LINE SHAPES OF THE EXOTIC cc¯ MESONS X(3872) AND Z±(4430) DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Meng Lu, B.S., M.S. ***** The Ohio State University 2008 Dissertation Committee: Approved by Eric Braaten, Adviser Richard J. Furnstahl Adviser Klaus Honscheid Graduate Program in Junko Shigemitsu Physics c Copyright by Meng Lu 2008 ABSTRACT The B-factory experiments have recently discovered a series of new cc¯ mesons, including the X(3872) and the first manifestly exotic meson Z±(4430). The prox- 0 0 imity of the mass of the X to the D∗ D¯ threshold has motivated its identification as a loosely-bound hadronic molecule whose constituents are a superposition of the 0 0 0 0 charm mesons pairs D∗ D¯ and D D¯ ∗ . Factorization formulas for its line shapes are derived by taking advantage of the universality of S-wave resonances near a 2-particle threshold and by including the effects from the nonzero width of D∗ meson and the inelastic scattering channels of the charm mesons. The best fit to the line shapes of + 0 0 0 X in the J/ψπ π− and D D¯ π channels measured by the Belle Collaboration corre- 0 0 sponds to the X being a bound state whose mass is just below the D∗ D¯ threshold. The differences between the line shapes of X produced in B+ decays and B0 decays as + + 0 0 0 0 well as in decay channels J/ψπ π−,J/ψπ π−π , and D D¯ π are further derived by taking into account the effects from the closeby channel composed of charged charm mesons. A more speculative application of the universality of S-wave resonances near a 2-particle threshold is to the Z±(4430), which is interpreted as a charm meson + ¯ 0 + ¯ 0 molecule composed of a superposition of D1 D∗ and D∗ D1. The small ratio of the + binding energy of the Z to the width of its constituent D1 is exploited to obtained + simple predictions for its line shapes in the channels ψ(2S)π and D∗D¯ ∗π. ii To my family iii ACKNOWLEDGMENTS It is difficult to overstate my gratitude to my advisor Dr. Eric Braaten for his guidance, advice, support, encouragement, and patience, which all together have made this dissertation possible. He has not only educated me with physics knowledge and techniques but also inspired me by his insight, rigor, enthusiasm, and focusing on physics research. It has been an honor and privilege to work with him. I would like to thank Dr. Jungil Lee, Dr. Masaoki Kusunoki, Dr. Dongqing Zhang, Daekyoung Kang, Huichao Song, Anastasios Taliotis, Evan S. Frodermann, and James C. Stapleton for collaborations and discussions. In particular, I am in debt to Dr. Masaoki Kusunoki, for his earlier research results on the X(3872) particle, and to Dr. Jungil Lee, for the collaboration on the research project on meson molecules. I would also like to thank all my teachers, friends, and classmates in Ohio State and Fudan for providing a stimulating and enjoyable environment in which I have learned and grown. My deepest gratitude goes to my parents Yuliang Lu and Yaqin Dong for their constant love and support. Last but not the least, I would like to thank Yuan Zhang for her love, understanding, and belief in me. This research was supported in part by the Department of Energy under grant DE-FG02- 91-ER4069. iv VITA June 4th, 1981 ............................. Born - Xuanhua, Hebei, China 2003 ........................................B.S. Fudan University 2006 ........................................M.S. The Ohio State University 2003-2004 ..................................University Fellow, The Ohio State University 2004-2006 ..................................Graduate Teaching Associate, The Ohio State University. 2006-current ................................Graduate Research Associate, The Ohio State University. PUBLICATIONS Research Publications E. Braaten and M. Lu, “Line Shapes of the Z(4430)”, arXiv:0712.3885 [hep-ph]. E. Braaten and M. Lu, “The Effects of Charged Charm Mesons on the Line Shapes of the X(3872)”, Phys. Rev. D, 77, 014029 (2008). E. Braaten and M. Lu, “Line Shapes of the X(3872)”, Phys. Rev. D, 76, 094028 (2007). E. Braaten and M. Lu, “Weakly-bound hadronic molecule near a 3-body threshold”, Phys. Rev. D, 76, 054010 (2007). E. Braaten and M. Lu, “Operator product expansion in the production and decay of the X(3872)”, Phys. Rev. D, 74, 054020 (2006). v Y. Xu, M. Lu and R. K. Su, “Extended Wronskian determinant approach and iterative solutions of one-dimensional Dirac equation”, Commun. Theor. Phys., 41, 859 (2004) FIELDS OF STUDY Major Field: Physics vi TABLE OF CONTENTS Page Abstract....................................... ii Dedication...................................... iii Acknowledgments.................................. iv Vita ......................................... v LIST OF TABLES x LIST OF FIGURES xi Chapters: 1. Introduction 1 1.1 Fundamental Particles and Their Interactions . ..... 1 1.2 OrdinaryHadrons............................. 5 1.3 ExoticHadrons .............................. 9 1.4 HadronicMolecules............................ 10 1.5 Recently Discovered cc¯ Mesons...................... 11 1.6 X(3872).................................. 17 vii 1.7 Z±(4430) ................................. 22 2. Charm Mesons 24 2.1 Isospin................................... 24 2.2 Charge Conjugation and G Parity.................... 26 2.3 MassesofCharmMesons. .. .. 29 2.4 Decay Widths of D∗ Mesons....................... 30 2.5 Energy-Dependent Width of Virtual D∗ ................. 33 3. Charm Meson Scattering 36 3.1 BasicScatteringFormalism . 36 3.2 Universality of S-wave Resonances Near Thresholds . 39 3.3 Scattering Channels for D∗D¯ ...................... 42 3.4 Scattering channels for D1D¯ ∗ ...................... 46 3.5 Single Neutral Scattering Channel . .. 48 3.6 Coupled Neutral and Charged Scattering Channels . .... 51 3.7 ConstraintsfromIsospinSymmetry . 54 0 0 4. Line shapes of X(3872) near D D¯ ∗ Threshold 60 4.1 Factorization Formulas with Neutral Channel . .... 60 4.2 The Mass and Width of the X(3872) .................. 65 4.3 Short Distance Decay of X(3872) .................... 66 4.4 Line Shapes of X(3872) in B Decays .................. 68 4.5 Fitstotheenergydistributions . 71 4.5.1 Experimentaldata ........................ 72 4.5.2 Theoreticalmodel. .. .. 75 viii 4.5.3 Fittingprocedure . .. .. 78 5. Line Shapes of X(3872) in DD¯ ∗ Threshold Region 81 5.1 Factorization Formulas with Coupled Neutral and Charged Channels 81 5.2 Constraintsfromisospinsymmetry . 85 5.3 Current-current factorization and heavy-quark symmetry....... 89 5.4 Line Shapes of X(3872) in B Decays .................. 90 5.5 RatiosofDecayRates .......................... 96 6. Line Shapes of Z±(4430) 100 6.1 Low-energycharmmesonscattering . 100 6.2 Lineshapes ................................ 102 7. Conclusion 107 Appendices: A. Universal Transition Amplitude for Particles with a Large Scat- tering Length 112 A.1 ThemodelandFeynmanrules. 112 A.2 Propagation for D1D2 betweeninteractions. 114 A.3 Universal Transition Amplitude f(E).................. 117 BIBLIOGRAPHY 120 ix LIST OF TABLES Table Page 1.1 Basic properties of some particles appearing in the thesis....... 8 1.2 Charmoniadiscoveredby1978. 13 1.3 New cc¯ mesons discovered since 2003 . 16 2.1 Basicpropertiesofcharmmesons. 25 + 4.1 Belle data on the J/ψπ π− energydistribution . 73 4.2 Belle data on the D0D¯ 0π0 energydistribution . 75 4.3 Parameters of line shapes determined by fitting Belle data ...... 79 x LIST OF FIGURES Figure Page + 1.1 Line shape of X(3872) in J/ψπ π− channel .............. 18 0 0 0 0 0 1.2 Line shape of X(3872) in D D¯ π and D∗ D¯ channels . 21 1.3 Line shape of Z±(4430) in ψ(2S) π± channel .............. 23 2.1 Energy-dependent widths of virtual D∗ mesons............. 35 3.1 Cartoons of the imaginary parts of 2-body scattering amplitude f(E) 43 4.1 Energy dependence of the short-distance decay rate of X(3872) ΓC (E) 67 4.2 Line shape of X(3872)inshort-distance decay mode. 69 4.3 Line shape of X(3872) in long-distance decay mode . 70 4.4 Smeared line shapes of X(3872)..................... 71 4.5 Data fitting for line shape of X(3872) in short-distance decay modes . 74 4.6 Data fitting for line shape of X(3872) in short-distance decay modes . 76 4.7 χ2 analysis of the fitting parameters for the line shapes . .. 78 5.1 Line shape of X(3872) in D0D¯ 0π0 channel ............... 92 + 0 5.2 Line shape of X(3872) in J/ψπ π−π channel............. 93 + 5.3 Line shape of X(3872) in J/ψπ π− channel .............. 95 5.4 Ratio of decay rates R as a function of scattering parameter 1/γ1 .. 99 xi 6.1 Line shapes of the Z+ .......................... 104 A.1 1-loop Feynman diagram for D D D D .............. 114 1 2 → 1 2 A.2 Geometricseriesofone-loopdiagrams. ... 117 xii CHAPTER 1 INTRODUCTION 1.1 Fundamental Particles and Their Interactions The modern theory for describing the elementary particles and their interactions is the standard model of particle physics. It is a relativistic quantum field theory, which means that it incorporates both quantum mechanics and special relativity and it is formulated in terms of quantum field operators that create and annihilate elementary particles. It is also a gauge theory with the gauge group SU(3) SU(2) U(1), which × × characterizes the symmetry properties of the interactions between the elementary particles. In the standard model of particle physics, the elementary particles that form matter in nature are classified into quarks and leptons. There are six types or flavors of quarks: up (u), down (d), strange (s), charm (c), bottom (b), and top (t) quark. Each quark has an antiparticle partner; they are labelledu ¯, d¯,s ¯,c ¯, ¯b, and t¯. The two lightest quarks u and d and their antiparticles are called light quarks in this thesis. The term heavy quarks is used to refer to c and b and their antiparticles in this thesis, although t is actually much heavier than b and c.