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Universal Four-Fermion Interaction in Lattice Effective Field Theory

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Universal Four-Fermion Interaction in Lattice Effective Field Theory Mississippi State University Scholars Junction Theses and Dissertations Theses and Dissertations 1-1-2015 Universal Four-Fermion Interaction in Lattice Effective Field Theory Kristopher Katterjohn Follow this and additional works at: https://scholarsjunction.msstate.edu/td Recommended Citation Katterjohn, Kristopher, "Universal Four-Fermion Interaction in Lattice Effective Field Theory" (2015). Theses and Dissertations. 4849. https://scholarsjunction.msstate.edu/td/4849 This Graduate Thesis - Open Access is brought to you for free and open access by the Theses and Dissertations at Scholars Junction. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of Scholars Junction. For more information, please contact [email protected]. Universal four-fermion interaction in lattice effective feld theory By Kristopher J. Katterjohn A Thesis Submitted to the Faculty of Mississippi State University in Partial Fulfllment of the Requirements for the Degree of Master of Science in Physics in the Department of Physics and Astronomy Mississippi State, Mississippi August 2015 Copyright by Kristopher J. Katterjohn 2015 Universal four-fermion interaction in lattice effective feld theory By Kristopher J. Katterjohn Approved: Gautam Rupak Lan Tai Moong (Major Professor) Dipangkar Dutta (Committee Member) R. Torsten Clay (Committee Member) Hendrik F. Arnoldus (Graduate Coordinator) R. Gregory Dunaway Professor and Dean College of Arts & Sciences Name: Kristopher J. Katterjohn Date of Degree: August 14, 2015 Institution: Mississippi State University Major Field: Physics Major Professor: Dr. Gautam Rupak Lan Tai Moong Title of Study: Universal four-fermion interaction in lattice effective feld theory Pages of Study: 42 Candidate for Degree of Master of Science In this thesis we study non-relativistic, low-energy, s-wave scattering in a four-body spin-1/2 fermion system. The scattering is caused by an attractive two-fermion contact interaction which is capable of producing a weakly bound state known as a dimer. This four-fermion system is used to study the scattering of a two-dimer system. Using Hybrid Monte Carlo methods we compute the ground state energies of the sys- tem on a lattice. Luscher’s fnite-volume formula and the Effective Range Expansion are used to calculate the dimer-dimer scattering length add and effective range rdd in terms of the fermion-fermion scattering length aff . Using these techniques we obtain the values add=aff = 0:60 +/- 0:04 and rdd=aff = 3:2 +/- 0:5. This scattering length shows excellent agreement with the numerous values in the literature. We also compare this effective range with the only currently known value in the literature. DEDICATION To my parents. ii ACKNOWLEDGEMENTS First, I would like to thank my thesis advisor, Dr. Gautam Rupak. He encouraged me to work independently, but he was always there to answer the many questions that I had. I could not have completed this without his help and guidance. I would also like to thank my committee members, Dr. R. Torsten Clay and Dr. Di- pangkar Dutta, for agreeing to be on my committee and for their comments and sugges- tions. In addition, I would like to thank the Department of Physics and Astronomy here at Mississippi State University for accepting me into the graduate program and for fnan- cially supporting my studies through teaching assistantships. I would also like to thank the various professors with whom I have taken classes. I have learned a lot about physics during my stay here. Most of the work in this thesis was done in collaboration with Dr. Dean Lee and Dr. Ser- dar Elhatisari of North Carolina State University. I have learned a lot by working with them. Computing resources were provided by the High Performance Computing centers at Mississippi State University, North Carolina State University and RWTH Aachen Univer- sity. Partial fnancial support was provided by the National Science Foundation. iii TABLE OF CONTENTS DEDICATION .................................... ii ACKNOWLEDGEMENTS .............................. iii LIST OF TABLES .................................. vi LIST OF FIGURES .................................. vii CHAPTER 1. INTRODUCTION .............................. 1 2. FORMALISM ................................ 6 2.1 Continuum Theory ......................... 6 2.2 Euclidean Time Projection ..................... 8 2.3 Lattice Theory ........................... 9 3. FINITE-VOLUME AND LATTICE METHODS ............. 16 3.1 Tuning the Lattice Coupling .................... 16 3.2 Luscher’s¨ Formula and Phase Shifts ................ 17 3.2.1 Luscher’s¨ Formula ...................... 17 3.2.2 Fermion-fermion Scattering Lengths on the Lattice .... 18 3.2.3 Dimer-dimer Momenta and Phase Shifts .......... 19 3.3 Hybrid Monte Carlo and Importance Sampling .......... 21 3.3.1 Markov Chain Monte Carlo ................. 21 3.3.2 Hybrid Monte Carlo ..................... 22 3.3.3 HMC Algorithm ....................... 23 3.3.4 Calculating the Ground State Energies ........... 25 3.3.5 HMC Effciency ....................... 26 4. DATA AND ANALYSIS .......................... 28 4.1 Numerical Parameters ....................... 28 iv 4.2 Data ................................. 28 4.2.1 Scatter Plot ......................... 29 4.2.2 Linear Fit with Errors in Both Coordinates ......... 34 5. CONCLUSIONS .............................. 37 REFERENCES .................................... 39 v LIST OF TABLES 3.1 Continuum and lattice fermion-fermion scattering lengths ......... 19 4.1 Comparison between HMC and exact matrix calculations .......... 29 4.2 Dimer-dimer lattice ERE parameters using scatter plots ........... 32 4.3 Dimer-dimer continuum scattering length and effective range ........ 34 4.4 Dimer-dimer lattice ERE parameters using a linear ft ............ 35 vi LIST OF FIGURES 3.1 Feynman diagrams for fermion-fermion scattering ............. 17 (1) 3.2 Fits forB = 5:0 MeV andL = 10 .................... 27 0 0 3.3 Comparison of HMC calculations withL t = 0 andL t =6 0 ......... 27 (1) 4.1 Scatter plot forB = 5:0 MeV with 21 cluster points ........... 31 4.2 Linear ft for add=aff using Eq. (4.3) ..................... 33 4.3 Linear ft forr dd=aff using Eq. (4.4) ..................... 33 4.4 Linear ft with errors in both directions shown ................ 36 vii CHAPTER 1 INTRODUCTION In this work we consider the scattering in a four-body system of spin-1/2 non-relativistic fermions interacting at low energy and producing a large scattering length. By “large” we mean that the scattering length is much larger in magnitude than the range of the interac- tion. We shall always work in “natural” units where~ = 1 = c. In the low energy regime s-wave scattering provides the dominant effects [1]. This is because, at a fxed energy, higher angular momentum corresponds to a more peripheral collision. The s-wave scattering amplitude A(p) for non-relativistic, low-energy, two-body scattering is given by 2π 1 A(p) = ; (1.1) µ p cot δff (p) − ip where p is the relative momentum between the fermions, µ is the effective mass of the fermions and δff (p) is the s-wave phase shift of the fermions. An analytical expansion of p cot δff (p), known as the Effective Range Expansion (ERE), is given by [2, 3, 4, 5] 1 1 2 4 p cot δff (p) = − + rff p + O(p ) ; (1.2) aff 2 where aff is known as the fermion-fermion scattering length and rff is known as the fermion- fermion effective range. Eq. (1.2) is a very general expansion that depends only on the 1 short-range nature of the interaction and on the analyticity of the scattering amplitude. It is common to keep only the frst two terms in the right-hand side of Eq. (1.2). This is then called the “shape-independent approximation formula” [1, 6, 7] or the “effective-range theory” [8]. For scattering at suffciently low momentum the scattering amplitude Eq. (1.1) is given by 2π 1 A(p) = ; (1.3) 1 µ − − ip aff which has a pole at p = i=aff in the complex p plane. Then the non-relativistic energy of this state is given by p2 −1 E = = : ff 2 (1.4) 2µ 2µaff This negative energy is expected because poles in the scattering amplitude correspond to bound states [1, 9, 10]. Here we are assuming that jrff j is of “natural size” or smaller. Also notice that a large scattering length yields a weakly bound state since Eff can be made arbitrarily small in magnitude (while staying negative) for suffciently large jaff j. In the case of Eq. (1.4), since the complex Eff plane is two-sheeted, we follow the usual convention of choosing the cut to lie along the positive real axis so that the frst sheet is the “physical” sheet [1, 7, 9]. Then these states with aff > 0 correspond to real bound states and states with aff < 0 correspond to so-called virtual bound states. Henceforth, by “dimer” we shall mean a real bound state between two fermions. There are a number of physical examples of systems which produce large scattering lengths when scattering at low energies. The scattering of neutrons and protons, which 2 are collectively known as nucleons, provide several examples of this. The neutron-neutron and neutron-proton scattering in the s-wave spin-singlet channel have scattering lengths of ann ∼ −19 fm [11] and anp ∼ −24 fm [12], respectively. Hence, these are virtual bound states. Neutron-proton scattering in the s-wave spin-triplet channel has a scattering length of anp ∼ +5 fm [13]. This real bound state is called a deuteron. In contrast, the range of these nucleon-nucleon interactions is set by the Compton wavelength of the pion, R ∼ 1=mπ ∼ 1:4 fm. In 1999, George Bertsch proposed the “Many-Body X Challenge” which asks for the ground state properties of many-body systems of spin-1/2 fermions interacting with a zero- range interaction and producing an infnite scattering length (the so-called unitary limit [1]). This can be considered a somewhat idealized model of a dilute neutron matter with an attractive two-body interaction. This also leads to the concept of a unitary Fermi gas where, at low momentum, the scattering amplitude becomes 2πi A(p) = ; (1.5) µp and the system is constrained by the need to conserve probability (unitarity).
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