On the Wilson coefficient of the penguin operator

Hai-Yang Cheng

doi:10.1103/PhysRevD.37.1908

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On the Wilson coefficient of the penguin operator

Hai-Yang Cheng

Physics Department, Indiana University, Bloomington, Indiana 47405

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Phys. Rev. D 37, 1908 – Published 1 April, 1988

DOI: https://doi.org/10.1103/PhysRevD.37.1908

Abstract

Recently an additional contribution to the real part of the Wilson coefficient function $c_{5}$ of the dominant penguin operator $O_{5}$ in K→2π decay was found by Bardeen, Buras, and Gérard. We show that in the effective-Hamiltonian approach used by Gilman and Wise with the ‘‘heavy’’-charm-quark approximation, the additional contribution to $c_{5}$ originates from the penguin diagrams with u-quark loops which are left after the charm quark is integrated out. For μ∼1 GeV (μ being the renormalization scale), the inclusion of the u-quark loop diagram amounts to adding a contribution (5/3 $α_{s}$ (μ)/12π to the standard estimate of $c_{5}$ . Penguin diagrams at the low-momentum scale are briefly discussed.

References (12)

W. A. Bardeen, A. J. Buras and J.-M. Gérard, Phys. Lett. B 180, 133 (1986); Nucl. Phys. B293, 787 (1987).
A. I. Vainshtein, V. I. Zakharov and M. A. Shifman, Zh. Eksp. Teor. Fiz. 72, 1275 (1977) [Sov. Phys. JETP 45, 670 (1977)].
F. J. Gilman and M. B. Wise, Phys. Rev. D 20, 2392 (1979); ibid. 27, 1128 (1983).
C. Bernard, T. Draper, G. Hockney, A. M. Rushton, A. Soni, Phys. Rev. Lett. 55, 2770 (1985).
The characteristic values of Wilson coefficients are (Ref. 3) $c_{1}$ = - 2.11, $c_{2}$ = 0.12, $c_{3}$ = 0.09, $c_{4}$ = 0.45, $c_{5}$ = - 0.025, $c_{6}$ = - 0.003 for μ = 0.86 GeV, $α_{s}$ = 0.75, $m_{t}$ = 40 GeV. Since the operator $O_{4}$ is pure Δ I = case 3 over 2, it does not contribute to penguin diagrams.
I wish to thank J. O. Eeg for pointing out that the same analytic result was also obtained in J. Finjord, Nucl. Phys. B181, 74 (1981); M. Galić, Report No. SLAC-PUB-2602, 1980 (unpublished).
If $k^{2}$ is timelike (i.e., $k^{2}$ > 0), there will be a dynamic phase induced in the penguin diagram, see M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. 43, 242 (1979).
It should be stressed that in the perturbative region of QCD, Figs. 1(c) and 1(d) should not be included in the analysis of BBG: the effect of the u quark is already taken into consideration in Eq. (1).
C. T. Hill and G. G. Ross [Phys. Lett. 94B, 234 (1980)] pointed out that a mu dependence in the penguin matrix elements arises from the momentum dependence of the quark masses. The usually quoted current-quark masses correspond to μ app 1 GeV. From our discussions below, it is necessary to include Fig. 1(d) to account for the full mu dependence of the matrix elements of penguin operators at the low-energy mu scale.
It is known that the perturbative evaluation of penguin diagrams is far from accounting for the Δ I = 1/2 rule. For recent estimates, see H. Y. Cheng, Phys. Rev. D 36, 2056 (1987); ibid. 37, 1338(E) (1988); and Ref. 12.
J. O. Eeg, Phys. Lett. 155B, 115 (1985); Phys. Lett. B 171, 103 (1986) Z. Phys. C 33, 227 (1986).
It certainly does not make sense to set mu as low as 0.2 GeV in the perturbative QCD calculations of the Wilson coefficients, as done in many early literature. The mu-dependence problem and its possible solution are discussed in the review of H. Y. Cheng, Report No. IUHET-132, 1987 (unpublished).

[c1] W. A. Bardeen, A. J. Buras and J.-M. Gérard, Phys. Lett. B 180, 133 (1986); Nucl. Phys. B293, 787 (1987).

[c2] A. I. Vainshtein, V. I. Zakharov and M. A. Shifman, Zh. Eksp. Teor. Fiz. 72, 1275 (1977) [Sov. Phys. JETP 45, 670 (1977)].

[c3] F. J. Gilman and M. B. Wise, Phys. Rev. D 20, 2392 (1979); ibid. 27, 1128 (1983).

[c4] C. Bernard, T. Draper, G. Hockney, A. M. Rushton, A. Soni, Phys. Rev. Lett. 55, 2770 (1985).

[c5] The characteristic values of Wilson coefficients are (Ref. 3) $c_{1}$ = - 2.11, $c_{2}$ = 0.12, $c_{3}$ = 0.09, $c_{4}$ = 0.45, $c_{5}$ = - 0.025, $c_{6}$ = - 0.003 for μ = 0.86 GeV, $α_{s}$ = 0.75, $m_{t}$ = 40 GeV. Since the operator $O_{4}$ is pure Δ I = case 3 over 2, it does not contribute to penguin diagrams.

[c6] I wish to thank J. O. Eeg for pointing out that the same analytic result was also obtained in J. Finjord, Nucl. Phys. B181, 74 (1981); M. Galić, Report No. SLAC-PUB-2602, 1980 (unpublished).

[c7] If $k^{2}$ is timelike (i.e., $k^{2}$ > 0), there will be a dynamic phase induced in the penguin diagram, see M. Bander, D. Silverman and A. Soni, Phys. Rev. Lett. 43, 242 (1979).

[c8] It should be stressed that in the perturbative region of QCD, Figs. 1(c) and 1(d) should not be included in the analysis of BBG: the effect of the u quark is already taken into consideration in Eq. (1).

[c9] C. T. Hill and G. G. Ross [Phys. Lett. 94B, 234 (1980)] pointed out that a mu dependence in the penguin matrix elements arises from the momentum dependence of the quark masses. The usually quoted current-quark masses correspond to μ app 1 GeV. From our discussions below, it is necessary to include Fig. 1(d) to account for the full mu dependence of the matrix elements of penguin operators at the low-energy mu scale.

[c10] It is known that the perturbative evaluation of penguin diagrams is far from accounting for the Δ I = 1/2 rule. For recent estimates, see H. Y. Cheng, Phys. Rev. D 36, 2056 (1987); ibid. 37, 1338(E) (1988); and Ref. 12.

[c11] J. O. Eeg, Phys. Lett. 155B, 115 (1985); Phys. Lett. B 171, 103 (1986) Z. Phys. C 33, 227 (1986).

[c12] It certainly does not make sense to set mu as low as 0.2 GeV in the perturbative QCD calculations of the Wilson coefficients, as done in many early literature. The mu-dependence problem and its possible solution are discussed in the review of H. Y. Cheng, Report No. IUHET-132, 1987 (unpublished).

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On the Wilson coefficient of the penguin operator

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