of < ψ bar ψ > of QED $_{2}$ when the approximation of quenching is removed, in agreement with some older machine work of E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B190, 734 (1981).
M.-E. Brachet and H. M. Fried, Phys. Lett. 103A, 309 (1984).
The integration is performed in detail for Eq. (3.7).

[c1] T. Grandou, H.-T. Cho and H. M. Fried, preceding paper, Phys. Rev. D 37, 946 (1988).

[c2] F. Guérin and H. M. Fried, Phys. Rev. D 33, 3039 (1986).

[c3] H. M. Fried and T. Grandou, Phys. Rev. D 33, 1151 (1986).

[c4] E. S. Fradkin, Nucl. Phys. 76, 588 (1966); E. S. Fradkin, U. Esposito and S. Termini, Riv. Nuovo Cimento 2, 498 (1970).

[c5] L. Durand and E. Mendel, Phys. Rev. D 26, 1368 (1982).

[c6] J. Schwinger, Phys. Rev. 82, 664 (1951).

[c7] The original representation discussed in Ref. 4 has here been converted to a functional integral over a proper-time-dependent vector. When $φ_{μ}$ (s) is represented as ${dx}_{μ}$ (s)/ds, the Fradkin representations for Green's functions and fermion determinants reduce to corresponding path-integral representations for these quantities.

[c8] All of the conventional eikonal models for all particle processes can be obtained by suitable Bloch-Nordsieck approximations to the exact Fradkin representation. Applications analogous to those of this paper (and Refs. 1–3) for problems of viscous Navier-Stokes fluids can be found in H. M. Fried and J. Tessendorf, J. Math. Phys. 25, 1144 (1984); and H. M. Fried, Phys. Fluids 28, 3220 (1985).

[c9] Estimates by simpler infrared methods of the renormalization-group beta function in its strong-coupling regimes have been discussed in a modified $φ^{4}$ interaction, and for scalar QED by H. M. Fried, Nucl. Phys. B169, 329 (1980); Phys. Rev. D 27, 2956 (1983).

[c10] See, for example, S. Coleman, Ann. Phys. (N.Y.) 101, 239 (1976), and other references quoted therein.

[c11] The estimates of Ref. 3 suggest a diminution on the order of 25% in the magnitude of < ψ bar ψ > of QED $_{2}$ when the approximation of quenching is removed, in agreement with some older machine work of E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B190, 734 (1981).

[c12] M.-E. Brachet and H. M. Fried, Phys. Lett. 103A, 309 (1984).

[c13] The integration is performed in detail for Eq. (3.7).

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Non-Abelian gauge invariance and the infrared approximation

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