Neutron-Deuteron Scattering and Three-Body Interactions

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 163

PHILIPPE MERMOD

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List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I J. Klug, J. Blomgren, A. Ataç, B. Bergenwall, A. Hildebrand, C. Johansson, P. Mermod, L. Nilsson, S. Pomp, U. Tippawan, K. Elmgren, N. Olsson, O. Jonsson, A.V. Prokoﬁev, P.-U. Renberg, P. Nadel-Turonski, S. Dangtip, P. Phansuke, M. Österlund, C. Le Brun, J.F. Lecolley, F.R. Lecolley, M. Louvel, N. Marie-Noury, C. Schweitzer, Ph. Eudes, F. Haddad, C. Lebrun, A.J. Koning, and X. Ledoux, "Elastic neutron scattering at 96 MeV from 12C and 208Pb", Phys. Rev. C 68, 064605 (2003).

II C. Johansson, J. Blomgren, A. Ataç, B. Bergenwall, S. Dangtip, K. Elmgren, A. Hildebrand, O. Jonsson, J. Klug, P. Mermod, P. Nadel-Turonski, L. Nilsson, N. Olsson, S. Pomp, A.V. Prokoﬁev, P.-U. Renberg, U. Tippawan, and M. Österlund, "Forward-angle neutron-proton scattering at 96 MeV", Phys. Rev. C 71, 024002 (2005).

III P. Mermod, J. Blomgren, B. Bergenwall, A. Hildebrand, C. Johansson, J. Klug, L. Nilsson, N. Olsson, M. Österlund, S. Pomp, U. Tippawan, O. Jonsson, A. Prokoﬁev, P.-U. Renberg, P. Nadel-Turonski, Y. Maeda, H. Sakai, and A. Tamii, "Search for three-body force effects in neutron-deuteron scattering at 95 MeV", Phys. Lett. B597, 243 (2004).

IV P. Mermod, J. Blomgren, A. Hildebrand, C. Johansson, J. Klug, M. Österlund, S. Pomp, U. Tippawan, B. Bergenwall, L. Nilsson, N. Olsson, O. Jonsson, A. Prokoﬁev, P.-U. Renberg, P. Nadel-Turonski, Y. Maeda, H. Sakai, and A. Tamii, "Evidence of three-body force effects in neutron-deuteron scattering at 95 MeV", Phys. Rev. C 72, 061002(R) (2005).

V P. Mermod, J. Blomgren, B. Bergenwall, C. Johansson, J. Klug, L. Nilsson, N. Olsson, A. Öhrn, M. Österlund, S. Pomp, U. Tippawan, P. Nadel-Turonski, O. Jonsson, A. Prokoﬁev, P.-U.

v Renberg, Y. Maeda, H. Sakai, A. Tamii, K. Amos, R. Crespo, and A. Moro, "95 MeV neutron scattering on hydrogen, deuterium, carbon and oxygen", submitted to Phys. Rev. C (2006).

Reprints were made with permission from the publishers.

vi Contents

1 Introduction ...... 1 1.1 Fundamental physics ...... 1 1.2 Nuclear applications ...... 2 1.3 The present work ...... 3 2 Examples of three-body forces ...... 5 2.1 Emotional interactions ...... 5 2.2 Nuclear interactions ...... 6 2.3 Strong interactions ...... 8 2.4 Electromagnetic interactions ...... 9 2.5 Gravitational interactions ...... 10 3 Theory of nuclear interactions ...... 11 3.1 Nucleon-nucleon potentials ...... 11 3.2 From nucleons to nuclei ...... 12 3.3 Many-body systems ...... 13 3.4 The off-shell problem ...... 13 3.5 Three-nucleon forces and effective ﬁeld theories ...... 14 4 Search for three-nucleon forces ...... 17 4.1 Three-nucleon force observables ...... 17 4.2 Neutron-deuteron scattering experiments at TSL ...... 19 4.3 Data analysis ...... 23 4.4 Results ...... 24 5 Conclusions and outlook ...... 31 6 Acknowledgements ...... 33 7 Summary in Swedish ...... 35 8 Bibliography ...... 37

vii

1. Introduction

Like a coin, physics has two sides. Physics research is driven by curiosity about nature (neophile side) as well as practical prospects (technophile side). This coin is in a state of constant flipping, since applications often motivate−and finance−fundamental research in particular fields, and theory can be used to predict quantities or phenomena which can lead to new or improved technologies. The present thesis is a good example of this duality. The neutron-deuteron scattering experiments on which it is based were designed to investigate fundamental interactions between three nucleons, but at the same time the information extracted from the experimental data may be useful for medical applications, and the work was carried out as part of a program on studies of neutron-induced nuclear reactions of importance for incineration of nuclear waste.

1.1 Fundamental physics All known interactions between physical objects can be interpreted in terms of four1 fundamental interactions: the electromagnetic interaction, the weak interaction, the strong interaction, and the gravitational interaction. Very often, the notion of force−which is what causes a change in an object−is mixed up with the notion of interaction since a force is necessarily due to an interaction (with another object). In this thesis, we are interested about the nuclear force, which is a manifestation of the strong interaction described phenomenologically as the effective interaction between nucleons (nucleons can be either protons or neutrons). Most theories of fundamental interactions involve hypothetically fundamental objects called elementary particles. In this approach, in principle, a total knowledge of the existence, properties and interactions of elementary particles allows to describe all physical phenomena: if the elementary particles are really fundamental, then they are the building blocks of every larger structure, and thus all structures must obey some complex (in practice, usu-

1 Traditionally, modern physicists have counted four interactions; however, electromagnetism and the weak interaction can be shown to be two aspects of a single electroweak interaction, and somewhat more speculatively, the electroweak interaction and the strong interaction can be combined using grand uniﬁed theories. How to combine the fourth interaction, gravity, with the other three is a topic of research in quantum gravity.

1 ally too complex to be computed exactly by human means) combination of the fundamental laws governing elementary particle dynamics. In quantum field theories, specific elementary particles (the gauge fields) act as mediators between other elementary particles. This exchange mechanism is used as a basis for the description of interactions between pairs of elementary particles. Three-body (as well as four-body, five-body, etc.) interactions are often overlooked in the description of physical systems, probably because, in nature, they tend to be insignificant compared to two-body (pairwise) interactions. As we will see in Chapter 2, quantum field theories of, e.g., electromagnetic and strong interactions predict new types of interactions which arise as soon as more than two objects are involved. These three-body interactions can be viewed as the coupling between the exchange particles themselves while they are mediating the interaction between the real objects. The present thesis primarily concerns experimental studies of three-body nuclear forces in neutron-deuteron scattering at 95 MeV. The primary aim is to provide data in the three-nucleon system for testing the models of three- nucleon forces. The work is put in a broader perspective by considering examples of three-body forces in other areas of physics.

1.2 Nuclear applications There are a number of new applications under development where neutrons of higher energies than in the traditional applications (nuclear power and nuclear weapons) play a significant role. The most important are transmutation of nuclear waste [1], medical treatment of tumors with fast neutrons [2], and the mitigation of single-event effects in electronics [3]. These applications would benefit from refinements of nuclear models, especially for neutron-induced nuclear reactions up to 1 GeV. None of these applications require precise data on neutron-deuteron scattering for their further development in a short-time perspective. In a longer term and a broader physics perspective, however, such data could be of large importance. It is an ultimate goal since long in nuclear physics to be able to describe heavy nuclei from simple fundamental forces only. At present, this goal is far from being fulfilled. In nuclear energy applications, precise knowledge of heavy nuclei, like uranium and plutonium, is required, but for these nuclei ab initio theories are not in a state where cross section predictions can meet the requirements for technological applications. Advanced nuclear theory can be used to describe some particularly favorable cases, but for most nuclei, specific models are constructed phenomenologically. The present work represents a step in the direction to establish a more fundamental treatment in the realm of applied nuclear physics. It has been es- tablished that nuclear theory based only on the nucleon-nucleon (two-body) interaction fail to describe even such a basic quantity as the binding energy

2 of A = 3 nuclei. With the introduction of three-nucleon forces, these problems can be remedied, and moreover, it seems that a combination of two- and three-nucleon forces can provide a fairly good description all the way up to A = 12. Thus, there is good hope that the quality of the description of many-body systems, like heavy nuclei, can progress signiﬁcantly already by improved understanding of three-nucleon forces. In the experiments of the present thesis, in addition to the neutron-deuteron scattering data, we obtained also differential cross sections for neutron scattering on carbon and oxygen at 95 MeV (see Paper V). Cancer therapy with fast neutrons may beneﬁt directly from the carbon and oxygen data, since these elements are abundant in biological tissue and the evaluation of the cell damage depends on the probability for the neutron to scatter inside the body.

1.3 The present work The present work was part of the experimental effort in neutron scattering pur- sued since 1994 by the Department of Neutron Research at Uppsala Univer- sity. The laboratory work was undertaken at The Svedberg Laboratory (TSL) in Uppsala. The papers (I-V) are interlinked and follow a consistent sequence. Paper I by Klug et al. is about elastic neutron scattering on carbon and lead at 96 MeV, the first fully analyzed data obtained with the SCANDAL2 detector setup. In Paper II by Johansson et al., neutron-proton (np) data at 96 MeV were obtained with essentially the same technique specifically adapted to the purpose of np scattering. When I joined the Uppsala neutron reaction research group in 2002, the data for Papers I and II were still under analysis. I provided a modest contribution in multiple scattering corrections. At that time, both the MEDLEY [4] and the SCANDAL [5] detector setups had proven their capabilities and could be used in a routine manner. This, and recent theoretical predictions about three-nucleon force effects in elastic neutron-deuteron (nd) scattering (see, e.g., Ref. [6]), motivated the nd scattering experiments presented in Pa- pers III, IV and V, in which I took active part at all stages. The MEDLEY nd data were analyzed first and published in summer 2004 (Paper III). Results of the SCANDAL experiments were obtained one year later (Paper IV) and constituted both a cross-check and an improvement of the MEDLEY results. When analyzing the SCANDAL nd data, the work previously done for Papers I and II was very useful in the sense that the same detectors were read out and the same type of corrections were applied. I have, however, almost completely re-engineered the analysis routines, partly because I found it convenient to do it in my own way and partly because the special conditions of nd scattering

2SCANDAL stands for SCAttered Nucleon Detection AssembLy.

3 demanded rather complex analysis procedures to obtain the required precision (details are given in Paper V). This allowed beneﬁcial consistency checks of the routines since by-products of the nd experiments, i.e., elastic scattering on carbon and on hydrogen, overlap with the results of Papers I and II, respectively. An introduction to the thesis is given in Chapter 1. In Chapter 2, I discuss three-body forces in a general context and I give examples of three-body forces outside the framework of nuclear physics. These personal contributions have a philosophical nature distinct from the experimental work described in the papers (in fact, except for three-nucleon forces, none of the three-body force examples cited in this thesis are depicted in the literature). Relevant three-nucleon interaction theory is outlined in Chapter 3. The nd scattering experiments and the search for three-nucleon force effects are outlined in Chap- ter 4. Finally, conclusions and an outlook are given in Chapter 5.

4 2. Examples of three-body forces

In most cases, the complex interactions that arise in systems of three objects or more are due to combinations of basic two-body interactions. In general, if the observed behavior of a system of more than two objects cannot be described by the two-body interactions between all possible pairs, the deviation is mainly due to a three-body force. The underlying mechanism, illustrated in Fig. 2.1, can be understood as follows: three-body forces are caused by an interaction with the interaction.

Figure 2.1: The interaction between two bodies is represented as a dashed line. On the right, the interaction is affected by the presence of the additional body (wavy line).

2.1 Emotional interactions As an analogy, if we identify bodies with human beings and forces with feel- ings, jealousy is a good example of a three-body force: it is not felt as long as only two persons are acting, but it can show up if a third person enters the scene (see Fig. 2.2). In this case, the three-body force can be responsible for strong effects upon the jealous individual: the jealousy can be expressed as an urge to ﬁght the concurrent or to gain the beloved one by any means. But the source of the jealousy has not much to do with the interaction with the concurrent himself nor the beloved one, but rather arises from what is exchanged between them.

5 Figure 2.2: Jealousy is a human example of a three-body force.

2.2 Nuclear interactions The nuclear force, or nucleon-nucleon (NN) interaction, is a particular aspect of the strong interaction which can be described very well phenomenologically in the framework of pion-exchange models (see section 3.1). As we will discuss in detail in Chapters 3 and 4, there are very good reasons to believe that the description of a system made of three nucleons or more is not complete if three-body forces are not taken into account (and, in principle, also four-body forces, ﬁve-body forces, etc.). Consistent models of three-nucleon (3N) interactions were proposed in the 1960s [7, 8, 9]. The main motivation was to reproduce the triton binding energy as accurately as possible. These early models contained already the basics of 3N interactions, illustrated by the diagrams (b) and (c) in Fig. 2.3. Models of 3N interactions, including the most recent ones, will be further discussed in section 3.5. The two-pion exchange 3N interaction represented in Fig. 2.3b is included in almost all models of 3N interactions and is often believed to be the dom- inant contribution. This effect was stated in the pioneering 1957 article by Fujita and Miyazawa [10] as follows: "Within a nuclear matter the potential between two nucleons takes a different shape due to many body forces.". More explicitly, due to the pion exchange between the ﬁrst two nucleons, one of the nucleons is changed, e.g., it is excited to a N∗ or a ∆ resonance, thus modifying the interaction with the third nucleon via another pion exchange. The 3N interaction shown in Fig. 2.3c, where a particle emitted by one nucleon interacts with the pion exchanged between the two other nucleons, is more intuitive. Since strong interactions conserve parity, the intermediate

6 Figure 2.3: Possible interactions between three nucleons. In (a), the simple two-pion exchange does not correspond to a three-body interaction, since it can be reduced to a succession of two-body interactions. The three other diagrams represent three-body interactions of different types. A typical two-pion-exchange 3N interaction is shown in (b), where one nucleon is excited between the absorption and the emission of the pions. In (c), one nucleon exchanges a scalar meson or a vector meson (double line) with a pion. In (d), there is a correlation between two nucleons while the pion is "in ﬂight".

7 Figure 2.4: Three-body interactions between quarks can arise due to gluon-gluon coupling. particle coupling to the pion cannot itself be a pion1. In a 1961 publication by Loiseau and Nogami [9], an interaction was proposed between the pion and a σ meson (spin 0 and even parity); in a 1965 article by Harrington [11], the particle was described as a two-pion resonance (also a scalar); and in a 1974 article by Yang [12], it was assumed to be a ρ meson (spin 1 and odd parity, or vector meson). The type of 3N interaction illustrated in Fig. 2.3d has been proposed recently (2000) by Canton and Schadow [13] as a new contribution that might help to reproduce experimental data which are sensitive to the spin structure of the 3N force (see section 4.1). It can be understood in the following way: a pion is emitted by a nucleon, and a contact interaction (without pion exchange) occurs between the two other nucleons before the pion is absorbed.

2.3 Strong interactions At a more fundamental scale, strong interactions are the interactions between quarks and gluons as described by quantum chromodynamics (QCD). Due to the fact that gluons, the mediators of the strong interaction, carry color charge, they can interact directly with each other. A gluon emitted by one quark can couple with another quark or with the gluon emitted by another quark. In the latter case, a three-body interaction can arise, as illustrated in Fig. 2.4. One would expect contributions such as the diagram in Fig. 2.4 to have some kind of effect inside baryons. However, there are serious problems if we try to perform calculations based on it. If all vertices are hard (high momentum transfer) then one can apply perturbation theory, but at this order (four vertices), many other diagrams must be taken into account, thus complicat- ing the calculations [14]. In a realistic case, however, the vertices are more likely to be soft (low momentum transfer) and then perturbation theory can-

1Pions are pseudoscalars with spin zero and odd parity, which forbids pion-pion vertices, but contact interactions between real pions are still possible.

8 Figure 2.5: Possible three-body interaction between electrons. The photons do not couple to other photons, but high-order diagrams allow the formation of a virtual fermion loop.

not be used (not even for two-body interactions) due to a large coupling. It is hard to imagine any kind of perturbative (and therefore calculable) process where three-quark forces would play a role. And in the case of baryons, it is in the completely non-perturbative regime where QCD must be replaced by an effective theory. Also, it is hard to think of any measurement that might be sensitive to such a contribution [15]. Anyway, although this process is perhaps irrelevant in practical cases, it is interesting from a philosophical point of view: it illustrates the fact that three- body forces should fundamentally exist in all quantum ﬁeld theories where the exchange particles can couple to each other.

2.4 Electromagnetic interactions The fundamental theory of electromagnetic interactions is quantum electrody- namics (QED), where the exchange particles are photons. Photons do not carry electric charge, and therefore they cannot couple to each other directly. How- ever, one can imagine a process through which a three-body force could also arise in electromagnetic interactions. Consider for example three electrons. One photon emitted by one electron can ﬂuctuate into a fermion-antifermion pair, and another photon emitted by another electron can couple to one of the fermions. At ﬁrst glance, one might think that the fermion-antifermion pair could annihilate into a photon which could couple to the third electron; however, such a process−with two photons going into one photon−is forbidden since electromagnetic interactions conserve charge conjugation (C-symmetry) and photons are C-odd [15]. But the process shown in Fig. 2.5, where two photons are re-emitted and couple to two of the electrons, is thinkable. However, it involves eight vertices, which makes such contributions negligible since the electromagnetic coupling constant is small (such a diagram appears only at high orders in a perturbative expansion).

9 2.5 Gravitational interactions Are there three-body forces in gravitational interactions? The most successful theory of gravity up to date is general relativity. General relativity is really non-linear and not even the two-body problem can be solved exactly [16]. In linearized general relativity, which is good enough for most purposes as long as gravity is not too strong, one considers only pairwise interactions between masses. This is an approximation, which is not expected to give accurate results in the vicinity of high energy densities. Gravity, as any other interaction, can, at least to some extent, be viewed as the exchange of particles of force, the gravitons. Interestingly, gravitons can couple to themselves. When gravity is strong, this can lead to a process in a three-body system which is not captured by the pairwise interactions, where a graviton leaves one of the bodies, splits into two, with one graviton going to each of the other bodies [17] (the same kind of three-body force process as illustrated in Fig. 2.4 in the case of QCD). Thus, clearly, there are three-body forces in quantum theories of gravity, as in all non-linear theories with cubic vertices for the force particles. The non-linear equations of general relativity are expected to automatically take everything into account. At large scales (typically in the neighborhood of black holes), one would expect the effects of three-body forces to play some role. When applying general relativity to perform calculations in many-body systems, however, approximations must be introduced. Having worked in the ﬁeld of nuclear physics and seen the advantages of introducing a three-nucleon potential in addition to the nucleon-nucleon interaction, I can imagine a similar phenomenological approach to tackle large-scale many-body gravitational problems: it might be advantageous to use linearized general relativity and introduce a three-body gravitational potential. The form of such a potential could possibly be based on a graviton-exchange theory. If high-precision ob- servations of, e.g., the dynamics of triple star systems can be made, they could be used to test the predictions with and without three-body forces.

10 3. Theory of nuclear interactions

The ultimate goal of nuclear physics is to understand the properties of atomic nuclei in terms of the basic interaction between pairs of nucleons. This implies the assumption that neutrons and protons are the fundamental building blocks of nuclear matter, which is not true since they are themselves known to consist of interacting quarks. However, the perturbative techniques of QCD cannot be used at the hadronic scale due to a too large coupling constant. We are therefore limited to (semi-)phenomenological models for nuclear interactions. If we want one consistent description for all reactions, thus avoiding a jungle of particular phenomenological cases, our best hope is to rely on the nucleon- nucleon (NN) interaction and gradually attempt to describe systems of two, three, and many nucleons.

3.1 Nucleon-nucleon potentials The interaction between two nucleons can be described quantum mechanically by considering each nucleon as a wave packet exposed to the potential V of the other nucleon. The evolution of the system is governed by the time-dependent Schrödinger equation. The cross section for nucleon-nucleon scattering from |ψ > |ψ > the initial state i to the final state f can be obtained from the scatter- =< ψ | |ψ > ing amplitude, which is derived from the matrix element Tif f V i . The usual technique to compute the matrix elements for a general collision is to express the eigenstates of the time-independent Schrödinger equation as the Lippmann-Schwinger equation [18]. The properties of the potential V have to be determined phenomenologically. The general form of the potential must be invariant under rotations, reflections, and time reversal, and can be written as the sum of six independent terms accounting for the central and tensor components of the nuclear force, the spin-spin interaction, and the spin-orbit, quadratic spin-orbit, and antisymmetric spin-orbit interactions. In 1935, Yukawa proposed that the nuclear interaction arises from the exchange of a massive scalar field [19], thus explaining its finite range. His theoretical work predicted the existence of mesons (unknown at the time) and at the same time provided a quantum field interpretation of the NN interactions, which can be used to derive the form of the NN potential. Shortly after the dis- covery of the pion in the late 1940s, the range of the nuclear force was conve- niently subdivided into three regions [20]: the phenomenological short-range

11 core (strongly repulsive at short distance), the dynamical intermediate-range dominated by two-pion exchange (which can be attractive), and the classical long-range interaction mediated by one pion (which falls quickly to zero with distance). In the 1960s, the models were refined by including heavy-meson exchange (in particular, vector mesons) as well as correlations between pions in two-pion exchange. Modern potentials are still based on pion-exchange models. The comprehensive NN models available in the 1980s [21, 22] gave a good overall description of the data but still they were not sufficiently accurate for reliable ab initio calculations. In 1993, the Nijmegen group presented a detailed partial- wave analysis1 of NN scattering [23]. It was in this context that the world pp and np database (published before December 1992) was scanned very criti- cally to eliminate the data that deviated too significantly. In the mid-1990s, high-precision NN potentials were developed with parameters adjusted to fit the large pp and np database surviving the selection of the Nijmegen group. The groups involved and the names of their potentials are, in chronological order: • Nijmegen group: Nijm1, Nijm2, and Reid93 potentials [24], • Argonne group: AV18 potential [25], • Bonn group: CD-Bonn potential [26, 27]. These potentials use about 45 parameters and fit the pre-scanned data base with a reduced χ2 close to one, for energies between 0 and 350 MeV. How- ever, if the data published after December 1992 are considered, the χ2 are not perfect anymore [28]. Large sets of pp data were published between 1993 and 1999, notably high-quality pp spin-correlation parameters from IUCF. It turns out that the IUCF data contains new information that calls for an improvement of the NN parameters, without contradicting the old data [28]. When considering np scattering differential cross sections, large sets of data were published after December 1992, but they are not always consistent with each other nor with the old pre-scanned data. Systematic effects affecting the shape or the absolute scale of the np angular distribution are probably responsible for these discrepancies. In Paper II, we insist on the importance of high-quality np data, in particular for a precise determination of the strength of the fundamental coupling of the pion to the nucleon, the πNN coupling constant.

3.2 From nucleons to nuclei With high-precision NN potentials at hand, it becomes possible to try the ab initio approach and describe systems of more than two nucleons from the basic interactions between nucleons. There are three major obstacles to overcome

1In a partial-wave analysis, the parametrization is not necessarily well founded on a theory, but rather chosen to optimize the ﬁt to the data.

12 before such models can become reliable: the difﬁculty to perform calculations in many-body systems, the so-called "off-shell problem", and the need for many-body forces.

3.3 Many-body systems Dealing with the spin components of the NN force in systems of more than two nucleons is a formidable numerical problem. The quantum-mechanical three-body problem can be solved exactly by using an NN potential in the Faddeev equations [29]. However, before powerful computers were available, it was very difﬁcult to obtain converged solutions, and approximations were necessary. Consequently, thirty years ago, there was typically a 10 − 20% uncertainty (∼ 1 − 2 MeV) in the triton binding energy, while a precision of ∼ 1% would be necessary to disentangle the physics [30]. Today, thanks to the ongoing advances in computational methods and resources, many few-nucleon properties can be reproduced accurately from ab initio calculations [31, 32]. Calculations in many-body systems are no longer an overwhelming obstacle.

3.4 The off-shell problem The momentum dependence of the NN force cannot be obtained from NN scattering data only, because in NN scattering (as well as in the deuteron), the energy of the two nucleons is conserved, i.e., it is the same before and after the interaction (they are "on the energy shell"). By contrast, in a many- body system, two nucleons may have different energies before and after they interact, i.e., their mutual interaction may be off the energy shell. Thus, the nuclear many-body problem does not have a unique solution. The off-shell NN interaction is empirically undetermined. Only theory can provide it. For- tunately, off-shell behavior can be derived from a relativistic meson-exchange ﬁeld theory, as has been done for the CD-Bonn potential [26]. The other potentials apply the so-called static/local approximation, which increases the tensor force off-shell [28]. Therefore, among the potentials presented in section 3.1, it is likely that the CD-Bonn potential has the most realistic off-shell behavior. The triton binding energy calculated with the CD-Bonn potential is about 0.4 MeV above the results obtained with the other potentials, and this discrepancy is probably due to different off-shell behavior [26].

13 3.5 Three-nucleon forces and effective field theories There is evidence that, in systems of more than two nucleons, three-nucleon (3N) forces (and maybe also four-nucleon forces, five-nucleon forces, etc.) play a significant role. Thus, we would like to include at least 3N forces in the description. In fact, in three-nucleon systems such as the triton and nucleon- deuteron scattering, it is possible to solve the Faddeev equations exactly with an additional 3N potential [6]. The challenge is to develop models of 3N interactions which are theoretically well-grounded and consistent with NN interactions. Originally, 3N interactions were investigated to solve the discrepancy in the binding energy of the triton, of typically 1 MeV between the experimental value and the expectations from NN interactions. The models of 3N interactions were built by successively adding new terms to the interaction amplitude. The first models in the early 1960s were of Fujita-Miyazawa [10] type. Their essence was in the part of the 3N interaction corresponding to two- pion exchange between three nucleons with the intermediate excitation of a ∆-resonance (see Fig. 2.3b on page 7). Heavier mesons (like the ρ) could replace the pions in two-pion exchange models, having a shorter range. Inter- actions of heavy bosons (such as σ, ρ, ω, and ππ correlations) with one of the exchanged pions, corresponding to Fig. 2.3c, may be added to contribute to the short-range 3N force [33]. The type of 3N interaction corresponding to Fig. 2.3d, where two nucleons interact while the pion is in flight, is a recent development [13] which is not yet included consistently with the other diagrams. It is believed to contribute to the spin dependence of the 3N force. In the late 1970s, it was realized that models of two-pion exchange 3N interactions should be constructed using chiral constraints, i.e., consistently with the (approximate) chiral symmetry of QCD. Two physically equivalent approaches were considered: one of them uses the so-called current algebra [34], and the other is based upon effective lagrangians in relativistic field theories [35]. In the field-theory approach, the lagrangians are not fundamental objects (they are "effective" or "phenomenological"), but they are quick and efficient tools for implementing chiral symmetry. Here is a list of modern 3N potentials, with short descriptions on how they are constructed.

• The Urbana 3N potential [36] does not consider chiral constraints and is based on the two-pion exchange process (or ρ exchange) with an intermediate ∆ (diagram 2.3b). An updated version is called Urbana IX [32]. • The Tucson-Melbourne 3N potential [34] uses current algebra for setting chiral constraints and is based on the two-pion exchange process (or ρ exchange) (diagram 2.3b). An updated version is called TM99 [37]. • The Brazil 3N potential [35] uses an effective lagrangian to implement chiral symmetry. It includes the two-pion exchange (or ρ exchange) process with an intermediate ∆ (diagram 2.3b) as well as processes mediated by

14 Figure 3.1: Three-nucleon interactions at next-to-next-to-leading order in chiral perturbation theory: two-pion exchange (left), one-pion exchange with NN contact (short- range) interaction (middle) and 3N contact interaction (right).

the interaction of a σ or ρ with a pion (diagram 2.3c). The more recent RuhrPot 3N potential [38] is constructed in a similar way. • The Padova-Vancouver 3N potential [39] is generated by the exchange of one pion in the presence of a NN correlation (diagram 2.3d). It does not yet contain other processes. An appropriate way to describe consistently both NN and 3N interactions is provided by tailored effective ﬁeld theories. This approach allows to unify the physics of nucleons and nuclei in connection with QCD [40]. In particular, chiral symmetry breaking can be analyzed in terms of chiral perturbation theory (CHPT). The theory uses perturbation techniques, providing a straight- forward way to improve the results by going to higher orders in a perturbative expansion, and also keeping theoretical uncertainties under control. In the past two decades, this framework has successfully been applied to a variety of low-energy reactions in the meson and baryon sectors. The original idea of applying CHPT to few-nucleon systems was formulated in the early 1990s by Weinberg [41, 42]. In few-nucleon systems, one has to deal with the non- perturbative nature of the nuclear force, which can be understood as contact interactions (without meson exchange). According to Weinberg [42], a resum- mation of certain classes of diagrams can be achieved via solving Faddeev- like equations with the effective nuclear potential derived using the CHPT technique. The ﬁrst model of this type (the Texas model) was proposed by Ordóñez and van Kolck [43, 44]. Subsequent developments by Epelbaum et al. allowed to perform a consistent analysis of nd scattering at next-to-next- to-leading order (NNLO, or third order) [45]. In parallel, nd phase shifts at low energy were extracted at NNLO by Bedaque et al. [46]. In CHPT, 3N forces appear at NNLO in the perturbative expansion, as illustrated by the diagrams in Fig. 3.1. These diagrams require some further explanation since they are not exactly of the same type as the 3N interactions introduced in section 2.2. In the CHPT diagrams, the NN interaction can be either mediated by a pion (πN vertices are represented as black dots) or due to a contact interaction which parametrizes the short-range physics (contact vertices are represented by crossings between the nucleon lines). Superposi-

15 tions of these vertices (black squares) are responsible for the 3N interactions. It is also possible to introduce the 3N-force contributions due to intermediate ∆ excitations (such as represented in Fig. 2.3b) in the CHPT representation. Such terms would arise already at NLO; however, the NNLO diagrams of Fig. 3.1 actually recover a large part of the terms arising from ∆ excitations, so that it is believed that the inclusion of ∆ intermediate states is not needed in the context of CHPT provided that the expansion is performed at an appropri- ately large order [47]. Fourth-order (NNNLO) calculations have been made recently in the two-nucleon system [48, 49], showing that the fourth order is necessary and sufﬁcient for a NN potential reliable up to 290 MeV, with an accuracy comparable to the one of the high-precision phenomenological potentials presented in section 3.1. With such a solid theoretical basis, accurate calculations in three-nucleon systems−which would naturally include both NN and 3N interactions−should become feasible in the near future.

16 4. Search for three-nucleon forces

As we have seen in the previous chapter, the theory of nuclear interactions predicts three-nucleon forces. It can be shown in the framework of CHPT (see section 3.5) that the NN interactions are more important that the 3N ones, which are more important than the four-nucleon interactions, and so on [42]. Still, the absolute strength of the 3N-force components cannot be predicted by theory: the unknown parameters must be determined experimentally. There are indications that 3N-force effects should be small, but not negligible. For instance, in the most simple three-nucleon system, the triton, it appears that the binding energy cannot be reproduced accurately by calculations with NN interactions alone. Relativistic Faddeev calculations with the CD-Bonn potential−which should have the correct off-shell behavior, see section 3.4−underbind the triton by 3.5% [26]. The inclusion of a 3N potential in the calculations does account for this discrepancy [50]. Now we must face the following problem: 3N forces are small compared with NN forces and thus they are difﬁcult to observe, but they still give a contribution comparable to, e.g., off-shell behavior effects, relativistic effects, or Coulomb effects. In an ab initio model of nuclei and nuclear reactions, all these effects must be correctly implemented. To disentangle them and study their details experimentally, we have to ﬁnd experimental observables which are especially sensitive to each particular phenomenon.

4.1 Three-nucleon force observables Three-nucleon forces can be best identiﬁed in three-nucleon systems. Systems of two neutrons and one proton, i.e., the triton and neutron-deuteron (nd) scattering states, are preferred since they are free of Coulomb interactions. In three-nucleon bound systems, 3H and 3He, as we have discussed, the binding energy is a very good 3N-force indicator, mainly because its experimental value is known to a very good accuracy. That is the reason why, in most theories, parameters related to the 3N interactions are ﬁxed from the triton binding energy. It is, however, interesting to note that other bound-system observables can be sensitive to 3N forces. For example, the description of the 3H and 3He charge form factors is also improved when 3N forces are considered [51]. In the low-energy limit, the nd and pd scattering lengths (or strengths of the interaction) show a sensitivity to 3N forces. In addition to the triton bind-

17 ing energy, the doublet nd scattering length can advantageously be used to fix the two parameters related to the 3N interactions at NNLO in tailored effective field theory approaches [45, 46]. A recent high-accuracy measurement of the coherent nd scattering length, which is almost equivalent to the doublet nd scattering length, allows to set tight constraints on model descriptions based on modern NN and 3N potentials [52]. It is observed that most models, although their description is improved when including 3N forces, fail to reproduce exactly the world-average experimental value of the nd scattering length. This has lead to the suggestion that additional diagrams may be needed in 3N potentials, where the additional parameters would be set to fit the nd scattering length in addition to the triton binding energy [53]. Spin observables in nd and pd scattering are interesting in the sense that they can be used to study the spin structure of 3N interactions. In particular, it has been observed since a long time [29] that the nucleon and deuteron vector analyzing powers in elastic nucleon-deuteron scattering below 30 MeV cannot be explained by means of NN calculations, and that an inclusion of existing 3N potentials does not remove this discrepancy [54]. In principle, it could be solved by a refinement of the 3N-interaction terms in CHPT [45]. Alternatively, a new class of diagrams was proposed recently [13] (see Fig. 2.3d on page 7) which are expected to affect the vector analyzing powers when implemented into a 3N potential. Another rich source of information is the nucleon-deuteron breakup reaction. In this case, pd is a much more practical choice than nd, since the complete kinematics of the reaction can be obtained by detecting the two outgoing protons. Effects of 3N forces in pd breakup can be studied in the intermediate energy range (65 − 250 MeV) if a sufficiently large fraction of the full phase space is measured [55, 56]. Recent pd breakup experimental results at 130 MeV show a clear preference for the predictions in which 3N forces are included [57]. Such an approach is further encouraged by the recent advances in including Coulomb interactions in the calculations [58]. Finally, 3N forces are expected to affect significantly the shape of the angular distribution in elastic nd and pd scattering at intermediate energies [6, 59]. The elastic nucleon-deuteron scattering differential cross section looks like two hills with a valley in between (see the drawing on the title page of this thesis). This is due to the basic NN scattering processes. However, if 3N forces are present, their contribution should scatter the incident nucleon in a much more isotropic way, since all three nucleons participate at once in the interaction, potentially absorbing an arbitrary fraction of the transverse momentum. Thus, the 3N forces are overwhelmed by NN forces in the regions of the hills, but they are on the same footing in the region of the valley. As a result, it turns out that 3N forces are expected to fill in the valley. While this effect is most clearly visible in the shape of the nucleon-deuteron angular distribution, it has also been observed that calculations without 3N forces underestimate the nd total cross section above 100 MeV [60].

18 The effect in the minimum of the differential cross section has been seen in numerous elastic pd scattering experiments at intermediate energies (see the references [23-29,32-34] of Paper V). However, the pd data sets do not always agree with each other. For instance, at 135 MeV, the theoretical predictions which include 3N forces agree very well with three japanese sets of experimental data [61], while they underestimate significantly the Groningen (Netherlands) data [62]. The Groningen data indicate a stronger effect than expected. To explain this, a possible candidate would be relativistic effects (which were not treated in the calculations), but a relativistic treatment performed without 3N forces indicates that such effects are not significant at this energy [61]. At 250 MeV, where a similar discrepancy is observed between the experimental data and the predictions with 3N forces for both pd and nd scattering, it is not excluded that relativistic effects are responsible for the excess [63, 64]. The nd data by Palmieri at 152 MeV agree well with non-relativistic calculations with 3N forces [65], thus supporting the picture in which, below 200 MeV, relativistic effects are small and an inclusion of a 3N potential in the calculations is necessary and sufficient to reproduce the differential cross section. The nd data at 95 MeV presented in Papers III, IV and V as well as in the next sections are fully consistent with this picture. These results have a double advantage. Firstly, elastic nd scattering is unaffected by the Coulomb interaction, and thus a comparison between nd and pd data, which has never been done before around 100 MeV, allows to verify the theoretical indications that Coulomb effects in pd scattering are not significant in the minimum of the angular distribution [66]. And secondly, at this energy, relativistic effects are expected to be negligible.

4.2 Neutron-deuteron scattering experiments at TSL The present elastic nd scattering experiments were performed at The Svedberg Laboratory (TSL) in Uppsala, using the neutron beam line simultaneously with the MEDLEY and SCANDAL setups (see Fig. 4.1). The TSL cyclotron produced a 98 MeV proton beam which was directed toward a lithium-7 target, producing quasi-monoenergetic neutrons with a peak energy of 95 MeV at zero degrees. The charged particles were removed by deﬂecting magnets and the remaining neutron beam was shaped by a set of collimators. The MEDLEY setup is a vacuum chamber of about 70 cm diameter containing up to eight detector telescopes each equipped with two silicon detectors and one CsI detector. Two ﬁssion-based neutron monitors are situated between the MEDLEY chamber exit foil and the SCANDAL setup. Each of the two SCANDAL arms can be seen as a set of twelve detector telescopes and can be used either to detect protons and deuterons directly or to detect neutrons by converting them to protons inside a converter scintillator detector. More information about the TSL neutron beam facility and the MEDLEY and

19 Figure 4.1: Overview of the Uppsala neutron beam facility before year 2004. The proton beam coming from the left stroke a lithium target and charged particles were de- ﬂected by magnets. The resulting neutron beam was collimated and available through the MEDLEY and SCANDAL setups.

Figure 4.2: Picture of the MEDLEY setup during the present experiment. The detector telescopes were placed in the forward hemisphere of the vacuum chamber. Each of them was equipped with two silicon detectors and one CsI detector.

20 Figure 4.3: Drawing of the SCANDAL experiment in deuteron detection mode. Seven targets (alternated CD2,CH2 and C) were used simultaneously. The setup was used one arm at a time, and recoil protons or deuterons were detected directly by one plastic scintillator, two drift chambers, another plastic scintillator and an array of twelve CsI crystals. A typical event is indicated.

Figure 4.4: Drawing of the SCANDAL experiment in neutron detection mode. The target was a cylinder containing water, heavy water or graphite. Scattered neutrons were detected by letting them pass through a veto plastic scintillator to reject charged particles, convert them into protons inside two thicker plastic scintillators, and tracking the protons with the same set of detectors as in Fig. 4.3. A typical event is indicated.

21 SCANDAL setups−which are shown in Figs. 4.2, 4.3 and 4.4 to illustrate the three present experiments−can be found in Ref. [4] (MEDLEY) and in Ref. [5] (SCANDAL) . The goal of the experiments was to measure of the full elastic nd scattering angular distribution. The simplest approach is to detect the recoil deuterons from deuterated polyethylene (CD2) and graphite (C) target foils (the background from carbon must be subtracted). The larger the deuteron scattering angle, the lower its energy, and therefore there is an angular limit beyond which the deuterons cannot be detected anymore (they are stopped in the material between their production and the detectors). Thus, one has to compromise between having a thick target which allows good statistics but a limited angular coverage and having a thin target which allows a large angular coverage but poorer statistics. With the MEDLEY setup, which is in vacuum and has a high sensitivity to low-energy particles, it was reasonable to choose a relatively thin CD2 target (0.280 mm thickness). On the other hand, the SCANDAL setup is comparatively large and massive since it is primarily designed to detect scattered neutrons. In the case where the recoil deuterons were detected, it was judicious to choose a relatively thick CD2 target (1.060 mm thickness), allowing good statistics in the limited angular range covered−which, fortu- nately, corresponds to the cross-section minimum, where 3N force effects are expected to be most clearly visible. This experiment is illustrated in Fig. 4.3, where seven targets (alternated CD2, C and CH2) were used simultaneously inside a multi-target device. The neutron scattering forward angular range, which was not covered with SCANDAL in deuteron mode, could be obtained by using SCANDAL in neutron detection mode. In this last experiment, illustrated in Fig. 4.4, we used water and heavy water contained in aluminum cans as targets, and detected the scattered neutrons by converting them to protons inside one of the converter scintillators and tracking the protons through the detectors. Thus, we used a convenient combination of three complementary experiments. With the ﬁrst experiment (deuteron detection with MEDLEY), we obtained a full angular coverage−although quite sparse−with statistics suf- ﬁcient to distinguish between presence or absence of 3N forces in the minimum. With the second experiment (deuteron detection with SCANDAL), we obtained high-accuracy data in the minimum, allowing to discriminate between the different models. With the third experiment (neutron detection with SCANDAL), we covered the forward angular range, thus allowing a consistency check of the shape of the whole angular distribution (versus the MED- LEY data). Although the effect we are looking for is really an effect in the shape of the angular distribution, the absolute normalization of the data is crucial, especially for the second experiment which covers only the minimum, where an error in the normalization could potentially fake or disguise the effect of a three-body force. For this reason, we took special care to design the experiments such as to allow an unambiguous internal normalization, by

22 measuring a known cross section at the same time as we measured nd scattering. For the two experiments in deuteron detection mode, the solution of choice was to use CH2 targets and normalize the data to np scattering. For the SCANDAL experiment in neutron mode, we could also measure np scattering but with poor accuracy. It proved to be a better technique to use a graphite target and normalize the data to the total elastic scattering cross section on carbon.

4.3 Data analysis The data analysis for the three experiments described in the previous section represents a major fraction of my working time as a PhD student, but I will only give a brief overview here, since all details are given in Paper V. The SCANDAL nd experiment where scattered neutrons were detected re- sembles the np experiment presented in Paper II. It was the most difficult to analyze, and the least significant as far as 3N forces are concerned, since it did not cover the cross-section minimum. Angular bins were defined (one for each CsI), by using the DCH tracking information and selecting the trajectories which intersected the crystals inside well-defined areas. The scintillator and CsI detectors were calibrated by detecting recoil protons from np scattering and associating the recorded pulse heights with the expected proton energies. Protons were identified in two-dimensional plots of the energy deposited in the CsI versus the energy deposited in the trigger scintillators. Signals from the scintillators allowed to identify in which detector the conversion occurred. The proton conversion angles and neutron scattering angles were obtained by tracing back the proton trajectories to one point in the converter scintillator plane. A maximum conversion angle of 10◦ was required in order to separate kinematically events converted in hydrogen from events converted in carbon. The neutron scattering energies were calculated by knowing the proton energies and conversion angles, and assuming np scattering inside the converter. The remaining steps to identify the elastic nd scattering events (including the subtraction of the oxygen and deuteron breakup backgrounds) were done by manipulating the neutron energy spectra obtained with the different targets. Elastic scattering on carbon (C target) was also identified in the spectra, which is important for the absolute normalization of the data; and, as a bonus, elastic scattering on oxygen (H2O and D2O targets) was also extracted from the spectra, as well as inelastic scattering to the low-lying collective excited states for both carbon and oxygen. Finally, corrections were applied, e.g., for dead time, multiple scattering inside the target, and the angular dependence of the conversion efficiency. For nd scattering, the systematic uncertainties per point were 10% to 20%, and were dominated by the uncertainties in the oxygen and breakup background subtraction procedures. In the regions were they over- lapped in angle, the data obtained with the two SCANDAL arms and the two

23 converters of each arm were combined, reducing both the statistical and systematic uncertainties per point since these four sets of data were analyzed separately. The analysis for the MEDLEY nd experiment and the analysis for the SCANDAL nd experiment where recoil deuterons were detected were similar. The principle is rather simple: essentially, all one has to do is to identify which events are induced by a 95 MeV neutron elastically scattered by a target deuteron and count them. Deuterons were identified by two-dimensional ∆E versus E plots; 95 MeV neutrons were selected by time-of-flight techniques; and the elastic events could be identified in the energy spectra. Background deuterons from carbon were taken care of by subtracting the spectra obtained with the graphite targets. With SCANDAL, it was important to take the effects of the multi-target device inefficiencies into account since they were angle-dependent and varied from one target plane to another: in particular, it could affect the relative normalization of the carbon spectra in the carbon background subtraction. For absolute normalization purposes, np data were obtained in the same way as for nd scattering. With SCANDAL, there was one more complication related to the accidental fact that the CH2 target foil was larger than the beam. This problem was circumvented by using the DCH tracking information to trace back the events to the CH2 target plane, which allowed to determine its effective area (the beam cross section area) with an accuracy of 2.25%. Corrections were applied for effects which could slightly affect the np and nd angular distributions, like the CsI detection efficiency and the contamination from low-energy neutrons in the neutron beam spec- trum. In the region of the nd cross-section minimum, the relative uncertainty per point was comparable for MEDLEY and SCANDAL (about ±4% statistical uncertainty and about ±5% systematic uncertainty). For both MEDLEY and SCANDAL, these uncertainties could be reduced by combining the sets of data taken on both sides of the beam; the SCANDAL measurement allowed to reach a better precision for testing 3N forces than the MEDLEY measurement because it was concentrated in the cross-section minimum, providing twice as much data points in this angular range.

4.4 Results The results for np scattering are shown in Fig. 4.5. We should not be especially impressed by the remarkable agreement with the high-accuracy Rahm et al. data [67] since the absolute scale of our data (with proton detection) was adjusted to agree with the Rahm et al. data. It is comforting, however, that the shape of our np angular distribution matches well both the previous data and the Nijmegen partial-wave analysis PWA93 [23] (high-precision NN potential predictions are not shown in the ﬁgure but they give very similar curves). Since the nd data were measured with the same technique and in essentially

24 the same conditions as the np data, we are confident that the shape of the nd angular distribution will be free of unexpected systematic errors. The nd final results are shown in Fig. 4.6. We observe a good agreement between the three experiments in the regions where they overlap. The data are in overall agreement with the expectations both in shape and absolute scale. In the minimum region, including a two-pion-exchange 3N potential in the calculations causes a 30% effect which fills the minimum (the dashed and dotted curves correspond to inclusions of the Tucson-Melbourne TM99 [37] and Urbana IX [32] 3N potentials, respectively, to be compared with the solid curve where 3N forces are not included). Our data actually fill the minimum as expected from theories of 3N interactions. In Fig. 4.7, by plotting the ratio between the nd data and the np data in the minimum region, the effects of 3N forces are definitely confirmed. In- deed, normalization errors (and some other sources of systematic errors) are cancelled out for this ratio. Thus, it is difficult to imagine what kind of experimental error might significantly affect this result. As a possible difference in detector behavior, I can identify the CsI efficiency and the multi-target efficiency, which have been taken into account. Other possible effects would be an inaccurate relative number of irradiated target nuclei (but this was determined with an accuracy better than 2.3%, see section 4.3), or the contamination of protons from deuteron breakup in nd scattering (but those were rejected by a particle identification cut). Relativistic effects, which were not included in the calculations, are expected to be negligible at this energy [68], and the relative differences in these effects for np and nd scattering must be even less important. Thus, the clear excess observed in the minimum must be due to either a three-body force or a totally unknown and unsuspected effect. Among the calculations using various NN and 3N potentials shown in Fig. 4.7, which are all considerably closer to the data when 3N forces are included, the best description (χ2 = 1.2) is obtained with the combination AV18 + Urbana IX. The present nd data have the advantage to be free of Coulomb force effects. The situation is different in elastic pd scattering, where, for studying 3N forces, one would like to be sure that Coulomb effects are negligible in the minimum region. The present nd data give us an opportunity to validate this assumption by providing the first comparison between elastic nd and pd scattering around 100 MeV (see Fig. 4.8). Moreover, a comparison in the forward angular range, where the Coulomb effects are expected to be strongest, would allow to test the models which attempt to include these effects in the calculations. Up do date, we cannot perform a very conclusive comparison because the only pd data available, by Chamberlain and Stern [69], do not have the required precision. New pd data at this energy are expected to be published soon and will allow a more detailed comparison [70].

25 Figure 4.5: The np differential cross section at 95 MeV. The black dots, black squares and open squares are results from the present experiments with MEDLEY, SCAN- DAL with proton detection, and SCANDAL with neutron detection, respectively. The data were normalized to the Rahm et al. data [67], shown as small triangles. The open triangles are the previous SCANDAL data in neutron mode from Paper II. The solid line is the Nijmegen partial-wave analysis PWA93 [23] (see section 3.1).

26 Figure 4.6: The nd differential cross section at 95 MeV. The MEDLEY data (black dots) and SCANDAL data with deuteron detection (black squares) were normalized to the np differential cross section using the Rahm et al. np data [67] as reference (see Fig. 4.5). The SCANDAL data with neutron detection (open squares) were normalized to the total elastic 12C(n,n) cross section (see Paper V for details). The solid curve was obtained with Faddeev calculations with the Argonne AV18 NN potential [25] without 3N forces, and the dashed and dotted curves were obtained with the same calculations using an additional 3N potential: the Tucson-Melbourne TM99 3N potential [37] and the Urbana IX 3N potential [32], respectively. The gray band corresponds to CHPT calculations at NNLO [45].

27 Figure 4.7: The ratio of the nd to the np differential cross sections at 95 MeV in the region of the cross-section minimum, as a function of the proton/deuteron detection angle. The dots and the squares are the MEDLEY and SCANDAL results (with deuteron or proton detection), respectively. The thin, middle-thick and thick curves were obtained with the AV18 [25], Nijm2 [24] and CD-Bonn [27] NN potentials, respectively. The dashed curves were obtained by including the TM99 [37] 3N potential in the nd calculations, and the dotted curve was obtained by including the Urbana IX [32] 3N potential. Note that the curves obtained with the AV18 and Nijm2 potentials are almost indistinguishable from each other; this is probably because they have similar off-shell behavior (see section 3.4).

28 Figure 4.8: Comparison between the nd and pd differential cross sections at 95 MeV. The dots are the complete set of the present nd data, and the triangles are pd data by Chamberlain and Stern [69]. With the present accuracy of the pd data, Coulomb force effects cannot be observed.

5. Conclusions and outlook

Given that the interactions between nucleons are sufficiently well understood, through the Faddeev formalism it is possible to predict the properties of three- nucleon systems. For example, one can calculate the shape of the angular distribution for 95 MeV neutrons elastically scattered from deuterons, taking into account all possible exchanges between the nucleons taken individually. By actually measuring this angular distribution we found that such a description does not reproduce exactly the experimental data: it predicts a deeper minimum than the data suggests. This is simply because three-body interactions have to be taken into account. This effect was foreseen by theory [6] before any measurement existed at 95 MeV−it was actually what motivated the experiments in the first place. After introducing a three-nucleon potential in the Faddeev equations, the theoretical predictions agree very well with our data. An excellent characterization of the physics of few-nucleon systems is obtained by a combination of modern NN and 3N potentials. Although far from being unique, the present case has the merit of providing a good illustration of this success: the elastic nd scattering differential cross section at 95 MeV is accurately described by calculations relying on NN and 3N potentials whose parameters are fixed solely by NN data and the triton binding energy. Chiral perturbation theory can also give reasonably good predictions, and provides a consistent framework for the description of few-nucleon systems. With a good understanding of the nuclear forces in three-body systems, as well as impressive advances in computer resources and computational techniques, it becomes gradually possible to unify the physics of nucleons and nuclei by trying to describe four-nucleon systems, and then five-nucleon systems, etc. The alpha particle binding energy is reproduced very accurately based on combined NN and 3N forces [71], suggesting that the role of four- nucleon forces is not significant. If four-nucleon forces are, indeed, negligible in nuclei, then it is reasonable to assume that forces of higher order (five- nucleon, etc.) are also negligible. In the near future, we will probably see major advances in this field of research: the behavior and properties of light isotopes could in principle be described by a fully consistent ab initio theory of nuclear interactions built upon the underlying two- and three-body interactions. In fact, recent models (in particular, the so-called ab initio no-core shell model (NCSM), see Refs. [72, 73] and references therein) are now able to give predictions for the binding energies and nuclear structures of light isotopes up to carbon. An inclusion of a 3N potential generally improves the description,

31 by increasing the binding energies and rearranging the level ordering and level spacing of the low-lying excitation spectra. Such an approach cannot be sustainable without constant reﬁnements of the structure of the NN and 3N potentials, which imply stringent comparisons with experimental data. For instance, spin observables such as analyzing powers call for a better understanding of the spin structure of the 3N forces. Ac- curate measurements of the scattering lengths, deuteron break-up, and elastic scattering observables in the nucleon-deuteron system provide a good testing ground for the 3N potentials. Concerning speciﬁcally elastic nd scattering, the data are scarce in the intermediate energy region and the present work represents an important contribution. As 3N forces are expected to be more accen- tuated with increasing energy and progress is being made in estimating the relativistic contributions, it would be natural to extend the present measurement up to higher energies. There has been a recent proposal for nd scattering experiments at TSL at 135 MeV, this energy being especially attractive as the pd scattering experiments performed at 135 MeV by two different groups [61, 62] differ by about 30%. Elastic nd scattering data might help to resolve this discrepancy. Above this energy (the maximum energy for the TSL neutron beam is 180 MeV), some upgrade of the CsI detectors is necessary, and such an upgrade is already in progress for MEDLEY.

32 6. Acknowledgements

I am extremely grateful to my supervisor Jan Blomgren for his positive spirit and dynamic guidance. Special thanks to Stephan Pomp and Udomrat Tippawan for precious contributions with the MEDLEY setup, and to Leif Nilsson for meticulously going through my articles. Thanks to Jan Källne and Nils Olsson for their interest and good advices. I am very thankful to all the other people who have been actively participating in my work during the last four years, including Bel Bergenwall, Anders Hjalmarsson, Cecilia Johansson, Joakim Klug, Pawel Nadel-Turonski, Pär Olsson, Angelica Öhrn, and Michael Österlund. Thanks to Susanne Söderberg for administration support, and to all INF colleagues for a pleasant working environment, among others, Peter Andersson, Erik Andersson Sundén, Maria Back, Jean-Christophe Bourselier, Sean Conroy, Göran Ericsson, Anna Flodin, Maria Gatu Johnson, Luca Giacomelli, Wolfgang Glasser, Joel Gustafsson, Moinul Habib, Masateru Hayashi, Carl Hellesen, Mikael Höök, John Loberg, Philip Magnusson, Christine Marklund, Emanuel Ronchi, Nils Sandberg, Henrik Sjöstrand, Janne Wallenius, Lovisa Wallin, Matthias Weiszﬂog, Gustav Wikström and Martin Wisell. If I had not been part of the AIM graduate research program, I could probably not have made an average of 1.25 transcontinental trips per year: many thanks to Erkki Brändas and Bo Thidé for their excellent leadership, as well as Marcus Dahlfors, Emma Hedlund, Roger Karlsson, Mattias Lantz, Joa Ljung- vall, Otasowie Osifo, Fredrik Robelius, Christofer Willman, and other AIM students for good companionship. I wish to thank Alexander Prokoﬁev and Per-Ulf Renberg for their remarkable work with the neutron beam monitoring, Olle Jonsson for the 7Li target handling, Anatoly Kolozhvari for making the data acquisition system work, as well as the other members of the TSL technical staff who were operating the cyclotron. I also want to thank Ib Koersner and Teresa Kupsc for computer support. I thank very much Yukie Maeda, Hideyuki Sakai and Atsushi Tamii for collaborating so closely with us and having participated in some of our experimental runs. I have also appreciated the precious collaboration with Kichiji Hatanaka and Nasser Kalantar-Nayestanaki. Fruitful discussions with Ulf Danielsson and Ruth Dürrer (gravity), and John Field and Gunnar Ingelman (QCD), were crucial for broadening my un-

33 derstanding of three-body forces beyond my expertise area. I am also very grateful to Kenneth Amos, Raquel Crespo, Evgeni Epelbaum, Walter Glöckle, Hiroyuki Kamada, Arjan Koning, Antonio Moro, and Henryk Witała for their enthusiastic supply of theoretical calculations. Finally, I want to express my deep gratitude to my parents, my brothers Alexandre and Kevin, and my wife Kristina for their love and support.

34 7. Summary in Swedish

Avhandlingens titel översatt till svenska är Neutron-deutronspridning och trekropparväxelverkan. Den handlar framför allt om tre experiment där vi mäter differentiella tvärsnitt för elastisk neutron-deutronspridning vid energin 95 MeV. Experimenten utfördes vid The Svedberglaboratoriet i Uppsala med detektorutrustningarna MEDLEY och SCANDAL. För att kunna täcka hela vinkelfördelningen med en bra precision har vi använt en kombination av två olika tekniker, där man detekterar antingen rekyldeutroner direkt (med både MEDLEY och SCANDAL) eller så detekterar man spridda neutroner som man konverterar till protoner i en plastscintillator (med SCANDAL). Absolutnormaliseringen bestämdes relativt till ett tvärsnitt som är välkänt och som mättes samtidigt som neutron-deutronspridning, i vårt fall neutron-protonspridning och elastisk neutronspridning på kol. Den experimentella metodiken och diskussion av resultaten redovisas i artiklar publicerade i internationella tidskrifter. Neutron-deutronsystemet består av tre nukleoner som växelverkar bara genom stark växelverkan, vilket underlätter studiet av trekropparkrafter. Växelverkan mellan två nukleoner kan påverka kraften på en tredje nukleon. Detta innebär att krafterna mellan tre nukleoner inte kan beskrivas exakt som summan av de individuella nukleon-nukleonväxelverkningarna, utan man behöver också att ta hänsyn till trekropparkrafter. Detta är inte unikt för stark växelverkan, utan det ﬁnns andra exampel inom fysik där trekropparkrafter kan uppstå. Egenskaper hos trenukleonsystemet kan beräknas genom användning av en nukleon-nukleonpotential (för att beskriva nukleon-nukleonväxelverkan) samt en trenukleonpotential (för att ta hänsyn till trekropparkrafter) i de så kallade Faddeevekvationerna. Sådana beräkningar visar att trekropparkrafter ökar det differentiella tvärsnittet för elastisk nukleon-deutronspridning med ungefär 30% i det vinkelområde där tvärsnittet är minst. Teoretiska förutsägelser som inkluderar trekropparkrafter stämmer bra öv- erens med våra experimentella data. Eftersom det är svårt att tänka sig någon annan effekt som skulle kunna modiﬁera vinkelfördelningen på samma sätt i minimumområdet, kan vi tolka resultatet så att vi faktiskt ser effekter av trekropparkrafter. Detaljerade undersökningar av trekropparväxelverkan mellan nukleoner kan vara av avgörande betydelse för en fundamental beskrivning av

35 atomkärnor. Att kunna beskriva nukleoner och kärnor i en och samma modell är ett viktigt mål i kärnfysik. Då slipper man använda fenomenologiska modeller som bara kan tillämpas i speciella fall. Nyligen har det blivit möjligt att beskriva lätta kärnor upp till A=12 (kol) ganska väl med beräkningar som baseras på nukleon-nukleon och tre-nukleonpotentialer. Genom att utveckla potentialerna kan den här metodiken sannolikt så småningom ge bättre och bättre resultat.

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