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Theses and Dissertations--Physics and Astronomy Physics and Astronomy

2021

ANALYZING THE EFFECT OF SECOND-CLASS CURRENTS ON ANALYZING THE EFFECT OF SECOND-CLASS CURRENTS ON

NEUTRON BETA DECAY OBSERVABLES AND THE EFFECT OF NEUTRON BETA DECAY OBSERVABLES AND THE EFFECT OF

THOMAS ROTATION ON THE RELATIVISTIC TRANSFORMATIONS THOMAS ROTATION ON THE RELATIVISTIC TRANSFORMATIONS

OF ELECTROMAGNETIC FIELDS OF ELECTROMAGNETIC FIELDS

Lakshya Malhotra University of Kentucky, lakshya9009@gmail.com Author ORCID Identifier:

https://orcid.org/0000-0002-0702-9645 Digital Object Identifier: https://doi.org/10.13023/etd.2021.049

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Recommended Citation Recommended Citation Malhotra, Lakshya, "ANALYZING THE EFFECT OF SECOND-CLASS CURRENTS ON NEUTRON BETA DECAY OBSERVABLES AND THE EFFECT OF THOMAS ROTATION ON THE RELATIVISTIC TRANSFORMATIONS OF ELECTROMAGNETIC FIELDS" (2021). Theses and Dissertations--Physics and Astronomy. 82. https://uknowledge.uky.edu/physastron_etds/82

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REVIEW, APPROVAL AND ACCEPTANCE REVIEW, APPROVAL AND ACCEPTANCE

The document mentioned above has been reviewed and accepted by the student’s advisor, on

behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of

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changes required by the advisory committee. The undersigned agree to abide by the statements

above.

Lakshya Malhotra, Student

Dr. Bradley Plaster, Major Professor

Dr. Christopher Crawford, Director of Graduate Studies

ANALYZING THE EFFECT OF SECOND-CLASS CURRENTS ON NEUTRONBETA DECAY OBSERVABLES AND THE EFFECT OF THOMAS ROTATION

ON THE RELATIVISTIC TRANSFORMATIONS OF ELECTROMAGNETICFIELDS

DISSERTATION

A dissertation submitted in partialfulfillment of the requirements forthe degree of Doctor of Philosophyin the College of Arts and Sciences

at the University of Kentucky

ByLakshya Malhotra

Lexington, Kentucky

Director: Dr. Bradley Plaster, Professor of PhysicsLexington, Kentucky

2021

Copyright c© Lakshya Malhotra 2021https://orcid.org/0000-0002-0702-9645

ABSTRACT OF DISSERTATION

ANALYZING THE EFFECT OF SECOND-CLASS CURRENTS ON NEUTRONBETA DECAY OBSERVABLES AND THE EFFECT OF THOMAS ROTATION

ON THE RELATIVISTIC TRANSFORMATIONS OF ELECTROMAGNETICFIELDS

The next generation of neutron beta decay measurements will attempt to analyzevarious beta decay observables of up to O(10−4) precision. At this level of experi-mental precision, the effects of second-class currents will contribute theoretical un-certainties to the interpretation of these measurements. Therefore, it is importantto investigate the effects of these second-class currents on decay parameters, such asthe Fierz interference term b which is linear in beyond standard model couplings.A maximum likelihood statistical framework and Rfit techniques are employed tostudy these second-class currents effects. Inputs to the Rfit technique are obtainedthrough combined global fits to Monte Carlo pseudo data which explore the energydependence of angular asymmetry coefficients.

This dissertation also investigates the effects of Thomas rotation in the relativistictransformation of electromagnetic fields. This may be important to future work oncalculating frequency shifts for relativistic spin-1/2 particles undergoing Larmor spinprecession in electromagnetic fields with small non-uniformities. We calculate theelectromagnetic field tensor for general three-dimensional successive boosts ~β and∆~β in the laboratory and the rest frame of an accelerated relativistic particle. Itis then compared with the electromagnetic field tensors obtained by a direct boost~β + δ~β. In the end, their consistency with Thomas rotation is analyzed.

KEYWORDS: neutron beta decay, physics beyond Standard model, special theoryof relativity, electromagnetic fields, Thomas rotation

Lakshya Malhotra

April 28, 2021

ANALYZING THE EFFECT OF SECOND-CLASS CURRENTS ON NEUTRONBETA DECAY OBSERVABLES AND THE EFFECT OF THOMAS ROTATION

ON THE RELATIVISTIC TRANSFORMATIONS OF ELECTROMAGNETICFIELDS

ByLakshya Malhotra

Bradley Plaster

Director of Dissertation

Christopher Crawford

Director of Graduate Studies

April 28, 2021

Dedicated to my mother Rama, and my brothers Abhishek and Ajnish.

ACKNOWLEDGMENTS

I would like to start by expressing my gratitude to my advisor Dr.Bradley Plaster.

I thank you and appreciate all the guidance and discussions you provided without

which this work would not have been possible. You have always been very supportive

and believed in my work, also giving me an opportunity to pursue my ideas. I am

also very grateful to you for supporting me during very difficult times.

I would also like to thank Dr.Tim Gorringe, Dr.Susan Gardner, and Dr.Steven

Yates for serving on my dissertation committee and seeing this process through.

Thank you to all the members of the “Plaster group”, Jared Brewington, Danielle

Schaper, Subash Nepal, Umit Coskun, Piya Amara Palamure, Mojtaba Behzadipour,

and Abel Manuel Lorente Campos, for the countless hours spent on the discussions

during group meetings and in-person that always provided new insights to my work.

Thank you to my friends, Shrushti Vora, Raghava Aditya, Laura Kelton, Alina

Aleksandrova, Ashkan Abtahi, Swetalina Maity, Shreya Patel, Rashika Gupta, and

Sudipto Das, for all the support and encouragement to make survival at the graduate

school possible.

Thank you to my roommates, Rishabh Shah and Aniruddha Shirodkar, for bearing

with me all those years.

Thank you to my family for their support and encouragement on this journey.

Most of all, thank you to my brother Abhishek Malhotra for his untiring support.

You have been the foundation of my success throughout my life, I would not be the

person I am today if it had not been for you. I could not have done it without you.

This material is based upon work supported by the U.S. Department of Energy,

Office of Science, Office of Nuclear Physics, under Award Number DE-SC0014622.

iii

TABLE OF CONTENTS

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Properties of the Neutron . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Recap of Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 All is not well with the Klein-Gordon equation . . . . . . . . . 21.2.3 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.3.1 Covariant form and the vector current . . . . . . . . 51.2.3.2 Spin Considerations . . . . . . . . . . . . . . . . . . 61.2.3.3 Helicity and Chirality . . . . . . . . . . . . . . . . . 7

1.2.4 S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Discrete Symmetries and CPT Theorem . . . . . . . . . . . . . . . . 10

1.3.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . . 121.3.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Electroweak Interactions and Standard Model of Particle Physics . . 141.4.1 Fermi Theory of β-decay . . . . . . . . . . . . . . . . . . . . . 15

1.4.1.1 Fermi and Gamow-Teller transitions . . . . . . . . . 171.4.2 Parity violation in weak interactions . . . . . . . . . . . . . . 17

1.4.2.1 K mesons and τ − θ puzzle . . . . . . . . . . . . . . 171.4.2.2 Wu et al. experiment . . . . . . . . . . . . . . . . . . 18

1.4.3 V − A law of the Weak interactions . . . . . . . . . . . . . . . 191.4.3.1 The CKM Matrix . . . . . . . . . . . . . . . . . . . . 191.4.3.2 Conserved Vector Current Hypothesis . . . . . . . . 20

1.4.4 Weak Interaction processes . . . . . . . . . . . . . . . . . . . . 211.4.4.1 Neutron β decay . . . . . . . . . . . . . . . . . . . . 211.4.4.2 Pion Decay . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Survey of Special Theory of Relativity . . . . . . . . . . . . . . . . . 221.5.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . 231.5.2 Addition of Velocities . . . . . . . . . . . . . . . . . . . . . . . 231.5.3 Matrix Representation of the Electromagnetic fields and Boost

Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5.4 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 2 Survey of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 302.1 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 30

iv

2.1.1 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 302.1.2 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . 312.1.3 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.1 Maximum likelihood analysis of neutron β-decay . . . . . . . . 33

2.3 The Rfit Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.1 Estimating SM parameters . . . . . . . . . . . . . . . . . . . . 352.3.2 Probing the SM . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Chapter 3 Neutron β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 The Matrix Element . . . . . . . . . . . . . . . . . . . . . . . 373.2 Differential Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1 Fierz Interference term b . . . . . . . . . . . . . . . . . . . . . 403.2.2 Angular Correlation Coefficients . . . . . . . . . . . . . . . . . 41

3.2.2.1 Spin-electron asymmetry coefficient A . . . . . . . . 423.2.2.2 Electron-antineutrino correlation coefficient a . . . . 43

3.2.3 Neutron Lifetime τ . . . . . . . . . . . . . . . . . . . . . . . . 433.2.3.1 Status of neutron lifetime measurements . . . . . . . 44

Chapter 4 nFitter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.1 High-level explanation . . . . . . . . . . . . . . . . . . . . . . 464.2 Unpolarized Neutron β Decay . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Monte Carlo Pseudodata . . . . . . . . . . . . . . . . . . . . . 474.2.2 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Polarized Neutron β Decay . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Workflow and Monte Carlo Pseudodata . . . . . . . . . . . . . 544.3.2 Single Parameter Fits on the Standard Model Data set . . . . 554.3.3 Global Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.3.1 Constraints on λ . . . . . . . . . . . . . . . . . . . . 614.3.3.2 Constraints on the Fierz term (b) . . . . . . . . . . . 624.3.3.3 Negligible correlation between λ and b . . . . . . . . 644.3.3.4 Fitting ymod without neutron lifetime (τ) . . . . . 644.3.3.5 Effects of including neutron lifetime τ on the global fits 664.3.3.6 Effect of number of the events in the Monte Carlo

pseudodata on the uncertainty of the Fierz term (b) . 684.3.3.7 Getting back the negligible correlation between λ and b 69

4.3.4 The “New Physics” data set . . . . . . . . . . . . . . . . . . . 71

Chapter 5 Thomas Rotation and its effects on the Electromagnetic Fields . 745.1 Transformations of the Electromagnetic Field Tensor . . . . . . . . . 74

5.1.1 Longitudinal-Transverse `t-frame . . . . . . . . . . . . . . . . 745.1.2 Laboratory xy-frame . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Validation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

v

5.3 Comparison with the previous work . . . . . . . . . . . . . . . . . . . 835.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.4.2 Longitudinal-Transverse `t-Frame . . . . . . . . . . . . . . . . 85

5.4.2.1 Direct Boosted Frame (~β + δ~β) . . . . . . . . . . . . 85

5.4.2.2 Successively Boosted Frame ~β and ∆~β . . . . . . . . 875.4.3 Laboratory xy-Frame . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.3.1 Direct Boosted Frame (~β + δ~β) . . . . . . . . . . . . 89

5.4.3.2 Successively Boosted Frame ~β and ∆~β . . . . . . . . 92

Chapter 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2 Observations and Inferences . . . . . . . . . . . . . . . . . . . . . . . 946.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

vi

LIST OF TABLES

1.1 Various Dirac field bilinears and their corresponding number of components. 101.2 Transformation of different objects under parity. . . . . . . . . . . . . . . 111.3 Transformation of different physical quantities under Charge Conjugation. 121.4 Transformation of different quantities under Time Reversal. . . . . . . . 14

4.1 Values of ∆χ2 corresponding to a given confidence level (CL) for m fitparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Fit results for the Rfit technique applied on the electron energy spectrumof the unpolarized β decay. The fitted value of b is defined by the loca-tion of the χ2

min, while the 68.3% CL is defined by ∆χ2 = 2.30 for twoparameters. We used the input b = 0. . . . . . . . . . . . . . . . . . . . . 53

4.3 Initial values for the simulation and fitting parameters. . . . . . . . . . . 564.4 Fit results for the differential decay distribution with b and N as the

fit parameters. The values in the parenthesis indicate the error valuescorresponding to ∆χ2 = 2.30 which corresponds to 68.3% CL. . . . . . . 57

4.5 Fit results for correlation coefficients from single parameter fits. The re-ported value comes from the numerical integration results discussed in [48].We use nbins = 1000, nevents = 1× 1010 and ndof = 999. . . . . . . . . . . 59

4.6 Fitted values of λ shown for different subsets of the observables xexp =Γ, a, A. We use the input λ = 1.2732 [55], nevents = 1 × 1010 andnbins = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7 Fit results of b shown for different subsets of the observables xexp =Γ, a, A. As shown in Figure 4.9, the electron energy spectrum puts themost stringent constraints on the Fierz term. In the fit scenarios where Γ isincluded in xexp, we used ymod = b,N as the fit parameters (whereN is the normalization). We use the input λ = 1.2732 [55], b = 0.0,nevents = 1× 1010 and nbins = 1000. . . . . . . . . . . . . . . . . . . . . . 63

4.8 Results of the global fits consisting of the observables (xexp = Γ, a, A).We employed fitting for ymod = λ, b,N where the fitted value is definedby the location of χ2

min and the definition of 68.3% CL is given by ∆χ2 =3.53 [Table 4.1]. We use the input λ = 1.2732 [55], b = 0.0, nevents =1× 1010 and nbins = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.9 Fitted values for λ, b from simultaneous fits of the observables xexp =Γ, a, A, τ to the Standard model data set. The fitted value of the pa-rameters are defined by the location of χ2

min. They correspond to a CL of68.3% which for a three parameter fit (m = 3) is defined by ∆χ2 = 3.53[Table 4.1]. We use the inputs λ = 1.2732 [55], b = 0, nevents = 1 × 1010

and nbins = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

vii

4.10 Fitted values for λ, b from simultaneous, and individual fits for differentsubsets of the observables xexp = Γ, a, A, τ to the New Physics dataset. Here the second-class currents are turned off (f3 = g2 = 0). Firsthalf of the table consists the case when the uncertainty in the neutronlifetime δτ is 1.0 second whereas the second half is when δτ = 0.1 seconds.The Pull values for the Fierz terms have been calculated with respect toits standard model value (b = 0). The fitted value of the parameterscorresponds to a CL of 68.3% which for a three parameter fit (m = 3) isdefined by ∆χ2 = 3.53 [Table 4.1]. We use the inputs λ = 1.2732 [55],nevents = 1× 1010 and nbins = 1000. . . . . . . . . . . . . . . . . . . . . . 72

4.11 Fitted values for λ, b from simultaneous fits of the observables xexp =Γ, a, A, τ to the new physics data set with Rfit technique. First half ofthe table consists the case when the uncertainty in the neutron lifetime δτis 1.0 second whereas the second half is when δτ = 0.1 seconds. The fittedvalue of the parameters corresponds to a CL of 68.3% which for a threeparameter fit (m = 3) is defined by ∆χ2 = 3.53 [Table 4.1]. Input valuesof λ = 1.2732 [55], bBSM = −0.00487, nevents = 1× 1010 and nbins = 1000. 73

5.1 Expressions for the indicated components of the electromagnetic field ten-sor after being transformed by the first boost ~β`t in the longitudinal-transverse frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Expressions for the indicated components of the electromagnetic field ten-sor after being transformed by the first boost ~βxy in the laboratory frame. 79

viii

LIST OF FIGURES

1.1 Four point interaction proposed by Fermi’s theory. All particles werethought to interact at the same space-time point. . . . . . . . . . . . . . 15

1.2 Feynman diagram for the β-decay of a neutron which takes place by theelectroweak interaction. At the quark level, a d quark converts to an uquark followed by the emission of a W− boson. Apart from a proton, anelectron and an electron-antineutrino are emitted. . . . . . . . . . . . . . 16

1.3 Results from Wu et al. experiment [14] showing the observed beta asym-metry from the emitted electrons. The curves show the different countingrates for two opposite orientations of the nucleus (magnetization direc-tion). It shows the existence of parity violation since the emission ofelectrons is more favored in the direction opposite to that of the nuclearspin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 General boost in three-dimensions. The dotted line represents the projec-tion of ~β on the xy-plane. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5 Rotation about z axis by an angle φ. The new x, y and z axes are calledthe x1, y1 and z1 axes, respectively. . . . . . . . . . . . . . . . . . . . . . 27

1.6 Second rotation about the y1 axis by an angle π2− θ. x, y, and z axes in

this new frame are called the x2, y2 and z2 axes, respectively. . . . . . . . 28

3.1 Kinetic energy distribution spectrum for the emitted electron in β-decay. 40

4.1 Histogram of the electron energy spectrum as a function of x, (ratio ofelectron energy to electron end point energy in β-decay) obtained fromsimulating 1× 108 neutron β-decay events with 500 energy bins. . . . . . 48

4.2 Normalized version of the Figure 4.1 which makes the area under thecurve to be unity. This allows us to have b as the only fit parameter forthe electron energy spectrum. . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Parameter space showing the variation of the χ2 with the Fierz term b.The 68.3% CL are defined by ∆χ2 = 2.30 [Table 4.1] which yielded thefitted value of the Fierz term b = 0.00066(75) with a χ2

min = 473.94. Weuse the inputs b = 0.0, nevents = 1×108, nbins = 500, second class currents:f3 = g2 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Variation of χ2 with b for a single parameter fit for the unpolarized β-decay after allowing second class currents to vary. The original parameterspace plotted along with the one obtained by varying the second classcurrents one at a time as well as both simultaneously with a step size of0.01; (a) f3 ∈ [−0.1, 0.1], g2 = 0, (b) g2 ∈ [−0.1, 0.1], f3 = 0, and (c)f3 ∈ [−0.1, 0.1], g2 ∈ [−0.1, 0.1]. We use nevents = 1× 108 and nbins = 500. 52

4.5 Histogram showing the differential decay distribution plotted as a functionof electron energy. The solid line (red) is a result of the fitting with b andN as the fit parameters. We use nbins = 1000 and nevents = 1× 1010. . . . 57

ix

4.6 Simulated data from the standard model dataset for the electron-asymmetryand electron-antineutrino correlation plotted as a function of electron en-ergy. The individual data points are shown in blue and the red solidline is the result for the single parameter fits with A and a being the fitparameters respectively. The parameter x is defined as the ratio of elec-tron’s energy to its end-point energy Eq. 4.3. We use nbins = 1000 andnevents = 1× 1010. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.7 Parameter space obtained from fitting for b and N in electron energyspectrum using Rfit. Panel (a) shows the parameter space for a given rangeof [-0.1, 0.1] but turning off either f3 or g2 to see their effects separately.Red curve involves both f3 and g2 varied simultaneously. Panel (b) showsthe parameter space for different ranges for both f3 and g2. The χ2 valueson the Y-axis in panel (b) are different from those in panel (a) because weobtained a much better χ2

min when we removed the first two energy binsof the histograms from the analysis. The step size for Rfit technique ineach of the cases is 0.01 with input b = 0.0. The 68.3% confidence levelsare defined by ∆χ2 = 2.30. We use nbins = 1000 and nevents = 1× 1010. . 59

4.8 Effects on the uncertainty in b as a function of the number of eventsobtained by fitting for both b and N in the electron energy spectrum withand without second-class currents. Both second-class currents f3 and g2are varied simultaneously in ranges described in the legend. Panel (b)shows the plot in panel (a) in log scale. ∆χ2 = 2.30 for 68.3% CL for eachcase and nbins = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.9 εS−εT space showing the constraints imposed by various observables. Dif-ferent subsets of xexp = Γ, a, A are considered. The plot for the globalfits of a,A,Γ is totally eclipsed with the one for only Γ. It looks likeΓ puts the most stringent constraints on the Fierz term b and therefore,on the new physics scalar and tensor couplings. However, Coulomb radia-tive corrections can also modify this as well but we omitted them in ouranalysis. We use λ = 1.2732 [55], nbins = 1000 and nevents = 1× 1010. . . 63

4.10 Parameter space for λ and b obtained from the global fits for Γ, a, A, τ.The input values for λ and b are 1.2732 [55] and 0.0, respectively fornevents = 1× 1010 and nbins = 1000. A 68.3% CL for three parameters (b,λ, and N) corresponds to ∆χ2 = 3.53. . . . . . . . . . . . . . . . . . . . 64

4.11 Parameter space for λ and b obtained from the global fits for Γ, a, Aafter letting second-class currents f3 and g2 vary simultaneously in ranges[−0.1, 0.1] and [−0.5, 0.5]. The input values for λ and b are 1.2732 [55]and 0.0, respectively for nevents = 1× 1010 and nbins = 1000. A 68.3% CLfor three parameters (b, λ, and N) corresponds to ∆χ2 = 3.53. . . . . . . 65

x

4.12 Parameter space showing the error ellipse with and without Rfit withxexp = Γ, a, A, τ. Both second-class currents f3 and g2 are variedsimultaneously in the ranges described in the legend. Panel (a) showsthe result when δτ = 0.1 seconds whereas panel (b) shows the case whenδτ = 1 second. The fitted value of b are defined by the location of theχ2min, while the 68.3% CL are defined by ∆χ2 = 3.53. We use the inputsλ = 1.2732 [55], b = 0.0, nevents = 1× 1010 and nbins = 1000. . . . . . . . 68

4.13 Variation in the uncertainty in b as a function of number of events obtainedby fitting b, N and λ in the global fits xexp = Γ, a, A, τ. Both second-class currents f3 and g2 are varied simultaneously in ranges described inthe legend. Panel (b) shows the plot in panel (a) in log scale. We use∆χ2 = 3.53 for 68.3% CL for each case, nevents = 1× 1010, and nbins = 1000. 69

4.14 Parameter space showing the error ellipse with and without Rfit withxexp = Γ, a, A, τ. Both second-class currents f3 and g2 are variedsimultaneously in ranges described in the legend. Panel (a) shows theresult when δτ = 0.1 seconds whereas panel (b) shows the case whenδτ = 1 second. As can be easily seen, letting the range of the second-class currents shrink brings back the original error ellipse with almostzero correlation. We use ∆χ2 = 3.53 for 68.3% CL for each case, andinputs λ = 1.2732 [55], b = 0.0, nevents = 1× 1010, and nbins = 1000. . . . 70

xi

Chapter 1 Introduction

Despite the success of the Standard model of particle physics (SM), it fails toexplain some important natural phenomena and hence, is considered incomplete. Forinstance – it does not explain gravity; it fails to unify gravity with quantum mechanics;it does not predict the masses of quarks or various coupling constants; it also fails toexplain why is there so much more matter than anti-matter in the universe. Theselimitations of the standard model motivate physicists to find the deviations from thepredictions of the theories or some new phenomenon that might help to answer theseunsolved questions.

Neutrons are an ideal candidate where one may test the robustness of the SM whileprobing for signatures of new physics. They react to all known forces and decay intoa proton, electron, and electron-antineutrino in a process known as neutron betadecay. This decay process offers us a powerful tool where studying the rate at whichit occurs and the angular correlations between the decay products provide insightinto this semi-leptonic decay.

The goal of this dissertation is twofold – first, it analyzes various neutron betadecay observables in context to the parameters like λ and the Fierz term b. It also triesto investigate the theoretical uncertainties arising from the ill-known weak hadronicsecond-class currents and to examine the limits they put on the ability to infer newphysics from neutron beta decay measurements. Secondly, it also presents a startingpoint for the development of a framework to calculate the electromagnetic field non-uniformities arising from the kinematical phenomenon of Thomas rotation.

The outline of this dissertation is as follows. Chapter 1 presents a literaturesurvey on the electroweak interactions and the special theory of relativity. A glimpseof some statistical ideas and concepts relevant to this work are presented in Chapter 2.Chapter 3 gives an overview of one special class of electroweak interactions, namely,neutron beta decay. The results of the maximum likelihood framework and Rfit onthe global fits to neutron beta decay observables are presented in Chapter 4. Chapter5 discusses the results obtained by including the Thomas rotation term in the Lorentztransformation of the electromagnetic field in the particles’ rest frame as well as thelaboratory frame. Finally, we conclude the dissertation with a brief summary inChapter 6.

1.1 Properties of the Neutron

According to the Quark model, the neutrons are composite particles with nonet charge (electrically neutral). They are composed of elementary particles calledquarks which are responsible for the internal structure of many classes of particles.For example, there are mesons that are composed of two quarks, and baryons whichare made up of three quarks. The quarks are charged particles and they come insix varieties – up(u), down(d), charm(c), strange(s), bottom(b), and top(t). Thesevarieties differ from each other on the basis of their mass and charge. Up, charm and

1

top are further grouped as up-type quarks whereas down, strange and bottom arecollectively referred to as down-type quarks. Up-type quarks have a fractional chargeof +(2/3)e1 while the down-type variety has a charge of −(1/3)e. The neutron,which is a baryon, consists of two down (d) quarks and one up (u) quark. The net-zero charge of the neutron can be seen from the fact that u contains +(2/3)e and dcontains −(1/3)e. The mass of the neutron as reported by the Particle Data Group(PDG) [2] is 939.5654133 MeV (mn > mp+me) with a lifetime of around 15 minutes.This difference in mass is one of the key factors why a neutron decays into a proton,an electron, and an electron-antineutrino. This is also one of the key areas of focusof this dissertation.

1.2 Recap of Quantum Field Theory

1.2.1 Fermions

Fermions are particles that carry half-integer spin. But there is more to that,fermions are particles whose physical states are totally anti-symmetric as per thespin-statistics theorem. According to this theorem, for a system of identical particles,the particles’ physical states have to follow

ψ(~r1, ~r2) = ±ψ(~r2, ~r1) (1.1)

where the plus sign is for bosons, and the minus sign is for fermions [1]. So far we haveseen all the particles which are thought of as the carriers of the fundamental forcesare bosons like the photon, gluon, W-boson, etc., whereas all the matter particles arefermions like the electron, neutron, proton, etc.

1.2.2 All is not well with the Klein-Gordon equation

One of the requirements of the relativistic formulation of quantum mechanics isthat the associated solutions be Lorentz invariant. We can expect to get a relativisticwave equation if we use the operator representation of various observables in therelativistic dispersion relationship which itself is Lorentz invariant [3, 4] :

E2 = (~p 2c2 +m2c4) (1.2)

Making operator substitutions yields the Klein-Gordon equation (E → E = i ∂∂t

,

~p→ p = −i~∇):

−∂2φ(x, t)

∂t2= −~∇2φ(x, t) +m2φ(x, t) (1.3)

in natural units (~ = c =1). The above equation can be written more compactly byintroducing

∂2 = ∂µ∂µ =∂2

∂t2− ~∇2 (1.4)

1 e refers to the magnitude of the charge of the electron, e = 1.602× 10−19 Coulombs.

2

which will express the Klein-Gordon equation in Lorentz invariant form:

(∂2 +m2)φ(x) = 0 (1.5)

Now, if we try to apply the continuity equation

dρ

dt+ ~∇ ·~j = 0 (1.6)

to the Klein-Gordon equation to calculate the probability density and current [5],

jµ(x) = iφ(x)∂µφ∗(x)− φ∗(x)∂µφ(x) (1.7)

it can be seen that for a plane wave solution (φ(~x, t) = Ne−iEt+i~p·~x = Ne−ip·x)2, thetemporal component of the probability current will be

ρ = j0 = 2 |N |2E (1.8)

From the dispersion equation Eq. 1.2, E can be positive or negative but for negativeE, ρ cannot be interpreted as a probability density since it does not make any senseto have negative probability! These apparent problems in the Klein-Gordon equationled to a search for an alternative formulation of relativistic quantum mechanics.

1.2.3 Dirac Equation

The requirement due to the energy-momentum dispersion relations in Eq. 1.2leads to the conclusion that any relativistic particle should follow the Klein-Gordonequation which is second order in both time and space derivatives. The Dirac equationis different in a way that it is linear in both time and space derivatives [3]. Inspired bythe dispersion relations, the most general form of the Dirac equation can be writtenas

Eψ = (~α · ~p+ βm)ψ (1.9)

In terms of operators, Eq. 1.9 becomes

i∂

∂tψ =

(−iαx

∂

∂x− iαy

∂

∂y− iαz

∂

∂z+ βm

)ψ (1.10)

We can actually place a strong constraint on the values of ~α and β by observingthat if Eq. 1.10 is followed by any relativistic particle, it should also follow theenergy-momentum relationship which, in turn, implies that it should also follow theKlein-Gordon equation. Understanding these parameters is important since these arethe starting point for Dirac matrices. To see that, we need to explore the relationshipbetween ~α and β which can be done by squaring Eq. 1.10:

−∂2ψ

∂t2=

(iαx

∂

∂x+ iαy

∂

∂y+ iαz

∂

∂z− βm

)(iαx

∂

∂x+ iαy

∂

∂y+ iαz

∂

∂z− βm

)ψ

(1.11)

2 xµ = (x0, ~x) = (t, ~x), pµ = (p0, ~p) = (E, ~p)p · x = gµνp

µxν = p0x0 − ~p · ~x = Et− ~p · ~x

3

which on simplification and comparing with the Klein-Gordon equation leads to thefollowing constraint equations:

α2x = α2

y = α2z = β2 = I,

αjβ + βαj = 0, (1.12)

αjαk + αkαj = 0 (j 6= k)

where I is the identity. It is clear from Eq. 1.12 that αi and β cannot be just plainreal numbers in order for the anticommutation relations to be satisfied. These objectshave to be matrices and we will show by simple mathematical arguments that theymust be even dimensional:

Tr(αi) = Tr(αiββ) = Tr(βαiβ) = −Tr(αiββ) = −Tr(αi) (1.13)

To see the eigenvalues of αi and β, we use the eigenvalue equation:

αiX = λX

X = λ2X (1.14)

Analogously, it is clear from Eq. 1.13 that Tr(A) = 0 where A = αx, αy, αz, β andλ = ±1. Using these it can be claimed that the only possible way a matrix witheigenvalues ±1 can have a zero trace if it is even dimensional. Moreover, it is clearfrom Eq. 1.12 that these matrices have to be hermitian:

αx = α†x, αy = α†y, αz = α†z, and β = β† (1.15)

Thus we can see the Pauli spin-matrices would have been perfect candidates for αiand β, but we need something whose basis vector is four-dimensional. Hence, thelowest dimension these αi and β can be is 4× 4. The Dirac Hamiltonian of Eq. 1.9 isa 4× 4 matrix of operators that must act on a four-component wavefunction, knownas a Dirac spinor [3–6],

ψ =

ψ1

ψ2

ψ3

ψ4

(1.16)

There are many representations in which we can represent Dirac matrices. In fact,any matrices which are generated by applying unitary transformations to αi and βin any representation will follow all the algebra satisfied by αi and β. This will makethose matrices an equally valid representation of the Dirac matrices. This makessense, since the Dirac equation does not depend on any specific representation, itspredictions will still be the same. One conventional choice of representation is theWeyl or Chiral representation which can be written as

β =

(0 II 0

), and αi =

(−σi 0

0 σi

)(1.17)

4

with

I =

(1 00 1

), σx =

(0 11 0

), σy =

(0 −ii 0

), and σz =

(1 00 −1

)(1.18)

Before proceeding further with the Dirac equation which is just set up by calculatingthe parameters αi and β, the probability density and current need to be evaluated. Itwill help to see the physical interpretation of this whole formalism of 4× 4 operators.The expressions for the probability density and current can be obtained using thesame procedure as for the Klein-Gordon equation using Eq. 1.7, the only differencebeing that the hermitian-transpose is used instead of complex conjugate: ψ∗ → ψ† =(ψ∗)T , using Eq. 1.7 and 1.9, we get the continuity equation [5]

~∇ · (ψ†~αψ) +∂(ψ†ψ)

∂t= 0 (1.19)

By comparing it with Eq. 1.6, the probability density and probability current for theDirac equation can be written as

ρ = ψ†ψ, and ~j = ψ†~αψ (1.20)

Unlike the Klein-Gordon equation, all the solutions of the Dirac equation have positiveprobability densities, which can be seen by writing the probability density as

ρ = ψ†ψ = |ψ1|2 + |ψ2|2 + |ψ3|2 + |ψ4|2 (1.21)

1.2.3.1 Covariant form and the vector current

A requirement for quantum field theory is that it should be Lorentz-invariant.As this should hold equally true for the Dirac theory, we need to express the Diracequation so that it manifests itself in a Lorentz-invariant form. This can be done bymultiplying Eq. 1.10 with β and simplifying

(iγµ∂µ −m)ψ = 0 (1.22)

where γµ are called the Dirac γ-matrices which are given by

γ0 ≡ β, γ1 ≡ βαx, γ2 ≡ βαy, and γ3 ≡ βαz (1.23)

Eq. 1.22 is referred to as the covariant form of the Dirac equation [3–6].The properties of the gamma matrices can be obtained easily from the properties

of the ~α and β matrices in the Eq. 1.12-1.15:

(γ0)2 = I,

(γk)2 = −I (k = 1, 2, 3), (1.24)

γµγν = −γνγµ (µ 6= ν)

The above equation can be written more succinctly in a form which might be familiar

γµ, γν ≡ γµγν + γνγµ = 2gµν (1.25)

5

Note that Eq. 1.24 and 1.25 hold regardless of the representation of the gammamatrices used. Hence, we can express the gamma matrices in the Weyl or Chiralrepresentation in a 2× 2 block form as

γ0 =

(0 II 0

), γi =

(0 σi

−σi 0

)(1.26)

It would be helpful to write the probability density and probability current in thecovariant form as well. Using the relationship between the γ and ~α, β matrices,Eq. 1.20 can be written as

jµ = (ρ,~j) = ψ†βγµψ = ψ†γ0γµψ (1.27)

jµ is a four-vector since it transforms under the Lorentz transformations in the sameway as other four-vectors like position and momentum. This allows us to write thecontinuity equation in Lorentz-invariant form,

∂µjµ = 0 (1.28)

Now, jµ can be further simplified if we introduce the adjoint spinor ψ [3–5], definedas

ψ = ψ†γ0

Using the adjoint spinor, the four-vector current can be written compactly as

jµ = ψγµψ (1.29)

1.2.3.2 Spin Considerations

From quantum mechanics, any physical observable has a corresponding operatorwhose time dependence can be calculated as

d

dtO = i 〈ψ| [H, O] |ψ〉 (1.30)

So if a physical observable commutes with the Hamiltonian of the system, that ob-servable will be a conserved quantity. This argument can be used for a free particlesatisfying the Dirac equation [3]. The Dirac Hamiltonian for the particle can bewritten from Eq. 1.9 as,

HD = ~α · ~p+ βm (1.31)

To see if the angular momentum of such a particle is conserved, the commutator

[HD,~L] yields

[HD,~L] = −i~α× ~p (1.32)

Therefore, the “orbital” angular momentum is not a conserved quantity, thus man-

dating the existence of another form of angular momentum (i.e. the spin~S) so that

6

the Hamiltonian commutes with the sum,~J =

~L +

~S which is the total angular

momentum of the system [3],

[HD,~J ] = 0 (1.33)

Note that, writing the total angular momentum simply as a sum of the orbital and the

spin angular momentum holds only for a non-relativistic fermion [4]. The operator~S

are 4× 4 matrices which can be defined in terms of Pauli spin-matrices as

~S =

1

2~Σ =

1

2

(~σ 00 ~σ

)=i

4~γ × ~γ (1.34)

Relativistically, the intrinsic angular momentum is given by the Pauli-Lubanski vectorWα which reduces to the ordinary three-dimensional angular momentum ~J in the restframe of the particle (~p = 0). W 2 and p2 are the operators which commute with allthe generators of the Poincare group [81].

1.2.3.3 Helicity and Chirality

The four component Dirac spinor Eq. 1.16 can be thought of as having 2 compo-nents

ψ =

(ψLψR

)These components are themselves two-component objects and are called Weyl spinors.Using this representation of the Dirac spinors we can substitute it into the Diracequation [3, 5, 6]:

(p0 + ~σ · ~p) ψL = mψR

(p0 − ~σ · ~p) ψR = mψL (1.35)

where p0 = i∂0 and ~p = −i~∇. Considering the special case of a massless particle, theequations for ψL and ψR decouple:

(p0 + ~σ · ~p) ψL = 0

(p0 − ~σ · ~p) ψR = 0 (1.36)

The above equation tells us that ψL and ψR are the eigenstates of the masslessDirac equation with ψL and ψR describing the left-handed particles and right-handedparticles, respectively. For massless particles Ep = | ~p | and the eigenvalues of p0 areindeed Ep. The above equation can be simplified as

~σ · ~p| ~p |

ψR = ψR

~σ · ~p| ~p |

ψL = −ψL (1.37)

7

It is worth noticing that the states ψL and ψR for the massless particles are eigenstatesof the helicity operator

h =~σ · ~p| ~p |

(1.38)

In order to generalize the formalism of the left- and right-handed wave functions, thechirality operator is used which is defined as

γ5 = iγ0γ1γ2γ3

=

(−I 00 I

)(1.39)

Now from basic matrix algebra, it can be easily shown that

γ5(ψL0

)= −

(ψL0

)γ5(

0ψR

)=

(0ψR

)(1.40)

which means an entirely left-handed spinor state is an eigenstate of γ5 with eigenvalue−1 and an entirely right-handed spinor state is then an eigenstate with eigenvalue+1 [3, 5]. Hence, in general, the Dirac equation predicts the existence of two typesof wave functions: left-handed and right-handed states. In fact, these left-handedand right-handed parts can be easily extracted from the wave function by using theprojection operators which are defined as

PL =1− γ5

2

PR =1 + γ5

2(1.41)

All of this is fine but it poses one question: aren’t helicity and chirality the samething? The answer to this follows from the discussion in this section: helicity andchirality are identical only for massless particles, but are, in general, not the same.A few important explanations follow from the discussion in this section:

• Helicity tells us whether the spin of the particle is parallel or anti-parallel tothe direction of motion. But it should be noted that the direction of motionof the particle is dependent on the frame of reference. This implies helicityalso depends on the frame of reference whereas there is no such condition forchirality [3–6].

• For the negative energy solutions of the Dirac equation, we can see that forthe massless particle, the dispersion relation is E = −|p0|. Using Eq. 1.36, asimilar eigenvalue equation for the helicity operator for anti-particles (negative

8

energy solutions) can be written:

~σ · ~p| ~p |

ψR = −ψR

~σ · ~p| ~p |

ψL = ψL (1.42)

This shows that right-handed anti-particle states have negative helicity andleft-handed anti-particle states have positive helicity as opposed to the case forleft-handed and right-handed particle states.

1.2.4 S-Matrix

Most of the physical processes in high-energy physics take place by the interactionof various particles with each other. Classically, this can be visualized as two well-separated particles which approach each other and then interact. Based on this ageneric Hamiltonian for this dynamical system can be written as

Htotal = Ha + Hb + Hint (1.43)

where Ha, Hb is the free Hamiltonian for particle a and b, respectively, and Hint istheir interaction Hamiltonian. The above equation can more or less be extended tothe case of high-energy physics with one major difference: most of the experimentshappen in the relativistic regime which is characterized by scattering. One can observethe interaction of two particles or beams with some initial states and analyze theprobability of obtaining a particular final state. This probability is often expressedin the form of a cross-section. A cross-section is expressed in terms of area; it is theeffective area of one particle or a beam observed by the other interacting particle,which subsequently becomes the final state.

How do we calculate the cross section then? It turns out that the recipe ofcalculating the cross section involves the time evolution of the initial states of theparticles3 for a long time4 with the time-evolution operator

e−iHintt

of the interacting part of the total Hamiltonian. These states are, then, projectedwith the resulting final states representing the desired set of final-state particles [5,6].Mathematically,5

out〈p′1p′2 · · ·|p1p2〉in = limT→∞

〈p′1p′2 · · ·| e−iHint(2T ) |p1p2〉 (1.44)

3 Eigenstates of the non-interacting Hamiltonian.4 To calculate the probability amplitudes during scattering we make an assumption that the initial

and final states are the eigenstates of the free Hamiltonian (without any interaction term). Inother words, we take the initial state at time t → −∞ and the final state at time t → ∞, to bethe eigenstates of only the free Hamiltonian H0 (H0 = Ha + Hb).

5 The time evolution due to the unitary operators from distant past to distant future resulted in afactor of 2T in the exponential because the overall strength of interactions is assumed to be timeindependent.

9

where pi and p′i are the momenta of the in and out states in the interacting theory [4].Thus, the in and out states are related to each other by the limit of the sequenceof unitary operators based on the interacting Hamiltonian. This limiting unitaryoperator is called the S -matrix. All the information about the types of interactionsfor two particles or colliding high energy beams of the particles is encoded in theS -matrix. The S -matrix is of some special relevance to the current work and it willbe discussed later.

1.3 Discrete Symmetries and CPT Theorem

Before going any further, we want to analyze the transformation properties ofvarious bilinear forms listed in Table 1.1 under Lorentz transformations. Any bilinearproduct ψAψ transforms under Lorentz transformations as

ψ′A ψ′ = ψ S[Λ]−1A S[Λ] ψ

where S[Λ] is a finite Lorentz transformation which can be written in terms of its gen-erators Sρσ and six numbers Ωρσ (representing boosts and rotations) as exp(1

2ΩρσS

ρσ)[6]. Using above equation and noting

S[Λ]† = γ0S[Λ]−1γ0

we can easily show that ψψ is a Lorentz scalar and ψγµψ is a 4-vector. In fact,for any 4 × 4 constant matrix Γ, ψΓψ can be decomposed in terms of sixteen 4 × 4basis matrices. These sixteen matrices constitute the Dirac bilinears as shown inTable 1.1. The bilinears with the prefix “pseudo” arise from the fact that under

Table 1.1: Various Dirac field bilinears and their corresponding number of compo-nents.

Bilinear Matrix form Componentsscalar (S) 1 1vector (V ) γµ 4tensor (T ) σµν = i

2[γµ, γν ] 6

pseudo vector (A) γµγ5 4pseudo scalar (P ) γ5 1

continuous Lorentz transformations they transform as a vector or scalar but theyhave an additional sign change under parity transformations. These will be studiedin a bit more detail in the coming sections but first, another important concept needsto be discussed: Symmetries!

Symmetries are one of the most powerful tools in physics as they usually providestrong clues about understanding a physical phenomenon whose governing principlesand equations are not known. There is a strong relationship between symmetriesand conservation laws. This was, in fact, proved by Emmy Noether [69] in the well

10

known Noether’s Theorem. It states that when a system possesses a continuous sym-metry there is a corresponding conservation law, which means a particular physicalquantity would be conserved. For instance, if a system is invariant under spatialtranslations, linear momentum will be a conserved quantity. This is one of the space-time symmetries. Similarly, for a complex scalar field, the Lagrangian is invariantunder the transformation φ→ eiαφ, this leads to particle number being an invariantquantity [4–6].

Besides continuous symmetries, there also exist discrete symmetries that are ofparticular interest to us. These are parity symmetry (P), time-reversal (T) and chargeconjugation (C). Three of the four fundamental forces of nature – the gravitational,the electromagnetic, and the strong forces – are symmetric with respect to P , C andT . But the fourth fundamental interaction called the weak interactions violate C andP separately, but preserves CP and T except for certain rare processes [4].

1.3.1 Parity

Parity transformation, P , is defined as the reversal of signs of the spatial coordi-nates [4, 6]. Mathematically,

P ~r P † = −~rP ~p P † = −~p (1.45)

This shows the action of the parity operator on position and linear momentum. Usingthis it can be easily shown that the angular momentum is indeed parity even,

P ~L P † = P (~r × ~p)P †

= −~r ×−~p (1.46)

= +~L

It can be seen from above that although angular momentum transforms like a normalvector under rotation, it transforms with an additional sign change under parity trans-formations as compared to normal vectors. These physical observables are known aspseudo-vectors or axial vectors. The transformations of different objects are discussedin Table 1.2.

Table 1.2: Transformation of different objects under parity.

Physical Observable Type Transformation Example

scalar Invariant Vector norm, ~a ·~bvector Variant Position vector, ~r

axial vector Invariant Angular Momentum, ~r × ~ppseudoscalar Variant Scalar triple product, ~a · (~b× ~c)

11

1.3.2 Charge Conjugation

The discrete symmetry of charge conjugation is also referred to as the particle-antiparticle symmetry C. Under charge conjugation, particles are transformed intotheir respective antiparticles. This transformation flips the sign of all the “charge-like” quantum numbers like electrical charge, lepton number, and baryon number.Table 1.3 shows the effect of charge conjugation on different physical quantities [4,6].

Table 1.3: Transformation of different physical quantities under Charge Conjugation.

Physical Quantity Transformation

Position (~r) Invariant: ~rC→ ~r

Momentum (~p) Invariant: ~pC→ ~p

Angular Momentum (~L) Invariant: ~LC→ ~L

Charge density (ρ) Variant: ρC→ −ρ

Current density ( ~J) Variant: ~JC→ − ~J

Electric field ( ~E) Variant: ~EC→ − ~E

Magnetic field ( ~B) Variant: ~BC→ − ~B

1.3.3 Time Reversal

The idea of time reversal can be explained very easily with the help of a dynamicalsystem: keep track of the motion of a system with a movie and rewind it. If thesystem is time-reversal invariant, then the dynamics of the system observed whilerewinding should also describe the possible time evolution of the system. In otherwords, if ~x(t) is a solution to the time evolution of the system then so is ~x(−t) [4–6].Mathematically,

T ~r T−1 = ~r

T ~p T−1 = −~p (1.47)

Again, from the above equations it follows that:

T ~LT−1 = T (~r × ~p)T−1

= ~r ×−~p (1.48)

= −~L

Classically, it is quite intuitive if we think about the action of time reversal on dynamicvariables, but in the realm of quantum mechanics, things get a little tricky. Let ustry to understand this with the help of position and linear momentum operators.

12

From quantum mechanics, the canonical conjugate variables follow the commutatorrelationship,

[~x, ~p] = i (1.49)

Applying time-reversal to Eq. 1.49 and using Eq. 1.48, it can be easily seen thatthis transformation is not consistent with commutation relations. To understand thisbetter, consider a physical system at t = 0 represented by a state |ψ(0)〉. Let us runthe state forward in time. Using recipes from quantum mechanics we know the timeevolution of a state is given by the application of the unitary operation [6]

|ψ(t)〉 = e−iHt |ψ(0)〉 (1.50)

Now suppose that we instead start with the time reversed state T |ψ(0)〉 and evolveit forward in time,

e−iHtT |ψ(0)〉 (1.51)

If the Hamiltonian itself is time reversal invariant, then the above equation should beequivalent to evolving |ψ(0)〉 backwards in time followed by a time reversal operation:

T eiHt |ψ(0)〉 = e−iHtT |ψ(0)〉 (1.52)

Expanding the above equation for infinitesimal time t, we get 6,7

T iH = −iHT (1.53)

If we assume the operator T to be linear8, then from the above equation we have

T H + HT = 0 (1.54)

This clearly is wrong and it’s easy to see why. Suppose we have an eigenstate |ψ〉 ofHamiltonian obeying H |ψ〉 = E |ψ〉. Using Eq. 1.52 we can write

HT |ψ〉 = −T H |ψ〉 = −ET |ψ〉 (1.55)

Physically this means every eigenstate with energy E must be accompanied by a timereversed eigenstate of energy −E which does not make any sense. This means theoperator T can not be a linear operator. It comes under a special class of operatorscalled anti-linear operators9,10 [70, 71]. Let us see if this property fixes the inconsis-tency in our commutation relations of Eq. 1.49. Applying the time reversal operationon the commutation relation:

T (~x ~p− ~p ~x)T−1 = ~T iT−1

T ~x ~pT−1 − T ~p ~xT−1 = −i−~x ~p+ ~p ~x = −i (1.56)

[~x, ~p] = i

6 ex = 1 + x+ x2

2! + · · ·7 T eiHt/~ ≈ T (1 + iHt/~)8 For a operator to be linear, it must follow Aα = αA for any complex number α.9 Anti-linear operators have a modified linear transformation rules: Bα = α∗B for any complex

number α.10 They are also called anti-unitary operators: BB† = B†B = 1.

13

Application of the time reversal operator on the factor i sandwiched between op-erators on the right-hand side makes it conjugated which, therefore, preserved ourcommutation relations. Table 1.4 shows the transformation of some physical quanti-ties under time reversal.

Table 1.4: Transformation of different quantities under Time Reversal.

Physical Quantity Transformation

Position (~r) Invariant: ~rT→ ~r

Momentum (~p) Variant: ~pT→ −~p

Charge density (ρ) Invariant: ρT→ ρ

Current density ( ~J) Variant: ~JT→ − ~J

Electric field ( ~E) Invariant: ~ET→ ~E

Magnetic field ( ~B) Variant: ~BT→ − ~B

It is worth noticing that all Lorentz invariant Quantum Field Theories are knownto be invariant under the combined operation of C, P, and T symmetries [4, 6]. Thisis known as the CPT theorem which states that CPT is believed to be a fundamentalsymmetry of nature. From this argument, one can infer that if CP symmetry isviolated in a physical process, that also implies T violation must also be there andvice versa.

1.4 Electroweak Interactions and Standard Model of Particle Physics

β-decay is a class of radioactive processes where unstable atoms attain stabilityby the emission of high energy electrons or positrons and neutrinos. β-decay happensin two ways – beta-minus decay, where a neutron in an atom’s nucleus turns into aproton, an electron and an electron-antineutrino and beta-plus decay, where a protonin an atom’s nucleus turns into a neutron, a positron and a neutrino. Originally,people assumed that it was just the emission of electrons which means that theenergy carried by the emitted electrons have discrete values. But the analysis ofthe kinetic energy distribution of the emitted electron revealed that they have acontinuous spectrum.

Only after the formulation of Fermi’s theory in 1934 the real understanding ofβ-decay came in. The puzzle of the continuous spectrum of the emitted electron inthe β-decay was solved after the postulation of the existence of a massless11 neutralparticle by Wolfgang Pauli [56]. Enrico Fermi succeeded in incorporating Pauli’s ideain his theory which led to the birth of the theory of electroweak interactions. Fermialso named these new particles “neutrinos”12. This new particle shares the reaction

11 They were thought massless when first postulated by Pauli, recent experiments revealed a smallmass of about 1.1eV [2].

12 Standard model actually predicts zero mass for the neutrinos.

14

energy with all other final state particles hence satisfying the conservation laws andmaking everything consistent [60].

Neutron β-decay is the primary topic of the current work and it will be discussedin more detail later.

1.4.1 Fermi Theory of β-decay

Fermi’s theory treated β-decay as a four-fermion contact interaction [10] [Figure1.1].

Figure 1.1: Four point interaction proposed by Fermi’s theory. All particles werethought to interact at the same space-time point.

This description of the electroweak interaction was formulated before the discoveryof parity violation, so it assumed the interactions would be of type vector-vector (asin the case of QED processes). In other words, the interaction contains two currentsof the vector type, ψγµψ [Table 1.1]. Using these properties, the matrix element forthe β-decay was expressed as

Mfi = GFgµν [ψpγµψn][ψeγ

νψν ] (1.57)

where GF is called the Fermi constant, which describes the strength of the electroweakinteraction and ψ represents the particle state [3].

Fermi assumed the interaction to be a four-fermion contact interaction i.e. allparticles interact at the same space-time point [57,58]. This means there is no prop-agator to mediate the interaction of the currents. Though at the quark level, thedecays proceed by the decay of a d-quark into an u-quark, and this decay is mediatedby the emission of a massive W boson, with mW = 80.379 GeV [2]. Figure 1.2 showsthe Feynman diagram for the decay process.

15

Figure 1.2: Feynman diagram for the β-decay of a neutron which takes place bythe electroweak interaction. At the quark level, a d quark converts to an u quarkfollowed by the emission of a W− boson. Apart from a proton, an electron and anelectron-antineutrino are emitted.

The mass of the W is much larger than the decay energy, and thus the W bosondecays nearly instantaneously into an electron and an electron-antineutrino.

For low energy weak interactions, where the momentum transfer13 is much lessthan the mass of the W boson (|q2| m2

W), the propagator can be approximated bya small constant term [3]

igµνm2

W

(1.58)

This means the interaction no longer has any q2 dependence. Hence, with the absenceof any q2 term, the weak charged-current interaction can be approximated in termsof a four-fermion contact interaction.

Fermi’s theory [57], although it worked well in predicting the continuous energyspectrum of the emitted electron, has one shortcoming: it assumed the interaction tobe of vector nature which would not permit the observed β-decay transitions where

13 Momentum transfer (q) is defined as the change in the four-momenta of initial and final states ateach vertex of the Feynmann diagram.

16

the spin of the decay products is different from the decaying nucleus [59]. This ledpeople to look beyond the standard vector interactions, in fact, the Hamiltonian ofFermi’s theory can be generalized by including all of the Dirac bilinears [7, 8, 57, 72]:

H = CS(ψpψn)(ψeψν) + CV (ψpγµψn)(ψeγµψν) + CP (ψpγ

5ψn)(ψeγ5ψν)

+ CA(ψpγµγ5ψn)(ψeγ

µγ5ψν) + CT (ψpσµνψn)(ψeσµνψν) (1.59)

where the coupling coefficients Ci, i = S, V, P,A, T characterize the strength ofeach interaction.

1.4.1.1 Fermi and Gamow-Teller transitions

To find the rate of β-decay, we have to find the transition amplitude of the decayfrom the initial state to the final state. Those allowed transitions with no angularmomentum transfer (∆J = 0) from initial to final states with no parity change arereferred to as Fermi transitions. Scalar and vector currents are associated with Fermitransitions. There are β-decay processes, however, where the allowed transitionstake place with angular momentum transfer of one (∆J = 0,±1, excluding 0+ →0+ transitions) with no parity change. Those transitions are called Gamow-Tellertransitions. These transitions are characterized by axial vector and tensor currents.This confirmed that the original vector current Hamiltonian proposed by Fermi wasnot enough to explain all the observed β-decay processes. The pseudoscalar termwould produce ∆J = 0 with a parity change [9], but the pseudoscalar term does notcontribute to the amplitude in the low energy limit [7, 10].

1.4.2 Parity violation in weak interactions

1.4.2.1 K mesons and τ − θ puzzle

In the late 1940s, the experiments revealed that although the τ+ and θ+ mesonshad the same mass and charge as that of the K+ mesons, they were very differenton the grounds of their intrinsic parity. The τ decayed into three pions and theθ decayed into two pions. This strongly suggested that the τ and θ are differentparticles [8, 11, 12].

In 1956, T.D. Lee and C.N. Yang suggested one way out of this difficulty: violationof parity conservation, so that the τ+ and θ+ are just different decay modes of the K+.All of this presented questions on the validity of the parity conservation in β-decaywhich was assumed to be a parity conserving weak interaction at that time. In fact,a careful analysis of β-decay by Lee and Yang showed that it indeed violates parityconservation [12]. They modified the interaction Hamiltonian by incorporating parity-

17

violating interactions which are shown by the terms with primed coefficients [12,13]:

H = (ψpψn)(CSψeψν + C ′Sψeγ5ψν)

+ (ψpγµψn)(CV ψeγµψν + C ′V ψeγ

µγ5ψν)

+1

2(ψpσµλψn)(CTψeσ

µλψν + C ′Tψeσµλγ5ψν) (1.60)

− (ψpγµγ5ψn)(CAψeγ

µγ5ψν + C ′Aψeγµψν)

+ (ψpγ5ψn)(CPψeγ

5ψν + C ′Pψeψν) + h.c.

1.4.2.2 Wu et al. experiment

Soon after Lee and Yang’s theory of violation of parity conservation, C.S. Wu, E.Ambler, R.W. Hayward, D.D. Hoppes and R.P. Hudson showed the first experimentalproof that parity conservation is indeed violated in the β-decay of atomic nuclei [14].They studied the decay of the oriented cobalt nucleus into nickel with the emission ofan electron and an antineutrino14. This decay is a Gamow-Teller transitions: paritystays the same but spin changes by one unit15. In the experimental setup, the cobaltnuclei were aligned in a strong magnetic field along the ±z direction.

Figure 1.3: Results from Wu et al. experiment [14] showing the observed beta asym-metry from the emitted electrons. The curves show the different counting rates fortwo opposite orientations of the nucleus (magnetization direction). It shows the exis-tence of parity violation since the emission of electrons is more favored in the directionopposite to that of the nuclear spin.

14 (60 ~Co→ 60 ~Ni + e− + νe)15 (JPCo = 5+, JPNi = 4+)

18

Hence, if parity was conserved in the weak interaction, the rate at which the elec-trons were emitted in some given direction should be identical to the rate of emittedelectrons in the opposite direction. Experimentally it was found that electrons had apreferential direction (predominantly opposite to the nuclear spin). This anisotropyin the direction of the emission of electrons was a clear indication of parity violation[see Figure 1.3]. It can be easily seen that during a reflection the direction of electronemission (vector) would be reversed while the direction of the magnetic field andnuclear spin (axial vector) remains the same. This experiment further solidified theclaim of Lee and Yang that the interaction Hamiltonian of the weak interaction cannot just include vector currents.

1.4.3 V − A law of the Weak interactions

The V − A law of weak interactions can be easily deduced if we try to write theprecise form of the interaction term in the Hamiltonian. From Eq. 1.57 we know thatthe leptonic terms in the Hamiltonian are of the form,

ψeOiψν (1.61)

where i = S, P, V,A, T are different Dirac bilinears. Nuclear β-decays are one classof weak interaction processes. Experimentally it was found that in nuclear β-decaysthe emitted electrons have mostly negative helicity and only a very small fraction ofpositive helicity electrons are present16. Besides helicity, we also need to put a moreserious constraint of Lorentz invariance on the interaction Hamiltonian. This willalso be the guiding principle for writing the correct form of the Hamiltonian [8].

1.4.3.1 The CKM Matrix

It has been observed in experiments that different weak interaction processeshave different coupling strengths for weak charged current vertices. For example, incontrast to an expected universal weak coupling for quarks, the measured decay ratefor17 K−(us) → µ−νµ is approximately 20 times smaller as compared to π−(ud) →µ−νµ. These observations were originally explained by Cabibbo [15]. In the Cabibbohypothesis, the weak interaction of the quarks is assumed to be of the same strengthas for the leptons18, but the weak eigenstates of the quarks differ from the masseigenstates of the electromagnetic and strong interactions.

The hadronic vertex of nuclear β-decay involves weak coupling between u and dquarks. Therefore, the matrix element of β-decay is suppressed by a factor of cosθc

19,where θc is known as the Cabibbo angle. This factor relates the weak eigenstates ofthe quarks to their mass eigenstates.

16 Helicity is frame dependent, these observations are with respect to the laboratory frame.17 Quarks are the elementary particles which are the fundamental building blocks of matter. Six

flavors of quarks exist up, down, strange, charm, bottom, and top.18 Elementary particles of half-integer spin (spin 1/2). There exist six types of leptons in 2 flavors

with 3 generations: electron, electron-neutrino, muon, muon-neutrino, tau, and tau-neutrino.19 cos θc is also referred to as Vud.

19

Ten years after the Cabibbo hypothesis, Kobayashi and Maskawa [61] generalizedthis idea by incorporating a third generation of quarks20. They described the weakinteractions of all the three generations of quarks in terms of a unitary matrix fa-mously known as the Cabibbo-Kobayashi-Maskawa (CKM) matrix. As an extensionto the Cabibbo hypothesis, it also relates the weak eigenstates of the quarks to theirmass eigenstates. d′s′

b′

=

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

dsb

(1.62)

The CKM matrix can be parameterized by three real mixing angles and one imaginaryphase [16]. This imaginary phase is responsible for CP-violating phenomena in flavorchanging processes in the Standard model.d′s′

b′

=

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23s13

dsb

(1.63)

where cij(sij) stand for cos θij(sin θij) (i, j = 1, 2, 3), angles θij relate the couplingsbetween each quark generation and δ refers to the complex phase. The relativestrength of the interaction is defined by the relevant element of the CKM matrix. Forexample, for β-decay, the weak charged current associated with the hadronic part inthe Hamiltonian can be modified as

igW

2√

2Vudψuγ

µ(CV − CAγ5)ψd (1.64)

In the Standard Model, the CKM matrix is unitary, V †V = I, which means that thefirst row of the matrix is bounded by the constraint

|Vud|2 + |Vus|2 + |Vub|2 = 1 (1.65)

Testing the unitarity of the CKM matrix is one of the probes to check the validity ofthe standard model [17].

1.4.3.2 Conserved Vector Current Hypothesis

Just like in the case of QED processes, the vector part of the weak interaction isassumed to be constant as was postulated by Feynman and Gell-Mann in 1958 [18].Up to O((md −mu)

2), the weak vector coupling constant (gV ) exhibits no deviationfrom unity as shown via the Ademollo-Gatto Theorem [19]. This conservation of thevector current in the weak interaction is known as the Conserved Vector Current(CVC) hypothesis. Hence, the value of gV is taken to be unity (gV = 1). For theaxial vector current, however, a similar conservation rule can not be assumed and thevalue of the axial vector coupling constant (gA) is estimated either experimentally orby numerical methods like lattice QCD.

20 charm, bottom, and top quarks had not yet been discovered when Cabibbo came up with histheory.

20

1.4.4 Weak Interaction processes

1.4.4.1 Neutron β decay

A free neutron is unstable. As mentioned in section 1.4, it decays into a proton,an electron, and an electron-antineutrino with an energy release of about 782 keV.As seen before, its decay is mediated by electroweak interactions which, to a goodassumption, are exclusively vector and axial vector in nature. Like mentioned pre-viously, the vector component of the weak interactions is conserved under the CVChypothesis. However, the axial vector component is not conserved which implies theaxial vector coupling constant CA in the weak hadronic current of Eq. 1.60 couldnot be accurately extracted from the β-decay of the complex nuclei. Only in the caseof a free neutron β-decay where the uncertainties due to the complex nuclear wavefunctions are absent, one can expect to evaluate it to high precision [20,21].

Various physical observables related to free neutron decay provide a testing groundfor the validity of the standard model (SM) and for the searches of beyond standardmodel physics (BSM). For instance, there is a discrepancy in the two approaches ofmeasuring neutron lifetime. The current world average reported by PDG [2] for theneutron lifetime is

τn = 879.4± 0.6 s.

This value differs significantly from the neutron lifetime value of τn = 887.7±1.2±1.9sreported by the “beam method” [22]. A precise measurement of neutron lifetime ispivotal in the understanding of the quark-level dynamics associated with β-decay. Itwill help in the determination of |Vud| and serve as a test for the unitarity of the CKMmatrix.

On the other hand, precise measurements of various correlation coefficients suchas electron asymmetry (A) and electron neutrino correlation (a) will help determinethe value of the axial vector coupling constant since they are directly related to CAand CV . These correlations are discussed in great detail in Chapter 3. A search forright-handed currents as well as for scalar and tensor currents in the weak interactionhave also been proposed.

1.4.4.2 Pion Decay

Charged pions (π±) are meson systems composed of ud and du states. They havea mass m(π±) ∼ 140 MeV and are the lightest mesons. Due to this, they can notdecay with strong interactions instead they decay with weak interactions21. Thereare three main decay modes of the π− which are mediated by the weak interaction:π− → e−νe, π

− → µ−νµ and π− → µ−νµγ.For the decay to electrons, the emitted electrons are highly-relativistic which

means for this two-body decay system, the decay rate would highly favor pion decay

21 π0 decays electromagnetically to conserve the lepton number.

21

to electrons22. However, it is observed in the experiments that charged pions decayalmost exclusively by π− → µ−νµ with a branching ratio of 99.988%. The decay rateto electrons is almost a factor of 10−4 smaller than the decay rate to muons.

This very small branching ratio for the decay to electrons is the manifestation ofthe weak interaction. The pion has zero spin, JP = 0−, therefore the two leptonswhich are emitted in opposite directions must have the same helicity states in therest frame of the pion. This would ensure the conservation of the angular momentumof the system. It should also be noted that the weak interactions only couple to left-handed chirality states of particles and right-handed chirality states of anti-particles.Since neutrinos are almost massless, their chirality states are equivalent to the he-licity states. Hence, the emitted electron-antineutrinos have positive helicity stateswhich also implies that the emitted electrons also have positive helicity owing to theconservation of the angular momentum. Because of the finite mass of the electrons,the helicity and chirality states for the emitted electrons are not equivalent. Hence,although the weak decay to the right-handed helicity states can occur, it may behighly suppressed [3].

After calculating the decay rates for both of the decay modes, we have

Γ(π− → e−νe)

Γ(π− → µ−νµ)=

[me(m

2π −m2

e)

mµ(m2π −m2

µ)

]2(1.66)

Since mµ/me ≈ 200, the decay mode π− → e−νe will be strongly suppressed ascompared to π− → µ−νµ. This explains the high branching ratio of the decay modeinvolving muons.

1.5 Survey of Special Theory of Relativity

The rest of this chapter is dedicated to the relativistic treatment of the electromag-netic fields and the way they transform when successive non-collinear Lorentz trans-formations are performed. Apart from these transformations being non-collinear, theorder in which these are applied is also important which is owed from the fact thatsuccessive Lorentz transformations do not commute with each other [24]:

[Si, Sj] = εijkSk

[Si, Kj] = εijkKk (1.67)

[Ki, Kj] = −εijkSk

where matrices Si, Ki are the generators of the Lorentz group (i = 1, 2, 3). Thematrices Si produce rotations in three dimensions, and the matrices Ki generateboosts.

22 For a general two-body decay a→ 1 + 2, the decay rate is given by

Γ =p

32π2m2a

∫|Mfi|2dΩ

where p is the momentum of daughter particles in the center-of-mass frame, M is the invariantmatrix element integrated over whole space [3].

22

It is evident from the above commutation relations that two successive Lorentztransformations are equivalent to a single Lorentz transformation with a three-dimensionalrotation. This three-dimensional rotation is particularly interesting to us. It arisesfrom a purely kinematical phenomenon known as Thomas precession. Its origin liesin the fact that whenever a particle undergoes an accelerated motion with at leastone component of acceleration perpendicular to the direction of motion, the particlewill experience Thomas precession [23, 24]. This phenomenon is important to con-sider, for instance, it affects the interactions of relativistic spin-1/2 particles withelectromagnetic fields. This is discussed in more detail in Chapter 5.

1.5.1 Lorentz Transformations

Consider two inertial reference frames Σ and Σ′ with a relative velocity ~v betweenthem. They are oriented in a way that the coordinate axes of Σ are parallel to Σ′

and Σ′ is moving in a direction which we assumed as the positive x as seen from Σ.The position 4-vector of Σ′ is related to the position 4-vector of Σ by the standardLorentz transformation equations

x′0 = γ(x0 − βx1)x′1 = γ(x1 − βx0)x′2 = x2 (1.68)

x′3 = x3

where

x0 = ct, x1 = x, x2 = y, x3 = z;

~β =~v

c, β = |~β|;

γ = (1− β2)−1/2 (Lorentz factor)

The generalization of Eq. 1.68 for the relative velocity of Σ′ in an arbitrary directionbut with the coordinate axes of the two frames still parallel to each other is given by:

x′0 = γ(x0 − ~β · ~x)

~x ′ = ~x+(γ − 1)

β2

(~β · ~x

)~β − γ~βx0 (1.69)

1.5.2 Addition of Velocities

Consider two inertial reference frames Σ and Σ′ such that the relative velocityof Σ′ with respect to Σ is ~v. A particle is moving in Σ′ such that its velocity withrespect to Σ′ is ~u′. The velocity of the particle with respect to Σ is then given by [24]

u‖ =u′‖ + v

1 + ~v·~u′c2

~u⊥ =~u′⊥

γv(1 + ~v·~u′c2

)(1.70)

23

where u‖ and ~u⊥ refer to the components of velocity parallel and perpendicular,respectively, to ~v.

It can be shown that the Lorentz factors of ~v, ~u, and ~u ′ are related to each otherby

γu = γvγu′

(1 +

~v · ~u ′

c2

)(1.71)

More generally, the velocity composition law for two arbitrary velocities can be writ-ten as [25–32]

~u⊕ ~v =~u+ ~v

1 + ~u·~vc2

+1

c2

(γu

γu + 1

)~u× (~u× ~v)

1 + ~u·~vc2

(1.72)

with

γu⊕v = γuγv

(1 +

~u · ~vc2

)(1.73)

where symbol ⊕ refers to the direct sum of the vector space of the velocity vectors.

1.5.3 Matrix Representation of the Electromagnetic fields and Boost Ma-trix

For the rest of the discussion, we will be using matrix methods to calculate Lorentztransformations as they are very convenient to use and are more explicit. All of theequations in Eq. 1.68 and 1.69 can easily be obtained by using the boost matricesfor Lorentz transformations. For example, for a boost along the x -axis, the boostmatrix [24] can be written as:

A =

γ −γβ 0 0−γβ γ 0 0

0 0 1 00 0 0 1

(1.74)

Hence, ct′

x′

y′

z′

=

γ −γβ 0 0−γβ γ 0 0

0 0 1 00 0 0 1

ctxyz

=

γ(ct− βx)γ(x− βct)

yz

(1.75)

For an arbitrary boost, the general form of the boost matrix A takes a form in whichthe matrix elements can be written as:

A00 = γ

A0i = Ai0 = −γβi (1.76)

Aij = Aji = δij + (γ − 1)βiβjβ2

24

where δij is the Kronecker delta. The general form of Eq. 1.74 can therefore bewritten as [24]:

A =

γ −γβx −γβy −γβz−γβx 1 + (γ − 1)β

2x

β2 (γ − 1)βxβyβ2 (γ − 1)βxβz

β2

−γβy (γ − 1)βyβxβ2 1 + (γ − 1)

β2y

β2 (γ − 1)βyβzβ2

−γβz (γ − 1)βzβxβ2 (γ − 1)βzβy

β2 1 + (γ − 1)β2z

β2

(1.77)

1.5.4 Set-up

To start with, consider two arbitrary boosts ~β and δ~β in three dimensions asshown in Figure 1.4.

~β = βxx+ βyy + βz z

δ~β = δβxx+ δβyy + δβz z (1.78)

In order to calculate the boost matrix for various boosts, we will apply a passivetransformation which will rotate our lab frame (xy) coordinate axes in such a way

that its x-axis is aligned with ~β. This rotated frame will hereafter be called thelongitudinal-transverse (`t) frame.

This whole transformation can be imagined as a product of two rotations: Thefirst rotation is about the z -axis by an angle φ which will align the x -axis along theprojection of the boost vector in the xy plane [see Figure 1.5]. The rotation matrixassociated with this rotation can be written as:

R1 =

1 0 0 00 cosφ sinφ 00 − sinφ cosφ 00 0 0 1

(1.79)

The second rotation is about the y1 axis [see Figure 1.6] by an angle π2−θ. The effect

of this rotation is that it aligns the x1 axis to the boost vector ~β. For the secondrotation, the rotation matrix is shown in Eq. 1.80.

R2 =

1 0 0 00 sin θ 0 cos θ0 0 1 00 − cos θ 0 sin θ

(1.80)

The overall effect of the two rotations can be combined in a single transformationmatrix R:

R = R2 ·R1 =

1 0 0 00 sin θ cosφ sin θ sinφ cos θ0 − sinφ cosφ 00 − cos θ cosφ − cos θ sinφ sin θ

(1.81)

25

Figure 1.4: General boost in three-dimensions. The dotted line represents the pro-jection of ~β on the xy-plane.

It is clear from Fig.1.4 that if:

~β = βxx+ βyy + βz z (1.82)

then

cos θ =βzλ1

; sin θ =η1λ1

; cosφ =βxη1

; sinφ =βyη1

(1.83)

where the parameters λ1 and η1 are defined in Chapter 5, Appendix section 5.4.1.Using Eq. 1.83, matrix R can be written as:

R =

1 0 0 0

0 βxλ1

βyλ1

βzλ1

0 −βyη1

βxη1

0

0 −βxβzη1λ1

−βyβzη1λ1

η1λ1

(1.84)

26

Figure 1.5: Rotation about z axis by an angle φ. The new x, y and z axes are calledthe x1, y1 and z1 axes, respectively.

As mentioned earlier, this rotation matrix will transform the lab frame coordinatesof any 4-vector (xy-frame) to its coordinates in the rotating frame also called thelongitudinal-transverse frame (`t-frame):

~β`t = R · ~βxy0β`txβ`tyβ`tz

= R ·

0βxyxβxyyβxyz

=

0λ100

(1.85)

where the superscripts `t and xy refer to the components in the longitudinal-transverseframe and laboratory frame, respectively, and for the sake of simplicity in notationwe assumed:

βxyx = βx, βxyy = βy, βxyz = βz (1.86)

27

Figure 1.6: Second rotation about the y1 axis by an angle π2− θ. x, y, and z axes in

this new frame are called the x2, y2 and z2 axes, respectively.

Similarly, the infinitesimal boost in the longitudinal-transverse frame is of the form:

δ~β`t = R · δ~βxy0δβ`txδβ`tyδβ`tz

= R ·

0

δβxyxδβxyyδβxyz

=

0λ2λ1λ6η1λ5η1λ1

(1.87)

where λ1, λ2, λ5, λ6 and η1 are defined in Chapter 5, Appendix section 5.4.1.To calculate the γ(~β+δ~β) in the `t-frame, we have:

(~β + δ~β)`t =

(λ1 +

λ2λ1

)x+

λ6η1y +

λ5η1λ1

z (1.88)

Keeping the terms linear in δβ, we get

|(~β + δ~β)`t|2 ≈ λ21 + 2λ2 (1.89)

28

Using the above equation we calculate:

γ(~β+δ~β) =(1− λ21 − 2λ2

)− 12

≈(1− λ21

)− 12

[1 +

λ21− λ21

](1.90)

Hence, γ(~β+δ~β) can be written as:

γ(~β+δ~β) ≈ γ(1 + γ2λ2) (1.91)

where γ = 1/√

1− λ21 is the Lorentz factor. Using Eq. 1.85, the boost matrix for

boost ~β`t can be calculated:

A(~β)`t =

γ −γλ1 0 0−γλ1 γ 0 0

0 0 1 00 0 0 1

(1.92)

Similarly, using Eq. 1.85 and 1.87, to first order in δβ, the boost matrix for the directboost ~β + δ~β in the `t-frame can be written as:

A(~β + δ~β)`t =

γ + γ3λ2 −γ(λ21+γ

2λ2)

λ1−γλ6

η1− γλ5η1λ1

−γ(λ21+γ2λ2)

λ1γ + γ3λ2

(γ−1)λ6η1λ1

(γ−1)λ5η1λ21

−γλ6η1

(γ−1)λ6η1λ1

1 0

− γλ5η1λ1

(γ−1)λ5η1λ21

0 1

(1.93)

29

Chapter 2 Survey of Statistics

2.1 Probability Distributions

Statistical distributions are just mathematical functions which give us the infor-mation about the possible values a random variable1 can take and how often theyoccur. Think of rolling of a die, each roll can take any integer value between 1 and6. If we roll the die several times, we can create a table of the number of times aparticular value has appeared. The table we obtain at the end would give us thedistribution of values.

In this case, the die would take a random discrete value on each roll, hence thesedistributions are called discrete distributions. Probability distributions can also becontinuous meaning the outcome will be a range of values following some continuousreal-valued function. An example of a continuous distribution is the spectrum ofenergy of the emitted electrons from the β decay of neutrons. The energy of theemitted electrons can take any value between 0 and about 781.5 keV. This idea ofdistributions can be further extended to certain mathematical functions which give usthe likelihood of the occurrence of the possible outcomes from an experiment. Thesefunctions are then called probability distributions.

Of the many probability distributions occurring in the analysis of particle and nu-clear physics experiments, three deserve special attention: the binomial distribution,the Poisson distribution and the Gaussian distribution [33].

2.1.1 Binomial Distribution

The binomial distribution is generally applied to the experiments where the totalnumber of observations n are fixed and are independent of each other. The outcomesfrom each observation are mutually exclusive and can be represented as either “suc-cess” or “failure” [33]. For each outcome, the probability p of the success is the sameand probability of failure q and success p are related to each other as

p = 1− q (2.1)

Suppose we toss a fair coin in the air. We know that it has 50% chance of landingheads up (H) and 50% chance that it will land tails up (T). Now suppose we toss twocoins at a time but now our sample space will consist of four different observations:HH, HT, TH, TT. This makes the probability for any particular permutation to be1/4. We can easily generalize these ideas for any number of observations by writing aprobability density function PB(k;n, p) which is defined as the probability of gettingk successes out of n observations with the success probability p. This probabilitydensity function (p.d.f.) is said to follow the binomial distribution and is given by

PB(k;n, p) =

(n

k

)pkqn−k =

n!

k!(n− k)!pk(1− p)n−k (2.2)

1 A statistical variable whose values are the outcomes of a random phenomenon.

30

The binomial distribution itself is not very interesting in nuclear and particle physicsbut we will see that both the Poisson distribution and Gaussian distribution are justthe limiting cases of the binomial distribution.

The mean and the standard deviation for the binomial distribution can also becalculated easily. Noting that for any discrete distribution the mean and the variance2

have the form

µ =n∑x=0

xP (x)

σ2 =n∑x=0

(x2P (x)− µ2) (2.3)

where P (x) is the probability distribution function. Using the above equations, themean and the standard deviation for the binomial distribution can be expressed as

µ = np

σ =√np(1− p) (2.4)

2.1.2 Poisson Distribution

Suppose we are performing an experiment in which we are given a radioactivesample and we want to calculate the number k of the nuclei which have decayed insome given time interval (say 15 minutes). If we continue performing this experimentfor many trials, we will almost certainly observe different values of k for differenttrials. Each nucleus decays at a random time, and in any random 15-minute interval,the number of decays may be different from the expected average number of decays inthat interval. This behavior is attributed to the intrinsic random nature of radioactivedecays.

Since the outcome of each event is mutually exclusive and independent, we can,of course, form a binomial distribution for the decay of the nucleus in the 15-minuteinterval: if there are n nuclei and the probability of decay of any nuclei is p then theprobability of the decay of k nuclei would be given by PB(k;n, p), Eq. 2.2. How-ever, there is one subtlety here: the number of nuclei (number of observations) areenormous and hence the probability of the decay for any one nucleus is exceptionallysmall! Under these limiting conditions, a binomial distribution probability functionasymptotically approaches to a Poisson distribution whose probability density func-tion, PP(k;λ), is given by

PP(k;λ) =λk

k!e−λ (2.5)

where the parameter λ is just the expected average number of decays [33–35]. We caneasily show that Eq. 2.3 actually follows from Eq. 2.2 after the limiting conditionshave been imposed. Using Eq. 2.3, the mean and the standard deviation for the

2 Standard deviation is defined as the square root of the variance.

31

Poisson distribution is given by

µ = λ

σ =√λ (2.6)

Most of the physical processes in nuclear physics follow Poisson distribution becausethey exactly replicate the limiting conditions mentioned above. A primary focus ofthis dissertation is the decay of a free neutron to a proton, an electron and an electron-antineutrino. This decay also follows Poisson statistics and we will see the role playedby it in the estimation of the uncertainties of the goodness-of-fit parameter later.

2.1.3 Gaussian Distribution

The Gaussian or Normal distribution is another special case of the binomialdistribution when the number of possible observations n becomes infinitely large andthe probability for each observation becomes large enough such that np 1 [33–35].It is also the limiting case of the Poisson distribution when λ becomes large. One ofthe characteristics of the Gaussian distribution is that the mean of the observationsfrom the sampling distribution gives the most probable estimate of the populationmean µ.

The probability density function for the Gaussian distribution is given by

PG(x, µ, σ) =1

σ√

2πexp

[−1

2

(x− µσ

)2]

(2.7)

The probability density function describes the probability of obtaining a value x in arandom observation from a distribution with mean µ and standard deviation σ.

The Gaussian distribution is probably the most important distribution in Statis-tics. For example, it is the basis of finding the significance levels in hypothesis testingand finding confidence intervals. Many frameworks like maximum likelihood are basedon the assumption that the underlying data follows a normal distribution.

2.2 Maximum Likelihood

Suppose we have a collection of N independent variables xi, where i = 1, 2, · · ·Nwhich are described by a probability density function P (xi;a) where a = a1, a2, · · · ,am is a set of m parameters whose values are unknown3. So the likelihood functionis a function of the unknown parameters and is given by the product of probabilitydensities evaluated at each of the N variables4

L(a) =N∏i=1

P (xi;a) (2.8)

3 These unknown parameters are often called the fit parameters.4 It should be noted that because the probability of observing any particular event is less than 1,

the product of all the probabilities will be very small. This might make the likelihood functionnumerically unstable. To get around this issue, it is preferred to use the logarithm of the likelihoodfunction and maximize it with respect to the fit parameters.

32

The maximum likelihood framework assumes those values of the parameters a1, a2, · · · ,am which maximize the likelihood function [33,35].

To make things a little less abstract, consider an example of a linear relationshipbetween two variables x and y such that the actual relationship is assumed to be:

y(x) = a01x+ a02 (2.9)

where a01 and a02 are the theoretical values of the fit parameters. The task of themaximum likelihood framework is to find the closest estimates a1 and a2 to thesetheoretical values which agree with them up to a given precision.

Assuming that the total number of measurements5 performed are N and that foreach independent variable xi, (i = 1, 2, · · · , N), the values of the measured parameteryexp(xi) follows a Gaussian distribution with standard deviation σi about the theoret-ical value ytheo(xi). So the probability P (xi) of obtaining the measured observationyexp(xi) is given by Eq. 2.7:

P (xi) =1

σi√

2πexp

[−1

2

(yexp(xi)− ytheo(xi)

σi

)2]

(2.10)

where ytheo(xi) is given by Eq. 2.9. The likelihood function for this probability densityfunction can be easily written using Eq. 2.8:

L(a1, a2) =N∏i=1

P (xi) =∏i

(1

σi√

2π

)exp

[−1

2

∑i

(yexp(xi)− ytheo(xi)

σi

)2]

(2.11)

Thus, the estimated values of these fit parameters a1 and a2 would maximize thelikelihood function. The first term in Eq. 2.11 is independent of the fit parameters,hence, maximizing the likelihood function is equivalent to minimizing the sum in theexponential. This sum is often called as goodness-of-fit parameter χ2:

χ2 =∑i

(yexp(xi)− ytheo(xi)

σi

)2

(2.12)

Hence, to find the best fitted values of parameters a01 and a02, one needs to minimizethe sum of the squares of the deviations weighted by 1/σ2

i .

2.2.1 Maximum likelihood analysis of neutron β-decay

In the context of the current work, the above N independent variables correspondto N energy bins discretizing the energy of the emitted electron. This allows us to an-alyze various energy dependent observables in the neutron β-decay such as the energyspectrum of the emitted electrons and the correlation coefficients. For convenience,the notation of the Rfit framework mentioned in [36, 64] will be used from now onwhich implies all the experimental measurements of the independent variables are

5 These measurements can be obtained from actual experiments or from simulations.

33

labelled as xexp,i. These experimental observables consist of binned-in-energy mea-surements of the differential decay distribution Γexp,i(Ee,j), the spin electron asymme-try correlation Aexp,i(Ee,j), the electron-antineutrino correlation aexp,i(Ee,j) and theneutron lifetime τexp,i (where the subscripts i and j refer to the particular experimentand the energy bin, respectively) [36, 37,64].

xexp = Γexp,1(Ee,1),Γexp,1(Ee,2), . . . , Aexp,1(Ee,1), Aexp,1(Ee,2), . . . ,

aexp,1(Ee,1), aexp,1(Ee,2), . . . , τexp,1, τexp,2, . . . (2.13)

Each of these experimental measurements are then compared with the correspondingtheoretical calculations xtheo,i. Each of these theoretical calculations are a functionof a set of Nmod parameters, called model parameters, a set of which is denotedby ymod. These Nmod model parameters can be further divided into two sets:Nfree, experimentally accessible free parameters of the model (like b, λ, Vud, f2, etc.)and Ncalc = Nmod − Nfree, the calculated parameters for which there have been noprior experimental measurements thereof, and which are not accessible in the currentexperiments (such as the second-class currents, f3 and g2).

xtheo(ymod) = Γtheo,1(Ee,1),Γtheo,1(Ee,2), . . . , Atheo,1(Ee,1), Atheo,1(Ee,2), . . . ,

atheo,1(Ee,1), atheo,1(Ee,2), . . . , τtheo,1, τtheo,2, . . . (2.14)

with the model parameters including the parameters defined in both the standardmodel (SM) and beyond the standard model (BSM)

ymod = λ, b, f2, f3, g2, Vud (2.15)

Assuming a sufficiently large number of beta-decay events in each energy bin, thedistribution may be considered as a Gaussian. With this assumption, it is reasonableto define the χ2 function in terms of the likelihood function for the ymod parameters.Using Eq. 2.11,

χ2(ymod) = −2 ln L(ymod) (2.16)

As discussed in [36, 64], the likelihood function can be further simplified by writingit as a product of an “experimental likelihood” and a “theoretical likelihood”. Theexperimental likelihood, Lexp is a function of both experimental, xexp, as well astheoretical parameters, xtheo, whereas the theoretical likelihood, Lcalc depends on thefree parameters ycalc

L(ymod) = Lexp(xexp, xtheo(ymod))Lcalc(ycalc) (2.17)

The experimental likelihood function is just like the usual likelihood function thathas been defined in Eq. 2.8 as the product of the probability densities evaluated ateach of the N independent variables xi. The individual probability densities maybe considered to follow Gaussian distributions because of the assumption of a largenumber of events per energy bin.

Although the theoretical likelihood function can also be defined as the productof the individual probability densities but the lack of knowledge about the ycalc

34

parameters prohibits the assumption of a Gaussian distribution for such parameters.Hence, these parameters can not directly contribute to χ2. One of the approaches ofanalyzing these parameters is the proposal of the Rfit technique [36, 64] and it willbe discussed in the next section.

2.3 The Rfit Technique

In this novel approach due to [36, 64], the theoretical likelihood Lcalc does notcontribute to the χ2 of the fit, while the ycalc parameters are permitted to varyfreely within some “allowed” ranges defined by the theoretical systematics evaluatedfor each of the ycalc,i. The restriction on the ycalc that they are bound to remainwithin their predefined allowed ranges is the most important assumption made in theRfit technique but all the values of these parameters in their predefined ranges aretreated on the same footing. However, this simple assumption is, indeed, a strongassumption in the sense that

• the output results obtained can be treated as valid only if these allowed rangesdo contain the true values of ycalc,i, and

• if the chosen predefined range is too big, then one may totally overlook a newdiscovery!

Using the proposals of the Rfit technique, the χ2 can be redefined as

χ2 =N∑i=1

(xexp,i − xtheo,i

σexp,i

)2

− 2 ln Lcalc(ycalc) (2.18)

where

−2 ln Lcalc(ycalc) ≡

0, ∀ ycalc,i ∈ [ycalc,i ± δycalc,i],∞, otherwise

Once the χ2 is set-up, two different types of analyses can be carried out:

(i) Estimating SM parameters, and

(ii) Probing the SM

2.3.1 Estimating SM parameters

In the first case, the SM is assumed to be valid and for metrological purposes,one can estimate the ymod parameters using some global fits on various observables.This can be done by allowing all the Nmod parameters to vary freely and finding acombination of them which yields the minimum value of χ2(ymod). The minimumvalue is denoted by χ2(ymod)min. To get the best possible estimates of the ymod

parameters, one can then calculate

∆χ2(ymod) = χ2(ymod)− χ2(ymod)min (2.19)

35

where χ2(ymod) is the χ2 value for a given set of model parameters ymod. Followingthe assumption that ymod are Gaussian distributed, the confidence levels P(ymod) canthen be easily obtained using:

P(ymod) = Prob(∆χ2(ymod), Ndof) (2.20)

where Prob(· · · ) denotes the probability of obtaining χ2 > ∆χ2(ymod) for Ndof degreesof freedom [36,37,64].

However, it may not be practically feasible to determine values for all of theymod parameters. In those cases, we take a Na (Na ≤ Nmod) parameters subsetya of the model parameters for which the confidence levels need to be estimated.For instance, in the context of the current work, ya = λ, b. The complementaryset of Nµ = Nmod −Na parameters is denoted by yµ.

Hence, the recipe for computing the confidence levels on the set of relevant param-eters is as follows: at each point in the parameter space formed by the parameters yafind the minimal value of the χ2 function by allowing the parameters yµ to freelyvary. This minimal value is denoted by χ2(ya; yµ)min. Hence, the offset-correctedχ2 can be calculated as

∆χ2(ya) = χ2(ya; yµ)min − χ2(ymod)min (2.21)

Using the above equation the confidence levels can then be obtained

P(ymod) = Prob(∆χ2(ya), Ndof) (2.22)

2.3.2 Probing the SM

So far it has been assumed that the SM is valid and hence, the minimizationscheme is used to estimate a subset or a complete set of ymod parameters. Thesame minimization procedure can, in principle, be used to test the validity of theSM according to the following scheme: using Monte Carlo simulations, values of allthe xexp variables are sampled using their corresponding theoretical expressionsxtheo and by allowing all of the ymod parameters to vary freely. After xexp areobtained, we can use the same prescription and scan the parameter space to find theminimum value of χ2 which will give us the “best” values of the ya parameters. Thisminimum value denoted by χ2(ymod)min could be further used to define a probabilityon the validity of SM [36,37,64]:

P(SM) ≤ Prob(χ2(ymod)min, Ndof) (2.23)

36

Chapter 3 Neutron β-decay

3.1 Overview

As mentioned in Chapter 1, neutron β-decay is the process in which a neutron istransformed into a proton, electron and electron anti-neutrino. The Hamiltonian ofthis process is described as a left-handed purely V −A interaction. Since this processviolates parity conservation, analyzing the rate at which this decay occurs and theangular correlation of various decay products plays a crucial role in the understandingof the standard model (SM). However, there are a variety of well-motivated novelapproaches which probe beyond standard model physics (BSM) and might falsify theV − A law for low energy weak interactions. These include, but are not limited to,finding right-handed currents and the possibility of the presence of scalar and tensorinteractions [13,17,38].

A starting point to study the β-decay process is to analyze the transition rate1

from the initial state to the final state of the decay products. Fermi’s golden ruleallows us to do that

W =2π

~|M|2ρ (3.1)

whereM is the matrix element and ρ is the density of states. The matrix element ofthe four point vector-vector interaction Eq. 1.57 given by Fermi’s theory can not befurther used because of the emergence of the V −A law as a consequence of violationof parity conservation.

3.1.1 The Matrix Element

The matrix element of polarized β-decay can be written as the product of hadronicand leptonic currents [21,39,41]

M =GFVud√

2J µLµ (3.2)

where Lµ is the usual weak interaction leptonic current ψeγν(1 − γ5)ψν and the

hadronic current J µ has six terms

J µ = ψp

[f1(q

2)γµ − if2(q2)

Mσµνqν +

f3(q2)

Mqµ + g1(q

2)γµγ5

−ig2(q2)

Mσµνγ5qν +

g3(q2)

Mγ5qµ

]ψn (3.3)

where σµν = i2[γµ, γν ], q = pn − pp is the momentum transfer and M is the neutron

mass.

1W = τ−1, transition rate is the inverse of the lifetime.

37

In neutron β-decay, the q2-dependent terms only appear in the higher orders ofrecoil expansion and hence they are not very significant. Also note that f1(0) = gV ,g1(0) = gA and these form-factors at zero momentum transfer appear in the leadingorder.

• gA and gV are weak axial vector and vector couplings2 and they are similar toCA and CV from Eq. 1.60 and 1.64, CA/CV = gA/gV = 1.2732. This ratio ofcoupling constants3 is known as λ

• f2(0) = f2 is the weak magnetism coupling constant [62]

• f3(0) = f3 is the induced scalar coupling constant

• g2(0) = g2 is the induced tensor coupling constant

• g3(0) = g3 is the induced psuedoscalar coupling constant

The weak magnetism coupling constant f2, under CVC it is given by (κp − κn)/2with κp(n) being the anomalous magnetic moment of the proton (neutron). The in-duced pseudoscalar term g3 is predicted to contribute to the electron energy spectrumat the order ε/(Rx) ≈ 10−4 [42, 63][Eq. 3.11]. These couplings, along with f1 andg1 are referred to as first-class currents. The remaining form factors f3 and g2 areknown as second-class currents. They are expected to be small and contribute to theelectron energy spectrum at order ε/(Rx) ≈ 10−4 [42, 63]. They are special becausethey violate G parity which is defined as

G = CeiπI2 (3.4)

It is a transformation defined by the rotation about the 2-axis of isospin (I) spaceby 180 followed by the charge conjugation (C). If G parity4 is conserved in weakinteractions then these second class currents would vanish [37,40].

Although the second-class currents f3 and g2 have not been probed experimen-tally probed in neutron β-decay, it is important to discuss the bounds on them inorder to understand the theoretical uncertainties posed by them [37]. Assuming theCVC hypothesis to be valid (i.e., gV = 1), the value of Vud can be extracted from themeasured ft values in superallowed 0+ → 0+ nuclear β-decay. This confirmed theuniversality of gV in these decays to a precision level of 1.3× 10−4 at 68% confidencelevel (CL) revealing a constraint of mef3/MgV = −(0.0011± 0.0013) [67] which im-plies f3 is constrained only to O(1). Decay correlation measurements coalesce tests ofthe CVC predictions of the weak magnetism form factor f2 with those which wouldlimit second-class currents [62]. There has not yet been a published measurement of

2 gV = 1 under CVC hypothesis.3 For all of the analysis in this dissertation, the values of parameters such as λ, M , M ′, me andVud are taken from PDG 2019 [37, 55] while the current PDG averages [2] are used only forpresentational purposes.

4 Those form factors which transform as GV µG−1 = V µ for vector currents and GAµG−1 = −Aµfor axial vector currents are classified as first-class currents and those which are transformedopposite are second-class currents.

38

f2 in neutron β-decay that could be estimated from the linear energy dependence ofthe angular correlation coefficients a, or A if g2 = 0 is assumed. However, the weakmagnetism factor can be ignored if we test the second-class currents by the compari-son of ft values in mirror transitions but an isospin-breaking additive correction fromthe axial form factor can arise. These experiments put the strongest empirical con-straints on second-class currents [73,74] although additional theoretical uncertaintiescan enter. A survey of nuclear β-decay data gives the limit |g2/f2| < 0.1 at 90% CL,yielding |g2| < 0.2 at 90% CL [75].

Some theoretical studies of g2 in context to the strangeness-changing transitionsexist. A bag model estimate gives g2/gV ∼ 0.3 [76] in |∆S| = 1 semileptonic transi-tions. More recently, nonzero second-class currents have been observed in quenchedlattice QCD calculations of form factors which appear in the hyperon semileptonic de-cay Ξ0 → Σ+`ν, yielding f3/gV = 0.14(9) and g2/gA = 0.68(18) [77]. Turning to thenucleon sector, we expect these estimates to be suppressed, crudely, by md/ms ∼ 0.1.This makes them nearly compatible in scale with the value for g2 determined usingQCD sum rule techniques, g2/gA = −0.0152± 0.0053 [78].

3.2 Differential Decay Rate

In the low-energy limit, the differential decay rate for neutron β-decay takes theform [13]

d3Γ

dEedΩedΩν

=1

2

F (±Z,Ee)(2π)5

peEe(E0 − Ee)2ξ ×[1 + b

me

Ee+ a

~pe · ~pνEeEν

+〈~σn〉 ·(A~peEe

+B~pνEν

+D~pe × ~pνEeEν

)](3.5)

where Ee(Eν) and ~pe(~pν) are respectively, the electron’s (antineutrino’s) total energyand momentum. a, A, B, and D are the angular correlation coefficients. b is calledthe Fierz interference term and under the standard model it vanishes. 〈~σn〉 is theneutron polarization and E0 is the electron endpoint energy, which is given by

E0 = M −M ′ − (M −M ′)2 −m2e

2M(3.6)

M being the neutron mass M = 939.5654133 MeV, M ′ = 938.2720813 MeV is themass of the proton and me = 0.5109989461 MeV [2] is the mass of the electron [seeFigure 3.1]. F (±Z,Ee) is the Fermi function which represents a distortion of theemitted electron wave by the Coulomb field of the residual point nucleus [21, 57, 72].This distortion manifests itself as a shape correction to the spectrum.

39

Figure 3.1: Kinetic energy distribution spectrum for the emitted electron in β-decay.

The parameter ξ is a function of the coupling constants of the weak HamiltonianEq. 1.60 as well as the amplitudes of the Fermi and Gamow-Teller transitions. Itdescribes the relative strengths of these transitions5 [13].

ξ = |MF|2(|CS|2 + |CV |2 + |C ′S|2 + |C ′V |2

)+|MGT|2

(|CA|2 + |CT |2 + |C ′A|2 + |C ′T |2

)(3.7)

Scalar and tensor interactions are particularly interesting, as probing for theirexistence at current experimental limits would indicate the presence of the physicsbeyond standard model. As a result, the hadronic matrix element in Eq. 3.3 can befurther generalized by adding the scalar and tensor currents which are

〈p(p′)|ψpψn |n(p)〉 = ψpgS(q2)ψn

〈p(p′)|ψpσµνψn |n(p)〉 = ψp

[gT (q2)σµν + g

(1)T (q2)(qµγν − qνγµ)

+g(2)T (q2)(qµP ν − qνP µ) + g

(3)T (q2)(γµ/qγ

ν − γν/qγµ)]ψn (3.8)

where P is the sum of neutron and proton momenta.

3.2.1 Fierz Interference term b

In the standard V − A theory at tree level, there does not exist any interactionwhich can shift or introduce any deviations in the electron energy spectrum. This

5 Terms with primed coefficients are the parity violating interactions as defined by Lee and Yang [12].

40

also explains why the energy spectrum of the emitted electrons is a natural place tolook for BSM interactions. The Fierz term is responsible for inducing any shifts ordeviations in the spectrum and is linear in the BSM couplings. This further solidifiesthe argument to study the electron energy spectrum. The Fierz term is composed ofboth Fermi and Gamow-Teller contributions [17] that can be expressed as

bF =CSCV

|CS|2 + |CV |2

bGT =CTCA

|CT |2 + |CA|2(3.9)

Both these terms contribute to the actual Fierz term (b = bF+bGT), this also explainsthat in the SM b vanishes because of the presence of scalar and tensor couplings CSand CT in Eq. 3.9.

3.2.2 Angular Correlation Coefficients

Taking into account this generalized matrix element and including the recoil orderterms, the differential decay rate for neutron β-decay becomes6 [41]

d3Γ

dEedΩeν

∝M4R4βx2(1− x)2Ξ

[1 + 3Rx+Rx

(4λ(1 + κp − κn)

1 + 3λ2

)−2R

(λ2 + λ+ λ(κp − κn)

1 + 3λ2

)− 4R

(λg2

1 + 3λ2

)+ 2

ε

Rx

(f3 − λg21 + 3λ2

)− ε

Rx

(1 + 2λ+ λ2 + 2λ(κp − κn)

1 + 3λ2

)][1 + bBSM

me

Ee+ Aβ cos θe (3.10)

+a1β cos θeν + a2β2 cos2 θeν

]where θe is the angle between the momentum of the electron and the polarization ofthe neutron, θeν is the electron-antineutrino opening angle and β = |~pe|/Ee. Thereare also kinematic factors which are defined as

ε =(me

M

)2, R =

E0

M, x =

EeE0

(3.11)

The parameter λ = gA/gV = CA/CV is just the ratio of the axial vector and vectorcoupling constants and is positive according to the sign conventions being used. λis an important factor in its right and it is one of the central parameter used inthe global fits in the current work7. It is a measure of how much each quark flavorcontributes to the nucleon spin. It appears in the prediction of energy consumptionin the fusion reaction, p+ p→ d+ νe + e+ going on in the sun. It also enters in theequations in the calculation of solar neutrino flux [20].

6 Ignoring the coefficients B and D in the current work.7 λ is being evaluated using numerical approaches like Lattice QCD which uses the nucleon matrix

elements of the axial vector current. Other methods of determining λ involve the measurementsof angular correlation coefficients such as a and A.

41

It is worth noticing that in the SM, the coefficients a and A are functions ofλ alone and in the absence of recoil order terms and interference effects, they areindependent of electron energy.

Neglecting the recoil-order terms, the terms Ξ and bBSM and the correlation co-efficients a0 and A0 can be defined in terms of both SM and BSM couplings [37,38]

Ξ = 1 + 3λ2 + (gSεS)2 + 3(4gT εT )2 (3.12)

bBSM =2(gSεS)− 6λ(4gT εT )

(1 + 3λ2) + (gSεS)2 + 3(4gT εT )2(3.13)

a0 =(1− λ2)− (gSεS)2 + (4gT εT )2

(1 + 3λ2) + (gSεS)2 + 3(4gT εT )2(3.14)

A0 =2λ(1− λ) + 2(4gT εT )2 + 2(gSεS)(4gT εT )

(1 + 3λ2) + (gSεS)2 + 3(4gT εT )2(3.15)

Note that we can retrieve the energy independent SM correlation coefficients if weturn off the BSM εS and εT couplings,

a0 =1− λ2

1 + 3λ2, A0 =

2λ(1− λ)

1 + 3λ2(3.16)

3.2.2.1 Spin-electron asymmetry coefficient A

Among all the correlation coefficients, the coefficient A has been measured togreater precision and most frequently due to the fact of its high sensitivity to λ.A is the angular correlation between the neutron spin and the electron momentum.Following [17,20], the probability that an electron is emitted with an angle of θe withrespect to the neutron spin polarization P is given by

W (Ee, θe) = 1 + β PA cos θe (3.17)

where β is just the ratio of the electron velocity to the speed of light.The correlation coefficient A can be expanded as the sum of the SM values and

the recoil order terms which can be written as [42,43]

A = A0 +1

(1 + 3λ2)2

ε

Rx

[4λ2(1− λ)(1 + λ+ 2f2) + 4λ(1− λ)(λg2 − f3)

]+R

[2

3[1 + λ+ 2(f2 + g2)] (3λ2 + 2λ− 1)

]+Rx

[2

3(1 + λ+ 2f2)(1− 5λ− 9λ2 − 3λ3) +

4

3g2(1 + λ+ 3λ2 + 3λ3)

](3.18)

42

3.2.2.2 Electron-antineutrino correlation coefficient a

The correlation a is defined as the correlation between the electron momentumand the antineutrino momentum. Measurements of a typically require very precisemeasurements of the proton energy spectrum and one of the factors that make thesemeasurements difficult is that a introduces a small shift in the spectrum. The pre-cision measurements of existing experiments gives the relative uncertainty of about2.6% [2]. However, apart from the electron asymmetry coefficient A, a is the nextpromising candidate for the measurement of λ since it offers almost the same sensi-tivity levels to λ as A [17]. Unlike A measurements, a measurements do not requireany polarimetry calibrations since it does not involve neutron polarization.

Similar to the correlation coefficient A, a can also be expanded as the sum of theSM values and the recoil order terms which can be written as [42,43]

a1 = a0 +1

(1 + 3λ2)2

4λ(1 + λ+ λ2 + λ3 + 2f2 + 2f2λ2)R

+(1 + 2λ− 2λ3 − λ4 + 4f2λ− 4f2λ3)

ε

Rx−[8λ(1 + 2f2 + λ2 + 2f2λ

2) + 3(1 + 3λ2)2]Rx

+[2(λ− λ3)g2 + 2(λ2 − 1)f3

] ε

Rx+ 8λ(1 + λ2)g2R

(3.19)

a2 =3(λ2 − 1)

1 + 3λ2Rx (3.20)

3.2.3 Neutron Lifetime τ

Another neutron β-decay observable of interest is the neutron lifetime. The totallifetime can be expressed as [17]

τ−1 =m5ec

4

2π3~7(g2V + 3g2A)f (3.21)

where gV (gA) are the usual vector (axial vector) form factors and f is the phase-spacefactor which includes the Fermi function, Coulomb corrections and various recoil orderterms [21]. It is defined as

f =

∫ E0

0

F (Z,Ee)peEe(E0 − Ee)2dEe (3.22)

where pe is the electron momentum. One important thing to notice is that the Eq.3.21 does not take the contribution from the radiative corrections8. Both the Coulombcorrections and radiative corrections are of O(α), though the Coulomb corrections are

8 Higher order (loop) contributions to basic tree-level processes. For instance, emission of extraphotons also known as bremsstrahlung in quantum electrodynamics.

43

much bigger [21, 65]. Improved calculations of the electroweak radiative correctionsand current high-precision measurements yield [44]

τ =4905.7(1.7) s

|Vud|2Ξ(3.23)

ignoring BSM couplings, Ξ becomes 1 + 3λ2. The constant term in the numeratoralso includes the corrections to the decay rate found by integrating it over the allowedphase space. The numerical constant in the numerator, however, does not account forthe corrections due to the second class currents of which only the contributions fromg2 are significant. Hence, including the second class currents in the neutron lifetimeyields

τ =4905.7 s

|Vud|2 [Ξ− (6.21× 10−3)g2λ](3.24)

3.2.3.1 Status of neutron lifetime measurements

The two different approaches that are being used to measure the neutron lifetime:Beam and Bottle techniques have their lifetime values significantly different from thePDG average [2]. This substantial difference between the lifetime values is not wellunderstood yet and several experimental efforts have been proposed which includeboth refinement of the existing approaches as well as new approaches.

It should be noted that the fitting techniques employed in this dissertation forparameter λ inferred from the various decay observables (specifically electron asym-metry A) should be compatible only with the τbottle. This is because τbeam is basedon the number of the protons detected obtained from the decay of the neutrons.

44

Chapter 4 nFitter Analysis

4.1 Overview

The central idea of this research is to probe physics beyond the standard model(BSM), particularly the V − A law of the electroweak interactions. In order to doso we need to push future neutron β-decay measurements to even higher precision(O(10−4)) [39, 45, 46]. One natural way to proceed is to analyze the energy depen-dence of the decay spectrum and various correlation coefficients associated with thedecay products. This analysis can be further extended by including the so called“second-class” currents and the limits to which they should be determined. Usingthis recipe, both experimental and theoretical uncertainties can be incorporated intothe measurements of the parameters which can be used to probe the beyond standardmodel.

This chapter is dedicated to the whole analysis workflow for nFitter. It describesthe whole data analysis pipeline which not only analyzes the data and infers the finalresults but also deals with the process of acquiring the data and cleaning it to makeit suitable for further analysis and numeric optimizations.

We begin by presenting the details of nFitter to the neutron beta decay pseudo-data generated by Monte Carlo simulations. We will show the impact of the simul-taneous fits to various neutron beta decay observables like the electron asymmetry(A), the electron-antineutrino correlation (a), the electron energy spectrum (Γ) andthe neutron lifetime (τ). We will also assess the effects of the second class currentsand the extent to which the theoretical uncertainties attributed by these second classcurrents can limit the validity of the Standard Model (SM).

We first performed a single parameter fit for the Fierz term b using unpolarizedneutron beta decay. We then performed the Rfit technique on the parameter spaceto see the constraints the second class currents put on the calculation of the Fierzterm. Analysis of unpolarized decay can be used to estimate the value of the Fierzterm to high precision since the estimated value would not contain the uncertaintycontributions from various correlations which are otherwise present in polarized neu-tron β-decay. But this would require us to have a precise value of lambda which, inturn, is dependent on the axial vector form factor (gA) and the neutron lifetime. Cur-rent lattice QCD calculations still have relatively large error bars on gA as comparedto the experimental measurements but they show a promise in reaching comparableprecision as the experimental measurements in the near future.

For the polarized beta decay, we performed the global fits using various β-decaycorrelations, electron energy spectrum and neutron lifetime. In this case we performedthree parameter fits on the Fierz term b, the ratio of the axial vector and vectorform factors λ and the normalization term for the electron energy spectrum N . Weemployed fitting for both b and λ here because the precision measurements of the Fierzterm from the correlation data is not possible without having a precise measurementof λ. Furthermore, since we are also fitting to the neutron lifetime, it is natural to

45

use λ as the fit parameter since both of them are highly correlated. Finally, we alsoassessed the validity of the SM by analyzing the effects of theoretical uncertainties.

4.1.1 High-level explanation

Following is the general sequence of the steps taken in the whole analysis workflow.At a high level, analysis of both unpolarized and polarized β-decay follow this samestrategy:

• Generate Monte Carlo pseudodata for a required number of neutron β-decayevents

• Pre-process and clean the data and plot histograms for various observables forthe whole energy range of the emitted electrons

• Fit for the parameters of interest ya and estimate error based on requiredconfidence levels

• Vary the second class currents simultaneously by performing the Rfit techniqueand analyze the effects of it on the parameter space

Apart from the similar general strategy, there are differences in how this analysisstrategy was implemented for unpolarized vs. polarized beta decay. However, theabove mentioned four steps explain almost all of the analysis of the unpolarizeddecay. Polarized β-decay is very rich from the analysis point of view and hence, itdeserves separate discussion of the implementation which will be discussed later.

Most of the analysis described in this work is based on the ROOT Data AnalysisFramework by CERN [47] and Python Data Science1 stack. Without these tools,much of this analysis would have been painstakingly difficult.

4.2 Unpolarized Neutron β Decay

In unpolarized neutron β-decay, all the terms in the differential decay distributionwhich involve the neutron spin are integrated out. Using the most general form forthe decay distribution Eq. 3.5 [13,37]

d3Γ

dEedΩedΩν

=1

(2π)5peEe(E0 − Ee)2ξ ×

[1 + b

me

Ee+ a

~pe · ~pνEeEν

+〈~σn〉 ·(A~peEe

+B~pνEν

+D~pe × ~pνEeEν

)](4.1)

we can easily see that the terms containing the correlations A, B and D will no longerappear. Including terms through next-to-leading order in the recoil expansion and

1 Various python libraries used for scientific computing as well as data science such as NumPy,Pandas, SciPy, Matplotlib.

46

also accounting for all six possible form factors, it will take the form [41]

d3Γ

dEedΩeν

∝M4R4βx2(1− x)2Ξ

[1 + 3Rx+Rx

(4λ(1 + κp − κn)

1 + 3λ2

)−2R

(λ2 + λ+ λ(κp − κn)

1 + 3λ2

)− 4R

(λg2

1 + 3λ2

)+ 2

ε

Rx

(f3 − λg21 + 3λ2

)(4.2)

− ε

Rx

(1 + 2λ+ λ2 + 2λ(κp − κn)

1 + 3λ2

)][1 + b

me

Ee

]where the different parameters are defined as follows

M = 939.5654133 MeV, me = 0.5109989461 MeV

Ξ = 1 + 3λ2, ε =(me

M

)2, R =

E0

M, x =

EeE0

(4.3)

where E0 is the electron end-point energy2. For the unpolarized β-decay, we are justinterested in the contribution of the electron energy spectrum to the uncertainty inthe Fierz term. Hence, we also integrated out the electron-antineutrino correlation(a) from the Eq. 4.1 to remove any contribution from the electron-antineutrinocorrelation to the uncertainty in b. The experimental values of the electron asymmetryand electron-antineutrino correlation are sensitive to b as shown in the followingequations [38,48,49]:

Aexp =1

2β

A

1 + bme

Ee

aexp =1

2β

a11 + bme

Ee+ 1

3a2β2

(4.4)

where a1 and a2 are the contribution of the SM and the energy dependent recoil-orderterms to the electron-antineutrino correlation Eq. 3.19 and 3.20.

4.2.1 Monte Carlo Pseudodata

To carry out the analysis, we employed Monte Carlo simulations to generate pseu-dodata by sampling the differential decay distribution Eq. 4.2 including the recoil-order terms. We generated 100 million simulated events with 500 energy bins each ofsize ∼1.56 keV. The current average from PDG 2019 [55] is assumed for λ (=1.2732).The CVC value of the weak magnetism factor3 f2 is assumed to generate the pseudo-data. We also assume zero values for the scalar εS and tensor εT couplings and othersmaller terms like the second class currents f3, g2.

2E0 = M −M ′ − (M−M ′)2−m2e

2M , and M ′ = 938.2720813 MeV is the mass of the proton.3 f2 = (κp − κn)/2 = 1.8529450

47

Figure 4.1: Histogram of the electron energy spectrum as a function of x, (ratio ofelectron energy to electron end point energy in β-decay) obtained from simulating1× 108 neutron β-decay events with 500 energy bins.

Carefully observing Eq. 4.2 shows that while performing Monte Carlo sampling onthe decay spectrum there should also be an additional parameter for the normalization(N) present in the fitting which will make the area under the plot unity. But lettingN be one of the ya parameters will increase the uncertainty in the fitted value of ourparameter of interest b which discourages the use of N as another fit parameter. Laterin the case of polarized β-decay, we will treat N as one of the fit parameters, sinceit will allow us to observe the agreement of results with [48]4. To remedy this issue,we normalized the differential decay distribution so that N = 1. The correspondinghistogram for the electron energy spectrum is shown in Figure 4.2.

4 [48] estimated the errors on parameters b, and λ and correlation coefficients such as a, A fromnumerical integration techniques. This served as the benchmark for us while considering the singleparameter fits in the case of polarized β-decay. It is discussed in more detail in section4.3.2.

48

Figure 4.2: Normalized version of the Figure 4.1 which makes the area under thecurve to be unity. This allows us to have b as the only fit parameter for the electronenergy spectrum.

4.2.2 Analysis and Results

After the data has been generated and pre-processed, we can use the recipes fromChapter 2 about the maximum likelihood framework for the estimation of the fitparameters. Specifically, we will numerically optimize the goodness-of-fit parameterχ2 which serves as our objective function to minimize. As shown in Eq. 2.12, χ2 isdefined by:

χ2 =∑i

(yexp(xi)− ytheo(xi)

σi

)2

where the sum is over all energy bins, ytheo(exp) are the theoretical(experimental5)values of the differential decay distribution function, and the σi are the error contri-butions from each energy bin following Poisson statistics6,7.

The “best values” for the parameters we want to estimate are calculated by min-imizing the χ2. The point of minimum χ2 which we call χ2

min gives us the best fittedvalue of the parameters of interest. It also allows us to calculate ∆χ2 = χ2 − χ2

min

which is used to calculate confidence intervals for the parameters of interest. Thevalue of ∆χ2 depends on the number of parameters being fitted, for instance, for agiven confidence level, a single parameter fit (implemented in the unpolarized β-decay)will have a smaller value of ∆χ2 as compared to the three parameter fits (implementedfor the case of polarized β-decay). This is shown for commonly used confidence levelsin Table 4.1 [35, 50].

5 Experimental data is analogous to the simulated data in our case.6 σi =

√ni where ni is the number of events in the ith energy bin.

7 Since the histograms are normalized, the errors have to be scaled the same way as the bin-content.

49

Table 4.1: Values of ∆χ2 corresponding to a given confidence level (CL) for m fitparameters.

CL (%) m = 1 m = 2 m = 368.27 1.00 2.30 3.5390. 2.71 4.61 6.2595. 3.84 5.99 7.82

95.45 4.00 6.18 8.03

For optimization, we implemented a grid-search algorithm which scans each pointin the parameter space and looks for χ2

min. This is shown in Figure 4.3. For smallerparameter spaces, this algorithm is quite fast but it suffers from high execution timesfor large parameter spaces because of the computational complexity. We also usedsome off-the-shelf fitting routines available from the ROOT framework8 [47].

The numerical results presented in this analysis serve a dual purpose:

1. They can be used to assess the validity of the SM in which we assume the inputvalue of b to be zero and fit for it to estimate the value to some pre-determinedprecision level. Here we employed the full energy range for Ee, meaning thekinetic energy of the electron is allowed to be in the range9: 0 ≤ Te ≤ T0.We assumed all the second class currents and new physics scalar and tensorcoefficients to be zero.

8 Specifically, we used the TMinuit package which comes with the standard ROOT installation.9 Although we are assuming the full energy range, in the real experiment only a subset of the energy

range is used. There will be limitations due to certain hardware thresholds and analysis cuts inthe lower-energy range.

50

Figure 4.3: Parameter space showing the variation of the χ2 with the Fierz term b.The 68.3% CL are defined by ∆χ2 = 2.30 [Table 4.1] which yielded the fitted valueof the Fierz term b = 0.00066(75) with a χ2

min = 473.94. We use the inputs b = 0.0,nevents = 1× 108, nbins = 500, second class currents: f3 = g2 = 0.

2. We can relax the assumption that second class currents are zero and applythe Rfit technique for the fitting of the Fierz term where ya = b andyµ = f3, g2. These yµ parameters are permitted to vary simultaneouslyover some prescribed range. The rest of the parameters: f2 and λ are assumedto have the same values as before.

51

(a)

(b)

(c)

Figure 4.4: Variation of χ2 with b for a single parameter fit for the unpolarizedβ-decay after allowing second class currents to vary. The original parameter spaceplotted along with the one obtained by varying the second class currents one at atime as well as both simultaneously with a step size of 0.01; (a) f3 ∈ [−0.1, 0.1],g2 = 0, (b) g2 ∈ [−0.1, 0.1], f3 = 0, and (c) f3 ∈ [−0.1, 0.1], g2 ∈ [−0.1, 0.1]. We usenevents = 1× 108 and nbins = 500.

52

Some explanation of the plots in Figure 4.4 is in order. The main observation thatfollows is the uncertainty in the value of b increases after letting the second classcurrents vary. This can be easily seen in the widening of the χ2 function whichis inherently convex. This is not very surprising since the second class currentsaccount for the theoretical uncertainties which means for a given confidence level inthe unpolarized β-decay, they will increase the error in the value of b. However, wenotice that the uncertainty in the value of b is almost insensitive to the value of f3.This is attributed by the fact that f3 appears in the term with ε/(Rx) which itselfis very small (ε/(Rx) ≈ 2.96 × 10−7). On the other hand, g2 is always scaled byR ≈ 1.37 × 10−3 as evident in Eq. 4.2. This accounts for the greater contributionfrom g2 in the fit. The results of the fit are summarized in Table 4.2.

Table 4.2: Fit results for the Rfit technique applied on the electron energy spectrumof the unpolarized β decay. The fitted value of b is defined by the location of theχ2min, while the 68.3% CL is defined by ∆χ2 = 2.30 for two parameters. We used the

input b = 0.

f3, g2 χ2min/ndof Fitted b(σ)

f3 ∈ [−0.1, 0.1], g2 = 0 473.944/499 0.00067(76)g2 ∈ [−0.1, 0.1], f3 = 0 473.943/499 0.00062(77)f3, g2 ∈ [−0.1, 0.1] 473.943/499 0.000635(785)

4.3 Polarized Neutron β Decay

For the case of the polarized β-decay, all the previously dropped terms containingneutron polarization are brought back. Using Eq. 4.1 and incorporating recoil orderexpansion terms, the differential decay distribution takes the form [41]

d3Γ

dEedΩeν

∝M4R4βx2(1− x)2Ξ

[1 + 3Rx+Rx

(4λ(1 + κp − κn)

1 + 3λ2

)−2R

(λ2 + λ+ λ(κp − κn)

1 + 3λ2

)− 4R

(λg2

1 + 3λ2

)+ 2

ε

Rx

(f3 − λg21 + 3λ2

)(4.5)

− ε

Rx

(1 + 2λ+ λ2 + 2λ(κp − κn)

1 + 3λ2

)][1 + bBSM

me

Ee+ Aβ cos θe

+a1β cos θeν + a2β2 cos2 θeν

]Note that we did not include correlation coefficients B, D into the decay distributionbecause they have been integrated out. To make the analysis more robust in termsof accounting for experimental uncertainties, global fits are employed which includethe electron energy spectrum Γ as well as the other observables like the correlationcoefficients a and A, and the neutron lifetime τ which will make xexp = Γ, a, A, τ.

53

As mentioned in Eq. 3.12-3.16, and Eq. 3.18-3.20, these parameters are up torecoil order terms are defined as [42–44]

A = A0 +1

(1 + 3λ2)2

ε

Rx

[4λ2(1− λ)(1 + λ+ 2f2) + 4λ(1− λ)(λg2 − f3)

]+R

[2

3[1 + λ+ 2(f2 + g2)] (3λ2 + 2λ− 1)

](4.6)

+Rx

[2

3(1 + λ+ 2f2)(1− 5λ− 9λ2 − 3λ3)

+4

3g2(1 + λ+ 3λ2 + 3λ3)

]

a1 = a0 +1

(1 + 3λ2)2

4λ(1 + λ+ λ2 + λ3 + 2f2 + 2f2λ2)R

+(1 + 2λ− 2λ3 − λ4 + 4f2λ− 4f2λ3)

ε

Rx(4.7)

−[8λ(1 + 2f2 + λ2 + 2f2λ

2) + 3(1 + 3λ2)2]Rx

+[2(λ− λ3)g2 + 2(λ2 − 1)f3

] ε

Rx+ 8λ(1 + λ2)g2R

a2 =3(λ2 − 1)

1 + 3λ2Rx (4.8)

τ =4905.7 s

|Vud|2 [Ξ− (6.21× 10−3)g2λ](4.9)

where the energy independent values of the correlation coefficients a0 and A0 can bewritten as [13]

a0 =(1− λ2)− (gSεS)2 + (4gT εT )2

(1 + 3λ2) + (gSεS)2 + 3(4gT εT )2,

A0 =2λ(1− λ) + 2(4gT εT )2 + 2(gSεS)(4gT εT )

(1 + 3λ2) + (gSεS)2 + 3(4gT εT )2(4.10)

and the Fierz term [13]

bBSM =2(gSεS)− 6λ(4gT εT )

(1 + 3λ2) + (gSεS)2 + 3(4gT εT )2(4.11)

in terms of SM and BSM (εS and εT ) couplings.

4.3.1 Workflow and Monte Carlo Pseudodata

The big picture remains the same as the case of unpolarized β-decay as men-tioned in section 4.1.1 and 4.2.1 with a few subtleties. Like mentioned in unpolarizedβ-decay, we employed Monte Carlo simulations to generate pseudodata by sampling

54

the polarized differential decay distribution including the correlation coefficients men-tioned in section 3.2.2. Using the theoretical expressions (xtheo) of the observables,the set of the experimental results (xexp) are sampled using the Monte Carlo simu-lations [see section 2.2.1]. The entire set of the model parameters for this case consistsof

ymod = λ, f3, g2, b, N (4.12)

where we determine confidence levels on only ya = λ, b,N parameters. Theremaining subset of ymod10 includes only the second class currents f3 and g2, andthey are categorized as yµ parameters.

In the case of polarized β-decay, we generated two different type of data sets [37].The first data set assumes zero values for both new physics scalar and tensor couplings(εS = εT = 0). We refer to this data set as the “Standard Model” data set. Wegenerated 10 different data files with event size ranging from 1 × 109 to 1 × 1010

events for this data set. This much volume of data is needed because of two reasons:First, the required precision level in the parameters of interest varies as the inverse ofsquare root of number of events (∝ 1/

√N) [48]. Second, it is important to understand

the effect of theoretical uncertainties caused by second class currents during Rfittechnique as the number of events approach infinity.

The value of λ = 1.2732 is taken as the current average from PDG 2019 [55] andthe CVC value of weak magnetism factor is assumed (f2 = (κp− κn)/2 = 1.8529450)and assuming zero values for other smaller terms like the second class currents f3, g2.

The second data set relaxes the restriction on the scalar and tensor couplings.We term it as the “New Physics” data set, here we assumed εS = εT = 0.001,(gS = gT ≈ 1) with everything else being identical to the earlier one. This is becausegSεS = gT εT = 0.001 corresponds to the most stringent limits on the scalar andtensor interactions which arise from 0+ → 0+ nuclear decays and Dalitz-plot studyof radiative pion decay (π → eνγ), respectively [38]. However, we generated only onedata file for this case with 1 × 1010 events since we are only interested in observingthe deviations of the ya parameters from their input values. These deviations arequantified in the terms of the “Pull” values11 which are analogous to the Z -scores instatistics.

As a final step of analysis, the parameters of the yµ parameter set, which consistsof the second class currents f3 and g2 are permitted to vary simultaneously in both ofthe data sets over some prescribed range, following the recipes of the Rfit technique[36,64].

4.3.2 Single Parameter Fits on the Standard Model Data set

Before jumping to the global fits for the ya parameters, we analyzed the ob-servables and fit parameters separately. Although it is not required for the actualanalysis, it is nevertheless performed for the sanity check. For checking we slightly

10 Other parameters like f2, g3 and Vud are assumed to take fixed values in which g3 = 0.11 Pull = (xfit − xinput)/σfit

55

modified our set of model parameters

ymod = ya = a,A, b, λ (4.13)

and checked for any under-estimation or over-estimation of the uncertainties of theseparameters. This will serve a dual purpose for the rest of the analysis: First, theuncertainty values obtained from these individual fits will serve as the baseline forthe global fits. Second, it will greatly help us to debug the code should there be anypossible bugs in the event-generator or analysis. The various initial values used inthe fitting are summarized in the Table 4.3.

Table 4.3: Initial values for the simulation and fitting parameters.

λ 1.2732b 0.0

nbins 1000nevents 1× 1010

bin-width 0.781 keV

We start by fitting for the Fierz term b, in which we generated Monte-Carlopseudodata for the differential decay distribution. The initial values of the inputparameters are listed in Table 4.3. It should be noted that while fitting for b usingthe differential decay distribution, we took the normalization N as the second fitparameter. This leads to the change in the definition of error for a given confidencelevel, for instance, if we want to evaluate the errors for a two parameter fit with 68.3%CL12, we can use Table 4.1 for the value of ∆χ2 [35, 50]:

∆χ2 = χ2 − χ2min = 2.30 (4.14)

Figure 4.5 shows the histogram of the neutron β-decay energy spectrum as well asthe fitted function.

12 In ROOT, 68.3% CL are given by 1-sigma deviation. In order to calculate them in TMinuit, weuse SetErrorDef(1).

56

Figure 4.5: Histogram showing the differential decay distribution plotted as a functionof electron energy. The solid line (red) is a result of the fitting with b and N as thefit parameters. We use nbins = 1000 and nevents = 1× 1010.

The results obtained from the fitting are summarized in Table 4.4. On inspecting

Table 4.4: Fit results for the differential decay distribution with b and N as the fitparameters. The values in the parenthesis indicate the error values corresponding to∆χ2 = 2.30 which corresponds to 68.3% CL.

Fit parameters Fitted value (σ)b 0.0001376(752)N 137300000(7000)χ2min 1040.05ndof 998

Prob(χ2 > χ2min) 0.173

the different error values, we observe that σb matches almost perfectly to the resultof numerical integration mentioned in [48] (σb = 7.5/

√N). It is important to note

though that the value of σb will only agree to the one specified in [48] if we calculate1-sigma standard deviation13 with only normalization and the Fierz term as the fitparameters Eq. 4.2.

13 ∆χ2 = 2.30 for 68.3% CL.

57

Moving on to the correlation coefficients, we obtained the “experimental asymme-tries” aexp and Aexp [37] by using the angles obtained from the Monte Carlo samplingof the polarized decay energy spectrum Eq. 4.5 and making suitable angle cuts.These experimental asymmetries are then analyzed in a typical “forward/backward”asymmetry measurement [37,66], where (compare with Eq. 4.4)

aexp ≡N(cos θeν > 0)−N(cos θeν < 0)

N(cos θeν > 0) +N(cos θeν < 0)

=1

2β

a11 + bme

Ee+ 1

3a2β2

(4.15)

Aexp ≡N(cos θe > 0)−N(cos θe < 0)

N(cos θe > 0) +N(cos θe < 0)

=1

2β

A

1 + bme

Ee

We observed a similar behavior for A, a and λ which again agrees with the resultsof [48]. We employed single parameter fits for the correlation coefficients a and A byincorporating next-to-leading order energy dependent recoil terms Eq. 4.6-4.7 andkeeping everything else fixed (λ = 1.2732 [55], f3 = g2 = 0, f2 = (κp − κn)/2 =1.8529450). The result of single parameter fits are shown in Figure 4.6.

(a) (b)Figure 4.6: Simulated data from the standard model dataset for the electron-asymmetry and electron-antineutrino correlation plotted as a function of electronenergy. The individual data points are shown in blue and the red solid line is theresult for the single parameter fits with A and a being the fit parameters respectively.The parameter x is defined as the ratio of electron’s energy to its end-point energyEq. 4.3. We use nbins = 1000 and nevents = 1× 1010.

The results of these fits have been summarized in the Table 144.5.

14 The fitted value of λ is obtained from the fitting it as a single parameter in the electron asymmetryA [Eq. 4.6].

58

Table 4.5: Fit results for correlation coefficients from single parameter fits. Thereported value comes from the numerical integration results discussed in [48]. We usenbins = 1000, nevents = 1× 1010 and ndof = 999.

Parameter Fitted value (σ) Reported value (σ)

A -0.118634(26) 2.7/√N

a -0.107224(26) 2.6/√N

λ 1.273146(72) ≈ 2.6σA = 7/√N

We also implemented Rfit technique on the single parameter fits for the electronenergy spectrum Γ to check the constraints imposed by the second class currents (f3and g2) on the Fierz term b. We ran Rfit for several different cases:

(a) vary f3 in [−0.1, 0.1] and g2 = 0

(b) vary g2 in [−0.1, 0.1] and f3 = 0

(c) vary both f3 and g2 simultaneously in [−0.1, 0.1]

(d) increase the interval size in which the second-class currents under Rfit can vary

The motivation for using these ranges for second-class currents is discussed in section3.1.1. The corresponding parameter space is shown in Figure 4.7.

(a) (b)Figure 4.7: Parameter space obtained from fitting for b and N in electron energyspectrum using Rfit. Panel (a) shows the parameter space for a given range of [-0.1, 0.1] but turning off either f3 or g2 to see their effects separately. Red curveinvolves both f3 and g2 varied simultaneously. Panel (b) shows the parameter spacefor different ranges for both f3 and g2. The χ2 values on the Y-axis in panel (b) aredifferent from those in panel (a) because we obtained a much better χ2

min when weremoved the first two energy bins of the histograms from the analysis. The step sizefor Rfit technique in each of the cases is 0.01 with input b = 0.0. The 68.3% confidencelevels are defined by ∆χ2 = 2.30. We use nbins = 1000 and nevents = 1× 1010.

59

We can infer a few things from Figure 4.7 which are also applicable to globalfits. First, f3 has almost negligible effect on the error in b as can be seen from thebroadening of the curve. This actually makes sense since the contribution of f3 isvery small because of the down-scaling due to the ε/(Rx) factor present as shown inEq. 4.5 [37]. On the other hand, g2 increases the uncertainty in b significantly whichcan also be explained in terms of the way it appears in the electron energy spectrum.Moreover, the effect of g2 is also slightly up-scaled by R [Eq. 4.5]. The combinedeffect of both f3 and g2 can be seen in the panel (b) in Figure 4.7. It shows that thetheoretical uncertainties due to second-class currents indeed increase the uncertaintyin the value of the Fierz term (b). We also see that the uncertainty on b gets aboutthree times bigger when we let both f3 and g2 vary in the range [−0.5, 0.5]. It alsoshows that bigger ranges for Rfit end up in even smaller χ2

min as it was for the casewhen both second-class currents were switched off.

We know that second-class currents contribute to the theoretical uncertaintiessince a very little is known about them and it might not be an appropriate as-sumption that their underlying probability distribution is Gaussian [37]. This makesit completely worthwhile to investigate them more and see how they affect otherymod parameters. In principle, we can get around this by having an infinite numberof simulation events and the value of fit parameters tend to approach their accuratevalue15. However, having an infinite number of simulation events is impractical, sowe want to see for what number of simulation events the uncertainty due to second-class currents approaches a saturation. This would give us the maximum numberof simulation events required to make the uncertainty due to second-class currentsalmost negligible. We implemented this by creating a series of data files containingsimulation events ranging from 1 billion to 10 billion [see section 4.3.1]. The resultsof performing Rfit technique on each of these data files is shown in Figure 4.8.

15 In statistics, this is exactly similar to finding the properties of a population using a samplingdistribution which is supposed to be a representative of a population.

60

(a) (b)Figure 4.8: Effects on the uncertainty in b as a function of the number of eventsobtained by fitting for both b and N in the electron energy spectrum with and withoutsecond-class currents. Both second-class currents f3 and g2 are varied simultaneouslyin ranges described in the legend. Panel (b) shows the plot in panel (a) in log scale.∆χ2 = 2.30 for 68.3% CL for each case and nbins = 1000.

It is evident from the above plot that for smaller ranges for Rfit the fractionalchange in the uncertainty in b is getting smaller as the number of simulated eventsincreases. However, for the case when both f3 and g2 are allowed to vary in therange [−0.5, 0.5], the error on b decreases at first and then from ∼3 billion events,it remains roughly constant. We believe that it might be because of the fact thatfor bigger ranges for second-class currents, the theoretical uncertainties imposed bythem totally dominates the statistical and other systematic uncertainties when thevolume of the simulated events surpass beyond a certain number.

4.3.3 Global Fits

We next studied global fits for the electron energy spectrum Γ, the electron-antineutrino correlation a, the electron asymmetryA and the neutron lifetime τ withλ, b,N as our ymod parameters for the standard model data set. We carried outseveral simulations to see the effect of each observable on the ymod parameters andthe results are presented in the subsections that follow.

4.3.3.1 Constraints on λ

We observed that fitted value of λ determined from the global fits of Γ, a, Ais strikingly different to its calculated value given by λ = gA/gV . This behavior isactually discussed in [37, 38] which explains that the appearance of a right-handedcoupling emergent from beyond standard model (BSM) physics might modify itsvalue. However, the unexpected value for λ appears when only the electron energyspectrum is used as the observable (xexp = Γ) with ymod = λ, b,N as the fitparameters. Inclusion of correlation coefficients in xexp seems to dilute this effectand the determined value of λ starts approaching the expected value. These results

61

Table 4.6: Fitted values of λ shown for different subsets of the observables xexp =Γ, a, A. We use the input λ = 1.2732 [55], nevents = 1× 1010 and nbins = 1000.

xexp Fitted λ(σλ) χ2min/ndof

Γ 1.57573(15610) 1037.63/997Γ, A 1.27317(7) 1983.14/1997Γ, a 1.27313(9) 2076.79/1997Γ, a, A 1.27316(6) 3020.03/2997

have been shown in Table 4.6. These results show that λ is much more sensitive to thecorrelation coefficients and the neutron lifetime16 than the electron energy spectrum.Because of this insensitivity to the Γ, for rest of the analysis about global fits, we arenot fitting for λ in the electron energy spectrum. In other words, λ is only includedas a fit parameter in the A, a, τ while Γ contains only b,N as fit parameters.

4.3.3.2 Constraints on the Fierz term (b)

Next, we find that the most stringent constraints for b also comes from the electronenergy spectrum. This is shown by considering the fact that b is very sensitive tothe new physics interactions since it depends on the new physics scalar and tensorcouplings, εS and εT , to linear order as defined in Eq. 4.11 [37,38]. We obtained thefit value for b corresponding to χ2

min and plotted the extracted value of scalar andtensor couplings in εS − εT space as shown in Figure 4.9.

16 To be discussed later.

62

Figure 4.9: εS − εT space showing the constraints imposed by various observables.Different subsets of xexp = Γ, a, A are considered. The plot for the global fits ofa,A,Γ is totally eclipsed with the one for only Γ. It looks like Γ puts the moststringent constraints on the Fierz term b and therefore, on the new physics scalarand tensor couplings. However, Coulomb radiative corrections can also modify thisas well but we omitted them in our analysis. We use λ = 1.2732 [55], nbins = 1000and nevents = 1× 1010.

It is evident from Figure 4.9, correlation coefficients alone put loose constraintson the Fierz term and when they are combined with the electron energy spectrumvia global fits, a strong constraint on the new physics scalar and tensor interactionsstarts to emerge. The electron energy spectrum, however, contributes the most tothis strong constraint as we can see the plot for the global fits (xexp = Γ, a, A)totally eclipsed by the one for the electron energy spectrum alone. The results of thefit are summarized in Table 4.7.

Table 4.7: Fit results of b shown for different subsets of the observables xexp =Γ, a, A. As shown in Figure 4.9, the electron energy spectrum puts the most strin-gent constraints on the Fierz term. In the fit scenarios where Γ is included in xexp,we used ymod = b,N as the fit parameters (where N is the normalization). Weuse the input λ = 1.2732 [55], b = 0.0, nevents = 1× 1010 and nbins = 1000.

xexp Fitted b(σb) χ2min/ndof

Γ -0.000053(752) 952.631/998A 0.003893(3607) 980.379/999a -0.005683(3951) 910.549/999a,A -0.000416(2664) 1894.129/1999Γ, a, A -0.000080 (724) 2846.778/2998

63

4.3.3.3 Negligible correlation between λ and b

Following the above two points we see that for the global fits, there is almostnegligible correlation between λ and b for the case when second-class currents arezero (f3 = g2 = 0). This can be easily explained with the help of the previous twosections: from Figure 4.9 it is evident that electron energy spectrum Γ alone putsthe most stringent constraints on b whereas λ is almost insensitive to Γ which iswhy we did not fit for λ in Γ. The presence of both λ and b in the expressions ofthe correlation coefficients resulted in a tiny correlation between them as shown inFigure 4.10.

Figure 4.10: Parameter space for λ and b obtained from the global fits for Γ, a, A, τ.The input values for λ and b are 1.2732 [55] and 0.0, respectively for nevents = 1×1010

and nbins = 1000. A 68.3% CL for three parameters (b, λ, and N) corresponds to∆χ2 = 3.53.

4.3.3.4 Fitting ymod without neutron lifetime (τ)

For the case of Rfit, we first removed the neutron lifetime from the global fitsand considered xexp = Γ, a, A. The motivation for doing this is to filter out theeffects of the neutron lifetime on the fit parameters. It is later added to xexp andthe results are discussed later. On performing Rfit we observed that the error ellipseis stretched hence increasing the uncertainty in the fit parameters. Interestingly,inclusion of the second-class currents due to Rfit also introduced a correlation in thefit parameters as compared to the case where second-class currents were switched off[see Figure 4.10]. The results of the Rfit technique are shown in Figure 4.11.

64

Figure 4.11: Parameter space for λ and b obtained from the global fits for Γ, a, Aafter letting second-class currents f3 and g2 vary simultaneously in ranges [−0.1, 0.1]and [−0.5, 0.5]. The input values for λ and b are 1.2732 [55] and 0.0, respectively fornevents = 1× 1010 and nbins = 1000. A 68.3% CL for three parameters (b, λ, and N)corresponds to ∆χ2 = 3.53.

This behavior of the parameter space can be again explained from the differentialdecay distribution Eq. 4.5. The terms involving second-class currents (especially g2)in the electron energy spectrum also involve λ. Even in the recoil order terms of thecorrelation coefficients Eq. 4.6 and 4.7, the second-class currents appear in the termswhich are polynomials of λ. All of these terms with second-class currents are furthersuppressed by the constant term R ≈ 10−3 and the energy dependent term ε/(Rx) ≈2.96× 10−7. Hence, letting these second-class currents vary in their predefined rangemake the terms involving them dominate over the other recoil order terms particularlyin the electron energy spectrum due to the presence of λ. Furthermore, the electronenergy spectrum also has a term involving the Fierz term b containing almost thesame factor me/Ee which make terms containing second-class currents and the Fierzterm to have almost the same scale17. The combined effect of all of this manifestsitself as a non-zero correlation between λ and b after the second-class currents areturned on. The results of the fit in Figure 4.11 are shown in Table 4.8.

17 From Eq. 4.3, R ≈ 0.00137; εRx =

(me

M

)·(me

Ee

)

65

Table 4.8: Results of the global fits consisting of the observables (xexp = Γ, a, A).We employed fitting for ymod = λ, b,N where the fitted value is defined by thelocation of χ2

min and the definition of 68.3% CL is given by ∆χ2 = 3.53 [Table 4.1].We use the input λ = 1.2732 [55], b = 0.0, nevents = 1× 1010 and nbins = 1000.

f3, g2 Fitted b(σb) Fitted λ(σλ) χ2min/ndof

f3, g2 = 0 0.000140(135) 1.27316 (11) 3020.1/2997f3, g2 ∈ [−0.1, 0.1] 0.000170(147) 1.27322 (23) 3020.07/2997f3, g2 ∈ [−0.5, 0.5] 0.000040(175) 1.27315 (22) 3020.12/2997

4.3.3.5 Effects of including neutron lifetime τ on the global fits

Putting back the neutron lifetime τ to the global fits, xexp = Γ, a, A, τ revealedvery interesting observations. First, it should be noted that the neutron lifetime doesnot have any energy dependence and that is why it only contributes as a single datapoint in the calculation of the goodness-of-fit parameter χ2. Secondly, the choice ofδτ has a significant impact in the uncertainties of the fit parameter. Smaller values ofδτ tend to increase the variance which can be seen from the broadening of the errorellipse as evident from Figure 4.12. This is because high precision for neutron lifetimeput large constraints on λ as can be seen from the following definition [37,44]:

τ =4905.7(1.7) s

|Vud|2 Ξ(4.16)

where the factor in the numerator is due to Fermi constant for muon decay as wellas the radiative corrections and phase space effects, the CKM matrix element Vud =0.97420 [2], and Ξ = 1 + 3λ2. The dependence of b on τ can also be explainedby observing the effects of new physics parameters on the standard model value ofneutron lifetime using the relation [49]:

τnτSMn

=1

1 + b 〈me

Ee〉

(4.17)

As noted in [37], Eq. 4.16 already embeds the small contributions due to variouscouplings such as gA, gV integrated over the available phase space but it does nottake into consideration the effect of the second-class currents, particularly g2 (effectof f3 is negligibly small). This effect can be included in the neutron lifetime as asmall correction term which is implemented by replacing Ξ with Ξ(1 + CgAg2) whereCgAg2 is the contribution due to gA and g2 couplings [21] and is defined as

CgAg2 = −6.21× 10−3λg2Ξ

(4.18)

After applying this correction term, the neutron lifetime takes the form

τ =4905.7 s

|Vud|2[Ξ− 6.21× 10−3g2λ](4.19)

66

The effect on the parameter space due to Rfit using this correction term on theneutron lifetime is shown in Figure 4.1218. Table 4.9 shows the value of the fitparameters with the corresponding errors for two different values of δτ .

Table 4.9: Fitted values for λ, b from simultaneous fits of the observables xexp =Γ, a, A, τ to the Standard model data set. The fitted value of the parameters aredefined by the location of χ2

min. They correspond to a CL of 68.3% which for a threeparameter fit (m = 3) is defined by ∆χ2 = 3.53 [Table 4.1]. We use the inputsλ = 1.2732 [55], b = 0, nevents = 1× 1010 and nbins = 1000.

δτ(s) Fitted b(σb) Pull b Fitted λ(σλ) Pull λ χ2min/ndof

f3 = g2 = 00.1 0.000140(140) (1.0σ) 1.27317(9) (−0.33σ) 3020.27/29981.0 0.000140(135) (1.03σ) 1.27316(10) (−0.40σ) 3020.1/2998

f3, g2 ∈ [−0.1, 0.1]0.1 0.000170(125) (1.36σ) 1.27320(16) (0) 3020.20/29981.0 0.000155(147) (1.05σ) 1.27321(22) (0.04σ) 3020.07/2998

f3, g2 ∈ [−0.5, 0.5]0.1 0.000040(115) (0.34σ) 1.27317(14) (−0.21σ) 3020.32/29981.0 0.000095(160) (0.59σ) 1.27315(22) (−0.23σ) 3020.12/2998

Table 4.9 shows the effect of the theoretical uncertainties as a result of the second-class currents. Without f3 and g2, we obtained a precision of around 1% on the valueof the Fierz term b which changes to around 3% when we allowed both f3 and g2 tovary in the interval [−0.5, 0.5].

18 It turns out the step size as well as the range for fitting N significantly affects the parameter spaceof b and λ as can be seen by the cutting out of the error ellipses. Smaller step size might help tofix this issue but it will take a lot of computational resources as well as the execution times wouldbe really long!

67

(a) (b)Figure 4.12: Parameter space showing the error ellipse with and without Rfit withxexp = Γ, a, A, τ. Both second-class currents f3 and g2 are varied simultaneouslyin the ranges described in the legend. Panel (a) shows the result when δτ = 0.1seconds whereas panel (b) shows the case when δτ = 1 second. The fitted value of bare defined by the location of the χ2

min, while the 68.3% CL are defined by ∆χ2 = 3.53.We use the inputs λ = 1.2732 [55], b = 0.0, nevents = 1× 1010 and nbins = 1000.

4.3.3.6 Effect of number of the events in the Monte Carlo pseudodata onthe uncertainty of the Fierz term (b)

Like in the case of single parameter fits, we also want to analyze the trend of theuncertainties of the fit parameters as we increase the number of events. We againemployed the same technique by creating Monte-Carlo pseudo-data for events rangingfrom 1 billion to 10 billion and perform Rfit using different ranges of the second-classcurrents. This case, however, is limited by the computational complexity for theRfit technique like the step size and the range for the fit parameters (b, λ,N). Forexample, choosing the step size for the fit parameters for Rfit more aggressivelymight not end up in the error ellipse at all whereas selecting values for the step sizemore conservatively make the execution time really long. Similarly, a lot of trial anderror goes into selecting the optimum range for the fit parameters which gives usthe expected results. Having a 4-dimensional parameter space involving χ2, b, λ,Nas well as two second-class currents make this optimization problem quite convolutedto handle. In any case, the result for the variation in the uncertainty of the fitparameters with the number of events in the presence of the second-class currents isshown in Figure 4.13.

68

(a) (b)Figure 4.13: Variation in the uncertainty in b as a function of number of eventsobtained by fitting b, N and λ in the global fits xexp = Γ, a, A, τ. Both second-class currents f3 and g2 are varied simultaneously in ranges described in the legend.Panel (b) shows the plot in panel (a) in log scale. We use ∆χ2 = 3.53 for 68.3% CLfor each case, nevents = 1× 1010, and nbins = 1000.

4.3.3.7 Getting back the negligible correlation between λ and b

Varying the second-class currents simultaneously in the Rfit tend to bring a non-zero association between λ and b. This is a consequence of the terms containingsecond-class currents in the recoil order terms in the electron energy spectrum andthe correlation coefficients a,A. We analyzed this association between λ and bfurther by letting the range of the second-class currents in Rfit to shrink. Lettingthe second-class currents to vary in a tiny interval started diminishing the correlationbetween λ and b and the error ellipse ended up transforming to the original ellipsewith negligible correlation. This is shown in Figure 4.14.

69

(a)

(b)Figure 4.14: Parameter space showing the error ellipse with and without Rfit withxexp = Γ, a, A, τ. Both second-class currents f3 and g2 are varied simultaneouslyin ranges described in the legend. Panel (a) shows the result when δτ = 0.1 secondswhereas panel (b) shows the case when δτ = 1 second. As can be easily seen, lettingthe range of the second-class currents shrink brings back the original error ellipsewith almost zero correlation. We use ∆χ2 = 3.53 for 68.3% CL for each case, andinputs λ = 1.2732 [55], b = 0.0, nevents = 1× 1010, and nbins = 1000.

This shows that second-class currents are indeed responsible for bringing up thecorrelation between the fit parameters. We can also infer from this that out ofxexp = Γ, a, A, τ, only the spectrum has terms which involve both λ, f3, g2

70

and b. Hence, most of the contribution to this correlation is coming out from thespectrum from the terms involving both g2 and b.

4.3.4 The “New Physics” data set

Apart from the “Standard Model” data set, we generated another data set con-sisting 10 billion simulated events for the measurement of the Fierz term. Followingthe analysis in [37], the only difference between these two data sets is the inclusion ofa non-zero value of the scalar and tensor couplings, namely, gSεS = gT εT = 1.0×10−3.These values correspond to the maximum constraints in the scalar and tensor cou-plings obtained from the analysis of 0+ → 0+ nuclear transitions [67] and the Dalitzplot study of the radiative pion decay [68], respectively. The most significant effect ofthese new physics couplings would be in the value of the Fierz term since it depends onthese scalar and tensor interactions to the linear order [Eq. 4.11]. Setting these inter-actions to non-zero will yield a non-zero value for the Fierz term (bBSM = −0.00487)which is the direct consequence of the beyond standard model physics.

We started with the global fits in which ya = λ, b,N and yµ is an emptyparameter set. We excluded the neutron lifetime τ and used xexp = Γ, a, A asthe set of the observables for the global fits. As we can see from the Table 4.10,except for the case where we used only the electron energy spectrum, these fits yieldreasonable values for χ2

min. However, it is evident from the table that the presence ofthe electron energy spectrum in the global fits revealed significant pulls [38,39] on thefitted values of the Fierz term. This makes sense since the electron energy spectrumput the most stringent constraints [section 4.3.3] on the Fierz term which is highlysensitive to the new physics scalar and tensor interactions.

71

Table 4.10: Fitted values for λ, b from simultaneous, and individual fits for differentsubsets of the observables xexp = Γ, a, A, τ to the New Physics data set. Herethe second-class currents are turned off (f3 = g2 = 0). First half of the table consiststhe case when the uncertainty in the neutron lifetime δτ is 1.0 second whereas thesecond half is when δτ = 0.1 seconds. The Pull values for the Fierz terms havebeen calculated with respect to its standard model value (b = 0). The fitted valueof the parameters corresponds to a CL of 68.3% which for a three parameter fit(m = 3) is defined by ∆χ2 = 3.53 [Table 4.1]. We use the inputs λ = 1.2732 [55],nevents = 1× 1010 and nbins = 1000.

xexp Fitted b Pull b Fitted λ Pull λ χ2min/ndof

Γ -0.00467 -62.53 - - 1109.97/998

δτ = 1.0 secondΓ, A, τ -0.00505 -67.60 1.27315 -0.68 2068.27/1998Γ, a, τ -0.00467 -62.60 1.27316 -0.43 2092.64/1998a,A, τ -0.00361 -2.52 1.27342 0.72 1940.97/1999Γ, a, A, τ -0.00467 -62.58 1.27319 -0.17 3051.49/2998

δτ = 0.1 secondsΓ, A, τ -0.00505 -68.24 1.27317 -0.53 2068.27/1998Γ, a, τ -0.00467 -63.11 1.27318 -0.32 2092.72/1998a,A, τ -0.00455 -9.68 1.27321 0.12 1941.45/1999Γ, a, A, τ -0.00467 -63.28 1.27319 -0.21 3051.48/2998

A closer inspection of the Table 4.10 also confirms that the uncertainty in the λis very sensitive to the value of the uncertainty in the neutron lifetime (δτ) as can beseen from the reduced pull values for λ for the case when δτ = 0.1 seconds. However,decreasing the uncertainty in the neutron lifetime further increases the pull valuesfor the Fierz term. This suggests some sort of association between the fit parametersand the neutron lifetime.

It should also be noted that λ shows small pull values which is in contrast withwhat is observed in [37,38]. This can be explained easily by observing that λ is fittedas the only fit parameter in the global fits in [37] whereas in each of the cases ofxexp discussed in the Table 4.10, we used ya = λ, b,N19. Hence, for 68.3%CL the error we obtained for λ would be much greater than the one obtained in [37]which will bring down the pull values for λ.

In the next step we performed the Rfit technique on this dataset where we allowthe second-class currents f3 and g2 to vary freely in some predefined interval. First,we analyzed the case where the uncertainty on the neutron lifetime δτ is 0.1 secondsand later we tried to implement the exact same recipes with δτ = 1 second. In bothof these cases, we tried two different intervals in which we can vary the second-classcurrents. The results of this analysis are summarized in Table 4.11.

19 More specifically, as mentioned in section 4.3.3, we fit for only b,N in the electron energyspectrum and fit for λ, b in the correlation coefficients and the neutron lifetime for global fits

72

Table 4.11: Fitted values for λ, b from simultaneous fits of the observables xexp =Γ, a, A, τ to the new physics data set with Rfit technique. First half of the table con-sists the case when the uncertainty in the neutron lifetime δτ is 1.0 second whereas thesecond half is when δτ = 0.1 seconds. The fitted value of the parameters correspondsto a CL of 68.3% which for a three parameter fit (m = 3) is defined by ∆χ2 = 3.53[Table 4.1]. Input values of λ = 1.2732 [55], bBSM = −0.00487, nevents = 1× 1010 andnbins = 1000.

f3, g2 Fitted b σb Fitted λ σλ χ2min/ndof

δτ = 1.0 second0 -0.00467 0.000135 1.2732 0.0001 3051.49/2998

[−0.5, 0.5] -0.00473 0.000140 1.27333 0.000088 3053.63/2998

δτ = 0.1 seconds0 -0.00467 0.000135 1.2732 0.00009 3051.49/2998

[−0.5, 0.5] -0.00485 0.000155 1.2733 0.000088 3053.93/2998

From Table 4.11, it is clear that the χ2min values for each of the cases are almost

equal to each other. We observed a similar behavior for the case of the global fitsfor standard model dataset [Table 4.9]. However, the obtained values of the χ2

min areworse as compared to their standard model dataset counterpart. As expected, theuncertainty in the Fierz term increases because of the second-class currents, it alsoresulted in large pull values with respect to the standard model value of the Fierzterm (b = 0).

Table 4.11 also revealed similar pull values for λ as Table 4.9. This suggests thatλ has almost negligible association with the new physics couplings, it does, however,has some association with the second-class currents which manifests itself due to thepresence of g2 in the expression of the neutron lifetime Eq. 4.19 as well as the recoilorder terms for the correlation coefficients Eq. 4.6 and 4.7.

(xexp = Γ, a, A, τ).

73

Chapter 5 Thomas Rotation and its effects on the Electromagnetic Fields

5.1 Transformations of the Electromagnetic Field Tensor

The central idea of this chapter is to investigate how the electromagnetic fieldstransform relativistically when there is accelerated motion. The motivation behindit concerns the impact of field non-uniformities on the particles’ Larmor precessionfrequencies and spin relaxation rates. Spin-1/2 particles in the presence of electro-magnetic fields undergo Larmor precession and the frequency associated with theLarmor precession is sensitive to the non-uniformities in the surrounding electro-magnetic fields. Considerations of such are then of paramount importance to theinterpretation of results from a precision measurement of a Larmor spin precessionfrequency. To our knowledge, the case of non-collinear boosts and their effects onthe electromagnetic field tensor had never before been published; thus, our work,published as [51], represents the first attempt to carry out these calculations.

As we already know from Chapter 1, a Lorentz transformation can be representedas a boost matrix. The non-uniform motion can also be broken down to boost matriceswith infinitesimal boost vector δ~β which is applied to the system in the time intervalδt. Because of the non-commuting nature of the Lorentz transformations, successiveLorentz transformations depend on the order in which they are applied. As statedearlier in Chapter 1, two successive Lorentz transformations are equivalent to a singleLorentz transformation followed by a three-dimensional rotation which is known asThomas rotation. The consequence of this coordinate rotation can be seen in thephysical quantities which can be represented by 4-vectors like electromagnetic fields,angular momentum, etc. It adds a correction term to these physical quantities whichis dependent on “Thomas” angular velocity (ωT ).

This work is particularly interested in the effects of the accelerated motions of therelativistic charged particles on the electromagnetic fields in their rest frame as well asthe lab frame. We will first inspect the transformations in the longitudinal-transverseframe (rest frame) of the particle and then apply the same transformations to the labframe.

5.1.1 Longitudinal-Transverse `t-frame

To see the effects on electromagnetic fields, we first need to bring the electromag-netic field tensor to the rotating `t-frame so that all the boosts and electromagneticfields are in the same frame to start with. The electromagnetic field tensor F µν inthe lab frame is given by [24]:

F µν =

0 −Ex −Ey −EzEx 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

(5.1)

For the rest of the chapter we will write F µν = F .

74

To get the field tensor in the `t-frame, we can apply the rotation matrix R fromEq. 1.84 on F µν :

F `t = R · F ·RT (5.2)

where the superscript T refers to matrix transpose. After plugging in the values of Ffrom Eq. 5.1 above and R, we can write F `t as:

F `t =

0 −κ1

λ1−κ2η1

− κ3η1λ1

κ1λ1

0 − κ6η1λ1

κ5η1

κ2η1

κ6η1λ1

0 −κ4λ1

κ3η1λ1

−κ5η1

κ4λ1

0

(5.3)

where

κ1 = βxEx + βyEy + βzEz

κ2 = βxEy − βyExκ3 = Ezβ

2x − βxβzEx + βy (βyEz − βzEy)

κ4 = Bxβx +Byβy +Bzβz

κ5 = Byβx −Bxβy

κ6 = Bzη21 − (Bxβx +Byβy) βz

To the electromagnetic field tensor obtained in Eq. 5.2, we will apply boost matrixfor the first successive boost ~β`t and the direct boost (~β+ δ~β)`t using the well-knownequation [24]:

F ′ = A · F · AT (5.4)

where F ′ and F are the electromagnetic field tensors in the boosted frame and thelab frame (or any inertial frame) respectively and A is the boost matrix.

For boost ~β`t, the transformation of F `t can be calculated using Eq. 1.92, 5.2 and5.3:

(F ′)`t = A(~β`t) · F `t · (A(~β`t))T

Table 5.1 shows the elements of the field tensor (F ′)`t after matrix multiplication.The electromagnetic fields in Table 5.1 are consistent with the standard field

transformation equations [24,25].

~E ′ = γ(~E + ~β × ~B

)− γ2

γ + 1~β(~β · ~E)

~B′ = γ(~B − ~β × ~E

)− γ2

γ + 1~β(~β · ~B) (5.5)

Similarly, for the direct boost (~β+ δ~β)`t, the electromagnetic field tensor transforma-tion is given by:

(F ′′)`t = A((~β + δ~β)`t) · F `t · A((~β + δ~β)`t)T (5.6)

75

Table 5.1: Expressions for the indicated components of the electromagnetic fieldtensor after being transformed by the first boost ~β`t in the longitudinal-transverseframe.

(F ′µν)`t

F ′10 βxEx+βyEy+βzEz

λ1

F ′20γ(−Bzη21+Bxβxβz+Byβyβz−Exβy+βxEy)

η1

F ′30γ(Byβxλ21−Bxβyλ21+β

2xEz−βxExβz−βyEyβz+β2

yEz)η1λ1

F ′32 Bxβx+Byβy+Bzβzλ1

F ′13γ(Byβx−Bxβy+β2

xEz−βxExβz−βyEyβz+β2yEz)

η1

F ′21γ(Bzη21−Bxβxβz−Byβyβz−β3

xEy+β2xExβy−βxβ2

yEy+Exβ3y−βxEyβ2

z+Exβyβ2z)

η1λ1

After simplification and keeping terms to linear order in δβ, (F ′′)`t can be calculatedand the detailed expressions of its elements are provided in the Appendix section5.4.2.

Because of the way we set up the problem, the electromagnetic field tensor de-scribed by the direct boost (~β + δ~β)`t already consists of rotations. To get the elec-tromagnetic fields which do not have any rotations (pure Lorentz boost) [24], we willuse the successive boosts. In particular, we will calculate a matrix AT :

AT = A(~β + δ~β) · A(−~β) (5.7)

For the `t-frame, AT looks like:

A`tT =

1 −γ2λ2

λ1−γλ6

η1− γλ5η1λ1

−γ2λ2λ1

1 (γ−1)λ6η1λ1

(γ−1)λ5η1λ21

−γλ6η1

− (γ−1)λ6η1λ1

1 0

− γλ5η1λ1

− (γ−1)λ5η1λ21

0 1

(5.8)

The matrix AT contains all the information regarding relativistic composition of ve-locities and Thomas rotation which can be seen if we write AT as [24]:

A`tT = A(∆~β)`t ·Rtom(∆~Ω)`t =(I −∆~β`t · ~K

)·(I −∆~Ω`t · ~S

)(5.9)

where:∆~β`t = successive boost with respect to frame with boost ~β;

∆~Ω`t =[(

γ−1β2

)~β`t × δ~β`t

]is the angle of rotation associated with Thomas rotation.

76

Matrices ~K and ~S are the generators of Lorentz boosts and rotations respectively:

K1 =

0 1 0 01 0 0 00 0 0 00 0 0 0

, K2 =

0 0 1 00 0 0 01 0 0 00 0 0 0

, K3 =

0 0 0 10 0 0 00 0 0 01 0 0 0

S1 =

0 0 0 00 0 0 00 0 0 −10 0 1 0

, S2 =

0 0 0 00 0 0 10 0 0 00 −1 0 0

, S3 =

0 0 0 00 0 −1 00 1 0 00 0 0 0

(5.10)

It can be easily shown that if the boosts and rotations are infinitesimal then:

A(∆~β) ·Rtom(∆~Ω) = Rtom(∆~Ω) · A(∆~β) (5.11)

Extracting the matrix form of A(∆~β`t) and Rtom(∆~Ω`t) from AT , we get:

A(∆~β)`t =

1 −γ2λ2

λ1−γλ6

η1− γλ5η1λ1

−γ2λ2λ1

1 0 0

−γλ6η1

0 1 0

− γλ5η1λ1

0 0 1

Rtom(∆~Ω)`t =

1 0 0 0

0 1 (γ−1)λ6η1λ1

(γ−1)λ5η1λ21

0 − (γ−1)λ6η1λ1

1 0

0 − (γ−1)λ5η1λ21

0 1

(5.12)

In order to find the electromagnetic fields due to pure Lorentz boosts, we calculatethe electromagnetic field tensor due to the successive boosts ~β`t and ∆~β`t:

(F ′′′)`t = A(∆~β)`t · A(~β)`t · F `t · (A(~β)`t)T · (A(∆~β)`t)T

= A(∆~β)`t · (F ′)`t · (A(∆~β)`t)T (5.13)

After simplification and keeping the terms which are linear in δβ, we get the matrix(F ′′′)`t whose elements are provided in the Appendix section 5.4.2. It should be notedthat since (F ′′′)`t and (F ′′)`t are different from each other by just a rotation, so (F ′′′)`t

can be obtained by operating an inverse Thomas rotation on (F ′′)`t. In fact, we usedthis as a check for verifying if the expressions of electromagnetic fields calculatedusing Eq. 5.12 are correct.

(F ′′′)`t = Rtom(−∆~Ω)`t · (F ′′)`t · (Rtom(−∆~Ω)`t)T (5.14)

77

5.1.2 Laboratory xy-frame

After getting the expressions of electromagnetic fields in the `t-frame obtainedby different boosts, we now calculate the electromagnetic fields by the same boostswith respect to the lab frame. The overall approach stays the same but all the boostmatrices are needed to be transformed in the xy-frame before being used to calculatethe electromagnetic field tensor. Another way of calculating the electromagnetic fieldtensor is to directly transform the field tensors obtained in `t-frame.

In order to calculate the electromagnetic field tensor for various boosts in the labxy-frame, we will just use the field tensor F as defined in Eq. 5.1. Since R is therotation matrix for passive coordinate transformations Eq. 1.84, we have:

R ·RT = RT ·R = I (5.15)

therefore we can write the electromagnetic tensors and boost matrices in the labxy-frame as:

F xy = RT · F `t ·RAxy = RT · A`t ·R (5.16)

After matrix multiplication, A(~β)xy can be written as:

A(~β)xy =

γ −γβx −γβy −γβz

−γβxγβ2

x+β2y+β

2z

λ21

(γ−1)βxβyλ21

(γ−1)βxβzλ21

−γβy (γ−1)βxβyλ21

β2x+γβ

2y+β

2z

λ21

(γ−1)βyβzλ21

−γβz (γ−1)βxβzλ21

(γ−1)βyβzλ21

β2x+β

2y+γβ

2z

λ21

(5.17)

which is in agreement with Eq. 1.77 if we substitute in

~β = ~βxy = βxx+ βyy + βz z

Using Eq. 5.3 and 5.16 we can calculate the electromagnetic field tensor (F ′)xy which

corresponds to the boost ~βxy as shown in Table 5.2. These components of the elec-tromagnetic field tensor can also be verified from the standard field transformationsas shown in Eq. 5.5.

Similarly, for the direct boost (~β + δ~β)xy, we can use Eq. 5.16 to calculate the

boost matrix. The detailed expression of A(~β + δ~β)xy is too long to write here but itshares the same features as [24] which can be seen if we let δβz = βy = βz = 0:

A(~β + δ~β)xy =

γ + γ3βxδβx −(γβx + γ3δβx) −γδβy 0

−(γβx + γ3δβx) γ + γ3βxδβx

(γ−1β2x

)βxδβy 0

−γδβy(γ−1β2x

)βxδβy 0 1

0 0 0 1

(5.18)

78

Table 5.2: Expressions for the indicated components of the electromagnetic fieldtensor after being transformed by the first boost ~βxy in the laboratory frame.

(F µν)xy

F ′10 γEx + γ (Bzβy −Byβz)− γ2βx(βxEx+βyEy+βzEz)

γ+1

F ′20 γEy + γ (Bxβz −Bzβx)− γ2βy(βxEx+βyEy+βzEz)

γ+1

F ′30 γEz + γ (Byβx −Bxβy)− γ2βz(βxEx+βyEy+βzEz)

γ+1

F ′32 γBx + γ (Eyβz − βyEz)− γ2βx(Bxβx+Byβy+Bzβz)

γ+1

F ′13 γBy + γ (Ezβx − βzEx)− γ2βx(Bxβx+Byβy+Bzβz)

γ+1

F ′21 γBz + γ (Exβy − βxEy)− γ2βx(Bxβx+Byβy+Bzβz)

γ+1

The above matrix is identical in form to the one shown in [24]. After calculating theboost matrix Eq. 5.18, we can again use Eq. 5.3 to calculate the electromagneticfield tensor in the direct boosted frame with respect to the laboratory whose detailedexpressions are provided in the Appendix section 5.4.3.

In order to calculate electromagnetic fields in the inertial frames which are boostedupon by pure Lorentz boosts (no rotation), we use successive boosts ~β and ∆~β. Forthat we have to calculate the expression of AxyT first as done in Eq. 5.7 which is:

AxyT =1 − 1

λ21(γλ3 + γ2βxλ2) − 1

λ21(γλ4 + γ2βyλ2) − 1

λ21(γλ5 + γ2βzλ2)

− 1λ21

(γλ3 + γ2βxλ2) 1 (γ−1)λ6λ21

− (γ−1)λ8λ21

− 1λ21

(γλ4 + γ2βyλ2) − (γ−1)λ6λ21

1 (γ−1)λ7λ21

− 1λ21

(γλ5 + γ2βzλ2)(γ−1)λ8

λ21− (γ−1)λ7

λ211

(5.19)

Using the value of AxyT from Eq. 5.19 and 5.9, we can again extract A(∆~β)xy and

79

Rtom(∆~Ω)xy:

A(∆~β)xy =1 − 1

λ21(γλ3 + γ2βxλ2) − 1

λ21(γλ4 + γ2βyλ2) − 1

λ21(γλ5 + γ2βzλ2)

− 1λ21

(γλ3 + γ2βxλ2) 1 0 0

− 1λ21

(γλ4 + γ2βyλ2) 0 1 0

− 1λ21

(γλ5 + γ2βzλ2) 0 0 1

(5.20)

Rtom(∆~Ω)xy =

1 0 0 0

0 1 (γ−1)λ6λ21

− (γ−1)λ8λ21

0 − (γ−1)λ6λ21

1 (γ−1)λ7λ21

0 (γ−1)λ8λ21

− (γ−1)λ7λ21

1

(5.21)

where λi, (i = 1, 2, · · · , 8) and η1 are defined in the Appendix section 5.4.1.Using Eq. 5.20 and 5.21, we can now calculate electromagnetic fields due to pure

Lorentz boosts whose detailed expressions are provided in the Appendix section 5.4.3.

(F ′′′)xy = A(∆~β)xy · A(~β)xy · F xy · (A(~β)xy)T · (A(∆~β)xy)T

= A(∆~β)xy · (F ′)xy · (A(∆~β)xy)T (5.22)

5.2 Validation of Results

All of the framework that we have constructed can be verified by two ways:

1. Verifying the form of boost matrices and electromagnetic field tensor for somespecial cases as discussed here [24].

2. Applying this whole formalism on a 4-vector like position.

For the first approach, in order to see the identical nature of results we will assumethe special case of βy = βz = δβz = 0. Applying this assumption on Eq. 1.92 and1.93 will give us:

A(~β)`t =

γ −γβx 0 0−γβx γ 0 0

0 0 1 00 0 0 1

A(~β + δ~β)`t =

γ + γ3βxδβx −(γβx + γ3δβx) −γδβy 0

−(γβx + γ3δβx) γ + γ3βxδβx

(γ−1βx

)δβy 0

−γδβy(γ−1βx

)δβy 1 0

0 0 0 1

(5.23)

80

Similarly, A`tT can be reduced to a familiar result [24]:

A`tT =

1 −γ2δβx −γδβy 0

−γ2δβx 1 (γ−1)δβyβx

0

−γδβy − (γ−1)δβyβx

1 0

0 0 0 1

(5.24)

In the lab xy-frame, we get the exact same results as Eq. 5.23 and 5.24 for the abovementioned special case. This makes perfect sense since letting βy = βz = δβz = 0would just make the original passive coordinate transformations redundant and boththe `t- and xy- frames will be identical.

To see if the matrix for Thomas rotation Rtom(∆~Ω) is correct we can directlycalculate it from its definition:

Rtom(∆~Ω) =(I −∆~Ω · ~S

)(5.25)

where ∆~Ω =[(

γ−1β2

)~β × δ~β

].

The Thomas rotation matrix calculated from the Eq. 5.25 using the correspondingrepresentations of the boost vectors in `t/xy -frames matches with Eq. 5.12, 5.20,and 5.21.

For the verification of Electromagnetic Field Tensors, we can calculate them indifferent ways. As an example, we calculated (F ′′′)xy using:

(F ′′′)xy = A(∆~β)xy · A(~β)xy · F xy · (A(~β)xy)T · (A(∆~β)xy)T

= Rtom(−∆~Ω)xy · (F ′′)xy · (Rtom(−∆~Ω)xy)T (5.26)

= RT · (F ′′′)`t ·R

All three equations yielded same results. Similar verification also holds for otherelectromagnetic field tensors involved.

Our second approach for verification is based on Ungar et al. [25,31,32] in which

we apply direct boost (~β+δ~β) and successive boosts ~β and ∆~β to a position 4-vector.We can check if the results are consistent and share the same overall features as theelectromagnetic field tensor. To see that we start with a general position 4-vectorin the lab frame and for simplicity, we ignore the time component in the position4-vector:

(r)xy =

0xyz

(5.27)

81

Transforming it in the `t- frame using the rotation matrix R from Eq. 1.84, we get:

(r)`t = R · rxy =

0

xβx+yβy+zβzλ1

yβx−xβyη1

zβ2x−xβzβx+βy(zβy−yβz)

η1λ1

(5.28)

We can calculate the expression of (r)`t transformed by the first successive boost ~β`t

using Eq. 1.92 in the same way we calculated the electromagnetic field tensor F µν :

(r′)`t = A(~β)`t · (r)`t =

−γ (xβx + yβy + zβz)

γ(xβx+yβy+zβz)

λ1

yβx−xβyη1

zβ2x−xβzβx+βy(zβy−yβz)

η1λ1

(5.29)

which is nothing but the standard Lorentz transformation of coordinates.Similarly, for the direct boost (~β+ δ~β)`t and successive boosts ~β`t and ∆~β`t, after

letting βz = δβz = 0 for simplicity, we have:

(r′′)`t = A(~β + δ~β)`t · (r)`t

=

−γ(xβ3x+(xδβxγ2+yβy+yδβy)β2

x+βy(xβy+(γ2−1)(yδβx+xδβy))βx+β2y(yδβyγ2+yβy+xδβx))

η21

(γ−1)(yβx−xβy)(βxδβy−βyδβx)+(xβx+yβy)η21((βxδβx+βyδβy)γ3+γ)η31

yβ3x−x(βy−(γ−1)δβy)β2

x+βy(yβy−(γ−1)(xδβx−yδβy))βx−β2y(xβy+y(γ−1)δβx)

η31

z

(r′′′)`t = A(∆~β)`t · A(~β)`t · (r)`t = A(∆~β)`t · (r′)`t

=

γ(− (xβx+yβy)(βxδβx+βyδβy)γ2

η21− xβx − yβy − (yβx−xβy)λ6

η21

)γ(xβx+yβy)(βxδβxγ2+βyδβyγ2+1)

η1

xγ2δβyβ2x+(βy(yδβy−xδβx)γ2+y)βx−βy(yβyδβxγ2+x)

η1

z

(5.30)

One way to check the validity of Eq. 5.29 and 5.30 is to show that the invariantinterval ds2 [24]:

ds2 = gµνdxµdxν = (dx0)2 − (dx1)2 − (dx2)2 − (dx3)2 (5.31)

82

remains the same, where gµν is the metric tensor:

gµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

(5.32)

In our case we are concerned with the invariance of

s2 = x20 − x21 − x22 − x23 (5.33)

To see if that is the case, we can apply Eq. 5.33 to (r′)`t, (r′′)`t and (r′′′)`t calculatedabove. The invariant

s2 = −x2 − y2 − z2

indeed stays the same for each case. This makes sense since we ignored the timecomponent.

Similar results can be obtained for the position 4-vector r in the lab xy -frameand it can be easily proved that the invariant does not change.

5.3 Comparison with the previous work

The work presented in this paper is another confirmation of the fact that twosuccessive boosts are not equal to a single direct boost only if the boosts are non-collinear. In fact, for non-collinear boosts (accelerated motion), just applying theusual electromagnetic field transformation equations will not result in the correctform of electromagnetic fields as Thomas rotation must be included.

Apart from the validations made in the previous section we will see if the elec-tromagnetic field tensors in the direct boosted frame ~β + δ~β and the successivelyboosted frames ~β and ∆~β are consistent with the Thomas rotation. To see that wecan take the difference between the corresponding elements of F ′′ and F ′′′ in boththe longitudinal-transverse `t and lab xy-frames.

After taking the difference of the electromagnetic field tensors F ′′ and F ′′′ wefound that

(F ′′)ij − (F ′′′)ij ∝ (γ − 1) (5.34)

for both `t- and xy-frames. This makes sense because both F ′′ and F ′′′ just differ byThomas rotation. Although taking the difference of F ′′ and F ′′′ is not very significantphysically it does show what we expected.

To our knowledge, this is the first time that someone has calculated the expressionsof the electromagnetic fields in the frames corresponding to general three-dimensionalnon-collinear boosts.

One application of this work concerns the calculation of shifts in the Larmorfrequency of highly relativistic particles moving through non-uniform magnetic andelectric fields. Such a formalism was developed for the motion of non-relativistic parti-cles [52–54]; however, this formalism is not directly applicable to relativistic particlesbecause the formalism assumes the electromagnetic fields are known in the particle

83

rest frame. For a highly relativistic particle undergoing acceleration (e.g., relativisticcharged particles stored by electromagnetic fields within a circular storage ring), onecan then apply the formalism developed here in this chapter to determine the electro-magnetic fields in an appropriate reference frame, where any residual motion of theparticle is then non-relativistic, and then proceed to calculate the frequency shifts asper the formalism of [52–54].

5.4 Appendix

5.4.1 Notation

We begin by recalling that the electromagnetic field tensor F µν is an anti-symmetricmatrix:

F µν =

0 −Ex −Ey −EzEx 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

(5.35)

Defining some dummy variables to make equations look cleaner:

λ1 =√β2x + β2

y + β2z

λ2 = βxδβx + βyδβy + βzδβz

λ3 = β2yδβx − βxβyδβy + βz (βzδβx − βxδβz)

λ4 = β2xδβy − βxβyδβx + βz (βzδβy − βyδβz)

λ5 = β2xδβz − βxβzδβx + βy (βyδβz − βzδβy)

λ6 = βxδβy − βyδβx (5.36)

λ7 = βyδβz − βzδβyλ8 = βzδβx − βxδβz

η1 =√β2x + β2

y

η2 =√β2y + β2

z

η3 =√β2x + β2

z

The detailed expressions of the electromagnetic fields (to the first order in δβ) is givenin the next two sections.

84

5.4.2 Longitudinal-Transverse `t-Frame

5.4.2.1 Direct Boosted Frame (~β + δ~β)

(F ′′)10 =1

λ1

[βxEx + βyEy + βzEz − λ3Exγ2 − λ4Eyγ2 − λ5Ezγ2 +Bzλ6γ

+Byλ8γ +Bxλ7γ +γ(γ − 1)η22δβxEx

λ21+γ(γ − 1)η23δβyEy

λ21+γ(γ − 1)η21δβzEz

λ21

−γ(γ − 1)βxβyδβyExλ21

− γ(γ − 1)βxβzδβzExλ21

− γ(γ − 1)βxβyδβxEyλ21

−γ(γ − 1)βyβzδβzEyλ21

− γ(γ − 1)βxβzδβxEzλ21

− γ(γ − 1)βyβzδβyEzλ21

](5.37)

(F ′′)20 = −γBzη1 +γBxβxβz

η1+γByβyβz

η1− γExβy

η1+γβxEyη1

− γ3Bzβxη1δβxλ21

+γ2δβxβyβzEz

η1− γ3Bzβyη1δβy

λ21+γ (γ2 − 1)Bxβ

2xβzδβx

η1λ21+γBxβxη1δβz

λ21

+γ (γ2 − 1)Byβxδβxβyβz

η1λ21− γBzβxδβxβ

2z

η1λ21− γBzβyδβyβ

2z

η1λ21+γ3Bxβxβ

2zδβz

η1λ21

+γ (γ2 − 1)Byβ

2yδβyβz

η1λ21+γ (γ2 − 1)Bxβxβyδβyβz

η1λ21− γ (γ2 − 1)Bzβzη1δβz

λ21

− (γ − 1)γ2βxδβxExβyη1

+γ3Byβyβ

2zδβz

η1λ21+γByβyη1δβz

λ21−γ3Exβ

2yδβy

η1

− γ2β2xExδβyη1

− γ3Exβyβzδβzη1

+γ3β2

xδβxEyη1

+γ3βxEyβzδβz

η1− γ2βxδβyβzEz

η1

+(γ − 1)γβ2

xExδβyη1λ21

− (γ − 1)γβxδβxExβyη1λ21

+(γ − 1)γ2βxβyδβyEy

η1

−(γ − 1)γδβxβ

2yEy

η1λ21+γ2δβxβ

2yEy

η1+

(γ − 1)γβxδβyβzEzη1λ21

− (γ − 1)γδβxβyβzEzη1λ21

+(γ − 1)γβxβyδβyEy

η1λ21(5.38)

85

(F ′′)30 =γByβxλ1

η1− γBxβyλ1

η1− γβxExβz

η1λ1− γβyEyβz

η1λ1+γEzη1λ1

+γ3Byβ

2xδβx

η1λ1

+γByδβxβ

2y

η1λ1− γBxβ

2xδβy

η1λ1−γ3Bxβ

2yδβy

η1λ1− γBzβzλ6

η1λ1+γ (γ2 − 1)Byβxβyδβy

η1λ1

− γ (γ2 − 1)Bxβxδβxβyη1λ1

− (γ − 1)γβ2xδβxExβz

η1λ31+γ3Byβxβzδβz

η1λ1− γ3Bxβyβzδβz

η1λ1

− γ2βxExδβzη1λ1

− (γ − 1)γ2β2xδβxExβz

η1λ1− (γ − 1)γ2βxExβyδβyβz

η1λ1− γ3βxExβ

2zδβz

η1λ1

+(γ − 1)γβxExδβzη1

λ31− (γ − 1)γβxExβyδβyβz

η1λ31− (γ − 1)γ2βxδβxβyEyβz

η1λ1

−(γ − 1)γ2β2

yδβyEyβz

η1λ1− γ2βyEyδβzη1

λ1+

(γ − 1)γβyEyλ5η1λ31

− γ3βyEyβ2zδβz

η1λ1

+(γ − 1)γβzδβzEzη1

λ31+γ3βyδβyEzη1

λ1+γ3βxδβxEzη1

λ1+γ2βxδβxβ

2zEz

η1λ1

− (γ − 1)γβyδβyβ2zEz

η1λ31+

(γ − 1)γ2βzδβzEzη31

λ31+

(γ − 1)γ2β3zδβzEzη1

λ31

− (γ − 1)γβxδβxβ2zEz

η1λ31+γ2βyδβyβ

2zEz

η1λ1(5.39)

(F ′′)13 = −γBxβyη1

+γByβxη1

− γβxExβzη1

− γβyEyβzη1

+ γEzη1 −γ3Bxβxδβxβy

η1

−γ3Bxβ

2yδβy

η1− γ3Bxβyβzδβz

η1+γ3Byβxβyδβy

η1+γ3Byβ

2xδβx

η1+γ3Byβxβzδβz

η1

− γ3βxExβ2zδβz

η1λ21+

(γ − 1)Bxβxδβxβyη1λ21

− (γ − 1)Bxβ2xδβy

η1λ21+

(γ − 1)Byδβxβ2y

η1λ21

+(γ − 1)Bzδβxβyβz

η1λ21− (γ − 1)Byβxβyδβy

η1λ21− (γ − 1)Bzβxδβyβz

η1λ21− γβxExδβzη1

λ21

− (γ2 − 1) γβ2xδβxExβz

η1λ21− (γ2 − 1) γβxExβyδβyβz

η1λ21−γ (γ2 − 1) β2

yδβyEyβz

η1λ21

− γ (γ2 − 1) βxδβxβyEyβzη1λ21

− γβyEyδβzη1λ21

− γ3βyEyβ2zδβz

η1λ21+γ3βyδβyEzη1

λ21

+γ3βxδβxEzη1

λ21+γβxδβxβ

2zEz

η1λ21+γ (γ2 − 1) βzδβzEzη1

λ21+γβyδβyβ

2zEz

η1λ21(5.40)

86

(F ′′)12 =γBzη1λ1

− γBxβxβzη1λ1

− γByβyβzη1λ1

+γExβyλ1

η1− γβxEyλ1

η1+

(γ − 1)Bxβ2xδβxβz

η1λ31

− γ3Bxβ2xδβxβz

η1λ1−γδβxβ

2yEy

η1λ1− γ3Byβxδβxβyβz

η1λ1+γ3Bzβxδβxη1

λ1+γ3Bzβyδβyη1

λ1

− γ3Bxβxβyδβyβzη1λ1

−γ3Byβ

2yδβyβz

η1λ1− γ3Bxβxβ

2zδβz

η1λ1+

(γ − 1)Bzβxδβxβ2z

η1λ31

+(γ − 1)Byβxδβxβyβz

η1λ31+

(γ − 1)Bxβxβyδβyβzη1λ31

+(γ − 1)Byβ

2yδβyβz

η1λ31

+(γ − 1)Bzβyδβyβ

2z

η1λ31− (γ − 1)Bxβxδβzη1

λ31− (γ − 1)Byβyδβzη1

λ31+γβxδβyβzEz

η1λ1

+γ (γ2 − 1) βxδβxExβy

η1λ1− (γ2 − 1) γβxβyδβyEy

η1λ1− γ3Byβyβ

2zδβz

η1λ1+γ3Bzβzδβzη1

λ1

+γ3Exβ

2yδβy

η1λ1+γ3Exβyβzδβz

η1λ1+γβ2

xExδβyη1λ1

− γ3β2xδβxEyη1λ1

− γ3βxEyβzδβzη1λ1

− (γ − 1)Bzβzδβzη1λ31

− γδβxβyβzEzη1λ1

(5.41)

(F ′′)32 =1

λ1

[Bxβx +Byβy +Bzβz − γExλ7 − γEyλ8 − γEzλ6 −

(γ − 1)Bzλ5λ21

−(γ − 1)Byλ4λ21

− (γ − 1)Bxλ3λ21

](5.42)

5.4.2.2 Successively Boosted Frame ~β and ∆~β

(F ′′′)10 =1

λ1

[βxEx + βyEy + βzEz + γ2Bzλ6 + γ2Byλ8 + γ2Bxλ7 + γ2βxExβyδβy

+γ2βxExβzδβz − γ2δβxExη22 + γ2βyEyβzδβz − γ2δβyEyη23 + γ2βxδβxβyEy

+γ2βyδβyβzEz + γ2βxδβxβzEz − γ2δβzEzη21]

(5.43)

(F ′′′)20 =2γByβyβz

η1− γBzη1 −

γExβyη1

+γβxEyη1

− γ3Bzη1λ2λ21

+γBzβzλ5η1λ21

− γ3Exβyλ2η1

+γ3βxEyλ2

η1+

2γByβyδβzη1λ21

+2γ (γ2 − 1)Byβxδβxβyβz

η1λ21+

2γ3Byβyβ2zδβz

η1λ21

+2γ (γ2 − 1)Byβ

2yδβyβz

η1λ21(5.44)

87

(F ′′′)30 =γByβxλ1

η1− γBxβyλ1

η1− γβxExβz

η1λ1− γβyEyβz

η1λ1+γEzη1λ1

− γBxβxλ6η1λ1

− γByβyλ6η1λ1

− γBzβzλ6η1λ1

+γ3Byβxλ2η1λ1

− γ3Bxβyλ2η1λ1

− γ3βxExβzλ2η1λ1

− γ3βyEyβzλ2η1λ1

+γ3Ezη1λ2

λ1(5.45)

(F ′′′)32 =1

λ1

[Bxβx +Byβy +Bzβz − γ2Exλ7 − γ2Eyλ8 − γ2Ezλ6 + γ2Bzβxδβxβz

+γ2Bxβxβyδβy + γ2Bzβyδβyβz − γ2Bxδβxη22 − γ2Byδβyη

23 − γ2Bzδβzη

21

+γ2Byβyβzδβz + γ2Bxβxβzδβz + γ2Byβxδβxβy]

(5.46)

(F ′′′)13 = −γBxβyη1

+γByβxη1

− γβxExβzη1

− γβyEyβzη1

+ γEzη1 −γ3Bxβyλ2

η1− γβxExλ5

η1λ21

+γ3Byβxλ2

η1− γ3βxExβzλ2

η1λ21− γ3βyEyβzλ2

η1λ21− γβyEyλ5

η1λ21+γ3Ezη1λ2

λ21− γβzEzλ5

η1λ21(5.47)

(F ′′′)21 =γBzη1λ1

− γBxβxβzη1λ1

− γByβyβzη1λ1

+γExβyλ1

η1− γβxEyλ1

η1+γ3Bzη1λ2

λ1

+γ3Exβyλ2η1λ1

− γ3Bxβxβzλ2η1λ1

− γ3Byβyβzλ2η1λ1

+γβxExλ6η1λ1

− γ3βxEyλ2η1λ1

+γβyEyλ6η1λ1

+γβzEzλ6η1λ1

(5.48)

88

5.4.3 Laboratory xy-Frame

5.4.3.1 Direct Boosted Frame (~β + δ~β)

(F ′′)10 = γBzβy − γByβz − γ2β2xEx +

1

λ21

[γEx

(γβ2

x + β2y + β2

z

)− (γ − 1)βxβyEy

−(γ − 1)βxβzEz − γ3Byβzλ2 + γ3Bzβyλ2 + γBzλ4 + γ3βxδβxExη22

+γ3Exβzδβzη22 − γ3βxβyδβyβzEz − γ3βxβ2

zδβzEz − γ3β2xδβxβyEy − γ3βxβ2

yδβyEy

−γByλ5 − γ3βxβyEyβzδβz − γ3β2xδβxβzEz + γ3Exβyδβyη

22 +

2(γ − 1)β2xExβyδβyλ21

+2(γ − 1)β2

xExβzδβzλ21

−(γ − 1)δβxβyEy

(−β2

x + β2y + β2

z

)λ21

+2(γ − 1)βxβyEyβzδβz

λ21

−(γ − 1)βxδβyEy

(β2x − β2

y + β2z

)λ21

+2(γ − 1)βxβyδβyβzEz

λ21− 2(γ − 1)βxδβxExη

22

λ21

−(γ − 1)δβxβzEz

(−β2

x + β2y + β2

z

)λ21

−(γ − 1)βxδβzEz

(β2x + β2

y − β2z

)λ21

](5.49)

(F ′′)20 = γBxβz − γBzβx − γ2βxExβy − γ2β2yEy − γ2βyβzEz +

1

λ21

[γ2β2

yEy + γEyη23

+γ3βxδβxEyη23 + γ3βyδβyEyη

23 + (γ − 1)γβxExβy + (γ − 1)γβyβzEz − γ3Bzβ

2xδβx

+(γ2 − 1

)γBxβxδβxβz −

(γ2 − 1

)γBzβxβyδβy − γBzδβxη

22 + γ3Bxβ

2zδβz + γBxδβzη

21

−(γ2 − 1

)γBzβxβzδβz + γ

(γ2 − 1

)Bxβyδβyβz − γ3βxExβyβzδβz + γ3Eyβzδβzη

23

−γ3β2xδβxExβy − γ3βxExβ2

yδβy − γ3βxδβxβyβzEz − γ3β2yδβyβzEz − γ3βyβ2

zδβzEz

−(γ − 1)βyδβzEz

(β2x + β2

y − β2z

)λ21

+2(γ − 1)β2

yEyβzδβz

λ21+

2(γ − 1)βxδβxβyβzEzλ21

−(γ − 1)δβxExβy

(−β2

x + β2y + β2

z

)λ21

+2(γ − 1)βxδβxβ

2yEy

λ21− 2(γ − 1)βyδβyEyη

23

λ21

−(γ − 1)βxExδβy

(β2x − β2

y + β2z

)λ21

−(γ − 1)δβyβzEz

(β2x − β2

y + β2z

)λ21

+2(γ − 1)βxExβyβzδβz

λ21

](5.50)

89

(F ′′)30 = γByβx − γBxβy − γ2β2zEz +

1

λ21

[γ3Byβ

2xδβx − (γ − 1)βxExβz − (γ − 1)βyEyβz

+γEz(β2x + β2

y + γβ2z

)+ γ3Byβxβyδβy − γ3Bxβ

2yδβy −

(γ2 − 1

)γBxβxδβxβy

+γByδβxη22 − γByβxβyδβy + γ3Byβxβzδβz − γ3Bxβyβzδβz + γBxβyβzδβz

−γBxδβyη23 − γByβxβzδβz − γ3β2

xδβxExβz − γ3βxExβyδβyβz − γ3βyEyβ2zδβz

−γ3βxExβ2zδβz − γ3βxδβxβyEyβz − γ3β2

yδβyEyβz + γ3βxδβxEzη21 + γ3βyδβyEzη

21

+γ3βzδβzEzη21 −

(γ − 1)δβxExβz(−β2

x + β2y + β2

z

)λ21

+2(γ − 1)βxExβyδβyβz

λ21

−(γ − 1)βxExδβz

(β2x + β2

y − β2z

)λ21

+2(γ − 1)βxδβxβyEyβz

λ21− 2(γ − 1)βzδβzEzη

21

λ21

−(γ − 1)δβyEyβz

(β2x − β2

y + β2z

)λ21

+2(γ − 1)βxδβxβ

2zEz

λ21+

2(γ − 1)βyδβyβ2zEz

λ21

−(γ − 1)βyEyδβz

(β2x + β2

y − β2z

)λ21

](5.51)

(F ′′)32 = γEyβz − γβyEz +1

λ21

[Bxβ

2x + γBxη

22 − (γ − 1)Byβxβy − (γ − 1)Bzβxβz

−γ3Byβ2xδβxβy − γ3Bzβ

2xδβxβz + γ3Bxβxδβxη

22 + γ3Bxβyδβyη

22 − γ3Byβxβ

2yδβy

−γ3Bzβxβyδβyβz + γ3Bxβzδβzη22 − γ3Bzβxβ

2zδβz − γ3Byβxβyβzδβz − γδβyEzη23

+γ3Eyβ2zδβz +

(γ2 − 1

)γβyδβyEyβz + γ

(γ2 − 1

)βxδβxEyβz + γEyδβzη

21

−γ3β2yδβyEz −

(γ2 − 1

)γβxδβxβyEz −

(γ2 − 1

)γβyβzδβzEz −

2(γ − 1)Bxβxδβxη22

λ21

+2(γ − 1)Bxβ

2xβzδβz

λ21−

(γ − 1)Bzδβxβz(−β2

x + β2y + β2

z

)λ21

+2(γ − 1)Bzβxβyδβyβz

λ21

+2(γ − 1)Byβxβyβzδβz

λ21+

2(γ − 1)Bxβ2xβyδβy

λ21−

(γ − 1)Bzβxδβz(β2x + β2

y − β2z

)λ21

−(γ − 1)Byβxδβy

(β2x − β2

y + β2z

)λ21

−(γ − 1)Byδβxβy

(−β2

x + β2y + β2

z

)λ21

](5.52)

90

(F ′′)13 = −γExβz + γβxEz +1

λ21

[Byβ

2y + γByη

23 − (γ − 1)Bzβyβz − (γ − 1)Bxβxβy

−γ3Bxβ2xδβxβy − γ3Bzβ

2yδβyβz − γ3Bxβxβ

2yδβy − γ3Bxβxβyβzδβz + γ3βxβzδβzEz

+γ3β2xδβxEz + γδβxEzη

22 − γβxβyδβyEz − γβxβzδβzEz − γ3Exβ2

zδβz − γExδβzη21−(γ2 − 1

)γβxδβxExβz −

(γ2 − 1

)γExβyδβyβz + γ3βxβyδβyEz − γ3Bzβyβ

2zδβz

+γ3Byβxδβxη23 + γ3Byβzδβzη

23 + γ3Byβyδβyη

23 −

(γ − 1)Bxβxδβy(β2x − β2

y + β2z

)λ21

+2(γ − 1)Byβxδβxβ

2y

λ21− 2(γ − 1)Byβyδβyη

23

λ21−

(γ − 1)Bzδβyβz(β2x − β2

y + β2z

)λ21

+2(γ − 1)Byβ

2yβzδβz

λ21+

2(γ − 1)Bzβxδβxβyβzλ21

−(γ − 1)Bzβyδβz

(β2x + β2

y − β2z

)λ21

−γ3Bzβxδβxβyβz +2(γ − 1)Bxβxβyβzδβz

λ21−

(γ − 1)Bxδβxβy(−β2

x + β2y + β2

z

)λ21

](5.53)

(F ′′)21 = γExβy − γβxEy +1

λ21

[Bzβ

2z + γBzη

21 − (γ − 1)Bxβxβz − (γ − 1)Byβyβz

+γ3Bzβxδβxη21 − γ3Bxβ

2xδβxβz − γ3β2

xδβxEy − γ3βxβyδβyEy − γ3βxEyβzδβz+γβxβyδβyEy + γβxEyβzδβz − γδβxEyη22 + γExδβyη

23 + γ

(γ2 − 1

)Exβyβzδβz

+γ(γ2 − 1

)βxδβxExβy − γ3Byβxδβxβyβz + γ3Bzβyδβyη

21 − γ3Byβ

2yδβyβz

−γ3Bxβxβyδβyβz + γ3Bzβzδβzη21 − γ3Bxβxβ

2zδβz − γ3Byβyβ

2zδβz + γ3Exβ

2yδβy

+2(γ − 1)Bzβyδβyβ

2z

λ21−

(γ − 1)Byδβyβz(β2x − β2

y + β2z

)λ21

− 2(γ − 1)Bzβzδβzη21

λ21

+2(γ − 1)Byβxδβxβyβz

λ21+

2(γ − 1)Bzβxδβxβ2z

λ21+

2(γ − 1)Bxβxβyδβyβzλ21

−(γ − 1)Bxβxδβz

(β2x + β2

y − β2z

)λ21

−(γ − 1)Bxδβxβz

(−β2

x + β2y + β2

z

)λ21

−(γ − 1)Byβyδβz

(β2x + β2

y − β2z

)λ21

](5.54)

91

5.4.3.2 Successively Boosted Frame ~β and ∆~β

(F ′′′)10 = γBzβy − γByβz − γ2β2xEx − γ2βxβzEz − γ2βxβyEy +

1

λ21

[γ2β2

xEx + γExη22

+γ2βxβyEy − γβxβyEy − γ2βxEzλ5 + γ2βxβzEz − γβxβzEz + γ2Byβxδβxβz

−γ2Bzβxδβxβy + γByβyδβyβz − γBzβyβzδβz − (γ − 1)γBxβxδβyβz

−γ3Byβzλ2 + γBzδβy(γβ2

x + β2z

)− γByδβz

(γβ2

x + β2y

)− γ2βxExλ3 − γ3βxβyEyλ2

+(γ − 1)γBxβxβyδβz − γ2βxEyλ4 − γ3βxβzEzλ2 + γ3Exη22λ2 + γ3Bzβyλ2

](5.55)

(F ′′′)20 = γBxβz − γBzβx − γ2βxExβy − γ2β2yEy − γ2βyβzEz +

1

λ21

[γ3Bxβyδβyβz

+γ(γ − 1)βxExβy + (γ − 1)γβyβzEz + γ3Bxβxδβxβz − γ3Bzβ2xδβx + γBzβxβzδβz

+(γ − 1)γByδβxβyβz − γBxβxδβxβz + γ2βxδβxβ2yEy − γ3β2

yδβyβzEz

+γEy(β2x + γβ2

y + β2z

)− γ3Bzβxβyδβy − γ2Bxβyδβyβz + γBxδβz

(β2x + γβ2

y

)−(γ − 1)γByβxβyδβz − γ3Bzβxβzδβz + γ3Bxβ

2zδβz − γ3β2

xδβxExβy

+γ2βxExβ2yδβy − γ2δβxExβyη22 + γ2βxExβyβzδβz − γ3βxExβyβzδβz

−γ2βyδβyEyη23 + γ3βyδβyEyη23 + γ2β2

yEyβzδβz + γ3Eyβzδβzη23 − γ2βyδβzEzη21

−γ3βxδβxβyβzEz − γ3βyβ2zδβzEz + γ2β2

yδβyβzEz + γ2βxδβxβyβzEz

−γBzδβx(γβ2

y + β2z

)+ γ2Bzβxβyδβy − γ3βxExβ2

yδβy + γ3βxδβxEyη23

](5.56)

(F ′′′)30 = γByβx − γBxβy − γ2βxExβz − γ2βyEyβz − γ2β2zEz +

1

λ21

[γ3Byβ

2xδβx

+(γ − 1)γβxExβz + (γ − 1)γβyEyβz + γEz(β2x + β2

y + γβ2z

)− γ3Bxβxδβxβy

+γByδβx(β2y + γβ2

z

)+ γBxβxδβxβy − (γ − 1)γBzδβxβyβz − γ3βxExβ2

zδβz

+γ3Byβxβyδβy − γ2Byβxβzδβz + (γ − 1)γBzβxδβyβz − γBxδβy(β2x + γβ2

z

)−γ3Bxβyβzδβz + γ3Byβxβzδβz + γ2Bxβyβzδβz − γ3β2

xδβxExβz

+γ2βxExβyδβyβz − γ2δβxExβzη22 + γ2βxExβ2zδβz − γ3βxExβyδβyβz

−2γ3β2yδβyEyβz − 2γ2δβyEyβzη

23 + γ2βxδβxβyEyβz + γ2βyEyβ

2zδβz

−γ3βyEyβ2zδβz + γ3βyδβyEzη

21 + γ3βzδβzEzη

21 + γ3βxδβxEzη

21 + γ2βxδβxβ

2zEz

−γ2βzδβzEzη21 − γ3Bxβ2yδβy + γ2βyδβyβ

2zEz − γ3βxδβxβyEyβz − γByβxβyδβy

](5.57)

92

(F ′′′)32 = γEyβz − γβyEz + γ3βyδβyEyβz − γ3β2yEyδβz + γ3δβyβ

2zEz − γ3βyβzδβzEz

+1

λ21

[−γ2Byβxλ4 + γ3Bxη

22λ2 − (γ − 1)Bzβxβz +Bx

(β2x + γ

(β2y + β2

z

))+(γ − 1)γβxExδβyβz − (γ − 1)γβxExβyδβz − γ2Bzβxλ5 − γ3Bzβxβzλ2

+γ2Eyδβz(β2x + γ

(β2y + β2

z

))− γ2δβyEz

(β2x + γ

(β2y + β2

z

))− γ2Bxβxλ3

+(γ − 1)γ2βxδβxEyβz − (γ − 1)Byβxβy − γ3Byβxβyλ2 − (γ − 1)γ2βxδβxβyEz]

(5.58)

(F ′′′)13 = γβxEz − γExβz − γ3βxδβxExβz + γ3β2xExδβz + γ3βxβzδβzEz +

1

λ21

[γ3Byη

23λ2

−γ2Bxβyλ3 − γ3Bxβxβyλ2 + γδβxβ2zEz − (γ − 1)Bxβxβy − (γ − 1)Bzβyβz

−γ2Byβyλ4 − γ2δβxβyEyβz − γ2Bzβyλ5 − γβxβyEyδβz − γ3Bzβyβzλ2

−(γ − 1)γ2Exβyδβyβz − γ3Exδβzη23 + γ2βxβyEyδβz + γ3β2xδβxEz + γ2δβxβ

2yEz

+γδβxβyEyβz + (γ − 1)γ2βxβyδβyEz +By

(γβ2

x + β2y + γβ2

z

)− γ2Exβ2

yδβz]

(5.59)

(F ′′′)21 = γ3βxδβxExβy − γ3β2xExδβy + γExβy + γ3δβxβ

2yEy − γβxEy − γ3βxβyδβyEy

+ γ3δβxβyβzEz − γ3βxδβyβzEz +1

λ21

[γ3Bzη

21λ2 − (γ − 1)Bxβxβz − γ2δβxEyβ2

z

−(γ − 1)Byβyβz +Bz

(γβ2

x + γβ2y + β2

z

)− γ3Bxβxβzλ2 − γ3Byβyβzλ2

+γ3Exδβyη21 − γ3δβxEyη21 − γ2Bxβzλ3 − γ2Byβzλ4 − γ2Bzβzλ5

+(γ − 1)γ2Exβyβzδβz − (γ − 1)γ2βxEyβzδβz + (γ − 1)γ2βzEzλ6 + γ2Exδβyβ2z

](5.60)

93

Chapter 6 Conclusion

6.1 Summary

One of the objectives of this research was to investigate various neutron β-decayobservables such as the electron energy spectrum Γ, the electron asymmetry A, theelectron-antineutrino correlation a and the neutron lifetime τ in order to analyzeparameters like the Fierz interference term b and the parameter λ (which in turnhelps us to study the axial vector form factor gA assuming the CVC hypothesisholds). We studied the sensitivity in global fits to these observables of theory errorsfrom ill-known matrix elements on the ability to identify new beyond standard model(BSM) physics in global fits to these observables.

We quantified the results from the global fits to Monte Carlo generated pseudo-data for the above mentioned observables under various scenarios and analyzed thestatistical impact that future improvements in the precision of neutron β-decay ob-servables should have on the assessment of the validity of the SM. We also discussedthe extent to which this assessment is limited by the theoretical uncertainties due topoorly known second-class currents, and used such studies to extract the precision towhich they should be established to eliminate that impact. All of these could serveas the tests of the well-known V − A law and at sufficient experimental resolutions,may yield violations of it. The work could serve as a baseline for future high precisionexperiments which plan to probe up to O(10−4) precision.

The second area of focus of this dissertation was to analyze the electromagneticfields in the rest frame as well as the laboratory frame of relativistic particles un-dergoing acceleration. Spin-1/2 particles in the presence of electromagnetic fieldsundergo Larmor precession and the frequency associated with the Larmor precessionis sensitive to the non-uniformities in the surrounding electromagnetic fields. For thecase in which these particles are moving in a highly-uniform magnetic field with smallnon-uniformities and a non-zero electric field, magnetic fields oriented in the trans-verse direction relative to the primary magnetic field direction can induce frequencyshifts in the particles Larmor spin precession frequencies. In a situation which em-ploys both electric and magnetic fields,there are, in general, two possible sources oftransverse magnetic fields in the particle rest frame: gradients in the magnetic field,leading to off-primary-axis transverse field components, and relativistic transforma-tions of electric fields. This is important in the interpretation of the results from theexperiments employing Larmor precession techniques such as experiments employingrelativistic particles stored in magnetic storage rings.

6.2 Observations and Inferences

• Analyzing neutron β-decay observables provides key insights to SM as well asBSM physics. Extraction of leading-order energy independent values of the an-gular correlation coefficients such as a, A and the neutron lifetime τ determine λ

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in the SM, which is an important parameter of the weak interaction. Measure-ments of the energy dependence in the electron energy spectrum and angularcorrelation coefficients give access to the Fierz term which is most sensitive tonew physics scalar and tensor interactions.

• Among different observables, the electron energy spectrum provides the moststringent constraints on the Fierz term b and hence to new physics scalar andtensor interactions. Including angular correlation coefficients does not seemto reinforce the constraint. However, additional uncertainties might arise dueto uncertainties in the Fermi function, which accounts for the electromagneticinteraction between the final-state particles. [21].

• The parameter λ cannot be determined to high precision from the electron en-ergy spectrum alone. In the SM, neutron beta decay observables such as theangular correlations (a and A) and the neutron lifetime τ can be used to deter-mine λ due to their high sensitivity to it. Determination of b cannot be donewithout pre-determining the value of gA using neutron lifetime measurementsor by simultaneously fitting for both b and λ.

• Second-class currents (SCCs) f3 and g2 tend to dilute the uncertainties in the fitparameters, but the theoretical uncertainties due to the SCCs can motivate anyattempt in falsifying the SM through the inclusion of a high-precision τ value.We propose for the determination of SCCs using lattice QCD techniques.

• Lastly, experiments employing Larmor spin precession techniques are sensitiveto small non-uniformities in electromagnetic fields, The effects due to Thomasprecession should be taken into consideration for such high precision exper-iments. This could potentially be quite relevant to a critical assessment ofresults.

6.3 Limitations

Coulomb radiative corrections to the differential decay distribution induces ashape correction to the electron energy spectrum in terms of a multiplicative fac-tor representing the Fermi function F (±Z,Ee) [37, 57, 72] [see Eq. 3.5]. Thesephase-space integrated Fermi function and corrections to it have been studied ingreat detail [21, 79]; we omit it, as well as the outer radiative corrections [80], in thegeneration of the Monte Carlo pseudodata for various observables. Future studiesextending the work beyond this dissertation should account for uncertainties in theFermi function and their impact on a critical assessment of high precision neutronbeta decay observables.

Copyright c© Lakshya Malhotra, 2021.

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Vita

Personal Information

Name: Lakshya MalhotraPlace of Birth: Panchkula (Haryana), India

Educational Institutions

M.S. PhysicsUniversity of KentuckyApril 2017

B.E. Aeronautical EngineeringPEC University of Technology, Chandigarh, IndiaMay 2013

Professional Positions

Graduate Research AssistantUniversity of KentuckyAugust 2017 – August 2019

Graduate Teaching AssistantUniversity of KentuckyAugust 2014 – July 2017, August 2019 – December 2020

Publications

Malhotra, L., Golub, R., Kraegeloh, E., Nouri, N., Plaster, B., Effect of ThomasRotation on the Lorentz Transformation of Electromagnetic fields. Sci. Rep. 10,5522 (2020). https://doi.org/10.1038/s41598-020-62082-z.

102