Properties of One- and Two-Nucleon Halo Nuclei in Effective Field Theory

A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulfillment of the requirements for the degree Doctor of Philosophy

Bijaya Acharya May 2015

© 2015 Bijaya Acharya. All Rights Reserved. 2

This dissertation titled Properties of One- and Two-Nucleon Halo Nuclei in Effective Field Theory

by BIJAYA ACHARYA

has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by

Daniel R. Phillips Professor of Physics and Astronomy

Robert Frank Dean, College of Arts and Sciences 3 Abstract

ACHARYA, BIJAYA, Ph.D., May 2015, Physics Properties of One- and Two-Nucleon Halo Nuclei in Effective Field Theory (160 pp.) Director of Dissertation: Daniel R. Phillips Halo nuclei have much larger sizes than other nuclei with a similar number of protons and neutrons. This is the result of the fact that the probability distribution of one or more nucleons in halo nuclei extends to much larger distances than that of the rest of the nucleons. Halo Effective Field Theory exploits the separation of scales between the size of the halo to that of the “core” to derive the correlations between low-energy observables as systematic expansions in the ratio of the size of halo to that of the core, with error estimates. We use this approach to calculate the reduced transition probability for E1 excitation of the halo neutron of 19C to the continuum up to next-to-next-to leading order in this expansion. By comparing the prediction with the Coulomb dissociation data from RIKEN, accurate results for the one-neutron separation energy and asymptotic normalization coefficient are extracted. Good agreement with data is also found for the longitudinal momentum distribution of the Coulomb dissociation cross section. We then derive universal relations for the matter radii of two-neutron halos and apply them to 22C to analyze its matter radius, which we use as input to constrain 21C and 22C energies. Even after accounting for the combined theoretical and 1 − σ experimental errors, the matter radius value imposes an upper bound of 100 keV in the two-neutron separation energy of 22C. We also derive universal relations for the E1 excitation of two-neutron halo nuclei and discover a simple scaling of the B(E1) spectrum with the three-body binding energy when all two-body subsystems are in the unitary limit. We compare our results with the data for 11Li, and make predictions for the outcome of an ongoing experiment on 22C. 4 Acknowledgments

This dissertation could not have been completed without the guidance, participation, suggestions and encouragement I received from many people over the last five years. I am grateful to my committee members, Profs. Daniel Phillips, Charlotte Elster, Prakash, Carl Brune and Alycia Stigall for being such wonderful mentors. I would like to express sincere appreciation to my advisor Prof. Daniel Phillips for introducing me to this exciting world of nuclear physics and for instilling in me the knowledge and skills needed to make contributions to it. I am grateful to Profs. Elster and Prakash, who have followed my research with keen interest and provided valuable suggestions. My interactions with Profs. Brune, Frantz, Hicks, Roche and Schiller at individual and group meetings, journal clubs and seminars helped me develop an understanding of the experiments that are relevant to my work in particular, and to nuclear physics in general. I will be forever indebted to the Physics and Astronomy faculty here at Ohio University for strengthening my academic foundation. I would like to thank my collaborators, Drs. Chen Ji, Philipp Hagen and Hans-Werner Hammer, for having the patience to work with me, and for sharing their valuable experience in the process. I am grateful to the nuclear theory group members, Arbin, Linda, Brian, Hadi, Xilin, Vasily, Mamun, Anton and Azamat; and my colleagues at the accelerator lab for invigorating discussions and constructive criticisms. Finally, I would like to thank all the staff at our department for their warmth, generosity and hospitality. 5 Table of Contents

Page

Abstract ...... 3

Acknowledgments ...... 4

List of Tables ...... 8

List of Figures ...... 9

1 Introduction ...... 14 1.1 Experimental studies ...... 16 1.1.1 Mass measurements ...... 17 1.1.2 Coulomb dissociation ...... 18 1.1.3 Measurement of charge radii in ion traps ...... 19 1.1.4 Matter radii from cross-section measurements ...... 19 1.1.5 Fragmentation reactions and momentum distributions ...... 21 1.2 Theories of halo nuclei ...... 21 1.2.1 The nuclear shell model ...... 21 1.2.2 Ab initio calculations with nucleons ...... 24 1.2.2.1 The no-core shell model (NCSM) ...... 27 1.2.2.2 The hyperspherical harmonics (HH) method ...... 28 1.2.2.3 Coupled cluster (CC) method ...... 28 1.2.3 Cluster models ...... 29 1.2.4 Halo effective field theory (Halo-EFT) ...... 30 1.2.4.1 Two-body systems in Halo-EFT ...... 30 1.2.4.2 Three-body systems in Halo-EFT ...... 33 1.3 This dissertation ...... 37

2 Effective Field Theory for Halo Nuclei: A Review of the Formalism ...... 39 2.1 Two-body systems ...... 41 2.2 Proton halo nuclei ...... 45 2.3 Electromagnetic observables in Halo-EFT ...... 48 2.4 Two-neutron halo with a spinless core ...... 52 2.5 Conclusion ...... 59

3 Coulomb Energy in a Square Well Potential Model ...... 61 3.1 Effective range for the square well ...... 63 3.2 Coulomb energy at first order in αem ...... 64 3.2.1 Errors in EFT calculation ...... 66 3.3 Coulomb energy to all orders in αem ...... 67 6

3.4 Conclusion ...... 71

4 Coulomb Dissociation of One-Neutron Halos: Applications to Carbon-19 . . . . 73 4.1 Coulomb dissociation cross section of Carbon-19 in Halo-EFT ...... 74 4.1.1 Extracting parameters from Coulomb dissociation data ...... 77 4.1.1.1 Angular distribution ...... 77 4.1.1.2 Relative energy spectrum ...... 78 4.1.2 Longitudinal momentum distribution ...... 82 4.2 Electric charge radii of Carbon-18 and Carbon-19 ...... 82 4.3 Conclusion ...... 83

5 Two-Neutron Halos: Matter Radii and Efimov States ...... 85 5.1 The Jacobi coordinates and momenta ...... 85 5.2 The three-body free states: plane waves and spherical waves ...... 86 5.2.1 The two-body T-matrices in Jacobi representation ...... 88 5.3 The bound state ...... 88 5.4 Two-neutron halo: identical neutrons and the permutation operator . . . . . 90 5.4.1 Matter radii ...... 93 5.5 Carbon-22 ...... 95 5.5.1 Matter radius constraints on binding energy ...... 95 5.5.1.1 EFT errors ...... 97 5.5.2 On the possibility of excited Efimov states ...... 98 5.6 Conclusion ...... 100

6 Universal Relations for the Dipole Excitation of Borromean Halo Nuclei . . . . . 101 6.1 The wavefunction for the three-body unbound state ...... 101 6.2 The dipole matrix element ...... 102 6.2.1 Evaluation of the matrix element by distortion of the integration paths108 6.3 The B(E1) spectrum ...... 110 6.3.1 Unitary Limit ...... 111 6.3.2 Lithium-11 ...... 112 6.3.3 Coulomb dissociation of Carbon-22 ...... 116 6.3.4 Momentum distributions ...... 117 6.4 The photonuclear sum rules ...... 119 6.4.1 Direct derivation of the sum rules for neutron halo nuclei . . . . . 122 6.4.2 Numerical evaluations of the sum rules for two-neutron halo nuclei 123 6.5 Conclusion ...... 127

7 Summary and Outlook ...... 128

References ...... 131

Appendix A: Coulomb Dissociation Cross Section ...... 147 7

Appendix B: Numerical Techniques and Errors ...... 150 8 List of Tables

Table Page

2 n,c 6.1 The value of bE and the maximum possible value of k0 for all the Zi (q, K; E) featuring in Eq. (6.23)...... 110 2 2 6.2 Power-law fits to mEB/(eZ) dB(E1)/dE in PWIA and with FSI included for various values of A...... 125 6.3 Integrated dipole strength in the unitary limit. The second and the third columns show, respectively, the integrated dipole strengths with dB(E1)/dE calculated in PWIA, and with FSI included...... 125 6.4 Non-energy-weighted and energy-weighted integrals of the dipole response function for A = 20 at several values of EB and Enc...... 126 √ √ B.1 A( mEB, mEB ; EB) evaluated at Z = 6, A = 20, Enc = 0.2 MeV by differentiation of the wavefunction as shown in Eq. (6.8) ...... 155 B.2 Dependence of In(Kn) on the integration path rotation angle for E = EB = 0.4 MeV, Z = 6, A = 20, with Enc fixed at 0.1 MeV...... 159 B.3 Dependence of Ic(Kc) on the integration path rotation angle for E = EB = 0.4 MeV, Z = 6, A = 20, with Enc fixed at 0.1 MeV...... 160 B.4 dB(E1)/dE calculated by numerically evaluating Eq. (6.32) with different number of Gauss-Legendre quadrature points. The calculation is performed at Z = 6, A = 20, EB = 0.4 MeV and Enc = 0.2 MeV...... 160 9 List of Figures

Figure Page

1.1 A low-mass region in the chart of nuclei (proton number versus neutron number). The black squares are stable nuclei. Several one-neutron, and two-neutron halo nuclei, along with the four-neutron halo nucleus 8He, are identified. Figure taken from Ref. [4]...... 15 1.2 The interaction radii of light nuclei versus the atomic number. Figure taken from Ref. [1]...... 20 1.3 The longitudinal momentum distributions of the fragments for 19C single- neutron removal on Beryllium target, measured at 57 MeV per nucleon at MSU [50], at 64 MeV per nucleon at RIKEN [47]; and on Carbon target, measured at 910 MeV per nucleon at GSI [45]. Figure from Ref. [47]...... 22 1.4 The nuclear interactions in χEFT at different powers of Q/Λχ = Mlo/Mhi. The solid lines represent nucleons, and the dashed lines represent pions. The small and large dots, squares and diamonds represent vertices that carry different powers of the nucleon momenta. Figure taken from Ref. [80]...... 26 1.5 Deuteron photodissociation cross section taken from Ref. [112]. References to experimental studies that obtained the data can be found in Ref. [114]. Dashed line is Halo-EFT calculation with one fit parameter, determined by minimizing χ2...... 32 1.6 E1 S factor for proton capture on 7Be as a function of the 7Be − p center of mass energy, taken from Ref. [129]. The green shaded band is the range of S (E) at leading order with scattering length varied within its 1σ error. The green error bar at 20 keV shows an estimate of the theory error from higher order terms. The solid black curve is the NCSM/RGM calculation of Ref. [85]. The dashed curve is a Halo-EFT calculation that uses the same set of inputs as Ref. [85]. There are five different sets of data from different experiments. . . . 34 1.7 The a−1K plane for the three-body problem. The allowed regions for three- atom scattering states and atom-dimer scattering states are labeled AAA and AD, respectively. The hatching indicates the threshold for scattering states. The y−axis represents the unitary limit. The thick curves labeled T are the Efimov states. The y−intercepts of these curves are in a geometric√ progression with a common ratio of approximately 2.2 instead of 22.7 ' 515.03 because a−1 and K are parametrized as a−1 = r cos ξ and K = r sin ξ, but r1/4 cos ξ and r1/4 sin ξ are plotted instead to cover large ranges of a−1 and K. Figure taken from Ref. [131]...... 35 1.8 Three-body force parameter H as a function of the cutoff Λ for particle-dimer scattering length a3 = 1.56a2, where a2 is the particle-particle scattering length, Λ∗ = 19.5/a2. The solid line is an approximate result obtained from an analysis of the ultraviolet behavior of the three-body equations and the dots are the numerical results. Figure taken from Ref. [132]...... 36 10

1.9 The various rms distances in the 11Li nucleus as a function of the 9Li−n energy (with negative values signifying that this subsystem is unbound), taken from Ref. [140]: distance between neutrons (top left), 9Li − n distance (top right), and distance of 9Li (bottom left) and that of the neutron (bottom right) from the 11Li center of mass. The LO results of Ref. [139] are represented by the solid black lines, with error bands represented by dashed lines. The NLO results are represented by the red dotted-dashed lines with purple error bands. The three- body binding energy and the two-body effective ranges were taken to be 247 keV and 1.4 fm respectively...... 37

2.1 Feynman diagrams contributing to the amplitude for the Lagrangian at LO. . . 41 2.2 The scattering amplitude for the Lagrangian in Eq. (2.8). The double (thick) line represents the bare (dressed) propagator of the d field...... 42 2.3 The scattering amplitude for a two-body system with Coulomb interaction in addition to the short-range interaction, as an expansion in αem. The dashed and the thin lines represent the particle propagators and the thick line represents the dressed propagator of the bound/virtual state...... 46 2.4 Non-perturbative resummation of TCS , represented by the shaded rectangle, as described in Eq. (2.32). The shaded ellipse represents propagation in the presence of Coulomb photons, as shown in the second row. The pure Coulomb amplitude involves the sum of all but the first diagram on the right-hand-side of the second row...... 47 2.5 Feynman diagrams contributing to the charge form factor. The dashed and the thin lines represent the core and the neutron propagators, respectively. The thick line represents the dressed propagator of the neutron halo nucleus. The diagram on the left (monopolar coupling) is suppressed compared to the one on the right (dipolar coupling). At LO, only the diagram on the right needs to be considered...... 49 2.6 Diagrams contributing to the amplitude for the photodissociation of the one- neutron halo into the core and the neutron. The dashed and the thin lines represent the core and the neutron propagators, respectively. The thick line represents the dressed propagator of the neutron halo nucleus. The dots represent diagrams which are higher order in the EFT expansion parameter. . . 50 2.7 Faddeev equations for the three-body scattering amplitudes. The solid and the dashed lines represent the propagators of the n and the c fields respectively, the thick gray line represents the dressed propagator for the d0 field and the thick black line represents the dressed propagator for the d1 field. The n − d1 scattering amplitudes are shown as black blobs and the coupled-channel ones as gray blobs...... 54 11

2.8 Faddeev equations for the three-body scattering amplitudes. The solid and the dashed lines represent the propagators of the n and the c fields respectively, the thick gray line represents the dressed propagator for the d0 field and the thick black line represents the dressed propagator for the d1 field. The c − d0 scattering amplitudes are shown as white blobs, and the coupled-channel ones as gray blobs...... 56 2.9 The expansion of the dressed propagator (thick blue line) of the t field in terms of the full T-amplitude...... 58 2.10 The expansion of the dressed propagator (thick blue line) of the t field as an infinite geometric series. The triple line represents the bare propagator. The blue blob is the t-particle irreducible c − d0 amplitude i.e. it does not contain any three-body contact interactions...... 58

3.1 Coulomb energy at first order in αem for square well (expressed in appropriate dimensionless units), LO calculation in EFT, and NLO calculation in EFT −1 versus the binding momentum in the units of R , at kcR = 0.1. The purple bands represent the EFT errors...... 68 3.2 The Coulomb energy to all orders in αem calculated from the square well model (expressed in appropriate dimensionless units), at LO in EFT, and at NLO in −1 EFT plotted against the binding momentum in the units of R for kcR = 0.1 . The purple bands represent the EFT errors...... 71

4.1 Contour plot of ∆χ2=1 for the angular distribution of the differential cross section in the aB-plane. (Published in Ref. [145].) ...... 78 4.2 Contour plot of ∆χ2=1 for the energy spectrum of the total cross section in the aB-plane. (Published in Ref. [145].) ...... 80 4.3 Contour plot of ∆χ2=1 for the combined data in the aB-plane (left) and the ar0-plane (right). (Published in Ref. [145].) ...... 81 4.4 The angular distribution of the differential cross section at a=7.5 fm and B=540 keV (dashed), and at a=7.75 fm and B=575 keV (solid), with data from Ref. [21] (left); and relative energy spectrum of the differential cross section at a=8.1 fm and B=580 keV (dashed), and at a=7.75 fm and B=575 keV (solid), with data from Ref. [174] (right). (Published in Ref. [146].) ...... 81 4.5 Longitudinal momentum distribution of the dissociation cross section on a 181 Ta target at 88 MeV/u for B=575 keV and r0=2.6 fm with R=13 fm (left); and with the normalization and the peak position fitted to data from Ref. [95](right). (Published in Ref. [145].) ...... 83

5.1 The dimensionless function f (0, 0; A), defined by Eq. (5.46), versus A. (Published in Ref. [146].) ...... 94 5.2 f (Enn/EB, Enc/EB; 20) versus (Enn/EB, Enc/EB). (Published in Ref. [146].) . . . 96 12 p 5.3 Plots of hR2i = 5.4 fm (blue, dashed), 6.3 fm (red, solid), and 4.5 fm (green, dotted), with their theoretical error bands, in the (EB, Enc) plane. (Published in Ref. [146].) ...... 98 5.4 The region in the (EB, Enc) plane that allows excited Efimov states (purple), and the region that encloses values consistent with the experimental rms matter radius of 5.4 ± 0.9 fm with same color-coding as in Fig. 5.3. (Published in Ref. [146].) ...... 99

6.1 Diagrammatic representation of the terms in Eq. (6.13): MPWIA (top, left), Mnn (top, middle), Mnc (top, right) and Mnnc (bottom row). The thin solid line represents the neutron, the dashed line represents the core, the thick dark line and the thick gray line represent the rescatterings in the nc and the nn subsystems respectively. The initial (bound) state is shown as a gray blob interacting with the photon. The white blob represents FSI in which all outgoing particles participate...... 105 6.2 Integration paths in the complex q plane (shown by the arrows) for K2 < bE (left) and for bE < K2 < 4bc2E/(4c2 − ab) (right). The dashed line is used to n,c indicate that the value of Zi (q, K; E) in the second Riemann sheet should be used...... 109 6.3 The universal dipole response functions in the unitary limit in PWIA (left) and with FSI included (right) for A = 1, 4, 9, 20 and 100...... 112 11 6.4 The dipole response spectrum for Li in the unitary limit and for EB = 0.369 MeV, Enc = 0.026 MeV and Enn = 0.118 MeV...... 114 11 6.5 The dipole response spectrum for Li for EB = 0.369 MeV and Enc = 0.026 MeV in PWIA, with rescatterings in the nn and in one nc subsystem in the final state, and with all LO S-wave FSI included...... 114 6.6 The dipole response spectrum for 11Li with all LO S-wave FSI included. The blue band is the error due to uncertainty in Enc and the purple band the EFT error.115 6.7 The dipole response spectrum for 11Li after folding the detector resolution (blue curve) with the theory error (purple band), and data from Ref. [22]. . . . . 116 22 6.8 The dipole response spectrum for C for EB = 50 keV, Enc = 10 keV (blue); EB = 50 keV, Enc = 100 keV (red) and EB = 70 keV, Enc = 10 keV (brown), with their EFT error bands...... 117 2 2 6.9 mEBd B(E1)/dEdKc versus Kc at different values of two-body scattering lengths in PWIA (dashed) and with FSI included (solid) at E = EB = 0.4 MeV, A = 20 and Z = 6...... 118 2 2 6.10 mEBd B(E1)/dEdKn versus Kn at different values of two-body scattering lengths in PWIA (dashed) and with FSI included (solid) at E = EB = 0.4 MeV, A = 20 and Z = 6...... 119 2 2 6.11 The dimensionless response function mEB/(eZ) dB(E1)/dE versus E/EB for A = 20 in PWIA in the unitary limit shown with a power law fit function. . . . 124 6.12 The integrands of the non-energy-weighted and the energy-weighted integrals used in the calculation of the values in Table 6.4...... 127 13

2 B.1 mEBhr i at A = 20 calculated from Eq. (5.44)...... 154 B.2 Real and imaginary parts of the integral in Eq. (6.15) obtained from rotation of the integration path and from subtraction method versus the external momentum Kn at Z = 6, A = 20, E = EB = 0.4 MeV and Enc = 0.1 MeV. The inset shows the percentage difference between the real parts obtained from the two methods. Differences in the imaginary parts are much√ smaller and are not shown. The rotation method can be used only up to Kn = mE...... 158 14 1 Introduction

Over the last few decades, experiments at radioactive ion beam facilities have revealed that several drip-line nuclei have a halo structure: clusterization into a “core” of normal nuclear density and one or more protons or neutrons whose probability distribution extends far beyond the range of nuclear interactions. (See Refs. [1, 2] for early reviews, and Ref. [3] for a recent one.) Halo nuclei owe their existence to quantum mechanics — quantum few-body systems bound weakly by short-range interactions can have wavefunctions that extend to distances much larger than the range of the interaction. A well known example is the deuteron, the bound state of a proton and a neutron, with a binding energy of about 2.23 MeV. Similarly, the halo structure arises in heavier nuclei as a result of very weak binding of one or more nucleons to the core. A low relative anglular momentum between the nucleon and the core is usually a prerequisite for halo formation because the angular momentum barrier tends to confine the nucleon wavefunction to a small region around the core. Therefore the wavefunctions of halo nuclei typically have large components of the configurations in which the last nucleon(s) are in low angular momentum states. For example, 11Be (1/2+) and 19C each have a neutron weakly bound to the 10Be and 18C cores, respectively, in an S-wave. Proton halo nuclei are rarer because the proton-core Coulomb barrier has a confining effect similar to that of the angular momentum barrier. 8B is a proton halo nucleus in which the proton and the 7Be core are in a P-wave state. 6He, 11Li and 22C are examples of two-neutron halos. These are Borromean three-body systems because all of the two-body subsystems, the dineutron, 5He, 10Li and 21C, are unbound. Most halo nuclei have only one bound state, with all the excited states lying in the continuum. Notable exceptions are 11Be, whose first excited (1/2−) state is also bound, and 17F, whose ground (5/2+) state does not exhibit halo structure but the first excited (1/2+) state has a proton halo structure. Figure 1.1 shows a region of the nuclear chart with several halo nuclei. 15

Figure 1.1: A low-mass region in the chart of nuclei (proton number versus neutron number). The black squares are stable nuclei. Several one-neutron, and two-neutron halo nuclei, along with the four-neutron halo nucleus 8He, are identified. Figure taken from Ref. [4].

Halo nuclei have a nucleon-core scattering length that is much larger than the range of the nuclear interaction between them. The long-distance/low-energy properties of halo nuclei can therefore be studied in Halo Effective Field Theory (Halo-EFT), which uses the nucleons and the core as degrees of freedom and provides an order by order expansion of physical observables in powers of the ratio of the interaction range to the scattering length [5–15]. This approach has several advantages. First, it reduces the complexity of the problem by exploiting the fact that, at sufficiently low energy, the nucleons present inside the core are unresolved. Second, by absorbing the effects of the short-distance physics into a small number of parameters via renormalization, this theory provides model-independent equations that encode the long-distance physics. Third, it provides a 16 systematic framework for estimating errors of this theory and calculating higher order corrections. This document is organized as follows. In the rest of this chapter, the experimental and theoretical studies on halo nuclei are reviewed. Some of the mathematical formalism of Halo-EFT compiled from existing literature is summarized in Chapter 2. In Chapter 3, the square well potential is used as a model for the nuclear interaction to study the Coulomb energy of an S-wave proton-core system. Halo-EFT is applied to study the Coulomb dissociation of 19C in Chapter 4, and the structure of 22C in Chapter 5. In Chapter 6, an extension of this EFT formalism to the Coulomb dissociation of two-neutron halos, with applications to 11Li and 22C, is presented. Chapter 7 contains a brief summary and outlook with an exploration of possible avenues to take this research further. Selected calculational and numerical techniques employed are discussed in the Appendix.

1.1 Experimental studies

Experiments on halo nuclei are challenging because they are highly unstable. They generally decay by β emission and have half lives of a small fraction of a second. Extensive experimental studies on halo nuclei were made possible by construction of rare isotope beam facilities that provide pure, intense, energetic beams of these short-lived nuclei. In a pioneering experiment [16, 17], Tanihata et al. measured the interaction cross section of He and Li isotopes on ordinary nuclear targets at a beam energy of 790 MeV per nucleon. An anomalous increase in the cross section was observed for 6He, 8He, and 11Li. Confirmation of the large spatial extension of the neutron wavefunction was obtained by Kobayashi et al., who measured the momentum distribution of 9Li resulting from the fragmentation of 11Li at a beam energy of 790 MeV per nucleon and found a narrow peak [18]. Strong peaks, arising from the disparity between the charge and matter distributions, that were predicted by Ref. [19], were seen in the low-energy spectrum of 17 the Coulomb dissociation cross section of 11Be [20], 19C [21] and 11Li [22]. A wider array of experimental techniques are now available. Here, some important classes of experiments performed on halo nuclei and some of their results are discussed.

1.1.1 Mass measurements

The separation energy of the halo nucleons is one of the key quantities that governs the low-energy properties of halo nuclei. It can be experimentally determined by measuring the masses of the halo nucleus and the core with a high precision such that the experimental errors in the masses measured are smaller than the size of the separation energy, which, for halo nuclei, is at least an order of magnitude smaller than the typical value of about 8 MeV for normal nuclei. The mass of the halo nucleus can be deduced from a precise measurement of the Q−value of a nuclear reaction in which the masses of all the other reactants and products are known. Reference [23] used this method to obtain a value of 295±35 keV for the two-neutron separation energy of 11Li from the 14C(11B,11Li)14O reaction. In Ref. [24], the charge to mass ratio, z, of several light exotic nuclei, including the halo nuclei 22C and 31Ne, were obtained based on the relationship between the magnetic rigidity and velocity, Bρ = γv/z, by directly measuring the radius of curvature of their path, ρ, and velocity, v, in a magnetic field of strength B. Another time of flight (TOF) technique, called radiofrequency spectrometry, which measures mass via the cyclotron resonance frequency, was used to measure the masses of the exotic nuclides 11Li and 11,12Be in Ref. [25]. Use of a Penning trap for laser spectroscopic measurement of isotope shift is the most precise mass determination method to date. (See Ref. [26] for a review of the general development of the technique, with an emphasis on applications to mass measurement of exotic nuclei.) For example, in 2008, the ion trap experiment at TRIUMF obtained a value 18 of 369.15(65) keV for the two-neutron separation energy of 11Li [27], with much improved precision compared to prior studies.

1.1.2 Coulomb dissociation

In addition to the giant dipole resonances, which are due to collective motion of neutrons and protons and occur at few tens of MeV for nuclei throughout the nuclear chart [28], the spectrum of the Coulomb dissociation cross section of halo nuclei has a peak, due to the motion of the core around the center of mass of the nucleus, at much lower energy. Coulomb dissociation experiments are designed to probe the enhancement of the Coulomb dissociation cross sections in halo nuclei at these low excitation energies. These experiments are performed by bombarding a highly charged stable target like 208Pb with secondary beams of halo nuclei, and making a kinematically complete detection of the outgoing fragments. Subtracting the background from the nuclear contribution to the break up is a key ingredient of Coulomb dissociation experiments. This is normally done by placing a cut to remove the events in which the center of mass of the break up fragments makes a large angle with the initial beam direction. (In the semiclassical approximation, this corresponds to placing a cut to remove small impact parameter events.) Coulomb dissociation of halo nuclei can be understood as E1 excitation to the continuum state due to interaction with the virtual photon field of the heavy target [29, 30]. Contributions from higher multipoles are smaller by several orders of magnitude for the beam energies of the order of 100 MeV per nucleon at which these experiments are typically performed [30]. As shown in Appendix A, assuming a direct reaction mechanism, there are simple relationships between energy, momentum and angular distributions of the Coulomb dissociation cross-section with the matrix element of the electric dipole operator between the bound and the continuum state halo 19 wavefunctions. Analytical and numerical tools for the calculation of such matrix elements are laid out in detail in this dissertation.

1.1.3 Measurement of charge radii in ion traps

One of the important recent developments in experimental studies of unstable nuclei is the laser spectroscopic measurement of nuclear charge radii of trapped ions. As for mass measurement, discussed earlier in Section 1.1.1, the observable actually measured in the experiment is the shift in the energies of electron transitions in isotopes. However, determination of the charge radii requires high precision and careful analysis because, for light nuclei, the mass shift, which is proportional to the difference between the masses of the nuclei, is several orders of magnitude larger than the field shift, which is proportional to the difference between the mean square charge radii of the nuclei. Therefore, to obtain the field shift, the mass shift has to be calculated theoretically and then subtracted from the total isotope shift measured. This method was employed to obtain a value of 2.054±0.014 fm for the rms charge radius of 6He [31], 2.068(11) fm for 8He [32], 2.467(37) fm for 11Li [33], and 2.463(16) for 11Be [34].

1.1.4 Matter radii from cross-section measurements

Nuclear sizes of halo nuclei are mainly determined from the measurement of their interaction and reaction cross sections on stable nuclear targets. In fact, the discovery of halo nuclei was made by the measurements of their interaction cross section [16, 17]. Interaction/reaction cross section of nuclei depend on the structure of the target and the kinematics of the scattering process. Glauber calculations [35–37] have been extensively used to model these dependences and isolate information about the structure of halo nuclei. In the optical limit assumption [38], it is assumed that the projectile and target nuclei present static density distributions whose geometric overlap determines the reaction cross section. (See Ref. [39] for a compilation of matter radii of unstable nuclei, extracted 20 both in the optical limit assumption and with different single particle densities assigned to the core nucleons and the halo nucleons, as prescribed by Refs. [40, 41].) Figure 1.2 shows the interaction radii RI, which is related to the interaction cross section by
2 σI = π RI(projectile) + RI(target) , for several isotopic chains in the low-mass region. Anomalous increases in the values are seen for 6He, 11Li, 11,14Be and 17B.

Figure 1.2: The interaction radii of light nuclei versus the atomic number. Figure taken from Ref. [1]. 21

1.1.5 Fragmentation reactions and momentum distributions

The width of the momentum distribution of fragments in a peripheral nucleus-nucleus collision, predicted by statistical models [42], given by

1/2 σ = σ0 [F(A − F)/(A − 1)] , where F and A are the mass number of the fragment and that of the incident nucleus, respectively, and σ0 ∼ 90 MeV/c, agrees with experiments within 10% for a wide range of nuclei. However, for halo nuclei, the observed width of the fragment momentum distribution is smaller than this by about an order of magnitude. This is because the Heisenberg’s uncertainty principle relates the spatial extension of the wavefunction to its momentum distribution — a large spatial distribution corresponds to a narrow momentum distribution. The presence of a narrow peak in the momentum distribution of the core fragment in neutron removal reactions has, therefore, been widely used (e.g. in Refs. [18, 43–49]) to establish the halo nature of neutron halo nuclei and to estimate the separation energy of the halo neutrons. If the neutron separation energy is known, the width and the shape of the momentum distribution is used to assign orbital angular momentum to the valence neutron — higher orbital angular momentum gives rise to a broader peak because the angular momentum barrier tends to confine the wavefunction in coordinate space. Figure 1.3 shows the longitudinal momentum distributions of the fragments from single-neutron knock-out of 19C on stationary Beryllium and Carbon targets at different beam energies.

1.2 Theories of halo nuclei

1.2.1 The nuclear shell model

The nuclear shell model assumes that, to a first approximation, the interaction of each nucleon in a nucleus with the rest of the nucleons can be represented by an average 22

Figure 1.3: The longitudinal momentum distributions of the fragments for 19C single- neutron removal on Beryllium target, measured at 57 MeV per nucleon at MSU [50], at 64 MeV per nucleon at RIKEN [47]; and on Carbon target, measured at 910 MeV per nucleon at GSI [45]. Figure from Ref. [47].

external field acting on the nucleon, such that the nuclear Hamiltonian can be written as 1

XA ∇2 ! Hˆ = − + u (r ) + residual two- and higher-body interactions, (1.1) 2m i i i=1 N where mN is the nucleon mass and ui(ri) are the single-particle potentials for each of the A nucleons [51]. In the original shell model, a nucleus at a given energy level was assumed to be a collection of noninteracting protons and neutrons, each of which occupies one of the single-particle states of the potential ui(ri) [52]. In this approach, the nuclear “magic numbers”, the number of neutrons or protons in a nucleus at which it is more stable than other nuclei with similar mass, arise naturally due to shell-closure — in analogy to the electrons in an atom. However, in more recent shell model calculations, which seek to obtain a precise quantitative description of a wide variety of nuclear properties, the shell

1 Throughout this dissertation, we adopt natural system of units with c = 1, ~c = 197.327 MeV fm. 23 model nuclear wavefunction can be a linear combination of several configurations in which the “valence” neutrons and protons occupy different single-particle states while the core remains inert [53]. Traditionally, the single particle potential is taken to be of the harmonic oscillator form with a spin-orbit interaction [51]. The radial wavefunctions of these single-particle states display a Gaussian asymptotic behavior — falling off as e−αr2 at large distance r. However, the long-distance wavefunctions of the neutron and the proton in a halo nucleus are, respectively, the Hankel functions and the Whittaker functions [54], which fall off more slowly, as e−βr, at large r. Larger basis sets are therefore required to obtain accurate results for properties that are sensitive to the long-distance part of the nuclear wavefunction. Most of the applications of the shell model to halo nuclei are thus limited to studying the energy levels and their configurations. A review of applications of the nuclear shell model to nuclei at the proton and neutron drip lines, including several halo nuclei, can be found in Ref. [55]. The β decay of 11Be to the various states of 11B was studied in Ref. [56]. Reference [57] calculated the electric quadrupole moments of light mirror nuclei in the shell model and found large enhancement for the proton halos 8B and 17F. In Ref. [58], Brown and Hansen investigated the properties of proton halo states in proton-rich 1s0d shell nuclei, 26,27P and 27S. The momentum distribution of proton-stripping reactions and the Coulomb energies were studied there. Ref. [59] calculated the E1 strengths of 12Be, 14Be and 13O, and found large B(E1) at low excitation energies for 12Be and 13O, in line with the low values of their respective neutron and proton separation energies. Ref. [60] used shell model calculations to describe the excitation energy spectra and the structures of the neutron-rich Carbon isotopes 15−19C. The one- and two-neutron knockout cross sections and their momentum distributions were also calculated for some of these isotopes. Ref. [61] derived the single-particle energies and the residual two-body interactions for neutron-rich Carbon isotopes 16−22C from 24 nucleon-nucleon (NN) interactions in chiral EFT. The ground state energies, excitation spectra and the strengths of M1 and E2 transitions were then calculated. In Ref. [62], the valence-space shell model interactions were derived from chiral EFT NN and three-nucleon (NNN) interactions for open-shell and neutron-rich Oxygen and Carbon isotopes. The energies of the ground and the low-lying excited states were calculated and compared with the data.

1.2.2 Ab initio calculations with nucleons

Ab initio methods seek to describe the properties of nuclei starting from the interactions between nucleons. The major challenge for such an approach is that Quantum chromodynamics (QCD), the fundamental theory of strong interactions, becomes nonperturbative in the regime of low-energy nuclear physics so that the interactions between nucleons cannot be derived from first principles. To overcome this challenge, the nuclear Hamiltonian is constructed with a sufficient number of parameters which are then fit to some of the existing data from NN scattering up to few-hundred MeV lab energy, and energy levels of light nuclei. The construction of this nuclear Hamiltonian is guided by the symmetries of QCD, namely invariance under translation and rotation, Lorentz invariance 2, antisymmetry under exchange, invariance under the transformations of parity and time reversal and (approximate) isospin symmetry. A phenomenological Hamiltonian can be written as

XA ∇2 X X   Hˆ = − + v r Oˆa + three- and higher-body interactions, (1.2) 2m a i j i j i=1 N i, j

where the indices i and j are summed over all A nucleons, which are separated from each

other by distances ri j. The scalar functions va contain fit parameters, with the index a

2 In practice, Galilean invariance is implemented; the relativistic corrections are small for nucleons in nuclei. 25 running over the various operators,

      ˆa 2 Oi j = 1, σi · σ j, τi · τ j, σi · σ j τi · τ j , (σi · rˆ) σ j · rˆ , L · S, L ...

In principle, there is an infinite set of such operators that are consistent with the symmetries mentioned above. However, only a finite linearly-independent subset is needed to fit to the low-energy NN scattering data for low partial waves. The approach of writing down the NN interaction as a linear combination of operators based on symmetry principles was followed in Refs. [63, 64]. Different combinations of operators have been used to construct different NN potentials, e.g. Reid [65], Nijmegen [66], Paris [67],

Urbana v14 [68], Argonne v14 [69], Bonn [70] and Argonne v18 [71]. NNN interactions are small, but are important to accurately describe the properties of nuclei heavier than the deuteron, for which NNN potentials like Urbana [72] and Illinois [73] are often used. At distances of several fermis, the shape of the NN interaction is mostly generated by one-pion exchange, whereas at intermediate range it can be understood as multiple-pion and meson exchanges, and at sub-fermi scale the interaction is strongly repulsive. One of the major recent advances in nuclear physics is the systematic and consistent treatment of NN, NNN and four-nucleon interactions in chiral effective field theory (χEFT) [74–79], which exploits the (broken) chiral symmetry of QCD to express the Lagrangian, written with nucleons and pions as explicit degrees of freedom, as a simultaneous expansion in the pion mass (Mπ) and the typical momentum scale of the phenomenon being studied

(Mlo). For Mlo smaller than Mhi ∼ 500 MeV, the interactions that are higher order in this expansion are less important than the ones that are lower order. Systematic improvements can be made by including more interactions, and estimates can be made for the contribution of the ones that are ignored. Figure 1.4 shows the hierarchy of nuclear interactions in χEFT. 26

Figure 1.4: The nuclear interactions in χEFT at different powers of Q/Λχ = Mlo/Mhi. The solid lines represent nucleons, and the dashed lines represent pions. The small and large dots, squares and diamonds represent vertices that carry different powers of the nucleon momenta. Figure taken from Ref. [80].

The strongly repulsive nature of χEFT interactions and most phenomenological potentials at short distances may become problematic for many-body calculations because it generates large matrix elements away from the diagonal when the potential is written in momentum representation, thus necessitating large basis sizes to accurately represent the many-body wavefunctions. Renormalization Group (RG) methods use unitary transformations to drive these matrix elements closer to zero, thus “softening” the short-distance interaction, and are sometimes employed before performing many-body calculations. An introduction to these methods with a review of applications to the NN system can be found in Ref. [81]. 27

More generally, the increasing complexity of the many-body problem as one tries to study heavier nuclei poses a challenge in applying ab initio methods to study halo nuclei. With the development of technology and algorithms, however, the “computable” region of the nuclear landscape is expanding. Here some of the existing many-body methods applied to study halo nuclei are discussed.

1.2.2.1 The no-core shell model (NCSM)

In the NCSM, the wavefuction of the nucleus, treated as a many-body system of all the A nucleons that constitute it, is expanded in a large harmonic-oscillator basis. As for the conventional shell model, the main difficulty for NCSM in describing halo nuclei is that the wavefunction and the basis states have very different long-distance behaviors, thereby requiring a large basis set to reproduce the correct asymptotics. In Ref. [82], the NCSM with realistic NN interactions and an NNN interaction was used to calculate the binding energies, excitation spectra and level configurations of several light nuclei, including the halo nuclei 6He and 11Be. The ground state energies, level-orderings and level-spacings were found to be substantially sensitive to the presence of the NNN interaction. However, the NNN interaction was dropped in Ref. [83], which studied 9Be and 11Be using the NCSM, to allow for the computational cost of using a larger basis set. The calculated energy levels of 11Be were found to be in reasonable agreement with data. However, the strength of the E1 transition between the two bound states of 11Be was underpredicted by a factor of about 20. In Ref. [84], NCSM was combined with the resonating group method (RGM) to study the scattering observables for several systems including 4He − n and 10Be − n, relevant to the halo nuclei 6He and 11Be. This approach was also used in Ref. [85] to study p + 7Be → 8B + γ. A comparison of the astrophysical S factor obtained there with experimental data and with a Halo-EFT calculation by Zhang et al. is shown in Fig. 1.6. The continuum states of 6He were studied using the 28

NCSM/RGM method in Ref. [86]. Several resonances were found, including the two lowest-lying ones observed at GANIL [87].

1.2.2.2 The hyperspherical harmonics (HH) method

In this method, the Schrodinger¨ equation is written in terms of the hyperspherical co-ordinates, consisting of a hyperradius and hyperangles. The solution of the hyperangular part of the free Hamiltonian are hyperspherical harmonics. The hyperangular part of the many-body wavefunction for the full Hamiltonian is expanded in a finite basis of these hyperspherical harmonics. The hyperradial part of the wavefunction is also expanded in some finite basis. Typically, this is a few-body method used for three- and four-body systems. However, extension to nuclei with higher number of nucleons is possible, albeit it requires a sophisticated algorithm to make the many-body wavefunction, expressed in hyperspherical coordinates, antisymmetric with respect to exchange of protons and neutrons. The ground state energy of 6He [88], and its matter and charge radii [89] have been calculated in this method with RG-evolved NN interactions from χEFT. In Ref. [88], the 6He ground state energy was found to be strongly dependent on the cutoff, Λ, of the RG-evolved Hamiltonian. In Ref. [89], the calculated matter radius was found to overlap with experimental result. However, the calculated value did not agree with the experimental result for the charge radius, which was known with a higher precision. The authors of Refs. [88, 89] attributed these discrepancies to the absence of NNN interactions in their calculations.

1.2.2.3 Coupled cluster (CC) method

Compared to HH and NCSM, this method can be used for a nucleus with much more nucleons, provided that the nucleus can be reached from a closed shell (or sub-shell) reference state with very few excitations. The proton halo nucleus 17F(1/2+), which can be easily reached from 16O(0+), is an example. Reference [90] used the CC method with 29

RG-evolved N3LO NN interactions from χEFT to calculate the spectra of the He isotopic chain, including the halo nuclei 6He and 8He. The results were found to be in agreement with the data within the expected accuracy of a calculation without NNN interactions. The binding energy of 17F(1/2+) was calculated in Ref. [91] and was found to be in agreement with the data. In Ref. [92], Hagen et al. used the CC method with χEFT interactions to study neutron-rich Ca isotopes.

1.2.3 Cluster models

Theoretical studies of halo nuclei are mostly done using potential models for the core-nucleon and the neutron-neutron interactions. This is because the wavefunction of the halo nucleons penetrates deeply into the classically forbidden region, and is therefore less sensitive to the details of the short-distance structure of the core as long as it is tuned to produce a wavefunction with the correct asymptotic form. The parameters of the potential are usually tuned to reproduce the experimental values for some physical quantities and then prediction is made for others. A Woods-Saxon potential was used to calculate the momentum distribution of 7Be and 10Be from fragmentation reaction of 8B and 11Be in Refs. [93, 94], and to deduce the neutron separation energy of 19C from its Coulomb dissociation cross section in Ref. [21]. Reference [95] used a Yukawa interaction to deduce the 19C neutron separation energy from its longitudinal momentum distribution data. In Ref. [96], electromagnetic strengths of proton and neutron halo nuclei were calculated using Woods-Saxon and square well models. The structure of 11Li was first interpreted as a deuteron-like system with a 9Li core weakly bound to a “dineutron” in Ref. [19]. Johannsen et al., by treating 11Li as a three-body system of two neutrons and a 9Li core interacting via short-range Gaussian interactions, were able to explain many of the results of existing experiments [97]. Reference [98] used a Woods-Saxon neutron-core potential and a density dependent 30 contact interaction between the two neutrons to study the dipole excitations in 11Li. References [99, 100] used a Woods-Saxon neutron-core potential and the GPT potential [101] between the neutrons to study the properties of the continuum wavefunction and the strengths of the ground state to continuum transitions in 6He. In Ref. [102], the constraints on the binding energy from a matter radius measurement of Ref. [103], and soft dipole excitations of 22C were calculated in a hyperspherical harmonics model that used a Woods-Saxon potential between 20C and the neutrons and a Gaussian interaction between the two neutrons. Reference [104] used the Faddeev formalism [105] with a renormalized zero-range potential model to study S-wave two-neutron halos and calculated the various mean square distances in the neutron-neutron-core system. In Ref. [104], these calculations were used to constrain the binding energy of 22C from the matter radius datum of Ref. [103].

1.2.4 Halo effective field theory (Halo-EFT)

This section is a review of the existing literature on applications of EFT to halo nuclei. A detailed derivation of certain aspects of the Halo-EFT formalism pertinent to this dissertation is given in Chapter 2.

1.2.4.1 Two-body systems in Halo-EFT

The scattering length, a, of a two-body system describes the low-energy scattering. In the limit of zero-energy, the scattering process is solely described by the scattering length. As the scattering energy of the particles increases from zero, the de Broglie wavelength of the two-body wavefunction shortens and starts to probe the details of the

−1 interaction. At momentum k . a , we can write down equations that are universally

valid, up to corrections of size r0/a, where r0 is the range of the interaction, for all particles regardless of their nature and that of the interaction between them. For example, 31 the binding energy, B, of an S-wave bound state is related to a by the equation r  2µBa2 = 1 + O 0 , (1.3) a where µ is the reduced mass of the two-body system. The fact that we can construct such dimensionless products/ratios of physical observables that have the same approximate

−1 −1 value for all systems as long as k, a  r0 is referred to as “universality”. Halo nuclei, along with ultra-cold atoms 3 and the nucleon-nucleon system, are examples of systems in nature for which the scattering lengths are much larger than the interaction ranges. These seemingly disparate systems, therefore, exhibit (approximately) universal features. In most situations, however, it is important to calculate the “beyond-universal” corrections in order to get agreement with experimental data. Since it yields universal results at LO and then systematically incorporates these beyond-universal corrections order by order as expansion in kr0 and r0/a, Halo-EFT is perhaps the best suited approach to study these systems. Halo-EFT was developed in Refs. [5–9], where the data for low energy NN scattering were reproduced. The electromagnetic form factors of the deuteron were calculated in Ref. [107]. In Ref. [108], the electric polarizability, the electric and magnetic formfactors of the deuteron and the capture cross section 4 for n + p → d + γ were studied. Reference [109] used an alternative “dibaryon” formalism [110] of the theory and presented simplified calculations of the deuteron electric form factor, the n + p → d + γ reaction and the Compton scattering of deuteron. References [111, 112] calculated higher order corrections to n + p → d + γ and obtained a cross section for sub-MeV center of mass energies to 1%. Figure 1.5 shows the deuteron photodissociation cross section, which is related to the cross section of n + p → d + γ by detailed balance, calculated in 3 The scattering length of trapped atoms can be manipulated by using an external magnetic field. Close to a Feshbach resonance, the value is extremely large. A detailed description of this technique, along with the general theory of Feshbach resonance, can be found in Ref. [106]. 4 This cross section is a key ingredient in estimating deuterium abundance in the universe and is therefore relevant to big-bang nucleosynthesis. 32

Ref. [112]. In Ref. [113], nucleon polarizabilities were extracted from deuteron Compton scattering with EFT errors smaller than the statistical errors of the fits, and were found to be in agreement with experimental data, which were scarce and had large uncertainties.

Figure 1.5: Deuteron photodissociation cross section taken from Ref. [112]. References to experimental studies that obtained the data can be found in Ref. [114]. Dashed line is Halo-EFT calculation with one fit parameter, determined by minimizing χ2.

Butler et al. calculated the cross section for the breakup of deuteron by neutrinos and antineutrinos in Halo-EFT [115]. Their result was used by the Sudbury Neutrino Observatory to assign theoretical error bars on the solar neutrino flux measurement [116, 117]. References [11, 12] extended Halo-EFT to include systems with resonant P-wave interactions and used it to study α − n scattering. These studies established the idea that the core, if bound much more tightly than the halo nucleons, can be treated as an explicit degree of freedom in an EFT. The photodissociation of the one-neutron halo nucleus 11Be was studied in Ref. [118]. The strength of the E1 transition between the 1/2+ and the 1/2− states was used as input to successfully predict the strength for bound state to 10Be − n continuum state transition with high accuracy. The reverse process, radiative capture of a 33 neutron, was studied in Refs. [119, 120] for n + 7Li → 8Li + γ. Reference [121] studied these processes for n + 14C → 15C + γ and found good agreement with the neutron-capture and Coulomb dissociation data. Kong and Ravndal extended Halo-EFT to include Coulomb interaction to all orders in the electromagnetic coupling constant [122, 123]. Reference [124]’s calculation of the

+ fusion process p + p → d + e + νe was instrumental in quantifying the uncertainty in the Gamow-Teller matrix element of this reaction [125]. αα scattering was studied using this theory in Ref. [126]. The charge form factor of an S-wave proton halo nucleus, and radiative proton capture on 16O (0+) to 17F (1/2+) were calculated in Ref. [127]. Zhang et al. combined Halo-EFT with ab initio calculations to study n + 7Li → 8Li + γ in Ref. [128], and its isospin mirror, p + 7Be → 8B + γ, which has a high astrophysical significance, in Ref. [129]. Figure 1.6 shows their result for the astrophysical S factor of the E1 contribution to the p + 7Be → 8B + γ reaction.

1.2.4.2 Three-body systems in Halo-EFT

Striking manifestations of universality can also be found in three-body systems. For simplicity we consider the case of spinless bosons, where all three particles are identical. In the unitary limit, i.e. when a → ±∞, if the interaction range goes to zero, Vitaly Efimov showed that there is an infinite sequence of bound states whose binding energies are in a geometric progression, with the density of states increasing as one moves from the deepest bound state towards the threshold [130]. While the ratio of two consecutive bound state energies is a universal number, approximately equal to 515.03, for all systems of three equal-mass particles in the unitary limit, the values of the energies themselves are not. An additional three-body parameter is needed to uniquely determine the energy of one of these bound states and thus determine the whole Efimov spectrum. This can be done by √ requiring the three-body binding momentum, K ≡ − m|E|, to have a value equal to some 34

Figure 1.6: E1 S factor for proton capture on 7Be as a function of the 7Be − p center of mass energy, taken from Ref. [129]. The green shaded band is the range of S (E) at leading order with scattering length varied within its 1σ error. The green error bar at 20 keV shows an estimate of the theory error from higher order terms. The solid black curve is the NCSM/RGM calculation of Ref. [85]. The dashed curve is a Halo-EFT calculation that uses the same set of inputs as Ref. [85]. There are five different sets of data from different experiments.

κ∗ for one of the infinite number of bound states. Physical observables for these bound states are related to κ∗ only through periodic functions of the logarithm of K/κ∗. If the scattering length takes finite values, with the interaction range still kept at zero, the bound state energies, although no longer in a geometric progression, are still related by discrete scaling laws, as can be seen in Fig. 1.7. It was Refs. [10, 132] that laid the foundation for Halo-EFT treatment of three-body systems by developing a consistent renormalization procedure at LO. An ultraviolet cutoff Λ was introduced to regularize the three-body equation. Independence of physical observables on Λ was achieved by promoting the three-body contact interaction to LO and by choosing the appropriate running of this coupling constant with Λ. At a given cutoff,

Λ∗, the short-distance physics of the three-body system determines the physical value of 35

Figure 1.7: The a−1K plane for the three-body problem. The allowed regions for three-atom scattering states and atom-dimer scattering states are labeled AAA and AD, respectively. The hatching indicates the threshold for scattering states. The y−axis represents the unitary limit. The thick curves labeled T are the Efimov states. The y−intercepts of these curves are in a√ geometric progression with a common ratio of approximately 2.2 instead of 22.7 ' 515.03 because a−1 and K are parametrized as a−1 = r cos ξ and K = r sin ξ, but r1/4 cos ξ and r1/4 sin ξ are plotted instead to cover large ranges of a−1 and K. Figure taken from Ref. [131].

the three-body force parameter, H(Λ∗), which is related to κ∗ described above and thus fixes the value of one of the Efimov states. As a consequence of universality, the properties of all the states in the Efimov spectrum can then be expressed at LO as functions of this three-body force parameter, the two-body scattering lengths, and the mass of the particles. Figure 1.8 shows the running of H(Λ) required to achieve independence of physical observables on Λ. The discrete scaling symmetry of the Efimov spectrum now appears as the log-periodic behavior of H(Λ). There is an infinite family of Λ values for which H(Λ) is the same. Λ may be chosen such that H(Λ) = 0. Higher order corrections were calculated for the system of three identical bosons in Refs. [13–15], for the nd system in Refs. [133–135], and for the pd system in Refs. [136–138]. 36

Figure 1.8: Three-body force parameter H as a function of the cutoff Λ for particle- dimer scattering length a3 = 1.56a2, where a2 is the particle-particle scattering length, Λ∗ = 19.5/a2. The solid line is an approximate result obtained from an analysis of the ultraviolet behavior of the three-body equations and the dots are the numerical results. Figure taken from Ref. [132].

In Refs. [139, 140], the matter form factors, rms matter radii, and possibility of excited Efimov states were studied for various S-wave two-neutron halo nuclei, including 11Li and 14Be. Figure 1.9 shows their results for rms matter radii of 11Li. Using the nα and nn interactions with parameters determined from scattering data, Ref. [141] solved the three-body dynamics of the 6He system, in which the nn interaction is predominantly S-wave whereas the nα interaction is dominant in the P-wave channel, in the Gamow shell model formalism. Reference [142] studied the charge form factor of two-neutron halo nuclei and made predictions for the charge radii of several nuclei. The possibility of 62Ca, which has not been produced yet but is potentially within the reach of next generation of radioactive ion beam facilities, to have excited Efimov states was explored in Ref. [92]. It was concluded that excited Efimov states possibly exist in the 62Ca system. Ji et al. studied 6He in Ref. [143]. The integral equations that describe the 37

Figure 1.9: The various rms distances in the 11Li nucleus as a function of the 9Li − n energy (with negative values signifying that this subsystem is unbound), taken from Ref. [140]: distance between neutrons (top left), 9Li−n distance (top right), and distance of 9Li (bottom left) and that of the neutron (bottom right) from the 11Li center of mass. The LO results of Ref. [139] are represented by the solid black lines, with error bands represented by dashed lines. The NLO results are represented by the red dotted-dashed lines with purple error bands. The three-body binding energy and the two-body effective ranges were taken to be 247 keV and 1.4 fm respectively.

bound state properties at leading order were renormalized, and the ground state wavefunction was computed.

1.3 This dissertation

The goals of this dissertation are to: 38

• make predictions for observables like matter radii of two-neutron halo nuclei and Coulomb dissociation cross sections for one-neutron halo nuclei so that comparisons can be made with existing data and predictions can be made for future experiments;

• combine efforts with the works in Refs. [92, 118–121, 127–129, 139, 140, 142–144] in extending the Halo-EFT formalism for calculation of new physical quantities like the Coulomb energy of proton halos and the dipole excitation strengths of Borromean halo nuclei;

• look for new aspects of the universal physics in the three-body system that emerges from large two-body scattering lengths.

The work included in this dissertation is as follows. In Ref. [145], we used the Coulomb dissociation data on 19C as input to determine the separation energy as well as the asymptotic normalization coefficient of the halo neutron. We then used these values to make prediction for the longitudinal momentum distribution of the fragments. In Ref. [146], we studied the universal relations between the rms matter radii and the two- and three-body energies in S-wave two-neutron halo nuclei. We then applied these findings to constrain the poorly known energies of 21C and 22C using experimental datum on the matter radius of 22C. In Ref. [147], which is yet to be published, we derive universal relations for the E1 dissociation of S-wave two-neutron halo nuclei. In the unitary limit, we discover a simple scaling of the dissociation spectrum with the three-body binding energy. We compare the results with data from Coulomb dissociation of 11Li and make predictions for the outcome of similar experiment on 22C. 39 2 Effective Field Theory for Halo Nuclei:AReview of

the Formalism

Effective field theories [148, 149] have been extensively used in nuclear and particle physics to study physical phenomena at low energy or long distances when the high-energy or short-distance substructures and interactions are unresolved. (Reference [150] is a classic review.) These theories are based on a simple yet powerful tenet: phenomena at low energy do not probe the details of the short-distance physics. Construction of an EFT for a physical system begins with writing down the most general Lagrangian consistent with all the symmetries of the system in terms of effective degrees of freedom that are relevant to describe physical phenomena in the energy regime being considered. The feasibility of this approach rests on our ability to classify the terms in this Lagrangian according to their importance, so that approximate results for physical observables can be calculated by retaining only a small number of these terms and systematic improvements in the approximation can be made by including more and more terms. This is done by exploiting the separation of scales; if the typical momentum of the phenomenon being studied, Mlo, is much smaller than Mhi, the momentum scale at which the details of the short-distance/high-energy physics begin to be resolved, it is possible to calculate physical observables order by order in powers of the ratio of Mlo to Mhi. This ratio also provides an estimate of the theory error due to the neglected terms in the

Lagrangian. A small value of this ratio facilitates calculation. When Mlo ∼ Mhi, the EFT expansion breaks down.

For a system of nucleons at Mlo much smaller than the pion mass, a pionless effective

field theory (EFT(π )) Lagrangian can be constructed with only nucleonic degrees of

freedom. The breakdown scale of this theory, Mhi, is then given by Mπ ∼ 140 MeV. The

−1 corresponding length scale Mhi ∼ 1.4 fm is approximately the range of the nuclear 40 interaction between nucleons. Consistent rules for power counting in Mlo/Mhi for this theory were developed in Refs. [5–9]. Although initially developed for nucleons, this theory can be applied to any system of nonrelativistic particles in which the S-wave scattering length is much larger than the interaction range. For example, the two-body scattering length of trapped atoms can be tuned to values much larger than the range of the van der Waals interaction between them by applying an external magnetic field. When applied to this system, Mhi is set by the range of the van der Waals interaction. In this light,

−1 this theory can be thought of as an expansion in r/a, where the interaction range, r ∼ Mhi ,

−1 and the scattering length, a ∼ Mlo . For halo nuclei, Mhi is set either by the size of the core or its first excitation energy, whichever gives a lower value. Since the size of a weakly bound two-body system can be approximated by the scattering length, Halo-EFT can also be thought of as an expansion in the ratio of the size of the core to that of the halo nucleus. This chapter contains a brief summary of the EFT formalism for non-relativistic particles whose short-range interactions generate large two-body S-wave scattering lengths. The results presented here were all originally derived elsewhere, by other authors, as indicated in the text. Two-body systems are discussed in the formalism of Ref. [110] in Section 2.1. The power counting rules and the subtraction scheme adopted in this section were developed in Refs. [5–9]. This theory can be applied to study the low energy properties of S-wave one-neutron halo systems. Section 2.2 contains EFT for nonrelativistic particles with a short-range interaction plus the long-range Coulomb interaction, developed in Ref. [122]. This theory has been applied to proton halo nuclei. To study electromagnetic properties of nuclei in Halo-EFT, the power-counting rules have to be extended to terms containing external photon fields. This was first done for the deuteron in Ref. [107]. The calculation of some of the electromagnetic observables are presented in Section 2.3. In Section 2.4, the EFT of two-neutron halos, systems of three non-relativistic particles, two of which are identical, is discussed. The formalism of 41

Ref. [142] is followed here. The renormalization technique used was developed in Ref. [10].

2.1 Two-body systems

In the absence of interactions, the Lagrangian density for a system of two non-relativistic particles of masses m0 and m1, represented by fields ψ0 and ψ1 respectively, is X ∇2 ! L(free) = ψ † i∂ + ψ . (2.1) x 0 2m x x=0,1 x The Euler-Lagrange equations for this Lagrangian are the free Schrodinger¨ equations,

∇2 ! i∂0 + ψx = 0; x = 0, 1. (2.2) 2mx

The particle propagators are given by

i iS (pµ) = ; x = 0, 1. (2.3) x p2 p0 − + i 2mx Only zero-range interactions are included at leading order. This is done by adding to the Lagrangian in Eq. (2.1), the term

κ 4−n L(int) = Cψ †ψ †ψ ψ , (2.4) 2 0 1 0 1

−  κ 4 n where C is the coupling constant and the factor of 2 ensures that the dimensionality of C stays equal to inverse-square of energy even when the theory is continued to n, the number of space-time dimensions, less than four. The amplitude for the scattering of the two particles is represented by the Feynman diagrams in Fig. 2.1.

Figure 2.1: Feynman diagrams contributing to the amplitude for the Lagrangian at LO. 42

Applying dimensional regularization, in Power Divergence Subtraction (PDS) scheme [5], this infinite series of diagrams can be summed exactly to give the amplitude

2π 1 T LO = , (2.5) µ 2π µCPDS + κ + ik where CPDS is the renormalized coupling constant, k is the relative momentum of the particles, and µ is the reduced mass. Matching to the effective range expansion (ERE) for S-wave scattering [151],

1 1 k cot(δ(k)) = − + r k2 − Pr 3k4 + O(k6), (2.6) a 2 0 0 at k → 0, we can relate the renormalized coupling constant to the scattering length,

2π/µ CPDS = . (2.7) 1/a − κ

Range effects can be introduced by adding derivative-carrying contact interactions to the Lagrangian. An alternative way to account for the non-zero range is through the auxiliary field formulation of Ref. [110]. The Lagrangian in this formulation is written as

X ∇2 ! " ∇2 # ! h i L(range) = ψ † i∂ + ψ + d† σ i∂ + + ∆ d − g d†ψ ψ + ψ†ψ†d , (2.8) x 0 2m x 0 2m 0 1 0 1 x=0,1 x where d is an auxiliary field that represents the bound/resonance/virtual state, m is the mass of this state and σ = −1. The LO Lagrangian discussed above can be recovered from Eq. (2.8) by setting σ = 0, and then integrating out the d field [152]. The amplitude of the scattering of the two particles can now be represented as shown in Fig. 2.2.

Figure 2.2: The scattering amplitude for the Lagrangian in Eq. (2.8). The double (thick) line represents the bare (dressed) propagator of the d field. 43

In dimensional regularization with the PDS scheme, the amplitude is g2D(E, 0), where E = k2/(2µ) is the energy in the two-body center of mass frame and i D ν i (p ) =  2  (2.9) − p − ν ∆ + σ p0 2m Σ2(p )

is the dressed propagator of the d field, where −iΣ2 is the ultraviolet divergent self-energy. In dimensional regularization, with the PDS scheme, the finite piece of the self-energy is r µg2 p2 − iΣ(R)(pν) = −i 2µ(−p + − i). (2.10) 2 2π 0 2m The divergence at δ → 0 is absorbed into the bare ∆ to obtain µg2 κ ∆(R) = ∆ + . (2.11) 2π δ We can relate ∆(R) and g to the scattering length and the effective range by the equations µg2 2π a = ; r = −σ , (2.12) 2π∆(R) 0 µ2g2 to obtain the scattering amplitude up to second order in the ERE of Eq. (2.6), 2π 1 T NLO = − . (2.13) µ − 1 1 2 − a + 2 r0k ik Using the completeness relation for the bound and unbound eigenstates of any Hamiltonian Hˆ , X Z d3q 1ˆ = |ψ ihψ | + |ψ(q)ihψ(q)|, (2.14) n n (2π)3 n the full Green’s function Gˆ(E), defined as 1/(E − Hˆ ), can be written as X 1 Z d3q 1 Gˆ(E) = |ψ i hψ | + |ψ(q)i hψ(q)|, (2.15) n n 3 q2 E + Bn (2π) − n E 2µ + i

where −Bn are the bound state energy eigenvalues. Using the expansion of Gˆ(E) in terms of the free Green’s functions, Gˆ0(E), and the potential, Vˆ ,

Gˆ(E) = Gˆ0(E) + Gˆ0(E)Vˆ Gˆ0(E) + Gˆ0(E)Vˆ Gˆ0(E)Vˆ Gˆ0(E) + ... (2.16)

= Gˆ0(E) + Gˆ0(E) Tˆ(E) Gˆ0(E), (2.17) 44

Eq. (2.15) gives the following relationship between the bound state wavefunction, |ψni and the operator Tˆ for any interaction.

ˆ ˆ−1 ˆ−1 lim (E + Bn) T(E) = G0 (−Bn) |ψnihψn| G0 (−Bn). (2.18) E→−Bn

The S-wave second order ERE amplitude in Eq. (2.13), therefore, yields a bound state wave function √ 1 hp|ψ i π A , n = 4 e 2 2 (2.19) p + γn p where γn = 2µBn is the real and positive square root of the equation

1 1 + r γ2 − γ = 0, (2.20) a 2 0 n n

and s 2γ Ae = n (2.21) 1 − r0γn is the one-dimensional asymptotic normalization coefficient (ANC) [153]. The corresponding coordinate-space wavefunction is

exp (−γnr) hr|ψni = Ae √ . (2.22) 4π r

The wavefunction renormalization of the dressed d propagator is given by the residue

of its pole at E → −Bn, 2πγ 1 Z n . d = 2 2 (2.23) µ g 1 − r0γn The LSZ reduction formula [154] replaces all dressed d propagators appearing as external √ legs in Feynman diagrams with a factor of Zd.

The leading order wavefunctions can be obtained by taking the limit r0 → 0 in Eqs. (2.21) and (2.23). Thus the dominant effect of including range correction is the modification of the normalization of the zero-range wavefunction by the factor of

− 1 (1 − r0γn) 2 [155]. 45

2.2 Proton halo nuclei

In this section, the theory necessary for a Halo-EFT treatment of proton halo nuclei is summarized. The formalism discussed here is due to Kong and Ravndal in Refs. [122, 123]. Extension of this theory to 8B, which has a proton and a 7Be core in P-wave was done in Ref. [129]. Here, only S-wave proton-core systems, like 17F(1/2+), are considered. To describe proton halo nuclei, the Lagrangian in Eq. 2.8 must be augmented with electromagnetic interactions. This is done by minimal substitution:

∂µ → Dµ = ∂µ + ieQAˆ µ, where Aµ is the photon field, Qˆ is the charge operator with eigenvalue 1 for the proton field and Z for the core field. This substitution generates terms which represent coupling of photons to the ψ0 and the ψ1 fields. In an EFT, one must also add all possible contact interactions that are consistent with the symmetries of the system and with gauge invariance. These interactions, however, do not appear at NLO. The contributions of the transverse photons do not appear either. It is therefore convenient to work in the Coulomb gauge, in which the Coulomb photons decouple from the transverse ones. When the Coulomb momentum, kc = Zαemµ, is much smaller than the relative momentum, p, between the particles, we can perform a simultaneous expansion in the electromagnetic coupling, αem, in addition to the EFT expansion. The amplitude for the scattering of the two particles due to an interaction consisting of the Coulomb and the short-range potential,     Tˆ = Vˆ + VˆC + Vˆ + VˆC Gˆ0Tˆ, (2.24)

0 ˆ 4π 0 2 where hk |VC|ki = µ kc/ (k − k) , can then be expanded as [54]   ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 T = VC + TS + VCG0TS + TS G0VC + TS G0VCG0TS + O VC , (2.25)

where TˆS is the amplitude due to the short-range interaction only, as calculated in Section 2.1. This equation can be represented by Feynman diagrams as shown in Fig. 2.3. 46

Figure 2.3: The scattering amplitude for a two-body system with Coulomb interaction in addition to the short-range interaction, as an expansion in αem. The dashed and the thin lines represent the particle propagators and the thick line represents the dressed propagator of the bound/virtual state.

If the Sommerfeld parameter, k η(k) ≡ c , (2.26) k is not much smaller than 1, the expansion shown in Eq. (2.25) and in Fig. 2.3 is either divergent or, at best, slowly convergent. To overcome this, we split the amplitude Tˆ into

the pure Coulomb amplitude, TˆC, and the Coulomb-distorted short-range amplitude, TˆCS , and use the Gellmann-Goldberger relation [156], so that

0 ˆ 0 ˆ 0 ˆ (−) ˆ (−) ˆ (+) hk |T|ki = hk |TC|ki + hk |TCS |ki = hχk0 |VC|ki + hχk0 |VS |ψk i, (2.27)

(−/+) (−/+) where |χk i and |ψk i are, respectively, the (incoming/outgoing) pure Coulomb and Coulomb-distorted short-range interaction eigenstates of momentum k. The co-ordinate space Coulomb wavefunction is

ηπ hr|χ(±)i = χ(±)(r) = exp(− )Γ(1 ± iη)Φ(∓iη, 1; ±ikr − ik.r) exp(ik.r), (2.28) k k 2

where Φ(a, b; x) is Kummer’s confluent hypergeometric function. At r = 0, the Coulomb wavefunction satisfies

2πη (±)∗ (±) 2 (∓)∗ (±) 2 ±2iσ0 χ 0 (0)χ (0) = ≡ C ; χ 0 (0)χ (0) = C e , (2.29) k k e2πη − 1 η k k η

2 where Cη is the Sommerfeld penetration factor and σ0 = Arg{Γ (1 + iη)} is the s-wave

Coulomb phase-shift. For a repulsive Coulomb force, the pure Coulomb amplitude, TC, is

analytic throughout the complex energy plane. So we only look at TCS for bound states. Defining the outgoing and incoming Coulomb Green’s function at energy E, 47

ˆ(±) ˆ ˆ GC (E) = 1/(E − HC ± i), where HC is the sum of the free Hamiltonian and the Coulomb interaction, with eigenstates defined by

k2 Hˆ |χ(±)i = |χ(±)i, (2.30) C k 2µ k

we can obtain the Coulomb analogue of the Lippmann-Schwinger equation,

(+) (+) ˆ(±) ˆ (+) |ψk i = |χk i + GC VS |ψk i. (2.31)

The Coulomb-distorted short-range amplitude can, therefore, be written as

∞ X  n (−) ˆ ˆ ˆ (+) TCS = hχk0 |VS GCVS |χk i. (2.32) n=0 Equation (2.32) can be depicted using Feynman diagrams as shown in Fig. 2.4.

Figure 2.4: Non-perturbative resummation of TCS , represented by the shaded rectangle, as described in Eq. (2.32). The shaded ellipse represents propagation in the presence of Coulomb photons, as shown in the second row. The pure Coulomb amplitude involves the sum of all but the first diagram on the right-hand-side of the second row.

Using the completeness relation of the Coulomb wavefunctions,

Z d3q |χ(±)ihχ(±)| = 1ˆ, (2.33) (2π)3 q q

the right hand side of Eq. (2.32) can be calculated in n spatial dimensions using the PDS scheme with the regulator κ, giving

2 2iσ0 2π Cηe TCS = − (R) , (2.34) 2π∆ 2 µ − C − πk − µg2 σ µ2g2 2kcH (η(k)) 48 where √ 2 " !#! (R) µg κ 1 3γ π κ ∆C ≡ ∆ + + 2kc − 1 + − log (2.35) 2π δ δ 2 2 kc is the renormalized residual mass, where γ is the Euler-Mascharoni constant, and

1 H(η) ≡ ψ (iη) + − log (iη) , (2.36) 2iη

where ψ(x) is the digamma function. From Eq. (2.34), we can identify the effective range parameters, µg2 2π a = ; r = −σ . (2.37) C (R) 0 µ2g2 2π∆C

Note that the effective range, r0 is unmodified at this order in EFT — only the scattering length and the unitarity term get modified due to the introduction of the Coulomb interaction [123].

2.3 Electromagnetic observables in Halo-EFT

This section contains some simple examples of calculations of electromagnetic observables in Halo-EFT for the simplest case — an S-wave one-neutron halo nucleus, described by the Lagrangian in Eq. (2.8) with the minimal substitution:

∂µ → Dµ = ∂µ + ieQAˆ µ, where Qˆ has eigenvalue 0 for the neutron field ψ0 and Z for the core field ψ1.

2 The electric formfactor, FE(q ) is defined as the three-dimensional Fourier transform of the electric charge distribution. The mean-square electric charge radius can be obtained by performing the Taylor’s series expansion of the formfactor around q2 = 0, yielding

2 2 ∂FE(q ) hrEi = −6 lim . (2.38) q2→0 ∂q2

The leading contributions to the electric formfactor come from the irreducible dA0d diagrams shown in Fig. 2.5. The diagram on the left contains not only the minimal

coupling of the A0 field to the d field, but also contains contact interactions generated by 49

† terms like LE0 d (∇ · E)d in the Lagrangian. According to naive dimensional analysis

(NDA), the coefficient LE0 carries two powers of Mlo/Mhi. However, modification to NDA to accommodate a shallow S-wave bound state takes this contribution one order higher in

3 Mlo/Mhi expansion. This diagram is therefore suppressed by a factor of (Mlo/Mhi) compared to the one on the right, which contributes at leading order.

Figure 2.5: Feynman diagrams contributing to the charge form factor. The dashed and the thin lines represent the core and the neutron propagators, respectively. The thick line represents the dressed propagator of the neutron halo nucleus. The diagram on the left (monopolar coupling) is suppressed compared to the one on the right (dipolar coupling). At LO, only the diagram on the right needs to be considered.

In the Breit frame, in which the photon carries momentum q and zero energy, with √ each of the d propagators appearing as external legs replaced by a factor of Zd, this

2 diagram is equal to −ieZFE(q ), where " #!−1 Z d3 p    m q2 F (q2) = 8πγ p2 + γ2 p + 1 + γ2 . (2.39) E 0 (2π)3 0 m 0 Evaluating the integral, and using the leading order expression for the wavefunction

renormalization, Zd, we get ! 2 2m0γ0 µ|q| FE(q ) = arctan . (2.40) µ|q| 2m0γ0 The mean square charge radius is then given by 1 µ2 h 2 i rE = 2 2 . (2.41) 2 m0γ0 50

This expression was obtained by treating the core as a point particle. The non-zero size of the core is accounted for by simply adding this result to the mean-square charge radius of the core. 1 µ2 h 2 i h 2 i rE = rE core + 2 2 . (2.42) 2 m0γ0 This prescription can be easily derived from the relationship between the charge density distributions of the one-neutron halo nucleus in the center of mass of the halo nucleus and in the center of mass of the core. For a one-neutron halo nucleus, the cross section for photodissociation (and that for its reverse process, radiative neutron capture) can be obtained by calculating the Feynman diagrams shown in Fig. 2.6. For processes dominated by E1 transitions, we need to only consider the terms involving the spatial derivatives acting on the core field [119]. This

ieZ generates a ψ0γψ0 vertex factor of  · k0, where k0 is the core momentum, and  is the m0 polarization vector of the photon. This is equivalent to the approach of Ref. [118], where

Ward identity was used to relate this coupling to that of A0 photons, which was then used to calculate E1 photon processes.

+++. . .

Figure 2.6: Diagrams contributing to the amplitude for the photodissociation of the one- neutron halo into the core and the neutron. The dashed and the thin lines represent the core and the neutron propagators, respectively. The thick line represents the dressed propagator of the neutron halo nucleus. The dots represent diagrams which are higher order in the EFT expansion parameter.

The first diagram in Fig. 2.6 is enhanced by Mhi/Mlo because the ψ0γψ0 vertex carries a factor of momentum. Conservation of angular momentum and parity requires that the

ψ0ψ1 vertex in the second diagram be a derivative interaction. In the absence of shallow 51

2 2 P-wave bound states or resonances, this diagram scales as Mlo/Mhi. The third diagram comes from including all gauge invariant contact interactions allowed by symmetry. Even

† m1 m0 the most important of these, LE1 d [(i m ∇ψ0)ψ1 − ψ0(i m ∇ ψ1)] · E + H.C., where H.C. represents the hermitian conjugate of the preceding term, only contributes at

3 3 Mlo/Mhi [118]. Up to, NNLO, the photodissociation amplitude is, therefore, given by the first diagram in the series. The squared amplitude is

e2Z2 1 |M|2 Z g2  · p 2 , = d 2 ( ) 2 (2.43) m0 k where, p is the momentum of the core, and the photon energy is given by

2 2 k = |k| ' (p + γ0) /(2µ) where k is the momentum of the photon. Note that we work in the neutron-core (or, equivalently, halo nucleus-photon) center-of-mass frame, and assume that all particles except the photon are nonrelativistic in this kinematics so that terms that are suppressed by the ratio of k to the mass of the nonrelativistic particles can be dropped. We align the z−axis with the direction of the photon so that the polarization vector lies on the xy−plane. The polarization-averaged squared amplitude is then

e2Z2 p2(1 − cos2 θ) 1 |M|2 Z g2 , = d 2 2 (2.44) m0 2 k where θ = pˆ · kˆ. The polarization-averaged photonuclear cross section is then given by [157] 5 Z +1 γ µ p 2 σE1 = d cos θ |M| . (2.45) 4π k −1 Using Eqs. (2.23) and (2.44), we obtain

4π γ Z2 p3 σγ = α 0 . (2.46) E1 − 2 3 3µ 1 r0γ0 m0 k This equation was first derived by Bethe and Longmire for the deuteron in Ref. [155].

The LO result, with r0 → 0, was derived in the earlier Ref. [158]. 5 Note that this equation looks different from the ones in Ref. [157]. This is because, in a nonrelativistic † theory, we define the free particle states by |ki = ak|0i, whereas Ref. [157] uses the relativistic normalization, √ † |ki = 2Ekak|0i, for all particles. 52

This photodissociation cross section is related to the Coulomb disscociation cross section, σ, discussed in Section 4.1, by

Z dω σ = N (ω, R) σγ (ω), (2.47) ω E1 E1

where ω is the photon energy and the function NE1(ω, R) is derived in Appendix A.

2.4 Two-neutron halo with a spinless core

Halo-EFT treatment of two-neutron halo nuclei was made possible by Ref. [10]’s discovery that a three-body contact interaction, unless it is precluded by Pauli principle, is required to regularize the ultraviolet behavior of this theory at leading order, and that this three-body force can be renormalized by using one three-body datum as input. We discuss this renormalization procedure in the context of two-neutron halo nuclei in the formalism of Refs. [139, 142]. Using the subscripts 0 and 1 for the core and the neutron respectively, with the two-body subsystem labeled by the subscript for the spectator particle, the LO Lagrangian can be written as [10, 142]

2 ! 2 ! † ∇ † ∇ † h † † † i L = n i∂0 + n + c i∂0 + c + ∆1d1d1 − g1 d1nc + c n d1 2m1 2m0 † g0 h † T T † i † h † † † i +∆0d d0 − √ d n Pn + (n Pn) d0 + Ωt t − h t cd0 + d c t , (2.48) 0 2 0 0

where n and c are the neutron and the core fields, and d j and t are the auxiliary fields

representing the two and three-body bound/resonant states. n and d1 are

two-component-spinor fields, whereas c, d0 and t are scalar fields. The symbol P denotes √ the spin-singlet projector, iσ2/ 2, such that      1       0 √  n↑ 1 T  2   
n Pn = n↑ n↓     = √ n↑n↓ − n↓n↑ . (2.49)  1    − √ 0  n↓ 2 2 53

† 0 2 The static field t gives a (cd0) cd0 coupling of (−ih = −ih /Ω), which is required to renormalize the three-body problem at LO. Equivalently, we could have defined a t field that couples with n and d1 instead [142]. Let Pµ be the total four-momentum of the three-body system in an arbitrary inertial frame of reference, M be the total mass, µ j is the reduced mass of the two-body subsystem labeled by the spectator particle j, m0 + m1 = M − m1 ≡ m1, and 2m1 = M − m0 ≡ m0. We

m j µ µ write the four-momenta of the particle propagators as M P − p and those of the d j

m j µ µ 6 propagators as M P + p . We define  2 m j P − p 0 m j 0 M p˜ j ≡ P − , (2.50) M 2m j

which is equal to p0 if the corresponding particle propagator is on-shell 7.

In this kinematics, the dressed propagators of the d j fields are given at leading order by

2π ! µ g2 m j µ µ j j iD j P + p = i (2.51) q  2 h i M 1 0 0 p 1 1 − 2µ j −E3 − p + p˜ + + − i a j j 2 m j m j

" # ! i p2 1 1 0 − 0 − = 2 t j E3 + p p˜ j + + i , (2.52) g j 2 m j m j P2 where E ≡ P0 − is the total energy of the system in the three-body center of mass 3 2M

frame, and t j(E) is the leading order two-body effective-range amplitude at energy E in

2 the two-body center of mass frame. This propagator has a pole at E = −1/(2µ ja j ) with residue, 2π Z j = 2 2 . (2.53) µ j g j a j

6 This does not cause any loss of generality because only the sum of the four momentum of the particle and that of the d j propagator is constrained by energy-momentum conservation. 7 This is true for all the internal lines because of the integral over the zeroth component of the loop momentum, and also for the external lines. 54

= + +

= + +

Figure 2.7: Faddeev equations for the three-body scattering amplitudes. The solid and the dashed lines represent the propagators of the n and the c fields respectively, the thick gray line represents the dressed propagator for the d0 field and the thick black line represents the dressed propagator for the d1 field. The n − d1 scattering amplitudes are shown as black blobs and the coupled-channel ones as gray blobs.

For virtual states, which have negative a j, we redefine Z j by its magnitude. For later convenience, we define

1 1 1 ≡ + . (2.54) eµ j m j m j

In this kinematics, the free three-body Green’s functions are

(1 − δ δ ) µ µ 0µ i0 j0 Gi j(P , p , p ) =  2 2 , (2.55) 0 0 0 0 p p0 p.p0 E3 + p − p˜ + p0 − p˜0 − − − + i i j 2µ j 2µi M−mi−m j where pµ and p0µ correspond to spectator i and the spectator j channels respectively.

µ µ µ Defining Ti j(P , p , p0 ) as the amplitude for scattering from a channel with spectator i to the one with spectator j, the equations depicted in Fig. 2.7 can now be written as:

µ µ µ 2 µ µ µ iT11(P , p , p0 ) =Z1(−ig1) iG11(P , p , p0 ) Z d3q p + Z (−ig )2iG (Pµ, pµ, qµ) (2π)3 1 1 11

2 µ µ µ p iD1(E3 − q /(2eµ1) + i, 0) iT11(P , q , p0 )/ Z1 Z d3q p √ + Z (−ig )iG (Pµ, pµ, qµ)(−i 2g ) (2π)3 1 1 10 0

2 µ µ µ p iD0(E3 − q /(2eµ0) + i, 0) iT01(P , q , p0 )/ Z0, (2.56) 55 and √ µ µ µ p µ µ µ p iT01(P , p , p0 ) = Z0(−i 2g0)iG01(P , p , p0 )(−ig1) Z1 Z d3q p √ + Z (−i 2g )iG (Pµ, pµ, qµ)(−ig ) (2π)3 0 0 01 1

2 µ µ µ p iD1(E3 − q /(2eµ1) + i, 0) iT11(P , q , p0 )/ Z1 Z d3q p + Z (−ih0) (2π)3 0

2 µ µ µ p iD0(E3 − q /(2eµ0) + i, 0) iT01(P , q , p0 )/ Z0. (2.57)

Defining the partial-wave projected amplitude by

µ µ µ X m m∗ (l) Ti j(P , p , p0 ) = 4π Yl (pˆ)Yl (pˆ0)Ti j (Ei j, p, p0), (2.58) l,m and the S-wave projected Green’s functions by

Z 1 (0) 1 µ µ µ Gi j (Ei j, p, p0) = d(pˆ.pˆ0) Gi j(P , p , p0 ) (2.59) 2 −1 " 2 2 # ! M − mi − m j M − mi − m j p p0 = −(1 − δi0δ j0) Q0 −Ei j + + − i , pp0 pp0 2µ j 2µi

0 0 0 0 where Ei j = E3 + p − p˜i + p0 − p˜0 j and Q0(z) = −Q0(−z) is the Legendre function of the second kind of order 0, which satisfies  1 |x + 1|  log , |x| > 1  Q (x − i) =  2 |x − 1| (2.60) 0  1 |x + 1| π  log + i , |x| < 1,  2 |x − 1| 2 we get the S-wave projected equations:

(0) 2 (0) T11 (E11, p, p0) = − Z1g1G11 (E11, p, p0) Z Λ dq q2 G(0) E p0 − p0, p, q t E − q2/ µ i + 2 11 ( 3 + ˜1 ) 1( 3 (2e1) + ) 0 2π (0) 0 0 T11 (E3 + p0 − p˜01, q, p0) s Z Λ 2 √ dq Z1g q2 1 G(0) E p0 − p0, p, q t E − q2/ µ i + 2 2 2 10 ( 3 + ˜1 ) 0( 3 (2e0) + ) 0 2π Z0g0

(0) 0 0 T01 (E3 + p0 − p˜01, q, p0), (2.61) 56

= + +

= + +

Figure 2.8: Faddeev equations for the three-body scattering amplitudes. The solid and the dashed lines represent the propagators of the n and the c fields respectively, the thick gray line represents the dressed propagator for the d0 field and the thick black line represents the dressed propagator for the d1 field. The c − d0 scattering amplitudes are shown as white blobs, and the coupled-channel ones as gray blobs.

and √ q q (0) 2 2 (0) T01 (E01, p, p0) = − 2 Z0g0 Z1g1G01 (E01, p, p0) s Z Λ 2 √ dq Z0g q2 0 G(0) E p0 − p0, p, q t E − q2/ µ i + 2 2 2 01 ( 3 + ˜0 ) 1( 3 (2e1) + ) 0 2π Z1g1

(0) 0 0 T11 (E3 + p0 − p˜01, q, p0) Z Λ dq H(Λ) q2 t E − q2/ µ i T (0) E p00 − p00, q, p0 , + 2 2 0( 3 (2e0) + ) 01 ( 3 + ˜ 1 ) 0 2π Λ (2.62)

2 2 0 2 h where H(Λ)/Λ = h /g0 = 2 . g0Ω Similarly, the Faddeev equations in Figure 2.8 are

√ q q (0) 2 2 (0) T10 (E10, p, p0) = − 2 Z0g0 Z1g1G10 (E10, p, p0) Z Λ dq q2 G(0) E p0 − p0, p, q t E − q2/ µ i + 2 11 ( 3 + ˜1 ) 1( 3 (2e1) + ) 0 2π (0) 0 0 T10 (E3 + p0 − p˜00, q, p0) s Z Λ 2 √ dq Z1g q2 1 G(0) E p0 − p0, p, q t E − q2/ µ i + 2 2 2 10 ( 3 + ˜1 ) 0( 3 (2e0) + ) 0 2π Z0g0

(0) 0 0 T00 (E3 + p0 − p˜00, q, p0), (2.63) 57 and

H(Λ) T (0)(E , p, p0) = − Z g2 00 00 0 0 Λ2 s Z Λ 2 √ dq Z0g q2 0 G(0) E p0 − p0, p, q t E − q2/ µ i + 2 2 2 01 ( 3 + ˜0 ) 1( 3 (2e1) + ) 0 2π Z1g1

(0) 0 0 T10 (E3 + p0 − p˜00, q, p0) Z Λ dq H(Λ) q2 t E − q2/ µ i + 2 2 0( 3 (2e0) + ) 0 2π Λ (0) 0 0 T00 (E3 + p0 − p˜00, q, p0). (2.64)

Equations (2.61) and (2.62), and Eqs. (2.63) and (2.64) form two sets of coupled integral equations if the incoming particle propagators are on-shell, i.e. if

0 0 0 0 p = p˜i ⇐⇒ Ei j = E3 + p0 − p˜0 j . A three-body observable is required as input to determine H(Λ), the strength of the three-body contact interaction for a given ultraviolet cutoff. For example, for a three-body bound state, H(Λ) can be fixed by requiring the dressed propagator of the t field to have a bound state pole at a known value, EB, of the three-body binding energy.

The wavefunction renormalization of the t field, Zt, is given by the residue of the pole of the dressed t propagator at E3 = −EB. Since the first and the second terms on the right hand side of the equation in Fig. 2.9 are regular at E3 = −EB,

2 Z Λ 2 Z Λ 2 h dp p dp0 p0 2 Zt = lim (E3 + EB) t0(E3 − p /(2µ0) + i) E →−E 2 2 2 2 e 3 B Ω g0 0 2π 0 2π T (0)(E , p, p0) 00 3 − 02 2 t0(E3 p /(2eµ0) + i). (2.65) Z0g0

We define At such such that

h2 1 H(Λ) Z = A2 = A2 (2.66) t 2 2 t 2 t Ω g0 Ω Λ 58

= + +

Figure 2.9: The expansion of the dressed propagator (thick blue line) of the t field in terms of the full T-amplitude.

= + +

Figure 2.10: The expansion of the dressed propagator (thick blue line) of the t field as an infinite geometric series. The triple line represents the bare propagator. The blue blob is the t-particle irreducible c − d0 amplitude i.e. it does not contain any three-body contact interactions.

(0) Solving Eqs. (2.63) and (2.64) for T00 (E3, p, p0), we can calculate At from Eq. (2.65) 8. Alternatively, we can use the equation in Fig. 2.10 to write

" !# i i h2 P2 T µ − 0 − T µ i (P ) = + 2 iΣ P i (P ), (2.67) Ω Ω g0 2M where iT is the dressed propagator and −iΣ is the self-energy of the t field, which is given by the sum of the bubbles in the second and the third terms on the right hand side of the equation in Fig. 2.10 9. This yields

1/Ω T (Pµ) = , (2.68) H(Λ) 1 − Σ(E ) Λ2 3

which, on expanding as a Laurent’s series in E3 around E3 = −EB, and requiring T to have a simple pole there, gives

Λ2 1 Λ2 − − Zt = 0 ; Σ( EB) = . (2.69) ΩH(Λ) Σ (−EB) H(Λ) √ 8 The external dressed t propagator gives a factor of Zt, which when combined with the factor of h/g0 2 gives AtH(Λ)/Λ . 9 2 2 Note that h /g0 is factored out of Σ to make it free from all factors of coupling constants. 59

Comparing Eqs. (2.65) and (2.69), we find an alternative way to calculate At, i.e. s Λ2 1 At = 0 . (2.70) H(Λ) |Σ (−EB)|

We have verified that Figs. 2.9 and 2.10 yield the same value for the wavefunction renormalization up to the accuracy of the numerical calculation. Two neutron halos can have a spectrum of two or more Efimov states if the neutron-core bound/virtual state energy is small enough (in magnitude). The spectrum can be found in a straight-forward manner using the equations derived in this section. For the given masses, two-body scattering lengths, and Λ and H(Λ) fixed to a given three-body input datum, one just needs to find the positions of the bound state poles of the dressed

(0) propagator, iT , or equivalently, those of the three-body scattering amplitudes, Ti j . One way to do this is to find the values of the three-body energies at which solutions to the homogeneous integral equations corresponding to Eqs. (2.61) and (2.62) (or Eqs. (2.63) and (2.64)) exist. This strategy was pursued in Ref. [139], where the possibility of known and suspected two-neutron halo nuclei to have an excited Efimov states was explored.

2.5 Conclusion

The equations derived in this chapter have only a small number of input parameters: the masses of the nucleons and the core, the neutron-neutron scattering length, and the separation energies of the halo nucleons. Of course, additional parameters enter the equations if we calculate higher order corrections to the results derived here. However, approximate results with estimates of the size of these corrections can be easily obtained while leaving open the possibility of improving the accuracy of the calculation as more and more input data become available. In the rest of this dissertation, we derive correlations between physical observables similar to the ones discussed above. We use experimental inputs for some of these observables and make predictions for others, with 60 anticipated errors from higher order terms in EFT, and compare to experimental data whenever available. 61 3 Coulomb Energy in a Square Well Potential Model

The role of the Coulomb interaction in nuclear structure is a well studied problem. (See, e.g. Reference [159].) The case of isobaric mirror nuclei, 13N and 13C, which are related to each other by interchange of protons and neutrons, is a classic example. These were first studied using the R−matrix formalism by Thomas [160] and Ehrman [161]. Both these nuclei can be well described in the nuclear shell model as a nucleon orbiting a

12 + 13 13 C core. The 1/2 state of N( C) is a proton (neutron) in a s1/2 orbit which is approximately 2.4 (3.1) MeV above the ground state, which is a 1/2− state with a proton

(neutron) in a p1/2 orbit. If we assume charge independence of nuclear forces and neglect effects of polarization of the 12C core, proton-neutron mass difference, magnetic interactions, etc., the difference of about −0.7 MeV between the excitation energies of 13N and 13C is due to the different values of the Coulomb energy, defined as the shift in energy

12 12 of the C−proton system due to the Coulomb interaction, Vc, between C and the proton, for the 1/2+ and 1/2− states of 13N. At first order in perturbation theory, the Coulomb energies of the two states of 13N are given by

∆E± = hφ±|Vc|φ±i, (3.1) where |φi is the unperturbed (i.e. 13C) state and the subscripts are used to label the 1/2+ and the 1/2− states. The difference in excitation energies of 13N and 13C is then

∆E+ − ∆E−, which was calculated in Ref. [162] to be −0.6 MeV. Physically, the fact that

+ ∆E− is larger than ∆E+ is a consequence of larger spatial extension of the 1/2 (S-wave) state compared to the 1/2− (P-wave) state of 13C for the same neutron separation energy. In 17F the 5/2+ state does not have a halo structure, whereas the 1/2+ state has a proton-halo structure. In the mirror nucleus 17O, neither the 5/2+ nor the 1/2+ states exhibit halo structure as a result of deeper binding due to the absence of the Coulomb interaction in the 16O−neutron system. The isobaric mirrors of neutron halo nuclei, on the 62 other hand, are low-lying proton-core resonances, e.g. 11Be has two neutron-halo states: the ground state (1/2+) and the first excited state (1/2−). The corresponding states in 11N are low-lying 10C−proton resonances. A challenge in calculating the Coulomb energy for such halo systems is that a perturbative expansion in Vc (or, equivalently, αem) converges more slowly for halo nuclei than for normal nuclei because of the low neutron-core binding energy. References [162, 163], however, calculated the Coulomb energies of these resonances using potential models. The experimental values of the positions and the widths of the 1/2+ and 1/2− resonances in 11N were well explained by the calculated values of Coulomb energies. The Coulomb energy of the S-wave proton-core state (1/2+) was found to be lower than that of the corresponding P-wave state (1/2+), consistent with the larger gap between the 1/2+ and 1/2− states of 11N than between those of 11Be.

Reference [162] also found that the higher order terms in Vc were small, contrary to what was a priori expected from the weak binding of the neutron in 11Be. However, these

higher order terms in Vc were found to be more important for the S-wave state than for the other states, not only for this pair of mirror nuclei, but also for the 17O − 17F pair. In this chapter, we calculate the Coulomb energy of an S-wave proton-core system whose isobaric mirror is a neutron halo nucleus. If we use only the data for the neutron halo as input, a model-independent result for the Coulomb energy can not be obtained. Thus the Coulomb shift is inherently dependent on the details of the short-range interaction. This is well known in the context of proton-proton scattering, where the scattering amplitude cannot be divided into strong and electromagnetic parts in a model-independent way [164–166]. In this chapter we use the square well model for the nuclear interaction to investigate these issues. We calculate the Coulomb energy for the square well potential, both at first order and to all orders in αem. We then derive the ERE parameters for the square well potential are repeat the analysis at LO and at NLO in EFT. In other words, we take the 63 result of the square well model as both input and output “data”. We emphasize that the EFT discussed in this chapter is an EFT for a nucleon-core system whose interaction in the absence of Coulomb is exactly described by a square well; it is different from the Halo-EFT discussed in the other chapters of this dissertation. The purpose of expanding an analytically-solvable model such as the square well in EFT is to test the convergence of the EFT by comparing the LO and NLO results with the full square well result.

3.1 Effective range for the square well

To begin with, we derive the effective range for a three-dimensional square well potential. Here, and throughout this chapter, we use the superscript (0) to denote quantities pertaining to the neutron-core system, and drop the superscript for the same quantities in the proton-core system. However, we saw in Section 2.2 that r0 is unmodified at NLO in EFT even after inclusion of the Coulomb interaction. We can, therefore, derive the effective range for just the neutron-core system, and use it for the proton-core system as well — the difference is higher order in EFT. In the center of mass frame, the three-dimensional square well potential,    −V, r ≤ R V(r) =  (3.2)   0, r > R, has the S-wave two-body radial wavefunction given by   C q(0)r, r ≤ R  sin u(r) =  (3.3)  (0) γ(0)(R−r)  C sin q R e 0 , r ≥ R.

(0) with binding momentum γ0 and r (0)  (0)2 q = 2µV − γ0 (3.4)

satisfying the continuity condition for the logarithmic derivative,

(0)  (0)  (0) q cot q R = −γ0 , (3.5) 64 using [54] Z ∞  2 !2  r  r0 = 2 dr  1 − − Cu0(r)  , (3.6)  a (0)  0 γ0 =0

where a is the scattering length and with the normalization C chosen such that Cu0(r) is equal to (1 − r/a) at r > R, i.e.

 R  p  C = 1 − csc 2µVR . (3.7) a

With this, the integral in Eq. (3.6) can be evaluated to give   p   R3  R2 cot 2µVR  p  r = R − + R 1 −  − cot2 2µVR  . (3.8) 0 2  p  3a a  2µVR 

(0) From Eqs. (3.4) and (3.5), we can show that, for γ0 R = 0, h  p   p   p i cot 2µVR / 2µVR − cot2 2µVR vanishes. Since this expression is an analytic

(0) (0) function of γ0 R around zero, it should at least be of order γ0 R. Using the ERE expansion, 1  2 = γ(0) + O γ(0) , (3.9) a 0 0 we get

 (0) 2   p   r γ0 R   cot 2µVR  p   3 0 − − (0)  − 2  (0) = 1 + 1 2γ0 R  p cot 2µVR  + O γ0 R . (3.10) R 3  2µVR 

At LO and NLO in the EFT, the effective range for the square well, r0, can, therefore, be substituted by the width of the well R; the difference is higher order in the EFT.

3.2 Coulomb energy at first order in αem

We now calculate the Coulomb energy for the square well model at first order in perturbation theory. We first obtain the exact result for the Coulomb energy as a function of the binding momentum for the square well potential. We then recalculate it at LO and NLO in EFT by fixing the EFT parameters to the square well result. 65

At first order in Vc (or αem), the Coulomb energy is the matrix element of the k proton-core Coulomb interaction, V = c , for the unperturbed (i.e. neutron-core) c µr wavefunction. In the square well model, this unperturbed wavefunction is given by Eq. (3.3). So R ∞ dr kc [u(r)]2 0 µr  2  ∆E = R ∞ + O αem . (3.11) dr [u(r)]2 0

Inserting Eqs. (3.4) and (3.3) in Eq. (3.11), we obtain

(0) (0) 2 (0) 2γ(0)R (0) − − 0 − kc Ci(2q R) + γ + log(2q R) 2 sin (q R)e Ei( 2γ0 R)  2  ∆E = 2 +O αem , (3.12) µR − sin(2q(0)R) sin (q(0)R) 1 2q(0)R + (0) γ0 R where Ci(x) and Ei(x) are, respectively, the cosine integral and the exponential integral

(0) functions [167]. We can simply replace γ0 , the binding momentum for the potential in

Eq. (3.2), by γ0, the binding momentum with the Coulomb interaction included, because

the difference is higher order in αem.

2 2γ0R kc −Ci(2qR) + γ + log(2qR) − 2 sin (qR)e Ei(−2γ0R)  2  ∆E = 2 + O αem , (3.13) µR 1 − sin(2qR) + sin (qR) 2qR γ0R where

p 2 q ≡ 2µV − γ0 . (3.14)

To calculate the Coulomb energy at first order in αem in EFT, we repeat the derivation with u(r) obtained from the EFT wavefunctions in Eq. (2.22). Equation (3.11) then gives a ∆E that is divergent. We regularize the integral by writing it in momentum space and by continuing to 3 − δ spatial dimensions with a regulator κ, to obtain, at LO,

 2δ Z 3−δ Z 3−δ 0 (0) κ d q d q 1 4πkc 1 ∆ELO = 8πγ0 − − 2 (2π)3 δ (2π)3 δ 02 (0)2 µ (q0 − q)2 q + γ0 1  2  2 + O αem . (3.15) 2 (0) q + γ0 66

Equation (3.15) has the same mathematical form as the two-loop ππ scattering diagram calculated by Gasser et al. in Ref. [168]. The result is

(0) kcγ h   i ∆E = 0 −4π log γ(0)/Λ + Υ (Λ) , (3.16) LO 2πµ 0 0

where the counterterm Υ0 at ultraviolet cutoff Λ cannot be calculated within the EFT. We

(0) −1 choose Λ = 1/R and fix Υ0 by matching ∆ELO to ∆E from Eq. (3.13) at some γ0  R . (0) Again, we can simply replace γ0 in Eq. (3.16) by γ0 because the difference is higher

order in αem. This yields

k γ   ∆E = c 0 −4π log (γ R) + Υ  + O α2 . (3.17) LO 2πµ 0 0 em

The NLO result for ∆E can be obtained by noting that the NLO wavefunction, p compared to the LO wavefunction, carries an extra factor of 1/ 1 − r0γ0. Therefore, the

Coulomb energy at first order in αem and at NLO in EFT is

1 kcγ0    2  ∆ENLO = −4π log (γ0R) + Υ0 + O α . (3.18) 1−γ0R 2πµ em

3.2.1 Errors in EFT calculation

Neglecting the effective range and higher order terms in the ERE of Eq. (2.6)

introduces a relative error of r0/a ∼ γ0R in the Coulomb energy calculated using the LO wavefunction. Inclusion of the effective range in the NLO calculation reduces this relative

2 error to (γ0R) . These errors reflect the failure of EFT to correctly reproduce the long-distance tail of the square well wavefunction, and can be systematically reduced by including higher order terms in the EFT expansion. However, since the EFT wavefunctions have the same functional form for all values of r, all the way to r → 0, an additional error is introduced by the discrepancy between the square well and the EFT wavefunctions at r < R. The resulting absolute error can be estimated by [169]

Z R kc 2 kc (0) dr [u(r)] . γ0 . (3.19) 0 µr µ 67

We estimate the error in the EFT calculation by adding the absolute values of these two errors. Figure 3.1 shows the plot of the Coulomb energies obtained from Eqs. (3.13), (3.17), and (3.18) along with the exact result to all orders in αem (from the next section), versus

−1 the binding momentum γ0 in units of R at kcR = 0.1. The constant Υ0 in Eqs. (3.17) and

(3.18) is chosen by matching each of these equations to Eq. (3.13) at γ0R = 0.01. We have expressed the Coulomb energy in the units of γ0/(2µR). The value of kc used is slightly lower than the typical values for light halo nuclei. The result at first order in αem is accurate only for γ0R  kcR. This is because the perturbation expansion in αem has better convergence for deeper bound states. However, this coincides with the region where the EFT has slow convergence. An accurate EFT calculation of the Coulomb energy of a proton-core system with a shallow-bound mirror nucleus therefore requires Coulomb to much higher orders in αem.

3.3 Coulomb energy to all orders in αem

To find the Coulomb energy for the square well model to all orders in αem and to all orders in EFT, we solve the Schrodinger¨ equation exactly for the potential   kc  −V + , rR

The radial wavefunction v(r) satisfies the radial Schrodinger¨ equation

1 d2v(r) − + [U(r) − E] v(r) = 0. (3.21) 2µ dr2

The wavefunction satisfying the boundary conditions required for a bound state is   A reiqr η q , − iqr , <  Φ(1 + ( ) 2; 2 ) r R v(r) =  (3.22)  −γ0r  B re Ψ(η(γ0), 2; 2γ0r), r>R, 68

Figure 3.1: Coulomb energy at first order in αem for square well (expressed in appropriate dimensionless units), LO calculation in EFT, and NLO calculation in EFT versus the −1 binding momentum in the units of R , at kcR = 0.1. The purple bands represent the EFT errors.

where Φ(a, b; x) and Ψ(a, b; x) are, respectively, Kummer’s and Tricomi hypergeometric functions [167]. Using the recurrence relations for the confluent hypergeometric functions, a Φ0(a, b; x) = Φ(a + 1, b + 1; x) (3.23) b and Ψ0(a, b; x) = −aΨ(a + 1, b + 1; x), (3.24)

the continuity of the logarithmic derivative at r = R gives Φ(2 + iη(q), 3; −2iqR) iq + [1 + iη(q)](−2iq) = 2Φ(1 + iη(q), 2; −2iqR)

Ψ(2 + η(γ0), 3; 2γ0R) − γ0 − 2γ0[1 + η(γ0)] , (3.25) Ψ(1 + η(γ0), 2; 2γ0R)

(0) which can be solved numerically to obtain q as a function of γ0. We can then express γ0

as a function of γ0 using Eqs. (3.4), (3.5) and (3.14). We can thus plot the Coulomb 69 energy to all orders in αem versus the physical binding momentum γ0. This can then be compared to the corresponding EFT result that we now derive. The T−matrix for the S-wave scattering of two particles interacting via the Coulomb interaction and a short-range interaction were calculated in Sec. 2.2. Bound/virtual states of the two-body system appear as poles in the T−matrix. For repulsive Coulomb interaction, the pure Coulomb amplitude, TC, in Section 2.2, is analytic throughout the complex energy plane. So we only need to look at the Coulomb-distorted short-range amplitude, TCS in Eq. (2.34), for the pole position. At leading order in EFT, using Eq. (2.35), " √ !#! 2π∆ κ 1 3γ π κ − − k − − − k H (η k ) . 2 + 2 c 1 + log 2 c ( ) = 0 (3.26) µg δ δ 2 2 kc

To cancel the divergences in the limit δ → 0, we let

∆ = ∆0 + kc∆1, (3.27)

with √ ! π∆ 1 3γ πκR − 1 − + 1 − + log ≡ Υ˜ (R). (3.28) µg2 δ 2 2 0

Noting that xψ(x) → −1 as x → 0 we can take the limit kc → 0 in Eq. (3.26) to get

2π∆(R) 2π∆ κ ≡ 0 + = γ(0). (3.29) µg2 µg2 δ 0

This gives ! (0) ˜ kc γ0 = 2kcΥ0(R) − 2kcψ − γ0 − 2kc log (γ0R) . (3.30) γ0

(0)2 2 (0) The Coulomb energy is then given by (γ0 − γ0 )/(2µ). Even if γ0 , the binding momentum of the neutron-core system, is experimentally known, we have an additional

parameter, Υ˜ 0(R), in Eq. (3.30) which cannot be fixed by using data on the neutron halo system alone. In order to calculate the Coulomb energy in a model-independent way, we therefore require an experimental datum for the proton-core system. The origin of this 70 difficulty is clearer in Eqs. (3.27), (3.28) and (3.29), and can be explained as follows. In order to cancel both the divergences that appear at zeroth order and at first order in kc, we were forced to split ∆ into two pieces, ∆0 and ∆1, which were both separately renormalized. The data for the neutron-core system can only be used to renormalize ∆0, leaving us with the necessity to find additional input to renormalize ∆1. If we were to do this in a model independent way, we would need experimental data for the proton-core system. Here we give up model-independence and determine Υ˜ 0(R) by matching to the exact solution for the square well plus Coulomb model. We perform this matching twice, once for the LO result and again for the NLO result.

2 We first extend Eq. (3.30) to NLO by including the effective range term, r0k /2, in Eq. (3.26). This would modify Eq. (3.29) to

2π∆ κ 1 2 0 + = γ(0) − r γ(0) . (3.31) µg2 δ 0 2 0 0

(0) Therefore, to obtain γ0 as a function of γ0 at NLO, we solve the quadratic equation,

2   (0) 1 (0) kc 1 2 γ − r0γ = 2kcΥ˜ 0(R) − 2kcψ − γ0 − 2kc log (γ0R) − r0γ , (3.32) 0 2 0 γ0 2 0

(0) and choose the root with γ0 R < 1. In Figure 3.2, we plot the Coulomb energies calculated in EFT from Eq. (3.30) and Eq. (3.32), along with their EFT error bands, as well as the one obtained from the numerical solution of Eq. (3.25), (the dotted curve in Fig. 3.1), against the binding

momentum γ0 in appropriate dimensionless units. We choose the constant Υ˜ 0 in

Eqs. (3.30) and (3.32) by matching the curves to the full square well result at γ0R = 0.01. A good agreement between the LO, the NLO and the square well results extends to

γ0R  0.01. The error bands of the EFT calculations provide an accurate estimate of the extent to which the higher order EFT terms can contribute. 71

Figure 3.2: The Coulomb energy to all orders in αem calculated from the square well model (expressed in appropriate dimensionless units), at LO in EFT, and at NLO in EFT −1 plotted against the binding momentum in the units of R for kcR = 0.1 . The purple bands represent the EFT errors.

3.4 Conclusion

The Coulomb energy of a two-body system with a short-range S-wave interaction is inherently sensitive to the details of the short-range interaction. If the short-range interaction in not completely known/solvable, which is the case for nuclear systems, the Coulomb energy can only be calculated by employing a model to approximate the short-distance physics. Since the square well potential yields simple analytic wavefunctions, it can be a useful toy-model to study the convergence pattern of EFT. Figs. 3.1 and 3.2 illustrate several key aspects of EFT calculations. The EFT predictions agree with the full result when the expansion parameter, γ0R . 0.5; the sizes of the EFT error bars grow when this expansion parameter increases. Although, higher order EFT 72 calculations may require additional input data to fix the unknown constants, they have smaller error bands and tend to agree more closely with the exact result. In this chapter, we fixed the low-energy constants in our EFT with the input data from square well model calculations. Accurate determination of the physical quantities of real nuclei require input from more realistic ab initio calculations, which are often unavailable, or from experimental data. We pursue the latter approach for the rest of this dissertation. In this approach, Halo-EFT bypasses the necessity of knowing the quantitative details of the short-distance interactions. But it can provide model-independent correlations between the long-distance/low-energy observables generated by these interactions. As expected from the weak binding between the neutron and the core in the model studied, the Coulomb energy calculated at first order in perturbation theory was found to significantly underpredict the Coulomb energy at γ0R  1. This is in contrast to the case of the deeply bound system like 13C − 13N, where most of the Coulomb energy could be explained at first order in the Coulomb potential. 73 4 Coulomb Dissociation of One-Neutron Halos:

Applications to Carbon-19

In this chapter, our results for the Coulomb dissociation of one-neutron halos with one S-wave neutron-core bound state in the absence of low lying P-wave resonances is presented. The analysis and the results in this chapter were published in Ref. [145]. Most of the material presented in this chapter has been reproduced verbatim from this publication. The nucleus 18C has a ground state with Jπ = 0+ [170]. Therefore, from the simplest shell-model picture, one would expect 19C to have a ground state Jπ of 5/2+. However, based on the observation of a narrow momentum peak and a large cross section, Ref. [43] suggested that the valence neutron in 19C is in an S-wave relative to the core, consistent with a shell-model calculation based on the Warburton-Brown effective interaction [171]. Reference [95] reported that the width of the longitudinal momentum distribution of 18C produced by the Coulomb break-up of 19C did not agree with a Yukawa potential model calculation which used the 19C neutron separation energy of 160 ± 110 keV, determined from prior mass measurements. Moreover, this value was found to be incompatible with the energy spectrum of the Coulomb dissociation cross section in Ref. [21] for all possible configurations. The authors, therefore, revised the neutron separation energy of 19C to 530 ± 130 keV by analyzing the angular dependence of the Coulomb dissociation differential cross section. With this value, the spectrum could be well reproduced for a

18 + dominant C (0 ) ⊗ 2s1/2 ground state configuration with a spectroscopic factor of 0.67, leading to the assignment of Jπ = 1/2+ to the 19C ground state. The tabulated value of 580 ± 90 keV for the one-neutron separation energy of 19C in Ref. [172] was evaluated by combining this result with the value of 650 ± 150 keV extracted from the inclusive longitudinal momentum distribution obtained in single-neutron knockout [50]. A 74 non-perturbative treatment of the Coulomb interaction by Banerjee and Shyam corroborated Nakamura et al.’s spin-parity and neutron separation energy assignments,

18 + albeit with a spectroscopic factor of 1 for the C (0 ) ⊗ 2s1/2 configuration [173]. Typel and Baur studied the experiment reported in Refs. [21, 174] and concluded that higher order electromagnetic effects are small [175]. In Ref. [176], Singh et al. analyzed some of the data in Refs. [21, 95] using a Woods-Saxon potential between the valence neutron and the core 10. A recent study hinted at the possibility that the first 5/2+ state of 19C is unbound [49]. The first excitation energy of 18C, 1620 ± 20 keV [170], is approximately three times larger than the neutron separation energy of 19C. The low-energy properties of 19C can then be studied in Halo-EFT with 18C core and the neutron as effective degrees of freedom and an expansion parameter whose rough estimate is provided by Mlo/Mhi ∼ Rcore/Rhalo, the ratio of the size of the core to that of the halo. Using the values deduced in Ref. [43], this ratio comes out to be 0.49, which is not particularly small. To reduce the theory error, we perform a next-to-next-to-leading order (N2LO) calculation of the Coulomb dissociation cross section, in which there are two undetermined parameters which need to be fitted to experimental input. We fit these parameters to the experimental cross section data. The fit parameters then provide predictions for the separation energy and the asymptotic normalization coefficient of the halo neutron.

4.1 Coulomb dissociation cross section of Carbon-19 in Halo-EFT

The electric dipole operator for a two-body system of reduced mass µ consisting of a neutral particle of mass m and a particle of mass M carrying charge eZ, is

µ OM = eZ rY M (rˆ) = eZ rY M (rˆ) . (4.1) 1 eff 1 M 1 10 We attempted to reproduce the results of Ref. [176] in detail, but were unable to do so, even when using the same input parameters. 75

In momentum representation, this operator can be written as

µ ∂ OM = ieZ Y M (pˆ) . (4.2) 1 M 1 ∂p

The reduced transition probability, B(E1) for the dipole transition from an initial state

with total angular momentum Ji to a final state with total angular momentum J f is defined as 1 X B(E1) = |hΦMi+M|OM|ΨMi i|2 (4.3) J f 1 Ji 2Ji + 1 MMi

For the transition from a bound state to a continuum state in which J f is not measured, we need to sum over all values of J f satisfying |Ji − 1| < J f < Ji + 1,

dB(E1) 1 X = |hΦMi+M|OM|ΨMi i|2. (4.4) d3 p J f 1 Ji 2Ji + 1 (2π)3 J f MMi

For a S-wave one-neutron halo, the initial state, |ΨMi i, is the S-wave wavefunction |Ψi Ji coupled to the spinor |χMi i, obtained by coupling the core and the neutron spinors. The Ji wavefunction of the final state, |ΦMi+Mi, is obtained by coupling this spinor to the tensor J f operator, Y M. The corresponding Clebsch-Gordan coefficients are 1 and CJi1J f 1 Mi M(Mi+M) respectively. In the Plane Wave Impulse Approximation (PWIA), |Φi = |pi. Using

2 E = p /2µ, for the wavefunction with binding momentum γ0, given by Eq. (2.19), we thus obtain,

dB(E1) 4 µ3 γ p3 = e2Z2 0 2 2  4 dE π M 1 − r0γ0 2 2 γ0 + p Z 1 X  2 ∗ CJi1J f dΩ Y M (p ˆ)Y M(p ˆ), (4.5) Mi M(Mi+M) pˆ 1 1 2Ji + 1 J f MMi which yields dB(E1) 12 µ3 γ p3 = Z2 0 , (4.6) 2 2 2  4 e dE π M 1 − r0γ0 2 2 γ0 + p which is independent of the units of e adopted. 76

The LO result, which can be obtained by setting r0 = 0, is equivalent to that of Ref. [177], where a zero-range potential model was used for the neutron-core interaction. Equation (4.6) was also derived in Ref. [121], where the authors extracted dB(E1)/dE by calculating the capture cross section for the E1 photon. The photodissociation cross section, which is related to the capture cross section by detailed balance [178], calculated in Section 2.3, yields the same expression for dB(E1)/dE. Since the shape parameter P only enters at N3LO, the amplitude in Eq. (2.13)

2 includes all effects up to N LO in the EFT expansion in powers of r0/a ∼ Rcore/Rhalo [5, 6, 9, 109]. The spectrum of 19C has been obtained from inelastic scattering on a proton target [179]. There is no evidence for a weakly bound state or a low-lying resonance in the P-wave 18C − n system. As discussed in Section 2.3, the final state interactions then appear only at N3LO. The contributions from the gauge-invariant contact terms are even more suppressed. Equation (4.6) therefore is good up to N2LO in our expansion parameter. Using Eq. (4.6) in the equations shown in Section A, we obtain the expression for the relative energy spectrum,

dσ 64π µ3 γ p3 = Z2α N (B + E, R) 0 , (4.7) 2 E1  4 dE 3 M 1 − r0γ0 2 2 γ0 + p angular distribution,

dσ 64π µ3 γ Z dn p3 = Z2α 0 E1 dE, (4.8) 2  4 dΩcm 3 M 1 − r0γ0 dΩcm 2 2 γ0 + p and the longitudinal momentum distribution,

dσ 32π µ2 γ Z ∞ p3 = αZ2 0 M (B + E, R) dp, (4.9) 2 E1  4 dpz 3 M 1 − r0γ0 |pz| 2 2 γ0 + p

2 of the Coulomb dissociation cross section at N LO in Halo-EFT. The functions NE1(ω, R),

ME1(ω, R), and nE1(ω) are defined in Section A. 77

4.1.1 Extracting parameters from Coulomb dissociation data

There are two parameters in Eqs. (4.7) and (4.8), r0 and γ0, which cannot be calculated within the theory. At this order in EFT, the 18C-n scattering length, a, is constrained by Eq. (2.20), and the ANC, Ae, by Eq. (2.21). We can thus pick any two of the four parameters γ0, a, r0, and Aeas the undetermined ones and fit those to experimental

2 input. The neutron separation energy, B, is then equal to γ0/2µ.

4.1.1.1 Angular distribution

The Coulomb dissociation of 19C into 18C + n on a 208Pb target at 67 A MeV was studied in Ref. [21]. With wave functions calculated from a Woods-Saxon potential, the neutron separation energy was determined to be 530 ± 130 keV from an analysis of the angular distribution of the 18C + n center of mass. By fitting the differential cross section given by Eq. (4.8) to the same data, we determine not only the neutron separation energy, but also the 18C-n scattering length. We convolve the differential cross section in Eq. (4.8) with the angular resolution of the detector and fit a and B to the data depicted in Figure 2 of Ref. [21]. To minimize the contribution of nuclear interaction to the break-up, we exclude the small impact parameter

2 2 (θ > 2.2 deg.) data points. A minimum χ of χmin = 1.23 per degree of freedom is obtained at B=540 keV and a=7.5 fm. In Fig. 4.1, we show the contour plot for

2 2 2 ∆χ ≡ χ − χmin=1 in the aB-plane. The projections of the contour on the coordinate axes give the 1-σ confidence intervals on the marginalized probability distributions of a and B. The 1-σ interval for the separation energy is found to be (480, 630) keV and that for the scattering length is (6.9, 8.0) fm. This is consistent with the value B = 530 ± 130 keV, determined by Nakamura et al. from the same data. 78

The contour plot in Fig. 4.1 was obtained by calculating the ∆χ2 on a two dimensional grid by varying a in steps of 0.1 fm and B in steps of 10 keV. Cubic Hermite interpolation [180] was used to generate ∆χ2 at other points in the aB plane. Since the interpolation simply returns the input values at the grid points, the error in interpolation does not appear in the extracted values of a and B up to the digits quoted.

!Χ2 # 1 0.65

0.60

eV " 0.55 M ! B

0.50

0.45 7.0 7.5 8.0 8.5 a !fm" Figure 4.1: Contour plot of ∆χ2=1 for the angular distribution of the differential cross section in the aB-plane. (Published in Ref. [145].)

4.1.1.2 Relative energy spectrum

Citing the sensitivity of Eq. (A.6) to the final-state interactions of 18C and n, Nakamura et al. did not use the energy spectrum of the total cross section in the determination of the neutron separation energy in Ref. [21]. However, we have shown in 79

Section 2.3 that, as long as the effective range parameters in the P-wave channel are natural, diagrams involving final-state interactions do not appear up to N2LO in our theory. We, therefore, proceed to fit the energy spectrum of the cross section given by Eq. (4.7) to the data in Figure 2(b) of Ref. [174]. Note that this data was obtained in the same experiment reported in Ref. [21], but with a different method to deal with the nuclear contribution to the break-up process: Ref. [21] subtracted a background scaled from 12C-target spectrum whereas Ref. [174] used a large value of the impact-parameter cut (R=30 fm). We convolve Eq. (4.7) with the spectral resolution of the detector and fit a and B to the data. We exclude the E > 1 MeV data points because these lie outside the domain of convergence of the EFT. A minimum χ2 of 1.7 per degree of freedom is obtained at B=580 keV and a=8.1 fm. The 1-σ intervals for B and a from Fig. 4.2, (510, 650) keV and (7.6, 8.4) fm, respectively, are consistent with the those determined from the angular distribution. Again, numerical errors in the calculation of ∆χ2 do not appear in the extracted values of a and B up to the digits quoted. The data in Ref. [21] and in Ref. [174] were obtained from the same experiment. However, we render the data sets independent by removing an overlapping region form our analysis. We can, then, simply add the χ2’s. The combined data has a minimum χ2 of 1.27 per degree of freedom at B=575 keV and a=7.75 fm. These correspond to a 18C-n effective range of 2.6 fm. From Fig. 4.3, the 1-σ confidence intervals for a, B and r0 are determined to be (7.4, 8.1) fm, (520, 630) keV, and (1.7, 3.2) fm, respectively. We revise the value of our EFT expansion parameter from the initial estimate of

Rcore/Rhalo ∼ 0.49 to the ratio r0/a = 0.33 of the best fit to the combined data set. Therefore, in addition to the statistical errors, all the parameters determined above have a

3 3 relative error of r0/a = 0.036 that comes from neglecting higher order terms in the EFT expansion. As discussed in Section 3.2.1, there is an additional EFT error that comes from 80

!Χ2 # 1 0.65

0.60

eV " 0.55 M ! B

0.50

0.45 7.0 7.5 8.0 8.5 a !fm" Figure 4.2: Contour plot of ∆χ2=1 for the energy spectrum of the total cross section in the aB-plane. (Published in Ref. [145].)

using the asymptotic wavefunctions even within the range of the interaction. Since the observable in question, σ, is proportional to the second radial moment, hr2i, this error also

3 enters at N LO [169]. The EFT extraction, therefore, gives values of B, a, and r0 of (575 ± 55(stat.) ± 20(EFT)) keV, (7.75 ± 0.35(stat.) ± 0.3(EFT)) fm, and

+0.6 (2.6−0.9(stat.) ± 0.1(EFT)) fm, respectively. Note that the EFT errors quoted here do not include uncertainties in the reaction theory discussed in Section A. In Fig. 4.4, we show the input data along with the best fits. The dashed line in the angular distribution is the fit to the small-angle data, the dashed line in the energy spectrum is the fit to the low-energy data, and the solid line is the fit to the combined data 81

2 2 !Χ # 1 !Χ # 1 8.2 0.64

0.62 8.0

0.60

" 7.8 eV "

0.58 fm ! M ! a

B 0.56 7.6 0.54

0.52 7.4

0.50 7.4 7.6 7.8 8.0 8.2 2.0 2.5 3.0

a !fm" r0 !fm" Figure 4.3: Contour plot of ∆χ2=1 for the combined data in the aB-plane (left) and the ar0-plane (right). (Published in Ref. [145].)

set. Good agreement is seen for both data sets. In particular, for the energy spectrum, the agreement extends well beyond the fit region.

250 0.7

" 0.6 . 200 sr eV " 0.5 M ns # 150 ns # 0.4 bar ! bar

! 0.3 cm 100 dE d # 0.2 # # 50 d Σ d Σ 0.1 0 0.0 0.0 0.5 1.0 1.5 2.0 0 1 2 3 4 Θ !deg." E !MeV" Figure 4.4: The angular distribution of the differential cross section at a=7.5 fm and B=540 keV (dashed), and at a=7.75 fm and B=575 keV (solid), with data from Ref. [21] (left); and relative energy spectrum of the differential cross section at a=8.1 fm and B=580 keV (dashed), and at a=7.75 fm and B=575 keV (solid), with data from Ref. [174] (right). (Published in Ref. [146].) 82

4.1.2 Longitudinal momentum distribution

In Ref. [95], the longitudinal momentum distribution of 18C after Coulomb dissociation of 19C on a 181Ta target was studied at a beam energy of 88 MeV/u. (See also the earlier Ref. [44].) The data can be compared to the prediction of Eq. (4.9), with the value of B determined in Section 4.1.1, up to an overall normalization factor (the experimental result is given in arbitrary units). Since the data is given as a function of the fragment momentum measured in the laboratory frame of reference, we need to apply a

Lorentz boost to pz. However this gives us a peak position which is different from that seen experimentally by about 2%, which appears to be consistent with the experimental uncertainty in the absolute energy calibration and particle energy loss in the target foil. We, therefore, fit the position and the height of the predicted peak to the data. (Note that this is different to what was done in Ref. [176], where the authors aligned their peak with the highest of the experimental data points.) In Fig. 4.5, we show the longitudinal momentum distribution of the Coulomb

181 break-up cross section given by Eq. (4.9) at B=575 keV and r0=2.6 fm for a Ta target. We used R=13 fm, obtained by adding the nuclear radii of the projectile and the target, and a small correction to account for the bending of the Rutherford trajectory. The width of the curve predicted by Eq. (4.9), with a separation energy of 575 keV, reinforcing the notion that the low-momentum part of the 19C wavefunction is dominated by a configuration with a loosely bound neutron and a 18C core in a relative S-wave.

4.2 Electric charge radii of Carbon-18 and Carbon-19

The difference between the mean-square electric charge radii of the 18C and 19C nuclei is related to the mean-square distance between the neutron and the 18C core by

2 2 2 µ 2 hr i19 − hr i18 = hr i. (4.10) E C E C M2 83

60 12 "

" 50 10 eV

units 40 M

# 8 b. ar

mb 30 ! ! 6 z z

dk 20 dp 4 # # d Σ d Σ 2 10

0 0 !100 !50 0 50 100 7280 7300 7320 7340 7360 pz !MeV" kz !MeV" Figure 4.5: Longitudinal momentum distribution of the dissociation cross section on a 181 Ta target at 88 MeV/u for B=575 keV and r0=2.6 fm with R=13 fm (left); and with the normalization and the peak position fitted to data from Ref. [95](right). (Published in Ref. [145].)

Using the NLO wavefunction in Eq. (2.22),

2 1/2 2 2 2 µ 2 µ Ae hr i19 − hr i18 = Ae = , (4.11) E C E C 2 3 7/2 2 3/2 4M γ0 2 M B

+0.02 2 which yields a value of 0.09−0.03 fm . Although the error quoted here also includes the EFT errors in the input parameters and those in Eq. (4.11), most of the error comes from the statistical uncertainties in the input parameters. Experimentally, because of its longer half-life, it should be easier to measure the charge radius of 18C than that of 19C. This result can be used to predict the charge radius for 19C as soon as the value for 18C is available.

4.3 Conclusion

An EFT analysis of data on the Coulomb dissociation of 19C from Refs. [21, 174] gives a neutron separation energy of (575 ± 55(stat.) ± 20(EFT)) keV, in agreement with the previously determined values of 530 ± 130 keV [21] and 580 ± 90 keV [172]. The width of the longitudinal momentum distribution obtained in Coulomb dissociation of 19C is then predicted by EFT and shows good agreement with data. 84

Furthermore, since the neutron separation energy and ANC, which determine these experimental observables, also constrain the effective-range parameters of 18C-n scattering we can infer that the 18C-n scattering length is (7.75 ± 0.35(stat.) ± 0.3(EFT)) fm and that

+0.6 the effective range is (2.6−0.9(stat.) ± 0.1(EFT)) fm. Based on these values, we predict that 19 18 +0.02 2 the mean square charge radius of C is larger than that of C by 0.09−0.03 fm . We reiterate that the theoretical uncertainty in the reaction theory that connects the E1 matrix element to scattering observables is not included in the error bars quoted here. Corrections due to higher order electromagnetic effects between the 208Pb target and the 19C beam is a subject for further investigation. However, these corrections are expected to be small [175, 181]. Since the expansion parameter inferred from the extracted values of the effective range and scattering length is ≈ 0.3 which is smaller than the nominal expansion

18 parameter, Rcore/Rhalo ≈ 0.49, obtained from an estimate of the size of C, the EFT expansion shows very good convergence — our N2LO calculation has uncertainties that are markedly better than the statistical precision of the data. The break-up of 19C has also been studied in experiments with light targets [45, 47, 49, 50]. In these studies the height of the peak of the momentum distribution was found to be of the order of a millibarn/MeV. Our result for the peak height suggests that Coulomb break-up accounts for roughly 10% of this cross section. Similar studies using a target with a higher charge could facilitate comparison. 85 5 Two-Neutron Halos:Matter Radii and Efimov States

The universal relation between matter radii of two-neutron halo nuclei with their two-neutron separation energy and the two-body scattering lengths is discussed in this chapter. We apply the relation to 22C, which was recently found to have an unusually large matter radius [103], even in comparison to other halo nuclei. Previous theoretical studies [102, 182, 183] showed that an extremely small separation energy of the halo neutrons is required in order to generate such large size. We use the experimental value of the rms matter radius of 22C to constrain its two-neutron separation energy by deriving the universal relation between the matter radius and this energy. Our analysis takes into account the anticipated error from the higher order terms in the Halo-EFT expansion. We study the implications of this constraint on the possibility of existence of excited Efimov states in this nucleus. These findings were published in Ref. [146], where most of the materials presented here already appeared.

5.1 The Jacobi coordinates and momenta

It is convenient to study three-body systems using Jacobi coordinates and momenta [184]. In this section, we define these momenta and derive equations for the transformation between the different sets of Jacobi momenta.

Let r1, r2 and r3 be the positions and k1, k2 and k3 the momenta of masses m1, m2 and m3 respectively. The Jacobi coordinates with m1 as the spectator are then r23 = r2 − r3

m2 m3 and r1,23 = r1 − r2 − r3. m2+m3 m2+m3

m3 m2 The corresponding momenta are p1 = k2 − k3 and m2+m3 m2+m3

m2+m3 m1 q1 = k1 − (k2 + k3). In the center of mass of the three-body system, m1+m2+m3 m1+m2+m3          m3 m2    p1  −  k2   =  m2+m3 m2+m3    . (5.1)       q1  −1 −1  k3 86

Similarly, the Jacobi momenta with m2 as the spectator are          m3    p2  1 k2   =  m1+m3    . (5.2)       q2  1 0 k3

Therefore          m1 m3(m1+m2+m3)    p2 − −  p1    m1+m3 (m2+m3)(m1+m3)      =     . (5.3)    m2    q2  1 −  q1 m2+m3

For a 2n halo, m2 = m3 = m, and m1 = Am so that                1 A+2       1    pc −  pn1   pc   1   qc     2 2A+2    ⇒    2      =       =     . (5.4)   − − A      − − A    qc 1 A+1 qn1 pn1 1 A+1 qn1 Similarly,                1 A+2       1    pc − −  pn2   pc  − −1  qc     2 2A+2    ⇒    2      =       =     . (5.5)    − A       A    qc 1 A+1 qn2 pn2 1 A+1 qn2 Therefore,                1 A(A+2)       1    pn2  − − 2  pn1  pn1   1  qn1     A+1 (A+1)    ⇒    A+1      =       =     . (5.6)    − 1      − − 1    qn2 1 A+1 qn1 pn2 1 A+1 qn2     From here on, we use the notation pi qi, q j and p j qi, q j to denote momenta related by such transformations.

5.2 The three-body free states: plane waves and spherical waves

The eigenstates of the Jacobi momentum operators, |pqii, satisfy orthonormality,

0 0 6 (3) 0
(3) 0
ihp q |pqii = (2π) δ p − p δ q − q , (5.7)

and completeness, Z d3 p Z d3q 1ˆ = |pqi hpq|. (5.8) (2π)3 (2π)3 i i

The partial-wave projected states, |pq; lmλµii, where l and m are the angular momentum quantum numbers of the pair and λ and µ are those of the spectator-pair system, are 87 defined by

2π2 2π2 hp0q0|pq; lmλµi = δ(p − p0) δ(q − q0) 4π Ym∗ pˆ0
Yµ qˆ0
. (5.9) i i p2 q2 l λ

These states are orthonormal, i.e.

2 2 0 0 0 0 0 0 2π 0 2π 0 hp q ; l m λ µ |pq; lmλµi = δ(p − p ) δ(q − q ) δ 0 δ 0 δ 0 δ 0 , (5.10) i i p2 q2 ll mm λλ µµ

and complete, i.e.

X Z ∞ dp p2 Z ∞ dq q2 1ˆ = |pq; lmλµi hpq; lmλµ|. (5.11) 2π2 2π2 i i lmλµ 0 0

From Equations (5.7), (5.8) and (5.9), we get

Z 3 Z 3 d q d q j hp0q0; l0m0λ0µ0|pq; lmλµi = i j i (2π)3 (2π)3 2 2 2π   0 2π 0  δ(p j qi, q j − p ) δ(q j − q ) 2 q2 p j qi, q j j 2π2   2π2  δ(pi qi, q j − p) δ(qi − q) 2 q2 pi qi, q j i

0 ∗ m0    µ   4π Yl0 pˆ j qi, q j Yλ0 qˆ j

m∗    µ 4π Yl pˆi qi, q j Yλ (qˆi) . (5.12)

For l = l0 = 0, this simplifies to

Z 1 0 0 0 0 δλλ0 δµµ0     jhp q ; 00λ µ |pq; 00λµii = d qˆi.qˆ j Pλ qˆi.qˆ j 2 −1 2 2π  0  δ(pi qqˆi, q qˆ j − p) 2  0  pi qqˆi, q qˆ j 2 2π  0  0 δ(p j qqˆi, q qˆ j − p ). (5.13) 2  0  p j qqˆi, q qˆ j 88

By changing the order in which the integrals are performed, we can get an alternative expression for this inner product,

1 Z Z hp0q0; 00λ0µ0|pq; 00λµi = dΩ dΩ j i 4π pˆi qˆi 2π2 δ(p0 − p (ppˆ , qqˆ ))Yµ (qˆ ) p02 j i i λ i 2 2π µ0 ∗   δ(q0 − q (ppˆ , qqˆ ))Y 0 qˆ (ppˆ , qqˆ ) . (5.14) q02 j i i λ j i i

5.2.1 The two-body T-matrices in Jacobi representation

If the two-body interactions are all in the S-wave channel,

∞ X (l) (0) tˆi = (2l + 1) tˆi Pl(cos θ) = tˆi , (5.15) l=0

where Pl(x) is the Legendre function of first kind of order l, at LO,

0 0 3 (3) 0
ihp q |tˆi|pqii = ti(E; q) (2π) δ q − q , (5.16)

where 2π 1 ti(E; q) = − q . (5.17) µi 1  q2  − + 2µi −E + − i ai 2eµi

Here E is the energy of the three-body system, µi and ai are the reduced mass and the

scattering length of the two-body subsystem labeled by spectator i, and eµi is the reduced mass of the spectator-pair system. From Eqs. (5.8), (5.9) and (5.16), we obtain

2 0 0 0 0 0 0 2π 0 hp q ; l m λ µ |tˆ|pq; lmλµi = t (E; q) δ(q − q )δ 0 δ 0 δ δ δ 0 δ 0 . (5.18) i i i i q02 λλ µµ l0 m0 l 0 m 0

5.3 The bound state

In this section, the Faddeev equations are derived for a bound state of three distinct particles with S-wave interactions. We then derive an expression for the S-wave three-body wavefunction in terms of these Faddeev components. 89

The three-body bound state, |Ψi, satisfies the Schrodinger¨ equation,    X3   ˆ ˆ  | i − | i H0 + Vi Ψ = EB Ψ , (5.19) i=1

with the boundary condition that hr jk ri, jk |Ψi → 0 when any of the Jacobi coordinates r jk

and ri, jk goes to infinity. In terms of the Gˆ0 operator,

X |Ψi = Gˆ0 Vˆ i|Ψi. (5.20) i

In terms of the Faddeev components [105], |ψii ≡ Gˆ0Vˆ i|Ψi, this can be written as

X |Ψi = |ψii. (5.21) i The Faddeev components satisfy

  X X 1 − Gˆ0Vˆ i |ψii = Gˆ0Vˆ i |ψ ji ⇒ |ψii = Gˆ0tˆi |ψ ji. (5.22) j,i j,i X It is customary to define the Faddeev spectator functions, |Fii ≡ |ψ ji, which satisfy the j,i Faddeev equations, X |Fii = Gˆ0 tˆj|F ji, (5.23) j,i X and are related to the bound state wavefunction by |Ψi = Gˆ0 tˆi|Fii. Equation (5.23) can i be written as

i X ihpq; lmλµ|Fii = G0 (p, q; −EB) ihpq; lmλµ|tˆj|F ji, (5.24) j,i where

i 1 G0 (p, q; E) = 2 2 . (5.25) E − p − q + i 2µi 2eµi Defining Z ∞ dp p2 F q lmλµ ≡ hpq lmλµ|F i, i( ; ) 2 i ; i (5.26) 0 2π 90 and using Eqs. (5.11), (5.13), and (5.18), we get the Faddeev equations in Jacobi representation for three distinguishable particles with S-wave interactions, X Z ∞ dq0 q02 1 Z 1   F q λµ q .q P q .q Gi p qq , q0q , q −E i( ; 00 ) = 2 d(ˆi ˆ j) λ(ˆi ˆ j) 0 i( ˆi ˆ j) ; B 2π 2 − j,i 0 1

0 0 t j(−EB; q )F j(q ; 00λµ). (5.27)

The three-body bound state wavefunction for a system with l = λ = 0 can be written in this representation as

i ihpq; 0000|Ψi = ihpq|Ψi = G0 (p, q; −EB) {ti(−EB; q)Fi(q; 0000) X 1 Z 1 + d(p ˆi.qˆ j)t j(−EB; q j(ppˆi, qqˆi))F j(q j(ppˆi, qqˆi); 0000)} 2 − j,i 1 (5.28) and is normalized such that hΨ|Ψi = 1. It can be calculated by solving the set of three coupled integral equations in Eq. (5.27).

5.4 Two-neutron halo: identical neutrons and the permutation operator

For a 2n halo, the above equations simplify further because of identicality of the two neutrons. The three-body wavefunction, including the spinor, should be antisymmetric under the exchange of the neutrons. In this LO calculation, we only consider S-wave interactions between the two neutrons; interactions in higher partial waves are suppressed and only appear in higher order calculations. The neutrons are therefore in a spin singlet state, which is antisymmetric under exchange of the neutrons. The spatial part of the three-body wavefunction, defined in Section 5.3, should then be symmetric 11. It is convenient to define the permutation operator, Pˆ that interchanges the labels on ˆ ˆ 2 the two neutrons, i.e. P |ψn1 i = |ψn2 i, and P = 1. Since the three-body wavefunction is 11 Note that this is different from what is usually done in the literature, e.g. Refs. [139, 143], where antisymmetrization is performed because the wavefunction and its Fadeev components are defined with the spin part included. For our purpose, it is useful to work with spatial wavefunctions so that the core- spin independence of the reduced matrix element of the dipole operator, which we calculate later in this dissertation, is manifest. 91 symmetric under this operation, Pˆ |Ψi = |Ψi, (5.29) the Faddeev component |ψci must also be symmetric, i.e.

Pˆ |ψci = |ψci. (5.30)

We now need only two Faddeev components: |ψci and one of |ψn1 i and |ψn2 i, which we

call |ψni; the third Faddeev component is given by Pˆ |ψni. It follows from the definitions in Section 5.1 that

Pˆ |pqic = |pqic, (5.31) and + −1 A(A + 2) 1 Pˆ |pqi = p − q p − q . (5.32) n 2 A + 1 (A + 1) A + 1 n The Faddeev equations can now be written as the following set of two coupled integral equations.

Z ∞ dq0 q02 1 Z 1 F q q.q0
n( ; 0000) = 2 d ˆ ˆ 0 2π 2 −1 n 0
0 0 {G0 π1(q, q ), q; −EB tn(−EB; q )Fn(q ; 0000)

n 0
0 0 + G0 π0(q, q ), q; −EB tc(−EB; q )Fc(q ; 0000)}, (5.33)

Z ∞ dq0 q02 1 Z 1 F q q.q0
c( ; 0000) = 2 2 d ˆ ˆ 0 2π 2 −1 c 0
0 0 G0 π2(q, q ), q; −EB tn(−EB; q )Fn(q ; 0000), (5.34)

where π0, π1 and π2 are the magnitudes of the momenta π0, π1, and π2 defined as

 A  π (q, q0) = q + q0, (5.35) 0 A + 1 ! 1 π (q, q0) = q + q0, (5.36) 1 A + 1 92 and 1 π (q, q0) = q + q0. (5.37) 2 2 Equations (5.38) and (5.39) can be regularized by introducing a cutoff Λ. A three-body contact interaction is then required to cancel the cutoff dependence at LO [10, 139, 140, 142, 146]. This gives

Z Λ dq0 q02 1 Z 1 F q q.q0
n( ; 0000) = 2 d ˆ ˆ 0 2π 2 −1 (" # H(Λ) Gn π (q, q0), q; −E
+ t (−E ; q0)F (q0; 0000) 0 1 B Λ2 n B n ) n 0
0 0 + G0 π0(q, q ), q; −EB tc(−EB; q )Fc(q ; 0000) , (5.38)

Z Λ dq0 q02 1 Z 1 F q q.q0
c( ; 0000) = 2 2 d ˆ ˆ 0 2π 2 −1 c 0
0 0 G0 π2(q, q ), q; −EB tn(−EB; q )Fn(q ; 0000), (5.39)

Λ has to be chosen large enough such that Fn and Fc are negligible at q > Λ, but small enough such that these integral equations can be solved accurately with reasonably √ 2 few mesh points. We choose Λ of the order of 10 mEB. For a given Λ, H(Λ) needs to be fixed by using one piece of three-body datum as input. We determine it by requiring the bound state to lie at −EB. More details about the numerical solution of Eqs. (5.38) and (5.39) can be found in Appendix B.2. The three-body wavefunction in the Jacobi representation with the spectator core is

c n chpq; 0000|Ψi =G0(p, q; −EB) tc(−EB; q)Fc(q; 0000) Z 1 o + d (pˆ.qˆ) tn(−EB; π2(q, p))Fn(π2(q, p); 0000) . (5.40) −1 93

Similarly, the three-body wavefunction in the Jacobi representation with the spectator neutron is

n n nhpq; 0000|Ψi = G0(p, q; −EB) tn(−EB; q)Fn(q; 0000) 1 Z 1 + d (pˆ.qˆ) tn(−EB; π1(q, p))Fn(π1(q, p); 0000) 2 −1 o + tc(EB; π0(q, p))Fc(π0(q, p); 0000) , (5.41)

A discussion of the numerical aspects of the calculation of the bound state wavefunction is given in Appendix B.2.

5.4.1 Matter radii

We now derive the expression for the matter radius of a two-neutron halo nucleus as a function of the three-body binding energy and the masses of the core and the neutrons in the unitary limit. Following Ref. [139], we first define the matter form factors,

Z dp p2 Z dq q2 1 Z 1 p F k2 y hpq | i hp q2 k2 − qky | i, x( ) = 2 2 d x ; 0000 Ψ x + 2 ; 0000 Ψ 2π 2π 2 −1 (5.42) where x = n, c, such that the mean-square distance of the neutron from the center of mass of the core and the other neutron, hr02i, can be extracted from

1 F (k2) = 1 − k2hr02i + ..., (5.43) n 6

and the mean-square distance of the core from the center of mass of the neutrons, hr2i, from 1 F (k2) = 1 − k2hr2i + .... (5.44) c 6 Based on geometrical arguments, we obtain the following formula for the mean-square

2 matter radius of a two-neutron halo in the point-like core approximation, hR0i: 2(A + 1)2 4A hR2i = hr02i + hr2i. (5.45) 0 (A + 2)3 (A + 2)3 94

2 At LO in Halo-EFT, the quantity mEBhR0i depends on all the variables featuring in the

Faddeev equations: Enn, Enc, EB and A. But it only depends on dimensionless ratios of these four parameters. We define the dimensionless function f (Enn/EB, Enc/EB; A), as [104]: ! 2 Enn Enc mEBhR0i ≡ f , ; A . (5.46) EB EB

2 We calculated mEBhR0i at the unitary limit, Enn = Enc = 0. It is plotted in Fig. 5.1 as a function of A. We would like to emphasize here that this result applies to any system of three non-relativistic particles, two of which are identical, in the limit of vanishing interaction range and infinite scattering length, regardless of their spin and statistics. These results are in qualitative agreement with those of Ref. [104], but our value of f is lower by approximately 15% for A = 20 in the unitary limit. This discrepancy is probably due to the incorrect renormalization procedure adopted while solving the Faddeev equations in Ref. [104]. (See Refs. [146, 185, 186] for further details.)

0.20

0.15 A "

0.10 0,0 ; ! f

0.05

0.00 0 5 10 15 20 A Figure 5.1: The dimensionless function f (0, 0; A), defined by Eq. (5.46), versus A. (Published in Ref. [146].) 95

5.5 Carbon-22

In Ref. [103], Tanaka et al. measured the reaction cross-section of 22C on a hydrogen target and, using Glauber calculations in the optical limit, deduced a 22C rms matter radius of 5.4 ± 0.9 fm. Their measurement implies that the two valence neutrons in 22C

preferentially occupy the 1s1/2 orbital and are weakly bound — a conclusion also supported by data on high-energy two-neutron removal from 22C [49]. Since 21C is unbound [187], this suggests that 22C is an S-wave Borromean halo nucleus with two neutrons orbiting a 20C core. The rms matter radius of 22C can be calculated at LO in Halo-EFT once the energies

of the neutron-core virtual state, Enc, the neutron-neutron virtual state, Enn, as well as the

12 22 binding energy, EB , are known. However, for C, neither Enc nor EB is well-known

[24, 172, 182, 188, 189]. We therefore use EFT to find constraints in the (EB, Enc) plane using Tanaka et al.’s value of the rms matter radius. In 2011, Yamashita et al. [183] attempted to analyze the experiment of Ref. [103] using the correlations between the energies and radii of halo nuclei derived in Ref. [104]. (See also Ref. [190] for a review.) The dependence of the rms matter radius of 22C on its binding energy was also explored by using a three-body model with a Woods-Saxon neutron-core potential and a density dependent contact interaction between the two neutrons [102]. Reference [182] made a similar attempt by employing a much simpler model for the structure of 22C.

5.5.1 Matter radius constraints on binding energy

The scaling function f shown in Fig. 5.1 can be calculated for any value of A. For 22C, which has A=20, Fig. 5.2 shows a three-dimensional plot of f in the

(Enn/EB, Enc/EB) plane. The disagreement with the results of Ref. [183] appears to be

12 The binding energy of 22C treated as a three-body system is equal to the two-neutron separation energy of the nucleus, S 2n. 96 worse at finite values of Enc and Enn than the 15% we found in the limit Enn = Enc = 0.

fHEnnEB, EncEB; 20L

0.04

0.03

0.02 0 0.010

2 2 E E EnnEB nc B 4 4

6 6

Figure 5.2: f (Enn/EB, Enc/EB; 20) versus (Enn/EB, Enc/EB). (Published in Ref. [146].)

Figure 5.2 gives us the results we need in order to set a model-independent constraint on the binding energies of 21C and 22C. First, though, we must take the finite matter radius of the 20C core into account. We do this by including that effect in our expression for the mean-square matter radius of the two-neutron halo:

A hR2i = hR2i + hR2i . (5.47) 0 A + 2 core

2 20 Here hR icore is the mean-square radius of the core, which we take from the C rms radius of (2.98 ± 0.05) fm measured by Ozawa et al. [191].

For all calculations in this dissertation we use the value of Enn obtained from

ann = (−18.7 ± 0.6) fm [192], except when we are working at the unitary limit, with

ann → ∞. 97

5.5.1.1 EFT errors

In addition to the EFT errors due to the LO two-body t-matrices used here, given by the ratio of the effective ranges to the respective scattering lengths, an additional EFT √ error enters this three-body calculation. This error can be quantified by 2mEB/Λ0,

where the break-down scale Λ0 can be estimated as the inverse of the size of the core. This new expansion parameter reflects the fact that there is an increased sensitivity of the three-body state to the short-distance interactions if the three-body binding is tighter. Since the neutron-neutron and the neutron-core effective ranges have roughly the same value as the 20C rms radius, we estimate the relative error of our calculation by √ √ √ 20 mEnn/Λ0, 2mEnc/Λ0 or 2mEB/Λ0, whichever is the largest. The nucleus C also has a bound 2+ state which lies 1.588 MeV above the ground state [193]. We have not used this energy scale in assessing the break-down scale of Halo-EFT for 22C, since the 2+ state can affect S-wave observables only via higher-dimensional operators which do not enter the calculation at next-to-leading order. √ 2 In Fig. 5.3, we plot the sets of (EB, Enc) values that give < R > = 4.5 fm, 5.4 fm and 6.3 fm, along with the theoretical error bands. All sets of EB and Enc values in the plotted region that lie within the area bounded by the edges of these bands give an rms matter radius within the combined (1-σ) experimental and theoretical error of the value Tanaka et al. extracted. The figure shows that, regardless of the value of the 20C − n virtual energy, Tanaka et al.’s experimental result puts a model-independent upper limit of 100 keV on the 2n separation energy of 22C. Since Yamashita et al. overpredict the matter radius for a given binding energy, their constraint on the maximum possible value of EB is about 20% weaker than ours. Our result for the maximum binding energy of 22C is a factor of two smaller than that found by Fortune and Sherr [182]. This indicates that their simple extension of the 98

p Figure 5.3: Plots of hR2i = 5.4 fm (blue, dashed), 6.3 fm (red, solid), and 4.5 fm (green, dotted), with their theoretical error bands, in the (EB, Enc) plane. (Published in Ref. [146].)

2 model-independent relationship between EB and hR0i that prevails in a one-neutron halo to two-neutron halos does not describe the three-body dynamics correctly.

5.5.2 On the possibility of excited Efimov states

Following Refs. [139, 140], we construct a region in the (EB, Enc) plane within which an excited Efimov state in 22C could occur. In Fig. 5.4 the purple region is that which allows at least one excited Efimov state above the ground state in an A=20 Borromean nucleus. In the same plot, we also show the boundary curves that enclose the sets of Enc and EB values which are consistent with an rms matter radius of 5.4 ± 0.9 fm once the theoretical errors are taken into account, i.e. those already displayed in Fig. 5.3. For a virtual state energy larger than a keV, the region that allows the existence of excited Efimov states does not overlap with the region in which Enc and EB are constrained by the matter radius for a 20C − n virtual state energy less than a keV. 13 Indeed, if the

13 The 22C radius can be computed accurately for 20C−n virtual state energies very close to threshold, but the computation of the existence of an Efimov state becomes numerically delicate near threshold. 99 trend in Fig. 5.4 continues to smaller values of Enc, Efimov states seem to be ruled out even for values of Enc 1 keV. However, an Efimov state is present if we take Enc = 0, as a consequence of the fact that there are two 20C−n pairs in the 22C system [131]. Recently, Mosby et al. analyzed the spectrum of the neutron decay of 21C obtained from single proton removal of 22N, and concluded that a 20C − n scattering length of magnitude less than 2.8 fm is required to explain the line shape [189]. Since the 21C system would need to be tuned very close to the unitary limit in order for 22C to have bound Efimov states, Tanaka et al.’s value for the matter radius rules out the existence of such states in 22C.

Figure 5.4: The region in the (EB, Enc) plane that allows excited Efimov states (purple), and the region that encloses values consistent with the experimental rms matter radius of 5.4 ± 0.9 fm with same color-coding as in Fig. 5.3. (Published in Ref. [146].) 100

5.6 Conclusion

We used Halo-EFT at leading order to examine the behavior of the rms matter radius

of S-wave Borromean halo nuclei in the (Enn/EB, Enc/EB) plane. We then applied these

22 results to C, and put constraints on the (EB, Enc) parameter space using the experimental value of the 22C matter radius. We derive stronger constraints than was done by prior studies [102, 182, 183] even with fewer assumptions about the structure of 22C. We use only the experimentally well-supported idea that it can be treated as a three-body system composed of 20C and two neutrons. Furthermore, our result incorporates the anticipated theoretical uncertainty of the leading-order Halo-EFT calculation based on this cluster picture. Even after this uncertainty, and the experimental (1-σ) error bar, are taken into

account we find that EB < 100 keV for all values of Enc. This rules out the possibility of an excited Efimov state in the 22C nucleus unless the 20C − n system has a virtual state with an energy much smaller than 1 keV. 101 6 Universal Relations for the Dipole Excitation of

Borromean Halo Nuclei

As discussed in Chapter 1, the Coulomb dissociation experiment is a powerful technique to study the structure of halo nuclei because of the enhancement of the cross section at low excitation energies. The reaction theory discussed in Appendix A is equally valid for two-neutron halo nuclei for the conditions in which the experiments are typically done. The spectrum of their dipole strength, B(E1), thus extracted from the Coulomb dissociation data, reveals information about the structure of the two-neutron halo nucleus. At LO in Halo-EFT, this spectrum is completely determined by the three-body binding energy and the two-body scattering lengths in the two-neutron halo nucleus. In this chapter, our derivation of the universal relation for the electric dipole excitation of S-wave Borromean halo nuclei to a continuum of two neutrons and a core is presented in detail. The results are compared with existing data on 11Li and predictions are made for an ongoing experiment on 22C. Our findings, combined with the forthcoming experimental data, can constrain the two-neutron separation energy of 22C and the virtual state energy of 21C. These results will appear in one or more publications in future [147].

6.1 The wavefunction for the three-body unbound state

As in Chapter 5, we work in Jacobi representation. The subscripts i and j are used to label the spectator particle. The subscript c is used if the spectator particle is the core and

n is used if one of the neutrons is the spectator. The two-body t-matrices tˆi and the three-body Green’s function Gˆ0 were defined in Chapter 5. To construct an unbound state three-body wavefunction, we need to add |Φi, the ˆ−1 solution of the homogeneous equation, G0 |Φi = 0, to the particular solution in Eq. (5.20) so that the overall solution, |Ψi, satisfies the appropriate boundary condition for scattering 102 when V1 = V2 = V3 = 0 [184, 194]. This gives

X X |Ψi = |Φi + Gˆ0 Vˆ i|Ψi = |Φi + |ψii, (6.1) i i where   X  X  |ψ i ≡ Gˆ Vˆ |Ψi = Gˆ Vˆ |Φi + Gˆ Vˆ |ψ i = Gˆ tˆ |Φi + |ψ i . (6.2) i 0 i 0 i 0 i j 0 i  j  j j,i The spectator functions now satisfy the Faddeev equations

X   |Fii = Gˆ0 tˆj |Φi + |F ji , (6.3) j,i and are related to the three-body wavefunction for the unbound state by

X |Ψi = |Φi + Gˆ0 tˆi (|Φi + |Fii) . (6.4) i 6.2 The dipole matrix element

0 For a two-neutron halo, the dipole operator is eZeffrY1(ˆr), where r is the distance of the core from the center of mass of the nn-pair and Zeff = 2Z/(A + 2). Using Eq. (5.11), its matrix element between the bound state |Ψini and the continuum state |Ψ f ii, can be written as

X Z ∞ dp p2 Z ∞ dq q2 M(E1) = hΨ |pq; lmλµi hpq; lmλµ|eZ rY0(ˆr) |Ψ i. (6.5) 2π2 2π2 f i i i eff 1 in lmλµ 0 0

0 The matrix element chpq; lmλµ|eZeffrY1(ˆr) |Ψini, for an initial state which has all subsystems in S-wave, can be written as

0 chpq; lmλµ|eZeffrY1(ˆr) |Ψini = −i δl0δm0δλ1δµ0 A(p, q; EB), (6.6)

where A(p, q; EB) can be calculated by Fourier-transforming the bound state wavefunction to coordinate space, which gives

1 Z ∞ Z ∞ A p, q E eZ r r3 j qr q0q02 j q0r hpq0 | i, ( ; B) = 3/2 eff d 1( ) d 0( ) c ; 0000 Ψin (6.7) π 0 0 103

0 or by writing the dipole operator itself in momentum representation, as ieZeffY1(ˆq) ∂/∂q, which yields 1 ∂ A(p, q; EB) = − √ eZeff chpq; 0000|Ψini. (6.8) 4π ∂q Note that the Kronecker deltas in Eq. (6.6) pick out a final state with odd parity and total angular momentum 1 when we evaluate the matrix element in Eq. (6.5).

In the Plane Wave Impulse Approximation (PWIA), |Ψ f ii = |Φi = |kcKcic, so that, using Eq. (5.9), we get

√   0 ˆ MPWIA(E1) = −i 2 π Y1 Kc A(kc, Kc; EB). (6.9)

Since we are dealing with systems in which the two-body interactions generate large S-wave scattering lengths, all S-wave final state interactions (FSI) contribute at LO [5–9]. Equation (6.5) therefore needs to be evaluated using Eq. (6.4) for the final-state wavefunction, with the unbound state spectator functions satisfying Eq. (6.3). Using

1 hpq; lmλµ|Gˆ tˆ|Φi = 4π5/2 δ(q − K ) Gi (p, q; E) t (E; q) Yµ∗(Kˆ )δ δ (6.10) i 0 i q2 i 0 i λ i l0 m0

and

Z ∞ dp0 p02 hpq lmλµ|Gˆ tˆ|F i Gi p, q E t E q δ δ hp0q lmλµ|F i, i ; 0 i i = 0( ; ) i( ; ) l0 m0 2 i ; i (6.11) 0 2π

where Kn is the momentum of the neutron in the three-body center of mass frame after fragmentation, along with Equations (5.14), (6.4), (6.6) and (6.9), and defining

Z 1 c 0
1 c 0
0
Ze2 q, q ; EB, E ≡ d (uˆ.vˆ) P1 (uˆ.vˆ) G0 π2(quˆ, q vˆ), q; E A π2(quˆ, q vˆ), q; EB ; 2 −1 (6.12) we can write (6.5) as

∗ ∗ ∗ ∗ ∗ M (E1) = MPWIA(E1) + Mnn(E1) + Mnc(E1) + Mnnc(E1), (6.13) 104

where √ Z ∞ dp p2 M∗ E i π t E K Y0 Kˆ A p, K E Gc p, K E nn( 1) = 2 c( ; c) 1 ( c) 2 ( c; B) 0( c; ) (6.14) 0 2π is the matrix element for the final state with rescatterings in the neutron-neutron subsystem only, and √ Z ∞ dq q2 M∗ E i π t E K Y0 Kˆ Zc (q, K E , E) nc( 1) = 2 n( ; n) 1 ( n) 2 e2 n; B (6.15) 0 2π is the matrix element for the final state with rescatterings in one neutron-core subsystem only, and Z ∞ dp p2 Z ∞ dq q2 M∗ E i A p, q E Gc p, q E t E q F K , K q nnc( 1) = 2 2 ( ; B) 0( ; ) c( ; ) c( c n; ; 0010) 0 2π 0 2π Z ∞ dq q2 Z ∞ dq q2 i n n t E q Zc (q, q E , E) F K , K q + 2 2 n( ; n)e2 n; B n( c n; n; 0010) 0 2π 0 2π (6.16) contains FSI in which all three particles participate. The terms in Eq. (6.13) can be represented diagrammatically as shown in Fig. 6.1. Unlike the bound state spectator functions of Eqs. (5.38) and (5.39), the unbound state spectator functions in Eq. (6.16) are obtained by solving the inhomogeneous Faddeev equations (Eq. (6.3)), √ 0 ˆ n Fn(Kc, Kn; q; 0010) =2 π tn(E; Kn) Y1 (Kn) Z1 (q, Kn; E) Z ∞ dq0 q02 Zn q, q0 E
t E q0 F K , K q0 + 2 1 ; n( ; ) n( c n; ; 0010) 0 2π √ 0 ˆ n + 2 π tc(E; Kc) Y1 (Kc) Z0 (q, Kc; E) Z ∞ dq0 q02 Zn q, q0 E
t E q0 F K , K q0 , + 2 0 ; c( ; ) c( c n; ; 0010) (6.17) 0 2π and √ 0 ˆ c Fc(Kc, Kn; q; 0010) =4 π tn(E, Kn) Y1 (Kn) Z2 (q, Kn; E) Z ∞ dq0 q02 Zc q, q0 E
t E q0 F K , K q0 , + 2 2 2 ; n( ; ) n( c n; ; 0010) (6.18) 0 2π 105

Figure 6.1: Diagrammatic representation of the terms in Eq. (6.13): MPWIA (top, left), Mnn (top, middle), Mnc (top, right) and Mnnc (bottom row). The thin solid line represents the neutron, the dashed line represents the core, the thick dark line and the thick gray line represent the rescatterings in the nc and the nn subsystems respectively. The initial (bound) state is shown as a gray blob interacting with the photon. The white blob represents FSI in which all outgoing particles participate.

where Z 1 n,c 0
1 n,c 0
Zi q, q ; E = d (uˆ.vˆ) P1 (uˆ.vˆ) G0 πi(quˆ, q vˆ), q; E . (6.19) 2 −1 However, solution of these integral equations at E > 0 is complicated by the fact that the integration kernels have singularities along the path of q0 at positions that depend on q — the so-called “moving” singularities which have the form [195]

2 02 0 |E − q − q + qq | log a b c , q2 q02 qq0 |E − a − b − c | where a, b and c are positive numbers with the dimension of mass. This problem has been dealt with using different methods in the past [196–202]. We use the integration-path distortion method of Refs. [196, 198], in which the singularities can be avoided by manipulating the intgral equations analytically. A quasi-analytic technique was developed in Refs. [203, 204], where the integrand was represented by a piecewise function using cubic spline interpolation and analytic formulas for the product of a cubic polynomial and a logarithm were then used. 106

We begin by expressing these spectator functions as

√ 0 ˆ Fx(Kc, Kn; q; 0010) =2 π tn(E; Kn) Y1 (Kn) Txn(q, Kn) √ 0 ˆ + 2 π tc(E; Kc) Y1 (Kc) Txc(q, Kc). (6.20)

where x = n, c. In order that Fn and Fc satisfy Eqs. (6.17) and (6.18), Tnx and Tcx must satisfy the following sets of coupled integral equations:

Z ∞ dq00 q002 T q, q0 Z q, q0 E
Zn q, q00 E t E q00 T q00, q0 nx( ) = nx ; + 2 1 ( ; ) n( ; ) nx( ) 0 2π Z ∞ dq00 q002 Zn q, q00 E
t E q00 T q00, q0 + 2 0 ; c( ; ) cx( ) (6.21) 0 2π and

Z ∞ dq00 q002 T q, q0 Z q, q0 E
Zc q, q00 E
t E q00 T q00, q0 , cx( ) = cx ; + 2 2 2 ; n( ; ) nx( ) (6.22) 0 2π

n n c where Znn = Z1 , Znc = Z0 , Zcn = 2Z2 and Zcc = 0. Note that these are different from the integral equations for the three-body T-matrices derived in Chapter 2 because the spin part are not included in Tnx and Tcx. We follow the method adopted by Hetherigton and Schick in Ref. [196] and evaluate

−iφ 0 0 −iφ Tnx and Tcx for q → |q|e , q → |q |e with a fixed argument φ in the interval (0, π/4). The integrals are now performed over q00 → |q00|e−iφ with q00 in the interval (0, Λ). Cauchy’s theorem ensures that this transformation is valid as long as the cutoff Λ is chosen large enough such that the contribution to the integral from the arc q00Λe−iθ, where θ goes √ 2 from φ to 0, is negligible. This is true for Λ of the order of 10 mEB, which we use in our calculation. The φ−independence of the result is numerically checked in Appendix B.4.

We can now rewrite (6.16) in terms of Tnx and Tcx, as 107

√ ∗ 0 ˆ Mnnc(E1) = i 2 π tn(E; Kn) Y1 (Kn) " ∞ ∞ Z dp p2 Z dq q2 A p, q E Gc p, q E t E q 2 2 ( ; B) 0( ; ) c( ; ) 0 2π 0 2π ( ∞ Z dq0 q02 Zc (q, K E) T q, q0 t E q0 Zn q0, K E
2 2 n; + 2 cn( ) n( ; ) 1 n; 0 2π ∞ ) Z dq0 q02 T q, q0 t E q0 Zc q0, K E
+ 2 2 cc( ) c( ; ) 2 n; 0 2π Z ∞ dq q2 Z ∞ dq q2 n n t E q Zc (q, q E , E) + 2 2 n( ; n) e2 n; B 0 2π 0 2π ( ∞ Z dq0 q02 Zn (q , K E) T q , q0 t E q0 Zn q0, K E
1 n n; + 2 nn( n ) n( ; ) 1 n; 0 2π ∞ )# Z dq0 q02 T q , q0 t E q0 Zc q0, K E
+2 2 nc( n ) c( ; ) 2 n; 0 2π √ 0 ˆ + i 2 π tc(E; Kc) Y1 (Kc) " ∞ ∞ Z dp p2 Z dq q2 A p, q E Gc p, q E t E q 2 2 ( ; B) 0( ; ) c( ; ) 0 2π 0 2π Z ∞ dq0 q02 T q, q0 t E q0 Zn q0, K E
2 cn( ) n( ; ) 0 c; 0 2π Z ∞ dq q2 Z ∞ dq q2 n n t E q Zc (q, q E , E) + 2 2 n( ; n) e2 n; B 0 2π 0 2π ( ∞ )# Z dq0 q02 Zn (q , K E) T q , q0 t E q0 Zn q0, K E
0 n c; + 2 nn( n ) n( ; ) 0 c; 0 2π (6.23)

Following the method of Ref. [198], we evaluate these integrals along paths in the fourth quadrant in the complex momentum plane as discussed in Section 6.2.1. This requires A(p, q; EB) to be evaluated at complex q and p, for which the bound state wavefunction, chpq; 0000|Ψini, has to be analytically continued to complex p and q. Some of the numerical techniques used to calculate the matrix elements in Eqs. (6.14), (6.15) and (6.23) are given in Appendix B.4. 108

6.2.1 Evaluation of the matrix element by distortion of the integration paths

We now investigate the nature of the singularities that appear in the integrand in Eq. (6.23). This is an extension of the aforementioned Hetherigton-Schick method to the case where the singularities involve real momenta, which can not be rotated in the complex plane. Cahill and Sloan performed this analysis for the case of three identical particles in Ref. [198].

n,c 0 The functions Zi (q, q ; E) have the form

Z 1 n,c 0
1 1 Z q, q ; E = dx P1(x) i q2 q02 qq0 x 2 −1 E − a − b − c + i " # ! c c q2 q02 = Q E − − + i , (6.24) qq0 1 qq0 a b

where q and q0 are real and positive momenta, and a, b, and c are positive numbers with

dimension of mass, and Q1 is the Legendre function of the second kind of degree 1,  x |x + 1|  −1 + log , |x| > 1, x ∈  R Q (x + i) =  2 |x − 1| (6.25) 1  x |x + 1| π  −1 + log − i x, |x| < 1, x ∈ .  2 |x − 1| 2 R

n,c 0 2 02 0 Zi (q, q ; E) thus has logarithmic branch points at E − q /a − q /b = ±qq /c. Unlike Eqs. (6.21) and (6.22), where we could stay in the logarithmic branch that is consistent with the E → E + i prescription by rotating both q and q0 into the fourth quadrant in the complex momentum plane, Eq. (6.23) contains integrals of type

Z ∞ 2 n,c I = dq q f (q) Zi (q, K; E) , (6.26) 0

where K is a physical momentum, which cannot be rotated into the complex plane. The

n,c branch points and associated branch cuts of Zi (q, K; E) in the q plane for real K are shown in Fig. 6.2. For K2 < bE, we can simply make the transformation q → q exp (−iφ) as before. However when bE < K2 < 4bc2E/(4c2 − ab), we get branch cuts in the fourth 109 quadrant of the complex q plane. Note that 4bc2E/(4c2 − ab) is the maximum possible value of all external momenta appearing in Eq. (6.23). We adopt the method used by Cahill and Sloan in Ref. [198]. We evaluate the integral in Eq. (6.26) along the path ABCDE of Fig. 6.2. In the segment BCD of this path, we take

n,c the value of Zi (q, K; E) in the second Riemann sheet. This gives  R  2 n,c 2  dq q f (q) Zi (q, K; E) , K < bE I =  CDE (6.27)  R R k0  dq q2 f (q) Zn,c (q, K; E) + dk k2 f (k) ∆Zn,c (k, K; E) , K2 > bE;  CDE i 0 i

n,c n,c where ∆Zi (k, K; E) is the difference between the values of Zi (k, K; E) at the two Riemann sheets, and s ! a a2K2 K2 k = K − − a − E . (6.28) 0 2c 4c2 b

Figure 6.2: Integration paths in the complex q plane (shown by the arrows) for K2 < bE (left) and for bE < K2 < 4bc2E/(4c2 − ab) (right). The dashed line is used to indicate that n,c the value of Zi (q, K; E) in the second Riemann sheet should be used.

Note that the last integral in Eq. (6.27) is performed along the real k axis. Table 6.1

2 n,c shows the value of bE and the maximum possible values of k0 for Zi (q, K; E). Although the function f (q) in Eq. (6.23) can include various three-body Green’s functions and 110

2 T-matrices, the values of bE and k0 are such that, as long as A ≥ 1, we never encounter any singularities while evaluating f (k) by performing all other integrals along the rotated path.

2 n,c Table 6.1: The value of bE and the maximum possible value of k0 for all the Zi (q, K; E) featuring in Eq. (6.23).

n,c 2 Zi bE maximum possible value of k0 n 2A 2 Z1 (q, K; E) A+1 mE (A+1)(A+2) mE c 2A2 Z2(q, K; E) mE (A+1)(A+2) mE n 2A A Z0 (q, K; E) A+1 mE A+2 mE

6.3 The B(E1) spectrum

For a core of spin J and projection M, we need to couple the spinor |χMi to the initial i Ji and the final state wavefunctions. Since two neutrons are in the spin singlet state, this simply gives an initial state wavefunction with spin Ji and projection M, and a final state wavefunction with angular momentum quantum numbers J f and M, where

Ji1J f |Ji − 1| ≤ J f ≤ Ji + 1. The associated Clebsch-Gordan coefficients are 1 and CM0M, respectively. The dipole response function B(E1) is then

3 X  2 d3k d3K 6B E CJi1J f |M E |2 c c , d ( 1) = M0M ( 1) 3 3 (6.29) 2Ji + 1 (2π) (2π) J f ,M

where M(E1) is given by Eq. (6.23). Evaluating the sum over J f and M, we get ! dB(E1) Z d3k Z d3K k2 K2(A + 2) = 3 c c |M(E1)|2 δ E − c − c . (6.30) dE (2π)3 (2π)3 m 4Am

By integrating over kc using the energy-conserving delta function, we can reduce the six-dimensional integral in Eq. (6.30) to a five-dimensional one. Furthermore, we can choose a coordinate system in which the direction of one of the vectors kc and Kc lies in 111

the zx−plane. This leaves an integral in four variables: Kc, Kˆ c · z, kˆc · z, and the azimuthal angle, φ, between the two vectors. We then have √ Z 4Am mE Z 1 Z 1 Z 2π dB(E1) 3m A+2 K K2 k Kˆ · z kˆ · z φ |M E |2. = 5 d c c c d( c ) d( c ) d ( 1) (6.31) dE 64π 0 −1 −1 0

Since the integrand is symmetric under kc → −kc, and since it depends on φ only through cos φ, the equation reduces to √ Z 4Am mE Z 1 Z 1 Z π dB(E1) 3m A+2 K K2 k Kˆ · z kˆ · z φ |M E |2, = 5 d c c c d( c ) d( c ) d ( 1) (6.32) dE 16π 0 −1 0 0 where we have used Z 2π Z π dφ f (cos φ) = 2 dφ f (cos φ), (6.33) 0 0 which is easily seen by making the transformation φ → 2π − φ.

2 The momentum distribution, d B(E1) , can be obtained by performing only the angular dEdKc

integrals. Similarly, a distribution in Kn can be obtained by writing the three-body phase space in terms of the Jacobi momenta with the neutron as the spectator. The integrals can be performed numerically as outlined in Appendix B.5. The EFT errors in this calculation are the same as those discussed in Sec. 5.5. However, an additional EFT error is introduced if the wavelength of the E1 photon is short enough to probe the short-distance substructures inside the core. We estimate this error by

p 2 ω hR icore, where ω = E + EB is the energy/momentum of the photon.

6.3.1 Unitary Limit

To begin with, we apply our LO EFT results to a two-neutron halo nucleus at the unitary limit. In this limit, not only the interaction ranges go to zero, but both the

two-body scattering lengths, anc and ann go to infinity. We define the dimensionless dipole

2 2 2 response function A /(eZ) m EB dB(E1)/dE, which is a function of dimensionless ratios

A and E/EB only in the unitary limit. In Fig. 6.3, we plot these universal dipole response functions. 112

0.1 0.1 0.09 0.08 A = 1 0.08 A = 1 0.07 A = 4 A = 4 A = 9 A = 9 0.06 0.06

dB(E1)/dE A = 20 dB(E1)/dE A = 20 2 2 B 0.05 A = 100 B A = 100

m E 0.04 m E 0.04 2 2 0.03 /(eZ) /(eZ) 2 0.02 2 0.02 A A 0.01 0 0 0 2 4 6 8 0 2 4 6 8 E/EB E/EB Figure 6.3: The universal dipole response functions in the unitary limit in PWIA (left) and with FSI included (right) for A = 1, 4, 9, 20 and 100.

Up to numerical errors, which would be unnoticeable in the figure, we obtain the same plots even when we vary EB by several orders of magnitude. Since the expression for dB(E1)/dE derived in Eq. (6.30) depends on EB in an intricate way, the fact that a simple scaling, which conforms with dimensional analysis, exists between dB(E1)/dE and EB is a non-trivial test of our calculations.

6.3.2 Lithium-11

We now apply this analysis to study the 11Li nucleus as a S-wave two-neutron halo with a 9Li core. The 9Li nucleus has a ground state Jπ of 3/2−, with a first excitation energy of 2.691 (5) MeV [205]. Its rms matter radius is 2.32 ± 0.02 fm [206]. When converted to momentum scale in the same manner as was done earlier for the 20C core inside the 22C nucleus, we obtain a momentum cutoff, Mhi of about 70 MeV. The ground state of the unbound nucleus 10Li is a virtual state at about 26 keV [207]. However, a low-lying P-wave resonance with a width of about 200 keV has been observed at about 400 keV in 9Li(d, p)10Li reaction [208]. 11Li, with a ground state Jπ of 3/2−, is believed to have a dominant configuration in which the two valence neutrons are in 113

S-wave around a 9Li core, with a strong admixture of 9Li − n P-wave configuration. This is supported by Ref. [209], which used shell model calculations to explain the β decay of 11Li to 11Be, and found about 50% component of 9Li − n S-wave. (See Ref. [205] for a compilation of data on 9,10,11Li). In 2006, Nakamura et al., by segregating the neutron-detector system into different walls and by imposing causality conditions on the neutron detection events, were able to separate “cross talk” of the neutron signal, and get a clean measurement of the 11Li Coulomb dissociation spectrum [22], which earlier experiments [210–212] had failed to do. Reference [213] compared the experimental data of Ref. [22] with a three-body potential model calculation by Esbensen and Bertsch [98], and found that the contribution of nn FSI was important to describe the data. A coupled channel three-body model [214] suggested that the nn and 9Li − n S-wave virtual states largely contribute to the shape and strength of the Coulomb dissociation cross section, and that the 9Li − n P-wave resonance hardly plays any role. We now use Halo-EFT to calculate the dipole response spectrum for the break up of 11Li into 9Li and neutrons at LO in the EFT. Note that there are only S-wave 9Li − n interactions included in this calculation. Figure 6.4 shows the B(E1) spectrum, expressed as a dimensionless ratio, for an

11 A = 11 nucleus in the unitary limit, and for Li at EB = 0.369 MeV, Enc = 0.026 MeV, and Enn = 0.118 MeV. B(E1) has smaller value at finite scattering lengths than in the unitary limit. This is because the mean square distance of the core from the three-body center of mass decreases as we go from unitary limit to finite scattering lengths. In Fig. 6.5, we show the 11Li B(E1) spectrum at the aforementioned values of two-and three-body energies, in PWIA and with the LO FSI partially and fully included. 114

11 Li at EB=369 keV, Enc=26 keV, Enn=118 keV 0.1 A = 9 nucleus in the unitary limit for arbitrary EB dB(E1)/dE 2 B

m E 0.05 2 /(eZ) 2 A

0 0 2 4 6 8 E/EB 11 Figure 6.4: The dipole response spectrum for Li in the unitary limit and for EB = 0.369 MeV, Enc = 0.026 MeV and Enn = 0.118 MeV.

1.5

PWIA only nn FSI included two-body FSI included all FSI included 1 /MeV) 2 fm 2

0.5 dB(E1)/dE (e

0 0 1 2 3 E (MeV) 11 Figure 6.5: The dipole response spectrum for Li for EB = 0.369 MeV and Enc = 0.026 MeV in PWIA, with rescatterings in the nn and in one nc subsystem in the final state, and with all LO S-wave FSI included. 115

The two-neutron separation energy of 11Li is known with high precision [27]. The 10Li virtual state energy, however, is not well constrained [207, 208, 215, 216].

9 Reference [215] determined that the Li − n scattering length, anc is less than −20 fm, which corresponds to Enc < 50 keV. Reference [208] quoted −24 fm < anc < −13 fm. The

NNDC value of Enc = 0.026 MeV [207] used in Fig. 6.5 corresponds to anc ' −30 fm. In Fig. 6.6, we show the all-FSI-included curve, along with the error bands from varying the 9Li − n scattering length from −50 fm to −13 fm (which corresponds to a range of 10 keV to 140 keV in Enc) and from the higher order terms in the EFT expansion. The error due to uncertainty in Enc is much smaller than the EFT error for this leading order calculation. L 1.5  MeV 2

fm 1.0 2 H e E

L d 0.5 1 E H B d 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 E MeV Figure 6.6: The dipole response spectrum for 11Li with all LO S-wave FSI included. The blue band is the error due to uncertainty in Enc and the purple band the EFT error. H L

Figure 6.7 shows the RIKEN data, read off from Ref. [22], for the B(E1) spectrum of 11Li along with our result, after it has been folded with the detector resolution. Within the uncertainty of a leading order calculation, a good agreement between the data and our prediction is seen. 116

L 1.5  MeV 2 1.0 fm 2 H e E

L d 0.5 1 H E B d 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 E MeV Figure 6.7: The dipole response spectrum for 11Li after folding the detector resolution (blue curve) with the theory error (purple band), and data from Ref. [22]. H L

6.3.3 Coulomb dissociation of Carbon-22

As measured in Ref. [103], the unusually large experimental rms matter radius of 22C, and the extremely low value of the two-neutron separation energy this indicates [146], motivates further studies of the two- and three-body energetics of this two-neutron halo system. In addition, the predominance of the S-wave neutron-core configuration in the wavefunction [49], and the absence of any evidence for a low-lying P-wave neutron-core resonances, makes 22C a good subject for the application of our theory. Figure 6.8 shows the dipole response spectrum for the break up of 22C into 20C and neutrons for different values of EB and Enc which lie within the 1 − σ confidence region we derived from the matter radius datum in Section 5.5. By comparing the results of a forthcoming experiment on the Coulomb dissociation of 22C [217] with this Halo-EFT prediction, we can obtain constraints for the two-neutron separation energy of 22C and the virtual state energy of 21C, and compare those with the constraints we derived [146] from the measurement of Ref. [103] (discussed in 117

22 Figure 6.8: The dipole response spectrum for C for EB = 50 keV, Enc = 10 keV (blue); EB = 50 keV, Enc = 100 keV (red) and EB = 70 keV, Enc = 10 keV (brown), with their EFT error bands.

Section 5.5). This will be a test of the consistency of the measured matter radius with the Coulomb dissociation data. Even though the curves in Fig. 6.8 have small EFT error bands as a result of the extremely weak binding of 22C, the fact that similar B(E1) spectra can arise from different sets of values of EB and Enc will lead to large uncertainties in the determination of these values individually.

6.3.4 Momentum distributions

We show in this section that the distribution of dB(E1)/dE in Kn, the momentum of the neutron in the center of mass of the fragments, is sensitive to the neutron-core interactions in the final state for large Kn when this two-body subsystem has small relative energy, and can therefore be used to determine the strengths of pairwise correlations. 118

2 2 In Fig. 6.9, mEBd B(E1)/dEdKc at A = 20 and Z = 6 is plotted versus Kc in PWIA and with FSI included. The FSI curve becomes highly sensitive to ann when Kc approaches √ √ max Kc = 2 mE A/(A + 2). This value corresponds to vanishing neutron-neutron relative

max momentum. For ann → ∞, at Kc = Kc , the momentum distribution goes to infinity.

0.0007

1/anc = -10 MeV, 1/ann = 0 (PWIA) 0.0006 1/anc = -10 MeV, 1/ann = 0 (FSI) 1/anc = 1/ann = -10 MeV (PWIA) /MeV)

2 0.0005 1/anc = 1/ann = -10 MeV (FSI) (e c 1/anc = -10 MeV, 1/ann = -3 MeV (FSI) 0.0004

0.0003 B(E1)/dEdK 2 d

2 0.0002 B

m E 0.0001

0 0 10 20 30 40 Kc (MeV) 2 2 Figure 6.9: mEBd B(E1)/dEdKc versus Kc at different values of two-body scattering lengths in PWIA (dashed) and with FSI included (solid) at E = EB = 0.4 MeV, A = 20 and Z = 6.

2 2 In Fig. 6.10, a similar behavior is observed for mEBd B(E1)/dEdKn when the √ max neutron carries most of the available energy, Kn ∼ Kn = 2(A + 1)mE/(A + 2), so that the neutron-core relative momentum becomes small.

max Although the momentum distributions in Figs. 6.9 and 6.10 are infinite at Kc = Kc

max and at Kn = Kn respectively, the B(E1) spectrum, and therefore the cross-section are still 119

0.002

1/anc = 0, 1/ann = -10 MeV (PWIA) 1/anc = 0, 1/ann = -10 MeV (FSI) 1/anc = 1/ann = -10 MeV (PWIA) /MeV) 2 1/anc = 1/ann = -10 MeV (FSI) (e n 1/anc = -3 MeV, 1/ann = -10 MeV (FSI)

0.001 B(E1)/dEdK 2 d 2 B m E

0 0 5 10 15 20 25 30 Kn (MeV) 2 2 Figure 6.10: mEBd B(E1)/dEdKn versus Kn at different values of two-body scattering lengths in PWIA (dashed) and with FSI included (solid) at E = EB = 0.4 MeV, A = 20 and Z = 6.

−1/2  max  max finite because the momentum distributions go as Kc,n − Kc,n , which, at Kc,n = Kc,n , are integrable singularities.

6.4 The photonuclear sum rules

The photonuclear sum rules allow us to study some basic properties of the ground state of a system without going into the details of the excited states [218, 219]. The Thomas-Reiche-Kuhn sum rule, commonly known as the energy-weighted sum rule (EWSR) was originally discovered for atomic excitations in Refs. [220–222], but has since proved extremely useful in nuclear systems too. The bremstrahlung sum rule or the non-energy-weighted sum rule (NEWSR) was developed for nuclei in Refs. [223, 224]. In this section, we begin by deriving these sum rules. The presentation here loosely follows 120 those of Refs. [219, 225]. We then calculate the integrals for the B(E1) spectrum we derived at LO in Halo-EFT and compare the results with those from the sum rules.

Let |0i be the ground state of the Hamiltonian Hˆ such that Hˆ |0i = E0|0i, and {|ni} be the excited states such that Hˆ |ni = En|ni. We define the response fuction, R(ω), for the transition from a ground state to any of the excited states by an operator Dˆ , as

∞ X 2 R(ω) = gn |hn|Dˆ |0i| δ(ω − En + E0), (6.34) n=1 where gn is the number of degenerate states at energy En. If the energy eigenvalues are continuous, the sum becomes an integral with gn representing the density of states. For an operator whose ground state expectation value, h0|Dˆ |0i, vanishes, we can include the

En = E0 term in the sum above, which allows us to use

X∞ gn |ni|hn| = 1ˆ. (6.35) n=0

p Using this completeness relation, and the expansion in binomial coefficients, Ck,

p X p p−k k (Hˆ − E0) = Ck Hˆ (−E0) , (6.36) k=0 we can derive Z p † p dω ω R(ω) = h0|Dˆ (Hˆ − E0) Dˆ |0i. (6.37)

In particular, for p = 0, we get the non-energy-weighted sum rule (NEWSR), Z dω R(ω) = h0|Dˆ †Dˆ |0i. (6.38)

And for p = 1, we get the energy-weighted sum rule (EWSR), which for an operator with Dˆ † = Dˆ , can be written as

Z 1 h h ii dω ω R(ω) = h0| Dˆ , Hˆ , Dˆ |0i. (6.39) 2 121

Equations (6.38) and (6.39) are derived in Ref. [219] for nuclei with point nucleons. Z A X X 3 The dipole operator is taken to be Dˆ ≡ zi = ziτi /2, where Z is the number of i=1 i=1 protons, A the number of nucleons, zi are the z−components of the positions ri of the

3 protons measured from the center of mass of the nucleus, and τi are their isospin projections with values 1 for the protons and −1 for the neutrons. If the Hamiltonian is such that [Dˆ , [Vˆ , Dˆ ]] = 0, the right hand side of Eq. (6.39) evaluates to NZ/(2mA), where N is the neutron number, and m is the proton (or nucleon) mass. For a three-dimensional A X 3 dipole operator, Dˆ ≡ riτi /2, as defined in Ref. [226], the right hand side of Eq. (6.39) i=1 is three times larger. Since it is customary to define B(E1) in terms of a dipole operator

M written as erY1 (ˆr), the EWSR for B(E1) of a nucleus with Z protons, N neutrons and A nucleons is 9 NZ e2 S (E1; A) = , (6.40) 1 4π A 2m as quoted in Ref. [225]. The EWSR for halo/cluster nuclei, can then be found by using Ref. [225]’s prescription,

S 1(E1; A1 + A2) = S 1(E1; A) − S 1(E1; A1) − S 1(E1; A2), (6.41) where S 1(E1; A1 + A2) is the sum rule of the system with clusters containing A1 and A2 nucleons treated as point charges, and S 1(E1; A1) and S 1(E1; A2) are the sum rules for the clusters themselves. For a one-neutron halo with a charge Z and a mass M for the core, and a neutron-core reduced mass µ, this gives Z ∞ dB(E1) 9 µ E E − E e2Z2 , d ( 0) = 2 (6.42) 0 dE 8π M

where E0 is the ground state energy, equal to −B in our notation of Chapter 4. This can be easily verified by using the expression for dB(E1)/dE from Eq. (4.6). For a two-neutron halo with a core of mass number A and charge Z, we obtain Z dB(E1) 9 1 2 dE (E − E ) = e2Z2 , (6.43) 0 dE 4π 2m A(A + 2) 122

where E0 is the energy of the core-neutron-neutron system, equal to −EB. This equation was derived in Ref. [102]. However, the weight E − E0 was erroneously replaced by E.

For a one-neutron halo, it is easy to show analytically that such an omission of E0 leads to an error of exactly one-sixth in the integral.

6.4.1 Direct derivation of the sum rules for neutron halo nuclei

For neutron halo nuclei, it is much simpler to derive the sum rules directly. For a one-neutron halo, the dipole operator can be taken to be

ˆ 0 p D = eZµ/M rY1 (ˆr) = eZµ/M 3/(4π) z, (6.44)

where r is the neutron core distance. In the absence of bound excited states, the excited states of the Hamiltonian

pˆ2 Hˆ = + Vˆ , (6.45) 2µ

form a continuum so that X Z d3 p g → . n (2π)3 En Equation (6.38) therefore gives

Z ∞ dB(E1) 3 µ2 E e2Z2 hr2i. d = 2 (6.46) 0 dE 4π M

And, since [ˆz, [p ˆ2, zˆ]] = 2, Eq. (6.39) gives Eq. (6.42). For a two-neutron halo, with the q ˆ 2 3 dipole operator given by, D = eZ A+2 4π r, where r is the position of the neutron-neutron center of mass relative to the core, and A is the ratio of the core mass to the neutron mass

2 such that the distance of the core from the center of mass of the halo nucleus is A+2 r. Equation (6.38) then gives

!2 Z dB(E1) 3 2 dE = e2Z2 hr2i, (6.47) dE 4π A + 2 123

and, since 1 A + 2 Hˆ = k2 + K2 + Vˆ , (6.48) m c 4Am c

where the Jacobi momentum kc commutes with r, Eq. (6.39) gives Eq. (6.43). Note that in these derivations, we have assumed that [Dˆ , [Vˆ , Dˆ ]] which does not hold for all the EFT interactions in general, but does so for the ones used in our calculations.

6.4.2 Numerical evaluations of the sum rules for two-neutron halo nuclei

We now evaluate the non-energy-weighted and the energy-weighted integrals of dB(E1)/dE , and test the sum rules. The right hand side of the NEWSR can be obtained by evaluating hr2i from Eqs. (5.42) and (5.44). This comparison can be done even for the

PWIA result since the plane-wave states, as eigenstates of the Hamiltonian Hˆ 0 = Hˆ − Vˆ , form a complete set of basis. But the PWIA result need not satisfy the EWSR because the derivation of Eq. (6.37) above for all p , 0 requires the final states to be eigenstates of the full Hamiltonian Hˆ . Once the final state interactions are included, however, both the NEWSR and the EWSR sum rule should be satisfied. Here we test the sum rules, first in the unitary limit with a wide range of A values. We begin by making power law fits to the dB(E1)/dE curves to find out whether the non-energy-weighted and the energy-weighted integrals converge or not. We do this both in the PWIA and with FSI included. We then repeat the analysis for finite scattering lengths at A = 20.

2 2 Fig 6.11 shows the plot of dimensionless response function mEB/(eZ) dB(E1)/dE versus E/EB for A = 20 in PWIA in the unitary limit. A fit to this curve, with the form

b −3 a(E/EB) with a = 1.0606 × 10 and b = −1.9708 is also shown in the figure.

2 2 Since the plot of mEB/(eZ) dB(E1)/dE versus E/EB depends on only one external parameter A, we repeat the analysis at several different values of A in Table 6.2. Assuming

2 2 that the fit function correctly approximates mEB/(eZ) dB(E1)/dE to E → ∞, we conclude the the NEWSR converges in PWIA, whereas EWSR does not converge since 124

0.000016

0.000014 calculated b fit, a x , with a = 0.001061 and b = 1.9708 0.000012

0.000010

dB(E1)/dE 0.000008 2 Z 2

/e 0.000006 2 B

0.000004 m E

0.000002

0.000000 20 40 60 80 100 E/EB 2 2 Figure 6.11: The dimensionless response function mEB/(eZ) dB(E1)/dE versus E/EB for A = 20 in PWIA in the unitary limit shown with a power law fit function.

the integrand of the energy-weighted integral falls off more slowly than E−1 at large E. We reiterate that the EWSR need not hold at PWIA. Once final state interactions are included, however, both NEWSR and EWSR converge because the integrands of both the energy-weighted and the non-energy weighted integrals fall off faster than E−1.

2 The dimensionless dipole strength, mEBB(E1)/(eZ) , is a universal function of A in

2 the unitary limit. Table 6.3 shows the comparison of the values of mEBB(E1)/(eZ) obtained by integrating the dipole response function and by using NEWSR with hr2i evaluated from from Eqs. (5.42) and (5.44) for different values of A. The energy-weighted integral, along with the value obtained from EWSR is also shown. The non-energy-weighted and the energy weighted integrals are obtained by numerical integration up to E/EB = 100, and by integrating the power-law extrapolant from

E/EB = 100 to E/EB → ∞. 125

2 2 Table 6.2: Power-law fits to mEB/(eZ) dB(E1)/dE in PWIA and with FSI included for various values of A.

b 2 2 b 2 2 A a(E/EB) fit to mEB/(eZ) dB(E1)/dE a(E/EB) fit to mEB/(eZ) dB(E1)/dE in PWIA with FSI

1 a = 1.2301 × 10−1, b = −1.9604 a = 3.5720 × 10−2, b = −2.4634 4 a = 1.8140 × 10−2, b = −1.9697 a = 1.3894 × 10−2, b = −2.3323 9 a = 4.6398 × 10−3, b = −1.9732 a = 9.4229 × 10−3, b = −2.3784 20 a = 1.0606 × 10−3, b = −1.9708 a = 1.7213 × 10−3, b = −2.3021 100 a = 4.6672 × 10−5, b = −1.9690 a = 7.9057 × 10−5, b = −2.3142

Table 6.3: Integrated dipole strength in the unitary limit. The second and the third columns show, respectively, the integrated dipole strengths with dB(E1)/dE calculated in PWIA, and with FSI included. R ∞ A mEB B(E1) mEB B(E1) mE B E / eZ 2 E E E dB(E1) 9 1 e2Z2 e2Z2 B ( 1) ( ) m 0 d ( + B) (eZ)2dE 4π A(A+2) in PWIA with FSI from NEWSR

1 5.55 × 10−2 6.25 × 10−2 5.65 × 10−2 4.5 × 10−1 2.39 × 10−1 4 8.32 × 10−3 1.01 × 10−2 8.43 × 10−3 8.4 × 10−2 2.98 × 10−2 9 2.14 × 10−3 2.76 × 10−3 2.19 × 10−3 2.0 × 10−2 7.23 × 10−3 20 5.03 × 10−4 6.57 × 10−4 5.12 × 10−4 5.1 × 10−3 1.63 × 10−3 100 2.24 × 10−5 2.96 × 10−5 2.27 × 10−5 2.2 × 10−4 7.02 × 10−5

In the unitary limit the non-energy-weighted integral overshoots the sum rule value by about 30% whereas the energy-weighted integral overshoots by about 200%. We now compare the results of energy-weighted and non-energy-weighted integrals

of B(E1) with the sum-rule values at finite scattering lengths for A = 20. We fix ann at

(−18.7 ± 0.6) fm and then pick several different values of EB and Enc. The integrals are 126

evaluated up to a large but finite cutoff. We then use a power-law extrapolant discussed above to obtain the integral to infinity. The results are shown in Table 6.4.

Table 6.4: Non-energy-weighted and energy-weighted integrals of the dipole response

function for A = 20 at several values of EB and Enc.  2 R ∞ E E B E / eZ 2 3 2 hr2i E E E dB(E1) 9 1 2 B nc ( 1) ( ) 4π A+2 0 d ( + B) (eZ)2dE 4π 2m A(A+2) (MeV) (MeV) (fm2) (fm2) (MeV fm2) (MeV fm2)

0.1 1.0 0.069 0.075 0.098 0.0675 0.1 0.6 0.075 0.082 0.102 0.0675 0.1 0.2 0.089 0.099 0.117 0.0675 0.2 0.2 0.053 0.059 0.106 0.0675 0.3 0.2 0.039 0.043 0.138 0.0675

For finite scattering lengths the non-energy weighted integral saturates only about 90% of the NEWSR value. The energy-weighted integral, on the other hand, overshoots

the sum rule value by roughly 50% for all the Enc and EB values chosen. Moreover the value of the energy-weighted integral seems to be dependent on Enc and EB, while the right hand side of the sum rule in Eq. (6.43) is not. Figure 6.12 shows the plots of the functions dB(E1)/dE and (E + EB)dB(E1)/dE that were integrated to evaluate the NEWSR and the EWSR respectively. Since the EWSR converges slowly, roughly half of the contribution to the energy-weighted integral comes from the E > 10EB region. The values calculated here could be sensitive to the goodness of the power law fits, which we have not analyzed in detail here. But it is worth mentioning that about 99% of the contribution to the non-energy weighted integral comes from the region E < 10EB. In this energy region, we have performed a detailed analysis of the errors in our calculation and concluded that the numerical inaccuracy is no more than about 1%. (See Appendix for details.) The deviation of the non-energy weighted integral from the sum rule value (about 10%) is therefore larger than can be accounted for by numerical errors. 127

7 2 )

E = 0.1 MeV, E = 0.2 MeV 2 E = 0.1 MeV, E = 0.2 MeV 6 B nc B nc

E = 0.2 MeV, E = 0.2 MeV fm E = 0.2 MeV, E = 0.2 MeV B nc 2 B nc

/MeV) 5 E = 0.3 MeV, E = 0.2 MeV E = 0.3 MeV, E = 0.2 MeV

2 B nc B nc E = 0.1 MeV, E = 0.6 MeV E = 0.1 MeV, E = 0.6 MeV

fm B nc B nc 2 4 E = 0.1 MeV, E = 1.0 MeV E = 0.1 MeV, E = 1.0 MeV B nc 1 B nc 3 ) dB(E1)/dE (e

2 B

1 dB(E1)/dE (e (E+E 0 0 0 1 2 3 4 0 1 2 3 4 E (MeV) E (MeV)

Figure 6.12: The integrands of the non-energy-weighted and the energy-weighted integrals used in the calculation of the values in Table 6.4.

6.5 Conclusion

We studied the dipole excitation of Borromean halo nuclei to continuum at LO in Halo-EFT. The results at this order are universal. In the unitary limit, we found that the

2 dimensionless quantity mEBdB(E1)/dE is a function of dimensionless ratios, E/EB and A only. We calculated dB(E1)/dE for finite scattering lengths and compared the result against experimental data for 11Li. Within the estimated theory error from neglected higher-order EFT terms, we obtained good agreement between the calculated and the experimental spectra. For 22C, our result is a prediction that can be compared to forthcoming experimental data. The effects of two-body correlation is seen in the distribution of dB(E1)/dE in the momentum of the third fragment. If experimental errors are small enough, the dB(E1)/dE spectrum and its momentum distribution data can be used along with our analysis to constrain the values of the 21C and 22C energies. 128 7 Summary and Outlook

In this dissertation, we studied the correlations between several properties of one- and two-nucleon halo nuclei in Halo-EFT, which uses the ratios between the short interaction range and the large two-body scattering lengths in these systems as an expansion parameter. We studied S-wave two-neutron halo nuclei at leading order in this expansion, in which these correlations are universal in the sense that they hold for a wide variety of systems regardless of the details of the short-range interactions. For one-nucleon halo nuclei, we performed higher order Halo-EFT calculations in order to take the range-dependent corrections into account. We calculated the Coulomb energy of a proton-core system by using a square well model for the nuclear interaction. Since the Coulomb-nuclear interference is inherently dependent on the model used for the nuclear interaction, a more detailed model of the short-range interaction than a square well is perhaps needed to get agreement with data. By using microscopic calculations to fix the values of all the required parameters, Halo-EFT can be used to connect the positions and the widths of the low-lying virtual states and resonances of proton-core systems like 11N with the neutron separation energies of their mirror nuclei. Such an approach is more advantageous than performing the entire calculation in the model because it is, presumably, much simpler to obtain a correct description of the low-energy physics of these systems in Halo-EFT than in any microscopic model. By an analysis of the experimental data on the Coulomb dissociation of 19C, we obtained the neutron separation energy and the asymptotic normalization coefficient of the halo neutron up to next-to-next-to-leading order in the ratio of the effective range to the scattering length of the 18C − n system. These parameters can be used to predict other properties of the 19C nucleus with high accuracy. For example, we have used our parameters to make a prediction for the longitudinal momentum distribution of the 129

Coulomb dissociation cross section. We have also predicted the difference between the mean-square charge radii of 19C and 18C. In the future, these parameters may be used to study the neutron-transfer reactions on 18C. We studied the universal relations between the matter radii and the binding energy of S-wave two-neutron halo nuclei. We used these relations, along with the experimental value of the matter radius, to constrain the two-neutron separation energy of 22C to a value less than 100 keV. We explored the consequences of this constraint on the possibility of occurrence of excited Efimov states in 22C, and found that an excited Efimov state cannot occur in 22C unless the virtual state energy of the unbound 21C nucleus is below 1 keV. A similar study of the correlation between the matter radius and the binding energy can be done for 6He which has a resonant P-wave interaction in the neutron-core subsystem. This can be done using the formalism and the results of Ref. [143]. The dipole response of S-wave Borromean halo nuclei halo nuclei was studied here. We found universal relation between the B(E1) spectrum and the two- and three-body energies. For 11Li, the predicted B(E1) spectrum agrees with the experimental result within the uncertainty of our leading order Halo-EFT calculation, whereas, for 22C, our result is a prediction that can be tested as soon as the experimental data is available. In order to study the Coulomb disscociation of 6He and to improve the accuracy of our 11Li calculation, this study has to be extended to include P-wave interactions, both in the initial and in the final state. If the two-neutron separation energy and the neutron-core virtual state energy of an S-wave two-neutron halo nucleus are known, our leading order calculations, predict its matter radius and dipole response function in a model-independent way. In the future, it will be interesting to look at how, and to what extent, range corrections modify our universal results for the properties of two-neutron halo nuclei. As more and more experimental data become available, it will become feasible to calculate these corrections 130 by fixing the low-energy constants of higher order EFT terms to these data. For example, for 22C, the parameters of a higher order EFT calculation can be constrained by using high-precision data on the charge radii of 22C and 20C, and from one- and two-neutron transfer reactions on 20C. These higher order calculations will be necessary if the matter radius and the B(E1) spectrum are measured with a high precision. 131 References

[1] I. Tanihata. Neutron halo nuclei. J.Phys., G22:157, 1996.

[2] A. S. Jensen, K. Riisager, D. V. Fedorov, and E. Garrido. Structure and reactions of quantum halos. Rev.Mod.Phys., 76:215, 2004.

[3] I. Tanihata, H. Savajols, and R. Kanungo. Recent experimental progress in nuclear halo structure studies. Prog.Part.Nucl.Phys., 68:215, 2013.

[4] J. Vervier, J. Aysto, H. Doubre, S. Gales, G. Morrison, et al. Nuclear physics in Europe: Highlights and opportunities. Proceedings, Workshop, Munich, Germany, April 23-26, 1997.

[5] D. B. Kaplan, M. J. Savage, and M. B. Wise. A New expansion for nucleon-nucleon interactions. Phys.Lett., B424:390, 1998.

[6] D. B. Kaplan, M. J. Savage, and M. B. Wise. Two nucleon systems from effective field theory. Nucl.Phys., B534:329, 1998.

[7] J. Gegelia. EFT and N N scattering. Phys.Lett., B429:227, 1998.

[8] M. C. Birse, J. A. McGovern, and K. G. Richardson. A Renormalization group treatment of two-body scattering. Phys.Lett., B464:169, 1999.

[9] U. van Kolck. Effective field theory of short range forces. Nucl.Phys., A645:273, 1999.

[10] P. F. Bedaque, H.-W. Hammer, and U. van Kolck. Renormalization of the three-body system with short range interactions. Phys.Rev.Lett., 82:463, 1999.

[11] C. A. Bertulani, H.-W. Hammer, and U. van Kolck. Effective field theory for halo nuclei. Nucl.Phys., A712:37, 2002.

[12] P. F. Bedaque, H.-W. Hammer, and U. van Kolck. Narrow resonances in effective field theory. Phys.Lett., B569:159, 2003.

[13] C. Ji, D. R. Phillips, and L. Platter. Beyond universality in three-body recombination: an Effective Field Theory treatment. Europhys.Lett., 92:13003, 2010.

[14] C. Ji, D. R. Phillips, and L. Platter. The three-boson system at next-to-leading order in an effective field theory for systems with a large scattering length. Annals Phys., 327:1803, 2012.

[15] C. Ji and D. R. Phillips. Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order. Few Body Syst., 54:2317, 2013. 132

[16] I. Tanihata, H. Hamagaki, O. Hashimoto, S. Nagamiya, Y. Shida, et al. Measurements of Interaction Cross-Sections and Radii of He Isotopes. Phys.Lett., B160:380, 1985.

[17] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, et al. Measurements of Interaction Cross-Sections and Nuclear Radii in the Light p Shell Region. Phys.Rev.Lett., 55:2676, 1985.

[18] T. Kobayashi, O. Yamakawa, K. Omata, K. Sugimoto, T. Shimoda, N. Takahashi, and I. Tanihata. Projectile fragmentation of the extremely neutron-rich nucleus 11Li at 0.79 GeV/nucleon. Phys.Rev.Lett., 60:2599, 1988.

[19] P. G. Hansen and B. Jonson. The neutron halo of extremely neutron-rich nuclei. EPL (Europhys.Lett.), 4:409, 1987.

[20] T. Nakamura, S. Shimoura, T. Kobayashi, T. Teranishi, K. Abe, N. Aoi, Y. Doki, M. Fujimaki, N. Inabe, N. Iwasa, et al. Coulomb dissociation of a halo nucleus 11Be at 72A MeV. Phys.Lett., B331:296, 1994.

[21] T. Nakamura, N. Fukuda, T. Kobayashi, N. Aoi, H. Iwasaki, et al. Coulomb dissociation of 19C and its halo structure. Phys.Rev.Lett., 83:1112, 1999.

[22] T. Nakamura, A. M. Vinodkumar, T. Sugimoto, N. Aoi, H. Baba, D. Bazin, N. Fukuda, T. Gomi, H. Hasegawa, N. Imai, M. Ishihara, T. Kobayashi, Y. Kondo, T. Kubo, M. Miura, T. Motobayashi, H. Otsu, A. Saito, H. Sakurai, S. Shimoura, K. Watanabe, Y. X. Watanabe, T. Yakushiji, Y. Yanagisawa, and K. Yoneda. Observation of strong low-lying E1 strength in the two-neutron halo nucleus 11Li. Phys.Rev.Lett., 96:252502, 2006.

[23] B. M. Young, W. Benenson, M. Fauerbach, J. H. Kelley, R. Pfaff, B. M. Sherrill, M. Steiner, J. S. Winfield, T. Kubo, M. Hellstrom,¨ N. A. Orr, J. Stetson, J. A. Winger, and S. J. Yennello. Mass of 11Li from the 14C(11B,11Li)14O reaction. Phys.Rev.Lett., 71:4124, 1993.

[24] L. Gaudefroy, W. Mittig, N. A. Orr, S. Varet, M. Chartier, P. Roussel-Chomaz, et al. Direct mass measurements of 19B, 22C, 29F, 31Ne, 34Na and other light exotic nuclei. Phys.Rev.Lett., 109:202503, 2012.

[25] C. Gaulard, C. Bachelet, G. Audi, C. Gunaut, D. Lunney, M. de Saint Simon, M. Sewtz, and C. Thibault. Mass measurements of the exotic nuclides 11Li and 11,12Be performed with the MISTRAL spectrometer. Nucl.Phys., A826:1, 2009.

[26] K. Blaum, Sz. Nagy, and G. Werth. High-accuracy penning trap mass measurements with stored and cooled exotic ions. J.Phys., B42:154015, 2009. 133

[27] M. Smith, M. Brodeur, T. Brunner, S. Ettenauer, A. Lapierre, R. Ringle, V. L. Ryjkov, F. Ames, P. Bricault, G. W. F. Drake, P. Delheij, D. Lunney, F. Sarazin, and J. Dilling. First penning-trap mass measurement of the exotic halo nucleus 11Li. Phys.Rev.Lett., 101:202501, 2008.

[28] B. L. Berman and S. C. Fultz. Measurements of the giant dipole resonance with monoenergetic photons. Rev.Mod.Phys., 47:713, 1975.

[29] C. A. Bertulani and G. Baur. Electromagnetic Processes in Relativistic Heavy Ion Collisions. Phys.Rept., 163:299, 1988.

[30] C. A. Bertulani. Theory and Applications of Coulomb Excitation. arXiv:0908.4307 [nucl-th], 2009.

[31] L.-B. Wang, P. Mueller, K. Bailey, G. W. F. Drake, J. P. Greene, D. Henderson, R. J. Holt, R. V. F. Janssens, C. L. Jiang, Z.-T. Lu, T. P. O’Connor, R. C. Pardo, K. E. Rehm, J. P. Schiffer, and X. D. Tang. Laser spectroscopic determination of the 6He nuclear charge radius. Phys.Rev.Lett., 93:142501, 2004.

[32] P. Mueller, I. A. Sulai, A. C. C. Villari, J. A. Alcantara-N´ u´nez,˜ R. Alves-Conde,´ K. Bailey, G. W. F. Drake, M. Dubois, C. Eleon,´ G. Gaubert, R. J. Holt, R. V. F. Janssens, N. Lecesne, Z.-T. Lu, T. P. O’Connor, M.-G. Saint-Laurent, J.-C. Thomas, and L.-B. Wang. Nuclear charge radius of 8He. Phys.Rev.Lett., 99:252501, 2007.

[33] R. Sanchez,´ W. Nortersh¨ auser,¨ G. Ewald, D. Albers, J. Behr, P. Bricault, B. A. Bushaw, A. Dax, J. Dilling, M. Dombsky, G. W. F. Drake, S. Gotte,¨ R. Kirchner, H.-J. Kluge, Th. Kuhl,¨ J. Lassen, C. D. P. Levy, M. R. Pearson, E. J. Prime, V. Ryjkov, A. Wojtaszek, Z.-C. Yan, and C. Zimmermann. Nuclear charge radii of 9,11Li: The influence of halo neutrons. Phys.Rev.Lett., 96:033002, 2006.

[34] W. Nortershauser, D. Tiedemann, M. Zakova, Z. Andjelkovic, K. Blaum, et al. Nuclear Charge Radii of 7,9,10Be and the one-neutron halo nucleus 11Be. Phys.Rev.Lett., 102:062503, 2009.

[35] R. J. Glauber. Cross-sections in deuterium at high-energies. Phys.Rev., 100:242, 1955.

[36] P. M. Fishbane and J. S. Trefil. Factorization as a simple geometrical property of composite systems. Phys.Rev.Lett., 32:396, 1974.

[37] V. Franco. Nuclear collisions and factorization. Phys.Rev.Lett., 32:911, 1974.

[38] W. Czyz and L. C. Maximon. High-energy, small angle elastic scattering of strongly interacting composite particles. Annals Phys., 52:59, 1969.

[39] A. Ozawa, T. Suzuki, and I. Tanihata. Nuclear size and related topics. Nucl.Phys., A693:32, 2001. Radioactive Nuclear Beams. 134

[40] J. S. Al-Khalili and J. A. Tostevin. Matter radii of light halo nuclei. Phys.Rev.Lett., 76:3903, 1996.

[41] J. S. Al-Khalili, J. A. Tostevin, and I. J. Thompson. Radii of halo nuclei from cross section measurements. Phys.Rev., C54:1843, 1996.

[42] A. S. Goldhaber. Statistical models of fragmentation processes. Phys.Lett., B53:306, 1974.

[43] D. Bazin, B. A. Brown, J. Brown, M. Fauerbach, M. Hellstrom, et al. One-Neutron Halo of 19C. Phys.Rev.Lett., 74:3569, 1995.

[44] F. M. Marqus, E. Liegard, N. A. Orr, J. C. Anglique, L. Axelsson, G. Bizard, W. N. Catford, et al. Neutrons from the breakup of 19C. Phys.Lett., B381:407, 1996.

[45] T. Baumann, H. Geissel, H. Lenske, K. Markenroth, W. Schwab, et al. Longitudinal momentum distributions of 16C, 18C fragments after one neutron removal from 17C, 19C. Phys.Lett., B439:256, 1998.

[46] D. Q. Fang, T. Yamaguchi, T. Zheng, A. Ozawa, M. Chiba, et al. One neutron halo structure in 15C. Phys.Rev., C69:034613, 2004.

[47] M. Chiba, R. Kanungo, B. Abu-Ibrahim, S. Adhikari, D. Q. Fang, N. Iwasa, K. Kimura, K. Maeda, S. Nishimura, T. Ohnishi, A. Ozawa, et al. Neutron removal studies on 19C. Nucl.Phys., A741:29, 2004.

[48] A. Ozawa, Y. Hashizume, Y. Aoki, K. Tanaka, T. Aiba, et al. One- and two-neutron removal reactions from 19,20C with a proton target. Phys.Rev., C84:064315, 2011.

[49] N. Kobayashi, T. Nakamura, J. A. Tostevin, Y. Kondo, N. Aoi, et al. One- and two-neutron removal reactions from the most neutron-rich Carbon isotopes. Phys.Rev., C86:054604, 2012.

[50] V. Maddalena, T. Aumann, D. Bazin, B. A. Brown, J.A. Caggiano, et al. Single neutron knockout reactions: Application to the spectroscopy of 16,17,19C. Phys.Rev., C63:024613, 2001.

[51] Heyde K. L. G. The Nuclear Shell Model. Springer Berlin Heidelberg, 1994.

[52] M. G. Mayer and J. H. D. Jensen. Elementary Theory of Nuclear Shell Structure. John Wiley and Sons, 1955.

[53] B.A. Brown and B. H. Wildenthal. Status of the nuclear shell model. Ann.Rev.Nucl.Part.Sci., 38:29, 1988.

[54] M. L. Goldberger and K. M. Watson. Collision theory. Dover Publications, Mineola, N.Y., 2004. 135

[55] B. A. Brown. The nuclear shell model towards the drip lines. Prog.Part.Nucl.Phys., 47:517, 2001.

[56] D. J. Millener, D. E. Alburger, E. K. Warburton, and D. H. Wilkinson. Decay scheme of 11Be. Phys.Rev., C26:1167, 1982.

[57] H. Kitagawa and H. Sagawa. Quadrupole-moments in mirror nuclei and proton halo. Phys.Lett., B299:1, 1993.

[58] B.A. Brown and P.G. Hansen. Proton halos in the 1s0d shell. Phys.Lett., B381:391, 1996.

[59] H. Sagawa, T. Suzuki, H. Iwasaki, and M. Ishihara. Low energy dipole strength in light drip line nuclei. Phys.Rev., C63:034310, 2001.

[60] E. C. Simpson and J. A. Tostevin. One- and two-neutron removal from the neutron-rich carbon isotopes. Phys.Rev., C79:024616, 2009.

[61] L. Coraggio, A. Covello, A. Gargano, and N. Itaco. Shell-model calculations for neutron-rich carbon isotopes with a chiral nucleon-nucleon potential. Phys.Rev., C81:064303, 2010.

[62] R. Jansen, G. J. Engel, G. Hagen, P. Navratil, and A. Signoracci. Ab Initio coupled-cluster effective interactions for the shell model: Application to neutron-rich oxygen and carbon isotopes. Phys.Rev.Lett., 113:142502, 2014.

[63] L. Wolfenstein and J. Ashkin. Invariance conditions on the scattering amplitudes for spin ½ particles. Phys.Rev., 85:947, 1952.

[64] S. Okubo and R. E. Marshak. Velocity dependence of the two-nucleon interaction. Annals Phys., 4:166, 1958.

[65] R. V. Reid Jr. Local phenomenological nucleon-nucleon potentials. Annals Phys., 50:411, 1968.

[66] M. M. Nagels, T. A. Rijken, and J. J. de Swart. A Low-Energy Nucleon-Nucleon Potential from Regge Pole Theory. Phys.Rev., D17:768, 1978.

[67] M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, J. Cote, et al. Parametrization of the Paris n n Potential. Phys.Rev., C21:861, 1980.

[68] I. E. Lagaris and V. R. Pandharipande. Phenomenological two-nucleon interaction operator. Nucl.Phys., A359:331, 1981.

[69] R. B. Wiringa, R. A. Smith, and T. L. Ainsworth. Nucleon Nucleon Potentials with and Without Delta (1232) Degrees of Freedom. Phys.Rev., C29:1207, 1984. 136

[70] R. Machleidt, K. Holinde, and C. Elster. The Bonn Meson Exchange Model for the Nucleon Nucleon Interaction. Phys.Rept., 149:1, 1987.

[71] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla. An Accurate nucleon-nucleon potential with charge independence breaking. Phys.Rev., C51:38, 1995.

[72] J. Carlson, V. R. Pandharipande, and R. B. Wiringa. Three-nucleon interaction in 3-body, 4-body, and infinite-body systems. Nucl.Phys., A401:59, 1983.

[73] S. C. Pieper, V. R. Pandharipande, R. B. Wiringa, and J. Carlson. Realistic models of pion exchange three nucleon interactions. Phys.Rev., C64:014001, 2001.

[74] S. Weinberg. Nuclear forces from chiral Lagrangians. Phys.Lett., B251:288, 1990.

[75] S. Weinberg. Effective chiral Lagrangians for nucleon-pion interactions and nuclear forces. Nucl.Phys., B363:3, 1991.

[76] U. van Kolck. Few nucleon forces from chiral Lagrangians. Phys.Rev., C49:2932, 1994.

[77] E. Epelbaum, W. Gloeckle, and U.-G. Meissner. Nuclear forces from chiral Lagrangians using the method of unitary transformation. Nucl.Phys., A671:295, 2000.

[78] D. R. Entem and R. Machleidt. Accurate charge dependent nucleon nucleon potential at fourth order of chiral perturbation theory. Phys.Rev., C68:041001, 2003.

[79] E. Epelbaum, W. Glockle, and U.-G. Meissner. The Two-nucleon system at next-to-next-to-next-to-leading order. Nucl.Phys., A747:362, 2005.

[80] R. Machleidt. Chiral effective field theory for nuclear forces: Achievements and challenges. EPJ Web Conf., 66:01011, 2014.

[81] S. K. Bogner, R. J. Furnstahl, and A. Schwenk. From low-momentum interactions to nuclear structure. Prog.Part.Nucl.Phys., 65:94, 2010.

[82] P. Navratil´ and W. E. Ormand. Ab initio shell model with a genuine three-nucleon force for the p-shell nuclei. Phys.Rev., C68:034305, 2003.

[83] C. Forssen,´ P. Navratil,´ W. E. Ormand, and E. Caurier. Large basis ab initio shell model investigation of 9Be and 11Be. Phys.Rev., C71:044312, 2005.

[84] S. Quaglioni and P. Navratil.´ Ab Initio many-body calculations of n − 3He, n − 4He, p − 3,4He and n − 10Be scattering. Phys.Rev.Lett., 101:092501, 2008.

[85] P. Navrtil, R. Roth, and S. Quaglioni. Ab initio many-body calculation of the radiative capture. Phys.Lett., B704:379, 2011. 137

[86] C. Romero-Redondo, S. Quaglioni, P. Navratil,´ and G. Hupin. 4He + n + n. Phys.Rev.Lett., 113:032503, 2014. [87] X. Mougeot, V. Lapoux, W. Mittig, N. Alamanos, F. Auger, B. Avez, D. Beaumel, Y. Blumenfeld, R. Dayras, A. Drouart, C. Force, et al. New excited states in the halo nucleus 6He. Phys.Lett., B718:441, 2012. [88] S. Bacca, A. Schwenk, G. Hagen, and T. Papenbrock. Helium halo nuclei from low-momentum interactions. Eur.Phys.J., A42:553, 2009. [89] S. Bacca, N. Barnea, and A. Schwenk. Matter and charge radius of 6He in the hyperspherical-harmonics approach. Phys.Rev., C86:034321, 2012. [90] G. Hagen, D. J. Dean, M. Hjorth-Jensen, and T. Papenbrock. Complex coupled-cluster approach to an ab-initio description of open quantum systems. Phys.Lett., B656:169, 2007. [91] G. Hagen, T. Papenbrock, and M. Hjorth-Jensen. Ab Initio computation of the 17F proton halo state and resonances in A =17 nuclei. Phys.Rev.Lett., 104:182501, 2010. [92] G. Hagen, P. Hagen, H.-W. Hammer, and L. Platter. Efimov physics around the neutron-rich 60Ca isotope. Phys.Rev.Lett., 111:132501, 2013. [93] H. Esbensen. Momentum distributions in stripping reactions of single-nucleon halo nuclei. Phys.Rev., C53:2007, 1996. [94] K. Hencken, G. Bertsch, and H. Esbensen. Breakup reactions of the halo nuclei 11Be and 8B. Phys.Rev., C54:3043, 1996. [95] D. Bazin, W. Benenson, B. A. Brown, J. Brown, B. Davids, et al. Probing the halo structure of 19,17,15C and 14B. Phys.Rev., C57:2156, 1998. [96] S. Typel and G. Baur. Electromagnetic strength of neutron and proton single-particle halo nuclei. Nucl.Phys., A759:247, 2005. [97] L. Johannsen, A. S. Jensen, and P. G. Hansen. The 11Li nucleus as a three-body system. Phys.Lett., B244:357, 1990. [98] H. Esbensen and G. F. Bertsch. Soft dipole excitations in 11Li. Nucl.Phys., A542:310, 1992. [99] B. V. Danilin, I. J. Thompson, J. S. Vaagen, and M. V. Zhukov. Three-body continuum structure and response functions of halo nuclei (i): 6He. Nucl.Phys., A632:383, 1998. [100] P. Banerjee, J. A. Tostevin, and I. J. Thompson. Coulomb breakup of two-neutron halo nuclei. Phys.Rev., 58:1337, 1998. 138

[101] D. Gogny, P. Pires, and R. De Tourreil. A smooth realistic local nucleon-nucleon force suitable for nuclear hartree-fock calculations. Phys.Lett., B32:591, 1970.

[102] S. N. Ershov, J. S. Vaagen, and M. V. Zhukov. Binding energy constraint on matter radius and soft dipole excitations of 22C. Phys.Rev., C86:034331, 2012.

[103] K. Tanaka, T. Yamaguchi, T. Suzuki, T. Ohtsubo, M. Fukuda, et al. Observation of a Large Reaction Cross Section in the Drip-Line Nucleus 22C. Phys.Rev.Lett., 104:062701, 2010.

[104] M. T. Yamashita, L. Tomio, and T. Frederico. Radii in weakly bound light halo nuclei. Nucl.Phys., A735:40, 2004.

[105] L. D. Faddeev. Scattering theory for a three particle system. Sov.Phys.JETP, 12:1014–1019, 1961.

[106] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga. Feshbach resonances in ultracold gases. Rev.Mod.Phys., 82:1225, 2010.

[107] D. B. Kaplan, M. J. Savage, and M. B. Wise. Perturbative calculation of the electromagnetic form factors of the deuteron. Phys.Rev., C59:617, 1999.

[108] J.-W. Chen, G. Rupak, and M. J. Savage. Nucleon-nucleon effective field theory without pions. Nucl.Phys., A653:386, 1999.

[109] S. R. Beane and M. J. Savage. Rearranging pionless effective field theory. Nucl.Phys., A694:511, 2001.

[110] D. B. Kaplan. More effective field theory for nonrelativistic scattering. Nucl.Phys., B494:471, 1997.

[111] J.-W. Chen and M. J. Savage. np → dγ for big bang nucleosynthesis. Phys.Rev., C60:065205, 1999.

[112] G. Rupak. Precision calculation of np → dγ cross-section for big bang nucleosynthesis. Nucl.Phys., A678:405, 2000.

[113] H. W. Griesshammer and G. Rupak. Nucleon polarizabilities from Compton scattering on the deuteron. Phys.Lett., B529:57, 2002.

[114] H. Arenhvel and M. Sanzone. Photodisintegration of the deuteron: a review of theory and experiment. Few-Body Systems Supplementum 3. Springer Publishing, Vienna, 1991.

[115] M. Butler, J.-W. Chen, and X. Kong. Neutrino deuteron scattering in effective field theory at next-to-next-to-leading order. Phys.Rev., C63:035501, 2001. 139

− [116] Q. R. Ahmad et al. Measurement of the rate of νe + d → p + p + e interactions produced by 8B solar neutrinos at the Sudbury Neutrino Observatory. Phys.Rev.Lett., 87:071301, 2001. [117] Q.R. Ahmad et al. Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory. Phys.Rev.Lett., 89:011301, 2002. [118] H.-W. Hammer and D. R. Phillips. Electric properties of the Beryllium-11 system in Halo EFT. Nucl.Phys., A865:17, 2011. [119] G. Rupak and R. Higa. Model-Independent Calculation of Radiative Neutron Capture on Lithium-7. Phys.Rev.Lett., 106:222501, 2011. [120] L. Fernando, R. Higa, and G. Rupak. Resonance Contribution to Radiative Neutron Capture on Lithium-7. Eur.Phys.J., A48:24, 2012. [121] G. Rupak, L. Fernando, and A. Vaghani. Radiative Neutron Capture on Carbon-14 in Effective Field Theory. Phys.Rev., C86:044608, 2012. [122] X. Kong and F. Ravndal. Proton proton scattering lengths from effective field theory. Phys.Lett., B450:320, 1999. [123] X. Kong and F. Ravndal. Coulomb effects in low-energy proton proton scattering. Nucl.Phys., A665:137, 2000. [124] M. Butler and J.-W. Chen. Proton proton fusion in effective field theory to fifth order. Phys.Lett., B520:87, 2001. [125] E. G. Adelberger, A. B. Balantekin, D. Balantekin, C. A. Bertulani, J.-W. Chen, et al. Solar fusion cross sections II: the pp chain and CNO cycles. Rev.Mod.Phys., 83:195, 2011. [126] R. Higa, H.-W. Hammer, and U. van Kolck. αα Scattering in Halo Effective Field Theory. Nucl.Phys., A809:171, 2008. [127] E. Ryberg, C. Forssn, H.-W. Hammer, and L. Platter. Effective field theory for proton halo nuclei. Phys.Rev., C89:014325, 2014. [128] X. Zhang, K. M. Nollett, and D. R. Phillips. Marrying ab initio calculations and Halo-EFT: the case of 7Li + n → 8Li + γ. Phys.Rev., C89:024613, 2014. [129] X. Zhang, K. M. Nollett, and D. R. Phillips. Combining ab initio calculations and low-energy effective field theory for halo nuclear systems: The case of 7Be + p → 8B + γ. Phys.Rev., C89:051602, 2014. [130] V. Efimov. Energy levels arising form the resonant two-body forces in a three-body system. Phys.Lett., B33:563–564, 1970. 140

[131] E. Braaten and H.-W. Hammer. Universality in few-body systems with large scattering length. Phys.Rept., 428:259, 2006.

[132] P. F. Bedaque, H.-W. Hammer, and U. van Kolck. The Three boson system with short range interactions. Nucl.Phys., A646:444, 1999.

[133] F. Gabbiani, P. F. Bedaque, and H. W. Griesshammer. Higher partial waves in an effective field theory approach to nd scattering. Nucl.Phys., A675:601, 2000.

[134] P. F. Bedaque, G. Rupak, H. W. Griesshammer, and H.-W. Hammer. Low-energy expansion in the three-body system to all orders and the triton channel. Nucl.Phys., A714:589, 2003.

[135] J. Vanasse. Fully Perturbative Calculation of nd Scattering to Next-to-next-to-leading-order. Phys.Rev., C88:044001, 2013.

[136] S. Koenig and H.-W. Hammer. Precision calculation of the quartet-channel p-d scattering length. Phys.Rev., C90:034005, 2014.

[137] J. Vanasse, D. A. Egolf, J. Kerin, S. Koenig, and R. P. Springer. 3He and pd Scattering to Next-to-Leading Order in Pionless Effective Field Theory. Phys.Rev., C89:064003, 2014.

[138] S. Koenig, H. W. Griesshammer, and H.-W. Hammer. The proton-deuteron system in pionless EFT revisited. J.Phys., G42:045101, 2015.

[139] D. L. Canham and H.-W. Hammer. Universal properties and structure of halo nuclei. Eur.Phys.J., A37:367, 2008.

[140] D. L. Canham and H.-W. Hammer. Range corrections for two-neutron halo nuclei in effective theory. Nucl.Phys., A836:275, 2010.

[141] J. Rotureau and U. van Kolck. Effective field theory and the Gamow shell model. Few Body Syst., 54:725–735, 2013.

[142] P. Hagen, H.-W. Hammer, and L. Platter. Charge form factors of two-neutron halo nuclei in halo EFT. Eur.Phys.J., A49:118, 2013.

[143] C. Ji, Ch. Elster, and D. R. Phillips. The 6He Nucleus in Halo EFT. Phys.Rev., C90:044004, 2014.

[144] E. Ryberg, C. Forssn, H.-W. Hammer, and L. Platter. Constraining Low-Energy Proton Capture on Beryllium-7 through Charge Radius Measurements. Eur.Phys.J., A50:170, 2014.

[145] B. Acharya and D. R. Phillips. Carbon-19 in Halo EFT: Effective-range parameters from Coulomb-dissociation experiments. Nucl.Phys., A913:103, 2013. 141

[146] B. Acharya, C. Ji, and D. R. Phillips. Implications of a matter-radius measurement for the structure of Carbon-22. Phys.Lett., B723:196–200, 2013.

[147] B. Acharya, P. Hagen, H. W. Hammer, and D. R. Phillips. To be published.

[148] S. Weinberg. Phenomenological Lagrangians. Physica, A96:327, 1979.

[149] E. Witten. Short Distance Analysis of Weak Interactions. Nucl.Phys., B122:109, 1977.

[150] H. Georgi. Effective field theory. Annual review of nuclear and particle science, 43:209, 1993.

[151] H. A. Bethe. Theory of the effective range in nuclear scattering. Phys.Rev., 76:38, 1949.

[152] J. Hubbard. Calculation of partition functions. Phys.Rev.Lett., 3:77, 1959.

[153] D. R. Phillips, G. Rupak, and M. J. Savage. Improving the convergence of N N effective field theory. Phys.Lett., B473:209, 2000.

[154] H. Lehmann, K. Symanzik, and W. Zimmermann. Zur formulierung quantisierter feldtheorien. Il Nuovo Cimento, 1:205–225, 1955.

[155] H. A. Bethe and C. Longmire. The effective range of nuclear forces ii. photo-disintegration of the deuteron. Phys.Rev., 77:647, 1950.

[156] M. Gell-Mann and M. L. Goldberger. The formal theory of scattering. Phys.Rev., 91:398, 1953.

[157] M. E. Peskin and D. V. Schroeder. An introduction to quantum field theory. Advanced book program. Westview Press Reading (Mass.), Boulder (Colo.), 1995. Autre tirage : 1997.

[158] H. Bethe and R. Peierls. Quantum Theory of the Diplon. Royal Society of London Proceedings Series A, 148:146, 1935.

[159] N. Auerbach. Coulomb effects in nuclear structure. Phys.Rept., 98:273, 1983.

[160] R. Thomas. An analysis of the energy levels of the mirror nuclei, 13C and 13N. Phys.Rev., 88:1109, 1952.

[161] J. Ehrman. On the displacement of corresponding energy levels of 13C and 13N. Phys.Rev., 81:412, 1951.

[162] N. Auerbach and N. Vinh Mau. About coulomb energy shifts in halo nuclei. Phys.Rev., C63:017301, 2000. 142

[163] S. Aoyama, N. Itagaki, K. Kato,¯ and K. Ikeda. Thomas-Ehrman effect on s1/2 and 10 p1/2 resonance states in C + p. Phys.Rev., C57:975, 1998. [164] P. Sauer. Can the charge symmetry of nuclear forces be confirmed by nucleon-nucleon scattering experiments? Phys.Rev.Lett., 32:626, 1974.

[165] T. Barford and M. C. Birse. Renormalization group approach to two-body scattering in the presence of long-range forces. Phys.Rev., C67:064006, 2003.

[166] J. Gegelia. Is the strong interaction proton proton scattering length renormalization scale dependent in effective field theory? Eur.Phys.J., A19:355, 2004.

[167] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions. Dover Publications, 1965.

[168] J. Gasser, V. E. Lyubovitskij, A. Rusetsky, and A. Gall. Decays of the π+ π− atom. Phys.Rev. D, 64:016008, 2001.

[169] D. R. Phillips and T. D. Cohen. Deuteron electromagnetic properties and the viability of effective field theory methods in the two nucleon system. Nucl.Phys., A668:45, 2000.

[170] F. Ajzenberg-Selove. Energy levels of light nuclei A = 18 − 20. Nucl.Phys., A475:1, 1987.

[171] E. K. Warburton and B. A. Brown. Effective Interactions for the Op1sOd nuclear shell model space. Phys.Rev., C46:923–944, 1992.

[172] G. Audi, A. H. Wapstra, and C. Thibault. The AME2003 atomic mass evaluation (II). Tables, graphs and references. Nucl.Phys., A729:337, 2002.

[173] P. Banerjee and R. Shyam. Structure of 19C from Coulomb dissociation studies. Phys.Rev., C61:047301, 2000.

[174] T. Nakamura, N. Fukuda, N. Aoi, H. Iwasaki, T. Kobayashi, T. Kubo, A. Mengoni, M. Notani, H. Otsu, H. Sakurai, S. Shimoura, T. Teranishi, Y. X. Watanabe, K. Yoneda, and M. Ishihara. Coulomb dissociation of halo nuclei. Nucl.Phys., A722:C301, 2003.

[175] S. Typel and G. Baur. Higher order effects in electromagnetic dissociation of neutron halo nuclei. Phys.Rev., C64:024601, 2001.

[176] P. Singh, R. Kumar, R. Kharab, and H. C. Sharma. Structural analysis of 19C through coulomb dissociation reactions. Nucl.Phys., A802:82, 2008.

[177] C. A. Bertulani and G. Baur. Coincidence cross sections for the dissociation of light ions in high-energy collisions. Nucl.Phys., A480:615, 1988. 143

[178] G. Baur, C. A. Bertulani, and H. Rebel. Coulomb dissociation as a source of information on radiative capture processes of astrophysical interest. Nucl.Phys., A458:188, 1986.

[179] Y. Satou, T. Nakamura, N. Fukuda, T. Sugimoto, Y. Kondo, N. Matsui, Y. Hashimoto, T. Nakabayashi, et al. Unbound excited states in 17,19C. Phys.Lett., B660:320, 2008.

[180] M. Grasselli and D. Pelinovsky. Numerical mathematics. Reprint of the 2008 ed. New Delhi: Narosa Publishing House. xiv, 668 p. , 2009.

[181] S. Typel and G. Baur. Scaling laws and higher-order effects in Coulomb excitation of neutron halo nuclei. Eur.Phys.J., A38:355, 2008.

[182] H. T. Fortune and R. Sherr. Binding energy of 22C. Phys.Rev., C85:027303, 2012.

[183] M. T. Yamasita, R. S. M. de Carvalho, T. Frederico, and L. Tomio. Constraints on two-neutron separation energy in the Borromean 22C nucleus. Phys.Lett., B697:90, 2011.

[184] W. Gloeckle. The quantum mechanical few-body problem. Texts and Monographs in Physics. Springer-Verlag, 1983.

[185] I. R. Afnan and D. R. Phillips. The Three body problem with short range forces: Renormalized equations and regulator independent results. Phys.Rev., C69:034010, 2004.

[186] C.-J. Yang, Ch. Elster, and D. R. Phillips. Subtractive renormalization of the NN scattering amplitude at leading order in chiral effective theory. Phys.Rev., C77:014002, 2008.

[187] M. Langevin, E. Quiniou, M. Bernas, J. Galin, J. C. Jacmart, F. Naulin, F. Pougheon, R. Anne, C. Dtraz, D. Guerreau, D. Guillemaud-Mueller, and A. C. Mueller. Production of neutron-rich nuclei at the limits of particles stability by fragmentation of 44 MeV/u 40Ar projectiles. Phys.Lett., B150:71, 1985.

[188] W. Horiuchi and Y. Suzuki. 22C: An s-wave two-neutron halo nucleus. Phys.Rev., C74:034311, 2006.

[189] S. Mosby, N. S. Badger, T. Baumann, D. Bazin, M. Bennett, J. Brown, G. Christian, P. A. DeYoung, J. E. Finck, M. Gardner, J. D. Hinnefeld, E. A. Hook, E. M. Lunderberg, B. Luther, D. A. Meyer, M. Mosby, et al. Search for 21C and constraints on 22C. Nucl.Phys., A909:69, 2013.

[190] T. Frederico, A. Delfino, L. Tomio, and M. T. Yamashita. Universal aspects of light halo nuclei. Prog.Part.Nucl.Phys., 67:939, 2012. 144

[191] A. Ozawa, O. Bochkarev, L. Chulkov, D. Cortina, H. Geissel, M. Hellstrm, M. Ivanov, R. Janik, K. Kimura, T. Kobayashi, et al. Measurements of interaction cross sections for light neutron-rich nuclei at relativistic energies and determination of effective matter radii. Nucl.Phys., A691:599, 2001. [192] D. E. Gonzalez Trotter, F. Salinas, Q. Chen, A. S. Crowell, W. Glockle, et al. New 1 Measurement of the S 0 Neutron-Neutron Scattering Length Using the Neutron-Proton Scattering Length as a Standard. Phys.Rev.Lett., 83:3788, 1999. [193] M. Stanoiu, D. Sohler, O. Sorlin, F. Azaiez, Zs. Dombradi,´ et al. Disappearance of the n = 14 shell gap in the carbon isotopic chain. Phys.Rev., C78:034315, 2008. [194] R. Pike and P. Sabatier, editors. Scattering: Scattering and inverse scattering in pure and applied science. Vol. 1, 2. Academic Press, 2002. [195] Schmidt E. W. and Ziegelmann H. The Quantum mechanical three-body problem. Pergamon Press, 1974. [196] J. H. Hetherington and L. H. Schick. Exact multiple-scattering analysis of low-energy elastic K–d scattering with separable potentials. Phys.Rev., 137:B935, 1965. [197] R. Aaron and R. D. Amado. Theory of the reaction n+d → n+n+p. Phys.Rev., 150:857, 1966. [198] R. T. Cahill and I. H. Sloan. Theory of neutron-deuteron breakup at 14.4 MeV. Nucl.Phys., A165:161, 1971. [199] N. M. Larson and J. H. Hetherington. Solution of the Faddeev integral equation without contour rotation. Phys.Rev., C9:699, 1974. [200] W. Gloeckle, H. Witala, D. Huber, H. Kamada, and J. Golak. The Three nucleon continuum: Achievements, challenges and applications. Phys.Rept., 274:107, 1996. [201] H. Kamada, Y. Koike, and W. Gloeckle. Complex energy method for scattering processes. Prog.Theor.Phys., 109:869, 2003. [202] Ch. Elster, W. Glockle, and H. Witala. A New Approach to the 3D Faddeev Equation for Three-Body Scattering. Few Body Syst., 45:1, 2009. [203] H. Liu, Ch. Elster, and W. Glockle.¨ Three-body scattering at intermediate energies. Phys.Rev., C72:054003, 2005. [204] H. Liu. Studies of the nuclear three-body system with three-dimensional Faddeev calculations. PhD Dissertation, Ohio Unviersity, 2005. [205] D. R. Tilley, J. H. Kelley, J. L. Godwin, D. J. Millener, J. E. Purcell, C. G. Sheu, and H. R. Weller. Energy levels of light nuclei. Nucl.Phys., A745:155, 2004. 145

[206] I. Tanihata, T. Kobayashi, O. Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda, and H. Sato. Measurement of interaction cross sections using isotope beams of Be and B and isospin dependence of the nuclear radii. Phys.Lett., B206:592, 1988.

[207] National Nuclear Data Center; Brookhaven National Laboratory. Chart of nuclides. http://www.nndc.bnl.gov/chart/ (accessed in 2014).

[208] H. B. Jeppesen, A. M. Moro, U. C. Bergmann, M. J. G. Borge, J. Cederkll, L. M. Fraile, H. O. U. Fynbo, J. Gmez-Camacho, H. T. Johansson, B. Jonson, M. Meister, T. Nilsson, G. Nyman, M. Pantea, K. Riisager, et al. Study of 10Li via the 9Li(2H,p) reaction at rex-isolde. Phys.Lett., B642:449, 2006.

[209] M. J. G. Borge, H. Fynbo, D. Guillemaud-Mueller, P. Hornshøj, F. Humbert, B. Jonson, T. E. Leth, et al. Elucidating halo structure by β decay: βγ from the 11Li decay. Phys.Rev., C55:R8, 1997.

[210] K. Ieki, D. Sackett, A. Galonsky, C. A. Bertulani, J. J. Kruse, W. G. Lynch, D. J. Morrissey, N. A. Orr, et al. Coulomb dissociation of 11Li. Phys.Rev.Lett., 70:730, 1993.

[211] D. Sackett, K. Ieki, A. Galonsky, C. A. Bertulani, H. Esbensen, J. J. Kruse, W. G. Lynch, D. J. Morrissey, N. A. Orr, et al. Electromagnetic excitation of 11Li. Phys.Rev., C48:118, 1993.

[212] M. Zinser, F. Humbert, T. Nilsson, W. Schwab, H. Simon, T. Aumann, M. J. G. Borge, L. V. Chulkov, J. Cub, Th. W. Elze, H. Emling, H. Geissel, D. Guillemaud-Mueller, et al. Invariant-mass spectroscopy of 10Li and 11Li. Nucl.Phys., A619:151, 1997.

[213] T. Nakamura. Soft E1 escitation of neutron halo nuclei. Nucl.Phys., A788:243c, 2007.

[214] Y. Kikuchi, T. Myo, K. Kato,¯ and K. Ikeda. Coulomb breakup reactions of 11Li in the coupled-channel 9Li + n + n model. Phys.Rev., C87:034606, 2013.

[215] M. Thoennessen, S. Yokoyama, A. Azhari, T. Baumann, J. Brown, A. Galonsky, P. Hansen, J. Kelley, R. Kryger, E. Ramakrishnan, and P. Thirolf. Population of 10Li by fragmentation. Phys.Rev., C59:111, 1999.

[216] M. Chartier, J. R. Beene, B. Blank, L. Chen, A. Galonsky, N. Gan, K. Govaert, P. G. Hansen, J. Kruse, V. Maddalena, M. Thoennessen, and R. L. Varner. Identification of the 10Li ground state. Phys.Lett., B510:24, 2001.

[217] T. Nakamura. Deformed halo nuclei probed by breakup reactions. J.Phys.Conf.Ser., 445:012033, 2013. 146

[218] E. Lipparini and S. Stringari. Sum rules and giant resonances in nuclei. Phys.Rept., 175:103, 1989.

[219] G. Orlandini and M. Traini. Sum rules for electron-nucleus scattering. Rept.Prog.Phys., 54:257, 1991.

[220] R. Ladenburg and F. Reiche. Absorption, zerstreuung und dispersion in der bohrschen atomtheorie. Naturwissenschaften, 11:584–598, 1923.

[221] W. Kuhn. Uber¨ die gesamtstrke der von einem zustande ausgehenden absorptionslinien. Zeitschrift fr Physik, 33:408, 1925.

[222] W. Thomas. Uber¨ die zahl der dispersionselektronen, die einem stationren zustande zugeordnet sind. (vorlaufige¨ mitteilung). Naturwissenschaften, 13:627, 1925.

[223] J. S. Levinger and H. A. Bethe. Dipole transitions in the nuclear photo-effect. Phys.Rev., 78:115, 1950.

[224] L. L. Foldy. Photodisintegration of the lightest nuclei. Phys.Rev., 107:1303, 1957.

[225] Y. Alhassid, M. Gai, and G. F. Bertsch. Radiative width of molecular-cluster states. Phys.Rev.Lett., 49:1482, 1982.

[226] D. Gazit, N. Barnea, S. Bacca, W. Leidemann, and G. Orlandini. Photonuclear sum rules and the tetrahedral configuration of 4He. Phys.Rev., C74:061001, 2006.

[227] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999. 147 Appendix A: Coulomb Dissociation Cross Section

Here we summarize some of the results of Refs. [29, 30] that are relevant for the case of E1 excitation of neutron halo nuclei upon small-angle scattering by a charged target at high beam energy. In the eikonal approximation, at first order in perturbation theory, the dissociation cross section is given by [29]

dσ Z2α2 =24π2 t ω2Z 2hri2 QdQ d3 p/(2π)3 γ2β2 e f f 01 X ∗ iM1−M2 χ Q χ∗ Q G /β G∗ /β Y M1 p Y M2 p , M1 ( ) M2 ( ) E1M1 (1 ) E1M2 (1 ) 1 ( ˆ) 1 ( ˆ) (A.1) M1 M2 where Zt is the charge of the target nucleus, ω is the excitation energy, Q is the momentum

p 2 exchanged between the projectile and the target nuclei, GE10(x) = −i4 π(x − 1)/3, √ GE11(x) = −GE1−1(x) = x 8π/3, hrilλ is the radial matrix element of the operator r between a bound state with orbital angular momentum l and a continuum state of orbital angular momentum λ, and

∞ ! Z ωb χM1 (Q) = db bJM1 (Qb)KM1 exp(iχ(b)), (A.2) 0 γβ where χ(b) is the eikonal phase. In the sharp-cutoff approximation,   0, if b ≤ R  exp(iχ(b)) =  (A.3)  1, otherwise, which gives Z ∞ ! 2 ωRx χM1 (Q) = R dx xJM1 (QRx)KM1 . (A.4) 1 γβ When expressed in terms of the reduced transition strength,

3 dB(E1) 2 d p = 3e2Z(1) hri2 , (A.5) dE e f f 01 (2π)3

and integrated overp ˆ and Q, Eq. (A.1) yields [29],

dσ 16π3 dB(E1) = α N (ω, R) , (A.6) dE 9 E1 e2dE 148

where ! Z2α β2   N (ω, R) = 2 t ξK (ξ)K (ξ) − ξ2 (K (ξ))2 − (K (ξ))2 , (A.7) E1 πβ2 0 1 2 1 0 ωR with ξ = γβ , is the virtual photon number for the E1 multipolarity, integrated over all impact parameters larger than R. To find the differential cross section of the center of mass of the dissociated fragments, it is convenient to replace Eq. (A.2) by the semiclassical approximation [30],

1 Z2Z2α2 |χ (Q)|2 = t cot2(θ /2)K2 (ξ ) , (A.8) M1 2 2 cm M1 0 Q 4Ecm where Ecm is the kinetic energy of the scattering nuclei in the center of mass frame, θcm is the Rutherford scattering angle, and ξ0 = ωb0/γβ, where

ZtZα b0 = cot(θcm/2) (A.9) 2Ecm

is the impact parameter for the Rutherford trajectory. Q and θcm are related by

Q = 2kcm sin(θcm/2), where kcm is the momentum of the beam (or the target), in the projectile-target center of mass frame. Equation (A.1) then yields

dσ 16π3 Z dn dB(E1) α E1 E, = 2 d (A.10) dΩcm 9 dΩcm e dE where 2 2 ! dnE1 Zt α ξ0 1 2 2 = K (ξ0) + K (ξ0) (A.11) 2 2 2 2 0 1 dΩcm 4π sin (θcm/2) β γ is the number of virtual E1 photons per unit solid angle. Using Eq. (A.9) to relate the solid angle to the impact parameter, we can integrate Eq. (A.11) over all impact parameters larger than R and get Eq. (A.7), i.e.

Z R dnE1 1 db0 = NE1(ω, R). (A.12) ∞ dΩcm db0/dΩcm Our use of Eqs. (4.8) and (A.11) is, therefore, equivalent to Nakamura et al.’s method of

replacing R in Eq. (A.7) by b0 and then differentiating it with respect to the solid angle to obtain the differential cross section. 149

The longitudinal momentum distribution of the dissociation cross section can be obtained from Eq. (A.1) by writing the volume element d3 p in cylindrical coordinates with the beam direction as the z-axis, and integrating over the transverse coordinates. This gives dσ 8π3 Z ∞ dB(E1) α M ω, R p. = E1( ) 2 d (A.13) dpz 9µ |pz| e dE The function

Z2α     M (ω, R) = t ξ2 (K (ξ))2 − (K (ξ))2 1 + 2P (p /p) − γ2 (1 − P (p /p)) E1 πγ2β2 1 0 2 z 2 z 2Z2α + t ξK (ξ)K (ξ) (1 − P (p /p)) (A.14) πβ2 1 0 2 z contains all required information about the kinematics of the scattering. 150 Appendix B: Numerical Techniques and Errors

An EFT calculation has an inherent theory error that comes from truncation of the EFT expansion. A key feature of any EFT is the ability to estimate this theory error. However, the usefulness of this ability is lost unless the numerical error in an EFT calculation is under control and is smaller than the theory error. Some of the numerical techniques used to obtain the results presented in this dissertation and estimates of numerical errors are discussed here.

B.1 Numerical integration

All numerical integrals in the limits (-1,1) in this dissertation are evaluated using the Gauss-Legendre quadrature rule [167],

Z 1 XN f (x) dx = wi f (xi) + δ(N), (B.1) −1 i=1

where {x1, x2 ... xN} are the N roots of PN(x), and

2 wi =   (B.2) 2 0 2 1 − xi [Pn(xi)] are the weights for the Gauss-Legendre quadrature [167]. For a given f (x) there is an optimum range of N values for which the error δ(N) is minimal. However, in practice, some numerical accuracy is often sacrificed for lower computational cost while choosing a value for N, since higher values require longer computing time in general. For arbitrary limits, we use the following transformation to make the integral amenable to Gauss-Legendre quadrature rule. ! Z b b − a Z 1 b − a a + b f (x) dx = f x + dx. (B.3) a 2 −1 2 2 Integrals in which the upperlimits are formally at infinity are truncated at a large but finite cutoff, Λ. To ensure that the error due to this truncation in under control, we vary Λ over a very wide range and check whether the integral has saturated. It might become necessary 151

to readjust the number of grid points during this variation. In Eqs. (5.38), (5.39), (5.40), (6.14), (6.15), (6.21), (6.22) and (6.23), the ranges of the integration variable q where the integrals receive most of their contribution is set by the two-body scattering lengths and (mainly) by the three-body energies, and can be easily estimated. We therefore split these integrals as follows.

Z Λ Z λ Z Λ f (q) dq = f (q) dq + f (q) dq, (B.4) a a λ

where λ is chosen such that the contribution of the second integral is small. We then take Λ  λ and use the mapping

Z Λ Z log Λ f (q) dq = g (ξ) dξ, (B.5) λ log λ   where g (ξ) = f eξ eξ. We then perform the integral on the right hand side using Gauss-Legendre quadrature rule. Since the interval (log λ, log Λ) is much smaller than the interval (λ, Λ), this mapping allows us to go to very large values of Λ without having to use a large number of grid points. In particular, this significantly reduces the computational cost of solving the Faddeev equations (Eqs. (5.38) and (5.39), and (6.21) and (6.22)), using the methods discussed in Appendix B.2 and B.3 respectively.

B.2 The three-body bound state: solution of Faddeev equations, calculation of the wavefunction and matter form factor

The angular integrations in Eqs. (5.38) and (5.39) are first performed numerically.

The integrals over the momenta are then converted to sums by using the grid {q1, q2, ...qN} to discretize q0. These grid points are chosen as discussed in Section B.1. The functions

Fn,c(q) are then evaluated at q lying on the same grid points. This gives a set of 2N coupled integral equations of the form

XN Fx(qi) = Kxy(qi, q j)Fy(q j), (B.6) i=1 152 where the labels x and y can each be either n or c. Kxy(qi, q j) contains factors of three-body Green’s function, two-body T-amplitude, and the quadrature weight, w j, associated with the point q j. Equations (5.38) and (5.39) can then be written as a single matrix equation, KF = F, where   F (q )  n 1     ···        Fn(qN) F =   (B.7)   Fc(q1)      ···      Fc(qN) and   K (q , q ) ··· K (q , q ) K (q , q ) ··· K (q , q )  nn 1 1 nn 1 N nc 1 1 nc 1 N     ··················        Knn(qN, q1) ··· Knn(qN, qN) Knc(qN, q1) ··· Knc(qN, qN) K =   (B.8)   Kcn(q1, q1) ··· Kcn(q1, qN) Kcc(q1, q1) ··· Kcc(q1, qN)      ··················      Kcn(qN, q1) ··· Kcn(qN, qN) Kcc(qN, q1) ··· Kcc(qN, qN) We use the subroutines in the LAPACK library [227] to find the eigenvalues and the eigenvectors of K. F is the eigenvector of K with eigenvalue 1. The three-body contact interaction, H(Λ), at the chosen cutoff, Λ, in Eq. (5.38) is fixed by requiring that the eigenvalue of 1 exists for the given three-body binding energy, EB. By numerical evaluation of Eqs. (5.38) and (5.39), we can find Fn,c(q) at a value of q that does not necessarily lie on the grid {q1, q2 ... qN}, for example, for the evaluation of the bound state wavefunctions in Eqs. (5.40) and (5.41). 153

B.2.1 Evaluation of matter radii

Finally, to calculate the matter radii , we write the formfactor, e.g. Equation (5.44), as

1 − F (k2) hr2i = 6 c + δ(k), (B.9) k2     where the first term on the right hand side is O k0 and δ(k) is O k2 at k < 1 fm−1. We need k to be small enough such that δ(k) is negligible compared to the preceding term.

2 However if k is too small numerical errors become large as Fc(k ) ∼ 1. In Fig. B.1, we

2 plot the dimensionless quantity mEBhr i calculated in this way for A = 20 in the unitary

limit at different values of EB versus the k values used. Up to numerical and truncation

(δ(k)) errors, this quantity should be independent of EB and k. We obtain a better than

0.1% precision for all values of EB shown, even when k is varied over five orders of magnitude. Since the bound state wavefunctions calculated to obtain the plots in Fig. B.1

depend on EB in a nontrivial manner, this also verifies that the bound state wavefunction derived is numerically accurate.

B.3 Integral equations for the unbound state

Equations (6.21) and (6.22), for x = n or c, can similarly be discretized and converted to a single matrix equation KT = T + Z, which can be solved for the 2N unknown entries of the column vector T. We use the subroutines in the LAPACK library [227] to solve the system of 2N linear equations. Since these equations have to be solved at E > 0, we make the transformation q00 → q00e−iφ, dq00 → dq00e−iφ where 0 < φ < π/4 as discussed in

0 Section 6.2.1 and discretize the integral to obtain Txy(q, q ) at the same grid points. Since, at E > 0, the kernel matrix is complex even for real momenta, this transformation does not add much to the computational cost. 154

0.28

0.27

0.26 > 2 0.25 < r B E = 0.02 MeV 0.24 B

m E EB = 0.04 MeV E = 0.06 MeV 0.23 B EB = 0.08 MeV 0.22 EB = 0.10 MeV

0.21 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 -1 k ( fm )

2 Figure B.1: mEBhr i at A = 20 calculated from Eq. (5.44).

B.4 Matrix element of the dipole operator

B.4.1 Evaluation of MPWIA

We calculate the matrix element in PWIA from Eq. (6.9) by evaluating A(kc, Kc; EB) using Eq. (6.8). Since the momentum-space ground state wavefunction is a slowly varying function of momentum, we can obtain an accurate value for the derivative from the simple forward difference method, ∂ hp q + h; 0000|Ψ i − hpq; 0000|Ψ i hpq; 0000|Ψ i = c in c in + δ(h). (B.10) ∂q c in h

At large h, the error δ(h) is due to truncation of the Taylor’s series of chp q + h; 0000|Ψini around q at first order in h. If h is too small, the difference between chp q + h; 0000|Ψini and chp q; 0000|Ψini has a large error due to loss of significant figures by subtractive cancellation. We infer from our analysis of the matter radius above (e.g. Figure B.1), that the numerical accuracy of the bound state wavefunction is such that we can let h be as 155 √ √ −6 small as 10 . In Table B.1, we show A( mEB, mEB ; EB) calculated at Z = 6, A = 20,

Enc = 0.2 MeV for a wide range of EB at different h values. The error δ(h) stays small even when h spans several orders of magnitude.

√ √ Table B.1: A( mEB, mEB ; EB) evaluated at Z = 6, A = 20, Enc = 0.2 MeV by differentiation of the wavefunction as shown in Eq. (6.8) . √ √ A( mEB, mEB ; EB) h (MeV) (e/MeV4)

EB = 0.1 MeV EB = 0.3 MeV EB = 1 MeV

10−6 -2.49171×10−5 -3.53072×10−6 -3.82708×10−7

10−5 -2.49171×10−5 -3.53072×10−6 -3.82708×10−7

10−4 -2.49172×10−5 -3.53072×10−6 -3.82708×10−7

10−3 -2.49176×10−5 -3.53076×10−6 -3.82710×10−7

B.4.2 Evaluation of Mnn

The integral in Eq. (6.14) has the following form.

Z ∞ f (p) Z ∞ f (p) f (k ) p P p − iπ c . d 2 2 = d 2 2 (B.11) 0 kc − p + i 0 kc − p 2kc The principal value integral can be expressed as

Z ∞ f (p) Z ∞ f (p) − f (k ) Z ∞ 1 P p p c f k P p . d 2 2 = d 2 2 + ( c) d 2 2 (B.12) 0 kc − p 0 kc − p 0 kc − p

If f (p) is an analytic function of at p = kc, then the integrand in the first term on the right hand side is a smooth function. It can be easily evaluated using the standard numerical methods like the one described in Section B.1. The second integral is formally zero. However, for a finite cutoff,

Z Λ 1 1 Λ + k P p c . d 2 2 = log (B.13) 0 kc − p 2kc Λ − kc 156

B.4.3 Evaluation of Mnc

Similarly, Eq. (6.15) can be written as √ ∗ 0 ˆ Mnc(E1) = i 2 π tn(E; Kn) Y1 (Kn) I (B.14) where I has the form Z ∞ Z 1 f (q, x) I = dq dx , 0 −1 x − x0 − i where − 2 − A+1 2 mE Kn 2A q x0 = . (B.15) qKn The singularity in the integrand can be avoided by using the Hetherington-Schick method of rotating the path of the q−integral into the fourth quadrant, as discussed in Section 6.2. √ √ For mE < Kn < (2A + 2)mE/(A + 2), however, the x−integral gives logarithmic branch cuts in the fourth quadrant in the complex q−plane. In this narrow interval, we perform the q−integral on real axis by using subtraction method to deal with the singularity. This can be done by writing the integral as

Z ∞ Z 1 f (q, x) Z ∞ Z 1 I = dq P dx + iπ dq dx f (q, x)δ(x − x0). 0 −1 x − x0 0 −1

The δ−function part evaluates to zero unless −1 < x0 < 1. This gives

Z ∞ Z 1 f (q, x) Z q+ dq P dx + iπ dq f (q, x0), 0 −1 x − x0 q− where r A 2A + 2 2 2
q− = Kc − Kc + mE − Kc (B.16) A + 1 A and  r  A  2A + 2  q = K + K2 + mE − K2
 . (B.17) + A + 1  c c A c  The principal value integral can be split into different parts to obtain

∞! Z q− Z Z 1 f (q, x) Z q+ Z 1 f (q, x) Z q+ + dq dx + dq P dx + iπ dq f (q, x ), − − 0 0 q+ −1 x x0 q− −1 x x0 q− 157

For q < q− and q+ < q, the integrand is smooth in x. The principal value integral in

q− < q < q+ can be calculated by the subtraction method discussed before:

Z q+ Z 1 Z q+ "Z 1 # f (q, x) f (q, x) − f (q, x0) |1 − x0| dq P dx = dq dx + f (q, x0) log . q− −1 x − x0 q− −1 x − x0 |1 + x0|

For the q−integral, however, the real part of the integrand still has integrable (logarithmic) singularities at q → q±. Therefore, care must be taken while evaluating them. For the integral in (q+, ∞) higher numerical accuracy is achieved for a given number of grid points by concentrating them at q±, which is automatically done by the exponential substitution

in Eq. (B.5). For the integral in (0, q−), we use

Z q− Z log 2q− ξ ξ dq f (q) = dξ f (2q− − e ) e 0 log q−

before discretizing the integral. This substitution makes a more efficient use of the

computational resources by concentrating the grid points near q−. In Fig. B.2, we compare the values obtained from the Hetherington-Schick method

and from the subtraction method for various values of Kn at Z = 6, A = 20,

E = EB = 0.4 MeV and Enc = 0.1 MeV. Good agreement is seen between the two methods √ in the range 0 < Kn < mE. Although a large percentage difference between the real

parts is seen at small Kn, the absolute difference is of the same order for all values of Kn. Moreover, phase-space integrated observables (like dB(E1)/dE) are more sensitive to the

values of the matrix element at larger Kn because the squared matrix element gets √ weighted by a larger phase space volume. In our calculations, for Kn > mE, we use the subtraction method since logarithmic branch cuts appear in the integrand at the fourth √ quadrant in the q−plane. For 0 < Kn < mE, we can avoid integrating the logarithmic singularities by using the rotation method instead. 158

Figure B.2: Real and imaginary parts of the integral in Eq. (6.15) obtained from rotation of the integration path and from subtraction method versus the external momentum Kn at Z = 6, A = 20, E = EB = 0.4 MeV and Enc = 0.1 MeV. The inset shows the percentage difference between the real parts obtained from the two methods. Differences in the imaginary parts√ are much smaller and are not shown. The rotation method can be used only up to Kn = mE.

B.4.4 Evaluation of Mnnc

The integrals in Eq. (6.23) are evaluated by distortion of the integration path as discussed in Section 6.2.1. In addition to numerical errors the truncation of the integrals at a large but finite cutoff, Λ, introduces an error due to the neglected pieces of the integrals at the arc of radius Λ in the complex momentum plane. Both these errors can be assessed 159

by evaluating the integrals over wide ranges of external momenta Kn and Kc at different

values of φ. We define In(Kn) and Ic(Kc) by rewriting Eq. (6.23) as √ h i ∗ 0 ˆ 0 ˆ Mnnc(E1) = i 2 π tn(E; Kn) Y1 (Kn) In(Kn) + tc(E; Kc) Y1 (Kc) Ic(Kc) . (B.18)

Table B.2 shows the real and imaginary parts of In(Kn) calculated at different values √ of φ for Kn = 0.1, 0.5, 0.9 and 1.3 mE. For 0.3 ≤ φ ≤ 0.6 radian, the percentage differences in both the real and the imaginary parts are of the order of 0.1%.

Table B.2: Dependence of In(Kn) on the integration path rotation angle for E = EB =

0.4 MeV, Z = 6, A = 20, with Enc fixed at 0.1 MeV.

−4 2 In(Kn) in units of 10 e/MeV with Kn equal to √ √ √ √ φ 0.1 mE 0.5 mE 0.9 mE 1.3 mE

(rad.) Re Im Re Im Re Im Re Im

0.1 -2.13 -4.187 -0.95 -14.23 -6.987 -35.56 -62.52 -40.35

0.2 0.765 -2.744 2.396 -15.00 -5.521 -34.48 -61.65 -41.78

0.3 0.790 -2.671 2.521 -14.98 -5.490 -33.34 -61.58 -41.82

0.4 0.789 -2.669 2.525 -14.98 -5.494 -33.33 -61.58 -41.82

0.5 0.788 -2.669 2.524 -14.97 -5.495 -33.33 -61.58 -41.81

0.6 0.788 -2.669 2.522 -14.97 -5.498 -33.33 -61.58 -41.80

0.7 0.787 -2.667 2.523 -14.97 -5.504 -33.32 -61.59 -41.87

Table B.3 shows the real and imaginary parts of Ic(Kc) calculated at different values √ of φ for Kc = 0.1, 0.5, 0.9 and 1.3 mE. For 0.3 ≤ φ ≤ 0.6 radian, the percentage differences in both the real and the imaginary parts are of the order of 0.1%.

B.5 Integrating over the momenta of the outgoing particles

Using N(Kc), N(Kˆ c · z), N(kˆc · z) and N(φ) to denote the number of Gauss-Legendre points used to evaluate Eq. (6.32), Table B.4 shows that N(Kc) ' 40, N(Kˆ c · z) ' 12, 160

Table B.3: Dependence of Ic(Kc) on the integration path rotation angle for E = EB =

0.4 MeV, Z = 6, A = 20, with Enc fixed at 0.1 MeV.

−5 2 Ic(Kc) in units of 10 e/MeV with Kc equal to √ √ √ √ φ 0.1 mE 0.5 mE 0.9 mE 1.3 mE

(rad.) Re Im Re Im Re Im Re Im

0.1 -0.993 1.585 -5.095 7.850 -12.25 18.99 -12.1 40.54

0.2 -1.213 1.848 -5.899 9.982 -9.49 21.56 -7.33 41.22

0.3 -1.229 1.858 -5.970 10.07 -9.44 21.76 -7.16 41.34

0.4 -1.231 1.858 -5.973 10.07 -9.44 21.77 -7.16 41.35

0.5 -1.231 1.858 -5.973 10.07 -9.44 21.77 -7.16 41.35

0.6 -1.230 1.858 -5.971 10.07 -9.44 21.77 -7.15 41.35

0.7 -1.228 1.858 -5.958 10.07 -9.41 21.77 -7.11 41.34

N(kˆc · z) ' 6 and N(φ) ' 6 is sufficient to get a 1% precision in dB(E1)/dE for a wide range of E.

Table B.4: dB(E1)/dE calculated by numerically evaluating Eq. (6.32) with different number of Gauss-Legendre quadrature points. The calculation is performed at Z = 6,

A = 20, EB = 0.4 MeV and Enc = 0.2 MeV.

2 2 E dB(E1)/dE in e fm /MeV for {N(Kc), N(Kˆc · z), N(kˆc · z), N(φ)} equal to (MeV) {40, 12, 6, 6} {80, 12, 6, 6} {40, 24, 6, 6} {40, 12, 12, 6} {40, 12, 6, 12}

−2 −2 −2 −2 −2 E = 0.1EB 2.6487 × 10 2.6486 × 10 2.6487 × 10 2.6487 × 10 2.6487 × 10

−1 −1 −1 −1 −1 E = EB 6.9398 × 10 6.9396 × 10 6.9398 × 10 6.9398 × 10 6.9398 × 10

−2 −2 −2 −2 −2 E = 10EB 4.2506 × 10 4.2514 × 10 4.2535 × 10 4.2510 × 10 4.2515 × 10 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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