Muonium-antimuonium Oscillations in an Extended Minimal Supersymmetric Standard Model

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Author: Boyang Liu

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Springer Theses Recognizing Outstanding Ph.D. Research

For further volumes: http://www.springer.com/series/8790

Aims and Scope The series ‘‘Springer Theses’’ brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic of should fall within the confines of Chemistry, Physics and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field.

Boyang Liu

Muonium–Antimuonium Oscillations in an Extended Minimal Supersymmetric Standard Model Doctoral Thesis accepted by Purdue University, West Lafayette, USA

123

Dr. Boyang Liu Chinese Academy of Sciences Institute of Physics P.O. Box 603 100190 Beijing People’s Republic of China e-mail: [email protected]

Supervisor Dr. Sherwin T. Love Department of Physics Purdue University 1396 Physics Building West Lafayette, IN 47907 USA e-mail: [email protected]

ISSN 2190-5053

e-ISSN 2190-5061

ISBN 978-1-4419-8329-9

e-ISBN 978-1-4419-8330-5

DOI 10.1007/978-1-4419-8330-5 Springer New York Dordrecht Heidelberg London Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover design: eStudio Calamar, Berlin/Figueres Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Supervisor’s Foreword

The results of a variety of neutrino oscillation experiments involving solar, atmospheric, accelerator and reactor neutrinos and anti-neutrinos have led to the conclusion that neutrinos produced in a well defined flavor eigenstate can be detected in a different flavor eigenstate after propagating a macroscopic distance. The simplest interpretation of this oscillation phenomenon is that the neutrinos have finite masses and that the neutrinos flavor eigenstates differ from their mass eigenstates. That is, the neutrino flavors mix. The existence of finite neutrino masses qualifies as the first empirical evidence for physics beyond the minimal Standard Model of particle physics. At present, there are still many questions regarding the nature of the neutrino masses which remain unanswered. For example, are neutrinos Dirac or Majorana fermions? It follows that the answers to neutrino related issues should add to the knowledge about the precise nature of the interactions which exist beyond the Standard Model. The existence of massive neutrinos opens the possibility of muonium–antimuonium oscillations provided the neutrinos are Majorana fermions. This process has been the subject of experimental search at the Paul Scherrer Institut (PSI) in Switzerland for a number of years. Thus far, only a lower bound has been placed on the muonium– antimuonium oscillation time scale. The PhD thesis of Boyang Liu involves the computation of the muonium–antimuonium oscillation time scale in a variety of Standard Model extensions. First he demonstrated its gauge invariance in the Standard Model modified only with the inclusion of right-handed neutrinos which were used to generate light neutrino masses via the see-saw mechanism. He explicitly displayed the gauge independence of the various 1-loop contributions. Next he considered the calculation of the muonium–antimuonium oscillation time scale in a supersymmetric (SUSY) extension of the Standard Model where there exist additional intermediate states which can contribute to the oscillation process thus possibly enhancing the rate to a range which is at or near to the current experimental bound. This thesis considers the Minimal Supersymmetric Standard Model (MSSM) modified only by the inclusion of right handed neutrino superfields while allowing for intra-generational lepton number violation. Using the current experimental limits on the SUSY partner masses, he found that the v

vi

Supervisor’s Foreword

computed oscillation time scale could indeed be lowered but it is still, in general, a couple of orders of magnitude above the current limit. Only in the special case when the mixing between the two Higgs doublets of the MSSM given by tan b is very small is the oscillation process sufficiently enhanced. In that case, the current experimental limit on the muonium–antimunium conversion can be used to bound tan b [ 10-7. October 2010

Sherwin T. Love

Preface

The electron and muon number violating muonium–antimuonium oscillation process in two different models is investigated. First, modifying the Standard Model only by the inclusion of singlet right-handed neutrinos and allowing for general renormalizable interactions producing neutrino masses and mixing, the leading order matrix element contribution to this process is computed in Rn gauge thereby establishing the gauge invariance to this order. To give a natural explanation of the smallness of the observed neutrino masses, the see-saw mechanism is explored resulting in three light Majorana neutrinos and three heavy Majorana neutrinos with mass scale MR MW. Present experimental limits set by the nonobservation of the oscillation process sets a lower limit on MR of roughly of order 600 GeV. Second, modifying the Minimal Supersymmetric Standard Model by the inclusion of three right-handed neutrino superfields and allowing only intrageneration lepton number violation but not inter-generation lepton number mixing, the muonium–antimuonium conversion can occur while the process l ? ec is forbidden. For a wide range of the parameters, the contributions to the muonium– antimuonium oscillation time scale are at least two orders of magnitude below the sensitivity of current experiments. However, if the ratio of the two Higgs field VEVs, tan b, is very small, there is a limited possibility that the contributions are large enough for the present experimental limit to provide an inequality relating tan b with the light neutrino mass scale mv which is generated by see-saw mechanism. The resultant lower bound on tan b as a function of mv is more stringent than the analogous bounds arising from the muon and electron anomalous magnetic moments as computed using this model. Beijing, May 2010

Boyang Liu

vii

Acknowledgments

I thank all the individuals that have been helpful and supportive to me both professionally and personally in the completion of this work and my life. I thank my major Professor, Sherwin Love, for his ideas, suggestions and profound professional insight. Everything that I learned from him has built a very solid scientific background for me. He is also such a nice person. I still remember that he helped me to revise my papers and postdoc application documents word by word. I really appreciate the time he shared with me. Many thanks to Professor Thomas Clark, Professor Tzee-Ke Kuo and Dr. Chi Xiong for all the valuable discussions with them. My thanks to my parents for supporting me in all my pursuits. They always encourage me to follow my heart and do the things I am really interested in. This precious love is an irreplaceable contribution to the completion of this thesis. Last, but by no means least, thanks to all my friends. Without them I would have such an enjoyable life in Purdue. I have probably left out many others, who deserve being mentioned here, to whom I apologize for the lack of space and my momentary lapse of memory.

ix

Contents

1

Introduction to Muonium–Antimuonium Oscillation . . . . . . . . . . .

1

2

Muonium–Antimuonium Oscillation in a Modified Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Supersymmetry and the Minimal Supersymmetric Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

The Muonium–Antimuonium Oscillation in the Extended Minimal Supersymmetric Standard Model . . . . . . . . . . . . . . . . . .

25

The Constraints from the Muon and Electron Anomalous Magnetic Moment Experiments . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Author Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3

4

5

6

xi

Chapter 1

Introduction to Muonium–Antimuonium Oscillation

The time-dependent oscillation between two distinct levels or particle species is an interesting quantum mechanical phenomenon which has been widely studied in many physical systems varying from a particle moving in a double-well potential 0 and B0 B 0 meson of the ammonia molecule to oscillations in the neutral K 0 K 1 systems. It was suggested roughly 60 years ago (Pontecorvo 1957) that there may be a spontaneous conversion between muonium and antimuonium resulting in an associated oscillation effect. Muonium ðMÞ is the Coulombic bound state of an is the Coulombic electron and an antimuon ðe lþ Þ, while antimuonium ðMÞ bound state of a positron and a muon ðeþ l Þ. Since it has no hadronic constituents, muonium is an ideal place to test electroweak interactions. Of particular interest is that such a muonium–antimuonium oscillation is totally forbidden within the Standard Model because the process violates the individual electron and muon number conservation laws by two units. Hence, its observation will be a clear signal of physics beyond the Standard Model. Since the initial suggestion, experimental searches have been conducted (Huber 1990; Matthias et al. 1991; Abela et al. 1996; Willmann et al. 1999) and a variety of theoretical models have been proposed which can give rise to such a muonium–antimuonium conversion. These include interactions which can be mediated by (a) a doubly charged Higgs boson Dþþ (Halprin 1982; Herczeg and Masiero 1992), which is contained in a left–right symmetric model, (b) massive Majorana neutrinos (Clark and Love 2004; Cvetic et al. 2005; Liu 2008), or (c) the s-sneutrino in an R-parity violation supersymmetric model (Halprin and Masiero 1993).

1

For a general review of present state of oscillation phenomena in neutral meson systems, see, 0 and for example Battaglia et al. (2002). For a general discussion of mixing in the neutral K 0 K 0 , see, for example Buras (2000). Analogous oscillation phenomena has also been B0 B speculated upon for the neutron–antineutron system. Here the mixing matrix elements arise from an underlying baryon number violating grand unified theory. For various discussions, see Glashow (1979); Marshak and Mohapatra (1980) and Kuo and Love (1980).

B. Liu, Muonium–Antimuonium Oscillations in an Extended Minimal Supersymmetric Standard Model, Springer Theses, DOI: 10.1007/978-1-4419-8330-5_1, Ó Springer Science+Business Media, LLC 2011

1

2

1 Introduction to Muonium–Antimuonium Oscillation

In Chap. 2, we focus on the muonium–antimuonium oscillations in a modified Standard Model which includes singlet right-handed neutrinos. There is now compelling evidence of the existence of neutrino oscillations from the experimental study of atmospheric and solar neutrinos (Fukuda et al. 1988; Ahmad et al. 2001, 2002; Eguchi et al. 2003; Ahn et al. 2003). That implies nonzero neutrino masses and mixing matrix elements. The size and nature of the neutrino mass and the associated mixing is still an open question subject to experimental determination and theoretical speculation (Altarelli and Feruglio 2002; Fogli et al. 2002; Gonzalez-Garcia and Nir 2003). One simple neutrino mass model is obtained by modifying the Standard Model by including singlet right-handed neutrinos and allowing for a general mass matrix for neutrinos. Left-handed neutrinos along with their charged leptonic partners are components of SUð2ÞL doublets and experience the weak interaction while any right-handed neutrinos are completely neutral under the Standard Model gauge group. The see-saw mechanism (Minkowski 1977; Yanagida 1979; Gell-Mann et al. 1979; Glashow 1980; Mohapatra and Senjanovic 1980) provides a natural explanation of the smallness of the three light Majorana neutrino masses, while ensuring that the other three Majorana neutrinos are heavy. Such a model could also lead to the muonium–antimuonium oscillation process. In order for there to be a nontrivial mixing between muonium and antimuonium, the individual electron and muon number conservation must be violated. Such a situation results provided the neutrinos are massive particles which mix amongst the various generations. This criterion can be met by the modified Standard Model and the e lþ and eþ l states could indeed mix. In Chaps. 3 and 4 we consider the muonium–antimuonium oscillation process in the Minimal Supersymmetric Standard Model (MSSM) extended by the inclusion of three righthanded neutrino superfields and where lepton flavor mixing is absent. This assumption automatically forbids the l ! ec decay, so that the muonium–antimuonium oscillation process can be used to constrain the model parameters. In order for there to be a nontrivial mixing between the muonium and antimuonium, the individual electron and muon number conservation must be violated by two units. Such a situation will result provided that the neutrinos are massive Majorana particles or the mass diagonal sneutrinos are lepton number violating scalar particles. The present experimental limit (Willmann et al. 1999) on the non-observation of muonium–antimuonium oscillation translates into a bound on the coupling constant of the effective Lagrangian of this oscillation process. Calculations show that for a wide range of the parameters, the contributions to the muonium–antimuonium oscillation time scale are at least two orders of magnitude below the sensitivity of current experiments. However, if the ratio of the two Higgs field VEVs, tan b, is very small, there is a limited possibility that the contributions are large enough for the present experimental limit to provide an inequality relating tan b with the light neutrino mass scale mm . In Chap. 5 the corrections of muon and electron anomalous magnetic moments in the Extended Minimal Supersymmetric Standard Model (EMSSM) are calculated. Then two lower bounds on tan b as a function of the light neutrino mass scale ml are generated by

1 Introduction to Muonium–Antimuonium Oscillation

3

the measurements of the muon and electron anomalous magnetic moments, respectively. Comparison shows that the bound from muonium–antimuonium oscillation is more stringent than ones from muon and electron anomalous magnetic moments. Finally, Chap. 6 presents the conclusions.

References R. Abela et al., Phys. Rev. Lett. 77, 1950 (1996) Q.R. Ahmad et al., Phys. Rev. Lett. 87, 071301 (2001), nucl-ex/0106015 Q.R. Ahmad et al., Phys. Rev. Lett. 89, 011301 (2002), nucl-ex/0204008 M.H. Ahn et al., Phys. Rev. Lett. 90, 041801 (2003), hep-ex/0212007 G. Altarelli and F. Feruglio, in Neutrino Mass, Springer Tracts in Modern Physics (G. Altarelli and K. Winter, eds.), hep-ph/0206077 (2002) M. Battaglia et al., The CKM Matrix and the Unitarity Triangle, in the Workshop on CKM Unitarity Triangle (CERN 2002–2003), Geneva, Switzerland, 13–16 Feb 2002, hep-ph/ 0304132 (2002) A. Buras, in International School of Subnuclear Physics: 38th Course: Theory and Experiment Heading for New Physics, Erice, 2000, hep-ph/0101336 T.E. Clark and S.T. Love, Mod. Phys. Lett. A19, 297 (2004) G. Cvetic, C.O. Dib, C.S. Kim and J.D. Kim, Phys. Rev. D 71, 113013 (2005) K. Eguchi et al., Phys. Rev. Lett. 90, 021802 (2003), hep-ex/0212021 G.L. Fogli, E. Lisi, A. Marrone, D. Montanino and A. Palazzo, Nucl. Phys. Proc. Suppl. 111, 106 (2002) Y. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1988), hep-ex/9807003 M. Gell-Mann, P. Ramond and R. Slansky in Supergravity (P. van Nieuwenhuizen and D. Freedman eds.) North-Holland, Amsterdam, 1979 S. Glashow, in Proceedings of Neutrino 79, International Conference on Neutrinos, Weak Interactions and Cosmology, Bergen, Norway, 1979, ed. by A. Haatuft and C. Jarlskog (Fysik Institutt, Bergen, 1980); S.L. Glashow, in Proceedings of the 1979 Cargese Institute on Quarks and Leptons (M. Levy et al., eds.) Plenum Press, New York 1980, p687 M.C. Gonzalez-Garcia and Y. Nir, Rev. Mod. Phys. 75, 345 (2003) A. Halprin, Phys. Rev. Lett. 48, 1313 (1982) A. Halprin and A. Masiero, Phys. Rev. D 48, R2987 (1993) P. Herczeg and R.N. Mohapatra, Phys. Rev. Lett. 69, 2475 (1992) T. Huber et al., Phys. Rev. D 41, 2709 (1990) T.K. Kuo and S.T. Love, Phys. Rev. Lett. 45, 93 (1980) B. Liu, Gauge Invariance of the Muonium–Antimuonium Oscillation Time Scale and Limits on Right-Handed Neutrino Masses (2008) R.E. Marshak and R.N. Mohapatra, Phys. Rev. Lett. 44, 1316 (1980); B. Matthias et al., Phys. Rev. Lett. 66, 2716 (1991) P. Minkowski, Phys. Lett. B67, 421 (1977) R.N. Mohapatra and G. Senjanovic. Phys. Rev. Lett. 44, 912 (1980) B. Pontecorvo, Sov. Phys. JETP 6, 429 (1957) L. Willmann et al., Phys. Rev. Lett. 82, 49 (1999) T. Yanagida, in Proceedings of the Workshop on the Unified Theories and Baryon Number in the Universe, Tsukuba, Japan, 1979, (O. Sawada and A. Sugamoto eds.), p95

Chapter 2

Muonium–Antimuonium Oscillation in a Modified Standard Model

2.1 Neutrino Masses and Mixings We modify the Standard Model only by the inclusion of singlet right-handed neutrinos and allowing for general renormalizable interactions. This implies nonzero neutrino masses and mixing matrix elements. The leptonic Yukawa interactions with the Higgs scalar doublet in the modified Standard Model take the form ! 3 X g ð0Þ ‘ ð0Þ ð0Þ D y ð0Þ / Lint ¼ pffiffiffi / ‘Ra mab mLb ‘La mab mRb þ H:C: ð2:1Þ 2M W a;b¼1 ð0Þ

ð0Þ

Here, ‘La and mLa are, respectively, the charged lepton and its associated neutrino ð0Þ

partner of the SUð2ÞL doublet, while mRa is the right-handed neutrino singlet. The superscript zero indicates weak interaction eigenstates so that the leptonic charged current interaction is 3 g l X ð0Þ ð0Þ LW ‘La cl mLa þ H:C: int ¼ pffiffiffiW 2 a¼1

ð2:2Þ

After spontaneous symmetry breaking, the mass term for the charged leptons takes the form L‘mass ¼

3 h X

ð0Þ

ð0Þ

ð0Þ

ð0Þ

‘Ra m‘ab ‘Lb þ ‘La m‘ ba ‘Rb

i

ð2:3Þ

a;b¼1

where m‘ is a 3 3 mass matrix. To diagonalize this matrix, one performs the biunitary transformation m‘ ¼ AR m‘diag ðAL Þy


ð2:4Þ

5

6

2 Muonium–Antimuonium Oscillation in a Modified Standard Model

where AR and AL are 3 3 unitary matrices and m‘diag is a diagonal matrix, the entries of which are the charged lepton masses. To implement this basis change, the charged lepton fields participating in the weak interaction are rewritten in terms of the mass diagonal fields as ð0Þ

‘La ¼

3 X

ALab ‘Lb ;

ð0Þ

‘Ra ¼

b¼1

3 X

ARab ‘Rb

ð2:5Þ

b¼1

On doing so, the mass term reads L‘mass ¼

3 X

m‘a ‘Ra ‘Lb þ ‘La ‘Rb

ð2:6Þ

a¼1

A general neutrino mass term resulting from renormalizable interactions takes the form ! ð0Þ 1 ð0Þ c ð0Þ 0 ðmD ÞT mL m þ H:C: ð2:7Þ Lmass ¼ ðmL Þ mR ð0Þ mD mR 2 ðmR Þc Note that the upper left 3 3 block in the neutrino mass matrix is set to zero. This block matrix involves only left-handed neutrinos and in the (modified) Standard Model its generation requires a nonrenormalizable mass dimension-five operator. Consequently, such a term will be ignored. For three generations of neutrinos, the six mass eigenvalues, mmA , are obtained from the diagonalization of the 6 6 matrix m

M ¼

0 mD

ðmD ÞT mR

ð2:8Þ

Since M m is symmetric, it can be diagonalized by a single unitary 6 6 matrix, U, as m ¼ U T M m U: Mdiag

ð2:9Þ

This diagonalization is implemented via the basis change on the original neutrino fields organized as the six-dimensional column vector ! ! ð0Þ ð0Þ mL ðmL Þc ð0Þ ð0Þ NL ¼ ; NR ¼ ð2:10Þ ð0Þ ð0Þ ðmR Þc mR to the new neutrino fields defined as ð0Þ

NL ¼ UNL ;

ð0Þ

NR ¼ U NR

ð2:11Þ

2.1 Neutrino Masses and Mixings

7

where NL ¼

mL ; ðmR Þc

NR ¼

ðmL Þc : mR

ð2:12Þ

The neutrino mass term then takes the form 6 6 h i X 1X Lmmass ¼ mmA mTA CmA þ mA CmTA ¼ mmA mA mA ; 2 A¼1 A¼1

ð2:13Þ

where mmA are the Majorana neutrino masses. Since a nonzero Majorana mass matrix mR does not require SUð2ÞL Uð1Þ symmetry breaking, it is naturally characterized by a much larger scale, MR , than the elements of the matrix mD the nontrivial values of which require SUð2ÞL Uð1Þ symmetry breaking and are thus expected to be somewhere in the order of the charged lepton mass to the W mass. Thus, one can take the elements of mD , characterized by a scale mD , to be much less than MR , the scale of the elements of mR . One then finds on diagonalization of the 6 6 neutrino mass matrix that three of the eigenvalues are crudely given by mma

m2D mD ; MR

a ¼ 1; 2; 3;

ð2:14Þ

while the other three eigenvalues are roughly mmi MR ;

i ¼ 4; 5; 6:

ð2:15Þ

This constitutes the so-called see-saw mechanism (Minkowski 1977; Yanagida 1979; Gell-Mann et al. 1979; Glashow 1979; Mohapatra and Senjanovic 1980) and provides a natural explanation of the smallness of the three light neutrino masses. Moreover, the elements of the mixing matrix are characterized by an MR mass dependence Uab Oð1Þ;

a; b ¼ 1; 2; 3

Uij Oð1Þ;

i; j ¼ 4; 5; 6 mD Uia Uai O ; a ¼ 1; 2; 3; i ¼ 4; 5; 6: MR

ð2:16Þ

Since the charged lepton mixing matrix is independent of MR , one finds that elements of the mixing matrix appearing in the charged current have the MR mass dependence Vab Oð1Þ; a; b ¼ 1; 2; 3 mD Vai O ; a ¼ 1; 2; 3; i ¼ 4; 5; 6 MR

ð2:17Þ

8


Inserting the transformations (2.5) and (2.11) in the interaction terms (2.1) and (2.2), and taking into account the mass matrix transformations (2.4) and (2.9), one obtains the explicit interactions of charged bosons with the leptons in their mass diagonal basis as 3 X 6 3 X 6 X g l X g mA VaA cl ‘La LW ‘La cl VaA mA pffiffiffiW þl int ¼ pffiffiffiW 2 2 a¼1 A¼1 a¼1 A¼1

ð2:18Þ

3 X 6 X g 1 c5 1 þ c5 ‘a VaA mla L/int ¼ pffiffiffi mA mmA / 2 2 2M W a¼1 A¼1 3 X 6 X g 1 þ c5 1 c5 mA VaA ‘a mmA pffiffiffi /þ mla 2 2 2M W a¼1 A¼1

ð2:19Þ

where VaA ¼

3 X ðA1 L Þab UbA

ð2:20Þ

b¼1

Note that the mixing matrix VaA satisfies the identities (Ilakovac and Pilaftisis 1995): 6 X

VaA VbA ¼ dab

ð2:21Þ

VaA VbA mmA ¼ 0

ð2:22Þ

A¼1 6 X A¼1

Identity (2.21) stems from the unitarity of matrices AL and U, while identity (2.22) is a consequence of the particular form of the neutrino mass matrix. In particular, it requires the Majorana mass term of the left-handed neutrinos be set to zero. A detailed proof of this latter identity is provided in Appendix A. The propagators in Rn gauge are (Cheng and Li 1988): Fermions :

iðcp þ mÞ m2 þ i

p2

i ðn 1Þpl pm W-bosons W : 2 glm þ 2 2 2 p MW þ i p nMW

Goldstone bosons / :

i 2 p2 nMW

ð2:23Þ

2.2 The Gauge Invariant T-Matrix Elements

9

2.2 The Gauge Invariant T-Matrix Elements of Muonium–Antimuonium Oscillation The lowest order Feynman diagrams accounting for muonium and antimuonium mixing are displayed in Fig. 2.1. We shall consistently employ the Rn gauge. The gauge invariance of the T-matrix element will be demonstrated by establishing its n independence. In Fig. 2.1, there are two neutrinos in the intermediate state for each graph, while every wavy line represents either a W boson or an Rn gauge charged erstwhile Nambu-Goldstone boson. Note that in the unitary gauge ðn ! 1Þ; the W boson propagator takes the form p p i ½g Ml 2 m . A theory with such a propagator has very bad power counting p2 M 2 þi lm W

W

convergence properties. As it turns out, the unitary gauge power counting divergent pieces in the W vector box diagrams vanish, as they must, after application of properties (2.21) and (2.22). Hence, when we calculate the T-matrix elements in Rn gauge, we will also apply properties (2.21) and (2.22) to establish the cancelation of the various terms in this case. As it turns out, graph (a) gives the same contribution as (b), as do graphs (c) and (d). Hence, we need only discuss the gauge invariant T-matrix elements of the

Fig. 2.1 Feynman graphs contributing to the muonium– antimuonium mixing. Each wavy line is either a W boson or an Rn gauge charged Nambu–Goldstone boson

(a)

(b)

(c)

(d)

10


Fig. 2.2 Feynman graphs of type (a) in Rn gauge, which are presented as one single graph (a) in Fig. 2.1

(a1)

(a2)

(a3)

(a4)

graphs (a) and (c). Figure 2.2 details explicitly the four separate graphs, which are represented by the single graph in Fig. 2.1. A straightforward application of the Rn gauge Feynman rules (Cheng and Li 1988) to the above graphs yields the T-matrix elements

X 6 X 6 g4 1 c5 l 1 c5 eð2Þ eð1Þ ðVlA VeA ÞðVlB VeB Þ l ð4Þc l ð3Þc Ta1 ¼ l 2 2 2 64p2 MW A¼1 B¼1 ( )# Z 1 " xA xB 2ðn 1Þ ðn 1Þ2 2 tþ dt 1þ t tþn ðt þ xA Þðt þ xB Þðt þ 1Þ2 4ðt þ nÞ2 0 ð2:24Þ Ta2 ¼ Ta3 ¼

g4 1 c5 l 1 c5 eð2Þ eð1Þ l ð4Þc l ð3Þc l 2 2 2 64p2 MW Z 6 6 1 XX xA xB ðVlA VeA ÞðVlB VeB Þ dt ðt þ xA Þðt þ xB Þðt þ 1Þðt þ nÞ 0 A¼1 B¼1

n1 ð2:25Þ tþ t2 4ðt þ nÞ


Ta4

11

X 6 X 6 g4 1 c5 l 1 c5 eð2Þ eð1Þ ¼ ðVlA VeA ÞðVlB VeB Þ l ð4Þc l ð3Þc l 2 2 2 64p2 MW A¼1 B¼1 # Z 1 " xA xB t2 dt ð2:26Þ ðt þ xA Þðt þ xB Þðt þ nÞ2 4 0

ð3Þ ¼ l ðp3 ; s3 Þ, l ð4Þ ¼ l ðp4 ; s4 Þ, eð1Þ ¼ eðp1 ; s1 Þ and eð2Þ ¼ eðp2 ; s2 Þ are where l m2

the spinors of the muons and electrons and xA ¼ MmA2 , A ¼ 1; . . .; 6: Note that in W

obtaining these results, we already applied properties (2.21) and (2.22) to eliminate various self-canceling terms. As such, the integrals in (2.24)–(2.26) are finite even in the n ! 1 limit. See Appendix B for detailed calculations of T-matrix elements. In order to discuss the n dependence in a manifest way, we rewrite these Tmatrix elements as ! 6 X 6 Z 1 X 1 1 þ f ðtÞ dtAðxA ; xB ; tÞ hðtÞ 2gðtÞ Ta1 ¼ tþn ðt þ nÞ2 A¼1 B¼1 0 ! 6 X 6 Z 1 X 1 1 dtAðxA ; xB ; tÞ hðtÞ þ gðtÞ ð2:27Þ Ta2 ¼ tþn ðt þ nÞ2 A¼1 B¼1 0 ! 6 X 6 Z 1 X 1 1 Ta3 ¼ dtAðxA ; xB ; tÞ hðtÞ þ gðtÞ tþn ðt þ nÞ2 A¼1 B¼1 0 ! 6 X 6 Z 1 X 1 Ta4 ¼ dtAðxA ; xB ; tÞ hðtÞ ðt þ nÞ2 A¼1 B¼1 0 where

g4 1 c5 l 1 c5 ð4Þc ð3Þcl ÞðVlB VeB Þ AðxA ; xB ; tÞ ¼ eð2Þ l eð1Þ ðVlA VeA l 2 2 2 64p2 MW

xA xB ðt þ xA Þðt þ xB Þ

ð2:28Þ

with t2 hðtÞ ¼ ; 4 Note that the

1 ðtþnÞ2

2

gðtÞ ¼

2

t þ t4 1 þ 2t þ t4 and f ðtÞ ¼ tþ1 ðt þ 1Þ2

ð2:29Þ

terms from the second and third graphs totally cancel against

1 the ones from the first and fourth graphs, while the tþn terms from the second and third graphs exactly cancel the one from the first graph. All n dependent contributions thus vanish and the only remaining piece is the term containing f ðtÞ

12


from the first graph, which is n independent. Hence, we have the gauge invariant T-matrix element for graphs of type (a)

g4 1c5 l 1c5 l ð3Þc Ta ¼ eð2Þ eð1Þ l ð4Þc l 2 2 2 64p2 MW Z 1 2 6 X 6 X 1þ2t þ t4 ðVlA VeA ÞðVlB VeB ÞxA xB dt ðt þxA Þðt þxB Þðt þ1Þ2 0 A¼1 B¼1 G2 M 2 ð3Þcl ð1c5 Þeð2Þ ½l ð4Þcl ð1c5 Þeð1Þ ¼ F 2W l 8p " # 6 6 X X 2 ðVlA VeA Þ SðxA Þþ ðVlA VeA ÞðVlB VeB ÞTðxA ;xB Þ ð2:30Þ A;B¼1;A6¼B

A¼1

Here, we have introduced the Fermi scale GF g2 pffiffiffi ¼ 2 2 8MW

ð2:31Þ

along with the Inami–Lim (Inami and Lim 1981) function SðxA Þ ¼

x3 11x2 þ 4x 4ð1 xÞ

We have also defined

TðxA ; xB Þ ¼ xA xB

2

3x3

lnðxÞ

ð2:32Þ

¼ TðxB ; xA Þ

ð2:33Þ

2ð1 xÞ3

JðxA Þ JðxB Þ xA x B

with JðxÞ ¼

x2 8x þ 4 4ð1 xÞ2

ln ðxÞ

3 1 4ð1 xÞ

ð2:34Þ

In a similar manner, the graph (c) in Fig. 2.1 represents the four graphs in Fig. 2.3. The T-matrix elements of these four graphs are ! 6 X 6 Z 1 X 1 1 Tc1 ¼ dtBðxA ; xB ; tÞ t 2 þ ~f ðxA ; xB ; tÞ tþn ðt þ nÞ2 A¼1 B¼1 0 ! 6 X 6 Z 1 X 1 1 dtBðxA ; xB ; tÞ t þ Tc2 ¼ ðt þ nÞ2 t þ n A¼1 B¼1 0 ! ð2:35Þ 6 X 6 Z 1 X 1 1 dtBðxA ; xB ; tÞ t þ Tc3 ¼ ðt þ nÞ2 t þ n A¼1 B¼1 0 ! 6 X 6 Z 1 X 1 Tc4 ¼ dtBðxA ; xB ; tÞ t ðt þ nÞ2 A¼1 B¼1 0


13

Fig. 2.3 Feynman graphs of type (c) in Rn gauge, which are presented as one single graph (a) in Fig. 2.1

where BðxA ; xB ; tÞ ¼

and

(c1)

(c2)

(c3)

(c4)

g4 1 c5 l 1 c5 2 eð2Þ eð1Þ ðVlA Þ2 ðVeB l ð4Þc l ð3Þc Þ l 4 2 2 64p2 MW m A mB xA xB ð2:36Þ 2 ðt þ xA Þðt þ xB Þ 4t þtþ2 ~f ðxA ; xB ; tÞ ¼ xA xB ðt þ 1Þ2

ð2:37Þ

In a similar fashion to the case for the graphs of type (a), all the n dependent terms again cancel against each other leaving only the n independent ~f ðxA ; xB ; tÞ term. Thus the type (c) T-matrix element is gauge invariant and is given by 6 6 2 X X G2F MW 2 l 2 ½ ð1 c Þeð2Þ l ð4Þc ð1 c Þeð1Þ ðV Þ ðVeB Þ l ð3Þc lA 5 l 5 8p2 A¼1 B¼1 ) pffiffiffiffiffiffiffiffiffi Z 1 ( x A xB 4t þ xA xB ðt þ 2Þ ð2:38Þ dt 2 ðt þ xA Þðt þ xB Þðt þ 1Þ2 0

Tc ¼

14


If xA ¼ xB , the relevant integral is pffiffiffiffiffiffiffiffiffi Z 1 xA xB 4t þ xA xB ðt þ 2Þ IðxA Þ ¼ dt 2 ðt þ xA Þðt þ xB Þðt þ 1Þ2 0 ¼

ðxA 4ÞxA ðxA 1Þ

2

þ

ðx3A 3x2A þ 4xA þ 4ÞxA 2ðxA 1Þ3

ln xA

ð2:39Þ

while for xA 6¼ xB , it takes the form pffiffiffiffiffiffiffiffiffi Z 1 xA xB 4t þ xA xB ðt þ 2Þ dt KðxA ; xB Þ ¼ 2 ðt þ xA Þðt þ xB Þðt þ 1Þ2 0 pffiffiffiffiffiffiffiffiffiLðxA ; xB Þ LðxB ; xA Þ ¼ xA xB xA xB

ð2:40Þ

with LðxA ; xB Þ ¼

4 xA xB xA ð2xB xA xB 4Þ þ ln xA 2ðxA 1Þ 2ðxA 1Þ2

ð2:41Þ

The T-matrix element of graph (c) is thus secured as Tc ¼

2 G2F MW ð4Þcl ð1 c5 Þeð1Þ ½l ð3Þcl ð1 c5 Þeð2Þ l 2 8p" # 6 6 X X 2 2 2 ðVlA VeA Þ IðxA Þ þ ðVlA Þ ðVeB Þ KðxA ; xB Þ

ð2:42Þ

A;B¼1;A6¼B

A¼1

2.3 The Effective Lagrangian Combining the various contributions, the T-matrix element can be reproduced using the gauge invariant effective Lagrangian given by: G ffiffiffi ½l cl ð1 c5 Þe cl ð1 c5 Þe l Leff ¼ pMM 2

ð2:43Þ

where " 6 6 2 X X GMM G2F MW 2 pffiffiffi ¼ ðV V Þ Sðx Þ þ ðVlA VeA ÞðVlB VeB ÞTðxA ; xB Þ lA A eA 16p2 A¼1 2 A;B¼1;A6¼B # 6 6 X X 2 2 2 ðVlA VeA Þ IðxA Þ ðVlA Þ ðVeB Þ KðxA ; xB Þ A¼1

A;B¼1;A6¼B

2.3 The Effective Lagrangian

15

" 6 2 X G2F MW 2 ¼ ðVlA VeA Þ ðSðxA Þ IðxA ÞÞ 16p2 A¼1 6 X

þ

# 2 2 ðVlA VeA ÞðVlB VeB ÞTðxA ; xB Þ ðVlA Þ ðVeB Þ KðxA ; xB Þ

A;B¼1;A6¼B

ð2:44Þ

2.4 Limit on MR Muonium (antimuonium) is a nonrelativistic Coulombic bound state of an electron and an anti-muon (positron and muon). The nontrivial mixing between the states is encapsulated in the effective muonium (jMiÞ and antimuonium (jMiÞ Lagrangian of Eq. (2.43) and leads to the mass diagonal states given by the linear combinations (see Appendix C for detailed derivation) 1 jM i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ð1 þ eÞjMi ð1 eÞjMi 2 2ð1 þ jej Þ

ð2:45Þ

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi MMM MMM pffiffiffiffiffiffiffiffiffiffiffiffi e ¼ pffiffiffiffiffiffiffiffiffiffiffiffi MMM þ MMM

ð2:46Þ

where

R hMj d 3 rLeff jMi MMM ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Mi hMjMihMj

R d 3 rLeff jMi hMj p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ MMM Mi hMjMihMj

ð2:47Þ

Since the neutrino sector is expected to be CP violating, these will be independent, complex matrix elements. If the neutrino sector conserves CP, with jMi CP conjugate states, then MMM ¼ MMM and ¼ 0. In general, the and jMi magnitude of the mass splitting between the two mass eigenstates is

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jDMj ¼ 2 Re MM M MMM ð2:48Þ Since muonium and antimuonium are linear combinations of the mass diagonal states, an initially prepared muonium or antimuonium state will undergo oscillations into one another as a function of time. The muonium–antimuonium oscillation timescale, sMM , is given by 1 ¼ jDMj: sMM

ð2:49Þ

16


We would like to evaluate jDMj in the nonrelativistic limit. A nonrelativistic reduction of the effective Lagrangian of Eq. (2.43) produces a local, complex effective potential G 3 ffiffiffi d ðrÞ Veff ðrÞ ¼ 8 pMM ð2:50Þ 2 Taking the muonium (antimuonium) to be in their respective Coulombic ground 1 1 , where aMM ¼ mred er=aMM states, /100 ðrÞ ¼ pffiffiffiffiffiffiffiffi 3 a is the muonium Bohr radius with mred ¼

me ml me þml

paMM

’ me the reduced mass of muonium, it follows that 1

sMM

’ 23 r/100 ðrÞjReVeff ðrÞj/ðrÞ100 jReG j 16jReGMM j 1 ffi j/100 ð0Þj2 ¼ pffiffiffi ¼ 16 pffiffiMM p 2 2 a3MM

ð2:51Þ

Thus, we secure an oscillation timescale 1 16jReGMM j pffiffiffi m3e a3 ’ sMM p 2

ð2:52Þ

The present experimental limit (Willmann et al. 1999) on the non-observation of muonium–antimuonium oscillation translates into the bound jReGMM j 3:0 103 GF where GF ’ 1:16 105 GeV2 is the Fermi scale. This limit can then be used to construct a crude lower bound on MR . For the case when the neutrino masses arise from a see-saw mechanism and taking mD to be of order MW , the MR dependence of GMM is obtained from Eq. (2.44) as: Case 1 : Case 2 : Case 3 :

4 G2F MW MR ln ; 2 MW MR 4 G2F MW MR jReGMM ln ; j 2 MW MR 6 G2F MW MR ln ; jReGMM j 4 MW MR

jReGMM j

A ¼ 1; 2; 3;

B ¼ 1; 2; 3

A ¼ 4; 5; 6;

B ¼ 4; 5; 6

A ¼ 1; 2; 3;

B ¼ 4; 5; 6

ð2:53Þ

Cases 1 and 2 give the same order MR dependence, while case 3 is suppressed by an additional factor of

2 MW : MR2

Hence, the term

4 G2F MW MR2

MR ln M gives the dominant W

contribution. We then roughly calculate a bound of MR as 4 G2F MW MR ln

3:0 103 GF 2 MW MR

ð2:54Þ

which has also been obtained in reference (Cvetic et al. 2005). Using MW ’ 80:4 GeV and GF ¼ 1:166 105 GeV2 , we finally secure MR 6 102 GeV

ð2:55Þ

2.4 Limit on MR

17

Note that this is just a rough estimate, since we are retaining only the dependence on MR while neglecting all numerical dependence on the mixing angles and CP violating phases in VaA .

References T.-P. Cheng and L.-F. Li, Gauge theory of elementary particle physics, 1988, p. 509. G. Cvetic, C.O. Dib, C.S. Kim and J.D. Kim, Phys. Rev. D 71, 113013 (2005). M. Gell-Mann, P. Ramond and R. Slansky in Supergravity, (P. van Nieuwenhuizen and D. Freedman eds) North-Holland, Amsterdam, 1979. S.L. Glashow, in Proceedings of the 1979 Cargese Institute on Quarks and Leptons (M. Levy et. al. eds.) Plenum Press, New York 1980, p. 687. A. Ilakovac and A. Pilaftisis, Nucl. Phys. B437 (1995) 491. T. Inami and C.S. Lim, Prog. Theor. Phys. 65, 297 (1981). P. Minkowski, Phys. Lett. B67, 421 (1977). R.N. Mohapatra and G. Senjanovic. Phys. Rev. Lett. 44, 912 (1980). L. Willmann et al, Phys. Rev. Lett 82, 49 (1999). T. Yanagida, in Proceedings of the Workshop on the Unified Theories and Baryon Number in the Universe, Tsukuba, Japan, 1979, (O. Sawada and A. Sugamoto eds), p. 95.

Chapter 3

Supersymmetry and the Minimal Supersymmetric Standard Model

3.1 Why Supersymmetry? 1. One reason that physicists explore supersymmetry (SUSY) is because it offers an extension to the more familiar space–time symmetries of quantum field theory. These symmetries are grouped into the Poincaré group and internal symmetries. The Coleman–Mandula theorem (Coleman and Mandula 1967) showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincaré group with a compact internal symmetry group or, if there is no mass gap, the conformal group with a compact internal symmetry group. In 1975, the Haag–Lopuszanski–Sohnius theorem (Haag et al. 1975) showed that considering symmetry generators that satisfy anticommutation relations allows for such nontrivial extensions of space–time symmetry. This extension of the Coleman–Mandula theorem prompted some physicists to study this wider class of theories. 2. One of the main motivations for SUSY comes from the quadratically divergent contributions to the Higgs mass squared. The quantum mechanical interactions of the Higgs boson causes a large renormalization of the Higgs mass and, unless there is an accidental cancelation or fine tuning, the natural size of the Higgs mass is the highest scale possible. However, taking all precision measurements together in a global fit, the current experiment infers that the Standard Model Higgs boson mass must be lighter than around 200 GeV (95% c.l.) (2006). This problem is known as the hierarchy problem (Weinberg 1976; Gildener 1976; Susskind 1979; t Hooft 1979). Consider a massive fermion loop correction to the propagator for the Higgs field, as shown in Fig. 3.1. If the Higgs field couples to a Dirac fermion f with a term in the Lagrangian kf Hf f , the leading correction is then given by


19

20

3 Supersymmetry and the Minimal Supersymmetric Standard Model

Fig. 3.1 Fermion loop contribution to the selfenergy of the Higgs boson

Dm2H ¼

jkf j2 2 ½K 2m2f lnðKUV =mf Þ þ 8p2 UV

ð3:1Þ

Here, KUV is an ultraviolet momentum cutoff used to regulate the loop integral. If the cutoff, KUV , is replaced by the Planck mass, Mplanck , the resulting correction would be some thirty degrees of magnitude larger than the experimental bound on the Higgs. Supersymmetry provides automatic cancelations between fermionic and bosonic Higgs interactions. For example, consider a scalar field S, which couples to Higgs with a Lagrangian term kS jHj2 jSj2 . Then the Feynman graph in Fig. 3.2 gives a correction Dm2H ¼

kS ½K2 2m2S lnðKUV =ms Þ þ 16p2 UV

ð3:2Þ

If each of the quarks and leptons of the Standard Model is accompanied by two complex scalars with kS ¼ jkf j2 , then the KUV contributions of Figs. 3.1, 3.2 will cancel. Supersymmetry relates the fermion and boson couplings in just this manner. Each Standard Model field gets a super partner with couplings to insure the cancelations. 3. Another motivation for supersymmetry existing at the weak scale is gauge coupling unification. The renormalization group evolution of the three gauge coupling constants of the Standard Model is sensitive to the particle content of the theory. These coupling constants do not quite meet together at a common energy scale if we run the renormalization group using the Standard Model. Supersymmetry actually allows for the unification of three other forces, strong, weak and electromagnetic as shown in Fig. 3.3 (Martin 1997). 4. The most general superpotential of MSSM contains terms where the baryon and the lepton numbers are violated. To present this, a new discrete symmetry, called R-parity can be introduced. It is a multiplicative quantum number where all the particles of the Standard Model have positive R-parity, while their superpartners have negative R-parity and the quantum number is given by R ¼ ð1Þ3ðBLÞþ2s

ð3:3Þ

for a particle with spin s and baryon and lepton number B and L. This would lead to a useful phenomenological result. The lightest s particle, called the LSP, must be absolutely stable. If it is electrically neutral, the LSP is an excellent candidate for dark matter (for a review, see Jungman et al. 1996).

3.2 Minimal Supersymmetric Standard Model

21

Fig. 3.2 Scalar loop contribution to the selfenergy of the Higgs boson

Fig. 3.3 Dashed lines denote the Standard Model couplings and solid lines denote the Minimal Supersymmetric Standard Model couplings

3.2 Minimal Supersymmetric Standard Model The Minimal Supersymmetric Standard Model (MSSM) is the supersymmetric extension of the Standard Model with a minimal particle content (Martin 1997). For each particle, there is a superpartner with the same internal quantum numbers, but with spin that differs by half a unit. All of the SM fermions have the property that the left-handed and right-handed components transform differently under the gauge groups. Only chiral multiplets can contain fermions in which left-handed components transform differently from their right-handed partners under a gauge group. Therefore in the MSSM, each of the fundamental particles must be in either a gauge or a chiral supermultiplet. The spin-0 superpartners of the SM fermions are appended with an s; for example, the electron superpartner is named selectron. Additionally, the superpartners are denoted with a ‘‘tilde’’; for example, the left-handed selectron is written ~eL . Note that the subscript on the selectron refers to the handedness of the SM partner of the selectron since the selectrons are spin-0. The left-handed and right-handed components of the fermions are separate two-component Weyl fermions with different gauge transformations. Therefore, they must have separate complex scalar partners. The gauge interactions of the scalar partners are the same as the corresponding SM fermion. The Higgs boson must reside in a chiral multiplet since it is spin-0. Actually, it turns out that we need two multiplets. This can be simply understood from the fact

22


that the superpotential is holomorphic, so the Higgs multiplet giving mass to the up-type quarks could not give mass to the down-type quarks. So to have masses for all the quarks, we need to have at least two Higgs fields, one with Y ¼ 12 and the other with Y ¼ 12. Another reason for needing two multiplets is that, if we just had one, then the electroweak gauge symmetry would suffer a triangle gauge anamoly. This lists all the chiral supermultiplets needed for MSSM and are summarized in Table 3.1. The vector bosons of the SM reside in gauge supermultiplets. Their fermionic superpartners are refered to as gauginos. The color interaction of QCD is mediated via gluons, and their superpartners are called gluinos. As before, a tilde is used to denote the superpartner. Table 3.2 summarizes the gauge supermultiplets of MSSM. _ indicate two-compoIn Tables 3.1, 3.2, the dotted and undotted indices, a, a, nent Weyl spinor fields. In the subsequent analysis, we will recast all the spin 12 fields as four-component Dirac spinor fields, which will be represented using the same symbols, but without the dotted and undotted ‘‘a’’s. For example, La is the m mLa Weyl representation of the left-handed neutrino field, while mL ¼ is the maL_ Dirac field. Based on SUð3Þ SUð2Þ Uð1Þ symmetry and supersymmetry and R-parity, we could construct the renormalizable Lagrangian of the MSSM as: I ¼ IYM þ IKin þ IMatter þ IBreaking ;

ð3:4Þ

where IKin ¼

Z

dx4 dh2 d h2 Qe2VQ Q þ U c e2VU U c þ Dc e2VD Dc þ Le2VL L þ Ec e2VE Ec ð3:5Þ þ HB e2VHB HB þ HTc e2VHT HTc

Table 3.1 Chiral supermultiplets of the MSSM Name Spin 0

Spin 12

SUð3ÞC ; SUð2ÞL ; Uð1ÞY

ðuLa dLa Þ

squarks and quarks (3 families)

Q Uc Dc

ð~ uL d~L Þ ~ uR d~

uaR_ dRa_

ð3; 2; 16Þ ð3; 1; 23Þ ð3; 1; 13Þ

sleptons, leptons (3 families) Higgs, higgsinos

L

ð~mL ~eL Þ

ðmLa eLa Þ

ð1; 2; 12Þ

Ec HT

~eR 0 ðhþ T hT Þ

eaR_ ~0 ð~ hþ Ta hTa Þ

ð1; 1; 1Þ ð1; 2; 12Þ

HB

ðh0B h BÞ

ð~ h0Ba ~h Ba Þ

ð1; 2; 12Þ

R

3.2 Minimal Supersymmetric Standard Model

23

Table 3.2 Gauge supermultiplets in the MSSM Names Spin 12 ~ ga ~0 ~ a W W a ~ Ba

Gluino, gluon Winos, W bosons Bino, B boson

IMatter ¼

Spin 1

SUð3ÞC ; SUð2ÞL ; Uð1ÞY

gl Wl Wl0 Bl

ð8; 1; 0Þ (1, 3, 0) ð1; 1; 0Þ

Z

h 0 0 dx4 dh2 mHTa HBa þ gffE Efc ab Laf0 HBb þ gffD Dcf ab Qaf0 HBb i 0 c Qaf0 þ H:C: þ gffU Ufc HTa

ð3:6Þ

All the above equations are expressed in superfields. Q; U c ; Dc ; L; Ec ; HT ; HB are U c ; L; E c; H T; H B are anti-chiral superfields. Chiral c; D all chiral superfields and Q; superfields and anti-chiral superfields have component structures as follows: pffiffiffi l ð3:7Þ E ¼ eihr hol ½~eðxÞ þ 2ha ea ðxÞ þ h2 FE ðxÞ pffiffiffi l Ec ¼ eihr hol ½~ec ðxÞ þ 2ha eca ðxÞ þ h2 FEc ðxÞ ð3:8Þ pffiffiffi a_ ¼ eihrl hol ½~e ðxÞ þ 2 E ha_ e ðxÞ þ h2 FE ðxÞ ð3:9Þ pffiffiffi c ¼ eihrl hol ½e~c ðxÞ þ 2 ha_ ec a_ ðxÞ þ h2 FE c ðxÞ ð3:10Þ E In Eq. (3.5) kI rA 1 þ g2 W A þ g1 Y 2 2 6

ð3:11Þ

VU ¼ g3 GI

kI 2 þ g1 Y 2 3

ð3:12Þ

VD ¼ g3 GI

kI 1 g1 Y 2 3

ð3:13Þ

VL ¼ g2 W A

rA 1 g1 Y 2 2

ð3:14Þ

VQ ¼ g3 GI

VE ¼ g1Y

ð3:15Þ

VHd ¼ g2 W A

rA 1 g1 Y 2 2

ð3:16Þ

VHu ¼ g2 W A

rA 1 g1 Y 2 2

ð3:17Þ

24


where GI ; W A and Y are vector superfields. In Wess–Zumino gauge, they have the component structure: 1 hGIl þ ihh GI ¼ hrl h kIG i h hhkIG þ hhhhDIG 2

ð3:18Þ

As mentioned earlier, if supersymmetry were an exact symmetry, particles and their SUSY partners would be degenerate in mass and we would have been able to observe the selectrons, photinos and gluinos by now. Since we have not able to do so, we know that SUSY is a broken symmetry. From a theoretical perspective, the symmetry must be spontaneously broken (SB), analogous to the electroweak symmetry in the SM. Many models of spontaneous symmetry breaking have indeed been proposed. These always involve extending the MSSM to include new particles and interactions at very high mass scales, and there is no consensus on exactly how this should be done. However, from a practical point of view, it is extremely useful to simply parameterize our ignorance of these issues by just introducing extra terms that softly break the supersymmetry explicitly in the effective MSSM Lagrangian. In doing so, one finds the breaking terms: 1 ~W ~ þ M1 B ~B ~ þ c:c:Þ IBreaking ¼ ðM3 ~ g þ M2 W 2 ~ u d~R ad QH ~ d ~eR ae LH ~ d þ c:cÞ ð~ uR au QH ~ L ~ y m2 Q ~ y m2 L ~~ uR m 2 c ~ u d~R m2 c d~ ~eR m2 c ~e Q Q

L

U

R

D

R

m2Hu Hu Hu m2Hd Hd Hd ðbHu Hd þ c:c:Þ:

E

R

ð3:19Þ

References S. Coleman, and J. Mandula, Phys. Rev. 159 (1967) 1251 E. Gildener, Phys. Rev. D 14, 1667 (1976) R. Haag, J. Lopuszan´ki, and M. Sohnius, Nucl. Phys. B 88, 257 (1975) G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rep. 267, 195 (1996) S.P. Martin, A supersymmetry Primer (1997). L. Susskind, Phys. Rev. D 20, 2619 (1979) G. t Hooft, in Recent developments in gauge theories, Proceedings of the NATO Advanced Summer Institute, Cargese 1979 (Plenum, 1980) S. Weinberg, Phys. Rev. D 13, 974 (1976) The ALEPH, DELPHI, L3, OPAL, SLD Collaborations, the LEP Electroweak Working Group, the SLD Electroweak and Heavy Flavour Groups, Phys. Rep. 427, 257 (2006), hep-ex/0509008

Chapter 4

The Muonium–Antimuonium Oscillation in the Extended Minimal Supersymmetric Standard Model

4.1 The Extended Minimal Supersymmetric Standard Model In order to implement the see–saw mechanism (Minkowski 1977; Yanagida 1979; Gell-Mann et al. 1979; Glashow 1980; Mohapatra and Senjanovic 1980) for neutrino masses, we consider an extension of the MSSM, where one adds three additional gauge singlet chiral superfields Nic (i ¼ e; l; s denotes the generation), whose h-component is a right-handed neutrino field, pffiffiffi Nic ¼ ~miR ðyÞ þ 2ha mR ðyÞa þ ha ha FNic ðyÞ; ð4:1Þ where

ha_ : yl ¼ xl þ iha rlaa_

ð4:2Þ

These SUð3Þ SUð2ÞL Uð1Þ singlet superfields are coupled to other MSSM superfields via the superpotential. We employ the most general R-parity conserving renormalizable superpotential so that the superpotential is 1 W ¼ lab HBa HTb þ ki ab Eic Lai HBb þ k0i ab HTa Lbi Nic þ MRi Nic Nic ; 2 while the relevant soft supersymmetry breaking terms are ¼ ðmiL~ Þ2 ~miL~miL þ ‘~iL ‘~iL ðmiR~ Þ2 ‘~iR ‘~iR ðmiN Þ2~miR~miR LEMSSM soft k0i Ai h0T ~miL~miR þ MRi Bi~miR~miR þ ki Ci h0B ‘~iL ‘~iR þ H:C: :

ð4:3Þ

ð4:4Þ

The interaction terms that contribute to the muonium–antimuonium oscillation and the electron and muon anomalous magnetic moments can be extracted from the Lagrangian of this extended Minimal Supersymmetric Standard Model (EMSSM) as g2 l ð4:5Þ ‘iL cl miL þ W þlmiL cl ‘iL ; LW int ¼ pffiffiffi W 2 B. Liu, Muonium–Antimuonium Oscillations in an Extended Minimal Supersymmetric Standard Model, Springer Theses, DOI: 10.1007/978-1-4419-8330-5_4, Ó Springer Science+Business Media, LLC 2011

25

26

4 The Muonium–Antimuonium Oscillation

~ ~ miL ~m W ~ ‘iL ; LW int ¼ ig2 ‘iL W ~ iL

~

LBint ~h

g2 i ~ 0 ~ ~0 0‘ ~ W ~ p ffiffi ffi ‘ W ; ‘ LW ¼ ‘ iL iL iL int iL 2 pffiffiffi g1 i ~ ~ ~ ‘iL þ 2g1 i ‘iR B ~ ‘~iR ‘~iR B ~ ‘iR ; ‘iL ‘~iL B ¼ pffiffiffi ‘iL B 2

mi mi ~ D ~ ~ ‘iL ~m ; ~ ~ h h ‘ þ þ ‘iR hB ~miL þ h~ ‘ m m iR iL iR B B B iL iR VB VT m m i ~0 ~ i ~0 ~ ‘iL hB ‘iR þ ‘~iR ~ ‘iR hB ‘iL þ ‘~iL ~h0B ‘iR : ¼ h0B ‘iL VB VB

LintB ¼ ~h0

LintB

ð4:6Þ ð4:7Þ ð4:8Þ ð4:9Þ ð4:10Þ

In the above equations, all the spin 12 fields are Dirac spinor fields. In particular, note that the field ~ h B has the Weyl field decomposition ! ~ h Ba ~ : ð4:11Þ hB ¼ a_ ~ hþ T The parameters VB and VT are the vacuum expectation values of the two Higgs fields: hh0B i ¼ VB and hh0T i ¼ VT : These VEVs are related to the known mass of the W boson and the electroweak gauge couplings as VB2 þ VT2 ¼ V 2 ¼

2 2MW ð174 GeVÞ2 ; g22

ð4:12Þ

while the ratio of the VEVs is traditionally written as tan b

VT : VB

ð4:13Þ

In the above, miD ¼ k0i VT are the Dirac mass parameters of neutrinos and mi are the lepton masses. Since the masses of electron and muon are small, the terms which have couplings proportional to mi =VB in interactions (4.9) and (4.10) are severely suppressed and will be ignored in the subsequent analysis.

4.2 Neutrino and Sneutrino Mass Eigenstates in the EMSSM The neutrino mass term can be extracted from the superpotential terms k0i ab HTa Lbi Nic and 12MRi Nic Nic in Eq. 4.3 as 0 mi 1 miL mi c D Lmass ¼ ðmiL Þ miR þ H:C: ð4:14Þ ðmiR Þc miD MRi 2

4.2 Neutrino and Sneutrino Mass Eigenstates in the EMSSM

27

Note that the upper left element in the neutrino mass matrix is zero. This element involves only left-handed neutrinos and in our EMSSM its generation requires a nonrenormalizable superpotential term. Consequently, we ignore this term. For every generation, the two mass eigenvalues, mmai ; are obtained from the diagonalization of the 2 2 matrix 0 miD mi M ¼ : ð4:15Þ miD MRi Since M mi is symmetric, it can be diagonalized by a single unitary 2 2 matrix, V i ; as ~mi ¼ V iT M~mi V i : Mdiag

ð4:16Þ

This diagonalization is implemented via the basis change as following c miL mi1 ðmiL Þc i i ðmi1 Þ ¼ V ; ¼ V : ðmiR Þc ðmi2 Þc miR mi2

ð4:17Þ

The neutrino mass term then takes the form i ¼ Lmmass

2 2 h i X 1X mmAi mTia Cmia þ mia CmTia ¼ mmai mia mia ; 2 a¼1 a¼1

ð4:18Þ

where mmAi are the Majorana neutrino masses. Since a nonzero Majorana mass parameter MRi does not require SUð2ÞL Uð1Þ symmetry breaking, it is naturally much bigger than the Dirac mass parameter miD whose nontrivial value does require SUð2ÞL Uð1Þ symmetry breaking. So doing, one finds on diagonalization of the 2 2 neutrino mass matrix that the two eigenvalues are crudely given by mm1i

ðmiD Þ2 miD ; MRi

mm2i MRi :

ð4:19Þ

This constitutes the so called see–saw mechanism (Minkowski 1977; Yanagida 1979; Gell-Mann et al. 1979; Glashow 1980; Mohapatra and Senjanovic 1980) and provides a natural explanation of the smallness of the three light neutrino masses. Moreover, the elements of the mixing matrix are characterized by an miD =MRi mi

dependence. We expand V i in power series of the matrix parameter ni ¼ MDi ; with R

the constraint ni 1: The form of V i to first order in ni can be estimated to be 1 ni i V ¼ : ð4:20Þ ni 1 The sneutrino masses are obtained by diagonalizing a 4 4 squared mass matrix. Here, it is convenient to define ~miL ¼ p1ffiffi2ð~miL1 þ i~miL2 Þ and ~miR ¼ p1ffiffi2ð~miR1 þ

28


i~miR2 Þ: Then, the sneutrino-squared mass matrix separates into CP-even and CPodd blocks (Yuval and Howard 1997), ! i / 1 i ~mi 2 1 Lmass ¼ /1 /i2 M~mi 2 /i2 ! ! 0 M~2mi þ /i1 1 i i ¼ /1 /2 ; ð4:21Þ 2 /i2 0 M~2mi where /ia ð~miLa M~2mi

¼

~miRa Þ and M~2mi consist of the following 2 2 blocks:

ðmiL~ Þ2 þ 12m2Z cos 2b þ ðmiD Þ2 miD ðAi l cot b MRi Þ

miD ðAi l cot b MRi Þ i 2 ðMR Þ þ ðmiD Þ2 þ ðmiN~ Þ2 2Bi MRi

! ;

ð4:22Þ with Ai and Bi are SUSY breaking parameters (cf. Eq. 4.4). Since the sneutrino mass matrix M~2mi is real and symmetric, it can be diagonalized by a real orthogonal 4 4 matrix, U i ; as M~2mi diag ¼ U iT M~2mi U i ; where U i is in a form as Ui ¼

Uþi 0

0 : Ui

This diagonalization is implemented via the basis change on /i1 and /i2 0 0 1 1 ~mi1 ~miL1 i B ~miR1 C B mi2 C /1 iB ~ C C ¼B @ ~miL2 A ¼ U @ ~mi3 A; /i2 ~miR2 ~mi4

ð4:23Þ

ð4:24Þ

ð4:25Þ

where ~mia are all real. Then the sneutrino mass term takes the form 4 1X i ¼ m~mi ~mia~mia ; L~mmass 2 a¼1 a

ð4:26Þ

where m~mai are the sneutrino mass eigenvalues. In the following derivation we assume that MRi is the largest mass parameter. Then, to the first order in 1=MRi ; the two light mass eigenvalues are roughly 1 2ðmiD Þ2 ðAi l cot b Bi Þ ; m~2mi1 ðmiL~ Þ2 þ m2Z cos 2b 2 MRi m~2mi3

ðmiL~ Þ2

1 2ðmiD Þ2 ðAi l cot b Bi Þ þ m2Z cos 2b þ ; 2 MRi

ð4:27Þ

4.2 Neutrino and Sneutrino Mass Eigenstates in the EMSSM

29

while the two heavy mass eigenvalues are m~2mi2 ðMRi Þ2 þ 2Bi MRi ;

ð4:28Þ

m~2mi4 ðMRi Þ2 2Bi MRi :

To avoid excessive complication in our calculations, we expand U i in powers of mi

the matrix parameter ni ¼ MDi : The form of U to first order of ni is R

0 Ui ¼

Uþi 0

0 Ui

B B ¼B @

1

ni

ni

1

1

0

0

1 ni

C C C: ni A 1

For every generation, the slepton mass term is given by 0 1 ! LL 2 LR 2 ðm Þ ðm Þ ~iL ~ ~ ‘ ‘ ‘ i i ‘~i A Lmass ¼ ‘~iL ‘~iR @ ; ‘~iR ðmLR Þ2 ðmRR Þ2 ‘~ ‘~ i

where

ð4:29Þ

ð4:30Þ

i

1 2 i 2 2 2 Þ ¼ ðm Þ þ m cos 2b sin h ðmLL ; W ~ Z L ‘~i 2

ð4:31Þ

Þ2 ¼ ki lVT þ ki Ci VB ; ðmLR ‘~i

ð4:32Þ

Þ2 ¼ ðmiR~ Þ2 m2Z cos 2b sin2 hW : ðmRR ‘~

ð4:33Þ

i

Since ki VB ¼ mi , the off diagonal matrix element, ðmLR Þ2 ; can be written as ‘~ i

Þ2 ðmLR ‘~i

¼ mi ðl tan b þ Ci Þ:

ð4:34Þ

Because the masses of electron and muon are very small compared with the sparticle mass scale, we ignore these off diagonal terms and consider ‘~iL and ‘~iR as mass eigenstates. Inserting the transformation (4.17) and (4.25) in the interaction terms (4.5–4.9) yields the explicit interactions in their mass basis: 2 g2 X i i p ffiffi ffi LW ¼ W l ‘iL cl V1a mia þ W þlmia V1a cl ‘iL ; int 2 a¼1 2 4 ig2 X g2 X ~ W i i ~ U1a ~ U3a ~mia þ pffiffiffi ~mia þ H:C:; ‘iL W ‘iL W ¼ pffiffiffi Lint 2 a¼1 2 a¼3

g2 i ~ 0 ~ ~0 ~ ~ 0 LW int ¼ pffiffiffi ‘iL W ‘iL ‘iL W ‘iL ; 2

ð4:35Þ

ð4:36Þ

ð4:37Þ

30


pffiffiffi g1 i ~ ~ ~ ~ ‘iL þ 2g1 i ‘iR B ~ ‘~iR ‘~iR B ~ ‘iR ; ‘iL ‘~iL B LBint ¼ pffiffiffi ‘iL B 2

ð4:38Þ

2 4 ~h mi X imiD X i i p ffiffi ffi ~ ‘iL h~ ‘iL h~ m mia þ H:C: LintB ¼ pffiffiffiD U þ B 2a ia B U4a~ 2VT a¼1 2VT a¼3

ð4:39Þ

4.3 The Muonium–Antimuonium Oscillation in the EMSSM The lowest order Feynman diagrams accounting for muonium and antimuonium mixing are displayed in Fig. 4.1. Graphs (a) and (b) are the non-SUSY contributions, which are mediated by Majorana neutrinos and W boson. The other graphs all involve SUSY partners. Graphs (c) and (d) are mediated by sneutrinos and wino, while graph (e) and (f) are mediated by sneutrinos and higgsino. The T-matrix elements of graphs (a) and (b) are Ta ¼ Tb ¼

g42 ½ lð3Þcl ð1 c5 Þeð2Þ ½ lð4Þcl ð1 c5 Þeð1Þ

2 512p2 MW 2 X 2 X l 2 e 2 ðV1a Þ ðV1b Þ Kðxmla ; xmeb Þ;

ð4:40Þ

a¼1 b¼1

ð3Þ ¼ l ðp3 ; s3 Þ; l where l ð4Þ ¼ l ðp4 ; s4 Þ; eð1Þ ¼ eðp1 ; s1 Þ and eð2Þ ¼ eðp2 ; s2 Þ are the spinors of the muons and electrons and xmia ¼

m2mia 2 ; MW

a ¼ 1; 2: The function

Kðxmla ; xmeb Þ takes the form pffiffiffiffiffiffiffiffiffiLðxA ; xB Þ LðxB ; xA Þ xA xB ; xA xB

ð4:41Þ

4 xA xB xA ð2xB xA xB 4Þ ln xA : þ ðxA 1Þ ðxA 1Þ2

ð4:42Þ

KðxA ; xB Þ ¼ with LðxA ; xB Þ ¼

The T-matrix elements of graphs (c) and (d) are Tc ¼ Td ¼

g42 ½ lð3Þcl ð1 c5 Þeð2Þ ½ lð4Þcl ð1 c5 Þeð1Þ

2 1024p2 MW ~ 2 X 2 2 X 4 X X l 2 l 2 e 2 e 2 ðU1a Þ ðU1b Þ Iðy~mla ; y~meb Þ ðU1a Þ ðU3b Þ Iðy~mla ; y~meb Þ a¼1 b¼1

a¼1 b¼3

4 X 2 X

! 4 X 4 X l 2 l 2 e 2 e 2 ðU3a Þ ðU1b Þ Iðy~mla ; y~meb Þ þ ðU3a Þ ðU3b Þ Iðy~mla ; y~meb Þ ;

a¼3 b¼1

a¼3 b¼3

ð4:43Þ

4.3 The Muonium–Antimuonium Oscillation in the EMSSM

31

Fig. 4.1 Feynman graphs contributing to the muonium– antimuonium mixing

(a)

(b)

(c)

(d)

(e)

(f)

where y~mia ¼

Iðx1 ; x2 Þ ¼

m~2mia ; 2 MW ~

Jðx1 Þ Jðx2 Þ ; x1 x 2

ð4:44Þ

ð4:45Þ

32


with JðxÞ ¼

x2 ln x x þ 1 : ðx 1Þ2

ð4:46Þ

Finally, the T-matrix elements of graphs (e) and (f) are Te ¼ Tf ¼

ðmlD Þ2 ðmeD Þ2 ½ lð3Þcl ð1 c5 Þeð2Þ ½ lð4Þcl ð1 c5 Þeð1Þ

1024VT4 p2 M~h2 B

2 X 2 2 X 4 X X l 2 l 2 e 2 e 2 ðU2a Þ ðU2b Þ Iðz~mla ; z~meb Þ ðU2a Þ ðU4b Þ Iðz~mla ; z~meb Þ a¼1 b¼1

a¼1 b¼3

! 4 X 2 4 X 4 X X l 2 l 2 e 2 e 2 ðU4a Þ ðU2b Þ Iðz~mla ; z~meb Þ þ ðU4a Þ ðU4b Þ Iðz~mla ; z~meb Þ ; a¼3 b¼1

a¼3 b¼3

ð4:47Þ where z~mia ¼

m~2mia : M~h2

ð4:48Þ

B

4.4 The Effective Lagrangian Combining all the T-matrix elements, we secure an effective Lagrangian which can be cast as: G ffiffiffi ½ Leff ¼ pMM lcl ð1 c5 Þe ½ lcl ð1 c5 Þe ; ð4:49Þ 2 where 2 X 2 X GMM g42 l 2 e 2 pffiffiffi ¼ ðV1a Þ ðV1b Þ Kðxmla ; xmeb Þ 2 2M 1024p 2 W a¼1 b¼1

g42 2 2048p2 MW ~ 2 X 4 X

2 X 2 X l 2 e 2 ðU1a Þ ðU1b Þ Iðy~mla ; y~meb Þ a¼1 b¼1

l 2 e 2 ðU1a Þ ðU3b Þ Iðy~mla ; y~meb Þ

a¼1 b¼3

þ

4 X 4 X a¼3 b¼3

4 X 2 X l 2 e 2 ðU3a Þ ðU1b Þ Iðy~mla ; y~meb Þ: a¼3 b¼1

! l 2 e 2 ðU3a Þ ðU3b Þ Iðy~mla ; y~meb Þ

ðmlD Þ2 ðmeD Þ2 2048VT4 p2 M~h2 B

4.4 The Effective Lagrangian

33

2 X 2 2 X 4 X X l 2 l 2 e 2 e 2 ðU2a Þ ðU2b Þ Iðz~mla ; z~meb Þ ðU2a Þ ðU4b Þ Iðz~mla ; z~meb Þ a¼1 b¼1

4 X 2 X

a¼1 b¼3

l 2 e 2 ðU4a Þ ðU2b Þ Iðz~mla ; z~meb Þ

a¼3 b¼1

þ

4 X 4 X

!

l 2 e 2 ðU4a Þ ðU4b Þ Iðz~mla ; z~meb Þ

:

a¼3 b¼3

ð4:50Þ

4.5 Estimate of the Effective Coupling Constant The present experimental limit (Willmann 1999) on the non-observation of muonium–antimuonium oscillation translates into the bound 3 jReGMM j 3:0 10 GF ;

ð4:51Þ

where GF ’ 1:16 105 GeV2 is the Fermi scale. This limit can then be used to construct some constraints on the parameters of this model. For simplicity, we set the neutrino Dirac masses meD ; mlD and the right-handed neutrino masses MRe ; MRl to some common mass scales mD and MR respectively. The light neutrino mass scale mm is of order m2D =MR ; while the heavy neutrino mass scale is of order MR : Using these assumptions and taking into account the mixing matrices approximations Eq. 4.20 and 4.29, we can simplify the effective coupling constant (4.50) to a more manageable approximated form. The contribution from graphs (a) and P P g4 l 2 e 2 is 1024p22 M2 2a¼1 2b¼1 ðV1a Þ ðV1b Þ Kðxmla ; xmeb Þ: With the limits of (b) in GMM W

m2 mml 1 ; mme 1 OðMDR Þ

l 2 e 2 and mml 2 ; mme 2 OðMR Þ; the contribution ðV1a Þ ðV1b Þ Kðxmla ; xmeb Þ can be approximated as m4 l 2 e 2 ðV11 Þ ðV11 Þ Kðxml1 ; xme1 Þ M2 MD 2 ln MmR M2 W ; R W D m8 l 2 l 2 e 2 e 2 ðV11 Þ ðV12 Þ Kðxml1 ; xme2 Þ; ðV12 Þ ðV11 Þ Kðxml2 ; xme2 Þ M4 MD 4 ln MmR M2 W ; ð4:52Þ D R W m4D l 2 MR e 2 ðV12 Þ ðV12 Þ Kðxml2 ; xme2 Þ M 2 M2 ln MW : R

W

Taking MR as the largest mass parameter, the first term and the third term are 2 Þ: comparable, while the second one is suppressed by a factor m4D =ðMR2 MW Therefore, the contribution from graphs (a) and (b) is roughly 2 X 2 X g42 g42 m4D MR l 2 e 2 ðV Þ ðV Þ Kðx ; x Þ ln : m m la eb 1b 1a 2 2 4 2 2 MW 1024p MW a¼1 b¼1 1024p MR MW

ð4:53Þ

34


The second term in Eq. 4.50 is the contribution of graph (c) and (d), in which the function Iðy~mla ; y~meb Þ is a decreasing function of y~mla and y~meb : It will be small for heavy sneutrinos. To see this, we employ the approximations (4.29) l ; U11

e U11 ;

l U33 ;

l U12 ;

e U12 ;

l U34 ;

e U33 Oð1Þ; mD e ; U34 O MR

ð4:54Þ

so that the terms involving heavy sneutrinos will get an extra suppression from the mixing matrix. Therefore, the contribution of graph (c) and (d) is dominated by the term that only includes the light sneutrinos so that

g42 2 2048p2 MW ~

2 X 2 2 X 4 X X l 2 l 2 e 2 e 2 ðU1a Þ ðU1b Þ Iðy~m ; y~meb Þ ðU1a Þ ðU3b Þ Iðy~m ; y~meb Þ a¼1 b¼1

a¼1 b¼3

4 X 2 4 X 4 X X l 2 l 2 e 2 e 2 ðU3a Þ ðU1b Þ Iðy~m ; y~meb Þ þ ðU3a Þ ðU3b Þ Iðy~mla ; y~meb Þ a¼3 b¼1

!

a¼3 b¼3

g42 2 2048p2 MW ~

Iðy~ml1 ; y~me1 Þ Iðy~ml1 ; y~me3 Þ Iðy~ml3 ; y~me1 Þ þ Iðy~ml3 ; y~me3 Þ : ð4:55Þ

Employing the squared-mass difference between the two light sneutrinos in Eq. 4.27, the above expression can be approximated as

g42 Iðy~ml1 ; y~me1 Þ Iðy~ml1 ; y~me3 Þ Iðy~ml3 ; y~me1 Þ þ Iðy~ml3 ; y~me3 Þ 2 2 2048p MW~

g42 o o ðy~ml1 y~ml3 Þðy~me1 y~me3 Þ Iðy~ml1 ; y~me1 Þ 2 2 oy~ml1 oy~me1 2048p MW~

Dm~2ml Dm~2me g42 o o 2 Iðy~ml1 ; y~me1 Þ; 2 2 2 2048p MW~ MW~ MW~ oy~ml1 oy~me1

ð4:56Þ

where the squared-mass differences are Dm~2ml ¼ Dm~2me

4ðmlD Þ2 ðAl l cot b Bl Þ ; MRl

4ðmeD Þ2 ðAe l cot b Be Þ ¼ : MRe

ð4:57Þ

Assuming Al ¼ Ae A and Bl ¼ Be B; the squared-mass differences of light muon sneutrinos and light electron sneutrinos are Dm~2ml ¼ Dm~2me Dm~2m ¼

4m2D ðA l cot b BÞ MR

ð4:58Þ

4.5 Estimate of the Effective Coupling Constant

35

so that Eq. 4.56 then simplifies to

Dm~2ml Dm~2me g42 o o 2 Iðy~ml1 ; y~me1 Þ 2 2 oy oy 2048p2 MW~ MW M ~ml1 ~me1 ~ ~ W g4 ðDm2 Þ2 o o 2 2 ~m 6 Iðy~ml1 ; y~me1 Þ: 2048p MW~ oy~ml1 oy~me1

ð4:59Þ

The contribution from graph (e) and (f) is not dominated by the terms involving only light sneutrinos even though Iðz~mla ; z~meb Þ is a decreasing function of z~mla and z~meb ; because these terms get suppressed by the mixing matrix. The terms including only light sneutrinos ~ml1 ; ~ml3 ; ~me1 ; ~me3 roughly gives l 2 l 2 e 2 e 2 Þ ðU21 Þ Iðz~ml1 ; z~me1 Þ ðU21 Þ ðU43 Þ Iðz~ml1 ; z~me3 Þ ðU21 l 2 l 2 e 2 e 2 ðU43 Þ ðU21 Þ Iðz~ml3 ; z~me1 Þ þ ðU43 Þ ðU43 Þ Iðz~ml3 ; z~me3 Þ 2 2 m4D Dm~ml Dm~me o o Iðz~ml1 ; z~me1 Þ MR4 M~h4 oz~ml1 oz~me1 B 1 ; O MR6

ð4:60Þ

while the terms including one light and one heavy sneutrino are roughly l 2 l 2 e 2 e 2 Þ ðU22 Þ Iðz~ml1 ; z~me2 Þ ðU21 Þ ðU44 Þ Iðz~ml1 ; z~me4 Þ ðU21 l 2 l 2 e 2 e 2 ðU43 Þ ðU22 Þ Iðz~ml3 ; z~me2 Þ þ ðU43 Þ ðU44 Þ Iðz~ml3 ; z~me4 Þ l 2 l 2 e 2 e 2 Þ ðU21 Þ Iðz~ml2 ; z~me1 Þ ðU22 Þ ðU43 Þ Iðz~ml2 ; z~me3 Þ þ ðU22 l 2 l 2 e 2 e 2 Þ ðU21 Þ Iðz~ml4 ; z~me1 Þ þ ðU44 Þ ðU43 Þ Iðz~ml4 ; z~me3 Þ ðU44 e 2 l 2 2 2 2 2 M 4 ~ MB 4 mD DM~ml Dm~me mD DM~me Dm~ml h B þ l l e 4 4 MR M MR MR M MRe B B 1 ; O MR6

ð4:61Þ

where DM~m2e and DM~m2e are the heavy sneutrino squared-mass differences DM~m2e ¼ 4Be MRe

and DM~m2l ¼ 4Bl MRl :

ð4:62Þ

Under our approximations, DM~m2e ¼ DM~m2l DM~m2 ¼ 4BMR :

ð4:63Þ

36


The terms including two heavy sneutrinos are roughly l 2 l 2 e 2 e 2 ðU22 Þ ðU22 Þ Iðz~ml2 ; z~me2 Þ ðU22 Þ ðU44 Þ Iðz~ml2 ; z~me4 Þ l 2 l 2 e 2 e 2 ðU44 Þ ðU22 Þ Iðz~ml4 ; z~me2 Þ þ ðU44 Þ ðU44 Þ Iðz~ml4 ; z~me4 Þ 6 DM~m2l DM~m2e M~hB M4 3MR6 B

ðDM~m2 Þ2 M2 B

3MR6 1 : O MR4

ð4:64Þ

Comparing the MR dependences of Eqs. 4.60, 4.61 and 4.64, we see that the dominant term is the one involving two heavy sneutrinos. Thus the contribution from graph (e) and (f) can be approximated as 2 X 2 2 X 4 X X ðmlD Þ2 ðmeD Þ2 l 2 l 2 e 2 e 2 ðU2a Þ ðU2b Þ Iðz~mla ;z~meb Þ ðU2a Þ ðU4b Þ Iðz~mla ;z~meb Þ 4 2 2 2048VT p M~h a¼1 b¼1 a¼1 b¼3 B ! 4 X 2 4 X 4 X X l 2 l 2 e 2 e 2 ðU4a Þ ðU2b Þ Iðz~mla ;z~meb Þ þ ðU4a Þ ðU4b Þ Iðz~mla ;z~meb Þ

a¼3 b¼1

a¼3 b¼3 4 2 2 4 4 2 2 m ðDM Þ g m ðDM~m Þ ð1 þ tan2 bÞ2 : D 4 ~m2 6 ¼ 2 D 4 6144VT p MR 24576p2 MR6 MW tan4 b

ð4:65Þ

Combining the various contributions, the effective coupling constant is thus roughly given by g42 m4D MR g42 ðDm~2m Þ2 o o jReGMM ln Iðy~ml1 ; y~me1 Þ j 4 6 1024p2 MR2 MW MW 2048p2 MW oy oy ~ ~me1 m l1 ~ g42 m4D ðDM~m2 Þ2 ð1 þ tan2 bÞ2 : ð4:66Þ 4 tan4 b 24576p2 MR6 MW The first term in Eq. 4.66 is the dominant contribution of graph (a) and (b), which contains intermediate neutrino and W boson. This contribution appears in the model in which all SUSY partners decoupled. The second term is the dominant contribution of graph (c) and (d), in which wino and sneutrino appear in the intermediate states. Finally, the third term is the dominant contribution of graph (e) and (f), with intermediate higgsino and sneutrinos lines. The second and third terms both depend on the sneutrino mass splitting. This reflects the intra-generation lepton number violating property of the muonium–Antimuonium oscillation process, because the sneutrino mass splitting is generated by the DL ¼ 2 operators


37

in the sneutrino mass matrix. To compare the relative sizes of these three terms, we use the current experimental limits of the neutrino and sparticle masses. The first terms in Eq. 4.66 has a factor m4D =MR2 ; which is the scale of the light neutrino mass square m2m ’ m4D =MR2 generated by see–saw mechanism. The experimental constraints on neutrino masses are summarized in reference (Vogel and Piepke 2008) as mm ðelectron basedÞ\225 eV; mm ðmuon basedÞ\0:19 MeV; mm ðtau basedÞ\18:2 MeV;

ð4:67Þ

mD MW ;

ð4:68Þ

For instance, assuming

mm ¼

m2D 1 eV; MR

ð4:69Þ

then the right-handed neutrino mass scale is about MR 1013 GeV:

ð4:70Þ

In this case, the first terms in Eq. 4.66 is roughly g42 m4D MR g42 m2m MR ln ¼ ln ’ 1:1 1029 GeV2 : 2 4 4 2 2 MW 1024p MW MW 1024p MR MW

ð4:71Þ

The second term in Eq. 4.66 depends on the light sneutrino squared-mass difference Dm~2m ; which can be written in terms of light sneutrino mass splitting Dm~m by Dm~2m ¼ 2m~m Dm~m ;

ð4:72Þ

where m~m is the mass scale of light sneutrinos. So doing the second term in Eq. 4.66 can be written as

g42 ðDm~2m Þ2 o o g42 m~2m ðDm~m Þ2 o o Iðy ; y Þ ¼ Iðy~ml1 ; y~me1 Þ: ~ ~ m m l1 e1 6 6 2 2 2048p MW~ oy~ml1 oy~me1 512p MW~ oy~ml1 oy~me1 ð4:73Þ

Yuval and Howard (1997) provide an upper limit on the sneutrino mass splitting by calculating the one-loop correction to the neutrino mass. Assuming that this correction is no larger than the tree result gives Dm~m 2 103 mm :

ð4:74Þ

38


Relaxing this absence of fine tuning constraint can substantially enhance the contribution of the graph (c) and (d). Taking the sneutrino mass splitting to be of the same order as sneutrino mass Dm~m m~m ;

ð4:75Þ

and m~ml ; m~me to be the common mass scale m~m gives y~ml1 y~me1 y~m ¼

m~2m : 2 MW ~

ð4:76Þ

Equation 4.73 can then be written as g4 m2 ðDm~m Þ2 o o g42 m~4m Iðy~ml1 ; y~me1 Þ 2 ~m 2 6 6 512p MW~ ~ml1 oy~me1 512p2 MW ~ o o Iðy~ml1 ; y~me1 Þ y~ml1 ;y~me1 ¼y~m : ð4:77Þ oy~ml1 oy~me1 m4 The function M6~m oyo~m oyo~m Iðy~ml1 ; y~me1 Þ y~ml1 ;y~me1 ¼y~m in Eq. 4.77 is a decreasing func~ W

l1

e1

tion of MW~ : In order to calculate the maximum contribution of graph (c) and (d), we use the experimental lower bound on MW~ : Many experimental searches for physics beyond the standard model have been conducted and provide various constraints on SUSY parameter space. Table 4.1 lists some of the constraints (Grivaz 2008). Fixing the wino mass to its lower limit in Table 4.1.

the contribution

g42 m~4m 512p2 M 6~ W

oyo~m

l1

MW~ ¼ 94:0 GeV; ð4:78Þ o oy~m Iðy~ml1 ; y~me1 Þ y~ml1 ;y~me1 ¼y~m as a function of sneutrino e1

mass scale m~m is shown in Fig. 4.2. When m~m ¼ 94:0 GeV; which is allowed by the experimental limit in Table 4.1, the contribution of graph (c) and (d) reaches its maximum so that g42 m~4m o o 10 Iðy ; y Þ GeV2 : ~ml1 ~me1 y~ml1 ;y~me1 ¼y~m 1:3 10 6 oy oy 512p2 MW ~ ~ m m l1 e1 ~

Table 4.1 Experimental lower limits on SUSY particle masses

ð4:79Þ

Sparticle

Lower limit (GeV)

~ v 1 ~ v01 ~m ~R l ~eL ~eR

94.0 46.0 94.0 94.0 107.0 73.0


39

Fig. 4.2 The contribution of graph (c) and (d) as a function of m~m when fixing the wino mass to its lower limit MW~

Finally, the third term in Eq. 4.66 depends on the heavy sneutrino squared-mass difference DM~m2 ¼ 2M~m DM~m ¼ 4BMR : Since we assume that MR is the largest mass scale, DM~m can’t be arbitrarily large. Taking parameter B one order of magnitude smaller than MR ; the heavy sneutrino mass splitting is DM~m

MR : 10

ð4:80Þ

The contribution of graph (e) and (f) can then be written as g42 m4D ðDM~m2 Þ2 ð1 þ tan2 bÞ2 g42 m2m ð1 þ tan2 bÞ2 ¼ : 4 4 tan4 b 614400p2 MW tan4 b 24576p2 MR6 MW

ð4:81Þ

When tan b is very small the contribution can get large and even reach the experimental limit Eq. 4.51. In this case, the experimental limit of muonium– antimuonium oscillation provides an inequality relating tan b and mm ; which is given by g42 m2m ð1 þ tan2 bÞ2 3:5 108 GeV2 : 4 614400p2 MW tan4 b

ð4:82Þ

This inequality translates into a lower bound of tan b for different light neutrino masses mm : tan b 3:7 107 ; tan b 3:7 108 ; tan b 3:7 109 ;

if mm ¼ 1 eV; if mm ¼ 102 eV; if mm ¼ 104 eV:

ð4:83Þ

The lower limit on tan b as a function of light neutrino mass scale mm is shown in Fig. 4.3. Notice that the ratio of the two Higgs VEVs tan b are related to the light

40


Fig. 4.3 The lower limit on tan b as a function of light neutrino mass scale mm provided by the muonium–antimuonium oscillation experiment. The area above the curve is allowed by the experiment results

neutrino masses in the above inequality, although the graph (e) and (f) do not involve any neutrinos in the intermediate states. This results since we are using a specific model where the neutrino masses are generated by see–saw mechanism mm Oðm2D =MR Þ: The sneutrino mixing matrix is approximated in term of mD =MR and the heavy sneutrino masses are also of order MR . If we take mD to be of order MW , the heavy sneutrino masses MR in the contribution of graph (e) and (f) can be expressed in term of the light neutrino mass scale mm : This explains the appearance of the parameter mm in the inequality Eq. 4.82. However, for non-infinitesimal values of tan b; this contribution is very small compared with the maximum of the second term in Eq. 4.66. For instance, taking the neutrino mass mm to be 1 eV and assuming tan b 104 ; the contribution of graph (e) and (f) is g42 m2m ð1 þ tan2 bÞ2 . 7:2 1018 GeV2 : 4 614400p2 MW tan4 b

ð4:84Þ

Thus, except for the case of very small tan b; the second term in Eq. 4.66 is the dominant contribution for a wide range of the parameters and its maximum is roughly two orders of magnitude below the sensitivity of the current experiments.

References P. Minkowski, Phys. Lett. B67, 421 (1977) T. Yanagida, In: Proceedings of the Workshop on the Unified Theories and Baryon Number in the Universe, Tsukuba, Japan, 1979, (O. Sawada and A. Sugamoto eds), p 95

References

41

M. Gell-Mann, P. Ramond and R. Slansky in Supergravity, (P. van Nieuwenhuizen and D. Freedman eds) North-Holland, Amsterdam, 1979 S.L. Glashow, In: Proceedings of the 1979 Cargese Institute on Quarks and Leptons (M. Levy et. al. eds.) Plenum Press, New York 1980, p 687 R.N. Mohapatra and G. Senjanovic. Phys. Rev. Lett. 44, 912 (1980). Y. Grossman and HE.Haber, Phys.Rev. Lett. 78, 3438 (1997). L. Willmann et al, Phys. Rev. Lett 82, 49 (1999) P. Vogel and A. Piepke, Phys. Lett. B667, 517 (2008) J.-F.Grivaz, Phys. Lett. B667, 1228 (2008)

Chapter 5

The Constraints from the Muon and Electron Anomalous Magnetic Moment Experiments

One has to be careful about other constraints on the model parameters. Examples of such potential constraints come from the measurements of the muon and electron anomalous magnetic moments. The correction to the muon anomalous magnetic moment in the model under consideration is found by calculating the one-loop graphs shown in Fig. 5.1. The muon anomalous magnetic moment contributed from the above graphs is ¼ aBSM l þ þ

g22 m2l 2 16p2 MW

l l 2 W ðV12 V12 Þ F ðxml2 Þ 2 4 X X l 2 C l 2 C ðU1a Þ F ðy~mla Þ þ ðU3a Þ F ðy~mla Þ

g22 m2l 2 32p2 MW ~

a¼1

ðmlD Þ2 m2l 32p2 VT M~h2 B

m2

~0 W

W

F ðxml2 Þ ¼

Z

1

dx 0

m2l~

a

M2~

a¼3

! 2 4 X X l 2 C l 2 C ðU2a Þ F ðz~mla Þ þ ðU4a Þ F ðz~mla Þ a¼1

g22 m2l F N ðsl~L Þ 2 32p2 MW ~0

where sl~a ¼ M2l~a , tl~a ¼

!

a¼3

g22 m2l N F ðtl~L Þ 32p2 MB2~

g21 m2l 8p2 MB2~

F N ðtl~R Þ;

ð5:1Þ

, and

B

4x2 ð1 þ xÞ 2xl x2 ðx 1Þ 2xml2 ð2x 3x2 þ x3 Þ ; ð5:2Þ xl x2 þ ð1 xl Þx þ xml2 ð1 xÞ

F C ðy~mla Þ ¼

2y~3mla 3y~2mla ð1 þ 2 ln y~mla Þ 6y~mla þ 1 ; 6ð1 y~mla Þ4


ð5:3Þ

43

44

5 Constraints from the Muon and Electron

F N ðsl~a Þ ¼

s3l~a 6s2l~a þ 3sl~a þ 6sl~a ln sl~a þ 2 : 6ð1 sl~a Þ4

ð5:4Þ

With assumption that MR is the largest mass scale, the dominant contribution of the graphs in Fig. 5.1 to aBSM is l aBSM l

g22 m2l m2D

g22 m2l m2D ð1 þ tan2 bÞ g22 m2l þ F C ðy~ml1 Þ þ F C ðy~ml3 Þ þ 2 2 2 2 M 2 tan2 b 2 2 12p MW MR 32p MW~ 96p2 MW R 2 2 2 2 g22 m2l g m g m 2 l 1 l F N ðsl~L Þ F N ðtl~L Þ 2 2 F N ðtl~R Þ: ð5:5Þ 2 2M2 32p2 MW 32p 8p MB~ ~0 ~ B

The second and the last three terms are all decreasing functions of slepton, chargino and neutralino masses. We can use the experimental bounds in Table 4.1 to calculate the maximum values of these terms yielding

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5.1 The Feynman graphs contributing to the muon anomalous magnetic moment beyond the Standard Model


g22 m2l 2 32p2 MW ~

45

F C ðy~ml1 Þ þ F C ðy~ml3 Þ 2:5 1010 ; g22 m2l 2 32p2 MW ~0

g22 m2l

F N ðsl~L Þ 1:9 1010 ; ð5:6Þ

F N ðtl~L Þ 1:9 1010 ; 2

32p2 MB~ g21 m2l 8p2 MB2~

F N ðtl~R Þ 2:1 1010 :

The maxima of these terms are all about one order of magnitude smaller than the present experimental bound on the contribution to al ¼ 12ðg 2Þ beyond the Standard Model (Melnikov and Arkady 2006): SM 9 dal ¼ aexp l al ¼ 2 10 :

ð5:7Þ

The first and third term both depend on the light neutrino mass scale mm and get suppressed. For instance, using the assumptions Eqs. (4.68) and (4.69), the first and third terms are g22 m2l m2D 2 M2 12p2 MW R

g22 m2l m2D ð1 þ tan2 bÞ 2 M2 96p2 MW R

tan2

b

¼

¼

g22 m2l m2m 4 12p2 MW

g22 m2l m2m ð1 þ tan2 bÞ 4 96p2 MW

tan2

b

8:7 1031 ; 1:1 1031

1 þ tan2 b : tan2 b

ð5:8Þ

Fig. 5.2 The lower limit on tan b as a function of light neutrino mass scale mm provided by the muon anomalous magnetic moment experiment. The area above the curve is allowed by the experiment results

46


The first term is negligible compared with the terms in Eq. (5.7). However, the third term can be large if tan b is very small. Therefore, the experimental bound on the muon magnetic moment will provide an inequality on tan b and mm , which is given by g22 m2l m2m ð1 þ tan2 bÞ 4 tan2 b 96p2 MW

2 109 :

ð5:9Þ

This inequality translates into a lower bound on tan b as a function of the light neutrino mass scale mm as shown in Fig. 5.2. The electron anomalous magnetic moment beyond the Standard Model is contributed by the six Feynman graphs displayed in Fig. 5.3. The experimental bound on the contribution to the electron anomalous magnetic moment beyond Standard Model is (Ellis et al. 1994) SM 11 dae ¼ aexp : e ae ¼ 1:4 10

ð5:10Þ

In analogy to the muon case, this experimental limit will also generate an inequality relation of mm and tan b given by

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5.3 The Feynman graphs contributing to the electron anomalous magnetic moment beyond the Standard Model


47

Fig. 5.4 The lower limit on tan b as a function of light neutrino mass scale mm provided by the electron anomalous magnetic moment experiment. The area above the curve is allowed by the experiment results

g22 m2e m2m ð1 þ tan2 bÞ 1:4 1011 : 4 tan2 b 96p2 MW

ð5:11Þ

This inequality is illustrated in Fig. 5.4. From Figs. 4.3, 5.2 and 5.4 we see that for the model we are considering, the muonium–antimuonium oscillation experiment gives a more stringent constraint on tan b than the muon and electron anomalous magnetic moment experiments.

References J.R. Ellis, M. Karliner, M.A. Samuel and E. Steinfelds, SLAC-PUB-6670, CERN-TH-7451-94, TAUP-2001-94, OSU-RN-293, hep-ph/9409376, 1994. K. Melnikov and A. Vainshtein, Theory of the Muon Anomalous Magnetic Moment, (Springer), 2006, p. 152.

Chapter 6

Conclusions

I have calculated the effective coupling constant of the muonium–antimuonium oscillation process in two different models. First, I modified the Standard Model by including three singlet right-handed neutrinos. Present experimental limits resulting from the non-observation of the oscillation process sets a lower limit on the right-handed neutrino mass scale MR roughtly of order 1 TeV. Second, the muonium–antimuonium oscillation was investigated in the Minimal Supersymmetric Standard Model extended by inclusion of three right-handed neutrino superfields where the required lepton flavor violation has its origin in the Majorana property of the neutrino and sneutrino mass eigenstates. For a wide range of the parameters, the contribution of the graphs mediated by the sneutrino and winos e is dominant. The maximum of this contribution to the effective coupling W constant is roughly two orders of magnitude below the sensitivity of current muonium–antimuonium oscillation experiments. However, there is very limited possibility that the contribution of the graphs mediated by sneutrinos and Higgsino e h B is dominant if tan b is very small. In this case, the contributions can even be large enough to reach the present experimental bound. Therefore, the experimental bound can provide an inequality on the model parameters, which can be translated into a lower bound on tan b as a function of light neutrino mass scale mm : The constraints from the muon and electron anomalous magnetic moments were also investigated. For this model, the muonium–antimuonium oscillation experiments give the most stringent constraints on the parameters.


49

Appendices

Appendix A. Proof of Identity (2.22) Using the definition of the mixing matrix P VaA ¼ 3c¼1 ðA1 L Þac UcA , one can write 6 X

6 3 X X VaA VbA m ¼ ðA1 L Þac UcA

A¼1

A¼1

!

c¼1

3 X

! ðA1 L Þbd UdA

d¼1

! 3 X 3 6 X X 1 ¼ ðAL Þac UcA mUdA ðA1 L Þbd c¼1 d¼1

mA

ðA:1Þ

A¼1

m where mmA are the diagonal elements of matrix Mdiag , m ÞAA : mmA ¼ ðMdiag

Consequently, we can express (A.1) takes the form 6 X

VaA VbA mmA ¼

A¼1

P6

A¼1

ðA:2Þ

UcA mmA UdA as a product of matrices and Eq.

3 X 3 X m T ðA1 Þ UM U ðA1 L ac diag L Þbd c¼1 d¼1

cd

ðA:3Þ

m Using Eq. (2.9), Mdiag ¼ U T M m U, it follows that m UT Mm ¼ UMdiag

ðA:4Þ

Substituting this result back into Eq. (A.3) then gives 6 X A¼1

VaA VbA mmA ¼

3 X 3 X m 1 ðA1 L Þac ðM Þcd ðAL Þbd

ðA:5Þ

c¼1 d¼1

51

52

Appendices

where M

m

¼

0 mD

ðmD ÞT mR

ðA:6Þ

Since c and d both run from 1 to 3, ðM m Þcd are the elements of the upper left 3 3 block of matrix (A.6), which is zero. Hence, we secure the identity 6 X

VaA VbA mmA ¼ 0

ðA:7Þ

A¼1

Appendix B. The Calculations of T-Matrix Elements of the Graphs in Fig. 2.1

B.1 T-Matrix Element of Graph (a1) in Fig. 2.2 The T-matrix element of graph (a1) is: Z 6 ig 4 d4 p X cp þ mA L ð3ÞclA 2 iTa1 ¼ pffiffiffi l V c ð2Þ 4 p m2A eA L 2 ð2pÞ A¼1 m 6 X cp þ mB 1 ðn 1Þp lm L ð4Þcq V 2 c V eL ð1Þ 2 g þ 2 l 2 2 p MW p m2B m eB p nMW B¼1 r 1 ðn 1Þp qr g þ 2 2 2 p MW p2 nMW

ðB:1Þ

We multiply out everything and write above equation in an explicit form: 4 X Z 6 X 6 g d4 p 1 ðVlA VeA ÞðVlB VeB Þ iTa1 ¼ pffiffiffi 4 2 2 2 2 2 2 A¼1 B¼1 ð2pÞ ðp mA Þðp m2B Þðp2 MW Þ 1 c5 1 c5 ½ lð3Þcl cpcr eð2Þ½ lð4Þcr cpcl eð1Þ 2 2 n1 1 c5 1 c5 ½ lð3Þcpcpcr eð2Þ½ lð4Þcr cpcp eð1Þ þ 2 2 2 2 p nMW n1 1 c5 1 c5 eð2Þ½ lð4Þcpcpcl eð1Þ þ 2 ½ lð3Þcl cpcp 2 2 2 p nMW ) ðn 1Þ2 1 c5 1 c5 þ eð2Þ½ lð4Þcpcpcp eð1Þ ½ lð3Þcpcpcp 2 2 2 2 ðp2 nMW Þ ðB:2Þ

Appendices

53

We list two useful equations as below: Z Z d4 p l m 2 glm d4p 2 2 p p f ðp Þ ¼ p f ðp Þ 4 ð2pÞ4 ð2pÞ4 1 c5 1 c5 eð2Þ½ lð4Þcm ca cl eð1Þ 2 2 1 c5 1 c5 eð2Þ½ lð4Þcl eð1Þ ¼ 4 ½ lð3Þcl 2 2

ðB:3Þ

½ lð3Þcl ca cm

ðB:4Þ

By using above two equations we can easily simplify the T-matrix element as 4 X 6 X 6 g 1 c5 1 c5 ðVlA VeA ÞðV VeB Þ½ lð3Þcl eð2Þ½ lð4Þcl eð1Þ iTa1 ¼ pffiffiffi 2 2 2 A¼1 B¼1 Z d4 p 1 2ðn 1Þ fp2 þ 2 p4 2 2 Þ2 p nMW ð2pÞ4 ðp2 m2A Þðp2 m2B Þðp2 MW þ

ðn 1Þ2 2 2 4ðp2 nMW Þ

p6 g

ðB:5Þ

The integral in Eq. (B.1) could be evaluated by doing Wick’s rotation. Z d4p 1 2 2 ð2pÞ4 ðp2 m2A Þðp2 m2B Þðp2 MW Þ ( ) 2ðn 1Þ ðn 1Þ2 2 4 6 p þ 2 p þ p 2 2 Þ2 p nMW 4ðp2 nMW Z djpE jjpE j3 X4 1 ¼ i 4 2 2 2 2 2 ðjpE j þ mA ÞðjpE j þ m2B ÞðjpE j2 þ MW ð2pÞ Þ ( ) 2 2ðn 1Þ ðn 1Þ jpE j2 þ jpE j4 þ jpE j6 2 2 2 Þ2 jpE j þW 4ðjpE j2 þ nMW

ðB:6Þ

where X4 is the four-dimensional solid angle, X4 ¼ 2p2 . Let

jPE j2 ¼ t; 2 MW

xA ¼

m2A ; 2 MW

xB ¼

m2B 2 MW

ðB:7Þ

The integral (B.6) could be written as

i 2 16p2 MW

Z

1

dt 0

1 ðt þ xA Þðt þ xB Þðt þ 1Þ2

(

2ðn 1Þ 3 ðn 1Þ2 4 t þ t2 þ t tþn 4ðt þ nÞ2

)

ðB:8Þ

54

Appendices

Note that the CKM matrix VaA satisfies an identity: 6 X

VaA VbA ¼ dab

ðB:9Þ

A¼1

This identity would help to reduce the power of the integrand in Eq. (B.8) by factor 2. We will rewrite the T-matrix element and manipulate the integrand of Eq. (B.8) into several parts. This way we could explicitly show that some parts vanish by using identity (B.9). 6 X 6 X g4 1 c5 l 1 c5 ½ l ð3Þc ðVlA VeA ÞðVlB VeB Þ eð2Þ½ l ð4Þc eð1Þ l 2 2 2 64p2 MW A¼1 B¼1 Z 1 " ðt þ xA Þðt þ xB Þ ðt þ xA ÞxB ðt þ xB ÞxA þ xA xB dt ðt þ xA Þðt þ xB Þðt þ 1Þ2 0 ( )# 2ðn 1Þ ðn 1Þ2 2 t 1þ tþ tþn 4ðt þ nÞ2

Ta1 ¼

6 X 6 X g4 1 c5 l 1 c5 eð2Þ½ l ð4Þc eð1Þ ½ l ð3Þc ðVlA VeA ÞðVlB VeB Þ l 2 2 2 64p2 MW A¼1 B¼1 Z 1 "( 1 xB xA dt 2 2 ðt þ xB Þðt þ 1Þ ðt þ xA Þðt þ 1Þ2 ðt þ 1Þ 0 ) ( )# xA xB 2ðn 1Þ ðn 1Þ2 2 þ 1þ t tþ tþn ðt þ xA Þðt þ xB Þðt þ 1Þ2 4ðt þ nÞ2

¼

ðB:10Þ Since 6 X A¼1

VlA VeA ¼ 0;

and

6 X

VlB VeB ¼ 0;

ðB:11Þ

B¼1

any part in Eq. (B.10) independent of xA or xB will vanish. We will be left with the parts which depend on both xA and xB . 6 X 6 X g4 1 c5 l 1 c5 eð2Þ½ l ð4Þc eð1Þ ½ l ð3Þc ðVlA VeA ÞðVlB VeB Þ l 2 2 2 64p2 MW A¼1 B¼1 ( )# Z 1 " xA xB 2ðn 1Þ ðn 1Þ2 2 tþ dt 1þ t tþn ðt þ xA Þðt þ xB Þðt þ 1Þ2 4ðt þ nÞ2 0

Ta1 ¼

ðB:12Þ

Appendices

55

B.2 T-Matrix Element of Graph (a2) in Fig. 2.2 ig 1 c5 þma ig 1 þ c5 ¼ ð3Þ pffiffiffi VlA cl pffiffiffi ðme 2 l VeA 4 2 2 2 p mA 2 2MW ð2pÞ A¼1 6 X 1 c5 ig 1 c5 1 þ c5 þmB ð4Þ pffiffiffi V ðml mA Þeð2Þ mB Þ 2 l 2 2 2 p m2B 2MW B¼1 ig 1 c5 1 ðn 1Þpl pm 1 eð1Þ 2 pffiffiffi VeB cm ½glm þ 2 2 2 2 2 2 p p M p nM 2 W W W 4 6 X 6 X g 1 ¼ pffiffiffi ðV VeA ÞðVlB VeB Þ 2 2 MW A¼1 B¼1 Z d4 p 1 4 ðp2 m2 Þðp2 M 2 Þðp2 M 2 Þðp2 2 Þ ð2pÞ B W W A 1 þ c5 1 c5 ð3Þcl ð3Þcl eð2Þ m2A l eð2Þ me l 2 2 1 þ c5 1 þ c ðn 1Þpl pm 5 ð4Þ ð4Þ ml l cð 1Þ m2B l cð 1Þ ½glm þ 2 2 2 2 p nMW Z

iTa2

6 d4 p X

ðB:13Þ We have Dirac’s equation: ðcp mÞuðpÞ ¼ 0

ðB:14Þ

uðpÞðcp mÞ ¼ 0

ðB:15Þ

Hence, 1 þ c5 1 þ c5 ð3Þcl cp eð2Þ ¼ l cp2 eð2Þ 2 2

ðB:16Þ

1 þ c5 1 þ c5 ð4Þcp4 cp cm eð1Þ ¼ l cm eð1Þ 2 2

ðB:17Þ

ð3Þcl cp me l ð4Þcp ml l

Since we ignore all the external momenta in our discussion, the above terms will vanish. We could simplify the T-matrix element as follows: 4 6 X 6 X g 1 1 c5 l 1 c5 iTa2 ¼ pffiffiffi eð2Þ½ l ð4Þc eð1Þ ½ l ð3Þc ðVlA VeA ÞðVlB VeB Þ l 2 2 2 2 MW A¼1 B¼1 Z 4 d p m2A m2B n1 ½1 þ p2 : 4 ðp2 m2 Þðp2 M 2 Þðp2 M 2 Þðp2 nM 2 Þ 2 nM 2 Þ 4ðp ð2pÞ B W W W A ðB:18Þ

56

Appendices

Doing Wick’s rotation and expressing the result in terms of t, xA and xB , we have 6 X 6 X g4 1 c5 1 c5 eð2Þ½ lð4Þcl eð1Þ ½ lð3Þcl ðVlA VeA ÞðVlB VeB Þ 2 2 2 2 64p MW A¼1 B¼1

Z 1 xA xB n1 2 ðB:19Þ dt tþ t ðt þ xA Þðt þ xB Þðt þ 1Þðt þ nÞ 4ðt þ nÞ 0

Ta2 ¼

B.3 T-Matrix Element of Graph (a3) in Fig. 2.2 It turned out graph (a3) has the same T-matrix element as (a2).

B.4 T-Matrix Element of Graph (a4) in Fig. 2.2

iTa4 ¼

4 Z 6 ig d4 p X 1 c5 1 þ c5 cp þ mA mA Þ 2 pffiffiffi ð3ÞVlA ðml V l 2 2 p m2A eA 2MW ð2pÞ4 A¼1 ðme

6 X 1 þ c5 1 c5 1 c5 1 þ c5 ð4ÞVlB ðml mA Þeð2Þ mB Þ l 2 2 2 2 B¼1

cp þ mB 1 þ c5 1 c5 1 mB Þeð1Þ V ðme 2 eB 2 2 2 2 2 2 p mB ðp nMW Þ

ðB:20Þ

Like what we did in Eqs. (B.16) and (B.17), we could ignore all the terms with ml or me in Eq. (B.20). Then we have 4 6 X 6 X g 1 c5 1 c5 iTa4 ¼ pffiffiffi eð2Þ½ lð4Þcl eð1Þ ½ lð3Þcl ðVlA VeA ÞðVlB VeB Þ 2 2 2M W A¼1 B¼1 Z d4 p m2A m2B p2 ðB:21Þ 2 2 4 Þ ð2pÞ4 ðp2 m2A Þðp2 m2B Þðp2 nMW By doing Wick’s rotation and expressing above equation in terms of t, xA and xB we can simplify the T-matrix element as 6 X 6 X g4 1 c5 l 1 c5 ½ l ð3Þc ðVlA VeA ÞðVlB VeB Þ eð2Þ½ l ð4Þc eð1Þ l 2 2 2 64p2 MW A¼1 B¼1 # Z 1 " xA xB t2 ðB:22Þ dt ðt þ xA Þðt þ xB Þðt þ nÞ2 4 0

Ta4 ¼

Appendices

57

B.5 T-Matrix Element of Graph (c1) in Fig. 2.3 The T-matrix element of graph (c1) is ( Z 6 ig 4 d4 p X mA 1 þ c5 c ðVlA Þ2 l ð3Þcl 2 c iTc1 ¼ pffiffiffi l ð4Þ 4 p m2A q 2 2 ð2pÞ A¼1

6 X

2 c ðVeB Þ e ð2Þcm

B¼1

p2

mB 1 c5 eð1Þ cr 2 2 mB

1 ðn 1Þpl pr ½glr þ 2 2 2 MW þ i p nMW

1 ðn 1Þpm pq mq þ ½g 2 2 þ i 2 p2 nMW p MW

p2

ðB:23Þ

We multiply out every thing and write above equation in an explicit form: ( 4 Z 6 X 6 g d4 p X mA mB 2 ðVlA Þ2 ðVeB Þ iTc1 ¼ pffiffiffi 4 2 2 Þ2 2 2 2 ð2pÞ A¼1 B¼1 ðp mA Þðp m2B Þðp2 MW 1 þ c5 c 1 c5 l ð4Þ½ec ð2Þcq cl eð1Þ ½ lð3Þcl cq 2 2 þ ½ lð3Þcl cp

1 þ c5 c 1 c5 n1 l ð4Þ½ec ð2Þcpcl eð1Þ 2 2 2 2 p nMW

1 þ c5 c 1 c5 n1 l ð4Þ½ec ð2Þcq cp eð1Þ 2 2 2 2 p nMW 2 )) 1 þ c5 c 1 c5 n1 c l ð4Þ½e ð2Þcpcp eð1Þ 2 þ ½ lð3Þcpcp 2 2 2 p nMW

þ ½ lð3Þcpcq

ðB:24Þ We list two useful equations as below: 1 þ c5 1 c5 u2 ½u3 cl cm u4 2 2 1 þ c5 1 c5 u2 ½u3 cm cl u4 ¼ ½u1 cl cm 2 2 1 þ c5 1 c5 u2 ½u3 u4 ¼ 4 ½u1 2 2 Z Z d4 p l m 2 glm d4p 2 2 p p f ðp Þ ¼ p f ðp Þ 4 4 ð2pÞ ð2pÞ4 ½u1 cl cm

ðB:25Þ ðB:26Þ

58

Appendices

By using above equations we can easily simplify the T-matrix element as 4 X 6 6 X g 1 þ c5 c 1 c5 2 l ð4Þ½ec ð2Þ eð1Þ ðVlA Þ2 mA ðVeB Þ mB ½ lð3Þ iTc1 ¼ pffiffiffi 2 2 2 A¼1 B¼1 ( Z d4 p 1 2ðn 1Þp2 4 þ 4 2 2 2 Þ p2 nMW ð2pÞ ðp2 m2A Þðp2 m2B Þðp2 MW !) ðn 1Þ2 p4 ðB:27Þ þ 2 Þ2 ðp2 nMW We could evaluate the integration in Eq. (B.27) by doing Wick’s rotation. ( !) Z d4 p 1 2ðn 1Þp2 ðn 1Þ2 p4 4þ 2 þ 2 2 Þ2 2 Þ2 p nMW ð2pÞ4 ðp2 m2A Þðp2 m2B Þðp2 MW ðp2 nMW Z djPE jjPE j3 X4 1 ¼i 2 Þ2 ð2pÞ4 ðjPE j2 þ m2A ÞðjPE j2 þ m2B ÞðjPE j2 þ MW ! 2ðn 1ÞjPE j2 ðn 1Þ2 jPE j4 4þ þ ðB:28Þ 2 2 2 jPE j2 þ nMW ðjPE j2 þ nMW Þ where X4 is the four-dimensional solid angle, X4 ¼ 2p2 . Let

jPE j2 ¼ t; 2 MW

xA ¼

m2A ; 2 MW

The integral (B.28) could be written as Z 1 ( i t dt 4 2 16p MW 0 ðt þ xA Þðt þ xB Þðt þ 1Þ2

xB ¼

m2B 2 MW

2ðn 1Þt ðn 1Þ2 t2 4þ þ tþn ðt þ nÞ2

ðB:29Þ

!)

ðB:30Þ Hence, we could eventually write the T-matrix element of graph (c1) as 6 6 X X g4 1 þ c5 c 1 c5 2 2 c ð2Þ ½ l ð3Þ ð4Þ½ e ðV Þ m ðVeB Þ mB l eð1Þ lA A 4 2 2 64p2 MW A¼1 B¼1 !) Z 1 ( t 2ðn 1Þt ðn 1Þ2 t2 dt 4þ þ tþn ðt þ nÞ2 ðt þ xA Þðt þ xB Þðt þ 1Þ2 0

Tc1 ¼

ðB:31Þ

Appendices

59

Note that the CKM matrix VaA satisfies an identity (Ilakovac et al. 1995): 6 X

Vlk Vl0 k mk ¼ 0;

ðB:32Þ

k

which is forced by the properties of the neutrino mass matrix of seesaw model. In seesaw model the generation of left-handed Majorana neutrino masses requires a mass dimension-five operator. Since the dimension-five operator is nonrenormalizable, we would like to suppress this mass part in seesaw model. This would generate the identity (B.32). We will show that this identity would help to reduce the power of the integrand in Eq. (B.31) by factor 2. We will rewrite the T-matrix element of graph (c1) and manipulate the integrand of Eq. (B.31) into several parts. This way we could explicitly show that some parts just vanish by using identity (B.32). 6 6 X X g4 1 þ c5 c 1 c5 2 2 c ð2Þ ½ l ð3Þ ð4Þ½ e ðV Þ m ðVeB Þ mB l eð1Þ lA A 4 2 2 64p2 MW A¼1 B¼1 Z 1 " 4t dt ðt þ xA Þðt þ xB Þðt þ 1Þ2 0

Tc1 ¼

þ 2ðn 1Þ 2

þ ðn 1Þ

ðt þ xA Þðt þ xB Þ ðt þ xA ÞxB ðt þ xB ÞxA þ xA xB ðt þ xA Þðt þ xB Þðt þ 1Þ2 ðt þ nÞ

tðt þ xA Þðt þ xB Þ tðt þ xA ÞxB tðt þ xB ÞxA þ xA xB t

#

ðt þ xA Þðt þ xB Þðt þ 1Þ2 ðt þ nÞ2

6 6 X X g4 1 þ c5 c 1 c5 2 2 c ð2Þ l eð1Þ ½ l ð3Þ ð4Þ½ e ðV Þ m ðVeB Þ mB lA A 4 2 2 64p2 MW A¼1 B¼1 Z 1 " 4t 1 dt þ 2ðn 1Þ 2 ðt þ xA Þðt þ xB Þðt þ 1Þ ðt þ 1Þ2 ðt þ nÞ 0

¼

þ

2

ðt þ xB Þðt þ 1Þ ðt þ nÞ

xA xB

xA ðt þ xA Þðt þ 1Þ2 ðt þ nÞ !

2

ðt þ xA Þðt þ xB Þðt þ 1Þ ðt þ nÞ

þ

xB

xB t ðt þ xB Þðt þ 1Þ2 ðt þ nÞ2 xA xB t

þ ðn 1Þ2

t 2

ðt þ 1Þ ðt þ nÞ2

xA t ðt þ xA Þðt þ 1Þ2 ðt þ nÞ2 !#

ðt þ xA Þðt þ xB Þðt þ 1Þ2 ðt þ nÞ2

ðB:33Þ

60

Appendices

We have 6 X A¼1

ðVlA Þ2 mA ¼ 0;

and

6 X

2 ðVeB Þ mB ¼ 0

ðB:34Þ

B¼1

Hence, any part of the integrand independent of xA or xB will vanish. We will be left with the parts which depend on both xA and xB . 6 6 X X g4 1 þ c5 c 1 c5 2 2 c ð2Þ l eð1Þ ½ l ð3Þ ð4Þ½ e ðV Þ m ðVeB Þ mB lA A 4 2 2 64p2 MW A¼1 B¼1 Z 1 " 4t 2ðn 1ÞxA xB dt þ 2 ðt þ xA Þðt þ xB Þðt þ 1Þ2 ðt þ nÞ ðt þ xA Þðt þ xB Þðt þ 1Þ 0 # ðn 1Þ2 xA xB t þ ðB:35Þ ðt þ xA Þðt þ xB Þðt þ 1Þ2 ðt þ nÞ2

Tc1 ¼

B.6 T-Matrix Element of Graph (c2) in Fig. 2.3 4 6 X 6 g 1 X 2 iTc2 ¼ pffiffiffi ðVlA Þ2 ðVeB Þ 2 2 MW A¼1 B¼1 Z d4 p ml mA 1 c5 mA 1 c5 c c l ð3Þ l ð3Þcp l ð4Þ l ð4Þ c c q q 2 2 p2 m2A ð2pÞ4 p2 m2A me m B c 1 þ c5 mb 1 c5 ð2Þcm c ð2Þcm cp eð1Þ 2 eð1Þ 2 2 e 2 e 2 2 p mB p mB

1 ðn 1Þpq pm 1 qm ðB:36Þ ½g þ 2 2 2 2 2 2 p MW p p nMW nMW Multiply every thing out, we have 4 6 X 6 g 1 X 2 ðVlA Þ2 ðVeB Þ mA mB iTc2 ¼ pffiffiffi 2 M 2 W A¼1 B¼1 Z d4 p 1 2 2 Þðp2 nMW Þ ð2pÞ4 ðp2 m2A Þðp2 m2B Þðp2 MW 1 c5 1 þ c5 lð3Þ cq lc ð4Þ½ec ð2Þcq eð1Þ ml me ½ 2 2 ml me ðn 1Þ 1 c5 1 þ c5 ½ lð3Þ cplc ð4Þ½ec ð2Þcp eð1Þ þ 2 2 2 2 p nMW 1 c5 1 c5 cq lc ð4Þ½ec ð2Þcq cp eð1Þ ml ½ lð3Þ 2 2

Appendices

61

ml ðn 1Þ 1 c5 1 c5 cplc ð4Þ½ec ð2Þcpcp eð1Þ ½ lð3Þ 2 2 2 p2 nMW 1 c5 1 þ c5 me ½ lð3Þcp cq lc ð4Þ½ec ð2Þcq eð1Þ 2 2 me ðn 1Þ 1 c5 1 þ c5 cplc ð4Þ½ec ð2Þcp eð1Þ ½ lð3Þcp 2 2 2 2 p nMW 1 c5 1 c5 cq lc ð4Þ½ec ð2Þcq cp eð1Þ þ ½ lð3Þcp 2 2

ðn 1Þ 1 c5 1 c5 c c cpl eð1Þ þ 2 ½ lð3Þcp ð4Þ½e ð2Þcpcp 2 2 2 p nMW

ðB:37Þ

We have Dirac’s equation: ðcp mÞuðpÞ ¼ 0

ðB:38Þ

uðpÞðcp mÞ ¼ 0

ðB:39Þ

Hence, ml ½ lð3Þ

1 c5 1 c5 lð3Þcp3 cq lc ð4Þ ¼ ½ cq lc ð4Þ 2 2

ðB:40Þ

1 þ c5 1 þ c5 eð1Þ ¼ ½ec ð2Þcq cp1 eð1Þ 2 2

ðB:41Þ

me ½ec ð2Þcq

Since we ignore all the external momenta in our discussion, the above terms will vanish. By the same token, we may neglect all the terms with ml or me in Eq. (B.37). Then, we have 4 6 X 6 g 1 X 2 iTc2 ¼ pffiffiffi ðVlA Þ2 ðVeB Þ mA mB 2 2 MW A¼1 B¼1 Z d4 p 1 4 ðp2 m2 Þðp2 m2 Þðp2 M 2 Þðp2 nM 2 Þ ð2pÞ B W W A 1 c5 1 c 5 cq lc ð4Þ½ec ð2Þcq cp eð1Þ ½ lð3Þcp 2 2

ðn 1Þ 1 c5 1 c5 c c cpl ð4Þ½e ð2Þcpcp eð1Þ ðB:42Þ ½ lð3Þcp þ 2 2 2 2 p nMW And it could be simplified further by Eqs. (B.25) and (B.26) as 4 6 X 6 X g 1 1 þ c5 c 1 c5 2 c ð2Þ l eð1Þ ½ l ð3Þ ð4Þ½ e ðVlA Þ2 ðVeB Þ mA mB iTc2 ¼ pffiffiffi 2 2 2 M 2 W A¼1 B¼1

Z d4 p 1 ðn 1Þ 2 p þ 2 Þðp2 nM 2 Þ 2 p2 nMW ð2pÞ4 ðp2 m2A Þðp2 m2B Þðp2 MW W ðB:43Þ

62

Appendices

As what we did for graph (c1) we can also write the T-matrix element in terms of t, xA and xB . 6 6 X X g4 1 þ c5 c 1 c5 2 l ð4Þ½ec ð2Þ eð1Þ ½ lð3Þ ðVlA Þ2 mA ðVeB Þ mB 4 2 2 2 64p MW A¼1 B¼1 Z 1 t ðn 1Þt2 ðB:44Þ tþ dt tþn ðt þ xA Þðt þ xB Þðt þ 1Þðt þ nÞ 0

Tc2 ¼

Using Eq. (B.32) to get rid of the vanishing parts, 6 6 X X g4 1 þ c5 c 1 c5 2 2 c ð2Þ l eð1Þ ½ l ð3Þ ð4Þ½ e ðV Þ m ðVeB Þ mB lA A 4 2 2 64p2 MW A¼1 B¼1 # Z 1 " xA xB ðn 1ÞxA xB t dt þ ðt þ xA Þðt þ xB Þðt þ 1Þðt þ nÞ ðt þ xA Þðt þ xB Þðt þ 1Þðt þ nÞ2 0

Tc2 ¼

ðB:45Þ

B.7 T-Matrix Element of Graph (c3) in Fig. 2.3 It turned out graph (c3) has the same T-matrix element as (c2).

B.8 T-Matrix element of graph (c4) in Fig. 2.3 6 X 6 ig 4 X 2 iTc4 ¼ pffiffiffi ðVlA Þ2 ðVeB Þ 2MW A¼1 B¼1 ( Z m2l mA d4 1 c5 c ml mA 1 þ c5 c l ð4Þ þ 2 l ð4Þ ½ lð3Þ ½ lð3Þcp 4 2 2 m2 2 2 p p m ð2pÞ A A ml mA 1 c5 c m3A 1 þ c5 c þ 2 l ð4Þ 2 l ð4Þ ½ lð3Þcp ½ lð3Þ 2 2 p m2A p m2A m2 mB 1 þ c5 me mB c 1 c5 2 e 2 ½ec ð2Þ eð1Þ þ 2 eð1Þ ½e ð2Þcp 2 2 p mB p m2B

mB me c 1 þ c5 m3B 1 c5 1 2 c 2 eð1Þ 2 eð1Þ ð 2 ½e ð2Þcp ½e ð2Þ Þ 2 2 2 p nMW p m2B p m2B ðB:46Þ

Appendices

63

Ignoring the external momenta, we can simplify above equation as 6 X 6 X ig 4 1 þ c5 c 1 c5 2 l ð4Þ½ec ð2Þ eð1Þ ½ lð3Þ ðVlA Þ2 ðVeB Þ iTc4 ¼ pffiffiffi 2 2 2MW A¼1 B¼1 Z d4 m3A m3B ðB:47Þ 4 2 2 Þ2 2 2 ð2pÞ ðp mA Þðp m2B Þðp2 nMW Doing Wick’rotation and expressing the T-matrix element in terms of t, mA and mB , we have 6 6 X X g4 1 þ c5 c 1 c5 2 2 c ð2Þ l eð1Þ ½ l ð3Þ ð4Þ½ e ðV Þ m ðVeB Þ mB lA A 4 2 2 64p2 MW A¼1 B¼1 Z 1 xA xB t dt ðB:48Þ ðt þ xA Þðt þ xB Þðt þ nÞ2 0

Tc4 ¼

Appendix C. General Formalism of Muonium–Anitmuonium Time Evolution Muonium (antimuonium) is a non-relativistic Coulombic bound state of an electron and an anti-muon (positron and muon). jMi (muonium) and jMi (antimuonium) are mixed by the weak interaction. Thus, in the presence of this type of interaction, the two states are inseparable; they form a basis for a twodimensional subspace. We may write, 1 1 1 ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jMi ¼ ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jMi ðC:1Þ 0 1 Mi hMjMi hMj Let jwðtÞi be an arbitrary state of such space, 1 1 ¼ jwðtÞi ¼ AðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jMi þ BðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jMi hMjMi hMjMi

AðtÞ BðtÞ

ðC:2Þ

The time evolution of wðtÞ is described by d wðtÞ ¼ HwðtÞ dt

ðC:3Þ

hijHjji i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Mij Cij 2 hijiihjjji

ðC:4Þ

i where

64

Appendices

In our two-dimensional space H has the matrix representation i H¼M C¼ 2

M11 2i C11 M21 2i C21

M12 2i C12 M22 2i C22

! ðC:5Þ

CPT invariance of H implies that M11 ¼ M22 M0 and C11 ¼ C22 C0 . Since by construction, both M and C are Hermitian, M0 and C0 are both real numbers, and M21 ¼ M12 and C21 ¼ C12 . Hence M0 2i C0 M12 2i C12 H¼ ðC:6Þ M12 2i C12 M0 2i C0 Diagonalizing (C.6), we find two mass eigen states: 1 jM i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1 þ eÞjMi ð1 eÞjMi 2ð1 þ jej2 Þ

ðC:7Þ

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi MMM MMM pffiffiffiffiffiffiffiffiffiffiffiffi e ¼ pffiffiffiffiffiffiffiffiffiffiffiffi MMM þ MMM

ðC:8Þ

where

MMM

R hMj d 3 rLeff jMi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Mi hMjMihMj

MMM

R d3 rLeff jMi hMj ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mi hMjMihMj

ðC:9Þ

The magnitude of the mass splitting between the two mass eigenstates is jDMj rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i ðM12 C12 ÞðM12 C12 Þ 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i i i C Þ M C ¼ ðM0 C0 Þ ðM12 C12 ÞðM12 2 2 2 2 12

i i Mþ Cþ ¼ ðM0 C0 Þ þ 2 2

Taking a difference of above two eigenvalues rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i i C Þ ðMþ M Þ ðCþ C Þ ¼ 2 ðM12 C12 ÞðM12 2 2 2 12

ðC:10Þ

ðC:11Þ

Then we have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i i C Þ jDMj ¼ 2Re ðM12 C12 ÞðM12 12 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2Re MMM MMM

ðC:12Þ

Appendices

65

Since muonium and antimuonium are linear combinations of the mass diagonal states, an initially prepared muonium or antimuonium state will undergo oscillations into one another as a function of time. The oscillation time is given 1 by sMM ¼ jDMj : , and we will see this oscillation time is related to jDMj as sMM According to (C.7), we have

jMi ¼

¼ jMi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ jej2 Þ 2ð1 þ eÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ jej2 Þ 2ð1 eÞ

ðjMþ i þ jM iÞ ðC:13Þ ðjMþ i jM iÞ

could be written in time evolution forms: Hence, jMi and jMi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ jej2 Þ iðM i C Þt i þ 2 þ e jMðtÞi ¼ jMþ ð0Þi þ eiðM 2C Þt jM ð0Þi 2ð1 þ eÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ jej2 Þ iðM i C Þt i þ 2 þ jMðtÞi ¼ e jMþ ð0Þi eiðM 2C Þt jM ð0Þi 2ð1 eÞ

ðC:14Þ

are given by The beam intensity for M and M ð1 þ jej2 Þ jhMþ ð0ÞjMþ ð0ÞijeCþ t þ jhM ð0ÞjM ð0ÞijeC t 2ð1 þ eÞ2 1 ðC:15Þ þ 2e2ðCþ þC Þt jþ ð0ÞjM ð0Þij cosðDMtÞ

jhMðtÞjMðtÞij ¼

and ð1 þ jej2 Þ jhMþ ð0ÞjMþ ð0ÞijeCþ t þ jhM ð0ÞjM ð0ÞijeC t 2ð1 eÞ2 1 ðC:16Þ 2e2ðCþ þC Þt jhMþ ð0ÞjM ð0Þij cosðDMtÞ

jhMðtÞj MðtÞij ¼

The period of the oscillation is T¼

2p jDMj

ðC:17Þ

Approximately the oscillation time scale is sMM ’

1 jDMj

ðC:18Þ

Author Biography Boyang Liu received his B.S. degree in Physics from the University of Science and Technology of China (USTC), Hefei, China, in 2003 and received his Ph.D. degree in physics from Purdue University, West Lafayette, IN, USA in 2010. He is currently working as a postdoc in the Institute of Physics, Chinese Academy of Sciences (CAS). His research interests include phenomenology of physics beyond the Standard Model, especially the supersymmetric extensions of the Standard Model and Brane World models, as well as neutrino physics, with an emphasis on models of neutrino masses, neutrino mixing and CP violation.

Reference A. Ilakovac, A. Pilaftisis, Nucl. Phys. B437 (1995) 491

67

Muonium-antimuonium Oscillations in an Extended Minimal Supersymmetric Standard Model

ISO10206 standard. Extended Pascal

An Extended Entity-Relationship Model - Fundamentals and Pragmatics

Dynamics of the standard model

Particles of the Standard Model

Supersymmetric Solitons

The Standard Model: A Primer

The Standard Model: A Primer

An Introduction to the Standard Model of Particle Physics

Oscillations in Ionized Gases

An Introduction to the Standard Model of Particle Physics

An Introduction to the Standard Model of Particle Physics

An Introduction to the Standard Model of Particle Physics

An introduction to the standard model of particle physics

Supersymmetric world

Oscillations in Neural Systems

Oscillations in Chemical Reactions

Supersymmetric Mechanics

Nonlinear oscillations

Effective Lagrangians for the Standard Model

Dynamics of the Standard Model, 2nd edition

Minimal Surfaces in R3

Meetings in Paris (Extended)

An Extended Entity-Relationship Model: Fundamentals and Pragmatics (Lecture Notes in Computer Science)

Model Theory: An Introduction

Neutrino Oscillations

Nonlinear Oscillations

Supersymmetric Mechanics: Attractors and Black Holes in Supersymmetric Gravity

Model theory: an introduction

Supersymmetric Solitons

Supersymmetric Mechanics

Muonium-antimuonium Oscillations in an Extended Minimal Supersymmetric Standard Model

ISO10206 standard. Extended Pascal

An Extended Entity-Relationship Model - Fundamentals and Pragmatics

Dynamics of the standard model

Particles of the Standard Model

Supersymmetric Solitons

The Standard Model: A Primer

The Standard Model: A Primer

An Introduction to the Standard Model of Particle Physics

Oscillations in Ionized Gases

An Introduction to the Standard Model of Particle Physics

An Introduction to the Standard Model of Particle Physics

An Introduction to the Standard Model of Particle Physics

An introduction to the standard model of particle physics

Supersymmetric world

Oscillations in Neural Systems

Oscillations in Chemical Reactions

Supersymmetric Mechanics

Nonlinear oscillations

Effective Lagrangians for the Standard Model

Dynamics of the Standard Model, 2nd edition

Minimal Surfaces in R3

Meetings in Paris (Extended)

An Extended Entity-Relationship Model: Fundamentals and Pragmatics (Lecture Notes in Computer Science)

Model Theory: An Introduction

Neutrino Oscillations

Nonlinear Oscillations

Supersymmetric Mechanics: Attractors and Black Holes in Supersymmetric Gravity

Model theory: an introduction

Supersymmetric Solitons

Supersymmetric Mechanics

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