LLow Energy Continuum and Lattice Effective Field Theories
Serdar ElhatisariA dissertation submitted to the Graduate Faculty ofNorth Carolina State Universityin partial fulfillment of therequirements for the Degree ofDoctor of PhilosophyPhysicsRaleigh, North Carolina2014 a r X i v : . [ nu c l - t h ] S e p Copyright 2014 by Serdar ElhatisariAll Rights Reserved
BSTRACT
In this thesis we investigate several constraints and their impacts on the short-range potentials inthe low-energy limits of quantum mechanics. We also present lattice Monte Carlo calculationsusing the adiabatic projection method.In the first part of the thesis we consider the constraints of causality and unitarity for particlesinteracting via strictly finite-range interactions. We generalize Wigner’s causality bound to thecase of non-vanishing partial-wave mixing. Specifically we analyze the system of the low-energyinteractions between protons and neutrons. We also analyze low-energy scattering for systemswith arbitrary short-range interactions plus an attractive 1 { r α tail for α ě
2. In particular, wefocus on the case of α “ { r tail. We arguethat a similar universality class exists for any attractive potential 1 { r α for α ě s -wave, p -wave,and d -wave phase shifts. For comparison, we also compute exact lattice results using Lanczositeration and continuum results using the Skorniakov-Ter-Martirosian equation. For our MonteCarlo calculations we use a new lattice algorithm called impurity lattice Monte Carlo. Thisalgorithm can be viewed as a hybrid technique which incorporates elements of both worldlineand auxiliary-field Monte Carlo simulations. EDICATION
Dedicated to my parents and siblings .i CKNOWLEDGEMENTS
It is with great pleasure that I take this opportunity to express my deepest appreciation to all ofthose who supported me in any respect during the completion of the thesis.First, I am very grateful to my advisor Dean Lee for his continuous supports, encouragements,endless helps and being a constant source of intellectual stimulation in every step of my studiesat NC State. I am very fortunate not only because I have had the opportunity to learn from hisbrilliant ideas but also because I have been involved in such a friendly scientific environmentprovided by his kindness.I wish to thank the Department of Physics at NC State University for taking such a nicecare of all students, and want to thank everyone in the department for establishing a verynice atmosphere. I am grateful to the Turkish Government Ministry of National Educationfor supporting me with a doctoral fellowship and the Department of Physics at NC StateUniversity for additional supports. I am also thankful to the North Carolina State University HighPerformance Computing center for extensive computer support and HPC Remedy Workgroup,specially Gary Howell, for valuable technical supports.I would like to thank Michelle Pine, Sebastian König and Shahin Bour who are members ofthe Lee Research Group. A special thank to Sebastian König for his collaborations and usefuldiscussions, and Michelle Pine for carefully reading and useful comments on the manuscript ofmy thesis. I also thank the physicist from the Nuclear Lattice Effective Field Theory collaborationfor their valuable discussions on my work. Specially I am grateful to Gautam Rupak for extensivediscussions on several aspects of the adiabatic projection method, and I thank Shahin Bour,Hans-Werner Hammer and Ulf-G. Meißner for discussions on the impurity Monte Carlo method.As a member of an internationally recognized group, I have had several opportunities totravel to many places and meet pioneers in the fields. I would like to thank Ruhr-Universitätiiochum for its hospitality during a couple of weeks in the summers of 2011, 2012 and 2013,and thank Evgeny Epelbaum and Hermann Krebs for useful discussions. I wish to thank theorganizers of the program “Light nuclei from first principles" in the Institute of Nuclear Theoryat the University of Washington for inviting me during two weeks of the program and thank theINT for its hospitality. Also I would like to thank the organizer of the Third UiO-MSU-ORNL-UT School on Topics in Nuclear Physics held at the Oak Ridge Laboratory for the financialsupport to attend the winter school.Last, but surely not least, I would like to thank all my friends for their unquestioning supportof my goals and dreams. I also wish to thank those whom I have met in the Triangle area formaking the last four years of my life more fun, special thanks to Ako ˘glu, Ay and Gökçe families,H. ˙Ibrahim Akyıldız, Murat An, Zubair Azad, Esra Öztürk, ˙I. ¸Safak Bayram, Shaan Qamar, EnisÜçüncü and Hasan Yıldırım. iii
ABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixChapter 1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 3 Lattice Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . Chapter 4 Neutron-proton scattering . . . . . . . . . . . . . . . . . . . . . . . . . iv.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Uncoupled Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Coupled Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Causality Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.5 Neutron-Proton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5.1 Uncoupled Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5.2 Coupled Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7 One-Pion Exchange Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.8 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Chapter 5 van der Waals interactions . . . . . . . . . . . . . . . . . . . . . . . . . (cid:96) . . . . . . . . . . . . . . . . . . 985.5 Impact on effective field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.6 Quantum defect theory and the modified effective range expansion . . . . . . . 1065.7 Causal range for single-channel scattering . . . . . . . . . . . . . . . . . . . . 1085.8 Causal range near a magnetic Feshbach resonance . . . . . . . . . . . . . . . . 1105.9 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Chapter 6 Impurity Lattice Monte Carlo and the Adiabatic Projection Method . .
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.1 Bessel and related Functions . . . . . . . . . . . . . . . . . . . . . . . . 155A.2 Coulomb wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . 156A.2.1 Regular solution f p ε , (cid:96) ; r γ q . . . . . . . . . . . . . . . . . . . 158v.2.2 Irregular solution.I h p ε , (cid:96) ; r γ q . . . . . . . . . . . . . . . . . . 159A.2.3 Irregular solution.II g p ε , (cid:96) ; r γ q . . . . . . . . . . . . . . . . . . 160A.3 van der Waals wave functions . . . . . . . . . . . . . . . . . . . . . . . . 165A.4 Low-energy expansions of the function terms in van der Waals wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169B.1 Wronskians of the wave functions . . . . . . . . . . . . . . . . . . . . . 169B.1.1 Single channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 169B.1.2 Coupled channels . . . . . . . . . . . . . . . . . . . . . . . . . . 172Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175C.1 Coupled-channel Parameterizations . . . . . . . . . . . . . . . . . . . . . 175C.2 Numerical Test using Delta-Function Shell Potentials in a coupled-channelsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184D.1 Finite-volume binding energy corrections and topological volume correc-tions for scattering with arbitrary (cid:96) . . . . . . . . . . . . . . . . . . . . . 184vi IST OF TABLES
Table 2.1 The integration constants ∆ n ,(cid:96) of a two-neutral-particles system for (cid:96) ď
2. . 22Table 2.2 Integration constants ∆ cn ,(cid:96) calculated from Eqs. (2.71)-(2.74) for (cid:96) ď γ ˜ h (cid:96) p i κ q in the Coulomb modified effective range expansionfor (cid:96) ď
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Table 3.1 Decomposition of the SO p q into the irreducible representations of theSO p , Z q for (cid:96) ď
6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Table 4.1 The eigenphase low energy parameters of uncoupled channels for neutron-proton scattering by the NijmII and the Reid93 interaction potentials. . . . . 67Table 4.2 The eigenphase low energy parameters of coupled channels for neutron-proton scattering by the NijmII and the Reid93 interaction potentials. . . . . 68Table 4.3 The eigenphase low energy mixing parameters of coupled channels forneutron-proton scattering by the NijmII and the Reid93 interaction potentials. 68Table 4.4 The causal ranges for uncoupled channels. . . . . . . . . . . . . . . . . . . 77Table 4.5 The causal ranges for coupled channels. . . . . . . . . . . . . . . . . . . . 77Table 4.6 The Cauchy-Schwarz ranges for coupled channels. . . . . . . . . . . . . . 77Table 4.7 The potential range dependence of the causal range in various channels. . . 88Table 5.1 Scattering parameters and causal ranges for s -wave scattering of Li , Na , and Cs pairs. The scattering data collection is taken from Ref. [66]. Incolumns I and IV the scattering data for Li are from Ref. [41], the scatteringdata for Na are from Refs. [40, 41], and data for
Cs are from Ref. [129].In column III the effective range parameters, R , are calculated analyticallyin Refs. [66, 75]. In column IV, the R are obtained from numerical calcula-tions. The scattering parameters in columns II and V are calculated usingEq. (5.37) and Eq. (5.39), and the causal ranges in column VI are obtainedfrom Eq. (5.40). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Table 6.1 Momentum of the dimer, (cid:126) p d , with p “ π { L . The total momentum of thesystem is zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Table 6.2 The exact and Monte Carlo results for the ground state and lowest lyingeven-parity energies in a periodic box of length La “ .
79 fm. The MonteCarlo results are obtained from the r ˆ M a p t qs ˆ adiabatic matrix. . . . . . . 130Table 6.3 The exact and Monte Carlo results for the energies of the lowest two odd-parity states in a periodic box of length La “ .
79 fm. The Monte Carloresults are obtained from the r ˆ M a p t qs ˆ adiabatic matrix. . . . . . . . . . . 132viiable C.1 Numerical results for scattering length and effective range in two-bodyinteraction by the delta function potentials. . . . . . . . . . . . . . . . . . . 180Table C.2 Numerical results for the mixing parameters in two-body interaction by thedelta function potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181viii IST OF FIGURES
Figure 4.1 The plot of r b (cid:96) p r q ´ r (cid:96) s{ r for neutron-proton scatteringvia the NijmII potential in the s ` (cid:96) j channel. . . . . . . . . . . . . . . . . 69Figure 4.2 The plot of f j ´ p r q as a function of r for neutron-proton scattering via theNijmII potential for j ď
3. Here f p r q is rescaled by a factor of 0 .
01 and f p r q is rescaled by a factor of 10 ´ . . . . . . . . . . . . . . . . . . . . . . 70Figure 4.3 The plot of g j ` p r q as a function of r for neutron-proton scattering via theNijmII potential for j ď
3. Here g p r q is rescaled by a factor of 0 .
1. . . . . 71Figure 4.4 We plot Re ”a f p r q g p r q ı , ´ Re ”a f p r q g p r q ı , and Re r h p r qs as func-tions of r for neutron-proton scattering in S - D coupled channel. . . . . 72Figure 4.5 We plot Re ”a f p r q g p r q ı , ´ Re ”a f p r q g p r q ı , and Re r h p r qs as func-tions of r for neutron-proton scattering in P - F coupled channel. Thefunctions are rescaled by a factor of 0 .
01. . . . . . . . . . . . . . . . . . . 74Figure 4.6 We plot Re ”a f p r q g p r q ı , ´ Re ”a f p r q g p r q ı , and Re r h p r qs as func-tions of r for neutron-proton scattering in D - G coupled channel. Thefunctions are rescaled by a factor of 0 .
01. . . . . . . . . . . . . . . . . . . 76Figure 4.7 Plot of the potential matrix elements W p r q “ V C p r q , W p r q “ W p r q “? V T p r q and W p r q “ V C p r q ´ V T p r q as a function of r in the S - D coupled channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 4.8 Plot of the model potential in the P channel. In this channel S “ T “ C π “ C A “ . C B “ . C Π “ C D “ C F “ C G “ ´ . m A “ . m π , m B “ . m π , m G “ . m π , C “ ´ . C “ . C “ .
933 and C “ ´ . R Gauss “ . R Exch. “ . D channel. In this channel S “ T “ C π “ ´ C A “ ´ . C B “ ´ . C Π “ C D “ C F “ C G “ . m A “ m π , m B “ m π , m G “ . m π , C “ . C “ ´ . C “ .
384 and C “ ´ . R Gauss “ . R Exch. “ . D channel. In this channel S “ T “ C π “ ´ C A “ ´ C B “ ´ C Π “ ´ C D “ ´ C F “ ´ C G “ . m A “ m D “ m π , m B “ m F “ m π , m G “ . m π , C “ . C “ ´ . C “ ´ .
455 and C “ . R Gauss “ . R Exch. “ . W r f , , f , sp r q , W r g , , g , sp r q , and W r g , , f , sp r q as a function of r for (cid:96) “ β “
50 (a.u.). . . . . . . . . . . . . . . . . 103Figure 5.2 (Color online) Plot of W r f , , f , sp r q , W r g , , g , sp r q , and W r g , , f , sp r q as a function of r for (cid:96) “ β “
50 (a.u.). . . . . . . . . . . . . . . . . 103ixigure 5.3 (Color online) Plot of W r f , , f , sp r q , W r g , , g , sp r q , and W r g , , f , sp r q as a function of r for (cid:96) “ β “
50 (a.u.). . . . . . . . . . . . . . . . . 104Figure 6.1 A segment of a worldline configuration on a 1+1 dimensional Euclideanlattice. See the main text for derivations of the reduced transfer-matrixoperators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Figure 6.2 The ground state energy is shown versus projection time t using either oneor four initial/final states in Panel (a) and the first two excited state energieswith even parity in Panel (b). For comparison we show the exact latticeenergies as dotted horizontal lines. . . . . . . . . . . . . . . . . . . . . . 129Figure 6.3 The lowest two odd parity energies as a function of Euclidean projec-tion time t . For comparison we show the exact lattice energies as dottedhorizontal lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Figure 6.4 The s -wave scattering phase shift versus the relative momentum betweenfermion and dimer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure 6.5 The p -wave scattering phase shift versus the relative momentum betweenfermion and dimer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Figure 6.6 The d -wave scattering phase shift versus the relative momentum betweenfermion and dimer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136x hapter Overview
Nuclear structure and reaction studies are at the heart of low energy nuclear physics research. Abroad goal of this research is to understand the basic interactions among fundamental particles.By fundamental particles we refer to the relevant degrees of freedom that can be probed by theenergy scales of a given system. In nuclear physics physical phenomena are observed over arange of different energy scales. In low energy nuclear physics the relevant degrees of freedomare nucleons and light mesons which mediate forces between nucleons, and physical observablesare insensitive to the details at high energies or equivalently short distances.Effective field theory (EFT) is a very general framework to understand physics. The generalidea is that a simple effective description of the physics can describe the relevant features ofa given system even if the details at short distances are disregarded. In low-energy nucleareffective field theory the detailed shape of the short-range nuclear forces are not important.Instead effective field theory organizes the nuclear interactions as an expansion in powers ofmomenta and other low energy scales such as the pion mass.Lattice effective field theory is a powerful numerical method which is formulated in theframework of effective field theory. The method is quite economical when one uses pionless1ffective field theory with the nucleons interacting via only local contact interactions. Recentdevelopments in lattice EFT allow one to study nuclear scattering and reactions. The adiabaticprojection method is a general framework for calculating scattering and reactions on the lattice.This method uses a set of initial cluster states and Euclidean time projection to construct alow-energy effective field theory for the cluster states.In this thesis we summarize work done on a number of topics in the framework of effectivefield theories during my Ph.D. study. We start with a brief review of scattering processes withstrictly finite-range interactions in Chapter 2. We also discuss the case where long-range forcesare present in addition to the short-range interactions, and we specifically consider Coulombinteractions.In Chapter 3 after a brief introduction we define the basic continuum and lattice formulationsof non-relativistic quantum mechanics. We then take a short detour to derive the connectionbetween the operator formalism in quantum mechanics and lattice Grassmann path integrals.We introduce the adiabatic projection method and describe the implementation of the methodin lattice effective field theory. We discuss the details of and some mathematical tools forscattering phase shift calculations at finite volume. Then we close this chapter by reviewingMonte Carlo simulations employed to compute observables in finite volumes with periodicboundary conditions.In Chapter 4 we consider the Wigner causality constraint and unitarity for the low-energyinteractions with strictly finite range. We derive the generalization of Wigner’s causality boundto the case of non-vanishing partial-wave mixing. As an application in nuclear physics, weanalyse the low-energy interactions of protons and neutrons. We investigate the constrains on therange of the interactions between neutrons and protons in effective field theory and universalityin many-body Fermi systems.In Chapter 5 we analyze low-energy scattering for arbitrary short-range interactions plus2ny attractive potential 1 { r α for α ě
2. This type of long-range force plays an important role inlow-energy atomic and nuclear physics. In particular, we consider the van der Waals interactionand derive the constraints of causality and unitarity. We also briefly discuss multichannel systemsnear a magnetic Feshbach resonance.In Chapter 6 we present scattering of composite particles using lattice Monte Carlo simula-tions with the adiabatic projection method. For our Monte Carlo simulations we introduce theimpurity lattice Monte Carlo algorithm. As an application we consider fermion-dimer elasticscattering in the limit of zero-range interactions. Then we present results for the scattering phaseshifts in the continuum and infinite volume limits as well as in finite volumes.3 hapter Scattering Theory
The scattering of two spinless particles is described by an incident plane wave along the z direction and spherical outgoing scattered wave with amplitude f as ψ p (cid:126) r q r Ñ8 ÝÑ p π q { „ e ipz ` f p p , θ q e ipr r , (2.1)where (cid:126) p is the outgoing relative momenta, (cid:126) r is the relative coordinates and θ is the anglebetween (cid:126) p and the z axis. The scattering amplitude is a non-trivial physical quantity in which allquantitative information about the scattering process is contained. An important constraint onthe scattering amplitude is unitarity. Unitarity requires that the sum of all outcome probabilitiesis one. In other word the normalization of the incoming wave must be preserved.In addition, the time evolution of any quantum mechanical system obeys causality as wellas unitarity. Causality requires that the cause of an event must occur before any resultingconsequences are produced. In non-relativistic quantum mechanics the causality constraintmeans that the outgoing wave may depart only after the incoming wave reaches the scattering4bject. The constraints of causality in quantum mechanics with finite range interactions werefirst studied by Wigner [174]. Phillips and Cohen [142] derived the causality bound on thescattering parameters of low energy effective field theories by considering the s -wave case andfinite-range interactions in three dimensions. Later, the causality bounds for arbitrary dimension d or arbitrary angular momentum (cid:96) were investigated in Refs. [94, 95].The scattering processes described above is an idealized system with the assumptions thatparticles are structureless. This simplified process is called single-channel since there is only onepossible final configuration which is the same as the initial one, a p b , a q b . In general, elementaryor composite particles have spin structures; and accordingly, the interaction potentials becomespin-dependent which makes the scattering processes more complicated. In these types ofsystem different possible final outcomes exist [164], and the scattering processes are called multi-channel scattering. See Ref. [100] for a detailed rigorous mathematical formulation of thesystem with short-range interactions, and the general multi-channel problems in the presence oflong-range potentials were studied in Ref. [49, 92]. In this and the following sections we will give brief reviews on several topics of the two-bodyelastic scattering problem. Following Refs. [136] with slightly changed notation, we will discussthe scattering of particles within the framework of the formal scattering theory in configurationspace. 5 .2.1 Single channel
We start with an idealized system and consider elastic scattering of two spinless particlesinteracting via a spherically symmetric potential in the center-of-mass frame. We use unitswhere ¯ h “
1. The free radial Schrödinger equation with energy E “ p {p µ q is „ ´ r ddr ˆ r ddr ˙ ` (cid:96) p (cid:96) ` q r R p p q (cid:96) p r q “ p R p p q (cid:96) p r q . (2.2)where (cid:96) is the orbital angular momentum. It is convenient to use the rescaled radial function u p p q (cid:96) p r q given by u (cid:96) p pr q “ r R p p q (cid:96) p r q , (2.3)then the Schrödinger equation describes the system is alternatively „ ´ d dr ` (cid:96) p (cid:96) ` q r u (cid:96) p pr q “ p u (cid:96) p pr q , (2.4)Two linearly independent (regular and irregular) solutions of Eq. (2.4) are the Riccati-Bessel S (cid:96) p pr q and Riccati-Neumann C (cid:96) p pr q functions, respectively, which are defined in terms of Besseland Neumann functions in Eq. (A.1) and Eq. (A.2). The asymptotic form of these functions forlarge arguments are S (cid:96) p ρ q | ρ |Ñ8 „ sin ˆ ρ ´ (cid:96) π ˙ , (2.5) C (cid:96) p ρ q | ρ |Ñ8 „ cos ˆ ρ ´ (cid:96) π ˙ . (2.6) By spinless particles we mean either exactly spinless fundamental particles or composite particles withzero-intrinsic angular momentum. Ψ ˘ “ p p ,(cid:126) r q “ ˆ µ p π ˙ { ÿ (cid:96) (cid:96) ÿ m “´ (cid:96) i (cid:96) ψ ˘ (cid:96) p pr q pr Y m (cid:96) p ˆ r q Y m (cid:96) ˚ p ˆ z q , (2.7)where µ is the reduced mass, Y m (cid:96) p ˆ ρ q are the spherical harmonics, ˆ r denotes the polar angles p θ , ϑ q of (cid:126) r , ψ ˘ (cid:96) p pr q is the radial part of the wave function. The solution with the superscript `{´ corresponds to the out/in asymptotic wave when we go to a time-dependent formalism.Using the Legendre polynomial, P (cid:96) p cos θ q “ π (cid:96) ` (cid:96) ÿ m (cid:96) “´ (cid:96) Y m (cid:96) p ˆ r q Y m (cid:96) ˚ p ˆ z q , (2.8)the total wave function is rewritten as Ψ ˘ p p ,(cid:126) r q “ ˆ µπ p ˙ { π r ÿ (cid:96) i (cid:96) p (cid:96) ` q P (cid:96) p cos θ q ψ ˘ (cid:96) p pr q . (2.9)Now inserting the total wave function in the Lippmann-Schwinger integral equation, we get thefollowing integral equation for the radial wave function, ψ ˘ (cid:96) p pr q “ S (cid:96) p pr q ` µ ż ż dr dr G ˘ (cid:96) p p ; r , r q W p r , r q ψ ˘ (cid:96) p pr q , (2.10)where W p r , r q is the interaction potential assumed to be a rotationally invariant operator, and G ˘ (cid:96) p p ; r , r q is the partial wave Green’s function which satisfies the differential equation „ ´ d dr ` (cid:96) p (cid:96) ` q r ´ p G ˘ (cid:96) p p ; r , r q “ ´ δ ` r ´ r ˘ . (2.11)7he Green’s function that satisfies Eq. (2.11) with suitable boundary conditions is defined interms of the Riccati-Bessel and Riccati-Henkel functions as G ˘ (cid:96) p p ; r , r q “ ´ p S (cid:96) p pr ă q h ˘ (cid:96) p pr ą q , (2.12)where r ă signifies the smaller of r and r and r ą is the larger, and h ˘ (cid:96) are the Riccati-Henkelfunctions defined in terms of the Riccati-Bessel and Neumann functions h ˘ (cid:96) p ρ q “ ˘ iS (cid:96) p ρ q ` C (cid:96) p ρ q , (2.13)and their asymptotic forms for large arguments are h ˘ (cid:96) p ρ q | ρ |Ñ8 „ e ˘ i p ρ ´ (cid:96) π q . (2.14)Eq. (2.10) is the solution of the following Schrödinger wave equation, „ ´ d dr ` (cid:96) p (cid:96) ` q r ψ ` (cid:96) p pr q ` µ ż dr W p r , r q ψ ` (cid:96) p pr q “ p ψ ` (cid:96) p pr q . (2.15)At the moment we do not impose any condition on the potential and postpone the discussion tillSection 2.3. At r Ñ 8 the asymptotic form of Eq. (2.10) has a formal solution written in termsof the incident wave, the scattered wave and the scattering matrix, S (cid:96) p p q , as ψ ` (cid:96) p pr q “ i “ h ´ (cid:96) p pr q ´ S (cid:96) p p q h ` (cid:96) p pr q ‰ , (2.16)where S (cid:96) p p q “ ´ i µ p ż ż dr dr S (cid:96) p pr q W p r , r q ψ ` (cid:96) p pr q , (2.17)8he scattering matrix (S-matrix) S (cid:96) p p q is a function of momentum and independent of r ;therefore, S (cid:96) p p q can be defined in terms of a real and momentum dependent phase angle δ (cid:96) p p q ,S (cid:96) p p q “ e i δ (cid:96) p p q . (2.18)Then Eq. (2.15) in the asymptotic limit, up to a normalization, becomes ψ ` (cid:96) p pr q „ sin ˆ pr ´ (cid:96) π ` δ (cid:96) p p q ˙ . (2.19)This scattered wave relative to Eq. (2.5) implies that the impact of the scattering event is tointroduce the shift δ (cid:96) p p q in the phase of the outgoing wave relative to the incoming wave. Theradial wave function is usually written in terms of the reaction matrix (K-matrix) rather thanthe S-matrix. The relation between the reaction and scattering matrix isK (cid:96) “ i p ´ S (cid:96) qp ` S (cid:96) q , (2.20)and the radial wave function in terms of the reaction matrix reads, up to a normalization, ψ ` (cid:96) p pr q „ S (cid:96) p pr q ` K (cid:96) p p q C (cid:96) p pr q . (2.21) In this section we make the scattering problem more complicated by considering that particleshave intrinsic spins. This brings some complications into the formalism introduced in thepreceding sections due to the fact that the spin is an additional degree of freedom. We considerthe scattering of two particles with total spin angular momenta s “ s ` s where s and s are the Sometimes it is called the reactance matrix [136]. (cid:96) and the spin angular momentum s are coupled to give the total angular momentum j . Denote the magnetic quantum number of j , (cid:96) and s in the z direction by M , m and m s , respectively. Since the Hamiltonian of the systemcommutes with the total angular momentum operator J in order to be rotationally invariant, theconserved quantities of the system are j and M . Therefore, depending on the values that s takes,the orbital angular momentum takes different values each of which corresponds to a differentradial wave function.Let us generalize the spherical harmonics for the system of particles with spin and definethe following functions Y Mj (cid:96) s p ˆ r q “ ÿ mm s C p (cid:96) s j , m m s M q Y m (cid:96) p ˆ r q χ sm s (2.22)and { Y Mj p (cid:96) sm s ; ˆ z q “ ´ i χ s ˚ m s ¨ Y Mj (cid:96) s p ˆ z q (2.23)where χ sm s is the normalized eigenfunction of the total spin, C p (cid:96) s j , m (cid:96) m s m q are the Clebsch-Gordan coefficients, and the dot signifies an inner product with respect to the internal coordinates.Since the individual particle spins are conserved, we will suppress them in the expressions. Thecompleteness relation of the Clebsch-Gordan coefficients is ÿ m (cid:96) m s C p (cid:96) s j , m (cid:96) m s m q C p (cid:96) s j , m (cid:96) m s m q “ δ j j δ mm . (2.24)Therefore, the total wave function is written as Ψ ˘ p sm s ; p ,(cid:126) r q “ ˆ µπ p ˙ { ÿ jM (cid:96)(cid:96) s ψ j p˘q (cid:96) s ,(cid:96) s p p , r q r Y Mj (cid:96) s p ˆ r q { Y M ˚ j p (cid:96) sm s ; ˆ z q . (2.25)10fter inserting Eq. (2.25) into the Lippmann-Schwinger integral equation, we obtain the generalform of the radial wave function, ψ j p˘q (cid:96) s ,(cid:96) s p p , r q “ S (cid:96) p pr q δ (cid:96)(cid:96) δ ss ` µ ÿ (cid:96) s ż ż d (cid:126) r d (cid:126) r G ˘ (cid:96) p p ; r , r q W (cid:96) s ,(cid:96) s p r , r q ψ j p˘q (cid:96) s ,(cid:96) s p p , r q , (2.26)where G ˘ (cid:96) is given in Eq. (2.12), and W (cid:96) s ,(cid:96) s is the spin-dependent interaction potential W (cid:96) s ,(cid:96) s p r , r q “ ż d Ω Y M ˚ j (cid:96) s p ˆ r q W p r , r q Y Mj (cid:96) s p ˆ r q . (2.27)The radial wave function ψ j p˘q (cid:96) s ,(cid:96) s is the solution of the coupled Schrödinger equation, „ ´ d dr ` (cid:96) p (cid:96) ` q r ψ j p`q (cid:96) s ,(cid:96) s ` µ ÿ (cid:96) s ż dr W (cid:96) s ,(cid:96) s p r , r q ψ j p`q (cid:96) s ,(cid:96) s “ p ψ j p`q (cid:96) s ,(cid:96) s . (2.28)Therefore, the S-matrix for the particles of the total spin s isS (cid:96) s ,(cid:96) s “ δ (cid:96)(cid:96) δ ss ´ i µ p ÿ (cid:96) s ż ż dr dr S (cid:96) p pr q W (cid:96) s ,(cid:96) s p r , r q ψ j p`q (cid:96) s ,(cid:96) s p p , r q . (2.29) In Eq. (2.18) we have naïvely defined the S-matrix without imposing any conditions on thepotential. However, in Eqs. (2.17) and (2.29) the unitarity condition we have discussed inSection 2.1 requires some constraints on the interaction potential.Throughout our analysis we assume that the interaction is sufficiently well-behaved at theorigin to admit the regular solution S (cid:96) p pr q . This assumption imposes the restriction that as11 Ñ ρ Ñ ψ (cid:96) p ρ q dd ρ ψ (cid:96) p ρ q “ . (2.30)In Ref. [21] it is proven that this condition is fulfilled by a class of potentials V p r q provided that ż R dr r ˇˇ V p r q ˇˇ ă 8 . (2.31)We also consider only energy independent interactions, and we assume that the interactionshave a finite range R . The finite-range condition implies that W p r , r q “ r ą R or r ą R . (2.32)These assumptions assure that the interaction can be written as a local potential, W p r , r q “ V p r q δ ` r ´ r ˘ . (2.33) Under our assumptions on the potential in the preceding section we can obtain the scatteringinformation by measuring Eq. (2.10) relative to the free solution in the asymptotic limit, ψ (cid:96) p r q “ S (cid:96) p pr q ` p f (cid:96) p p q h ` (cid:96) p pr q , (2.34)where f (cid:96) p p q is the partial wave amplitude f (cid:96) p p q “ S (cid:96) p p q ´ ip “ p (cid:96) p (cid:96) ` cot δ (cid:96) p p q ´ ip (cid:96) ` . (2.35)12or the system of particles interacting with a finite-range potential, p (cid:96) ` cot δ (cid:96) p p q is describedby the well-known power series expansion around p , the so called effective range expansion [99,18], p (cid:96) ` cot δ (cid:96) p p q “ ´ a (cid:96) ` r (cid:96) p ` P (cid:96) p ` Q (cid:96) p ` O p p q , (2.36)where a (cid:96) is the scattering length , r (cid:96) is the effective range , and coefficients in higher order termsof p are the shape parameters. The effective range formula for the multi channel scatteringproblem is of the following form [46, 149, 20, 19, 102], ÿ m , n p mm r K ´ s m n p n n “ ´ a mn ` r mn p ` P mn p ` Q mn p ` O p p q , (2.37)where ˆK is the multi channel reaction matrix, p is the diagonal momentum matrix p “ ¨˚˚˚˚˝ p j ´ s ` . . . p j ` s ` ˛‹‹‹‹‚ , (2.38) a mn is the scattering length matrix, r mn is the effective range matrix, and P mn and Q mn are thefirst two lowest shape parameter matrices.In the cases where particles are interacting via long-range forces the power series or theconvergence (analyticity) of Eq. (2.36) is spoiled. Then the function p (cid:96) ` cot δ (cid:96) p p q needsspecial treatment to modify the expansion to make it an analytic function of p . For instance,for a class of potentials falling off as e ´ mr at large distances, the potential introduces a branchcut starting at p “ ´ m { m is the mass of the particle which mediates the interaction,see Ref. [90] for the detailed proof. Nevertheless, the expansion is still converges in a circle ofradius m { p plane.13or the system of charged particles, a modified effective range expansion is used to deal withthe Coulomb tails [39, 18, 21]. For long-range potentials of the form of 1 { r α with α ą α “ α ą α “ In this section we return to the idealized system of two spinless particles. Here we consider thesystem of two-particle interacting via a spherically symmetric potential with finite-range R . Thissystem is described by the radial Schrödinger equation „ ´ d dr ` (cid:96) p (cid:96) ` q r ´ p U p p q (cid:96) p r q ` µ ż R dr W p r , r q U p p q (cid:96) p r q “ . (2.39)where µ is the reduced mass, and W p r , r q is the finite-range potential, W p r , r q “ r ą R or r ą R . Therefore, for r ą R with an arbitrary normalization the solution of Eq. (2.39) is writtenas U p p q (cid:96) p r q “ p (cid:96) r cot δ (cid:96) p p q S (cid:96) p pr q ` C (cid:96) p pr qs . (2.40)14or later convenience we define the rescaled Riccati-Bessel and Riccati-Neumann functions by s (cid:96) p p , r q “ p ´ (cid:96) ´ S (cid:96) p pr q and c (cid:96) p p , r q “ p (cid:96) C (cid:96) p pr q . (2.41)Insertion of the rescaled functions and the effective range expansion, Eq. (2.40) is rewritten asan expansion in powers of p , U p p q (cid:96) p r q “ u ,(cid:96) p r q ` u ,(cid:96) p r q p ` u ,(cid:96) p r q p ` u ,(cid:96) p r q p ` O p p q , (2.42)where u p r q ’s are defined in terms of the effective range expansion parameters by u ,(cid:96) p r q “ ´ a (cid:96) s ,(cid:96) p r q ` c ,(cid:96) p r q , (2.43) u ,(cid:96) p r q “ r (cid:96) s ,(cid:96) p r q ´ a (cid:96) s ,(cid:96) p r q ` c ,(cid:96) p r q , (2.44) u ,(cid:96) p r q “ P (cid:96) s ,(cid:96) p r q ` r (cid:96) s ,(cid:96) p r q ´ a (cid:96) s ,(cid:96) p r q ` c ,(cid:96) p r q , (2.45) u ,(cid:96) p r q “ Q (cid:96) s ,(cid:96) p r q ` P (cid:96) s ,(cid:96) p r q ` r (cid:96) s ,(cid:96) p r q ´ a (cid:96) s ,(cid:96) p r q ` c ,(cid:96) p r q . (2.46)and the functions s n ,(cid:96) p r q and c n ,(cid:96) p r q are given in Appendix A.1. We consider two particles interacting at long distances in addition to the short-range potential.The example we consider in detail is the system of two particles carrying electric charges eZ and eZ . The radial Schrödinger equation (2.39) has a Coulomb potential term γ { r , „ ´ d dr ` (cid:96) p (cid:96) ` q r ` γ r ´ p V p p q (cid:96) p r q` µ ż R dr W p r , r q V p p q (cid:96) p r q “ , (2.47)15here γ “ µα Z Z . Here V p p q (cid:96) p r q is the rescaled radial wave function of a two-body system ofcharged particles, V p p q (cid:96) p r q “ r ¨ R p p q (cid:96) p r q . (2.48)We choose a normalization such that, for r ą R , V p p q (cid:96) p r q is V p p q (cid:96) p r q “ p (cid:96) C η ,(cid:96) ” cot ˜ δ (cid:96) p p q ˆ F p p q (cid:96) p r q ` G p p q (cid:96) p r q ı , “ p (cid:96) ` C η ,(cid:96) cot ˜ δ (cid:96) p p q ˆ f (cid:96) p p , r q ` g (cid:96) p p , r q , (2.49)where F p p q (cid:96) p r q and G p p q (cid:96) p r q are the regular and irregular Coulomb wave functions, and the relatedfunctions f (cid:96) p p , r q and g (cid:96) p p , r q are defined for later convenience as f (cid:96) p p , r q “ p (cid:96) ` C η ,(cid:96) F p p q (cid:96) p r q , (2.50) g (cid:96) p p , r q “ p (cid:96) C η ,(cid:96) G p p q (cid:96) p r q “ ˜ g (cid:96) p p , r q ` ” γ ˜ h (cid:96) p p q ´ ip (cid:96) ` C η ,(cid:96) ı f (cid:96) p p , r q . (2.51)See Appendix A.2 for the functions f (cid:96) p p , r q , g (cid:96) p p , r q and ˜ g (cid:96) p p , r q . The factor C η ,(cid:96) is given by C η ,(cid:96) “ (cid:96) Γ p (cid:96) ` q (cid:96) ź s “ p s ` η q C η , , (2.52)and the function ˜ h (cid:96) p p q is defined as˜ h (cid:96) p p q “ p p q (cid:96) Γ p (cid:96) ` q | Γ p (cid:96) ` ` i η q | | Γ p ` i η q | „ ψ p i η q ` i η ´ log p i η q , (2.53)16here η “ γ p , (2.54) C η , “ πη e πη ´ , (2.55)and ψ p z q is the digamma function. Using these new expressions given in Eqs.(2.50)-(2.55), wecan rewrite Eq. (2.49) as, V p p q (cid:96) p r q “ ” p (cid:96) ` C η ,(cid:96) ´ cot ˜ δ (cid:96) p p q ´ i ¯ ` γ ˜ h (cid:96) p p q ı ˆ f (cid:96) p p , r q ` ˜ g (cid:96) p p , r q , (2.56)The expression in square brackets is the modified Coulomb effective range expansion [21], p (cid:96) ` C η ,(cid:96) ´ cot ˜ δ (cid:96) p p q ´ i ¯ ` γ ˜ h (cid:96) p p q “ ´ a c (cid:96) ` r c (cid:96) p ` P c (cid:96) p ` Q c (cid:96) p ` O ` p ˘ . (2.57)Finally, the Coulomb wave function is written as an expansion in powers of p , V p p q (cid:96) p r q “ v ,(cid:96) p r q ` v ,(cid:96) p r q p ` v ,(cid:96) p r q p ` v ,(cid:96) p r q p ` O p p q , (2.58)17here v p r q ’s are written in terms of the modified Coulomb effective range expansion parameters, v ,(cid:96) p r q “ ´ a c (cid:96) f ,(cid:96) p r q ` g ,(cid:96) p r q , (2.59) v ,(cid:96) p r q “ r c (cid:96) f ,(cid:96) p r q ´ a c (cid:96) f ,(cid:96) p r q ` g ,(cid:96) p r q , (2.60) v ,(cid:96) p r q “ P c (cid:96) f ,(cid:96) p r q ` r c (cid:96) f ,(cid:96) p r q ´ a c (cid:96) f ,(cid:96) p r q ` g ,(cid:96) p r q , (2.61) v ,(cid:96) p r q “ Q c (cid:96) f ,(cid:96) p r q ` P c (cid:96) f ,(cid:96) p r q ` r c (cid:96) f ,(cid:96) p r q ´ a c (cid:96) f ,(cid:96) p r q ` g ,(cid:96) p r q . (2.62) In this section we follow the steps in Ref. [95] and derive the Wronskian integral formula for atwo-particle system. We consider two solutions of Eq. (2.39) for momentum p a and p b , „ ´ d dr ` (cid:96) p (cid:96) ` q r ´ p a U a p r q ` µ ż R dr W p r , r q U a p r q “ , (2.63) „ ´ d dr ` (cid:96) p (cid:96) ` q r ´ p b U b p r q ` µ ż R dr W p r , r q U b p r q “ , (2.64)where we use the shorthand notation U a p r q “ U p p a q (cid:96) p r q and U b p r q “ U p p b q (cid:96) p r q . We multiplyEq. (2.63) by V b p r q on the left, Eq. (2.64) by V a p r q , and then subtracting the resulting equationsyields U a p r q U b p r q ´ U b p r q U a p r q ` ` p b ´ p a ˘ U b p r q U a p r q“ µ ż R dr “ U a p r q W p r , r q U b p r q ´ U b p r q W p r , r q U a p r q ‰ . (2.65)18ntegrating Eq. (2.65) from radius ρ to some radius r ě R , we get r U a U b ´ U b U a s| r ρ ` ` p b ´ p a ˘ ż R dr U b p r q U a p r q“ µ ż R ρ dr ż R dr “ U a p r q W p r , r q U b p r q ´ U b p r q W p r , r q U a p r q ‰ . (2.66)Now for the right hand side of this equation using the condition given in Eq. (2.30), we finallyobtain the Wronskian integral formula, W r U p p b q (cid:96) , U p p a q (cid:96) sp r q p b ´ p a “ ż r U p p a q (cid:96) p r q U p p b q (cid:96) p r q dr . (2.67)In the low energy regime, when we set p a “ p b “ p , Eq. (2.67) reads W r u ,(cid:96) , u ,(cid:96) sp r q ` p W r u ,(cid:96) , u ,(cid:96) sp r q ` p W r u ,(cid:96) , u ,(cid:96) sp r q` p W r u ,(cid:96) , u ,(cid:96) sp r q ´ ż r ” U p p q (cid:96) p r q ı dr ` O p p q “ . (2.68)In Appendix B.1 the Wronskians of the functions u n p r q are given in terms of the scatteringparameters.The integral terms can be rearranged and rewritten as ż r ” U p p q (cid:96) p r q ı dr “ ż ” U p p q (cid:96) p r q ı dr ´ ż r ” U p p q (cid:96) p r q ı dr . (2.69)Since Eq. (2.42) is the solution of the function U p p q (cid:96) p r q for r ą R , it can be used in the second19ntegral of the right hand side. Inserting Eq. (2.42) and Eq. (B.2)-(B.5) into Eq. (2.68) we get ż ” U p p q (cid:96) p r q ı dr “ ´ r (cid:96) ` ∆ ,(cid:96) ´ ` P (cid:96) ´ ∆ ,(cid:96) ˘ p ´ ˆ Q (cid:96) ´ ∆ ,(cid:96) ´ ∆ ,(cid:96) ˙ p ` O p p q , (2.70)where ∆ n ,(cid:96) are integration constants and calculated from the following equations, ∆ ,(cid:96) “ ddr b ,(cid:96) p r q ` ż r dr “ u ,(cid:96) p r q ‰ , (2.71) ∆ ,(cid:96) “ ddr b ,(cid:96) p r q ` ż r dr “ u ,(cid:96) p r q u ,(cid:96) p r q ‰ , (2.72) ∆ ,(cid:96) “ ddr b ,(cid:96) p r q ` ż r dr “ u ,(cid:96) p r q u ,(cid:96) p r q ‰ , (2.73) ∆ ,(cid:96) “ ddr b ,(cid:96) p r q` ż r dr “ u ,(cid:96) p r q ‰ ´ ż r dr “ u ,(cid:96) p r q u ,(cid:96) p r q ‰ , (2.74)where the b n ,(cid:96) p r q functions are given in Appendix B.1. Eq. (2.67)-(2.74) are valid for Coulomb case. In that case, U p p q (cid:96) p r q and u n ,(cid:96) p r q are replaced by V p p q (cid:96) p r q and v n ,(cid:96) p r q , respectively. In addition, a superscript c is used in the scattering parameters to denote the Coulomb scatteringparameters ( a c (cid:96) , r c (cid:96) , P c (cid:96) and Q c (cid:96) ). .6 Loosely bound systems The bound state wave function with momentum p “ i κ in the asymptotic region is ψ (cid:96) p r q “ i (cid:96) A h p q (cid:96) p i κ r q (2.75)where A is the asymptotic normalization coefficient (ANC) and h p q (cid:96) is the Riccati Hankelfunction. The bound state solution is normalized according to ż “ ψ (cid:96) p r q ‰ dr “ . (2.76)Furthermore, for the bound state regime, we havecot δ (cid:96) p i κ q “ i , (2.77)and the effective range expansion reads p´ q (cid:96) κ (cid:96) ` “ a (cid:96) ` r (cid:96) κ ´ P (cid:96) κ ` Q (cid:96) κ ` O p κ q , (2.78)21 able 2.1 The integration constants ∆ n ,(cid:96) of a two-neutral-particles system for (cid:96) ď (cid:96) ∆ ,(cid:96) ∆ ,(cid:96) ∆ ,(cid:96) ∆ ,(cid:96) κ ´ κ κ ψ (cid:96) p r q and U p i κ q (cid:96) p r q can be obtained, for r ą R , U p i κ q (cid:96) p r q “p i κ q (cid:96) r cot δ (cid:96) p i κ q ˆ S (cid:96) p i κ r q ` C (cid:96) p i κ r qs“p i κ q (cid:96) r i ˆ S (cid:96) p i κ r q ` C (cid:96) p i κ r qs “ p i κ q (cid:96) h p q (cid:96) p i κ r q “ κ (cid:96) A ψ (cid:96) p r q (2.79)Inserting Eq. (2.79) into the integral term of Eq. (2.70) we obtain the following expression forthe ANC A (cid:96) « κ (cid:96) c ´ r (cid:96) ` ∆ ,(cid:96) ` ` P (cid:96) ´ ∆ ,(cid:96) ˘ κ ´ ´ Q (cid:96) ´ ∆ ,(cid:96) ´ ∆ ,(cid:96) ¯ κ , (2.80)where integration constants ∆ n ,(cid:96) are given in Table 2.1. An alternative expression can be obtainedby eliminating the effective range r (cid:96) using Eq. (2.78) A (cid:96) « κ (cid:96) ` b a (cid:96) ´ p´ q (cid:96) κ (cid:96) ` ` ∆ ,(cid:96) κ ` ` P (cid:96) ´ ∆ ,(cid:96) ˘ κ ´ ` Q (cid:96) ´ ∆ ,(cid:96) ´ ∆ ,(cid:96) ˘ κ . (2.81)This expression agrees with Eq.(11) of Ref. [108] up to the given order.22 .6.1.2 Coulomb case The bound state Coulomb wave function with binding momentum κ in the asymptotic region is ψ c (cid:96) p r q “ A c W ´ i η ,(cid:96) ` p κ r q (2.82)where A c is the Coulomb-ANC and W ´ i η ,(cid:96) ` is the Whittaker function. The bound state solutionis normalized according to ż “ ψ c (cid:96) p r q ‰ dr “ . (2.83)In the bound state regime, we have cot ˜ δ (cid:96) p i κ q “ i . (2.84)Therefore, the wave function becomes, for r ą R , V p i κ q (cid:96) p r q “p i κ q (cid:96) C η ,(cid:96) ” i F p i κ q (cid:96) p r q ` G p i κ q (cid:96) p r q ı “p i κ q (cid:96) C η ,(cid:96) e i σ (cid:96) e ´ i π p (cid:96) ` i η q W ´ i η ,(cid:96) ` p κ r q , (2.85)where σ (cid:96) is the Coulomb phase shift, e i σ (cid:96) “ Γ p (cid:96) ` ` i η q Γ p (cid:96) ` ´ i η q . (2.86)23he wave function V p i κ q (cid:96) p r q in terms of the bound state Coulomb wave function reads V p i κ q (cid:96) p r q “ κ (cid:96) A c ˜ C η ,(cid:96) ψ c (cid:96) p r q , (2.87)where ˜ C η ,(cid:96) “ (cid:96) Γ p (cid:96) ` ` i η q Γ p (cid:96) ` q . (2.88)Finally, the ANC can be written as | A c (cid:96) | « κ (cid:96) ˜ C η ,(cid:96) ” ´ r c (cid:96) ` ∆ c ,(cid:96) ` ´ P c (cid:96) ´ ∆ c ,(cid:96) ¯ κ ´ ´ Q c (cid:96) ´ ∆ c ,(cid:96) ´ ∆ c ,(cid:96) ¯ κ ı { . (2.89)Integration constants ∆ cn ,(cid:96) for (cid:96) ď Table 2.2
Integration constants ∆ cn ,(cid:96) calculated from Eqs. (2.71)-(2.74) for (cid:96) ď (cid:96) ∆ c ,(cid:96) ∆ c ,(cid:96) ∆ c ,(cid:96) ∆ c ,(cid:96) γ γ γ γ
108 11270 γ γ γ γ γ A c can be found by eliminating the effective range parameter r c (cid:96) in Eq. (2.89) using the Coulomb modified effective range expansion in the bound state regime, γ ˜ h (cid:96) p i κ q “ ´ a c (cid:96) ´ r c (cid:96) κ ` P c (cid:96) κ ´ Q c (cid:96) κ ` O ` k ˘ , (2.90)24here γ ˜ h (cid:96) p i κ q are given in Table 2.3 for (cid:96) ď
2. We find
Table 2.3
The function γ ˜ h (cid:96) p i κ q in the Coulomb modified effective range expansion for (cid:96) ď (cid:96) γ ˜ h (cid:96) p i κ q ´ κ γ ` κ γ ´ κ γ ` O p κ q ´ γκ ` κ γ ´ κ γ ` O p κ q ´ γ κ ` γκ ´ κ γ ` O p κ q| A c (cid:96) | « κ (cid:96) ` ˜ C η ,(cid:96) ” a c (cid:96) ` γ ˜ h (cid:96) p i κ q ` ∆ c ,(cid:96) κ ` ´ P c (cid:96) ´ ∆ c ,(cid:96) ¯ κ ´ ´ Q c (cid:96) ´ ∆ c ,(cid:96) ´ ∆ c ,(cid:96) ¯ κ ı { (2.91)This expression matches with Eq. (85) of Ref. [107] [cf. Eq.(5.87) of Ref.[105]] up to the givenorder and Eq. (21) of Ref. [159] for (cid:96) “
2. The relation between our convention of the effectiverange parameters and the convention of Ref. [159] is˜ A c “ ˆ (cid:96) (cid:96) ! Γ p (cid:96) ` q ˙ A c . (2.92)25 hapter Lattice Effective Field Theory
In the first theoretical descriptions initiated by Yukawa [178] the strong nuclear forces betweennucleons are mediated by massive bosons called mesons. Phenomenological models whichwere only based on one-boson-exchange (OBE) well described the strong interactions at largedistances [34, 64, 134]. Later, efforts were made to construct highly sophisticated potentialmodels in order to improve the shape of the potentials in intermediate ranges [98, 115, 162, 176,126]. For more on potential models and a historical review see Ref. [128, 127].At the same time attempts were taken to describe the strong interactions between nucleonswithin the framework of quantum field theory (QFT), and a breakthrough came with thediscovery of quarks [84]. Quarks are elementary particles in the Standard Model and they areconstituents of hadrons. The theory of the strong interactions is called quantum chromodynamics(QCD). The fundamental degrees of freedom in QCD are gluons as well as quarks. Quarks havesix different flavors (up, down, strange, charmed, top, bottom) and three colors (red, green, blue).Colors are the charges of quarks, and the strong interactions are governed by a non-abelian26auge theory with the SU(3)-color group [69, 70]. The eight generators of the SU(3) groupcorrespond with the eight gluons.The running coupling constant of the strong interactions makes it possible to carry outcalculations of observables perturbatively at higher energies. However, the same feature of thetheory causes a breakdown in perturbative treatments at low energies. This clearly manifests thenecessity of non-perturbative methods in order to predict observables from QCD. An elegantmethod was proposed by Wilson [175]. He formulated lattice gauge theory on a discretized space-time lattice, which is commonly known as lattice QCD (LQCD), and this method gave access tostudy QCD in the low energy limit or at large distances using numerical methods. Therefore,LQCD has become a powerful approach to probe the structure of hadrons using quarks andgluons as degrees of freedom [8, 67]. Also, LQCD has been used for studying elastic scatteringof meson-mesons [114, 1, 121, 6, 4, 7], meson-baryon [72], and baryon-baryon [9, 3, 97, 5].Another direct approach to access the low-energy regime of QCD is effective field theory(EFT) which is based upon the seminal work of Weinberg [171]. This idea is rooted in a generalconcept of separation of scales. Physical processes and observables are well defined in energyscales relevant to the dynamics of the system. In the low energy limit of QCD since quarks andgluons are strongly confined in hadrons by color charge forces, the relevant degrees of freedomat large distance scales are hadrons, instead of quarks and gluons. Therefore, a scale separationis inevitable here and it is indeed the key point of EFT. The spectrum of hadrons shows a visiblelarge gap between the masses of light mesons ( π , π ˘ ) and the masses of nucleons ( N ) andheavier mesons. The EFT formulation sets a soft scale Q at the mass scale of light mesons anda hard scale Λ at the scale of the nucleon mass. Then using EFT one can perform systematiccalculations as an expansion in powers of a small parameter Q { Λ . This formulation is known aschiral effective field theory ( χ EFT).The hard scale Λ also corresponds to the spontaneous chiral symmetry breaking scale. Chiral27ymmetry is a symmetry of QCD associated with the smallness of the light quark masses. Inthe limit of zero quark mass, one can do independent unitary transformations of the left andthe right components of the quarks. Chiral EFT enforces the fact that chiral symmetry must bemanifested in the phenomenology of hadrons at low energies. Chiral EFT was first applied tothe elastic scattering of ππ [81] and π N [82]. Later, its applicability to the nuclear structure andinteractions was derived by Weinberg [172, 173]. Refs. [127] and [56] provide detailed reviewson the subject.In such systems where momenta is smaller than the pion mass the pionless effective fieldtheory ( { π EFT) is a more economic and efficient formulation to use. In the { π EFT pions areintegrated out and the interactions are only local contact interactions between dynamicalnucleons [167, 11, 12, 13, 38, 145, 96]. For example, the deuteron binding momentum is γ d “
45 MeV which is much smaller than the lightest pion mass m π “
140 MeV. This clearlysuggests here that the pion mass can be considered as the hard scale here since the relevantenergy scale is much lower than the pion mass.Interactions derived from EFT mentioned above have been combined with powerful numer-ical methods to study low energy nuclear physics from first principles. This growing field isknown as lattice effective field theory (lattice EFT). Lattice EFT was formulated on discretizedspace-time from the chiral EFT [23]. Ref. [118] provides a detailed review of lattice EFT fromthe { π EFT and chiral EFT. In the last decade lattice EFT methods have proven its successeswith significant contributions made to nuclear structure studies [22, 24, 62, 57]. Some recentsuccesses of lattice EFT are the ab initio calculation of Hoyle state of carbon-12 [63], which isthe states that is responsible for the carbon-12 production in the stars, and ab initio calculationof the spectrum and structure of O [60]. Also, very recently these calculations have beenextended to medium mass nuclei [116]. Nuclear reaction calculations from lattice EFT wereinitiated by Ref. [154]. 28 .2 The path integral
Following Dirac’s pioneer work [48] that suggested that there is a connection between theexponent of the classical action e iS r q p t qs and the transition amplitude of a quantum mechanicalparticle at two points, Feynman was the first who incorporated classical Lagrangian approachesinto quantum mechanics [65]. With his work, Feynman reformulated quantum mechanics andquantum field theory by the so called path integral (PI) method. The PI formulation has broughtparticular advantages in quantum field theories and become very a important tool for numericaltechniques in quantum systems.Starting with the time evolution operator e ´ iHt in the Hamiltonian formalism, the transitionamplitude of a quantum mechanical particle from an initial point q I to a final point q F is definedby x q F | e ´ iHt | q I y , (3.1)where | q y denote the complete set of states in the Dirac bra-ket notation 1 “ ş dq | q y x q | . Now,we want to obtain an expression from Eq. (3.1) in a path integral form. To achieve this wesplit the time t into L t equal segments α t “ t { L t and rewrite the evolution operator e ´ iHt as L t products of e ´ iH α t . By insertion of the completeness relations between those segmentedoperators, Eq. (3.1) becomes the products of the transition amplitudes (the propagators) at two29oints over a small time segment α t , x q F | e ´ iHt | q I y “ ż . . . ż dq dq . . . dq L t ´ dq L t ´ ˆ x q F | e ´ iH α t | q L t ´ y x q L t ´ | e ´ iH α t | q L t ´ y . . .. . . ˆ x q | e ´ iH α t | q y x q | e ´ iH α t | q y x q | e ´ iH α t | q I y . (3.2)An individual propagator for the Hamiltonian H “ ˆ p m ` V p ˆ q q which describes a particle in apotential V p ˆ q q is of the following form, x q n ` | e ´ iH α t | q n y “ x q n ` | e ´ i ˆ p m α t | q n y e ´ iV p q n q α t (3.3) “ c ´ im πα t e i m α t ´ qn ` ´ qn α t ¯ e ´ iV p q n q α t . (3.4)This is an infinitesimal transition amplitude which describes the particle’s evolution from q n ` to q n . In Eq. (3.3) we use the Baker-Campbell-Hausdorff formula , and from Eq. (3.3) toEq. (3.4) we use the state | p y which is the eigenstate of ˆ p and whose normalization is such that ş d p π | p y x p | “
1. Plugging in the individual propagator given by Eq. (3.4) into Eq. (3.2) we get x q F | e ´ iHt | q I y “ ˆ ´ im πα t ˙ L t { ż L t ´ ź n “ dq n exp i α t N ´ ÿ k “ « m ˆ q k ` ´ q k α t ˙ ´ V p q k q ff+ , (3.5)and in the continuum limit α t Ñ x q F | e ´ iHt | q I y “ ż D q p t q e iS r q p t qs , (3.6) e A ` B ` r A , B s` ... “ e A e B ż D q p t q “ ˆ ´ im πα t ˙ L t { ˜ L t ´ ź n “ dq n ż ¸ (3.7)and S r q p t qs is the action defined in terms of the classical Lagrangian L p q , q , t q “ m q ´ V p q q ,exp iS r q p t qs “ exp i ż dtL p q , q , t q . (3.8)In this formalism the classical Lagrangian is the fundamental quantity. Eq. (3.6) is the FeynmanPI formulation which reformulates the quantum mechanical amplitude as the integral over allpossible paths weighted by e iS .The weighting function in the PI has an oscillatory nature. For later convenience and thefavor of numerical methods to be employed we want to suppress these oscillations and desirethe weighting function to be positive semi-definite and non-oscillating. Therefore, rotation tothe Euclidean time direction it Ñ τ is a crucial step to obtain the Euclidean action S E , S E r q p τ qs “ ż d τ L E p q , q , τ q “ ż d τ ” m q ` V p q q ı , (3.9)and the Euclidean time formulation of the PI becomes x q F | e ´ iHt | q I y “ x q F | e ´ H τ | q I y “ ż D q p τ q e ´ S E r q p τ qs . (3.10)The PI formulations derived above hold for any quantum system as well as quantum fieldtheory. Eq. (3.10) can be rearranged according to the following table, This is the so called Wick rotation. The integrand of the action is rotated from the Re t axis to the Im t in thecomplex t -plane. p τ q Ñ φ p (cid:126) r q S E r q p τ qs Ñ S E p φ q ż D q p τ q Ñ ż D φ and the PI formulation for fields becomes x φ F p (cid:126) r q| e ´ H τ | φ I p (cid:126) r qy “ ż D φ e ´ S E p φ q . (3.11)Here φ is the field amplitude which is the dynamical variable of quantum field theory, and theeuclidean action S E p φ q is defined in terms of the Lagrangian in the non-relativistic limit ofquantum field theory with L E “ ´ φ : BB τ φ ´ m p ∇ φ : q ¨ p ∇ φ q´ ż d (cid:126) r φ : p (cid:126) r q φ p (cid:126) r q V p (cid:126) r ´ (cid:126) r q φ : p (cid:126) r q φ p (cid:126) r q (3.12)with the density operator ρ p (cid:126) r q “ φ : p (cid:126) r q φ p (cid:126) r q . (3.13)The Hamiltonian of the Schrödinger field equation is H “ ż d (cid:126) r „ π p (cid:126) r q BB τ φ p (cid:126) r q ´ L E “ H ` H V , (3.14)32here π p (cid:126) r q “ B L BpB φ {B τ q is the momentum density conjugate to φ p (cid:126) r q , H is the free Hamiltonian H “ m ż d (cid:126) r ∇ φ : p (cid:126) r q ¨ ∇ φ p (cid:126) r q , (3.15)and H V is the interaction term H V “ ż ż d (cid:126) r d (cid:126) r ρ p (cid:126) r q V p (cid:126) r ´ (cid:126) r q ρ p (cid:126) r q . (3.16) The quantizations of bosonic fields are based upon the commutation relation, while only theanti-commutation relation yields a consistent theory for the fermionic fields. See Refs.[179, 141]for details and comprehensive discussions on the topic.The anti-commuting variables that we are in need of are Grassmann variables. In order tostudy the PI for fermions we necessarily reconsider Eqs.(3.11)-(3.14) in terms of Grassmannvariables. Therefore, we discuss some basic properties of Grassmann variables, and theirintegration and differentiation are introduced.Let η k for l “ , , ..., N be a set of Grassmann variables which satisfy the anti-commutationrelation η k η l ` η l η k ” t η k , η l u “ , (3.17)for any k and l . Eq. (3.17) imposes that η k “
0. Assuming that Grassmann variables can beexpanded in a Taylor series, the most general function of a Grassmann variable has the form f p η q “ a ` b η where a and b are ordinary numbers.33he left and right differentiation of Grassmann variables are defined as ÝÑBB η k η l “ ´ η l ÐÝBB η k “ δ kl , (3.18)and using this we can write ÝÑBB η k η l η k “ ´ η l “ η l η k ÐÝBB η k . (3.19)We use the standard notation for Grassmann variables so that the integration can be written by ż d η k “ ż d η k η k “ ´ ż η k d η k “ . (3.20)A complex Grassmann variable can be written as a combination of two real Grassmannvariables, θ “ η k ` i η l . The properties of the complex Grassmann variables can be defined byusing the properties of real Grassmann variables above, θ “ θ ˚ “ θ ˚ θ “ i r η k , η l s , (3.21)where θ ˚ “ η k ´ i η l . Ref. [163] provides detailed discussions on Grassmann algebra and someinteresting physics applications of Grassmann variables. In this section by following Ref. [118, 117] we introduce a lattice formalism in which spacetimeis a discretized periodic cubic lattice with L ˆ L t points. In the lattice formalism of ourdiscussion, the lattice spatial spacing is denoted by a , and the lattice temporal spacing is a t .34lso, we introduce α t “ a t { a as the ratio of temporal lattice spacing to spacial lattice spacing.Here we use dimensionless parameters and physical quantities in lattice units multiplied by theappropriate power of a .We consider two-component fermions interacting via zero-range potentials, and we call thetwo components Ò and Ó spins. The PI for fermions is defined by the anti-commuting Grassmannvariables on lattice, Z “ ż »– ź n t ,(cid:126) n , s “Ò , Ó d θ s p n t ,(cid:126) n q d θ ˚ s p n t ,(cid:126) n q fifl e ´ S r θ , θ ˚ s . (3.22)Grassmann variables are periodic along the spatial direction, θ p (cid:126) n ` L , n t q “ θ p (cid:126) n , n t q θ ˚ p (cid:126) n ` L , n t q “ θ ˚ p (cid:126) n , n t q , (3.23)and anti-periodic in the temporal direction, θ p (cid:126) n , n t ` L t q “ ´ θ p (cid:126) n , n t q θ ˚ p (cid:126) n , n t ` L t q “ ´ θ ˚ p (cid:126) n , n t q . (3.24)The non-relativistic lattice action is defined by S r θ , θ ˚ s “ ÿ n t t S t r θ , θ ˚ , n t s ` S H r θ , θ ˚ , n t s ` S V r θ , θ ˚ , n t su , (3.25)where S t and S H contain temporal hopping and spatial hopping terms of the free lattice actionrespectively, S t r θ , θ ˚ , n t s “ ÿ s “Ò , Ó ÿ (cid:126) n “ θ ˚ s p n t ` ˆ0 ,(cid:126) n q θ s p n t ,(cid:126) n q ´ θ ˚ s p n t ,(cid:126) n q θ s p n t ,(cid:126) n q ‰ , (3.26)35 H r θ , θ ˚ , n t s “ α t m ÿ s “Ò , Ó ÿ (cid:126) n ÿ l “ θ ˚ s p n t ,(cid:126) n q “ θ s p n t ,(cid:126) n q ´ θ s p n t ,(cid:126) n ` ˆ l q ´ θ s p n t ,(cid:126) n ´ ˆ l q ‰ . (3.27)Here ˆ0 denotes the lattice unit vector in the forward temporal direction. We can also write S H r θ , θ ˚ , n t s S H r θ , θ ˚ , n t s “ α t H Ò r θ s , θ ˚ s , n t s ` α t H Ó r θ s , θ ˚ s , n t s , (3.28)and the interaction term S V r θ , θ ˚ , n t s as S V r θ , θ ˚ , n t s “ α t H V r θ , θ ˚ , n t s , (3.29)where H s r θ s , θ ˚ s , n t s “ m ÿ (cid:126) n ÿ l “ θ ˚ s p n t ,(cid:126) n q “ θ s p n t ,(cid:126) n q ´ θ s p n t ,(cid:126) n ` ˆ l q ´ θ s p n t ,(cid:126) n ´ ˆ l q ‰ , (3.30)and H V r θ , θ ˚ , n t s “ C ÿ (cid:126) n θ ˚Ò p n t ,(cid:126) n q θ Ò p n t ,(cid:126) n q θ ˚Ó p n t ,(cid:126) n q θ Ó p n t ,(cid:126) n q . (3.31)In the last equation C is the coupling strength of the zero-range potential. The Grassmann formalism given by Eq. (6.10) is convenient for deriving the lattice Feynmanrules. On the other hand, the operator formalism or so called transfer matrix formalism ismore convenient for numerical calculations. Therefore, in this section we review the connectionbetween the PI formulation and the operator formalism in quantum mechanics [42].36s a first step we analyze the connection in quantum mechanics and rewrite Eq. (3.5) for theEuclidean time lattice, Z “ ż D q p α t q exp ´ α t L t ´ ÿ n “ « m ˆ q n ` ´ q n α t ˙ ` V p q n q ff+ . (3.32)For finite lattice of L t sites this expression can be rewritten in the form of Z “ ż L t ´ ź n “ dq n M n ` , n . (3.33)where the matrix M n ` , n is the transfer matrix, M n , n “ ˆ m πα t ˙ { exp « ´ α t m ˆ q n ´ q n α t ˙ ´ α t V p q n q ff . (3.34)From Eqs. (3.3) and (3.4), we already know that M n , n describes the evolution of particles overone lattice spacing in the temporal direction M n , n “ x q n ` | e ´ α t H | q n y . (3.35)It is should be noted that the n , n subscripts are not the matrix indices of the transfer matrix, butrather the coordinates of the particle.Now we turn to the exact correspondence between the Grassmann path integral and thetransfer matrix formalism. For the moment let us consider a single component fermion and use b and b : to denote fermion anti-commuting creation and annihilation operators, respectively, t b , b u “ t b : , b : u “ , t b , b : u “ . (3.36)37or any function f p b , b : q the exact relation between the Grassmann path integral and the transfermatrix formalism is given by [44],Tr ” : f p b , b : q : ı “ ż d θ ˚ d θ e θ ˚ θ f p θ ˚ , θ q , (3.37)where the symbol : : signifies normal ordering. Normal ordering rearranges operators betweensthe symbol : : such that all annihilation operators are on the right and creation operators are onthe left. Using anti-periodicity of Grassmann fields in temporal direction, i.e. θ p q “ ´ θ p q ,Eq. (3.37) can be rewritten as a path integral over a short time interval,Tr ” : f p b , b : q : ı “ ż d θ ˚ p q d θ p q e θ ˚ p qr θ p q´ θ p qs f p θ ˚ p q , θ p qq , (3.38)This can be applied to the product of any normal-ordered functions of different componentfermion creation and annihilation operators, which leads to the following exact correspondencebetween the PI integral and operator formalism [43, 44],Tr ” : f L t ´ r b : s p (cid:126) n q , b s p (cid:126) n qs : ¨ ¨ ¨ : f r b : s p (cid:126) n q , b s p (cid:126) n qs : ı “ ż »– ź n t ,(cid:126) n , s “Ò , Ó d θ ˚ s p n t ,(cid:126) n q d θ s p n t ,(cid:126) n q fifl e ´ S t r θ , θ ˚ s L t ´ ź n t “ f n t r θ ˚ s p n t ,(cid:126) n q , θ s p n t ,(cid:126) n qs , (3.39)Therefore the transfer matrix formulations of the path integral given in Eq. (3.22) has the thefollowing form, Z p L t q “ Tr “ ˆ M L t ‰ , (3.40)38here ˆ M is the normal-ordered transfer matrix operator,ˆ M “ : exp « ´ α t ˆ H ´ α t C ÿ (cid:126) n ˆ ρ Ò p (cid:126) n q ˆ ρ Ó p (cid:126) n q ff : . (3.41)Here ˆ H is the free non-relativistic lattice Hamiltonian in terms of anti-commuting creation andannihilation operators ˆ H “ ˆ H Ò ` ˆ H Ó , (3.42)where ˆ H s “ m ÿ ˆ l “ ÿ (cid:126) n ” b : s p (cid:126) n q b s p (cid:126) n q ´ b : s p (cid:126) n q b s p (cid:126) n ` ˆ l q ´ b : s p (cid:126) n q b s p (cid:126) n ´ ˆ l q ı , (3.43)and ˆ ρ s are the lattice density operators,ˆ ρ s p (cid:126) r q “ b : s p (cid:126) r q b s p (cid:126) r q . (3.44) s signify the spin component of fermions and ˆ l denotes the spatial lattice unit vectors. The adiabatic projection method is a general procedure for calculating scattering and reactionson the lattice. The main tools of the method are initial cluster states of the system. By clusterswe mean either a single particle or a composite state of several particles. The method constructsa low energy effective theory for clusters, and in the limit of large Euclidean time projectionthese cluster states will span the low-energy subspace of the Hamiltonian.The initial cluster states can be parameterized by either the initial spatial separations [143]39r alternatively the relative momentum between clusters. The latter reduces the number ofrequired initial states, and it is quite advantageous to adopt for improving the efficiency ofthe calculations. Let us use | Ψ (cid:126) ρ y to denote a set of initial cluster states where (cid:126) ρ stands for theparameters chosen to define the state, i.e., (cid:126) R and (cid:126) p . The dressed cluster states are formed byprojecting the states | Ψ (cid:126) ρ y in the Euclidean time, | Ψ (cid:126) ρ y t “ e ´ ˆ Ht | Ψ (cid:126) ρ y (3.45)Now the adiabatic projection method uses these dressed cluster states to calculate the matrixelements of observables such as the Hamiltonian and the transfer matrix.The dressed cluster states are generally non-orthogonal, and as a result of this the methodinvolves calculating a norm matrix. As an example in the following we consider the calculationof the adiabatic matrix representation of the Hamiltonian operator ˆ H by following the procedureused in Ref. [143]. Let us define the dual state t p Ψ (cid:126) ρ | written as a linear functional, t p Ψ (cid:126) ρ | u y “ ÿ (cid:126) ρ t p Ψ (cid:126) ρ | Ψ (cid:126) ρ y t t x Ψ (cid:126) ρ | u y , (3.46)such that the dual state t p Ψ (cid:126) ρ | satisfies that t @ Ψ (cid:126) ρ ˇˇ u D “ (cid:126) ρ ñ t p Ψ i | u y “ (cid:126) ρ (3.47) t p Ψ (cid:126) ρ | Ψ (cid:126) ρ y t “ δ (cid:126) ρ (cid:126) ρ . (3.48)The inner product of the propagated initial and final state is the norm matrix, r N t s (cid:126) ρ (cid:126) ρ “ t x Ψ (cid:126) ρ | Ψ (cid:126) ρ y t . (3.49)40nd the inner product of the propagated initial and dual state defines t p Ψ (cid:126) ρ | Ψ (cid:126) ρ y t “ r N ´ t s (cid:126) ρ (cid:126) ρ . (3.50)Therefore, the adiabatic matrix of the Hamiltonian ˆ H projected onto the set of dressed clusterstates, r ˆ H at s (cid:126) ρ (cid:126) ρ “ ÿ (cid:126) ρ t p Ψ (cid:126) ρ | Ψ (cid:126) ρ y t t x Ψ (cid:126) ρ | ˆ H | Ψ (cid:126) ρ y t “ r N ´ t s (cid:126) ρ (cid:126) ρ t x Ψ (cid:126) ρ | ˆ H | Ψ (cid:126) ρ y t . (3.51)By using a similarity transformation, we can define the Hermitian adiabatic Hamiltonian as r ˆ H at s (cid:126) ρ (cid:126) ρ “ r N ´ { t s (cid:126) ρ (cid:126) ρ t x Ψ (cid:126) ρ | ˆ H | Ψ (cid:126) ρ y t r N ´ { t s (cid:126) ρ (cid:126) ρ . (3.52)ˆ H at is the two-body adiabatic Hamiltonian describing the scattering and reactions betweeninteracting clusters, and the calculations become systematically more accurate as the projectiontime t is increased. See Ref. [143] for detailed analysis on an estimate of the residual error as afunction of the projection time. In the finite-volume calculation, we compute the volume dependent energy spectrum of thesystem. However, the information about the short-range interaction potentials between twoclusters is encoded in the scattering phase shifts. Therefore, in the following we review themathematical tools and methods that we use in order to determine the scattering phase shift in41nite volume calculations.
In the discretized lattice the rotational symmetry cannot be explored using an arbitrary rotationangle since the SO p q rotational symmetry of continuum space is broken to the finite rotationalgroup SO p , Z q . The cubic rotational group SO p , Z q which is also known as the octahedralgroup consists of 24 rotations about the x , y and z axes. Since a finite rotation can be obtainedby a set of infinitesimal rotations about an axis, the rotation operator R ˆ n p φ q of the SO p q alsodefines elements of the SO p , Z q group for a rotation by φ “ m π { n axis, where m isinteger and n denotes the axes. Therefore, the angular momentum operators L x , L y and L z in theSO p , Z q group are defined by R ˆ n p φ q “ exp p´ iL ˆ n φ q (3.53)It is clear that the eigenvalues of L ˆ n are integers modulo 4.The 2 (cid:96) ` p q group. Under the SO p , Z q group these representations are reducible inmost cases and they break up into the five irreducible representations denoted by A , T , E , T and A . Examples for the decompositions of the orbital angular momentum eigenstates (cid:96) ď p , Z q group are given in Table 3.1 [101] Lüscher’s method [122, 124] is a well-known tool used to determine elastic phase shifts fortwo-body scattering from the volume dependence of two-body scattering states in a periodiccubic box. The method has been extended to higher partial waves, two-body systems in moving42 able 3.1
Decomposition of the SO p q into the irreducible representations of the SO p , Z q for (cid:96) ď SO p q SO p , Z q (cid:96) “ A (cid:96) “ T (cid:96) “ E ‘ T (cid:96) “ T ‘ T ‘ A (cid:96) “ A ‘ T ‘ E ‘ T (cid:96) “ T ‘ T ‘ E ‘ T (cid:96) “ A ‘ T ‘ E ‘ T ‘ T ‘ A frames, multi-channel scattering cases, and scattering of particles with spin [152, 125, 71, 119,30, 32, 33, 31]. Lüscher’s framework has also been successfully applied to the determinationof resonance parameters [15], and recently this technique has been applied to moving framecalculations [51, 89]. See Ref. [16, 130, 50, 120, 52] for further studies on the extraction ofresonance properties at finite volume. We note also recent work on improving lattice interactionsin effective field theories using Lüscher’s method [55].In the following we summarize how Lüscher’s method relates the s -wave scattering phaseshift to two-body energy levels in a periodic cubic box. Later we come back to this discussionfor higher angular momentum in Section 6.7.1.We consider a two-body system in a periodic box of length L . The relation between the s -wave scattering phase shift and the two-particle energy levels in the center of mass frame isdefined by p cot δ p p q “ ? π L Z , p η q (3.54)43here η “ ˆ Lp π ˙ , (3.55)and Z , p η q is the three-dimensional zeta function, Z , p η q “ ? π lim Λ Ñ8 «ÿ (cid:126) n θ p Λ ´ (cid:126) n q| (cid:126) n | ´ η ´ π Λ ff . (3.56)Alternatively, we can evaluate the zeta function using exponentially-accelerated expression [125] Z , p η q “ π e η p η ´ q ` e η ? π ÿ (cid:126) n e ´| (cid:126) n | | (cid:126) n | ´ η ´ π ż d λ e λ η λ { ˜ λ η ´ ÿ (cid:126) n e ´ π | (cid:126) n | { λ ¸ . (3.57) While Lüscher’s method is very powerful at low energies, the method is limited to calculationsof scattering phase shifts below the inelastic threshold. Also, the phase shifts obtained usingLüscher’s method depend crucially on an accurate calculation and analysis of finite-volumeenergy levels. Furthermore, since the eigenstates of the angular momentum for (cid:96) ą p , Z q , there is no one-to-one correspondencebetween the finite-volume energy spectrum and the phase shifts [125]. Therefore, the scatteringphase shift calculations become more and more difficult at higher angular momentum.Borasoy at al. [23] proposed how to compute phase shifts for non-relativistic fundamentalparticles on the lattice without encountering some of the difficulties mentioned above. Thismethod uses a spherical boundary condition on the lattice. A hard wall boundary of radius R wall
44s imposed on the relative separation of the two particles which removes the periodic latticeeffects between particles.For two particles interacting via a finite-range R potential, the radial part of the solution atvalues r ą R is described by Eq. (2.40) which vanishes at r “ R wall ,cot δ (cid:96) p p q S (cid:96) p pR wall q ` C (cid:96) p pR wall q “ . (3.58)Here p is the relative momentum of the particles, and it can be determined from the energy E “ p { µ with the reduced mass µ . Therefore, the scattering phase shifts can be computed bythe following expression, cot δ (cid:96) p p q “ cot ´ „ ´ C (cid:96) p pR wall q S (cid:96) p pR wall q . (3.59)See Ref. [23, 118] for the detailed discussion on the topic and for the case of partial-wavemixing. In the preceding sections we have derived two elegant approaches which describe evolutions ofparticles between two space-time points in quantum systems. In principle numerical solutionsfor Eqs. (6.10) or (3.40) are not impossible, but it is computationally very expensive, perhaps notpractical, due to the multi-dimensional integral in the PI formula and massive matrix operationsin the operator formalism. In the presence of such difficulties Monte Carlo methods are the mostpowerful and commonly used techniques to approximate physical observables.45onte Carlo techniques are based on the idea of simulating particle configurations by usingrandom numbers as dynamics of the method. The fact that makes Monte Carlo techniques veryadvantageous is that among all states of the physical system only a small fraction are chosen atrandom according to some weighting function P p c q in order to estimate mean values of physicalquantities. Suppose O be an observable that we want to estimate from Monte Carlo simulations,then the approximated expectation value (the estimator) of O is the average over some subsetstates M of the complete states of the system , x O y M “ ř Mi “ O p c i q P p c i q ř Mi “ P p c i q , (3.60)where O p c i q is the value of the observable for the configuration c i . The neglected subset statesintroduce some statistical errors in Monte Carlo simulations which are suppressed by the size ofthe sample. The chosen subset states in Eq. (3.60) determines the accuracy of the estimator and the perfor-mance of Monte Carlo simulations. One powerful technique that improves the accuracy of thesimulations is importance sampling .Importance sampling is implemented by selecting states with the probability function P p c q in Eq. (3.60). Then Eq. (3.60) takes the following simpler form x O y M “ M M ÿ i “ O p c i q , (3.61) The path integrals in Euclidean time are computed using a larger samples weighted by the exponentialBoltzmann factor, P p c q “ e ´ β E p c i q . The estimator is simply the average of the selected configurations. σ “ d ř Ni “ r O p c i q ´ x O y N s N ´ . (3.62)This expression indicates that the statistical error can be easily kept under control by the sizeof the sample used in simulations, and the estimator can be made as accurate as desired byincreasing the number of samples. Importance sampling has the key role of reducing the variance. However, if we randomlygenerate states, then most of them would be rejected and the computation time would be mostlywasted in such simulations. In order to make computations more efficient, we need a techniquethat generates some random set of states according to the distribution P p c q . This is done using a Markov process , which we now explain.Suppose the probability distribution is time dependent and at time t the system is in a state A with probability P p c A , t q . Let Ω p A Ñ B q be the probability that generates a state B at a latertime t ` ∆ t , then the evolution of P p c A , t q is P p c A , t q ´ P p c A , t ` ∆ t q “ ÿ B ‰ A r P p c B , t q Ω p B Ñ A q ´ P p c A , t q Ω p A Ñ B qs , (3.63)where Ω p B Ñ A q is the transition probability for selecting the state A from the state B . The mostimportant features of the transition probability for a Markov process are that;• it is independent of time,• it does not depend on any prior state that the system in before the state B ,47 Ω p A Ñ B q ě ř B Ω p A Ñ B q “ ergodicity .After the simulation runs for many steps, the system comes to an equilibrium,lim t Ñ8 P p c , t q » P p c q , (3.64)and the Markov process eventually generates successive states with the probability P p c q . Ifwe repeat this Markov process in the simulation to generate successive states, we construct a Markov chain which is a set of states each of which is selected with the probability P p c q .A sufficient condition which ensures that in equilibrium all states are generated accordingto the probability distribution P p c q is the condition of detailed balance . The detailed balancerequires that each term in the sum in Eq. (3.63) must be zero . This will then satisfy Eq. 3.64, P p c B , t q Ω p B Ñ A q “ P p c A , t q Ω p A Ñ B q . (3.65)This condition also ensures that the transition rate into any state is equal to the transition rateout of the same state. See Ref.[135] for rigorous proofs and detailed discussions. .6.4 Metropolis algorithm The detailed balance equation (3.65) imposes a constraint on the transition probability from astate A to another state B by Ω p A Ñ B q Ω p B Ñ A q “ P p c B , t q P p c A , t q “ e ´ β r E p c B q´ E p c A qs . (3.66)where the equilibrium distribution is chosen to be the Boltzmann distribution, P p c q “ P e ´ β E p c q .Therefore, the states of the desired Markov chain are distributed according to the probabilitydistribution P e ´ β E p c q . The most popular method for generating such successive states byrespecting the detailed balance condition is the Metropolis algorithm [132]. In the following wegive a simple recipe for the Metropolis algorithm. Step 1:
Initially start with an arbitrarily configuration C i . Step 2:
Generate a proposed configuration C p from the configuration C i . Step 3:
Select a random number r P r , q . Step 4:
Accept the proposed configuration if r ă e ´ β r E p C p q´ E p C i q s and set C i “ C p . Otherwiseleave C i the same. Step 5:
Repeat
Step 2-4 .If the proposed configuration has a lower energy, E p C p q ´ E p C i q ď
0, then the Metropolisalgorithm always accept that configuration. If a configuration with higher energy is proposed,then it is only accepted with some probability given in
Step 4 .49 .6.5 Sign problem
Before ending this chapter we would like to discuss the sign problem that Monte Carlo methodssuffer from simulating fermions. The main difficulty of simulations with fermions is the signcancellation due to the identical particle permutations. When we perform calculations bysampling configurations, the fluctuation in the signs associated with Fermi-Dirac statistics resultsin significant cancellations. In order for rigorous expression let us consider the calculation ofthe expectation value of a physical observable O , x O y “ ř c s p c q P p c q O p c q ř c s p c q P p c q ” x s p c q O yx s p c qy , (3.67)where c is the number of configurations, s p c q “ ˘ P p c q is the magnitude of theweight. Since the average sign appears as the denominator in Eq. (3.67), it is crucial that thecancellation is tolerable. The average sign over c configurations is x s p c qy “ ř c s p c q P p c q ř c P p c q “ exp p´ t ∆ E q , (3.68)where ∆ E “ E phy ´ E bos is the difference between the physical ground state energy ( E phy ) andthe ground state energy ( E bos ) due to the bosonic ensemble. Since ∆ E ą hapter Neutron-proton scattering
Chiral effective field theory describes the low-energy interactions of protons and neutrons.If one neglects electromagnetic effects, the long range behavior of the nuclear interactionsis determined by pion exchange processes. See Ref. [166, 14, 58, 59] for reviews on chiraleffective field theory. But there are also systems of interest where momenta smaller than the pionmass are relevant. In such cases it is more economical to use pionless effective field theory withonly local contact interactions involving the nucleons. The pionless formulation is theoreticallyelegant since the theory at leading order is renormalizable and the momentum cutoff scale canbe arbitrarily large [167, 11, 12, 13, 38, 145, 96]. This allows an elegant connection with theuniversal low-energy physics of fermions at large scattering length and other systems such asultracold atoms [29, 88].For local contact interactions the range of the interactions are set by the momentum cutoffscale for the effective theory. There are rigorous constraints for strictly finite-range interactionsset by causality and unitarity. Some violations of unitarity can relax these constraints if one51orks at finite order in perturbation theory or includes unphysical propagating modes withnegative norm. However at some point one must accurately reproduce the underlying unitaryquantum system by going to sufficiently high order in perturbation theory or decoupling theeffects of propagating unphysical modes.The time evolution of any quantum mechanical system obeys causality and unitarity. Causal-ity requires that the cause of an event must occur before any resulting consequences are produced,and unitarity requires that the sum of all outcome probabilities equals one. In the case of non-relativistic scattering, these constraints mean that the outgoing wave may depart only afterthe incoming wave reaches the scattering object and must preserve the normalization of theincoming wave. In this chapter we discuss the constraints of causality and unitarity for finiterange interactions. Specifically we consider neutron-proton scattering in all spin channels up to j “ ∆ t “ d δ dE . (4.1)If d δ { dE is negative, the outgoing wave is produced earlier than that for the non-interactingsystem. However the incoming wave must first arrive in the interacting region before theoutgoing wave can be produced. For each partial wave, (cid:96) , this puts an upper bound on theeffective range parameter, r (cid:96) , in the effective range expansion, p (cid:96) ` cot δ (cid:96) p p q “ ´ a (cid:96) ` r (cid:96) p ` O p p q . (4.2)We introduce the effective range expansion in Eq. (2.36). In this chapter we truncate the series52t p .Phillips and Cohen [142] derived the causality bound for the s -wave effective range parameterfor finite-range interactions in three dimensions. Constraints on nucleon-nucleon scatteringand the chiral two-pion exchange potential was considered in Ref. [140], and correlationsbetween the scattering length and effective range have been explored for one-boson exchangepotentials [36]. Same authors studied the relationship between the scattering length and effectiverange for the van der Waals interaction [35, 151]In Refs. [94, 95] the causality and unitarity bounds for finite-range interactions were ex-tended to an arbitrary number of space-time dimensions or value of angular momentum. Acomplementary discussion based upon conformal symmetry and scaling dimensions can befound in Ref. [137]. Also the interactions with attractive and repulsive Coulomb tails were firstconsidered in Ref. [107].Let R be the range of the interaction. For the case d “
3, it was found that the effectiverange parameter must satisfy the upper bound [94, 95] r (cid:96) ď b (cid:96) p r q “ ´ Γ p (cid:96) ´ q Γ p (cid:96) ` q π ´ r ¯ ´ (cid:96) ` ´ (cid:96) ` a (cid:96) ´ r ¯ ` π Γ p (cid:96) ` q Γ p (cid:96) ` q a (cid:96) ´ r ¯ (cid:96) ` , (4.3)for any r ě R . This inequality can be used to determine a length scale, R b , which we call thecausal range, r (cid:96) “ b (cid:96) p R b q . (4.4)The physical meaning of R b is that any set of interactions with strictly finite range that reproducesthe physical scattering data must have a range greater than or equal to R b .In this chapter we extend the causality bound to the case of two coupled partial-wave53hannels. For applications to nucleon-nucleon scattering the relevant coupled channels are S - D , P - F , D - G , etc. As we will show, there is some modification of the effectiverange bound in Eq. (4.3) due to mixing. For total spin j we show that the lower partial-wavechannel (cid:96) “ j ´ r j ´ ď b j ´ p r q ´ q Γ p j ` q Γ p j ` q π ´ r ¯ ´ j ´ , (4.5)where q is the first term in the expansion of the mixing angle ε j in the Blatt-Biedenharneigenphase convention [20], tan ε j p p q “ q p ` q p ` O p p q . (4.6)We note that the last term in Eq. (4.5) is negative semi-definite and diverges as r Ñ
0. Fromthis observation we make the general statement that non-vanishing partial-wave mixing isinconsistent with zero-range interactions. We will explore in detail the consequences of thisresult as it applies to nuclear effective field theory.We also derive a new causality bound associated with the mixing angle itself. Using theCauchy-Schwarz inequality we derive a bound for the parameter q in the expansion Eq. (4.6).This leads to another minimum interaction length scale, which we call the Cauchy-Schwarzrange, R C-S . We use the new causality bounds to determine the minimum causal and Cauchy-Schwarz ranges for each s ` (cid:96) j channel in neutron-proton scattering up to j “
3. Since the longrange behavior of the nuclear interactions is determined by pion exchange processes, one expects R b „ R C-S „ m ´ π “ . We analyze in this section the channels with only one partial wave, (cid:96) . We summarize theresults obtained Section 2.5 [cf. in Ref. [95]]. For simplicity, we will assume throughout thecalculations that the interaction has finite range R , and we use units where ¯ h “
1. For thetwo-body system the rescaled radial wave function U p p q (cid:96) p r q satisfies the radial Schrödingerequation, „ ´ d dr ` (cid:96) p (cid:96) ` q r ´ p U p p q (cid:96) p r q ` µ ż R W p r , r q U p p q (cid:96) p r q dr “ . (4.7)55e write W p r , r q for the non-local interaction potential as a real symmetric integral operator.As we discuss in Section 2.3 we assume that the potential has finite range R which requires that W p r , r q “ r ą R or r ą R .In Eq. (2.68) taking the limit p Ñ
0, we obtain that for any r ą R the effective range satisfiesthe following relation, r (cid:96) “ b (cid:96) p r q ´ ż r ” U p q (cid:96) p r q ı dr , (4.8)where b (cid:96) p r q is b (cid:96) p r q “ a (cid:96) π Γ ` (cid:96) ` ˘ Γ ` (cid:96) ` ˘ ´ r ¯ (cid:96) ` ´ a (cid:96) (cid:96) ` ´ r ¯ ´ Γ ` (cid:96) ´ ˘ Γ ` (cid:96) ` ˘ π ´ r ¯ ´ (cid:96) ` . (4.9)Since the wave function is real and the integral term in Eq. (4.8) is positive semi-definite, thisequation puts an upper bound on the effective range, r (cid:96) ď b (cid:96) p r q . This relation and causalitybound are analyzed in Ref. [95] for arbitrary dimension or angular momentum (cid:96) . In this section we derive the general wave functions for spin-triplet scattering with mixingbetween orbital angular momentum (cid:96) “ j ´ (cid:96) “ j `
1. The coupled-channel wave functionssatisfy the following coupled radial Schrödinger equations, „ ´ d dr ´ p ` j p j ´ q r U p p q j ´ p r q` µ ż R r W p r , r q U p p q j ´ p r q ` W p r , r q V p p q j ` p r qs dr “ , (4.10)56 ´ d dr ´ p ` p j ` qp j ` q r V p p q j ` p r q` µ ż R r W p r , r q U p p q j ´ p r q ` W p r , r q V p p q j ` p r qs dr “ . (4.11)Here the non-local interaction potentials are represented by a real symmetric 2 ˆ W p r , r q , W p r , r q “ ¨˚˝ W p r , r q W p r , r q W p r , r q W p r , r q ˛‹‚ . (4.12)In Eq. (4.10)-Eq. (4.11) the U p p q j ´ p r q corresponds with the spin-triplet (cid:96) “ j ´ V p p q j ` p r q is for the spin-triplet (cid:96) “ j `
1. These wave functions are the rescaled form of the radialwave functions. In the non-interacting region r ě R the coupled radial Schrödinger equationsreduce to the free radial Schrödinger equations „ ´ d dr ´ p ` j p j ´ q r U p p q j ´ p r q “ , (4.13) „ ´ d dr ´ p ` p j ` qp j ` q r V p p q j ` p r q “ . (4.14)The solutions of these differential equations are the Riccati-Bessel functions, U p p q j ´ p r q “ A S j ´ p pr q ` B C j ´ p pr q , (4.15) V p p q j ` p r q “ A S j ` p pr q ` B C j ` p pr q . (4.16)where A , and B , are amplitudes associated with incoming and outgoing waves, respectively.More details regarding the Riccati-Bessel functions are given in Appendix A.1. The relation57etween incoming and outgoing wave amplitudes is B “ K A , (4.17)K is the reaction matrix and is defined in terms of the unitary scattering matrix S by Eq. (2.20).Therefore, the Eq. (4.17) is written as ˜ B “ S ˜ A , (4.18)where ˜ A , and ˜ B , are rescaled amplitudes associated with incoming and outgoing waves. Fortwo coupled channels the 2 ˆ ˆ ˆ U S d “ U S U ´ “ ¨˚˝ e i δ α e i δ β ˛‹‚ , (4.19)that contains one real parameter ε , U “ ¨˚˝ cos ε sin ε ´ sin ε cos ε ˛‹‚ . (4.20) δ α p p q and δ β p p q are the two phase shifts, and ε p p q is the mixing angle. The S-matrix explicitly58s S “ ¨˚˝ e i δ α cos ε ` e i δ β sin ε cos ε sin ε ´ e i δ α ´ e i δ β ¯ cos ε sin ε ´ e i δ α ´ e i δ β ¯ e i δ α sin ε ` e i δ β cos ε ˛‹‚ . (4.21)The eigenvalue equation S | X y “ λ | X y results in eigenvalues λ “ e i δ α and λ “ e i δ β , withcorresponding eigenstates, | X y “ ¨˚˝ cos ε sin ε ˛‹‚ and | X y “ ¨˚˝ ´ sin ε cos ε ˛‹‚ , (4.22)which satisfy the orthogonality condition x X | X y “ . (4.23)We can write Eq. (4.18) as ¨˚˝ ˜ B α ˜ B β ˜ B α ˜ B β ˛‹‚ “ ¨˚˝ S S S S ˛‹‚¨˚˝ ˜ A α ˜ A β ˜ A α ˜ A β ˛‹‚ , (4.24)where the matrices ˜ A and ˜ B are˜ A “ ¨˚˝ e ´ i δ α cos ε ´ e ´ i δ β sin ε e ´ i δ α sin ε e ´ i δ β cos ε ˛‹‚ , (4.25)˜ B “ ¨˚˝ e i δ α cos ε ´ e i δ β sin ε e i δ α sin ε e i δ β cos ε ˛‹‚ . (4.26)We now define some additional notation. We write all α -state phaseshifts δ α p p q as δ j ´ p p q β -state phaseshifts δ β p p q as δ j ` p p q . The notation is appropriate since in the p Ñ α -state is purely (cid:96) “ j ´ β -state is purely (cid:96) “ j `
1. We also drop thesuperscript p in the wave functions. We choose the normalization of the wave function to bewell-behaved in the zero-energy limit. Using the relations S j ˘ p pr q ÝÑ as p Ñ ? π p pr q j ˘ ` ´ j ¯ ´ Γ p j ˘ ` { q , (4.27) C j ˘ p pr q ÝÑ as p Ñ p pr q ´ j ¯ ? π j ˘ Γ p j ˘ ` { q , (4.28)and removing an overall phase factor, we get wave functions of the form U α p r q “ cos ε j p p q p j ´ r cot δ j ´ p p q S j ´ p pr q ` C j ´ p pr qs , (4.29) V α p r q “ sin ε j p p q p j ´ r cot δ j ´ p p q S j ` p pr q ` C j ` p pr qs , (4.30) U β p r q “ ´ sin ε j p p q p j ` r cot δ j ` p p q S j ´ p pr q ` C j ´ p pr qs , (4.31) V β p r q “ cos ε j p p q p j ` r cot δ j ` p p q S j ` p pr q ` C j ` p pr qs . (4.32)For later convenience we define s (cid:96) p p , r q “ p ´ (cid:96) ´ S (cid:96) p pr q , (4.33) c (cid:96) p p , r q “ p (cid:96) C (cid:96) p pr q . (4.34) s (cid:96) p p , r q and c (cid:96) p p , r q are given in Appendix A.1. Plugging in these function into Eq. (4.29)-60q. (4.32) we obtain U α p r q “ cos ε j p p qr p j ´ cot δ j ´ p p q s j ´ p p , r q ` c j ´ p p , r qs , (4.35) V α p r q “ sin ε j p p qr p j ` cot δ j ´ p p q s j ` p p , r q ` p ´ c j ` p p , r qs , (4.36) U β p r q “ ´ sin ε j p p qr p j ` cot δ j ` p p q s j ´ p p , r q ` p c j ´ p p , r qs , (4.37) V β p r q “ cos ε j p p qr p j ` cot δ j ` p p q s j ` p p , r q ` c j ` p p , r qs . (4.38)The multi channel effective range expansion formula is given in Eq. (2.37), and here wetruncate the expansion at p and write the two-channel effective range expansion for the totalspin angular momentum s “ ÿ m , n p mm r K ´ s m n p n n “ ´ a mn ` r mn p ` O p p q , (4.39)where a mn is the scattering length matrix, r mn is the effective range matrix, and p mn is thediagonal momentum matrix diag p p j ´ { , p j ` { q . The two-channel effective range expansion inthe Blatt and Biedernharn parameterization is ÿ m m n n p mm U m m r K ´ s m n r U ´ s n n p n n “ ´ a mn ` r mn p ` O p p q (4.40)where a mn “ diag p a j ´ , a j ` q , r mn “ diag p r j ´ , r j ` q , and we drop the indices the reaction andunitary matrices for the sake of simplicity. In addition we get an analytic expansion for thetangent mixing angle [19] tan ε j p p q “ q p ` q p ` O ´ p ¯ (4.41)61ith mixing parameters q and q . Now using Eq. (4.41) we obtain the following final forms ofwave functions for r ě R , U α p r q “ ´ a j ´ s , j ´ p r q ` c , j ´ p r q` p ! r j ´ s , j ´ p r q ´ a j ´ s , j ´ p r q ` c , j ´ p r q ) ` O p p q , (4.42) V α p r q “ q c , j ` p r q ` p ! q c , j ` p r q ` q c , j ` p r q ) ` O p p q , (4.43) U β p r q “ q a j ` s , j ´ p r q` p ! q a j ` s , j ´ p r q ´ q r j ` s , j ´ p r q ` q a j ` s , j ´ p r q ) ` O p p q , (4.44) V β p r q “ ´ a j ` s , j ` p r q ` c , j ` p r q` p ! r j ` s , j ` p r q ´ a j ` s , j ` p r q ` c , j ` p r q ) ` O p p q . (4.45)As in the single channel case, the tool that we use to derive the causality bound is theWronskian identity. Through the derivation we recall assumption on the potential in Section 2.3.We assume that the potential is not singular at the origin and regular solutions of the Schrödingerequations U p r q and V p r q for two different values of momenta, p a and p b , satisfylim ρ Ñ ` U b p ρ q U a p ρ q “ lim ρ Ñ ` U a p ρ q U b p ρ q “ , (4.46)lim ρ Ñ ` V b p ρ q V a p ρ q “ lim ρ Ñ ` V a p ρ q V b p ρ q “ . (4.47)62ollowing the procedure given in Section. 2.5.3, for γ “ α , β states we obtain p p a ´ p b q ż r r U a γ p r q U b γ p r q ` V a γ p r q V b γ p r qs dr “ W r U a γ p r q , U b γ p r qs ` W r V a γ p r q , V b γ p r qs , (4.48)and for the combination of α and β states, we get p p a ´ p b q ż r r U a α p r q U b β p r q ` V a α p r q V b β p r q ` U b α p r q U a β p r q ` V b α p r q V a β p r qs dr “ W r U a α p r q , U b β p r qs ` W r U a β p r q , U b α p r qs ` W r V a α p r q , V b β p r qs ` W r V a β p r q , V b α p r qs . (4.49)The Wronskian of the α -state wave functions and the β -state wave functions for the non-interacting region r ě R are given in Appendix B.1.2.In Eq. (4.48), we set p a “ p “ p b Ñ
0. In the region r ě R we obtainthe following relations for the effective range parameters, r j ´ “ b j ´ p r q ` q W r c p r q , c p r qs j ` ´ ż r ˆ” U p q α p r q ı ` ” V p q α p r q ı ˙ dr , (4.50) r j ` “ b j ` p r q ` q a j ` W r s p r q , s p r qs j ´ ´ ż r ˆ” U p q β p r q ı ` ” V p q β p r q ı ˙ dr . (4.51)Here b j ¯ are b j ¯ p r q “ a j ¯ W r s p r q , s p r qs j ¯ ` a j ¯ W r c p r q , s p r qs j ¯ ` a j ¯ W r s p r q , c p r qs j ¯ ` W r c p r q , c p r qs j ¯ , (4.52)63hich reduce to the form b j ¯ p r q “ a j ¯ π Γ ` j ¯ ` ˘ Γ ` j ¯ ` ˘ ´ r ¯ p j ¯ q` ´ a j ¯ j ¯ ` ´ r ¯ ´ Γ ` j ¯ ´ ˘ Γ ` j ¯ ` ˘ π ´ r ¯ ´ p j ¯ q` . (4.53)In Eq. (4.49), we set p a “ p “ p b Ñ
0. In the region r ě R weobtain q a j ` “ d j p r q ´ ż r ” U p q α p r q U p q β p r q ` V p q α p r q V p q β p r q ı dr . (4.54)Here d j p r q is d j p r q “ ´ q a j ´ a j ` W r s p r q , s p r qs j ´ ` q W r c p r q , c p r qs j ` ` q a j ` ! W r c p r q , s p r qs j ´ ´ W r c p r q , s p r qs j ` ) , (4.55)and this can be written as d j p r q “ ´ q a j ´ a j ` π Γ ` ` j ˘ Γ ` ` j ˘ ´ r ¯ j ` ` q a j ` p j ´ qp j ` q r ´ q Γ ` j ` ˘ Γ ` j ` ˘ π ´ r ¯ ´ j ´ . (4.56)All of equations derived here have been numerically checked using a simple potential model.The numerical calculations using delta-function shell potentials with partial-wave mixing havebeen performed, and details are given in Appendix. C.2.64 .4 Causality Bounds The terms in the integrals in Eq. (4.50) and Eq. (4.51) are positive semi-definite since the wavefunctions are real. Therefore Eq. (4.50) and Eq. (4.51) place upper bounds for the effectiverange r j ´ and r j ` respectively. As noted in the introduction, these upper bounds result fromthe causality and unitarity in the quantum scattering problem. Our results are extensions ofsingle-channel results in Ref. [142] for the s -wave in three dimensions and in Ref. [94] forarbitrary angular momentum and arbitrary dimensions.The causality bounds for the lower and higher partial-wave effective ranges are r j ´ ď b j ´ p r q ´ q Γ p j ` q Γ p j ` q π ´ r ¯ ´ j ´ , (4.57) r j ` ď b j ` p r q ` q a j ` π Γ p j ` q Γ p j ` q ´ r ¯ j ` . (4.58)We note that the effective range bounds are modified due to partial-wave mixing. The causalityupper bound for r j ´ is lowered by the negative term on the right hand side of Eq. (4.57), whilethe causality upper bound for the higher partial-wave is increased by the term on the righthand side of Eq. (4.58). When q is nonzero and we take the limit of zero range interactions,Eq. (4.57) tells us that r j ´ is driven to negative infinity for any j . We conclude that the physicsof partial-wave mixing requires a non-zero range for the interactions in order to comply with theconstraints of causality and unitarity. In Ref. [94] a similar negative divergence in the effectiverange parameter was found for single-channel partial waves with (cid:96) ą
0. What is interesting hereis that the negative divergence of the effective range occurs already in the S channel due topartial-wave mixing.We note that the integral terms in Eq. (4.50), Eq. (4.51) and Eq. (4.54) are closely related.65nalysis of these equations using the Cauchy-Schwarz inequality provides another usefulrelation for the coupled-channel wave functions. For real functions f p r q , f p r q , g p r q and g p r q ,the Cauchy-Schwarz inequality is ´ ż r f p r q f p r qs »—– f p r q f p r q fiffifl dr ¯´ ż r g p r q g p r qs »—– g p r q g p r q fiffifl dr ¯ ě ˇˇˇ ż r f p r q g p r q ` f p r q g p r qs dr ˇˇˇ . (4.59)When we apply the inequality to our coupled wave functions, we get f j ´ p r q g j ` p r q ě “ h j p r q ‰ , (4.60)where f j ´ p r q “ b j ´ p r q ´ q Γ p j ` q Γ p j ` q π ´ r ¯ ´ j ´ ´ r j ´ , (4.61) g j ` p r q “ b j ` p r q ` q a j ` π Γ p j ` q Γ p j ` q ´ r ¯ j ` ´ r j ` , (4.62)and h j p r q “ d j p r q ´ q a j ` . (4.63)This inequality is used to define a Cauchy-Schwarz range, R C-S , as the minimum r for eachcoupled channel where Eq. (4.57), Eq. (4.58) and Eq. (4.60) hold. We now apply our causality bounds to physical neutron-proton data. In this study, we use thelow energy neutron-proton scattering data (0-350 MeV) from the NN data base by the Nijmegen66roup [161]. Tables 4.1- 4.3 show the low-energy threshold parameters in the eigenphaseparameterization for the NijmII and the Reid93 potentials. These parameters are calculatedusing the results obtained in Ref. [165] for the low-energy threshold parameters of the nuclearbar parameterization and relations between eigenphase and nuclear bar parameterizations givenin Appendix C.1. Using these numbers we analyze Eq. (4.50), Eq. (4.51) and Eq. (4.54), as wellas causality bounds for the uncoupled channels.
Table 4.1
The eigenphase low energy parameters of uncoupled channels for neutron-proton scatteringby the NijmII and the Reid93 interaction potentials.
Channel a (cid:96) [fm (cid:96) ` ] r (cid:96) [fm ´ (cid:96) ` ]NijmII (Reid93) NijmII (Reid93) S -23.727 (-23.735) 2.670 (2.753) P P -2.468 (-2.469) 3.914 (3.870) P D -1.389 (-1.377) 14.87 (15.04) D -7.405 (-7.411) 2.858 (2.851) F F We start with channels of a single uncoupled partial wave. Since there is no mixing betweendifferent partial-waves, we evaluate Eq. (4.50) and Eq. (4.51) with zero mixing angle, and weobtain the following equation for the effective range r (cid:96) “ b (cid:96) p r q ´ ż r ” U p q (cid:96) p r q ı dr , (4.64)67 able 4.2 The eigenphase low energy parameters of coupled channels for neutron-proton scattering bythe NijmII and the Reid93 interaction potentials.
Channel a (cid:96) [fm (cid:96) ` ] r (cid:96) [fm ´ (cid:96) ` ]NijmII (Reid93) NijmII (Reid93) S D P -0.2844 (-0.2892) -11.1465 (-10.7127) F D -0.1449 (-0.177) 288.428 (198.528) G Table 4.3
The eigenphase low energy mixing parameters of coupled channels for neutron-protonscattering by the NijmII and the Reid93 interaction potentials.
Mixing angle q [fm ] q [fm ]NijmII (Reid93) NijmII (Reid93) ε ε -5.65752 (-5.5325) 65.8602 (64.2979) ε b (cid:96) is given in Eq. (4.9). These solutions were derived by Hammer and Lee [95] forarbitrary dimension and angular momentum.Here, we analyze the causality bound of the effective range for (cid:96) ď r b (cid:96) p r q ´ r (cid:96) s for all of uncoupled channels with (cid:96) ď
3. The physical region corresponds with r b (cid:96) p r q ´ r (cid:96) s ě s -wave scattering b p r q “ a r ´ a r ` r , (4.65)68 / [ b l ( r )- r l ] [f m - l + ] r [fm] S P P P D D F F Figure 4.1
The plot of r b (cid:96) p r q ´ r (cid:96) s{ r for neutron-proton scattering via the NijmIIpotential in the s ` (cid:96) j channel. for p -wave, b p r q “ r a ´ r a ´ r , (4.66)for d -wave, b p r q “ a r ´ a r ´ r , (4.67)for f -wave, b p r q “ r a ´ r a ´ r , (4.68)and for g -wave, b p r q “ r a ´ r a ´ r . (4.69)69 .5.2 Coupled Channels We now analyze channels with coupled partial waves. We plot Eq. (4.61) and Eq. (4.62) for allcoupled channels with j ď
3. The physical region correspond both f j ´ p r q ě g j ` p r q ě -1 0 1 2 3 4 0 1 2 3 4 5 6 f J - ( r ) [f m - J + ] r [fm] S P D Figure 4.2
The plot of f j ´ p r q as a function of r for neutron-proton scattering via the NijmII potentialfor j ď
3. Here f p r q is rescaled by a factor of 0 .
01 and f p r q is rescaled by a factor of 10 ´ . g J + ( r ) [f m - J - ] r [fm] D F G Figure 4.3
The plot of g j ` p r q as a function of r for neutron-proton scattering via the NijmII potentialfor j ď
3. Here g p r q is rescaled by a factor of 0 . S - D Coupling.
We consider Eq. (4.50) - Eq. (4.55) for the S - D coupled channel. We evaluate the Wronskiansfor j “ b p r q ´ q r ´ r “ ż r ˆ” U p q α p r q ı ` ” V p q α p r q ı ˙ dr , (4.70) b p r q ` q r a ´ r “ ż r ˆ” U p q β p r q ı ` ” V p q β p r q ı ˙ dr , (4.71) d p r q ´ q a “ ż r r U α p r q U β p r q ` V α p r q V β p r qs dr . (4.72)71 p r q and b p r q are given in Eq. (4.65) and in Eq. (4.67), respectively, and d p r q is d p r q “ ´ q a a r ` q a r ´ q r . (4.73)Using the scattering parameters in Tables 4.2 - 4.3, we plot Eq. (4.70), Eq. (4.71) andEq. (4.72) as functions of r . In Figure 4.4 we show the physical region where the causality -15-10-5 0 5 10 15 0 1 2 3 4 5 6 7 8 [f m - ] r [fm] Re[ h ( r )] -15-10-5 0 5 10 15 0 1 2 3 4 5 6 7 8 [f m - ] r [fm] Re[ √ f g ] -15-10-5 0 5 10 15 0 1 2 3 4 5 6 7 8 [f m - ] r [fm] -Re[ √ f g ] Figure 4.4
We plot Re ”a f p r q g p r q ı , ´ Re ”a f p r q g p r q ı , and Re r h p r qs as functions of r forneutron-proton scattering in S - D coupled channel. bounds f p r q ě g p r q ě
0, and f p r q g p r q ě h p r q , are satisfied. Here we have f p r q “ a r ´ a r ` r ´ q r ´ r , (4.74)72 p r q “ a r ´ a r ´ r ` q r a ´ r , (4.75) h p r q “ ´ q a a r ` q a r ´ q r ´ q a . (4.76) P ´ F Coupling.
In the P - F coupled channel Eq. (4.50) - Eq. (4.55) take the following forms, b p r q ´ q r ´ r “ ż r ˆ” U p q α p r q ı ` ” V p q α p r q ı ˙ dr , (4.77) b p r q ` q a r ´ r “ ż r ˆ” U p q β p r q ı ` ” V p q β p r q ı ˙ dr , (4.78) d p r q ´ q a “ ż r r U α p r q U β p r q ` V α p r q V β p r qs dr . (4.79) b p r q and b p r q are defined in Eq. (4.66) and Eq. (4.68), respectively, and d p r q is d p r q “ ´ q a a r ` q a r ´ q r . (4.80)The causality bounds are f p r q ě g p r q ě
0, and f p r q g p r q ě h p r q , where f p r q “ r a ´ r a ´ r ´ q r ´ r , (4.81) g p r q “ r a ´ r a ´ r ` q a r ´ r , (4.82) h p r q “ ´ q a a r ` q a r ´ q r ´ q a . (4.83)In Figure 4.5 we show the physical region for the P - F coupled channel wave functions.73 [f m - ] r [fm] Re[ h ( r )] -4-3-2-1 0 1 2 3 4 1.5 2 2.5 3 3.5 4 4.5 5 [f m - ] r [fm] Re[ √ f g ] -4-3-2-1 0 1 2 3 4 1.5 2 2.5 3 3.5 4 4.5 5 [f m - ] r [fm] - Re[ √ f g ] Figure 4.5
We plot Re ”a f p r q g p r q ı , ´ Re ”a f p r q g p r q ı , and Re r h p r qs as functions of r forneutron-proton scattering in P - F coupled channel. The functions are rescaled by a factor of 0 . .5.2.3 D ´ G Coupling.
For j “
3, the D and G channels are coupled. In this case Eq. (4.50)-Eq. (4.55) read b p r q ´ q r ´ r “ ż r ˆ” U p q α p r q ı ` ” V p q α p r q ı ˙ dr , (4.84) b p r q ` q a r ´ r “ ż r ˆ” U p q β p r q ı ` ” V p q β p r q ı ˙ dr , (4.85) d p r q ´ q a “ ż r r U α p r q U β p r q ` V α p r q V β p r qs dr . (4.86)Here d p r q is d p r q “ ´ q a a r ` q a r ´ q r . (4.87)The causality bounds are again f p r q ě g p r q ě
0, and f p r q g p r q ě h p r q , where f p r q “ r a ´ r a ´ r ´ q r ´ r , (4.88) g p r q “ r a ´ r a ´ r ` q a r ´ r , (4.89) h p r q “ ´ q a a r ` q a r ´ q r ´ q a . (4.90)We show plots for the D - G channel in Figure 4.6. In this section we present the results for the causal and Cauchy-Schwarz ranges, R b , and R C-S .We use the NijmII scattering data for neutron-proton scattering presented above. In Table 4.475 [f m - ] r [fm] Re[ h ( r )] -3-2-1 0 1 2 3 3 3.5 4 4.5 5 5.5 6 [f m - ] r [fm] Re[ √ f g ] -3-2-1 0 1 2 3 3 3.5 4 4.5 5 5.5 6 [f m - ] r [fm] - Re[ √ f g ] Figure 4.6
We plot Re ”a f p r q g p r q ı , ´ Re ”a f p r q g p r q ı , and Re r h p r qs as functions of r forneutron-proton scattering in D - G coupled channel. The functions are rescaled by a factor of 0 .
76e show results for the causal range for all uncoupled channels by setting r (cid:96) “ b (cid:96) p r q . (4.91)In Table 4.5 we determine the causal range for all coupled channels using Eqs. 4.61 - 4.62. Also,we find the Cauchy-Schwarz ranges shown in Table 4.6 using Eq. (4.60). Table 4.4
The causal ranges for uncoupled channels.
Channels S P P P D D F F R b [fm] 1 .
27 0 .
31 3 .
07 0 .
23 3 .
98 4 .
91 1 .
88 1 . Table 4.5
The causal ranges for coupled channels.
Channels S D P F D G R b [fm] 1 .
29 1 .
20 2 .
23 1 .
73 4 .
03 3 . Table 4.6
The Cauchy-Schwarz ranges for coupled channels.
Channels S - D P - F D - G R C-S [fm] 1 .
29 4 .
65 5 . f j ´ p r q g j ` p r q ě “ h j p r q ‰ , (4.92)where f j ´ p r q “ b j ´ p r q ´ q Γ p j ` q Γ p j ` q π ´ r ¯ ´ j ´ ´ r j ´ , (4.93) g j ` p r q “ b j ` p r q ` q a j ` π Γ p j ` q Γ p j ` q ´ r ¯ j ` ´ r j ` , (4.94) h j p r q “ d j p r q ´ q a j ` , (4.95) b j ¯ p r q “ a j ¯ π Γ ` j ¯ ` ˘ Γ ` j ¯ ` ˘ ´ r ¯ p j ¯ q` ´ a j ¯ j ¯ ` ´ r ¯ ´ Γ ` j ¯ ´ ˘ Γ ` j ¯ ` ˘ π ´ r ¯ ´ p j ¯ q` , (4.96) d j p r q “ ´ q a j ´ a j ` π Γ ` ` j ˘ Γ ` ` j ˘ ´ r ¯ j ` ` q a j ` p j ´ qp j ` q r ´ q Γ ` j ` ˘ Γ ` j ` ˘ π ´ r ¯ ´ j ´ . (4.97)We note that the leading power of r in g j ` p r q is1 a j ` π Γ ` j ` ` ˘ Γ ` j ` ` ˘ ´ r ¯ j ` . (4.98)78his has a very small numerical prefactor multiplying a ´ j ` r j ` . For j “ { j “ { j “ { r is large compared with p a j ` q {p j ` q . If we neglect this term, then the term with the leadingpower of r on the left hand side of Eq. (4.92) is the same as that on the right hand side,1 a j ´ π Γ ` j ´ ` ˘ Γ ` j ´ ` ˘ ´ r ¯ j ` ¨ q a j ` π Γ p j ` q Γ p j ` q ´ r ¯ j ` “ « ´ q a j ´ a j ` π Γ ` ` j ˘ Γ ` ` j ˘ ´ r ¯ j ` ff . (4.99)As a result the curves for f j ´ p r q g j ` p r q and “ h j p r q ‰ are approximately parallel for large r until the term that we have neglected becomes significant. These nearly parallel trajectoriesinflate the value of the Cauchy-Schwarz range r “ R C-S where the two curves cross.For the S - D coupled channel we find R C-S is about the same size as the Comptonwavelength of the pion, m ´ π “ . j ą P - F channel we have R C-S “ .
65 fm. And for the D - G coupled channel we find R C-S “ .
68 fm. These values are surprisingly large in comparison with m ´ π . We note that there are some channels where the causal range R b is also quite large. By definition R C-S ě R b and so the Cauchy-Schwarz range will then also be large. The causal range is theminimum value for r such that f j ´ p r q “ b j ´ p r q ´ q Γ p j ` q Γ p j ` q π ´ r ¯ ´ j ´ ´ r j ´ ě g j ` p r q “ b j ` p r q ` q a j ` π Γ p j ` q Γ p j ` q ´ r ¯ j ` ´ r j ` ě q “ R b occur when the effective range parameter is positive or near zero.See for example the causal ranges for the D , D , D , and G channels. What happens isthat the function f j ´ p r q or g j ` p r q remains negative with a rather small slope until r becomesquite large. The small slope is again associated with the fact that the term with the highestpower of r has a small numerical prefactor.The range of the interaction plays the dominant role in setting the causal range. In thelanguage of local potentials, this is the radius at which the magnitude of the potential isnumerically very small. However there is also some influence of the exponential tail of thepotential upon the causal range.In all channels where the causal range is unusually large, D , D , D , and G , we findthat the tail of the one-pion exchange potential is attractive. At smaller radii, the potentialcrosses over at some classical turning point to become repulsive. See for example Figures 2 - 4in Ref. [162].The detailed mechanism requires further study, but it appears that this geometry can cause anear-threshold wavepacket to reflect before reaching the classical turning point, thus mimickinga longer range potential. However some fine tuning is needed to produce a large causal range,as there is no enhancement in the S and S channels and a smaller amount of enhancement inthe P channel.There seems to be no such enhancement of the causal range in the P , P , and D channelswhere the tail of the potential is repulsive. In fact, the causal range for the P and P channels80re unusually small. This appears be related to quantum tunneling into the inner region wherethe potential is attractive.In the following analysis we will investigate the importance of the tail of the one-pionexchange potential plays in setting the causal range, R b . We show that even though the one-pionexchange potential is numerically small at distances larger than 5 fm, chopping off the one-pion exchange tail at such distances produces a non-negligible effect. The one-pion exchangepotential tail appears to be the source of the large values for R b in higher partial waves wherethe central one-pion exchange tail is attractive.If we neglect electromagnetic effects, then the neutron-proton interaction potential at largedistances is governed by the one-pion exchange (OPE) potential, which in configuration space is V OPE p r q “ V C p r q ` S V T p r q . (4.102)Here V C p r q is the central potential, V C p r q “ g π N π ˆ m π M N ˙ p (cid:126) τ ¨ (cid:126) τ qp (cid:126) σ ¨ (cid:126) σ q e ´ m π r r , (4.103) V T p r q is the tensor potential, V T p r q “ g π N π ˆ m π M N ˙ p (cid:126) τ ¨ (cid:126) τ q ´ ` m π r ` p m π r q ¯ e ´ m π r r , (4.104)and S is the tensor operator, S “ p (cid:126) σ ¨ ˆ r qp (cid:126) σ ¨ ˆ r q ´ (cid:126) σ ¨ (cid:126) σ . (4.105)Here m π is the pion mass, M N is the nucleon mass, and g π N “ . j “ W p r , r q “ ¨˚˝ V C p r q ? V T p r q? V T p r q V C p r q ´ V T p r q ˛‹‚ δ p r ´ r q . (4.106)In Figure (4.7) we plot W p r q “ V C p r q , W p r q “ W p r q “ ? V T p r q , and W p r q “ V C p r q ´ V T p r q in the S - D coupled channel. -1-0.500.51 0 1 2 3 4 5 W ( r , r , ) [f m - ] r [fm] W ( r ) W ( r ) W ( r) Figure 4.7
Plot of the potential matrix elements W p r q “ V C p r q , W p r q “ W p r q “ ? V T p r q and W p r q “ V C p r q ´ V T p r q as a function of r in the S - D coupled channel. To demonstrate the origin of large causal ranges found in Table 4.4 and Table 4.5, we willpresent some simple but illustrative numerical examples. For each channel we add a shortrange potential to the one-pion exchange potential in order to reproduce the physical low-energy82cattering parameters. The specific model we use for the short range potential is not important toour general analysis nor is it the most economical. We choose a simple scheme which consistsof three well-defined functions in three different regions and which is continuously differentiableeverywhere. The potential has the form V p r q “ V Gauss p r q θ p R Gauss ´ r q ` V Spline p r q θ p r ´ R Gauss q θ p R Exch. ´ r q` V Exch. p r q θ p r ´ R Exch. q , (4.107)where θ is a unit step function.The short-range part is a Gaussian function V Gauss p r q “ C G e ´ m G r . (4.108)The intermediate-range part of the potential is a cubic spline use to connect the short- andlong-range regions, V Spline p r q “ C ` C r ` C r ` C r . (4.109)The long-range part consists of the usual one-pion exchange potential together with two addi-tional heavy meson exchange terms, V Exch. p r q “ V π , A , BC p r q ` S V π , D , FT p r q . (4.110)The central part of the potential is composed of Yukawa functions V π , A , BC p r q “ g π N π ˆ m π M N ˙ " C π e ´ m π r r ` C A e ´ m A r r ` C B e ´ m B r r * , (4.111)83nd the tensor part of the potential has the form V π , D , FT p r q “ g π N π ˆ m π M N ˙ " C Π „ ` m π r ` p m π r q e ´ m π r r ` C D „ ` m D r ` p m D r q e ´ m D r r ` C F „ ` m F r ` p m F r q e ´ m F r r * . (4.112)Here C π “ p (cid:126) S ´ qp (cid:126) T ´ q , C Π “ (cid:126) T ´ g π N “ . m π “
140 MeV, and M N “ . P channel. Figure 4.9 shows the potential in the D channel, and Figure 4.10 shows the potentialin the D channel.After having recovered the physical low-energy scattering parameters, we now multiply anadditional step function to the potential, V p r q Ñ V p r q θ p R ´ r q , (4.113)which removes the tail of the potential beyond range R . We then recalculate the low-energyscattering parameters with this modification. The results are shown in Table 4.7 for the P , D , and D channels. The causal ranges for the D and D channels are quite large for thephysical scattering data, 4 . . R “ . . V ( r ) [f m - ] r [fm] P Figure 4.8
Plot of the model potential in the P channel. In this channel S “ T “ C π “ C A “ . C B “ . C Π “ C D “ C F “ C G “ ´ . m A “ . m π , m B “ . m π , m G “ . m π , C “ ´ . C “ . C “ .
933 and C “ ´ . R Gauss “ . R Exch. “ . V ( r ) [f m - ] r [fm] D Figure 4.9
Plot of the model potential in the D channel. In this channel S “ T “ C π “ ´ C A “ ´ . C B “ ´ . C Π “ C D “ C F “ C G “ . m A “ m π , m B “ m π , m G “ . m π , C “ . C “ ´ . C “ .
384 and C “ ´ . R Gauss “ . R Exch. “ . V ( r ) [f m - ] r [fm] D Figure 4.10
Plot of the model potential in the D channel. In this channel S “ T “ C π “ ´ C A “´ C B “ ´ C Π “ ´ C D “ ´ C F “ ´ C G “ . m A “ m D “ m π , m B “ m F “ m π , m G “ . m π , C “ . C “ ´ . C “ ´ .
455 and C “ . R Gauss “ . R Exch. “ . able 4.7 The potential range dependence of the causal range in various channels. (cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)
Causal range Potential range R b P R b D R b D R b in higher partial waves where the central one-pion exchange tail is attractive. In this study we have derived the constraints of causality and unitarity for neutron-protonscattering for all spin channels up to j “
3. We have defined and calculated interaction lengthscales which we call the causal range, R b , and the Cauchy-Schwarz range, R C-S . The causalrange is the minimum value for r such that the causal bounds, f j ´ p r q “ b j ´ p r q ´ q Γ p j ` q Γ p j ` q π ´ r ¯ ´ j ´ ´ r j ´ ě , (4.114) g j ` p r q “ b j ` p r q ` q a j ` π Γ p j ` q Γ p j ` q ´ r ¯ j ` ´ r j ` ě , (4.115)are satisfied. For uncoupled channels these bounds simplify to the form f (cid:96) p r q “ g (cid:96) p r q “ b (cid:96) p r q ´ r (cid:96) ě . (4.116)88or coupled channels the Cauchy-Schwarz range is the minimum value for r satisfying thecausal bounds as well as the Cauchy-Schwarz inequality, f j ´ p r q g j ` p r q ě “ h j p r q ‰ . (4.117)If one reproduces the physical scattering data using strictly finite range interactions, then therange of these interactions must be larger than R b and R C-S . From these bounds we have derivedthe general result that non-vanishing partial-wave mixing cannot be reproduced with zero-rangeinteractions. As the range of the interaction goes to the zero, the effective range for the lowerpartial-wave channel is driven to negative infinity.This finding has consequences for pionless effective theory where the range of the interac-tions is set entirely by the value of the cutoff momentum. If the cutoff momentum is too high,then it is impossible to obtain the correct threshold physics in coupled channels without violatingcausality or unitarity. In some channels we find that the causal range and Cauchy-Schwarzrange are as large 5 fm. We have shown that these large values are driven by the tail of theone-pion exchange potential. In these channels the problems will be even more severe, andthe cutoff momentum will need to be rather low in order to reproduce the physical scatteringdata in pionless effective field theory. How low this cutoff momentum must be depends on theparticular regularization scheme.We should note that all of these mixing observables are non-vanishing only when onereaches higher orders in the power counting expansion, and there is no direct impact on pionlesseffective field theory calculations at lower orders. See, for example, Ref. [38] for details onpower counting in pionless effective field theory. In the zero-range limit, the term which drives89he negative divergence of the effective range parameter r j ´ is ´ q Γ p j ` q Γ p j ` q π ´ r ¯ ´ j ´ . (4.118)At leading order there is no divergence since there is no partial wave mixing and q “
0. Ifhigher-order terms are iterated non-perturbatively as in Ref. [167], then the divergence appearsat order Q , the first order at which q is non-vanishing. If higher-order terms are iteratedorder-by-order in perturbation theory, then the term in Eq. (4.118) appears at order Q . Thisis one order higher than the analysis presented in Ref. [38], and we predict that zero-rangedivergences in r j ´ will first appear at this order.It important to note that if one works order-by-order in perturbation theory, then the con-straints of causality and unitarity always appear somewhat hidden. At every order in the effectivefield theory calculation there are new operator coefficients which appear and are determinedby matching to physical data. There are no obstructions to setting these operator coefficients toreproduce physical values.It is only when one iterates the new interactions, i.e., by solving the Schrödinger equation,that non-linear dependencies on the operator coefficients appear. In this case one finds thatthe constraints of causality and unitarity give necessary conditions for keeping the operatorcoefficients real. Once we fix the regularization, the bound corresponds with branch cuts of theeffective theory when viewed as a function of physical scattering parameters.These branch cuts cannot be seen at any finite order in perturbation theory. However anearby branch point may spoil the convergence of the perturbative expansion. In this context, ourcausality and unitarity bounds can be viewed as setting physical constraints for the convergenceof perturbative calculations in pionless effective field theory.If the cutoff is taken too high, a branch cut develops which jeopardizes the convergence of the90erturbative calculation. Similarly if one does calculations using dimensional regularization, thenthe renormalization scale sets the scale at which the infrared and ultraviolet physics are regulated[85]. Similar problems with perturbative convergence would arise if the renormalization scale istaken too high.There is much theoretical interest in the connection between dilute neutron matter and theuniversal physics of fermions in the unitarity limit [86, 25, 61, 117, 87, 177, 131]. In the limit ofisospin symmetry our analysis of the isospin triplet channels can be applied to neutron-neutronscattering in dilute neutron matter. In this study we have shown there are intrinsic lengthscales associated with the causal range and the Cauchy-Schwarz range. When the averageseparation between neutrons is smaller than these length scales, one expects non-universalbehavior controlled by the details of the neutron-neutron interactions. For the S channel, R b “ . P channel, R b “ . F channel, R b “ . P - F mixing, we find R C-S “ . P - F mixing will becomenon-universal at lower densities than the S interactions. In particular the densities where P superfluidity is expected to occur will be well beyond this universal regime.91 hapter van der Waals interactions Low-energy universality appears when there is a large separation between the short-distancescale of the interaction and the physically relevant long-distance scales. Some well-knownexamples include the unitarity limit of two-component fermions [138, 103, 180, 2, 113] and theEfimov effect in three-body and four-body systems [53, 54, 11, 144, 96, 168, 47, 93, 83, 110].See Refs. [29, 88] for reviews of the subject and literature. There have been many theoreticalstudies of low-energy phenomena and universality for interactions with finite range. Thesestudies have direct applications to nuclear physics systems such as cold dilute neutron matter orlight nuclei such as the triton and alpha particle. To a good approximation, the van der Waalsinteractions between alkali-metal atoms can also be treated as a finite-range interaction.However, there are some differences. For potentials with an attractive 1 { r α tail and α ą s -wave scattering phase shift near threshold has been formulated in Ref. [133]. For α ą s -wave Feshbach resonance were derived in Ref. [156]using coupled-channel calculations. Analytical expressions for the s -wave scattering length and92ffective range for two neutral atoms and α “ (cid:96) ě (cid:96) ě
1, amodified version of effective range theory known as quantum-defect theory is needed [75, 79].Furthermore, scattering parameters of magnetically tunable multichannel systems have beenstudied in the context of multichannel quantum-defect theory [146, 80]. See Ref. [155] for avery recent development of multichannel quantum-defect theory for higher partial waves. Thereis also growing empirical evidence that there exists a new type of low-energy universality thatties together all interactions with an attractive 1 { r tail. This might seem surprising since thereis no such analogous behavior for interactions with a Coulomb tail. In this chapter we derivethe theoretical foundations for this van der Waals universality at low energies by studying thenear-threshold behavior and the constraints of causality. We also show that this universalityextends to any power-law interaction 1 { r α with α ě ´ C { r . We define the van der Waals length scale, β , as β “ p µ C q , (5.1)where µ is the reduced mass of the scattering particles. For simplicity we use atomic units (a.u.)throughout our discussion. So, in particular, we set ¯ h “
1. In Refs. [35, 151] it was noticed thatan approximate universal relationship exists between the effective range and inverse scatteringlength for s -wave scattering in many different pairs of scattering alkali-metal atoms. If we write93 as the scattering length and R as the effective range, the relation is R « β Γ p { q π ´ β A ` πβ Γ p { q A . (5.2)This approximate relation becomes exact for a pure ´ C { r potential. What is surprisingabout Eq. (5.2) is that the van der Waals length β dominates over other length scales whichcharacterize the short-distance repulsive force between alkali-metal atoms. This approximateuniversality suggests there is some separation of scales between the van der Waals length β and the length scales of the short-range forces. This separation of scales will become moretransparent later in our analysis when we determine the coefficients of the short-range ˆK-matrix.It would be useful to exploit the separation of scales as an effective field theory with an explicitvan der Waals tail plus contact interactions. In this chapter, we discuss the constraints on such avan der Waals effective field theory.We note that a similar dominance of the van der Waals length β has been discovered for thethree-body parameter in the Efimov effect [17, 169, 170]. In the analysis here we focus only ontwo-body systems. However, our analysis should be useful in developing the foundations forvan der Waals effective field theory. This in turn could be used to investigate the Efimov effectand other low-energy phenomena in a model-independent way. An extension of our analysismay be useful to understand the recently observed universality of the three-body parameter fornarrow Feshbach resonances [150].The organization of this chapter is as follows. We first discuss the connection betweencausality bounds and effective field theory. Next we consider asymptotic solutions of theSchrödinger equation. After that, we derive causality bounds for the short-range ˆK-matrix andconsider the impact of these results on van der Waals effective field theory. Then we discussquantum-defect theory and calculate causal ranges for several examples of single-channel s -wave94cattering in alkali-metal atoms. We also consider the constraints of causality near magneticFeshbach resonances. We then conclude with a summary and discussion. For an effective field theory with local contact interactions, the range of the interactions iscontrolled by the momentum cutoff scale. Problems with convergence can occur if the cutoffscale is set higher than the scale of the new physics not described by the effective theory. Itis useful to have a quantitative measure of when problems may or may not appear, and this iswhere the causality bound provides a useful diagnostic tool. For each scattering channel we usethe physical scattering parameters to compute the causal range, R b , which is the minimum rangefor the interactions consistent with the requirements of causality and unitarity and discussedin details in Chapter 4. For any fixed cutoff scale, the causality bound marks a branch cut ofthe effective theory when viewed as a function of physical scattering parameters, see Chapter 4and Ref. [107]. The coupling constants of the effective theory become complex for scatteringparameters violating the causality bound. These branch cuts do not appear in perturbation theory,but they can spoil the convergence pattern of the perturbative expansion.Wigner was the first to recognize the constraints of causality and unitarity for two-bodyscattering with finite-range interactions [174]. The time delay of a scattered wave packet isgiven by the energy derivative of the phase shift, ∆ t “ d δ { dE . It is clear that the incomingwave packet must first reach the interacting region before the outgoing wave packet can leave.So the causality bound can be viewed as a lower bound on the time delay, ∆ t . When appliedto wave packets near threshold, the causality bound becomes an upper bound on the effectiverange parameter.A brief historical review on the analysis of the constraints of causality and universality is95iven in the introduction of Chapter 4. Also in that chapter we have presented the first study ofcoupled-channel systems with partial-wave mixing We consider a system of two spinless particles interacting via a spherically symmetric finite rangepotential in the center-of-mass frame. In addition to the non-singular finite-range interactionsparameterized by W p r , r q , we assume that there is a long-range local van der Waals potential ´ C { r for r ą R . The van der Waals length scale β was defined in Eq. (5.1). As noted in theIntroduction, we use atomic units where ¯ h “
1. The radial Schrödinger equation is „ d dr ´ (cid:96) p (cid:96) ` q r ` β r θ p r ´ R q ` p U p p q (cid:96) p r q “ µ ż R dr W p r , r q U p p q (cid:96) p r q . (5.3)The step function θ p r ´ R q cuts off the long-range potential at distances less than R . Thisensures that we satisfy the regularity condition discussed in Eq. (2.30) and avoids mathematicalproblems associated with unregulated singular potentials [68]. The general form of the solutionsfor Eq. (5.3) has been discussed by Gao in Ref. [78].In order to simplify some of the more lengthy expressions to follow, we introduce dimen-sionless rescaled variables r s “ r { β , p s “ β p , and ρ s “ {p r s q . In the outer region, r ą R , theSchrödinger equation reduces to „ d dr ´ (cid:96) p (cid:96) ` q r ` β r ` p U p p q (cid:96) p r q “ „ d dr s ´ (cid:96) p (cid:96) ` q r s ` r s ` p s U p p q (cid:96) p r q “ . (5.5)96he exact solutions for Eq. (5.5) have been studied in detail in Ref. [76] using the formalism ofquantum-defect theory [157, 91, 92].The van der Waals wave functions F (cid:96) and G (cid:96) are linearly independent solutions of Eq. (5.5).In order to write these out we first need several functions defined in Appendix A.3. The vander Waals wave functions F (cid:96) and G (cid:96) can be written as summations of Bessel and Neumannfunctions, F (cid:96) p p , r q “ r { s x (cid:96) p p s q ` y (cid:96) p p s q « x (cid:96) p p s q ÿ m “´8 b m p p s q J ν ` m p ρ s q´ y (cid:96) p p s q ÿ m “´8 b m p p s q N ν ` m p ρ s q ff , (5.6) G (cid:96) p p , r q “ r { s x (cid:96) p p s q ` y (cid:96) p p s q « x (cid:96) p p s q ÿ m “´8 b m p p s q N ν ` m p ρ s q` y (cid:96) p p q ÿ m “´8 b m p p s q J ν ` m p ρ s q ff . (5.7)The function x (cid:96) is defined in Eq. (A.52), and y (cid:96) is defined in Eq. (A.53). For m ě b m is given in Eq. (A.47), while b ´ m is given in Eq. (A.48). The offset ν appearing in theorder of the Bessel functions is given by the solution of Eq. (A.51) in Appendix A.3. Fornotational convenience, however, we omit writing the explicit p s dependence of ν . Let us define δ p short q (cid:96) p p q to be the phase shift of the van der Waals wave functions due to the scattering fromthe short-range interaction. The normalization of U p p q (cid:96) p r q is chosen so that, for r ą R , U p p q (cid:96) p r q “ F (cid:96) p p , r q ´ tan δ p short q (cid:96) p p q G (cid:96) p p , r q . (5.8)Our van der Waals wave functions are related to the functions f c (cid:96) and g c (cid:96) defined of Ref. [78] by97he normalization factors F (cid:96) “ f c (cid:96) {? G (cid:96) “ ´ g c (cid:96) {?
2. Henceforth, we write all expressionsin terms of the short-range reaction matrixK (cid:96) p p q “ tan δ p short q (cid:96) p p q , (5.9)which is related to the short-range scattering matrix viaS (cid:96) “ e i δ p short q (cid:96) “ i ´ K (cid:96) i ` K (cid:96) . (5.10)For any finite-range interaction, K (cid:96) is analytic in p and can be calculated by matchingsolutions for r ď R and r ą R at the boundary. It can be written in compact form asK (cid:96) “ W p U p p q (cid:96) , F p p q (cid:96) q W p U p p q (cid:96) , G p p q (cid:96) q ˇˇˇˇˇ r “ R , (5.11)where U p p q (cid:96) is the solution of Eq. (5.3) that is regular at the origin, and W denotes the Wronskianof two functions, W p f , g q “ f g ´ f g . K -matrix K (cid:96) In this section we derive causality bounds for the short-range K-matrix. For this we need toexpand the wave function U p p q (cid:96) p r q in powers of p . The steps we follow are analogous to thoseused in Chapter 4 [cf. in Refs. [94, 95, 107]]. We first expand K (cid:96) ,K (cid:96) “ tan δ p short q (cid:96) p p q “ ÿ n “ K (cid:96), n p n . (5.12)98he first two terms K (cid:96), and K (cid:96), are analogous to the inverse scattering length and effectiverange parameters in the usual effective range expansion. The higher-order terms can be regardedas analogs of the shape parameters. Next we expand the van der Waals wave functions in powersof p , F (cid:96) p p , r q “ f (cid:96), p r q ` f (cid:96), p r q p ` O p p q , (5.13) G (cid:96) p p , r q “ g (cid:96), p r q ` g (cid:96), p r q p ` O p p q . (5.14)In the following, we define ν “ p (cid:96) ` q , which corresponds to the value of ν at threshold. Using the low-energy expansions in Ap-pendix A.4, we find that the coefficients in Eq. (5.13) are f (cid:96), p r q “ r { s J ν p ρ s q (5.15)and f (cid:96), p r q “ Γ p ν q Γ p ν ´ q Γ p ν ` q Γ p ν q β r { s “ J ν ´ p ρ s q ` N ν p ρ s q ‰ ´ Γ p ν q Γ p ν ` q Γ p ν ` q Γ p ν ` q β r { s “ J ν ` p ρ s q ´ N ν p ρ s q ‰ . (5.16)Similarly, the coefficients in Eq. (5.14) are g (cid:96), p r q “ r { s N ν p ρ s q (5.17)99nd g (cid:96), p r q “ Γ p ν q Γ p ν ´ q Γ p ν ` q Γ p ν q β r { s “ N ν ´ p ρ s q ´ J ν p ρ s q ‰ ´ Γ p ν q Γ p ν ` q Γ p ν ` q Γ p ν ` q β r { s “ N ν ` p ρ s q ` J ν p ρ s q ‰ . (5.18)Using Eq. (5.8), we can now express U p p q (cid:96) p r q as an expansion in powers of p . For r ą R , wehave U p p q (cid:96) p r q “ f (cid:96), p r q´ K (cid:96), g (cid:96), p r q` p “ f (cid:96), p r q ´ K (cid:96), g (cid:96), p r q ´ K (cid:96), g (cid:96), p r q ‰ ` O p p q . (5.19)We now consider two solutions of the Schrödinger equation U p p a q (cid:96) p r q and U p p b q (cid:96) p r q withmomenta p a and p b , respectively. We have „ d dr ´ (cid:96) p (cid:96) ` q r ` β r θ p r ´ R q ` p a U p p a q (cid:96) p r q “ µ ż R dr W p r , r q U p p a q (cid:96) p r q , (5.20) „ d dr ´ (cid:96) p (cid:96) ` q r ` β r θ p r ´ R q ` p b U p p b q (cid:96) p r q “ µ ż R dr W p r , r q U p p b q (cid:96) p r q . (5.21)Following the same steps as in Section 2.5.3, we obtain the Wronskian integral formula W r U p p b q (cid:96) , U p p a q (cid:96) sp r q p b ´ p a “ ż r U p p a q (cid:96) p r q U p p b q (cid:96) p r q dr , (5.22)100or any r ą R . Using Eq. (5.19) for momenta p a and p b we find W r U p b q (cid:96) , U p a q (cid:96) sp r q p b ´ p a “ W r f (cid:96), , f (cid:96), sp r q ´ K (cid:96), (cid:32) W r g (cid:96), , f (cid:96), sp r q ` W r f (cid:96), , g (cid:96), sp r q ( ` K (cid:96), W r g (cid:96), , g (cid:96), sp r q ´ K (cid:96), W r g (cid:96), , f (cid:96), sp r q ` O p p a , p b q . (5.23)In the Wronskian integral formula Eq. (5.22), we set p a “ p b Ñ
0. With thewave function at zero energy written as U p q (cid:96) , the result is K (cid:96), “ b vdW (cid:96) p r q ´ π ż r ” U p q (cid:96) p r q ı dr , (5.24)where b vdW (cid:96) p r q “ π W r f (cid:96), , f (cid:96), sp r q ` π K (cid:96), W r g (cid:96), , g (cid:96), sp r q´ π K (cid:96), (cid:32) W r g (cid:96), , f (cid:96), sp r q ` W r f (cid:96), , g (cid:96), sp r q ( . (5.25)The Wronskians appearing in Eq. (5.25) can be written out explicitly as W r f (cid:96), , f (cid:96), sp r q “ β ρ s ν p ν ´ q “ J ν ´ p ρ s q J ν p ρ s q ´ J ν ´ p ρ s q ‰ ` β ρ s ν p ν ` q “ J ν ` p ρ s q J ν p ρ s q ´ J ν ` p ρ s q ‰ ` β ρ s p ν ´ qp ν ` q „ J ν ` p ρ s q J ν ´ p ρ s q ´ J ν p ρ s q ` πρ s , (5.26)101 r g (cid:96), , g (cid:96), sp r q “ β ρ s ν p ν ´ q “ N ν ´ p ρ s q N ν p ρ s q ´ N ν ´ p ρ s q ‰ ` β ρ s ν p ν ` q “ N ν ` p ρ s q N ν p ρ s q ´ N ν ` p ρ s q ‰ ` β ρ s p ν ´ qp ν ` q „ N ν ` p ρ s q N ν ´ p ρ s q ´ N ν p ρ s q ` πρ s , (5.27)and W r g (cid:96), , f (cid:96), sp r q “ W r f (cid:96), , g (cid:96), sp r q“ β ρ s ν p ν ´ q (cid:32) J ν ´ p ρ s q r N ν ` p ρ s q ´ N ν ´ p ρ s qs ´ N ν p ρ s q r J ν p ρ s q ´ J ν ´ p ρ s qs ( ´ β ρ s ν p ν ` q (cid:32) J ν ` p ρ s q r N ν ` p ρ s q ´ N ν ´ p ρ s qs ´ N ν p ρ s q r J ν ` p ρ s q ´ J ν p ρ s qs ( . (5.28)The fact that the integral on the right-hand side of Eq. (5.24) is positive semidefinite sets anupper bound on the short-range parameter K (cid:96), . We find that K (cid:96), ď b vdW (cid:96) p r q (5.29)for any r ą R . In this section we discuss the impact of our causality bounds for an effective field theory withshort-range interactions and an attractive 1 { r tail. In Figure 5.1 we plot the (cid:96) “ W r f , , f , s , W r g , , g , s , and W r g , , f , s for β “
50 (a.u.). Figure 5.2 and Figure 5.3 showthe analogous plots for (cid:96) “ (cid:96) “
2, respectively.102 r [a.u.] L = 0 W[ f , f ]( r )W[ g , g ]( r )W[ f , g ]( r )= W[ g , f ]( r ) Figure 5.1 (Color online) Plot of W r f , , f , sp r q , W r g , , g , sp r q , and W r g , , f , sp r q as a function of r for (cid:96) “ β “
50 (a.u.). -4-2 0 2 4 6 8 10 0 10 20 30 40 50 r [a.u.] L = 1 W[ f , f ]( r )W[ g , g ]( r )W[ f , g ]( r )= W[ g , f ]( r ) Figure 5.2 (Color online) Plot of W r f , , f , sp r q , W r g , , g , sp r q , and W r g , , f , sp r q as a function of r for (cid:96) “ β “
50 (a.u.). r [a.u.] L = 2 W[ f , f ]( r )W[ g , g ]( r )W[ f , g ]( r )= W[ g , f ]( r ) Figure 5.3 (Color online) Plot of W r f , , f , sp r q , W r g , , g , sp r q , and W r g , , f , sp r q as a function of r for (cid:96) “ β “
50 (a.u.).
We note that all of the Wronskian functions in Figures 5.1–5.3 vanish in the limit r Ñ r (cid:96) , satisfies the upper bound r (cid:96) ď b (cid:96) p r q , (5.30)where the function b (cid:96) p r q is given in Eq. (4.9) [cf. Eq. (60) in Ref. [95]] b (cid:96) p r q “ ´ Γ p (cid:96) ´ q Γ p (cid:96) ` q π ´ r ¯ ´ (cid:96) ` ´ (cid:96) ` a (cid:96) ´ r ¯ ` π Γ p (cid:96) ` q Γ p (cid:96) ` q a (cid:96) ´ r ¯ (cid:96) ` , (5.31)and a (cid:96) is the scattering length. As already discussed in Section 4.5.1 near r “ (cid:96) p r q is b (cid:96) p r q “ ´ Γ p (cid:96) ´ q Γ p (cid:96) ` q π ´ r ¯ ´ (cid:96) ` ` O p r q . (5.32)We see that b (cid:96) p r q diverges to negative infinity as r Ñ (cid:96) ě
1. The causality bound on r (cid:96) alsodrives r (cid:96) to negative infinity for (cid:96) ě , r (cid:96) ď ´ Γ p (cid:96) ´ q Γ p (cid:96) ` q π ´ r ¯ ´ (cid:96) ` ` O p r q . (5.33)For an effective field theory with local contact interactions, the range of the interactions arecontrolled by the momentum cutoff scale. No matter the values for a (cid:96) and r (cid:96) , it is not possibleto take the momentum cutoff scale arbitrarily high without violating the causality bound forchannels with angular momentum (cid:96) ě
1. For finite-range interactions with an additional attractiveor repulsive Coulomb tail, one finds the same leading behavior [107] b Coulomb (cid:96) p r q “ ´ Γ p (cid:96) ´ q Γ p (cid:96) ` q π ´ r ¯ ´ (cid:96) ` ` O p r ´ (cid:96) q , (5.34)i.e., the only difference in the causality bound relation for the Coulomb-modified effective rangeis the subleading O p r ´ (cid:96) q pole term which is absent in the purely finite-range case. Hence,also for an effective field theory with contact interactions and long-range Coulomb tail, it isnot possible to take the momentum cutoff scale arbitrarily high for (cid:96) ě b vdW (cid:96) p r q at r “ { r interaction. For aneffective field theory with contact interactions and van der Waals tail, the causality bound doesnot impose convergence problems as long as K (cid:96), is less than or equal to zero. This holds true for To get this analogy, we use here the normalization of the Coulomb-modified effective range expansion foundin Eq. (28) of Ref. [107] and insert it in Eqs. (64), (A.6), and (A.7) of the same paper, which give the explicitexpressions for the Coulomb-modified causality bound functions for (cid:96) “ , ,
2. The statement for arbitrary (cid:96) thenfollows by generalization. (cid:96) . There is no constraint from causality and unitarity preventing one from taking the cutoffmomentum to be arbitrarily large. The key difference between the van der Waals interactionand the Coulomb interaction is that, when extended all the way to the origin, the attractive 1 { r interaction is singular and the spectrum is unbounded below. An essential singularity appears at r “
0, and both van der Waals wave functions F (cid:96) and G (cid:96) vanish at the origin.These exact same features appear in any attractive 1 { r α interaction for α ą { r interaction when thecoupling constant is strong enough to form bound states. The key point is that in the zero-rangelimit of these attractive singular potentials, the spectrum of bound states extends to arbitrarilylarge negative energies. As a consequence, the scattering wave functions above threshold mustvanish at the origin in order to satisfy orthogonality with respect to all such bound-state wavefunctions localized near the origin. In all of these cases the function b vdW (cid:96) p r q remains finite as r Ñ (cid:96) . We conclude that for an effective field theory with contact interactions andattractive singular power-law interactions, we can take the cutoff momentum arbitrarily largefor any (cid:96) without producing a divergence in the coefficient K (cid:96), of the short-range K-matrix. Up to now we have been discussing the short-range phase shift of K-matrix for scattering relativeto the van der Waals wave functions F (cid:96) and G (cid:96) . For power-law interactions 1 { r α with α ą p . For the van der Waals interactions in the (cid:96) “ p , and so the scattering parameters a and r are well defined, but coefficients in higher order terms of the expansion are not. For (cid:96) “ p and so only the scattering length a is welldefined. For (cid:96) ě { r potentials, one defines an offset for the phaseshift [79], η (cid:96) “ π p ν ´ ν q . (5.35)The modified effective range expansion is then p (cid:96) ` cot p δ (cid:96) ` η (cid:96) q “ ´ A (cid:96) ` R (cid:96) p ` O ` p ln p ˘ , (5.36)where A (cid:96) and R (cid:96) are the generalized scattering length and effective range parameters. Thesedefinitions coincide with the usual scattering length a (cid:96) for (cid:96) “ , r (cid:96) for (cid:96) “
0. The generalized scattering length and effective range can be written in terms of theshort-range K-matrix parameters as A (cid:96) “ π β (cid:96) ` (cid:96) ` r Γ p (cid:96) ` q Γ p (cid:96) ` qs „ p´ q (cid:96) ´ K (cid:96), (5.37)and R (cid:96) “ ´ (cid:96) ` Γ ` (cid:96) ` ˘ Γ ` (cid:96) ` ˘ β ´ (cid:96) ´ π “ K (cid:96), p´ q (cid:96) ´ ‰ »– β ´ K (cid:96), ` ¯ (cid:96) ` (cid:96) ´ ´ K (cid:96), fifl . (5.38)From these results we see that the short-range parameter K (cid:96), appears in combination with β . But in nearly all single-channel scatterings between pairs of alkali-metal atoms, from the107ollowing equation, K (cid:96), “ β (cid:96) ` (cid:96) ´ $&% ` « p´ q (cid:96) ´ A (cid:96) (cid:96) ` Γ ` (cid:96) ` ˘ Γ ` (cid:96) ` ˘ π β (cid:96) ` ff ´ ,.- ` R (cid:96) A (cid:96) (cid:96) Γ ` (cid:96) ` ˘ Γ ` (cid:96) ` ˘ π β (cid:96) ` « p´ q (cid:96) ´ A (cid:96) (cid:96) ` Γ ` (cid:96) ` ˘ Γ ` (cid:96) ` ˘ π β (cid:96) ` ff ´ , (5.39)one quantitatively finds that K (cid:96), is at least one order of magnitude smaller than β . Thisseparation of scales is the reason for the approximate universality found in Refs. [35, 151].The dominance of β n over the subleading coefficients K (cid:96), n in Eq. (5.12) for n ě (cid:96) [77]. This producesa surprisingly rich class of universal physics for single-channel van der Waals interactionswhere K (cid:96), n is negligible compared to β n for all (cid:96) , and K (cid:96), is approximately the same for all (cid:96) . Therefore β and the s -wave scattering length will determine, to a good approximation, thethreshold scattering behavior for all values of (cid:96) . We have shown that for negative K (cid:96), ď
0, the range R of the short-range interaction can betaken all the way down to zero. But when K (cid:96), is positive, there is a constraint on R and we use108q. (5.29) to determine the causal range R b , K (cid:96), “ b vdW (cid:96) p R b q . (5.40)As pointed out in Ref. [107], one can show a priori that b vdW (cid:96) p r q is a monotonically increasingfunction of r . Therefore, if a real solution to Eq. (5.40) exists, then it is unique. If, however,there is no real solution, then there is no constraint on the interaction range and we define R b tobe zero. For an effective field theory with contact interactions and van der Waals tail, the cutoffmomentum can be made as large as „ { R b before the causality bound is violated.In the following analysis we extract the single-channel s -wave effective range parameters a and r for several different pairs of alkali-metal atoms Li, Na, and
Cs in singlet andtriplet channels. The data is taken from Refs. [40, 41, 129, 66, 75]. The reduced masses for Li , Na , and Cs are µ “ . , , Li , Na , and Cs are C “ , , (cid:96) “ A “ a and R “ r . The resultsfor the scattering parameters and causal ranges are given in columns II, V and VI of Table 5.1.The discrepancies in R are due to the fact that in the analytic studies in Refs. [66, 75] K , is neglected, while the numerical calculations of Refs. [40, 41, 129] include the short-rangecontribution from K , .In column V of Table 5.1, we present an approximate range for K , for each atomic pairusing the values for R in columns III and IV. Since K , is positive, we cannot go all the way tothe zero-range limit. However, in each case K , is at least one order of magnitude smaller than β . Although we cannot take the zero-range limit, the causal ranges are small in comparison to Note that K (cid:96), has the dimension of an area (in the appropriate atomic units). able 5.1 Scattering parameters and causal ranges for s -wave scattering of Li , Na , and Cs pairs.The scattering data collection is taken from Ref. [66]. In columns I and IV the scattering data for Liare from Ref. [41], the scattering data for Na are from Refs. [40, 41], and data for
Cs are fromRef. [129]. In column III the effective range parameters, R , are calculated analytically in Refs. [66,75]. In column IV, the R are obtained from numerical calculations. The scattering parameters incolumns II and V are calculated using Eq. (5.37) and Eq. (5.39), and the causal ranges in column VIare obtained from Eq. (5.40). I II III IV V VIAtoms State β A K , R R K , R b Li– Li Σ g „
124 7 „ Li– Li Σ u „
17 3 „ Na– Na Σ g „
86 4 „ Na– Na Σ u „
13 16 „ Cs– Cs Σ g „
146 7 „ β . In each case R b is less than one-third the size of β . Hence one can probe these interactionsin a van der Waals effective field theory with cutoff momentum up to roughly three times 1 { β without violating the causality bound. In Ref. [74] the multichannel problem of scattering around a magnetic Feshbach resonance isreduced to a description by an effective single-channel K-matrix that depends on the appliedmagnetic field B . The behavior around the resonance is described by several parameters. B ,(cid:96) isthe position of the resonance, while g res parametrizes the width of the Feshbach resonance. K bg (cid:96) is a background value for the K-matrix, and the scale d B ,(cid:96) is introduced to define a dimensionless110agnetic field. We write the effective single-channel K-matrix asK eff (cid:96) p p , B q “ ´ K bg (cid:96) „ ` g res p β ´ g res p B s ` q , (5.41)with B s “ ` B ´ B ,(cid:96) ˘ d B ,(cid:96) . (5.42)The parametrization given above corresponds to Eq. (18) in Ref. [74]. Note that we have changedthe notation slightly and are using a different sign convention.By expanding the right-hand side of Eq. (5.41) in p , it is straightforward to determine theK-matrix expansion parameters K (cid:96), and K (cid:96), . A short calculation yields that K eff (cid:96), “ ´ K bg (cid:96) ˆ ` B s ` ˙ , (5.43) K eff (cid:96), “ β K bg (cid:96) g res p B s ` q . (5.44)As noted in Ref. [74], the parameters K bg (cid:96) and g res are constrained by the conditionK bg (cid:96) g res ă . (5.45)From this we directly see that K eff (cid:96), given by Eq. (5.44) is always negative. From the causalitybound in Eq. (5.29) it follows that where this effective single-channel description is applicableand correctly captures the entire energy dependence of the short-range K-matrix, the causalrange will be zero when the interaction is tuned close to a Feshbach resonance.111 .9 Summary and discussion In this chapter we have analyzed two-body scattering with arbitrary short-range interactions plusan attractive 1 { r tail. We derived the constraints of causality and unitarity for the short-rangeK-matrix, K (cid:96) “ tan δ p short q (cid:96) p p q “ ÿ n “ K (cid:96), n p n . (5.46)For any r larger than the range of the short-range interactions, R , we find that K (cid:96), satisfies theupper bound K (cid:96), ď b vdW (cid:96) p r q , (5.47)where b vdW (cid:96) p r q is b vdW (cid:96) p r q “ π W r f (cid:96), , f (cid:96), sp r q ` π K (cid:96), W r g (cid:96), , g (cid:96), sp r q´ π K (cid:96), " W r g (cid:96), , f (cid:96), sp r q ` W r f (cid:96), , g (cid:96), sp r q * , (5.48)and the Wronksians are given in Eq. (5.26), Eq. (5.27), and Eq. (5.28).In clear contrast with the case for only finite-range interactions which was the subject ofChapter 4 or with Coulomb tails [107], the function b vdW (cid:96) p r q does not diverge but rather vanishesas r Ñ (cid:96) . When K (cid:96), ď
0, there is no constraint derived from causality and unitarity thatprevents the use of an effective field theory with zero-range contact interactions plus an attractive1 { r tail. This holds true for any angular momentum value (cid:96) . For the phenomenologicallyimportant case of a multichannel system near a magnetic Feshbach resonance, the effectivevalue for K (cid:96), is negative and so the short-range interaction can be taken to have zero range.The van der Waals interaction is qualitatively different from the Coulomb interaction where b Coulomb (cid:96) p r q diverges for (cid:96) ě
1. The key difference is that both van der Waals wave functions112 (cid:96) and G (cid:96) vanish at the origin. This phenomenon also occurs for an attractive 1 { r α interactionfor α ą { r interactionwhen the coupling constant is strong enough to form bound states. For an effective field theorywith contact interactions and attractive singular power-law tail, the cutoff momentum can bemade arbitrarily large for any (cid:96) without producing a divergence in the coefficient K (cid:96), of theshort-range K matrix.When K (cid:96), is positive, there is a lower bound on the range of the short-range interactions.We define the causal range R b as this minimum value for the range, given by the condition K (cid:96), “ b vdW (cid:96) p R b q . (5.49)We have analyzed several examples of s -wave scattering in alkali-metal atoms in Table 5.1. Wefind that the K (cid:96), is at least one order of magnitude smaller than β . As a result we find that thecausal ranges are small in comparison with β .In summary, we find that β dominates over distance scales parametrizing the short-rangeinteractions. The origin of this van der Waals universality can be explained by two facts. The firstfact is the phenomenological observation that, in single-channel scattering between alkali-metalatoms, there is a significant separation between the typical length scales of the short-distancephysics and β . This can be seen by the small size of the short-range parameter K (cid:96), comparedwith β . As Gao has shown, this also leads to the approximate universal relation that K (cid:96), is thesame for all (cid:96) [77]. Therefore, to a good approximation, β and the s -wave scattering lengthwill determine the threshold scattering behavior for all values of (cid:96) . For the multichannel casenear a magnetic Feshbach resonance, we find that the effective K eff (cid:96), is no longer negligible.However, K eff (cid:96), is negative, and this means that there is no constraint from causality preventingthe zero-range limit for the short-distance interactions.113he second fact underlying the van der Waals universality is that the zero-range limit ofshort-distance interactions is well behaved with regard to scattering near threshold. We note,however, that there is still no scale-invariant limit for (cid:96) ě β goes to zero. This can be seen from the β ´ (cid:96) ` behaviorwith negative coefficient for (cid:96) ě { r tail and contact interactions. Similarly, one can also construct effective fieldtheories for other attractive singular potentials 1 { r α for α ě
2. These effective field theoriescould be used to investigate the Efimov effect and other low-energy phenomena in a model-independent way. 114 hapter Impurity Lattice Monte Carlo and theAdiabatic Projection Method
The adiabatic projection method is a general framework for calculating scattering and reactionson the lattice. The method constructs a low-energy effective theory for clusters which becomesexact in the limit of large Euclidean projection time. Previous studies of this method [154, 143]have used exact sparse matrix methods. In this work we demonstrate the first applicationusing Monte Carlo simulations. As we will show, the adiabatic projection method significantlyimproves the accurate calculation of finite-volume energy levels. As we also will show, thefinite-volume energy levels must be calculated with considerable accuracy in order to determinethe scattering phase shifts using Lüscher’s method. We give a short summary of Lüscher’smethod later in our discussion.The goal of this analysis is to benchmark the use of lattice Monte Carlo simulations withthe adiabatic projection method. The example we consider in detail is fermion-dimer scattering115or two-component fermions and zero-range interactions. Our calculation also corresponds toneutron-deuteron scattering in the spin-quartet channel at leading order in pionless effectivefield theory. In our interacting system there are two components for the fermions. We call thetwo components up and down spins, Ò and Ó . The bound dimer state is composed of one Ò andone Ó , and our fermion-dimer system consists of two Ò and one Ó . While s -wave scattering hasbeen considered previously [28, 26, 148, 143], we will present the first lattice calculations of p -wave and d -wave fermion-dimer scattering.As discussed in Ref. [143], the adiabatic projection method starts with a set of initial clusterstates. By clusters we mean either a single particle or a bound state of several particles. Inour analysis here we consider fermion-dimer elastic scattering where there are two clusters. InRef. [143], the initial fermion-dimer states were parameterized by the initial spatial separationbetween clusters, (cid:126) R . The initial cluster states can be written explicitly as | (cid:126) R y “ ÿ (cid:126) n b :Ò p (cid:126) n q b :Ó p (cid:126) n q b :Ò p (cid:126) n ` (cid:126) R q | y , (6.1)where the spatial volume is a periodic cubic box of length L in lattice units. The initial states arethen projected using Euclidean time to form dressed cluster states, | (cid:126) R y t “ e ´ ˆ Ht | (cid:126) R y . (6.2)The adiabatic method uses these dressed cluster states to calculate matrix elements of theHamiltonian and other observables. The result is a low-energy effective theory of interactingclusters which becomes systematically more accurate as the projection time t is increased. Anestimate of the residual error is derived in Ref. [143].For our calculations here we follow the same general process except that we build the initial116luster states in a different manner. Instead of working with the relative separation betweenclusters, we work with the relative momentum between the clusters. We find that this changeimproves the efficiency of the Monte Carlo calculation by reducing the number of requiredinitial states. The new technique involves first constructing a dimer state with momentum (cid:126) p using Euclidean time projection and then multiplying by a creation operator for a second Ò particle with momentum ´ (cid:126) p . For example, we can write the initial fermion-dimer state explicitlyas | (cid:126) p y “ ˜ b :Ò p´ (cid:126) p q e ´ ˆ Ht ˜ b :Ò p (cid:126) p q ˜ b :Ó p (cid:126) q | y . (6.3)From these states we produce dressed cluster states by Euclidean time projection, | (cid:126) p y t “ e ´ ˆ Ht { | (cid:126) p y . (6.4)We then proceed in the same manner as in Ref. [143] and calculate the matrix elements of theHamiltonian in the basis of the dressed cluster states.For our Monte Carlo simulations we introduce a new algorithm which we call the impuritylattice Monte Carlo algorithm. Credit for developing this algorithm is to be shared with Ref. [27],where applications to impurities in many-body systems are being investigated using the samemethod. It can be viewed as a hybrid algorithm in between worldline and auxiliary-field MonteCarlo simulations. In worldline algorithms, the quantum amplitude is calculated by samplingparticle worldlines in Euclidean spacetime. In auxiliary-field Monte Carlo simulations, theinteractions are recast as single particle interactions, and the quantum amplitude is computedexactly for each auxiliary field configuration. In impurity Monte Carlo, we handle the impuritiesusing worldline Monte Carlo simulations while all other particles are treated using the auxiliary-field formalism. Furthermore, the impurity worldlines themselves are acting as additionalauxiliary fields felt by other particles in the system. We have found that for our system of two Ò Ó particles, impurity lattice Monte Carlo method is computationally superior to othermethods such as the auxiliary-field Monte Carlo due to its speed and efficiency as well as controlover sign oscillations. We will derive the formalism of impurity Monte Carlo simulations indetail in our discussion here.The organization of this chapter is as follows. We first start with the basic continuum andlattice formulations of our interacting system with zero-range two-component fermions. Wethen take a short detour to derive the connection between normal-ordered transfer matrices andlattice Grassmann actions. Using our dictionary between lattice Grassmann actions and quantumoperators, we derive the transfer matrix induced by a given single impurity worldline. We thendescribe the implementation of the adiabatic projection method and the details of our MonteCarlo simulations for computing finite-volume energy levels.In order to determine scattering phase shifts, we then discuss Lüscher’s finite-volume method.As part of this discussion we discuss for the first time, the character of topological volumecorrections for fermion-dimer scattering in the p -wave and d -wave channels. By topologicalvolume corrections, we are specifically referring to momentum-dependent finite-volume cor-rections of the dimer binding energy [28, 45]. Previous studies looking at topological volumecorrections had only considered s -wave scattering [28, 26, 148, 143]. The extension to higherpartial waves is given in the appendix. We then conclude with a comparison of Monte Carloresults as well as exact lattice calculations and continuum calculations. We consider a three-body system of two-component fermions with equal mass, m Ò “ m Ó “ m .We consider the limit of large scattering length between the two components where the theinteraction range of the fermions is taken to be negligible. We start with the free non-relativistic118amiltonian, ˆ H “ m ÿ s “Ò , Ó ż d (cid:126) r (cid:126) ∇ b : s p (cid:126) r q ¨ (cid:126) ∇ b s p (cid:126) r q , (6.5)In the low-energy limit the interaction can be simplified as a delta-function interaction betweenthe two spin components,ˆ H “ m ÿ s “Ò , Ó ż d (cid:126) r (cid:126) ∇ b : s p (cid:126) r q ¨ (cid:126) ∇ b s p (cid:126) r q ` C ż d (cid:126) r ˆ ρ Ò p (cid:126) r q ˆ ρ Ó p (cid:126) r q , (6.6)where ˆ ρ Ò , Ó p (cid:126) r q are density operators, ˆ ρ Ò p (cid:126) r q “ b :Ò p (cid:126) r q b Ò p (cid:126) r q , (6.7)ˆ ρ Ó p (cid:126) r q “ b :Ó p (cid:126) r q b Ó p (cid:126) r q . (6.8)The ultraviolet physics of this zero-range interaction must be regulated in some manner. In ourcase the lattice provides the needed regularization. We denote the spatial lattice spacing as a andthe temporal lattice spacing as a t . We will write all quantities in lattice units, which are physicalunits multiplied by the corresponding power of a to render the combination dimensionless. Weuse the free non-relativistic lattice Hamiltonian defined in Eqs. (3.42) and (3.43) and the contactinteraction potential is ˆ V “ C ÿ (cid:126) n ˆ ρ Ò p (cid:126) n q ˆ ρ Ó p (cid:126) n q . (6.9)Here ˆ l denotes a lattice unit vector in one of the spatial directions, ˆ l “ ˆ1 , ˆ2 , ˆ3. The unknowninteraction coefficient C is tuned to reproduce the desired binding energy of the dimer at infinitevolume. 119 .3 Lattice path integrals and transfer matrices For our Monte Carlo simulations and exact lattice calculations we use the transfer matrixformalism introduced in Chapter 3. It is convenient to collect some of the important formulasgiven in Chapter 3. As it is already discussed, the Grassmann path integral has the form Z “ ż »– ź n t ,(cid:126) n , s “Ò , Ó d θ ˚ s p n t ,(cid:126) n q d θ s p n t ,(cid:126) n q fifl e ´ S r θ , θ ˚ s , (6.10)where S r θ , θ ˚ s is the non-relativistic lattice action and defined in Section 3.4, and θ ˚ s and θ s areanti-commuting Grassmann variables. Our lattice action can be decomposed into three parts.While the Grassmann formalism is convenient for deriving the lattice Feynman rules, thetransfer matrix formalism is more convenient for numerical calculations.Therefore, we make theconnection between the two formulations, and we use the following exact relation between theGrassmann path integral formula and the transfer matrix formalism [43, 44]. For any function f ,Tr ” : f L t ´ r a : s p (cid:126) n q , a s p (cid:126) n qs : ¨ ¨ ¨ : f r a : s p (cid:126) n q , a s p (cid:126) n qs : ı “ ż »– ź n t ,(cid:126) n , s “Ò , Ó d θ ˚ s p n t ,(cid:126) n q d θ s p n t ,(cid:126) n q fifl e ´ ř nt S t r θ , θ ˚ , n t s L t ´ ź n t “ f n t r θ ˚ s p n t ,(cid:126) n q , θ s p n t ,(cid:126) n qs , (6.11)where the symbol : : signifies normal ordering. Normal ordering rearranges all operators so thatall annihilation operators are moved to the right and creation operators are moved to the leftwith the appropriate number of anticommutation minus signs. Then the desired transfer matrixformulation of the path integral is Z “ Tr “ ˆM L t ‰ , (6.12)120here ˆ M is the normal-ordered transfer matrix operator,ˆ M “ : exp « ´ α t ˆ H ´ α t C ÿ (cid:126) n ˆ ρ Ò p (cid:126) n q ˆ ρ Ó p (cid:126) n q ff : . (6.13)Here ˆ H is the free lattice Hamiltonian given in Eq. (3.42). In this section we derive the formalism for impurity lattice Monte Carlo for a single impurity. Inimpurity Monte Carlo the impurities are treated differently from other particles. The assumptionis that there are only a small number of impurities and these can be sampled using worldlineMonte Carlo without strong fermion sign oscillation problems from antisymmetrization. In ourcase there is exactly one Ó particle, and we treat this as a single impurity for our system.Let us consider the occupation number basis, ˇˇˇ χ Ò n t , χ Ó n t E “ ź (cid:126) n b :Ò p (cid:126) n q ı χ Ò nt p (cid:126) n q ” b :Ó p (cid:126) n q ı χ Ó nt p (cid:126) n q + | y (6.14)where χ sn t p (cid:126) n q counts the occupation number on each lattice site at time step n t and has valueswhich are either 0 or 1. Let us define the Grassmann functions, X p n t q “ ź (cid:126) n ” e θ ˚Ò p n t ,(cid:126) n q θ Ò p n t ,(cid:126) n q e θ ˚Ó p n t ,(cid:126) n q θ Ó p n t ,(cid:126) n q ı , (6.15)and M p n t q “ e ´ S H r θ , θ ˚ , n t s e ´ S V r θ , θ ˚ , n t s . (6.16)The transfer matrix element between time steps n t and n t ` x χ Ò n t ` , χ Ó n t ` | ˆ M | χ Ò n t , χ Ó n t y“ ź (cid:126) n $&%« ÝÑBB θ ˚Ó p n t ,(cid:126) n q ff χ Ó nt ` p (cid:126) n q « ÝÑBB θ ˚Ò p n t ,(cid:126) n q ff χ Ò nt ` p (cid:126) n q ,.- X p n t q M p n t qˆ ź (cid:126) n $&%« ÐÝBB θ Ò p n t ,(cid:126) n q ff χ Ò nt p (cid:126) n q « ÐÝBB θ Ó p n t ,(cid:126) n q ff χ Ó nt p (cid:126) n q ,.-ˇˇˇˇˇˇ θ “ θ “ θ Ò “ θ Ó “ . (6.17)This result can be verified by checking the different possible combinations for the occupationnumbers. Since we have only one Ó particle, the right hand side is nonzero only if ÿ (cid:126) n χ Ó n t p (cid:126) n q “ ÿ (cid:126) n χ Ó n t ` p (cid:126) n q “ . (6.18)We now derive the transfer matrix formalism for one spin- Ó particle worldline in a mediumconsisting of an arbitrary number of spin- Ò particles. The impurity worldline is to be consideredfixed. To provide a simple visual representation of the worldine, we draw in Figure 6.1 anexample of a single-particle worldline configuration on a 1+1 dimensional Euclidean lattice.We now remove or “integrate out" the impurity particle from the lattice action. We considerfirst the case when the Ó particle hops from (cid:126) n to some nearest neighbor site. In other words, χ Ó n t p (cid:126) n q “ χ Ó n t ` p (cid:126) n ˘ ˆ l q “ l . In this case we have A χ Ò n t ` , χ Ó n t ` ˇˇˇ ˆ M ˇˇˇ χ Ò n t , χ Ó n t E “ ź (cid:126) n $&%« ÝÑBB θ ˚Ò p n t ,(cid:126) n q ff χ Ò nt ` p (cid:126) n q ,.- { X p n t q { M (cid:126) n ˘ ˆ l ,(cid:126) n p n t q ź (cid:126) n $&%« ÐÝBB θ Ò p n t ,(cid:126) n q ff χ Ò nt p (cid:126) n q ,.-ˇˇˇˇˇˇ θ “ θ Ò “ , (6.19)122 igure 6.1 A segment of a worldline configuration on a 1+1 dimensional Euclidean lattice. See themain text for derivations of the reduced transfer-matrix operators. where { X p n t q “ ź (cid:126) n ” e θ ˚Ò p n t ,(cid:126) n q θ Ò p n t ,(cid:126) n q ı , (6.20)and { M (cid:126) n ˘ ˆ l ,(cid:126) n p n t q “ ´ α t m ¯ exp ! ´ α t H Ò r θ s , θ ˚ s , n t s ) . (6.21)Next we consider the case when χ Ó n t p (cid:126) n q “ χ Ó n t ` p (cid:126) n q “ A χ Ò n t ` , χ Ó n t ` ˇˇˇ ˆ M ˇˇˇ χ Ò n t , χ Ó n t E “ ź (cid:126) n $&%« ÝÑBB θ ˚Ò p n t ,(cid:126) n q ff χ Ò n t p (cid:126) n q ,.- { X p n t q { M (cid:126) n ,(cid:126) n p n t q ź (cid:126) n $&%« ÐÝBB θ Ò p n t ,(cid:126) n q ff χ Ò nt p (cid:126) n q ,.-ˇˇˇˇˇˇ θ “ θ Ò “ , (6.22)where { M (cid:126) n ,(cid:126) n p n t q “ ˆ ´ α t m ˙ exp ´ α t H Ò r θ s , θ ˚ s , n t s ´ α t C ´ α t m θ ˚Ò p n t ,(cid:126) n q θ Ò p n t ,(cid:126) n q + . (6.23)From these Grassmann lattice actions with the impurity integrated out, we can write downthe corresponding transfer matrix operators. When the impurity makes a spatial hop, the reducedtransfer-matrix operator is ˆ { M (cid:126) n ˘ ˆ l ,(cid:126) n “ ´ α t m ¯ : exp ” ´ α t ˆ H Ò ı : . (6.24)When the impurity worldline remains stationary the reduced transfer-matrix operator isˆ { M (cid:126) n ,(cid:126) n “ ˆ ´ α t m ˙ : exp « ´ α t ˆ H Ò ´ α t C ´ α t m ρ Ò p (cid:126) n q ff : . (6.25)We note that these reduced transfer matrices are just one-body operators on the linear space of Ò particles. 124 .5 Adiabatic Projection Method In this section we describe our application of the adiabatic projection method using a set ofcluster states constructed in momentum space. As already described in Eq. (6.3) in a simplifiednotation, we let | Ψ (cid:126) p y be the fermion-dimer initial state with relative momentum (cid:126) p , | Ψ (cid:126) p y “ ˜ b :Ò p´ (cid:126) p q ˆ M L t ˜ b :Ò p (cid:126) p q ˜ b :Ó p (cid:126) q | y , (6.26)where we use the transfer matrix operator ˆ M given in Eq. (6.13) for some number of time steps L t . The purpose of this time propagation is to allow the dimer to bind its constituents beforeinjecting an additional Ò particle. In this part of the calculation we in fact increase the attractiveinteractions between the two spins to allow them to form the bound dimer faster. We find thatthis trick increases the computational efficiency on large lattice systems. The dressed clusterstates are defined as | Ψ (cid:126) p y L t { “ ˆ M L t { | Ψ (cid:126) p y , (6.27)for some even number L t , and the overlap between dressed cluster states is Z (cid:126) p (cid:126) p p L t q “ x Ψ (cid:126) p | ˆ M L t | Ψ (cid:126) p y . (6.28)For large L t we can obtain an accurate representation of the low-energy spectrum of ˆ M bydefining the adiabatic transfer matrix as r ˆ M a p L t qs (cid:126) p (cid:126) p “ ÿ (cid:126) p Z ´ (cid:126) p (cid:126) p p L t q Z (cid:126) p (cid:126) p p L t ` q . (6.29)125lternatively we can also construct a symmetric version of the adiabatic transfer matrix as r ˆ M a p L t qs (cid:126) p (cid:126) p “ ÿ (cid:126) p ,(cid:126) p Z ´ { (cid:126) p (cid:126) p p L t q Z (cid:126) p (cid:126) p p L t ` q Z ´ { (cid:126) p (cid:126) p p L t q . (6.30)Either form will produce exactly the same spectrum. As with any transfer matrix, we interpretthe eigenvalues λ i p L t q of the adiabatic transfer matrix as energies using the relations e ´ E i p L t q α t “ λ i p L t q , E i p L t q “ ´ α t ´ log λ i p L t q . (6.31)The exact low-energy eigenvalues of the full transfer matrix ˆ M will be recovered in the limit L t Ñ 8 .As a special case, one can simply restrict the adiabatic projection calculation to a singleinitial momentum state, for example, (cid:126) p “
0. In that case the adiabatic transfer matrix is just thescalar ratio Z (cid:126) p (cid:126) p p L t ` q{ Z (cid:126) p (cid:126) p p L t q . (6.32)However we find that the energy calculations are significantly more accurate and converge muchfaster with increasing L t when using a set of several initial cluster states. The reduced transfer matrices ˆ { M (cid:126) n ,(cid:126) n in Eq. (6.24) and (6.25) are one-body operators on the linearspace of Ò particles. Therefore we can simply multiply the reduced transfer matrices together. Itis perhaps worthwhile to note that the (cid:126) n ,(cid:126) n subscripts are not the matrix indices of the reducedtransfer matrix, but rather the coordinates of the Ó particle that was integrated out. The matrixindices of ˆ { M (cid:126) n ,(cid:126) n are being left implicit. 126he Euclidean time projection can be written as a sum over worldline configurations of the Ó particle. As a convenient shorthand we writeˆ { M r L t st (cid:126) n j u “ ˆ { M (cid:126) n Lt ,(cid:126) n Lt ´ . . . ˆ { M (cid:126) n ,(cid:126) n , (6.33)where (cid:126) n j denotes the spatial position of the spin- Ó particle at time step j . The projectionamplitude for cluster states | Ψ (cid:126) p y and | Ψ (cid:126) p y is then Z (cid:126) p (cid:126) p p L t q “ ÿ (cid:126) n ,...,(cid:126) n Lt x Ψ (cid:126) p | ˆ { M r L t st (cid:126) n j u | Ψ (cid:126) p y . (6.34)The states | Ψ (cid:126) p y and | Ψ (cid:126) p y defined in Eq. (6.26) are constructed using single particle creationoperators, and so the amplitude Z (cid:126) p (cid:126) p p L t q is just the determinant of a 2 ˆ L t projection steps in between someof the creation operators. This gives us the following structure, Z (cid:126) p (cid:126) p p L t q “ ÿ (cid:126) n ,...,(cid:126) n Lt ÿ (cid:126) n ,...,(cid:126) n L t ÿ (cid:126) n ,...,(cid:126) n L t det M ˆ , (6.35)where (cid:126) n “ (cid:126) n L t , (cid:126) n L t “ (cid:126) n , and M ˆ “ »—– x (cid:126) p | ˆ { M r L t st (cid:126) n j u ˆ { M r L t st (cid:126) n j u ˆ { M r L t st (cid:126) n j u | (cid:126) p y x (cid:126) p | ˆ { M r L t st (cid:126) n j u ˆ { M r L t st (cid:126) n j u | ´ (cid:126) p yx´ (cid:126) p | ˆ { M r L t st (cid:126) n j u ˆ { M r L t st (cid:126) n j u | (cid:126) p y x´ (cid:126) p | ˆ { M r L t st (cid:126) n j u | ´ (cid:126) p y fiffifl . (6.36)The calculation of Z (cid:126) p (cid:126) p p L t q has now been recast as a problem of computing the determinant ofthe matrix M ˆ over all possible impurity worldlines. We use a Markov chain Monte Carloprocess to select worldline configurations. The Metropolis algorithm is used to accept or rejectconfigurations with importance sampling given by the weight function | Z (cid:126) p (cid:126) p p L t q | , where (cid:126) p is one127f the initial momenta.We now benchmark our results for the low-energy spectrum calculated using adiabaticprojection and the impurity Monte Carlo method. We compare with exact lattice results computedusing the Lanczos iterative eigenvector method with a space of „ L basis states. Althoughexact lattice results provide a useful benchmark test for the three-particle system, the extensionto larger systems is computationally not viable due to exponential scaling in memory and CPUtime. In contrast, the impurity Monte Carlo calculation does scale well to much larger systems.In fact, many-body impurity systems are currently being studied in Ref. [27]. Table 6.1
Momentum of the dimer, (cid:126) p d , with p “ π { L . The total momentum of the system is zero. n (cid:126) p d x p , , y x , p , y x , , p y x p , ´ p , y x p , , ´ p y x , p , ´ p y In our lattice calculations we take the particle mass to be the average nucleon mass, 938 . C is tuned to obtain the deuteron energy, ´ . L periodic cubic volume with spatial lattice spacing a “ .
97 fm. The values of L usedwill be specified later. In the temporal direction we use L t time steps with a temporal latticespacing a t “ .
31 fm/ c .Let N be the number of initial/final states. We choose the initial dimer momenta, (cid:126) p d , asshown in Table 6.1. In all cases the total momentum of the three-particle system is set to zero. Welabel and order the various possible dimer momenta with index n “ , ¨ ¨ ¨ , N . We then constructthe corresponding N ˆ N adiabatic matrix, r ˆ M a p L t qs nn , and obtain the N low-lying energy states128f the finite-volume system. There is no restriction on the choice of N . Therefore, so long as thenumerical stability of the matrix calculations is under control, it is advantageous to maximizethe number N . While constructing a large adiabatic matrix requires more computational time, itsignificantly accelerates the convergence with the number of projection time steps, L t . −1.3−1.2−1.1−1−0.9−0.8 0 0.2 0.4 0.6 0.8 1 E [ M e V ] t [MeV −1 ] E +0 [ M^ a (t)] −1.3−1.2−1.1−1−0.9−0.8 0 0.2 0.4 0.6 0.8 1 E [ M e V ] t [MeV −1 ] E +0 [ M^ a (t)] −1.3−1.2−1.1−1−0.9−0.8 0 0.2 0.4 0.6 0.8 1 E [ M e V ] t [MeV −1 ] E +0 −1.3−1.2−1.1−1−0.9−0.8 0 0.2 0.4 0.6 0.8 1 E [ M e V ] t [MeV −1 ] E +0 (a) E [ M e V ] ’ t [MeV -1 ] E +2 E [ M e V ] ’ t [MeV -1 ] E +1 E [ M e V ] ’ t [MeV -1 ] 3 3.5 4 4.5 5 5.5 6 6.5 0 0.2 0.4 0.6 0.8 1 E [ M e V ] ’ t [MeV -1 ] 3 3.5 4 4.5 5 5.5 6 6.5 0 0.2 0.4 0.6 0.8 1 E [ M e V ] ’ t [MeV -1 ] (b) Figure 6.2
The ground state energy is shown versus projection time t using either one or four ini-tial/final states in Panel (a) and the first two excited state energies with even parity in Panel (b). Forcomparison we show the exact lattice energies as dotted horizontal lines. In Fig. 6.2(a) we compare the ground state energies using r ˆ M a p t qs ˆ and r ˆ M a p t qs ˆ adiabatic matrices. We are plotting the energies versus projection time t “ L t a t . The resultsshown are obtained using a lattice box of length L a “ .
79 fm, while the number of time stepsis varied over a range of values to extrapolate to the limit t Ñ 8 . We use a simple exponential129nsatz to extrapolate away the residual contribution from higher-energy states, E i p t q “ E i p8q ` c i e ´ ∆ E i t ` ¨ ¨ ¨ . (6.37)As can be seen clearly in the figure, the r ˆ M a p t qs ˆ results converge with a significantly fasterexponential decay than the r ˆ M a p t qs ˆ results. This is consistent with the derivation in Ref. [143]that the energy gap ∆ E i in Eq. (6.37) is increased by including more initial states. The cor-responding extrapolated ground state energies obtained from the r ˆ M a p t qs ˆ and r ˆ M a p t qs ˆ adiabatic matrices are ´ . p q MeV and ´ . p q MeV, respectively.
Table 6.2
The exact and Monte Carlo results for the ground state and lowest lying even-parity energiesin a periodic box of length La “ .
79 fm. The Monte Carlo results are obtained from the r ˆ M a p t qs ˆ adiabatic matrix. E ` [MeV] E ` [MeV] E ` [MeV]Exact ´ . . . ´ . p q . p q . p q In Figs. 6.2(a) and (b) we plot the lowest lying even-parity energies as a function of Euclideanprojection time t . To be able to calculate the two excited states in Fig. 6.2(b) we use seveninital/final states and construct a 7 ˆ ˆ ˆ E [ M e V ] t [MeV -1 ] E -1 E [ M e V ] t [MeV -1 ] E -0 E [ M e V ] t [MeV -1 ] 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 E [ M e V ] t [MeV -1 ] 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 E [ M e V ] t [MeV -1 ] Figure 6.3
The lowest two odd parity energies as a function of Euclidean projection time t . For com-parison we show the exact lattice energies as dotted horizontal lines. In Fig. 6.3 we present the energies for the lowest two states with odd parity. In order tocalculate these odd parity energies we use five initial/final states and construct a 5 ˆ In this section we present lattice results for the fermion-dimer elastic scattering phase shiftsfor angular momentum up to (cid:96) “ able 6.3 The exact and Monte Carlo results for the energies of the lowest two odd-parity states ina periodic box of length La “ .
79 fm. The Monte Carlo results are obtained from the r ˆ M a p t qs ˆ adiabatic matrix. E ´ [MeV] E ´ [MeV]Exact 2 .
509 7 . . p q . p q for explaining the finite-volume calculations, we first briefly review Lüscher’s method for the s -wave scattering of two particles in Section 3.5.2. In the following, we extend the previousdiscussion to higher partial waves (cid:96) ď
2. Since the phase shifts depend crucially on an accuratecalculation and analysis of finite-volume energy levels, we also discuss in the appendix somecorrections which are due to modifications of the dimer binding energy at finite volume.
Lüscher [122, 124] is a well-known technique for extracting elastic phase shifts for two-bodyscattering from the volume dependence of two-body continuum states in a cubic periodic box.Lüscher’s relation between scattering phase shifts and two-body energy levels in a cubic periodicbox has the following forms [122, 124, 125] p (cid:96) ` cot δ (cid:96) p p q “ $’’’’’’’&’’’’’’’% ? π L Z , p η q for (cid:96) “ , ˆ π L ˙ ηπ { Z , p η q for (cid:96) “ , ˆ π L ˙ π { „ η Z , p η q ` Z , p η q for (cid:96) “ . (6.38)132here η “ ˆ Lp π ˙ . (6.39)Here Z (cid:96), m p η q are the generalized zeta functions [122, 124], Z (cid:96), m p η q “ ÿ (cid:126) n | (cid:126) n | (cid:96) Y (cid:96), m p ˆ n q| (cid:126) n | ´ η , (6.40)and Y (cid:96), m p ˆ n q are the spherical harmonics. We can evaluate the zeta functions using exponentially-accelerated expressions [125]. For (cid:96), m “ Z , p η q “ π e η p η ´ q ` e η ? π ÿ (cid:126) n e ´| (cid:126) n | | (cid:126) n | ´ η ´ π ż d λ e λ η λ { ˜ λ η ´ ÿ (cid:126) n e ´ π | (cid:126) n | { λ ¸ , (6.41)and for arbitrary (cid:96) and m , Z (cid:96), m p η q “ ÿ (cid:126) n | (cid:126) n | (cid:96) Y (cid:96), m p ˆ n q| (cid:126) n | ´ η e ´ Λ p| (cid:126) n | ´ η q ` ż Λ d λ ´ πλ ¯ (cid:96) ` { e λ η ÿ (cid:126) n | (cid:126) n | (cid:96) Y (cid:96), m p ˆ n q| (cid:126) n | ´ η e ´ π | (cid:126) n | { λ . (6.42) We now use our lattice results for the finite-volume energies and use Eq. (6.38) to determinethe elastic phase shifts. We compute phase shifts using data from the impurity Monte Carlocalculations as well as the exact lattice energies using the Lanczos method. The fermion-dimersystem that we are considering corresponds exactly to neutron-deuteron scattering in the spin-133 δ [ D e g r ee ] p [MeV] Quartet- s Lattice MC -80-70-60-50-40-30-20-10 0 0 10 20 30 40 50 60 70 80 δ [ D e g r ee ] p [MeV] Quartet- s Lattice
Exact -80-70-60-50-40-30-20-10 0 0 10 20 30 40 50 60 70 80 δ [ D e g r ee ] p [MeV] Quartet- s -80-70-60-50-40-30-20-10 0 0 10 20 30 40 50 60 70 80 δ [ D e g r ee ] p [MeV] Quartet- s STM LO -80-70-60-50-40-30-20-10 0 0 10 20 30 40 50 60 70 80 δ [ D e g r ee ] p [MeV] Quartet- s Breakup
Figure 6.4
The s -wave scattering phase shift versus the relative momentum between fermion anddimer. δ [ D e g r ee ] p [MeV] Quartet- p Lattice MC δ [ D e g r ee ] p [MeV] Quartet- p Lattice
Exact δ [ D e g r ee ] p [MeV] Quartet- p δ [ D e g r ee ] p [MeV] Quartet- p STM LO δ [ D e g r ee ] p [MeV] Quartet- p Breakup
Figure 6.5
The p -wave scattering phase shift versus the relative momentum between fermion anddimer. δ [ D e g r ee ] p [MeV]-8-7-6-5-4-3-2-1 0 0 20 40 60 80 100 120 140 δ [ D e g r ee ] p [MeV] Quartet- d Lattice MC -8-7-6-5-4-3-2-1 0 0 20 40 60 80 100 120 140 δ [ D e g r ee ] p [MeV] Quartet- d Lattice
Exact -8-7-6-5-4-3-2-1 0 0 20 40 60 80 100 120 140 δ [ D e g r ee ] p [MeV] Quartet- d STM LO -8-7-6-5-4-3-2-1 0 0 20 40 60 80 100 120 140 δ [ D e g r ee ] p [MeV] Quartet- d Breakup
Figure 6.6
The d -wave scattering phase shift versus the relative momentum between fermion anddimer. ´ . T -matrix is T (cid:96) p k , p q “ ´ πγ mpk Q (cid:96) ˆ p ` k ´ mE ´ i ` pk ˙ ´ π ż dq qp T (cid:96) p k , q q a q { ´ mE ´ i ` ´ γ Q (cid:96) ˆ p ` q ´ mE ´ i ` pq ˙ , (6.43)where γ is the dimer binding energy, E “ p {p m q ´ γ { m is the total energy, and Q (cid:96) is theLegendre function of the second kind, Q (cid:96) p a q “ ż ´ dx P (cid:96) p x q x ` a . (6.44)The scattering phase shifts can be calculated from the on-shell T -matrix formula, T (cid:96) p p , p q “ π m p (cid:96) p (cid:96) ` cot δ (cid:96) ´ ip (cid:96) ` . (6.45)We show results for the s -wave, p -wave and d -wave phase shifts in Fig. 6.4, 6.5 and 6.6respectively. The square points indicate the data from the lattice Monte Carlo simulations, thecircular points are the exact lattice calculations, and the solid lines are a fit of the exact latticedata using an effective range expansion, p (cid:96) ` cot δ (cid:96) p p q “ ´ a (cid:96) ` r (cid:96) p ` O p p q . (6.46)The dashed lines are leading order results from the STM calculation. The dotted vertical lines137ndicate the inelastic breakup threshold of the dimer. The range of lattice box sizes is L ď
16 forthe exact lattice and L ď p -and d -wave channels. In this chapter we have presented the adiabatic projection method and its first application usingMonte Carlo methods. The adiabatic method is a general framework for studying scatteringand reactions on the lattice. The method constructs a low-energy effective theory for clusters,and in the limit of large Euclidean projection time the description becomes exact. In previousstudies [154, 143] the initial cluster states were parameterized by the initial spatial separationsbetween clusters. In this study we have used a new technique which parameterizes the cluster138tates according to the relative momentum between clusters. This new approach is crucial fordoing calculations with a small number of initial states in order to improve the efficiency of theMonte Carlo calculations. The system we have analyzed in detail here is fermion-dimer elasticscattering for two-component fermions interacting via zero-range attractive interactions.For our calculations we have introduced a new Monte Carlo algorithm which we callimpurity lattice Monte Carlo. This can be seen as a hybrid algorithm in between worldline andauxiliary-field Monte Carlo simulations. In impurity Monte Carlo we use worldline Monte Carlofor the impurities, and these impurity worldlines are acting as additional auxiliary fields in thesimulation of the other particles. By using the impurity lattice Monte Carlo algorithm, we havefound significant improvement over more standard auxiliary-field Monte Carlo calculations.In addition to greater speed and efficiency of the calculations, we also found a reduction offermonic sign oscillations, and this has greatly improved the resulting accuracy.We have found that the adiabatic projection method with impurity Monte Carlo enableshighly accurate calculations of the finite-volume energy levels of the fermion-dimer system.From these energy levels we have used Lüscher’s method to present the first lattice calculationsof p -wave and d -wave phase shifts for fermion-dimer elastic scattering. In addition to findingexcellent agreement between Monte Carlo and exact lattice phase shifts, we have also foundgood agreement with continuum STM caculations of neutron-deuteron elastic scattering in thespin-quartet channel at leading order in pionless effective field theory.Our results show that the adiabatic projection method with Monte Carlo simulations is a vi-able approach to calculating elastic phase shifts. The method can be applied in a straightforwardmanner to other two-cluster scattering systems. One area where more work is needed is that ourapplication of Lüscher’s method does not account for inelastic breakup processes. Another areathat needs improvement is that Lüscher’s method has too much sensitivity to small changes inthe finite-volume energy levels. For these reasons we are now working to develop new methods139hich incorporates more information from the adiabatic projection wavefunction in order toextract scattering information in a more robust manner.140 IBLIOGRAPHY [1] S. Aoki, M. Fukugita, S. Hashimoto, K-I. Ishikawa, N. Ishizuka, Y. Iwasaki, K. Kanaya,T. Kaneko, Y. Kuramashi, V. Lesk, M. Okawa, Y. Taniguchi, A. Ukawa, and T. Yoshié. i “ Phys. Rev. D , 67:014502, Jan2003.[2] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. Hecker Denschlag, andR. Grimm. Crossover from a molecular bose-einstein condensate to degenerate fermi gas.
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PPENDICES ppendix A A.1 Bessel and related Functions S (cid:96) p r q and C (cid:96) p r q are Riccati-Bessel and Riccati-Neumann functions, which are defined in termsof the Bessel and Neumann functions as S (cid:96) p x q “ c π x J (cid:96) ` p x q “ ? π ´ x ¯ (cid:96) ` ÿ n “ i n Γ p n ` q Γ p n ` (cid:96) ` q ´ x ¯ n , (A.1) C (cid:96) p x q “ ´ c π r N (cid:96) ` p x q“ ? π ´ x ¯ ´ (cid:96) Γ p´ (cid:96) ` q Γ p (cid:96) ` q ÿ n “ i n Γ p n ` q Γ p n ´ (cid:96) ` { q ´ x ¯ n . (A.2)We define the following functions s (cid:96) p p , r q and c (cid:96) p p , r q in terms of Riccati-Bessel and Riccati-Neumann functions, s (cid:96) p p , r q “ p ´ (cid:96) ´ S (cid:96) p pr q “ ÿ n “ ? π i n p n Γ p n ` q Γ ` n ` (cid:96) ` ˘ ´ r ¯ n ` (cid:96) ` , (A.3)155 (cid:96) p p , r q “ p (cid:96) C (cid:96) p pr q “ ÿ n “ i n p n ? π Γ p´ (cid:96) ` q Γ p (cid:96) ` q Γ p n ` q Γ p n ´ (cid:96) ` { q ´ r ¯ n ´ (cid:96) . (A.4)The relations in Eqs. (A.3) and (A.4) with Eqs. (A.1) and (A.2) indicate that s (cid:96) p p , r q and c (cid:96) p p , r q can be written in powers of p , s (cid:96) p p , r q “ ÿ n “ s n ,(cid:96) p r q p n ` and c (cid:96) p p , r q “ ÿ n “ c n ,(cid:96) p r q p n ` , (A.5)where s n ,(cid:96) p r q and c n ,(cid:96) p r q are s n ,(cid:96) p r q “ ? π i n Γ p n ` q Γ ` n ` (cid:96) ` ˘ ´ r ¯ n ` (cid:96) ` , (A.6) c n ,(cid:96) p r q “ i n ? π Γ ` ´ (cid:96) ` ˘ Γ ` (cid:96) ` ˘ Γ p n ` q Γ ` n ´ (cid:96) ` ˘ ´ r ¯ n ´ (cid:96) . (A.7) A.2 Coulomb wave functions
We consider radial wave function V p p q (cid:96) p r q that satisfies the radial Schrödinger equation for r ą R , „ d dr ´ (cid:96) p (cid:96) ` q r ´ γ r ` p V p p q (cid:96) p r q “ , (A.8)where µ is the reduced mass, and γ “ µα Z Z . Defining r γ “ ´ γ r { ε “ p { γ Eq. (A.8)can be written as « d dr γ ´ (cid:96) p (cid:96) ` q r γ ` r γ ` ε ff χ “ , (A.9)156he solution to this differential equation has the form [158] of χ p ε , (cid:96) ; r γ q “ ÿ q “ ε q »– q ÿ p “ α q ´ p χ q p (cid:96), r γ q fifl , (A.10)as linearly independent functions χ q p (cid:96), r γ q , χ q p (cid:96), r γ q “ $’’’’’’’&’’’’’’’% q ÿ p “ q C q , p φ p p (cid:96), r γ q q ą , ´ p (cid:96) ` q φ ` φ q “ , φ p (cid:96), r γ q q “ , (A.11)where φ p p (cid:96), r γ q “ a P p p (cid:96), r γ q ` b Q p p (cid:96), r γ q , (A.12)and C q , p “ $’’’’’&’’’’’% ´p (cid:96) ` p q C q ´ , p ´ ` C q ´ , p ´ p for 2 q ď p ď q , q “ p “ , q ą p and p ą q . (A.13)The coefficient α p in Eq. (A.11) and the constants a and b in Eq. (A.12) are to be determineddepending on the choice of normalization of the function χ p ε , (cid:96) ; r γ q . The functions P p p (cid:96), r γ q and Q p p (cid:96), r γ q are defined in terms of Bessel and Modified Bessel functions P p p (cid:96), r γ q “ $’&’% p r γ q p p ` q{ J (cid:96) ` ` p p a r γ q for r γ ą , p´ q (cid:96) ` ` p p´ r γ q p p ` q{ I (cid:96) ` ` p p a ´ r γ q for r γ ă , (A.14)157nd Q p p (cid:96), r γ q “ $’&’% p r γ q p p ` q{ Y (cid:96) ` ` p p a r γ q for r γ ą , π p´ q (cid:96) ` ` p p´ r γ q p p ` q{ K (cid:96) ` ` p p a ´ r γ q for r γ ă . (A.15) A.2.1 Regular solution f p ε , (cid:96) ; r γ q Here the normalization of the Coulomb wave functions is chosen as the same as that of Ref. [21].From the functions introduced in the previous section, the regular Coulomb wave function iswritten as a convergent expansion in ε , f p ε , (cid:96) ; r γ q “ ÿ q “ ε q »– q ÿ p “ α q ´ p χ p f q p p (cid:96), r γ q fifl , (A.16)and this function is equivalent to f (cid:96) p p , r q in Eq. (2.56). Comparing to the regular function ofBollé and Gesztesy [21], we find a “ b “ α q as α q “ $’’’’’’&’’’’’’% p (cid:96) ` q ! p´ γ q (cid:96) ` for q “ r γ ą , p´ q ´ (cid:96) p (cid:96) ` q ! p´ γ q (cid:96) ` for q “ r γ ă , q ě . (A.17)We find the first few functions of the expansion of f (cid:96) p p , r q in power of p that we use inEq. (2.59)–(2.62) are f ˘ ,(cid:96) p r q “ p (cid:96) ` q ! p˘ γ q (cid:96) ` { ? r J p˘q (cid:96) ` p ?˘ γ r q , (A.18)158 ˘ ,(cid:96) p r q “ ´ p (cid:96) ` q !3 ? r p˘ γ q (cid:96) ` { ” p (cid:96) ` q J p˘q (cid:96) ` p ?˘ γ r q ˘ ?˘ γ r J p˘q (cid:96) ` p ?˘ γ r q ı , (A.19) f ˘ ,(cid:96) p r q “ p (cid:96) ` q !90 ? r p˘ γ q (cid:96) ` { ” p r γ ` (cid:96) ` (cid:96) ` q J p˘q (cid:96) ` p ?˘ γ r q˘p (cid:96) ` q?˘ γ r J p˘q (cid:96) ` p ?˘ γ r q ı , (A.20) f ˘ ,(cid:96) p r q “ ´ p (cid:96) ` q !5670 ? r p˘ γ q (cid:96) ` { ! “ (cid:96) ` (cid:96) ` (cid:96) p r γ ` q ` p r γ ` q ‰ J p˘q (cid:96) ` p ?˘ γ r q˘?˘ γ r ` (cid:96) ` (cid:96) ` r γ ` ˘ J p˘q (cid:96) ` p ?˘ γ r q ) , (A.21)where J p´q n p x q is the Bessel function of the first kind J n p x q and J p`q n p x q is the modifiedBessel fucntion of the first kind I n p x q . A.2.2 Irregular solution.I h p ε , (cid:96) ; r γ q The first irregular Coulomb wave function is obtained setting a “ b “ h p ε , (cid:96) ; r γ q “ A p ε , (cid:96) q »– M ÿ q “ ε q χ p h q q p (cid:96), r γ q ` O ` ε M ` ˘fifl , (A.22)159here χ p h q q p (cid:96), r γ q “ $’’’’’’’&’’’’’’’% q ÿ p “ q C q , p Q p p (cid:96), r γ q q ą , ´ p (cid:96) ` q Q ` Q q “ , Q p (cid:96), r γ q q “ , (A.23)and A p ε , (cid:96) q “ (cid:96) ź p “ p ` p ε q “ (cid:96) ÿ n “ ε n σ n ,(cid:96) . (A.24)However, this irregular solution h p ε , (cid:96) ; r γ q is not a desired function since it is not analytic in ε . A.2.3 Irregular solution.II g p ε , (cid:96) ; r γ q The second irregular Coulomb wave function which is analytic in ε is defined as a linearcombination of h p ε , (cid:96) ; r γ q and f p ε , (cid:96) ; r γ q , g p ε , (cid:96) ; r γ q “ ´ h p ε , (cid:96) ; r γ q ´ A p ε , (cid:96) q B p ε , q f p ε , (cid:96) ; r γ q , (A.25)where B p ε , (cid:96) q “ π ψ ˆ i ? ε ` (cid:96) ` ˙ ` π ψ ˆ i ? ε ´ (cid:96) ˙ ´ π log ˆ i ? ε ˙ ´ i exp ´ π ? ε ¯ ´ “ επ $&% (cid:96) ÿ p “ p ` p ε ` ˆ ` ε ` ε ` ε ` O p ε q ˙,.- . (A.27)160n Eq. (A.25) B p ε , (cid:96) “ q is set according to the normalization which gives ˜ g (cid:96) p p , r q “ g p ε , (cid:96) ; r γ q .Let us define A p ε , (cid:96) q B p ε , q “ N ÿ n “ ε n ω n ,(cid:96) ` O p ε N ` q , (A.28)then the convergent expression is written as g p ε , (cid:96) ; r γ q “ ÿ q “ ε q »– q ÿ p “ β q ´ p χ p g q p p (cid:96), r γ q fifl , (A.29)where χ p g q p p (cid:96), r γ q “ $’’’&’’’% σ ,(cid:96) χ p h q q p (cid:96), r γ q for q “ , min p q ,(cid:96) q ÿ m “ σ m ,(cid:96) χ p h q q ´ m p (cid:96), r γ q ´ q ÿ m “ ω m ,(cid:96) χ p f q q ´ m p (cid:96), r γ q for q ą , (A.30)and β i “ $’’’’’’’&’’’’’’’% ´ π p´ γ q (cid:96) p (cid:96) ` q ! for i “ r γ ą , ´ π p´ γ q (cid:96) p (cid:96) ` q ! for i “ r γ ă R “ ´ r γ , i ě . (A.31)In the following we give the first few functions of the expansion of ˜ g (cid:96) p p , r q in power of p that we use in Eq. (2.59)–(2.62, g p˘q ,(cid:96) p r q “ ˘ p˘ γ q (cid:96) ` { p (cid:96) ` q ! ? r N p˘q (cid:96) ` p ?˘ r γ q , (A.32)161 p˘q , p r q “ r N p˘q p ?˘ r γ q ¯ ? r p˘ γ q { J p˘q p ?˘ r γ q , (A.33) g p˘q , p r q “ r ” ˘ p r γ ´ q N p˘q p ?˘ r γ q¯ p r γ ´ q N p˘q p ?˘ r γ q?˘ r γ ¯ J p˘q p ?˘ r γ q?˘ r γ ı , (A.34) g p˘q , p r γ q “ ?˘ r γ ” ˘ ?˘ r γ p γ r ´ q N p˘q p ?˘ r γ q¯ p γ r ´ q N p˘q p ?˘ r γ q ¯ J p˘q p ?˘ r γ q ı , (A.35) g p˘q , p r q “ γ ! ¯ p´ γ r q N p˘q p ?˘ r γ q˘ p γ r ` qp˘ γ r q { N p˘q p ?˘ r γ q˘ γ r J p˘q p ?˘ r γ q ¯ ?˘ γ r J p˘q p ?˘ r γ q ) , (A.36) g p˘q , p r q “ r { p˘ γ q { ! ` γ r ´ ˘ J p˘q p ?˘ r γ qp˘ γ r q { ˘ p γ r p γ r ´ q ´ q J p˘q p ?˘ r γ qp γ r q ˘ r γ r p γ r ` q ´ s N p˘q p ?˘ r γ q¯ ?˘ γ r p γ r ´ q N p˘q p ?˘ r γ q ) , (A.37)162 p˘q , p r q “ r p˘ γ r q { ! ˘ p γ r ´ q J p˘q p ?˘ r γ q˘ ?˘ γ r p γ r p γ r ` q ´ q K p ? r γ q` ?˘ γ r p γ r ´ q J p˘q p ?˘ r γ q˘ p γ r ´ γ r ´ γ r ` q N p˘q p ?˘ r γ q ) , (A.38) g p˘q , p r γ q “ r γ ! p˘ γ r q p γ r ` q N p˘q p ? r γ q´ p γ r ` qp˘ γ r q { N p˘q p ?˘ r γ q´ ´ γ r ` γ r ` γ r ´ γ r ´ γ r ` γ r ` ¯ J p˘q ` ? r γ ˘ p˘ γ r q { ´ ´ γ r ` γ r ´ γ r ´ γ r ` γ r ` ¯ J p˘q p ?˘ r γ q γ r ) , (A.39)163 p˘q , p r γ q “ r γ ! ´ p˘ γ r q { J p˘q p ?˘ r γ q` ?˘ γ r r γ r p ´ γ r q ` s N p˘q p ? r γ q˘ γ r r γ r p γ r ` q ´ s N p˘q p ?˘ r γ q´ ´ γ r ´ γ r ´ γ r ` γ r ` γ r ¯ J p˘q p ?˘ r γ qp γ r q ´ p γ r ` γ r ´ γ r ´ γ r ` q J p˘q p ?˘ r γ qp˘ γ r q { ) , (A.40) g p˘q , p r γ q “ r γ ! ´ ` γ r ´ γ r ` ˘ J p˘q p ?˘ γ r q γ r ´ ´ γ r ` γ r ´ γ r ` γ r ` ¯ J p˘q p ?˘ γ r qp˘ γ r q { ` ?˘ γ r ´ ´ γ r ` γ r ` γ r ´ ¯ N p˘q p ?˘ γ r q˘ γ r ´ γ r ´ γ r ´ γ r ` ¯ N p˘q p ?˘ γ r q ) , (A.41)where N p´q n p x q stands for π { N n p x q and N p`q n p x q is the modified Bessel function of the second kind K n p x q .164 .3 van der Waals wave functions In this section we derive the van der Waals wave functions F (cid:96) and G (cid:96) , following the steps inRef. [76]. We first redefine the radial function as U (cid:96) p r q “ ? r s Z p ρ s q . This rearrangement putsEq. (5.5) into the form of an inhomogeneous Bessel equation, L ν Z p ρ s q “ „ ρ s d d ρ s ` ρ s dd ρ s ´ ν ` ρ s Z p ρ s q “ ´ p s Z p ρ s q ρ s , (A.42)with ν “ p (cid:96) ` q . The idea, introduced in Ref. [37], is now to consider Z ν p ρ s q as a series expansion of solutions, Z p ρ s q “ ÿ n “ p ns ϕ p n q p ρ s q , (A.43)and to use perturbation theory to obtain a solution for Z ν p ρ s q . Substituting Eq. (A.43) intoEq. (A.42) leads to an infinite number of differential equations, L ν ϕ p q p ρ s q ` p s „ L ν ϕ p q p ρ s q ` ρ s ϕ p q p ρ s q ` p s „ L ν ϕ p q p ρ s q ` ρ s ϕ p q p ρ s q ` ¨ ¨ ¨ “ . (A.44)The zeroth-order differential equation is homogenous, while all other orders are inhomogeneous.This procedure generates a secular perturbation in all inhomogeneous differential equations aswell as driving terms. The secular terms here refer to the solutions of the zeroth-order differentialequation, which are Bessel functions.Following Ref. [76], we introduce a function Z ν p ρ s q which has an expansion in terms of165essel functions with momentum-dependent coefficients, Z ν p ρ s q “ ÿ m “´8 b m p p s q J ν ` m p ρ s q . (A.45)We insert this as an ansatz into Eq. (A.42) with ν yet to be determined. Here J n denotescollectively the Bessel and Neumann functions, J n and N n . Substitution of Eq. (A.45) intoEq. (A.42) yields a three-term recurrence relation for the b m functions with ´8 ă m ă 8 , rp ν ` m q ´ ν s b m p p s q ` p s p m ` ν ´ q b m ´ p p s q ` p s p m ` ν ` q b m ` p p s q “ . (A.46)Solving these equations for b m p p s q yields b m p p s q “ p´ q m ´ p s ¯ m Γ p ν q Γ p ν ´ ν ` q Γ p ν ` ν ` q Γ p ν ` m q Γ p ν ´ ν ` m ` q Γ p ν ` ν ` m ` q c m p ν q (A.47)and b ´ m p p s q “ p´ q m ´ p s ¯ m Γ p ν ´ m ` q Γ p ν ´ ν ´ m q Γ p ν ` ν ´ m q Γ p ν ` q Γ p ν ´ ν q Γ p ν ` ν q c m p´ ν q (A.48)for m ě
0. The functions c m p˘ ν q are defined as c m p˘ ν q “ m ´ ź s “ Q p˘ ν ` s q b p p s q , (A.49)where Q p ν q is given by Q p ν q “ ´ p s p ν ` qrp ν ` q ´ ν o sp ν ` qrp ν ` q ´ ν o s Q p ν ` q . (A.50)166he coefficient b p p s q only determines the overall normalization and is simply set to one in thefollowing. Eq. (A.46) for m “ ν in the order of the Bessel functions. Wedetermine ν using the constraint p ν ´ ν q ´ Q p´ ν q ν p ν ´ qrp ν ´ q ´ ν s p s ´ Q p ν q ν p ν ` qrp ν ` q ´ ν s p s “ . (A.51)In general there are several roots which become complex beyond a critical scaled momentum p s , and one must be careful to choose the physical solution. For a detailed discussion of thispoint, see Refs. [76, 79].Choosing either J n “ J n or J n “ N n already yields a pair of linearly independent solutions.However, in order to get a pair with energy-independent normalization as r s Ñ x (cid:96) p p s q “ cos η (cid:96) ÿ m “´8 p´ q m b m p p s q ´ sin η (cid:96) ÿ m “´8 p´ q m b m ` p p s q (A.52)and y (cid:96) p p s q “ sin η (cid:96) ÿ m “´8 p´ q m b m p p s q ` cos η (cid:96) ÿ m “´8 p´ q m b m ` p p s q , (A.53)with η (cid:96) “ π p ν ´ ν q . Combining everything, we arrive at the van der Waals wave functions, F (cid:96) p p , r q “ r { s x (cid:96) p p s q ` y (cid:96) p p s q « x (cid:96) p p s q ÿ m “´8 b m p p s q J ν ` m p ρ s q ´ y (cid:96) p p s q ÿ m “´8 b m p p s q N ν ` m p ρ s q ff , (A.54) G (cid:96) p p , r q “ r { s x (cid:96) p p s q ` y (cid:96) p p s q « x (cid:96) p p s q ÿ m “´8 b m p p s q N ν ` m p ρ s q ` y (cid:96) p p q ÿ m “´8 b m p p s q J ν ` m p ρ s q ff . (A.55)167 .4 Low-energy expansions of the function terms in van derWaals wave functions In this appendix we expand all functions relating to the van der Waals wave functions in powersof momentum. We first consider ν , the shift in the order of the Bessel functions in Eq. (5.6) andEq. (5.7). Using Eq. (A.51) in Appendix A.3, we find ν “ ν ´ ν p ν ´ qp ν ´ q p s ` O p p s q , (A.56)where ν “ p (cid:96) ` q{
4. Using the expansion in Eq. (A.47), Eq. (A.48), and Eq. (A.50), we get b m p p s q “ p´ q m Γ p ν q Γ p ν ` q m ! Γ p ν ` m q Γ p ν ` m ` q ´ p s ¯ m ` O p p m ` s q (A.57)and b ´ m p p s q “ Γ p ν ´ m ` q Γ p ν ´ m q m ! Γ p ν ` q Γ p ν q ´ p s ¯ m ` O p p m ` s q (A.58)for m ě
0. Substituting these expressions into Eq. (A.52) and Eq. (A.53) we obtain x (cid:96) p p s q “ ` O p p s q , (A.59) y (cid:96) p p s q “ ´ „ Γ p ν q Γ p ν ´ q Γ p ν ` q Γ p ν q ` Γ p ν q Γ p ν ` q Γ p ν ` q Γ p ν ` q ´ p s ¯ ` O p p s q . (A.60)168 ppendix B B.1 Wronskians of the wave functions
B.1.1 Single channel
Here we calculate Wronskians of the wave function, U p p q (cid:96) p r q , and Wronskians of combinationsof the s n ,(cid:96) p r q and c n ,(cid:96) p r q functions. The Wronskian of U p p q (cid:96) p r q for the non-interacting region r ą R is W r U p p a q (cid:96) p r q , U p p b q (cid:96) p r qs “p p a ´ p b q W r u ,(cid:96) , u ,(cid:96) sp r q` p p a ´ p b q W r u ,(cid:96) , u ,(cid:96) sp r q` p p a p b ´ p b p a q W r u ,(cid:96) , u ,(cid:96) sp r q` p p a ´ p b q W r u ,(cid:96) , u ,(cid:96) sp r q ` O p p a ` p b q , (B.1)where W r u ,(cid:96) , u ,(cid:96) sp r q “ r (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q ` b ,(cid:96) p r q , (B.2)169 r u ,(cid:96) , u ,(cid:96) sp r q “ P (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q ` b ,(cid:96) p r q , (B.3) W r u ,(cid:96) , u ,(cid:96) sp r q “ Q (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q ` b ,(cid:96) p r q , (B.4) W r u ,(cid:96) , u ,(cid:96) sp r q “ b ,(cid:96) p r q . (B.5)It should be noted that W r f , g s “ ´ W r g , f s . The functions b n ,(cid:96) p r q are defined in terms ofWronskians of combinations of the s n ,(cid:96) p r q and c n ,(cid:96) p r q functions by b ,(cid:96) p r q “ a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q ´ a (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q´ a (cid:96) W r c ,(cid:96) , s ,(cid:96) sp r q ` W r c ,(cid:96) , c ,(cid:96) sp r q , (B.6) b ,(cid:96) p r q “ ´ r (cid:96) a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q ` r (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q` a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q ´ a (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q´ a (cid:96) W r c ,(cid:96) , s ,(cid:96) sp r q ` W r c ,(cid:96) , c ,(cid:96) sp r q , (B.7) b ,(cid:96) p r q “ W r c ,(cid:96) , c ,(cid:96) sp r q ` P (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q ` a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q´ a (cid:96) W r c ,(cid:96) , s ,(cid:96) sp r q ´ a (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q ` r (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q´ P (cid:96) a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q ´ r (cid:96) a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q , (B.8)170 ,(cid:96) p r q “ W r c ,(cid:96) , c ,(cid:96) sp r q ` ˆ r (cid:96) ` P (cid:96) a (cid:96) ˙ W r s ,(cid:96) , s ,(cid:96) sp r q ´ r (cid:96) a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q` r (cid:96) W r c ,(cid:96) , s ,(cid:96) sp r q ` r (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q ` a (cid:96) W r s ,(cid:96) , s ,(cid:96) sp r q` P (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q ´ a (cid:96) W r c ,(cid:96) , s ,(cid:96) sp r q ´ a (cid:96) W r s ,(cid:96) , c ,(cid:96) sp r q . (B.9)We now calculate Wronskians of all possible combinations of s p r q , s p r q , c p r q and c p r q functions. We find W r s ,(cid:96) , c ,(cid:96) s “ ´ , (B.10) W r s ,(cid:96) , c ,(cid:96) sp r q “ ´ r ` (cid:96) , (B.11) W r s ,(cid:96) , c ,(cid:96) sp r q “ r ` (cid:96) , (B.12) W r s ,(cid:96) , c ,(cid:96) sp r q “ r (cid:96) p (cid:96) ` q ´ , (B.13) W r s ,(cid:96) , s ,(cid:96) sp r q ´ π Γ ` ` (cid:96) ˘ Γ ` ` (cid:96) ˘ ´ r ¯ ` (cid:96) , (B.14) W r c ,(cid:96) , c ,(cid:96) sp r q “ ´ Γ ` ´ ` (cid:96) ˘ Γ ` ` (cid:96) ˘ π ´ r ¯ ´ (cid:96) . (B.15)171 .1.2 Coupled channels Here we calculate Wronskians of the U p r q and V p r q wave functions. Wronskians of U α p r q and V α p r q for the non-interacting region r ě R are W r U a α p r q , U b α p r qs “ p p a ´ p b q ! r j ´ W r s p r q , c p r qs j ´ ` a j ´ W r s p r q , s p r qs j ´ ` a j ´ W r c p r q , s p r qs j ´ ` a j ´ W r s p r q , c p r qs j ´ ` W r c p r q , c p r qs j ´ ) ` O p p a q ` O p p b q , (B.16) W r V a α p r q , V b α p r qs “ p p a ´ p b q q W r c p r q , c p r qs j ` ` O p p a q ` O p p b q . (B.17)Wronskians of the β -state wave functions are W r U a β p r q , U b β p r qs “ p p a ´ p b q q a j ` W r s p r q , s p r qs j ´ ` O p p a q ` O p p b q , (B.18) W r V a β p r q , V b β p r qs “ p p a ´ p b q ! r j ` W r s p r q , c p r qs j ` ` a j ` W r s p r q , s p r qs j ` ` a j ` W r c p r q , s p r qs j ` ` a j ` W r s p r q , c p r qs j ` ` W r c p r q , c p r qs j ` ) ` O p p a q ` O p p b q . (B.19)172ronskian of the combinations of the α and β -states are W r U a α p r q , U b β p r qs “ q a j ` W r c p r q , s p r qs j ´ ´ p a ! q a j ´ a j ` W r s p r q , s p r qs j ´ ´ q a j ` W r c p r q , s p r qs j ´ ) ` p b ! q a j ´ a j ` W r s p r q , s p r qs j ´ ´ q a j ` W r s p r q , c p r qs j ´ ` q r j ` W r s p r q , c p r qs j ´ ´ q a j ` W r s p r q , c p r qs j ´ ) ` O p p q , (B.20) W r U a β p r q , U b α p r qs “ ´ q a j ` W r c p r q , s p r qs j ´ ` p b ! q a j ´ a j ` W r s p r q , s p r qs j ´ ´ q a j ` W r c p r q , s p r qs j ´ ) ´ p a ! q a j ´ a j ` W r s p r q , s p r qs j ´ ´ q a j ` W r s p r q , c p r qs j ´ ` q r j ` W r s p r q , c p r qs j ´ ´ q a j ` W r s p r q , c p r qs j ´ ) ` O p p q , (B.21) W r V a α p r q , V b β p r qs “ ´ q a j ` W r c p r q , s p r qs j ` ´ p a ! q a j ` W r c p r q , s p r qs j ` ` q a j ` W r c p r q , s p r qs j ` ´ q W r c p r q , c p r qs j ` ) ` p b ! q r j ` W r c p r q , s p r qs j ` ´ q a j ` W r c p r q , s p r qs j ` ` q W r c p r q , c p r qs j ` ) ` O p p q , (B.22)173 r V a β p r q , V b α p r qs “ q a j ` W r c p r q , s p r qs j ` ` p b ! q a j ` W r c p r q , s p r qs j ` ` q a j ` W r c p r q , s p r qs j ` ´ q W r c p r q , c p r qs j ` ) ´ p a ! q r j ` W r c p r q , s p r qs j ` ´ q a j ` W r c p r q , s p r qs j ` ` q W r c p r q , c p r qs j ` ) ` O p p q . (B.23)174 ppendix C C.1 Coupled-channel Parameterizations
The scattering matrix in terms of the eigenphase parameters was given in Eq. (4.21). Thescattering matrix in terms of the nuclear bar parameters isS “ ¨˚˝ e i ¯ δ α cos 2 ¯ ε ie i p ¯ δ α ` ¯ δ β q sin 2 ¯ ε ie i p ¯ δ α ` ¯ δ β q sin 2 ¯ ε e i ¯ δ β cos 2 ¯ ε ˛‹‚ . (C.1)Here δ α , δ β and ε are the nuclear bar phase shifts and mixing angle [160]. The relationsbetween the eigenphase and the nuclear bar parameters aresin p δ α ´ δ β q “ sin 2 ε sin 2 ε , (C.2) δ α ` δ β “ δ α ` δ β , (C.3)tan 2 ε “ tan 2 ε sin p δ α ´ δ β q . (C.4)The two-channel effective range expansion is defined slightly differently in the eigenphase175nd the nuclear bar parameterizations. In the eigenphase parameterization, ÿ m m n n p mm U m m r K ´ s m n r U ´ s n n p n n “ ´ a mn ` r mn p ` O p p q , (C.5)and in the nuclear bar parameterization, ÿ m n p mm r K ´ s m n p n n “ ´ a mn `
12 ¯ r mn p ` O p p q , (C.6)where p mn is the diagonal momentum matrix diag p p j ´ { , p j ` { q . Therefore, by straightforwardcalculations we find the following relations among the threshold scattering parameters, a α “ ¯ a α , (C.7) r α “ ¯ r α ` q ¯ q ¯ a α ` ¯ q ¯ r β ¯ a α , (C.8) a β “ ¯ a β ´ ¯ q ¯ a α , (C.9) r β “ ¯ r β , (C.10) q “ ¯ q ¯ a α , (C.11) q “ ` ¯ a β ¯ a α ´ ¯ q ˘ ` ¯ a α ¯ q ` ¯ r β ¯ q ˘ a α . (C.12)For the uncoupled channels q and q are zero, and these relations become a α “ ¯ a α , r α “ ¯ r α , a β “ ¯ a β , and r β “ ¯ r β . 176 .2 Numerical Test using Delta-Function Shell Potentials ina coupled-channel system As an example to test the equalities in Eq. (4.50), Eq. (4.51) and Eq. (4.54), we consider thescattering of two spin- particles with a delta-function shell potential and partial-wave mixing, W p r , R q “ ¨˚˝ C C C C ˛‹‚ δ p r ´ R q . (C.13)The coupled radial Schrödinger equations become ´ d U p r q dr ´ k U p r q ` µ C δ p r ´ R q U p r q ` µ C δ p r ´ R q V p r q “ , (C.14) ´ d V p r q dr ´ k V p r q ` r V p r q ` µ C δ p r ´ R q U p r q ` µ C δ p r ´ R q V p r q “ . (C.15)The interaction potentials are non-vansihing only at r “ R , and everywhere else the wavefunctions of particles are free wave solutions. We split the space in two regions, r ą R and r ă R .Solutions for the region r ą R are the same as functions in Eq. (4.29)-(4.32). For r ă R thesefunctions must satisfy the boundary conditions at the origin. After normalization, the solutionsfor the region r ą R are U II α p r q “ cos ε p k q k J ´ ” cot δ J ´ p k q S J ´ p kr q ` C J ´ p kr q ı , (C.16) V II α p r q “ sin ε p k q k J ´ ” cot δ J ´ p k q S J ` p kr q ` C J ` p kr q ı , (C.17) U II β p r q “ ´ sin ε p k q k J ` ” cot δ J ` p k q S J ´ p kr q ` C J ´ p kr q ı , (C.18) V II β p r q “ cos ε p k q k J ` ” cot δ J ` p k q S J ` p kr q ` C J ` p kr q ı , (C.19)177nd for the region r ă R , U I α p r q “ A p k q cos ε p k q k J ´ S J ´ p kr q , (C.20) V I α p r q “ B p k q sin ε p k q k J ´ S J ` p kr q , (C.21) U I β p r q “ ´ D p k q sin ε p k q k J ` S J ´ p kr q , (C.22) V I β p r q “ E p k q cos ε p k q k J ` S J ` p kr q . (C.23)Here A p k q , B p k q , D p k q and E p k q are amplitudes to be determined by boundary conditions.At the boundary between two regions we have U II p R q “ U I p R q , (C.24) V II p R q “ V I p R q . (C.25)In addition, by integrating Eq. (C.14) and Eq. (C.15) around r “ R , we have ´ ˆ dU p r q dr ˙ R ` η R ´ η ` µ C U p R q ` µ C V p R q “ , (C.26) ´ ˆ dV p r q dr ˙ R ` η R ´ η ` µ C V p R q ` µ C U p R q “ . (C.27)Taking η Ñ
0, we obtain two more boundary conditions,lim η Ñ ´ dU II p r q dr ˇˇˇ p R ` η q ´ dU I p r q dr ˇˇˇ p R ´ η q ¯ “ µ C U p R q ` µ C V p R q , (C.28)lim η Ñ ´ dV II p r q dr ˇˇˇ p R ` η q ´ dV I p r q dr ˇˇˇ p R ´ η q ¯ “ µ C V p R q ` µ C U p R q . (C.29)Next we use these four boundary conditions to find phase shifts and mixing parameters as wellas all unknown amplitudes. After substituting wave functions in Eq. (C.16)-(C.23) into these178oundary conditions, we get the following equations for the J ´ A p k q “ cot δ J ´ p k q ` C J ´ p kR q S J ´ p kR q , (C.30) B p k q “ cot δ J ´ p k q ` C J ` p kR q S J ` p kR q , (C.31)cot δ J ´ p k q S J ´ p kR q ` C J ´ p kR q ´ A p k q S J ´ p kR q“ µ C A p k q S J ´ p kR q ` µ C tan ε p k q B p k q S J ` p kR q , (C.32)tan ε p k q ” cot δ J ´ p k q S J ` p kR q ` C J ` p kR q ´ B p k q S J ` p kR q ı “ µ C tan ε p k q B p k q S J ` p kR q ` µ C A p k q S J ´ p kR q , (C.33)and following equations for the J ` D p k q “ cot δ J ` p k q ` C J ´ p kR q S J ´ p kR q , (C.34) E p k q “ cot δ J ` p k q ` C J ` p kR q S J ` p kR q , (C.35)tan ε p k q ” cot δ J ` p k q S J ´ p kR q ` C J ´ p kR q ` D p k q S J ´ p kR q ı “ µ C tan ε p k q D p k q S J ´ p kR q ´ µ C E p k q S J ` p kR q , (C.36)cot δ J ` p k q S J ` p kR q ` C J ` p kR q ´ E p k q S J ` p kR q“ µ C E p k q S J ` p kR q ´ µ C tan ε p k q D p k q S J ´ p kR q . (C.37)179n the following, we present the results from numerical calculations for j ď
3. For numericalsolutions we use free parameters µ , C , C , C , R and r to calculate some numerical data forthe scattering phase shifts and mixing angle. Then we use some fitting procedure to determine thescattering lengths, effective ranges and mixing parameters. We simply fit the data to Eq. (4.40)and (4.41). It is clear that there is an abundance of free parameters which we can use to determinethe scattering parameters. However, we set these free parameters such values that we can obtainvery nice fits to the data. C.2.0.1 Example 1. S - D Coupling.
Our first example is the S - D coupled channel corresponding with j “
1. We performnumerical calculations using 2 µ C “ ´ . µ C “ ´ .
28 MeV, 2 µ C “ ´ . R “ . Table C.1
Numerical results for scattering length and effective range in two-body interaction by thedelta function potentials.
Channel a L [fm L ` ] r L [fm ´ L ` ] S D -0.326 -36.876 P F -0.039 -287.01 D G r “ . k “ .
335 MeV, we obtain b p r q ´ q r ´ r “ .
577 fm , (C.38)180 able C.2 Numerical results for the mixing parameters in two-body interaction by the delta functionpotentials.
Mixing angle q [fm ] q [fm ] ε ε ε ż R ´“ U I α p r q ‰ ` “ V I α p r q ‰ ¯ dr ` ż rR ´“ U II α p r q ‰ ` “ V II α p r q ‰ ¯ dr “ .
575 fm . (C.39)We also find b p r q ` q r a ´ r “ .
51 fm ´ , (C.40)agrees with2 ż R ˆ” U I β p r q ı ` ” V I β p r q ı ˙ dr ` ż rR ˆ” U II β p r q ı ` ” V II β p r q ı ˙ dr “ .
13 fm ´ . (C.41) C.2.0.2 Example 2. P - F Coupling.
The second example is the coupled channel P - F with J “
2. We use 2 µ C “ ´ .
759 MeV,2 µ C “ ´ .
28 MeV, 2 µ C “ ´ .
36 MeV and R “ . a , a , r , q and q shown in Table C.1 and Table C.2. For r “ .
42 fm and k “ .
33 MeV we get b p r q ´ q r ´ r “ .
188 fm ´ , (C.42)181hich agrees within numerical precision with2 ż R ´“ U I α p r q ‰ ` “ V I α p r q ‰ ¯ dr ` ż rR ´“ U II α p r q ‰ ` “ V II α p r q ‰ ¯ dr “ .
187 fm ´ . (C.43)We also find b p r q ` q r a ´ r “ . ˆ fm ´ , (C.44)which agrees with2 ż R ˆ” U I β p r q ı ` ” V I β p r q ı ˙ dr ` ż rR ˆ” U II β p r q ı ` ” V II β p r q ı ˙ dr “ . ˆ fm ´ . (C.45) C.2.0.3 Example 3. D - G Coupling.
The last example is the D - G coupled channel with J “
3. We use 2 µ C “ ´ .
255 MeV,2 µ C “ ´ .
28 MeV, 2 µ C “ ´ .
27 MeV and R “ . a , a , r , a , q and q are calculated numerically and indicated in Table C.1 and Table C.2. For r “ . k “ .
33 MeV the results for the D channel are b p r q ´ q r ´ r “ .
256 fm ´ , (C.46)which agrees within numerical precision with2 ż R ´“ U I α p r q ‰ ` “ V I α p r q ‰ ¯ dr ` ż rR ´“ U II α p r q ‰ ` “ V II α p r q ‰ ¯ dr “ .
88 fm ´ . (C.47)182or r “ .
93 fm and k “ . G channel are b p r q ` q a r ´ r “ . ˆ fm ´ , (C.48)which agrees with2 ż R ˆ” U I β p r q ı ` ” V I β p r q ı ˙ dr ` ż rR ˆ” U II β p r q ı ` ” V II β p r q ı ˙ dr “ . ˆ fm ´ . (C.49)183 ppendix D D.1 Finite-volume binding energy corrections and topologi-cal volume corrections for scattering with arbitrary (cid:96)
In order to apply Lüscher’s finite-volume method with maximal accuracy, we consider alsofinite-volume corrections to the binding energy of the dimer. The finite-volume correction totwo-body s -wave binding energies was derived in Ref. [123] and extended to arbitrary angularmomentum in Ref. [106, 109]. There has also been significant work towards understandingthree-body binding energy corrections at finite volume [112, 111].It was noticed in Ref. [28] that the finite-volume corrections to the dimer binding energyis dependent on the motion of the dimer. This fact has been used to cancel out finite-volumecorrections to the binding energy [45]. The dimer motion induces phase-twisted boundaryconditions on the dimer’s relative-coordinate wavefunction. These effects are called topologicalvolume corrections and were found to have an effect on the finite-volume analysis for scatteringof the dimer. The study of topological volume corrections were carried out for s -wave scatteringin Ref. [28, 26] and further applied in Ref. [148, 143]. In the following we show the extensionto general partial wave (cid:96) . 184he general solution of the Helmholtz equation has the form of ψ p p (cid:126) r q “ ÿ (cid:96), m c (cid:96), m p p q G (cid:96), m p (cid:126) r , p q . (D.1)The functions G (cid:96), m p (cid:126) r , p q form a linearly independent complete basis set and are defined as G (cid:96), m p (cid:126) r , p q “ Y (cid:96), m p ∇ q G p (cid:126) r , p q . (D.2)Here Y (cid:96), m are the solid spherical harmonic polynomials and defined in terms of the sphericalharmonics as Y m (cid:96) p (cid:126) r q “ r (cid:96) Y m (cid:96) p θ , φ q , (D.3)and G p (cid:126) r , p q is the periodic Green’s function solution to the Helmholtz equation for (cid:96), m “ G , p (cid:126) r , p q “ G p (cid:126) r , p q “ L ÿ (cid:126) k e i π L (cid:126) k ¨ (cid:126) r ´ π L (cid:126) k ¯ ´ p . (D.4)Using Eq. (B1) of Ref. [124], we have G (cid:96), m p (cid:126) r , p q “ r (cid:96) Y m (cid:96) p θ , φ q ˆ r BB r ˙ (cid:96) G p (cid:126) r , p q . (D.5)Inserting Eq. (D.4) into Eq. (D.5), we write the asymptotic form of the scattering wave functionas u (cid:96) p r q “ C ÿ (cid:126) k | (cid:126) k | (cid:96) e i π L (cid:126) k ¨ (cid:126) r ´ π L (cid:126) k ¯ ´ p , (D.6)185here C is the normalization coefficient. The derivation of the topological volume correctionsfor the s -wave scattering of two composite particles A and B is given in Ref. [28, 26, 148]. Herewe focus on the fermion-dimer scattering and derive the topological volume corrections forhigher partial waves.In this analysis we take the continuum limit. We let the total momentum of the fermionplus dimer system to be zero and let p be the magnitude of the relative momentum between thefermion and dimer. Let E d ,(cid:126) p8q be the dimer energy at infinite volume and m d be the dimermass. Then the fermion-dimer energy at infinite volume, E df p p , , is E df p p ,
8q “ p m d ` p m ` E d ,(cid:126) p8q . (D.7)As in previous studies of fermion-dimer scattering on the lattice [28, 26, 148, 143], we calculatethe effective dimer mass of the dimer on the lattice by computing the dimer dispersion relation.Now we let E df p p , L q be the finite-volume energy of the fermion-dimer system. FollowingRef. [28, 26, 148], we can compute the expectation value E df p p , L q “ ş d r u ˚ (cid:96) p r q ˆ Hu (cid:96) p r q ş d r | u (cid:96) p r q| “ N (cid:96) k max ÿ (cid:126) k | (cid:126) k | (cid:96) p m d ` E d ,(cid:126) k p L q ´ (cid:126) k ´ η ¯ , (D.8)where E d ,(cid:126) k p L q is the finite-volume energy of the dimer with momentum (cid:126) k , N (cid:96) is defined as N (cid:96) “ k max ÿ (cid:126) k | (cid:126) k | (cid:96) ´ (cid:126) k ´ η ¯ ´ , (D.9)and η “ ´ Lp π ¯ . For (cid:96) ą Λ characterizing the range of the fermion-dimer interactions.186he corresponding maximum index value k max scales as Λ L {p π q .Let ∆ E d ,(cid:126) p L q “ E d ,(cid:126) p L q ´ E d ,(cid:126) p8q be the finite-volume energy shift of the dimer energy inits rest frame, and ∆ E d ,(cid:126) k p L q “ E d ,(cid:126) k p8q ´ E d ,(cid:126) k p L q be the finite-volume energy shift of the dimerenergy with momentum (cid:126) k . One can show that [28], ∆ E d ,(cid:126) k p L q ∆ E d ,(cid:126) p L q “ ÿ i “ cos p π k i α q . (D.10)Using Eq. (D.7), (D.8), and (D.10), we can now write the fermion-dimer energy correction atfinite volume as E df p p , L q ´ E df p p ,
8q “ τ (cid:96) p η q ∆ E d ,(cid:126) p L q , (D.11)where τ (cid:96) p η q is the topological factor, τ (cid:96) p η q “ N (cid:96) k max ÿ (cid:126) k | (cid:126) k | (cid:96) ř i “ cos p π k i α q ´ (cid:126) k ´ η ¯ , (D.12)with α “ m {p m ` m d q “ {
3. Due to the short distance behavior of the momentum modesummations for (cid:96) ą
0, we find that the topological phase factor τ (cid:96) p η q is suppressed by the latticelength L , τ (cid:96) ą p η q “ O ` L ´ ˘ . (D.13)In other words, the topological volume correction for (cid:96) ą L relativeto the (cid:96) “ (cid:96) “ , but neglected the correctionsfor (cid:96) ą
0. We find that this prescription gives good agreement with the continuum infinite-volume187TM results for partial waves (cid:96) “ ,,
0. We find that this prescription gives good agreement with the continuum infinite-volume187TM results for partial waves (cid:96) “ ,, ,,