Renormalization of Yang-Mills theory in the Abelian gauge

Hyunsoo Min; Taehoon Lee; P. Y. Pac

doi:10.1103/PhysRevD.32.440

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Renormalization of Yang-Mills theory in the Abelian gauge

Hyunsoo Min, Taehoon Lee, and P. Y. Pac

Department of Physics, Seoul National University, Seoul, 151, Korea

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Phys. Rev. D 32, 440 – Published 15 July, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.440

Abstract

Renormalization of pure Yang-Mills theory is studied in the so-called Abelian gauge, a special bilinear gauge with a favored role given to a certain Abelian subgroup from the full non-Abelian gauge group. Renormalization in this gauge exhibits some unusual features—notably, different wave-function renormalizations are necessary for gauge fields with different ‘‘Abelian’’ charges and quartic ghost interaction terms are generated as renormalization counterterms. Despite these difficulties we show, through a careful analysis of the Becchi-Rouet-Stora transformation properties of the effective action, that Yang-Mills theory can be consistently renormalized in this gauge. Possible physical applications of this type of gauge (and its generalization which renders a favored role to a certain non-Abelian subgroup from the given gauge group) are noted briefly.

References (28)

For a review, see J. C. Taylor, Gauge Theories of Weak Interactions (Cambridge University, London, 1976); B. W. Lee, in Methods in Field-Theory, edited by R. Balian and J. Zinn-Justin (North-Holland, Amsterdam, 1976).
Beyond perturbation theory, gauge fixing becomes highly nontrivial [see V. N. Gribov, Nucl. Phys. B139, 1 (1978)]. In this paper we shall ignore such difficulties arising at the nonperturbative level.
See J. Zinn-Justin, in Trends in Elementary Particle Theory, edited by H. Rollnik and K. Dietz (Springer, Berlin, 1975).
G. 't Hooft, Nucl. Phys. B190, [FS3], 455 (1981).
Analogous phenomenon occurs also when one adopts the background gauge-fixing term or adopts the axial gauge. For renormalization of YM theory with the background gauge-fixing term, see, for instance, L. F. Abbott, Nucl. Phys. B185, 189 (1981). Renormalization in the axial gauge is discussed, for instance, in W. Konetschny and W. Kummer, ibid. B124, 145 (1977).
Here, magnetic monopoles should be included as topological singularities (see Ref. 4). But such magnetic monopoles may be ignored in discussions concerning only short-distance properties of the theory.
C. Lee and P. Y. Pac, Phys. Rev. D 30, 2628 (1984).
Some relevant works in this direction are S. Weinberg, Phys. Lett. 91B, 51 (1980); K. Fujikawa, Phys. Rev. D 7, 393 (1973).
G. 't Hooft, Nucl. Phys. B79, 276 (1974); A. M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 20, 430 (1974) [JETP Lett. 20, 194 (1974)].
S. Joglekar, Phys. Rev. D 10, 4095 (1974).
L. Alvarez-Gaumé and L. Baulieu, Nucl. Phys. B212, 255 (1983).
K. Shizuya, Nucl. Phys. B109, 397 (1976). In this work, the BRS invariance of the action has not been used explicitly for the analysis.
C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42, 127 (1975).
Our metric convention is that with signature (---+).
G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972); C. G. Bollini and J. J. Giambiagi, Nuovo Cimento B12, 20 (1972); J. F. Ashmore, Lett. Nuovo Cimento 4, 289 (1972).
To one-loop order, the well-known Yang-Mills beta function can be read off from the first expression in Eq. (5). This should be so due to the Abelian subgroup gauge invariance.
J. A. Dixon and J. C. Taylor, Nucl. Phys. B78, 552 (1974); H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D 12, 3159 (1975); S. D. Joglekar and B. W. Lee, Ann. Phys. (N.Y.) 97, 160 (1976); C. Lee, Phys. Rev. D 14, 1078 (1976).
Vanishing vacuum angle is assumed in this paper.
Here, g and alpha correspond to tree-level or renormalized values.
This may be ascribed to the fact that, in the ghost and antighost sector, the Abelian gauge transformation (17) has significantly different structure from the Abelian subgroup portion of the BRS transformation (14). (Note that χ δ ω app θ .)
Note that, according to our definition, there are also l-loop contributions to F ${hat}_{(l - 1)}$ resulting from the action $S^{star (l - 1)}$ itself. They are of course included in F ${hat}_{(l - 1)}^{l}$ , and this should explain why we have written ( $S^{star (l)}$ - $S^{star (l - 1)}$ ) | ${}_{l ~ l o o p}$ (rather than $S^{star (l)}$ | ${}_{l ~ l o o p}$ ) in the left-hand side of Eq. (29).
Of course, with gauge fields scaled differently depending on respective Abelian charges.
B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5, 3121 (1972); ibid. 5, 3137 (1972); G. 't Hooft and M. Veltman, Nucl. Phys. B50, 318 (1972); see also reviews cited in Ref. 1.
Suitable regularization of various quantities is implicitly assumed in Eq. (39).
We are here ignoring well-known infrared difficulties on the (perturbative) physical mass shell.
C. Lee, Ref. 17; W. S. Deans and J. A. Dixon, Phys. Rev. D 18, 113 (1978).
H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974).
J. Zinn-Justin, Nucl. Phys. B246, 246 (1984).

[c1] For a review, see J. C. Taylor, Gauge Theories of Weak Interactions (Cambridge University, London, 1976); B. W. Lee, in Methods in Field-Theory, edited by R. Balian and J. Zinn-Justin (North-Holland, Amsterdam, 1976).

[c2] Beyond perturbation theory, gauge fixing becomes highly nontrivial [see V. N. Gribov, Nucl. Phys. B139, 1 (1978)]. In this paper we shall ignore such difficulties arising at the nonperturbative level.

[c3] See J. Zinn-Justin, in Trends in Elementary Particle Theory, edited by H. Rollnik and K. Dietz (Springer, Berlin, 1975).

[c4] G. 't Hooft, Nucl. Phys. B190, [FS3], 455 (1981).

[c5] Analogous phenomenon occurs also when one adopts the background gauge-fixing term or adopts the axial gauge. For renormalization of YM theory with the background gauge-fixing term, see, for instance, L. F. Abbott, Nucl. Phys. B185, 189 (1981). Renormalization in the axial gauge is discussed, for instance, in W. Konetschny and W. Kummer, ibid. B124, 145 (1977).

[c6] Here, magnetic monopoles should be included as topological singularities (see Ref. 4). But such magnetic monopoles may be ignored in discussions concerning only short-distance properties of the theory.

[c7] C. Lee and P. Y. Pac, Phys. Rev. D 30, 2628 (1984).

[c8] Some relevant works in this direction are S. Weinberg, Phys. Lett. 91B, 51 (1980); K. Fujikawa, Phys. Rev. D 7, 393 (1973).

[c9] G. 't Hooft, Nucl. Phys. B79, 276 (1974); A. M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 20, 430 (1974) [JETP Lett. 20, 194 (1974)].

[c10] S. Joglekar, Phys. Rev. D 10, 4095 (1974).

[c11] L. Alvarez-Gaumé and L. Baulieu, Nucl. Phys. B212, 255 (1983).

[c12] K. Shizuya, Nucl. Phys. B109, 397 (1976). In this work, the BRS invariance of the action has not been used explicitly for the analysis.

[c13] C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42, 127 (1975).

[c14] Our metric convention is that with signature (---+).

[c15] G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972); C. G. Bollini and J. J. Giambiagi, Nuovo Cimento B12, 20 (1972); J. F. Ashmore, Lett. Nuovo Cimento 4, 289 (1972).

[c16] To one-loop order, the well-known Yang-Mills beta function can be read off from the first expression in Eq. (5). This should be so due to the Abelian subgroup gauge invariance.

[c17] J. A. Dixon and J. C. Taylor, Nucl. Phys. B78, 552 (1974); H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D 12, 3159 (1975); S. D. Joglekar and B. W. Lee, Ann. Phys. (N.Y.) 97, 160 (1976); C. Lee, Phys. Rev. D 14, 1078 (1976).

[c18] Vanishing vacuum angle is assumed in this paper.

[c19] Here, g and alpha correspond to tree-level or renormalized values.

[c20] This may be ascribed to the fact that, in the ghost and antighost sector, the Abelian gauge transformation (17) has significantly different structure from the Abelian subgroup portion of the BRS transformation (14). (Note that χ δ ω app θ .)

[c21] Note that, according to our definition, there are also l-loop contributions to F ${hat}_{(l - 1)}$ resulting from the action $S^{star (l - 1)}$ itself. They are of course included in F ${hat}_{(l - 1)}^{l}$ , and this should explain why we have written ( $S^{star (l)}$ - $S^{star (l - 1)}$ ) | ${}_{l ~ l o o p}$ (rather than $S^{star (l)}$ | ${}_{l ~ l o o p}$ ) in the left-hand side of Eq. (29).

[c22] Of course, with gauge fields scaled differently depending on respective Abelian charges.

[c23] B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5, 3121 (1972); ibid. 5, 3137 (1972); G. 't Hooft and M. Veltman, Nucl. Phys. B50, 318 (1972); see also reviews cited in Ref. 1.

[c24] Suitable regularization of various quantities is implicitly assumed in Eq. (39).

[c25] We are here ignoring well-known infrared difficulties on the (perturbative) physical mass shell.

[c26] C. Lee, Ref. 17; W. S. Deans and J. A. Dixon, Phys. Rev. D 18, 113 (1978).

[c27] H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974).

[c28] J. Zinn-Justin, Nucl. Phys. B246, 246 (1984).

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