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Kaon Physics: CP Violation and Hadronic Matrix Elements

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Kaon Physics: CP Violation and Hadronic Matrix Elements Kaon Physics: CP Violation and Hadronic Matrix Elements Mar´ıa Elvira G´amiz S´anchez Departamento de F´ısica Te´orica y del Cosmos arXiv:hep-ph/0401236v1 29 Jan 2004 Universidad de Granada October 2003 · · Contents 1 Introduction 5 1.1 OverviewoftheStandardModel. .. 7 1.1.1 TheScalarSector............................ 9 1.1.2 CP Symmetry Violation in the Standard Model . .. 12 2 CP Violation in the Kaon System 15 2.1 CP Violation in K0 K¯ 0 Mixing: Indirect CP Violation . 15 − 2.2 εK intheStandardModel ........................... 17 2.3 CP Violation in the Decay: Direct CP Violation . ...... 18 ′ 2.4 εK intheStandardModel ........................... 20 1 2.4.1 ∆I = 2 Contribution .......................... 23 3 2.4.2 ∆I = 2 Contribution .......................... 24 ′ 2.4.3 Discussion on the Theoretical Determinations of εK ......... 25 2.5 Direct CP Violation in Charged Kaons K 3π Decays........... 27 → 3 The Effective Field Theory of the Standard Model at Low Energies 31 3.1 Lowest Order Chiral Perturbation Theory . ..... 31 3.2 Next-to-Leading Order Chiral Lagrangians . ....... 33 3.3 Leading Order Chiral Perturbation Theory Predictions . .......... 38 3.4 CouplingsoftheLeadingOrderLagrangian . ..... 40 4 Charged Kaons K 3π CP Violating Asymmetries at NLO in CHPT 43 → 4.1 Numerical Inputs for the Weak Counterterms . ..... 44 4.1.1 Counterterms of the NLO Weak Chiral Lagrangian . ... 45 4.2 CP-Conserving Observables . .. 46 4.2.1 Slope g .................................. 47 4.2.2 Slopes h and k ............................. 47 4.2.3 DecayRates............................... 49 4.3 CP-Violating Observables at Leading Order . ...... 50 4.3.1 CP-Violating Asymmetries in the Slope g ............... 50 4.3.2 CP-Violating Asymmetries in the Decay Rates . .... 51 4.4 CP-Violating Observables at Next-to-Leading Order . .......... 52 4.4.1 FinalStateInteractionsatNLO . 52 1 2 CONTENTS 4.4.2 Results on the Asymmetries in the Slope g .............. 55 4.4.3 Results on the Asymmetries in the Decay Rates . ... 57 4.5 ComparisonwithEarlierWork. .. 57 5 QCD Short-Distance Constraints and Hadronic Approximations 61 5.1 Basics of the Model and Two-Point Functions . ..... 63 5.1.1 General ................................. 65 5.1.2 ChiralLimit............................... 68 5.1.3 BeyondtheChiralLimit . 71 5.2 Three-PointFunctions . 74 5.2.1 The Pseudoscalar-Scalar-Pseudoscalar Three-Point Function and the Scalar Form Factor 5.2.2 The Vector-Pseudoscalar-Pseudoscalar Three-Point Function and the Vector Form Factor 5.2.3 The Scalar-Vector-Vector Three-Point function . ........ 79 5.2.4 The Pseudoscalar–Vector–Axial-vector Three-Point Function . 80 5.2.5 The Pseudoscalar–Axial-Vector–Scalar Three-Point Function . 82 5.3 ComparisonwithExperiment . 83 5.4 Difficulties in Going Beyond the One-Resonance Approximation ...... 84 5.5 A General Problem in Short-Distance Constraints in Higher Green Functions 85 5.6 Applications: Calculation of BˆK ........................ 87 5.7 An Example of OPE Green’s Function Calculation . ...... 89 6 Hadronic Matrix Elements of the Electroweak Operators Q7 and Q8 93 6.1 The Q7 and Q8 Operators ........................... 95 6.2 Exact Long–Short-Distance Matching at NLO in αS ............. 97 6.2.1 The Q7 Contribution.......................... 97 6.2.2 The Q8 Contribution.......................... 99 6.2.3 Sum ................................... 102 6.3 BagParameters................................. 103 T 2 6.4 The ΠLR(Q ) Two-Point Function and Integrals over It . 105 (0−3) 2 6.5 The Scalar–Pseudo-Scalar Two-Point Function ΠSS+PP (Q ) ....... 111 6.6 Numerical Results for the Matrix-Elements and Bag Parameters . 114 6.7 Comparisonwithearlierresults . .... 117 6.7.1 RˆoleofHigherDimensional Operators . 119 ′ 7 Application to εK 121 ′ 7.1 εK withaDifferentSetofInputParameters . 124 8 Conclusions 125 T 2 (0−3) conn 2 A OPE of ΠLR(Q ) and ΠSS+PP (Q ) 129 A.1 Calculation of the Corrections of (a2) to the Dimension Six Contribution to ΠT (Q2)129 O LR A.1.1 Renormalization Group Analysis . 129 (0) (1) A.1.2 Calculation of the Constants Ci and Fi .............. 132 CONTENTS 3 A.1.3 The constants pij ............................ 133 2 (0−3) conn 2 A.2 Calculation of the Corrections of (a ) to the Dimension Six Contribution to ΠSS+PP (Q )134 A.2.1 Renormalization Group AnalysisO . 134 ˜(0) ˜(1) A.2.2 Calculation of the Constants Ci and Fi .............. 135 A.2.3 Calculation of the pij. ......................... 136 B Analytical formulas for K 3π decays 137 → B.1 K 3π AmplitudesatNLO.......................... 137 B.2 The→ Slope g and ∆g atLOandNLO ..................... 139 B.3 The Quantities A 2 and ∆ A 2 atLOandNLO ............... 140 | | | | B.4 FinalStateInteractionsatNLO . 144 B.4.1 Notation................................. 145 B.4.2 Final State Interactions for K+ π+π+π− .............. 146 → B.4.3 Final State Interactions for K+ π0π0π+ .............. 147 B.4.4 Integrals.................................→ 148 B.4.5 Analytical Results for the Dominant FSI Phases at NLO . ..... 150 C Some Short-Distance Relations for Three-point Functions 153 4 CONTENTS Chapter 1 Introduction CP symmetry violation was discovered several decades ago in neutral kaon decays [1]. Effects of CP symmetry breaking have been also recently observed in B meson decays [2, 3], but kaon physic continues being an exceptional ground to study this kind of phenomena. The analysis of the parameters that describe CP violation constitutes a great source of information about flavour changing processes and, in the Standard Model, they can provide us with information about the worst known part of the Lagrangian, the scalar sector, where CP violation has its origin. ′ The main CP-violating parameter in the kaon decays is εK/εK, for which there exist ′ −3 very precise experimental measures Re (εK/εK) = (1.66 0.16) 10 -see Chapter 2 for definitions and discussions. Experimental data for this± quantity× and another CP– violating parameters –as well as other quantities related with different fundamental aspects in particle physics as flavour changing neutral currents or quark masses– are based on the analysis of observables involving hadrons, that interact through strong interactions. In particular, these observables are governed in a decisive way by the non-perturbative regime of Quantum Chromodynamics (QCD), the theory describing strong interactions. At low energies –in the non-perturbative regime– QCD is not completely understood, so it is not easy to get theoretical predictions for the hadronic matrix elements involved in these processes without large uncertainties. Any improvement in the calculation of these hadronic matrix elements would thus been fundamental in the understanding of experimental CP-violating results. The goal is twofold, on one hand we pretend to analyze theoretically with great precision some observables in the Standard Model, specially those related to the CP-violating sector which is embedded in the scalar sector, the worst known part of the Standard Model lagrangian. On the other hand, the precise knowledge of the Standard Model predictions for CP-violating observables can serve to unveil the existence of new physics and check the validity of certain models beyond the Standard Model. The main problem is dealing with strong interactions at intermediate energies. At very low and high energies we can use Chiral Perturbation Theory (CHPT) [4, 5] and perturbative QCD respectively, that are well established theories. They let us do reliable calculations using next orders in the expansion as an estimate of the error associated to them. Several methods and approximations have been developed in order to try to suitably 5 6 Chapter 1: Introduction describe the intermediate region. Any reliable calculation give a good matching between the long- and short- distance regions. The most important of these methods are listed in Chapter 5 and their predictions for several CP-violating parameters are commented along the different chapters of this Thesis. The basic objects that are needed for the description of the low energy physics are the two-quark currents and densities Green’s functions. The couplings of the strong CHPT la- grangian are coefficients of the Taylor expansion in powers of masses and external momenta of some of these Green’s functions. While other order parameters needed in kaon physics such as BK , G8, Im GE or G27 are obtained by also doing the appropriate identification of the coefficients in the Taylor expansion of integrals of this kind of Green’s functions over all the range of energies. See Section 2.4.1 of Chapter 2 for more details. A good description of Green’s functions would thus provide predictions for the parameters we are interested in. A description of a general method to perform a program of this kind is given in Chapter 5 and some possible applications of it are discussed in the conclusions. This Thesis is organized in two parts. In the first part, Chapters 1-3, we describe the framework in which the Thesis has been developed and establish the definitions and notation necessary in the second part. In the next section of this chapter we give an overview of the Standard Model, ex- plaining the different sectors in which it can divided and the grade of knowledge we have about each of them. We define the symmetry CP and discuss in which conditions it can be measured experimentally. Discussions about the value of the SM parameters ms and V calculated in [6, 7] are also provided in this chapter. The Chapter 2 is a more de- | us| tailed description of the theory involved in the CP violation in K decays, in which we define the main CP-violating observables and outline the theoretical calculation of these parameters. The present experimental and theoretical status of direct CP violation in the kaon decays is also given. In Chapter 3 we introduce CHPT, collect the Lagrangians at leading and next-to-leading order in this theory and discuss the values of the couplings of these Lagrangians.
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