A Mathematica based Version of the CKMfitter Package

More documents

Recommendations

Info

24 Chapter 4. A Mathematica based Version of the CKMfitter Package 4.5.4 DecayBagParameters The Mathematica based version of CKMfitter avoids the usage of global variables. However, there are some model parameters, e. g. the decay constants and bag parameters of Bd- and Bs-mesons: fBs , fBs fBd , Bs , Bs Bd , (4.4) which are used in different theory packages. Thus, they need to be “globalized”, through initializing them in a separate package, which will be sourced from the other theory packages. The DecayBagParameters package is currently loaded from the BBbarKKbarMixing package as well as from the LeptonicDecay package. 4.6 Look-Up Tables Since there is not always the possibility to express a measurement in terms of a value and an error, the experimental likelihood, translated into a χ 2 -contour, can also be used as an input. It is available in a discrete look-up table, where values in-between the discrete LUT entries are calculated through interpolation. The Mathematica based CKMfitter version provides the possibility of a cubic spline interpolation and uses a separate FORTRAN based function to load the LUTs during the fit. 4.6.1 Cubic Spline Interpolation Piecewise interpolation using low order polynomials leads to a global continuous interpolating function, but is in general not continuous at the interval boundaries. This is avoided by using cubic spline interpolation, which gives back a smooth interpolating function. The goal of cubic spline interpolation is to get an interpolation formula that is smooth in the first, and continuous in the second derivative, both within an interval and at its boundaries. A cubic spline interpolation function s(x) to the sampling points x0 < x1 < . . . < xn−1 < xn and the corresponding sampling values yj (j = 0, 1, 2, . . . , n) is defined by [25]: • s (xj) = yj for (j = 0, 1, . . . , n); • s (x) is for x ∈ [xi, xi+1] (i = 0, . . . , n − 1) a polynomial of at most of order 3; • s (x) ∈ C 2 ([x0, xn]); • s ′′ (x0) = s ′′ (xn) = 0.
4.6. Look-Up Tables 25 In the Mathematica based CKMfitter version, polynomials of third order y(x) = a + bx + cx 2 + dx 3 (4.5) are used as cubic spline interpolation functions, where a, b, c and d are the spline coefficients. They are calculated in a separate notebook using a public available routine coded by Joseph M. Herrmann [26]. Its result is a six column look-up table in ASCII format, which is used as input file for the fit. The column order of the LUT is given in Table 4.4 and the number of lines is written in the first row. There is also an intrinsic Mathematica function called SplineFit, which generates a spline function object. Due to the fact that the generated spline function doesn’t depend directly on the observable values, it cannot be used in CKMfitter to create the LUTs. Column: 1 2 3 4 5 6 Content: value of observable χ 2 a b c d 4.6.2 The Tableau Function Table 4.4: LUT column order The numerical contribution to the χ 2 -function of variables with LUT input is calculated by separate FORTRAN functions in the file specialfunctions.f. The function TABLEAU provides the χ 2 -contribution, where DTABLEAUO2 calculates the contributions to its gradients 3 . During the first call of TABLEAU, the different LUTs are loaded using the separate subroutine LoadLUT. The functions TABLEAU and DTABLEAUO2 are called during the fit for each prediction value of the fit parameter. Using the spline coefficients of the next lower entry in the LUT, the contribution to the χ 2 -function or its gradient is calculated. The source code of these functions is available in Appendix D. It is possible that the predicted value of the fit lies outside of the LUT boundaries. In this case, the TABLEAU function will be constantly continued using the first (last) value of the LUT, if the prediction is smaller (larger) than the lower (upper) bound. The gradient, obtained by DTABLEAUO2 is set to zero. This can lead to critical effects at the boundaries. A possibility to avoid such critical effects is to enlarge the LUTs periodically, if this is allowed by the variable. Examples for these cases are the UT angles α and γ (see Section 5.1). 3 DTABLEAUO2 (“O2” means over two) calculates the half of the gradient contribution.
Page 1 and 2: A Mathematica based Version of the
Page 3: Kurzfassung Das CKMfitter Software-
Page 6 and 7: vi CONTENTS 6 New Physics Beyond th
Page 8 and 9: viii LIST OF FIGURES B.6 Constraint
Page 10 and 11: x LIST OF TABLES
Page 12 and 13: 2 Chapter 1. Introduction Unfortuna
Page 14 and 15: 4 Chapter 2. Theory the bosons of t
Page 16 and 17: 6 Chapter 2. Theory VCKM is a compl
Page 18 and 19: 8 Chapter 2. Theory 2.4 Neutral Mes
Page 20 and 21: 10 Chapter 2. Theory b u,c,t d 0 B
Page 22 and 23: 12 Chapter 2. Theory CP violation i
Page 24 and 25: 14 Chapter 3. CKMfitter The quantit
Page 26 and 27: 16 Chapter 3. CKMfitter
Page 28 and 29: 18 Chapter 4. A Mathematica based V
Page 40 and 41: 30 Chapter 5. Probing the Standard
Page 60 and 61: 50 Chapter 6. New Physics Beyond th
Page 68 and 69: 58 Chapter 7. Conclusions and Persp
Page 70 and 71: 60 Appendix A. The Inami-Lim Functi
Page 72 and 73: 62 Appendix B. Additional Figures o
Page 74 and 75: 64 Appendix B. Additional Figures o
Page 76 and 77: 66 Appendix C. Testjob } globalMinS
Page 78 and 79: 68 Appendix D. Source Code else k =
Page 80 and 81: 70 Appendix D. Source Code do i = 1
Page 82 and 83: 72 Appendix E. User Guide E.2 Datac
Page 84 and 85:
74 Appendix E. User Guide scanMin T
Page 86 and 87:
76 Appendix E. User Guide E.3 Input
Page 88 and 89:
78 Appendix E. User Guide
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
98 BIBLIOGRAPHY [12] L. Wolfenstein
Page 110 and 111:
100 BIBLIOGRAPHY [43] A. G. Akeroyd
Page 112 and 113:
102 BIBLIOGRAPHY [74] T. Inami and
Page 115:
Erklärung Hiermit versichere ich,
show all

A Mathematica based Version of the CKMfitter Package

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?