Nuclear Thermodynamics with Chiral Low-Momentum Interactions

Technische UniversitätMünchen Physik-Department Institut fürTheoretische Physik T39 Univ.-Prof. Dr. W. Weise

Diploma Thesis by Corbinian Wellenhofer

December 2012

Supervisors: Prof. Dr. Norbert Kaiser Dr. Jeremy W. Holt Prof. Dr. Wolfram Weise

Abstract

Several decades after its historical emergence (Rutherford, Geiger and Marsden discovered the nucleus in 1909, and the existence of the proton and neutron were established about ten and twenty years later, respectively) nuclear physics has experienced a significant revival through the development of chiral effective field theory in the end of the last century. In particular the description of interacting nuclear many-body systems (nuclear matter) has benefitted to a large extent from this modern theory of nucleons and pions. Nuclear interactions emerge in an intricate way as a long-distance phenomenon of quantum chromodynamics, the fundamental theory of the strong interaction. In recent years, a new approach to treat the nonperturbative behaviour of these interactions has been initiated: the construction of effective low-momentum potentials with the use of renormalization group methods. This approach allows a perturbative treatment of nuclear matter, which we employ in this thesis. In the following thesis, we investigate the thermodynamics of nuclear matter using chiral low-momentum interactions. The effective low-momentum nucleon-nucleon potential Vlow-k(Λ), characterized by a momentum cut-off Λ, is combined with three-nucleon forces from chiral perturbation theory. In detail, we calculate the free energy per particle for both isospin-symmetric nuclear matter and neutron matter by considering contributions up to second order in many-body perturbation theory. From the free energy per particle we determine other thermodynamical quantities, such as the pressure, the entropy per particle and the chemical potential, and study the liquid-gas phase transition of isospin-symmetric nuclear matter. A main purpose of this work is to examine the model-dependence of results in perturbative nuclear matter calculations. For that reason we use different low-momentum two-body interactions (with the three-body interactions adjusted to them) in our calculations, and analyze the differences of the respective results. Finally we extend the calculations to the case of isospin-asymmetric nuclear matter with vary- ing proton-to-neutron ratio.

“The more we learn about the world, and the deeper our learning, the more conscious, speciﬁc, and articulate will be our knowledge of what we do not know, our knowledge of our ignorance”

- Karl Popper

Contents

1. Introduction1

2. From Quantum Chromodynamics to Chiral Effective Field Theory4 2.1. QCD Lagrangian ...... 4 2.2. Running Coupling Constant, Asymptotic Freedom, Confinement ...... 6 2.3. Global Low-Energy Symmetries and Hadron Spectrum ...... 8 2.4. Spontaneous Chiral Symmetry Breaking ...... 11 2.5. Effective Field Theory Program ...... 12 2.6. Realization of Chiral Symmetry ...... 13 2.6.1. Transformation Properties of Pion Fields ...... 13 2.6.2. Transformation Properties of Nucleon Field ...... 16 2.7. Chiral Effective Lagrangian ...... 16 2.7.1. Relativistic Pion and Pion-Nucleon Lagrangians ...... 17 2.7.2. Heavy Baryon Formalism ...... 18 2.7.3. Nucleon Contact Terms ...... 19 2.7.4. Organization by Index of Interaction ...... 20

3. From Chiral Lagrangians to Nuclear Potentials 22 3.1. Chiral Perturbation Theory and Power Counting ...... 22 3.2. Hierarchy of Nuclear Forces ...... 23 3.3. Construction of Chiral NN Potential ...... 25 3.3.1. Diagrammatic Amplitudes ...... 25 3.3.2. Bethe-Salpeter Equation ...... 27 3.3.3. Chiral NN Potential with Regulator Function ...... 28 3.4. General Properties of NN Potentials ...... 28

4. Thermodynamics of Many-Particle Systems 30 4.1. Zero Temperature Formalism ...... 31 4.2. Finite Temperature Formalism ...... 35 4.3. Diagrammatic Analysis ...... 40 4.4. Free Energy per Particle and Zero Temperature Limit ...... 43

5. Chiral Low-Momentum Interactions 47 5.1. Nonperturbative Features of NN Potentials ...... 47 5.2. Construction of Low-Momentum Potentials ...... 48 5.3. Chiral 3N Forces and Low Energy Constants ...... 51 5.3.1. Genuine 3N Force from ChEFT ...... 51 5.3.2. Iterated NN Interaction and Contact Terms ...... 52 Contents

6. Evaluation of Many-Body Diagrams for Nuclear Matter 54 6.1. Zeroth Order: Free Fermi Gas Contribution ...... 54 6.1.1. Relativistic Corrections at Zero Temperature ...... 54 6.1.2. Relativistic Corrections at Finite Temperature ...... 55 6.2. First Order Contribution from NN Potential ...... 59 6.2.1. Properties of Two-Body Matrix Elements ...... 60 6.2.2. Partial Wave Representation at Zero Temperature ...... 62 6.2.3. Generalization to Finite Temperatures ...... 64 6.3. Second Order Normal Diagram ...... 64 6.3.1. Partial Wave Representation at Finite Temperature ...... 65 6.3.2. Speciﬁcation to Zero Temperature ...... 67 6.4. Anomalous Contributions ...... 69

7. Nuclear and Neutron Matter Equation of State 72 7.1. Tests of Numerical Algorithm ...... 72 7.2. Results for Neutron Matter ...... 76 7.2.1. Free Energy per Particle with Diﬀerent Two- and Three-Body In- teractions ...... 76 7.2.2. Derived Thermodynamic Quantities ...... 79 7.3. Results for Isospin-Symmetric Nuclear Matter ...... 82 7.3.1. Free Energy per Particle with Diﬀerent Two- and Three-Body In- teractions ...... 82 7.3.2. Physical Free Energy per Particle, Critical Temperature, Liquid- Gas Phase Transition ...... 85 7.3.3. Derived Thermodynamic Quantities ...... 87 7.4. Discussion of Anomalous Contributions ...... 89

8. Isospin-Asymmetric Nuclear Matter 93 8.1. Grand-Canonical Potential Density for Two Species ...... 93 8.2. Free Energy per Particle and Zero Temperature Limit ...... 97

9. Discussion and Conclusion 100

A. Notation and Conventions 102

B. Maxwell Construction and Critical Temperature 104

C. Chiral Three-Nucleon Interaction 106 C.1. Three-Body Kernels ...... 106 C.2. Isospin-Asymmetric Nuclear Matter ...... 107

D. Test Interaction: Scalar-Isoscalar Boson Exchange 108 D.1. Isospin-Symmetric Nuclear Matter ...... 109 D.2. Isospin-Asymmetric Nuclear Matter ...... 110

E. Test Interaction: Pseudoscalar Boson Exchange 111 E.1. Isospin-Symmetric Nuclear Matter ...... 111 E.2. Isospin-Asymmetric Nuclear Matter ...... 113 Contents

F. Bibliography 114

Acknowledgements 119 1. Introduction

Nuclear matter is defined as the idealized system of a very large number of protons and neutrons interacting only by strong nuclear forces, i.e. electromagnetic interactions of −3 protons are not taken into consideration. It represents the low-density (ρ . 0.35 fm ) and low-temperature (T . 100 MeV) phase of Quantum Chromodynamics (QCD) - the phase where quarks are confined into hadrons and chiral symmetry is spontaneously broken. The aim of this thesis is to investigate the thermodynamic properties of nuclear matter using the modern effective field theory (EFT) approach to nuclear forces. This is moti- vated by the fact that the nuclear equation of state (EoS) has a wide range of applications; for instance to astrophysics (neutron stars [LP01], neutron star mergers [TBB11], core- collapse supernovae [Mez05]), the in-medium chiral condensate [FKW12b] and heavy-ion collisions [NCG+10]. A further application is to better understand selected bulk properties of heavy nuclei. Regarding applications in astrophysics, neutron stars can be said to constitute a pri- mary example. Infinite neutron matter is thought to be a good model for the matter that appears in the inner crust and the outer core of neutron stars. By solving the Tolman- Oppenheimer-Volkoff equation with a specific EoS one can obtain the mass (as a function of the radius) of a neutron star which emerges from that EoS. The recent observation of a two-solar-mass neutron star [DPR+10] provides important constraints on the EoS of neutron star matter - constraints which are best met by a description based on ordinary nuclear matter, without any exotic components (cf. Figure 1.1).

The modern theory of nuclear forces arises from Chiral Effective Field Theory (ChEFT), which allows for a systematic construction of nuclear two- and three-body potentials. In detail, one orders the Feynman diagrams arising from the Lagrangian of ChEFT with respect to increasing powers of the chiral expansion parameter Q/Λχ, where Q is a soft scale in ChEFT and Λχ signifies the scale associated with the breaking of chiral symmetry. This organization scheme is called chiral perturbation theory (ChPT). The non-perturbative short-distance part of the nucleon-nucleon potential VNN can be integrated out by evolving VNN to a low-momentum potential Vlow-k(Λ), which becomes strongly perturbative for momentum cut-offs around Λ ' 2 fm−1. At low momenta Vlow-k(Λ) produces the same two-nucleon observables as the bare potential VNN. In combination with chiral three-nucleon forces also selected three-nucleon and four-nucleon ob- 3 3 4 servables (the binding energies of H, He and He) are reproduced by Vlow-k(Λ). In this thesis we calculate the nuclear equation of state by applying many-body perturbation theory (MBPT) to the nuclear many-body system. In MBPT, thermodynamic quantities are obtained by an expansion in terms of the interaction Hamiltonian, which is composed of the nuclear two- and three-body potentials, i.e. in our case the low- momentum potential Vlowk(Λ) and the chiral three-nucleon potential V3N at next-to-next- to-leading order (NNLO). We also perform calculations with the bare chiral nucleon-

1 1. Introduction

nucleon potential VNN, multiplied with a regulator function, in order to examine the model-dependence of this approach to nuclear thermodynamics.

The thesis is divided into the following parts:

• In the second chapter we present some basic elements of QCD and ChEFT. The approach here is to start from QCD as the underlying fundamental theory and construct the eﬀective theory of nuclear forces, i.e. ChEFT. In doing so we investigate and make use of the two essential properties of low-energy QCD - color conﬁnement and spontaneous chiral symmetry breaking.

• The third chapter deals with the construction of nuclear two- and three-body potentials in ChEFT. After reviewing the power counting formalism necessary for ChPT, we show the Feynman diagrams (up to order ν = 4 in the chiral expansion parameter) that correspond to interactions between nucleons. Subsequently we compile the nucleon-nucleon potential VNN from these diagrams, and examine its properties. The construction of the three-nucleon potential V3N is straightforward, as only three diﬀerent diagrams contribute when we restrict ourselves to ν = 3. • In the fourth chapter we give an introduction to the formalism of many-body perturbation theory and its application to nuclear matter. We consider both the zero temperature and the ﬁnite temperature formalism, and discuss the relation between both.

• In chapter ﬁve, we show how matrix elements of the low-momentum potential Vlow-k(Λ) in the partial-wave basis can be constructed with renormalization group methods. We also discuss the non-perturbative features of the chiral NN potential, and show that these featues are absent in Vlow-k(Λ). In addition, we investigate the role of induced and genuine three-body interactions in nuclear matter.

• With Vlow-k(Λ) and the chiral three-nucleon potential V3N, our framework - chiral low-momentum interactions - for the investigation of nuclear thermodynamics is complete. In chapter six, we then evaluate the different contributions to the free energy per particle of nuclear matter in terms of partial wave matrix-elements of Vlow-k(Λ). • In chapter seven, we present numerical results for different thermodynamic quantities of isospin-symmetric nuclear matter and neutron matter. The free energy per particle in particular is calculated using three different models: Vlow-k(Λ) with two different cut-offs, and the bare chiral nucleon-nucleon potential VNN. In each case the chiral three-body potential V3N is adjusted to the two-body potential such that the binding energies of 3H, 3He and 4He remain unchanged.

• In chapter eight, we extend the formalism to the case of isospin-asymmetric nuclear matter.

• Finally, a discussion of the obtained results, a summary of the present work and an outlook to possible extensions and further applications of the methods used in this thesis are given in the last chapter.

2 1. Introduction

Figure 1.1.: Neutron star mass-radius relation (taken from [DPR+10]) for several equations of state. The horizontal bands show constraints from neutron star measurements, such as the newly discovered two-solar-mass star J1614-2230. Any EoS which leads to a line that does not intersect the upper band is ruled out.

3 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

During the last three decades of the past century it became clear that Quantum Chromo- dynamics (QCD) is the fundamental theory of the strong interaction (color force). QCD is a non-abelian gauge theory with gauge group SU(3)c; as such it describes the dynamics 1 of quarks (spin- 2 fermions) and massless gauge bosons, the gluons. In the high-energy region the QCD coupling strength g is weak, which allows for a perturbative treatment of the quark and gluon interactions. However, at energies and momenta Q . 1 GeV QCD becomes nonperturbative; related to this are two phenomena, confinement of quarks and gluons into hadrons (in particular nucleons and pions) and spontaneous chiral symmetry breaking. These properties of QCD in the low-energy regime necessitate the use of an effective low-energy description in terms of the relevant degrees of freedom (nucleons and pions) and their effective interactions - this is provided by Chiral Effective Field Theory (ChEFT). In this chapter we sketch the path from QCD to ChEFT. We begin with a discussion of the QCD Lagrangian in section 2.1. Section 2.2 deals with three crucial properties of QCD, namely the renormalization group flow of its parameters, the asymptotic freedom at high energies, and the property of confinement at low energies. In sections 2.3 and 2.4 we investigate the symmetries of low-energy QCD, in particular chiral symmetry, and discuss consequences of spontaneous chiral symmetry breaking. The remaining three sections 2.5, 2.6 and 2.7 deal with the formulation of ChEFT, in particular with the construction of the chiral effective Lagrangian.

2.1. QCD Lagrangian

The QCD Lagrangian can be constructed from the Lagrangian for free quarks by applying the gauge principle to the internal symmetry group SU(3)c. It reads 1 L =q ¯(iγµD − m) q − Tr G Gµν , (2.1) QCD µ 2 µν where for each ﬂavor f ∈ {u, d, s, c, b, t} the quark ﬁeld has three color components, qf = (qf,r, qf,b, qf,g), and m = diag(mf) is the quark mass matrix. Moreover, γµ are the Dirac matrices, and Dµ is the gauge covariant derivative

a Dµ := ∂µ − igAµ ,Aµ := taAµ , (2.2)

a λa that incorporates the eight gluon ﬁelds Aµ, a ∈ {1,..., 8}, and the generators ta := 2 of SU(3)c (λa are the Gell-Mann matrices). The Lagrangian LQCD is invariant under local a gauge transformations U(x) = exp(−itaΘ (x)), under which the quark and gluon ﬁelds

4 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory transform as

qf(x) → U(x) qf(x) , (2.3) i A (x) → U(x) A (x) + ∂ U †(x) . (2.4) µ µ g µ

The second formula can be made more explicit by Taylor expanding around Θa(x) = 0: 1 Aa (x) → Aa (x) − ∂ Θa(x) + f abcΘ (x)Aµ(x) + O(Θ2) , (2.5) µ µ g µ b c where f abc are the structure constants of su(3), the Lie algebra of SU(3). The ﬁrst two terms on the right-hand side of Eq. (2.5) deﬁne the gauge transformation in an abelian theory (like Quantum Electrodynamics (QED)), whereas the third term is characteristic of the non-abelian case. Besides the quark-gluon part of the QCD Lagrangian, there is also an entirely gluonic sector: 1 1 L = − Tr G Gµν = − Ga Gµν , (2.6) glue 2 µν 4 µν a

a where the gluon ﬁeld strength tensor Gµν is given by

a a a abc Gµν := ∂µAν − ∂νAµ + gf Aµ,bAν,c . (2.7)

Again, the third term would be absent in an abelian gauge theory. It gives rise to gluonic self-interactions, i.e. pure gluon vertices which appear already at leading order in g as shown in Figure 2.1.

Figure 2.1.: Three-gluon vertex (order g) and four-gluon vertex (order g2)

The principle of stationary action applied to LQCD yields the Euler-Lagrange equations (the classical equations of motion) for the quark and gluon ﬁelds,

µ (iγµD − mf) qf(x) = 0 , (2.8) µ a abc µ a ∂ Gµν + gf Ab (x)Gµν,c(x) = −g jν (x) , (2.9)

a a where jµ(x) =q ¯(x)γµt q(x) is the Noether (color) current of quarks. The second term on the left-hand side of Eq. (2.9) is a distinctive property of a non-abelian gauge theory and renders QCD a highly nonlinear ﬁeld theory.

5 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

2.2. Running Coupling Constant, Asymptotic Freedom, Conﬁnement

QCD belongs to the class of so-called “renormalizable” quantum field theories. These are theories in which the divergences occuring at all orders in perturbation theory (via loop diagrams) can be removed by a redefinition of the parameters and the fields. This procedure is called renormalization. The renormalizability of non-abelian gauge theories was first proven by ’t Hooft and Veltman [tHV72] using the technique of dimensional regularization. In this regularization scheme the divergent loop integrals are evaluated in D = 4 − 2 space-time dimensions, which allows to extract the singularities as poles in . Because the coupling constant is dimensionless the change to D dimensions requires the modification

g → g µ , (2.10) where µ is a mass scale. The theory can then be made ﬁnite by reexpressing all results according to the renormalization prescription

g0 = Zg g µ , m0 = Zm m , (2.11) µ p µ p A0 = ZA A , q0 = Zq q . (2.12) Here, a subscript 0 refers to the unrenormalized (bare) quantities, and no subscript to the renormalized quantities. Zg, Zm, ZA and Zq are called renormalization constants; they are chosen such that, at a given order in perturbation theory, the divergent terms get cancelled. Thus, they are themselves divergent quantities. All bare quantities are independent of µ, whereas the renormalized parameters and fields, as well as the renormalization constans, depend explicitely on µ. In this way, renormalization leads to finite results in perturbation theory, but gives all results a µ-dependence. From the µ-independence of g0 and m0 one can derive the renormalization group equations (RGE) for g(µ) and m(µ): dg 1 dZ = − g + β(g) , β(g) = −g g , (2.13) d ln µ Zg d ln µ dm 1 dZ = −γ(g) m , γ(g) = m . (2.14) d ln µ Zm d ln µ Here, β(g) and γ(g) are called renormalization group functions. In particular, β(g) governs the µ-dependence of the coupling constant g(µ), or equivalently that of the QCD fine- 2 structure constant αs = g /4π. At one-loop level it is given by

g3 11N − 2N β(g) = −β , β = c f , (2.15) 0 16π2 0 3 where Nc is the number of colors and Nf denotes the number of ﬂavors that have to be considered at a given energy scale. Introducing a scale parameter µ0, this leads to

dg2 g4 = − (2.16) µ2 β0 2 . d ln 2 16π µ0

6 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

This equation can now be integrated, yielding the leading-order result of the running QCD coupling, 4π 16π2 α (µ) = ,Λ2 := µ2 exp − . (2.17) s 2 QCD 0 2 µ β0g(µ0) β0 ln 2 ΛQCD This equation defines the value of the QCD coupling at a given energy scale µ, based on its value at some fixed point µ0 which has to be adjusted to experimental data. For Nf = 6 and Nc = 3 the coefficient β0 > 0 is positive, and consequently the running coupling decreases logarithmically as energy increases, or distance decreases. This property, asymptotic freedom, was first derived by Gross and Wilczek [GW73a, GW73b] and Politzer [Pol73]. Asymptotic freedom is a peculiar property of non-abelian gauge theories; it entails that at high enough energies quarks behave like weakly interacting point particles.

The decrease of αs(µ) at high energies contrasts with its behaviour at large distances; in the case of momenta transfers Q . 1GeV, or distances & 1 fm, the coupling becomes strong (i.e. αs ∼ 1) and perturbation theory breaks down. This is the regime of nonperturbative QCD, whose noticeable feature is color confinement - quarks and gluons are confined into color-singlet bound-states. These bound-states are called hadrons; they appear mainly in two species 1: bound-states of quark-antiquark (qq¯) pairs with compen- sating color and anticolor charge (mesons), and baryons, bound-states of three quarks with complementary colors (red, blue and green). In both cases the resulting state is a color-singlet. Confinement implies that quarks and gluons cannot exist as asymptotic states, and indeed, no experimental efforts so far have lead to the observation of isolated quarks [B+12]. The absence of free quarks in experiments can be explained with the following picture: as one tries to seperate a pair of quarks from each other by increasing the distance between them the gluonic interaction effectively forms narrow strings of color charge between the two quarks, which tend to hold them together, and at a certain point break, leading to the formation of quark-antiquark pairs. The original two quarks and the qq¯ pairs then again form hadrons. In particle accelerators this phenomenon - hadronization - is seen in the form of so-called jets: large numbers of baryons and mesons moving in the same direction. Since it is an intrinsically nonperturbative feature of QCD, confinement cannot be explained in terms of the RGE-increase of αs(µ) for decreasing energy scales µ. Actual calculations showing the confinement of quarks where first done by Wilson [Wil74] in a latte formulation of QCD 2. Shortly thereafter, Mandelstam, ’t Hooft and Nambu inde- pendently put forward the idea of a dual Meissner effect as the underlying mechanism 1 The possibility of other bound-states has been contemplated (cf. [KZ07]): e.g. glueballs (bound- states with no constituent quarks), pentaquarks and tetramesons. Exotic (non-qq¯) mesons have been observed [B+12], whereas the existence of other exotic hadrons is still hypothetical. 2 Over time, this approach, Lattice Quantum Chromodynamics (LQCD), has developped into a discipline of its own. Benefitting from the continuous increase of computer power, the lattice spacing, which serves as a natural UV-cutoff in LQCD, could be taken small enough to fall within the perturbative region, such that the lattice formulation constitutes a good approximation of the continuum theory. A crucial feature of confinement is the emergence of a distance-independent force between two color carriers. This has been measured in LQCD, i.e. the effective potential between (static) quarks is found to exhibit a linear growth V (r) = Kr at large distance r ≥ 0.25 fm (cf. [Sim96]).

7 2. From Quantum Chromodynamics to Chiral Effective Field Theory for color confinement by postulating that the low-energy vacuum is in a chromomagnetic Higgs phase (cf. [Nam74], [tH78], [Man75]). This idea is inspired by the phenomenon called the Meissner effect in type-II superconductors, which leads the formation of magnetic flux tubes 3, implying a constant force between magnetic poles (cf. Figure 2.2). As in QCD the confined objects are color charges, the flux tubes 4 must be chromoelec- tric rather than chromomagnetic (i.e. the low-energy vacuum is a superinsulator of color rather than superconductor of color). Qualitative evidence for this mechanism of confinement comes from investigations of the large Nc approximation of QCD [tH74a], and from the study of QCD-type theories in lower space-time dimensions [tH74b]. Later in the seventies, Polyakov [Pol77] showed that also instantons can lead to confinement. These ideas have been developed further, and also other confinement schemes have been proposed (an early review is given by Mandelstam [Man80]), yet up to now no analytic proof exists that QCD is indeed a confining theory (cf. [JW00]).

S N

Figure 2.2.: Illustration of a ﬂux tube between two magnetic monopoles in a type-II superconductor.

With the color charge being permanently confined at large distance the effective interaction between hadrons has to be a residual color interaction, similar to the van der Waals force between neutral molecules, which emerges as a large scale phenomenon from QED. A description of these interactions in terms of quarks and gluons would be ex- tremely complicated and not very practical. Instead, an effective field theory that uses more appropriate degrees of freedom is needed.

2.3. Global Low-Energy Symmetries and Hadron Spectrum

In addition to being invariant under (local) gauge transformations, and the obvious invariance under Poincar´eand CPT-transformations, QCD exhibits a number of (approximate) global symmetries. As the quark-gluon coupling grows with increasing distances, the masses of the two lightest quarks, i.e. mu = 5.1 ± 0.9 MeV and md = 9.3 ± 1.4 MeV at µ = 1 GeV (cf. [Leu96]), become less important (for comparison: the nucleon mass is 5 MN ' 938.918 MeV ). It is therefore of interest to consider the quark sector of the QCD

3 This effect was first predicted by Abrikosov [Abr57]. These flux tubes are therefore also known as Abrikosov vortices. 4 Or, more precise, the non-abelian analogue of the flux tubes in a superconductor associated with the abelian QED. 5 The significant higher mass of all hadrons compared to mass of their constituents demonstrates two noteworthy aspects of the strong interaction: Firstly, that almost all of their mass arises from the gluonic interaction; and secondly the presence of positive binding energy, which is a distinctive feature of a confining theory.

8 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

Lagrangian, restricted to up and down quarks, in the limit of vanishing quark masses:

X µ µ LQCD = (¯qf,L i γ Dµ qf,L +q ¯f,R i γ Dµ qf,R) , (2.18) f∈{u,d}, mf=0 f=u,d

1 where the left- and right-handed components of the quark ﬁelds, qL/R = 2 (1 ∓ γ5) q, have been separated. This Lagrangian has a global U(2)L × U(2)R symmetry in ﬂavor space, which can be decomposed as

U(2)L × U(2)R = U(1)L × U(1)R × SU(2)L × SU(2)R . (2.19)

Here, a subscript L means that quark ﬁelds of left-handed chirality transform under the fundamental representation of the group, while right-handed ﬁelds are invariant, and analogously for subscripts R. For example, using the Weyl-representation of Dirac spinors, a transformation under SU(2)L is given by:

τ q SU(2)L exp iΘ 0 q q = L −−−−→ 2 L , (2.20) qR 0 1 qR where τ = (τ1, τ2, τ3) are the Pauli matrices, and Θ = (Θ1,Θ2,Θ3) three angles. The abelian part of U(2)L × U(2)R has the alternative decomposition U(1)A × U(1)V . The subscript V denotes a vector symmetry, i.e. a symmetry that does not distinguish between the left- and right-handed components of the quark field, while the subscript A labels an axial symmetry, which treats fields with different chirality in a different way:

U(1)V U(1)A q −−−→ exp (iΘ) q , q −−−→ exp (iΘγ5) q . (2.21)

The group

Gχ := SU(2)L × SU(2)R (2.22) is generally called the chiral symmetry group. A transformation under its diagonal subgroup SU(2)V is given by:

SU(2) τ q −−−−→V exp iΘ q . (2.23) 2

An example of a transformation under an element g ∈ Gχ ∧ g∈ / SU(2)V is given by:

gγ τ q −−→5 exp iΘ γ q . (2.24) 2 5 Note that the class of transformations deﬁned in Eq. (2.24) are not closed under the group operation. In general, if we write Gχ = SU(2)V × SU(2)A, then SU(2)A does not constitute a group because SU(2)V is not a normal subgroup of Gχ.

It is easy to see that U(1)V is a symmetry of LQCD even for nonvanishing quark masses. Since mu and md are in fact nonzero, U(1)A and chiral symmetry Gχ are only approximate symmetries of the full QCD Lagranian. Regarding the low-energy regime they are very important, as established in the beginning of this section. However, even if the Lagrangian

9 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory and therefore the classical equations of motions possess these approximate symmetries, they can be broken in the quantum theory.

To begin with, U(1)V is also a symmetry in the quantum theory. It leads to the conservation of the baryon number B 6. The axial symmetry U(1)A however is not a symmetry of QCD even for mu = md = 0. It is violated by quantum corrections, i.e. the well-known triangle diagrams, which contribute to the pion decays π0 → γγ and π0 → e+e−γ. This property was ﬁrst derived by Adler, Bell and Jackiw [Adl69, BJ69] and is generally referred to as the axial anomaly.

SU(2)V (often called ﬂavor symmetry) is an exact symmetry of LQCD also for nonvanishing quark masses if one assumes mu and md to be equal, whereas the full SU(2)L ×SU(2)R is broken by mu/d 6= 0 (cf. [Hel10] pp. 12-13). In the low-energy regime of QCD the lightest quark masses are negligible (the neglect of their masses is called the chiral limit of QCD). But is SU(2)L × SU(2)R really a symmetry of the low-energy QCD spectrum? To answer this question, we consider the Noether currents associated with the groups SU(2)L and SU(2)R: τ a ja =q ¯ (x)γ q (x) , ∂µja = 0 , (2.25) L,µ L µ 2 L L,µ τ a ja =q ¯ (x)γ q (x) , ∂µja = 0 . (2.26) R,µ R µ 2 R R,µ It is now convenient to introduce the vector and axial vector currents: τ a ja = ja + ja =q ¯(x)γ q(x) , ∂µja = 0 , (2.27) V,µ R,µ L,µ µ 2 V,µ τ a ja = ja − ja =q ¯(x)γ γ q(x) , ∂µja = 0 . (2.28) A,µ R,µ L,µ µ 5 2 A,µ

If Gχ were a symmetry not only of the Lagrangian, but also of the vacuum state |0i, then a a the corresponding Noether charges QV and QA would annihilate the vacuum: Z τ a Qa |0i = 0 ,Qa = d3x q†(x) q(x) , (2.29) V V 2 Z τ a Qa |0i = 0 ,Qa = d3x q†(x)γ q(x) . (2.30) A A 5 2 This is known as the Wigner-Weyl realization of chiral symmetry with an invariant vacuum. The realization given by Eqs. (2.29) and (2.30) would have certain consequences for a a the hadron spectrum. Because QV and QA have opposite parity and commute with the + QCD Hamiltonian HQCD, for every positive-parity eigenstate |Φ i of HQCD with energy a + E one could construct a state |αi = QA |Φ i with the same energy but opposite parity:

a + a + HQCD |αi = HQCDQA Φ = QAHQCD Φ = E |αi , (2.31) a −1 + P |αi = PQAP P Φ = − |αi . (2.32) | {z } a | {z+ } −QA +|Φ i

6 In the standard model U(1)V is broken by nonperturbative eﬀects related to static solutions of the electroweak equations of motions (so-called sphalerons, cf. [KRS85]).

10 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

Consequently, hadrons should organize themselves into parity doublets. Such a pattern is, however, not observed in nature; the pseudoscalar π-meson (mπ ' 140 MeV) for example has no scalar partner of equal mass. Hence, the Wigner-Weyl realization cannot be correct. a Since the flavor symmetry SU(2)V is observed, it follows that QA |0i= 6 0. This is called the Nambu-Goldstone realization of chiral symmetry. It implies that the chiral symmetry group Gχ is broken down to its subgroup SU(2)V by the nonperturbative QCD vacuum - an effect commonly referred to as spontaneous chiral symmetry breaking (SChSB). This has profound consequences on our effective description of the nuclear interactions.

2.4. Spontaneous Chiral Symmetry Breaking

With chiral symmetry being spontaneously broken the Goldstone theorem [Gol61, GSW62] comes into play:

If there is a continuous symmetry group G (with generators Qi, i = 1, ..., NG) under which the Lagrangian is invariant, but the vacuum |0i is invariant only under a subgroup H ⊆ G (with generators Qi, i = 1, ..., NH ), then there must exist NG − NH massless states with zero spin. These Nambu-Goldstone bosons then carry the quantum numbers of the “broken” charges a Qi, i = NH + 1, ..., NG. In our case, because QA are axial charges, they are the three lightest pseudoscalar mesons - the pions π+, π−, π0. Because chiral symmetry is also explicitely broken by the nonvanishing masses of the up- and down-quarks, pions have a nonzero, but relatively small mass 7 compared to other hadrons. Furthermore, at small momenta they are weakly interacting particles; this can be seen by considering the action n of HQCD on a state |π i with n pions:

n a1 an a1 an HQCD |π i ∼ HQCD (QA ··· QA ) |0i = (QA ··· QA ) HQCD |0i = 0 . (2.33)

Thus, in the case of vanishing momenta, the n-pion state is energetically degenerate with the vacuum |0i. This is due to spontaneous chiral symmetry breaking being accompanied by a qualitative rearrangement of the vacuum - it is now populated by scalar quark- antiquark pair ﬂuctuations. This means that the quark-antiquark correlatorqq ¯ (scalar quark density) has a nonvanishing vacuum expectation value, the so-called chiral or quark condensate:

h0 | qq¯ | 0i ≡ hqq¯ i = hq¯uqui + hq¯dqdi , (2.34) with up-quark ﬁelds qu(x) and down-quark ﬁelds qd(x). Because the scalar quark density mixes left- and right-handed components

qq¯ =q ¯LqR +q ¯RqL , (2.35) a nonvanishing chiral condensate signals the breaking of chiral symmetry. It therefore represents an order parameter of SChSB in the sense that hqq¯ i= 6 0 signiﬁes the phase with broken symmetry, and hqq¯ i = 0 amounts to the restoration of chiral symmetry (cf.

7 For this reason the pions are often more correctly referred to as pseudo Nambu-Goldstone bosons. We do not pay attention to this nicety and disregard the additional preﬁx.

11 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

[TW01] 147-151).

Note that because the chiral condensate also breaks U(1)A there would be another Nambo-Goldstone boson, the η0-meson. This isoscalar meson has however a signiﬁcantly larger mass, mη0 ' 958 MeV, than the pions; this is explained by the breaking of U(1)A due to the axial anomaly, which prevents it from being a Nambo-Goldstone boson.

There exists an argument by Casher and Banks [Cas79, BC80] that states that if quarks are confined, then chiral symmetry must be spontaneously broken. This does not apply for the converse, i.e. SChSB implying confinement. The exact relation between confinement and SChSB is however not yet fully understood.

2.5. Eﬀective Field Theory Program

To summarize, we have learned that pions are the Nambu-Goldstone bosons associated with SChSB, and that confinement implies that hadrons are the relevant degrees of freedom of low-energy QCD. These insights, together with the weakness of pionic interactions, are the fundaments for the construction of an effective field theory of low-energy QCD. The effective field theory of the nuclear interactions in particular must be a theory of nucleons and pions. The strategy for the construction of an effective field theory (EFT) was first formulated by Weinberg in [Wei79]. In the context of an EFT of low-energy QCD it consists of the following steps:

• Identify the soft and hard scales, and the appropriate degrees of freedom of low- energy QCD (see below).

• Identify the relevant symmetries of low-energy QCD and investigate if and how they are broken (cf. section 2.3).

• Construct the most general Lagrangian consistent with these symmetries and symmetry breaking patterns → sections 2.6 and 2.7.

• Design an organizational scheme that can distinguish between more and less important terms - a low-momentum expansion → section 3.1.

• Using this expansion, calculate Feynman diagrams for a given problem under consideration of the desired accuracy → sections 3.2 and 3.3.

The hard scale of nuclear physics is ﬁxed by the scale of chiral-symmetry breaking Λχ ' 1 GeV. The soft scale is then given by a small momentum below Λχ, typically of the order of magnitude of the pion mass Q ∼ mπ ' 140 MeV. We have already identiﬁed the relevant degrees of freedom, pions and nucleons, and we have seen that for low momenta pionic interactions are weak. At the energy regime set by the soft scale the internal structure of nucleons and pions is not resolved; we can treat them as point particles.

The ﬁrst two steps of the EFT program are therefore achieved. The next step consists

12 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

of the construction of the effective Lagrangian Leff. Chiral symmetry SU(2)L × SU(2)R and its spontaneous breaking dictate the form of Leff. Hence, before constructing Leff, we must find out how pion and nucleon fields transfrom under these symmetries.

2.6. Realization of Chiral Symmetry

Chiral transformations of the pion ﬁelds π+, π−, π0 cannot be realized by a (linear) representation of the chiral group Gχ. This follows from the fact that the Lie algebra of Gχ = SU(2)L × SU(2)R is isomorphic to that of SO(4), whose smallest nontrivial representation is four-dimensional. We have, however, only three coordinates at our disposal (the triplet of pion ﬁelds).

Instead of a representation we therefore consider a general realization of Gχ of the form

g∈Gχ π −−−→ φg(π) , φg(0) = 0 ⇔ g ∈ SU(2)V . (2.36)

As shown in [CWZ69] the condition that only the subgroup SU(2)V leaves the origin 0 invariant guarantees that the group realization becomes linear when restricted to SU(2)V (called the stability group in this context). Under the full SU(2)L × SU(2)R the pion fields transfrom nonlinearly. This is in accordance with the fact that Gχ is broken by the chiral condensate. A transformation under g ∈ Gχ ∧ g∈ / SU(2)V does not constitute a symmetry transformation; instead it corresponds to an excitation of the vacuum. As pions are degenerate with the vacuum, they have to transform in a complicated nonlinear way. In section 2.6.1 we show how such a realization of Gχ on pion fields can be constructed. In section 2.6.2 we then discuss the transformation behaviour of the nucleon field.

2.6.1. Transformation Properties of Pion Fields General Discussion We consider a physical system described by an effective Lagrangian that is invariant under a symmetry group G, while the vacuum of the system is invariant only under the subgroup H. The Nambu-Goldstone bosons associated with this spontaneous symmetry breaking, we call them pions, are described by a multi-component vector π, where each entry πi is a differentiable real function on Minkowski space M. Let Mπ be the vector space of these fields,

3 Mπ ≡ {π : M → R πi : M → R (diﬀerentiable)} . (2.37)

A transformation of these ﬁelds under the symmetry G is deﬁned by a mapping φ which uniquely associates with each pair (g, π) ∈ G × Mπ an element φg(π) ∈ Mπ with the properties

• φe(π) = π ∀ π ∈ Mπ , e ≡ identity element of G .

• φg1 ◦ φg2 (π) = φg1g2 (π) ∀ g1, g2 ∈ G, ∀ π ∈ Mπ .

13 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

This mapping deﬁnes an operation of the group G on Mπ, which will however, because of the missing linearity condition φg(λπ) = λφg(π), not constitute a representation of G, but a realization of G on Mπ. As mentioned above, we require in addition that the origin be invariant under transformations restricted to the subgroup H:

φh(0) = 0 ∀ h ∈ H. (2.38)

Following [Sch03] we now show that there exists an isomorphism ξ between Mπ and the set of all left cosets {gH g ∈ G} (also known as the quotient space G/H),

ξ : Mπ → G/H , π 7→ gH , (2.39) where for one element g of G, the set gH = {gh h ∈ H} deﬁnes the left coset of g, one element of G/H. From Eq. (2.38) it follows that

φgh(0) = φg ◦ φh(0) = φg(φh(0)) = φg(0) ∀ g ∈ G , h ∈ H. (2.40)

Moreover, the mapping φ is injective with respect to the cosets, which can be seen by 0 0 considering two elements g and g of G, with g ∈/ gH. Assuming that φg(0) = φg0 (0) leads to

0 = φe(0) = φg−1g(0) = φg−1 (φg(0)) = φg−1 (φg0 (0)) = φg−1g0 (0) . (2.41)

By Eq. (2.38) this implies that g−1g0 ∈ H, or equivalently g0 ∈ gH, which contradicts the assumption. This shows that the mapping

G/H → Mπ , gh˜ 7→ φgh˜ (0) ∀ g˜ ∈ G , h ∈ H (2.42) is bijective, and we can for each π ∈ Mπ identify a corresponding cosetgH ˜ by the relation φgh˜ (0) = π. The isomorphism between pion ﬁelds and left cosets is thus given by:

ξ(π) =gH ˜ ⇔ φgh˜ (0) = π . (2.43)

The transformation behaviour of the pion ﬁelds is now uniquely determined (up to an appropriate choice of parametrization) by the tranformation of its coset representation, i.e.

φ φ g g 0 Mπ / Mπ π / π . (2.44) _ _ ξ ξ ξ ξ g g G/H / G/H gH˜ / ggH˜

Application to Chiral Symmetry In the case of the chiral symmetry group we have

Gχ = SU(2)L × SU(2)R = {(L, R) L ∈ SU(2) ,R ∈ SU(2)} , (2.45)

H = SU(2)V = {(V,V ) V ∈ SU(2)} , (2.46) where (X,Y ) speciﬁes the transformation behaviour of the left-handed, qL → XqL, and the right-handed quark ﬁelds, qR → Y qR.

14 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

The left coset of an elementg ˜ ∈ Gχ,gH ˜ = {(LV,˜ RV˜ ) V ∈ H}, can be uniquely characterized through the SU(2)-matrix U = R˜L˜†: gH˜ = (LV,˜ RV˜ ) = (LV,˜ R˜L˜†LV˜ ) = (1, R˜L˜†)(LV,˜ LV˜ ) ⇒ gH˜ = (1,U)H. (2.47) | {z } ∈H The transformation of U under g = (L, R) is given by multiplication in the left coset ggH˜ = (L, RU)H = (1,RUL†)(L, L)H = (1,RUL†)H. (2.48)

g Hence, U −→ U 0 = RUL†, and it suﬃces to specify an isomorphic mapping of the pion ﬁelds to the space of unitary matrices U with determinant one. An often used parametrization is iτ · π U = exp , (2.49) fπ where τ = (τ1, τ2, τ3) are the Pauli matrices, fπ is the pion decay constant, and

 √1 (π+ + π−) 2 i π = √i (π+ − π−) (2.50)  2 i  π0 is the vector which collects the pion ﬁelds. The group operation φ is then given by

φ φ g g 0 Mπ / Mπ π / π , (2.51) _ _ ξ ξ ξ ξ g g † MU / MU U / RUL where MU is the space of U-matrices (which is not a vector space):

MU = {U : M → SU(2) x 7→ U(x)} . (2.52)

When we restrict ourselves to transformations under the subgroup SU(2)V the pions should transform linearly. This can be checked by expanding U in a power series:

∞ α ∞ α X i τ · π X i V τ · π V † VUV † = V V † = . (2.53) f f α=0 π α=0 π

† Hence it is τ · π → V (τ · π) V , which is a linear representation of SU(2)V . That pions transform nonlinearly under g ∈ Gχ ∧ g∈ / SU(2)V can for example be seen by considering the transformation given by Eq. (2.23), which, using the Weyl- representation of Dirac spinors, can be decomposed as

gγ τ gγ τ q −−→5 exp −iΘ q ≡ A q , q −−→5 exp iΘ q ≡ A q . (2.54) L 2 L L L R 2 R R R † Hence, it is U → ALUAR under that particular transformation, and by expanding U as in Eq. (2.53) we do not obtain a linear transformation of pion ﬁelds, but instead a nonlinear one.

15 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

2.6.2. Transformation Properties of Nucleon Field There are many possibilies to realize chiral symmetry for the nucleon field (cf. [Geo09] 93-96). They all lead to the same physics. One particular convenient choice is given by √ √ Ψ → K(L, R, U)Ψ = LU †R†R UΨ, (2.55) where Ψ = (p, n) is the nucleon field, with isospin components given by the proton Dirac field p and the neutron Dirac field n. Because K depends on the pion-field matrix U(π(x)) the above equation defines a local transformation law.

To see more explicitly how Ψ transforms under SU(2)L × SU(2)R we consider an infinitesimal transformation K = exp(iγτ ), and use the parametrizations L = exp(i(α − β)τ ) and R = exp(i(α + β)τ ), where α and β are infinitesimal. Using the Baker- Campbell-Hausdorff (BCH) formula 1 1 1 h i ex ey = ez , z = x + y + [x, y] − [x, y], x − y − [x, y], x, y + ..., (2.56) 2 12 24 we find the following relation:

β × π (β × π) × π × π π4 γ = α − + 3 + O( ) . (2.57) 2fπ 6fπ

This shows that for a general transformation under SU(2)L × SU(2)R the transformed nucleon field is a nonlinear function of pion fields. In contrast, for a transformation under SU(2)V with β = 0 we obtain K = V (up to 3 order π , for infinitesimal transformations).√ This can be shown to√ be true also in the † † † † general case by observing that ζ1 := VU V ∈ SU(2) and ζ2 := V U V ∈ SU(2) both 2 † † lead to the same expression, ζ1/2 := VU V ∈ SU(2), when squared. The square root of an SU(2)-matrix is unique up to a sign, therefore it is ζ1 = ±ζ2. Using the parametrization V = exp(iατ ) one can show by explicit calculation that it is indeed ζ1 = ζ2. We therefore have √ √ K(V,V,U) = V U †V †V U = V. (2.58)

Hence, Ψ transforms linearly as an isospin doublet under SU(2)V . Flavor symmetry between up- and down-quarks has become isospin symmetry between protons and neutrons.

2.7. Chiral Eﬀective Lagrangian

The eﬀective Lagrangian can be decomposed as

Leff = Lππ + LπN + LNN , (2.59) where the first part specifies the dynamics of pions in the abscence of nucleons, the second part describes nucleon-pion interactions, and the third part consists of nucleon contact- interactions. Leff contains all possible terms which are consistent with the symmetries of low-energy QCD. As such, it involves an unlimited number of free parameters, which are

16 2. From Quantum Chromodynamics to Chiral Effective Field Theory called low-energy constants (LECs). Regarding the derivation of Leff we keep the presentation brief. More details can for example be found in [Sch03], [BKM95] and [Eck95]. The freedom of choice regarding the parametrization of U(x) by pion fields is related to a certain arbitrariness of several coefficients in the pion-field expanded Lagrangians 8. This effects only off-shell amplitudes, whereas physical on-shell amplitudes are invariant under a change of parametrization (cf. [ME11]). We present the Lagrangians that follows from our choice of parametrization, Eq. (2.49).

2.7.1. Relativistic Pion and Pion-Nucleon Lagrangians Pion Sector Because pionic interactions vanish for zero momentum we expect them to come in powers of ∂µU, and only even powers are allowed because of Lorentz invariance:

(2) (4) Lππ = Lππ + Lππ + ... (2.60) Here, the superscript denotes the number of derivatives or pion mass insertions - the so-called chiral dimension d. The leading order (LO) ππ Lagrangian is given by [GL84]

f 4 L(2) = π tr ∂ U∂µU † + m2 (U + U †) , (2.61) ππ 4 µ π where the ﬁrst term constitutes the most general Gχ invariant Lagrangian with minimal number of derivatives, and the second term (the mass term) breaks Gχ explicitely down to SU(2)V . Expanding U in a power series and leaving out constant terms we get 1 1 L(2) = ∂ π · ∂µπ − m2 π2 ππ 2 µ 2 π 1 1 1 π π π µπ π2 π µπ 2 π4 π6 − ( · ∂µ )( · ∂ ) − ∂µ · ∂ + 2 mπ + O( ) . (2.62) 6fπ 6fπ 24fπ

Pion-Nucleon Sector

We have seen in section 2.6.2 that nucleons transform locally under SU(2)L × SU(2)R. Hence, we need a chirally covariant derivative: i π π π4 Dµ = ∂µ + Γµ ,Γµ = 2 τ · ( × ∂µ ) + O( ) , (2.63) 4fπ where Γµ is the so-called chiral connection. The relativistic πN Lagrangian (at leading order) is then given by [GSS88]: g L(1) = Ψ¯ iγµD − M + A γµγ u Ψ, (2.64) πN µ N 2 5 µ where MN is the nucleon mass (in the chiral limit), Ψ is the Dirac ﬁeld representing the nucleon, and gA is the axial-vector strength, which can be measured in neutron β-decay, 8 i.e. the coeﬃcients corresponding to multi-pion vertices in Eq. (2.62), Eq. (2.66), and in terms of higher order.

17 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

− n → p e νe, gA ' 1.26 [BKM95]. In addition to the chiral derivative and the mass term, the above Lagrangian also includes a coupling term which involves the so-called vielbein uµ, which is given by 1 1 1 π π π π π2 π π4 uµ = − τ · ∂µ + 3 (τ · )( · ∂µ ) + 3 (τ · ∂µ ) + O( ) . (2.65) fπ 6fπ 6fπ The explicit form of Eq. (2.64) is then

1 g (1) ¯ µ µ π π A µ π LπN = Ψ iγ ∂µ − MN − 2 γ τ · ( × ∂µ ) − γ γ5τ · ∂µ + ... Ψ. (2.66) 4fπ 2fπ

The term proportional to gA/2fπ is the pseudo-vector coupling of one pion to the nucleon, 2 and the quadratic term proportional to 1/4fπ is known as the Weinberg-Tomozawa coupling. At chiral dimension two the pion-nucleon Lagrangian is given by

4 (2) X ¯ (2) LπN = ci ΨOi Ψ, (2.67) i=1 where ci are the ﬁrst LECs; they have to be determined by experiment. The various (2) operators Oi are such that they represent all terms at chiral dimension two that are consistent with chiral symmetry and Lorentz invariance. For the explicit form of these operators we refer to [ME11] and references therein.

2.7.2. Heavy Baryon Formalism The relativistic treatment of nucleons leads to certain problems (cf. [Sch03] p. 190), which can be avoided by treating nucleons as heavy static sources (“extreme non-relativistic limit”). We begin by reparametrizing the nucleon four-momentum:

µ µ µ p = MN v + l , (2.68)

µ µ µ where v is the relativistic four-velocity satisfying v vµ = 1, and l a small residual µ momentum, i.e. v lµ MN. We define the projection operators 1 ± γ vµ P ± = µ ,P + + P − = 1 , (2.69) v 2 v v where P + projects on the large component of the nucleon field, which can be seen by considering its Fourier expansion X Z Ψ(x) = d3p u(x; ~p,σ, τ) a†(~p,σ, τ) , l = 1, 2, 3, 4 , (2.70) στ where σ is the spin projection quantum number along the z-axis , τ the isospin projection quantum number, and a†(~p,σ, τ) the operator which creates a nucleon with momentum ~p and quantum numbers σ and τ. The coefficient u(x; ~p,σ, τ) is given by the plane wave solution of the free Dirac equation

µ (iγ ∂µ − MN)u(x) = 0 . (2.71)

18 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

In the Dirac representation it is given by (we omit the isospin part): r µ p0 + MN χs −ip xµ u(x) = ~σ·~p e , (2.72) χs 2MN p0+MN

p 2 9 where p0 = ~p + MN, ~σ = (σ1, σ2, σ3) are the Pauli matrices , and χs is the Pauli- 1 + − spinor for spin- 2 . P projects on the large upper component of u(x), whereas P on the lower component ∝ ~σ · ~p which is small in the static limit. Deﬁning the velocity dependent ﬁelds

µ µ i MN v xµ + i MN v xµ − N = e Pv Ψ , h = e Pv Ψ (2.73) we can write the nucleon ﬁeld as

µ Ψ = e−i MN v xµ (N + h) . (2.74)

We want to rewrite the Lagrangian in Eq. (2.66) using this decomposition. The Euler- (1) Lagrange equations corresponding to LπN allow to express the small component h in −1 terms of N. These corrections enter at second order and are suppressed by a factor MN . ˘(1) Hence, the heavy baryon projected (indicated by the breve) ﬁrst order Lagrangian LπN is given by Eq. (2.66) with the substitution

µ Ψ → e−i MN v xµ N. (2.75)

Assuming that vµ = (1, 0, 0, 0) this leads to (cf. [ME11] pp. 13-14)

1 g L˘(1) =N¯ i∂ − τ · (π × ∂ π) − A τ · (~σ · ∇~ )π πN 0 4f 2 0 2f π π (2.76) g h i h i A π π ~ π π2 ~ π − 3 (τ · ) · (~σ · ∇) + τ · (~σ · ∇) + ... N, 12fπ where the ellipsis stand for terms involving four or more pion ﬁelds.

2.7.3. Nucleon Contact Terms Nucleon contact terms are needed to compensate the divergences arising from loop- diagrams, and to parametrize the unresolved short-distance dynamics of the nuclear forces. Invariance under parity transformations restricts the nucleon contact interactions to involve only even powers of derivatives: ˘ ˘(0) ˘(2) ˘(4) LNN = LNN + LNN + LNN + ... (2.77)

The lowest order two-nucleon contact Lagrangian is given by 1 1 L˘(0) = − C NN¯ NN¯ − C (N~σN¯ ) · (N~σN¯ ) , (2.78) NN 2 S 2 T

9 We use an alternative notation to distinguish from the isospin Pauli matrices τ in the Lagrangian.

19 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

where CS and CT are further LECs which have to be determimed by ﬁts to experimental NN data. At the next order there are also three-nucleon contact terms, and contact terms with additional pions: h i ˘(2) cD ¯ ¯ ~ 1 ¯ ¯ ¯ LNN = − (NN)N τ · (~σ · ∇)π N − cE(NN)(Nτ N) · (Nτ N) + ..., (2.79) 4fπ 2 where the ellipsis represent further two-nucleon contact interactions. The part propor- tianal to the constant cD gives rise to a vertex involving two in- and outgoing nucleon lines as well as one pion line, while the cE part represents a pure three-nucleon contact interaction. These two terms contribute to the leading order chiral three-nucleon forces (cf. section 3.2).

2.7.4. Organization by Index of Interaction For the construction of nuclear forces it is useful to organize the terms in the effective Lagrangian according to the so-called index of interaction n ∆ ≡ d + − 2 , (2.80) 2 where d is the chiral dimension defined in the beginning of section 2.7.1, and n is the number of nucleon fields. The leading order Lagrangian in this organization scheme reads 1 1 L˘∆=0 = ∂ π · ∂µπ − m2 π2 2 µ 2 π 1 1 1 π π π µπ π2 π µπ 2 π4 + 2 ( · ∂µ )( · ∂ ) − 2 ∂µ · ∂ + 2 mπ 6fπ 6fπ 24fπ g 1 ¯ A ~ π π π + N i∂0 − τ · (~σ · ∇) − 2 τ · ( × ∂0 ) N 2fπ 4fπ g h i g h i ¯ A π π ~ π A π2 ~ π − N 3 (τ · ) · (~σ · ∇) + 3 τ · (~σ · ∇) N 12fπ 12fπ 1 1 − C NN¯ NN¯ − C (N~σN¯ ) · (N~σN¯ ) + ... (2.81) 2 S 2 T To clarify some notational aspects: The Pauli-matrix vector ~σ is always multiplied with another quantity that has an explicit vector arrow, and the Pauli-matrices in the vector ~σ act on the spinor part of the “heavy” nucleon field N, which is represented by the two upper components of the nucleon Dirac field Ψ; hence, they are associated with the 1 spin-operator ~s = 2~σ. The product of a boldfaced τ Pauli-matrix vector with a pion field vector π leads to

1 + − i + − 0 τ · π = τ1 √ (π + iπ ) + τ2 √ (π − iπ ) + τ3π 2 2 √ + ) 0 = 2(τ−π + τ+π + τ3π . (2.82) 1 Hence, the operators τ± = 2 (τ1 ∓ iτ2) and τ3 are in the isospin-basis of the pion ﬁeld, {τ+, τ−, τ0} = {χ1, χ−1, χ0} (where χs are the Pauli-spinors for spin-1). They act on the isospin components of N = (p, n) in the following way:

τ− |pi = |ni τ− |ni = 0 (2.83)

τ+ |pi = 0 τ+ |ni = |pi (2.84)

τ3 |pi = |pi τ3 |ni = − |ni (2.85)

20 2. From Quantum Chromodynamics to Chiral Eﬀective Field Theory

This guarantees isospin conservation; for instance in a (N → Nπ)-type vertex where the initial nucleon is a proton and the ﬁnal nucleon a neutron, the outgoing pion has to be a π+.

−1 The subleading Lagrangians are (with the MN -corrections from the heavy-baryon approximation included)

~ 2 h i ˘∆=1 ¯ ∇ igA ←− −→ L = N − τ · ~σ · (∇∂0π − ∂0π∇) 2MN 4MNfπ i h←− −→i π ~ π π ~ π − 2 τ · ∇ · ( × ∇ ) − ( × ∇ ) · ∇ N 8MNfπ h 2c g2 1 ¯ 2 1 2 π2 A π π (2.86) + N 4c1mπ − 2 mπ + c2 − 2 (∂0 · ∂0 ) fπ 8MN fπ c 1 1 i 3 π µπ ijk abc i a iπb kπc + 2 (∂µ · ∂ ) − c4 + 2 σ τ (∂ )(∂ ) N fπ 4MN 2fπ cD ¯ ¯h i 1 ¯ ¯ ¯ − (NN)N τ · (~σ · ∇~ )π N − cE(NN)(Nτ N) · (Nτ N) + ..., 4fπ 2

˘∆=2 (4) (3) (2) L = Lππ + LπN + LNN , (2.87)

˘∆=3 (4) L = LNN . (2.88)

The ellipsis represent terms that are irrelevant for the construction of nuclear forces.

At this stage the ﬁrst three steps in the construction of an EFT of nuclear interactions have been accomplished. We still need to relate the terms in these Lagrangians to a low-momentum expansion scheme, and ﬁnd the relevant Feynman diagrams for the problem of our choice, i.e. the determination of nuclear forces. These tasks are tackled in the next chapter.

21 3. From Chiral Lagrangians to Nuclear Potentials

In the previous chapter we arrived at the Lagrangian of ChEFT, Leff, which is constructed in a way that it constitutes the most general Lagrangian consistent with the symmetries of low-energy QCD. ChEFT is a so-called non-renormalizable theory; it involves non- renormalizable (also called irrelevant) operators and is therefore UV-unstable. Moreover, the Lagrangian Leff has infinitely many terms, and gives rise to an unlimited number of interactions between pions and nucleons. This means that the usual renormalization procedure cannot be applied; the theory has to be renormalized order by order (in a given regularization scheme), leading to an unlimited number of counterterms which have to be determined by experiment. However, regarding its applicability in the low-energy regime, the non-renormalizability of ChEFT does not constitute a problem. With the hard scale being set by Λχ (chiral symmetry breaking scale) and the soft scale by Q (small momentum or pion mass) it is suggestive to organize contributions to a given problem in increasing powers of Q/Λχ ν (called the chiral expansion parameter). Then at a given order (Q/Λχ) only a finite number of diagrams contribute. Thus, for calculations in the low-energy regime up to a certain degree of accuracy, ChEFT is a well-controlled theory. Its limited range of applicability and limited accuracy 1 are what makes it an effective theory. In conclusion, we now need to develop a scheme which orders Feynman diagrams in terms of powers of the chiral expansion parameter Q/Λχ. This scheme is presented in section 3.1. It allows to construct nuclear interactions in a systematic way, as shown in section 3.2. Three-nucleon forces emerge naturally in this perturbative scheme - they are a consequence of our low-energy approximation. For calculations in nuclear many- body systems we need to construct potentials from these diagrams. An outline of the construction of the nucleon-nucleon (NN) potential is given in section 3.3. The basic properties of the NN potential are discussed in the final section.

3.1. Chiral Perturbation Theory and Power Counting

The systematic ordering of Feynman diagrams in terms of powers of the chiral expansion parameter Q/Λχ is called chiral perturbation theory (ChPT). To determine the power ν of a given diagram, we have to consider its diﬀerent pieces. From the rules of covariant perturbation theory it follows that

• an intermediate nucleon line counts as Q−1,

1 To be precise, the property of limited accuracy is not peculiar to eﬀective ﬁeld theories. Since the expansion in terms of powers of the coupling constant g constitutes an asymptotic series with formally zero radius of convergence, also renormalizable theories have limited accuracy (cf. [FHS12]).

22 3. From Chiral Lagrangians to Nuclear Potentials

• an intermediate pion line counts as Q−2, • each derivative in any vertex counts as Q, • each four-momentum integration counts as Q4. Using this dimensional analysis and some topological identities one can derive the following formula for the chiral power of a diagram involving A nucleons (cf. [ME11] pp. 16-18): X ν = −2 + 2A − 2C + 2L + ∆i . (3.1) i In this formula, C denotes the number of separately connected pieces, and L the number of loops in the diagram. ∆i is the interaction index defined by Eq. (2.80), and i refers to the different vertices in a diagram. For the NN interaction (C = 1, A = 2) this formula collapses to the simpler expression X ν = 2L + ∆i , (3.2) i while for the 3N interaction (C = 1, A = 3) we get X ν = 2 + 2L + ∆i . (3.3) i The general formula (3.1) can be also written in a more concise form: X 3 ν = −2 + κ , κ = d − n + π − 4 , (3.4) i i i 2 i i i where again di is the number of derivatives or pion-mass insertions, ni the number of nucleon fields, and πi the number of pion fields at the vertex i. Up to the additional constant −4 the quantity κi is then equal to the canonical field dimension of the vertex and gives the inverse mass dimension of the corresponding coupling constant. In order for chiral perturbation theory to work, the chiral power of a diagram should increase with every additional vertex. Therefore the effective Lagrangian must contain only non- renormalizable interactions κi > 1, which is guaranteed by the spontaneously broken chiral symmetry [Epe07].

3.2. Hierarchy of Nuclear Forces

With the chiral power counting formulas given above we can now systematically construct diagrams for nuclear inteactions. For a diagram involving a given number of nucleons we start with the same number of in- and out-going nucleon lines and connect these lines in all possible ways allowed by the Lagrangians given by Eqs. (2.81) to (2.88). At a given order in the chiral expansion there is then only a ﬁnite number of diagrams. Up to ν = 4 there exist the following contributions to two-, three- and four-nucleon forces: (0) (2) (3) (4) VNN = VNN + VNN + VNN + VNN + ... (3.5) (3) (4) V3N = V3N + V3N + ... (3.6) (4) V4N = V4N + ... (3.7)

23 3. From Chiral Lagrangians to Nuclear Potentials

NN Force 3N Force 4N Force

LO 0 (Q/Λχ)

NLO 2 (Q/Λχ)

NNLO 3 (Q/Λχ) + ...

N3LO + ... 4 (Q/Λχ) + ... + ...

Figure 3.1.: Hierarchy of nuclear forces in ChPT. Solid lines represent nucleons, and dashed lines pions. Tiny dots, large solid dots, large solid squares and large crossed squares denote vertices with ∆ = 0, 1, 2 and 4, respectively. All diagrams except the planar box diagram (the ﬁfth at NLO) are irreducible diagrams (in the sense that they cannot be separated by cutting only nucleon- lines). The reducible part of the planar box diagram becomes apparent by considering its time-ordered components (cf. Figure 3.3).

(ν) (ν) Here, VNN denotes nucleon-nucleon amplitudes at order ν, V3N three-nucleon amplitudes, and so on. Contributions at order ν = 1 vanish due to parity and time-reversal invariance. Moreover, there are already 3N diagrams at ν = 2, but these cancel each other (cf. [ME11] pp. 18-19).

The diagrams in Figure 3.1 can be used to construct nuclear potentials VNN, V3N and V4N. These potentials are needed for the perturbative approach to the nuclear many-body problem (discussed in chapter4). They are hierarchically ordered (with respect to their appearance at diﬀerent orders in ν) in the following way:

VNN > V3N > V4N . (3.8)

In that way chiral EFT provides a qualitative explanation of the observed hierarchy of nuclear N-body forces. In this thesis we restrict ourselves to two- and three-body interactions, the latter up 3 to order (Q/Λχ) . Thus, we have exactly three diagrams to consider for the three-body interaction. The corresponding amplitudes in momentum space immediately give the

24 3. From Chiral Lagrangians to Nuclear Potentials

potential V3N, i.e.

X (3) V3N = V3N . (3.9) diagrams In the case of the NN interaction there is however a vast number of diagrams to consider.

3.3. Construction of Chiral NN Potential

In this section we outline the construction of the chiral NN potential. In the first subsection we give the amplitudes VNN for (some of) the different NN diagrams. These are then used in the second subsection to define and construct the NN potential VNN, starting from the Bethe-Salpeter equation. Finally, in the third subsection we regularize VNN with a Gaussian regulator function.

3.3.1. Diagrammatic Amplitudes The numerous contributions to the NN interaction can be distinguished as follows:

VNN = Vc.t. + V1π + V2π + V3π ... (3.10) where Vc.t. denotes the contribution from contact diagrams, and Vnπ contributions that come from diagrams that involve the exchange of n pions. In detail, we have

(0) (2) (4) Vc.t. = Vc.t. + Vc.t. + Vc.t. + ... (3.11) (0) (2) (3) V1π = V1π + V1π + V1π + ... (3.12) (2) (3) V2π = V2π + V2π + ... (3.13) (4) V3π = V3π + ... (3.14) where the superscript denotes the order with respect to the chiral expansion parameter.

Contact Interactions The leading order (LO) contact Lagrangian , Eq. (2.78), leads to the following NN contact-potential:

Vc.t. = CS + CT ~σ1 · ~σ2 . (3.15) For the next-to-leading order (NLO) and third order (N3LO) contact interactions we refer to [ME11].

One-Pion Exchange Regarding the one-pion exchange (1PE) contributions there is just one diagram at leading order. It involves twice the pseudo-vector coupling in Eq. (2.81), which leads to the momentum-space amplitude

2 0 gA (~σ1 · ~q )(~σ2 · ~q ) V1π(~p,~p ) = − 2 τ 1 · τ 2 2 2 , (3.16) 4fπ q + mπ

25 3. From Chiral Lagrangians to Nuclear Potentials where ~p and ~p 0 are the ﬁnal and inital momentum of the interacting nucleons in the center-of-mass system (CMS), and ~q = ~p 0 − ~p is the momentum transfer. At NLO the one-pion exchange potential gets renormalized due to several one-loop graphs and counter term insertions.

2 Figure 3.2.: One-pion exchange diagrams at order (Q/Λχ)

The ﬁrst two diagrams at the top of Figure 3.2 renormalize the nucleon propagator and the next two the pion propagator. The four diagrams at the bottom renormalize the pion-nucleon coupling constant. In the one-loop diagrams all vertices are from the leading order Lagrangian L˘∆=0 given in Eq. (2.81), whereas the counter-term insertions stem from L˘∆=2. At NNLO and N3LO further corrections arise; they renormalize various LECs, the 2 pion-mass, and the axial-vector strength gA , but do not generate any pion-nucleon form- factors. Therefore the one-pion exchange contributions maintain the structure given by Eq. (3.16) up to fourth order.

Two-Pion Exchange and Beyond Two-pion exchange (2PE) starts at NLO and involves loop diagrams which have to be regularized. An explicit calculation of the NLO diagrams can be found in [ME11] pp. 63-77. The corresponding momentum-space amplitude in the center-of-mass frame has the general form:

0 V2π(~p,~p ) =VC + τ 1 · τ 2WC

+ (VS + τ 1 · τ 2WS) ~σ1 · ~σ2 + (VLS + τ 1 · τ 2WLS) −iS~ · (~q × K~ )

+ (VT + τ 1 · τ 2WT )(~σ1 · ~q )(~σ2 · ~q ) h i h i + (VσL + τ 1 · τ 2WσL) ~σ1 · (~q × K~ ) ~σ2 · (~q × K~ ) , (3.17)

0 ~ 1 0 where ~q := ~p − ~p is the momentum transfer, K := 2 (~p + ~p ) the average momentum, ~ 1 and S = 2 (~σ1 + ~σ2) is the total spin operator. For on-shell scattering, Vα and Wα, α ∈ {C, S, LS, T, σL}, can be expressed as functions of q = |~q| and K = |K~ |, only (cf. [ME11] p. 22). Three-pion exchange (3PE) starts at N3LO; at this order it involves only vertices from the leading order Lagrangian L˘∆=0 (tiny dots) - these contributions however are negligible (cf. [ME11] p. 26, and references therein).

2 At NLO this correction is known as the Goldberger-Treimann discrepancy; it leads to gA → g˜A = 1.290.

26 3. From Chiral Lagrangians to Nuclear Potentials

3.3.2. Bethe-Salpeter Equation Having obtained the amplitudes of the relevant diagrams we can now proceed to define the NN potential VNN. We start with the covariant Bethe-Salpeter (BS) equation T = W + W GT, (3.18) where T is the invariant T -matrix of the two-nucleon scattering process, W the sum of the amplitudes of all connected irreducible two-nucleon diagrams, and G is (−i) times the relativistic two-nucleon propagator. Note that W involves also off-shell contributions, i.e. diagrams in which at least one of the inital or final nucleons does not satisfy the relativistic mass-energy relation. The nucleon-nucleon-potential VNN can be related to the quantities in the BS-equation via the following three-dimensional reduction:

T = VNN + VNN gT , (3.19)

VNN = W + W (G − g)VNN , (3.20) where g is a covariant three-dimensional propagator which preserves relativistic elastic unitarity. The equivalence of the BS-equation and the above two equations can be easily veriﬁed:

VNN + VNN gT = W + W (G − g)VNN + W gT + W (G − g)VNN gT

= W + W G (VNN + VNN gT ) −W g (VNN + VNN gT ) +W gT . (3.21) | {z } | {z } T T In the heavy-baryon formalism relativity and relativistic off-shell effects are accounted for by the 1/MN expansion of the lower component field h. The NN potential is then approximately given by (cf. [ME11] pp. 34-35, and references therein)

VNN ≈ W (on-shell) + W1πGW1π − V1πgV1π (3.22) ˜ ≈ Vc.t. + V1π + V2π + W1πGW1π − V1πgV1π , (3.23)

3 ˜ where we neglected 3PE contributions and diagrams above N LO. Moreover, V2π denotes the 2PE amplitude without the planar box diagram; this diagram is given by the expression W1πGW1π. Finally, V1πgV1π is the iterated 1PE contribution, which is equal to the reducible part of the planar box diagram (cf. Figure 3.3). Thus, Eq. (3.23) is equivalent to

0 VNN ≈ Vc.t. + V1π + V2π , (3.24) 0 where V2π is the sum of all irreducible 2PE contributions.

= + + + + +

Figure 3.3.: Planar box diagram: Feynman diagram (left-hand side of the equal sign) and corresponding time-ordered graphs (right-hand side). The ﬁrst two graphs constitute irreducible topologies, whereas the remaining four are reducible.

27 3. From Chiral Lagrangians to Nuclear Potentials

3 The potential VNN satisfies the relativistic Lippman-Schwinger equation (3.19) , but it can be modified in such a way that it satisfies the non-relativistic K-matrix equation (cf. [ME11] pp. 35-36)

Z d3q V (~p 0, ~q )K (~q, ~p ) K (~p 0, ~p ) = V (~p 0, ~p ) + M − NN NN , NN NN N (2π)3 p2 − q2 where the dashed integral denotes the Cauchy principal value. This equation will be used in chapter5 to deﬁne the low-momentum NN potential.

3.3.3. Chiral NN Potential with Regulator Function ChPT is a low-momentum expansion which is valid only for momenta below the chiral 0 symmetry breaking scale, i.e. Q < Λχ ' 1 GeV. In order to restrict VNN(~p , ~p) to momenta below Λχ one multiplies it with a regulator function

0 0 0 VNN(~p , ~p ) → VNN(~p , ~p )f(p , p) , (3.25) where f(p0, p) usually has the following form: " # p 2n p0 2n f(p0, p) = exp − − . (3.26) ΛR ΛR

Two-nucleon observables obviously depend on the value of ΛR. This unwanted dependence can however be balanced by the low energy constants that parametrize the short-distance + behaviour of VNN. A recent work by Coraggio et al. [CHI 12] also supports the independence of results in many-body problems on the precise form of the regulator (for values of ΛR around 450 MeV). The potential published by Entem and Machleidt [EM03] has a regulator with n = 2 and ΛR = 500 MeV. This potential is the one we use; it is from now on referred to as the 3 bare chiral N LO NN potential, or simply VNN.

3.4. General Properties of NN Potentials

The previous sections outlined the modern construction of nuclear forces. Before the es- tablishment of chiral eﬀective ﬁeld theory in nuclear physics, so called phenomenological potentials were used 4 - i.e. potentials adjusted to experimental data and constructed under consideration of basic symmetry principles. In this last section of the chapter we review these symmetries and how they restrict the form of the nucleon-nucleon potential.

The symmetry principles and their eﬀect on the form of the NN potential

VNN ≡ VNN(~r1~k1~σ1τ 1, ~r2~k2~σ2τ 2) (3.27) are the following:

3 This equation is also known as the Blankenbecler-Sugar equation (cf. [Mac89]). 4 In the 1990’s a number of phenomenological NN potentials, e.g. the Argonne AV18, the CD-Bonn and the Reid93 potential, have been constructed. These high-precision potentials are still used today.

28 3. From Chiral Lagrangians to Nuclear Potentials

• Translational invariance: VNN(~r;~k1~σ1τ 1,~k2~σ2τ 2), where ~r := ~r1 − ~r2

• Galilei invariance: VNN(~r~p; ~σ1τ 1, ~σ2τ 2), where ~p := ~p1 − ~p2

• Rotational invariance: VNN(~r~p; ~σ1τ 1, ~σ2τ 2) = VNN(~r −~p; −~σ1τ 1, −~σ2τ 2)

• Invariance under particle exchange: VNN(~r~p; ~σ1τ 1, ~σ2τ 2) = VNN(~r~p; ~σ2τ 2, ~σ1τ 1)

• Invariance under parity: VNN(~r~p; ~σ1τ 1, ~σ2τ 2) = VNN(−~r −~p; ~σ1τ 1, ~σ2τ 2)

• Invariance under time-reversal: VNN(~r~p; ~σ1τ 1, ~σ2τ 2) = VNN(~r −~p; −~σ1τ 1, −~σ2τ 2)

• Isospin symmetry: VNN can only involve operators that are scalars in isospin-space, i.e. {1, τ 1 · τ 2}

† • Hermiticity: VNN = VNN Okuba and Marshak [OM58] showed that the most general form of the NN potential allowed by these symmetries is given by

VNN(r) = VC (r) + V1(r) + V2(r) , (3.28) where VC (r) is the central part of the potential, in the sense that it involves only spherically symmetric terms. It can be dissected as follows:

VC (r) = V0(r) + Vσ(r) ~σ1 · ~σ2 + Vτ (r) τ 1 · τ 2 + Vστ (r)(~σ1 · ~σ2)(τ 1 · τ 2) . (3.29)

V1(r) is compromised of terms which are proportional to the spin-orbit operator ~` · S~,

V1(r) = VLS(r) ~` · S~ + VLSτ (r)(~` · S~)(τ 1 · τ 2) , (3.30)

~ ~ 1 where ` = ~r × ~p is the orbital angular momentum operator, and S = 2 (~σ1 + ~σ2). The last part V2(r) contains tensor forces. The most familiar one is given by the operator 3 S = (~σ · ~r)(~σ · ~r) − ~σ · ~σ , (3.31) 12 r2 1 2 1 2 which corresponds to the “usual” tensor force

VT (r)S12 + VT τ (r)S12 τ 1 · τ 2 . (3.32)

Note how the diﬀerent operator parts of the potential (i.e. contact, spin-orbit, tensor) appear in the amplitudes generated by 1PE and 2PE, Eqs. (3.16) and (3.17). In section 6.2.1 we make use of the decomposition of V provided here to investigate the properties NN E ~ of the potential matrix element, hij | VNN | kli (where |ii ≡ kστ are the 1-particle states).

29 4. Thermodynamics of Many-Particle Systems

In the previous chapter we outlined the construction of nuclear two- and three-body potentials. With these potentials we can now leave the framework of chiral eﬀective ﬁeld theory and proceed with the study of the many-nucleon system. For this purpose, we recall the methods of many-particle quantum mechanics, i.e. the non-relativistic occupation number formalism. The Hamiltonian of the many-nucleon system is given by

H = H0 + VNN + V3N , (4.1) where H0 is the unperturbed Hamiltonian, representing a non-interacting many-nucleon system, and VNN and V3N constitute the interaction part of H. H0, VNN and V3N are operators in the second-quantized form; they are given by the equations Z 3 ˆ† ˆ H0 = d x ψ (~x) H0 ψ(~x) , (4.2) 1 Z Z V = d3x d3x0 ψˆ†(~x) ψˆ†(~x 0) V (~x,~x 0) ψˆ(~x 0) ψˆ(~x) , (4.3) NN 2 NN 1 Z Z Z V = d3x d3x0 d3x00 ψˆ†(~x) ψˆ†(~x 0) ψˆ†(~x 00) V (~x,~x 0, ~x 00) ψˆ(~x 00) ψˆ(~x 0) ψˆ(~x) , (4.4) 3N 6 3N

0 0 00 where H0 is the kinetic energy operator, and VNN(~x,~x ) and V3N(~x,~x , ~x ) are the nuclear two- and the three-body potentials in the coordinate space representation. ψˆ(~x) and ψˆ†(~x) are field operators, the first called a destruction operator due to its property of annihilating a particle at ~x, the second (the adjoint of the destruction operator) correspondingly called a creation operator, as it creates the same particle when acting on the vacuum |0i. Because 1 1 we are dealing with nucleons, i.e. spin- 2 and isospin- 2 fermions, the field operators have four components, which we label with Greek indices:  ˆ  ψ1(~x)  ˆ  ˆ ψ2(~x) ˆ ψ(~x) =  ˆ  = ψα(~x) , α = 1, 2, 3, 4 . (4.5) ψ3(~x) ˆ ψ4(~x) The field operators obey the canonical anticommutation relations

ˆ ˆ† 0 0 ψα(~x) , ψβ(~x ) + = δαβ δ(~x − ~x ) , (4.6) ˆ ˆ 0 ˆ† ˆ† 0 ψα(~x) , ψβ(~x ) + = ψα(~x) , ψβ(~x ) + = 0 . (4.7) The goal of this chapter is to derive expressions for the ground-state energy (at zero temperature) and the free energy per particle (at ﬁnite temperature) corresponding to the

30 4. Thermodynamics of Many-Particle Systems nuclear many-body system described by the Hamiltonian in Eq. (4.1). For this purpose, many-body perturbation theory can be applied, assuming that the interaction part of the Hamiltonian can be considered a small correction to the free Hamiltonian H0. Whether this condition is fulﬁlled shall not be of concern in this chapter but will be discussed in the next chapter. Here we assume, without further justiﬁcation, that VNN and V3N can be treated perturbatively.

The perturbative treatment of many-body systems at finite temperature with the use of quantum field theory methods originates from a paper by Matsubara [Mat55]. It was shown therein that one can express the grand-canonical potential density as a series of statistical averages of an increasing number of field operators. This represents a generalization of many-body perturbation theory at zero temperature. We discuss the T = 0 case in the first section of this chapter, and move on to finite temperatures in the second. The third section summarizes the results of the first two sections, and depicts them in terms of diagrams. Finally, in the fourth section we show how other thermodynamic quantities, in particular the free energy per particle, can be derived from the grand-canonical potential density.

4.1. Zero Temperature Formalism

The ground-state energy of the system is equal to the expectation value of the interacting Hamiltonian

E = hHi = hH0i + hVNNi + hV3Ni , (4.8) where the expectation value of some operator A is given by hΨ | A | Ψ i hAi ≡ 0 0 , (4.9) hΨ0 | Ψ0i with |Ψ0i being the ground-state of the interating system. In order to avoid having to solve the Schr¨odingerequation for |Ψ0i we want to express hHi in terms of the ground- state of the non-interacting system |Φ0i, whose properties are known. The connection between these states is given by the Gell-Mann and Low theorem, which expresses the ground-state of the interacting system as 1 ˆ |Ψ0i U(0, ±∞) |Φ0i = D E . (4.10) hΦ0 | Ψ0i ˆ Φ0 U(0, ±∞) Φ0 The states in this equation are in the interaction picture representation. Hence, their time development is given by

−iH(t−t0) |Ψ(t)i = e |Ψ(t − t0)i , |Ψi ∈ {|Ψ0i , |Φ0i} . (4.11)

1 A caveat: To be precise, the Gell-Mann and Low theorem gives the above connection for the state |Ψ0i that develops adiabatically from |Φ0i, as the interaction is switched on. This need not be the ground-state of the interacting system, which does not always have a perturbation series in terms of the interaction. A simple example of such a scenario is for example a perfect Fermi gas in a uniform magnetic ﬁeld (cf. [FW72] pp. 61,254,289). However, we will not pay attention to this subtlety in the present section. The conditions under which the adiabatically evolved state is indeed the ground-state of the interacting system become apparent in the end of section 4.4.

31 4. Thermodynamics of Many-Particle Systems

ˆ The operator U(t, t0) is formally given by the equation

ˆ iH0t −iH(t−t0) −iH0t0 U(t, t0) = e e e , (4.12) where the index corresponds to the interaction being switched on and off adiabatically. In order to produce this feature the Hamiltonian has to be modified according to H = −|t| H0 + e V, where is infinitesimal and V shorthand for both the two-body and the three-body interaction part of the Hamiltonian H. ˆ It is easy to see that U(t, t0) satisfies the differential equation ∂ i Uˆ (t, t ) = V Uˆ (t, t ) , (4.13) ∂t 0 I 0 where VI(t) is the interaction part of the Hamiltonian in the interaction picture. The representation of some operator A in that picture is defined as

iH0t −iH0t AI(t) ≡ e A e . (4.14)

For comparison, the Heisenberg representation of the same operator is given by

iHt −iHt AH(t) ≡ e A e . (4.15)

The two pictures are related in the following way: ˆ ˆ AH(t) = U(0, t) AI(t) U(t, 0) . (4.16)

ˆ The solution of Eq. (4.13) with boundary condition U(t0, t0) = 1 is

∞ Z t Z t X ν 1 Uˆ (t, t ) = (−i) dt ··· dt e−(|t1|+...+|tν |) T [V (t ) ···V (t )] , (4.17) 0 ν ! 1 ν I 1 I ν ν=0 t0 t0 where the symbol T [...] denotes the time ordered product of (fermionic) operators. Its eﬀect on a sequence of time-dependent operators is that of ordering them in terms of descending values of t, and multiplying the whole expression with a factor (−1)P , where P is the number of permutations needed to restore the original ordering. For instance, the time-ordered product of two operators A(t) and B(t) is deﬁned by ( A(t)B(t0) t > t0 T [A(t)B(t0)] = , (4.18) −B(t0)A(t) t < t0 and accordingly for three or more operators. With the use of Eq. (4.17) and the Gell-Mann and Low theorem it is possible to express the expectation value of some Heisenberg operator AH (t) in terms of |Φ0i (cf. [FW72] pp. 82-84):

∞ Z t Z t hΨ0 | AH (t) | Ψ0i 1 X ν 1 = D E hΦ0| (−i) dt1 ··· dtν hΨ0 | Ψ0i ˆ ν ! t t Φ0 U(t, t0) Φ0 ν=0 0 0 (4.19)

−(|t1|+ ... +|tν |) × e T [VI (t1) ···VI (tν)AI (t)] |Φ0i .

32 4. Thermodynamics of Many-Particle Systems

This defines a perturbative expansion of the expectation value hAi in terms of the interaction Hamiltonian V. So in order to determine hHi and thereby the ground-state energy of the interacting system we must evaluate an expression similar to Eq. (4.19). Because the operator structure of V is made up of field operators, as specified in Eqs. (4.3) and (4.4), this boils down to the problem of determining how to evaluate the expectation value in the non-interacting ground-state of the time ordered product of field operators, i.e. D E ˆ† ˆ 0 0 Φ0 T [ψ (~x1t1) ··· ψIα (~xn t )] Φ0 . (4.20) Iα1 n n In order to find out how the field operators act on the non-interacting ground-state we consider their decomposition in momentum space:

ˆ X ψα(~x) = Ψα(~x~pλ)a ˆ(~pλ) , (4.21) ~pλ ˆ† X † † ψα(~x) = Ψα(~x~pλ)a ˆ (~pλ) , (4.22) ~pλ where the index λ is used for the spin and isospin degrees of freedom. The coeﬃcients † † Ψα(~x~pλ) are the single-particle wave functions, anda ˆ (~pλ) anda ˆ(~pλ) are creation and destruction operators in momentum space. Because the ground-state is characterized by the occupation of all states with momenta below the Fermi momentum κF , and none above, these operators are subject to the following relations:

† aˆ(~pλ) |Φ0i = 0 ∀p > κF anda ˆ (~pλ) |Φ0i = 0 ∀p < κF . (4.23)

ˆ Thus the ﬁeld operator ψα(~x) can be uniquely seperated in two parts

ˆ ˆ(+) ˆ(−) ψα(~x) = ψα (~x) + ψα (~x) , (4.24) where the (+) sign indicates that all momentum components of the operator lie outside the Fermi sphere, while a (−) operator has only momentum components inside of it. Thus (+) ψˆα (~x) annihilates the non-interacting ground-state

ˆ(+) ψα (~x) |Φ0i = 0 . (4.25)

Correspondingly, the decomposition of the adjoint operator is given by

ˆ† ˆ(+)† ˆ(−)† ψα(~x) = ψα (~x) + ψα (~x) , (4.26)

(−)† but with ψˆα (~x) now being the destruction part

ˆ(−)† ψα (~x) |Φ0i = 0 . (4.27)

The situation resembles that in relativistic quantum field theory, where the quantum fields are separated in positive (+) and negative (−) frequency parts which act on the (free) vacuum in the same way as the above defined (+) and (−) field operators act on the non-interacting ground-state (cf. [Wei95] pp. 192-195). Hence, in order to evaluate Eq. (4.20), we can make use of Wick’s theorem, which states that the ground-state average of

33 4. Thermodynamics of Many-Particle Systems the time-ordered product of a given sequence of ﬁelds operators is equal to the sum over all fully contracted versions of this sequence, i.e. D E ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Φ0 T [ψ1ψ2ψ3ψ4 ···] Φ0 = ψ1ψ2ψ3ψ4 ··· + ψ1ψ2ψ3ψ4 ··· + ..., (4.28)

ˆ ˆ ˆ† where we used the shorthand notation ψj ∈ {ψIα(~xt), ψIα(~xt)}. The only nonvanishing type of contraction is that of a creation operator with a destruction operator 2, D E ˆ ( ) ˆ† ( 0 0) ≡ T [ ˆ ( ) ˆ† ( 0 0)] =: 0 ( 0 0) (4.29) ψIα ~xt ψIβ ~x t Φ0 ψIα ~xt ψIβ ~x t Φ0 iGαβ ~xt,~x t ,

0 0 0 where iGαβ(~xt,~x t ) is the free single-particle Green function or propagator. From the decompositions (4.24) and (4.26) together with the annihilation relations (4.25) and (4.27) it follows that  D E ˆ(+)( ) ˆ(+)†( 0 0) 0 D E  Φ0 ψIα ~xt ψIβ ~x t Φ0 t > t T [ ˆ ( ) ˆ† ( 0 0)] = (4.30) Φ0 ψIα ~xt ψIβ ~x t Φ0 D E .  − ˆ(−)†( 0 0) ˆ(−)( ) 0  Φ0 ψIβ ~x t ψIα ~xt Φ0 t > t

Remembering that for (+) operators the momentum space decomposition is restricted by p > κF , and for (−) operators by p < κF we can write down the explicit formula for the propagator 3

1 X 0 0 iG0 (~xt,~x 0t0) = δ ei~p ·(~x−~x ) e−iεp(t−t ) Θ(t − t0)Θ(p − κ ) − Θ(t0 − t)Θ(κ − p) , αβ αβ V F F ~p (4.31)

2 where εp = p /2MN denotes the kinetic energy of a single particle with momentum p. For the momentum dependent Heavyside step functions that define the unperturbed Fermi ¯ surface we shall from now on use the abbreviations Θp ≡ Θ(κF − p) and Θp ≡ Θ(p − κF ). The Θp-part of the propagator corresponds to the lower part of Eq. (4.30), in which a ˆ(−) particle in the filled Fermi sea is destroyed at {~x,t} by ψIα (~xt) and subsequently recreated ˆ(−)† 0 0 0 0 by ψIβ (~x t ) at {~x t }. Thus, it is suitable to interpret this consecutive annihilation and creation process as the propagation of a hole in the Fermi sea. Correspondingly, in the Θp-case we speak of the propagation of a particle. In fact, the decomposition of field operators in Eqs. (4.24) and (4.26) defines a canonical transformation to particle (+) and hole (−) operators. This observation concludes this section. We are now in possession of all the tools necessary to compute the ground-state energy to a given order in many-body perturbation theory. Let us proceed with the finite temperature case.

2 In the rest of this section we limit ourselves to the case where the operator on the right-hand side of the contraction is a creation operator. The other case follows from adjungation. 3 The factor δαβ arises from the completeness relation of the spin and isospin part of the single-particle wavefunctions. The factor V −1 together with the sum over plane waves comes from the (proper normalized) spatial part of the wavefunctions and the interaction picture representation of the time propagation operator exp(iH0t).

34 4. Thermodynamics of Many-Particle Systems

4.2. Finite Temperature Formalism

In treating a system of nucleons at ﬁnite temperatures, it is most convenient to use the grand-canonical ensemble, because it allows for the possibility of a variable number of particles. We deﬁne the grand-canonical Hamiltonian as

K := H0 − µN +VNN + V3N , (4.32) | {z } K0 where the chemical potential µ and the number operator N = R d3x ψˆ†(~x) ψˆ(~x) have been introduced. A statistical ensemble in the grand-canonical description is characterized by the statistical operator% ˆ, which is deﬁned as

%ˆ = Z−1 e−βK , (4.33) where β = 1/T is the inverse temperature and Z is the grand partition function which is deﬁned as Z = Tr e−βK. The trace Tr denotes the sum over the complete set of states that correspond to the statistical ensemble with macroscopic parameters β and µ:

−βK X X −βK Tr e = ··· n1 . . . n∞ e n1 . . . n∞ , (4.34) n1 n∞ with ni being equal to the number of particles occupying the state |ii. The basic quantity from which all thermodynamic properties of the system in equilib- rium follow is the grand-canonical potential density Ω, which is given by 1 Ω = − ln Z . (4.35) βV In order to ﬁnd a perturbative expansion of this quantity we proceed to an analytic continuation to imaginary time (a so-called Wick rotation):

t → −iτ , τ ∈ R . (4.36)

The ﬁnite temperature interaction picture representation AI(τ) and the ﬁnite temperature Heisenberg picture representation AH(τ) of an operator A are then given by substituting H → K and t → −iτ in Eqs. (4.14) and (4.14):

K0τ −K0τ AI(τ) ≡ e A e , (4.37) Kτ −Kτ AH(τ) ≡ e A e . (4.38)

Similarly, for the representations of the adjoint operator A† we get

† K0τ † −K0τ AI (τ) ≡ e A e , (4.39) † Kτ † −Kτ AH(τ) ≡ e A e . (4.40)

† † Note that, contrary to the zero temperature case, AI (τ) and AH(τ) are not the adjoints of AI(τ) and AH(τ). The two pictures are again related by a simple equation:

(†) ˆ (†) ˆ AH (τ) = U (0, τ) AI (τ) U (τ, 0) , (4.41)

35 4. Thermodynamics of Many-Particle Systems

ˆ where the operator U (τ, τ0) is given by

ˆ K0τ −K(τ−τ0) −K0τ0 U (τ, τ0) = e e e . (4.42)

Diﬀerentiation with respect to τ yields the following relation:

∂ ˆ ˆ U (τ, τ0) = VI U(τ, τ0) , (4.43) ∂τ where we again use the shorthand notation V = VNN + V3N. The above equation is of ˆ exactly the same form as Eq. (4.13). Because also U (τ0, τ0) = 1 we can immediately write down the solution ∞ Z τ Z τ ˆ X ν 1 U (τ, τ0) = (−1) dτ1 ··· dτν Tτ [VI(τ1) ···VI(τν)] , (4.44) ν ! ν=0 τ0 τ0 where Tτ is the imaginary-time ordering operator which arranges the operators inside the square brackets so that τ is decreasing from left to right, and (as in the zero temperature case) multiplies the whole expression with a factor (−1)P . Returning to Eq. (4.35), we observe that with e−βK = e−βK0 Uˆ(β, 0) the above ex- ˆ pression for U (τ, τ0) leads to a more explicit expression for the grand-canonical potential density: 1 Ω = − ln Tr e−βK0 Uˆ(β, 0) βV ∞ Z β Z β 1 X ν 1 = − ln (−1) dτ ··· dτ Tr e−βK0 T [V (τ ) ···V (τ )] . (4.45) βV ν ! 1 ν τ I 1 I ν ν=0 0 0 Restricting ourselves to terms up to second order in ν, this becomes

1 Z β 1 Z β Z β −βK0 −βK0 0 −βK0 0 Ω = − ln Tr e − dτ Tr e VI(τ) + dτ dτ Tr e Tτ [VI (τ)VI(τ )] . βV 0 2 0 0 (4.46)

A simple algebraic conversion leads to

1 1 Z β 1 Z β Z β −βK0 0 0 Ω = − ln Tr e − ln 1 − dτ Tr% ˆ0VI(τ) + dτ dτ Tr% ˆ0Tτ [VI (τ)VI(τ )] , βV βV 0 2 0 0 | {z } | {z } =:Ω0 =:ΩV (4.47)

−1 −βK0 where the statistical operator of the non-interacting system,% ˆ0 = Z0 e (with Z0 = Tr e−βK0 ) has been introduced. The above equation explicitly separates the interaction part of grand-canonical potential density from the contribution corresponding to the non- interacting system, Ω0. Furthermore, the interaction part ΩV contains only traces over operators weighted by% ˆ0. These expressions are called ensemble averages. For the average of some operator A in the non-interacting ensemble we write

hAi0 ≡ Tr% ˆ0A . (4.48)

36 4. Thermodynamics of Many-Particle Systems

Using this notation and employing the Taylor expansion of the logarithm, the expression for ΩV becomes  β β β β  Z Z 2 Z Z 1 1 1 0 0 3 ΩV =  dτ hVI(τ)i + dτ hVI(τ)i − dτ dτ hTτ [VI (τ)VI(τ )]i + O(V ) . βV 0 2 0 2 0 0 0 0 0 (4.49) Hence, we have derived a perturbative expansion of the grand-canonical potential density in terms of the interaction Hamiltonian V. The only remaining task is now to find out how to evaluate the ensemble average of the Tτ -product of a sequence of field operators, i.e. D E ˆ† ˆ 0 0 Tτ [ψIα (~x1τ1) ··· ψIαn (~xn τn)] (4.50) 1 0 There exists a finite temperature version of Wick’s theorem that allows to deal with Eq. (4.50) in the same way as with the ground-state expectation value in the section above. We will now go through a succinct derivation of that theorem, based on the more detailed proof in [FW72] pp. 232-241. Again we make use of the momentum space decomposition of the field operators given by Eqs. (4.21) and (4.22). The equation of motion of the momentum space operators in the finite temperature interaction picture,a Î(~pλ; τ) and † aÎ (~pλ; τ), is given by ∂ ∂ aˆ(†)(~pλ; τ) = eK0τ aˆ(†)(~pλ) e−K0τ = eK0τ [K , aˆ(†)(~pλ)] e−K0τ . (4.51) ∂τ I ∂τ 0 −

Because H0 gives the total kinetic energy present in the non-interacting system, and K0 = H0 − µN , the commutator [K0 , aˆ(~pλ)]− can be readily identiﬁed with −(εp − µ)ˆa(~pλ). With that, Eq. (4.51) can be easily integrated and we arrive at:

−(εp−µ)τ aˆI(~pλ; τ) =a ˆ(~pλ) e . (4.52)

† † Similarly, with [K0 , aˆ (~pλ)]− = (εp − µ)ˆa (~pλ), the solution of the equation of motion of thea ˆ†-type operators reads:

† † (εp−µ)τ aˆI (~pλ; τ) =a ˆ (~pλ) e . (4.53) Thus, we can write down an explicit formula for the ﬁeld operators in the interaction picture:

ˆ X −(εp−µ)τ ψIα(~xτ) = Ψα(~x~pλ)a ˆ(~pλ) e , (4.54) ~pλ

ˆ† X † † (εp−µ)τ ψIα(~xτ) = Ψα(~x~pλ)a ˆ (~pλ) e . (4.55) ~pλ These equations allow to reformulate the ensemble average (4.50) in terms of the momentum space operators. Without loss of generality consider now the case were the ﬁeld operators in the ensemble average are already imaginary-time ordered. This gives (in shorthand notation) D ˆ ˆ E D ˆ ˆ E X X Tτ [ψ1 ··· ψn] = ψ1 ··· ψn = ··· χj1 ··· χjn haˆj1 ··· aˆjn i0 . (4.56) 0 0 j1 jn

37 4. Thermodynamics of Many-Particle Systems

−(εj −µ)τ † (εj −µ)τ ˆ Here, χj denotes Ψj e or Ψj e , and ψj as well asa ˆj can be both a creation ˆ ˆ ˆ† † operator or a destruction operator, i.e. ψj ∈ {ψIα(~xτ), ψIα(~xτ)} anda ˆj ∈ {aˆ(~pλ), aˆ (~pλ)}.

We can anticommute the ﬁrst operatora ˆj1 successively to the right

haˆj1 ··· aˆjn i0 = h[ˆaj1 , aˆj2 ]+aˆj3 ··· aˆjn i0 − haj2 [ˆaj1 , aˆj3 ]+ ··· aˆjn i0 + ...

± haˆj2 aˆj3 ··· [ˆaj1 , aˆjn ]+i0 ∓ haˆj2 aˆj3 ··· aˆjn aˆj1 i0 . (4.57) The anticommutators are given by

† 0 0 [â(~pλ) , aˆ (~p λ )]+ = δλλ0 δ~p~p 0 , (4.58) 0 0 † † 0 0 [â(~pλ) , aˆ(~p λ )]+ = [â (~pλ) , aˆ (~p λ )]+ = 0 , (4.59) and may therefore be taken outside the ensemble average. Furthermore, setting τ = β in the equation of motion of the momentum space operators, Eq. (4.51), and inserting the solutions, Eqs. (4.52) and (4.53), we obtain the relation

ξj (εj −µ)β aˆj%ˆ0 =% ˆ0aˆj e , (4.60) where ξj = 1 ifa ˆj is a creation operator and ξj = −1 if it is a destruction operator.

Remembering that h...i0 = Tr% ˆ0 ... , and using the cyclic property of the trace, the last term can be rewritten as

ξj1 (εj1 −µ)β ∓ Tra ˆj1 %ˆ0aˆj2 aˆj3 ··· aˆjn = ∓ e Tr% ˆ0aˆj1 aˆj2 aˆj3 ··· aˆjn , (4.61) which, when applied to Eq. (4.57), leads to the important result

[ˆaj1 , aˆj2 ]+ [ˆaj1 , aˆj3 ]+ haˆj1 ··· aˆjn i0 = ξ (ε −µ)β haˆj3 ··· aˆjn i0 − ξ (ε −µ)β haˆj2 aˆj4 ··· aˆjn i0 + ... 1 ± e j1 j1 1 ± e j1 j1

[ˆaj1 , aˆjn ]+ ± ξ (ε −µ)β aˆj2 ··· aˆjn−1 0 . (4.62) 1 ± e j1 j1 This suggests to deﬁne a contraction for the momentum space creation and destruction operators:

[ˆaj , aˆj0 ]+ aˆjaˆj0 ≡ . (4.63) 1 ± eξj (εj −µ)β In the particular case of only two operators in the ensemble average, Eq. (4.62), shows that also aˆjaˆj0 = haˆjaˆj0 i0. The only nonvanishing contractions are those of a creation operator and a destruction operator both having the exact same values of j = {~pλ}:

† † [â (~pλ) , aˆ(~pλ)]+ 1 aˆ (~pλ)â(~pλ) = = =: np , (4.64) 1 ± e(εp−µ)β 1 ± e(εp−µ)β † † [â(~pλ) , aˆ (~pλ)]+ 1 aˆ(~pλ)â (~pλ) = = =:n ¯p . (4.65) 1 ± e−(εp−µ)β 1 ± e−(εp−µ)β

The sign in these expressions depends only on the number of operators in haˆj1 ··· aˆjn i0 =

Tr %0Tτ [ˆaj1 ··· aˆjn ]. Utilizing the fact that the number operator commutes with% ˆ0 and again using the cyclic property of Tr we have

Tr %0NTτ [âj1 ··· aˆjn ] = Tr N %0Tτ [âj1 ··· aˆjn ] = Tr %0Tτ [âj1 ··· aˆjn ]N . (4.66)

38 4. Thermodynamics of Many-Particle Systems

Hence, it does not matter whether N acts before or after the Tτ -ordered product of 4 momentum space operators (in the interaction picture, i.e.a Î(~pλ; τ) ) on the ket-states in the ensemble average, as defined in Eq. (4.34). Consequently, the number of particles specified by these states is left unchanged, meaning there has to be an equal number of creation and destruction operators. Thus, the total number of operators must be even and we obtain the positive sign in Eqs. (4.64) and (4.65). The quantity np thereby becomes the Fermi-Dirac distribution and we have the relationn ¯p = 1 − np. Returning to the original expression, the ensemble average of field operators in Eq. (4.56), we define a field operator contraction as

 D ˆ ˆ E P  ψ1ψ2 = χj1 χj2 aˆj1 aˆj2 τ1 > τ2 D E  0 ˆ ˆ ˆ ˆ j1j2 ψ1ψ2 ≡ Tτ [ψ1ψ2] = D E . (4.67) 0 − ˆ ˆ = P ˆ ˆ  ψ2ψ1 χj1 χj2 aj2 aj1 τ2 > τ1 0 j1j2

The application of Eq. (4.62) then yields

D ˆ ˆ ˆ ˆ E D ˆ ˆ ˆ ˆ E D ˆ ˆ ˆ ˆ E Tτ [ψ1ψ2ψ3 ··· ψn] = Tτ [ψ1ψ2ψ3 ··· ψn] + Tτ [ψ1ψ2ψ3 ··· ψn] + ... 0 0 0 D ˆ ˆ ˆ ˆ E + Tτ [ψ1ψ2ψ3 ··· ψn] , (4.68) 0 In each term on the right-hand side the contraction may be taken out of the ensemble average, leaving a structure similar to the expression on the left-hand side. The same analysis again applies, and we therefore conclude that the non-interacting ensemble average of a Tτ -product of field operators is equal to the sum over all possible contractions of the field operators. This is the finite temperature version of Wick’s theorem 5:

D ˆ ˆ ˆ ˆ E ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Tτ [ψ1ψ2ψ3ψ4 ··· ] = ψ1ψ2ψ3ψ4 ··· + ψ1ψ2ψ3ψ4 ··· + ... (4.69) 0

Finally, to conclude this section, we deﬁne the free ﬁnite temperature propagator as D E 0 0 0 ˆ ˆ† 0 0 Gαβ(~xτ, ~x τ ) := − Tτ [ψIα(~xτ)ψIβ(~x τ ) .] (4.70) 0 Eqs. (4.64), (4.65) and (4.67) show that its explicit form is

1 0 0 0 0 0 X i~p ·(~x−~x ) −(εp−µ)(τ−τ ) 0 0 G (~xτ, ~x τ ) = δαβ e e Θ(τ − τ )n ¯p − Θ(τ − τ) np . (4.71) αβ V ~p

Hence, we have reobtained the “particle-hole” picture from the zero temperature case, with the momenta of holes now being distributed by np and the ones of particles byn ¯p.

4 ˆ ˆ ˆ ˆ Here, we return to the general case where Tτ [ψ1 ··· ψn] 6= ψ1 ··· ψn, cf. Eq. (4.56). 5 A remark: In contrast to Wick’s theorem at zero temperature, which is an operator identity and therefore valid for arbitrary matrix elements, the ﬁnite temperature version deals explicitely with the ensemble average of operators, thereby constituting merely an algebraic identity.

39 4. Thermodynamics of Many-Particle Systems

4.3. Diagrammatic Analysis

We have now obtained perturbative expressions for the ground-state energy E, Eq. (4.19), and for the grand-canonical potential density Ω, Eqs. (4.45) to (4.49) 6. In each case, the diﬀerent contributions in the perturbation series can be evaluated with the use of the respective version of Wick’s theorem. We are now going to represent these contributions in terms of diagrams. Regarding the two-body interaction Hamiltonian VNN we restrict ourselves to contributions up to second order. As the three-body interactions are of higher order in the chiral expansion parameter (cf. section 3.2), we allow only ﬁrst order contributions from V3N.

Because we evaluate ground-state expection values, or ensemble averages, all diagrams have to be closed, i.e. there are no external lines. Furthermore, in the zero temperature case the denominator in Eq. (4.19) leads to the cancellation of all disconnected diagrams (cf. [PvS95] pp. 90-99). For finite temperatures this cancellation is achieved in a different way, namely by additional terms in the perturbation series, like the second term in Eq. (4.49). These terms, which arise from the Taylor expansion of the logarithm, have no other effect than restricting to connected diagrams. Thus, in both cases the perturbation series is equal to the sum over linked clusters, i.e. the sum of all connected closed diagrams. All terms in the perturbation series for E appear also in the series for Ω, in their finite temperature version. As the evaluation of these terms is done in a similar way in both cases (via Wick’s theorem), all diagrams that appear at zero temperature occur as well in the finite temperature case. However, at finite temperatures there are also diagrams that do not appear at zero temperature - the so-called anomalous contributions. These stem from certain combinations of contractions which are not allowed in the zero temperature 0 case. For instance, in the case of the second order NN contribution ∼ [VNN]I(τ)[VNN]I(τ ) the field operators can be contracted in the following way:

ˆ† ˆ† 0 ˆ 0 ˆ ˆ† 0 ˆ† 0 0 ˆ 0 0 ˆ 0 ψI (~xτ)ψI (~x τ)ψI(~x τ)ψI(~xτ)ψI (~yτ )ψI (~y τ )ψI(~y τ )ψI(~yτ ) . (4.72)

The contraction that involves the operator on the far left and the one that involves the one on the far right are free of complications. Because they connect operators whose imaginary-time variables have the same value, imaginary-time ordering has no eﬀect on their sequence. Thus we see from Eq. (4.70) that both of them yield a momentum integral over a hole degree of freedom:

ˆ† ˆ ψI (~xτ)ψI(~xτ) nk1 , (4.73) ˆ† 0 ˆ 0 ; ψI (~yτ )ψI(~yτ ) nk3 . (4.74) ;

6 It should be noted here that the contribution associated with the free Hamiltonian K0, Ω0 in Eq. (4.47), has not the form of an ensemble average of ﬁeld operators. Hence, it has to be treated in a diﬀerent way; we postpone the evaluation of Ω0 to section 6.1.2.

40 4. Thermodynamics of Many-Particle Systems

The other two contractions however lead to

( 0 ˆ† 0 ˆ 0 0 nk2 τ > τ ψ (~x τ)ψI(~y τ ) , (4.75) I n¯ τ 0 > τ ; k2 ( 0 n¯k0 τ > τ ˆ ( 0 ) ˆ†( 0 0) 2 (4.76) ψI ~x τ ψI ~y τ 0 . nk0 τ > τ ; 2 0 0 3 0 i(~k2−~k )~x Because the d x -integral over the plane wave parts ∼ e 2 furnishes a factor δ~ ~ 0 , k2k2 the expression resulting from these contractions contains a momentum integral over both a hole and a particle degree of freedom, distributed by nk2 andn ¯k2 , respectively. For zero temperature, where instead of Fermi-Dirac distributions we have Heavyside step functions, the range of such an integral is zero.

When we now determine all possible contractions, or equivalently all linked clusters, it turns out that several topologically nonequivalent diagrams lead to identical expressions except for the order of particles in the potential matrix elements. These diagrams can be combined by multiplying the two- and three-body potentials with the antisymmetrization operator A , which is given by X A = sign(α)Pα , (4.77) α where the sum runs over all possible permutations Pα of particles in the two- |iji and three-nucleon state |ijki, respectively. In the case of the NN potential it is A = 1 − P12 and in the case of the 3N-potential A = (1 − P12)(1 − P13 − P23) (cf. [Kai12b]). The diagrams (up to second order in VNN and first order in V3N) that remain after this modification are shown in Figure 4.1. The analytical expressions for the first order NN and the second order normal NN contribution to the grand-canonical potential density are (cf. [TFS08] and [FW72] pp. 273-274) 1 Z d3k Z d3k Ω = tr tr 1 2 n n h12 |(1 − P )V | 12i , (4.78) 1,NN 2 σ1,τ1 σ2,τ2 (2π)3 (2π)3 k1 k2 12 NN 4 Z 3 ! 1 Y d ki 3 Ω = − tr (2π) δ ~k + ~k − ~k − ~k 2,n 8 σi,τi 3 1 2 3 4 i=1 (2π)

nk1 nk2 n¯k3 n¯k4 − n¯k1 n¯k2 nk3 nk4 2 × |h12 |(1 − P12)VNN| 34i| , (4.79) ε3 + ε4 − ε1 − ε2 where hij | (1 − P12)VNN | kli is the potential matrix element of the two-nucleon states |iji and |kli. The corresponding contributions to the ground-state energy (per volume) are ¯ given by the substitutions np → Θp andn ¯p → Θp. The zero temperature limit of Ω1,NN 7 and Ω2,n however is given by the substitution of Θ(µ − εp) and Θ(εp − µ) for np andn ¯p. The same applies for the ﬁrst order 3N-contributions, whose explicit form is discussed in

7 Note that these are not the Heavyside step functions for the unperturbed Fermi-surface (as in our deﬁnitions at the end of section 4.1). This is because µ is the chemical potential of the fully interacting system; the Fermi momentum that is associated with the zero temperature limit of this system is deﬁned by εinteracting = µ (and corresponds to the interacting zero temperature ground-state), which κF

is not equal to the one corresponding to the free system, εκF = µ0.

41 4. Thermodynamics of Many-Particle Systems

k k k1 k2 k1 k3 1 3 k4 k2

(a) (1,NN) (b) (1,3N) (c) (2,n) k3

k2 k1

(d) (2,a)

Figure 4.1.: First order (a) NN and (b) 3N, and second order NN - (c) normal and (d) anomalous - contributions (antisymmetrized interactions). An arrow pointing

downwards symbolizes a hole-line, giving a factor nki , while an arrow point-

ing upwards symbolizes a particle-line, giving a factorn ¯ki . The wiggly lines symbolize an interaction mediated by VNN, whereas the double-wiggly lines one by V3N. section 5.3. However, the situation is diﬀerent in the case of the second order anomalous diagram, which does not exist at zero temperature, yet also does not vanish in the T → 0 limit:

3 Z 3 ! β Y d ki Ω = − tr n n n¯ n 2,a 2 σi,τi 3 k1 k2 k2 k3 i=1 (2π)

× h12 |(1 − P12)VNN| 12i h23 |(1 − P12)VNN| 23i . (4.80)

To see that this expression does not vanish in the zero temperature limit we use the relation

∂np = −βnpn¯p , (4.81) ∂εp which allows to rewrite Eq. (4.80) in the following way:

3 Z 3 ! 1 Y d ki ∂nk Ω = tr n 2 n 2,a 2 σi,τi 3 k1 ∂ε k3 i=1 (2π) k2

× h12 |(1 − P12)VNN| 12i h23 |(1 − P12)VNN| 23i . (4.82)

T →0 Because np −−−→ Θ(µ − εp), the factor ∂np/∂εp reduces to −δ(µ − εp) in the zero temperature limit. Moreover, we can bracket together the matrix-elements 8, which leads

8 This rearrangement is allowed because the matrix elements are invariant under the combined interchange of the single-particle states in the bra- and ket-state (cf. section 3.4, or ﬁrst part of section 6.4).

42 4. Thermodynamics of Many-Particle Systems to Z 3 T →0 1 d k2 Ω2,a −−−→− trσ ,τ δ(µ − εk ) 2 2 2 (2π)3 2 Z 3 Y d kα × trσ ,τ Θ(µ − εk ) hα2 |(1 − P12)VNN| α2i , (4.83) α α (2 )3 α α∈{1,3} π which is obviously nonzero. The derivation of the free energy per particle from the grand- canonical potential density in the next section makes clear how this issue is resolved.

4.4. Free Energy per Particle and Zero Temperature Limit

Free Energy and Other Thermodynamic Quantities The free energy per particle is given by a Legendre transformation of the grand-canonical potential density with respect to the chemical potential: 1 F¯(ρ, T ) = Ω(µ, T ) + µ . (4.84) ρ Hence, in order to ﬁnd the free energy per particle, we need the chemical potential as a function of the particle density, µ(ρ, T ). The particle density ρ is by a standard thermodynamic relation deﬁned as the derivative of Ω with respect to the chemical potential. Making use of the perturbative expansion of the grand-canonical potential density we arrive at:

∂Ω ∂Ω0 ∂Ω1 ∂Ω2 ρ(µ, T ) = − = − − − − ..., (4.85) ∂µ T ∂µ T ∂µ T ∂µ T where we collected all ﬁrst order terms (NN and 3N) in Ω1, and the second order diagrams (2,n) and (2,a) in Ω2. It has been shown by Kohn and Luttinger [KL60] that the inversion of Eq. (4.85) can be performed by expanding µ as a perturbation series

µ = µ0 + µ1 + µ2 + ..., (4.86) where all terms are formally of the same order as the respective Ω terms, i.e. µ0 ∼ Ω0, µ1 ∼ Ω1, and so on. The lowest order term µ0 is ﬁxed by the condition that it is equal to the chemical potential of a non-interacting system with the same particle density ρ as the fully interacting system, i.e.

∂Ω0 ρ = − , (4.87) ∂µ µ0 which tells us that the density is of order zero in that ordering scheme. The higher order terms can be determined iteratively by inserting Eq. (4.86) into the density equation

43 4. Thermodynamics of Many-Particle Systems

(4.85), subsequently expanding each term around µ0 and then solving the equation order by order. Regarding the ﬁrst term µ1 this leads to:

∂Ω1/∂µ µ1 = − 2 2 . (4.88) ∂ Ω0/∂µ µ0 The Legendre transformation in (4.84) can then be performed in an analogous manner, namely by using Eq. (4.86) and expanding every term around µ0. Keeping only terms up to second order this results in:

2 2 ! ¯ 1 ∂Ω0 µ1 ∂ Ω0 ∂Ω1 F (ρ) = Ω0(µ0) + (µ1 + µ2) + + Ω1(µ0) + µ1 + Ω2(µ0) 2 2 ρ ∂µ µ0 ∂µ µ0 ∂µ µ0

+ µ0 + µ1 + µ2 . (4.89)

We see that the two terms involving µ2 cancel each other. Using Eq. (4.88) we arrive at:

2 ! ¯ 1 1 (∂Ω1/∂µ) F (ρ) = µ0 + Ω0(µ0) + Ω1(µ0) + Ω2(µ0) − . (4.90) 2 2 2 ρ ∂ Ω0/∂µ µ0 Because at second order only contributions from NN interactions were considered we neglect the 3N-part in the last term in Eq. (4.90) (which is of second order in the expansion in terms of V). The free energy per particle is then given by:

" 2 #! ¯ 1 1 (∂Ω1,NN/∂µ) F (ρ) = µ0 + Ω0(µ0) + Ω1(µ0) + Ω2,n(µ0) + Ω2,a(µ0) − . 2 2 2 ρ ∂ Ω0/∂µ µ0 (4.91)

The second term in the square bracket (including the sign) is from here on called the anomalous derivative term (ADT). From the free energy per particle F¯, the pressure P , the chemical potential µ, the entropy per particle S¯ and the internal energy per particle U¯ follow by standard thermodynamic relations:

∂F¯(ρ, T ) P (ρ, T ) = ρ2 , (4.92) ∂ρ ∂F¯(ρ, T ) µ(ρ, T ) = F¯(ρ, T ) + ρ , (4.93) ∂ρ ∂F¯(ρ, T ) S¯(ρ, T ) = − , (4.94) ∂T U¯(ρ, T ) = F¯ + T S¯(ρ, T ) . (4.95)

Zero Temperature Limit The square bracket in Eq. (4.91), i.e. the second order anomalous contribution plus the anomalous derivative term, vanishes in the T → 0 limit. To show this, we deﬁne Z 3 d kα Fα := trσ ,τ Θk hα2 |(1 − P12)VNN| α2i , (4.96) α α (2π)3 α

44 4. Thermodynamics of Many-Particle Systems

which is a function of ~k2, σ2 and τ2. Together with Eq. (4.83) this allows to write the zero temperature limit of Ω2,a(µ0) as Z 3 T →0 1 d k2 Y Ω2,a(µ0) −−−→− trσ ,τ δ(εκ − εk ) Fα . (4.97) 2 2 2 (2 )3 F 2 π α∈{1,3} | {z } 2 = F1

Note that because all terms are evaluated at µ0, the chemical potential of the non- interacting system, we have now the Heavyside step function for the unperturbed Fermi sphere Θkα ≡ Θ(κF − kα), and εκF = µ0 instead of µ in the delta-function. The explicit form of the ﬁrst-order derivative term is the following (cf. Eq. (4.78)): Z 3 Z 3 ∂Ω1,NN 1 d k1 d k2 ∂nk1 ∂nk2 = trσ1,τ1 trσ2,τ2 nk2 + nk1 h12 |(1 − P12)VNN| 12i . 2 (2 )3 (2 )3 ∂µ µ0 π π ∂µ ∂µ µ0 (4.98)

T →0 With ∂np/∂µ −−−→ δ(εκ − εp) and by making use of Eq. (4.96) we can write the zero µ0 F temperature limit of the above expression as Z 3 ∂Ω1,NN T →0 1 d k2 −−−→ trσ2,τ2 δ(εκ − εk2 ) F1 2 (2 )3 F ∂µ µ0 π 3 3 + {1, εk1 , trσ1,τ1 , d k1} ↔ {2, εk2 , trσ2,τ2 , d k2} . (4.99)

Because the matrix element is invariant under the interchange 1 ↔ 2 in the bra- and ket-state (cf. section 3.4), the two summands in the above equation are equal, and we obtain Z 3 ∂Ω1,NN T →0 d k2 −−−→ trσ2,τ2 δ(εκ − εk2 ) F1 . (4.100) (2 )3 F ∂µ µ0 π To conclude the proof that the square bracket in Eq. (4.91) vanishes in the zero temper- 2 2 ature limit we need the ∂ Ω0/∂µ term. The derivation of the zeroth order contribution to the grand-canonical potential density, Ω0, has been omitted so far; its precise form is investigated in section 6.1.2. We need to borrow only one result from there, Eq. (6.10), which reads (written in a slightly diﬀerent way) Z 3 ∂Ω0 d k = − trσ,τ nk . (4.101) (2 )3 ∂µ µ0 π µ0 With the T → 0 rules developed above it follows immediately that 2 Z 3 ∂ Ω0 T →0 d k −−−→− trσ,τ δ(εκ − εk) . (4.102) 2 (2 )3 F ∂µ µ0 π With the following deﬁnition of an average over the unperturbed Fermi surface (for some momentum-dependent quantity X(k))

R d3k trσ,τ (2π)3 X(k) δ(εκF − εk) h X(k) iF ≡ (4.103) R d3k trσ,τ (2π)3 δ(εκF − εk)

45 4. Thermodynamics of Many-Particle Systems we can bring the zero temperature limit of the sum of the two anomalous terms into the compact form

2 Z 3 1 (∂Ω1,NN/∂µ) T →0 h 2 2i d k Ω2,a(µ0) − −−−→− F − hF1i trσ,τ δ(εκ − εk) 2 2 2 1 F F (2 )3 F ∂ Ω0/∂µ µ0 π 2 Z d3k F F = − 1 − h 1iF trσ,τ 3 δ(εκF − εk) F (2π) ≤ 0 . (4.104)

For spherically symmetric Fermi surfaces and interactions that are invariant under rota- tions, which are both the case here, this is obviously zero.

The foregoing cancellation is one specific example of a general theorem due to Kohn, Luttinger and Ward [LW60, KL60], which states that under the two conditions stated above the T → 0 limit of the free energy per particle F¯ always gives the energy per particle E¯, as calculated in the zero temperature formalism. This is not obvious at all, because, as indicated in the beginning of section 4.1, the zero temperature formalism calculates the energy of the interacting system that is in the state which adiabatically evolves from the non-interacting ground-state. The T → 0 limit of the finite temperature formalism however always yields the true ground-state of the interacting system. In the case where the Kohn-Luttinger-Ward theorem does not apply, this qualitative difference between the two formalisms manifests itself in the form of the non-vanishing zero temperature limit of the sum of the additional terms that appear in the finite temperature case, i.e. at second order the (2,a)-contribution and the anomalous derivative term 9, which then makes up the difference between the energy of the interacting ground-state and the adiabatically evolved state. Note that Eq. (4.104) shows that the ground-state energy is indeed lower than the energy of the abiabatically evolved state.

9 Since the zero temperature formalism predates the ﬁnite temperature one, these terms have been dubbed anomalous, which has now become the convention [FW72].

46 5. Chiral Low-Momentum Interactions

In the last chapter it was shown how to apply the chiral two- and three-body potentials to the nuclear matter system, and how to calculate thermodynamic quantities using many-body perturbation theory. This was based on the assumption that these potentials constitute a small correction to the non-interacting system. The chiral NN potential however exhibits certain nonperturbative features, which are discussed in section 5.1. The usual approach to this problem is to introduce an effective interaction, the Brueckner G-matrix, which resums in-medium particle-particle scattering (cf. [HJKO95]). This resummation however makes it difficult to treat particle-particle and particle-hole correlations on an equal footing [NBS04]. An alternative strategy consists in integrating out the high-momentum components of the chiral NN potential; using renormalization group (RG) methods, VNN can be evolved to an effective low-momentum potential Vlow-k(Λ), which exhibits strong perturbative behaviour for momentum cut-offs Λ around 2 fm−1. The low-momentum potential reproduces two-nucleon observables, i.e. NN scattering phase shifts and the deuteron binding energy with similar accuracy as the input potential VNN. We discuss the construction and the properties of Vlow-k(Λ) in section 5.2. By construction, Vlow-k(Λ) is much softer than the input potential VNN, making it more suitable for many-body perturbation theory. However, the properties of nuclear matter are significantly affected by many-nucleon forces, which raises the question whether a consistent evolution of three-nucleon forces, four-nucleon-forces, etc. is required for an accurate description of nuclear many-body systems 1. This problem is addressed in section 5.3.

5.1. Nonperturbative Features of NN Potentials

Nonperturbative behaviour in the two-nucleon channel arises from several sources. First, in order to ﬁt phase shifts using NN potentials with large cut-oﬀs, it is necessary to introduce a strongly repulsive short-range interaction (the so-called hard-core of the potential). Secondly, the tensor part of the potential is highly singular at short distances, and third there is the presence of low-energy bound states or nearly-bound states. All of these nonperturbative features are also present in phenomenological high-precision potentials, such as the CD-Bonn or the AV18 potentials (cf. Figure 5.1a), or potentials obtained from lattice simulations (cf. Figure 5.1b).

On can investigate the nonperturbative behaviour of the NN potential by looking at

1 Chiral four-nucleon interactions are not considered in this thesis. They have been found to give rise to an appreciable contribution to the nuclear equation of state [Kai12a]. This contribution is however expected to be reduced by other higher order corrections, e.g. by subleading three-body forces.

47 5. Chiral Low-Momentum Interactions the Born series of the K-matrix for nucleon-nucleon scattering:

KNN(E) = VNN + VNN G0(E) VNN + VNN G0(E) VNN G0(E) VNN + ..., (5.1) where G0(E) is the free nucleon propagator

1 QκF G0(E) = (free-space) ,G0(E) = (in-medium) . (5.2) E − H0 E − H0

QκF is the Pauli-blocking operator that allows scattering only to unoccupied states above the Fermi momentum κF . In momentum space and in a given partial wave the ﬁrst and second Born terms are given by

(1) 0 0 KNN(p , p; E) = VNN(p , p) , (5.3) ∞ M Z V (p0, q)Q (q, K)V (q, p) K(2) (p0, p; E) = N − dq q2 NN κF NN , (5.4) NN 2π2 E − q2 0

~ ~ ~ 0 ~ 0 where K = |k1 + k2| = |k1 + k2| is the total momentum in the two nucleon system, and ~ ~ 0 ~ 0 ~ 0 p = |k1 − k2| and p = |k1 − k2| the relative momenta of the in- and out-going nucleons, respectively. In the free-space case QκF has to be replaced with 1. Again, the dashed integral denotes the Cauchy principal value. The strongly repulsive core in the nucleon-nucleon potential leads to strong high- momentum matrix elements. As a direct consequence, higher Born terms become large. This has been explicitely calculated by Bogner et al. [BSFN05], who found that the second order Born term is several times larger than the ﬁrst-order contribution over all considered momenta (free-space and in-medium). The nonperturbative properties of VNN can be ascertained in a rigorous way through an analysis of the spectrum of the operator G0(E) VNN. As proven by Weinberg [Wei63], the Born series converges at energy E if and only if the eigenvalues satisfy

|ην(E)| < 1 ∀ν . (5.5)

As discussed by Bogner et al., the strong hard-core and the strong tensor force generate at least one large negative eigenvalue that causes the Born series to diverge. In the free-space scenario the largest positive eigenvalue is ην+ (Bd) = 1, which corresponds to the deuteron pole. This pole however is strongly suppressed in the in-medium case for suﬃciently high densities; Pauli-blocking drives ην+ into a perturbative regime, i.e. ην+ (Bd) . 0.7 for −1 −1 ρ ' 0.01 fm and ην+ (Bd) . 0.4 for ρ ' 0.07 fm [BSFN05]. We conclude that in nuclear matter the nonperturbative behaviour of the NN interaction due to bound and nearly-bound states is absent.

5.2. Construction of Low-Momentum Potentials

The remaining two perturbative features of the nuclear force, the hard-core scattering and the tensor force contributions, both result from the short-distance part of the chiral NN potential in coordinate space. The momentum space representation is given by the

48 5. Chiral Low-Momentum Interactions

(a) Phenomenological NN potentials (b) NN potential from Lattice QCD

Figure 5.1.: Phenomenological NN potentials (taken from [IAH07]) and NN potential from lattice QCD (taken from [Aok10]). Both ﬁgures show the central part of the 1 potential in the singlet S-wave VC (r)[ S0], whereas the lattice potentials also 3 include the central part in the triplet S-wave VC (r)[ S1], and the tensor part of the potential VT (r). One can clearly see the nonperturbative behaviour of both the central and the tensor part at short distances. Figure 5.1b also depicts the dominant contributions in diﬀerent areas, i.e. pion-exchange in the long-range part and two-pion exchange as well as exchange of heavier mesons in the intermediate part of the potential. The short-range part of VC is entirely dominated by the hard-core.

Fourier transform of VNN(r):

∞ 4π Z V (q) = dr r V (r) sin(qr) , (5.6) NN q NN 0 where q = |~k0 −~k| denotes the momentum exchange in the two-nucleon scattering process in the center-of-mass system 2. For large values of q the Fourier integral “sees” more of the short-distance part of VNN(r). Thus, the nonperturbative part of the coordinate space potential is incorporated in the high-momentum part of VNN(q). The idea is now to evolve the chiral NN potential to a low-momentum potential Vlow-k by using renormalization group (RG) methods. The RG evolution separates the details of the short-distance dynamics, while incorporating their eﬀects on low-energy phenomena, thereby removing the remaining nonperturbative behaviour of the NN potential.

We start with the half-on-shell K-matrix equation for the bare chiral NN potential

2 ~ ~ ~ In the center-of-mass system the momenta of the interacting nucleons are given by k1 = −k2 =: k and ~ 0 ~ 0 ~ 0 k1 = −k2 =: k .

49 5. Chiral Low-Momentum Interactions

3 in a given partial wave:

∞ M Z V (p0, q)K (q, p; p2) K (p0, p; p2) = V (p0, p) + N − dq q2 NN NN . (5.7) NN NN 2π2 p2 − q2 0

We write (p0, p; p2) to indicate that the in-going nucleons with relative momentum p = |~p | are on-shell, while the out-going nucleons with relative momentum p0 = |~p 0| are not. Next we deﬁne a low-momentum version of this equation by imposing a cut-oﬀ Λ:

Λ M Z V (p0, q)K (q, p; p2) K (p0, p; p2) = V (p0, p) + N − dq q2 low-k low-k . (5.8) low-k low-k 2π2 p2 − q2 0 We demand the matching condition

0 2 0 2 0 Klow-k(p , p; p ) = KNN(p , p; p ) ∀ p , p < Λ , (5.9) in order to ensure that Vlow-k gives the same low-momentum two-nucleon observables as the full potential VNN. This means that the low-momentum K-matrix in Eq. (5.8) does not depend on the cut-off Λ. Differentiation with respect to Λ then leads to the following flow-equation for Vlow-k:

d M V (p0,Λ)K (Λ, p; Λ2) V (p0, p) = N low-k low-k . (5.10) dΛ low-k 2π2 p 2 1 − Λ Note that the K-matrix in the right-hand side of Eq. (5.10) is left side half-on-shell, as opposed to the K-matrices in Eqs. (5.7) and (5.8), and therefore not RG-invariant. Because Eq. (5.10) is asymmetric with respect to p and p0, it generates a non-Hermitian Vlow-k. This deﬁciency can be cured by working with a symmetrized version of Eq. (5.10):

0 2 0 2 ! d 0 MN Vlow-k(p ,Λ)Klow-k(Λ, p; Λ ) Klow-k(p ,Λ; Λ )Vlow-k(Λ, p) Vlow-k(p , p) = + , dΛ 4π2 1 − p 2 p0 2 Λ 1 − Λ (5.11) which preserves the on-shell K-matrix (cf. [BKS03a] pp. 15-16). As an alternative to the derivation based on half-on-shell K-matrix equivalence, this flow-equation can also be derived using the Kuo-Lee-Ratcliff folded diagram technique, or with similarity transformation methods (for both cf. [BKC01], and references therein). To obtain Vlow-k one can now integrate the RG equation numerically with VNN as the large-cutoff initial condition. A more practical approach however is the calculation of VNN with so-called Lee-Suzuki model space methods (cf. [BKC01, BKS03a]).

The matching condition (5.9) guarantees that the low-momentum potential Vlow-k(Λ) reproduces two-nucleon observables. This has been checked in detail by Bogner et al.; + in [BKS 03b] it is shown that Vlow-k(Λ) produces the same NN scattering phase shifts

3 In fact, one can start from an arbitrary high precision nucleon-nucleon potential. For cut-oﬀs Λ below −1 + 2.1 fm the low-momentum potential Vlow-k becomes model-independent [BKS 03b].

50 5. Chiral Low-Momentum Interactions as the original input potential, and in [BSFN05] that it reproduces the deuteron pole ην+ (Bd) = 1 in the free-space Born series. In addition, they find that in the in-medium case the deuteron pole is even more supressed for decreasing cut-offs. More striking is however the result that the largest negative eigenvalue ην− becomes perturbative for cut- −1 −1 offs Λ . 5 fm . At a cut-off around Λ ' 2 fm it is |ην− (Bd)| . 0.1, which indicates that the Born series converges rapidly. We conclude that the remaining two sources of nonperturbative behaviour of the nuclear force, the hard-core and the strong tensor force, have been removed by reparametrizing the short distance part of the NN potential in terms of the effective potential Vlow-k. Thus, Vlow-k should converge significantly faster in perturbative many-body calculations than the chiral VNN.

5.3. Chiral 3N Forces and Low Energy Constants

When applied to the calculation of low-energy two-nucleon observables Vlow-k is by construction physically equivalent to the full potential VNN (for energies below the cut-off Λ). However, at second order in the many-body formalism the NN potential gets iterated also in three-nucleon systems. Since Vlow-k is constructed within the NN system, three-body forces due to degrees of freedom missing in ChEFT (i.e. the “genuine” chiral 3N force) as well as due to the truncation of the intermediate nucleon lines in 3N scattering diagrams to low momenta (“induced” 3N forces) are neglected. The question whether and how this issue can be rendered compatible with the RG-evolution of only the two-body force is the topic of this section. In the first part of the section we discuss the genuine chiral 3N force at N3LO in the context of the many-body formalism. In detail, we evaluate the three-body contribution to the grand-canonical potential density, Ω1,3N, and discuss the values of the low-energy constants that parametrize the chiral 3N force. The second part then examines the occurance of iterated NN interactions in the many- body formalism. Two types of diagrams appear in the iteration: ladder diagrams in the two-body channel and consecutive NN scattering (in different NN subsystems) in the three-body channel. Both types of diagrams contribute to the second order contributions Ω2,n and Ω2,a.

5.3.1. Genuine 3N Force from ChEFT At NNLO in ChPT there are three diﬀerent irreducible 3N diagrams (cf. Figure 3.1): a pure contact diagram which is proportional to the LEC cE, a contact-1PE diagram proportional to cD, and a 2PE diagram that depends on c1, c3 and c4. The corresponding interactions are all of order ∆ = 1 in the interaction index (cf. section 2.7.4); their explicit form is given by the Lagrangian in Eq. (2.86). The corresponding many-body diagrams are obtained by closing the external legs of the chiral interaction diagrams. In the case of the 2PE diagram this can be done in two qualitatively diﬀerent ways. Thus, there arise two many-body diagrams from it: the Hartree- and the Fock-diagram (cf. Figure 5.2).

51 5. Chiral Low-Momentum Interactions

(a) cE (b) cD (c) Hartree (d) Fock

Figure 5.2.: Contributions to Ω1,3N from chiral 3N Forces at NNLO

Accordingly, there are four contributions to Ω1,3N, which can be written in the compact form ∞ ∞ ∞ Z p Z p Z p Ω = dp 1 dp 2 dp 3 K n n n , (5.12) 1,3N 1 2π2 2 2π2 3 2π2 3 p1 p2 p3 0 0 0

(cE ) (cD) (H) (F ) where K3 = K3 + K3 + K3 + K3 , with each part corresponding to the respective diagram in Figure (5.2). The explicit form of these terms is given in Appendix C.1. The three LECs appearing in the Hartree- and Fock-type 2PE interactions appear also in NN force diagrams. Hence, they can be determined by fitting to NN scattering phase 3 shifts. Using the regularized chiral N LO two-body-potential VNN (cf. section 3.3) Entem −1 −1 and Machleidt [EM03] obtained the values c1 = −0.81 GeV , c3 = −3.2 GeV and −1 c4 = 5.4 GeV . The other two LECs, cD and cE, can then be constrained by fitting to properties of few-nucleon systems. Gazit et al. [GQN09] obtained cD = −0.20 and cE = −0.205 by fitting to the triton half-life and the binding energies of A = 3 nuclei. These values for c1, c3 , c4, cD and cE will be referred to as the Machleidt LECs in the remainder of the thesis. Another set of ci LECs has been determined by the Nijmegen group [RTFdS99] in an partial-wave analysis of the proton-proton interaction. They obtained the values c1 = −1 −1 −1 −0.76 GeV , c3 = −4.78 GeV and c4 = 3.96 GeV , which we refer to as the Nijmegen LECs.

5.3.2. Iterated NN Interaction and Contact Terms At second order in the many-body system the NN interaction can be iterated in two diﬀerent ways: on the one hand one can generate a two-nucleon ladder diagram, on the other hand one can connect three diﬀerent nucleon lines (cf. Figure 5.3). Both of

(a) Second order ladder (b) Iterated NN force for diagram 3N system

Figure 5.3.: Diﬀerent possibilites of iterating the NN interaction.

52 5. Chiral Low-Momentum Interactions

these interactions contribute to Ω2, i.e. by closing the external legs in Figure 5.3a one obtains the (2,n)-diagram, and from Figure 5.3b we get the (2,n)- or the (2,a)-diagram (depending on how the external legs are closed). Figure 5.3b however corresponds to a three-nucleon scattering process; the relative momentum corresponding to the out-going nucleon on the left-hand side of the diagram and the internal line labelled by “k” has to be integrated over. When the interaction is mediated by Vlow-k this momentum cannot go beyond the cut-off Λ. Because this diagram (and of course also 3N diagrams of higher order) is not taken account of by the matching condition Eq. (5.9), the values of three- 3 3 body observables calculated using Vlow-k (e.g. the binding energies of H and He) have a remaining dependence on the cut-off Λ. In three-nucleon systems also the chiral three- body interaction comes into play. Hence, for a consistent treatment of three-body systems, a full RG-evolution of the combined two- and three-nucleon potential would be needed, which is however not yet available [BSFN05]. As a substitution for this combined evolution Nogga et al. [NBS04] have refitted the chiral NNLO three-body potential V3N to be consistent Vlow-k. This approach is based on the idea that the cut-off dependence of the diagram in Figure 5.3b can be balanced by introducing an “induced” 3N force. Nogga et al. assessed the size of induced 3N forces by examining the cut-off dependence of 3N and 4N binding energies (calculated by solving the Faddeev-Yakubovsky equations with only the two-body Vlow-k). They found that the 3N forces due to the truncation to low momenta are of the same order as the genuine chiral three-body forces 4. Based on this observation they then used the Nijmegen LECs that parametrize the two-pion exchange part of V3N and matched the short and intermediate 3 3 distance part given by the cD- and cE-contributions to binding energies of H, He and 4He 5. They obtained the following values:

−1 cE(Λ) = −0.625 cD(Λ) = −2.062 for Λ = 2.1 fm , (5.13) −1 cE(Λ) = −0.822 cD(Λ) = −2.785 for Λ = 2.3 fm . (5.14)

To what extent this substitution works for calculations in many-body systems is not clear at this stage. One goal of the thesis is to investigate the quality of this approach in the context of perturbative thermodynamics of nuclear matter.

4 It should be noted that in effective field theory approaches the effects of the truncation to small cut-offs are inseparable from those of missing degrees of freedom like the ∆ [NBS04]. 5 Nogga et al. used the chiral 3N force multiplied with a sharp regulator of the form exp − (p/Λ)8, + where Λ is the cut-off taken from Vlow-k (cf. [NBS04, ENG 02]). In our expression for the three- body grand-canonical potential, Eq. (C.1), all momenta are formally integrated over from 0 to ∞ (numerically this is realized by evaluating every pi-integral up to some cut-off x, i.e. pi ∈ [0, x], which is then gradually increased, x → x + δx, until the relative size of the increase of Ω1,3N is below a −7 fixed value, e.g. 10 %). This difference affects the size of Ω1,3N only for high temperatures. Our neutron matter results (cf. section 7.2.1, comparison with results obtained by Tolos et al.) show that at T = 10 MeV the difference due to the regulator is negligible. Nevertheless, the dependency of the three-body contribution Ω1,3N on the regulator is something which should be further investigated.

53 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Having established the framework of nuclear thermodynamics with chiral low-momentum interactions we can now proceed to evaluate the diﬀerent contributions to the free energy per particle in Eq. (4.91). We begin with the derivation of the free grand-canonical potential density Ω0(µ0) in section 6.1 and proceed with the partial wave representation of the expressions for the NN diagrams (Ω1,NN, Ω2,n and Ω2,a) derived in chapter4 in sections 6.2, 6.3 and 6.4. The anomalous derivative term follows immediately from Ω0 and Ω1,NN. The three-body contribution Ω1,3N has already been discussed in section 5.3. In each section, we derive also the corresponding zero temperature quantities, i.e. the diﬀerent contributions to the energy per particle E¯.

6.1. Zeroth Order: Free Fermi Gas Contribution

When all interactions among the nucleons are neglected, the description of nuclear matter is equivalent to that of a free Fermi gas. In its momentum space decomposition the free Hamiltonian is given by

X † H0 = εp aˆ~p,σ,τ aˆ~p,σ,τ , (6.1) ~p,σ,τ where εp is the kinetic energy of a nucleon with momentum ~p. Starting from the relativistic mass-energy relation and expanding around p = 0 we obtain q 2 4 6 8 2 2 p 1 p 1 p p εp = p + MN − MN = − 3 + 5 + O 7 . (6.2) 2MN 8 MN 16 MN MN The ﬁrst term, the nonrelativistic kinetic energy, is counted as order zero in this expansion. In this section, we use Eq. (6.2) to derive relativistic corrections to the free grand- canonical potential density Ω0(µ0), which are then compared to the results obtained from the description of a free Fermi gas with fully relativistic kinematics. The purpose of this endeavor is to show that (most of) these corrections are negligibly small, and that it is justiﬁed to use non-relativistic Fermi-Dirac distributions instead of relativistic ones (which is demanded from consistency considerations since we use the non-relativistic many-body formalism to take account of nuclear interactions).

6.1.1. Relativistic Corrections at Zero Temperature The particle number N is given by the expectation value of the number operator: X X N = hN i = hΦ0 | nˆ~p,σ,τ | Φ0i = Θ(κF − p) . (6.3) ~p,σ,τ ~p,σ,τ

54 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Using the continuum limit to evaluate the sum over momenta ~p we get: X V Z 2 N = d3p Θ(κ − p) = V κ3 . (6.4) (2π)3 F 3π2 F σ,τ This allows to write down the expression for the particle density: N 2 ρ = = κ3 . (6.5) V 3π2 F We see that at zero temperature the particle density of the free nucleon gas in the ground- state depends solely on the Fermi momentum κF . In particular, the formula (6.5) is rel- ativistically exact, i.e. there are no relativistic corrections for the particle density. The energy of the free Fermi gas is given by the expectation value of H0: V X Z E = hΦ | H | Φ i = d3p ε . (6.6) 0 0 0 0 (2π)3 p σ,τ ~p≤κF By inserting the expansion (6.2) and using the density formula (6.5) we obtain the following non-relativistic approximation of the energy per particle:

2 4 6 ¯(nonrel+1st+2nd) 3 κF 3 κF 1 κF E0 = − 3 + 5 . (6.7) 10 MN 56 MN 48 MN The leading term together with the ﬁrst relativistic correction is already a very good −1 −1 approximation to the full relativistic formula, Eq. (6.6). For κF ∈ [0.50 fm ; 1.80 fm ] −1 with stepsize ∆κF = 0.01 fm the mean relative error of the approximative formula ¯(nonrel+1th) E0 is 0.03 %, while the leading term alone leads to a mean relative error of 0.84 % (cf. Figure 6.2).

6.1.2. Relativistic Corrections at Finite Temperature At ﬁnite temperature the starting point is the grand partition function in the case of one fermionic degree of freedom j:

−β(H0−µ0N ) −β(εj −µ0) Zj = tr e = 1 + e , (6.8) where the trace runs over states where j is either occupied or not. The grand partition Q function of the non-interacting many-particle system is then Z = j Zj, where the index j runs over the spin, isospin and momentum degrees of freedom. With this, the free grand-canonical potential density is given by:

∞ 1 Y 1 2 Z Ω = − ln Z = − dp p2 ln 1 + e−β(εp−µ0) . (6.9) 0 βV j β π2 j 0

The particle density is given by the derivative of Ω0 with respect to the chemical potential: ∞ ∞ Z −β(ε−µ0) Z ∂Ω0 2 2 e 2 2 ρ = − = 2 dp p −β(ε −µ ) = 2 dp p np . (6.10) ∂µ0 π 1 + e p 0 π 0 0

55 6. Evaluation of Many-Body Diagrams for Nuclear Matter

This is the formula that was anticipated in the second part of section 4.4.

As a consistency check we can now calculate the zero temperature limit of the free energy per particle Ω F¯ = µ + 0 , (6.11) 0 0 ρ which should lead to the expression (6.7) for the energy per particle. Making use of L’Hˆopital’srule we obtain

∞ Z L0H 2 2 lim Ω0 = lim dp p (εp − µ0) np . (6.12) T →0 T →0 π2 0

The Fermi-Dirac distribution np becomes Θ(κF − p) in the zero temperature limit (and the chemical potential µ0 becomes the Fermi energy εκF ). We therefore have

κ Z F 2 2 lim Ω0 = dp p (εp − εκ ) , (6.13) T →0 π2 F 0

2 3 lim ρ = κF . (6.14) T →0 3π2 Comparing with Eq. (6.11) we ﬁnd that the above two equations give the correct zero ¯ T →0 ¯ ¯ temperature limit F0 −−−→ E0, where E0 is given by Eqs. (6.5) and (6.6).

Non-relativistic approximations to the free grand-canonical potential density can be obtained by making use of the identity

∞ ∞ Z Z f 0(k) k3 dk k2 ln f(k) = − dk , (6.15) f(k) 3 0 0 which, when applied to Eq. (6.9), leads to

∞ 2 Z Ω = − dp p3 n ∂ ε . (6.16) 0 3π2 p p p 0

Using the expansion of the kinetic energy εp, Eq. (6.2), the grand-canonical potential density becomes

∞ ∞ ∞ Z 4 Z 6 Z 8 (nonrel+1st+2nd) 2 p 1 p 1 p Ω0 = − 2 dp np + 2 dp 3 np − 2 dp 5 np . (6.17) 3π MN 3π MN 4π MN 0 0 0 where the ﬁrst term is the non-relativistic approximation and the other terms are relativistic corrections. In order verify that Eq. (6.17) deﬁnes a valid approximation of the relativistic formula given by Eq. (6.9) we need to check whether it gives the correct zero temperature limit.

56 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Note that at this stage the Fermi-Dirac distributions in Eq. (6.17) and Eq. (6.10) are so far still in their fully relativistic form, with the kinetic energy εp given by the relativistic 0 mass-energy relation. An expansion around an arbitrary approximation εp leads to

0 0 0 0 2 np = np + npn¯pβ(εp − εp) + O(∆εp) , (6.18)

0 0 where np is the Fermi-Dirac distribution with εp instead of εp. Using L’Hôpital’srule it T →0 0 can be shown that np −−−→ np, which means that corrections to np do not affect the zero temperature limit. 0 Now take εp to be given by the approximation defined in Eq. (6.2). This means that the zero temperature prescription np → Θ(κF − p) is altered in the sense that the Fermi momentum is now given by 0 = instead of = . Hence, we obtain for the → 0 εκF µ0 εκF µ0 T limit of the approximation given by Eq. (6.17):

2 4 6 T =0 κF 1 κF 1 κF µ0 −−→ − 3 + 5 , (6.19) 2MN 8 MN 16 MN ∞ −1 Z 4 6 8 Ω0 T =0 2 3 2 p 1 p 1 p −−→ 2 κF dp − 2 + 2 3 − 2 5 Θ(κF − p) ρ 3π 3π MN 3π MN 4π MN 0 2 4 6 κF κF κF = − + 3 − 5 . (6.20) 5MN 14MN 24MN

At every order in κF this leads to the correct zero temperature limit, Eq. (6.7), which leads to the conclusion that Eq. (6.17) can indeed be used to deﬁne a non-relativistic approximation of Eq. (6.9), but the kinetic energy in the Fermi-Dirac distributions must be approximated to the same order in the expansion deﬁned by Eq. (6.2) as Ω0. In the zero temperature limit a given approximation of Ω0 then leads to the corresponding ¯ approximation of E0 as given in Eq. (6.7). In particular, when we want to use non-relativistic Fermi-Dirac distributions, i.e. 0 = εκF κ2 F , the present expansion scheme tells us that we must not use any relativistic corrections 2MN (nonrel) to Ω0 . A counterexample to this demand is given by Fritsch et al. [FKW02], who have constructed the following non-relativistic approximation:

∞ ∞ Z 4 Z 6 (nonrel+FKW1) 2 p 1 p Ω0 = − 2 dp np − 2 dp 3 np . (6.21) 3π MN 4π MN 0 0 With non-relativistic Fermi-Dirac distributions this leads to the zero temperature limit (nonrel+1st) E0 . This can be extended to second order by adding the additional term

∞ Z 8 (FKW2) 1 p Ω0 = 2 dp 5 np , (6.22) 8π MN 0

(2nd) nonrel+FKW1 which leads to E0 in the T → 0 limit. Since Ω0 already agrees very well with the full relativistic expression, Eq. (6.9), (cf. Figures 6.1 and 6.2) we did not check whether this term leads to an improvement in accuracy. It suﬃces to conclude that it is

57 6. Evaluation of Many-Body Diagrams for Nuclear Matter possible to construct a consistent non-relativistic approximation that is suﬃciently accu- nonrel+FKW1 rate and uses non-relativistic Fermi-Dirac distributions, Ω0 , which we will use in our calculations.

Free Fermi Gas (Nucleons): Rel. Cor. - Full Plot 40

0 F/N [MeV]

-10 0th -20 2nd [FKW02] REL -30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ρ [1/fm3]

¯ Figure 6.1.: Zeroth order contribution to the free energy per particle, F0(ρ, T ), calculated with the relativistic formula, Eq. (6.9), (”REL”) and with the approximations (nonrel) (nonrel+1st+2nd) (nonrel+FKW1) Ω0 (“0th”), Ω0 (“2nd”) and Ω0 (“[FKW02]”). We used temperatures T [MeV] ∈ {5, 10, 15, 25}, where the curve at the bottom (red) corresponds to T = 25 MeV, the next (purple) to T = 15 MeV, and so on. The curve at the top (blue-gray) corresponds to zero temperature; it shows the relativistic energy per particle given by Eq. (6.6), (“REL”), and its approximations given by Eq. (6.7). At this resolution only the deviation of the non-relativistic approximation with no corrections (“0th”, dashed lines) from “REL” (solid lines) is visible.

58 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Free Fermi Gas (Nucleons): Rel. Cor. - Closeup 40

20 F/N [MeV]

5 0.295 0.3 0.305 0.31 0.315 0.32 ρ [1/fm3]

Figure 6.2.: Closeup of Figure 6.1. Still, only the deviation of the “0th” approximations is visible. In the zero temperature case the discrete points are obtained from the “1st” approximation; in the ﬁnite temperature case these are obtained from the “[FKW02]” approximation.

6.2. First Order Contribution from NN Potential

For zero temperature the (1,NN)-contribution is given by (cf. Eq. (4.78))

E 1 Z d3k Z d3k 1,NN = tr tr 1 2 h12 |(1 − P )V | 12i . (6.23) V 2 σ1,τ1 σ2,τ2 (2π)3 (2π)3 12 low-k ~ ~ |k1|≤κF |k2|≤κF E ~ ~ Here, |ii = kiσiτi is a one-nucleon state characterized by the momentum ki, the spin projection quantum number σi and the isospin projection quantum number τi. Further- more, P12 is the permutation operator which interchanges the two nucleons in the state |12i. For the potential Vlow-k in combination with 1 − P12, i.e. the antisymmetrized ˜ nucleon-nucleon-interaction, we from now on use the notation Vlow-k := (1 − P12)Vlow-k.

As Vlow-k is given in terms of partial waves, in order to evaluate Eq. (6.23) we need to perform a partial wave expansion of the two-nucleon states in the matrix element. In the case of the (1,NN)- and the (2,a)-contribution the matrix elements are diagonal in terms of two-particle states |iji. This is not the case for the second order normal diagram. Both cases can however be treated in an analogous manner. In general, the matrix element after the partial wave expansion will be of the form D E ˜ J,`1,`2,S,T p1J`1ST Vlow-k p2J`2ST ≡ p1 V p2 , (6.24) where p1 and p2 are the absolute values of the relative momenta of the nucleons in the respective two-particle states, J is the total angular momentum quantum number, `1 and `2 are the orbital angular momentum quantum numbers of the two-nucleon-states, and S

59 6. Evaluation of Many-Body Diagrams for Nuclear Matter and T are the total spin and total isospin quantum numbers. In order to bring the matrix elements in the form given by Eq. (6.24), starting from |12i E ~ being given in terms of one-particle states kστ in the original expression, Eq. (6.23), the following steps have to be accomplished (in this order):

• Coupling of single particle (iso)spins ~si (~τi) to the total (iso)spin S~ = ~s1 + ~s2 (T~ = ~τ1 + ~τ2)

• Change of the integration variables ~ki (the one-particle momenta) such that the matrix element is given in terms of relative momentum states |~p1i and |~p2i

• Expansion of the plane-wave states |~p i in spherical-wave states |p`mi

• Coupling of the total spin S~ and orbital angular momentum ~` to the total angular momentum J~

These transformations lead to two-particle states of the form |pJM`ST ti, where M is the eigenvalue of Jz = J~· eˆz, and t is the two-particle isospin projection quantum number. As the matrix elements do not depend on M and t, these numbers are omitted in Eq. (6.24).

6.2.1. Properties of Two-Body Matrix Elements The matrix elements, when evaluated in terms of two particle states |pJM`ST ti, have the following properties:

• They are independent of M and t

• The values of J, M, S, T and t are conserved

• The `-values in the bra- and the ket-state obey ∆` = |`1 − `2| ∈ {0, 2}

• The sum ` + S + T has to be odd

Conservation of J and M for instance follows immediately from rotational invariance, i.e. invariance under transformations generated by the total angular momentum operator J~. Moreover, charge conservation leads to conservation of t, as otherwise protons could be transformed into neutrons. Conservation of T follows from the fact that only isoscalar operator like 1 and τ1 · τ2 were allowed in order to guarantee charge independence of the NN potential. To see that the matrix element has to be diagonal with respect to S and to determine what values of `1 and `2 are possible, a bit more of an eﬀort has to be put forth. First of all these numbers depend on each other. This is due to the permutation operator P12, space spin isospin which can be decomposed as P12 = P12 P12 P12 . The spin part of a two-nucleon state is symmetric under exchange of the particles in the triplet case (where S = 1) and antisymmetric in the singlet case (where S = 0), and likewise for the isospin part. Thus, spin isospin S T it can be written P12 P12 = (−1) (−1) . The eﬀect of the spatial exchange operator space P12 is equivalent to that of the parity operator P, which, when acting on spherical wave

60 6. Evaluation of Many-Body Diagrams for Nuclear Matter

` states, gives P = (−1) (cf. [Sak94] p. 267). So in conclusion the operator 1 − P12 gives, when acting on two-nucleon states in the spherical wave basis: ( `+S+T 2 for ` + S + T odd 1 − P12 → 1 − (−1) = , (6.25) 0 for ` + S + T even

In the matrix element (6.24) the exchange operator can act either on the bra- or the ket-state, resulting in diﬀerent values of `, and of S if spin were not conserved. Hence, we need to know how these numbers can be changed by the interaction. The central and the spin-orbit part of the potential leave both spin and angular momentum invariant, so only the parts involving tensor operators matter in that regard. Let us consider the “usual” + tensor operator S12, which can be written in the alternative form (cf. [BDO 06]) r 24π S = [~s ⊗ ~s ](2) · Y (2)(ˆr) , (6.26) 12 5 1 2 where [. ⊗ .](k) means the coupling of two single particle operators in the brackets to a tensor of rank k, and (. · .) denotes the coupling of the rank 2 spin operator with the rank 2 spherical harmonic to a scalar in total angular momentum space. This decomposition of S12 allows to analyze the spin and the angular momentum part of the matrix element seperately by means of the Wigner-Eckart theorem (cf. [Sak94] p. 253), which states (k) that the matrix elements of a spherical tensor Tq with rank k and projection quantum number q satisfy

(k) 0 (k) 0 0 0 0 0 ` T ` `m T ` m = h` m kq | `m` ki √ , (6.27) q 2` + 1 where the Clebsch-Gordan coeﬃcient on the right hand side corresponds to the coupling of `0 and k to get `. The matrix element on the right-hand side is independent of m, m0 and q. From the requirement that the Clebsch-Gordan coeﬃcient be nonvanishing follows the triangular relation

|`0 − k| ≤ ` ≤ `0 + k . (6.28)

Applied to the spatial part of S12, where we have k = 2, this relation restricts the diﬀerence of the angular momentum quantum numbers to satisfy ∆` = |` − `0| ≤ 2. When the Wigner-Eckart theorem is applied to the spin-part of S12, where also k = 2, the corresponding triangular relation for the spin quantum numbers S and S0 can be fulﬁlled only if both are equal. This follows from the rules of addition of angular momenta, which state that when S = 0 and S0 = 1 the coupling of S0 to k would restrict S to lie in {1, 2, 3}. The possibility of having S = 1 and S0 = 0 is forbidden by a similar restriction. Therefore the matrix element is diagonal in S. But conservation of spin eliminates the two possibities of having ∆` = 1, because in that case the condition (6.25) together with conservation of T would lead to ∆S = 1. So altogether we have ∆` ∈ {0, 2}, ∆S = 0 and ∆T = 0 and, as a direct consequence, ` + S + T is odd in the bra-state of the matrix element if and only if it is odd also in the ket-state. Therefore it does not matter whether the permutation operator acts on the bra- or the ket-state.

61 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Finally, from the Wigner-Eckart theorem it follows also that the matrix element cannot depend on M and t. In the case of the total angular momentum numbers J and M, the Wigner-Eckart theorem states that the only dependence of the matrix element on M is through the Clebsch-Gordan coeﬃcient hJMkq | JMJki. Because the conservation of J and M allows only operators of rank k = 0 in total angular momentum space this Clebsch-Gordan coeﬃcient is unity. Thus no depedency on M remains, and also none on t, using similar arguments.

After all we are now able to tabulate the allowed matrix elements:

channel 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 J 011012223334 445 556 ` 001111222333 444 555 S 010111011011 011 011 T 100111000011 100 011

Table 6.1.: Matrix elements V J,`1,`2,S,T allowed by the selection rules discussed in this section (only the ﬁrst eighteen channels are tabulated). Bold faced numbers represent the coupled channels, in which there are additional matrix elements with (`1, `2) ∈ {(`, ` + 2), (` + 2, `), (` + 2, ` + 2)} and otherwise unchanged quantum numbers. These are exactly the cases when S = 1 and J = ` + 1, because only then ` and S can also be coupled to the same value of J when ` is increased by two. In all the other cases `1 = `2 = ` is the only allowed case.

6.2.2. Partial Wave Representation at Zero Temperature The necessary steps to transform Eq. (6.23) such that is can be evaluated in terms of the partial-wave matrix elements (6.24) will now be taken. To begin with, the spin-trace gives, when evaluated in terms of eigenstates of the total spin S~: D E X D E tr ˜ = ˜ σ1,σ2 12 Vlow-k 12 σ1σ2 Vlow-k σ1σ2 σ1σ2 D E X X 0 0 ˜ 0 0 = hσ1σ2 | Sςi hS ς | σ1σ2i Sς Vlow-k S ς σ1σ2 SςS0ς0 D E X ˜ = Sς Vlow-k Sς . (6.29) Sς In the last step the (second) orthogonality relation of Clebsch-Gordan coeﬃcients was P 0 0 used: h | i h | i = 0 0 . Similarely, the isospin-trace leads to: σ1σ2 σ1σ2 Sς S ς σ1σ2 δSS δςς D E X D E tr ˜ = ˜ (6.30) τ1,τ2 12 Vlow-k 12 T t Vlow-k T t . T t Here and before in the spin case only the relevant part (the (iso)spin part) of the state |12i is written out.

62 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Next we change the integration variables to the half-total and the half-relative momen- 1 ~ 1 ~ ~ 1 ~ ~ tum which are deﬁned as: K = 2 (k1 + k2) and ~p = 2 (k1 − k2). With the Jacobian of this transformation being J = 8 we get:

1 Z d3k Z d3k D E Z d3p Z d3K D E 1 2 ~k ~k V˜ ~k ~k → 4 ~p V˜ ~p . (6.31) 2 (2π)3 (2π)3 1 2 low-k 1 2 (2π)3 (2π)3 low-k ~ ~ ~ |k1|≤κF |k2|≤κF |~p|≤κF |K±~p |≤κF

The integration region for K~ is the volume of two identical spherical caps which are given by the intersection of a sphere of radius κF by two parallel planes whose distance from the center of the sphere is set by ~p and −~p. We therefore have

Z hπ 2π d3K = 2 (3a2 + h2) = (κ − p)2(2κ + p) , (6.32) 6 3 F F

|K~ ±~p|≤κF

p 2 2 where a = κF − p is the radius of the base of the caps and h = κF − p is the height of the caps.

Now we need to expand the two-nucleon states in spherical-wave states according to P ` ∗ the relation |~p i = 4πi Y`m(ˆp) |p`mi. This leads to: `,m

κF Z D E Z 0 Z D E 3 ˜ 2 2 X `−` ∗ 0 ˜ 0 0 d p ~p Vlow-k ~p = (4π) dp p i dΩ Y`m(ˆp)Y`0m0 (ˆp ) p`m Vlow-k p` m `m`0m0 |~p|≤κF 0 | {z } δ``0 δmm0 κF Z D E 2 2 X ˜ = (4π) dp p p`m Vlow-k p`m . (6.33) 0 `m

Finally, coupling the orbital angular momentum ~` with the total spin S~ gives D E X ˜ X X 0 0 `mSς Vlow-k `mSς = h`mSς | JM`Si hJ M `S | `mSςi `mSς `mSς JMJ0M 0 D E ˜ 0 0 × JM`S Vlow-k J M `S D E X ˜ = JM`S Vlow-k JM`S . (6.34) JM`S Again the second orthogonality relation of Clebsch-Gordan coeﬃcients was used. Because the matrix element does not depend on M, the sum over M gives a factor 2J + 1. When combining the particular results derived above (the four steps listed in the last

1 We use the half-total and half-relative momentum instead of simply the total and relative momentum in order to have the Heavyside step functions (or, at ﬁnite temperature, the Fermi-Dirac distributions)

in the simple form Θ|K~ ±~p| (or n|K~ ±~p|).

63 6. Evaluation of Many-Body Diagrams for Nuclear Matter section) we get for the energy per particle: κ ZF 3! ¯ 4 2 3p 1 p X J,`,`,S,T E1,NN = dp p 1 − + (2J + 1)(2T + 1) p V p . π 2κF 2 κF 0 J,`,S (6.35) The factor 2T +1 comes from the sum over t, which can be carried out because the matrix element does not depend on t.

6.2.3. Generalization to Finite Temperatures For ﬁnite temperature the single-particle momenta are not restricted to lie inside the Fermi sphere anymore, but instead are distributed according to the Fermi-Dirac distributions: Z d3k Z d3k Z d3k Z d3k 1 2 → 1 2 n n . (6.36) (2π)3 (2π)3 (2π)3 (2π)3 k1 k2 ~ ~ |k1|≤κF |k2|≤κF Changing to half-total and half-relative momentum then leads to: 1 Z d3k Z d3k D E Z d3p Z d3K D E 1 2 n n ~k ~k V˜ ~k ~k → 4 n n ~p V˜ ~p . 2 (2π)3 (2π)3 k1 k2 1 2 low-k 1 2 (2π)3 (2π)3 |K~ −~p | |K~ +~p | low-k |~p|≤Λ (6.37) The absolute value of the half-relative momentum must not exceed the low-momentum ~ 3 cut-oﬀ Λ. Furthermore, with ϑK = ^(~p, K), the azimuthal part of the d K-integral is trivial. Moreover, the integral over the polar angle is solveable:

1 Z ln(1 + eη+2x) − ln (e2x + eη) d cos ϑ n n = =: F(p, K) (6.38) K |K~ −~p| |K~ +~p| x (e2η −1) −1

2 2 where x = β Kp and η = β K +p − µ . 2MN 2MN These are the only modiﬁcations needed to get Ω1,NN from E1,NN. The grand-canonical potential density is then given by:

Λ ∞ Z Z 4 2 2 X J,`,`,S,T Ω1,NN = dp p dKK F(p, K) (2J + 1)(2T + 1) p V p . (6.39) π3 0 0 J,`,S 6.3. Second Order Normal Diagram

The expression for the second order normal contribution to the grand-canonical potential density is (cf. Eq. (4.79))

4 Z 3 ! 1 Y d ki 3 Ω = − tr (2π) δ ~k + ~k − ~k − ~k 2,n 8 σi,τi 3 1 2 3 4 i=1 (2π) ¯ ¯ − ¯ ¯ D E 2 nk1 nk2 nk3 nk4 nk1 nk2 nk3 nk4 ˜ × 12 Vlow-k 34 . (6.40) ε3 + ε4 − ε1 − ε2

64 6. Evaluation of Many-Body Diagrams for Nuclear Matter

In the case where the energy denominator is zero (when ε1 + ε2 = ε3 + ε4) also the numerator consisting of the diﬀerence of Fermi-Dirac distributions vanishes, leaving the integrand bounded. This can be seen by using the identity

β(εp−µ) n¯p = np e (6.41) which allows to write

2βµ β(ε3+ε4) β(ε1+ε2) nk1 nk2 n¯k3 n¯k4 − n¯k1 n¯k2 nk3 nk4 = nk1 nk2 nk3 nk4 e e − e . (6.42)

We will see that this property (the cancellation of the pole) is maintained in the partial wave expansion.

6.3.1. Partial Wave Representation at Finite Temperature In principle, the (2,n)-contribution (6.40) can be treated in the same manner as in the (1,NN)-case. The most obvious diﬀerence with respect to the partial wave expansion of the matrix elements is that here we have two of them instead of only one. In this regard, using the orthogonality relation of Clebsch-Gordan coeﬃcients twice, the spin-traces become: D ED E X D ED E ˜ ˜ = ˜ 0 0 ˜ trσ1,σ2,σ3,σ4 12 Vlow-k 34 34 Vlow-k 12 Sς Vlow-k Sς Sς Vlow-k Sς . Sςς0 (6.43)

In the case of the isospin-traces the additional condition ∆t = 0 leads to: D ED E X D ED E ˜ ˜ = ˜ ˜ trτ1,τ2,τ3,τ4 12 Vlow-k 34 34 Vlow-k 12 T t Vlow-k T t T t Vlow-k T t . T t (6.44)

The new integration variables are the half-total and half-relative momenta corresponding to the states |12i, K~ 1 and ~p1, and |34i, K~ 2 and ~p2, leading to a Jacobian J = 64. With these new variables the δ-function in the original expression (6.40) becomes δ(2K~ 1 −2K~ 2), which allows to eliminate one half-total momentum integral. Naming the remaining half- total momentum K~ , the diﬀerence of Fermi-Dirac distributions becomes

F(p1, p2, K, ϑ1, ϑ2) := n ~ n ~ n¯ ~ n¯ ~ − n¯ ~ n¯ ~ n ~ n ~ , |K+~p1| |K−~p1| |K+~p2| |K−~p2| |K+~p1| |K−~p1| |K+~p2| |K−~p2| (6.45)

~ where ϑ1/2 = ^(~p1/2, K), so contrary to the (1,NN)-case the Fermi-Dirac distributions have an angular dependence. With K~ in z-direction these angles become the polar angles 3 3 in the d pi-integrals and we can integrate out the angular part of the d K-integral, which gives a factor 4π. As a consequence of this additional dependence on the polar part of the half-relative momenta, the spherical harmonics that arise from the expansion of the two matrix elements in terms of spherical-wave states do not vanish as in the (1,NN)-case, i.e. as in Eq.

65 6. Evaluation of Many-Body Diagrams for Nuclear Matter

(6.33) 2. In detail, we have:

Λ Λ Z Z D ED E Z Z Z Z 3 3 ˜ ˜ 4 2 2 d p1 d p2 ~p1 Vlow-k ~p2 ~p2 Vlow-k ~p1 = (4π) dp1 p1 dΩ1 dp2 p2 dΩ2

|~p1|≤Λ |~p2|≤Λ 0 0 0 0 X `2−`1 ` −` ∗ ∗ × i i 1 2 Y (ˆp1) Y` ,m (ˆp2) Y 0 0 (ˆp2) Y`0 ,m0 (ˆp1) `1,m1 2 2 `2,m2 1 1 0 0 0 0 `1m1`2m2`1m1`2m2 D ED E ˜ 0 0 0 ˜ 0 0 × p1`1m1 Vlow-k p2`2m2 p1`2m2 Vlow-k p2`1m1 (6.46)

Because the matrix elements are not diagonal with respect to the `’s and m’s, also the Clebsch-Gordan coefficients that appear when the angular momenta are coupled to the total spin S~ survive. In the (1,NN)-case this cancellation automatically made the matrix element diagonal with respect to the J and M quantum numbers. Therefore we now have to make explicit use of J- and M-conservation, what leads to the simplification: D ED E X ˜ 0 0 0 ˜ 0 0 ... J1M1`1S Vlow-k J2M2`2S J2M2`2S Vlow-k J1M1`2S 0 0 0 0 J1M1J2M2J1M1J2M2 D ED E X ˜ 0 0 0 ˜ 0 0 = ... JM`1S Vlow-k JM`2S J M `2S Vlow-k J M `2S (6.47) JMJ0M 0 The Clebsch-Gordan coefficients, which have been left out in the above equation, are of the form hJM`S | `mSςi. Because they vanish for m 6= (M −ς) we can substitute M −ς for 0 m, thereby eliminating the sum over the mi’s (and the mi’s). With this substitution the remaining two azimuthal integrals, which concern only the spherical harmonics Y`m(ˆpi), lead to

2π 2π Z Z ∗ ∗ dϕ1 dϕ2 Y (ϑ1, ϕ1)Y` ,(M−ς )(ϑ2, ϕ2)Y 0 (ϑ2, ϕ2)Y`0 ,(M−ς )(ϑ1, ϕ1) `1,(M−ς1) 2 2 `2,(M−ς2) 1 1 0 0 2 = (2 ) 0 Y ( )Y ( )Y 0 ( )Y 0 ( ) (6.48) π δMM `1,(M−ς1) ϑ1 `2,(M−ς2) ϑ2 `2,(M−ς2) ϑ2 `1,(M−ς1) ϑ1 ,

imϕ where Y`,m(ϑ) are the spherical harmonics without the azimuthal part e . So only one of the M quantum numbers survives, and it receives an additional constraint, being now subject to the restriction |M| ≤ min(J, J 0). Altogether we then have: 3

2 It would be possible to get rid of the spherical harmonics related to one of the two matrix elements by using a coordinate system in which one of the two half-relative momenta is in z-direction, but then the angular dependence in (6.45) would involve two polar angles and one azimuthal angle. Thus, an additional azimuthal integral would have to be evaluated numerically. Because we aim for the expression which leads to easiest numerical evaluation, this possibility is neglected here. 3 The factor 32/π2 arises from the factor 1/(212π9) in the original expression (6.40), the factor 64 from the Jacobian, (4π)4 from the spherical-wave expansion, (2π)2 from the integrals over the azimuthal angles ϕ1 and ϕ2, a factor 4 from the two permutation operators and a factor 1/8 from the scaling of the transformed δ-function. The nucleon mass MN stems from the energy denominator.

66 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Λ 1 Λ 1 ∞ Z Z Z Z Z 32 2 2 2 F(p1, p2, K, ϑ1, ϑ2) Ω2,n = − 2 MN dp1 p1 d cos ϑ1 dp2 p2 d cos ϑ2 dKK 2 2 π p2 − p1 0 −1 0 −1 0 X X X ` −` `0 −`0 J,` ,` ,S,T D J0,`0 ,`0 ,S,T E 2 1 1 2 1 2 2 1 × i i p1 V p2 p2 V p1 0 0 0 S J,`1,`2 J ,`1,`2 min(J,J0) S S X X X × Ξ(ϑ1, ϑ2) . (6.49)

M=−min(J,J0) ς1=−S ς2=−S

The function Ξ(ϑ1, ϑ2) collects the spherical harmonics and the remaining Clebsch-Gordan coeﬃcients:

( ) = (2 + 1)Y ( )Y ( )Y 0 ( )Y 0 ( ) Ξ ϑ1, ϑ2 T `1,(M−ς1) ϑ1 `2,(M−ς2) ϑ2 `2,(M−ς2) ϑ2 `1,(M−ς1) ϑ1

× h`1(M − ς1)Sς1 | JM`1Si hJM`2S | `2(M − ς2)Sς2i 0 0 0 0 0 0 × h`2(M − ς2)Sς2 | J M`2Si hJ M`1S | `1(M − ς1)Sς1i 0 0 × Θ (min (`1, `1) ± (M − ς1)) Θ (min (`2, `2) ± (M − ς2)) . (6.50)

The Heavyside step functions (the Θ’s) in Ξ(ϑ1, ϑ2) arise from the substitution of the m quantum numbers with M − ςi, which brings an additional constraint on the sum over M and the sum over the ςi’s, due to mi ∈ [−`i, `i]. 2 2 The term 1/(p1 − p2) originates from the energy denominator in Eq. (6.40). It has a pole for p1 = p2, but it can be shown that in that case F(p1, p1, K, ϑ1, ϑ2) = 0. Using again the identity (6.41) we rewrite the F-function as

2 2 2 K p1 p2 2β 2M −µ β M β M F(p1, p1, K, ϑ1, ϑ2) = n ~ n ~ n ~ n ~ e N e N − e N . |K1+~p1| |K1−~p1| |K1+~p2| |K1−~p2| (6.51)

This expression vanishes for p1 = p2, so again the integrand is bounded.

6.3.2. Speciﬁcation to Zero Temperature In order to obtain the zero temperature expression for the second order normal contribution we have to substitute the Fermi-Dirac distributions in Eq. (6.45) with Heavyside step functions: ¯ ¯ ¯ ¯ F(p1, p2, K, ϑ1, ϑ2) = Θ ~ Θ ~ Θ ~ Θ ~ − Θ ~ Θ ~ Θ ~ Θ ~ . |K+~p1| |K−~p1| |K+~p2| |K−~p2| |K+~p1| |K−~p1| |K+~p2| |K−~p2| (6.52)

These can be absorbed into the boundaries of the integrals. For instance, the two “hole” Heavyside step functions in the ﬁrst term in Eq. (6.52) give rise to the following cascade of conditions on the integral variables:

67 6. Evaluation of Many-Body Diagrams for Nuclear Matter

( p1 ≤ κF Θ ~ Θ ~ ⇒ 2 2 2 , (6.53) |K+~p1| |K−~p1| κF −K −p1 −α1 ≤ cos ϑ1 ≤ α1 , where α1 := 2Kp1 q 2 2 α1 ≥ 0 ⇒ p1 ≤ κF − K , (6.54) 2 2 κF − p1 ≥ 0 ⇒ K ≤ κF . (6.55)

The two “particle” Heavyside step functions (in the same term) then lead to the additional conditions:

2 2 2 ¯ ¯ K + p2 − κF Θ ~ Θ ~ ⇒ −β2 ≤ cos ϑ2 ≤ β2 , where β2 := , (6.56) |K+~p2| |K−~p2| 2Kp1 q 2 2 β2 ≥ 0 ⇒ p2 ≥ κF − K , (6.57) 2 2 κF − p2 ≥ 0 ⇒ K ≤ κF . (6.58)

In these inequalities, the momentum p2 has no upper limit. However, since we are working with Vlow-k, it is restricted to values below the cut-oﬀ Λ. To summarize, Eqs. (6.53) to (6.58) imply the following restrictions on the integration regions:

q q h i 2 2 2 2 ˜ ˜ K ∈ [0, κF ] , p1 ∈ 0, κF − K , p2 ∈ κF − K ,Λ , ϑ1 ∈ [−α˜1, α˜1] , ϑ2 ∈ −β2, β2 , (6.59)

˜ where we deﬁnedα ˜i := min(αi , 1) and βi := min(βi , 1). In the exact same manner the second set of Heavyside step functions in the F-function results in the conditions

q q h i 2 2 2 2 ˜ ˜ K ∈ [0, κF ] , p1 ∈ κF − K ,Λ , p2 ∈ 0, κF − K , ϑ1 ∈ −β1, β1 , ϑ2 ∈ [−α˜2, α˜2] . (6.60)

Thus we have two separate contributions to ground-state energy (per volume) with diﬀer- ent integration regions, given by the conditions (6.59) and (6.60), respectively. Mutually renaming p1 and p2 in the second contribution leads to identical momentum integrals in both contributions, brings along an additional minus sign (in the second contribution) 2 2 ˜ ˜ due to the 1/(p2 − p1)-term, and leads to the interchanges β1 ↔ β2 andα ˜2 ↔ α˜1 (in the second contribution). By mutually renaming ϑ1 and ϑ2 we then obtain equivalent conditions on the integral boundaries of both contributions. With the particle density at zero temperature, Eq. (6.5), the expression for the energy per particle is then given by:

68 6. Evaluation of Many-Body Diagrams for Nuclear Matter

√ 2 2 κ κF −K Λ α˜ β˜ Z F Z Z Z 1 Z 2 ¯ 48 2 2 2 1 E2,n = − 3 MN dKK dp1 p1 dp2 p2 d cos ϑ1 d cos ϑ2 2 2 κF √ p2 − p1 0 0 2 2 −α˜1 −β˜2 κF −K X X X ` −` `0 −`0 J,` ,` ,S,T D J0,`0 ,`0 ,S,T E 2 1 1 2 1 2 2 1 × i i p1 V p2 p2 V p1 0 0 0 S J,`1,`2 J ,`1,`2 min(J,J0) S S X X X × Ξ(ϑ1, ϑ2) + Ξ(ϑ2, ϑ1) . (6.61)

M=−min(J,J0) ς1=−S ς2=−S

The contribution related to Ξ(ϑ1, ϑ2) is obviously the same as the one related to Ξ(ϑ2, ϑ1). Hence, we can leave out the second contribution and instead multiply the whole expression by a factor 2.

6.4. Anomalous Contributions

Second Order Anomalous Diagram The second order anomalous contribution is given by (cf. Eq. (4.80))

3 Z 3 ! β Y d ki Ω = − tr n n n¯ n 2,a 2 σi,τi 3 k1 k2 k2 k3 i=1 (2π) D E ˜ × h12 |(1 − P12)Vlow-k| 12i 23 Vlow-k 23 . (6.62) This can be written in an alternative form: Z 3 Z 3 β d k2 Y d kα Ω2,a = − trσ ,τ nk n¯k trσ ,τ nk hα2 |(1 − P12)Vlow-k| α2i . 2 2 2 (2 )3 2 2 α α (2 )3 α π α∈{1,3} π (6.63)

Changing to total spin states and making use of ∆S = 0 the spin-trace in the square bracket becomes D E X X D E tr α ˜ α = h | i h 0 | i ˜ (6.64) σα 2 Vlow-k 2 σασ2 Sς Sς σασ2 Sς Vlow-k Sς , σα Sςς0 whereas with ∆T = 0 and ∆t = 0 the isospin-trace gives

D E X X 2 D E tr α ˜ α = |h | i| ˜ (6.65) τα 2 Vlow-k 2 T t τατ2 T t Vlow-k T t . τα T t The last two equations, together with the fact that the matrix element only depends on relative momenta, make explicitly clear that the step from Eq. (6.62) to Eq. (6.63) (which boils down to the exchange of 1 and 2 in the matrix element) is allowed. 4 After the change of the integral variables to ~k := ~k2 and ~p := ~kα − ~k2 the momentum

4 To be precise, we would have to write ~pα, but as the α-index becomes obsolete later we omit it here.

69 6. Evaluation of Many-Body Diagrams for Nuclear Matter

D E ˜ part of the matrix element in the square bracket is ~p Vlow-k ~p . The Fermi-Dirac distribution in the square bracket has an angular dependence after this transformation, i.e. → . With = ( ~) the expansion of this matrix element in spherical- nkα n|~p−~k| ϑ ^ ~p, k wave states leads to

Λ +1 2π Z D E 0 Z Z Z 3 ˜ 2 X `−` 2 ∗ d p ~p Vlow-k ~p = 16π i dp p d cos ϑ dϕY`m(ˆp)Y`0m0 (ˆp) `m`0m0 −1 0 |~p|≤Λ 0 | {z } Y`m(ϑ)Y`0m0 (ϑ)δmm0 D E ˜ 0 0 × p`m Vlow-k p` m . (6.66) Following this, the coupling of spin and angular momentum to total angular momentum gives (taking into account ∆J = 0 and ∆M = 0) D E X ˜ 0 0 X X 0 0 0 `mSς Vlow-k ` mSς = h`mSς | JM`Si hJM` S | ` mSς i ``0mSςς0 ``0mSςς0 JM D E ˜ 0 × JM`S Vlow-k JM` S . (6.67) Substituting m with M −ς in the first Clebsch-Gordan coefficient and M −ς0 in the second we on the one hand get rid of the sum over m and on the other hand obtain a factor δςς0 which eliminates one of the ς’s. The product of the two square brackets with α = 1 and α = 3 can of course be transformed into the square of one square bracket, with no index whatsoever. The final result is then given by 5

∞ 1 1  Λ Z 2 2 Z 16 2 X X 2 X `−`0 D J,`,`0,S,T E Ω2,a = − β dk k nkn¯k  dp p i p V p π2 1 1 J,`,`0,S 0 σk=− 2 τk=− 2 0 2 (6.68) 1 J S  Z X X × d cos ϑ n|~p−~k| Ξ(ϑ) , −1 M=−J ς=−S where, after substituting σα with ς − σk (we write σk instead of σ2 to take account of our new integral variables) in Eq. (6.64) and τα with t − τk in Eq. (6.65), the Ξ-function is given by: 1 Ξ(ϑ) = Θ ± (ς − σ ) |hSς | (ς − σ ) σ i|2 2 k k k T X 1 2 × Θ ± (t − τ ) |hT t | (t − τ ) τ i| 2 k k k t=−T 0 0 × Y`,(M−ς)(ϑ)Y`0,(M−ς)(ϑ) h`(M − ς)Sς | JM`Si hJM` S | ` (M − ς)Sςi × Θ (min (`, `0) ± (M − ς)) . (6.69)

5 The factor 16/π2 comes from the factor 1/(210π9) in the original expression (6.62), the (4π)4 from the 2 spherical-wave expansion, the (2π) from the dϕ1- and dϕ3-integral, the 4π from the integral over the spherical angle Ωk and the factor 4 from the two permutation operators.

70 6. Evaluation of Many-Body Diagrams for Nuclear Matter

Anomalous Derivative Term From Eq. (6.10) it follows immediately that

∞ ∂2Ω (µ) 2 Z 0 = − dp M n . (6.70) ∂µ2 π2 N p 0

We also need the derivative of Ω1,NN with respect to the chemical potential. The only µ dependent part in Eq. (6.39) is the function F(p, K), i.e.

Λ ∞ Z Z Ω1,NN 4 2 2 ∂F(p, K) X J,`,`,S,T = dp p dKK (2J + 1)(2T + 1) p V p , (6.71) ∂µ π3 ∂µ 0 0 J,`,S

The µ-derivative of F(p, K) is given by

∂F(p, K) β ln(1 + eη+2x) − ln (e2x + eη) eη (1 − e4x) = 2 + , (6.72) ∂µ x (eη − e−η)2 (e2η −1) (eη+e2x ) (1 + eη+2x) where x and η are the same as in Eq. (6.38). Evaluating Eqs. (6.70) and (6.71) separately we then get the anomalous derivative term,

2 ¯ 1 (∂Ω1,NN/∂µ) FADT = − . (6.73) 2 2 2 ρ ∂ Ω0/∂µ µ0

71 7. Nuclear and Neutron Matter Equation of State

The numerical evaluation of the different contributions to the free energy per particle has been carried out using Fortran 95. All integrals were evaluated using the three-point Gauss-Legendre quadrature rule (cf. [AS65]). The numerical algorithm was tested using two model interactions; we present the test results in section 7.1. In section 7.2 we discuss the results for neutron matter. The free energy per parti- ¯ cle F (ρ) was calculated using Vlow-k with two different cut-offs, and also with the chiral potential from section 3.3.3. The pressure, chemical potential, entropy per particle and the internal energy per particle were then calculated by approximating the derivatives of F¯(ρ, T ) with respect to the density or the temperature with a finite difference approximation. In section 7.3 we then discuss the equation of state of isospin-symmetric nuclear matter. For temperatures below a critial value Tc isospin-symmetric nuclear matter occurs in two different phases, namely as a gas (for small ρ) and as a liquid (for large ρ). In the liquid-gas coexistence region the free energy per particle exhibits non-physical behaviour. The physical free energy per particle is obtained using the Maxwell construction. ¯ ¯ In section 7.4 we examine the role of the two anomalous contributions F2,a and FADT. It turned out that these contributions are responsible for an “anomalous” crossing of pressure isotherms, which however could be a numerical artifact.

7.1. Tests of Numerical Algorithm

The different contributions to Ω derived in the above sections (in this section referred to as Ωpw) involve a number of momentum integrals that have to be evaluated numerically. In order to test the numerical algorithm we consider two model interactions which are of a form that allows to calculate the different contributions to Ω directly (referred to as Ωanalytic), without expanding the matrix elements in partial waves. The first test interaction is given by the scalar-isoscalar boson exchange (abbreviated by ISOSC)

2 gs VISOSC = − 2 2 , (7.1) ms + q where gs = 1.0 is the coupling constant and ms = 700 MeV the mass of the exchange particle. Since this potential does not contain any tensor operators only matrix elements that are diagonal in ` are allowed in the partial wave representation. In order to test also coupled channels we therefore need another test interaction, one with tensor structure. We use the pseudoscalar boson exchange (abbreviated by MODPI)

72 7. Nuclear and Neutron Matter Equation of State

2 g (~σ1 · ~q )(~σ2 · ~q )(~τ1 · ~τ2) VMODPI = − , (7.2) 2 2 2 (mπ + q ) with coupling constant g = 2.5 and mπ = 300 MeV. The different contributions to Ωanalytic are given in AppendixD (for ISOSC) and in AppendixE (for MODPI). Because for the test interactions a momentum cut-off Λ is missing there is no upper bound in the (half-)relative momentum integrals (in the Ωpw-terms). Numerically this is no problem in the case of the (1,NN)-diagram, the anomalous derivative term and the (2,a) diagram, because there the part of the integrand associated with the matrix elements involves only Fermi-Dirac distribution of the form nK~ ±~p which vanish rapidly for high values of p. In contrast, the behaviour of the (2,n)-integrand is governed by the F-function given by Eq. (6.45), which involves alson ¯-type distributions. For high values of p2 (i.e. εp µ) the second part of F vanishes due to the factor n ~ n ~ , but 2 |K+~p2| |K−~p2| the factorn ¯ ~ n¯ ~ in the first part is then very close to unity. Thus high values |K+~p2| |K−~p2| of p2 can give a noticeable contribution to the end results, and similarly also high values of p1 (but not simultaneously, as then all n-type distributions vanish). This makes the evaluation of the (2,n)-diagram with these test interactions a rather tedious endeavor, which is the reason why only a few values were calculated in that case. The main loop parameter in the numerical algorithm is the “finite temperature Fermi momentum” κ defined by p κ = sign(µ0) 2µ0MN . (7.3)

Using Eq. (4.87) the data Ω(κ) can then be used to plot Ω vs. ρ. In Figures 7.1, 7.2, 7.3 and 7.4 the results from the direct computation are compared with the results using the partial wave representation of the matrix elements. The quantities σISOSC and σMODPI are the averaged relative errors |∆Ω/Ωanalytic|, where ∆Ω = Ωpw − Ωanalytic is the diﬀerence between the values obtained from the partial wave representation (“pw”) with respect to the direct results (“analytic”). Because ∆Ω scales with Ω these quantities give a good description of the achieved accuracy. For the (2,n)- diagram σISOSC and σMODPI were calculated with the eight points that can be found in the plot. In the other cases the given σ’s were actually calculated using 100 points, but for the sake of clarity only a few of them are included in the plots. The anomalous derivative term and the (2,a)-contribution cancel each other almost exactly at T = 5 MeV for these model interactions 1. One reason for the comparatively larger error of the MODPI results is the fact that the ISOSC-matrix elements vanish earlier, i.e. at lower relative momenta, leading to higher numerical precision. In the case of the (2,n)- and the (2,a)-diagram an additional reason is the worsening of the convergence of the partial expansion due to the appearance of coupled channel matrix elements. Nevertheless, the achieved errors are satisfyingly small for both model interactions.

1 This rough statement is made more qualitative in section 7.4, where the cancellation of the anomalous contributions is examined thoroughly.

73 7. Nuclear and Neutron Matter Equation of State

Test (1,NN)-diagram, T=5 MeV, nw=18 3.5 ISOSC analytic ISOSC pw 3 MODPI analytic MODPI pw 2.5 ] 3 2 σ MODPI=0.18% 1.5 [MeV/fm Ω 1 σ ISOSC=0.05% 0.5

0 0 0.05 0.1 0.15 0.2 0.25 ρ [1/fm3]

Figure 7.1.: Ω1,NN calculated with the scalar-isoscalar boson exchange potential (“ISOSC”) and the pseudoscalar boson exchange potential (“MODPI”). The temperature was set to T = 5 MeV and only the ﬁrst eighteen partial waves (“nw=18”) were taken into account (exactly those listed in Table 6.1).

Test (2,n)-diagram, T=5 MeV, nw=18 0

-2

] -4 σ 3 MODPI=1.65% -6 σ [MeV/fm ISOSC=0.47% Ω -8

ISOSC analytic [keV/fm3] -10 ISOSC pw [keV/fm3] MODPI analytic MODPI pw -12 0 0.05 0.1 0.15 0.2 0.25 ρ [1/fm3]

−3 Figure 7.2.: Ω2,n with test interactions. In the ISOSC case the Ω-axis has units keV fm .

74 7. Nuclear and Neutron Matter Equation of State

Test anom. deriv. term, T=5 MeV, nw=18 5 ISOSC analytic [10 keV/fm3] 4.5 ISOSC pw [10 keV/fm3] MODPI analytic 4 MODPI pw 3.5 ] 3 3 2.5 σ =0.01% [MeV/fm 2 ISOSC Ω 1.5 1 σ 0.5 MODPI=0.57% 0 0 0.05 0.1 0.15 0.2 0.25 ρ [1/fm3]

02 00 Figure 7.3.: The anomalous derivative term ΩADT := Ω1,NN/(2Ω0 ) with test interactions. In the ISOSC case the Ω-axis has units 10 keV fm−3.

Test (2,a)-diagram, T=5 MeV, nw=18 0 -0.5 σ MODPI=0.62% -1 -1.5 ] 3 -2 σ ISOSC=0.03% -2.5

[MeV/fm -3 Ω -3.5 3 -4 ISOSC analytic [10 keV/fm ] ISOSC pw [10 keV/fm3] -4.5 MODPI analytic MODPI pw -5 0 0.05 0.1 0.15 0.2 0.25 ρ [1/fm3]

Figure 7.4.: Ω2,a with test interactions. Again in the ISOSC case the Ω-axis has units 10 keV fm−3.

75 7. Nuclear and Neutron Matter Equation of State

7.2. Results for Neutron Matter

The modification of the different many-body contributions to produce the equation of state (EoS) of neutron matter is straightforward. The restriction to neutrons is achieved by 1 considering only particles with isospin τ = − 2 . Regarding the free Fermi gas contribution this means that the factor 2 from the isospin-trace is absent, and we get 1 1 Ωneutron = Ωnuclear , ρneutron = ρnuclear . (7.4) 0 2 0 2 The two-particle isospin is now restricted to T = 1 and t = −1. This leads to the following modifications

neutron 1 nuclear Ω1,NN = Ω1,NN , (7.5) 3 T =1 neutron 1 nuclear Ω2,n = Ω2,n , (7.6) 3 T =1

Ωneutron = Ωnuclear . (7.7) 2,a 2,a 1 T =1,τk=− 2

1 The factor 3 in the case of the (1,NN) and the (2,a) diagram comes from the factor (2T +1) in Eq. (6.39) and Eq. (6.50), respectively. The Pauli-principle forbids a three-neutron vertex. Therefore the cE-diagram and the cD-diagram do not exist in neutron matter. Also the contribution related to c4 is forbidden. The three-body contribution in neutron matter is then given by

Ωneutron = Ωnuclear , (7.8) 1,3N 1,3N 1 (H) 1 (F ) K3= 12 K3 + 6 K3 ,cE =cD=c4=0

1 1 where the factors 12 and 6 are related to the missing possibility of identifying hole-lines with protons in the Hartree- and the Fock-diagram in Figure 5.2.

7.2.1. Free Energy per Particle with Different Two- and Three-Body Interactions ¯ First we calculated the free energy per particle F (ρ) using Vlow-k(Λ) together with the chiral three-body potential V3n (with c1 and c3 given by the Nijmegen LECs). The results are plotted in Figure 7.5. We considered two different cut-offs, Λ = 2.1 fm−1 and Λ = 2.3 fm−1, and obtained no significant change in F¯(ρ). Comparing the T = 10 MeV curve to the one calculated by Tolos et al. [TFS08], who calculated the EoS of neutron matter using a similar model 2, we see that for densities above ρ ' 0.1 fm−3 our values for F¯ are slightly smaller (the difference is ∆F¯ ' 1 MeV at ρ = 0.15 fm−3). This is a consequence ¯ of our additional relativistic correction to the zeroth order contribution F0 (cf. Figure 6.2).

2 In detail, they used Vlow-k in combination with the leading order chiral 3N interaction, multiplied with the regulator mentioned in a footnote in section 5.3.2.

76 7. Nuclear and Neutron Matter Equation of State

Neutron Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 30

20 Λ=2.1 fm-1 vs. Λ=2.3 fm-1 10

-10

F/N [MeV] -20

-30 T = 0 MeV T = 5 MeV T = 10 MeV -40 T = 15 MeV T = 25 MeV -50 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ρ [1/fm3]

Figure 7.5.: Free energy per particle in neutron matter calculated with Vlow-k(Λ) and the chiral three-body potential V3n with Nijmegen LECs, for two diﬀerent cut- oﬀs; Λ = 2.1 fm−1 (solid lines) and Λ = 2.3 fm−1 (dashed lines). The curves overlap at this resolution.

Nijmegen Machleidt

c1 c3 c1 c3 ρ[fm−3] Hartree Fock Hartree Fock Hartree Fock Hartree Fock T = 5 MeV 0.05 0.11 -0.05 1.06 -0.45 0.12 -0.05 0.71 -0.30 0.10 0.39 -0.14 4.98 -2.11 0.41 -0.15 3.34 -1.41 0.15 0.77 -0.28 12.17 -5.13 0.82 -0.29 8.15 -3.43 T = 25 MeV 0.05 0.08 -0.02 1.41 -0.63 0.08 -0.02 0.94 -0.41 0.10 0.30 -0.10 5.83 -2.60 0.32 -0.11 3.91 -1.72 0.15 0.62 -0.21 13.48 -5.89 0.67 -0.22 9.03 -3.94

¯neutron Table 7.1.: Size of the diﬀerent contributions to F1,3N in units [MeV]. The −1 −1 Nijmegen are c1 = −0.76 GeV , c3 = −4.78 GeV , whereas the −1 −1 Machleidt LECs are c1 = −0.81 GeV , c3 = −3.2 GeV .

¯ We calculated F (ρ) also using the bare chiral two-body potential, VNN, together with V3n with the Machleidt LECs. The results are plotted in Figure 7.6 (dotted curve). One sees that the obtained values for F¯ are increasingly smaller for increasing densities than the one obtained with Vlow-k, independent of the temperature. This is mainly caused by the diﬀerent size of the three-body contribution. Some values are given in Table 7.1; ¯ one sees that the the size of F1,3N is almost independent of T , and that the contribution

77 7. Nuclear and Neutron Matter Equation of State

proportional to c3 is the largest. We also give some values of the two-body contributions in Table 7.2. One sees that the individual anomalous contributions are sizeable in each case, but their sum is small. For increasing temperature and density the size of the overall anomalous contribution ¯ ¯ increases. Comparing the results obtained with Vlow-k and VNN, we see that F1,NN, F2,a ¯ ¯ and FADT are larger when calculated with Vlow-k, whereas F2,n is larger when calculated with VNN. The size of the total second order contribution is smaller when calculated with ¯ Vlow-k, whereas the first order NN contribution F1,NN is larger. This signifies the expected increase in perturbative quality (with respect to the many-neutron system) resulting from the RG-evolution of VNN to low momenta. However, despite these differences, the value of the total two-body contributions is almost independent of the potential that is used. This is reflected in the dashed curve in Figure 7.6, which is obtained by using VNN in combination with V3N with the Nijmegen LECs.

Neutron Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 30

20 Λ=2.1 fm-1 vs. N3LO 10

-10

F/N [MeV] -20

-30 T = 0 MeV T = 5 MeV T = 10 MeV -40 T = 15 MeV T = 25 MeV -50 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ρ [1/fm3]

−1 Figure 7.6.: Free energy per particle in neutron matter, calculated with Vlow-k(2.1 fm ) and V3N as in Figure 7.5 (solid lines), with the chiral VNN in combination with V3N with the same LECs (Nijmegen) as for Vlow-k (dashed lines), and with VNN and V3N with Machleidt LECs (dotted lines).

78 7. Nuclear and Neutron Matter Equation of State

−1 Vlow-k(2.1 fm ) VNN ρ[fm−3] 1,NN 2,n 2,a ADT 1,NN 2,n 2,a ADT T = 5 MeV 0.05 -8.53 -1.41 -7.01 6.94 -7.21 -3.02 -4.76 4.70 0.10 -15.45 -1.44 -14.15 13.86 -12.44 -5.55 -8.23 8.02 0.15 -21.49 -1.31 -20.49 19.84 -16.42 -7.48 -9.98 9.56 T = 25 MeV 0.05 -6.80 -2.02 -2.86 2.73 -4.86 -4.45 -1.48 1.33 0.10 -13.02 -2.87 -8.38 7.94 -8.84 -8.02 -3.76 3.27 0.15 -18.78 -3.15 -14.58 13.69 -12.08 -11.03 -5.59 4.70

Table 7.2.: Contributions to F¯neutron[MeV] from two-body forces, calculated −1 with Vlow-k(2.1 fm ) and with the bare chiral VNN.

7.2.2. Derived Thermodynamic Quantities Here, we present results for the pressure, the chemical potential, the entropy per particle and the internal energy per particle, using the formulas from the end of the ﬁrst part of section 4.4 and the above results for the free energy per particle calculated with −1 Vlow-k(2.1 fm ).

Pressure and Chemical Potential In order to obtain the pressure P (ρ, T ) and the chemical potential µ(ρ, T ) from the free energy per particle F¯(ρ, T ), the derivative of F¯ with respect to ρ has to be calculated numerically. This means that it is necessary to make ρ the main loop parameter instead of κ, which can be done by inverting the relation ρ(κ) using linear interpolation. The derivative can then be calculated using a ﬁnite diﬀerence approximation:

∂F¯(ρ) −F¯(ρ + 2∆ρ) + 8F¯(ρ + ∆ρ) − 8F¯(ρ − ∆ρ) + F¯(ρ − 2∆ρ) ≈ . (7.9) ∂ρ 12∆ρ

We use ∆ρ = 0.0025 fm−3. The smoothness of the obtained curves serves as a natural tool for assessing the accuracy of this approximation.

In Figure 7.7 one sees that the P (ρ) T =0 and P (ρ) T =5 MeV curves cross each other at ρ ' 0.12fm−3. This is caused by the dispersion of the corresponding F¯(ρ)-curves in the high-density regime. By the thermodynamic relations (cf. [Sch04] pp. 84-92)

∂P α 1 ∂V 1 ∂V = , where α = , κT = − (7.10) ∂T V κT V ∂T P,N V ∂P T,N we learn, using the fact that the isothermal compressibility κT has to be greater than zero to guarantee the stability of the system, that this crossing implies a negative coeﬃcient of thermal expansion α in a particular subregion of T [MeV] ∈ [0, 5] (similar to the anomalous density increase of water for temperatures up to 3.984 ◦C).

79 7. Nuclear and Neutron Matter Equation of State

Regarding Figure 7.8, we observe that at equal chemical potential the interacting neutron gas is denser than the non-interacting. This is clear as the nuclear force acts pre- dominantly in an attractive way.

Neutron Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 3.5

3 -1 2.5 Λ=2.1 fm ] 3 2

1.5 P [MeV/fm 1 T=0MeV T=5MeV 0.5 T=10MeV T=15MeV T=25MeV 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ρ [1/fm3]

Figure 7.7.: Pressure P (ρ, T ) in neutron matter, calculated using the results for F¯(ρ, T ) −1 with Vlow-k(2.1 fm ) and Nijmegen LECs.

Neutron Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 60

50 Λ -1 40 =2.1 fm

20 [MeV] µ 10

0 T=0MeV T=5MeV T=10MeV -10 T=15MeV T=25MeV -20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ρ [1/fm3]

Figure 7.8.: Chemical potential µ(ρ, T ) in neutron matter, obtained from F¯(ρ, T ) with −1 Vlow-k(2.1 fm ) and Nijmegen LECs. The dashed lines show the chemical potential of the non-interacting system, µ0. The zero temperature curves show the analogous zero temperature quantities, i.e. the interacting and free Fermi energies, interacting and . εκF εκF

80 7. Nuclear and Neutron Matter Equation of State

Entropy per Particle and Internal Energy per Particle In principle, one could derive analytic expressions for S¯(ρ, T ) and U¯(ρ, T ) by differentiat- ing the relevant terms (the Fermi-Dirac distributions, and in the case of the (2,a)-diagram also the 1/T factor) in the different contributions to F¯(ρ, T ). However, as these expressions become rather involved in the case of the second-order contributions, we instead used a finite difference approximation (like the one in Eq. (7.9), but with respect to T instead of ρ; we use ∆T = 0.1 MeV) to calculate the temperature derivative of F¯(ρ, T ). At ρ → 0 the interactions between neutrons vanish and the classical limit exp(µ0/T ) 1 is approached. Thus, the interal energy per particle, shown in Figure 7.9, comes close to the value 3T/2 which corresponds to the equation of state of a classical ideal gas. In Figure 7.10 we show the entropy per particle S¯(ρ, T ). One sees that the curves do not cross each other; this is in accordance with the required positivity of the heat capacity CV (cf. [Sch04] pp. 90-91):

∂S C = V ≥ 0 . (7.11) ∂T V,N T

Neutron Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 40

30 Λ=2.1 fm-1 25

U/N [MeV] 15

10 T=0MeV T=5MeV T=10MeV 5 T=15MeV T=25MeV 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ρ [1/fm3]

Figure 7.9.: Internal energy per particle U¯(ρ, T ) in neutron matter, obtained from F¯(ρ, T ) −1 with Vlow-k(2.1 fm ) and Nijmegen LECs. The zero temperature curve shows the energy per particle E¯(ρ, T ).

81 7. Nuclear and Neutron Matter Equation of State

Neutron Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 6 T=5MeV T=10MeV 5 T=15MeV T=25MeV Λ=2.1 fm-1 4

3 S/N

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ρ [1/fm3]

Figure 7.10.: Entropy per particle S¯(ρ, T ) in neutron matter, calculated using the results ¯ −1 for F (ρ, T ) with Vlow-k(2.1 fm ) and Nijmegen LECs.

7.3. Results for Isospin-Symmetric Nuclear Matter

To calculate the equation of state of isospin-symmetric nuclear matter we proceed in principle in the same way as for neutron matter (but leave out the isospin restrictions). The main diﬀerence is that below a certain critical temperature Tc nuclear matter appears in two diﬀerent phases, as a liquid for high densities, and as a gas for low densities. In the intermediate density region (the liquid-gas coexistence region) the free energy per particle F¯(ρ, T ), plotted vs. the volume per particle υ = 1/ρ, leads to a concave curve. This however would imply a negative isothermal compressibility κT , which violates the stability of the system. The physical equation of state can be obtained by applying the Maxwell construction.

7.3.1. Free Energy per Particle with Different Two- and Three-Body Interactions In neutron matter we found no significant cut-off dependence of the two-body contribution ¯ ¯ ¯ ¯ ¯ FNN = F1,NN+F2,n+F2,a+FADT. However, in the case of isospin-symmetric nuclear matter there is a small, but visible dependence on the cut-off Λ (cf. Figure 7.11). This is caused by the larger size of the two-body contributions due to the missing isospin restrictions and ¯ overall factors (cf. Eqs. (7.5), (7.6) and (7.5)). Also the three-body contribution F3N is considerably larger in isospin-symmetric nuclear matter. In Table 7.3 some selected values of the additional cE(Λ), cD(Λ) and c4 contributions are given. One sees that the sum of the two cut-off dependent contact contributions is approximately cut-off independent. Figure 7.11 also shows some results obtained by Fiorilla et al. [FKW12a] in a different framework (in-medium ChPT with contact terms fitted to selected properties of nuclear matter at zero temperature). For the intermediate temperatures they approximately agree with our results, yet for T = 25 MeV they deviate significantly from our results

82 7. Nuclear and Neutron Matter Equation of State

−3 (but the curves seem to approach each other for ρ . 0.05 fm ). For zero temperature we see agreement only in the low-density regime. This means that we find also a different ¯ saturation point (the minimum of the zero temperature curve), namely E0 = −12.6 MeV, −3 −1 ρ0 ' 0.114 fm , κF 0 ' 1.19 fm = 236 MeV, which deviates from the empirical value ¯ −3 −1 + E0 = −16.0 MeV, ρ0 ' 0.17 fm , κF 0 ' 1.36 fm = 269 MeV [LLS 06]. Similar to the neutron matter case, the finite temperature curves display a singular behaviour for ρ → 0; this is a generic feature shared with other approaches to nuclear matter. However, at very low densities the nuclear-matter description vanishes anyway - nucleons cluster together, forming nuclei. In Table 7.4 we show selected values of the two-body contributions. Again we observe the increased convergence behaviour of the many-body perturbation series with Vlow-k.

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 30

20 Λ=2.1 fm-1 vs. Λ=2.3 fm-1 10 0 -10 -20 F/N [MeV] -30 T = 0 MeV -40 T = 5 MeV T = 10 MeV -50 T = 15 MeV T = 25 MeV -60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ρ [1/fm3]

Figure 7.11.: Free energy per particle in isospin-symmetric nuclear matter calculated with Vlow-k(Λ) and the chiral three-body potential V3n with Nijmegen LECs, for two diﬀerent cut-oﬀs; Λ = 2.1 fm−1 (solid lines) and Λ = 2.3 fm−1 (dashed lines). The single points are taken from the results obtained by Fiorilla et al. [FKW12a].

83 7. Nuclear and Neutron Matter Equation of State

Nijmegen Machleidt Λ = 2.1 fm−1 Λ = 2.3 fm−1 −3 ρ[fm ] cE cD cE cD c4 cE cD c4 T = 5 MeV 0.10 1.37 -1.02 1.78 -1.37 -0.31 0.44 -0.10 -0.23 0.20 5.42 -4.44 7.13 -6.00 -1.51 1.78 -0.43 -1.11 0.30 12.20 -10.46 16.04 -14.13 -3.76 4.00 -1.01 -2.76 T = 25 MeV 0.05 1.36 -1.19 1.78 -1.61 -0.42 0.44 -0.12 -0.31 0.10 5.43 -4.85 7.13 -6.55 -1.76 1.78 -0.47 -1.29 0.15 12.21 -11.07 16.05 -14.95 -4.12 4.00 -1.07 -3.02

Table 7.3.: Size of additional three-body contributions in isospin-symmetric nuclear matter, in units [MeV], calculated with the Nijmegen c4 = −1 3.96 GeV and the cut-oﬀ contact LECs (cE(Λ) = −0.625 and −1 cD(Λ) = −2.062 for Λ = 2.1 fm , cE(Λ) = −0.822 and cD(Λ) = −1 −2.785 for Λ = 2.3 fm ), and with the Machleidt LECs cE = −1 −0.205, cE = −0.2, c4 = 5.4 GeV . The contributions related to ¯ nuclear c1 and cE are given by the values in Table7 .1 with F (ρ) = 6F¯(ρ/2)neutron in the Hartree- and by F¯(ρ)nuclear = 3F¯(ρ/2)neutron in the Fock-case.

Figure 7.12) shows the free energy per particle calculated using VNN in combination −3 with V3N with Machleidt LECs. We see that for densities ρ & 0.1 fm the obtained curves (dotted lines) differ significantly from the ones obtained using Vlow-k plus V3N with Nijmegen LECs. Again, this is caused by the different LECs used with V3N, which lead to a considerable difference in the size of the dominant c3-contribution. In neutron matter this effect was suppressed by a factor 6, but here it causes the curves to spread to a large extent. In contrast, we find again (as in the neutron matter case) that the two-body contributions are almost model-independent (cf. Figure 7.12, dashed lines). The size of the three-body contribution Ω1,3N however depends strongly on the fitting-scheme used −3 to determine c3. For example, at T = 5 MeV and ρ = 0.3 fm the difference of the c3-contributions calculated with the Nijemgen and the Machleidt LECs is ∆c3 = (1 − Machleidt Nijmegen) ¯Nijmegen = 19 02 MeV, while the difference of all other contribu- c3 /c3 F1,3N;c3 . Nijmegen Machleidt tions, when calculated with Vlow-k + V3N and VNN + V3N , is ∆rest = 3.64 MeV, of which ∆NN = 0.87 MeV is the difference of two-body contributions only. We conclude that fitting c3 to two-nucleon phase shifts in different schemes leads to significantly different results in perturbative nuclear matter calculations. Because the Nijmegen-results agree much better with the results obtained by Fiorilla et al. we proceed with them instead of the ones calculated with Machleidt LECs.

84 7. Nuclear and Neutron Matter Equation of State

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 30

20 Λ=2.1 fm-1 vs. N3LO 10 0 -10 -20 F/N [MeV] -30 T = 0 MeV -40 T = 5 MeV T = 10 MeV -50 T = 15 MeV T = 25 MeV -60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ρ [1/fm3]

Figure 7.12.: Free energy per particle in isospin-symmetric nuclear matter, calculated with −1 Vlow-k(2.1 fm ) and V3N as in Figure 7.5 (solid lines), with the chiral VNN in combination with the same V3N (dashed lines), and with VNN and V3N with Machleidt LECs (dotted lines).

−1 Vlow-k(2.1 fm ) VNN ρ[fm−3] 1,NN 2,n 2,a ADT 1,NN 2,n 2,a ADT T = 5 MeV 0.10 -22.44 -10.42 -44.42 43.33 -17.48 -15.07 -27.49 27.03 0.20 -40.76 -11.25 -95.32 91.96 -29.71 -23.23 -46.31 44.76 0.30 -56.48 -10.06 -137.90 131.46 -38.94 -27.97 -55.88 52.82 T = 25 MeV 0.10 -18.19 -8.47 -20.51 19.34 -11.51 -15.41 -8.37 7.42 0.20 -34.43 -11.86 -58.26 54.33 -20.87 -25.74 -21.21 18.12 0.30 -49.14 -12.83 -98.35 90.75 -28.43 -32.98 -31.48 25.80

Table 7.4.: Contributions to F¯nuclear[MeV] from two-body forces, calculated −1 with Vlow-k(2.1 fm ) and with the bare chiral VNN.

7.3.2. Physical Free Energy per Particle, Critical Temperature, Liquid-Gas Phase Transition ¯ As pointed out in the beginning of the section, the concave region [υ1, υ2] of F (υ) (at ﬁxed temperature T < Tc) is unphysical (cf. Figure B.1 in AppendixB). The reason for

85 7. Nuclear and Neutron Matter Equation of State this is that concavity of F¯(υ) implies that

∂2 ∂ ρ ¯ υ υ 2 F ( ) = − P ( ) = < 0 . (7.12) ∂υ ∂υ κT

Hence, the isothermal compressibility κT would be negative in that region. This problem can be ﬁxed by applying the Maxwell construction, which means that we have to ﬁnd values υ1 = 1/ρ1 and υ2 = 1/ρ2 which satisfy (cf. [Sch04] 249-253) ¯ ¯ F (υ1) − F (υ2) = P0 (υ2 − υ1) , (7.13)

∂F¯(υ) ∂F¯(υ) P0 = − = − . (7.14) υ υ ∂ υ1 ∂ υ2

The physical free energy per particle is then obtained by substituting the part of F¯(υ) ¯ between υ1 and υ2 (the concave part) with the linear function f(υ) = P0(υ − υ1) + F (υ1). ¯ For zero temperature there are only two points at which E(υ) has the same slope, υ2 = ∞ ¯ and the minimum of E(υ), located at υ1. Thus, it is P0 = 0 in that case, and the energy ¯ per particle has the constant value E(υ1) for υ ≥ υ1. In the other cases one has to numerically solve Eqs. (7.13) and (7.14). The results are shown in Figure 7.13. We also calculated the free energy per particle for additional intermediate temperatures in order to narrow down the critical temperature. We find Tc ' 16.0 MeV (cf. Figure B.2 in AppendixB), which falls into the empirical range 15 MeV . Tc . 20 MeV deduced from multifragmentation and fission measurements [KOB+08]. The reason why the physical free energy per particle differs from the analytical free energy per particle is that for temperatures below Tc nuclear matter exists in two phases, namely as a liquid in the high-density region [0, υ1], and as a gas in the low-density region [υ2, ∞] (above Tc there is only the gas phase). Similar to a classical van der Waals gas, nuclear matter splits into two phases in the intermediate region [υ1, υ2], i.e. the liquid state with density ρ1, and the gas state with density ρ2. The pressure and the chemical potential remain constant in this interval, P (ρ ∈ [ρ2, ρ1]) = P0 and µ(ρ ∈ [ρ2, ρ1]) = µ0.

86 7. Nuclear and Neutron Matter Equation of State

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 30 T=0MeV 20 T=5MeV -1 T=10MeV Λ=2.1 fm T=15MeV 10 T=16MeV T=17MeV 0 T=25MeV

-10

F/N [MeV] -20

-30

-40

-50 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ρ [1/fm3]

Figure 7.13.: Physical free energy per particle in isospin-symmetric nuclear matter, obtained by the Maxwell construction (for T < Tc) and with Vlow-k(Λ) and the chiral three-body potential V3n with Nijmegen LECs.

7.3.3. Derived Thermodynamic Quantities The analytical pressure P (ρ, T ), shown in Figure 7.14, and the analytical chemical potential µ(ρ, T ), shown in Figure 7.15, can be calculated with the same methods already used in the neutron matter case. As already mentioned, in the liquid-gas coexistence region (transition region) the physical pressure and chemical potential are constant and determined by the Maxwell construction. Thus, the transition region is projected onto a single point in the P − µ diagram, Figure 7.16. Since the Maxwell construction does ¯ not preserve the curvature of F (υ) at the borders {υ1, υ2} of the transition region, both

P (ρ) T and µ(ρ) T (the physical quantities) are not diﬀerentiable at these points. More- over, when the density ρ(T < Tc) is plotted vs. the pressure, or vs. the chemical potential, one obtains a discontinuous curve. This renders the transition from the liquid to the gas state a ﬁrst-order phase transition in the Ehrenfest sense. The phase transition termi- nates at the critical point indicated by the dot in Figures 7.14, 7.15 andd 7.16. It is given by the following critical values of thermodynamic quantites (temperature, pressure, −3 chemical potential, density): Tc ' 16.0 MeV, Pc ' 0.244 MeV fm , µc ' −29.4 MeV and −3 ρc ' 0.045 fm

Furthermore, we observe (in Figure 7.14) the crossing of the P (ρ) T =0 curve with −3 the P (ρ) T =5 MeV curve at ρ ' 0.11 fm , and also with the P (ρ) T =10 MeV curve at ρ ' 0.15 fm−3. This eﬀect was already present (but less marked) in neutron matter.

87 7. Nuclear and Neutron Matter Equation of State

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 3 T=0MeV T=5MeV 2.5 T=10MeV T=15MeV 2 T=16MeV T=17MeV ] 3 T=25MeV 1.5

1 P [MeV/fm 0.5 Λ=2.1 fm-1 0

-0.5 0 0.05 0.1 0.15 0.2 ρ [1/fm3]

Figure 7.14.: Pressure P (ρ, T ) in isospin-symmetric nuclear matter, calculated using the ¯ −1 results for F (ρ, T ) with Vlow-k(2.1 fm ) and Nijmegen LECs. The non- physical behaviour of the analytic isotherms in the liquid-gas coexistence region is depicted by the dashed lines (for T < Tc ' 16.0 MeV). The physical pressure is calculated using the Maxwell construction. The dot indicated the critical point.

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT

80 T=0MeV T=5MeV T=10MeV -1 60 T=15MeV Λ=2.1 fm T=16MeV T=17MeV 40 T=25MeV

[MeV] 20 µ

-20

-40 0 0.05 0.1 0.15 0.2 0.25 ρ [1/fm3]

Figure 7.15.: Chemical potential µ(ρ, T ) in isospin-symmetric nuclear matter, obtained ¯ −1 from F (ρ, T ) with Vlow-k(2.1 fm ) and Nijmegen LECs. The dashed lines at T > Tc show the non-physical behaviour in the transition region. The dotted lines show the chemical potential of the non-interacting system, µ0. The T = 0 curves show the analogous zero temperature quantities, i.e. the interacting and free Fermi energies, interacting and . The dot indicates εκF εκF the critical point.

88 7. Nuclear and Neutron Matter Equation of State

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 7 T=0MeV 6 T=5MeV Λ -1 T=10MeV =2.1 fm T=16MeV 5 T=17MeV T=25MeV ]

3 4

P [MeV/fm 2

-1 -30 -25 -20 -15 -10 -5 0 5 µ [MeV] Figure 7.16.: Pressure isotherms (in isospin-symmetric nuclear matter) as functions of ¯ −1 the chemical potential µ, obtained from F (ρ, T ) with Vlow-k(2.1 fm ) and Nijmegen LECs. The dashed double-valued region of the curves at temperatures below Tc ' 16.0 MeV corresponds to the non-physical behaviour of the nuclear equation of state in the transition region. In this region the actual pressure and chemical potential are constant and determined by the Maxwell construction. The dot indicates the critical point.

7.4. Discussion of Anomalous Contributions

In the last two sections we have observed the crossing of the P (ρ) curve with the T =0 pressure isotherms P (ρ) T =5 MeV (and also P (ρ) T =10 MeV in the case of isospin-symmetric nuclear matter; cf. Figures 7.7 and 7.14). As noted above, this crossing implies a negative coeﬃcient of thermal expansion α. In order to examine this eﬀect more closely one should calculate the free energy per particle F¯(ρ, T ) for temperatures below T = 5 MeV. However, the numerical code for the (2,a)-contribution becomes unstable for T . 2 MeV (in its present form). This might be due to the expression nkn¯k/T , which becomes critical (regarding numerics) in the zero temperature limit. ¯ As can be seen in Figure 7.17 (dashed lines), the overall anomalous contribution Fanom = ¯ ¯ F2,a + FADT ist still relatively large for T = 2.5 MeV. However, when the two anomalous ¯ ¯ contributions are neglected, the curves for F (ρ)|T =0 and F (ρ)|T =2.5 MeV come very close to each other (cf. solid lines in Figure 7.17). Figure 7.18 shows that the pressure isotherms obtained from these curves do not cross the P (ρ) T = 0 curve. Hence, the crossing of ¯ pressure isotherms is entirely generated by the two anomalous contributions, F2,a and ¯ FADT.

89 7. Nuclear and Neutron Matter Equation of State

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n) 30 20

10 Λ=2.1 fm-1 0 -10 -20 F/N [MeV] -30 T=0MeV T=2.5MeV -40 T=5MeV T=10MeV -50 T=15MeV T=25MeV -60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ρ [1/fm3]

Figure 7.17.: Free energy per particle F¯(ρ, T ) in isospin-symmetric nuclear matter (analytical results, i.e. without application of the Maxwell construction), obtained ¯ −1 from F (ρ, T ) with Vlow-k(2.1 fm ) and Nijmegen LECs. The solid lines show ¯ ¯ the results without the anomalous contributions, F2,a and FADT, whereas the ¯ ¯ ¯ dashed lines show F (ρ, T ) with F2,a and FADT included.

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n) 3 T=0MeV T=2.5MeV 2.5 T=5MeV T=10MeV 2 T=15MeV T=25MeV ] 3 1.5

1 P [MeV/fm 0.5 Λ=2.1 fm-1 0

-0.5 0 0.05 0.1 0.15 0.2 ρ [1/fm3]

Figure 7.18.: Pressure P (ρ, T ) in isospin-symmetric nuclear matter (analytical results, i.e. without application of the Maxwell construction), calculated using the re- ¯ −1 sults for F (ρ, T ) (without the anomalous contributions) with Vlow-k(2.1 fm ) and Nijmegen LECs.

90 7. Nuclear and Neutron Matter Equation of State

The Kohn-Luttinger-Ward theorem requires that the sum of the two anomalous contri- ¯ ¯ ¯ butions Fanom = F2,a +FADT (at fixed µ0) vanishes smoothly in the zero temperature limit. Table 7.5 shows selected values of the anomalous contributions calculated with different −1 interactions, i.e. Vlow-k(2.1 fm ), the bare chiral VNN, and the two test interactions from section 7.1, VMODPI and VISOSC (both by direct computation as well as using the partial wave representation). We also show the values of the quantity ∆anom, which we define as

|F¯ + F¯ | := 2,a ADT (7.15) ∆anom 1 ¯ ¯ . 2 |F2,a| + |FADT|

Thus, ∆anom constitutes a measure for the quality of the cancellation of the two anomalous contributions. √ In Table 7.5 one sees that for fixed κ = sign(µ0) 2µ0MN the sum of the two anomalous contributions decreases with decreasing temperature, independent of the used interaction and method of computation. Moreover, all results obtained using the partial wave repre- ¯ sentation show that Fanom increases when κ goes to larger values (whith the temperature 3 being fixed) . Comparing the results for the test interactions, we find that ∆anom is about an order of magnitude larger when one uses VMODPI instead of VISOSC, indepedent of the values of κ and T and the method of computation. Ignoring numerical errors, this would lead to the conclusion that the quality of the cancellation is reduced when coupled channels are allowed. With Vlow-k and VNN the quantity ∆anom is further increased (compared to the “VMODPI” results), which would indicate that the quality of the cancellation is even more reduced for realistic NN potentials. ¯ The partial wave results for Fanom and ∆anom, calculated with VISOSC, deviate slightly from the results obtained by direct computation, but are of the same order of magnitude 4. However, for large κ and large T the agreement is decreased in the case of the pseudoscalar boson exchange, VMODPI. This is accordance with the precision tests of the Fortran code for Ω2,a and ΩADT (cf. Figures 7.3 and 7.4). If we extrapolate the accuracy findings for the MODPI potential to the case of the realistic NN potentials, Vlow-k and ¯ 5 VNN, the values Fanom could be significantly reduced within the error bars σMODPI(κ, T ) . We conclude that the cancellation of the anomalous contributions is a delicate subject ¯ regarding the numerical evaluation. As can be seen in Tables 7.2 and 7.4, both F2,a and ¯ FADT are individually large for realistic NN interactions. The accuracy of the numerical algorithm is probably not high enough to produce a reasonably correct value of the total ¯ anomalous contribution Fanom. To improve on this issue it would be of advantage to have low-momentum test interactions, i.e. interactions which allow direct computation and are restricted in momentum space. This would allow a more direct comparison to the results ¯ with Vlow-k and VNN. One should than aim to produce the same Fanom to high accuracy with both the direct computation and the computation in the partial wave representation.

3 This behaviour is however not present in the results for VMODPI at T = 5 MeV obtained by direct computation. 4 −1 For κ = 1.55 fm and T = 25 MeV the partial wave results for F¯anom have a positive sign. This is expected to be a numerical artifact due to the high quality of the cancellation in the ISOSC case. 5 However, due to the restriction in momentum space (i.e. due to the cut-oﬀ Λ for Vlow-k and due to the regulator in the case of the bare chiral NN potential) the numerical accuracy (for high temperatures) is expected to be increased in the case of the realistic NN potentials.

91 7. Nuclear and Neutron Matter Equation of State

κ[fm−1] T = 5 MeV T = 25 MeV ¯ ¯ ¯ ¯ ¯ ¯ 0.85 FADT F2,a Fanom ∆anom FADT F2,a Fanom ∆anom

Vlow-k 23.29 -23.54 -0.25 1.08 % 33.52 -35.72 -2.20 6.35 % VNN 12.70 -12.84 -0.14 1.12 % 12.22 -13.91 -1.70 12.98 % VMODPI 0.6249 -0.6277 -0.0028 0.46 % 1.952 -1.915 -0.037 1.90 % −7 −5 VISOSC 0.01388 -0.01388 −2.1 · 10 0.0015 % 0.04670 -0.04672 −1.5 · 10 0.033 % MODPI 0.6248 -0.6282 -0.0034 0.54 % 1.946 -1.952 -0.0060 0.31 % ISOSC 0.01388 -0.01388 −7.7 · 10−7 0.0055 % 0.04668 -0.04670 −1.3 · 10−5 0.027 % ¯ ¯ ¯ ¯ ¯ ¯ 1.55 FADT F2,a Fanom ∆anom FADT F2,a Fanom ∆anom

Vlow-k 122.69 -126.88 -4.19 3.35 % 102.73 -111.74 -9.01 8.40 % VNN 50.17 -52.51 -2.34 4.56 % 28.16 -34.12 -5.96 19.14 %

VMODPI 7.052 -6.949 -0.103 1.47 % 7.022 -6.84 -0.181 2.613 % −6 −4 VISOSC 0.15338 -0.15338 1.6 · 10 0.001 % 0.1939 -0.1937 2.6 · 10 0.154 % MODPI 7.023 -7.024 −5.7 · 10−4 0.008 % 7.039 -7.077 -0.038 0.54 % ISOSC 0.15337 -0.15338 −4.1 · 10−6 0.0027 % 0.1939 -0.1940 −6.1 · 10−5 0.032 %

¯ ¯ ¯ ¯ ¯ Table 7.5.: Anomalous contributions F2,a, FADT, and their sum Fanom = F2,a +FADT (all in units MeV) in isospin-symmetric nuclear matter, calculated using diﬀerent interactions. The rows labelled “MODPI” and “ISOSC” show the results from the test interactions obtained by direct computation, i.e. without expanding the matrix elements in partial waves. The quantity ∆anom is the relative ¯ 1 ¯ ¯ size of Fanom with respect to 2 (|F2,a| + |FADT|). The densities ρ(κ, T ) corresponding to the two values of the “ﬁnite temperature Fermi momentum” κ are ρ(0.85 fm−1, 5 MeV) ' 0.048 fm−3, ρ(1.55 fm−1, 5 MeV) ' 0.25 fm−3, ρ(0.85 fm−1, 25 MeV) ' 0.14 fm−3 and ρ(1.55 fm−1, 25 MeV) ' 0.33 fm−3.

92 8. Isospin-Asymmetric Nuclear Matter

In chapter7 we have investigated the equations of state of neutron matter and isospin- symmetric nuclear matter. These were both situations that featured only one specific particle species, in a trivial way in the first case and because protons and neutrons are treated as identical particles in the second. However, if we want to consider nuclear matter with a proton-neutron ratio that lies in between these two exceptional cases, it is necessary to explicitly distinguish between protons and neutrons. The different densities of protons and neutrons are taken into account by the introduction of the two independent chemical momenta, µp for protons and µn neutrons. As a consequence there are now two different Fermi-Dirac distributions:

p p β(εk−µp) nk := e , (8.1) n n β(εk−µn) nk := e . (8.2) We begin by showing how the diﬀerent contributions to the grand-canonical potential density have to be modiﬁed to apply for isospin-asymmetric nuclear matter in section 8.1. In section 8.2 we then evaluate the resulting expression for the free energy per particle ¯ F (ρp, ρn,T ), and demonstrate the cancellation of the anomalous contributions in the zero temperature limit.

8.1. Grand-Canonical Potential Density for Two Species

In contrast to the cases of isospin-symmetric nuclear matter and neutron matter, where there was only one particle species to consider, we now need to distinguish between proton p n states (distributed by nk) and neutron states (distributed by nk) in the diﬀerent contributions to the grand-canonical potential density. This is applied straightforward to the free Fermi gas contribution Ω0, Eq. (6.17), as it involves only one Fermi-Dirac distribution. p n We consequently have two diﬀerent zeroth order terms Ω0 and Ω0 .

Regarding the higher order contributions however one must explicitly identify proton- and neutron-lines in the respective diagrams. In the case of the ﬁrst order NN contribution we have

p p n n p n

Ω1,NN = + + 2 × , (8.3) where the factor 2 in front of the third diagram denotes that there are two contributions of that kind, namely the one displayed above and its reflection with interchanged proton- and neutron-lines. p Besides leading to different Fermi-Dirac distributions, i.e. nk for every proton-line and n nk for every neutron-line, the different contributions to Eq. (8.3) differ only in terms

93 8. Isospin-Asymmetric Nuclear Matter of the two-particle isospin quantum numbers T and t. For instance, the ﬁrst diagram 1 1 leads to the coupling of two proton states |pi = 2 , τ = 2 , which gives a two-particle state |ppi = |T = 1, t = 1i (only the isospin-part of the states is written out). Similarly 1 1 the third diagram involves two neutrons |ni = 2 , − 2 , leading to a two-particle state |nni = |1, −1i. In contrast, the two remaining contributions involve the coupling of a proton with a neutron state, i.e. |pni in the case of the diagram shown above and |npi in the case of the diagram with interchanged proton and neutron-lines. The isospin-part of these states is given by: 1 |pni = √ |1, 0i + |0, 0i , (8.4) 2 1 |npi = √ |1, 0i − |0, 0i . (8.5) 2 D E ˜ From conservation of T it follows that the matrix element 12 Vlow-k 12 is the same for both cases. Hence, both diagrams lead to the same expression, which makes the symbolic 2 in Eq. (8.3) a genuine factor 2. This factor is then cancelled by the factor 1/2 coming from the matrix element which involves the square of the normalization factor of the states (8.4) and (8.5), respectively. In conclusion the (1,NN)-contribution is given by Eq. (6.39), with the following changes:

J,`,`,S,T n|K~ −~p|n|K~ +~p|(2T + 1) p V p p p p n n n J,`,`,S,T =1 → n n + n n + n n p V p |K~ −~p| |K~ +~p| |K~ −~p| |K~ +~p| |K~ −~p| |K~ +~p| p n J,`,`,S,T =0 + n n p V p . (8.6) |K~ −~p| |K~ +~p|

As an immediate check we observe that in the isospin-symmetric nuclear matter limit, where µp = µn and protons and neutrons are treated as identical partices, the factor 2T + 1 is reproduced, as there are three contributions in the T = 1 case and only one in the T = 0 case.

In the case of the second order normal diagram the interacting states are |12i and |34i. Diagramatically these states correspond to the coupling of either the two hole- or the two particle-lines. There are then six diﬀerent diagrams which lead to matrix elements that conserve T and t:

p p p p n n n n p n p n Ω2,n = + + 4 × . (8.7)

The factor 4 in front of the third diagram denotes that there are four permutations of proton- and neutron-lines which preserve T and t, namely the interchange of the two hole-lines or the interchange of the two particle-lines, or both. D E 2 ˜ Regarding the partial wave expansion of the squared matrix element 12 Vlow-k 34 ,

94 8. Isospin-Asymmetric Nuclear Matter the ﬁrst two diagrams give the same result, i.e.

2 D ˜ E  pp Vlow-k pp  J,` ,` ,S,T =1 D J0,`0 ,`0 ,S,T =1 E 1 2 2 1 D E 2 → . V . . V . . (8.8) ˜ nn Vlow-k nn 

The other four diagrams involve the mixed states given by Eq. (8.4) and Eq. (8.5). There are two diﬀerent sorts of matrix elements, on the one hand those where both states are of the same kind D E 2 pn V˜ pn low-k  1 2 D E 2 → |h1, 0 | ... | 1, 0i + h0, 0 | ... | 0, 0i| ˜ 4 np Vlow-k np  1 ...,T =1 ...,T =1 ...,T =0 ...,T =0 → . V . . V . + . V . . V . 4 ...,T =1 ...,T =0 + 2 . V . . V . , (8.9) and on the other hand those where the two states are diﬀerent D E 2 pn V˜ np low-k  1 2 D E 2 → |h1, 0 | ... | 1, 0i − h0, 0 | ... | 0, 0i| ˜ 4 np Vlow-k pn  1 ...,T =1 ...,T =1 ...,T =0 ...,T =0 → . V . . V . + . V . . V . 4 ...,T =1 ...,T =0 − 2 . V . . V . . (8.10)

These four contributions all lead to two proton and two neutron Ferm-Dirac distributions in the F-function given in Eq. (6.45). Because the Ξ-function defined in Eq. (6.50) 1 obeys Ξ(±ϑ1, ±ϑ2) = Ξ(ϑ1, ϑ2) and the integrals over the angles ϑ1 and ϑ2 run over all possible values it makes no difference which of the first two Fermi-Dirac distributions in the F-function corresponds to the proton or, respectively, the neutron. In other words, p n x x p n x x n p x x n p x x F(...) = n1n2n¯3n¯4 −n¯1n¯2n3n4 leads to the same results as F(...) = n1n2n¯3n¯4 −n¯1n¯2n3n4, and likewise for the second two Fermi-Dirac distributions belonging to the states |3i and |4i. Hence, the four contributions to the third diagram can be combined. The mixed terms in Eq. (8.9) and Eq. (8.10) then cancel each other and factor 1/4 vanishes due to the terms where both matrix elements have the same total isospin number appearing four times. The required modifications with respect to the expression for the symmetric case

1 m The ϑ -dependence of Ξ is given by the terms Y (ϑ )Y 0 (ϑ ). With P (cos ϑ) being i `i,(M−ςi) i `i,(M−ςi) i ` m either symmetric or antisymmetric under ϑ → −ϑ, what can be seen immediately from P` (cos ϑ) = m m dm (−1) (sin ϑ) d(cos ϑ)m (Pl(cos ϑ)), these terms are symmetric under ϑi → −ϑi.

95 8. Isospin-Asymmetric Nuclear Matter

(6.49) are therefore:

J,`,`,S,T D J0,`0 ,`0 ,S,T E 2 1 F(...) (2T + 1) . V . . V . → F(...) 1=p,2=p,3=p,4=p + F(...) 1=n,2=n,3=n,4=n + F(...) 1=p,2=n,3=p,4=n J,` ,` ,S,T =1 D J0,`0 ,`0 ,S,T =1 E 1 2 2 1 × . V . . V . D 0 0 0 E J,`1,`2,S,T =0 J ,` ,` ,S,T =0 + F(...) . V . . V 2 1 . . (8.11) 1=p,2=n,3=p,4=n Again the isospin-symmetric nuclear matter limit is reproduced.

The second order anomalous diagram involves three one-particle states |1i, |2i and |3i. Contrary to the case of the (2,n)-diagram the matrix elements are now again diagonal with respect to these states, as was the case in the (1,NN)-contribution. Therefore there are no restrictions coming from the conservation of the two-particle isospin quantum numbers T and t, and the identification of the three states |1i, |2i and |3i with either protons |pi or neutrons |ni leads to eight different diagrams. Restricting ourselves first to the case where |2i is a proton, we have: p n

p p p p Ω2,a = + |2i=|pi

p n

p p + n + n . (8.12)

The matrix elements corresponding to these diagrams are:

 D ˜ ED ˜ E  pp Vlow-k pp pp Vlow-k pp   D ED E  ˜ ˜ D ED E  pp Vlow-k pp pn Vlow-k pn ˜ ˜ 12 Vlow-k 12 23 Vlow-k 23 → D ED E . (8.13)  np V˜ np pp V˜ pp  low-k low-k  D ED E  ˜ ˜  np Vlow-k np pn Vlow-k pn The second and third diagrams give the same contribution because the respective matrix elements coincide and in both cases there is one proton and one neutron Fermi-Dirac distribution related to the outer states |1i and |3i. In the symmetric case we have seen that the outer states give otherwise identical contributions, so it does not matter which is the proton- and which the neutron-state. After performing the partial wave expansion the contribution from the ﬁrst diagram can be written symbolically as:

...,T =1 ...,T =1 p p D1 = . V . t=1 . V . t=1 n1n3 . (8.14)

96 8. Isospin-Asymmetric Nuclear Matter

The combined contribution from the second and third diagram is:

...,T =1 ...,T =0 ...,T =1 p n D2 + D3 = . V . t=0 + . V . t=0 . V . t=1 n1n3 . (8.15) In the fourth diagram both two-particle states consist of one proton and one neutron. The symbolic expression reads:

1 ...,T =1 ...,T =0 D4 = . V . + . V . 4 t=0 t=0 ...,T =1 ...,T =1 n n × . V . t=0 + . V . t=0 n1n3 . (8.16) These four contributions can be combined in one single symbolic expression: 4 X Y ...,T =1 p 1 ...,T =1 ...,T =0 n Di = . V . n + . V . + . V . n . t=1 α 2 t=0 t=0 α i=1 α∈{1,3} (8.17)

The other four diagrams (where |2i is a neutron) lead to a similar expression. Altogether we have, compared to the symmetric nuclear matter case given by Eq. (6.68):

1 2 2 " T # X X 1 2 n n¯ ... n Θ ± (t − τ ) |hT t | (t − τ ) τ i| ... k k |~p−~k| 2 k k k 1 t=−T τk=− 2 2 p p p 1 n → n n¯ ... n ... + ... n ... + ... k k ~p−~k T =1,t=1 ~p−~k T =1,t=0 T =0,t=0 | | 2 | | 1 τk= 2 2 n n n 1 p + n n¯ ... n ... + ... n ... + ... . k k ~p−~k T =1,t=−1 ~p−~k T =1,t=0 T =0,t=0 | | 2 | | 1 τk=− 2 (8.18)

p n ~ For nx = nx, x ∈ {k, |~p − k|}, the expression on the left side of the arrow and the expression on the right side coincide. This can be seen immediately through the observation that the only values of T , t and τk where the squared Clebsch-Gordan coeﬃcient in the left-hand side of the equation is nonvanishing are the ones appearing in the right-hand side. For t = 1 the squared Clebsch-Gordan coeﬃcient is equal to one and for t = 0 it gives a factor of one half, independent of the values of T and τk.

What is not yet accounted for are the contributions to the two-species grand-canonical potential density coming from the the chiral thee-nucleon interaction. Here it suﬃces to identify particle lines with neutrons and protons without looking at isospin values of coupled states. The resulting expressions are given in Appendix C.2.

8.2. Free Energy per Particle and Zero Temperature Limit

Having clariﬁed the change of the Ω-terms in the isospin-asymmetric case, we need to ﬁnd out how to calculate the free energy per particle. The analogue of Eq. (4.84) in the

97 8. Isospin-Asymmetric Nuclear Matter case of two diﬀerent particle species (protons p and neutrons n) is 1 F¯(ρ , ρ ,T ) = Ω(µ , µ ,T ) + (x µ + x µ ) , (8.19) p n ρ p n p p n n

ρp ρp with xn = ρ , xn = ρ and ρ = ρp + ρn. In order to perform this Legendre transformation we proceed similarly to the symmetric case, starting with the expansion of the chemical momenta µp and µn:

p ∂Ω0 µp = µp,0 + µp,1 + µp,2 , with µp,0 ﬁxed by ρp = − , (8.20) ∂µp µp,0 n ∂Ω0 µn = µn,0 + µn,1 + µn,2 , with µn,0 ﬁxed by ρn = − . (8.21) ∂µn µn,0

The subsequent expansion of each term on the right-hand side of Eq. (8.19) around µp,0 and µn,0 then leads to:

1 F¯(ρ , ρ ) =x µ + x µ + Ωp(µ ) + Ωn(µ ) + Ω (µ , µ ) + Ω (µ , µ ) p n p p n n ρ 0 p,0 0 n,0 1,NN p,0 n,0 2,n p,0 n,0 " 2 2 #! 1 (∂Ω1,NN/∂µp) 1 (∂Ω1,NN/∂µn) + Ω2,a(µp,0, µn,0) − p − . 2 2 2 2 2 n 2 ∂ Ω0 /∂µp µp,0,µn,0 ∂ Ω0 /∂µn µn,0,µn,0 (8.22) Hence, in isospin-asymmetric matter there are two anomalous derivative terms, 2 1 (∂Ω1,NN/∂µp) ΩADT1 = − p , (8.23) 2 2 2 ∂ Ω0 /∂µp µp,0,µn,0 2 1 (∂Ω1,NN/∂µn) ΩADT2 = − . (8.24) 2 2 n 2 ∂ Ω0 /∂µn µn,0,µn,0

We have to verify that they cancel the second-order anomalous contribution Ω2,a in the zero temperature limit. From Eq. (8.6) we see that the numerator in the expression for ΩADT1 is given by " # ∂Ω ∂ ∂ 1,NN =Ωpp np np + np np 1,NN K~ −~p K~ +~p K~ −~p K~ +~p ∂µp ∂µp | | | | | | ∂µp | | " # ∂ + Ωpn np nn , (8.25) 1,NN K~ −~p K~ +~p ∂µp | | | |

pp pn where Ω1,NN denotes the {T = 1, t = 1} part of Ω1,NN, Ω1,NN denotes the combined {T = 1, t = 0} and {T = 0, t = 0} part, and the square brackets depict the terms relevant for the µp-derivative. The ﬁrst term on the right-hand side of Eq. (8.25) is a pure proton object. A similar term appeared also in the second part of section 4.4, where it was shown that the summands in the square bracket lead to identical expressions. Thus, we have returned to the case of one particle species, and observe that pp 2 ppp 1 ∂Ω1,NN/∂µp T →0 Ω (µp,0, µn,0) − − p −−−→ 0 , (8.26) 2,a 2 2 2 ∂ Ω0 /∂µp µp,0,µn,0

98 8. Isospin-Asymmetric Nuclear Matter

ppp where Ω2,a is the {T = 1, t = 1} part of Ω2,a. The second term on the right-hand side of Eq. (8.25) involves two particle species, but only one term in the square brackets. Recalling again section 4.4 we can write the zero temperature limit of this term as

pn Z 3 ∂Ω1,NN T →0 1 d k2 p p n −−−→ trσ2 δ(ε − ε ) F , (8.27) 2 (2 )3 κF k2 1 ∂µ µp,0,µn,0 π

n where F1 is the function deﬁned in Eq. (4.96) restricted to neutrons. The {T = 1, t = 0} and {T = 0, t = 0} parts of the second order anomalous contributions have an additional factor 1/2 from the isospin Clebsch-Gordan coeﬃcients. The zero temperature limit of the sum of these terms is therefore given by (cf. Eq. (4.97)):

Z 3 ppn T →0 1 d k2 p p n2 Ω2,a (µp,0, µn,0) −−−→− trσ2,τ2 δ(εκ − ε ) F1 . (8.28) 4 (2π)3 F k2

This shows that also the mixed-species part of ΩADT1 is cancelled by the corresponding part of Ω2,a in the zero temperature limit. The cancellation of ΩADT2 with the remaining parts of Ω2,a then follows from the analogous form of the respective terms. Hence, we have obtained again the relation T →0 F¯ −−−→ E¯ in the isospin-asymmetric case.

99 9. Discussion and Conclusion

In this thesis we have investigated the thermodynamic properties of nuclear matter by using many-body perturbation theory together with the effective low-momentum potential Vlowk(Λ) and the leading order chiral three-body potential V3N. This approach constitutes an explicit application of the modern theory of nuclear physics (and low-energy QCD in general), chiral effective field theory, to the nuclear many-body problem. In particular, Vlowk(Λ) can be derived in terms of a renormalization group evolution of the chiral two-body potential VNN to momenta below a cut-off Λ. The RG-evolution conserves observables from two-body scattering processes, but does not address three-body processes, which are however an integral part of many-nucleon systems. Hence, for a consistent low-momentum theory of nuclear matter a combined evolution of two- and three-body forces would be required, which is however not yet available. As a replacement for this full many-body RG-evolution we used V3N with the c1, c3 and c4 low-energy constants determined by the Nijmegen group and cE(Λ) and cD(Λ) fitted to selected three- and four-nucleon observables for every value of Λ (Nijmegen LECs). In addition, we calculated the equation of state of isospin-symmetric nuclear matter and of neutron matter also with the bare chiral two-body potential VNN (multiplied with a regulator function) together with V3N with the low-energy constants determined in a different fitting scheme (Machleidt LECs).

Discussion of Results In contrast to neutron matter, which is unbound and appears only in the gas phase, isospin-symmetric nuclear matter is a bound system and for temperature below the critical tempeature Tc it undergoes a phase transition from the gas state to the liquid state when the density is increased. As a result of this, the analytical free energy per particle ¯ F (T < Tc) leads to a concave curve when plotted vs. the volume per particle υ. This behaviour is unphysical. In order to obtain the physical free energy per particle one has to apply the Maxwell construction. Our results show only a very small model-independence of the total two-body con- ¯ ¯ ¯ ¯ tribution to the free energy per particle, FNN = F1,NN + F2,n + F2,a + FADT, both in isospin-symmetric nuclear matter and in neutron matter. The expected increase regarding the convergence behaviour of the many-body perturbation series with Vlowk(Λ) instead ¯ of VNN manifests itself in the relative size of F1,NN and the second order contribution, ¯ ¯ ¯ ¯ F2,NN = F2,n + F2,a + FADT. Despite this difference, the total two-body contribution remains almost the same, independent of the potential used. The size of the three-body contribution F1,3N on the other hand depends strongly on the fitting scheme used to determine the LECs, in particular on the value of c3, which gives the dominant part of F1,3N. In neutron-rich matter this effect is suppressed and becomes visible only in the high-density regime. When the number of protons is increased, i.e. in particular in isospin-symmetric nuclear matter, F1,3N becomes large and the results

100 9. Discussion and Conclusion

obtained with c3 from the Nijmegen LECs deviate largely form the ones with c3 from the Machleidt LECs. We have seen that the Nijmegen results agree much better with results from other authors. We have seen in both the nuclear matter and the neutron matter case that the pressure isotherms P (ρ) for low temperatures (T . 5 MeV in neutron matter, and T . 10 MeV T in isospin-symmetric nuclear matter), plotted over the density, cross the P (ρ) T =0 curve. This crossing implies a negative coeﬃcient of thermal expansion α, and is entirely gen- ¯ ¯ ¯ erated by the sum of the two anomalous contributions, Fanom = F2,a + FADT. We have ¯ seen that in order to calculate the correct value of Fanom very high numerical accuracy is required.

Conclusion and Outlook The main aim of this thesis was to assess the model-depedence of perturbative many-body calculations for nuclear matter. Our investigations show that model-independent results in nuclear many-body systems can be obtained using chiral low-momentum interactions only if the low-energy constants that parametrize the three-body force are fixated. Using different sets of LECs, albeit leading to consistent results in few-body systems, has a large impact on results in isospin-symmetric nuclear matter. As the overall size of the three- body contribution (and also the size of two-body contributions) is significantly reduced in neutron matter, this effect is weakened in that case. From these findings it can be deduced that the role of three-body forces (both induced and genuine) in nuclear matter needs further investigation. Ultimately, a combined evolution of two- and three-body forces on the basis of the Faddeev equations would be needed. Regarding the crossing of pressure curves due to the anomalous contributions, at this stage it cannot be decided whether this effect is due to a numerical artifact or really a consequence of our model (although it probably is the former). Further investigations in that regard are necessary, for example with the use of additional test interactions. Finally, as pointed out in the introductory chapter of the thesis, in order to test the phenomenological quality of our results one should solve the Tolman-Oppenheimer-Volkoff equation with the equation of state calculated in this thesis. As a result of the low- momentum approach, our model provides a sensible description of neutron-rich matter only for intermediate densities 1. Hence, in order to describe the matter that appears in neutron stars, it has to extended by an equation of state which is valid for higher densities, such as the one calculated by Akmal, Pandharipande and Ravenhall [APR98] with sophisticated many-body techniques.

1 In the low-density region (i.e. the outer crust of neutron stars), the nuclear matter approach generally breaks down, independent of the use of the use of low-momentum interactions. Here, a description in terms of neutron-rich nuclei has to be applied (cf. [BP75]).

101 A. Notation and Conventions

• We work in units ~ = c = kB = 1

• The nucleon mass is MN ' 938.918 MeV, the neutron mass Mn ' 939.565 MeV and the proton mass Mp ' 938.272 MeV • In the zero temperature formalism:

The Fermi momentum is denoted by κF The Heavyside step function for the unperturbed Fermi-surface is written as ( 1 for κF ≥ p Θp ≡ Θ(κF − p) := 0 for κF < p ¯ We use the abbreviation Θp ≡ Θ(p − κF ) • In the ﬁnite temperature formalism: The inverse temperature is denoted by β The chemical potential of the interacting system is denoted by µ

The chemical potential of the non-interacting system is denoted by µ0 √ 1 The “ﬁnite temperature Fermi momentum” is κ = sign(µ0) 2µ0MN −1 β(εp−µ) The Fermi-Dirac distribution is np := e +1 , where εp is the kinetic energy of a particle with momentum ~p

We use the abbreviationn ¯p ≡ (1 − np) • The spherical harmonics are s (2` + 1) (` − m)! Y (ϑ, ϕ) := P m(cos ϑ) eimϕ . (A.1) `,m 4π (` + m)! `

They satisfy the orthonormality condition

π 2π Z Z ∗ dΩY`,m(ϑ, ϕ) Y`0,m0 (ϑ, ϕ) = δ``0 δmm0 . (A.2) ϑ=0 ϕ=0

imϕ The spherical harmonics without the azimuthal part e are denoted by Y`,m(ϑ). E ~ • 1-particle states: kστ , where

1 This quantity is usually not used in the literature. It is introduced here only for convenience regarding the numerical algorithm, where it serves as the main loop parameter.

102 A. Notation and Conventions

σ is the spin projection quantum number along the z-axis τ is the isospin projection quantum number • 2-particle states in the plane wave basis: |~pSςT ti, where ~p is the (half)-relative momentum of the two particles S is the principal spin quantum number ς is the spin projection quantum number along the z-axis T is the 2-particle isospin quantum number t is the 2-particle isospin projection quantum number • 2-particle states in the spherical wave basis: |p`mSςT ti, where ` is the quantum number associated with the squared orbital angular momen- 2 tum operator ~`

m is the quantum number associated with the operator `z • 2-particle states in the total angular momentum representation: |pJM`ST ti, where J is the quantum number associated with the squared total angular momentum 2 2 operator J~ = (~` + S~)

M is the quantum number associated with the operator Jz • The spherical wave states are normalized as

0 0 0 1 1 0 hp`m|p ` m i = δ 0 δ 0 δ(p − p ) . (A.3) 4π2 p2 `` mm

In this convention spherical wave states |p`mi and plane wave states |~p i have the h − 3 i same dimension MeV 2 , and the partial wave expansion of a plane wave state is given by

X ` ∗ |~p i = 4πi Y`m(ˆp) |p`mi . (A.4) `,m

103 B. Maxwell Construction and Critical Temperature

In Figure B.1 we show the free energy per particle F¯ (in isospin-symmetric nuclear matter) as a function of the volume per particle υ = 1/ρ, together with the linear functions f(υ,T ) from the Maxwell construction. These double-tangents eliminate the concave ¯ region [υ1, υ2] of F ; the values of their slopes and the boundaries of the concave regions ¯ of F (υ) T are given in Table B.1. The critical temperature Tc ' 16.0 MeV is signaled by T →Tc the vanishing of the concave region, i.e. υ1 −−−→ υ2. It can be more easily determined by looking at the corresponding pressure isotherms P (υ) T , where the transition region is identiﬁed with the region of positive slope (cf. Figure B.2).

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 0

-10

-20

-30

-40 T=0MeV

F/N [MeV] T=5MeV T=10MeV -50 T=13MeV T=15MeV Λ -1 -60 T=16MeV =2.1 fm T=17MeV T=25MeV -70 0 10 20 30 40 50 60 70 υ [fm3]

Figure B.1.: Free energy per particle in isospin-symmetric nuclear matter as a function of ¯ the volume per particle, F (υ), calculated with Vlowk(Λ) and the chiral three- body potential V3n with Nijmegen LECs. The dashed lines are the double tangents from the Maxwell construction.

104 B. Maxwell Construction and Critical Temperature

3 3 −3 T [MeV] υ1[fm ] υ2[fm ] P0[MeV fm ] 0 8.89 ∞ 0 5 8.85 1328.7 0.0021 10 9.87 167.01 0.0447 13 11.33 67.82 0.1174 15 14.63 35.29 0.1993

¯ Table B.1.: Concave region [υ1, υ2] of F (υ) for diﬀerent temperatures, and slope ∂υf(υ) = −P0 of the double-tangents f(υ).

Nuclear Matter: (0)+(1,NN)+(1,3N)+(2,n)+(2,a)+ADT 0.27 T=15.8MeV 0.265 T=15.9MeV Λ=2.1 fm-1 T=16.0MeV T=16.1MeV 0.26 T=16.2MeV ]

3 0.255

0.25

P [MeV/fm 0.245

0.24

0.235

0.23 16 18 20 22 24 26 28 υ [fm3]

Figure B.2.: Pressure isotherms (in isospin-symmetric nuclear matter) as functions of the ¯ volume per particle, F (υ), calculated with Vlowk(Λ) and the chiral three-body potential V3n with Nijmegen LECs.

105 C. Chiral Three-Nucleon Interaction

The contributions to Ω1,3N can be written as

∞ ∞ ∞ Z p Z p Z p Ω = dp 1 dp 2 dp 3 K n n n , (C.1) 1,3N 1 2π2 2 2π2 3 2π2 3 p1 p2 p3 0 0 0

(cE ) (cD) (H) (F ) where K3 = K3 + K3 + K3 + K3 , with each part corresponding to the respective diagram in Figure (5.2).

C.1. Three-Body Kernels

There are four diﬀerent three-body kernels K3 giving a contribution to Eq. (C.1). They are given by the following equations:

(cE ) 12cE K3 = − 4 p1p2p3 , (C.2) fπ Λχ

2 2 2 ! (cD) 3gAcD mπ mπ + (p1 + p2) K = p3 p1p2 − ln , (C.3) 3 4 2 2 fπ Λχ 4 mπ + (p1 − p2)

2 " 2 2 (H) 3gA 2 mπ + (p1 + p2) K = p3 2 (c3 − c1) m ln − 4c3p1p2 (C.4) 3 4 π 2 2 fπ mπ + (p1 − p2) 4 1 1 + (c3 − 2c1) m − , (C.5) π 2 2 2 2 mπ + (p1 + p2) mπ + (p1 − p2)

2 (F ) gA h 2 c3 i K3 = 4 3c1mπH(p1)H(p2) + − c4 X(p1)X(p2) + (c3 + c4) Y (p1)Y (p2) . fπ p3 2 (C.6)

The functions H(p1), X(p1) and Y (p1) in the Fock-contribution are:

2 2 2 2 2 p3 − p1 − mπ mπ + (p1 + p3) H(p1) = p1 + ln , 2 2 4p3 mπ + (p1 − p3) 2 2 2 mπ mπ + (p1 + p3) X(p1) = 2p1p3 − ln , 2 2 2 mπ + (p1 − p3) 2 2 2 2 2 2 2 2 p1 2 2 2 3 (p1 − p3 + mπ) + 4mπp3 mπ + (p1 + p3) Y (p1) = 5p − 3p − 3m + ln . 3 1 π 2 2 2 4p3 16p3 mπ + (p1 − p3)

106 C. Chiral Three-Nucleon Interaction

C.2. Isospin-Asymmetric Nuclear Matter

In the case of isospin-asymmetric nuclear matter the above contributions have to be modiﬁed in the following way:

1 K(cE )n n n → K(cE ) np nn np + nn , (C.7) 3 p1 p2 p3 3 2 p1 p2 p3 p3 1h i K(cD)n n n → K(cE ) np + 2nn np nn + nn + 2np nn np , (C.8) 3 p1 p2 p3 3 6 p1 p1 p2 p3 p1 p1 p2 p3 1 K(H)n n n → K(H) np np + 4np nn + nn nn np + nn . (C.9) 3 p1 p2 p3 3 12 p1 p2 p1 p2 p1 p2 p3 p3

The part of the Fock-contribution corresponding to the low-energy constants c1 and c3 changes according to:

(c −F ) (c −F ) 1h i K 1/3 n n n → K 1/3 np np + 2nn nn np + 2np np + 2nn nn nn . 3 p1 p2 p3 3 6 p1 p2 p1 p2 p3 p1 p2 p1 p2 p3 (C.10)

The part associated with c4 however has to be changed in the following way:

1h i K(c4−F )n n n → K(c4−F ) np + 2nn np nn + 2np + nn nn np . (C.11) 3 p1 p2 p3 3 6 p1 p1 p2 p3 p1 p1 p2 p3

107 D. Test Interaction: Scalar-Isoscalar Boson Exchange

The interaction part of the grand-canonical potential density, ΩV , obtained by using the ISOSC potential given in Eq. (7.1), can be written as

∞ ∞ Z p Z p Ω = dp 1 dp 2 K n n V 1 2π2 2 2π2 2 p1 p2 0 0 ∞ ∞ ∞ Z p Z p Z p + dp 1 dp 2 dp 3 n n n (K + A) . (D.1) 1 2π2 2 2π2 3 2π2 p1 p2 p3 3 0 0 0 The two- and three-body kernels are given by (1a) (1b) (H) (F ) K2 = K2 + K2 +K2 + K2 , (D.2) | {z } (1) K2 (H) (F ) K3 = K3 + K3 . (D.3) (1a) (1b) (H) (F ) Here, K2 and K2 correspond to the ﬁrst-order contribution Ω1,NN, whereas K2 , K2 , (H) (F ) K3 and K3 belong to the second order normal contribution Ω2,n (cf. Figure E.1e). The other term in the triple-integral in Eq. (D.1), A, gives the the second order anomalous contribution Ω2,a. The expressions for the kernels K and for A are given in Appendix D.1 for the case of isospin-symmetric nuclear matter, and in Appendix D.2 for the case of isospin-asymmetric nuclear matter.

For the anomalous derivative term we need the µ-derivative of Ω1,NN:

∞ ∞ ∂Ω ∂ Z p Z p 1,NN = dp 1 dp 2 K(1) n n (D.4) ∂µ ∂µ 1 2π2 2 2π2 2 p1 p2 0 0 ∞ ∞ Z p Z p ∂n =2 dp 1 dp 2 K(1) n p2 . (D.5) 1 2π2 2 2π2 2 p1 ∂µ 0 0

MN Using the identity ∂µnp = − ∂p np and performing partial integration this expression 2 p2 2 2 simpliﬁes to:

∞ ∞ Z Z (1) ∂Ω1,NN 2MN p1 1 ∂K2 = dp1 2 dp2 2 np1 np2 . (D.6) ∂µ p2 2π 2π ∂p2 0 0

108 D. Test Interaction: Scalar-Isoscalar Boson Exchange

(1a) (1b) (H) (F ) (a) K2 (b) K2 (c) K2 (d) K2

(H) (F ) (e) K3 (f) K3 Figure D.1.: Diagrammatic representations of the diﬀerent contributions corresponding to the two- and three-body kernels in Eqs. (D.2) and (D.3). A nucleon-line

with a double-slash corresponds to a medium-insertion, bringing a factor npi , where pi is the corresponding momentum.

D.1. Isospin-Symmetric Nuclear Matter

The diﬀerent two- and three-body kernels in Eqs. (D.2) and (D.3) are given by

(1a) 2 p1p2 K2 = −8gs 2 , (D.7) ms

g2 m2 + (p2 + p2)2 K(1b) = s ln s 1 2 , (D.8) 2 2 2 2 2 2 ms + (p1 − p2)

g4M m2 + (p2 + p2)2 K(H) = − s N ln s 1 2 , (D.9) 2 2 2 2 2 4πms ms + (p1 − p2)

p1+p2 2ms 4 Z (F ) gs MN arctan 2x − arctan x K2 = − dx 2 , (D.10) 2πms 1 − 2x |p1−p2| 2ms

√ p2+p3 2 2 2 Z 1 1 p1 R + (p3 + ms − p1) h (F ) 4 √ K3 = gs MN dq 2 2 ln √ , (D.12) ms + q R 2 2 2 p1 R + (p1 − ms − p3) h |p2−p3|

109 D. Test Interaction: Scalar-Isoscalar Boson Exchange where R and h are given by 2 2 22 2 2 2 R = ms + p1 − p3 + 4ms p3 − h (D.13) 1 h = p2 − p2 − q2 . (D.14) 2q 2 3 The (2,a)-contribution is given by the following expression:

4 2 2 2 2 ! 2 2 2 2 ! gs 8p3 1 ms + (p1 + p2) 8p1 1 ms + (p3 + p2) A = β n¯p − + ln − + ln . 2 2 2 2 2 2 2 2 2 2 2 2 ms 2p2 ms + (p1 − p2) ms 2p2 ms + (p3 − p2) (D.15)

D.2. Isospin-Asymmetric Nuclear Matter

For isospin-asymmetric nuclear matter the following changes have to be applied to the isoscalar-scalar boson exchange kernels: 1 K(1a)n n → K(1a) np + nn np + nn , (D.16) 2 p1 p2 2 4 p1 p1 p2 p2 1 K(1b)n n → K(1b) np np + nn nn , (D.17) 2 p1 p2 2 2 p1 p2 p1 p2

1 K(H)n n → K(H) np + nn np + nn , (D.18) 2 p1 p2 2 4 p1 p1 p2 p2

1 K(F )n n → K(F ) np np + nn nn , (D.19) 2 p1 p2 2 2 p1 p2 p1 p2

1 K(H)n n n → K(H) np np + nn nn np + nn , (D.20) 3 p1 p2 p3 3 12 p1 p2 p1 p2 p3 p3 1 K(F )n n n → K(F ) np np np + nn nn nn . (D.21) 3 p1 p2 p3 3 2 p1 p2 p3 p1 p2 p3 The second order anomalous contribution A has to be modiﬁed according to g4 8p p A → s 1 3 p + n p ¯p + 2 ¯p p + n np1 np2 np3 β 4 np1 np1 np2 np2 np2 np2 np3 np3 2 ms 2p m2 + (p + p )2 − 3 ln s 1 2 p p ¯p + n n ¯p p + n 2 2 2 np1 np2 np2 np1 np2 np1 np3 np3 p2ms ms + (p1 − p2) 1 m2 + (p2 + p2)2 m2 + (p2 + p2)2 + ln s 1 2 ln s 3 2 np np n¯p np + nn nn n¯n nn . 2 2 2 2 2 2 2 2 2 p1 p2 p2 p3 p1 p2 p2 p3 8p2 ms + (p1 − p2) ms + (p3 − p2) (D.22) The two anomalous derivative terms (cf. Eqs. (8.23) and (8.24)) follow from the equations

∂Ω1 ∂Ω1 = , (D.23) ∂µp ∂µ n n → 1 np +2nn np p1 p2 6 ( p1 p1 ) p2

∂Ω1 ∂Ω1 = . (D.24) ∂µn ∂µ n n → 1 nn +2np nn p1 p2 6 ( p1 p1 ) p2

110 E. Test Interaction: Pseudoscalar Boson Exchange

Analogous to the case of the test-interaction being the scalar-isoscalar boson exchange, the contributions to the grand-canonical potential density can be written as double- and triple- integrals over two- and three-body kernels, as in Eq. (D.1). The form of the potential corresponding to the pseudoscalar boson exchange, Eq. 7.2, tells us that because in the diagram corresponding to K(1a) there is no momentum exchange between the nucleons, this contributions vanishes. Hence, besides the (2,a)-diagram, there are only the ﬁve diagrams depicted in Figure E.1 to consider.

(1) (H) (F ) (a) K2 (b) K2 (c) K2

(H) (F ) (d) K3 (e) K3 Figure E.1.: Diagrams associated with the respective two- and three-body kernels in the case of the interaction being the pseudoscalar boson exchange. As in Figure D.1, a double-slash denotes a medium-insertion.

E.1. Isospin-Symmetric Nuclear Matter

The two- and three-body kernels corresponding to the diagrams in Figure E.1 are given by ! 3 m2 m2 m2 + (p + p )2 K(1) = g2 π − π + ln π 1 2 , (E.1) 2 2 2 2 2 2 2 2 mπ + (p1 + p2) mπ + (p1 − p2) mπ + (p1 − p2)

g4M m4 m4 5m2 K(H) = N π − π − π 2 2 2 2 2 2 8πmπ 2 2 2 + ( + ) mπ + (p1 + p2) mπ + (p1 − p2) mπ p1 p2 ! (E.2) 5m2 15 m2 + (p + p )2 + π − ln π 1 2 , 2 2 2 2 mπ + (p1 − p2) 4 mπ + (p1 − p2)

111 E. Test Interaction: Pseudoscalar Boson Exchange

3g4M m2 m2 2m2 K(F ) = N π − π + π 2 2 2 2 2 2 2 8πmπ 4mπ + (p1 + p2) 4mπ + (p1 − p2) 2mπ + (p1 + p2) 2 2 2 2 2 2m 2m + (p1 + p2) 4m + (p1 − p2) − π + ln π π 2 2 2 2 2 2 2mπ + (p1 − p2) 2mπ + (p1 − p2) 4mπ + (p1 + p2) (E.3)

p1+p2 2mπ ! Z 1 + 4x2 + 8x4 + dx (arctan 2x − arctan x) , (1 + 2x2)3 |p1−p2| 2mπ

( 2 2 ! (F ) 4 1 1 mπ + (p1 + p3) p1 + p3 p3 − p1 K =3g MN ln − + 3 2 2 2 2 2 2 2p3 2p3 mπ + (p1 − p3) mπ + (p1 + p3) mπ + (p1 − p3) ! p − p p + p 1 m2 + (p + p )2 × 3 2 − 2 3 + ln π 2 3 2 2 2 2 2 2 mπ + (p2 − p3) mπ + (p2 + p3) 2p3 mπ + (p2 − p3) p2+p3 Z q2 4m2 hp (m2 + p2 − p2) + dq π 1 π 1 3 2 2 2 2 2 2 (mπ + q ) R mπ + (p1 + p3) mπ + (p1 − p3) |p2−p3| √ 2 2 2  ) p1 R + (p3 + mπ − p1) h 1 2 2 2 2 2 2 + 3/2 mπ + p1 + p3 − 2h + p1 − p3 ln √  , R 2 2 2 p1 R + (p1 − mπ − p3) h (E.5) where R and h are given by

2 2 22 2 2 2 R = mπ + p1 − p3 + 4mπ p3 − h , (E.6) 1 h = p2 − p2 − q2 . (E.7) 2q 2 3 The second order anomalous contribution is given by

4 2 2 2 2 ! 9g 1 mπ mπ mπ + (p1 + p2) A =β n¯p − + ln (E.8) 2 2 2 2 2 2 2 2 8 p2 mπ + (p1 + p2) mπ + (p1 − p2) mπ + (p1 − p2) ! m2 m2 m2 + (p + p )2 × π − π + ln π 3 2 . (E.9) 2 2 2 2 2 2 mπ + (p3 + p2) mπ + (p3 − p2) mπ + (p3 − p2) The anomalous derivative term is given by the analogue of the expression for the scalar- (1) isoscalar boson exchange, Eq. (D.6), with the kernel K2 given by Eq. (E.1).

112 E. Test Interaction: Pseudoscalar Boson Exchange

E.2. Isospin-Asymmetric Nuclear Matter

In the case of isospin-asymmetric nuclera matter the above contributions change according to: 1 K(1)n n → K(1) np np + nn nn + 4np nn , (E.10) 2 p1 p2 2 6 p1 p2 p1 p2 p1 p2

1 K(H)n n → K(H) np np + nn nn + 10np nn , (E.11) 2 p1 p2 2 12 p1 p2 p1 p2 p1 p2

1 K(F )n n → K(F ) 8np nn − np np − nn nn , (E.12) 2 p1 p2 2 6 p1 p2 p1 p2 p1 p2

1 K(H)n n n →K(H) np np + 4np nn + +nn nn np 3 p1 p2 p3 3 12 p1 p2 p1 p2 p1 p2 p3 (E.13) p p n p n n n + np1 np2 + 4np1 np2 + np1 np2 np3 ,

1 K(F )n n n →K(F ) 2np nn + 2nn np − nn nn nn 3 p1 p2 p3 3 6 p1 p2 p1 p2 p1 p2 p3 (E.14) p n n p p p p + 2np1 np2 + 2np1 np2 − np1 np2 np3 .

Regarding the second order anomalous contribution A, the following changes have to be applied:

1 n n n¯ n → np np np n¯p + 4nn n¯n + nn nn nn n¯pn + 4np n¯p p1 p2 p2 p3 18 p1 p3 p2 p2 p2 p2 p1 p3 p2 p2 p2 p2 p n p p n n + np1 np3 np2 n¯p2 + n p2 n¯p2 . (E.15)

The anomalous derivative term changes as in the ISOSC case (cf. Eq. (D.23)).

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118 Acknowledgements

In these final lines I want to express my thankfulness and my appreciation to all those who have inspired and supported me during my work on this thesis, and during my studies in Munich and Christchurch. I want to thank Jeremy W. Holt for offering great help and advice during my work on this thesis. I benefitted greatly from him acquainting me with my topic and helping me with various physical and programming issues at any time. Our numerous discussions, his support regarding the evaluation of the second order normal and anomalous contributions, and his readiness to help me debugging the Fortran codes were vital for the realization of this thesis. I want to thank Norbert Kaiser for guiding me through my thesis and for introducing me to the fine arts of nuclear physics. I appreciated very much his helpful advice and the open door to his office at any time of day. Also for his readiness to discuss any problem which puzzled me and for his patience regarding my last-minute strategy of producing results I want to thank him, and of course for him going through my Fortran codes with me before my seminar talk, when the time was running out and the bugs seemed to hide. I want to thank Wolfram Weise for familiarizing me with the world of the strong interactions in his QCD lecture, and for offering me to work at T39. Also for his helpful and encouraging comments during my seminar talk I want to thank him. Finally, I thank all the people at T39 who made me enjoy the time of my thesis very much. I thank Salvatore Fiorilla, who also helped me a lot regarding many physical and computational aspects of my work, for being a communicative and entertaining office mate. Also thanks to Max Duell and Michael Altenbuchinger, my office mates at the beginning of my thesis and now proud inhabitants of the “coffee office”, who together with Salvatore and myself shared and cultivated a severe addiction to caffeine. Also Thomas Hell, Alexander Laschka, Matthias Drews, Robert Lang, Nino Bratovic, Stefan Petschauer, Bertram Klein, Paul Springer and again Jeremy Holt I want to thank for numerous at-work and beyond-work conversations and activities. Last but not least I thank my friends and my family, in particular my mother - without her support my studies would probably have been correlated with starvation.