<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>L</mi></mrow></msub></mrow></math>→<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mi mathvariant="normal">−</mi></mrow></msup></mrow></math> decays for large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></math>

Claudio O. Dib; Isard Dunietz; Frederick J. Gilman

doi:10.1103/PhysRevD.39.2639

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$K_{L}$ → $π^{0}$ $l^{+}$ $l^{-}$ decays for large $m_{t}$

Claudio O. Dib, Isard Dunietz, and Frederick J. Gilman

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309

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Phys. Rev. D 39, 2639 – Published 1 May, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.2639

Abstract

We discuss the decays $K_{L}$ → $π^{0}$ $l^{+}$ $l^{-}$ when $m_{t}$ is large. Unlike the case of $K_{L}$ →ππ, CP violation in the decay amplitude itself is comparable to that which comes from the mass matrix. We study the CP-violating effects, including strong-interaction (QCD) corrections to the amplitudes which arise from one-loop diagrams. Short-distance contributions from diagrams that involve a W and a Z or two W’s as well as from those with a photon and a W are important when $m_{t}$ ≳ $M_{W}$ .

References (39)

J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. 13, 138 (1964).
L. Wolfenstein, Phys. Rev. Lett. 13, 562 (1964).
M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
H. Burkhardt et al., Phys. Lett. B 206, 169 (1988).
M. Woods et al., Phys. Rev. Lett. 60, 1695 (1988).
M. A. Shifman, in Lepton and Photon Interactions, proceedings of the International Symposium on Lepton and Photon Interactions at High Energies, Hamburg, West Germany, 1987, edited by W. Bartel and R. Rückl [Nucl. Phys. B, Proc. Suppl. 3, 289 (1987)].
F. J. Gilman, in International Symposium on the Production and Decay of Heavy Flavors, Stanford, California, 1987, edited by E. Bloom and A. Fridman [Ann. N.Y. Acad. Sci. 535, 211 (1988)].
G. Altarelli and P. J. Franzini, Report No. CERN-TH-4914/87, 1987 (unpublished).
F. J. Gilman and M. B. Wise, Phys. Lett. 83B, 83 (1979); Phys. Rev. D 20, 2392 (1979).
For a review of the theoretical estimates of CP violation in B decay and the experimental possibilities, see K. J. Foley et al.., in Experiments, Detectors, and Experimental Areas for the Supercollider, proceedings of the Workshop, Berkeley, California, 1987, edited by R. Donaldson and M. G. D. Gilchriese (World Scientific, Singapore, 1988), p. 701; in Proceedings of the Workshop on High Sensitivity Beauty Physics at Fermilab, Batavia, Illinois, 1987, edited by A. J. Slaughter, N. Lockyer, and M. Schmidt (Fermilab, Batavia, 1988).
M. Baker and S. L. Glashow, Nuovo Cimento 25, 857 (1962). The argument can be extended to apply to an intermediate virtual $Z^{0}$ or the local operator generated by two virtual intermediate W's.
J. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B109, 213 (1976).
F. J. Gilman and M. B. Wise, Phys. Rev. D 21, 3150 (1980).
A. I. Vainshtein et al., Yad. Fiz. 24, 820 (1976) [Sov. J. Nucl. Phys. 24, 427 (1976)]; V. V. Flambaum, ibid. 22, 661 (1975) [22, 340 (1975)]; D. V. Nanopoulos and G. G. Ross, Phys. Lett. 56B, 279 (1975); L. E. Ibanez et al., Phys. Rev. D 21, 1428 (1980).
E. Witten, Nucl. Phys. B122, 109 (1977).
See, for example, the report on K physics by the group led by R. Bock and L. Littenberg, in Proceedings of the Summer Study on High Energy Physics in the 1990s, Snowmass, Colorado, 1988 (unpublished).
A preliminary report of this work was given by C. Dib, in Proceedings of the Summer Study on High Energy Physics in the 1990s (Ref. 16); and F. J. Gilman, lectures at the Heavy Flavor Symposium, Beijing, China, 1988, Report No. SLAC-PUB-4736, 1988 (unpublished). A brief version is found in C. O. Dib, I. Dunietz, and F. J. Gilman, Phys. Lett. (to be published). Other recent work on this subject is found in J. Flynn and L. Randall, Report No. LBL-26310, 1988 (unpublished).
G. Ecker, A. Pich and E. de Rafael, Phys. Lett. B 189, 363 (1987); Nucl. Phys. B303, 665 (1988).
J. F. Donoghue, B. R. Holstein and G. Valencia, Phys. Rev. D 35, 2769 (1987).
L. M. Sehgal, Phys. Rev. D 38, 808 (1988). See also T. Morozumi and H. Iwasaki, Report No. KEK-TH-206, 1988 (unpublished).
A discussion of the size of the B amplitude and of a mistake in the calculation of Morozumi and Iwasaki (Ref. 20) is found in Dib (Ref. 17) and in J. Flynn and L. Randall, Report No. LBL-26008, 1988 (unpublished).
E. de Rafael (private communication). See also G. Ecker et al.., Report No. CERN-TH-5185/88, 1988 (unpublished).
A. Pich, talk at the Workshop on Rare K Decays and CP Violation, Brookhaven, New York, 1988 (unpublished); Report No. CERN-TH-5194, 1988 (unpublished).
Particle Data Group, M. Aguilar-Benitez et al., Phys. Lett. 170B, 1 (1986).
In a chiral-perturbation-theory calculation, G. Ecker, A. Pich and E. de Rafael, Nucl. Phys. B291, 692 (1987), obtain two values, 0.25 and 2.5.
We use the technique and notation of Ref. 9.
This was done previously for the closely related process b -> ${se}^{+}$ $e^{-}$ by B. Grinstein, M. J. Savage, and M. B. Wise, Report No. LBL-25014, 1988 (unpublished).
The relation of our choice of operators to those in Ref. 27, after an appropriate change of quark labels to account for the different processes under consideration, is $C_{7 V}$ = $c_{8}$ ${}^{(GSW)}$ / 2 pi and $C_{7 A}$ = $c_{9}$ ${}^{(GSW)}$ / 2 pi. For the ``electromagnetic penguin'' contribution, the coefficient $F_{1}$ of Ref. 29 is related by $F_{1}$ ( $x_{i}$ ) = 2 π& ${Ctilde}_{7 V, i}$ .
T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); ibid. 65, 1772(E) (1981). We have repeated this calculation and are in agreement with it and the corresponding one in Ref. 27.
The 7 times 7 matrix is given in Appendix B of Ref. 13. There is an error in the two entries $γ_{37}$ and $γ_{47}$ of the anomalous-dimensions matrix which describe the mixing of the ``penguin'' operators $Q_{3}$ and $Q_{4}$ with $Q_{7}$ ; they should be increased by case 8 over 9 and case 8 over 3, respectively. We thank J. Flynn and L. Randall for pointing this out to us. This changes the numerical results very little (see text), and has no effect on the discussion here, which neglects this mixing anyway.
J. O. Eeg and I. Picek, Phys. Lett. B 214, 651 (1988).
Vainshtein et al. (Ref. 14); V. A. Novikov et al., Phys. Rev. D 16, 223 (1977).
C. Jarlskog, Phys. Rev. Lett. 55, 1839 (1985); Z. Phys. C 29, 491 (1985).
O. W. Greenberg, Phys. Rev. D 32, 1841 (1985); D.-D. Wu, ibid. 33, 860 (1986); J. D. Bjorken and I. Dunietz, ibid. 36, 2109 (1987).
See also the discussion in Ref. 13, where long-distance contributions are estimated.
In this regard, we proceed similarly to J. Ellis and J. S. Hagelin, Nucl. Phys. B217, 189 (1983), in calculating the QCD corrections to K -> π ν ν bar.
L. K. Gibbons et al., Phys. Rev. Lett. 61, 2661 (1988).
G. D. Barr et al., Phys. Lett. B 214, 303 (1988).
We thank L. Littenberg for calling our attention to this consequence of the asymptotic behavior of the contributions as $m_{t}$ -> inf. See L. Littenberg, Phys. Rev. D (to be published).

[c1] J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. 13, 138 (1964).

[c2] L. Wolfenstein, Phys. Rev. Lett. 13, 562 (1964).

[c3] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).

[c4] H. Burkhardt et al., Phys. Lett. B 206, 169 (1988).

[c5] M. Woods et al., Phys. Rev. Lett. 60, 1695 (1988).

[c6] M. A. Shifman, in Lepton and Photon Interactions, proceedings of the International Symposium on Lepton and Photon Interactions at High Energies, Hamburg, West Germany, 1987, edited by W. Bartel and R. Rückl [Nucl. Phys. B, Proc. Suppl. 3, 289 (1987)].

[c7] F. J. Gilman, in International Symposium on the Production and Decay of Heavy Flavors, Stanford, California, 1987, edited by E. Bloom and A. Fridman [Ann. N.Y. Acad. Sci. 535, 211 (1988)].

[c8] G. Altarelli and P. J. Franzini, Report No. CERN-TH-4914/87, 1987 (unpublished).

[c9] F. J. Gilman and M. B. Wise, Phys. Lett. 83B, 83 (1979); Phys. Rev. D 20, 2392 (1979).

[c10] For a review of the theoretical estimates of CP violation in B decay and the experimental possibilities, see K. J. Foley et al.., in Experiments, Detectors, and Experimental Areas for the Supercollider, proceedings of the Workshop, Berkeley, California, 1987, edited by R. Donaldson and M. G. D. Gilchriese (World Scientific, Singapore, 1988), p. 701; in Proceedings of the Workshop on High Sensitivity Beauty Physics at Fermilab, Batavia, Illinois, 1987, edited by A. J. Slaughter, N. Lockyer, and M. Schmidt (Fermilab, Batavia, 1988).

[c11] M. Baker and S. L. Glashow, Nuovo Cimento 25, 857 (1962). The argument can be extended to apply to an intermediate virtual $Z^{0}$ or the local operator generated by two virtual intermediate W's.

[c12] J. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B109, 213 (1976).

[c13] F. J. Gilman and M. B. Wise, Phys. Rev. D 21, 3150 (1980).

[c14] A. I. Vainshtein et al., Yad. Fiz. 24, 820 (1976) [Sov. J. Nucl. Phys. 24, 427 (1976)]; V. V. Flambaum, ibid. 22, 661 (1975) [22, 340 (1975)]; D. V. Nanopoulos and G. G. Ross, Phys. Lett. 56B, 279 (1975); L. E. Ibanez et al., Phys. Rev. D 21, 1428 (1980).

[c15] E. Witten, Nucl. Phys. B122, 109 (1977).

[c16] See, for example, the report on K physics by the group led by R. Bock and L. Littenberg, in Proceedings of the Summer Study on High Energy Physics in the 1990s, Snowmass, Colorado, 1988 (unpublished).

[c17] A preliminary report of this work was given by C. Dib, in Proceedings of the Summer Study on High Energy Physics in the 1990s (Ref. 16); and F. J. Gilman, lectures at the Heavy Flavor Symposium, Beijing, China, 1988, Report No. SLAC-PUB-4736, 1988 (unpublished). A brief version is found in C. O. Dib, I. Dunietz, and F. J. Gilman, Phys. Lett. (to be published). Other recent work on this subject is found in J. Flynn and L. Randall, Report No. LBL-26310, 1988 (unpublished).

[c18] G. Ecker, A. Pich and E. de Rafael, Phys. Lett. B 189, 363 (1987); Nucl. Phys. B303, 665 (1988).

[c19] J. F. Donoghue, B. R. Holstein and G. Valencia, Phys. Rev. D 35, 2769 (1987).

[c20] L. M. Sehgal, Phys. Rev. D 38, 808 (1988). See also T. Morozumi and H. Iwasaki, Report No. KEK-TH-206, 1988 (unpublished).

[c21] A discussion of the size of the B amplitude and of a mistake in the calculation of Morozumi and Iwasaki (Ref. 20) is found in Dib (Ref. 17) and in J. Flynn and L. Randall, Report No. LBL-26008, 1988 (unpublished).

[c22] E. de Rafael (private communication). See also G. Ecker et al.., Report No. CERN-TH-5185/88, 1988 (unpublished).

[c23] A. Pich, talk at the Workshop on Rare K Decays and CP Violation, Brookhaven, New York, 1988 (unpublished); Report No. CERN-TH-5194, 1988 (unpublished).

[c24] Particle Data Group, M. Aguilar-Benitez et al., Phys. Lett. 170B, 1 (1986).

[c25] In a chiral-perturbation-theory calculation, G. Ecker, A. Pich and E. de Rafael, Nucl. Phys. B291, 692 (1987), obtain two values, 0.25 and 2.5.

[c26] We use the technique and notation of Ref. 9.

[c27] This was done previously for the closely related process b -> ${se}^{+}$ $e^{-}$ by B. Grinstein, M. J. Savage, and M. B. Wise, Report No. LBL-25014, 1988 (unpublished).

[c28] The relation of our choice of operators to those in Ref. 27, after an appropriate change of quark labels to account for the different processes under consideration, is $C_{7 V}$ = $c_{8}$ ${}^{(GSW)}$ / 2 pi and $C_{7 A}$ = $c_{9}$ ${}^{(GSW)}$ / 2 pi. For the ``electromagnetic penguin'' contribution, the coefficient $F_{1}$ of Ref. 29 is related by $F_{1}$ ( $x_{i}$ ) = 2 π& ${Ctilde}_{7 V, i}$ .

[c29] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); ibid. 65, 1772(E) (1981). We have repeated this calculation and are in agreement with it and the corresponding one in Ref. 27.

[c30] The 7 times 7 matrix is given in Appendix B of Ref. 13. There is an error in the two entries $γ_{37}$ and $γ_{47}$ of the anomalous-dimensions matrix which describe the mixing of the ``penguin'' operators $Q_{3}$ and $Q_{4}$ with $Q_{7}$ ; they should be increased by case 8 over 9 and case 8 over 3, respectively. We thank J. Flynn and L. Randall for pointing this out to us. This changes the numerical results very little (see text), and has no effect on the discussion here, which neglects this mixing anyway.

[c31] J. O. Eeg and I. Picek, Phys. Lett. B 214, 651 (1988).

[c32] Vainshtein et al. (Ref. 14); V. A. Novikov et al., Phys. Rev. D 16, 223 (1977).

[c33] C. Jarlskog, Phys. Rev. Lett. 55, 1839 (1985); Z. Phys. C 29, 491 (1985).

[c34] O. W. Greenberg, Phys. Rev. D 32, 1841 (1985); D.-D. Wu, ibid. 33, 860 (1986); J. D. Bjorken and I. Dunietz, ibid. 36, 2109 (1987).

[c35] See also the discussion in Ref. 13, where long-distance contributions are estimated.

[c36] In this regard, we proceed similarly to J. Ellis and J. S. Hagelin, Nucl. Phys. B217, 189 (1983), in calculating the QCD corrections to K -> π ν ν bar.

[c37] L. K. Gibbons et al., Phys. Rev. Lett. 61, 2661 (1988).

[c38] G. D. Barr et al., Phys. Lett. B 214, 303 (1988).

[c39] We thank L. Littenberg for calling our attention to this consequence of the asymptotic behavior of the contributions as $m_{t}$ -> inf. See L. Littenberg, Phys. Rev. D (to be published).

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$K_{L}$ → $π^{0}$ $l^{+}$ $l^{-}$ decays for large $m_{t}$

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KL→π0l+l− decays for large mt

Abstract

References (39)

$K_{L}$ → $π^{0}$ $l^{+}$ $l^{-}$ decays for large $m_{t}$