LINE SHAPES of the EXOTIC C¯C MESONS X(3872)

Home , Exotic hadron, Exotic meson

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

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. There are also six types of

leptons: the three charged leptons, electron (e−), muon (µ−), tau (τ −), and the three

associated neutrinos νe, νµ, and ντ . The neutrinos are nearly massless. Each lepton

1 + + + has an antiparticle partner. They are labelled e , µ , τ ,ν ¯e,ν ¯µ, andν ¯τ . The six quarks can be organized into three families or generations, each of which is a doublet

consisting of an up-type quark (u, c, or t) and a down-type quark (d, s, or b). The quark doublets are (u,d), (c,s), and (t, b). The leptons can also be organized into three generations of doublets: (νe, e−), (νµ,µ−), and (ντ , τ −).

Elementary particles can carry electric charges which are usually expressed in units of the proton charge. The up-type and down-type quarks have electric charges of + 2 and 1 , respectively. The electron, muon, and tau have electric charge of 1. 3 − 3 − The neutrinos are electrically neutral. The antiparticles have the opposite electric charges as the particles. Elementary particles can have intrinsic angular momentum.

Quantum mechanics restricts the intrinsic angular momentum of particles to discrete values labelled by a quantum number J called the spin, whose possible values are

J =0, 1 , 1, 3 , . The quarks and leptons all have spin J = 1 . 2 2 ··· 2 Quarks and leptons interact through the four fundamental interactions known in nature, i.e. the gravitational, weak, electromagnetic and strong interactions. All particles interact through gravitational interaction, which is many orders of magnitude weaker than the other three interactions. The best description of gravitational interaction currently available is Einstein’s general theory of relativity, which is a low-energy

eﬀective theory of gravity that ignores quantum eﬀects. Constructing a quantum

theory of gravity consistent with general relativity is an active ﬁeld of research, with

the leading candidate being String Theory. Quarks and leptons interact via the weak

+ interaction, which is mediated by three massive spin-1 particles called W , W − and

Z0. The weak interaction is the only interaction that can alter the ﬂavors of quarks.

The quarks and the charged leptons also interact via the electromagnetic force, which

2 is mediated by a massless spin-1 particle, the photon (γ). The strength of the interaction is proportional to the electric charges of the interacting particles. Glashow,

Salam, Weinberg, and others developed the uniﬁed gauge theory that combines the electromagnetic and weak interaction into the electroweak interaction, which is based on the gauge group SU(2) U(1). × The strongest of the four fundamental interaction is the strong interaction. The theory of the strong interaction is Quantum Chromodynamics (QCD), which is a gauge theory based on the symmetry group SU(3). Only the quarks and antiquarks interact through the strong interaction. A quark carries any one of the three color charges, red, blue, and green, while an antiquark carries any one of three complimentary color charges. The strong force is mediated by massless spin-1 particles called gluons (g), which couple to the color charge of the quarks. The parameters of QCD are the QCD coupling constant αs and the masses of the six ﬂavors of quarks. Unlike leptons, isolated quarks which are color charged have never been observed in experiments.

The strong interaction always binds quarks into composite particles that have no net color charge called hadrons. This phenomenon is called color conﬁnement. The interaction between the quarks in hadrons becomes arbitrarily weak at ever shorter distance. This property called asymptotic freedom allows perturbative calculations in the high-energy regime where the QCD coupling constant αs is small. On the other hand, the interaction between the quarks becomes increasingly strong at ever larger distances or lower energies. In the strongly-interacting low-energy regime of QCD,

αs is suﬃciently large that nonperturbative approaches are required. This makes the calculation of hadronic properties directly from QCD a diﬃcult task.

3 Lattice QCD is a mathematically well-deﬁned computational approach to QCD.

It implements the gauge theory of quarks and gluons on a discretized spacetime lattice of finite size. In Lattice QCD, hadronic properties are calculated by evaluating path integrals over the spacetime lattice using Monte Carlo methods. In principle, the numerical predictions given by Lattice QCD can be systematically improved by decreasing the lattice spacing and increasing the size of the lattice. However, the computing power that is required grows dramatically as the masses of the quarks are decreased. Present computing capabilities are insufficient to calculate hadronic properties for the physical values of the light (u and d) quark masses. Thus, in practice, it is necessary to use larger light quark masses and extrapolate towards the physical values. The HPQCD Collaboration has recently shown that by using advanced algorithms certain hadronic properties can finally be reproduced to within few percent errors of the experimental values [1].

Another approach to calculating hadronic properties is to use an eﬀective ﬁeld theory that exploits the separation of physical scales for a particular type of hadron.

This is a partially phenomenological approach in which certain eﬀects of the details of QCD are parameterized by constants that can be tuned to their measured values.

An eﬀective ﬁeld theory can be used to expand observables in powers of a small ratio of scales that occur in the hadron. Examples are

chiral perturbation theory, which describes hadrons interacting through the ex- • change of pions and involves an expansion in powers of the pion momentum and

the pion mass,

4 heavy-quark eﬀective theory (HQET), which describes hadrons containing one • heavy quark (c or b) and involves an expansion in inverse powers of the heavy

quark mass,

non-relativistic QCD (NRQCD), which describes hadrons containing a heavy • quark (c or b) and its antiquark and involves an expansion in powers of the

relative velocity of the heavy quarks.

1.2 Ordinary Hadrons

A cluster of quarks, antiquarks and gluons with overall neutral color charge is

called a color singlet. QCD implies that only color-singlet clusters of quarks, anti-

quarks, and gluons can have ﬁnite energy and can therefore exist as isolated composite

particles. The hadrons are color-singlet clusters that are either bound states, which

are stable with respect to decays by the strong interaction, or resonances, which can

decay into two or more hadrons but have long enough lifetimes for their properties to be measured. The hadrons are generally classiﬁed into baryons, antibaryons, and

mesons. Ordinary baryons are color-singlet clusters of three quarks. For example,

the proton (p) and the neutron (n) are composed of uud and udd, respectively. Or-

dinary antibaryons are color-singlet clusters of three antiquarks. For example, the

antiproton (¯p) and the antineutron (¯n) are composed ofu ¯u¯d¯ andu ¯d¯d¯, respectively.

Ordinary mesons are color-singlet clusters of a quark and an antiquark. For example,

+ 0 the pions π , π−, and π are composed of ud¯, du¯, and (uu¯ + dd¯)/√2, respectively.

The Review of Particle Physics [2, 3, 4], which will be referred to simply as the

Review in the remainder of this thesis, is a comprehensive review of the ﬁeld of particle

physics that is published biannually by the Particle Data Group (PDG) Collaboration.

5 In this review, the Summary Tables of Particle Properties lists the well-established

particles that have been observed and measured by multiple experiments. The longer

Particle Listings include additional particles, some of which have not been conﬁrmed.

There are about one hundred well-established mesons and about ﬁfty well-established

baryons listed in the Summary Tables.

In the Review, mesons are categorized into several types according to their con-

stituent quark ﬂavors. Mesons composed of the light quarks u,u ¯, d and d¯ are called

light unﬂavored mesons 1. Mesons composed of an s ors ¯ and a light quark are called

strange mesons or K mesons. Mesons composed of a c orc ¯ and a light quark are

called charm mesons or D mesons. Mesons containing a b or ¯b and a light quark are

called bottom mesons or B mesons.

Another type of meson is quarkonium, which is a bound state of a quark and

an antiquark of the same ﬂavor, i.e. an unﬂavored meson. Quarkonium is to QCD

as positronium, i.e. the electron-positron bound state, is to Quantum Electrodynam-

ics (QED), the quantum ﬁeld theory for electromagnetism. Due to the small masses

of u, d and s quarks, quarkonium states of uu¯, dd¯ and ss¯ appear often as quantum

mechanical mixtures, such as η uu¯ + dd¯ 2ss¯ /√6. In the Review, they are listed ≈ − among the light unflavored mesons along with ud¯and du¯ mesons. Such quantum mechanical mixtures do not occur for charmonium (cc¯) and bottomonium (b¯b) due to the significantly heavier masses of c and b quarks. In the Review, charmonia are listed under cc¯ mesons and bottomonia are listed under b¯b mesons. The first charmonium state J/ψ was discovered in 1974 at SLAC [5] and Brookhaven [6]. The first bottomonium state Υ(1S) was discovered in 1977 at Fermilab [7]. Toponium (tt¯) does

1Here the “unﬂavored” refers to the absence of the heavier quark ﬂavors s, c, b, and t.

6 not really exist due to the fact that the top quark is very heavy and decays very fast

through the weak interaction into a W + boson and a b quark before a bound state can be formed. Thus the only heavy quarkonia are the charmonium and bottomonium.

Ever since its discovery, charmonium has been an important system for improving our understanding of the strong interaction, which is the least understood sector of the standard model of particle physics. Charmonium and bottomonium are the simplest hadrons. They exhibit the color confinement character and other nonperturbative aspects of QCD. At the same time, the QCD coupling constant at the scale of the charm quark mass is small enough that some aspects of charmonium can be calculated using perturbation theory. The two constituent charm quarks in a charmonium are sufficiently heavy that they are fairly non-relativistic, which allows them to be described by an effective field theory called Non-Relativistic QCD (NRQCD). Other commonly used approaches to study charmonium include various constituent quark potential models and QCD sum rules. An extensive interplay between theoretical, numerical, and experimental analysis has been particularly beneficial in understanding charmonium.

The Review lists the single-particle properties of a hadron. They include its mass m and its decay width Γ, which is related to its lifetime τ by Γ = ~/τ. The spin

quantum number J of a hadron denotes its total angular momentum due to the spins

of its constituents and the orbital angular momentum between them. Hadrons are

also characterized by the parity quantum number P , which determines its behavior

under a parity transformation i.e. spatial reﬂection. The possible values of P are

+1 (even) or 1 (odd). P = +1 if the quantum state is invariant under a parity − transformation, and P = 1 if it changes sign. Mesons with no net ﬂavor are also −

7 meson quark content mass (MeV) width (MeV) J P (C) Light unﬂavored mesons 0 1 6 + π uu¯ dd¯ 134.98 7.86 10− 0− √2 − × + 14 π /π− ud/d¯ u¯ 139.57 2.53 10− 0− × Strange mesons 0 16 0 0 KS : 7.3514 10− K /K¯ ds/¯ ds¯ 497.648 × 0− 0 14 K : 1.287 10− L × + 14 K−/K¯ su¯/us¯ 493.677 5.31 10− 0− × Charm mesons 0 0 9 D /D¯ cu/u¯ c¯ 1864.5 1.606 10− 0− × + 10 D /D− cd/d¯ c¯ 1869.3 6.329 10− 0− × Bottom mesons 0 0 10 B /B¯ d¯b/bd¯ 5279.4 4.302 10− 0− × + 10 B /B− u¯b/bu¯ 5279.0 4.018 10− 0− ×

Table 1.1: Basic properties of some of the particles appearing in this thesis. The masses and widths are their central values [2]. A complete listing of charm mesons is presented separately in Table 2.

characterized by the charge conjugation quantum number C, which determines its behavior under charge-conjugation transformation, i.e. the interchange of particles with their antiparticles. C = +1 (even) if the quantum state is invariant under a charge-conjugation transformation and C = 1 (odd) if it changes sign. The spin − and parity (and charge conjugation) quantum numbers of a hadron are generally denoted by J P (or J P C ). In Table 1.2, the basic properties of some of the hadrons that appear in this thesis are listed.

8 1.3 Exotic Hadrons

Most of the well-established hadrons in the Summary Tables of the Review can be identiﬁed as ordinary mesons (qq¯) or ordinary baryons (qqq). However, besides the ordinary hadrons, QCD in principle also allows other color-singlet clusters of quarks, antiquarks, and gluons to exist as bound states or resonances. Hadrons that can not be classiﬁed as ordinary baryons, antibaryons, or mesons are referred to as exotic hadrons. Some of the simplest possibilities for exotic hadrons are the following:

glueball or gluonium (gg, ggg), which comprises only gluons, •

hybrid meson (qqg¯ ), which comprises a valence quark-antiquark pair and one or • more gluons,

tetraquark meson (qqq¯q¯), which comprises two quarks and two antiquarks. •

pentaquark baryon (qqqqq¯), which comprises four quarks and one antiquark. •

These exotic hadrons could be compact color-singlet clusters of their constituent quarks and gluons, or the constituents could be organized into subclusters. In the case of tetraquark mesons, some of the possibilities for subclusters are

meson molecule, in which the subclusters are two color-singlet qq¯ pairs or equiv- • alently a pair of ordinary mesons,

diquark-antidiquark meson, in which the subclusters are a color-antitriplet di- • quark (qq) and an color-triplet antidiquark (¯qq¯).

QCD is a completely well-deﬁned theory for strong interaction. Given the values of αs and the quark masses, it gives predictions for the complete spectrum of hadrons.

9 Thus it predicts whether or not the exotic hadrons exist. However, QCD is so complicated in the nonperturbative low-energy regime that the predictions are not yet known.

1.4 Hadronic Molecules

Another category of hadrons is hadronic molecule, which is a composite particle whose constituents themselves are mesons and baryons. The molecules that are fa- miliar from chemistry consist of atoms bound together by a residual force arising from the incompletely cancellation of the electromagnetic forces between the electrically- charged constituents of the electrically-neutral atoms. In a similar fashion, hadronic molecules consist of hadrons bound together by a residual force arising from the incomplete cancellation of the strong forces between the colored quarks inside the color-singlet hadrons. The electromagnetic force is suﬃciently simple that the properties of the simplest molecules, such as the hydrogen molecule H2, can be calculated from ﬁrst principles. The strong force of QCD is much more complicated, and therefore the residual force that might bind constituent hadrons into hadronic molecules are not understood.

The nuclei of atoms can be viewed as hadronic molecules composed of protons and neutrons. Since their constituents are ordinary baryons, nuclei can be regarded as baryonic molecules. The simplest baryonic molecule is the deuteron, which consists of a proton and a neutron. Nuclei are generally not considered to be exotic hadrons, but any other type of hadronic molecule could be considered exotic. One type of exotic hadronic molecule is baryonium, which is composed of an ordinary baryon and its antibaryon. Another type of exotic hadronic molecule is meson molecule, which

10 is composed of two ordinary mesons. Since a meson has constituents qq¯, a meson molecule has quark constituents qqq¯ q¯. Thus it is a special case of a tetraquark meson.

Meson molecules have been sought after for a long time. For example, the spin-0 mesons a0(980) and f0(980) have been proposed as candidates for KK¯ molecules [8, 9].

The existence of charm meson molecules, whose constituents are two charm mesons, was predicted not long after the discovery of the charm quark in 1974 [10, 11]. Interest in this possibility has been revived by the recent discovery of a series of new cc¯ mesons, some of which may be exotic mesons. To date, no meson molecules have been unambiguously identiﬁed. In this thesis, evidence is added that one of the recently discovered cc¯ mesons can be identiﬁed as a charm meson molecule.

1.5 Recently Discovered cc¯ Mesons

Mesons containing c andc ¯ are called cc¯ mesons. Since the quark ﬂavors of the c andc ¯ in a cc¯ meson cancel, they are sometimes called mesons with “hidden charm”.

A cc¯ meson whose only constituents are a c and ac ¯ is called charmonium. The study of charmonium began with the discovery of J/ψ in November, 1974. It was discovered

+ simultaneously in e e− annihilation experiments, where it was observed as a dramatic resonant peak in the cross section [5], and in p Be collision experiments, where it − + was observed through its decays into e e− [6]. The only states that are directly

+ accessible in e e− annihilation experiments are those with the photon’s quantum

P C number J = 1−−, such as J/ψ and ψ(2S). Some states with other quantum numbers were discovered by using the radiative decays of J/ψ and ψ(2S) into a photon and a lighter charmonium state. By 1978, the well-established charmonium states were

11 six 1−− states: J/ψ, ψ(2S), ψ(3770), ψ(4040), ψ(4160), and ψ(4415), •

+ the lowest 0− state η , • c

three states χ (1P ), χ (1P ), χ (1P ) with quantum numbers 0++, 1++, and • c0 c1 c2 2++, respectively.

In the constituent quark model, the charmonium states can be labeled by the

2S+1 spectroscopic notation n LJ where n, S, L, and J are the radial excitation, total spin, orbital angular momentum, and total angular momentum quantum numbers, respectively. The orbital angular momentum number is an integer L = 0, 1, 2, , ··· but they are conventionally labeled by S,P,D, . The total spin quantum number ··· takes values S =0 or1, so 2S + 1 is either 1 (spin-singlet) or 3 (spin-triplet). For given values of L, the radial excitation quantum number takes values n = 1, 2, . ··· The parity quantum number P and charge conjugation quantum number C of a quarkonium can be related to L and S in a simple way [12]:

P = ( 1)L+1, (1.1a) − C = ( 1)L+S. (1.1b) −

Table 1.5 is a complete listing of charmonia discovered by 1978 and their properties. In the following twenty-ﬁve years, no new charmonium states were added to the

Summary Tables of the Review. In the Particle Listings, the only charmonium states

2 added were the ηc(2S), which was reported in 1982 [14], and the hc(1P ) [15], which was reported by experiments in 1986.

2The ψ(3836) was reported in 1994 [13]. It appeared in the Particle Listings from 1996 to 2006 when it was ﬁnally removed.

12 P C 2S+1 name J n LJ mass (MeV) width (MeV) discovery + 1 η (1S) 0− 1 S 2980.4 1.2 25.5 3.4 1978 c 0 ± ± 3 J/ψ(1S) 1−− 1 S 3096.916 0.011 0.0934 0.0021 1974 1 ± ± χ (1P ) 0++ 13P 3414.76 0.35 10.4 0.7 1977 c0 0 ± ± χ (1P ) 1++ 13P 3510.66 0.07 0.89 0.05 1975 c1 1 ± ± χ (1P ) 2++ 13P 3556.20 0.09 2.06 0.12 1976 c2 2 ± ± 3 ψ(2S) 1−− 2 S 3686.093 0.034 0.337 0.013 1975 1 ± ± 3 ψ(3770) 1−− 1 D 3771.1 2.4 23.0 2.7 1977 1 ± ± 3 ψ(4040) 1−− 3 S 4039 1 0.00086 0.00007 1975 1 ± ± 3 ψ(4160) 1−− 2 D 4153 3 0.00083 0.00007 1977 1 ± ± 3 ψ(4415) 1−− 4 S 4421 4 0.00058 0.00007 1977 1 ± ±

Table 1.2: A listing of charmonium states discovered by 1978. The masses and widths are taken from the 2006 edition of the Review [2].

The last ﬁve years (2003 – 2008) has witnessed an unexpected renaissance of charmonium spectroscopy, thanks to experiments at B factories and charm factories.

+ B factories are e e− annihilation experiments at 10.54 GeV, which is the energy of the

Υ(4S) bottomonium state. The Υ(4S) decays predominantly into a pair of B mesons,

+ 0 0 either B B− or B B¯ . Thus these experiments create large numbers of B mesons, which is why they are called B factories. Since one of the favored decay channels of the bottom quark is b csc¯, B mesons can decay into a cc¯ meson and a strange (or K) → meson. Therefore, B factories can be exploited to study cc¯ mesons. The B factories at the KEK laboratory in Japan and the SLAC laboratory in the United States began operations at the turn of the century. The experimental collaborations that built and operate the detectors at these two B factories are called the Belle Collaboration

13 + and the BaBar Collaboration, respectively. Charm factories are e e− annihilation experiments at energies in the charmonium region. When operated at energy 3.77

GeV, they produce the ψ(3770) charmonium state, which decays predominantly into

0 0 + a pair of charm mesons, either D D¯ or D D−. Thus they can produce large numbers

of charm mesons. The charm factories can also produce any of the 1−− charmonium

states by operating at the energy of that state. There are charm factories at the

Institute of High Energy Physics in Beijing and at Cornell University in Ithaca, New

York. The experimental collaborations that operate the detectors at these charm

factories are the BES Collaboration and the CLEOc Collaboration, respectively.

The renaissance of charmonium spectroscopy started with the discovery of a cc¯

meson called X(3872) in September, 2003 by the Belle Collaboration. Since then, a

series of new cc¯ states have been discovered, including some that have been identiﬁed

as charmonium states and others that are good candidates for exotic mesons. Since

2003, four new well-established cc¯ mesons have been added to the Summary Tables

of the Review:

X(3872), added in 2004 [3]. •

χ (2P ) (also known as Z(3930)), added in 2004 [3]. It has quantum numbers • c2 P C ++ 3 J =2 and is identiﬁed as the 2 P2 charmonium state.

P C + η (2S), added in 2006 [2]. It has quantum numbers J =0− and is identiﬁed • c 1 as the 2 S0 charmonium state.

h (1P ), added in the 2007 partial update [16]. It has quantum numbers J P C = • c + 1 1− and is identiﬁed as the 1 P1 charmonium state.

14 X(4260) (also known as Y (4260)), added in the 2007 partial update [16]. It has • P C quantum numbers J =1−−.

There are also two new cc¯ states that require conﬁrmation and therefore appear only

in the Particle Listings:

X(3945) (also known as Y (3940)), added in 2006 [2], •

X(3940), added in the 2007 partial update [16]. •

Moreover, four additional cc¯ states have been reported in the literature as of May,

2008:

X(4160) [17], •

Y (4360) and Y (4660) with quantum numbers 1−− [18], •

Z±(4430) [19]. •

In Table 1.5, a complete listing of the new cc¯mesons discovered since 2003 is presented.

Among these recently discovered cc¯ mesons, only ηc(2S), hc(1P ), and χc2(2P ) have been established as conventional charmonium states. Some of the others may also be missing charmonium states. However, many of them either do not ﬁt well into the predicted charmonium spectrum or have decay modes that are not expected for charmonia. Therefore, various exotic mesons have been proposed as candidate interpretations for them. The strongest candidates for exotic mesons are X(3872),

X(4260), and Z±(4430). The X(3872) is the main subject of this thesis. Its experimentally observed properties are described below in Section 1.6. The Z±(4430) will also be discussed in this thesis. Its experimentally observed properties are described in Section 1.7.

15 J P C name mass (MeV) width (MeV) decay modes discovery + 0− ηc(2S) 3637 4 14 7 KKπ¯ , γγ Belle 12/2003 [20] ± ± +10 +20 + 1−− X(4260) 4264 12 83 17 J/ψππ, J/ψK K− BaBar 6/2005[21] − − + Y (4360) 4361 13 74 18 ψ(2S)π π− Belle 7/2007 [18] ± ± 3 + Y (4660) 4664 12 48 15 ψ(2S)π π− BaBar/Belle 7/2007 [18] ± ± + 0 1 − hc(1P ) 3525.93 0.27 < 1 J/ψπ , η γ CLEO 5/2005 [23] ± c c ++ 2 χc2(2P ) 3929 5 2 29 10 2 γγ, DD¯ Belle 12/2005 [24] ± ± ± ± 1++ X(3872) 3871.4 0.6 < 2.3 J/ψ 2π, J/ψ 3π, J/ψγ, D0D¯ 0π0 Belle 9/2003 [25] ± ? X(3940) 3943 6 6 < 52 DD¯ ∗, DD¯, J/ψω Belle 7/2005 [24] ± ± X(3945) 3943 11 13 87 22 26 J/ψω Belle 4/2008 [26]

16 ± ± ± ± +113 ¯ X(4160) 4156 29 139 65 D∗D∗ Belle 8/2007 [17] ± − +18+30 Z±(4430) 4433 4 2 45 13 13 ψ(2S)π± Belle 7/2008 [19] ± ± − −

Table 1.3: A listing of new cc¯ mesons discovered since 2003. The masses and widths of the listed particles are taken from the 2007 partial update of the Review [16] or from the literature. The bold-faced particles are included in the Summary Tables. 1.6 X(3872)

The X(3872) is a hadronic resonance near 3872 MeV, whose discovery was announced by the Belle Collaboration in July 2003 [25]. The X(3872) was observed as

+ a narrow peak in the J/ψπ π− invariant mass distribution from the exclusive decay

+ + + 4 process B K J/ψπ π−. The mass of the resonance was measured to be →

M = 3872.0 0.6 0.5. (1.2) X ± ±

The resonance is suﬃciently narrow that only an upper bound on the width was obtained:

ΓX < 2.3 MeV (90% CL). (1.3)

The resonance peak is shown in Figure 1.1. The resonance peak is interpreted as the production of X(3872) by the process B± K± + X followed by its decay → + into J/ψπ π−. The product of the branching fractions for these two processes was measured to be

+ + + 5 Br B K + X Br X J/ψπ π− = (1.30 0.025 0.014) 10− . (1.4) → → ± ± × The X(3872) was later conﬁrmed by the CDF [28] and D0 Collaborations [29] in proton-antiproton (pp¯) collision experiments and by the BaBar Collaboration [30] in

B meson decay experiments. The combined measurement of the mass of the X(3872) from the four experiments is [16]

M = 3871.4 0.6 MeV. (1.5) X ± 4Here and below, when two error bars on an experimental measurement are given, the ﬁrst is for statistical errors and the second one is for systematic errors.

17 400 2

200 Events/0.010 GeV/c 0 0.40 0.60 0.80 1.00 1.20 M(π+π-J/ψ) - M(J/ψ) (GeV/c2)

Figure 1.1: Discovery of the X(3872) by the Belle Collaboration: the invariant mass + + + + distribution of J/ψπ π− in the decay B K J/ψπ π−. The smaller peak at ap- → proximately 0.8 GeV above the J/ψ mass is identified with the production of X(3872) + + + from B K X(3872) followed by its decay by X(3872) J/ψπ π−. The statistical significance→ of the observed peak is more than 10σ.→ This figure is taken from Ref. [27].

In May 2005, the Belle Collaboration presented evidence for the decay of X into

+ 0 J/ψπ π−π [31]. The measured branching ratio relative to the discovery decay mode

was + 0 Br[X J/ψπ π−π ] → =1.0 0.4 0.3. (1.6) Br[X J/ψπ+π ] ± ± → − The invariant mass distribution of the three pions in this decay mode is dominated by

the virtual ω resonance [31]. Since J/ψ and ω both have isospin I = 0, the decay mode

+ 0 + + J/ψπ π−π has isospin I = 0. In the discovery decay mode J/ψπ π−, the π π−

invariant mass distribution was observed to peak close to the upper kinematic limit,

which is consistent with originating from the decay of a virtual ρ0 resonance. Since

18 0 + ρ has isospin I = 1, the decay mode J/ψπ π− has isospin I = 1. The comparable

+ 0 + branching fractions of X(3872) in the decay modes J/ψπ π−π and J/ψπ π−, which

have isospins 0 and 1, respectively, indicates large violations of isospin symmetry.

In May 2005, the Belle Collaboration also presented evidence for the decay of

the X into J/ψγ [31], which was later conﬁrmed by the BaBar Collaboration [32].

Since both J/ψ and γ are odd under charge conjugation, the decay mode J/ψγ must

be even under charge conjugation. Thus the X(3872) must have charge conjugation

quantum number C = +.

+ An analysis by the Belle Collaboration of the decays of X into J/ψπ π− [33]

indicates that the angular correlations between ﬁnal-state particles rules out J P C =

++ + + 0 and 0− . The invariant mass distribution of the π π− system peaks at near

its upper kinematic limit, which favors S-wave over P -wave as the relative orbital

angular momentum between the ﬁnal-state dipion and J/ψ. This strongly disfavors

P C + + J = 1− and 2− assignments for the X. The accumulated evidence strongly

favors a J P C =1++ assignment for the X(3872), although the 2++ possibility is not

ruled out [33].

The Belle Collaboration reported recently the ratio of the branching fractions for

the production of X(3872) in decays of B0 and B+ [34]:

B [B0 K0X(3872)] → = 0.94 0.24 0.10. (1.7) B [B+ K+X(3872)] ± ± → This value is consistent with unity as one might naively expect from exact isospin

symmetry. The BaBar Collaboration also reported the ratio with a signiﬁcantly

smaller central value [35]:

B [B0 K0X(3872)] → = 0.41 0.24 0.05. (1.8) B [B+ K+X(3872)] ± ± → 19 The upper limit for the ratio drawn from this measurement is 0.73 at 90% CL.

In June 2006, the Belle Collaboration announced the discovery of a near-threshold enhancement in the D0D¯ 0π0 system from the decay B KD0D¯ 0π0 at an energy a → few MeV higher than the mass of the X(3872) [36]. The peak in the threshold

enhancement was at the energy 5

D0D¯ 0π0 +0.3 MX = 3875.9 0.7 0.7MeV, (1.9) ± −

This result was conﬁrmed by the BaBar Collaboration [38], who expressed it as a

0 0 0 0 threshold enhancement in the D∗ D¯ system in the decay B KD∗ D¯ . The peak → in the threshold enhancement was at the energy

D∗0D0 +0.7 MX = 3875.1 0.5 0.5 MeV. (1.10) − ±

For both experiments, the diﬀerence between the measured peak mass of the threshold

enhancement and the mass of X(3872) is more than 4 standard deviations higher

than the measured mass of the X(3872). The observed threshold enhancements are

shown in Figure 1.2. Both experiments ﬁnd the location of the peak of the threshold

0 0 0 0 0 enhancement in the D D¯ π or D∗ D¯ channel to be approximately 4 MeV higher

than the mass MX of the X(3872) given in Eq. (1.5), which was measured in the

+ J/ψπ π− decay channel.

0 0 The proximity of the mass of the X(3872) to the D∗ D¯ energy threshold has

motivated its identiﬁcation as a weakly-bound molecule whose constituents are a

0 0 0 0 superposition of the charm meson pairs D∗ D¯ and D D¯ ∗ [39, 40, 41, 42]. In Chapter

3, it will be shown that the establishment of the quantum numbers of the X(3872)

5The central value here has been increased by 0.7 MeV to take into account a subsequent accurate 0 measurement of the D mass by the CLEOc Collaboration [37] and the systematic error from the D0 mass has been omitted.

20 as 1++ makes this conclusion unavoidable. In Chapter 4 and 5, the implications for

the line shapes of the X(3872) from its identiﬁcation as a charm meson molecule

are discussed. A natural explanation is given for the diﬀerence between the mass

+ 0 0 0 measured in the J/ψπ π− decay channel and the one measured in the D D¯ π (or

0 0 D∗ D¯ ) decay channel.

2 16 (e) X(3872) 14 *0 All D D0 modes 12 10 8

Events/2 MeV/c 6 4 2 0 3.88 3.9 3.92 3.94 3.96 3.98 4 *0 D D0 Invariant Mass (GeV/c2)

0 0 0 0 0 Figure 1.2: Threshold enhancements in D D¯ π slightly above D∗ D¯ threshold. The upper panel shows the invariant mass distribution of D0D¯ 0π0 seen by Belle Collaboration. The statistical significance of the observed peak is 6.4σ. The figure is 0 0 taken from Ref. [36]. The lower panel shows the invariant mass distribution of D∗ D¯ seen by BaBar Collaboration. The statistical significance of the observed peak is 4.9σ. The figure is taken from Ref. [38].

21 1.7 Z±(4430)

The Z+(4430) is a hadronic resonance near 4.43 GeV discovered by the Belle Col- laboration in summer 2008 [19]. It was observed as a resonance peak in the ψ(2S)π± invariant mass distribution from the exclusive decay process B Kψ(2S)π±, as → shown in Figure 1.3. The ﬁt using a Breit-Wigner resonance yields the peak mass:

M = 4433 4 2 MeV, (1.11) Z ± ± and a relatively narrow width:

+18+30 ΓZ = 45 13 13 MeV. (1.12) − −

The resonance peak in Figure 1.3 is interpreted as the production of Z+(4430) by

B+ K0Z+ followed by its decay into ψ(2S)π+. The product of the branching → fractions for these two processes was determined to be

0 5 Br B K∓Z±(4430) Br Z± ψ(2S)π± = (4.1 1.0 1.4) 10− . (1.13) → → ± ± × P The spin and parity quantum numbers J of the Z±(4430) have not been measured. Since it is a meson with net electric charge, it does not have a charge conjugation eigenvalue. The decay products of ψ(2S) and π+ have isospins I = 0 and 1, respectively. This implies that the Z± has isospin 1 and must be the charged members

+ 0 of an isospin triplet (Z ,Z ,Z−).

The Z+ decays into ψ(2S)π+, where ψ(2S) is a charmonium meson whose constituents are a charm quark and antiquark (cc¯) and π+ is a charged pion whose constituents are a light quark and antiquark (ud¯). Thus the Z+ must have quark

22 30

20 Events/0.01 GeV

0 3.8 4.05 4.3 4.55 4.8 ι M(π+ψ ) (GeV)

Figure 1.3: The discovery of Z+(4430) by the Belle Collaboration: the invariant mass distribution of ψ(2S)π± in the decay B Kψ(2S)π± [19]. The peak near 4430 MeV is identified with the production of Z+(4430)→ from B+ K0Z+ followed by its decay by Z+ ψ(2S)π+. The statistical significance of the→ observed peak is 6.5σ. The → figure is taken from Ref. [19].

content ccu¯ d¯, which means that it is a manifestly exotic tetraquark meson. If the existence of the Z±(4430) is conﬁrmed by the BaBar Collaboration, it will be the ﬁrst well-established exotic hadron. However its structure will still be an open question.

The mass of the Z±(4430) is close to the threshold for the charm meson pair D1D¯ ∗.

This proximity has motivated the interpretation of the Z+(4430) as a charm meson

+ ¯ 0 + ¯ 0 molecule whose constituents are a superposition of D1 D∗ and D∗ D1. In Chapter

6, simple consequences of this interpretation for the line shapes of the Z±(4430) are discussed.

23 CHAPTER 2

CHARM MESONS

Charm mesons refer to ordinary mesons composed of either a charm quark and

a light antiquark (cq¯) or an charm antiquark and a light quark (qc¯), where the light

quark q is either the up quark (u) or the down quark (d). Charm mesons are sometimes

called mesons with “open charm” since they carry the ﬂavor of the charm quark. In

this chapter, we discuss the single-particle properties of charm mesons. The quark

contents, J P quantum numbers, masses, and decay widths of the charm mesons are listed in Table 2.

2.1 Isospin

The masses of the light quarks u and d are much smaller than the masses of any hadrons. As a consequence, the strong interactions have an approximate symmetry under unitary transformations that mix u and d. These transformations form the isospin symmetry group SU(2). The isospin doublets formed of light quarks are

u d¯ , , (2.1) d −u¯ where the minus sign in front of d¯ ensures that the two isospin doublets share the same set of Clebsh-Gordan coeﬃcients. The upper and lower components of an isospin

24 particle quark content J P mass (MeV) width (MeV) 0 0 9 D /D¯ cu/u¯ c¯ 0− M = 1864.5 0.4 1.606 10− 0 ± × + 10 D /D− cd/d¯ c¯ 0− M = 1869.3 0.4 6.329 10− 1 ± × 0 0 D∗ /D¯ ∗ cu/u¯ c¯ 1− M 0 = 2006.7 0.4 < 2.1 ∗ ± + D∗ /D∗− cd/d¯ c¯ 1− M 1 = 2010.0 0.4 0.096 0.022 ∗ ± ± D0/D¯ 0 cu/u¯ c¯ 1+ M = 2422.3 0.4 20.4 1.7 1 1 D1 ± ± + + D /D− cd/d¯ c¯ 1 2423.4 3.1 25 6 1 1 ± ± 0 0 + D∗ /D¯ ∗ cu/u¯ c¯ 2 M ∗ = 2461.1 1.6 43 4 2 2 D2 ± ± + + D∗ /D∗− cd/d¯ c¯ 2 2459 4 29 5 2 2 ± ±

Table 2.1: Basic properties of charm mesons. The masses and widths are taken from the 2006 edition of R.P.P. [2]. There is only an experimental upper bound on the 0 width of D∗ . The theoretical prediction of its width, which is given in Eq. (2.31), is 65.5 15.4 keV. ±

doublet correspond to the following isospin quantum states:

I = 1 , I =+ 1 2 3 2 . (2.2) |I = 1 , I = 1 i | 2 3 − 2 i The isospin symmetry of the u and d quarks can be extended to hadrons containing

light quarks as constituents. The proton (p) and neutron (n) are identiﬁed as the two

states of an isospin doublet: p . (2.3) n + 0 The three pions π , π , and π− can be identiﬁed as the three states of an isospin

triplet:

π+ ud¯ − π0 = 1 uu¯ dd¯ , (2.4)    √2 −  π− du¯    

25 where the three components of the triplet correspond to the following isospin states:

I =1, I = +1 | 3 i I =1, I = 0 . (2.5)  3  |I =1, I = 1i | 3 − i   The Charm Meson Doublets

The charm quark c and the charm antiquarkc ¯ have isospin 0. One can therefore attach them to the isospin doublets of light quarks in Eq. (2.1) to obtain isospin doublets of qc¯ and cq¯, which are the quark contents of charm mesons:

uc¯ cd¯ , − . (2.6) dc¯ cu¯

P The isospin doublets of charm mesons with J =0− are

D¯ 0 D+ , − 0 . (2.7) D− D

P The isospin doublets of charm mesons with J =1− are

0 + D¯ ∗ D∗ , − 0 . (2.8) D∗− D∗ 2.2 Charge Conjugation and G Parity

Charge conjugation Cˆ is an operator that changes quarks into their antiquarks and vice versa. The phase convention for charge conjugation applied to quarks in this thesis is such that

Cqˆ = +¯q, (2.9a)

Cˆq¯ = + q. (2.9b)

Thus the operator satisﬁes Cˆ2 = 1.

26 Ordinary mesons composed a quark and an antiquark of the same ﬂavor are eigenstates of the charge conjugation transformation. For example, the J/ψ, which is composed of cc¯, has charge conjugation C = 1 (odd). The π0, which has quantum − P C + 0 P C numbers J =0− , and the ρ , which has quantum numbers J =1−−, are composed of admixtures of uu¯ and dd¯. Their charge conjugation quantum numbers are

C = +1 (even) and 1 (odd), respectively: − Cπˆ 0 = +π0, (2.10a)

Cρˆ 0 = ρ0. (2.10b) − Charge conjugation transform a meson with net quark ﬂavor into its antiparticle up to multiplication by a phase factor of +1 or 1. If the ﬂavored meson is in an − isospin multiplet with a neutral meson that is an eigenstate of Cˆ, a natural phase convention consistent with the convention set by Eqs. (2.9a) is to choose the phase factor to be the same as the charge conjugation eigenvalue of the neutral component.

For example, for the charged pions,

Cˆ π± =+ π∓, (2.11) and for the charged ρ mesons

Cˆ ρ± = ρ∓. (2.12) − For the charm mesons, we adopt the same phase conventions as for the light unﬂavored mesons with the same J P quantum numbers. The charge conjugation

P for charm mesons with J = 0− has the same sign as that for the pions given in

Eqs. (2.10a) and (2.11):

CDˆ 0 = +D¯ 0, (2.13a)

+ CDˆ = +D−. (2.13b)

27 P The charge conjugation transformation for charm mesons with J =1− has the same sign as that for the ρ mesons given in Eqs. (2.10b) and (2.12):

0 0 CDˆ ∗ = D¯ ∗ , (2.14a) − + CDˆ ∗ = D∗−. (2.14b) −

Moreover, it is possible to construct charm meson pairs which are eigenstates under

charge conjugation. A D∗D¯ pair and a D1D¯ ∗ pair with even charge conjugation are constructed in Section 3.3.

As the isospin can be viewed as an analogue of the spin, a discrete symmetry called G parity can be identiﬁed in analogy to the parity symmetry in conﬁguration

space. The G parity operator Gˆ can be deﬁned as the action of the charge conjugation

operator Cˆ succeeded by an isospin rotation relative to the original y axis by π:

iπTˆy Gˆ = e− C,ˆ (2.15)

where Tˆy is the rotation matrix for y axis in isospin space. G parity will be used

in Section 3.4 when we consider the scattering of D1 and D∗ mesons. The G parity

transformations for these charm mesons are

0 GDˆ ∗ = D∗−, (2.16a) − + 0 GDˆ ∗ = D¯ ∗ , (2.16b) − ˆ 0 GD1 = +D1−, (2.16c)

ˆ + ¯ 0 GD1 = +D1. (2.16d)

28 2.3 Masses of Charm Mesons

Masses

In this section, we introduce concise notations for the masses of charm mesons

0 + and pions. We denote the masses of the spin-0 charm mesons D and D by M0 and

0 + M1, respectively. We denote the masses of the spin-1 charm mesons D∗ and D∗ by

0 + M 0 and M 1, respectively. The masses of the pions π and π are denoted by m0 ∗ ∗

and m1, respectively. The subscripts on these masses indicate the absolute value of the electric charge of the mesons. The values of these masses are listed in Table 2.

Reduced Masses

We also introduce concise notations for simple combinations of the masses that are useful for expressing kinematics compactly. We denote the reduced mass of a spin-1 charm meson and a spin-0 charm meson by

M iMj M ij = ∗ , (i, j =0, 1). (2.17) ∗ M i + Mj ∗ We denote the reduced mass of a spin-0 charm meson and a pion by

Mimj mij = , (i, j =0, 1). (2.18) Mi + mj

Mass Diﬀerences

The isospin splittings between the charm meson masses are

M M = 4.79 0.10 MeV, (2.19a) 1 − 0 ±

M 1 M 0 = 3.3 0.57 MeV. (2.19b) ∗ − ∗ ±

+ 0 0 The energy splitting between the D∗ D− and D∗ D¯ thresholds is

ν (M 1 + M1) (M 0 + M0)=8.08 0.12 MeV. (2.20) ≡ ∗ − ∗ ± 29 The splittings between the charm meson masses in Eq. (2.19a), rather than the indi-

vidual masses of charm mesons in Table 2, have been used to calculate ν to obtain smaller error bars.

It is also useful to introduce shorthand notations for the diﬀerences between the

D∗ masses and Dπ thresholds:

δijk = M i Mj mk, (i, j, k =0, 1). (2.21) ∗ − −

The diﬀerences between the D∗ masses and the thresholds for Dπ states with the same electric charge are

δ = 7.14 0.07 MeV, (2.22a) 000 ± δ = 2.23 0.12 MeV, (2.22b) 011 − ± δ = 5.85 0.01 MeV, (2.22c) 101 ± δ = 5.66 0.10 MeV. (2.22d) 110 ±

0 The negative value for δ011 implies that D∗ is kinematically forbidden to decay into

+ D π−.

2.4 Decay Widths of D∗ Mesons

We denote the partial decay widths for D∗ Dπ by Γ ijk, where the subscripts → ∗ i, j, and k take values 0 and 1 and indicate the absolute value of the electric charge

of the D∗, D, and π mesons, respectively. Similarly, the partial decay widths for

D∗ Dγ are denoted by Γ iiγ . → ∗

The charged charm meson D∗± has a measured width of Γ [D∗±]= 96 22 keV. ± + 0 + + 0 + The decay modes of D∗ are D π , D π , and D γ. The measured branching

30 fractions are

+ 0 + Br[D∗ D π ] = (67.7 0.5)%, (2.23a) → ± + + 0 Br[D∗ D π ] = (30.7 0.5)%, (2.23b) → ± + + Br[D∗ D γ] = (1.6 0.4)%. (2.23c) → ±

+ The partial decay widths for D∗ are obtained by multiplying the branching fractions

+ by the width of D∗ :

+ 0 + Γ 101 Γ[D∗ D π ] = 65.0 14.9 keV , (2.24a) ∗ ≡ → ± + + 0 Γ 110 Γ[D∗ D π ] = 29.5 6.8 keV , (2.24b) ∗ ≡ → ± + + Γ 11γ Γ[D∗ D γ ] = 1.5 0.5 keV. (2.24c) ∗ ≡ → ±

0 There is only an upper bound on the width of the neutral charm meson D∗ :

0 0 0 0 0 Γ [D∗ ] < 2.1 MeV [2]. The decay modes of D∗ are D π and D γ. Their measured

branching fractions are

0 0 0 Br[D∗ D π ] = (61.9 2.9)%, (2.25a) → ± 0 0 Br[D∗ D γ] = (38.1 2.9)%. (2.25b) → ±

The T-matrix elements for the decay D∗ Dπ is proportional to the dot product → of the 3-momentum ~k of the π and the polarization vector ~ǫ(λ) of the D∗:

3 g [D∗ Dπ]= i Dπ D∗ ~k ~ǫ(λ). (2.26) T → − 2 fπ h | i · q The coeﬃcient has been expressed as a product of a constant, that has been denoted

as 3/2g/f , and a Clebsch-Gordan coeﬃcient Dπ D∗ . These Clebsch-Gordan π h | i p

31 coeﬃcients are proportional to the inner products of the isospin states of Dπ and D∗:

0 + + 1 1 1 1 D π D∗ = , , 1, +1 , + = 2/3, (2.27a) h | i h 2 − 2 | 2 2 i − + 0 + 1 1 1 1 p D π D∗ = , + , 1, 0 , + =+ 1/3, (2.27b) h | i h 2 2 | 2 2 i 0 0 0 1 1 1 1 p D π D∗ = , , 1, 0 , = 1/3. (2.27c) h | i h 2 − 2 | 2 − 2 i − p The partial width for the decays D∗ Dπ are obtained by squaring the T-matrix → element in Eq. (2.26), averaging over the spin states labeled by λ, and integrating

over the phase space of the Dπ. The resulting partial widths are

2 + 0 + 2√2g 5/2 3/2 Γ 101 Γ[D∗ D π ] = 2 m01 δ101 , (2.28a) ∗ ≡ → 3πfπ 2 + + 0 √2g 5/2 3/2 Γ 110 Γ[D∗ D π ] = 2 m10 δ110 , (2.28b) ∗ ≡ → 3πfπ 2 0 0 0 √2g 5/2 3/2 ∗ Γ 000 Γ[D D π ] = 2 m00 δ000 . (2.28c) ∗ ≡ → 3πfπ

The reduced masses mij and the mass differences δ ijk, which are defined in Eqs. (2.18) ∗ and (2.21), respectively, account for the differences in phase space for the various

decay modes. Γ 101 has an extra factor of 2 from isospin symmetry. Since δ011 < 0, ∗ 0 + D∗ does not decay into D π−.

The numerical value of the coupling constant g/fπ can be obtained either by comparing the experimental value for the partial width Γ 101 in Eq. (2.24a) with the ∗ theoretical value in Eq. (2.28a), or by comparing Eqs. (2.24b) and (2.28b) for Γ 110. ∗ + 0 + The more accurate one of the two determinations comes from the decay D∗ D π : →

4 3/2 g/f = (2.82 0.32) 10− MeV− . (2.29) π ± ×

0 0 0 Inserting this value into Eq. (2.28c), we can predict the partial width for D∗ D π : →

0 0 0 Γ 000 Γ[D∗ D π ] = 40.5 9.3 keV. (2.30) ∗ ≡ → ± 32 By dividing this prediction with the experimental value for the branching fraction

0 0 0 for D∗ D π in Eq. (2.25a), one obtains a prediction for the total width of the → 0 D∗ :

0 Γ[D∗ ]=65.5 15.4 keV. (2.31) ±

0 0 The prediction for the partial width for the decay D∗ D γ is obtained by multi- → plying this by the branching fraction in Eq. (2.25b):

0 0 Γ 00γ Γ[D∗ D γ] = 25.0 6.2 keV. (2.32) ∗ ≡ → ±

2.5 Energy-Dependent Width of Virtual D∗

In the treatment of the line shapes of the X(3872), it will be necessary to system-

atically account for the energy dependence of the width of a virtual D∗. A virtual

particle is one whose energy in its rest frame is not equal to its mass. Virtual particles

arise as intermediate states in quantum mechanical processes. The partial widths for

D∗ Dπ are fairly sensitive to the mass of the D∗, since they scale like the 3/2 power →

of the energy diﬀerence between the D∗ mass and the Dπ threshold. The sensitivity

of the width of the D∗ to its mass implies a sensitivity of the width of a virtual D∗

to its energy.

0 + A virtual D∗ (or D∗ ) with energy M 0 + E (or M 1 + E) can be considered as ∗ ∗

a D∗ whose rest energy diﬀers from its physical mass by the energy E. The width

of the virtual particle varies with E. We denote the energy-dependent widths of the

+ 0 D∗ and D∗ by Γ 1(E) and Γ 0(E), respectively. If E is small compared to mπ, ∗ ∗ | | these energy-dependent widths can be obtained simply by scaling the physical partial

33 widths for the decays D∗ Dπ: →

0 0 0 0 0 3/2 Γ 0(E) = Γ[D∗ D γ]+Γ[D∗ D π ] [(δ000 + E)/δ000] θ(δ000 + E) ∗ → → 5/2 3/2 +2(m11/m00) [(δ011 + E)/δ000] θ(δ011 + E) , (2.33a)

+ + + + 0 3/2 Γ 1(E) = Γ[D∗ D γ]+Γ[D∗ D π ] [(δ110 + E)/δ110] θ(δ110 + E) ∗ → → + 0 + 3/2 +Γ[D∗ D π ] [(δ + E)/δ ] θ(δ + E). (2.33b) → 101 101 101

The energy dependence of the decay widths into Dγ is ignored, because the photon energy and the phase space for the decays D∗ Dγ do not vary signiﬁcantly in the →

D∗D¯ threshold region. In Fig. 2.1, the energy-dependent widths Γ 0(E)andΓ 1(E ν) ∗ ∗ − as functions of E are plotted. The oﬀset ν 8.1 MeV in Γ 1(E ν) was chosen so ≈ ∗ − that Γ 0(E) and Γ 1(E ν) are the relevant widths for a D∗D¯ system consisting of ∗ ∗ − 0 0 a D∗ and a D¯ with total energy E relative to the D∗ D¯ threshold. Thus Γ 0(E) ∗ 0 + reduces to Γ[D∗ ] at E =0andΓ 1(E ν) reduces to Γ[D∗ ] at E = ν. The physical ∗ − 0 + widths Γ[D∗ ] and Γ[D∗ ] are shown in Fig. 2.1 as data points with error bars. At

0 0 + the D∗ D¯ threshold, the energy-dependent width of the D∗ is Γ 1( ν) 1.5 keV. ∗ − ≈ The individual terms in Eqs. (2.33) have obvious interpretations as energy-dependent

+ 0 partial widths for decays of D∗ and D∗ . We can deﬁne energy-dependent branching fractions by dividing these terms by Γ 1(E) or Γ 0(E). For example, the energy- ∗ ∗ 0 0 0 + + 0 dependent branching fractions for D∗ D π and D∗ D π are → →

0 0 0 Γ[D∗ D π ] 3/2 Br000(E) = → [(δ000 + E)/δ000] θ(δ000 + E), (2.34a) Γ 0(E) ∗ + + 0 Γ[D∗ D π ] 3/2 Br110(E) = → [(δ110 + E)/δ110] θ(δ110 + E). (2.34b) Γ 1(E) ∗

34 0.3

0.2 (MeV) *1 Γ , *0

Γ 0.1

0 -10 0 10 E (MeV)

0 Figure 2.1: The energy-dependent widths Γ 0(E) and Γ 1(E ν) for a virtual D∗ + ∗ ∗ − with energy M 0 + E and a virtual D∗ with energy M 1 + E ν, respectively, as ∗ ∗ − functions of E. The points with error bars at E = 0 and E = ν indicate the central 0 + values and uncertainties of the physical widths of D∗ and D∗ , respectively.

35 CHAPTER 3

CHARM MESON SCATTERING

In this chapter, we discuss aspects of the scattering of charm mesons that are relevant to the interpretation of the X(3872) and Z±(4430) as weakly-bound charm meson molecules.

3.1 Basic Scattering Formalism

In this section, the basic scattering formalism that is required to describe the low-energy scattering of two charm mesons is summarized.

Consider two distinguishable particles with masses m1 and m2 and reduced mass m12 = m1m2/(m1 + m2). The elastic scattering of two such particles with opposite momenta ~k and total kinetic energy E = ~2k2/m can be described by a station- ± 12 ary wave function ψ(r) that depends on the separation vector r of the two particles.

As r , the wave function approaches the sum of a plane wave and an outgoing → ∞ spherical wave:

eikr ψ(r)= eikz + f (θ) . (3.1) k r

This equation deﬁnes the scattering amplitude fk(θ), which depends on the scattering angle θ and the wave number k. The diﬀerential cross section dσ/dΩ can be expressed

36 in terms of the scattering amplitude as

dσ = f (θ) 2 . (3.2) dΩ | k |

The elastic scattering cross section σ can be obtained by integrating the diﬀerential

cross section over the entire 4π solid angle.

The partial-wave expansion resolves the scattering amplitude fk(θ) into contri-

butions from partial waves with deﬁnite angular momentum quantum number L by expanding it in terms of Legendre polynomials of cos θ:

1 ∞ f (θ)= (2L + 1)c (k)P (cos θ) . (3.3) k k L L L=0 X The coeﬃcients in Eq. (3.3) are constrained by unitarity to satisfy c (k) 1. Uni- | L | ≤ tarity guarantees that the total probability for all quantum processes is 1.

The unitarity constraints can be taken into account automatically by expressing

the coeﬃcients in terms of phase shifts δL(k):

iδL(k) cL(k)= e sin δL(k) . (3.4)

If there are no inelastic 2-body channels, the phase shifts δL(k) are real-valued. If

there are inelastic channels, the phase shifts can be complex-valued with positive

imaginary parts. The expression for the scattering amplitude in Eq. (3.3) written in

terms of the phase shifts is

∞ 2L +1 f (θ)= P (cos θ) . (3.5) k k cot δ (k) ik L L=0 L X − The expression for the elastic cross section integrated over the scattering angle is then

∞ (elastic) 4π 2 Im δL(k) 2 σ (E)= (2L + 1)e− sin δ (k) , (3.6) k2 | L | L=0 X 37 2 2 where E = ~ k /m12 is the total kinetic energy of the two particles. If there are

no inelastic scattering channels, this expansion for the elastic cross section can be

simpliﬁed by using the fact that the phase shift δL(k) are real:

4π ∞ σ(elastic)(E)= (2L + 1) sin2 δ (k) . (3.7) k2 L XL=0 The optical theorem relates the total cross section to the forward-scattering limit

(θ 0) of the scattering amplitude: → 4π σ(total)(E)= Im f (θ = 0) . (3.8) k k

If there are no inelastic scattering channels, the total cross section on the left side

of Eq. (3.8) is the elastic cross section in Eq. (3.7). If there are inelastic scattering

channels, the total cross section is the sum of the elastic and inelastic cross sections.

If the particles have short-range interactions, the phase shift δL(k) approaches zero like k2L+1 in the low-energy limit k 0 [43]. Thus S-wave (L = 0) scattering, → which is isotropic, dominates in the low-energy limit. At suﬃciently low energies,

2 the S-wave phase shift δ0(k) can be expanded in powers of k [44]. The expansion is called the eﬀective-range expansion and is conventionally expressed in the form

k cot δ (k)= 1/a + 1 r k2 + ... . (3.9) 0 − 2 s

The first two terms define the scattering length a and the S-wave effective range rs,

respectively. The inverse of scattering length is conventionally denoted as γ = 1/a.

The scattering length a can be equivalently deﬁned by the low-energy limit of the

scattering amplitude:

f (θ) a as k 0 . (3.10) k −→ − →

38 The absolute value of the scattering length can be determined by measuring the low-energy limit of the elastic cross section:

σ(elastic)(E) 8π a 2 as E 0 . (3.11) −→ | | →

To determine the sign of the scattering amplitude requires more complicated measurements involving interference eﬀects. If there are no inelastic channels at E = 0, the scattering length a is real-valued.

3.2 Universality of S-wave Resonances Near Thresholds

Particles with short-range interaction and large scattering length have universal features that are not sensitive to the structure of the particles and the details of their short-range interactions [45]. For two-particle scatterings, the eﬀective range rs can be used as a quantitative estimate of the range of the interaction. It determines a natural low-energy length scale r and a natural energy scale ~2/(m r2) for the | s| 12 s two-particle scattering. If

the scattering length a is much larger than the natural length scale r , and • | s|

the total kinetic energy E = ~2k2/(2m ) in the center-of-mass (CM) frame of • 12 ~2 2 the two particles is much smaller than the natural energy scale /(m12rs ), then the eﬀective range expansion in Eq. (3.9) is dominated by the contribution from

L = 0 partial wave, i.e. S-wave, which is the inverse scattering length γ =1/a:

k cot δ (k) γ . (3.12) 0 ≈ −

The higher order corrections to it are suppressed by powers of r /a 1. The partial | s |≪ wave expansion in Eq. (3.3) can be reduced to an expression that only depends on the

39 inverse scattering length γ and the total kinetic energy E up to corrections suppressed

by r /a : | s | 1 f(E)= . (3.13) γ i√2m E − − 12 The diﬀerential cross section in Eq. (3.2) can be simpliﬁed to

dσ 1 = 2 . (3.14) dΩ γ +2m12E

The cross section is obtained by integrating over the solid angle 4π. The wave function

for two-particle scattering states at long distances is a universal function of the scat-

tering length and the separation r. For r much larger than the natural low-energy

length scale r , the stationary wave function in the center-of-mass frame for two | s| 2 2 particles in an S-wave (L = 0) state with energy E = ~ k /(2m12) is

1 ψ(r)= sin [kr arctan(ka)] . (3.15) r −

The expression of scattering amplitude f(E) given in Eq. (3.13) can be generalized

to an expression deﬁned for complex values of the energy E:

1 f(E)= , (3.16) γ + κ(E) − where

κ(E)=( 2m E iε)1/2. (3.17) − 12 −

If E is real, κ(E) is real and positive for E < 0 and it is pure imaginary with a negative imaginary part for E > 0:

κ(E) = 2m E (E < 0), − 12 p = i 2m E (E > 0). (3.18) − 12 p 40 A derivation of the expression of f(E) given in Eq. (3.16) is included in Appendix A.

If γ > 0, the amplitude f(E) in Eq. (3.16) has a pole at a negative value of the energy E:

E = γ2/(2m ), (3.19) pole − 12

This pole indicates the existence of a stable bound state, whose binding energy is

2 much smaller than the natural low-energy scale ~ /(m12rs). The binding energy EX has a universal form depending only on the inverse scattering length up to corrections suppressed by r /a : | s | ~2γ2 EX = . (3.20) 2m12 The wave function of the bound state at separations r r is also universal: ≪| s|

1 r/a ψ (r)= e− . (3.21) X r

The size of the bound state can be estimated by the mean separation of the constituents:

r = a/2. (3.22) h i

The optical theorem in Eq. (3.8) can be generalized to other processes that produce

the two particles with variable energy E relative to the m1 + m2 threshold. As will

be justiﬁed in detail in Section 4.1, the energy distribution is proportional to the

imaginary part of f(E). If a > 0, Imf(E) has a delta-function contribution at

E = Epole in addition to a contribution above the threshold that is proportional to

√E:

πγ 2 2 Im f(E)= δ E + γ /(2m12) + θ(E) f(E) 2m12E, (γ > 0). (3.23) m12 | | p

41 The delta function at E < 0 results in a sharp resonance peak in the invariant mass distribution below the threshold. The √E term results in an enhancement above the

threshold. This energy distribution is illustrated in the upper panel of Figure 3.2.

If γ < 0, the pole in the amplitude f(E) is not on the real E axis, but on the

second sheet of the complex variable E:

E = ei2πγ2/(2m ). (3.24) pole − 12 This pole indicates the existence of a virtual state. A virtual state is not a physical

state, however it can also have a resonance peak in its invariant mass distribution.

The imaginary part of the amplitude is nonzero only above the threshold:

Im f(E)= θ(E) f(E) 2 2m E, (γ < 0). (3.25) | | 12 p This energy distribution is illustrated in the lower panel of Figure 3.2.

3.3 Scattering Channels for D∗D¯

In this section, we identify the two-body scattering channels for charm mesons

that are relevant to the interpretation of X(3872) as a charm meson molecule.

An important feature of the X(3872) is that its mass is extremely close to the

0 0 D∗ D¯ threshold. The CLEO Collaboration has recently made a precise measurement

0 0 0 of the D mass [37]. Taking this into account, the D∗ D¯ threshold is

M 0 + M0 = 3871.81 0.36 MeV. (3.26) ∗ ±

The PDG value for the mass MX of the X(3872) in Eq. (1.5) comes from combining

+ measurements of X in the J/ψπ π− decay mode [2]. The diﬀerence between the

0 0 PDG value for MX and the D∗ D¯ energy threshold given in Eq. (3.26) is

MX (M 0 + M0)= 0.6 0.6 MeV. (3.27) − ∗ − ± 42 Im f(E)

E Im f(E)

Figure 3.1: Cartoons of the imaginary part of the scattering amplitude Imf(E) of a S-wave resonance state near a 2-particle threshold for γ > 0 (upper panel) and γ < 0 (lower panel).

43 The negative central value in Eq. (3.27) is compatible with X(3872) being a shallow

0 0 bound state composed of charm mesons D∗ D¯ , i.e. a charm meson molecule. The

0 0 diﬀerence between the the D∗ D¯ threshold and the resonance peak mass of the

D0D¯ 0π0 threshold enhancement observed by Belle Collaboration [32] and that of the

0 0 D∗ D threshold enhancement observed by BaBar Collaboration [35] are

D0D¯ 0π0 +0.3 MX (M 0 + M0) = 4.1 0.7 1.6 MeV, (3.28a) − ∗ ± − D∗0D¯ 0 MX (M 0 + M0) = 4.3 1.1 0.5 MeV. (3.28b) − ∗ ± ±

D0D¯ 0π0 The result in Eq. (3.28a) is obtained from MX measured in Ref. [36] by subtracting 2M0 +(M 0 M0) and by dropping the error bar associated with M0. The ∗ − positive central values in Eqs. (3.28) are compatible with the X(3872) being a virtual

0 0 state of the charm mesons D∗ D¯ .

0 0 The fact that the mass of X(3872) is extremely close to the D∗ D¯ energy thresh-

0 0 0 0 old indicates that it has a resonant coupling to D∗ D¯ and its charge conjugate D D¯ ∗ .

+ The mass of the X(3872) is also fairly close to the D∗ D− energy threshold for the

+ + charged charm meson pairs D∗ D− and D D∗−, which are only about 8.1 MeV higher as given in Eq. (2.20). This implies that the X(3872) may also have a nonnegligible

+ + coupling to scattering channels composed of D∗ D− and D D∗−.

The established spin and parity quantum numbers J P =1+ of the X(3872) constrain the quantum numbers of the charm meson pairs D∗D¯ or DD¯ ∗. Since the D∗

P and D¯ have spin and parity quantum numbers J = 1− and 0−, respectively, their total spin is 1. For their total angular momentum quantum number to be 1, which is the spin of the X(3872), their orbital momentum quantum number must be L = 0.

Thus the X(3872) has an S-wave coupling to the charm mesons. The parity quantum

2+L number of a D∗D¯ or DD¯ ∗ charm meson pair is P =( 1) . The two factors of 1 − − 44 L are from the intrinsic parities of D and D∗ mesons, and the factor of ( 1) is from the − orbital factor of the spatial wavefunction of the charm meson pair. The two charm mesons in an S-wave (L = 0) state has even parity, which is in accordance with the parity of the X(3872).

The established charge conjugation parity C = + for the X implies that it can only couple to charm meson pairs that are even under charge conjugation. From the behaviors of charm mesons under charge conjugation given in Section 2.2, the behavior of charm meson pairs D∗D¯ under charge conjugation can be deduced:

0 0 0 0 Cˆ(D∗ D¯ ) = D D¯ ∗ , (3.29a) − 0 0 0 0 Cˆ(D D¯ ∗ ) = D∗ D¯ , (3.29b) − + + Cˆ(D∗ D−) = D D∗− , (3.29c) − + + Cˆ(D D¯ ∗−) = D∗ D− . (3.29d) −

The linear combinations that are even under change conjugation are:

0 1 0 0 0 0 (D∗D¯) + D∗ D¯ D D¯ ∗ , (3.30a) + ≡ √2 − 1 1 + + (D∗D¯) + D∗ D− D D∗− . (3.30b) + ≡ √2 − ¯ i The superscript i on (D∗D)+ is the absolute value of the electric charge of the charm

¯ 0 ¯ 1 mesons. We will refer to the charm meson scattering channels (D∗D)+ and (D∗D)+

as the neutral D∗D¯ channel and charged D∗D¯ channel, respectively.

The channels with isospin quantum numbers I = 0 and I = 1 can be formed

as the antisymmetric and symmetric linear combinations of the neutral and charged

45 channels, respectively:

I=0 1 0 1 (D∗D¯) = (D∗D¯) (D∗D¯) , (3.31a) + √2 + − + I=1 1 0 1 (D∗D¯) = (D∗D¯) +(D∗D¯) . (3.31b) + √2 + + ¯ 0 Therefore, the charm meson pair (D∗D)+ can also be written as a linear combination of isospin states

0 1 I=0 I=1 (D∗D¯) = + (D∗D¯) +(D∗D¯) , (3.32a) + √2 + +

1 1 I=0 I=1 (D∗D¯) = (D∗D¯) (D∗D¯) . (3.32b) + − √2 + − + 3.4 Scattering channels for D1D¯ ∗

In this section we identify the two-body scattering channels for charm mesons that

are relevant to the interpretation of Z±(4430) as a charm meson molecule.

The recently observed cc¯ meson Z±(4430) is a manifestly exotic meson. Its mea-

sured mass given in Eq. (1.11) is close to the energy threshold for the charm mesons

+ ¯ 0 ¯ D∗ and D1, which have quark content cd andcu ¯ and spin-parity quantum num-

P + bers J = 1− and 1 . The diﬀerence between charm meson masses are measured

more accurately than the masses themselves. Combining the PDG values of the mass

diﬀerences [2] with the recent precise measurement of the D0 mass by the CLEO

+ ¯ 0 Collaboration [37], the D∗ D1 threshold is at

MD0 + M 1 = 4432.2 0.9 MeV. (3.33) 1 ∗ ±

0 This is not a sharp threshold because the D1 has a width [2]

Γ[D0]=20.4 1.7 MeV. (3.34) 1 ±

46 + ¯ 0 0 + The D1 D∗ threshold may diﬀer by a few MeV from the D1D∗ in Eq. (3.33), but

0 the diﬀerence is negligible compared to Γ[D1]. The diﬀerence between the threshold in Eq. (3.33) and the measured mass of the Z±(4430) in Eq. (1.11) is

MZ MD0 + M 1 = +1 4 MeV. (3.35) − 1 ∗ ± The central value is positive corresponding to a virtual state, but the error bars are compatible with a bound state. The fact that the mass of Z±(4430) is fairly close

to the D1D¯ ∗ energy threshold indicates that it may have resonant coupling to charm

meson pairs D1D¯ ∗. The resonance is in an S-wave channel only if the spin-parity

P + + + quantum numbers of Z± are J =0 , 1 , or 2 .

+ In the discovery decay mode of Z±(4430), the decay products are ψ(2S)π . The

+ G isospin and G-parity quantum numbers of ψ(2S) and π are I =0− and 1−, respec-

tively. Thus the Z+ has quantum numbers IG =1+. The G parity for the Z implies

that it can only couple to charm meson pairs D1D¯ ∗ that are even under G parity.

From the trasformations of charm mesons under G parity given in Eq. (2.16d), the

transformations of charm meson pairs D1D¯ ∗ under G parity can be deduced:

+ 0 + 0 Gˆ(D∗ D¯ ) = D D¯ ∗ , (3.36a) 1 − 1 + 0 + 0 Gˆ(D D¯ ∗ ) = D∗ D¯ , (3.36b) 1 − 1 ˆ 0 0 ¯ 0 G(D1D∗−) = +D1D∗ , (3.36c)

ˆ 0 ¯ 0 0 G(D1D∗ ) = +D1D∗− . (3.36d)

47 The linear combinations that have even G parity form an isospin triplet:

+ 1 + 0 + 0 (D D¯ ∗) = D∗ D¯ D D¯ ∗ , (3.37a) 1 √2 1 − 1 0 1 0 0 0 0 + + (D D¯ ∗) = D D¯ ∗ D∗ D¯ + D D∗− D∗ D− , (3.37b) 1 2 1 − 1 1 − 1 1 0 0 (D D¯ ∗)− = D D∗− D∗ D− . (3.37c) 1 √2 1 − 1 The superscript on (D1D¯ ∗) is the total electric charge of the charm meson pair.

3.5 Single Neutral Scattering Channel

0 0 In this section, we consider the scattering of the charm mesons D∗ and D¯ in

0 0 the energy interval near the D∗ D¯ threshold. We will use approximations that are valid when E, which is the total energy of the charm mesons in the center-of-mass

0 0 (CM) frame relative to the D∗ D¯ threshold, is small compared to ν, the splitting

0 0 + before D∗ D¯ and D∗ D− energy thresholds. Since ν 8 MeV, this restricts the ≈ 0 0 energy E to within a few MeV of the D∗ D¯ threshold. The relative momenta of the

1/2 charm mesons is required to be small compared to (2M 00ν) 125 MeV. This is ∗ ≈ numerically comparable to the pion mass: m 135 MeV. π ≈ 0 0 The presence of the X(3872) resonance so close to the D∗ D¯ threshold with

0 0 0 0 quantum numbers that allow S-wave couplings to D∗ D¯ and D D¯ ∗ indicates that it is necessary to treat the interaction that produces the resonance nonperturbatively in order to take into account the constraints of unitarity. The resonance is in the

¯ 0 neutral channel (D∗D)+. In this section, the eﬀect from scattering in the charged

¯ 1 ¯ 0 channel (DD∗)+ is neglected compared to scattering in the neutral channel (DD∗)+.

Since the kinetic energy is so low, the scattering will be predominantly S-wave.

48 We express the transition amplitude (E) for the scattering of nonrelativistically A ¯ 0 normalized charm mesons in the channel (D∗D)+ in the form

2π (E)= f(E). (3.38) A M 00 ∗ The f(E) is the conventional nonrelativistic scattering amplitude as given in Eq. (3.16):

1 f(E)= , (3.39) γ κ(E) − − where

1/2 κ(E)=( 2M 00E iε) . (3.40) − ∗ − This expression for the low-energy scattering amplitude with an S-wave threshold

resonance is compatible with unitarity. If γ is complex, the imaginary part of the scattering amplitude f(E) in Eq. (3.39) satisﬁes:

Im f(E)= f(E) 2 Im [γ κ(E)] . (3.41) | | −

If γ was a real constant and if κ(E) was given by the simple expression in

Eq. (3.40), the scattering amplitude f(E) in Eq. (3.39) would satisfy the constraints

of unitarity for a single-channel system exactly. For positive real values of the energy

E, Eq. (3.41) would simply be the optical theorem for this single-channel system:

2 Im f(E)= f(E) 2M 00E, (E > 0). (3.42) | | ∗ p The left side is the imaginary part of the T-matrix element for elastic scattering in the

¯ 0 (D∗D)+ channel. The right side is the cross section for elastic scattering multiplied

1/2 by [E/(2M 00)] . ∗ ¯ 0 0 However, scattering in (D∗D)+ channel cannot be exactly unitary, because D∗

has a nonzero width and because the charm mesons have inelastic scattering channels.

49 0 0 0 0 0 0 The inelastic channels include D D¯ π and D D¯ γ, which are related to D∗ decays,

+ + 0 as well as all the decay modes of X(3872), which include J/ψπ π−, J/ψπ π−π , and

0 0 J/ψγ. The eﬀects of decays of D∗ and D¯ ∗ can be taken into account by replacing

κ(E) in the amplitudes in Eqs. (3.47) by

κ(E)= 2M 00[E + iΓ 0(E)/2]. (3.43) − ∗ ∗ p 0 0 At the D∗ D¯ threshold E = 0, the energy-dependent width Γ 0(E) reduces to the ∗ 0 physical width Γ[D∗ ]. The expression for κ(E) in Eq. (3.43) requires a choice of

branch cut for the square root. If E is real, an explicit expression for κ(E) that

corresponds to the appropriate choice of branch cut can be obtained by using the

identity

1/2 2M[E + iΓ/2] = √M E2 +Γ2/4 E − − p p 1/2 i E2 +Γ2/4+ E . (3.44) − p The eﬀects of inelastic scattering channels other than DDπ¯ and DDγ¯ can be

taken into account by replacing the real parameter γ by a complex parameter with a

positive imaginary part. The expression in Eq. (3.42) for the imaginary part of the

amplitude f(E) can now be interpreted as the optical theorem for the multi-channel

¯ 0 system consisting of (D∗D)+, and all the inelastic scattering channels. The right side

¯ 0 can be interpreted as the total cross section for scattering in the (D∗D)+ channel

1/2 multiplied by (2M 00E) /(4π). The term proportional to Im γ can be interpreted as ∗ the contribution from the inelastic scattering channels. This interpretation requires

Im γ > 0. The term proportional to Im κ(E) includes a contribution that reduces to −

the right side of Eq. (3.42) in the limits Im γ 0 and Γ 0 0. It can be interpreted → ∗ → 0 0 0 0 as due to elastic scattering into D∗ D¯ or D D¯ ∗ . If Re γ > 0, the term proportional

50 to Im κ(E) also includes a contribution that reduces to the right side of Eq. (3.23) in − the limits Im γ 0 and Γ 0 0. It can be interpreted as due to scattering into the X → ∗ → 0 0 resonance. The D∗ or D¯ ∗ produced by the elastic scattering process will eventually

0 0 decay. Similarly, the constituent D∗ or D¯ ∗ in the X resonance will eventually decay.

Thus the ultimate ﬁnal states corresponding to the term proportional to Im κ(E) in − 0 0 0 0 0 + 0 0 + Eq. (3.41) can be identiﬁed as D D¯ π and D D¯ γ, and also D D¯ π− and D D−π if the energy E exceeds the threshold δ =2.2 MeV. | 011| 3.6 Coupled Neutral and Charged Scattering Channels

In this section, we consider scattering of the charm mesons D∗D¯ in the D∗D¯

0 0 threshold region, which we take to be within 10 or 20 MeV of the D∗ D¯ threshold.

The results of Section 3.5 are generalized to the system consisting of the two coupled

¯ 0 ¯ 1 channels (D∗D)+ and (D∗D)+ deﬁned by Eqs. (3.30).

¯ 0 ¯ 1 The amplitudes for transitions between the channels (D∗D)+ and (D∗D)+ can be expressed as a 2 2 matrix (E), i, j 0, 1 . We ﬁrst write down a general × Aij ∈ { } expression for the transition amplitudes for S-wave scattering in the two channels that is compatible with unitarity in this two-channel system. A convenient way to parameterize these amplitudes is to express the inverse of the matrix of amplitudes

(E) in the form Aij

1 1 √M 00 0 γ00 + κ(E) γ01 (E)− = ∗ − − A 2π 0 √M 11 γ01 γ11 + κ1(E) ∗ − − √M 00 0 ∗ , (3.45) × 0 √M 11 ∗

1/2 1/2 where κ(E)=( 2M 00E iε) , κ1(E)=( 2M 11(E ν) iε) , and E is the total − ∗ − − ∗ − − 0 0 energy of the charm mesons in the center-of-mass (CM) frame relative to the D∗ D¯

51 threshold. The parameterization of the inverse matrix in Eq. (3.45) was chosen so

that the analytic expressions for the entries of (E) would be as simple as possible. Aij

It is convenient to deﬁne a matrix fij(E) of scattering amplitudes by

2π ij(E)= fij(E). (3.46) A M iiM jj ∗ ∗ p The entries of the matrix fij(E) are

2 1 γ01 − f00(E) = γ00 + κ(E) , (3.47a) − − γ11 + κ1(E) − 1 [ γ00 + κ(E)][ γ11 + κ1(E)] − f01(E) = γ01 + − − , (3.47b) − γ01 1 γ2 − f (E) = γ + κ (E) 01 . (3.47c) 11 − 11 1 − γ + κ(E) − 00

If the parameters γ00, γ01, and γ11 are complex, the imaginary parts of the scattering amplitudes in Eq. (3.47) satisfy

Im f (E) = f (E) f ∗ (E) Im [γ κ(E)] + f (E) f ∗ (E) Im [γ κ (E)] ij i0 j0 00 − i1 j1 11 − 1

+ fi0(E) fj∗1(E)+ fi1(E) fj∗0(E) Im [γ01] . (3.48) Since Imfij(E) is real, an alternative form for this unitarity equation can be obtained

by taking the complex conjugate of the right side.

There are two limits in which the amplitudes f01(E) and f11(E) vanish and f00(E)

reduces to the single-channel amplitude in Eq. (3.39). The ﬁrst limit is ν + , → ∞ which corresponds to increasing the energy gap between the two thresholds. In this

case, f (E) reduces to Eq. (3.39) with γ = γ . The second limit is γ ,γ 00 00 01 11 → ∞ 2 with γ01/γ11 ﬁxed, which corresponds to decreasing the interaction strength between

the two channels. In this case, f00(E) again reduces to Eq. (3.39) but with γ =

γ γ2 /γ . 00 − 01 11 52 If γ00, γ01, and γ11 were all real constants and if κ(E) and κ1(E) were given by

the simple expressions after Eq. (3.48), the amplitudes fij(E) in Eqs. (3.47) would

satisfy the constraints of unitarity for this two-channel system exactly. For positive

real values of the energy E, the expressions in Eqs. (3.48) for the imaginary parts of

the amplitudes f00(E) and f11(E) are just the optical theorems for this two-channel

system:

2 2 Im f00(E) = f00(E) 2M 00E + f01(E) 2M 11(E ν) θ(E ν),(3.49a) | | ∗ | | ∗ − − 2p 2p Im f11(E) = f01(E) 2M 00E + f11(E) 2M 11(E ν) θ(E ν).(3.49b) | | ∗ | | ∗ − − p p The left sides of Eqs. (3.49a) and (3.49b) are proportional to the imaginary parts

¯ 0 ¯ 1 of the T-matrix elements for elastic scattering in the (D∗D)+ and (D∗D)+ channels,

respectively. The ﬁrst and second terms on the right side of each equation are pro-

¯ 0 ¯ 1 portional to the cross sections for scattering into the (D∗D)+ and (D∗D)+ channels,

respectively. In the region E < 0, the imaginary parts of f00(E) and f11(E) may also

have delta function contributions analogous to the one in Eq. (3.23).

¯ 0 ¯ 1 However, scattering in the (D∗D)+ and (D∗D)+ channels cannot be exactly uni-

0 + tary, because the D∗ and D∗ have nonzero widths and because the charm mesons

have inelastic scattering channels. The inelastic channels include DDπ¯ and DDγ¯ ,

which are related to D∗ or D¯ ∗ decays, as well as all the decay modes of X(3872),

+ + 0 0 which include J/ψπ π−, J/ψπ π−π , and J/ψγ. The eﬀects of decays of D∗ and

0 D¯ ∗ can be taken into account by replacing κ(E) in the amplitudes in Eqs. (3.47) by

+ the expression in Eq. (3.43). Similarly the eﬀects of decays of D∗ and D∗− can be

taken into account by replacing κ1(E) in the amplitudes in Eqs. (3.47) by

κ1(E)= 2M 11[E ν + iΓ 1(E ν)/2]. (3.50) − ∗ − ∗ − p 53 + At the D∗ D− threshold E = ν, the energy-dependent width Γ 1(E ν) reduces ∗ − + to the physical width Γ[D∗ ]. The eﬀects of inelastic scattering channels other than

DDπ¯ and DDγ¯ can be taken into account by replacing the real parameters γ00, γ01,

and γ11 by a complex parameter with a positive imaginary part. The expression in

Eq. (3.48) for the imaginary part of the amplitude fii(E) can now be interpreted as the

¯ 0 ¯ 1 optical theorem for the multi-channel system consisting of (D∗D)+, (D∗D)+, and all

the inelastic scattering channels. The right side can be interpreted as the total cross

¯ i 1/2 section for scattering in the (D∗D)+ channel multiplied by (2M iiE) /(4π). The ∗

terms proportional to Imκ(E) and Imκ1(E) are the cross sections for scattering into

¯ 0 ¯ 1 the (D∗D)+ and (D∗D)+ channels, respectively. The terms proportional to Imγ00,

Imγ01, and Imγ11 are the cross sections for inelastic channels. This interpretation

requires Imγ00 > 0 and Imγ11 > 0.

3.7 Constraints from Isospin Symmetry

The approximate isospin symmetry of QCD imposes constraints on the scattering

of charm mesons. Deviations from isospin symmetry can be treated as small pertur-

bations except at low energies that are comparable to the isospin splittings between

hadron masses, which in the case of charm hadrons are less than 5 MeV. In strong

interaction processes, isospin-symmetry-violating eﬀects come primarily from hadron

mass diﬀerences. Exact isospin symmetry would require the masses of the charged

charm mesons to be equal to those of their neutral counterparts, which implies ν =0

and M 11 = M 00. It would also require the inverse matrix of amplitudes in Eq. (3.45) ∗ ∗ to be diagonal in the isospin basis. These conditions can be expressed as

γ00 + κ(E) γ01 γ0 + κ(E) 0 U − − U † = − , (3.51) γ γ + κ(E) 0 γ + κ(E) − 01 − 11 − 1 54 where γ0 and γ1 are the inverse scattering lengths in the isospin-symmetry limit for the I = 0 and I = 1 channels, respectively, and U is the unitary matrix associated with the transformation between the charged/neutral basis in Eqs. (3.30) and the isospin basis in Eqs. (3.31). The conditions in Eq. (3.51) imply

γ00 = (γ1 + γ0)/2, (3.52a)

γ = (γ γ )/2, (3.52b) 01 1 − 0

γ11 = (γ1 + γ0)/2. (3.52c)

The constraints on the amplitudes fij(E) from the approximate isospin symmetry of

QCD are obtained by inserting these values for the parameters into Eqs. (3.47).

In terms of the parameters γ0 and γ1, the scattering amplitudes in Eqs. (3.47)

reduce to

(γ + γ )+2κ (E) f (E) = − 0 1 1 , (3.53a) 00 D(E) γ γ f (E) = 1 − 0 , (3.53b) 01 D(E) (γ + γ )+2κ(E) f (E) = − 0 1 , (3.53c) 11 D(E)

where the denominator is

D(E)=2γ γ (γ + γ )[κ(E)+ κ (E)]+2κ (E)κ(E). (3.54) 1 0 − 1 0 1 1

The unitarity conditions in Eq. (3.48) can be written

Im f (E) = f (E) f ∗ (E) Im [γ + γ 2κ(E)] /2 ij i0 j0 1 0 −

+f (E) f ∗ (E) Im [γ + γ 2κ (E)] /2 i1 j1 1 0 − 1

+ f (E) f ∗ (E)+ f (E) f ∗ (E) Im [γ γ ] /2. (3.55) i0 j1 i1 j0 1 − 0 55 0 0 If there is a bound state or virtual state near the D∗ D¯ threshold with complex

energy Epole, the denominator D(E) given in Eq. (3.54) vanishes at that energy. If we deﬁne a variable γ by

γ = κ(Epole), (3.56)

the equation D(Epole) = 0 can be expressed as

γκ (E ) 1 (γ + γ ) [γ + κ (E )] + γ γ =0. (3.57) 1 pole − 2 1 0 1 pole 1 0

The variable γ can be identiﬁed with the inverse scattering length introduced in

Eq. (3.39). The energy Epole is given approximately by Eq. (4.13). If we neglect the

small diﬀerence between κ1(Epole) and κ1(0), one can obtain an approximate solution

of Eq. (3.57) for γ0 in terms of γ1 and γ:

γ κ (0) + γ γ 2κ (0)γ γ 1 1 1 − 1 . (3.58) 0 ≈ 2γ κ (0) γ 1 − 1 − 0 0 If the energy E is within a few MeV of the D∗ D¯ threshold, the scattering am-

plitudes in Eqs. (3.53) can be simpliﬁed. If the small diﬀerence between κ1(E) and

κ1(0) is neglected, the denominator D(E) given in Eq. (3.54) reduces to

D(E) [γ + γ 2κ (0)][ γ + κ(E)] . (3.59) ≈ − 1 0 − 1 −

In the numerators, κ(E) and γ can be neglected compared to κ1(0), γ0, and γ1. The scattering amplitudes then reduce to

f (E) f(E), (3.60a) 00 ≈ γ0 γ1 f10(E) − f(E), (3.60b) ≈ γ1 + γ0 2κ1(0) γ +−γ f (E) 1 0 f(E), (3.60c) 11 ≈ γ + γ 2κ (0) 1 0 − 1

56 where f(E) is the single-channel scattering amplitude in Eq. (3.39). If γ is neglected compared to κ1(0) and γ1, the expression for γ0 in Eq. (3.58) reduces to

γ κ (0) γ 1 1 . (3.61) 0 ≈ 2γ κ (0) 1 − 1

Using this expression to eliminate γ0 in favor of γ1, the coeﬃcient of f(E) in the amplitudes fij(E) in Eqs. (3.60) can be factored into a term that depends on the channel i and a term that depends on the channel j:

f (E) c f(E) c , (3.62) ij ≈ i j where the coeﬃcients ci are given by

c0 = 1, (3.63a) γ c = 1 . (3.63b) 1 −γ κ (0) 1 − 1

The values of the two independent parameters γ0 and γ1 could be calculated using

potential models for heavy mesons with pion-exchange interactions. As pointed out

by Tornqvist in 1993, these models indicate that there should be D∗D¯ bound states

near threshold in several I = 0 channels, including the S-wave 1++ channel, but not

in any of the I = 1 channels [46]. Tornqvist could not predict whether the 1++ state was just barely bound or not quite bound, because his results depended on an ultraviolet cutoﬀ whose value was estimated to be the same as the corresponding ultraviolet cutoﬀ for the two-nucleon system [46]. He also could not predict whether

0 0 + the state would be closer to the D∗ D¯ threshold or the D∗ D− threshold, because his calculations were carried out in the isospin symmetry limit. With the discovery of the X(3872), the ambiguity associated with the ultraviolet cutoﬀ can be removed by

57 using the observed binding energy of the X(3872) to tune the value of the ultraviolet

cutoﬀ. One can then use the meson potential model to predict the binding energies

of other heavy meson molecules in both the charm sector and the bottom sector [47].

In the absence of explicit calculations of the parameters γ0 and γ1, one can still

use results of the meson potential model calculations in Ref. [46] to get some idea

of the likely values of these parameters. The bound state near threshold with I =0

and J P C = 1++ arises from the eﬀects of coupled S-wave and D-wave channels. In

the S-wave channel, the pion-exchange potential is not deep enough to give a bound

state. The D-wave interaction provides just enough additional attraction to obtain a

bound state very near threshold. Thus we expect γ to be signiﬁcantly smaller than | 0| the natural scale mπ. The sign of γ0 could be either positive or negative. The meson potential model calculations in Ref. [46] indicate that there is no bound state with

I = 1 and J P C = 1++. The pion-exchange potential has the opposite sign as in the

I = 0 case, so it is repulsive. We therefore expect γ1 to be positive and comparable

to or larger than the natural scale mπ. In particular, γ1 should be much larger than

γ . Given an estimate of γ , an estimate of γ is actually superﬂuous because it can | 0| 1 0 be determined using Eq. (3.58).

The scattering amplitudes fij(E) in Eqs. (3.53) simplify if the parameter γ1 is

assumed to be large compared to κ1(0). The denominator D(E) given in Eq. (3.54) reduces to

D(E) γ [ 2γ + κ(E)+ κ (E)] . (3.64) ≈ − 1 − 0 1

The scattering amplitudes reduce to

1 1 1 fij(E) − . (3.65) ≈ 2γ + κ (E)+ κ(E) 1 1 − 0 1 − ij

58 The matrix projects onto the I = 0 channel. The denominator in Eq. (3.64) vanishes at Epole. If the small diﬀerence between κ1(Epole) and κ1(0) is neglected, we get an approximate expression for γ0 in terms of the variable γ deﬁned by Eq. (3.56):

κ (0) + γ γ 1 . (3.66) 0 ≈ 2

59 CHAPTER 4

0 0 LINE SHAPES OF X(3872) NEAR D D¯ ∗ THRESHOLD

0 0 In this chapter, the line shapes of the X(3872) are studied in the D∗ D¯ threshold

0 0 region, which extends to a few MeV from the threshold. In the D∗ D¯ threshold region, it is suﬃcient to take into account explicitly a single scattering channel for neutral charm mesons. The analysis presented in this chapter was published in Physics

Review D [48]. Here I present a slightly modiﬁed version to be consistent with this dissertation.

4.1 Factorization Formulas with Neutral Channel

If X(3872) is a loosely bound charm meson molecule, the separation of its constituents is of the same magnitude as the large scattering length a. Large separation of the constituents of a charm meson molecule will suppress its decay modes that occur at short distances through quark-quark and quark-gluon interactions. Such decay

+ + 0 modes of X(3872) that have been observed are J/ψπ π−, J/ψπ π−π , and J/ψγ.

These decay modes can occur if the two constituents of the X(3872) are close to each other and are suppressed by the large scattering length. Therefore, they are called the short-distance decay modes of X(3872). The decay modes of the charm meson molecule that occur through the decays of its constituents are not suppressed by the

60 large scattering length. Such decay modes of X(3872) that have been observed are

D0D¯ 0π0 and D0D¯ 0γ. These decay modes receive contributions from decays of the

0 0 0 0 0 constituent D∗ (or D¯ ∗ ), whose predominant decay modes are D π and D γ. These decay modes are called long-distance decay modes of X(3872) in this thesis. All the

decay modes of X(3872) are inelastic scattering channels for the charm mesons D0

0 and D∗ . The dominant eﬀects of these inelastic scattering channels on charm meson scattering can be taken into account through simple modiﬁcations of the variables γ and κ(E) in the resonant amplitude f(E) in Eq. (3.16).

If a set of particles C has total quantum numbers that are compatible with those of the X(3872) resonance and if the total energy E of these particles is near the

0 0 D∗ D¯ threshold, then there will be a resonance in the channel C. The line shape of

X(3872) in the channel C is the diﬀerential decay rate for X C as a function of → the total energy E of the particles in C. The line shapes of the X(3872) for energy

0 ¯ 0 ¯ 0 E close to the D∗ D threshold require considering the resonant channel (D∗D)+ defined in Eq. (3.30a). Throughout this chapter, effects from the charged channel ¯ 1 D∗D + defined in Eq. (3.30b) are ignored. The resulting approximation to the line shapes is good provided that E is small compared to the energy ν 8.1 MeV of the ≈ + D∗ D− threshold. This approximation is not necessarily bad for larger values of E.

The line shapes may not be predicted accurately in this region but the approximation

0 0 predicts correctly that they are small compared to in the D∗ D¯ threshold region.

If the width of the X is suﬃciently small, there is a clear qualitative diﬀerence between the line shapes of X in its short-distance and long-distance decay modes.

We ﬁrst consider the D0D¯ 0π0 decay mode, which has a contribution from the decay

0 0 0 0 of a constituent D∗ . If the X was a charm meson molecule, its line shape in D D¯ π

61 0 0 would consist of a Breit-Wigner resonance below the D∗ D¯ threshold and a threshold

0 0 enhancement above the D∗ D¯ threshold. If the X was a virtual state, there would

0 0 only be the threshold enhancement above the D∗ D¯ threshold. We next consider

0 decay modes that have no contributions from the decay of a constituent D∗ , such as

+ J/ψπ π−. If the X was a charm meson molecule, its line shape in such a decay mode

0 0 would be a Breit-Wigner resonance below the D∗ D¯ threshold. If the X was a virtual

0 0 state, there would only be a cusp at the D∗ D¯ threshold. Increasing the width of

the X provides additional smearing of the line shapes. This makes the qualitative

diﬀerence between the line shapes of a charm meson molecule and a virtual state

less dramatic. To discriminate between these two possibilities therefore requires a

quantitative analysis.

We consider the inclusive diﬀerential decay rate of X(3872) into all channels that

¯ 0 + are enhanced by the (D∗D)+ resonance. The optical theorem for the width of the B

is: 1 Γ[B+]= Im [B+ B+], (4.1) −MB A → where [B+ B+] is the one-meson-irreducible forward amplitude for B+. This A → amplitude has contributions from intermediate states consisting of a K+ recoiling

0 0 against sets of particles whose invariant mass M 0 + M0 + E is near the D∗ D¯ ∗ threshold. There is resonant enhancement for small E if the particles are accessible

¯ 0 from the (D∗D)+ channel. The resonant contributions to the forward amplitude can

+ be expressed as a loop integral over the 4-momentum PK of the K :

4 + + d PK K+ K+ i + + res[B B ]= 4 B f(E) B 2 2 . (4.2) A → − (2π) C C PK mK + iε Z −

62 There is an implicit restriction of the integral to the region of small E. The expression inside the parentheses takes into account the amplitude for the creation of charm

¯ 0 mesons in the channel (D∗D)+, the resonant propagation of the pair of charm mesons, and the amplitude for their annihilation. Factorization has been used to express it as the product of a long-distance factor and two short-distance factors. The long-

¯ 0 distance factor f(E) is the scattering amplitude for elastic scattering in the (D∗D)+ channel given in Eq. (3.16) with κ(E) given in Eq. (3.40). The short-distance factors

K+ + + CB+ depend on the 4-momenta PB and PK of the B and K , but they are insensitive

2 2 to the small energy E deﬁned by (PB PK) =(M 0 + M0 + E) . The short-distance − ∗ factors can therefore be simpliﬁed by setting E = 0. Using the Cutkosky cutting rules, the resonant contribution to the imaginary part of the forward amplitude can be written

3 + + d PK K+ K+ + + ∗ Im res[B B ]= 3 B Im f(E)( B ) . (4.3) A → − (2π) 2EK C C Z Again there is an implied restriction of the integral to the region of small E. The contribution to the width of B+ from its decay into K+ and the X(3872) resonance can be obtained by inserting Eq. (4.3) into Eq. (4.1). The distribution in the invariant mass M = M 0 + M0 + E of the resonance can be obtained by inserting the identity ∗

1= d4P δ4(P P P ) dM 2 δ(M 2 P 2 ). (4.4) R B − K − R − R Z Z

Changing the order of integration and using EB EK > 0 and E M 0 + M0, this − | |≪ ∗ can be written

3 M 0 + M0 d PR 4 4 1= ∗ dE (2π) δ (P P P ). (4.5) π (2π)32E B − K − R Z Z R

63 Upon inserting this into Eq. (4.3), we obtain a factorization formula for the inclusive

energy distribution summed over all resonant channels:

dΓ + + K+ [B K + resonant] = 2Γ + Im f(E). (4.6) dE → B

The short-distance factor is a positive real constant:

3 3 K+ M 0 + M0 d Pk d PK 4 4 K+ 2 Γ + = ∗ (2π) δ (P P P ) + . (4.7) B 2πM (2π)32E (2π)32E B − k − R |CB | B Z R Z R A more explicit expression for the short-distance factor can be obtained by using

K+ Lorentz invariance to express the short-distance factor + in the form CB

K+ K+ + = C + P (ǫ ∗ )∗, (4.8) CB B B · D

+ ∗ 0 K where ǫD is a polarization vector for the D∗ [49, 50] and CB+ is a constant with dimensions of inverse mass. Evaluating the phase space integral and summing over

0 the D∗ spins, we get

3/2 2 K+ λ (MB, mK , M 0 + M0) K+ + ∗ + ΓB = 2 3 CB . (4.9) 64π (M 0 + M0)MB ∗

The optical theorem in Eq. (3.41) can be used to resolve the inclusive resonant

rate in Eq. (4.6) into two terms according to whether they have Imγ or Imκ(E) as

a factor. We interpret the term proportional to Imγ as the contribution from all

short-distance decay channels C. The imaginary part of γ can be expressed as a sum

over those decay channels:

Imγ = ΓC (E). (4.10) C X We have allowed for the possibility that the dependence of some of the short-distance

C 0 0 factors Γ (E) on the energy E may not be negligible in the D∗ D¯ threshold region.

64 Thus the energy distribution in a speciﬁc short-distance channel C can be expressed

dΓ + + K+ 2 C [B K + C]=2Γ + f(E) Γ (E). (4.11) dE → B | |

The term in Eq. (4.6) proportional to Imκ(E) can be interpreted as the contribution

0 0 0 0 0 from channels that correspond to D∗ D¯ or D D¯ ∗ followed by the decay of the D∗

0 0 0 0 or D¯ ∗ . This term can be resolved into the contributions from the channels D D¯ π ,

+ 0 0 + 0 0 D D¯ π−, D D−π , and D D¯ γ by multiplying it by the energy-dependent branching

1 1 fractions Br000(E), 2 Br011(E), 2 Br011(E), and Br00γ (E), which add up to 1. A simple expression for Imκ(E) can be obtained by using the identity in Eq. (3.44). The

resulting expression for the energy distribution in the D0D¯ 0π0 channel is

dΓ + + 0 0 0 K+ 2 [B K + D D¯ π ]=2Γ + f(E) dE → B | | 1/2 2 2 M 00 E +Γ 0(E) /4+ E Br000(E), (4.12) × ∗ ∗ h p i where Br000(E) is given in Eq. (2.34a).

The energy distributions of the X resonance in the decays B0 K0 +X are given → by expressions identical to those in Eqs. (4.6), (4.11), and (4.12) except that the short-

K+ K0 distance constant ΓB+ is replaced by ΓB0 . Thus the line shapes for X(3872) produced in B+ decays and B0 decays are predicted to be identical in the region within a few

0 0 MeV of the D∗ D¯ threshold.

4.2 The Mass and Width of the X(3872)

The scattering amplitude f(E) given by Eq. (3.16) with κ(E) deﬁned in (3.43) is a double-valued function of the complex energy E with a square-root branch point and a pole. If we neglect the energy dependence of the width Γ 0(E), the branch ∗

65 0 point is near E = iΓ[D∗ ]/2 and the position of the pole is − 2 γ i 0 Epole Γ[D∗ ]. (4.13) ≈ −2M 00 − 2 ∗ If Re γ > 0, the pole is on the physical sheet of the energy E. It can be expressed in the form

E = E iΓ /2, (4.14) pole − X − X

where EX and ΓX are given by

2 2 EX (Re γ) (Im γ) /(2M 00), (4.15a) ≈ − ∗ 0 ΓX Γ[D∗ ] + 2(Re γ)(Im γ)/M 00. (4.15b) ≈ ∗

If E > 0 and Γ E , the state X is a resonance whose line shape in the region X X ≪ X E E E is a Breit-Wigner resonance centered at energy E with full width | − X |≪ X − X at half maximum ΓX . IfΓX is not small compared to EX , one can choose to deﬁne the binding energy and the width of X to be given by the expressions for EX and ΓX

in Eqs. (4.15), but they should not be interpreted literally. If Re γ < 0, the pole at the

energy Epole in Eq. (4.13) is on the second sheet of the energy E and it corresponds to a virtual state. In this case, the expressions for EX and ΓX in Eqs. (4.15) have no simple physical interpretations.

4.3 Short Distance Decay of X(3872)

In Ref. [52], the decay rates of X(3872) into J/ψ plus pions and photons were

calculated under the assumption that these decays proceed through couplings of the

X(3872) to J/ψ and the vector mesons ρ0 and ω. The results of Ref. [52] can be

used to calculate the dependence of the factor ΓC (E) in Eq. (4.11) on the energy

+ + 0 0 E for C = J/ψπ π−, J/ψπ π−π , J/ψπ γ, and J/ψγ. The normalization of this

66 12

(0) 8 0 C

Γ − π + π

(E) / ψ π C J/

Γ 4 + − J/ψ π π

0 -20 -10 0 10 20 E (MeV)

Figure 4.1: Energy dependence of the ﬁnal-state factors ΓC (E) for the channels C = + + 0 C C J/ψπ π− and J/ψπ π−π . The factors Γ (E)/Γ (0) as functions of the energy E 0 0 relative to the D∗ D¯ threshold are shown as solid curves. The dashed curves are the corresponding factors used in Ref. [51].

factor, which we can take to be ΓC (0), can only be determined by measurements of the branching fraction of X(3872) into the ﬁnal state C. In Fig. 4.1, we plot the

C + + 0 energy dependence of the factors Γ (E) for C = J/ψ π π− and J/ψπ π−π . The

+ 0 larger variations for J/ψ π π−π are due to the width of ω being much smaller than that of ρ0. Very close to the threshold, the ﬁnal state factors can be approximated by the expressions

+ − ΓJ/ψ π π (E) (1 + a E)Γψ2π, (4.16a) ≈ 2 0 + − 0 ΓJ/ψ π π π (E) (1 + a E)Γψ3π, (4.16b) ≈ 3 0

67 1 1 where the coeﬃcients are a2 =0.0175 MeV− and a3 =0.115 MeV− The approxima-

tions in Eqs. (4.16a) and (4.16b) are accurate to within 1% for 8.5 MeV

and Γπ+π−π0J/ψ(E)/g in Ref. [51]. The additional dependence on E from the coupling

of the resonances to pions and from the integration over the phase space of the pi-

ons was not take into account. In the region E < 1 MeV, the factors in Ref. [51] | | C C + analogous to Γ (E)/Γ (0) diﬀer from ours by less than 1.4% for C = J/ψπ π− and

+ 0 by less than 3.7% for C = J/ψπ π−π . As illustrated in Fig. 4.1, the diﬀerences

are more substantial when E is 10 MeV or larger and they are particularly large for | | + 0 J/ψπ π−π .

4.4 Line Shapes of X(3872) in B Decays

0 0 In Figs. 4.2 and 4.3, we illustrate the line shapes for X(3872) near the D∗ D¯

0 threshold. We take into account the D∗ width, but we neglect the eﬀect on the

line shapes of inelastic scattering channels for the charm mesons. We show the line

shapes for three values of γ: +34, 0, and 34 MeV. For γ = +34 MeV, the peak of − the resonance is at E = 0.6 MeV, which is the central value of the measurement − in Eq. (3.27). In Fig. 4.2, we show the line shapes in a short-distance decay mode,

+ 0 + such as J/ψπ π−π or J/ψπ π−. The line shape is given by Eq. (4.11). We have

neglected the energy-dependence of the factor ΓC (E). The relative normalizations of

the curves for the three values of γ are determined by using the same short-distance

K+ C factors ΓB+ and Γ . For γ = +34 MeV, which corresponds to a bound state, the line shape is dominated by the Breit-Wigner resonance near E = 0.6 MeV. For γ = 34 − −

68 /d E Γ d

-2 -1 0 1 2 E (MeV)

0 0 Figure 4.2: The line shapes near the D∗ D¯ threshold for X(3872) decaying into a + + 0 short-distance channel, such as J/ψπ π− or J/ψπ π−π . The line shapes are shown for three values of γ: +34 MeV (solid line), 0 (dotted line), and 34 MeV (dashed line). −

MeV, which corresponds to a virtual state, the line shape has a cusp near E = 0

MeV. In Fig. 4.3, we show the line shapes in the D0D¯ 0π0 channel. The line shape is given by Eq. (4.12). The relative normalizations of the curves for the three values of

K+ γ are determined by using the same short-distance factor ΓB+ . For γ = +34 MeV, which corresponds to a bound state, the dominant features of the line shape are a

Breit-Wigner resonance near E = 0.6 MeV and a threshold enhancement for E > 0. − For γ = 34 MeV, which corresponds to a virtual state, the line shape has only the − threshold enhancement.

69 /d E Γ d

-2 -1 0 1 2 E (MeV)

0 0 0 0 0 Figure 4.3: The line shapes near the D∗ D¯ threshold for X(3872) in the D D¯ π channel. The line shapes are shown for three values of γ: +34 MeV (solid line), 0 (dotted line), and 34 MeV (dashed line). −

In an actual measurement, the line shapes shown in Figs. 4.2 and 4.3 would be smeared by the eﬀects of experimental resolution. In the Belle discovery paper, the

+ X(3872) signal in J/ψπ π− was ﬁt by a Gaussian with width 2.5 MeV, which is

compatible with the experimental resolution [25]. For the D0D¯ 0π0 enhancement near threshold, the signal was ﬁt by a Gaussian with width 2.24 MeV, which is again compatible with the experimental resolution [31]. In Fig. 4.4, we illustrate the eﬀects of experimental resolution by smearing the line shapes in Figs. 4.2 and 4.3 by a

Gaussian with width 2.5 MeV. At the values γ = 48, 34, 0, 34, and 48 MeV, − − the diﬀerences between the positions of the peaks in the D0D¯ 0π0 channel and the

short-distance channel are 0.45, 0.82, 2.0, 2.4, and 3.2 MeV, respectively. Thus, after

70 2 10

1 10

0 /dE 10 Γ d

-1 10

-2 10-20 -10 0 10 20 E (MeV)

Figure 4.4: The line shapes of X(3872) smeared by a Gaussian function with width 2.5 MeV to mimic the eﬀects of experimental resolution. The smeared line shapes in the short-distance channel (solid lines) and in the D0D¯ 0π0 channel (dashed lines) are separately normalized so that the curve for γ = 0 has a maximum value of 1. At E = 0 MeV, the order of both the solid lines and the dashed lines from top to bottom is γ = 48, 34, 0, 34, and 48 MeV. − −

+1.0 taking into account the eﬀects of experimental resolution, the 4.7 1.8 MeV diﬀerence − between the masses in Eqs. (3.28) and (3.27) is compatible with a positive value of γ and X(3872) being a loosely-bound charm meson molecule.

4.5 Fits to the energy distributions

+ 0 0 0 The line shapes of X(3872) in the J/ψπ π− and D D¯ π decay channels have been measured by the Belle and BaBar Collaborations for the production processes

B K +X. In this section, we ﬁt the Belle measurements for the production process → B+ K+ + X to a theoretical model for the line shapes that takes into account the → 71 0 width of the constituent D∗ as well as inelastic scattering channels for the charm mesons.

4.5.1 Experimental data

The Belle and BaBar Collaborations have both measured the energy distribution

+ + + + of J/ψπ π− near the X(3872) resonance for the decay B K +J/ψπ π− [25, 30]. → The Babar data has larger error bars, so we will consider only the Belle data. The

+ Belle data on the J/ψπ π− energy distribution is given in Fig. 2b of Ref. [25]. The

ﬁgure shows the number of events per 5 MeV bin as a function of MJ/ψ π+π− from

3820 MeV to 3920 MeV. If we use the CLEO measurement of the D0 mass and the

0 0 PDG value for the D∗ D mass diﬀerence, this corresponds to E extending from − 51.8 MeV to +48.2 MeV. In Ref. [51], Hanhart et al. used only the 8 bins extending − from 21.8 MeV to +18.2 MeV. They subtracted the linear experimental background − shown in Fig. 2b of Ref. [25] to get the 8 data points given in Table 4.1. Our values for the energies at the centers of the bins diﬀer from those in Ref. [51] by 0.2 MeV, − because they used the PDG values for the masses M0 and M 0 instead of using the ∗

CLEO value for M0 and the PDG value for M 0 M0. In our analysis, we have chosen ∗ − to omit the ﬁrst data point in Table 4.1 so that the data points are more symmetric about E = 0. We will use a theoretical model for the energy distribution that is essentially 0 at E = 19.3 MeV, so including this point would simply increase the − χ2 by a constant 0.57. The 7 data points used in our analysis are plotted in Fig. 4.5.

Note that there are only two bins in which the data diﬀers from 0 by signiﬁcantly more than one error bar.

72 E N ∆N 19.3 2.10 2.78 − − 14.3 1.10 2.89 − 9.3 −1.21 3.04 −4.3− 9.60 4.83 − 0.7 24.56 6.46 5.7 1.47 2.99 − 10.7 1.57 2.99 15.7 −0.58 3.25 − last 7 bins 28.23 10.54 20.7 0.42 3.46

+ Table 4.1: Belle data on the J/ψπ π− energy distribution: numbers of events N and their uncertainties ∆N in 5 MeV bins centered at the energies E. The numbers of events N were obtained from Fig. 2b of Ref. [25] by subtracting the linear experimental background. We used only the last 7 data points in our analysis.

The Belle Collaboration has measured the energy distribution of D0D¯ 0π0 near the X(3872) resonance for the decay B K + D0D¯ 0π0 [36]. The D0D¯ 0π0 energy → distribution is shown in Fig. 2a of Ref. [36]. The ﬁgure shows the events per 4.25

MeV bin as a function of M 0 ¯ 0 0 2M 0 M 0 from 0 MeV to 76.5 MeV. This D D π − D − π corresponds to E extending from 7.14 MeV to +69.36 MeV. In Fig. 2a of Ref. [36], − the data for B+ K+ + X and B0 K0 + X are combined in the same plot. The → → energy distributions for B+ decay and B0 separately were presented at the ICHEP

2006 conference [53]. In Ref. [51], Hanhart et al. used only the data for B+ → K+ + X in 5 bins extending from 2.89 MeV to +18.36 MeV. They subtracted the − combinatorial background to obtained the data points for those 5 bins. To account for the remaining experimental background, which is an increasing function of E, they added a background term to the theoretical expression for dΓ/dE and determined its coeﬃcient by ﬁtting to the data. That background term is weakly constrained

73 800 40 600 400 30 200 0 20 -5 -3 -1 1

events per 5 MeV bin 0

-20 -10 0 10 20 E (MeV)

+ + + Figure 4.5: Number of events per 5 MeV bin for B K + J/ψπ π− as a function + →0 0 of the total energy E of J/ψπ π− relative to the D∗ D¯ threshold. The data points, which are given in Table 4.1, were obtained by subtracting the linear experimental background from the data of the Belle Collaboration in Ref. [25]. The theoretical curves are the diﬀerential number distributions dN/dE in Eq. (4.19) multiplied by the 5 MeV bin width. The three curves correspond to the global minimum of χ2 2 (dashed curve), the local minimum of χ (solid curve), and the ﬁt ABelle of Ref. [51] (dotted curve). The inset shows the peaks of the distributions.

by the 5 data points. We have therefore chosen to subtract the total experimental

background instead of only the combinatorial background. The resulting data points

are given in Table 4.2. Our values for the energies at the centers of the bins diﬀer

from those in Ref. [51] by 0.34, because they used the PDG values for the masses − 0 0 M0 and M 0 to determine the D∗ D¯ threshold instead of using the CLEO value for ∗

M0 and the PDG value for M 0 M0. The ﬁrst data point in Table 4.2 was omitted ∗ − in the analysis of Ref. [51], because it would have given a constant contribution to

χ2 of 0.17. In our analysis, we have chosen to include this data point even though

74 E N ∆N 5.015 0.42 1.03 − 0.765 0.90 1.38 −3.485 11.58 4.13 7.735 1.35 2.82 11.985 1.50 3.30 16.235 0.89 3.09 − all 6 bins 14.86 6.96

Table 4.2: Belle data on the D0D¯ 0π0 energy distribution: numbers of events N and their uncertainties ∆N in 4.25 MeV bins centered at the energies E. The numbers of events N were obtained from Ref. [53] by subtracting the total experimental background. Only the last 5 data points were used in the analysis of Ref. [51].

its effect on our analysis is negligible. The 6 data points used in our analysis are plotted in Fig. 4.6. Note that there is only one bin in which the data differs from 0 by significantly more than one error bar.

The analysis of Ref. [51] also used the Belle measurement of the branching ratio

+ 0 + for decays of X(3872) into J/ψπ π−π and J/ψπ π− [31]:

+ 0 Br[X J/ψπ π−π ] → =1.0 0.4 0.3. (4.17) Br[X J/ψπ+π ] ± ± → − + 0 The signal region for J/ψπ π−π included only energies E within 16.5 MeV of 3872

MeV. Using the CLEO value for M0 and the PDG value for M 0 M0, this corresponds ∗ − to 16.3 MeV

+ We summarize our model for the X(3872) lines shapes in the channels J/ψπ π−,

+ 0 0 0 0 + + 0 J/ψπ π−π , and D D¯ π channels. The J/ψπ π− and J/ψπ π−π energy distributions are given by Eq. (4.11), while the D0D¯ 0π0 energy distribution is given in

75 20

events per 4.25 MeV bin 0 -20 -10 0 10 20 E (MeV)

Figure 4.6: Number of events per 4.25 MeV bin for B+ K+ +D0D¯ 0π0 as a function 0 0 0 0 →0 of the total energy E of D D¯ π relative to the D∗ D¯ threshold. The data points, which are given in Table 4.2, were obtained by subtracting the total experimental background from the data of the Belle Collaboration in Ref. [36]. The theoretical curves are the diﬀerential number distributions dN/dE in Eq. (4.20) multiplied by the 4.25 MeV bin width. The three curves correspond to the global minimum of χ2 2 (dashed curve), the local minimum of χ (solid curve), and the ﬁt ABelle of Ref. [51] (dotted curve).

Eq. (4.12). The scattering amplitude f(E) is given by Eq. (3.16) with Im γ replaced

+ − + − 0 by ΓJ/ψ π π (E)+ΓJ/ψ π π π (E):

+ − + − 0 1 f(E)= Re γ iΓJ/ψ π π (E) iΓJ/ψ π π π (E)+ κ(E) − . (4.18) − − − h i + − + − 0 The energy dependence of the functions ΓJ/ψ π π (E) and ΓJ/ψ π π π (E) are shown

+ − as solid lines in Fig. 4.1. Their normalizations are determined by Γψ2π ΓJ/ψ π π (0) 0 ≡ + − 0 and Γψ3π ΓJ/ψ π π π (0), which we treat as adjustable parameters. The function 0 ≡

κ(E) in Eq. (4.18) is given in Eq. (3.43). The functions Γ 0(E) and Br000(E) are ∗

76 given in Eqs. (2.33a) and (2.34a), respectively. Thus our model has 4 adjustable real

K+ ψ2π ψ3π parameters: Re γ, ΓB+ , Γ0 , and Γ0 . To translate the diﬀerential rates dΓ/dE into numbers of events in the Belle experiment, we follow the prescription used in Ref. [51]. The diﬀerential number of

+ J/ψπ π− events at the energy E is

+ − J/ψ π π + ~ dN + Nobserved τ[B ]/ [J/ψπ π−] = dE Br[B+ K+ + X] Br[X J/ψπ+π ] → → − dΓ + + + [B K + J/ψπ π−]. (4.19) ×dE →

J/ψ π+π− For the number of observed events, we use the central value in Ref. [25]: Nobserved = 35.7. For the product of branching fractions in the denominator, we use the central

5 0 0 0 value in Ref. [25], which is 1.3 10− . The diﬀerential number of D D¯ π events at × the energy E is

0 0 0 dN N D D¯ π τ[B+]/~ dΓ [D0D¯ 0π0]= observed [B+ K+ + D0D¯ 0π0]. (4.20) dE Br[B+ K+ + D0D¯ 0π0] dE → → D0D¯ 0π0 For the number of observed events, we use the central value in Ref. [36]: Nobserved = 17.4. For the branching fraction in the denominator, we use the central value in

4 Ref. [36], which is 1.02 10− . × Following Ref. [51], we take into account the Belle measurement of the branching ratio in Eq. (4.17) through a constraint on the parameters of our theoretical model.

We demand that

+ + + 0 Γ[B K + J/ψπ π−π ; E < 16.5 MeV] → | | =1.0. (4.21) Γ[B+ K+ + J/ψ π+π ; E < 16.5 MeV] → − | | ψ3π ψ2π This constraint determines the ratio Γ0 /Γ0 as a function of the parameters Re γ

ψ2π and Γ0 . If we combine the errors in Eq. (4.17) in quadrature, the error bar on the

right side of Eq. (4.21) is 0.5. We ignore this error bar and constrain the ratio in ± 77 20

16 2

χ 14

8 -60 -40 -20 0 20 40 60 Reγ (MeV)

Figure 4.7: The minimum χ2 for the 13 data points in Figs. 4.5 and 4.6 as a function 2 K+ ψ2π of Re γ. The χ has been minimized with respect to the parameters ΓB+ , Γ0 , and ψ3π 2 Γ0 subject to the constraint in Eq. (4.21). The dot is the χ for the ﬁt ABelle of Ref. [51].

Eq. (4.21) to be 1.0 so that our results can be compared more directly with those in

Ref. [51].6

4.5.3 Fitting procedure

One of the most important issues is whether the data on the X(3872) is compatible with it being a bound state (corresponding to Re γ > 0) or whether it must be a virtual state (corresponding to Re γ < 0) as advocated in Ref. [51]. We therefore analyze the Belle data by ﬁxing the parameter Re γ and minimizing the χ2 with respect to

K+ ψ2π ψ3π the other 3 adjustable parameters ΓB+ , Γ0 , and Γ0 subject to the constraint in

6In Ref. [51], the partial widths on the right side of Eq. (4.21) were integrated over the larger region E < 20 MeV, but this difference has a negligible effect on the analysis. | | 78 K+ 14 ψ2π ψ3π 2 fit Reγ Γ + 10 Γ Γ χ B × 0 0 global minimum +57.2 3.7 0.66 0.77 8.4 local minimum +16.1 8.3 2.4 2.2 10.1 A of Ref. [51] 48.9 38 0.93 0.71 19.1 Belle −

Table 4.3: Parameters (in units of MeV) and values of χ2 for three ﬁts to the subtracted Belle data: the global minimum of χ2, the local minimum of χ2, and the ﬁt ABelle of Ref. [51].

Eq. (4.21). The χ2 is the sum of 13 terms corresponding to the last 7 data points in

Table 4.1 and the 6 data points in Table 4.2. The constraint in Eq. (4.21) determines

ψ3π ψ2π 2 the ratio Γ0 /Γ0 for ﬁxed Re γ. In Fig. 4.7, we show the minimum value of χ with

K+ ψ2π respect to variations of the two remaining parameters ΓB+ and Γ0 as a function of Re γ. The global minimum is χ2 = 8.4 at Re γ = 57.2 MeV. There is also a local

minimum at Re γ = 16.1 MeV with χ2 = 10.1. Since Re γ > 0 for both the global

minimum and the local minimum, these ﬁts correspond to a bound state. For the

2 sake of comparison, the ABelle ﬁt of Ref. [51] gives χ = 19.1. The real part of the

inverse scattering length for this fit is Re γ = 48.9 MeV. Since this is negative, this − 2 fit corresponds to a virtual state. In Fig. 4.7, the value of χ for the ABelle fit is shown

as a dot that lies just above the line.

In Table 4.3, we list the parameters and the values of χ2 for the global minimum,

the local minimum, and the ﬁt ABelle of Ref. [51]. For the ﬁt ABelle, Reγ is the real part

ψ2π ψ3π − of the inverse scattering length, and Γ0 and Γ0 are the values of Γπ+π J/ψ(E)/g

and Γπ+π−π0J/ψ(E)/g at E = 0.

+ 0 0 0 The line shapes of X(3872) in the J/ψπ π− and D D¯ π decay channels for

various ﬁts are shown together with the Belle data in Figs. 4.5 and 4.6, respectively.

79 The line shapes corresponding to the local minimum, the global minimum, and the

ﬁt ABelle of Ref. [51] are shown as solid, dashed, and dotted lines, respectively. The data points in these ﬁgures should be compared to the average values of the curves over the appropriate bins centered on the data points. The global minimum of χ2

+ 0 0 0 is somewhat pathological in that for both J/ψ π π− and D D¯ π the integral of the line shape over a bin is largest not in the bin with the highest data point, but in the next lower bin. For the local minimum of χ2, the integrated line shape gives a good

+ ﬁt to the data point in the highest bin for J/ψπ π− but it is smaller than the data

point in the highest bin for D0D¯ 0π0 by more than two standard deviations.

80 CHAPTER 5

LINE SHAPES OF X(3872) IN DD¯ ∗ THRESHOLD REGION

In this chapter, the line shapes of the X(3872) are studied in the DD¯ ∗ threshold

0 0 region, which extends to 10 or 20 MeV from the D∗ D¯ threshold. In the DD¯ ∗

threshold region, it is necessary to take into account explicitly two scattering channels,

one involving the neutral charm mesons and the other involving the charged charm

mesons. The analysis presented in this chapter was published in Physics Review D

[54]. I present here a slightly modiﬁed version to be consistent with this dissertation.

5.1 Factorization Formulas with Coupled Neutral and Charged Channels

The factorization formulas in Section 4.1 can be generalized to the two-channel

case. We begin by generalizing the forward amplitude in Eq. (4.2). We have to take

into account the possibility of resonant scattering between any pair of the charged

and neutral channels. The amplitude can be written as

4 1 1 + + d PK K+,i K+,j i [B B ]= + f (E) + . (5.1) Ares → − (2π)4 CB ij CB P 2 m2 + iε i=0 j=0 K K Z X X − Following the same path as in Section 4.1, we ultimately arrive at a factorization formula for the inclusive energy distribution summed over all resonant channels: 1 1 dΓ + + K+,ij [B K + resonant] = 2 Γ + Im f (E). (5.2) dE → B ij i=0 j=0 X X 81 The short-distance factors are

3 3 K+,ij M 0 + M0 d PR d PK 4 4 Γ + = ∗ (2π) δ (P P P ) B 2πM (2π)32E (2π)32E B − K − R B Z R Z K K+,i K+,j + ( + )∗. (5.3) ×CB CB

K+,00 K+,11 K+,01 The short-distance factors ΓB+ and ΓB+ are positive real constants, while ΓB+ =

K+,10 (ΓB+ )∗ is a complex constant. Thus there are four independent real constants associated with the B+ K+ transitions. These constants satisfy the Schwarz inequality → 2 K+,01 K+,00 K+,11 Γ + Γ + Γ + . (5.4) B ≤ B B

The optical theorem in Eq. (3.55) can be used to resolve the inclusive resonant rate in Eq. (5.2) into four terms according to whether they have Imγ0, Imγ1, Imκ(E), or Imκ1(E) as a factor. We interpret the terms proportional to Imγ0 and Imγ1 as the contributions from short-distance decay channels C. The imaginary parts of γ0 and

γ1 can be expressed as sums over those decay channels:

C Imγ0 = Γ0 (E), (5.5a) C X C Imγ1 = Γ1 (E). (5.5b) XC The factorization formula for the energy distribution in a speciﬁc short-distance decay channel C is

dΓ [B+ K+ + C]= dE → 1 1 K+,ij C Γ + [f (E) f (E)][f ∗ (E) f ∗ (E)] Γ (E) B i0 − i1 j0 − j1 0 i=0 j=0 X X 1 1 K+,ij C + ΓB+ [fi0(E)+ fi1(E)][fj∗0(E)+ fj∗1(E)] Γ1 (E). (5.6) i=0 j=0 ! X X The terms in Eq. (5.2) proportional to Imκ(E) and Imκ1(E) also have simple interpretations. We interpret the term proportional to Imκ(E) as the contribution

82 0 0 0 0 0 from channels that correspond to D∗ D¯ or D D¯ ∗ followed by the decay of the D∗

0 or D¯ ∗ . We can resolve this term into the contributions from the individual channels

0 0 0 + 0 0 + 0 0 D D¯ π , D D¯ π−, D D−π , and D D¯ γ by multiplying it by the energy-dependent

1 1 branching fractions Br000(E), 2 Br011(E), and 2 Br011(E), and Br00γ (E), which add up to 1. For example, the line shape of X in the D0D¯ 0π0 decay mode is

1 1 dΓ + + 0 0 0 K+,ij [B K + D D¯ π ]=2 Γ + f (E)f ∗ (E) dE → B i0 j0 i=0 j=0 X X 1/2 2 2 M 00 E + (Γ 0(E)/2) + E Br000(E), (5.7) × ∗ ∗ h p i where Br000(E) is given in Eq. (2.34a). We interpret the term in Eq. (5.2) propor-

+ tional to Imκ1(E) as the contribution from channels that correspond to D∗ D− or

+ + D D∗− followed by the decay of the D∗ or D∗−. We can resolve this term into

+ 0 0 + + 0 the contributions from the individual channels D D−π , D D−π , D D¯ π−, and

+ D D−γ by multiplying it by energy-dependent branching fractions. For example,

+ 0 the line shape of X in the D D−π decay mode is

1 1 dΓ + + + 0 K+,ij [B K + D D−π ] = 2 Γ + f (E)f ∗ (E) dE → B i1 j1 i=0 j=0 X X 1/2 2 2 M 11 (E ν) + (Γ 1(E ν)/2) + E ν Br110(E), (5.8) × ∗ − ∗ − − h p i where Br110(E) is given in Eq. (2.34b). The expressions for the line shapes in the

0 + 0 + decay channels D D−π and D−D π are more complicated because they receive

0 0 0 0 contributions from channels that correspond to D∗ D¯ or D D¯ ∗ as well as channels

+ + that correspond to D∗ D− or D D∗−.

0 0 If the energy E is very close to the D∗ D¯ threshold, the two-channel factorization formulas in Eqs. (5.2), (5.6), and (5.7) should reduce to the single-channel factorization formulas in Eqs. (4.6), (4.11), and (4.12). For the factorization formula for

83 B+ K+ + D0D¯ 0π0 in Eq. (5.7), this can be veriﬁed by inserting the expressions in →

Eq. (3.62) for the scattering amplitudes fij(E) at small E. The factorization formula

K+ reduces to Eq. (4.12) with the short-distance factor ΓB+ given by

1 1 K+ K+,ij Γ + Γ + c c∗. (5.9) B ≈ B i j i=0 j=0 X X Similarly, the factorization formula for B+ K++C in Eq. (5.6) reduces to Eq. (4.11) → K+ C with ΓB+ given by Eq. (5.9) and Γ (E) given by

1 c 2 1+ c 2 ΓC (E) | − 1| ΓC (E)+ | 1| ΓC (E). (5.10) ≈ 2 0 2 1

To see that the two-channel factorization formula in Eq. (5.2) for the inclusive reso-

nant rate reduces to Eq. (4.6), we express the imaginary part of fij(E) in a form that

is compatible with the Cutkosky cutting rules:

Imfij(E)= ci f(E) (Imcj)+ ci (Imf(E)) cj∗ + (Imci) f ∗(E) cj∗. (5.11)

Since c0 = 1, it has no imaginary part. The expression for c1 in Eq. (3.63b) is a

function of γ1 and κ1(0) only. The imaginary part of κ1(0) is suppressed relative

+ to its real part by a factor of Γ[D∗ ]/ν. We expect γ1 to have a real part that is

comparable to or larger than mπ, so the imaginary part of γ1 should also be small relative to its real part. Thus the only term on the right side of Eq. (5.11) that is not suppressed is the one with the factor Imf(E). Inserting that term into the

K+ factorization formula in Eq. (5.2), we ﬁnd that it reduces to Eq. (4.6) with ΓB+ given by Eq. (5.9).

The factorization formulas for the energy distributions simplify if the parameter

γ1 is assumed to be large compared to κ1(0). The scattering amplitudes fij(E) in

Eq. (3.53) reduce to the expressions in Eq. (3.65). The two-channel factorization

84 formula in Eqs. (5.2), (5.6), and (5.7) all reduce to the single-channel factorization

formulas in Eqs. (4.6), (4.11), and (4.12) with the scattering amplitude f(E) replaced

by the expression for f00(E) given in Eq. (3.65). By using Eq. (3.66) to eliminate γ0

in favor of γ, the scattering amplitude reduces to

1 f(E) . (5.12) ≈ γ + κ(E)+ κ (E) κ (0) − 1 − 1 The short-distance factor for B+ K+ transitions reduces to → 1 1 K+ i+j K+,ij Γ + ( 1) Γ + . (5.13) B ≈ − B i=0 X XJ=0 ¯ i The sums project the (D∗D)+ channels onto isospin 0. The short-distance factor for the short-distance decay channel reduces to

ΓC (E) 2ΓC (E). (5.14) ≈ 0

C The coeﬃcient of Γ1 (E) goes to zero in this limit. Thus the decay of X into ﬁnal

+ states C with total isospin quantum number I = 1, such as J/ψπ π−, are suppressed in the large-γ1 limit.

5.2 Constraints from isospin symmetry

In this section, the approximate isospin symmetry of QCD will be exploited to constrain the scattering formalism for coupled neutral and charged scattering channels of charm mesons developed in Section 3.6. Since the short-distance factors only involve momenta of order mπ and larger, isospin-violating eﬀects can be neglected in these factors. Thus isospin symmetry can be used to constrain the short-distance factors. At the quark level, the transitions B K +D∗D¯ and B K +DD¯ ∗ proceed → → through two operators in the eﬀective weak Hamiltonian: the charged current operator

85 ¯bγµ(1 γ )c cγ¯ (1 γ )s and the neutral current operator ¯bγµ(1 γ )s cγ¯ (1 γ )c. − 5 µ − 5 − 5 µ − 5 These operators are both isospin singlets. Thus isospin symmetry is respected by

these transitions. It can therefore be used to relate the short-distance coeﬃcients

+ 0 K ,i + + K ,i + for the B K transition to the short-distance coefficients 0 for the CB → CB B0 K0 transition. Since B+ and B0 form an isospin doublet and K+ and K0 form → K0,i K+,i an isospin doublet, the coefficients 0 and + are related by Clebsch-Gordan CB CB coefficients:

K0,0 K+,1 0 = + , (5.15a) CB −CB K0,1 K+,0 0 = + . (5.15b) CB −CB

K0,ij This implies that the short-distance constants ΓB0 in the factorization formulas

+ 0 0 K ,ij for B K transitions are related to the corresponding constants Γ + in the → B factorization formulas for B+ K+ transitions by →

K0,00 K+,11 ΓB0 = ΓB+ , (5.16a)

K0,01 K+,01 ΓB0 = (ΓB+ )∗, (5.16b)

K0,11 K+,00 ΓB0 = ΓB+ . (5.16c)

Thus the short-distance constants associated with the B+ K+ and B0 K0 → → transitions are determined by four independent real constants.

C Isospin symmetry also constrains the short-distance factors ΓI (E) associated with

decays of X into short-distance decay modes. It implies that for a decay channel C

with deﬁnite isospin quantum number I =0 or I = 1, only the term with the factor

ΓI (E) contributes. An example of a decay channel with isospin quantum number

+ 0 + 0 I = 0 is J/ψπ π−π , assuming that the π π−π comes from the decay of a virtual

+ ω. An example of a decay channel with isospin quantum number I = 1 is J/ψπ π−,

86 + 0 assuming that the π π− comes from the decay of a virtual ρ . We will give the factorization formulas for short-distance decay channels with deﬁnite isospin quantum number I = 0 and I = 1 for both B+ K+ transitions and B0 K0 transitions. → → For a short-distance decay channel C with deﬁnite isospin quantum number I = 0,

+ 0 such as J/ψπ π−π , the energy distribution in Eq. (5.6) reduces to

dΓ + + K+,00 2 [B K + C] = 4 Γ + γ κ (E) dE → B | 1 − 1 | K+,01 2Re[Γ + (γ κ (E))(γ κ(E))∗] − B 1 − 1 1 − C K+,11 2 Γ0 (E) +Γ + γ κ(E) , (5.17a) B | 1 − | D(E) 2 | | dΓ 0 0 K+,00 2 [B K + C] = 4 Γ + γ κ(E) dE → B | 1 − | K+,01 2Re[Γ + (γ κ(E))(γ κ (E))∗] − B 1 − 1 − 1 C K+,11 2 Γ0 (E) +Γ + γ κ (E) . (5.17b) B | 1 − 1 | D(E) 2 | | For a short-distance decay channel C with deﬁnite isospin quantum number I = 1,

+ such as J/ψπ π−, the energy distribution in Eq. (5.6) reduces to

dΓ + + K+,00 2 [B K + C] = 4 Γ + γ κ (E) dE → B | 0 − 1 | K+,01 +2Re[Γ + (γ κ (E))(γ κ(E))∗] B 0 − 1 0 − C K+,11 2 Γ1 (E) +Γ + γ κ(E) , (5.18a) B | 0 − | D(E) 2 | | dΓ 0 0 K+,00 2 [B K + C] = 4 Γ + γ κ(E) dE → B | 0 − | K+,01 +2Re[Γ + (γ κ(E))(γ κ (E))∗] B 0 − 0 − 1 C K+,11 2 Γ1 (E) +Γ + γ κ (E) . (5.18b) B | 0 − 1 | D(E) 2 | | In Eqs. (5.17b) and (5.18b), we have used the isospin symmetry relations in Eqs. (5.16)

K0,ij K+,ij to express the short-distance coeﬃcients ΓB0 in terms of ΓB+ .

87 ¯ 1 The eﬀects of the charged charm meson channel (D∗D)+ on the line shapes of

+ + 0 X(3872) in the decays B K + J/ψπ π− and B K + J/ψπ π−π have been → → discussed recently by Voloshin [55]. Voloshin made conceptual errors by ignoring

¯ 0 ¯ 1 resonant scattering between the (D∗D)+ and (D∗D)+ channels and ignoring the constraints of isospin symmetry on the transitions B K. In Voloshin’s paper, our →

parameters γ0 and γ1 are denoted by κ0 and κ1 and the analogs of our functions κ(E)

and κ (E) are denoted by ik and κ . Voloshin took into account the constraints of 1 − n c + + 0 isospin symmetry associated with the J/ψπ π− and J/ψπ π−π in the ﬁnal state.

His results for the energy distributions can be expressed in the form

2 dΓ + 0 γ1 κ1(E) J/ψ ω [B K + J/ψπ π−π ] = − Φ , (5.19a) dE → D(E)

2 dΓ + γ0 κ1(E) J/ψ ρ [B K + J/ψ π π−] = − Φ , (5.19b) dE → D(E)

1/2 1/2 where κ(E)=( 2M 00E iε) and κ1(E)=( 2M 11 (E ν) iε) . The − ∗ − − ∗ − − normalizing factors ΦJ/ψ ρ and ΦJ/ψ ω can presumably be diﬀerent for B+ decays and

B− decays, although this was not stated explicitly in Ref. [55]. The line shapes

+ however were predicted to be the same for B decays and B− decays. Voloshin’s

results in Eqs. (5.19) correspond to speciﬁc choices for the short-distance factors

K+,ij ΓB+ in our general factorization formulas in Eqs. (5.17) and (5.18). In the case of B+ decays, his results in Eqs. (5.19) are consistent with our factorization formulas

K+,00 in Eqs. (5.17a) and (5.18a) if ΓB+ is the only nonzero short-distance factor for the B K transition. In the case of B0 decays, his results in Eqs. (5.19) are consistent → K+,11 K0,00 with our factorization formulas in Eqs. (5.17b) and (5.18b) if ΓB+ = ΓB0 is the only such nonzero factor. However these conditions for B+ decays and B0 decays are

inconsistent. Thus Voloshin’s results are incompatible with the constraints of isospin

88 symmetry associated with the B K transitions. The primary conceptual error in → Ref. [55] was the assumption that there is a resonance in the amplitude only if the

0 B K transition creates the charm mesons in the neutral channel (D∗D¯) . However → + there is also a resonant contribution coming from the B K transition creating → ¯ 1 charm mesons in the charged channel (D∗D)+ followed by the resonant scattering of

the charm mesons into the neutral channel. A second conceptual error in Ref. [55] was

the failure to take into account the constraints of isospin symmetry on the amplitudes

for the B K transition. → 5.3 Current-current factorization and heavy-quark symmetry

In Ref. [50], it was pointed out that the combination of a standard current-current

factorization approximation and heavy-quark symmetry could be used to simplify the

factorization formulas associated with the X resonance in B K transitions. In → the standard current-current factorization approximation, the matrix elements of the

relevant terms in the eﬀective weak Hamiltonian are expressed as products of matrix

elements of currents:

µ KD∗D¯ ¯bγ (1 γ )c cγ¯ (1 γ )s B h | − 5 µ − 5 | i ≈ µ D¯ ¯bγ (1 γ )c B KD∗ cγ¯ (1 γ )s 0 , (5.20a) h | − 5 | ih | µ − 5 | i µ KD∗D¯ ¯bγ (1 γ )s cγ¯ (1 γ )c B h | − 5 µ − 5 | i ≈ µ K ¯bγ (1 γ )s B D∗D¯ cγ¯ (1 γ )c 0 . (5.20b) h | − 5 | ih | µ − 5 | i

The D∗ and D¯ in the ﬁnal state can equally well be replaced by D and D¯ ∗. The

matrix element of the charged current ¯bγµ(1 γ )c in Eq. (5.20a) is nonzero only if − 5 + the D¯ contains the same light quark as the B. In the case of a B , the D¯ or D¯ ∗ must

89 0 0 0 be D¯ or D¯ ∗ . In the case of a B , the D¯ or D¯ ∗ must be D− or D∗−. As pointed out

in Ref. [50], heavy-quark symmetry implies that the matrix element of the neutral

currentcγ ¯ (1 γ )c in Eq. (5.20b) vanishes at the D∗D¯ threshold. Thus this matrix µ − 5

element is suppressed in the D∗D¯ threshold region. Putting these two observations

together, we conclude that the current-current factorization approximation together

with heavy quark symmetry puts strong constraints on the matrix elements of the

eﬀective weak Hamiltonian. It implies that in B+ K+ transitions, the formation → of the X(3872) resonance is dominated by the creation of charm mesons at short

0 0 0 distances in the neutral channel (D∗D¯) . Similarly, in B K transitions, the + → formation of the X(3872) resonance is dominated by the creation of charm mesons

¯ 1 at short distances in the charged channel (D∗D)+. These statements imply that the

K+,1 K0,0 K+,0 K0,1 short-distance coeﬃcients + = 0 are suppressed relative to + = 0 . CB −CB CB −CB This suppression leads to a hierarchy in the short-distance factors associated with

B K transitions in the factorization formulas: →

K+,11 K+,01 K+,00 Γ + Γ + Γ + . (5.21) B ≪| B |≪ B 5.4 Line Shapes of X(3872) in B Decays

K+,11 K+,01 K+,00 If we assume that Γ + and Γ + are negligible compared to Γ + , the B | B | B expressions for the line shapes of X(3872) become rather simple. For a short-distance

+ 0 decay channel C with deﬁnite isospin quantum number I = 0, such as J/ψπ π−π , the energy distributions in Eqs. (5.17) reduce to

2 dΓ + + K+,00 γ1 κ1(E) C [B K + C] 4Γ + − Γ (E), (5.22a) dE → ≈ B D(E) 0

2 dΓ 0 0 K+,00 γ1 κ(E) C [B K + C] 4Γ + − Γ (E). (5.22b) dE → ≈ B D(E) 0

90 For a short-distance decay channel C with deﬁnite isospin quantum number I = 1,

+ such as J/ψπ π−, the energy distributions in Eqs. (5.18) reduce to

2 dΓ + + K+,00 γ0 κ1(E) C [B K + C] 4Γ + − Γ (E), (5.23a) dE → ≈ B D(E) 1

2 dΓ 0 0 K+,00 γ0 κ(E) C [B K + C] 4Γ + − Γ1 (E). (5.23b) dE → ≈ B D(E)

For the D0D¯ 0π0 channel, the energy distribution in Eqs. (5.7) from the B+ K+ → transition and its analog from the B0 K0 transition reduce to → 2 dΓ + + 0 0 0 K+,00 γ1 + γ0 2κ1(E) [B K + D D¯ π ] 2Γ + − dE → ≈ B D(E)

1/2 2 2 M 00 E +Γ 0(E) /4+ E Br000(E ), (5.24a) × ∗ ∗ h i 2 dΓ 0 0 0p0 0 K+,00 γ1 γ0 [B K + D D¯ π ] 2Γ + − dE → ≈ B D(E)

1/2 2 2 M 00 E +Γ 0(E) / 4+ E Br000(E). (5.24b) × ∗ ∗ h p i Note that the line shapes in Eqs. (5.22), (5.23), and (5.24) are determined by the

parameters γ0 and γ1 or, equivalently, γ and γ1. The relative normalizations of the

rates from the B0 K0 transition and from the B+ K+ transition are also → →

determined by γ and γ1.

In Figs. 5.2, 5.3, and 5.1, we illustrate the line shapes in the D∗D¯ threshold region

0 for X(3872) produced by B K transitions. We take into account the D∗ width, → but we neglect the eﬀect on the line shapes of inelastic scattering channels for the

charm mesons. For simplicity, we show only the line shapes for the limiting case

γ . Thus the denominators D(E) can be approximated by Eq. (3.64) and 1 → ±∞ numerator factors such as γ κ (E) or γ κ(E) can be approximated by γ . The 1 − 1 1 − 1 parameters γ and γ are related by the pole equation 2γ γ κ (E ) = 0, where 0 0 − − 1 pole

Epole is given in Eq. (4.13). If we take γ0 to be real, then γ has an unphysical negative

91 /d E Γ d

-12 -8 -4 0 4 8 12 E (MeV)

Figure 5.1: The line shapes in the D∗D¯ threshold region for X(3872) produced by a B+ K+ or B0 K0 transition and decaying into D0D¯ 0π0. The line shapes are shown→ for γ = →and three values of γ: +34 MeV (solid line), 0 (dotted line), and 1 ±∞ 34 MeV (dashed line). − imaginary part. We therefore take γ to be real and use the pole equation to determine

the complex parameter γ0:

1 2 0 γ0 = 2M 11ν +(M 11/M 00)γ + iM 11(Γ[D∗ ] Γ 1( ν)) + γ . (5.25) 2 ∗ ∗ ∗ ∗ − ∗ − p We show the line shapes for three real values of γ: +34, 0, and 34 MeV. The cor- −

responding values of γ0 have real parts 82 MeV, 63 MeV, and 48 MeV, respectively.

Their imaginary parts are all approximately 0.00012 MeV, which is completely negligible. For γ = +34 MeV, the peak of the resonance is at E = 0.6 MeV, which is − the central value of the measurement in Eq. (3.27).

+ 0 In Fig. 5.2, we show the line shapes in the short-distance decay mode J/ψπ π−π .

The line shapes, which are the same for X produced by a B+ K+ or B0 K0 → → transition, are given in Eqs. (5.22). The relative normalizations of the curves for the

92 /d E Γ d

-12 -8 -4 0 4 8 12 E (MeV)

Figure 5.2: The line shapes in the D∗D¯ threshold region for X(3872) produced by a + + 0 0 + 0 B K or B K transition and decaying into J/ψπ π−π . The line shapes are shown→ for γ =→ and three values of γ: +34 MeV (solid line), 0 (dotted line), 1 ±∞ and 34 MeV (dashed line). −

K+,00 three values of γ are determined by using the same short-distance factors ΓB+ and

+ − 0 ΓJ/ψ π π π . In Fig. 5.3, we show the line shapes in the short-distance decay mode

+ J/ψπ π−. The line shapes are given in Eqs. (5.23). The upper and lower panels show the line shapes produced by B+ K+ and B0 K0 transitions, respectively. → → The line shapes from the B+ K+ transition have approximate zeros near +6 MeV, → while the line shapes from the B0 K+ transition have approximate zeros near → 2 MeV. The relative normalizations of all six curves are determined by using the − + K ,00 J/ψ π+π− same short-distance factors ΓB+ and Γ . In Fig. 5.1, we show the line shapes in D0D¯ 0π0. The line shapes, which are the same for X produced by a B+ K+ → or B0 K0 transition, are given in Eqs. (5.24). The relative normalizations of the → curves for the three values of γ are determined by using the same short-distance factor

K+,00 ΓB+ .

93 Fig. 5.3 illustrates the fact that the line shape of the X(3872) may depend not only

on the decay channel but also on the production mechanism for the resonance. The

+ diﬀerence between the line shapes in the J/ψπ π− decay channel for X produced by B+ K+ and B0 K0 transitions is particularly dramatic because of the → → approximate zeros in the line shapes. These approximate zeros are general features of the line shapes in Eqs. (5.23). If the imaginary parts of γ0 and κ1(E) are neglected, the numerator factor γ κ (E) 2 in the energy distribution in Eq. (5.23a) has a zero | 0 − 1 | 0 0 + between the D∗ D¯ and D∗ D− thresholds. If γ κ (0) γ , the approximate | |≪| 1 |≪| 1| expression for γ in Eq. (3.58) reduces to κ (0)/2. The zero is therefore near 3 ν 6.1 0 1 4 ≈

MeV. If the imaginary parts of γ0 and κ(E) are neglected, the numerator factor

2 0 0 γ κ(E) in the energy distribution in Eq. (5.23b) has a zero below the D∗ D¯ | 0 − | threshold. If γ κ (0) γ , the zero is near 1 ν 2.0 MeV. In the case of | |≪| 1 |≪| 1| − 4 ≈ − B+ decays, the approximate zero forces the line shape to be narrower on the trailing edge of the resonance. In the case of B0 decays, the approximate zero forces the line shape to be narrower on the leading edge of the resonance.

To deduce the implications of the current-current factorization approximation and

0 0 heavy quark symmetry in the D∗ D¯ threshold region, we consider the general factorization formulas in Eqs. (5.22), (5.23), and (5.24) for κ(E) small compared to κ1(0).

By neglecting the small diﬀerence between κ1(E) and κ1(0), the denominator D(E) in

Eq. (3.54) reduces to the expression in Eq. (3.59). The factorization formulas for the

D0D¯ 0π0 channel in Eqs. (5.24) reduce to the form of the single-channel factorization

formula in Eq. (4.12), with the short-distance constants for the B K transition →

94 /d E Γ d

-12 -8 -4 0 4 8 12 E (MeV) /d E Γ d

-12 -8 -4 0 4 8 12 E (MeV)

Figure 5.3: The line shapes in the D∗D¯ threshold region for X(3872) produced by a + B K transition and decaying into J/ψπ π−. The line shapes are diﬀerent for X produced→ by a B+ K+ transition (upper panel) and a B0 K0 transition (lower panel). The line shapes→ are shown for γ = and three→ values of γ: +34 MeV 1 ±∞ (solid lines), 0 (dotted lines), and 34 MeV (dashed lines). −

given by

K+ K+, 00 Γ + Γ + , (5.26a) B ≈ B 2 K0 K+, 00 γ1 γ0 Γ 0 Γ + − . (5.26b) B ≈ B γ + γ 2κ (0) 1 0 1 −

If γ is neglected compared to γ1 and κ1(0), the approximate solution for γ0 is given

in Eq. (3.61). Inserting this solution, the last factor in Eq. (5.26b) reduces to c 2, | 1|

95 where the coeﬃcient c1 is given in Eq. (3.63b). The factorization formulas for the short-distance decay modes in Eqs. (5.22) and (5.23) all reduce to the form of the single-channel factorization formula in Eq. (4.11), if we neglect the diﬀerence between

κ1(E) and κ1(0) and if we neglect κ(E) compared to γ0 and γ1. From Eqs. (5.22a)

and (5.24a) and from Eq. (5.22b) and (5.24b), we can obtain expressions for the

short-distance constants associated with the transition of X into an I = 0 ﬁnal state

+ 0 C, such as J/ψπ π−π :

γ κ (0) 2 ΓC (E) 2 1 − 1 ΓC (E). ≈ γ + γ 2κ (0) 0 1 0 − 1 2 γ1 C 2 Γ (E). (5.27) ≈ γ γ 0 1 0 −

These two expressions agree if we insert the approximate solution for γ0 in Eq. (3.61).

From Eqs. (5.23a) and (5.24a) and from Eqs. (5.23b) and (5.24b), we can obtain the expressions for the short-distance constants associated with the transition of X into

+ an I = 1 ﬁnal state, such as J/ψπ π−:

γ κ (0) 2 ΓC (E) 2 0 − 1 ΓC (E) ≈ γ + γ 2κ (0) 1 1 0 − 1 2 γ0 C 2 Γ1 (E). (5.28) ≈ γ γ 1 0 −

These two expressions agree if we insert the approximate solution for γ0 in Eq. (3.61).

The expressions for ΓC in Eqs. (5.27) and (5.28) are both consistent with the general

formula in Eq. (5.10).

5.5 Ratios of Decay Rates

We now consider the ratios of the line shapes and the decay rates for the processes

0 0 + + + + 0 B K C and B K C for the decay modes C = J/ψπ π−, J/ψπ π−π , and → → 96 D0D¯ 0π0. The short-distance factors associated with the initial and ﬁnal states in

Eqs. (5.22), (5.23), and (5.24) cancel in the ratio of the line shapes:

dΓ [B0 K0C] /dE γ κ(E) 2 → = 1 − , (5.29a) dΓ [B+ K+C] /dE γ κ (E) + − 0 1 1 → C=J/ψ π π π −

d B0 K0C /dE γ κ E 2 Γ [ ] 0 ( ) + → + = − , (5.29b) dΓ [B K C] /dE γ0 κ1(E) C=J/ψ π+π− → − 0 0 2 dΓ [B K C] /dE γ0 + γ 1 → = − . (5.29c) d B+ K+C /dE γ γ κ E Γ [ ] 0 0 0 0 + 1 2 1( ) → C=D D¯ π −

0 ¯ 0 In the D∗ D threshold region, these ratios all reduce to the same value:

dΓ [B0 K0C] /dE γ 2 → = 1 , (5.30) dΓ [B+ K+C] /dE γ κ (0) 1 1 → −

where we have approximated κ1(E) by κ1(0) and neglected γ and κ(E) with respect

to κ1(0) and γ1.

One can deﬁne the ratio of the partial decay rates for processes B0 K0X and → B+ K+X as the integral of the line shape in a speciﬁc decay channel C over an → appropriate region:

Γ [B0 K0X] dE w(E)dΓ [B0 K0C] /dE → → . (5.31) Γ [B+ K+X] ≡ dE w(E)dΓ [B+ K+C] /dE → C R →

R The integration region is speciﬁed by the weighting factor w(E). The weighting factor can also be used to take into account the experimental resolution. In general, the

ratio in Eq. (5.31) depends on the decay channel C and the weighting factor w(E).

However if w(E) is chosen so that most of the contributions to the integrals come

0 0 from the D∗ D¯ threshold region, the ratio of line shapes reduces to the constant in

Eq. (5.30) for all decay modes C:

Γ [B0 K0X] γ 2 R → 1 . (5.32) ≡ Γ [B+ K+X] ≈ γ κ (0) 1 1 → −

97 In Fig. 5.4, the ratio R is plotted as a function of the scattering parameter 1/γ1.

The ratio approaches 1 for γ κ (0) and it approaches as γ κ (0). A | 1| ≫ 1 ∞ 1 → 1 measurement of R can be used to determine γ1. Given a value of R, Eq. (5.32) has two solutions for γ1. The most plausible solution is the one satisfying 1/γ1 < 1/κ1(0).

+ The ratio R in the decay channel J/ψ π π− has been measured by both the

Belle and BaBar Collaborations. Their results, which are given in Eqs. (1.7) and

(1.8), are only marginally compatible with each other. Since the measurements were obtained by integrating the line shapes over intervals whose widths are about 20

MeV, the simple formula for R in Eq. (5.32) does not obviously apply. It may be applicable if the observed line shapes are actually dominated by production in the

0 0 D∗ D¯ threshold region with broadening due to the energy resolution. The values of γ1 obtained by ﬁtting the expression for the ratio R in Eq. (5.32) to the central values of R measured by the Belle and BaBar Collaborations are γ 3982 MeV 1 ≈ − and 222 MeV, respectively. More accurate measurements of this ratio could be used − to determine the scattering parameter γ1.

98 2

1.5

R 1

0.5

0 -0.02 -0.01 0 0.01 0.02 γ -1 1/ 1 (MeV )

Figure 5.4: The ratio of decay rates for the processes B0 K0X and B+ K+X in → → Eq. (5.32) is plotted as a function of the scattering parameter 1/γ1. The two branches of the curve approach inﬁnity as γ approaches κ (0) 125 MeV. 1 1 ≈

99 CHAPTER 6

LINE SHAPES OF Z±(4430)

In this chapter, the formalism developed for deriving line shapes of the X(3872)

is applied to the case of the manifestly exotic cc¯ meson Z±(4430), assuming that

Z±(4430) has quantum numbers consistent with a charm meson molecule composed

of linear combinations of charm meson pairs D1D∗. Simple features of the line shapes

of Z± that follow from its identiﬁcation as a weakly-bound S-wave charm meson molecule are presented. The analysis presented in this chapter was published in

Physical Review D [56]. I present here a slightly modiﬁed version to be consistent with this dissertation.

6.1 Low-energy charm meson scattering

P P The J quantum numbers of the Z±(4430) have not been determined. If the J quantum numbers are 0−, 1−, or 2−, it has an S-wave coupling to D1D¯ ∗. The tiny binding energy in Eq. (3.35) then implies that the Z+(4430) is an S-wave threshold resonance with universal properties that include simple features of its line shapes.

The amplitude for elastic scattering of a D D¯ ∗ pair in a resonant S-wave channel A 1 can be written as

= (2π/M1 )f(E) , (6.1) A ∗ 100 where M1 is the reduced mass for the D1D¯ ∗ pair. The scattering amplitude f(E) ∗ depends only on the energy diﬀerence E between the invariant mass of the charm

mesons and the D1D¯ ∗ threshold. If there is a threshold S-wave resonance, the scat-

tering amplitude for energy E near the threshold has the simple universal form given

in Eq. (3.16), where γ is the inverse scattering length and κ(E) is deﬁned as

1/2 κ(E)=( 2M1 E iǫ) . (6.2) − ∗ −

The universal expression in Eq. (3.16) is valid when γ is large compared to r , | | | s|

where rs is the S-wave eﬀective range. Since the long-range interactions of D1 and

D¯ ∗ are dominated by pion exchange, we expect r . 1/m . Thus the universal | s| π 2 region E < 2/(M 1rs ) should at least include the interval E < 36 MeV. The | | ∗ | | simple expression for κ(E) with a branch point at E = 0 applies if the constituents

are stable. If the width Γ1 of D1 is taken into account, the branch point should be at

E = iΓ /2. Thus an expression for κ(E) that takes into account the width of the − 1

constituent D1 is [48]

κ(E)= 2M1 (E + iΓ1/2). (6.3) − ∗ p If E is real, a more explicit expression for κ(E) with the appropriate choice of branch cut can be obtained by using the identity in Eq. (3.44).

The universal expression for the scattering amplitude given by Eqs. (3.16) can be motivated by unitarity. The imaginary part of f(E) can be written as

Imf(E)= f(E) 2 Im [γ κ(E)] . (6.4) | | −

1/2 If γ is real and if Γ1 = 0, Im[γ κ(E)] reduces to (2M1 E) θ(E). In this case, − ∗ Eq. (6.4) is simply the optical theorem that expresses the exact unitarity of elastic scattering in the S-wave channel. The eﬀects of inelastic scattering can be taken into

101 account through the variables Imγ and Γ1. Since the dominant decay modes of the

D1 are D∗π [57], the variable Γ1 takes into account inelastic scattering channels of the form D∗D¯ ∗π. The variable Imγ takes into account other inelastic scattering channels

+ that do not involve the decay of a constituent of Z (4430), such as ψ′π, J/ψπ, D∗D¯

+ and D∗D¯ ∗. These are short-distance decay modes for Z (4430), because they proceed through intermediate states in which the D1 and D¯ ∗ have a separation that is small compared to 1/ γ . The imaginary part of γ can be partitioned into contributions | | from the individual short-distance channels:

C Imγ = C Γ . (6.5) P 6.2 Line shapes

To derive universal expressions for the line shapes of the Z+ in the decay B+ → K0 + Z+, we start from the optical theorem for the inclusive decay into K0 plus particles that couple to the Z+ resonance. If the diﬀerence E between the invariant mass

0 of the resonating particles and the D1D¯ ∗ threshold is small enough, the invariant mass distribution can be written as dΓ B+ K0 + resonant = 2ΓK Imf(E) , (6.6) dE → B

K where ΓB is a constant with dimension of energy that takes into account the probability for the constituents of Z+ to be created in the transition B+ K0. The energy → distribution in Eq. (6.6) can be decomposed into the line shapes for individual reso-

nant channels by inserting Eqs. (6.4) and (6.5). The line shape in the short-distance

+ decay mode ψ′π is

dΓ + 0 + K 2 ψ′π B K + ψ′ π = 2Γ f(E) Γ . (6.7) dE → B | | 102 This line shape applies equally well to any other short-distance decay mode C, with

ψ′π C + the constant Γ replaced by Γ . The line shape of Z in the D∗D¯ ∗π channels can be obtained by inserting Eq. (6.4) into Eq. (6.6) and using the expression for Imκ(E) in Eq. (3.44):

dΓ + 0 ¯ K 2 2 2 1/2 1/2 B K + D∗D∗π = 2ΓB f(E) M1 (E +Γ1/4) + E , (6.8) dE → | | ∗ p ¯ + + + ¯ 0 0 0 ¯ 0 + ¯ 0 where D∗D∗π denotes D∗ D∗−π or D∗ D∗ π or D∗ D∗ π . Since D1 decays pri-

+ ¯ 0 0 2 1 + marily into D∗−π and D∗ π with branching fractions 3 and 3 and D1 decays pri-

+ 0 0 + 1 2 marily into D∗ π and D∗ π with branching fractions 3 and 3 [57], the line shapes in

any of the three individual D∗D¯ ∗π channels can be obtained by multiplying Eq. (6.8)

by 1/3. The line shapes are given by Eqs. (6.7) and (6.8) only if the Z+(4430) has

P J =0−,1−,or2−. The spin J determines the correlations between the polarizations

of the two spin-1 constituents of Z+.

We proceed to illustrate the line shapes of the Z+(4430). We will assume that

+ decays of Z (4430) into D∗D¯ ∗π dominate over the short-distance decays. We can

2 therefore neglect Imγ in the resonance factor f(E) in the line shapes for ψ′π and | |

D∗D¯ ∗π in Eqs. (6.7) and (6.8). Thus the only unknown parameters are Reγ and

K ψ′π the normalization factors ΓB and Γ . We identify the binding energy EZ with the

negative of the position of the peak in the line shape for ψ′π. It satisﬁes the equation

Reγ 2 2 1/2 1/2 EZ = (EZ +Γ1/4) + EZ . (6.9) 2√M1 ∗ + + The line shapes of Z in ψ′π and D∗D¯ ∗π are shown in Fig. 6.1 for three values of

the binding energy: E = +3, 0 and 3 MeV, which correspond to Reγ = +53.9, 0, Z − ′ and 72.0 MeV, respectively. The normalization factors ΓK and Γψ π were chosen so − B that the two line shapes have the same peak value for EZ = 0. These same factors

103 2

1.5

/ dE 1 Γ d 0.5

0 -60 -40 -20 0 20 40 60 E (MeV)

+ + Figure 6.1: Line shapes of the Z in the ψ′π decay channel (solid lines) and in the D∗D¯ ∗π decay channel (dashed lines) for three values of the binding energy: EZ = +3 MeV (upper lines), 0 MeV (middle lines), and 3 MeV (lower lines). The line − shapes are normalized so that their maximum values are 1 for EZ = 0.

were then used for E = 3 MeV. The peaks in the line shape are most dramatic Z ±

if EZ is positive. The line shapes in ψ′π are nearly symmetric about the peak at the energy E . If E < 0, the line shape is sometimes referred to as a “cusp” [58] − Z Z + because it has a discontinuity in slope at E =0ifΓ1 = 0. In the case of Z , the cusp

is completely smoothed out by the width of the D1, as is evident in the lowest solid

line in Fig. 6.1. The line shapes in D∗D¯ ∗π have a peak at a higher energy than for

ψ′π and they also have a broad shoulder on the high energy side of the peak. This

behavior can be attributed to a threshold enhancement in the production of D1D¯ ∗

and D∗D¯ 1 that overlaps with the resonance.

104 To quantify the behavior of the line shapes in Eqs. (6.7) and (6.8), we can ex-

ploit the fact that the binding energy EZ is small compared to the width Γ1 of the constituent. Eq. (6.9) for EZ can be solved as an expansion in powers of Reγ whose

1/2 leading term is [Γ1/(8M1 )] Reγ. The expansion can be inverted to give Reγ as an ∗ expansion in EZ /Γ1. To ﬁrst order in EZ /Γ1, the energy at which the line shape for

D∗D¯ ∗π has its maximum is

∗ ∗ ED D¯ π 0.289Γ 2.03 E . (6.10) max ≈ 1 − Z

Since E Γ , the diﬀerence between the locations of the peaks in the D∗D¯ ∗π and | Z|≪ 1

ψ′π line shapes is approximately Γ /√12 6 MeV. The line shapes in Eqs. (6.7) and 1 ≈ (6.8) are not Breit-Wigner resonances, so they can not be characterized simply by a width ΓZ. A simple way to quantify their widths is to give the energies at which the line shapes decrease to half of their maxima. For the ψ′π line shape, the half-maxima are at the energies

′ Eψ π 0.866Γ 3.42 E , (6.11a) + ≈ 1 − Z ψ′π E 0.866Γ1 +0.16 EZ . (6.11b) − ≈ − The full width at half maximum is approximately √3 Γ 35 MeV, which is con- 1 ≈ + sistent with the measured width of Z in Eq. (1.12). For the D∗D¯ ∗π line shape, the half-maxima are at the energies

∗ ∗ ED D¯ π 3.017Γ 15.37 E , (6.12a) + ≈ 1 − Z D∗D¯ ∗π E 0.371Γ1 1.08 EZ . (6.12b) − ≈ − − The full width at half-maximum is approximately 3.4Γ 69 MeV, which is about 1 ≈ twice the width in ψ′π. The right half-maximum is about 4 times as far from the peak energy as the left half-maximum.

105 The universal scattering amplitude in Eq. (3.16) also applies to the X(3872) if it is a D∗D¯ molecule [48]. The difference between the lifetimes of the constituents of the X and Z± leads to a significant difference in their line shapes. In the case of

0 the X, the D∗ has a width of about 65 keV, which is small compared to the 8 MeV

0 0 + splitting between the D∗ D¯ and D∗ D− thresholds. Since the binding energy of X

0 0 0 0 relative to the D∗ D¯ threshold is small compared to 8 MeV, it is a D∗ D¯ molecule

+ with a very small D∗ D− component. The line shapes of the X are given by simple universal formulas analogous to those in Eqs. (6.7) and (6.8) only if the energy is

0 0 within a few MeV of the D∗ D¯ threshold. For energies more than a few MeV from the threshold, one must take into account the resonant coupling to the charged charm

+ meson channel D∗ D− [55, 54]. The width of the D∗ provides less smearing than that of the D1, so it is possible for the line shape for the X to have a two-peaked structure consisting of a resonance and a threshold enhancement.

In summary, the hypothesis that the Z± is a weakly-bound S-wave charm meson molecule implies that its line shape in D∗D¯ ∗π should peak at a higher energy and have a larger width than its line shape in ψ′π. This hypothesis requires the Z± to

P have quantum numbers J =0−, 1−, or 2−. If the quantitative predictions for these line shapes are conﬁrmed, it would provide strong support for this interpretation of

Z±(4430).

106 CHAPTER 7

CONCLUSION

In this dissertation, we have deduced the behavior of the line shapes of the exotic cc¯ mesons X(3872) and Z±(4430) that follow from the identiﬁcation of these mesons as weakly-bound charm meson molecules. In the case of the X(3872), the proximity of its

0 0 measured mass to the D∗ D¯ energy threshold and the establishment of its quantum numbers as J P C = 1++ imply unambiguously that it is a charm meson molecule. In the case of the Z±(4430), the proximity of its mass to the D1D¯ ∗ threshold motivates its identiﬁcation as a charm meson molecule. This interpretation of Z±(4430) remains

P speculative, because it relies on its quantum numbers being J =0−, 1−, or 2−.

In Chapter 3, we discussed aspects of the scattering of charm mesons that are relevant to the interpretation of X(3872) and Z±(4430) as charm meson molecules.

We exploited the fact that if there is an S-wave resonance state near a two-particle energy threshold, the scattering length a is much larger than the eﬀective range rs.

This separation of length scales implies universal properties that depend only on the scattering length a but are otherwise insensitive to details of interactions at distances small compared to a . Many of these universal properties can be expressed in terms | | of the universal transition amplitude f(E) in Eq. (3.16), which depends only on the

107 inverse scattering length γ and the total energy E of the charm mesons in their center-of-momentum frame.

0 0 In Chapter 4, the line shapes of X(3872) within a few MeV of the D∗ D¯ threshold

¯ 0 were derived from an expression for the resonant scattering amplitude in the (D∗D)+

0 channel that takes into account the D∗ width and inelastic charm meson scattering channels. The line shapes were expressed in the form of factorization formulas, in which short-distance eﬀects and long-distance eﬀects are separated into multiplicative

+ 0 factors. The line shape for a short-distance decay mode, such as J/ψπ π−π or

+ 0 0 0 J/ψπ π−, and for D D¯ π are given in Eqs. (4.11) and (4.12), respectively. The

diﬀerence in these line shape can explain the diﬀerence between the masses of X(3872)

+ 0 0 0 measured in the J/ψπ π− and D D¯ π decay modes. A quantitative analysis of data

from the Belle Collaboration on the line shapes for these two decay modes was carried

out. The result indicates that the existing data favor the X(3872) to be a bound state

of charm mesons, but a virtual state is not excluded. When more extensive data on

the line shapes of the X(3872) in various decay channels and for various production

processes becomes available, it should be possible to determine conclusively whether

the X(3872) is a bound state or a virtual state.

In Chapter 5, the region of validity for the line shapes for the X(3872) was ex-

tended to the entire D∗D¯ threshold region by taking into account the resonant cou-

¯ 0 ¯ 1 pling between the (D∗D)+ and (D∗D)+ channels. By taking into account isospin

symmetry at high energies, the coupled-channel scattering amplitudes were expressed

in terms of two parameters: the I = 0 and I = 1 inverse scattering lengths γ0

and γ1. Since the line shapes are very sensitive to the inverse scattering length γ

0 0 in the D∗ D¯ channel, γ and γ1 are a convenient choice of independent scattering

108 parameters. Isospin symmetry was also taken into account in the short-distance factors in the factorization formulas. In the case of production of the X resonance in

B K transitions, isospin symmetry reduces the short-distance factors to four inde- → K+,00 K+,11 K+,01 pendent real constants: ΓB+ , ΓB+ , and the real and imaginary parts of ΓB+ . The current-current factorization approximation together with heavy quark symmetry were exploited to further simplify the factorization formulas for the line shapes of

X(3872). The short-distance constants associated with the B K transitions reduce → K+,00 to a single real constant ΓB+ . The line shapes for an I = 0 decay channel, an I =1 decay channel, and D0D¯ 0π0 are given in Eqs. (5.22), (5.23), and (5.24), respectively.

The line shapes produced by the B+ K+ transition are diﬀerent from the line → shapes produced by the B0 K0 transition. The parameter γ could be determined → 1 phenomenologically from ratios of rates for B0 K0 + X and B+ K+ + X. When → →

γ1 is determined, the predictive power of the line shape formulas will be dramatically increased.

The predictions of the line shapes of X(3872) could be further improved by taking

0 0 0 0 into account pion interactions explicitly. The system consisting of D∗ D¯ , D D¯ ∗ ,

0 0 0 0 0 and D D¯ π states with energies near the D D¯ ∗ threshold can be described by a nonrelativistic eﬀective ﬁeld theory as described in Ref. [59]. The simplest such theory

¯ 0 0 has S-wave scattering in the (D∗D)+ channel and π couplings that allow the decay

0 0 0 D∗ D π . Fleming, Kusunoki, Mehen, and van Kolck developed power-counting → rules for this effective field theory and showed that the pion couplings can be treated perturbatively [60]. They used the effective field theory to calculate the decay rate for

X(3872) D0D¯ 0π0 to next-to-leading order in the pion coupling. In applying this → eﬀective ﬁeld theory to the line shapes of the X(3872), one complication that will be

109 0 0 encountered is infrared singularities at the D∗ D¯ threshold that are related to the

0 0 0 decay D∗ D π . This problem has been analyzed in a simpler model with spin-0 → particles and momentum-independent interactions [59]. The problem was solved by a resummation of perturbation theory that takes into account the perturbative shift

0 0 of the D D¯ ∗ threshold into the complex energy plane because of the nonzero width

0 of the D∗ .

In Chapter 6, the formalism associated with the universality of S-wave resonances near a 2-particle threshold was applied to the manifestly exotic cc¯ meson Z±(4430).

The Z+(4430) was assumed to be a charm meson molecule whose constituents are

+ ¯ 0 + ¯ 0 P an S-wave superposition of D1 D∗ and D∗ D1. This assumption requires the J quantum numbers of the Z±(4430) to be 0−, 1−, or 2−. The small ratio of the

+ binding energy of the Z to the width Γ1 of its constituent D1 was exploited to predict

+ properties of its line shapes. Its full width at half maximum in the channel ψ′π was predicted to be approximately √3Γ 35 MeV, which is consistent with the measured 1 ≈ + width of the Z . The Z±(4430) should also decay into D∗D¯ ∗π through decay of its

constituent D1. The peak in the line shape for D∗D¯ ∗π was predicted to be at a higher

+ energy than the peak in the line shape for ψ′π by about Γ /√12 6 MeV. The line 1 ≈

shape in D∗D¯ ∗π was predicted to be broader and asymmetric, with a shoulder on the high energy side that can be attributed to a threshold enhancement in the production of D1D¯ ∗.

The universality of S-wave resonances near a two-particle threshold implies that the X(3872) is a weakly-bound charm meson molecule and therefore a cc¯ tetraquark

meson. This identiﬁcation is not yet universally accepted, but the results presented

in this thesis should allow this case to be made more convincing. The ﬂavors of

110 the decay products of the Z±(4430) reveal that it is also a cc¯ tetraquark meson. It may also be a weakly-bound charm meson molecule, but this identiﬁcation is more speculative than the case of the X(3872). The existence of the X(3872) and Z±(4430) suggest that there may also be cc¯ tetraquark mesons in other channels. It also implies that there must be a b¯b tetraquark meson with a much larger binding energy [61, 62].

The reason for this is that a bound state requires a balance between kinetic energy, which is positive, and potential energy, which can be negative. Due to heavy quark symmetry, replacing a c quark by a more massive b quark should decrease the kinetic energy without signiﬁcantly changing the potential energy. Therefore, the binding energy of a bottom meson molecule should be larger than that of the corresponding charm meson molecule. Thus the discovery of the X(3872) and the Z±(4430) is just the beginning of the spectroscopy of tetraquark mesons.

111 APPENDIX A

UNIVERSAL TRANSITION AMPLITUDE FOR PARTICLES WITH A LARGE SCATTERING LENGTH

In this appendix, we derive the universal transition amplitude f(E) for two nonrelativistic particles interacting through a short-range interaction with a large scattering length.

A.1 The model and Feynman rules

Since the transition amplitude is universal, it can be derived using any model with short-range interactions and a large scattering length. We will use a particularly simple model in which the two particles D1 and D2 are spin-0 particles with a point- like interaction. We denote their masses by m1 and m2. The reduced mass of D1 and

D2 is

m12 = m1m2/(m1 + m2). (A.1)

The particles D1 and D2 can be described by a nonrelativistic quantum ﬁeld theory with two complex scalar ﬁelds D1(x) and D2(x). The free terms in the Lagrangian are 7

∂ 1 2 = D† i m + D . (A.2) Lfree i ∂t − i 2m ∇ i i=1, 2 i X 7The superscript on a ﬁeld indicates its complex conjugate. † 112 The interaction term in the Lagrangian is

= λ D†D†D D . (A.3) Lint − 0 1 2 1 2

The coupling constant λ , which has mass dimension 2, takes into account short- 0 −

range interactions. The subscript on λ0 emphasizes that it is a bare coupling constant that requires renormalization. We assume there is a ﬁne-tuning of the short-range interactions that makes the D1D2 scattering length large. This implies that the

D1D2 interaction with coupling constant λ0 must be treated nonperturbatively. The

Feynman rules for this model are:

Non-relativistic propagators for the particles D , i =1, 2: • i

0 i ∆i(k , ~k ) . (A.4) | | ≡ k0 m ~k 2 (2m )+ iε − i − i The propagators are represented by solid lines labelled by 1 or 2.

D†D†D D vertex factor: • 1 2 1 2

iλ . (A.5) − 0

The vertex is represented by a solid dot.

Loop momentum integral: • d4q . (A.6) (4π)4 Z The 4-momentum is conserved at each vertex and any 4-momentum (q0, ~q )

that is not determined by that of the external lines is integrated over with this

measure.

113 2 2

1 1

Figure A.1: Feynman diagram for the amplitude iL(E) for the propagation of D1 and D2 between interactions. The open dots indicate that the D1†D2†D1D2 vertex factors are omitted.

A.2 Propagation for D1D2 between interactions

A basic ingredient in the calculation of the D1D2 transition amplitude is the

amplitude iL(E) for the propagation of the D1D2 pair between interactions. This

amplitude is represented by the Feynman diagram in Fig. A.1. Using the Feynman

rules, the expression for this amplitude in the rest frame of the D1D2 pair is

d4q iL(E) ∆ (q0, q) ∆ (E q0, q), (A.7) ≡ (2π)4 2 1 − Z where E is the total energy ﬂowing into the D1D2 loop. The more explicit form for the loop integral is

d3q dq0 i iL(E) = (2π)3 2π q0 m ~q 2/ (2m )+ iε Z Z − 2 − 2 i . (A.8) ×E q0 m ~q 2/ (2m )+ iε − − 1 − 1 The q0 integral is evaluated using contour integration with the contour closed in

the upper half-plane of the complex variable q0. The contour encloses a pole at

E m ~q 2/(2m ) + iε. This reduces the amplitude to an integral over the 3- − 1 − 1 momentum ~q:

d3q 1 iL(E) = i , (A.9) (2π)3 E ~q 2/ (2m )+ iε Z 12 − 12 114 where the variable E = E m + m is total energy of the D D pair relative to the 12 − 1 2 1 2 m1 + m2 energy threshold. It is convenient to introduce a momentum variable κ(E):

κ(E) 2m E iε. (A.10) ≡ − 12 12 − p This variable is real if E12 < 0 and pure imaginary if E12 > 0:

κ(E) = 2m12E12 (E12 < 0) (A.11a) p = i 2m E (E > 0). (A.11b) − 12 12 12 p Since the integrand in Eq. (A.9) is isotropic, the integral can be reduced to an integral over the magnitude q:

2 m ∞ q iL(E) = i 12 dq . (A.12) − π2 q2 + κ2(E) Z0 This integral is inﬁnite because of an ultraviolet divergence associated with the upper limit of integration. In order to proceed further, it is necessary to implement regularization to make the integral ﬁnite. We will discuss two alternative regularizations.

Momentum cutoﬀ regularization

With momentum cutoﬀ regularization, the upper limit for the divergent integral

in Eq. (A.12) is replaced by an ultraviolet momentum cutoﬀ Λ that is much larger

than the momentum scale κ(E) in the integrand. The resulting expression for the | | integral in Eq. (A.12) is m Λ q2 iL(E) = i 12 dq . (A.13) − π2 q2 + κ2(E) Z0 This integral can be evaluated analytically. Taking the limit Λ κ(E) , we obtain ≫| | a simple result: m L(E) = 12 Λ π κ(E) . (A.14) − π2 − 2 The ultraviolet divergence in iL(E) has been isolated in a term that is linear in Λ.

115 Dimensional regularization

With dimensional regularization, a 3-dimensional integral over the momentum is made ﬁnite by analytically continuing it to 3 2ǫ dimensions. The analytic continu- − ation can be carried out using the prescription

3 2ǫ ǫ 3 2ǫ d − k lim (4π)− 2 ǫ µ , (A.15) ~ ≡ ǫ 0 (2π)3 2ǫ Zk → − Z − where µ is the renormalization scale and (z)a is the Pochhammer symbol:

Γ(z + a) (z) = . (A.16) a Γ(z)

The function of ǫ in the prefactor is designed to simplify the analytic expressions for dimensionally regularized integrals by cancelling the eﬀects of the analytic contin- uation of angular integrals. This makes the expression for the integral of a scalar function of ~k particularly simple:

1 2ǫ ∞ 2 2ǫ f( ~k ) lim µ dk k − f(k). (A.17) ~ | | ≡ ǫ 0 2π2 Zk → Z0 Using this prescription, the divergent integral in Eq. (A.12) can be rewritten as

2 2ǫ m ∞ k iL(E) = i 12 lim dk − . (A.18) − π2 ǫ 0 k2 + κ2(E) → Z0 This integral can be evaluated analytically and the result is

2ǫ m12µ 1 2ǫ L(E)= [κ(E)] − . (A.19) 2π cos(ǫπ)

1 This expression has a pole at ǫ = 2 : m µ L(E) 12 as ǫ 1 . (A.20) −→ −2π2(ǫ 1 ) → 2 − 2 The pole plays the role of the linear ultraviolet divergence in the expression for L(E)

in Eq. (A.14) using the momentum cutoﬀ regularization. The result for L(E) using

116 + + + · · ·

th Figure A.2: The connected Green’s function i (E) for D1D2 D1D2 at 0 order in g can be obtained by summing a geometricA series of one-loop→ diagrams.

dimensional regularization is obtained by analytically continuing the expression in

Eq. (A.19) to 3-dimensional space by setting ǫ = 0 :

m L(E)= 12 κ(E). (A.21) 2π

This is equal to the ﬁnite part of the result from the momentum cutoﬀ regularization given in Eq. (A.14).

A.3 Universal Transition Amplitude f(E)

The amputated connected Green’s function i (E) for D D D D can be A 1 2 → 1 2 calculated nonperturbatively by summing the diagrams in Fig. A.2 to all orders in the bare coupling constant λ0 :

i (E)= iλ + iλ2L(E) λ3L(E)2 + (A.22) A − 0 0 − 0 ···

This inﬁnite sum is a geometric series, so it can be evaluated analytically:

i i (E)= − . (A.23) A 1/λ L(E) 0 −

117 Momentum cutoﬀ regularization

If we use the expression for L(E) in Eq. (A.14) given by momentum cutoﬀ regu-

larization, the denominator of i (E) in Eq. (A.23) becomes A m m 1/λ L(E)= 1/λ + 12 Λ 12 κ(E), (A.24) 0 − 0 π2 − 2π where κ(E) is as deﬁned in Eq. (A.10). We can obtain a ﬁnite limit as Λ if → ∞

the bare coupling constant λ0 depends on Λ in such a way as to cancel the explicit

divergence. The expression in parentheses in Eq. (A.24) can approach an arbitrary

constant in this limit:

m m 1/λ + 12 Λ 12 γ as Λ , (A.25) 0 π2 −→ π2 →∞ where γ is an arbitrary parameter with the dimension of momentum. A simple expression for the bare coupling constant with the correct limiting behavior is

π2 λ = . (A.26) 0 m2 (Λ γ) 12 − The limiting expression for the Green’s function in Eq. (A.23) as Λ is then →∞ 2π 1 (E)= . (A.27) A m γ + κ(E) 12 − Dimensional regularization

With dimensional regularization, the expression for L(E) in Eq. (A.21) is auto-

matically ﬁnite. We can express the bare coupling constant λ0 in the form

π2 λ0 = 2 , (A.28) −m12γ where γ is a parameter with the dimension of momentum. Inserting this in Eq. (A.23),

we obtain the same result in Eq. (A.27) for the transition amplitude.

118 Using two diﬀerent regularization schemes, we have arrived at the same universal expression in Eq. (A.27) for the transition amplitude. The physical interpretation of the arbitary parameter γ can be obtained from the T -matrix element for two particle scattering. The total kinetic energy for a pair of particles with momentum +~p and

~p is E = p2/(2m ). Evaluating the amplitude in Eq. (A.27) at this energy, we − 12 12 obtain 2π 1 (p)= . (A.29) T −m12 γ + ip The partial wave expansion for the general two-particle scattering amplitude is given in Eq. (3.5). The T-matrix element corresponding to S-wave (L = 0) scattering is

2π 1 (p)= , (A.30) T m p cot δ (p) ip 12 0 − where δ0(p) is the S-wave phase shift. Inserting the eﬀective range expansion in

Eq. (3.9) and comparing the two expressions for the T -matrix element in Eqs. (A.29)

and (A.30), we can identify γ as the inverse scattering length 1/a.

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