Master formulae for ΔF=2 NLO-QCD factors in the Standard Model and beyond

Andrzej J. Buras; Sebastian Jäger; J. örg Urban

doi:10.1016/S0550-3213(01)00207-3

Master formulae for ΔF=2 NLO-QCD factors in the Standard Model and beyond

Andrzej J. Buras, Sebastian Jäger, J. örg Urban

Chair of Theoretical Elementary Particle Physics

Technical University of Munich

Research output: Contribution to journal › Article › peer-review

197 Scopus citations

Abstract

We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients C_i(μ_t) and C_i(μ), with μ_t=O(m_t) and μ<μ_t, of the ΔF=2 dimension six four-quark operators Q_i in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements 〈Q_i(2 GeV)〉 and 〈Q_i(μ_K)〉 with μ_K<2 GeV are also presented. The formulae are given in the NDR scheme. The strongest renormalization-group effects are found for the operators with the Dirac structures (1-γ₅)⊗(1+γ₅) and (1-γ₅)⊗(1-γ₅). We calculate the matrix elements 〈 K̄⁰|Q_i|K⁰〉 in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences ΔM_K and ΔM_B and the CP-violating parameter ε_K in terms of the non-perturbative parameters B_i and the Wilson coefficients C_i(μ_t). The latter summarize the dependence on new physics contributions.

Original language	English
Pages (from-to)	600-624
Number of pages	25
Journal	Nuclear Physics, Section B
Volume	605
Issue number	1-3
DOIs	https://doi.org/10.1016/S0550-3213(01)00207-3
State	Published - 2 Jul 2001

Keywords

12.38.Bx
12.38.Cy
12.60.-i
14.40.Aq
14.40.Nd

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10.1016/S0550-3213(01)00207-3

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title = "Master formulae for ΔF=2 NLO-QCD factors in the Standard Model and beyond",

abstract = "We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients Ci(μt) and Ci(μ), with μt=O(mt) and μ<μt, of the ΔF=2 dimension six four-quark operators Qi in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements 〈Qi(2 GeV)〉 and 〈Qi(μK)〉 with μK<2 GeV are also presented. The formulae are given in the NDR scheme. The strongest renormalization-group effects are found for the operators with the Dirac structures (1-γ5)⊗(1+γ5) and (1-γ5)⊗(1-γ5). We calculate the matrix elements 〈 {\=K}0|Qi|K0〉 in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences ΔMK and ΔMB and the CP-violating parameter εK in terms of the non-perturbative parameters Bi and the Wilson coefficients Ci(μt). The latter summarize the dependence on new physics contributions.",

keywords = "12.38.Bx, 12.38.Cy, 12.60.-i, 14.40.Aq, 14.40.Nd",

author = "Buras, {Andrzej J.} and Sebastian J{\"a}ger and Urban, {J. {\"o}rg}",

note = "Funding Information: This work has been supported in part by the German Bundesministerium f{\"u}r Bildung und Forschung under the contract 05HT9WOA0. ",

year = "2001",

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doi = "10.1016/S0550-3213(01)00207-3",

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T1 - Master formulae for ΔF=2 NLO-QCD factors in the Standard Model and beyond

AU - Buras, Andrzej J.

AU - Jäger, Sebastian

AU - Urban, J. örg

N1 - Funding Information: This work has been supported in part by the German Bundesministerium für Bildung und Forschung under the contract 05HT9WOA0.

PY - 2001/7/2

Y1 - 2001/7/2

N2 - We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients Ci(μt) and Ci(μ), with μt=O(mt) and μ<μt, of the ΔF=2 dimension six four-quark operators Qi in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements 〈Qi(2 GeV)〉 and 〈Qi(μK)〉 with μK<2 GeV are also presented. The formulae are given in the NDR scheme. The strongest renormalization-group effects are found for the operators with the Dirac structures (1-γ5)⊗(1+γ5) and (1-γ5)⊗(1-γ5). We calculate the matrix elements 〈 K̄0|Qi|K0〉 in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences ΔMK and ΔMB and the CP-violating parameter εK in terms of the non-perturbative parameters Bi and the Wilson coefficients Ci(μt). The latter summarize the dependence on new physics contributions.

AB - We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients Ci(μt) and Ci(μ), with μt=O(mt) and μ<μt, of the ΔF=2 dimension six four-quark operators Qi in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements 〈Qi(2 GeV)〉 and 〈Qi(μK)〉 with μK<2 GeV are also presented. The formulae are given in the NDR scheme. The strongest renormalization-group effects are found for the operators with the Dirac structures (1-γ5)⊗(1+γ5) and (1-γ5)⊗(1-γ5). We calculate the matrix elements 〈 K̄0|Qi|K0〉 in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences ΔMK and ΔMB and the CP-violating parameter εK in terms of the non-perturbative parameters Bi and the Wilson coefficients Ci(μt). The latter summarize the dependence on new physics contributions.

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Master formulae for ΔF=2 NLO-QCD factors in the Standard Model and beyond

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