Topics in Theory and Experiment in Relativistic Heavy-Ion Physics
A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy
by Jeremy Alford May 2015
c Copyright All rights reserved Except for previously published materials Dissertation written by Jeremy Alford BS, Rochester Institute of Technology, 2004 MA, Kent State University, 2009 Ph.D., Kent State University, 2015
Approved by
Michael Strickland, Associate Professor, Ph.D., Department of Physics, Doctoral Advisor Declan Keane, Professor, Ph.D., Department of Physics, Doctoral Advisor Veronica Dexheimer, Assistant Professor, Ph.D., Department of Physics Antal Jakli, Professor, Ph.D., Liquid Crystal Institute John L. West, Professor, Ph.D., Department of Chemical Physics
Accepted by
James Gleeson, Chair, Ph.D., Department of Physics James L. Blank, Dean, Ph.D., College of Arts and Sciences Contents
TABLE OF CONTENTS ...... iii
LIST OF FIGURES ...... vi
LIST OF TABLES...... x
ACKNOWLEDGEMENTS ...... xi
1 Introduction 1 1.1 Heavy-Ion Collisions ...... 1 1.2 Structure of Dissertation ...... 2 1.3 Natural Units ...... 2
2 Quantum Chromodynamics 4 2.1 Confinement ...... 6 2.2 Running Coupling ...... 7 2.3 Lagrangian ...... 7 2.4 Finite Temperature Effects ...... 9
3 Strong Magnetic Field Effects on Quarkonia 13
4 Pseudomomentum 15 4.1 Two Coupled Particles in a Constant Magnetic Field ...... 17 4.2 Two Particles With Equal and Opposite Charge ...... 19 4.3 Heavy-Light System ...... 20 4.4 Particle-Antiparticle Pair ...... 21 4.5 Pseudomomentum and Kinetic COM Momentum ...... 21 4.6 The Motional Stark Effect ...... 22
iii 5 Heavy Quark Model 24 5.1 Heavy Quark Effective Theory ...... 24 5.2 Heavy Quark Lagrangian ...... 25 5.3 Heavy Quark Hamiltonian ...... 32
6 Harmonic Oscillator 38 6.1 Analytic Solution ...... 38 6.2 Sudden Approximation ...... 41
7 Numerical Solution 49 7.1 Solving the Schr¨odinger Equation ...... 51 7.2 Finding the Ground State ...... 52 7.3 Finding the Excited States ...... 52 7.4 Center-of-Mass Kinetic Energy Subtraction ...... 53
8 Vacuum Results 55 8.1 Bottomonia ...... 55 8.2 Charmonia ...... 55
9 Spin-Mixing 58
10 Final Results 64 10.1 Bottomonia ...... 66 10.2 Charmonia ...... 71 10.3 Discussion ...... 76 10.4 What is a Particle? ...... 77
11 Search for Three-body Hypertriton Decay 80 11.1 History of the Hypertriton ...... 80 11.2 Decay Channels ...... 81
iv 11.3 Experimental Search ...... 81 11.4 Previous Measurements ...... 82
12 The STAR Experiment 84 12.1 Time Projection Chamber ...... 85 12.2 Time of Flight ...... 86
13 Identification of Hypertriton Candidates 87 13.1 Background Subtraction ...... 88
14 Locating the Decay Vertex 90
15 Particle Identification 93 15.1 Deuteron Selection ...... 93 15.2 Time-of-Flight Correction ...... 95
16 Background Suppression 99 16.1 TMVA ...... 100 16.2 Simulation ...... 101
17 Yields 105
18 Other Measurements Considered 109 18.1 Embedding ...... 110
19 Summary 112
References 114
v LIST OF FIGURES
Figure 1. Schematic of confinement. The number of lines between the quark (q) and antiquark (¯q) represent the strength of the attractive force. As theq ¯ are forced to separate the force becomes stronger until there is enough energy to create a new qq¯ pair...... 7
Figure 2. Running coupling of the strong force...... 8
Figure 3. Debye screening in classical electrodynamics. Positively charged particles (red) will attract negatively charged particles (blue) from the medium reducing the force between the positively charged particles. The dashed circles represent the Debye screening length ...... 10
Figure 4. The Cornell potential in vacuum and finite temperature. In QCD with 6 flavors,
7g3 β is related to the coupling constant gs by β = − 16π2 . Figure taken from [1] . . 11
Figure 5. a) Rest masses of 1s bottomonium states. b) Rest masses of 1p bottomonium states...... 65
Figure 6. a) Rest masses of 1s charmonia states. b) Rest masses of 1p charmonia states.66
Figure 7. a) Rest masses of the spin-mixed 1s bottomonium states. b) Rest masses of the unmixed degenerate spin-triplet 1s bottomonium states ...... 68
Figure 8. Rest masses of the non-degenerate spin-mixed 1p bottomonium states...... 68
Figure 9. Rest masses of the degenerate spin-mixed 1p bottomonium states ...... 69
Figure 10. a) Rest masses of the unmixed degenerate 1s bottomonium states. b) Rest masses of the unmixed non-degenerate 1p bottomonium states...... 70
Figure 11. a) Rest masses of the spin-mixed 1s charmonium states. b) Rest masses of the unmixed degenerate spin-triplet 1s charmonium states ...... 73
vi Figure 12. Rest masses of the non-degenerate spin-mixed 1p charmonium states...... 74
Figure 13. Rest masses of the degenerate spin-mixed 1p charmonium states ...... 74
Figure 14. a) Rest masses of the unmixed degenerate 1s charmonium states. b) Rest masses of the unmixed non-degenerate 1p charmonium states...... 75
Figure 15. Left: Decay length measurements of the lambda hyperon and hypertriton. The insert shows the χ2 analysis used to obtain the lifetime measurement. Right: Measured lifetimes and theory calculation of the hypertriton. Figure taken from [2]...... 83
Figure 16. Energy loss per centimeter in the TPC vs. particle momenta. The upper band are deuterons, the middle band are protons, and the bottom band are pions. . .85
Figure 17. Schematic of rotation technique used to estimate the combinatorial background.
The transverse momentum (pt) vector and decay vertex position (rt) vectors for a daughter track are rotated about the primary vertex (PV) in the plane transverse to the beam direction (arbitrary scale). Different versions of this technique may use different daughter tracks and different rotation angles...... 89
Figure 18. Left: Hypertriton invariant mass distribution (red) with combinatorial background (blue) estimated by rotating the pion tracks. Right: Hypertriton invariant mass distribution (red) with combinatorial background (blue) estimated by rotating the deuteron tracks...... 89
Figure 19. Schematic of Λ → p+π− decay. The decay is assumed to take place at the midpoint of the line segment representing the distance of closest approach between the daughter particles. A two-body hypertriton decay may be obtained by replacing the proton with a helium-3 nucleus ...... 91
vii Figure 20. Schematic of 3-body hypertriton decay. The decay is assumed to take place at the center of a triangle formed by the midpoints of the line segments representing the distances of closest approach between each pair of daughter particles...... 91
Figure 21. 2D histograms showing deuteron selection for the three matter data sets used. The red box indicates the locations of the track cuts. The antimatter data from the 2011 run at 200 GeV is similar to the corresponding matter data set but with tighter cuts on the horizontal axis. See text for exact cuts ...... 96
Figure 22. Masses of daughter particles with (blue) and without (red) the TOF correction97
Figure 23. Reconstructed hypertriton mass spectrum (red) and estimated combinatorial back- ground (blue) using the TOF correction. This plot was made using the 2011 run, 200 GeV (matter) data set ...... 98
Figure 24. Left: Definition of the pointing angle θ. Right: Distribution of θ. Hypertriton candidates with a large pointing angle cannot be real ...... 100
Figure 25. Transverse momentum vs. the distance of closest approach to the primary vertex of the daughter particles for real and simulated data...... 102
Figure 26. Distance of closest approach between pairs of daughters vs. the reconstructed hypertriton transverse momentum for real and simulated data. Due to limited data storage, no candidate with a DCA larger than 1 cm was saved ...... 102
Figure 27. Distances of closest approach between different daughter particles plotted against each other for real and simulated data. Here we see a weak correlation in the simulated data between the inter-particle DCAs involving the pion but it is not strong enough to define a new cut without significantly reducing the number of accepted hypertriton candidates ...... 103
viii Figure 28. Distance between the decay vertex and the midpoint of the inter-particle DCA vs. the reconstructed hypertriton transverse momentum for real and simulated data...... 103
Figure 29. Top: Distribution of real and simulated deuteron momentum transverse to hyper- triton momentum. Middle: Distribution of real and simulated proton momentum transverse to hypertriton momentum. Bottom: Distribution of real and simulated pion momentum transverse to hypertriton momentum. All data sets have similar distributions for these three quantities. The red vertical lines indicate the locations of the cuts ...... 104
Figure 30. Background-subtracted reconstructed hypertriton invariant mass spectrum with a bin-width of 4 MeV/c2. The blue arrow shows the location of the hypertriton mass (2.991 GeV/c2). There is a narrow peak in the data at this point for the 2011 run data (bottom row). For the 2010 run data (top row), there is a narrow peak located 4 MeV/c2 higher than the known hypertriton mass ...... 106
Figure 31. Background-subtracted reconstructed hypertriton invariant mass spectrum with a bin-width of 4/3 MeV/c2. The blue arrow shows the location of the hypertriton mass (2.991 GeV/c2). There is a narrow peak in the data located 4/3 MeV/c2 higher than this point for the 2011 run data (bottom row) and 8/3 MeV/c2 higher than this point for the 2010 run data (top row) ...... 107
Figure 32. Reconstructed lambda invariant mass spectrum (with combinatorial background) for the decay Λ → p + π−. The large peak is centered on the lambda hyperon mass of 1.116 GeV/c2 ...... 108
ix LIST OF TABLES
Table 1. Unit Conversions ...... 3
Table 2. Expectation Values...... 50
Table 3. Vacuum Bottomonia Results ...... 56
Table 4. Vacuum Charmonia Results ...... 57
Table 5. Vacuum Eigenstates ...... 59
Table 6. Modified Eigenstates ...... 60
Table 7. Bottomonia Eigenstates with Moderate Magnetic Field ...... 67
Table 8. Bottomonia Eigenstates with Strong Magnetic Field ...... 67
Table 9. Charmonia Eigenstates with Moderate Magnetic Field ...... 72
Table 10. Charmonia Eigenstates with Strong Magnetic Field...... 72
Table 11. Eigenstates with Infinite Magnetic Field ...... 78
Table 12. Data Statistics ...... 84
Table 13. Decay Vertex Cuts ...... 92
x Acknowledgements
First, I would like to thank my advisors, Michael Strickland and Declan Keane for their guidance and support. Their generosity and understanding made it possible for me to pursue my education without sacrificing my family life. Although he was never officially my research advisor, Spiros Margetis also provided me with much useful advice. Thanks are also due to the many people who helped me with this research. Specifically, I would like to mention Amilkar Quintero, who did most of the TMVA work; Jinhui Chen, who studied the two-body hypertriton decay and provided me with many valuable suggestions; Yuhui Zhu, who also studied the three-body hypertriton decay and provided useful discussions; Jonathan Bouchet for his help with the hypertriton simulation; and Patrick Huck for his help with the time- of-flight correction. Finally, I would like to thank my family for their enduring faith in me during this long journey, especially my wife, Alana Criswell, and son, Bowen Criswell.
xi 1 Introduction
1.1 Heavy-Ion Collisions
A physics professor once said, “The most primitive way to learn how a locomotive works is to force two of them to crash at high speed and examine the pieces.” This is the basic idea behind relativistic heavy-ion collisions. We learn about nuclear matter by accelerating nuclei to near the speed of light, forcing them to collide, then studying the particles produced. The fundamental theory describing the interaction between nucleons is quantum chromodynamics (QCD). It is a very complicated theory making it very difficult to do most calculations from first principles. Instead, we must rely on lattice calculations and phenomenological models to understand heavy-ion collisions. The hot dense matter formed in these collisions is believed to be very similar to the matter that filled the universe immediately after the big bang. The energy involved in the collision is enough to create new particles that were not present in the colliding nuclei. The newly created particles may be matter or antimatter. Many of the particles produced are never observed directly because they are unstable and quickly decay into lighter particles. In addition to producing many exotic particles, heavy-ion collisions also provide a medium to test predictions of QCD. Because the nuclei are moving at relativistic speeds, Lorentz contraction turns approxi- mately spherical nuclei into an oblate spheroid in the lab frame. Most of the collisions are not head-on, so some of the nucleons, called spectators, are not involved in the collision. At high energy the participant nucleons break into their constituent partons and form a quark- gluon plasma. As the plasma expands and cools, particles are released in all directions.
1 Large particle detectors are placed near the point of the collision. The information from the released particles is used to reconstruct the collision event. In addition to producing particles from the energy of the collision, a short-lived but very intense magnetic field is expected to be generated in the quark-gluon plasma. To fully understand particle production, one must account for these magnetic effects.
1.2 Structure of Dissertation
Over the course of my research I worked on several projects, some experimental and some theoretical. The first part of this dissertation describes a theoretical exploration of the effects of very strong magnetic fields on quarkonia, specifically, bottomonia and charmonia. After the concept of pseudomomentum is introduced and the model Hamiltonian is derived, the results of a numerical calculation with no background magnetic field are presented and com- pared to experimental results. The same numerical algorithm is used to calculate quarkonia energy levels for various values of magnetic field strength neglecting the interaction between the particle spin and the background magnetic field. The spin-field interaction is accounted for and the final results are presented. The second part describes an experimental search for a particular three-body decay of hypertriton particles, the nuclei of an unstable heavy isotope of hydrogen. Hypertritons are produced in heavy-ion collisions but decay into smaller particles so they cannot be observed directly. Instead we must look for the decay products and reconstruct the original parent particle. If enough hypertritons are found, the expected lifetime can be calculated and compared to previous calculations.
1.3 Natural Units
Before making any measurement, one must first choose an appropriate set of units. There is a set of units commonly used by physicists which is called “natural” because it simplifies many calculations by setting fundamental constants, such as the speed of light and Planck’s
2 constant, equal to one. This may seem impossible since, in SI units, the speed of light is a very large number and Planck’s constant is a very small number but the conversion is no more difficult than converting to any other set of units. The conversion factors are listed in Table 1. In natural units, everything has units of energy raised to an integer power. The factors of the speed of light and Planck’s constant are not always written but can easily be replaced using dimensional analysis.
Unit Conversions SI units Natural units Energy 1.6022 × 10−10 J 1 GeV Mass 1.7827 × 10−27 kg 1 GeV/c2 Time 6.5822 × 10−25 s 1 GeV−1~ Distance 1.9733 × 10−16 m 1 GeV−1~c Magnetic Field 1.6904 × 1020 Gauss 1 GeV2(e~c2)−1
Table 1: Conversions between SI and natural units. Formally, the natural units should include factors of Planck’s constant and the speed of light but, in practice, they are usually not written.
3 2 Quantum Chromodynamics
There are many similarities between QCD and the theory of quantum electrodynamics (QED). Instead of two degrees of freedom (positive and negative charges) like QED, QCD has three degrees of freedom, called color charges, which are arbitrarily labeled red, blue, and green. Like QED, the force between different charges is attractive, the force between like charges is repulsive and when all degrees of freedom are combined the overall charge is neutral. Antimatter particles have the opposite electric charge relative to their matter partners. The color charges for antimatter are antired, antiblue and antigreen. When two different color charges are combined, they form the antimatter version of the third color charge. When a color charge is combined with an anticolor charge of the same color, the result is color neutral; this is the case for quarkonia. Under normal circumstances (low temperature, low pressure, ...), quarks always combine in ways that are color neutral. There are only two types of color neutral particles composed of quarks: mesons and baryons. Mesons are bound states of a quark and an antiquark and baryons are bound states of three quarks of different colors. The strong force between quarks is transmitted by gauge bosons called gluons which are analogous to photons from QED. Another important difference between QCD and QED, known as confinement, is the fact that, unlike electric charges and photons, free quarks and gluons are not found in nature. QCD was first proposed in 1963 by Gell-Mann and Zweig [4]. By this time, QED had already been established and there was much experimental data to be explained. The light mesons had the right quantum numbers to be bound states of quarks and antiquarks and baryons were interpreted as bound states of three quarks. The correct electric charges
4 and other quantum numbers were obtained by assuming three flavors of quarks (up, down, strange). The others (charm, bottom, top) were discovered later. Theoretically, there could be many more but they would have to be more massive than the top quark or they would have already been discovered. Nature has already pushed the limit of weirdness with the existence of the top quark, which is about 175 times more massive than the proton, so it seems unlikely that heavier quarks will ever be discovered. To get the correct electric charges of baryons the up, charm and top quarks must have a charge of +2/3e and the down, strange and bottom quarks have a charge of −1/3e, where e is the elementary charge. Now the nucleon charges can be easily explained if the proton is made up of two up quarks and a down quark and the neutron is two down quarks and an up quark. Quarks have spin 1/2, so they must obey Fermi-Dirac statistics but to satisfy the observed baryon spectrum the quark wavefunction must be symmetric under the interchange of spin and flavor quantum numbers. For example, the ∆++ is a spin-3/2 nucleon with charge +2. This particle can be easily understood as a bound state of three up quarks with their spins parallel and no orbital angular momentum but without another quantum number it would not obey the proper spin statistics. The additional quantum number is provided by the color charge. Since there are three quarks in a baryon and three color charges, the baryon wavefunction can be totally symmetric in spin and flavor but totally antisymmetric in color making it totally antisymmetric overall and satisfying the Fermi-Dirac statistics. This is how the color charge was originally identified as the proper degree of freedom to describe the strong nuclear force. Mathematically, we say QED belongs to the group U(1) and there is one type of photon that forms the generator for the group. QCD belongs to the group SU(3) with eight types
2 of gluons as the generators. The number of generators is given by Nc − 1, where Nc is the number of different charges in the theory. QED is an Abelian gauge theory and QCD is a non-Abelian gauge theory. They both have a vector potential and different components of
5 the potential commute in QED but not in QCD.
2.1 Confinement
Confinement is the physical manifestation of the non-Abelian nature of QCD. It is a result of gluons having a color charge causing them to couple to each other and quarks. Confinement does not happen in QED because photons do not have an electric charge. Confinement can be understood qualitatively by imagining the quarks are connected by dense field lines that form a string. The string is described as a color flux tube with constant cross-sectional area and constant energy density. As the quarks are separated, the force holding them together becomes stronger and it becomes energetically favorable to break the string and create a new pair of quarks, see Figure 1. Mathematically, confinement can be understood in terms of the Cornell potential,
4 α V = − s + σr , (1) Cornell 3 r
where αs is the strong coupling constant, σ is the string tension and r is the distance between the quarks. The first term is like the Coulomb potential from classical electrodynamics with a different coupling strength 1. The second term comes from the non-Abelian part of QCD and is responsible for confinement. It can be understood as the potential energy stored in the string which, due to the constant energy density, increases linearly with length. From Eq. (1) it is easy to see that, when the quark separation is small, the Coulomb part dominates and the theory becomes almost identical to QED but, for large quark separation, the linear part dominates and the quarks are never able to become free. Despite much effort, a rigorous mathematical proof of confinement does not exist [5]. However, simple potential models like the one above describe the observed particle spectra very well.
1The factor of 4/3 is called the color factor and is a result of the existence of six different color charges (including anti-colors). The color factor is 4/3 for color-singlet states and −1/6 for color-octet states.
6 Figure 1: Schematic of confinement. The number of lines between the quark (q) and anti- quark (¯q) represent the strength of the attractive force. As the qq¯ are forced to separate the force becomes stronger until there is enough energy to create a new qq¯ pair.
2.2 Running Coupling
The coupling constant in QCD, αs, is not constant. As the distance between interacting quarks decreases, so does the strength of the coupling constant. Since probing the interaction at shorter distances requires higher energy it is also correct to say that the strength of the coupling constant decreases with increasing energy. This is known as asymptotic freedom and is a prediction of perturbative QCD. Figure 2 shows the remarkable agreement between theoretical prediction and experimental results. Asymptotic freedom is also a result of the non-Abelian nature of QCD. In fact, any renomalizable quantum field theory must have non-Abelian gauge fields to be asymptotically free [6].
2.3 Lagrangian
The QCD Lagrangian satisfies the charge and parity symmetries and quark flavor conserva- tion. Although QCD is much more complicated than QED, its Lagrangian can be written in a way that looks like the QED Lagrangian,
1 L = Ψ(¯ iγ Dµ − m)Ψ − F F µν , (2) QCD µ 4 µν
7 Sept. 2013 decays (N3LO) s(Q) Lattice QCD (NNLO) DIS jets (NLO) 0.3 Heavy Quarkonia (NLO) – e+e jets & shapes (res. NNLO) Z pole fit ( N3LO) (–) pp –> jets (NLO) 0.2
0.1 QCD s(Mz) = 0.1185 ± 0.0006 1 10 100 1000 Q [GeV] Figure 2: Running coupling of the strong force. with the covariant derivative defined as
a Dµ = ∂µ − igτaAµ , (3) and the gluon field strength tensor defined as
a a b c a Fµν = ∂µAν − ∂νAµ + gsfabcAµAν τ , (4) where the Greek indices run from 0 to 3, Roman indices run from 1 to 8 and repeated indices are summed over. The γµ are Dirac matrices and Ψ is the Dirac spinor for the quark field. A sum over quark flavors is implied. The τ a matrices satisfy the commutation relations
a b abc a a [τ , τ ] = if τc and are related to the Gell-Mann matrices by τ = λ /2. The Gell-Mann matrices play the same role in QCD as the Pauli spin matrices in QED. The f abc parameters, which are not present in QED, are called structure constants. The parameter gs is sometimes
2 called the coupling constant and is related to αs by gs = 4παs. The field tensor can be defined in terms of a commutation relation between covariant
8 derivatives,
igsFµν = [Dµ,Dν] . (5)
There is a conserved color current, Jµ, analogous to the electric current from QED, related to the field tensor by
ν D Fµν = gJµ . (6)
These are Maxwell’s equations for the strong interaction. Unlike Maxwell’s equations for electrodynamics, these equations are non-linear, making a general solution impossible. These equations are written in a way that makes the similarity to QED obvious but it is important to remember that QCD contains matrix-valued currents,
a Jµ = Jµτa , (7)
a Fµν = Fµντa . (8)
2.4 Finite Temperature Effects
So far we have only considered quarks interacting with each other in a vacuum. In heavy-ion collisions, the quarks are interacting in a very hot and dense medium so we should consider the effects of finite temperature. Like most states of matter, a collection of quarks will expand and become less dense at high temperatures. The temperature dependence of the average momentum of a quark in medium is given by
R d3p 1 2 −p/T 3 p e h~p 2i = (2π) 2Ep R d3p 1 −p/T 3 e (2π) 2Ep = T 2 + ... . (9)
9 The first term is obtained by considering only the full relativistic case, Ep = |~p|. The other terms are of order m/T which are negligible at high temperature. Since |~p| ∼ ph~p 2i ∼ T , the running coupling, αs, decreases with increased momentum. At very high temperature the quarks overcome confinement and form a quark-gluon plasma. To understand how quarks interact in the hot, dense medium it is useful to first un- derstand how electrically charged particles behave in an ionized medium. If a positively ion is placed in a medium with free electrons, the electrostatic force will cause some of the electrons to gather near the ion, reducing it’s effective charge at large distances (see Figure
3). The result is that the Coulomb potential becomes modified by a factor of e−r/λD , where
λD ∼ gs/T is the Debye screening length. This effectively turns the long-range Coulomb force into a short-range force. The screening effect in QCD is similar to Debye screening in classical electrodynamics. The medium screens the color charge, lowering the effective coupling strength. The linear part of the Cornell potential is modified to allow the quarks to become deconfined, see Figure 4. In QCD, the screening effect reduces the tension in the color flux tube connecting two quarks.
Figure 3: Debye screening in classical electrodynamics. Positively charged particles (red) will attract negatively charged particles (blue) from the medium reducing the force between the positively charged particles. The dashed circles represent the Debye screening length.
10 Figure 4: The Cornell potential in vacuum and finite temperature. In QCD with 6 flavors, 7g3 β is related to the coupling constant gs by β = − 16π2 . Figure taken from [1].
11 Theory: Quarkonia in a Very Strong Magnetic Field 3 Strong Magnetic Field Effects on Quarkonia
People have been aware of magnets since ancient times. They were used for navigation centuries before any scientific understanding of electromagnetic fields. Now, more than a century after learning how to generate magnetic fields and how they affect charged particles, magnetic fields are important components of many technological innovations but there is still much to be learned about how they affect matter. One of the first experiments on magnetic field effects on matter were conducted over one hundred years ago by Pieter Zeeman where he showed that an external magnetic field affected the spectrum of light emitted by a flame [7, 8, 9]. Some of the most interesting effects are only measurable when the field strength is very high. Such high fields were once thought to not exist in nature but new evidence suggests they may be generated in high-energy heavy-ion collisions. They may also be present in a class of compact stars called magnetars. During early times after non-central heavy ion
2 18 collisions one expects B ∼ mπ ∼ 10 Gauss at energies probed by the Relativistic Heavy
2 19 Ion Collider (RHIC) and B ∼ 15 mπ ∼ 1.5 × 10 Gauss at Large Hadron Collider (LHC) energies [10, 11, 12, 13, 14, 15] and in the interior of magnetars the central magnetic fields are on the order of 1018-1019 Gauss [16]. For comparison, the strength of a typical refrigerator magnet is 50 Gauss and 1.6 × 105 Gauss is enough to levitate a frog [17]. There has been much work in recent years related to the chiral magnetic effect [10]. It has been shown how to self-consistently take into account this effect through Berry curvature flux in the presence of a magnetic field [18, 19]. The existence of such high magnetic fields has also prompted many research groups to study how the finite temperature deconfinement
13 and chiral phase transitions are affected by the presence of a strong background magnetic field. These studies have included direct numerical investigations using lattice QCD [20, 21, 22, 23, 24] and theoretical investigations using a variety of methods including, for example, perturbative QCD studies, model studies, and string-theory inspired anti-de Sitter/conformal field theory (AdS/CFT) correspondence studies [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. We will consider the effects of magnetic fields on heavy quarkonium states, focusing on s and p charmonium and bottomonium states. The physics of quantum mechanical bound states in a background magnetic field is complicated by the fact that in a background magnetic field the center-of-mass (COM) momentum is not a conserved quantity due to the breaking of translational invariance by the vector potential. Instead, one must take into account the Lorentz force on the constituents and construct a quantity called the COM pseudomomentum [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. However, in practice one finds that, even after expressing the Hamiltonian in terms of the pseudomomentum, it is not possible to factorize the Hamiltonian into free COM motion plus decoupled internal motion. As a result, the spectrum of a bound state in a background magnetic field depends on the COM momentum of the system. The first theoretical consideration of motional effects was by Lamb [53] and, as we will show, this effect is related to the motional Stark effect. Heavy quarkonium is a nice test bed for QCD since heavy quark states are dominated by short distance physics and can be treated using heavy quark effective theory [65]. Based on such effective theories of QCD, non-relativistic quarkonium states can be reliably described.
Their binding energies are much smaller than the quark mass mq ΛQCD (q = c, b), and their sizes are much larger than 1/mq. Since the velocity of the quarks in the bound state is small (v c), quarkonium can be understood in terms of non-relativistic potential models such as the Cornell potential which can be derived directly from QCD using effective field theory [66, 67, 68].
14 4 Pseudomomentum
The effects of strong magnetic fields on meson masses have been investigated in Ref. [69]. There it is shown that the mass does not become negative as was previously suggested in Ref. [70]. The concept of pseudomomentum is introduced and the author claims its use allows the center-of-mass motion to be factored out of the Hamiltonian but we will see later that this is not the case. As one would expect from classical electrodynamics, the center-of-mass motion is very important for mesons in a magnetic field. We begin with the basics by introducing the pseudomomentum in the context of a single classical non-relativistic charged spin one-half particle in a background magnetic field. Unlike the particle momentum, the pseudomomentum is conserved since it takes into account the Lorentz force on the particle. The classical non-relativistic Hamiltonian for a particle in a constant magnetic field can be written
1 H = [p − qA(r)]2 + V (r) − µ · B + m , (10) 2m where m is the rest mass of the particle and we assume B(x) = (0, 0,B) which, in symmetric
1 1 gauge, can be expressed in terms of the vector potential A(r) = 2 B × r = 2 B(−y, x, 0). We can apply Hamilton’s equations to derive the equation of motion
∂H − =p ˙i , ∂ri ∂H =r ˙i . (11) ∂pi
15 The second Hamilton equation gives mr˙i = pi −qAi, which allows us to solve for the canonical
momentum, pi = mr˙i + qAi. Using this, we can evaluate the full time-derivative of the canonical momentum
∂Ai drj ∂Ai p˙i = mr¨i + q + , ∂t dt ∂rj
∂Ai = mr¨i + qr˙j , (12) ∂rj
where, in going from the first to second line, we have used the fact that the vector potential is static in the case under consideration. The right hand side of the first Hamilton equation gives
∂H 1 ∂A ∂V − = (p − qA(r)) · q − , ∂ri m ∂ri ∂ri
∂Aj ∂V = qr˙j − . (13) ∂ri ∂ri
Equating the two sides, we obtain
∂Aj ∂Ai ∂V mr¨i = qr˙j − qr˙j − . (14) ∂ri ∂rj ∂ri
Using v × B = r˙ × (∇ × A) = ∇(r˙ · A) − (r˙ · ∇)A, we can rewrite this as
m¨r = qr˙ × B − ∇V. (15)
In the case that the there is no potential, V = 0, we have only the Lorentz force acting on the particle m¨r = qr˙ × B , (16) which shows that the momentum is not conserved in a constant magnetic field, as expected;
16 however, we can introduce a quantity which is conserved called the pseudomomentum, K,
K = mr˙ + qB × r , q = p + B × r , 2 = p + qA , (17) such that the equation of motion can be expressed as
d K = 0 . (18) dt
4.1 Two Coupled Particles in a Constant Magnetic Field
We next consider the case of two particles subject to a translationally invariant potential in non-relativistic quantum mechanics. We will follow closely the treatment found in Ref. [62]. The Hamiltonian operator for two particles in a constant magnetic field can be written as
1 2 1 2 H = [p1 − q1A(r1)] + [p2 − q2A(r2)] + V (r1 − r2) − µ · B + m1 + m2 , (19) 2m1 2m2
where µ = µ1 + µ2 is the sum of the two particles’ magnetic moments and B(x) = (0, 0,B),
1 1 which can be expressed in terms of the vector potential A(r) = 2 B × r = 2 B(−y, x, 0) th in symmetric gauge. As usual, pi = −i∇ is the momentum operator for the i particle. As in the previous section, one finds that the COM momentum of the system is no longer conserved. This is due to the breaking of translational invariance by the vector potential (changing the origin changes A). In order to preserve translational invariance in a constant magnetic field, an additional gauge transformation is required. This can be achieved by introducing the generalized pseudomomentum operator [62]
2 X ∂ Z rj ∂A K = −i − q · dr , (20) k ∂x j ∂x j=1 jk 0 k
17 where k = 1, 2, 3 denotes cartesian components and j = 1, 2 indexes each of the particles. Integrating, and discarding a constant, one obtains
2 X K = (pj − qjAj + qjB × rj) . (21) j=1
1 In the gauge used herein, we have A(r) = 2 B × r, which allows us to simplify this to
2 X 1 K = p + q B × r , j 2 j j j=1
2 X = (pj + qjAj) . (22) j=1
This is the generalization of the one-particle case obtained in the previous section. One can verify explicitly that the pseudomomentum operator commutes with the Hamiltonian
[K, H] = 0 . (23)
One can also compute the commutator of two components of K,
2 ! X [Kk, Kl] = −iεklmBm qj , (24) j=1 which means that one will only be able to determine all components of K simultaneously for a system with no net electric charge.
18 4.2 Two Particles With Equal and Opposite Charge
In this section we specialize to the case that q1 = −q2 = q. To proceed, we introduce center of mass and relative coordinates
m r + m r R = 1 1 2 2 , M
r = r1 − r2 , (25)
where M = m1 + m2. As is standard, we can express the individual positions as
mr r1 = R + r , m1 mr r2 = R − r , (26) m2
where mr = m1m2/M is the reduced mass. This allows us to simplify the pseudomomentum operator
2 X 1 K = p + q B × r , j 2 j j j=1 ∂ ∂ 1 = −i + + qB × (r1 − r2) , ∂r1 ∂r2 2 ∂ 1 = −i + qB × r . (27) ∂R 2
Since the system is neutral, the full two-particle eigenfunctions, Φ, of the Hamiltonian are simultaneous eigenfunctions of all components, Ki, of the pseudomomentum with eigenvalues
Ki. This allows us to factorize the full wavefunction
1 Φ(R, r) = exp i K − qB × r · R Ψ(r) ≡ φ(R, r)Ψ(r) , (28) 2
which satisfies KjΦ = KjΦ by construction.
19 Expanding out the two-particle Hamiltonian, one finds the “relative” Hamiltonian
K2 q p2 q 1 1 Hrel = − (K × B) · r + + − B · (r × p) 2M M 2mr 2 m1 m2 2 q 2 + (B × r) + V (r) − µ · B + m1 + m2 , (29) 8mr where p = −i∇ is the relative momentum operator and one has the new eigenvalue equa- tion HrelΨ(r) = EΨ(r). Note that, unlike the case without the external field, the energy eigenvalue, E, depends on the value of K through coupling in the second term and not only through the term K2/2M.
4.3 Heavy-Light System
In the limit that m2 → ∞ while holding m1 fixed, we have M → ∞ and mr = m1 ≡ m and we obtain
p2 q q2 H = − B · (r × p) + (B × r)2 + V (r) − µ · B + m , (30) rel 2m 2m 8m
1 where we have discarded the infinite constant m2 in this case. Recalling that A = 2 B × r = 1 2 2 2 2 2 2 2 B(−y, x, 0), one has (B × r) = B ρ , where ρ = x + y . Using B · (r × p) = (B × r) · p =
ρBpφ = −iB∂φ we obtain
1 i ∂ mω2 H = − ∇2 + ω + c ρ2 + V (r) − µ · B + m , (31) rel 2m 2 c ∂φ 8
where φ is the azimuthal angle and ωc = qB/m. This is the standard non-relativistic Hamiltonian for a spin-one-half particle subject to a potential V and an external magnetic field.
20 4.4 Particle-Antiparticle Pair
For a bound state consisting of a particle-antiparticle pair we have m1 = m2 = m, M = 2m, and mr = m/2. In this case, the relative Hamiltonian simplifies to
2 2 2 K q ∇ q 2 Hrel = − (K × B) · r − + (B × r) + V (r) − µ · B + M. (32) 2M M 2mr 8mr
Next, we decompose K = Kxxˆ + Kyyˆ + Kzzˆ and simplify the expression above to obtain
2 2 2 2 K qB qB ∇ q B 2 Hrel = + Kxy − Kyx − + ρ + V (r) − µ · B + M. (33) 2M 4mr 4mr 2mr 8mr
4.5 Pseudomomentum and Kinetic COM Momentum
We now derive a general relation between the pseudomomentum and kinetic center-of-mass momentum. The COM kinetic momentum of the system is given by
X ∂ P = −i − q A , kinetic ∂r j j j j ∂ 1 = −i − qB × r . (34) ∂R 2
Therefore, we have R R Φ∗ −i ∂ − 1 qB × rΦ hP i = R r ∂R 2 . (35) kinetic R R ∗ R r Φ Φ Using ∂ 1 − i Φ = K − qB × r Φ , (36) ∂R 2 one finds
hPkinetici = K − qB × hri . (37)
21 4.6 The Motional Stark Effect
One can argue that all magnetic field effects are relativistic effects since the existence of magnetic fields is essentially a relativistic effect. With this in mind, let’s start with the state in the rest frame and boost to another inertial frame. If there is only a magnetic field in the z-direction of rest frame, the boosted frame will have electric and magnetic fields. If the boost is along the x-axis in the positive direction, the fields are given by
E = −γvB ˆy ,
B = γB ˆz , (38)
− 1 v2 2 where v is the speed of the boosted frame relative to the original frame and γ = 1 − c2 . In the non-relativistic limit γ = 1. For a boost in an arbitrary direction in the non-relativistic limit the fields transform according to
E = −v0 × B0 ,
B = B0 , (39)
where v0 is the velocity of the boosted (unprimed) frame relative to the original (primed) frame. A particle at rest in the original frame will have a velocity of v = −v0 in the boosted frame. Using the non-relativistic expression for velocity, the electric field in the boosted frame becomes 1 E = P × B , (40) M
where P is the particle’s momentum in the boosted frame and M is the particle’s mass. Using equation (36) the momentum can be expressed in terms of the pseudomomentum giving,
1 1 E = K − qB × r × B . (41) M 2
22 The electric field gives an additional interaction of the form
HStark = −qE · r q q2 = − (K × B) · r + ((B × r) × B) · r M 2M q q2 = − (K × B) · r + (B × r)2 . (42) M 2M
For a particle-antiparticle pair, M = 4mr and
2 q q 2 HStark = − (K × B) · r + (B × r) . (43) M 8mr
These are the second and fourth terms in equation (32).
23 5 Heavy Quark Model
The relatively large mass of heavy quarks (charm, bottom, and top) allows a non- relativistic treatment. The purpose of this section is not to provide a rigorous derivation of the Hamiltonian used to study heavy quarkonia, but rather give a simple, but incomplete, explanation of the interaction terms. The notation has been kept as general as possible so the results can be used for QED or QCD.
5.1 Heavy Quark Effective Theory
Physics is often simplified by considering extreme cases such as very low or very high mass. The large mass of charm and bottom quarks suggests a non-relativistic treatment may be sufficient. There are several approaches to heavy-quark physics in the literature. They typi- cally involve choosing a scale and integrating out the irrelevant degrees of freedom [71]. Most effective field theories use either Feynman diagrams or Wilson loops to match coefficients in the Lagrangian to a perturbative theory [72, 73, 74]. Some of the methods in the literature use an expansion in inverse powers of the speed of light (c) but this becomes difficult to interpret in a system of units where c = 1. In classical non-relativistic quantum chromo- dynamics (NRQCD), the Dirac Lagrangian is simplified by factoring out the time evolution factor and expanding in powers of c−1 [75]. For a review of heavy quarkonium effective field theories see Ref. [76]. There is no matching calculation in the derivation of the Hamiltonian used here, so there is no need for Feynman diagrams or Wilson loops. There is no explicit scale chosen and nothing has been integrated out. A non-relativistic approximation is made by using the fact
24 that at speeds much less than the speed of light a particle’s kinetic energy is much less than the rest mass. Instead of expanding in powers of c−1, the Dirac Lagrangian is expanded in inverse powers of the quark mass. The Lagrangian is simplified by adding higher-order terms and redefining the quark fields to eliminate all but one of the time derivatives. The Hamiltonian is obtained from the Lagrangian.
5.2 Heavy Quark Lagrangian
Starting with the Lagrangian for a Dirac field [75],
L = Ψ(¯ iγ · D − m)Ψ , (44) with
Dµ = ∂µ − igAµ , (45) and
ϕ Ψ = , (46) χ where ϕ is the quark spinor and χ is the antiquark spinor, we have
† i ϕ 1 0 iD0 − m −iσ Di ϕ L = (47) i χ 0 −1 −iσ Di iD0 − m χ
† † i † i † = ϕ (iD0 − m) ϕ − ϕ iσ Di χ − χ iσ Di ϕ + χ (iD0 + m) χ . (48)
25 From the Dirac equation, (iγ · D − m) Ψ = 0, we have
i iD0 − m −iσ Di ϕ = 0 , (49) i iσ Di −iD0 − m χ or
i (iD0 − m) ϕ − iσ Diχ = 0 , (50)
i iσ Diϕ − (iD0 + m) χ = 0 . (51)
Rearranging these equations gives
−1 i ϕ = (iD0 − m) iσ Di χ , (52)
−1 i χ = (iD0 + m) iσ Di ϕ . (53)
Substituting these expressions into the Dirac Lagrangian gives
† † i −1 i L = ϕ (iD0 − m) ϕ − ϕ iσ Di (iD0 + m) iσ Di ϕ
† i −1 i † −χ iσ Di (iD0 − m) iσ Di χ + χ (iD0 + m) χ . (54)
In the non-relativistic limit, the eigenvalue of iD0 is approximately m for particles and −m
iD0−m for antiparticles. Therefore, the operator 2m has a small positive eigenvalue when acting
iD0+m on ϕ and the operator 2m has a small negative eigenvalue when acting on χ. Now, the −1 operator (iD0 ± m) can be expanded in a Taylor series,
1 iD ∓ m −1 (iD ± m)−1 = 0 ± 1 0 2m 2m 1 iD ∓ m = ± 1 ∓ 0 + ... . (55) 2m 2m
26 The heavy quark-antiquark Lagrangian may now be written as
† † L = ϕ (iD0 − m) ϕ + χ (iD0 + m) χ 1 iD − m −ϕ† iσiD 2 − iσiD 0 iσiD + ... ϕ 2m i i 4m2 i 1 iD + m +χ† iσiD 2 + iσiD 0 iσiD + ... χ . (56) 2m i i 4m2 i
i j ij ij ij ijk To proceed, we will need the identity σ σ = δ − σ with σ = −i σk.
i 2 ij ij iσ Di = − δ − σ DiDj 1 = −D~ 2 + σij [D ,D ] 2 i j g = −D~ 2 + i σijG 2 ij g = −D~ 2 + ijkσ G 2 k ij = −D~ 2 − g~σ · B.~ (57)
k 1 ijk Where we have used igGµν = [Dµ,Dν] and the chromo-magnetic field B = − 2 Gij. Now, the Lagrangian may be written
1 g L = ϕ† iD + D~ 2 + ~σ · B~ − m ϕ 0 2m 2m 1 g +χ† iD − D~ 2 − ~σ · B~ + m χ + .... (58) 0 2m 2m
The second-order terms in this expansion are
1 1 ϕ† iσiD (iD − m) iσiD ϕ + χ† iσiD (iD + m) iσiD χ . (59) 4m2 i 0 i 4m2 i 0 i
27 To evaluate these we will need
i i i j iσ Di (iD0) iσ Di = −iσ σ DiD0Dj i = − σiσj (D ([D ,D ] + D D ) + ([D ,D ] + D D ) D ) 2 i 0 j j 0 i 0 0 i j i = − σiσj (D (−igE + D D ) + (igE + D D ) D ) 2 i j j 0 i 0 i j 1 h i 1 = g σ · E,~ σ · D~ + { iσiD 2 , iD } . (60) 2 2 i 0
h ~ ~ i Where we have used the chromo-electric field Ei = Gi0. Now we need to evaluate σ · E, σ · D .
h i ~ ~ i j j i i σ · E, σ · D = σ σ EiDj − σ σ DjE
~ ~ ~ ~ ij ji = E · D − D · E − σ EiDj + σ DjEi
~ ~ ~ ~ ijk = E · D − D · E + i σk{Ei,Dj} = E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ . (61)
The second-order quark term can now be written as
1 1 ϕ† iσiD (iD − m) iσiD ϕ = ϕ† iσiD (iD ) iσiD − m iσiD 2 ϕ 4m2 i 0 i 4m2 i 0 i i 1 1 = ϕ† g E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ 4m2 2 1 + { iσiD 2 , iD } − m iσiD 2 ϕ 2 i 0 i 1 1 = ϕ† g E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ 4m2 2 1 − {D~ 2 + g~σ · B,~ iD } + mD~ 2 + mg~σ · B~ ϕ . (62) 2 0
28 The second-order antiquark term can be obtained by making the substitutions ϕ → χ and m → −m. We now have the heavy quark-antiquark Lagrangian to second order in 1/m2.
1 g L = ϕ† iD + D~ 2 + ~σ · B~ − m ϕ 0 2m 2m g + ϕ† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ ϕ 8m2 1 − ϕ† {D~ 2 + g~σ · B,~ iD } − 2m D~ 2 + g~σ · B~ ϕ 8m2 0 1 g +χ† iD − D~ 2 − ~σ · B~ + m χ 0 2m 2m g + χ† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ χ 8m2 1 − χ† {D~ 2 + g~σ · B,~ iD } + 2m D~ 2 + g~σ · B~ χ . (63) 8m2 0
This can be written in a more compact way using commutation and anticommutation relations.
1 2 L = ϕ† iD − i~σ · D~ − m ϕ 0 2m g h i + ϕ† ~σ · E,~ ~σ · D~ ϕ 8m2 1 2 + ϕ†{ i~σ · D~ , (iD − m)}ϕ 8m2 0 1 2 +χ† iD + i~σ · D~ + m χ 0 2m g h i + χ† ~σ · E,~ ~σ · D~ χ 8m2 1 2 + χ†{ i~σ · D~ , (iD + m)}χ . (64) 8m2 0
29 Factoring out a factor of m and rearranging some of the terms gives
!2 iD 1 i~σ · D~ iD L = mϕ† 0 − 1 + { , 0 − 1 } ϕ m 2 2m m
1 2 g h i +mϕ† − i~σ · D~ + ~σ · E,~ ~σ · D~ ϕ 2m2 8m3 !2 iD 1 i~σ · D~ iD +mχ† 0 + 1 + { , 0 + 1 } χ m 2 2m m
1 2 g h i +mχ† i~σ · D~ + ~σ · E,~ ~σ · D~ χ . (65) 2m2 8m3
iDi Every term in this Lagrangian is of second-order or lower in m . In the non-relativistic
† iDi limit, ϕ m ϕ 1, so we can shift the Lagrangian by a negligible amount by adding terms
iDi of higher order in m . L → L + ∆L where
!2 !2 m i~σ · D~ iD i~σ · D~ ∆L = ϕ† 0 − 1 ϕ 4 2m m 2m !2 !2 m i~σ · D~ 1 2 g h i i~σ · D~ + ϕ† − i~σ · D~ + ~σ · E,~ ~σ · D~ ϕ 4 2m 2m2 8m3 2m !2 m i~σ · D~ 1 2 g h i + ϕ†{ , − i~σ · D~ + ~σ · E,~ ~σ · D~ }ϕ 2 2m 2m2 8m3 !2 !2 m i~σ · D~ iD i~σ · D~ + χ† 0 + 1 χ 4 2m m 2m !2 !2 m i~σ · D~ 1 2 g h i i~σ · D~ + χ† i~σ · D~ + ~σ · E,~ ~σ · D~ χ 4 2m 2m2 8m3 2m !2 m i~σ · D~ 1 2 g h i + χ†{ , i~σ · D~ + ~σ · E,~ ~σ · D~ }χ . (66) 2 2m 2m2 8m3
30 Now the Lagrangian reduces to
iD 1 2 g h i L = mϕ˜† 0 − 1 − i~σ · D~ + ~σ · E,~ ~σ · D~ ϕ˜ m 2m2 8m3 iD 1 2 g h i +mχ˜† 0 + 1 + i~σ · D~ + ~σ · E,~ ~σ · D~ χ˜ , (67) m 2m2 8m3
with
!2 1 i~σ · D~ ϕ˜ = 1 + ϕ 2 2m
!2 1 i~σ · D~ χ˜ = 1 + χ . (68) 2 2m
This procedure is very similar to the quark field redefinition used in Ref. [75]. Substituting 2 h i i~σ·D~ ~ ~ the expressions for 2m and ~σ · E, ~σ · D gives
1 g L =ϕ ˜† iD − m + D~ 2 + ~σ · B~ ϕ˜ 0 2m 2m g + ϕ˜† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ ϕ˜ 8m2 1 g +˜χ† iD + m − D~ 2 − ~σ · B~ χ˜ 0 2m 2m g + χ˜† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ χ˜ . (69) 8m2
Neglecting the fast degrees of freedom, this is the same as the NRQCD Lagrangian from Eq. (10) of Ref. [75] but without the undetermined coefficients.
31 5.3 Heavy Quark Hamiltonian
The Hamiltonian is defined in terms of the Lagrangian by [77]
† † H =ϕ ˜ (i∂0)ϕ ˜ +χ ˜ (i∂0)χ ˜ − L 1 g =ϕ ˜† −gA + m − D~ 2 − ~σ · B~ ϕ˜ 0 2m 2m g − ϕ˜† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ ϕ˜ 8m2 1 g +˜χ† −gA − m + D~ 2 + ~σ · B~ χ˜ 0 2m 2m g − χ˜† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ χ˜ . (70) 8m2
For convenience, let’s separate the Hamiltonian into quark and antiquark parts.
H = Hϕ + Hχ 1 g H =ϕ ˜† −gA + m − D~ 2 − ~σ · B~ ϕ˜ ϕ 0 2m 2m g − ϕ˜† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ ϕ˜ 8m2 1 g H =χ ˜† −gA − m + D~ 2 + ~σ · B~ χ˜ χ 0 2m 2m g − χ˜† E~ · D~ − D~ · E~ + i~σ · E~ × D~ − D~ × E~ χ˜ . (71) 8m2
~ Focusing on the quark part, let’s express D in terms of the momentum operator, ~p = −iO~ , ~ and the potential, Aµ = (A0, −A). In the following, we will use a convention where the O~ operator acts only on the field immediately to it’s right and the ~p operator acts on everything to it’s right.
D~ 2 ϕ˜ = i~p + igA~ · i~p + igA~ ϕ˜
= −~p · ~p ϕ˜ − g~p · A~ ϕ˜ − gA~ · ~p ϕ˜ − g2A~ · A~ ϕ˜ 2 ~ ~ 2 ~ 2 = −~p + igO~ · A − 2gA · ~p − g A ϕ˜ . (72)
32 E~ · D~ − D~ · E~ ϕ˜ = E~ · i~p + igA~ ϕ˜ − i~p + igA~ · E~ ϕ˜
~ ~ ~ ~ ~ ~ ~ = iE · ~p ϕ˜ + igE · A ϕ˜ − O~ · E ϕ˜ − iE · ~p ϕ˜ − igA · E ϕ˜ ~ ~ ~ ~ ~ = −O~ · E + ig E · A − A · E ϕ˜ . (73)
E~ × D~ − D~ × E~ ϕ˜ = E~ × i~p + igA~ ϕ˜ − i~p + igA~ × E~ ϕ˜
~ ~ ~ ~ ~ ~ ~ = iE × ~p ϕ˜ + igE × A ϕ˜ − O~ × E ϕ˜ + iE × ~p ϕ˜ − igA × E ϕ˜ ~ ~ ~ ~ ~ ~ = 2iE × ~p − O~ × E + ig E × A − A × E ϕ˜ . (74)
The quark part of the Hamiltonian may now be written as
1 ig g g2 H =ϕ ˜† − gA + m + ~p 2 − ~ · A~ + A~ · ~p + A~ 2 ϕ 0 2m 2mO m 2m g g ig2 − ~σ · B~ + ~ · E~ − E~ · A~ − A~ · E~ 2m 8m2 O 8m2 g ig g2 + ~σ · E~ × ~p + ~σ · ~ × E~ + ~σ · E~ × A~ − A~ × E~ ϕ˜ . (75) 4m2 8m2 O 8m2
1 ijk Using Ei = Gi0 and Bi = 2 Gjk, we can express the chromo-electric and chromo-magnetic fields in terms of the potential,
i G ϕ˜ = − [D ,D ]ϕ ˜ µν g µ ν i = − ((∂ − igA )(∂ − igA ) − (∂ − igA )(∂ − igA ))ϕ ˜ g µ µ ν ν ν ν µ µ i = − [∂ , ∂ ] − ig [∂ ,A ] + ig [∂ ,A ] − g2 [A ,A ] ϕ˜ g µ ν µ ν ν µ µ ν i = − −igϕ∂˜ A + igϕ∂˜ A − g2 [A ,A ]ϕ ˜ g µ ν ν µ µ ν
= −ϕ˜ (∂µAν − ∂νAµ) + ig [Aµ,Aν]ϕ ˜ . (76)
33 This gives,
Ei ϕ˜ = −ϕ˜ (∂iA0 − ∂0Ai) + ig [Ai,A0]ϕ ˜ ∂ h i E~ = −~ A − A~ − ig A,~ A . (77) O 0 ∂t 0 and
1 B ϕ˜ = ijk (−ϕ˜ (∂ A − ∂ A ) + ig [A ,A ]ϕ ˜) i 2 j k k j j k ~ ~ ~ ~ B = O~ × A + igA × A. (78)
h ~ i ~ ~ In QED A, A0 = 0 and A × A = 0, but because of the non-Abelian nature of QCD, different components of the potential do not commute, so we must keep these terms. Sub- stituting these expressions for E~ and B~ into the ~σ · B~ and ~σ · E~ × ~p terms in the quark part of the Hamiltonian gives
1 ig g g2 H =ϕ ˜† − gA + m + ~p 2 − ~ · A~ + A~ · ~p + A~ 2 ϕ 0 2m 2mO m 2m g ig2 g ig2 − ~σ · ~ × A~ − ~σ · A~ × A~ + ~ · E~ − E~ · A~ − A~ · E~ 2m O 2m 8m2 O 8m2 g g ∂ ig2 h i − ~σ · ~ A × ~p − ~σ · A~ × ~p − ~σ · A,~ A × ~p 4m2 O 0 4m2 ∂t 4m2 0 ig g2 + ~σ · ~ × E~ + ~σ · E~ × A~ − A~ × E~ ϕ˜ . (79) 8m2 O 8m2
Keeping only the terms up to first order in g gives
1 ig g g g H =ϕ ˜† − gA + m + ~p 2 − ~ · A~ + A~ · ~p − ~σ · ~ × A~ + ~ · E~ ϕ 0 2m 2mO m 2m O 8m2 O g g ∂ ig − ~σ · ~ A × ~p − ~σ · A~ × ~p + ~σ · ~ × E~ ϕ˜ . (80) 4m2 O 0 4m2 ∂t 8m2 O
34 The QED interactions can be included by making the substitutions
QED QCD gA0 → qA0 + gA0 ,
gA~ → qA~QED + gA~QCD , (81)
e Where q is the electric charge of the quark. If we neglect the terms of order m2 , the Hamil- tonian can be written as
1 2 ig H =ϕ ˜† − qAQED − gAQCD + m + ~p − qA~QED − ~ · A~QCD ϕ 0 0 2m 2mO g q g g + A~QCD · ~p − ~σ · ~ × A~QED + ~ · E~ QCD − ~σ · ~ AQCD × ~p m 2m O 8m2 O 4m2 O 0 g ∂ ig − ~σ · A~QCD × ~p + ~σ · ~ × E~ QCD ϕ˜ . (82) 4m2 ∂t 8m2 O
~QED ~ If there is a strong background magnetic field, O~ × A = Bbackground is very large. If there QED QCD is not a strong background electric field, qA0 will be small compared to gA0 . If we neglect the QCD vector potential, the Hamiltonian simplifies to
1 2 q H =ϕ ˜† − gAQCD + m + ~p − qA~QED − ~σ · B~ ϕ 0 2m 2m background g g − ~ 2AQCD − ~σ · ~ AQCD × ~p ϕ˜ . (83) 8m2 O 0 4m2 O 0
Assuming a Cornell potential,
2α 1 σ AQCD = s − r , (84) 0 3g r 2g
the gradient and Laplacian of the QCD potential can be calculated.
2α 1 σ 1 ~ AQCD = s + ~r, O 0 3g r3 2g r 4α 1 ~ 2AQCD = s , (85) O 0 3g r3
35 where αs and σ are tunable parameters and ~r is the position vector of the quark relative to the antiquark. The derivatives are taken with respect to the position of the quark. The results for the antiquark are the same except for the sign of the gradient. The quark part of the Hamiltonian is now
2α 1 σ 1 2 q H =ϕ ˜† − s + r + m + ~p − qA~QED − ~σ · B~ ϕ 3 r 2 2m 2m background α 1 1 α 1 σ 1 − s − s + ~σ · ~r × ~p ϕ˜ , (86) 6m2 r3 2m2 3 r3 4 r
and the antiquark part is
2α 1 σ 1 2 q H =χ ˜† − s + r − m − ~p − qA~QED − ~σ · B~ χ 3 r 2 2m 2m background α 1 1 α 1 σ 1 − s + s + ~σ · ~r × ~p χ˜ . (87) 6m2 r3 2m2 3 r3 4 r
These are actually Hamiltonian densities. Integrating over all space gives the full Hamilto- nians,
2α 1 σ 1 2 q H = − s + r + m + ~p − qA~QED − ~σ · B~ ϕ 3 r 2 2m 2m background α 1 1 α 1 σ 1 − s − s + ~σ · ~r × ~p, (88) 6m2 r3 2m2 3 r3 4 r 2α 1 σ 1 2 q H = − s + r + m + ~p − qA~QED + ~σ · B~ χ 3 r 2 2m 2m background α 1 1 α 1 σ 1 − s + s + ~σ · ~r × ~p. (89) 6m2 r3 2m2 3 r3 4 r
Adding these equations together gives the full Hamiltonian for the quark-antiquark system,
1 2 1 2 4α 1 q H = ~p − qA~QED + ~p + qA~QED − s + σr − (~σ + ~σ ) · B~ 2m ϕ ϕ 2m χ χ 3 r 2m ϕ χ background α 1 1 α 1 σ 1 − s − s + (~σ · ~r × ~p − ~σ · ~r × ~p ) + 2m , (90) 3m2 r3 2m2 3 r3 4 r ϕ ϕ χ χ
36 where q is the electric charge of the quark and the subscripts ϕ and χ have been used to distinguish between quark and antiquark operators. The first two terms are the kinetic energy and electromagnetic terms, the next two terms are the Cornell potential and the last term on the first line is the interaction between the background magnetic field and the spin of the quark-antiquark system. The first term on the second line has the same form as the tensor interaction term used in Ref. [78], but with a different coefficient, and the next term has the same form as the spin-orbit term used in Ref. [78], but with a different coefficient. These coefficients are different because, in this derivation, additional terms of the same forms have been neglected. The last term is just the rest mass of the quark and antiquark. There are some important differences between the Hamiltonian in Ref. [78] and the one derived above. In this derivation, the tensor aspects of the QCD potential were ignored by treating it as a scalar. Also, the vector part of the QCD potential is probably not negligible ~ ~QED ~ and contributes to the spin-orbit interaction. By assuming O × Aϕ = Bbackground the spin- spin interaction was lost. As mentioned earlier, in this chapter, I only intended to present a simple derivation of the heavy quark Hamiltonian. In practice, I will keep all relevant terms, including the spin-orbit and spin-spin interactions.
37 6 Harmonic Oscillator
6.1 Analytic Solution
We now specialize to the case that the potential is harmonic, in which case the wave func- tions and energy levels can be obtained analytically. Some of the results contained in this subsection were first obtained explicitly by Herold et al. [62]. We repeat the derivation here in order to use them as a basis for discussion of the COM-momentum dependence of the energy. We also use this case as a check for our numerics since it can be solved analytically. Using the general relative Hamiltonian for a particle-antiparticle pair (32) and V (x) =
1 2 1 2 2 2 2 2 kx = 2 mrω0(x + y + z ) we have
2 2 2 K ∇ 1 2 ωc 2 2 ωcKy ωcKx 1 2 2 Hrel = − + mr ω0 + (x + y ) − x + y + mrω0z − µ · B + M, 2M 2mr 2 4 4 4 2 K2 ∇2 1 1 = − + a(x2 + y2) − bx + cy + dz2 − µ · B + M, (91) 2M 2mr 2 2
2 2 where ωc = qB/mr, µ = µ1 + µ2, a = mr(ω0 + ωc /4), b = ωcKy/4, c = ωcKx/4, and
2 d = mrω0. We can rewrite the third, fourth, and fifth terms using
" # 1 1 b 2 c 2 1 a(x2 + y2) − bx + cy = a x − + y + − (b2 + c2) . (92) 2 2 a a 2a
We can simplify things further by making use of a constant coordinate shiftx ¯ ≡ x − b/a and
38 y¯ ≡ y + c/a.
2 2 K ∇ 1 2 2 1 2 1 2 2 Hrel = − + a(¯x +y ¯ ) + dz − (b + c ) − µ · B + M (93) 2M 2mr 2 2 2a
which suggests that we use cylindrical coordinates withx ¯ = ρ cos φ,y ¯ = ρ sin φ, and z = z.
After this, the eigenvalue equation HrelΨ = EΨ becomes
∇2 1 1 K2 b2 − + aρ2 + cz2 Ψ(r) = E − + + µ · B + M Ψ(r) . (94) 2mr 2 2 2M 2a
Factorizing the relative wavefuction as Ψ(r) = ei`φZ(z)ψ(ρ), we find
∂2 1 ∂ |`|2 − − + + α4ρ2 ψ = 2m λψ , (95) ∂ρ2 ρ ∂ρ ρ2 r
2 √ p 2 2 2 2 2 where α = mra = mr ω0 + ωc /4, λ = E − Ez − K /2M + (b + c )/2a + µ · B + M and
Ez is the eigenvalue of the separated z-equation
∂2 − + γ4z2 Z = 2m E Z, (96) ∂z2 r z
1/4 √ where γ = (mrc) = mrω0 which has a solution
1 2 2 − 2 γ z Z = N e Hnz (γz), (97)
where Hnz (γz) are the Hermite polynomials, and energy eigenvalue
1 E = n + ω . (98) z z 2 0
Convergence as ρ → ∞ requires
2 r 2 α 2 ωc λ = (2n⊥ + 1 + |`|) = (2n⊥ + 1 + |`|) ω0 + . (99) mr 4
39 Solving for E, we obtain the energy eigenvalues for the system
r K2 ω2(K2 + K2) 1 ω2 E = − c x y + n + ω +(2n +1+|`|) ω2 + c −m ·B+M. K,n⊥nz` 2 2 z 0 ⊥ 0 r 2M 32mr(ω0 + ωc /4) 2 4 (100) We can now write the full two-particle wave function
|`| i`φ − 1 γ2z2 − 1 α2ρ2 |`| 2 2 i K− 1 qB×r ·R Φ (R, r) = N ρ e e 2 e 2 H (γz)L (α ρ )e ( 2 ) , (101) K,n⊥nz` nz n⊥
where N is a normalization constant and
qB ωc = , mr r ω2 α2 = m ω2 + c , r 0 4 ωc β = 2 2 , 4mr(ω0 + ωc /4) 2 γ = mrω0 ,
2 2 2 ρ = (x − βKy) + (y + βKx) , y + βK φ = arctan x . (102) x − βKy
Center-of-mass Kinetic Momentum
Using this, we can analytically compute the relationship between the pseudomomentum and the COM kinetic momentum of the state. Using Eq. (37) and
c b B × r = B(−y, x, 0) = B −ρ sin φ + , ρ cos φ + , 0 , (103) a a
one finds in this case qBc qBb hP i = K − xˆ − yˆ . (104) kinetic a a
40 Plugging in the definitions of a, b, and c we obtain
2 2 4ω0 4ω0 hPkinetici = 2 2 Kx, 2 2 Ky,Kz , (105) 4ω0 + ωc 4ω0 + ωc
As we can explicitly see from this expression, the components of the kinetic COM momentum do not directly correspond to the pseudomomentum components. In the next section this formula is derived in a different manner by assuming a time-dependent magnetic field which turns on rapidly.
6.2 Sudden Approximation
Here we explore what happens to a system which suddenly has a magnetic field turned on. We will model this as being instantaneous in order to simplify the treatment and restrict our attention to a linear combination of 3d harmonic oscillator eigenstates since it is possible to make much more analytic progress in this case. We start by positing that for t < 0 there is no magnetic field and that the system is subject only to an internal harmonic interaction, in which case the full state can be decomposed in terms of the no-magnetic-field eigenstates
(0) Φk X (0) (0) −iEk t Φ(t) = ckΦk e t < 0 , (106) k where k collects all relevant quantum numbers and the sum represents a sum over discrete quantum numbers and integration for continuous quantum numbers. For t ≥ 0 we can
(1) expand in terms of the eigenstates in the presence of the magnetic field Φm
(1) X (1) −iEm t Φ(t) = dmΦm e t ≥ 0 , (107) m
At t = 0, we match the coefficients which requires
X (1) (0) dn = ckhΦn |Φk i . (108) k
41 Pure state for t < 0
(1) (0) If the state for t < 0 is a pure state with ck = δkm, we obtain dn = hΦn |Φm i. We now turn to the computation of the overlap integrals necessary for the case at hand. The t < 0 states are
(0) (0) |`0| i`0φ − 1 γ2(ρ2+z2) |`0| 2 2 iP·R 2 0 Φ 0 0 0 (R, r) = N ρ e e Hnz (γz)L 0 (γ ρ )e , (109) P,n⊥nz` n⊥ where s γ|`0|+3/2 n0 ! N (0) = √ ⊥ , (110) n0 3/2 0 0 0 2 z π nz!(|` | + n⊥)! and the t ≥ 0 states are
(1) (1) |`| i`φ˜ − 1 γ2z2 − 1 α2ρ˜2 |`| 2 2 i K− 1 qB×r ·R Φ (R, r) = N ρ˜ e e 2 e 2 H (γz)L (α ρ˜ )e ( 2 ) , (111) K,n⊥nz` nz n⊥ with
qB ωc = , mr r ω2 α2 = m ω2 + c , r 0 4 2 γ = mrω0 ,
2 2 2 ρ˜ = (x − βKy) + (y + βKx) , y + βK φ˜ = arctan x , x − βKy ωc β = 2 2 , (112) 4mr(ω0 + ωc /4) and s α|`|+1γ1/2 n ! N (1) = √ ⊥ . (113) 2nz π3/2 nz!(|`| + n⊥)!
42 The six-dimensional overlap integral in relative cylindrical coordinates becomes
Z ∞ Z 2π Z ∞ Z (0) (1) 3 |`0| |`| i(`0φ−`φ˜) −γ2z2 − 1 (γ2ρ2+α2ρ˜2) dn = N N ρ dρ dφ dz d R ρ ρ˜ e e e 2 0 0 −∞ |`0| 2 2 |`| 2 2 i(P−K+ 1 qB×r)·R 0 2 ×Hnz (γz)Hnz (γz)L 0 (γ ρ )Ln (α ρ˜ )e . (114) n⊥ ⊥
1 1 1 Using 2 qB × r = 2 qB(−y, x, 0) = 2 qBρ(− sin φ, cos φ, 0) and the orthonormality of the Hermite polynomials we can perform the z and Z integrations. Using the exponential, we can further perform the X and Y integrations. The remaining two integrals are evaluated in cartesian coordinates. The result is
2 2 |`0| |`| i(`0φ−`φ˜) − 1 (γ2ρ2+α2ρ˜2) |`0| 2 2 |`| 2 2 d = N˜ δ 0 δ(P − K )ρ ρ˜ e e 2 L (γ ρ )L (α ρ˜ ) , n nm nznz z z n0 n⊥ |q|B ⊥ (115) where
s 0 √ 0 n ! n ! ˜ 3 (0) (1) nz 2 |`|+1 |` |+1 ⊥ ⊥ Nnm = (2π) N N π 2 nz!/γ = 2(2π) α γ 0 0 , (|` | + n⊥)! (|`| + n⊥)! 2 2 ρ2 = x2 + y2 = (P − K )2 + (P − K )2 , qB x x y y y P − K φ = arctan = arctan x x , x Ky − Py 2 2 ρ˜2 = (λK − P )2 + (P − λK )2 , qB y y x x P − λK φ˜ = arctan x x , (116) λKy − Py
2 2 2 2 1 with λ ≡ (8ω0 + ωc )/(8ω0 + 2ωc ), which satisfies 2 ≤ λ ≤ 1. Note that the above definitions ˜ only apply for the probability amplitude dn. Forρ ˜ and φ in the wavefunction, we need to use the definitions in Eq. (112). Gaussian Wave Packet as Initial Condition
Let’s consider that the initial condition is not a pure state but instead a Gaussian linear
43 combination
X (0) (0) −iEk t Φ(t) = ckΦk e , (117) k where k = (`, kz, k⊥, P). We will assume that the system is in a well-defined internal state
0 0 0 (` , nz, n⊥) but has a spread in COM momentum,
r 3/2 8π −(P−P0)2/(2σ2) ck = δ`0`δn0k δn0 k e . (118) σ3 z z ⊥ ⊥
P (1) (0) In this case, the coefficient dn is more complicated, dn = m cmhΦn |Φm i; however, we can use the pure state result obtained previously
2 (1) (0) 2 |`0| |`| i(`0φ−`φ˜) − 1 (γ2ρ2+α2ρ˜2) |`0| 2 2 |`| 2 2 hΦ |Φ i = N˜ δ 0 δ(P −K )ρ ρ˜ e e 2 L (γ ρ )L (α ρ˜ ) , n m nm nznz z z n0 n⊥ |q|B ⊥ (119)
0 0 0 with m = (` , nz, n⊥, P) and n = (`, nz, n⊥, K). Time evolution of the center-of-mass kinetic momentum
We consider next the evolution of the COM kinetic momentum after the magnetic field is applied. We seek to evaluate hPkinetici = hΦ(t)|Pkinetic|Φ(t)i for t > 0.
(1) (1) X ∗ (1) (1) −i(En −Em )t hΦ(t)|Pkinetic|Φ(t)i = dmdnhΦm |Pkinetic|Φn ie , (120) m,n
0 0 0 0 where m = (` , nz, n⊥, K ), n = (`, nz, n⊥, K), and
∞ ∞ ∞ X X X X Z d3K0 ≡ , (2π)3 m 0 0 0 nz=0 ` =−∞ n⊥=0 ∞ ∞ ∞ X X X X Z d3K ≡ . (121) (2π)3 n nz=0 `=−∞ n⊥=0
44 Evaluating the expectation value in the summation gives,
(1) (1) (1) (1) hΦm |Pkinetic|Φn i = hΦm |K − qB × r|Φn i c b = Kδ − qBhΦ(1)| −ρ˜sin φ˜ + , ρ˜cos φ˜ + , 0 |Φ(1)i , (122) mn m a a n
3 3 0 where δ = δ 0 δ 0 δ 0 δ 0 , δ 0 ≡ (2π) δ (K − K), and we remind the reader that mn ` ` nznz n⊥n⊥ K K K K 2 2 a = mr(ω0 + ωc /4), b = ωcKy/4, c = ωcKx/4. Considering the second term, we have
c b −hΦ(1)| ρ˜sin φ˜ |Φ(1)i + δ , hΦ(1)| ρ˜cos φ˜ |Φ(1)i + δ , 0 . (123) m n a mn m n a mn
+ (1) iφ˜ (1) − (1) −iφ˜ (1) To proceed, we first consider Jmn ≡ hΦm |ρ˜ e |Φn i and Jmn ≡ hΦm |ρ˜ e |Φn i. Using
+ − equations 111 - 113 we evaluate Jmn and Jmn for various values of `. For ` ≥ 0
0 0 0 + δK Kδnznz δ` ,`+1 h p √ i J = δn0 n n⊥ + ` + 1 − δn0 ,n −1 n⊥ . (124) mn α ⊥ ⊥ ⊥ ⊥
For ` ≤ −1
0 0 0 √ + δK Kδnznz δ` ,`+1 h p i J = δn0 n n⊥ − ` − δn0 ,n +1 n⊥ + 1 . (125) mn α ⊥ ⊥ ⊥ ⊥
For ` ≥ 1
0 0 0 √ − δK Kδnznz δ` ,`−1 h p i J = δn0 n n⊥ + ` − δn0 ,n +1 n⊥ + 1 . (126) mn α ⊥ ⊥ ⊥ ⊥
For ` ≤ 0
0 0 0 − δK Kδnznz δ` ,`−1 h p √ i J = δn0 n n⊥ − ` + 1 − δn0 ,n −1 n⊥ . (127) mn α ⊥ ⊥ ⊥ ⊥
With these, we have determined
1 hΦ(1)| ρ˜sin φ˜ |Φ(1)i = (J + − J − ) ≡ S , m n 2i mn mn mn 1 hΦ(1)| ρ˜cos φ˜ |Φ(1)i = (J + + J − ) ≡ C . (128) m n 2 mn mn mn
45 To evaluate hΦ(t)|Pkinetic|Φ(t)i we will need
∞ ∞ ∞ ∞ Z (1) (1) X X X X ∗ ± −i(En −Em )t dmdnJmn e 0 n =0 0 0 K `=−∞ ⊥ ` =−∞ n⊥=0 ∞ ∞ h 2 √ 2 i 1 X X ∗ p iα t/mr ∗ −iα t/mr = d d±`,n n⊥ + ` + 1 e − d d±`,n n⊥ e α ±(`+1),n⊥ ⊥ ±(`+1),n⊥−1 ⊥ `=0 n⊥=0 ∞ ∞ h 2 √ 2 i 1 X X ∗ p −iα t/mr ∗ iα t/mr + d d∓`,n n⊥ + ` e − d d∓`,n n⊥ + 1 e , α ∓(`−1),n⊥ ⊥ ∓(`−1),n⊥+1 ⊥ `=1 n⊥=0 (129) where we have used Eq. (100).
∗ ∗ 0 0 To proceed we note that d d` ,n = d 0 0 d−`,n⊥ . Now we have, after some work, `,n⊥ ⊥ −` ,n⊥
∞ ∞ ∞ ∞ Z (1) (1) X X X X ∗ + − −i(En −Em )t dmdn(Jmn ± Jmn) e 0 n =0 0 0 K `=−∞ ⊥ ` =−∞ n⊥=0 ∞ ∞ 2 X X ∗ ∗ p = (d d−`−1,n ± d d`+1,n ) n⊥ + ` + 1 α −`,n⊥ ⊥ `,n⊥ ⊥ `=0 n⊥=0 √ −(d∗ d ± d∗ d ) n + 1 cos(α2t/m ) .(130) −`,n⊥+1 −`−1,n⊥ `,n⊥+1 `+1,n⊥ ⊥ r
R 2 Using d`,n⊥ = d P⊥ m`,n⊥ with
r 3/2 2 ˜ 0 2 2 N 8π 2 −(Kz−P ) /(2σ ) m = δ 0 e z `,n⊥ (2π)3 σ3 |q|B nznz
0 2 2 0 0 ˜ 1 2 2 2 2 |`0| −(P⊥−P⊥) /(2σ ) |` | |`| i(` φ−`φ) − 2 (γ ρ +α ρ˜ ) 2 2 |`| 2 2 ×e ρ ρ˜ e e L 0 (γ ρ )Ln (α ρ˜ ) , (131) n⊥ ⊥
s 0 ˜ 2 |`|+1 |`0|+1 n⊥! n⊥! N = 2(2π) α γ 0 0 , (132) (|` | + n⊥)! (|`| + n⊥)!
46 and a recurrence relation for the Laguerre polynomials, one obtains
∞ ∞ ∞ X X X Z d3K c b hP i = d∗ d K − qB , , 0 kinetic (2π)3 n n a a nz=0 n⊥=0 `=−∞ ∞ ∞ ∞ Z 3 Z 2 X X X d K ∗ 2 ˜ ˜ − 2qB cos(α t/mr) d d P⊥ m`,n ρ˜ − sin φ , cos φ , 0 . (133) (2π)3 `,n⊥ ⊥ nz=0 n⊥=0 `=0
Focusing on the second term, we need to evaluate
∞ ∞ ∞ Z 3 Z X X X d K ∗ 2 ˜ ˜ d d P⊥ m`,n ρ˜ − sin φ , cos φ , 0 . (134) (2π)3 `,n⊥ ⊥ nz=0 n⊥=0 `=0
The summation over nz and integration over Kz can be done analytically. Next, we ˜ change integration variables from (K⊥, P⊥) to (ρ, φ, ρ,˜ φ) using equations (116) and use the
completeness of the Laguerre polynomials to eliminate the summation over n⊥. Now, one of the integrals overρ ˜ and the summation over ` can be done analytically. The remaining five integrals are evaluated numerically and found to converge to zero. We now have
∞ ∞ ∞ X X X Z d3K c b hP i = d∗ d K − qB , , 0 kinetic (2π)3 n n a a nz=0 n⊥=0 `=−∞ 2 ∞ ∞ ∞ Z 3 ∞ ∞ ∞ Z 3 4ω0 X X X d K ∗ X X X d K ∗ = 2 2 3 dndnK⊥ +z ˆ 3 dndnKz . 4ω0 + ωc (2π) (2π) nz=0 n⊥=0 `=−∞ nz=0 n⊥=0 `=−∞ (135)
Again, the summation over nz and integration over Kz can be done analytically. We change variables, use the completeness of the Laguerre polynomials, and perform one of the integrals overρ ˜. Now, we use the completeness of the azimuthal modes and do one of the
47 integrals over φ˜.
2 2 0 Z Z β 2(|`0|+1) 4ω0 n⊥! hPkinetici = γ 2 2 0 0 ρdρ dφ πσ 4ω0 + ωc (|` | + n⊥)! 2 Z Z 0 2 2 |`0| 0 2 2 2|` | −γ ρ 2 2 ˜ −(P⊥−P⊥) /σ ×ρ e L 0 (γ ρ ) ρd˜ ρ˜ dφ K⊥e n⊥ 2 0 Z Z β 2(|`0|+1) 0 n⊥! +z ˆ γ Pz 0 0 ρdρ dφ πσ (|` | + n⊥)! 2 Z Z 0 2 2 |`0| 0 2 2 2|` | −γ ρ 2 2 ˜ −(P⊥−P⊥) /σ ×ρ e L 0 (γ ρ ) ρd˜ ρ˜ dφ e . (136) n⊥
Using Px = −β(λy − y˜), Py = β(λx − x˜), Kx = −β(y − y˜), Ky = β(x − x˜) and the orthogonality of the Laguerre polynomials, the remaining integrals can be done analytically. The final result is
2 2 4ω0 0 4ω0 0 0 hPkinetici = 2 2 Px , 2 2 Py ,Pz . (137) 4ω0 + ωc 4ω0 + ωc
Recall from Eq. (118), P0 is the central value in the initial COM momentum distribution of the system. By comparing the above equation to Eq. (105) we see hPkinetici is independent of the way in which the background magnetic field is turned on and the COM momentum distribution of the initial state is centered on the pseudomomentum.
48 7 Numerical Solution
In some cases, such as the case of a harmonic interaction, the energies and wave functions can be solved for analytically; however, in most cases this is not possible. In these cases it is necessary to solve the Schr¨odingerequation numerically. In practice, we can subtract out any terms which are independent of the position from the Hamiltonian. In addition, if the potential still possesses azimuthal symmetry, we can set either Kx or Ky to zero by rotating the coordinate system appropriately. We choose herein to set Ky to zero. The resulting Hamiltonian which is used in the numerical solutions is then of the form
2 2 2 0 ∇ qB q B 2 Hrel = − + Kxy + ρ + V (r) . (138) 2mr 4mr 8mr
After numerical solution using Eq. (138), the constant terms can be added back in manually in order to obtain the full energy eigenvalues. For the charmonium and bottomonium states considered here we use a Cornell potential plus spin-spin, spin-orbit, and tensor interactions, each with a separate potential
4 α V (r) = − s + σr + hS · S i V (r) + V (r) hL · Si + V (r) hTi . (139) 3 r 1 2 s LS T
The expectation value hS1 · S2i reduces to -3/4 for the singlet state and 1/4 for the triplet states. The expectation values hL · Si and hTi are zero for all singlet and s-wave states.
Their values for triplet states with ` > 0 are listed in Table 2. For the spin potential Vs(r),
49 we use a form found from fits to the charm spin-spin potential in lattice studies [79]
−βr Vs(r) = γe . (140)
For the spin-orbit and tensor potentials we use the form used by Barnes, Godfrey and Swanson [78], 1 2α σ V (r) = s − , (141) LS m2 r3 2r
1 4α V (r) = s . (142) T m2 r3
Where m is the quark mass (1.29 GeV for charm and 4.70 GeV for bottom), αs is the QCD fine structure constant, σ is the string tension, and r is the distance between the quark and antiquark.
Expectation Values
State hL · Si hTi ` j = ` + 1 ` − 6(2`+3) 1 j = ` -1 6 `+1 j = ` − 1 −(` + 1) − 6(2`−1)
Table 2: Expectation values for the spin-orbit and tensor interactions.
For charmonia, the constants γ and β above were fit to lattice data in Ref. [79]. They found γ = 0.825 GeV and β = 1.982 GeV. In this paper we allow for variation of γ. For both charm and bottom states we will hold β fixed to the value from Ref. [79], but we adjust the amplitude γ in order to reproduce the experimentally measured splittings using Eq. (139) as the interaction potential. We present the resulting parameter sets and the corresponding B = 0 spectra of bottomonium states in Table 3 and charmonia states in Table 4. For the bottom system we present a single “tuning” which reproduces all states through the Υ(3s) with a maximum error of 0.37%. For the charm system we present a single “tuning” which reproduces all states through the Jψ(3s) with a maximum error of 3.1%.
50 We note that the interaction potential and the non-derivative terms in Eq. (138) can be combined into a “pseudopotential” of the form
2 2 qB q B 2 4 αs −βr Vpseudo(r) = Kxy + ρ − + σr + hS1 · S2iγe 4mr 8mr 3 r 1 2α σ 1 4α + s − hL · Si + s hTi . (143) m2 r3 2r m2 r3
7.1 Solving the Schr¨odingerEquation
To solve the resulting Schr¨odingerequation we use the finite difference time domain method [80, 81, 82]. Here we briefly review the technique. To determine the wave functions of bound quarkonium states, we must solve the time-independent Schr¨odinger equation for the relative wave function
ˆ HrelΨυ(r) = Eυ Ψυ(r) , (144)
on a three-dimensional lattice in coordinate space. The index υ on the eigenfunctions, φυ,
and energies, Eυ, represents a list of all relevant quantum numbers. To obtain the time- independent eigenfunctions, we start with the time-dependent Schr¨odinger equation
∂ i Ψ(x, t) = Hˆ Ψ(x, t) , (145) ∂t rel
which can be solved by expanding in terms of the eigenfunctions, Ψυ(r):
X −iEυt Ψ(r, t) = cυΨυ(r)e . (146) υ
If one is only interested in the lowest energy states (ground state and first few excited states) an efficient way to proceed is to transform (145) and (146) to Euclidean time using a Wick
51 rotation, τ ≡ it: ∂ Ψ(r, τ) = −Hˆ ψ(r, τ) , (147) ∂τ rel and
X −Eυτ Ψ(r, τ) = cυΨυ(r)e . (148) υ
For details of the discretizations used etc. we refer the reader to Refs. [81].
7.2 Finding the Ground State
By definition, the ground state is the state with the lowest energy eigenvalue, E0. Therefore, at late imaginary time the sum over eigenfunctions (148) is dominated by the ground state eigenfunction
−E0τ lim Ψ(r, τ) → c0Ψ0(r)e . (149) τ→∞
As a consequence, one can obtain the ground state wavefunction, φ0, and energy, E0, by solving Eq. (147) starting from a random three-dimensional wavefunction, Ψinitial(r, 0), and evolving forward in imaginary time. The initial wavefunction should have a nonzero overlap with all eigenfunctions of the Hamiltonian; however, due to the damping of higher-energy eigenfunctions at sufficiently late imaginary times we are left with only the ground state,
Ψ0(r). Once the ground state wavefunction (or any other wavefunction) is found, we can compute its energy eigenvalue via
hΨ |Hˆ |Ψ i R d3x Ψ∗ Hˆ Ψ E (τ → ∞) = υ υ = υ υ . (150) υ R 3 ∗ hΨυ|Ψυi d x ΨυΨυ
7.3 Finding the Excited States
The basic method for finding excited states is to first evolve the initially random wavefunction to large imaginary times, find the ground state wavefunction, Ψ0, and then project this state out from the initial wavefunction and re-evolve the partial-differential equation in imaginary
52 time. However, there are (at least) two more efficient ways to accomplish this. The first is to record snapshots of the 3d wavefunction at a specified interval τsnapshot during a single evolution in τ. After having obtained the ground state wavefunction, one can go back and extract the excited states by projecting out the ground state wavefunction from the recorded snapshots of Ψ(r, τ) [80, 81]. An alternative way to select different excited states is to impose a symmetry condition on the initially random wavefunction which cannot be broken by the Hamiltonian evolution [81]. For example, one can select the first p-wave excited state by anti-symmetrizing the initial wavefunction around either the x, y, or z axes. In the non-spherical case this method can be used to separate the different excited state polarizations in the quarkonium system and to determine their energy eigenvalues with high precision.
7.4 Center-of-Mass Kinetic Energy Subtraction
Since the energy of a particle-antiparticle state in the presence of a magnetic field has a non-trivial dependence on the pseudomomentum quantum number K, one has to specify the precise manner in which the energy associated with the COM motion is subtracted
2 from the total energy. Our prescription for doing this is to subtract hPkinetici /2M where
M = m1 + m2 = 2mq from the total energy with hPkinetici computed via Eq. (37). As a concrete example, let’s return to the case of a harmonic interaction. As demon- strated in the previous section, this can be computed analytically in the case of a harmonic
2 interaction. Taking Eq. (100) and subtracting hPkinetici /2M with hPkinetici given in Eq. (105) we obtain
hP i2 E˜ = E − kinetic , K,n⊥nz` K,n⊥nz` 2M 2ω2ω2(K2 + K2) r 2 c 0 x y 1 2 ωc = 2 2 2 + nz + ω0 + (2n⊥ + 1 + |`|) ω0 + − µ · B + M. M(ωc + 4ω0) 2 4 (151)
53 As we can see from this expression, as B → 0 the dependence of the COM-subtracted energy on the COM pseudomomentum vanishes as it should; however, for non-vanishing background magnetic field, there is still a residual dependence on the components of the pseudomomen- tum which are perpendicular to the background magnetic field. In the case of the harmonic interaction, we are able to obtain the answer analytically. The harmonic interaction was also calculated using the numerical method described above and found to agree with the analyt- ical solution within machine precision. In cases other than the simple harmonic interaction, it may not be possible to obtain analytic expressions. Absent analytic expressions for the energy and necessary expectation values, one must perform the subtraction described in this section numerically by computing hPkinetici using the wavefunctions obtained.
54 8 Vacuum Results
8.1 Bottomonia
In Table 3 we compare bottomonia experimental data [3] and the bottomonium state masses computed using the model potential specified in Eq. (139). The model results were computed on a lattice size of 2563 with lattice spacing of a = 0.1 GeV−1. The parameters used were
2 mb = 4.7 GeV, γ = 0.318 GeV, β = 1.982 GeV, αs = 0.315443, and σ = 0.210 GeV . With the inclusion of the spin-orbit and tensor interactions we are able to reproduce the splitting between the 1p states. There is remarkable agreement with experimental data. For every state studied, the predicted mass is well within one percent of the measured mass.
8.2 Charmonia
In Table 4 we compare charmonia experimental data [3] and the charmonium state masses computed using the model potential specified in Eq. (139). The model results were com- puted on a lattice size of 2563 with lattice spacing of a = 0.2 GeV−1. The parameters used
2 were mb = 1.29 GeV, γ = 2.06 GeV, β = 1.982 GeV, αs = 0.234, and σ = 0.174 GeV . With the inclusion of the spin-orbit and tensor interactions we are able to reproduce the splitting between the 1p states. The agreement with experimental data is not as good as the bottomonia case but still very good. The difference between predicted masses and experi- mental data is about three percent or less. It should not be surprising that the model works better for bottomonia because the larger mass of the bottom quark means the nonrelativistic approximation is more valid.
55 Vacuum Bottomonia Results State Name Exp. [3] Model Rel. Err. 1 1 S0 ηb(1S) 9.398 GeV 9.398 GeV 0.001% 3 1 S1 Υ(1S) 9.460 GeV 9.461 GeV 0.01% 3 1 P0 χb0(1P ) 9.859 GeV 9.835 GeV 0.24% 3 1 P1 χb1(1P ) 9.893 GeV 9.871 GeV 0.22% 3 1 P2 χb2(1P ) 9.912 GeV 9.886 GeV 0.26% 1 1 P1 hb(1P ) 9.899 GeV 9.862 GeV 0.37% 1 2 S0 ηb(2S) 9.999 GeV 9.977 GeV 0.22% 3 2 S1 Υ(2S) 10.023 GeV 9.999 GeV 0.24% 3 2 P0 χb0(2P ) 10.232 GeV 10.212 GeV 0.02% 3 2 P1 χb1(2P ) 10.255 GeV 10.247 GeV 0.08% 3 2 P2 χb2(2P ) 10.269 GeV 10.261 GeV 0.08% 1 2 P1 hb(2P ) - 10.242 GeV - 1 3 S0 ηb(3S) - 10.347 GeV - 3 3 S1 Υ(3S) 10.355 GeV 10.360 GeV 0.05%
Table 3: Comparison of experimentally measured particle masses from Ref. [3] for the bottomonium system with “bottom-tuned” model predictions obtained using the potential model specified in Eq. (139). The parameters used were mb = 4.7 GeV, γ = 0.318 GeV, 2 β = 1.982 GeV, αs = 0.315443, and σ = 0.210 GeV . The uncertainty in the experimentally determined values is less than 0.04%. The case of no experimental data is indicated with a dash.
These masses are for rest states with no background magnetic field. In addition to a small change in mass due to the magnetic field, the interaction with the magnetic moment of the particle-antiparticle pair causes states with no spin component along the axis of the magnetic field to become mixed. A similar thing happens to positronium [83, 84]. This spin- mixing effect needs to be understood before the masses in a magnetic field can be accurately calculated.
56 Vacuum Charmonia Results State Name Exp. [3] Model Rel. Err. 1 1 S0 ηc(1S) 2.981 GeV 2.989 GeV 0.27% 3 1 S1 J/ψ(1S) 3.097 GeV 3.102 GeV 0.17% 3 1 P0 χc0(1P ) 3.415 GeV 3.408 GeV 0.20% 3 1 P1 χc1(1P ) 3.511 GeV 3.450 GeV 1.7% 3 1 P2 χc2(1P ) 3.556 GeV 3.445 GeV 3.1% 1 1 P1 hc(1P ) 3.525 GeV 3.429 GeV 2.7% 1 2 S0 ηc(2S) 3.639 GeV 3.590 GeV 1.3% 3 2 S1 J/ψ(2S) 3.686 GeV 3.650 GeV 0.97% 3 2 P0 χc0(2P ) - 3.853 GeV - 3 2 P1 χc1(2P ) - 3.906 GeV - 3 2 P2 χc2(2P ) 3.927 GeV 3.907 GeV 0.51% 1 2 P1 hc(2P ) - 3.888 GeV - 1 3 S0 ηc(3S) - 4.034 GeV - 3 3 S1 J/ψ(3S) - 4.079 GeV -
Table 4: Comparison of experimentally measured particle masses from Ref. [3] for the charmonium system with “charm-tuned” model predictions obtained using the potential model specified in Eq. (139). The parameters used were mc = 1.29 GeV, γ = 2.06 GeV, 2 β = 1.982 GeV, αs = 0.234, and σ = 0.174 GeV . The uncertainty in the experimentally determined values is less than 0.04%. The case of no experimental data is indicated with a dash.
57 9 Spin-Mixing
To understand the spin-mixing effect, we need to isolate the magnetic moment from the rest of the Hamiltonian and write it as a matrix. In general, one can write,
ˆ ˆ H = H0 − µ · B , (152)
ˆ where H0 contains all terms which depend on the spatial coordinates and
µ = µq + µq¯
− + = g µqSq + g µqSq¯ 1 = gµ (σ− − σ+) , (153) 2 q
where µq = Q/2mq is the quark magneton and g is the Land´e g-factor. In going from the second to third lines, we have used g− = −g+ = g. Herein, we ignore effects of the anomalous magnetic moment and take g = 2. The coupled spin states to be considered are
|11i = |↑↑ i ,
|1−1i = |↓↓ i , 1 |10i = √ |↑↓ i + |↓↑ i , 2 1 |00i = √ |↑↓ i − |↓↑ i . (154) 2
In the case of cc¯ states, the 1s triplet and singlet states correspond to the J/ψ and the
58 ¯ ηc, respectively. For bb states the 1s triplet and singlet states correspond to the Υ(1s) and
ηb, respectively. Without a spin-spin interaction, these states would be degenerate. With a spin-spin interaction, the triplet and singlet states split. With a spin-orbit interaction, the 1p triplet states split. In vacuum, the charmonium 1s splitting is approximately ∆E = 113 MeV and for bottomonium it is approximately ∆E = 63 MeV. In the presence of a
Vacuum Eigenstates
State Abbreviation |j mj ` si rep. |` m` s msi representation 1 1 S0 |ψ0i |0 0 0 0i |0 0 0 0i 3 1 S1 |ψ1i |1 1 0 1i |0 0 1 1i 3 1 S1 |ψ2i |1 − 1 0 1i |0 0 1 − 1i 3 1 S1 |ψ3i |1 0 0 1i |0 0 1 0i 13P |ψ i |0 0 1 1i √1 |1 1 1 − 1i − √1 |1 0 1 0i + √1 |1 − 1 1 1i 0 4 3 3 3 1 1 P1 |ψ5i |1 1 1 0i |1 1 0 0i 1 1 P1 |ψ6i |1 − 1 1 0i |1 − 1 0 0i 1 1 P1 |ψ7i |1 0 1 0i |1 0 0 0i 13P |ψ i |1 0 1 1i √1 |1 1 1 − 1i − √1 |1 − 1 1 1i 1 8 2 2 13P |ψ i |1 1 1 1i √1 |1 1 1 0i − √1 |1 0 1 1i 1 9 2 2 13P |ψ i |1 − 1 1 1i − √1 |1 − 1 1 0i + √1 |1 0 1 − 1i 1 10 2 2 3 1 P2 |ψ11i |2 2 1 1i |1 1 1 1i 3 1 P2 |ψ12i |2 − 2 1 1i |1 − 1 1 − 1i 13P |ψ i |2 1 1 1i √1 |1 1 1 0i + √1 |1 0 1 1i 2 13 2 2 13P |ψ i |2 − 1 1 1i √1 |1 − 1 1 0i + √1 |1 0 1 − 1i 2 14 2 2 q 13P |ψ i |2 0 1 1i √1 |1 1 1 − 1i + 2 |1 0 1 0i + √1 |1 − 1 1 1i 2 15 6 3 6
Table 5: Lowest 16 eigenstates of the Hamiltonian for B = 0. These states will be used as a basis for the Hamiltonian. The second column shows a simplified notation where the states are labeled in order of increasing energy.
magnetic field there is mixing between some of these spin states. Taking the orbital angular
momentum into account and using the |` m` s msi representation, one can easily verify that
+ − (σz − σz )|` m` 1±1i = 0 ,
+ − (σz − σz )|` m` 10i = 2 |` m` 00i ,
+ − (σz − σz )|` m` 00i = 2 |` m` 10i . (155)
59 Modified Eigenstates
|` m` s msi representation |j mj ` si representation µ · B|ψ0i −gµqB|0 0 1 0i −gµqB|ψ3i µ · B|ψ1i 0 0 µ · B|ψ2i 0 0 µ · B|ψ3i −gµqB|0 0 0 0i −gµqB|ψ0i 1 1 µ · B|ψ4i gµqB √ |1 0 0 0i gµqB √ |ψ7i 3 3 µ · B|ψ i −gµ B|1 1 1 0i −gµ B √1 |ψ i + √1 |ψ i 5 q q 2 9 2 13 µ · B|ψ i −gµ B|1 − 1 1 0i gµ B √1 |ψ i − √1 |ψ i 6 q q 2 10 2 14 q µ · B|ψ i −gµ B|1 0 1 0i gµ B √1 |ψ i − 2 |ψ i 7 q q 3 4 3 15 µ · B|ψ8i 0 0 µ · B|ψ i −gµ B √1 |1 1 0 0i −gµ B √1 |ψ i 9 q 2 q 2 5 µ · B|ψ i gµ B √1 |1 − 1 0 0i gµ B √1 |ψ i 10 q 2 q 2 6 µ · B|ψ11i 0 0 µ · B|ψ12i 0 0 µ · B|ψ i −gµ B √1 |1 1 0 0i −gµ B √1 |ψ i 13 q 2 q 2 5 µ · B|ψ i −gµ B √1 |1 − 1 0 0i −gµ B √1 |ψ i 14 q 2 q 2 6 q 2 q 2 µ · B|ψ15i −gµqB 3 |1 0 0 0i −gµqB 3 |ψ7i
Table 6: Effect of µ · B acting on the basis states. States with no spin component along the axis of the magnetic field are mixed. States with a spin component along the axis of the magnetic field are not mixed.
From this we see that there is no magnetic field effect on the |` m` 1±1i states but there
will be mixing between the |` m` 00i and |` m` 10i states. The states listed in Table 3 and
Table 4 can be written in the |` m` s msi representation using a Clebsch-Gordan expansion (Table 5). Then Eqs. (153) and (155) can be applied to determine how the states are mixed.
The resulting states are then converted back to the |j mj ` si representation (Table 6).
To determine the effect of the mixing, we use the |j mj ` si states as a basis to write the
60 Hamiltonian as a matrix,
E0 0 0 χ 000 000000000
0 E1 00000 000000000 0 0 E1 0000 000000000 χ 0 0 E1 000 000000000 χ 0 0 0 0 E2 0 0 − √ 0 0 0 0 0 0 0 0 3 χ χ 0 0 0 0 0 E3 0 0 0 √ 0 0 0 √ 0 0 2 2 χ χ 0 0 0 0 0 0 E3 0 0 0 − √ 0 0 0 √ 0 2 2 χ q 2 0 0 0 0 − √ 0 0 E3 0 0 0 0 0 0 0 χ 3 3 H = (156) 0 0 0 0 0 0 0 0 E4 0 0 0 0 0 0 0 χ 0 0 0 0 0 √ 0 0 0 E4 0 0 0 0 0 0 2 χ 0 0 0 0 0 0 − √ 0 0 0 E4 0 0 0 0 0 2 0000000 0000 E5 0 0 0 0 0000000 00000 E5 0 0 0 χ 0 0 0 0 0 √ 0 0 0 0 0 0 0 E5 0 0 2 χ 0 0 0 0 0 0 √ 0 0 0 0 0 0 0 E5 0 2 q 2 0 0 0 0 0 0 0 3 χ 0 0 0 0 0 0 0 E5 where
χ = gµqB, (157) and δ0 n . |ψni = . , (158) δ15 n are the basis states (δij = 1 for i = j, 0 otherwise). Five of the eigenstates and their corresponding eigenvalues are trivial and two can be solved analytically. The other nine must be solved numerically. Eliminating the states with trivial solutions and separating the 1s and 1p states, we have
E χ 0 H1s = (159) χ E1
61 and
χ E2 0 0 − √ 0 0 0 0 0 3 0 E 0 0 √χ 0 √χ 0 0 3 2 2 0 0 E 0 0 − √χ 0 √χ 0 3 2 2 χ q 2 − √ 0 0 E3 0 0 0 0 χ 3 3 χ H1p = 0 √ 0 0 E 0 0 0 0 . (160) 2 4 0 0 − √χ 0 0 E 0 0 0 2 4 χ 0 √ 0 0 0 0 E5 0 0 2 0 0 √χ 0 0 0 0 E 0 2 5 q 2 0 0 0 3 χ 0 0 0 0 E5
The five trivial eigenstates are not affected by the external magnetic field because their total spin is either parallel or antiparallel to the field. These states are two of the 1s triplet states, one of the 1p triplet states with j = 1, and two of the 1p triplet states with j = 2. Their energies are given by
H0|ψ1i = E1|ψ1i
H0|ψ2i = E1|ψ2i
H0|ψ8i = E3|ψ8i
H0|ψ11i = E5|ψ11i
H0|ψ12i = E5|ψ12i . (161)
In most cases, the above eigenvalue equations need to be solved numerically. For the states that get mixed by the magnetic field, a different notation is needed because each mixed state is a combination of various |ψni. The convention used here is |ψni represents a pure basis state and |Ψni represents a generalized state that may be pure or mixed. |Ψni = |ψni only for the unmixed states listed above or when there is no external magnetic field.
62 The eigenstates and energies for the 1s states can be found analytically by diagonalizing the H1s matrix. The eigenstates are
1 |Ψ i = (|ψ i + |ψ i) , 0 p 2 0 − 3 1 + − 1 |Ψ i = (|ψ i + |ψ i) , (162) 3 p 2 0 + 3 1 + +
where
s ∆E 2χ 2 = 1 ± 1 + , (163) ± 2χ ∆E
and ∆E = E1 − E0 is the energy difference between the 1s singlet and triplet states. The corresponding energies of the eigenstates are
1 q E = E + E − (∆E)2 + 4χ2 , Ψ0 2 0 1 1 q E = E + E + (∆E)2 + 4χ2 . (164) Ψ3 2 0 1
The eigenstates and energies for the 1p states can be found numerically by diagonalizing the
H1p matrix for given values of magnetic field strength and COM kinetic momentum. The results for bottomonia and charmonia for various values of magnetic field strength and COM kinetic momentum are given in the next section.
63 10 Final Results
The diagonal elements of the H1s and H1p matrices are found by numerically solving the Schr¨odingerequation using the reduced Hamiltonian (138), adding the constant terms and subtracting the kinetic energy. A different matrix must be used for each value of magnetic field strength and kinetic momentum. The exact mixing of the basis states depends on the strength of the magnetic field, the vacuum energy of the state and the kinetic momentum. The 1s states that get mixed by magnetic field become mixtures of two states and the 1p states that get mixed by the magnetic field become mixtures of three states. The mixing between states becomes stronger with increased magnetic field strength. The masses of the
2 hPkinetici eigenstates with nonzero center-of-mass momentum are calculated by subtracting 2M from the total energy. The |Ψni notation of the previous section is used with a superscript b to denote the bottomonia states and a superscript c to denote the charmonia states. The results for bottomonia and charmonia states with no center-of-mass momentum are shown in Figures 5 and 6, respectively. The magnetic field effects on the charmonia states are much stronger than the effects on the bottomonia states. In both cases, the 1p states are affected more by the magnetic field than the 1s states. The labels on the left indicate the names of the states with no background magnetic field. For the unmixed states these labels are valid for any value of eB but for the mixed states they should only be used when eB = 0. In Figure 6, two of the curves labeled hc (red circles and blue squares) appear to be degenerate for all values of eB. Actually, these states are only degenerate when there is no background magnetic field but the difference is smaller than the resolution of the plot.
When there is a background magnetic field, the 1s singlet state is mixed with the ms = 0
64 state and the 1p states become various combinations of the χb0, χb1 and χb2 states and the
χb1, hb and χb2 states. The effects of spin-mixing and nonzero COM kinetic momentum are discussed in more detail below. It is important to understand that, for the mixed states, the change in mass is partially due to the mixing between states and not just a shift in the energy eigenvalue.
(a) (b) 9.47 9.9 Υ 9.46 9.89 χb2 9.45 mixed, nondegenerate 9.88 9.44 unmixed, degenerate hb mixed, degenerate 9.87 9.43 unmixed, nondegenerate χb1 9.86 9.42 mass [GeV] mass [GeV] 9.85 9.41 ηb 9.84 9.4 χb0 9.39 9.83 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2]
Figure 5: a) Rest masses of 1s bottomonium states. b) Rest masses of 1p bottomonium states.
65 (a) (b) 3.5 3.15 3.48 J/ψ 3.1 3.46 χ mixed, nondegenerate c2 unmixed, degenerate h 3.05 3.44 c mixed, degenerate χc1 unmixed, nondegenerate mass [GeV] mass [GeV] 3.42 η 3 c χc0 3.4 2.95 3.38 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2]
Figure 6: a) Rest masses of 1s charmonia states. b) Rest masses of 1p charmonia states.
10.1 Bottomonia
The eigenstates and energies of bottomonium with no center-of-mass momentum for eB = 0.1 GeV2 and eB = 0.3 GeV2 are listed in Tables 7 and 8, respectively. The 1s states are not
b b b listed because they can be obtained analytically and the |Ψ8i, |Ψ11i and |Ψ12i states are not
b b b b listed because these states do not get mixed. In other words, |Ψ8i = |ψ8i, |Ψ11i = |ψ11i and
b b 2 |Ψ12i = |ψ12i. For eB = 0.1 GeV , all of 1s and 1p states get over 87% of their contribution from the corresponding vacuum state. Even at eB = 0.3 GeV2, the most strongly mixed state gets about 60% from the vacuum state. The results for all 1s and 1p bottomonium states as a function of magnetic field strength for various values of the center-of-mass momentum are shown in Figures 7 - 10. The label on the left side of the plot indicates the corresponding vacuum state. In the presence of a magnetic field, the original vacuum state gets mixed with other vacuum states. For the 1s triplet states, a superscript has been added to Υ to indicate the z-component of the spin. The mass of the state increases with magnetic field strength for all states with center-of- mass momentum greater than 2 GeV. Some of the low-momentum states have a decreasing mass with increasing magnetic field strength, but this trend is not expected to continue
66 indefinitely.
Bottomonia Eigenstates with Moderate Magnetic Field Eigenstate Energy Eigenstate decomposition b b b b |Ψ4i 9.834 GeV −0.9886|ψ4i + 0.1499|ψ7i + 0.01667|ψ15i b b b b |Ψ5i 9.859 GeV 0.9153|ψ5i + 0.3661|ψ9i + 0.1679|ψ13i b b b b |Ψ6i 9.859 GeV 0.9153|ψ6i − 0.3661|ψ10i + 0.1679|ψ14i b b b b |Ψ7i 9.861 GeV 0.1498|ψ4i + 0.9634|ψ7i + 0.2222|ψ15i b b b b |Ψ9i 9.873 GeV 0.3467|ψ5i − 0.9283|ψ9i + 0.1344|ψ13i b b b b |Ψ10i 9.873 GeV 0.3467|ψ6i + 0.9283|ψ10i + 0.1344|ψ14i b b b b |Ψ13i 9.887 GeV −0.2051|ψ5i + 0.06485|ψ9i + 0.9766|ψ13i b b b b |Ψ14i 9.887 GeV −0.2051|ψ6i − 0.06485|ψ10i + 0.9766|ψ14i b b b b |Ψ15i 9.887 GeV −0.01724|ψ4i − 0.2221|ψ7i + 0.9749|ψ15i
Table 7: Eigenstates and the corresponding energies for bottomonium with eB = 0.1 GeV2 b b b and zero center-of-mass momentum. The state |Ψ15i has higher energy than |Ψ13i and |Ψ14i but the difference is within the accuracy of the model.
Bottomonia Eigenstates with Strong Magnetic Field Eigenstate Energy Eigenstate decomposition b b b b |Ψ4i 9.830 GeV −0.9052|ψ4i + 0.4065|ψ7i + 0.1237|ψ15i b b b b |Ψ5i 9.847 GeV 0.8110|ψ5i + 0.4965|ψ9i + 0.3095|ψ13i b b b b |Ψ6i 9.847 GeV 0.8110|ψ6i − 0.4965|ψ10i + 0.3095|ψ14i b b b b |Ψ7i 9.858 GeV 0.4139|ψ4i + 0.7775|ψ7i + 0.4736|ψ15i b b b b |Ψ9i 9.878 GeV 0.3053|ψ5i − 0.8104|ψ9i + 0.5000|ψ13i b b b b |Ψ10i 9.878 GeV 0.3053|ψ6i + 0.8104|ψ10i + 0.5000|ψ14i b b b b |Ψ13i 9.896 GeV −0.4990|ψ5i + 0.3111|ψ9i + 0.8088|ψ13i b b b b |Ψ14i 9.896 GeV −0.4990|ψ6i − 0.3111|ψ10i + 0.8088|ψ14i b b b b |Ψ15i 9.896 GeV −0.09638|ψ4i − 0.4799|ψ7i + 0.8720|ψ15i
Table 8: Eigenstates and the corresponding energies for bottomonium with eB = 0.3 GeV2 b b b and zero center-of-mass momentum. The state |Ψ15i has higher energy than |Ψ13i and |Ψ14i but the difference is within the accuracy of the model.
In most cases, the mass of the state increases with increasing field strength. The most notable exception is the zero-momentum 1s singlet state, where the mass at eB = 0.3 GeV2
b is about 0.05% smaller than the vacuum mass. The mass of the zero-momentum |Ψ4i state
2 b decreases by about 0.02% at eB = 0.3 GeV and the masses of the zero-momentum |Ψ5i
b 2 and |Ψ6i states decrease by about 0.1% at eB = 0.3 GeV . The mass of the 1s singlet
67 (a) (b) 9.56 9.56 9.54 = 8 GeV 9.52 kinetic 9.48 > mass [GeV] 0 > mass [GeV] b Υ 9.51 b 3 9.46 2 9.5 9.44 9.49 > , | Ψ > , | Ψ b 9.42 b 0 1 9.48 ηb | Ψ 9.4 | Ψ 9.47 Υ± 9.38 9.46 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] Figure 7: a) Rest masses of the spin-mixed 1s bottomonium states. b) Rest masses of the unmixed degenerate spin-triplet 1s bottomonium states. (a) (b) 10.05 10.15 = 0 GeV = 2 GeV = 4 GeV = 8 GeV 9.95 4 7 9.95 | Ψ 9.85 | Ψ χb0 9.9 χb1 9.8 9.85 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] Figure 8: Rest masses of the non-degenerate spin-mixed 1p bottomonium states. state with center-of-mass momentum of 2 GeV also appears to decrease slightly. The mass b of the |Ψ4i state with center-of-mass momentum of 2 GeV has a minimum of 9.83 GeV at 2 b b eB = 0.03 GeV . The masses of the |Ψ5i and |Ψ6i states with center-of-mass momentum of 2 b 2 GeV reach a minimum value of 9.86 GeV at eB = 0.18 GeV . The mass of the |Ψ7i state zero-momentum has a minimum of 9.86 GeV at eB = 0.195 GeV2 and, with center-of-mass momentum of 2 GeV, it has a minimum of 9.86 GeV at eB = 0.03 GeV2. All of the low-momentum states are weakly affected by the magnetic field. As expected, the magnetic field effects become much more important for high momentum states. For the 68 (a) (b) 10.15 10.2 = 0 GeV 10.1 = 2 GeV = 4 GeV = 8 GeV 10 b b 6 10 10 9.95 9.95 > , | Ψ > , | Ψ b b 5 9.9 9 9.9 | Ψ h χb1 | Ψ b 9.85 9.85 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] (c) 10.25 10.2 b = 8 GeV 10.1 kinetic 15 10.05 > , | Ψ b 10 14 9.95 χb2 > , | Ψ 9.9 b 13 9.85 | Ψ 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] Figure 9: Rest masses of the degenerate spin-mixed 1p bottomonium states. 1s states, the mass increase is less than one percent at eB = 0.3 GeV2, even for the highest momentum studied. For the 1p states, the mass increase is about three percent at eB = 0.3 GeV2 for the highest momentum case. 69 (a) (b) 10.2 10.25 = 0 GeV 10.15 = 4 GeV 10.1 = 8 GeV b 10.05 10 12 > mass [GeV] 10 b 8 9.95 > , | Ψ 9.95 | Ψ b 9.9 11 χb2 hb 9.9 | Ψ 9.85 9.85 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] Figure 10: a) Rest masses of the unmixed degenerate 1s bottomonium states. b) Rest masses of the unmixed non-degenerate 1p bottomonium states. 70 10.2 Charmonia The eigenstates and energies of charmonium with zero center-of-mass momentum for eB = 0.1 GeV2 and eB = 0.3 GeV2 are listed in Tables 9 and 10, respectively. Like the bottomonia c c c tables, the 1s, |Ψ8i, |Ψ11i and |Ψ12i states are not listed. The spin-mixing effect is much stronger for the charmonium states. From Table 9 one can see that even for a relatively small magnetic field strength (eB = 0.1 GeV2), five of the nine states get the strongest contribution from a state other than the corresponding vacuum eigenstate. This doesn’t happen to the bottomonium states until sometime after the magnetic field strength exceeds eB = 0.3 GeV2. The results for the charmonium states with nonzero center-of-mass momentum are shown in Figures 11 - 14. The label on the left side of the plot indicates the corresponding vacuum state. For the 1s triplet states a superscript has been added to J/ψ to indicate the z- component of the spin. In the presence of a magnetic field, the original vacuum state gets mixed with other vacuum states. Qualitatively, they are the same as the bottomonium results. The values of the center-of-mass momentum considered here are less than those considered for the bottomonium states because of the lower mass of the charm quark. A reliable calculation of the mass at higher momentum would require a relativistic treatment. The charmonia states show the same qualitative behave as the bottomonia states. Like the lowest bottomonia state, the mass of the zero-momentum 1s singlet state decreases with increasing magnetic field strength but the effect is much more pronounced. The mass of this state is about 3% less relative to the vacuum mass at eB = 0.3GeV 2. The masses of the zero- c c c 2 momentum |Ψ4i, |Ψ5i and |Ψ6i states also decrease by about 3% at eB = 0.3 GeV . The mass of the 1s singlet state with center-of-mass momentum of 2 GeV decreases by about 2% at eB = 0.3 GeV2. At center-of-mass momentum of 4 GeV, this state has a minimum mass 2 c of 2.985 GeV at eB = 0.1 GeV . The |Ψ4i state with center-of-mass momentum of 2 GeV has a minimum mass of 3.75 GeV at eB = 0.19 GeV2 and for center-of-mass momentum of 4 2 c GeV there is a minimum mass of 3.40 GeV at eB = 0.06 GeV . The masses of the |Ψ5i and 71 Charmonia Eigenstates with Moderate Magnetic Field Eigenstate Energy Eigenstate decomposition c c c c |Ψ4i 3.380 GeV −0.6255|ψ4i − 0.6682|ψ7i + 0.4028|ψ15i c c c c |Ψ5i 3.390 GeV 0.7679|ψ5i − 0.4353|ψ9i − 0.4699|ψ13i c c c c |Ψ6i 3.390 GeV 0.7679|ψ6i + 0.4353|ψ10i − 0.4699|ψ14i c c c c |Ψ7i 3.426 GeV −0.7377|ψ4i + 0.3384|ψ7i − 0.5842|ψ15i c c c c |Ψ9i 3.452 GeV −0.04589|ψ5i + 0.6943ψ9i − 0.7182|ψ13i c c c c |Ψ10i 3.452 GeV 0.04589|ψ6i + 0.6943|ψ10i + 0.7182|ψ14i c c c c |Ψ13i 3.496 GeV 0.6389|ψ5i + 0.5731|ψ9i + 0.5132|ψ13i c c c c |Ψ14i 3.496 GeV 0.6389|ψ6i − 0.5731|ψ10i + 0.5132|ψ14i c c c c |Ψ15i 3.490 GeV −0.2541|ψ4i + 0.6625|ψ7i + 0.7046|ψ15i Table 9: Eigenstates and the corresponding energies for charmonium with eB = 0.1 GeV2 c and zero center-of-mass momentum. Unlike the bottomonia case, the state |Ψ15i has lower c c energy than the |Ψ13i and |Ψ14i states. Charmonia Eigenstates with Strong Magnetic Field Eigenstate Energy Eigenstate decomposition c c c c |Ψ4i 3.312 GeV −0.490|ψ4i − 0.7046|ψ7i + 0.5134|ψ15i c c c c |Ψ5i 3.322 GeV 0.7304|ψ5i − 0.4783|ψ9i − 0.4876|ψ13i c c c c |Ψ6i 3.322 GeV −0.7304|ψ6i − 0.4783|ψ10i + 0.4876|ψ14i c c c c |Ψ7i 3.456 GeV 0.8035|ψ4i − 0.1363|ψ7i + 0.5795|ψ15i c c c c |Ψ9i 3.487 GeV −0.01037|ψ5i + 0.7061|ψ9i − 0.7081|ψ13i c c c c |Ψ10i 3.487 GeV 0.01037|ψ6i + 0.7061|ψ10i + 0.7081|ψ14i c c c c |Ψ13i 3.632 GeV −0.6829|ψ5i + 0.5222|ψ9i + 0.5108|ψ13i c c c c |Ψ14i 3.632 GeV 0.6829|ψ6i − 0.5222|ψ10i + 0.5108|ψ14i c c c c |Ψ15i 3.625 GeV 0.3383|ψ4i − 0.6964|ψ7i − 0.6329|ψ15i Table 10: Eigenstates and the corresponding energies for charmonium with eB = 0.3 GeV2 c and zero center-of-mass momentum. Unlike the bottomonia case, the state |Ψ15i has lower c c energy than the |Ψ13i and |Ψ14i states. c |Ψ6i states with center-of-mass momentum of 2 GeV reach a minimum value of 3.39 GeV at eB = 0.18 GeV2, for center-of-mass momentum of 4 GeV there is a minimum mass of 3.42 GeV at eB = 0.07 GeV2 and for center-of-mass momentum of 8 GeV there is a minimum 2 c mass of 3.43 GeV at eB = 0.03 GeV . The mass of the |Ψ7i state zero-momentum has a minimum of 3.42 GeV at eB = 0.06 GeV2 with center-of-mass momentum of 2 GeV it has a minimum of 3.43 GeV at eB = 0.04 GeV2 and with center-of-mass momentum of 2 GeV it has a minimum of 3.43 GeV at eB = 0.02 GeV2. 72 (a) (b) 3.4 4.1 3.35 4 = 1.5 GeV = 1.5 GeV 3.2 kinetic 3.7 kinetic 3.15 3.6 3.1 3.5 > mass [GeV] > mass [GeV] c c 0 3.05 3 3.4 | Ψ η | Ψ 3 c 3.3 2.95 3.2 J/ψ0 2.9 3.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] (c) 4.1 4 = 1.5 GeV 3.7 kinetic > mass [GeV] 3.6 c 2 3.5 3.4 > , | Ψ c 1 3.3 | Ψ 3.2 J/ψ± 3.1 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] Figure 11: a) Rest masses of the spin-mixed 1s charmonium states. b) Rest masses of the unmixed degenerate spin-triplet 1s charmonium states. At high momentum, the mass increases faster than the bottomonia case. For the highest momentum 1s states, the mass increase is between 10% and 20% at eB = 0.3 GeV2. For the highest momentum 1p states the mass increase is over 50% at eB = 0.3 GeV2. 73 (a) (b) 4.8 = 0 GeV 4.6 = 0.5 GeV = 1 GeV 4.4 = 1.5 GeV 4.2 | Ψ 3.6 | Ψ 3.5 χc1 χc0 3.4 3.2 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] Figure 12: Rest masses of the non-degenerate spin-mixed 1p charmonium states. (a) (b) 5 = 0 GeV 5 = 0.5 GeV = 1 GeV = 1.5 GeV 4.5 > , | Ψ hc χ > , | Ψ c c1 3.5 3.5 c 5 9 | Ψ | Ψ 3 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] (c) 5.5 > mass [GeV] 5 c 13 3 | Ψ 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] Figure 13: Rest masses of the degenerate spin-mixed 1p charmonium states. 74 (a) (b) 5.5 5.5 | Ψ h χ 3.5 c c 3.5 c2 11 | Ψ 3 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 eB [GeV2] eB [GeV2] Figure 14: a) Rest masses of the unmixed degenerate 1s charmonium states. b) Rest masses of the unmixed non-degenerate 1p charmonium states. 75 10.3 Discussion Based on these findings one can estimate the effect of strong magnetic fields on bottomonium and charmonium production in the LHC heavy ion collisions (eB ∼ 0.3 GeV2). The cross sections for quarkonium production from both gluon-gluon fusion and quark-antiquark an- nihilation both scale (to leading order) as M −2, where M is the mass of the state. Assuming that we need only build in the mass correction in order to account for the magnetic field, the maximal effect on bottomonium is estimated to be on the order of 2% for 1s states and 6% for 1p states. For charmonium, the maximal effect is estimated to be between 17% and 31% for 1s states and on the order of 60% for 1p states. To fully understand particle production in relativistic heavy-ion collisions, these magnetic effects need to be considered. The higher mass due to the background magnetic field will reduce the probability of these particles being produced and may help explain the suppression of certain particles, like J/psi, observed in recent experiments. In all cases the mass increases very rapidly with magnetic field strength at high center-of- mass momentum. This suggests that high-momentum heavy quarkonia are rare when strong magnetic fields are present. Although only bottomonia and charmonia were considered here, the same procedure is applicable to toponia as well. Actually, the high mass of the top quark makes it the ideal candidate for the non-relativistic treatment used here. The masses for toponia were not calculated because of the lack of experimental data for comparison. Some of the masses were observed to decrease with increased magnetic field strength. Other states have a minimum mass at a magnetic field strength less than 0.3 GeV2. The states with a continuously decreasing mass when the field strength is less than 0.3 GeV2 are expected to have a minimum mass for a stronger magnetic field. A more detailed model is needed to investigate this. It would be very surprising if nature allowed the mass to drop to zero or become negative. It has already been shown that the ρ meson mass does not vanish in a strong magnetic field as previously thought [85]. In principle, one might expect all states to have the same qualitative behavior and this may the case. The states that appear to have 76 a continuously increasing mass may actually have a minimum that is just under the vacuum mass for a very weak magnetic field. It may seem strange that the mass depends on the momentum since we should be able to Lorentz boost to the rest frame of the particle. If we boost to the rest frame the kinetic momentum effects vanish, however, because of the presence of a magnetic field, the boost creates an electric field. The electric field makes it necessary to include additional terms in the Hamiltonian. These additional terms are exactly the kinetic momentum effects. This is called the motional Stark effect and was discussed earlier. The more general case with an electric field was not considered because, unlike strong magnetic fields, strong steady-state electric fields are not known to exist in nature. Perhaps the most interesting effect of strong magnetic fields is the mixing of different spin states. This effect is usually ignored because it is very weak unless the magnetic field is very strong. In some cases, if the field is strong enough, the vacuum state contributes nothing to the corresponding mixed state. The limiting behavior for an arbitrarily strong magnetic field is listed in Table 11. In all cases, the eigenstates |Ψni reduce to the basis states |ψni in the limit B → 0. 10.4 What is a Particle? In addition to the fundamental quarks, leptons and gauge bosons, bound states of these particles are also called particles, for example, nucleons, atoms and molecules. Fundamental, as well as composite particles are typically identified by their mass. In principle, any excited state of a composite particle will have a higher mass relative to the ground state but for large particles, like atoms and molecules, the mass of the excited state is only slightly larger than the mass of the ground state. For example, the difference between the mass of a hydrogen atom in an excited state and the mass of a hydrogen atom in the ground state is seven orders of magnitude smaller than the rest mass of a hydrogen atom in the ground state. The situation is very different in quarkonium because the coupling strength is much stronger and 77 Eigenstates with Infinite Magnetic Field Eigenstate Eigenstate decomposition |Ψ i √1 |ψ i − √1 |ψ i 0 2 0 2 3 |Ψ i √1 |ψ i + √1 |ψ i 3 2 0 2 3 |Ψ i − √1 |ψ i + √1 |ψ i + √1 |ψ i 4 6 4 2 7 3 15 |Ψ i √1 |ψ i + 1 |ψ i + 1 |ψ i 5 2 5 2 9 2 13 |Ψ i √1 |ψ i − 1 |ψ i + 1 |ψ i 6 2 6 2 10 2 14 q |Ψ i 2 |ψ i + √1 |ψ i 7 3 4 3 15 |Ψ i − √1 |ψ i + √1 |ψ i 9 2 9 2 13 |Ψ i √1 |ψ i + √1 |ψ i 10 2 10 2 14 |Ψ i − √1 |ψ i + 1 |ψ i + 1 |ψ i 13 2 5 2 9 2 13 |Ψ i − √1 |ψ i − 1 |ψ i + 1 |ψ i 14 2 6 2 10 2 14 |Ψ i − √1 |ψ i − √1 |ψ i + √1 |ψ i 15 6 4 2 7 3 15 Table 11: Mixed eigenstates for bottomonium or charmonium with an arbitrarily strong magnetic field and zero center-of-mass momentum. The energy is not listed here because it is also arbitrarily large. the rest mass of the ground state is much smaller. For example, the difference between the mass of bottomonium in the first excited state and mass of bottomonium in the ground state is 0.67% of the rest mass of bottomonium in the ground state. For charmonium, the mass difference is 3.8%. In the early days of particle physics, different states of quarkonium were naturally iden- tified as different particles because they clearly had different masses. These masses are the eigenvalues of the Hamiltonian with no center-of-mass momentum. If we define a particle as an eigenstate of the Hamiltonian, changing the Hamiltonian could change the definitions of particles. Including a strong external magnetic field in the Hamiltonian changes the eigenvalues and eigenstates. The new eigenstates are combinations of the eigenstates of the Hamiltonian with no magnetic field. It seems inaccurate to say the magnetic field simply changes the masses of the particles. It actually changes the particles themselves. As the strength of the magnetic field increases, the Hamiltonian eigenstates change continuously and so do the precise definitions of the particles associated with those eigenstates. 78 Experiment: A Three-Body Decay of Hypertriton 11 Search for Three-body Hypertriton Decay A hypernucleus is any bound state of nucleons and at least one hyperon (a baryon with non-zero strangeness quantum number). The hypertriton is the lightest, and therefore the most common, hypernucleus. It is a bound state of a proton, neutron, and Λ hyperon. The Λ hyperon, a bound state of an up quark, down quark, and strange quark, is electrically neutral and 19% heavier than a neutron. Information on the binding energy and lifetime of the hypernucleus can help us understand the hyperon-nucleon interaction which will help distinguish between competing models of compact stars [86]. 11.1 History of the Hypertriton In 1952, the first hypernucleus was observed in an emulsion experiment conducted by Mirian Danysz and Jerzy Pniewski [87]. It was first described as a “hyperfragment” because it was produced when a primary cosmic ray collided with a nucleus in a detector material with similar properties as photographic emulsion, flown on a high-altitude balloon. The newly produced fragment appeared to create new fragments after almost coming to rest. Since a slow moving fragment cannot create new fragments by colliding with another nucleus, the result was interpreted as the decay of the hyperfragment. This observation opened up the possibility that bound systems of nucleons and hyperons might exist. Qualitatively, these new systems are very similar to nuclei but it is important to understand that the physics is different because the hyperons have a non-zero strangeness quantum number allowing it to share quantum states with nucleons that would be impossible for another nucleon because of the Pauli exclusion principle. 80 11.2 Decay Channels It has been suggested that the hypertriton may be described as a bound state of deuteron and Λ hyperon but that has not yet been established. In the three-body decay studied here, 3 − the hypertriton decays into a deuteron, proton and pion (ΛH→ d + p + π ). We can also 3 ¯ ¯ + study the decay of an antihypertriton into the corresponding antiparticles (Λ¯ H → d+p ¯+π ). In a two-body decay of the lambda hyperon, it decays into a pion and proton (Λ → p + π−). It is tempting to think that the three-body decay may actually be two successive two-body decays, the hypertriton decaying into a deuteron and lambda hyperon immediately followed by the lambda hyperon decaying into a proton and pion. This type of compound decay cannot happen because it would violate the conservation of energy. The combined mass of the decay products must be less than the mass of the parent particle. The mass of the hypertriton is 2.991 GeV/c2, the mass of the lambda hyperon is 1.116 GeV/c2 and the mass of the deuteron is 1.876 GeV/c2. The mass of the lambda hyperon plus the mass of the deuteron is slightly more than the mass of the hypertriton so the hypertriton cannot decay into a lambda hyperon and deuteron. If the hypertriton is a bound state of a deuteron and lambda hyperon and the three-body decay is a result of the hyperon decaying into a proton and pion, the hyperon decay must occur while it is still bound to the deuteron. 3 It is also possible for the hypertriton to decay into a deuteron, neutron and pion (ΛH→ d+n+π−). This decay is much harder to detect experimentally because most modern particle detectors are designed to detect charged particles making it difficult to detect neutrons. 11.3 Experimental Search To identify a particular decay in an accelerator experiment, information must be gathered on all potential decay products for each event (primary collision) and all possible combinations of those particles must be analyzed. For the three-body decay of hypertriton, track informa- tion for all deuterons, protons, and pions are stored for each event. Each deuteron is matched with each proton produced in the same event and each of those deuteron-proton pairs is then 81 matched with each pion from the same event. The total number of three-particle combina- tions for a given event is equal to the number deuterons times the number of protons times the number of pions in that event. Obviously, only a small number of those combinations are from real hypertriton decays. The same procedure is used to locate antihypertritons with all the particles replaced with their corresponding antiparticles. 11.4 Previous Measurements There has already been much work on the 2-body decay of hypertriton [88]. The first estimates of the hypertriton lifetime were published in 1964 [89]. Most studies have found the hypertriton lifetime to be slightly less than the lifetime of a free Λ hyperon. The most recent measurements put the hypertriton lifetime at 182 picoseconds and the Λ hyperon lifetime at 263 picoseconds. The left panel of Figure 15 shows the results for a recent measurement of the decay lengths of the lambda hyperon and hypertriton. The right panel shows the results for the hypertriton lifetime measurements from various experiments over the past 50 years along with a prediction from theory. 82 Figure 15: Left: Decay length measurements of the lambda hyperon and hypertriton. The insert shows the χ2 analysis used to obtain the lifetime measurement. Right: Measured lifetimes and theory calculation of the hypertriton. Figure taken from [2]. 83 12 The STAR Experiment The data for this analysis was collected at the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Lab (BNL). The Solenoidal Tracker at RHIC (STAR) is one of two collider experiments at BNL. RHIC is very versatile. It is capable of colliding protons and a variety of heavy ions at a wide range of energies. The data used for this analysis are gold-gold collisions at 39 GeV and 200 GeV per nucleon pair. The 39 GeV data was taken during the 2010 run of the experiment and the 200 GeV data was taken during the 2010 and 2011 runs of the experiment. Only data taken during the 2011 run is used for the analysis of the antihypertriton decay. Table 12 shows the statistics for all data sets. Data Statistics Data Set Millions of Events 2010 run, 39 GeV 229 2010 run, 200 GeV 329 2011 run, 200 GeV 603 Table 12: Statistics for all data sets used for this analysis. The gold ions are accelerated around the two circular beam pipes in opposite directions until they are traveling very close to the speed of light and forced to collide. The colliding nucleons produce a very hot and dense quark-gluon plasma [90, 91, 92, 93, 94]. Many particles are produced as the plasma expands and cools. Generally, lighter particles are produced in greater numbers. Some of these particles are hypertritons and some of these hypertritons decay into a deuteron, proton and pion. The hypertritons mostly decay while traversing the STAR detector, and in many cases, the detector provides a good measurement of the 84 daughter tracks. The STAR detectors measure the energy and momentum of these particles but determining which ones came from hypertriton decays and which ones came from other sources requires much more effort. 12.1 Time Projection Chamber The Time Projection Chamber (TPC) is the main component of the detector system at STAR [95]. It is a large cylinder, 4.2 meters long and 4 meters in diameter, concentric with the beam axis, and filled with a mixture of 90% argon and 10% methane gas. It covers the full range of azimuthal angle and up to 1.8 units of pseudorapidity (a function of the polar angle). It is surrounded by a large solenoidal magnet capable of producing fields up to 1 Tesla. It identifies particles by measuring their curvature in the magnetic field and energy loss as they traverse the gas in the TPC. It is capable of measuring particle momenta between 100 MeV/c and 30 GeV/c. Figure 16 shows the energy loss vs. the particle momenta for deuterons, protons, and pions. The paths of particles passing through the gas are reconstructed with sub-millimeter precision. Figure 16: Energy loss per centimeter in the TPC vs. particle momenta. The upper band are deuterons, the middle band are protons, and the bottom band are pions. 85 12.2 Time of Flight The Time of Flight detector is a barrel around the outside of the TPC. As the name implies, it measures the time it takes for a particle to travel from the collision point of the gold nuclei to the detector. If the distance traveled during this time is known, the speed of the particle can easily be calculated. Combined with the momentum measurement from the TPC, the TOF provides excellent particle identification. Knowing the speed and momentum, the mass of the particle is given by r p v2 m = 1 − , (165) v c2 where p is the particle momentum, v is the particle speed, and c is the speed of light. 86 13 Identification of Hypertriton Candidates Hypertriton candidates are identified by calculating the invariant mass of the recon- structed parent particle. The invariant mass is given by Einstein’s mass-energy relation, E2 − ~p 2 = m2 . (166) E and ~p are the total energy and momentum, respectively, of the daughter particles. E = Ed + Ep + Eπ , (167) ~p = ~pd + ~pp + ~pπ . The momentum of each daughter particle can be measured directly with the STAR detectors. The energy of each daughter particle is determined by Einstein’s mass-energy relation using the known mass and measured momentum. The mass of the hypertriton candidate can then be calculated. Each event produces many deuterons, protons and pions but it is a challenge to de- termine which ones are produced by a hypertriton decay and which ones are produced by another mechanism. For each event, the invariant mass is calculated for all combinations of a deuteron, proton and pion. Combinations that come from actual three-body hypertriton decays have an invariant mass very close to the known value of 2.991 GeV/c2. Other combi- nations form a distribution known as the combinatorial background. The hypertriton signal is extracted from the distribution of invariant mass values by estimating the combinatorial 87 background and subtracting it from the data. 13.1 Background Subtraction There are many techniques for estimating the combinatorial background. The technique used here is called the rotated background. This technique works by rotating the deuteron transverse momentum vector and the decay vertex position vector by 180 degrees about the primary vertex in the transverse plane, see Figure 17. The effect of these rotations is that accepted hypertriton candidates appear at different places in the invariant mass distribution causing the signal to get smeared. The final result is a new reconstructed hypertriton mass distribution that is normally almost identical to the original except that the sharp peak centered at the hypertriton mass is removed; see right-hand panel of Figure 18. Once the combinatorial background is estimated, it is subtracted from the original invariant mass distribution to obtain the background-subtracted invariant mass distribution. For a two-body decay, it doesn’t matter which particle is used for the rotation but, as usual, for a three-body decay things are more complicated. Since many of the proton-pion pairs in the combinatorial background come from Λ decays, using the proton or pion for the rotation technique results in an estimated background that is much smaller than the actual background because hypertriton candidates that consist of a proton and pion from a real Λ decay and an uncorrelated deuteron are eliminated from the estimated background. Figure 18 shows the results of rotating the pion track and rotating the deuteron track. Rotating the proton track gives a similar result as rotating the pion track. 88 Figure 17: Schematic of rotation technique used to estimate the combinatorial background. The transverse momentum (pt) vector and decay vertex position (rt) vectors for a daughter track are rotated about the primary vertex (PV) in the plane transverse to the beam direction (arbitrary scale). Different versions of this technique may use different daughter tracks and different rotation angles. Figure 18: Left: Hypertriton invariant mass distribution (red) with combinatorial back- ground (blue) estimated by rotating the pion tracks. Right: Hypertriton invariant mass distribution (red) with combinatorial background (blue) estimated by rotating the deuteron tracks. 89 14 Locating the Decay Vertex In general, a particle decay is found by identifying a point where daughter candidate tracks converge. Most tracks converge on the primary vertex but for particle decays, the daughter tracks converge away from the primary vertex forming a secondary, or decay vertex. A cut can be placed on the distance between these two vertices to increase the probability of the secondary vertex being the result of a real particle decay. It is very simple to locate the decay vertex for a two-body decay such as Λ → p + π+. It is the midpoint of the line segment connecting the particle tracks along their distance of closest approach (DCA), see Figure 19. The only parameter which needs to be subject to a cut is the DCA, d12. The situation is more complicated for a three-body decay. There are now three DCA parameters, d12, d13, and d23. The midpoints of the line segments associated with these parameters form the vertices of a triangle and the decay vertex is defined as the center of the triangle, see Figure 20. In addition to the three DCA parameters, there are three more parameters, d1, d2, and d3, needed to limit the size of the triangle. These parameters are defined as the distance from the decay vertex to the each of the midpoints associated with the DCA parameters. 90 Figure 19: Schematic of Λ → p + π− decay. The decay is assumed to take place at the midpoint of the line segment representing the distance of closest approach between the daughter particles. A two-body hypertriton decay may be obtained by replacing the proton with a helium-3 nucleus. Figure 20: Schematic of 3-body hypertriton decay. The decay is assumed to take place at the center of a triangle formed by the midpoints of the line segments representing the distances of closest approach between each pair of daughter particles. 91 Decay Vertex Cuts Data Set 2010 Run, 39 GeV 2010 Run, 200 GeV 2011 Run, 200 GeV d12 0.8 cm 0.8 cm 0.8 cm d13 0.8 cm 0.8 cm 0.8 cm d23 0.8 cm 0.8 cm 0.8 cm d1 1.5 cm 1.0 cm 1.5 cm d2 1.5 cm 1.0 cm 1.5 cm d3 1.5 cm 1.0 cm 1.5 cm Table 13: Cuts used to identify the decay vertex. 92 15 Particle Identification This analysis used the Time Projection Chamber (TPC) and Time of Flight (TOF) de- tectors at STAR. The distance between the point where a particle entered the TOF detector and the primary collision vertex is measured as well as the amount of time it took for the particle to travel that distance. This information is combined with the momentum of the particle, measured by the TPC detector, to identify the deuterons. Another way to identify particles is to measure the amount of energy lost by the particle as it travels through the TPC detector. Different particles lose energy at different rates. Protons and pions are iden- tified by requiring this energy loss to be within two standard deviations of the mean value for protons and pions, respectively. Before the specific particle identification algorithms are used, each track must pass a set of quality cuts to ensure the data is reliable. For all tracks it is required that |η| < 1 and pT < 1.5 GeV/c, where η is the pseudorapidity and pT is the transverse momentum. In addition to track cuts, it is also required that the primary collision vertex be less than 30 cm from the center of the detector system. These cuts ensure that all the tracks are well within the acceptance of the STAR detectors. For pion tracks it is also required that the distance of closest approach to the primary vertex be greater than 1.5 cm. This is because there are many pions coming directly from the primary vertex but pions from a real hypertriton decay are most likely to originate away from the primary vertex. 15.1 Deuteron Selection One way to identify different particles is to measure the amount of energy deposited in the dE TPC and compare that to what is expected for various possible particle masses. ( dx )ex is 93 dE the measured amount of energy deposited in the TPC detector and ( dx )m is the amount predicted by modeling the detector. For the 2010 run, 200 GeV data set it is required that ( dE ) log dx ex < 0.1 , (168) dE ( dx )m and for the 2010 run, 39 GeV dataset it is required that ( dE ) log dx ex < 0.2 . (169) dE ( dx )m For the 2011 run, 200 GeV dataset it is required that dE ( dx )ex −0.15 < log dE < 0.1 . (170) ( dx )m Another way to identify particles is to calculate the mass by combining information from the TOF detector and the momentum measured in the TPC. The momentum of a daughter particle is given by mβ p = , (171) p1 − β2 where β is the ratio of the speed of the particle to the speed of light. If the decay length is not too long, the speed of the particle can be approximated by calculating the distance from the primary vertex to the point where the particle makes contact with the TOF detector and dividing by the measured time for the particle to travel between those two points. This approximation can be improved by accounting for the fact that the particle did not originate from the primary vertex (see section 15.2). With this information it is easy to calculate a mass for the detected particle and compare it to a known particle mass. For the 2010 run, 94 39 GeV and 2011 run, 200 GeV (matter) data sets it is required that M 0.9 < ex < 1.1 , (172) M where Mex is the measured mass and M is the known mass of the deuteron, 1.8756 GeV. For the 2010 run, 200 GeV data set it is required that M 0.93 < ex < 1.07 , (173) M and for the 2011 run, 200 GeV (antimatter) dataset it is required that M 0.95 < ex < 1.03 . (174) M By combining the above methods of particle identification, a very clean deuteron sample was obtained, see Figure 21. The precise locations of these cuts were tuned to maximize the signal to noise ratio. The cuts used for the 2010 run, 200 GeV data are not centered on the peak like the other data sets because it is more likely for deuterons from real hypertriton decays to have an above-average value of Mex. This is probably a result of the way Mex is calculated, see section 15.2. 15.2 Time-of-Flight Correction The time-of-flight calculation can be improved by correcting the length of the path traveled. Since the vast majority of detected particles come from the primary collision vertex, the default time of flight calculation assumes the detected particle traveled from that vertex to the point it entered the TOF detector. A particle from a decay does not originate from the primary vertex, so for these particles, the time of flight calculation would have an additional error without a suitable correction and therefore the resulting mass calculation would also need a correction. If the location of the decay vertex is known, the calculation can be 95 Figure 21: 2D histograms showing deuteron selection for the three matter data sets used. The red box indicates the locations of the track cuts. The antimatter data from the 2011 run at 200 GeV is similar to the corresponding matter data set but with tighter cuts on the horizontal axis. See text for exact cuts. corrected by replacing the primary vertex with the decay vertex. The new mass calculation tends to shift the distribution to lower values, see Figure 22. The TOF correction was not used for this analysis because it is incompatible with the background rotation technique used to estimate the background. To get an accurate estimate of the background it is important for the data to be analysed in the same way with or without rotating the tracks. The correction reduces the total number of hypertriton candidates by eliminating part of the combinatorial background. If the correction is applied to the non- rotated tracks but not applied to the rotated tracks the estimated background will be too high. Rotating the deuteron tracks changes the dynamics in a way that is inconsistent with the TOF correction. The correction depends on the point where the particle entered the detector, but rotating the track means reversing the direction of the momentum. If a 96 Figure 22: Masses of daughter particles with (blue) and without (red) the TOF correction. particle actually had the rotated momentum it would enter the TOF detector at a different place. This inconsistency causes more hypertriton candidates to be removed, resulting in an estimate for the combinatorial background that is too small, see Figure 23. The TOF correction could still be applied to the proton and pion tracks, since these do not get rotated when estimating the combinatorial background, but doing so did not change the final results so use of the TOF correction was abandoned. 97 Figure 23: Reconstructed hypertriton mass spectrum (red) and estimated combinatorial background (blue) using the TOF correction. This plot was made using the 2011 run, 200 GeV (matter) data set. 98 16 Background Suppression Naturally, a three-body decay produces a much larger combinatorial background than a two-body decay. The number of hypertriton candidates in the two-body decay is constrained by the relatively small number of helium-3 nuclei but deuterons and protons are much more common. The number of real hypertriton decays is much smaller than the number deuteron- proton-pion combinations so the hypertriton signal would be lost without ways to suppress the combinatorial background. An easy way to suppress some of the background is to put cuts on the topology of the decay. For all data sets, the distance of closest approach of the reconstructed hypertriton track to the primary vertex must be less than 2 cm and the decay length of the hypertriton must be greater than 1 cm. For the 2011 run data sets, there is an additional requirement that the invariant mass of the proton-pion pairs must be less than 1.116 GeV/c2. This cut is designed to eliminate hypertriton candidates where the proton and pion are the decay products of a real Λ hyperon because these particles cannot also be the products of a real hypertriton decay. This cut was only effective for the 2011 run data because this data set is larger and has better particle identification than the others. Applying this cut to other data sets resulted in a significant reduction of the hypertriton signal. Any real hypertriton must come from the primary vertex but many of the reconstructed hypertriton candidate tracks do not. If the momentum of the hypertriton candidate (the vector sum of the daughter momenta) does not point away from the primary vertex the hypertriton candidate must not be real. It may be possible for some kind of rescattering process to produce a hypertriton away from the primary vertex but this effect would be very 99 small and impossible to detect. The combinatorial background can be significantly reduced by placing a cut on the pointing angle, θ, defined as the angle between the secondary vertex position and the reconstructed hypertriton momentum vectors (see Figure 24). For all 2010 run data sets it is required that θ < 0.2 radians and for all 2011 run data sets it is required that θ < 0.6 radians. It is necessary to use a tighter cut for the 2010 run data because of the greater chance of particle misidentification associated with that data set. Also, the fact that the data from the 2010 runs contain about half as many events as the data from the 2011 run makes identifying the hypertriton signal more difficult so reducing the combinatorial background becomes more important. For the 2011 data, a tight cut on θ was not needed to reduce the combinatorial background so a wide cut was used to increase statistics. Figure 24: Left: Definition of the pointing angle θ. Right: Distribution of θ. Hypertriton candidates with a large pointing angle cannot be real. 16.1 TMVA TMVA is a software package designed to analyze many variables simultaneously [96]. It works by looking for differences between the signal and the combinatorial background. It then generates data cuts based on these differences. In the end, the cuts established by TMVA were no better at distinguishing real hypertriton decays from the combinatorial background than cuts established by other means. 100 16.2 Simulation A simulation of the 3-body hypertriton decay was performed using 105 events with one hypertriton in each event. Dynamic variables such as transverse momentum, decay length, etc. were measured and compared to real data. For the simulation, it was assumed that the transverse momentum was uniformly distributed between 1 GeV/c and 5 GeV/c and the pseudorapidity was uniformly distributed between η = −1 and η = 1. The simulation did not include TOF detector so there was a small difference in the deuteron selection. In most cases, the simulated data was very similar to the real data. Although there were some differences, they were not significant enough to motivate additional cuts. A few of these distributions are shown if Figures 25 - 28. The only significant difference found is the component of the daughter momentum transverse to the reconstructed deuteron momentum which has a smaller upper bound for the simulated data than for the real data. By placing a cut on the real data at the upper bound of the simulated data, some of the hypertriton candidates in combinatorial background are eliminated without reducing the number of real hypertriton candidates. Figure 29 shows the distributions for each of the three daughters. Based on these plots from simulation, cuts were placed at 230 MeV/c for the deuteron momentum transverse to the hypertriton, 220 MeV/c for the proton momentum transverse to the hypertriton, and 110 MeV/c for the pion momentum transverse to the hypertriton. These cuts were used for all data sets. 101 Figure 25: Transverse momentum vs. the distance of closest approach to the primary vertex of the daughter particles for real and simulated data. Figure 26: Distance of closest approach between pairs of daughters vs. the reconstructed hypertriton transverse momentum for real and simulated data. Due to limited data storage, no candidate with a DCA larger than 1 cm was saved. 102 Figure 27: Distances of closest approach between different daughter particles plotted against each other for real and simulated data. Here we see a weak correlation in the simulated data between the inter-particle DCAs involving the pion but it is not strong enough to define a new cut without significantly reducing the number of accepted hypertriton candidates. Figure 28: Distance between the decay vertex and the midpoint of the inter-particle DCA vs. the reconstructed hypertriton transverse momentum for real and simulated data. 103 Figure 29: Top: Distribution of real and simulated deuteron momentum transverse to hyper- triton momentum. Middle: Distribution of real and simulated proton momentum transverse to hypertriton momentum. Bottom: Distribution of real and simulated pion momentum transverse to hypertriton momentum. All data sets have similar distributions for these three quantities. The red vertical lines indicate the locations of the cuts. 104 17 Yields The background-subtracted reconstructed hypertriton invariant mass spectra for the four data sets used are shown in Figure 30. Plots with the background included are not shown because, without the subtraction, it is difficult to see a difference between the estimated combinatorial background and the real data. The results are 76 ± 33 hypertritons for the 39 GeV data, 85±27 hypertritons for the 200 GeV data from the 2010 run, 123±50 hypertritons for the 2011 run data using ordinary matter and, 55 ± 25 hypertritons for the 2011 run data using antimatter. The same results are shown in Figure 31 with finer binning. The broad bins (Figure 30) are 4 MeV/c2 wide and the narrow bins (Figure 31) are exactly one-third the width of the broad bins. All data sets have a very narrow peak at, or very close to, the known hypertriton mass of 2.991 GeV/c2. The peak in the spectrum from the 2010 run, 39 GeV data set with fine-binning has a Gaussian shape with a width of 4 MeV/c2 (3 bins) at half max. The peak in the spectrum from the 2010 run, 200 GeV data set is 8/3 MeV/c2 (2 bins) wide but does not have a clear Gaussian shape. The peaks in the spectra from the 2011 run data sets are no more than 4/3 MeV/c2 (1 bin) wide. There appears to be a systematic difference between the 2010 run and 2011 run data sets causing the peaks in the Run 10 data to be shifted 4 MeV/c2 (1 bin) to the right. This shift is present for both data sets from the 2010 run. Small shifts of this magnitude are expected, based on the known systematic uncertainties in the measurements. As a check of the method used to construct the invariant mass spectrum, the same method was used for the lambda hyperon decaying into a proton and pion. The results are shown in Figure 32. Here, the background is included because the combinatorial background is much 105 smaller making it easy to distinguish from the real data. There is sharp Gaussian shaped peak centered on the lambda hyperon mass of 1.116 GeV/c2. It is possible to get a much sharper peak with a relatively smaller combinatorial background but Figure 32 was made using a small subset of the available data. Figure 30: Background-subtracted reconstructed hypertriton invariant mass spectrum with a bin-width of 4 MeV/c2. The blue arrow shows the location of the hypertriton mass (2.991 GeV/c2). There is a narrow peak in the data at this point for the 2011 run data (bottom row). For the 2010 run data (top row), there is a narrow peak located 4 MeV/c2 higher than the known hypertriton mass. 106 Figure 31: Background-subtracted reconstructed hypertriton invariant mass spectrum with a bin-width of 4/3 MeV/c2. The blue arrow shows the location of the hypertriton mass (2.991 GeV/c2). There is a narrow peak in the data located 4/3 MeV/c2 higher than this point for the 2011 run data (bottom row) and 8/3 MeV/c2 higher than this point for the 2010 run data (top row). 107 Figure 32: Reconstructed lambda invariant mass spectrum (with combinatorial background) for the decay Λ → p + π−. The large peak is centered on the lambda hyperon mass of 1.116 GeV/c2. 108 18 Other Measurements Considered Like all unstable particles, hypertritons obey an exponential decay, −t/τ N(t) = N0e , (175) where N(t) is the number of hypertritons after some time, t, N0 is the number of hypertritons at t = 0 and τ is the characteristic lifetime of the hypertriton. The lifetime of an individual hypertriton candidate is calculated by dividing the decay length (distance between the pri- mary and secondary vertices) by βγ, where β is the reconstructed speed of the hypertriton − 1 and γ = (1 − β2) 2 . The relativistic factor, γ, is needed to account for time dilation. Due to the relativistic speed of the hypertriton, its lifetime in the lab frame is much longer than its lifetime in the rest frame. The characteristic lifetime can be calculated using the least χ2 method. The hypertriton sample is separated into n lifetime bins. The expected number of hypertritons in the ith bin is Z ti+1 N0xi = N(t)dt ti −ti/τ −ti+1/τ = N0τ e − e . (176) With this, χ2 is defined as n 2 X N0xi − yi χ2 = , (177) σ i=1 i 109 th where yi is the measured number of hypertritons in the i lifetime bin and σi is the uncer- 2 tainty associated with yi. To minimize χ we require ∂χ2 = 0 . (178) ∂N0 From this it follows Pn xiyi i=1 σ2 N = i . (179) 0 2 Pn xi i=1 σi Equation (177) now becomes 2 n x Pn xiyi X j i=1 σ2 yj χ2 = i − . (180) 2 Pn xi σj j=1 σj i=1 σi 2 Since xi is a function of τ and all other quantities are measured, we now have χ as a function of τ. The most likely value for τ is the one that minimizes χ2. Although the lifetime calculation was never completed, it is likely that the error would be very large given the difficulty of separating real hypertriton three-body decays from the combinatorial background. When separated into just three lifetime bins, the hypertriton signal becomes lost in the noise. A reliable calculation would require a very large data set that would allow the use of tighter cuts to obtain a purer hypertriton sample while keeping enough candidates to get accurate lifetime statistics. 18.1 Embedding Getting an accurate measurement of the characteristic lifetime is slightly more complicated than what was described above. Due to detector efficiencies, the relative numbers of hy- pertritons in the lifetime bins may not be correct which would distort the calculation. To correct these numbers, simulated hypertriton decays are mixed with, or embedded in, the 110 real data. Embedding is a general technique needed for many types of analysis. The number of embedded particles must be kept very small compared with the number of real particles. The simulated data is processed in the exact same way as the real data and the number of simulated hypertriton decays that survive to the final analysis is compared the number originally embedded for each lifetime bin. The ratio of these numbers is the efficiency for that lifetime bin which is used to correct the measured values. Since the lifetime calculation was never completed and there was no other measurement that required embedding, such as a production cross-section, there was no reason to do embedding for this analysis. 111 19 Summary The topics discussed here are a small subset of the rich field of relativistic heavy-ion collisions. There has been no discussion of jets or collective motion which are two very active areas of research in this field. Instead, I have focused on theoretical strong magnetic field effects on quarkonia and an experimental search for a three-body decay of the hypertriton, a short-lived isotope of hydrogen. Magnetic fields have unexpected effects on matter and will continue to fascinate us for a long time. Their effects on quarkonia are important for the phenomenology of heavy-ion collisions and forces us to reconsider the precise definitions of particles. We have learned that the brief but intense magnetic field generated in heavy-ion collisions may be enough to produce measurable changes to meson spectra. Changes to the quarkonia state happen in a smooth continuous way regardless of how quickly the magnetic field increases. For a strong enough magnetic field, the masses of quarkonia increase but some states reach a minimum before the mass begins its steady increase with increasing magnetic field strength. It may be that all quarkonia states have a minimum mass for some critical value of magnetic field strength but for some states the minimum mass is very shallow and occurs at a small critical value. The presence of a background magnetic field leads to off-diagonal terms in the Hamiltonian causing different vacuum eigenstates to be mixed. There are a few important caveats to be mentioned. The intense magnetic fields produced in relativistic heavy-ion collisions are not uniform and exist for a very brief time ( 10−15 s). Extending this analysis to high momentum states will require a more careful treatment of higher order effects. The effects of vacuum polarization ([97],[98]) and finite temperature have 112 been completely ignored. Although the heavy mass of the charm and bottom quarks allow for a non-relativistic treatment, some relativistic corrections may be necessary, especially for charmonia and the states with high COM momentum. The biggest challenge to identifying particle decays is distinguishing real decays from the combinatorial background. The problem becomes more difficult as the number of de- cay products increase and their masses decrease. A precise measurement of the hypertriton lifetime and binding energy is important for our understanding of the hyperon-nucleon inter- action. A precise lifetime measurement requires a large sample, especially for the three-body decay. Most lifetime measurements completed so far are consistent with theory but have large uncertainties. The topics discussed here have important implications for astrophysics as well as heavy- ion collisions. The equation of state for the dense nuclear matter found inside compact stars may be sensitive to hyperon physics. The importance of hyperons in astrophysics is still under appreciated and poorly understood. 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