Density-Dependent Neutron-Neutron Interaction from Subleading Chiral Three-Neutron Forces
DDensity-Dependent Neutron-NeutronInteraction from Subleading ChiralThree-Neutron Forces Lukas TreuerPhysik-Department T39, Technische Universität München, 85748 Garching b. München,Germany
E-Mail: [email protected]
Abstract
Three-nucleon forces are an essential ingredient for an accurate description of nuclearfew- and many-body systems. However, implementing them directly in many-bodycalculations is technically very challenging. Thus, there is a need for an efficientapproximation method. By closing one nucleon line to a loop, it is possible to deriveeffective in-medium nucleon-nucleon interactions that represent the underlying three-nucleon forces, as constructed in Chiral Effective Field Theory. Since three-neutronforces are equally as important for the computation of the equation of state for pureneutron matter, this work applies the aforementioned approach to the subleading chiralthree-neutron forces, in particular the short-range terms and relativistic corrections.It is shown in this work that, while many contributions to the in-medium neutron-neutron interaction are - apart from a constant factor - identical to the terms in isospin-symmetric matter, some differ drastically. Moreover, previously vanishing terms yieldnow non-zero contributions. As a result of this work, density-dependent in-mediumneutron-neutron potentials are now available for the implementation in nuclear many-body calculations, either in closed analytical form, or requiring at most one numericalintegration. Bachelor’s thesis in physics, Technische Universität München, September 2020 a r X i v : . [ nu c l - t h ] S e p ontents
1. Introduction 12. Topologies of One-Loop Diagrams and the In-Medium Neutron-NeutronInteraction 3
3. Resulting Contributions to the In-Medium Neutron-Neutron Interaction 14
4. Conclusion and Outlook 30Appendix A. Loop-Functions 32 Γ ν Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322. γ ν Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333. G ν Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354. K ν Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
References 38 . Introduction
Three-nucleon (3N) forces allow for a deeper understanding of the strong interactionsand they provide more precise predictions for nuclear many-body systems, thus playingan important role not only in nuclear physics, but in astrophysics as well. Particularly innuclear few-body systems, 3N-forces represent an essential ingredient for the accuratedescription of experimental data and basic nuclear phenomena, such as binding energyper particle or the saturation density of nuclear matter [1].The 3N-interactions are constructed using Chiral Effective Field Theory (ChEFT), whichgoverns the low-energy regime of the quantum field theory of the fundamental stronginteraction, namely Quantum Chromo Dynamics. This is achieved by making use ofspontaneous and explicit breaking of chiral symmetry, induced by a non-zero scalarquark condensate and non-vanishing current quark masses, respectively. Thus, therelevant degrees of freedom are the nucleons on the one hand side, and the Goldstonebosons of the spontaneously broken chiral symmetry on the other hand side. Thelatter comprise the light pseudoscalar meson octet, or the tree pions in the case of twoquark-flavours [1, 2].In the low-momentum expansion of chiral EFT, 3N-forces are not present up to next-to-next-to-leading order (N LO). Up to that order, the NN-interaction consists of effectivezero-range contact terms and longer-range components through one-pion- (1 π ) andtwo-pion- (2 π ) exchanges [1].When implementing 3N-forces in nuclear many-body systems, however, it is compu-tationally very difficult to calculate their contributions directly. Therefore, a simplermethod was developed in ref. [1], where the inclusion of 3N-forces is done via a density-dependent nucleon-nucleon (NN) interaction. Since this involves the construction of aneffective in-medium potential V med , it is only a good approximation when dealing withmany-body systems that can be viewed in the thermodynamic limit. The systems areassumed to be at zero temperature, thus restricting the occupied states of nucleons tobelow the Fermi surface.The approach of constructing V med from chiral 3N-forces was not only implementedfor the leading chiral 3N-forces [1], but for the subleading contributions, as well. Thesubleading 3N-forces were divided into short-range terms and relativistic corrections[3], intermediate-range terms [4], as well as long-range terms [5]. Specifically, in the1 . Introduction mentioned articles, the method of consructing V med was executed for isospin-symmetricspin-saturated nuclear matter. The results are thus applicable to nuclear systems withthe same number of protons and neutrons, and fully paired spins.However, up to that point, the contributions of subleading chiral 3n-forces to an effec-tive in-medium interaction in pure neutron matter had not been calculated. Hence, theaim of this work is to begin the application of the previously mentioned approximationmethod to spin-saturated neutron matter. Specifically, this work presents the results fordensity-dependent neutron-neutron (nn) interactions arising from short range termsand relativistic corrections of the subleading chiral three-neutron (3n) forces, originallyderived in ref. [2] for three nucleons.Adapting them accordingly to represent only three-neutron interactions, the subleadingchiral 3N-forces used as starting point are taken from ref. [3] for the sake of consistencyand comparability, as some corrections and specifications have been made with respectto ref. [2].In perspective, the results presented in this work will be useful for implementationin nuclear many-body calculations, in order to deepen our understanding of systemssuch as neutron stars, as well as to gain more insight into some physical parameters ofnuclear matter, e.g. the isospin-asymmetry energy and its slope as a function of nucleardensity. These can be obtained from the equation of state of pure neutron matter andsymmetric nuclear matter [6].Beginning the main section of this work, the topologies involved in the interaction, aswell as the method of calculation are detailed in chapter 2. The results for in-mediumnn-potentials are presented together with the 3n-forces they stem from in chapter 3.Following the accompanying discussions, a summarizing conclusion and an outlookto future studies are given in chapter 4. Finally, all the relevant loop-functions aredefined and specified in the appendix, either in closed analytical form, or throughone-parameter integrals. 2 . Topologies of One-Loop Diagrams andthe In-Medium Neutron-NeutronInteraction In order to discuss the effective in-medium potentials presented in chapter 3, it ishelpful to first introduce the general form of the 3n-interaction and to understandthe involved topologies, as well as the method of calculation. To that end, thoroughexplanations are given in this chapter. That includes the derivation of the basic identitiescharacterizing the situation in pure neutron matter, related to isospin operators.For the sake of comprehensibility, this chapter is divided into two sections:Section (2.1) deals with the fundamental interaction scheme of the considered 3n-forcesand the subsequently appearing in-medium topologies, while the technicalities ofderiving the effective nn-potentials are explained in section (2.2). n n n π srFigure 2.1.: Generic 3n-interaction: π -exchange (dashed line) between the first andsecond neutron, short-range interaction (wiggly line) between neutron 2and neutron 3Beginning with the generic form of the 3n-interaction shown in figure (2.1), it is impor-tant to familiarize oneself with the relevant interaction mechanism. The three neutrons3 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction interact via the exchange of one pion (or two pions) and an effective zero-range contactcoupling containing a momentum-independent propagator. Also, there exist caseswhere the symbolic diagram in fig. (2.1) is interpreted as two-pion-exchange. Inparticular, this applies to the 3n-interactions presented in subsection (3.3.2).However, contrary to 3N-forces, only the neutral pion π - within the pion-triplet - is in-volved in 3n-forces. In other words, the strong nuclear interaction in isospin-symmetricmatter can be executed via the exchange of neutral and charged pions, whereas onlythe neutral pion can mediate the residual strong interaction between neutrons, due tocharge conservation.After considering the generic 3n-interaction, the focus now lies on the appearing topolo-gies when calculating the in-medium potential. Their diagrammatic structure is shownin figs. (2.2) - (2.5), where mirror images - in other words contributions arising from theexchange of the external neutron lines n ↔ n - have to be added. In some cases thissimply yields a factor of 2. Thus, the relevant topologies are introduced here, whereastheir computational intricacies are detailed in the following section.Beginning with the term "topology" itself, whose meaning somewhat differs dependingon the field and topic. It will henceforth denote the equivalent diagrammatic structureor meaning of Feynman diagrams, and as such the underlying terms in the transitionamplitude, constructed from the Feynman rules of the appropriate quantum fieldtheory - in this case ChEFT.Fundamentally, one obtains the in-medium nn-potential by closing one of the threeneutron lines to an in-medium loop, leading to the four distinct topologies. Whendoing this, it is critical to follow the appropriate Feynman rules, that is, integratingover the respective neutron four-momentum, and in the case of fermionic loops, takingthe trace over the neutron’s spin states. Additionally, there appears a minus-sign forclosed fermion lines.First of all, it is possible to close one neutron line to itself, which can be seen in fig.(2.2). Such contributions from self-closings are henceforth denoted as V ( ) med .Secondly, one obtains two different kinds of vertex corrections, through a short-rangeinteraction on the one hand, and through pion exchange on the other hand, giving riseto the pieces V ( ) med and V ( ) med , respectively. These topologies are visible in figs. (2.3) and(2.4).Lastly, the fourth topology describes double exchanges, whose structure is shown infig. (2.5), yielding the contribution V ( ) med . 4 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction n n π sr n n π sr n n π srFigure 2.2.: Feynman diagrams from self-closings yielding V ( ) med . Graphs arising fromthe exchange n ↔ n are not shown and have to be added.n n n π sr n n n π srFigure 2.3.: Feynman diagrams from short-range vertex corrections yielding V ( ) med .Graphs arising from the exchange n ↔ n are not shown and have tobe added.n n n π sr n n n π srFigure 2.4.: Feynman diagrams from pionic vertex corrections yielding V ( ) med . Graphsarising from the exchange n ↔ n are not shown and have to be added.5 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction n n n π sr n n n π srFigure 2.5.: Feynman diagrams from double exchanges yielding V ( ) med . Graphs arisingfrom the exchange n ↔ n are not shown and have to be added.In these Feynman diagrams, it is possible to see two parallel slashes on the neutronlines within the loop. This denotes the medium insertion of the neutron propagatorcoming from the filled Fermi sea. Its origin and form are described in the next section.Having specified the basic mode of 3n-interaction and the topologies arising fromclosing one neutron line to a loop, the construction of the in-medium nn-potential fromsubleading chiral 3n-forces is outlined in the next section.6 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction In this section, the calculational methods employed in this work are laid out andexplained, supplementing background information when it is required.A basic component of the aforementioned approach to the treatment of 3n-forces is theneutron propagator. As one now considers in-medium interactions, it is required toreplace the free neutron propagator by its in-medium counterpart, visible in expression(2.1) below. It describes a particle propagating freely outside the Fermi sphere (leftterm), while a hole propagates within the Fermi sphere (right term) - as detailed onpage 26 of ref. [7]. Reinstalling a factor i with respect to ref. [7], the in-medium neutronpropagator takes on the form i θ ( | (cid:126) l | − k n ) l − E ( (cid:126) l ) + i (cid:101) + i θ ( k n − | (cid:126) l | ) l − E ( (cid:126) l ) − i (cid:101) , (2.1)where θ ( x ) is the usual Heaviside step function, and l , (cid:126) l are the components of theneutron four-momentum - the neutron energy and its three-momentum. Furthermore, k n denotes the neutron Fermi momentum, which is related to the neutron density by ρ n = k n /3 π . E ( (cid:126) l ) = (cid:126) l /2 M is the kinetic energy with M =
940 MeV the neutron mass[8], and (cid:101) is an infinitesimal positive parameter to properly treat the poles.However, by employing the identity1 x ± i (cid:101) = P x ∓ i πδ ( x ) , (2.2)it is possible to derive another representation, which is particularly useful in thefollowing computations. Here, P stands for the Cauchy principal value, and δ ( x ) denotes the Dirac delta-function. Thus, after multiplying with the imaginary unit, thein-medium neutron propagator reads: il − E ( (cid:126) l ) + i (cid:101) − πδ ( l − E ( (cid:126) l )) θ ( k n − | (cid:126) l | ) −→ il + i (cid:101) − πδ ( l ) θ ( k n − | (cid:126) l | ) . (2.3)In the second line, the heavy baryon limit M → ∞ was employed. It is equivalent tothe scaling l (cid:119) | (cid:126) l | , such that terms of order O ( (cid:126) l ) can be neglected.The first term describes a free propagation and does not yield any k n -dependentcontributions, thus being irrelevant henceforth. The crucial second part leads to density-dependent terms when performing the necessary loop integral ( π ) − (cid:82) d l , and it7 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction is called the in-medium insertion. Making use of the delta- and theta-functions, onefinally obtains the Fermi sphere integral present in all calculations to obtain the effectivenn-interaction, − (cid:90) | (cid:126) l | < k n d l ( π ) . (2.4)After covering the medium insertion as the salient feature for computations, it is nowtime to consider the kinematical variables and operators constituting the 3n-interactions,and effective nn-potential.Within the explicit form of the 3N-forces taken from ref. [3], a number of quantitiesappear, namely the in- and outgoing particle momenta (cid:126) p i , (cid:126) p i (cid:48) , as well as the spin-and isospin-vector operators (cid:126) σ i , (cid:126) τ i , where i =
1, 2, 3 denotes the respective nucleon.Furthermore, the three-momentum-transfers, defined as (cid:126) q i = (cid:126) p i (cid:48) − (cid:126) p i , are encounteredfrequently. All of the appearing 3N-force terms are expressed by these variables. Also,using conservation of momentum, it is easy to verify that the momentum-transferssatisfy the constraint (cid:126) q + (cid:126) q + (cid:126) q = (cid:126) p = − (cid:126) p ≡ (cid:126) p , (cid:126) p (cid:48) = − (cid:126) p (cid:48) ≡ (cid:126) p (cid:48) holds.Additionally, elastic on-shell scattering is assumed, meaning that | (cid:126) p | = | (cid:126) p (cid:48) | ≡ p ,leading to the restraints on the momentum-transfer modulus 0 ≤ q ≤ p , where q ≡ | (cid:126) q | = p sin ( θ /2 ) = p (cid:112) ( − z ) with z = cos θ and θ the CM scattering angle.Having defined the kinematical quantities, it is now possible to understand the variousoperators appearing in the effective nn-potential, constructed from spin-vector operatorsand momenta of the involved two neutrons.Beginning with the spin-operators, there are 5 distinct terms, which do not changewithin the medium when compared to their counterparts in free space,1, (cid:126) σ · (cid:126) σ , (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q , i ( (cid:126) σ + (cid:126) σ ) · ( (cid:126) q × (cid:126) p ) , (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) . (2.5)In the given order, the first four are called central, spin-spin, tensor and spin-orbitterms. The last term is related to the quadratic spin-orbit operator by the following8 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction identity: (cid:126) σ · ( (cid:126) q × (cid:126) p ) (cid:126) σ · ( (cid:126) q × (cid:126) p ) = q (cid:18) p − q (cid:19) (cid:126) σ · (cid:126) σ + (cid:18) q − p (cid:19) (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q − q ( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) . (2.6)This decomposition can be checked by making use of the Levi-Civita symbol identity: (cid:101) ijk (cid:101) lmn = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ il δ im δ in δ jl δ jm δ jn δ kl δ km δ kn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.7)where (cid:101) ijk and δ ij denote the Levi-Civita symbol and the Kronecker delta, respectively.As a side-note, many of the calculations require the application of the following usefulidentities for products of Pauli spin matrices: (cid:126) σ · (cid:126) a (cid:126) σ · (cid:126) b = (cid:126) a · (cid:126) b + i (cid:126) σ · ( (cid:126) a × (cid:126) b ) , (2.8) (cid:101) ijk σ j σ k = i σ i , (2.9)where (cid:126) a , (cid:126) b are arbitrary three-vectors.Concerning the occurring isospin-vector operators within the in-medium NN-potentialin isospin-symmetric nuclear matter, these only encompass the two possible structures1 and (cid:126) τ · (cid:126) τ , (2.10)which are identical to the ones for the NN-potential in free space.This leads directly into the key difference between isospin-symmetric matter, involvingequal parts protons and neutrons, and pure neutron matter. That is, the substitutions (cid:126) τ i · (cid:126) τ j → i , j =
1, 2, 3 and (cid:126) τ · ( (cid:126) τ × (cid:126) τ ) → (cid:104) n | (cid:126) τ i | n (cid:105) = − , (2.11)9 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction one obtains (cid:104) n n | (cid:126) τ i · (cid:126) τ j | n n (cid:105) = − · − = (cid:104) n n | (cid:126) τ · ( (cid:126) τ × (cid:126) τ ) | n n (cid:105) = − · − × − = ρ n = k n /3 π . Notice the difference to thenucleon density in isospin-symmetric matter, ρ = k n /3 π , due to a missing isospin-multiplicity factor of 2 for the two Fermi seas of protons or neutrons.Lastly, throughout this work, the same sign-convention is followed as in ref. [3], suchthat at tree-level, the one-pion-exchange nn-potential is given by V π = − ( g A /2 f π ) × ( m π + q ) − (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q . Here, g A = f π = m π =
135 MeV the neutral pion mass [9]. In contrast to ref. [3], m π is set to be theneutral pion mass instead of the average mass among the pion triplet, since the chargedpions are not involved, as previously explained.After covering the basic ingredients needed in this work, it is now time to move on tothe calculation of Fermi sphere integrals.During the derivation of effective nn-potentials, whose results are presented in thefollowing chapter, one encounters Fermi sphere integrals over even functions10 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron InteractionF ( s ) = F ( − s ) , where s = ( (cid:126) l + (cid:126) p ) = p + l + l pz , and z = cos α with α beingthe angle between (cid:126) l and (cid:126) p . Making use of F ( s ) being even, these can be reduced toone-parameter integrals in the following way: (cid:90) | (cid:126) l | < k n d l π F ( s ) = k n (cid:90) dl l (cid:90) − dz F ( s ) = k n (cid:90) dl lp p + l (cid:90) | p − l | ds s F ( s )= k n (cid:90) dl lp p + l (cid:90) p − l ds s F ( s ) = p + k n (cid:90) p − k n ds s F ( s ) k n (cid:90) | s − p | dl lp = p + k n (cid:90) p − k n ds s p (cid:2) k n − ( s − p ) (cid:3) F ( s ) . (2.14)The crucial step consists of dropping the absolute magnitude at | p − l | , which is allowedsince the antiderivative of s F ( s ) is an even function.Furthermore, Fermi sphere integrals involving tensorial factors { l i , l i l j , l i l j l k } appearfrequently as well. These are solved by making use of the symmetry regarding theexchange of indices, and constructing the appropriate general form of the integralin terms of tensorial factors built from δ ij and p i on the one hand, and scalar loop-functions on the other hand. Finally, one contracts the integral with the prefactors ofthe loop-functions, thus obtaining linear equations. Adhering to this method, one canfind the following reduction formulae for integrals: (cid:90) | (cid:126) l | < k n d l π F ( s ) { l i , l i l j } = p + k n (cid:90) p − k n ds s p (cid:2) k n − ( s − p ) (cid:3) F ( s ) × { χ p i , χ δ ij + χ p i p j } , (2.15)which are employed throughout this work, and are equally applicable to (cid:126) p → (cid:126) p (cid:48) . Theweighting functions have been determined through the previously outlined method11 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction and read: χ = p ( s + sp − p − k n ) , χ = p (cid:2) k n − ( s − p ) (cid:3) ( s + sp + p − k n ) , χ = p (cid:2) k n + k n ( p − sp − s ) + ( s − p ) ( s + sp + p ) (cid:3) . (2.16)Using these expressions, and plugging in a pion propagator (cid:2) m π + ( (cid:126) l + (cid:126) p ) (cid:3) − for F ( s ) ,one can obtain the analytical expressions for the employed loop-functions Γ ν ( p , k n ) , γ ν ( p , k n ) , ν =
1, . . . , 5 , which are given in the appendix. In the case of the tensorialfactor l i l j l k ( ν =
4, 5 ) , the loop-functions are obtained by constructing projection opera-tors out of their prefactors, and contracting them with the Fermi sphere integral. Thus,these loop-functions are calculated directly, without first computing the polynomialweighting functions; the underlying methodology is of course identical.Apart from these loop-functions, two other kinds are used as well, G ν ( p , q , k n ) and K ν ( p , q , k n ) , ν =
1, 2, 3. Their analytical construction is limited to the reduction to aradial integral by using the Feynman parametrization,1 AB = (cid:90) dx (cid:2) xA + ( − x ) B (cid:3) , (2.17)where A , B each denote different pion propagators. One executes the angular- and x -integrals to obtain a purely radial integral, and lastly solves a system of linear equa-tions. Although they are not analytical, the functions G ν and K ν are given in a formwhich requires just a one-parameter numerical integration.For the sake of readability and notational simplicity, the arguments of the loop-functionswill be suppressed henceforth.Armed with the basic tools to evaluate loop integrals and the knowledge of the ap-pearing interaction topologies, it is now possible to understand the procedure of thenecessary calculations, as well as the underlying physics of the approach employed inthe derivation of the effective in-medium nn-potential.First, the calculations for self-closings are illustrated in more detail. Performing theFermi sphere integral arising from closing the neutron-loop and making use of thein-medium insertion, it is important to remember the Feynman rules for fermionicloops, leading to an additional factor of minus one. Also, it is necessary to take the traceover the respective neutron spin, meaning that contributions containing the spin-vectors12 . Topologies of One-Loop Diagrams and the In-Medium Neutron-Neutron Interaction vanish, as (cid:126) σ is traceless.Terms involving the respective momentum-transfer yield zero as well, due to momen-tum conservation at every vertex demanding (cid:126) q i =
0, where the index i refers to theclosed neutron line. Thus, using the previously mentioned relation (cid:126) q + (cid:126) q + (cid:126) q = (cid:126) q j = − (cid:126) q k , the indices j , k standing for the external neutrons.Hence, as there are no (cid:126) l -dependent terms, the Fermi sphere integral yields a factor of ρ n /2. In the end, one has to add the individual contributions - though, in most casesonly one of the neutron loops leads to a non-zero term - and make the appropriateadjustments to the indices, (
2, 3 ) → (
1, 2 ) when closing n or 3 → n .As final step, the result arising from the mirror diagram is added to the previouslydetermined one.With respect to each other, the three remaining contributions exhibit very similar ap-proaches to their calculation.One begins by assigning the appropriate momenta ± (cid:126) p , ± (cid:126) p (cid:48) , ± (cid:126) l to the (cid:126) p i , (cid:126) p i (cid:48) andthus derive the expressions for the corresponding momentum-transfers (cid:126) q i . To theend of simplifying calculations, it is advantageous to choose + (cid:126) l as three-momentumat the medium insertion for short-range vertex corrections, and − (cid:126) l for pionic vertexcorrections and double exchanges. As the Fermi sphere is invariant under (cid:126) l ↔ − (cid:126) l , onehas the freedom to choose the more convenient option.After doing this for both types of Feynman diagrams with a set constellation of n and n , it is advisable to check that the derived (cid:126) q i satisfy (cid:126) q + (cid:126) q + (cid:126) q =
0. Subsequently,one writes down the Fermi sphere integrals for the given interaction, plugging in thederived expressions above, and taking care to arrange the spin operators correspondingto the order they are applied to the neutron line. Then, one reassigns the neutronindices, namely (
2, 3 ) → (
1, 2 ) for pionic vertex corrections and 3 → n ↔ n from the initial calculations, and only add them to V med inthe end.Thus, after explaining the employed tools and necessary information, it is now possibleto present the resulting in-medium nn-potentials in the following chapter, togetherwith the underlying calculations to obtain them.13 . Resulting Contributions to theIn-Medium Neutron-Neutron Interaction In this chapter, the results for contributions to the in-medium nn-potential V med ,obtained by closing one neutron line of the subleading chiral 3n-forces, are givenexplicitly. They are expressed in terms of the loop-functions defined in the appendix,and discussed subsequently.First, the method is applied to the short-range one- and two-pion-exchange-contacttopologies in sections (3.1) and (3.2) respectively. After that, the contributions arisingfrom relativistic corrections are presented in section (3.3), divided further into a one-pion-exchange-contact topology, as well as a two-pion-exchange topology. Starting with the 1 π -exchange-contact topology, there are two contributions to the3n-interaction: V n = − g A C T m π π f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q m π + q (3.1)and V n = g A C T m π π f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q m π + q , (3.2)taken from eqs. (3) and (4) of ref. [3], respectively. The operator substitution is per-formed by making use of both isospin identities mentioned in the previous chapter.In the interactions above, C T denotes a low-energy constant assigned to the leadingspin-dependent NN-contact interaction [2].As the terms are opposite in sign and identical otherwise, they add up to zero, yieldingno net contribution within the 1 π -exchange-contact topology. This is consistent withref. [2], where these 3N-forces vanish by total antisymmetrization, and ref. [3] in whichthe contributions to the NN-potential in isospin-symmetric matter cancel within eachpartial wave. 14 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction This section on the 1 π -exchange-contact topology is followed by the contributionsarising from the 2 π -exchange-contact topology interactions, which are discussed on thefollowing pages. 15 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction Within the 2 π -exchange-contact topology, two contributions have been derived in ref.[2] for the 3N-force. The first one, adapted to the 3n-case, reads: V n = − g A C T π f π (cid:126) σ · (cid:126) σ (cid:104) m π + ( m π + q ) A ( q ) (cid:105) , (3.3)where A ( q ) = q arctan q m π (3.4)the relevant pion-loop function. These expressions are taken from eqs. (8) and (9) ofref. [3], after making use of the identity (cid:126) τ · (cid:126) τ = A ( ) = m π , (3.5)one obtains the contribution V ( ) med = − g A C T m π k n π f π (cid:126) σ · (cid:126) σ = g A C T m π k n π f π . (3.6)While the analogous term in isospin-symmetric nuclear matter was zero due to avanishing isospin trace, this is evidently not the case in pure neutron matter.Furthermore, (cid:126) σ · (cid:126) σ = − (cid:126) σ · (cid:126) σ = − V ( ) med contains only a spin-spin coupling term, and does not depend on q = p ( − z ) and thus the scattering angle in the CM frame, leading to a total spin of S =
0. Thiscan be demonstrated by employing the partial wave projection formulae in eqs. (6) - (9)of ref. [10], and making use of the orthogonality of ordinary Lergendre polynomials.This approach yields the conditions ( LSJ ) = ( ) , ( ) , with L denoting the totalorbital angular momentum, and J the total angular momentum. The second of theaforementioned options is inapplicable, however, since fulfillment of L + S + I = odd is demanded, in order to account for an antisymmetric total fermion wave-functionunder exchange of nucleons. In this case, as the total isospin satisfies I = L + S = even . Hence, using regularspectroscopic notation, eq. (3.6) contributes to the S + L J = S partial wave only,16 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction where the spins fulfill (cid:126) σ = − (cid:126) σ , effectively leading to an isotropic interaction.Next, the contribution from short-range vertex corrections - as seen in fig. (2.3) - reads: V ( ) med = g A C T k n π f π (cid:104) m π + ( m π + q ) A ( q ) (cid:105) , (3.7)which is identical to the result in eq. (10) of ref. [3], provided the appropriate adaptionto neutron matter (cid:126) τ · (cid:126) τ → V ( ) med + V ( ) med = g A C T π f π (cid:126) σ · (cid:126) σ (cid:26) m π k n ( p + k n − m π )+ ( p + k n ) (cid:20) k n p ( k n + m π ) − m π + k n + pk n − p (cid:21) arctan p + k n m π + ( p − k n ) (cid:20) k n p ( k n + m π ) + m π − k n + pk n + p (cid:21) arctan p − k n m π + m π p ( p − k n + m π ) ln 4 m π + ( p + k n ) m π + ( p − k n ) (cid:27) , (3.8)which, following the same argumentation as for eq. (3.6), contributes only to the S -wave.Thus, it is possible to add the three contributions acting only in the S partial-wave toobtain: 17 . Resulting Contributions to the In-Medium Neutron-Neutron InteractionV ( + + ) med = − g A C T π f π (cid:26) m π k n ( p − k n − m π )+ ( p + k n ) (cid:20) k n p ( k n + m π ) − m π + k n + pk n − p (cid:21) arctan p + k n m π + ( p − k n ) (cid:20) k n p ( k n + m π ) + m π − k n + pk n + p (cid:21) arctan p − k n m π + m π p ( p − k n + m π ) ln 4 m π + ( p + k n ) m π + ( p − k n ) (cid:27) , (3.9)where a new notation was employed for the sake of readability, defining V ( + + ) med ≡ V ( ) med + V ( ) med + V ( ) med . Additionally, it is interesting to note that eq. (3.8) is identical tothe result in eq. (11) of ref. [3], provided the appropriate substitution 3 + (cid:126) τ · (cid:126) τ → π -exchange-contact topology isgiven in eq. (12) of ref. [3], and after adaption to the 3n-case it reads: V n = g A C T π f π (cid:26) (cid:126) σ · (cid:126) σ (cid:20) m π − m π m π + q + ( m π + q ) A ( q ) (cid:21) + [ (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q − q (cid:126) σ · (cid:126) σ ] A ( q ) (cid:27) . (3.10)In this particular case, it is convenient to add the contributions of self-closings fromthe second and third neutron line (here denoted as V ( (cid:48) ) med ) and the short-range vertexcorrections, to obtain V ( (cid:48) ) med + V ( ) med = g A C T k n π f π (cid:20) m π m π + q − m π − ( m π + q ) A ( q ) (cid:21) , (3.11)18 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction since V ( (cid:48) ) med is canceled by a term present in V ( ) med .In the same way as with eq. (3.9), it is advantageous to subsequently add the contri-butions arising from closing the first neutron line to itself (here denoted as V ( (cid:48)(cid:48) ) med ), thepionic vertex corrections, and double exchanges. Making use of the notation employedin eq. (3.9), this yields V ( (cid:48)(cid:48) + + ) med = g A C T π f π (cid:26) m π k n (cid:18) p + k n − m π (cid:19) + (cid:20) k n p (cid:18) m π + k n (cid:19) + (cid:18) m π + m π k n + k n (cid:19) + pk n + p (cid:18) k n − m π (cid:19) − p (cid:21) arctan p + k n m π + (cid:20) k n p (cid:18) m π + k n (cid:19) − (cid:18) m π + m π k n + k n (cid:19) + pk n + p (cid:18) m π − k n (cid:19) + p (cid:21) arctan p − k n m π + m π p (cid:20) ( p − k n ) − m π (cid:21) ln 4 m π + ( p + k n ) m π + ( p − k n ) (cid:27) . (3.12)Here, the fact that the terms involving the factor q in eq. (3.10) cancel each other whenadding V ( ) med and V ( ) med was already made use of, leaving a contribution which onlycontributes to the S partial-wave. This holds true for V ( (cid:48)(cid:48) ) med as well, hence the additionto the combined contribution in eq. (3.12).The notable contrast in length compared to the result presented in eq. (14) of ref. [3]stems from the cancellation of terms proportional to ( (cid:126) l + (cid:126) p ) A ( | (cid:126) l + (cid:126) p | ) + ( (cid:126) l + (cid:126) p (cid:48) ) × A ( | (cid:126) l + (cid:126) p (cid:48) | ) inside the Fermi sphere integrals for V ( ) med and V ( ) med .The integrals appearing in this section have been computed using eq. (2.15), derived inthe previous chapter.This concludes the section on the 2 π -exchange-contact topology. Thus, as the last -albeit the most expansive - entry within the chapter on results of V med , contributionsarising from the leading relativistic corrections to the chiral 3n-interaction are presentedin the following section. 19 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction In this section, contributions to the in-medium nn-potential from the leading relativisticcorrection terms to chiral 3n-forces are presented. These expressions arise from 1/ M -corrections to the leading π nn and ππ nn vertices on the one hand, and retardationeffects on the other hand.Moreover, the 3n-force expressions treated in this section are subdivided into the1 π -exchange-contact topology of subsection (3.3.1) with contributions proportional to g A C T , S / M f π , and the 2 π -exchange topology of subsection (3.3.2) with contributionsproportional to g A / M f π . Here, C S denotes another low-energy constant, associatedwith the lowest-order spin-independent NN-contact interaction [2].One should note that the following calculations and expressions are significantlymore complex than the previous ones, as they - in most cases - involve Fermi sphereintegrals over one or multiple pion propagators, and spin-operator orderings have tobe considered before changing neutron indices. Within this topology, there are two V N terms, both of which are proportional to (cid:126) τ · (cid:126) τ and only consist of terms containing either (cid:126) σ or (cid:126) q . In isospin-symmetric nuclearmatter, this leads to vanishing contributions from self-closings ( V ( ) med ) due to tr (cid:126) τ = (cid:126) σ = (cid:126) q = (cid:126) τ · (cid:126) τ → V ( ) med ) and double exchanges( V ( ) med ). The contribution from pionic vertex corrections ( V ( ) med ) are reduced by a factorof 3, due to the dot product of identical isospin-vector operators (cid:126) τ · (cid:126) τ = V n = − g A M f π m π + q (cid:110) C T (cid:104) i (cid:126) σ · ( (cid:126) p + (cid:126) p (cid:48) )( (cid:126) σ × (cid:126) σ ) · (cid:126) q + (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q + i (cid:126) σ · (cid:126) q ( (cid:126) σ × (cid:126) σ ) · ( (cid:126) p + (cid:126) p (cid:48) ) (cid:105) + C S (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q (cid:111) , (3.13)and arises out of the 1/ M correction to the π nn vertex ( π n coupling) and 4n-contactvertex (nn-contact interaction). 20 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction As previously mentioned, and contrary to the 3N-case, the contribution stemming fromself-closings (of the middle n-line) does not vanish for pure neutron matter, yielding V ( ) med = g A C T k n π M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) qm π + q . (3.14)On the other hand, short-range vertex corrections give rise to the contribution V ( ) med = g A k n π M f π ( C T − C S ) (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) qm π + q , (3.15)which is proportional to the 1 π -exchange nn-interaction potential in momentum space[11], and linear in the neutron density ρ n = k n /3 π .Next, when considering pionic vertex corrections, one can derive V ( ) med = g A π M f π C T (cid:26)(cid:20) p ( Γ − Γ ) + q ( Γ + Γ ) + Γ (cid:21) (cid:126) σ · (cid:126) σ − ( Γ + Γ ) (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q + ( Γ − Γ )( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) (cid:27) + g A π M f π ( Γ + Γ ) (cid:104) ( C S − C T ) i ( (cid:126) σ + (cid:126) σ ) · ( (cid:126) q × (cid:126) p ) − C S q (cid:105) , (3.16)with double exchanges giving rise to V ( ) med = g A π M f π (cid:126) σ · (cid:126) σ (cid:26) C T (cid:20) q ( Γ + Γ ) − p ( Γ + Γ + Γ ) (cid:21) − ( C T + C S ) Γ (cid:27) + g A π M f π (cid:110) ( C S − C T )( Γ + Γ ) (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q + (cid:104) C T ( Γ + Γ + Γ ) − C S ( Γ + Γ + Γ ) (cid:105) ( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) (cid:111) + g A π M f π C T (cid:26) ( Γ + Γ ) (cid:20) q − i ( (cid:126) σ + (cid:126) σ ) · ( (cid:126) q × (cid:126) p ) (cid:21) − Γ − p ( Γ + Γ + Γ ) (cid:27) . (3.17)21 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction Besides the already known spin-spin (cid:0) ∼ (cid:126) σ · (cid:126) σ (cid:1) and tensor-type (cid:0) ∼ (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q (cid:1) components, both contributions contain a spin-orbit term (cid:0) ∼ i ( (cid:126) σ + (cid:126) σ ) · ( (cid:126) q × (cid:126) p ) (cid:1) anda part that involves the quadratic spin-orbit operator according to eq. (2.6), namely oneproportional to (cid:0) ∼ ( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) (cid:1) .The loop-functions Γ ν , ν =
1, . . . , 5 are defined as Fermi sphere integrals over a singlepion propagator, and explained in more detail, as well as given explicitly in the ap-pendix. They involve arctangents and logarithms, and depend on the pion mass m π ,in conjunction with the modulus of the CM on-shell neutron momentum p , and theneutron Fermi momentum k n . This holds true for all following loop-functions, as well.The second interaction within the 1 π -exchange-contact topology, arising from therespective retardation corrections, reads: V n = g A M f π (cid:126) σ · (cid:126) q ( m π + q ) (cid:110) (cid:126) q · (cid:126) q ( C S (cid:126) σ · (cid:126) q + C T (cid:126) σ · (cid:126) q )+ iC T ( (cid:126) σ × (cid:126) σ ) · (cid:126) q ( (cid:126) p + (cid:126) p (cid:48) + (cid:126) p + (cid:126) p (cid:48) ) · (cid:126) q (cid:111) , (3.18)which visibly includes a squared pion propagator, and is taken from eq. (27) of ref. [3].Closing the middle neutron line to itself leads to the non-vanishing contribution V ( ) med = − g A C T k n q π M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q ( m π + q ) , (3.19)while short-range vertex corrections yield V ( ) med = g A k n q π M f π ( C S − C T ) (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q ( m π + q ) , (3.20)both of which are linear in the neutron density ρ n .Computing the results for pionic vertex corrections and double exchanges, one obtainsthe following comparably long expressions:22 . Resulting Contributions to the In-Medium Neutron-Neutron InteractionV ( ) med = g A C S q π M f π (cid:104) Γ + Γ − m π ( γ + γ ) (cid:105) + g A C T π M f π (cid:26) ( γ + γ ) (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q + (cid:20) Γ − k n + m π ( Γ − m π γ − γ )+ q ( γ + γ ) + (cid:18) q − p (cid:19) (cid:0) m π ( γ + γ ) + γ + γ − Γ − Γ (cid:1)(cid:21) (cid:126) σ · (cid:126) σ + (cid:20) Γ + Γ + Γ − m π ( γ + γ + γ ) − ( γ + γ )+ (cid:18) q − p (cid:19) ( γ + γ + γ + γ ) (cid:21) ( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) (cid:27) , (3.21)23 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction and V ( ) med = g A π M f π ( C T − C S ) (cid:0) γ + γ (cid:1) (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q + g A C S π M f π (cid:26)(cid:104) Γ − ( m π + q ) γ − q γ (cid:105) (cid:126) σ · (cid:126) σ + (cid:20) Γ + Γ + Γ − m π ( γ + γ + γ ) − q ( γ + γ + γ + γ ) (cid:21) × ( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) (cid:27) + g A C T π M f π (cid:26) k n + m π ( m π γ − Γ ) + q (cid:104) m π ( γ + γ ) − Γ − Γ (cid:105) + (cid:20)(cid:18) p + q (cid:19) (cid:16) m π ( γ + γ ) + γ + γ − Γ − Γ (cid:17) + k n − Γ + m π ( γ + m π γ − Γ ) (cid:21) (cid:126) σ · (cid:126) σ + (cid:20) (cid:16) m π ( γ + γ + γ ) − Γ − Γ − Γ (cid:17) + ( γ + γ )+ (cid:18) p − q (cid:19) ( γ + γ + γ + γ ) (cid:21) ( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) (cid:27) .(3.22)The present calculation has revealed typing errors q /8 → q /4 in the last line of eq.(29) and − Γ → − Γ − Γ in the fourth line of eq. (30) in ref. [3], which are correctedin the expressions (3.21) and (3.22), respectively.Furthermore, new loop-functions γ ν , ν =
1, . . . , 5 - defined by Fermi sphere integralsover a squared pion propagator - are introduced, and the corresponding analyticalexpressions can be found in the appendix.This concludes the subsection on relativistic corrections to the subleading chiral 3n-force in the 1 π -exchange-contact topology. Subsequently, the 2 π -exchange topology isdiscussed in the next subsection. 24 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction The corrections arising from the 2 π -exchange topology detailed in this subsectionare yet more computationally demanding, as different pion propagators are involvedwithin the same three-neutron interaction V n .However, of the four contributions seen in eqs. (31), (34), (37) and (41) of ref. [3],only the latter two remain to be considered, as the former two are proportional to theisospin-vector scalar triple product (cid:126) τ · ( (cid:126) τ × (cid:126) τ ) , which vanishes in the neutron-onlycase. Thus, there are no contributions to the in-medium nn-interaction from 1/ M - andretardation corrections to a 2 π -exchange through two π nn vertices ( π n couplings) anda ππ nn (Weinberg-Tomozawa) vertex.As the aforementioned isospin-vector scalar triple product is also present within thetwo surviving terms, the contributions arising from the corresponding 3n-interactionsare noticeably different to the isospin-symmetric case. Nevertheless, when consideringdouble exchanges, the results overlap clearly, as the contributions from the scalar tripleproduct lead to a prefactor of (cid:126) τ · (cid:126) τ in symmetric nuclear matter, and the remainingterms differ from the 3n-result only by the previously explained factor of 3 due to ascalar product of two identical isospin vectors.Also, the results for self-closings differ merely by a factor of 1/2 from those in isospin-symmetric matter. In the latter case, the nucleon loop involves two realizations of thethird component of isospin, namely protons and neutrons.Additionally, it is important to keep in mind that in ref. [3], both of the pionic vertexcorrections have been added to obtain the result for V ( ) med , as it was particularly conve-nient for the now vanishing interaction terms. This was due to their symmetry underthe nucleon exchange 1 ↔
3, yielding the same result for either of the vertex corrections.However, since this symmetry is not applicable to the remaining two 3n-interactionterms, this reduced notation is not employed in this work.Beginning the presentation of results with the 3N-interaction given in eq. (37) of ref.[3], arising from the 1/ M -corrections to the π NN vertices, the relevant term adapted tothe three-neutron case reads: V n = g A M f π (cid:126) σ · (cid:126) q ( m π + q )( m π + q ) (cid:26) (cid:126) σ · (cid:126) q (cid:104) i (cid:126) σ · (cid:0) (cid:126) q × ( (cid:126) p + (cid:126) p (cid:48) ) (cid:1) + q (cid:105) + i (cid:126) σ · ( (cid:126) p + (cid:126) p (cid:48) ) (cid:126) σ · ( (cid:126) q × (cid:126) q ) (cid:27) . (3.23)25 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction This leads to the following contribution from self-closing the middle neutron line: V ( ) med = − g A k n q π M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q ( m π + q ) , (3.24)which - as was previously the case as well - is linear in the neutron density ρ n = k n /3 π .Calculating both types of pionic vertex corrections, one obtains the following contribu-tions to the in-medium nn-potential: V ( ) med = g A π M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) qm π + q (cid:104) p ( Γ − Γ ) + q ( Γ + Γ + Γ ) − Γ (cid:105) , (3.25)and V ( ) med = g A π M f π m π + q (cid:26)(cid:104) k n + ( q − p )( Γ + Γ + Γ ) − m π ( Γ + Γ ) (cid:105) × (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q + q ( Γ + Γ + Γ ) (cid:0) (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) (cid:1)(cid:27) . (3.26)Finally, the evaluation of the in-medium loop in the diagram representing doubleexchanges yields V ( ) med = g A π M f π (cid:26) k n − q ( Γ + Γ ) + m π (cid:104) ( m π + q ) G − Γ (cid:105) + (cid:104) G − Γ + Γ + ( m π + p + q ) G + ( p − q ) (cid:0) G + G (cid:1) − m π G (cid:105) i ( (cid:126) σ + (cid:126) σ ) · ( (cid:126) q × (cid:126) p ) + (cid:0) G + G (cid:1) (cid:126) σ · ( (cid:126) q × (cid:126) p ) (cid:126) σ · ( (cid:126) q × (cid:126) p ) (cid:27) ,(3.27)which now explicitly includes a term proportional to the quadratic spin-orbit opera-tor (cid:126) σ · ( (cid:126) q × (cid:126) p ) (cid:126) σ · ( (cid:126) q × (cid:126) p ) . Here, it appears directly in the underlying computation,whereas it was previously only present through the operator ( (cid:126) σ · (cid:126) p (cid:126) σ · (cid:126) p + (cid:126) σ · (cid:126) p (cid:48) (cid:126) σ · (cid:126) p (cid:48) ) after further decomposition.As with the previous loop-functions, the new G ν , ν =
0, . . . , 3 - given by Fermi sphereintegrals over two different pion propagators - are detailed in the appendix, and exhibitan additional dependence on the momentum-transfer modulus q . However, unlike the26 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction loop-functions introduced up to this point, the G ν cannot be given in closed analyticalform, but involve a one-parameter integral that needs to be solved numerically.The last 3n-interaction term covered in this work arises from retardation corrections tothe consecutive 2 π -exchange and it is given by V n = g A M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q ( m π + q ) ( m π + q ) (cid:110) − ( (cid:126) q · (cid:126) q ) + i (cid:126) σ · ( (cid:126) q × (cid:126) q )( (cid:126) p + (cid:126) p (cid:48) + (cid:126) p + (cid:126) p (cid:48) ) · (cid:126) q (cid:111) , (3.28)where eq. (41) of ref. [3] was adapted to the three-neutron case.Following the same methods as before, the contribution V ( ) med = g A k n q π M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q ( m π + q ) (3.29)can be obtained for self-closings, which again, is linear in the neutron density ρ n = k n /3 π .On the other hand, the two types of pionic vertex corrections give rise to V ( ) med = g A q π M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q ( m π + q ) (cid:20) k n − m π ( Γ + Γ ) − q ( Γ + Γ + Γ + Γ ) − ( Γ + Γ ) (cid:21) , (3.30)and V ( ) med = g A π M f π (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) qm π + q (cid:26) k n + m π ( γ + m π γ − Γ ) − Γ + q (cid:104) m π ( γ + γ + γ ) + ( γ + γ ) − Γ − Γ − Γ (cid:105) + ( p − q ) (cid:20) q ( γ + γ + γ + γ ) + γ + γ (cid:21) + (cid:18) p − q (cid:19) (cid:104) m π ( γ + γ ) − Γ − Γ (cid:105)(cid:27) . (3.31)Lastly, only the contribution due to double exchanges remains, which for the sakeof clarity and readability, is divided into the results arising from the first ( V ( (cid:48) ) med ) and27 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction second line ( V ( (cid:48)(cid:48) ) med ) of eq. (3.28), respectively.After some tedious calculations, one obtains: V ( (cid:48) ) med = g A π M f π (cid:26) q ( Γ − γ − q γ ) − k n + ( m π + q ) (cid:0) Γ − q γ (cid:1) − ( m π + q ) (cid:0) γ + G (cid:1) + ( m π + q ) K + (cid:20) q ( γ + γ ) − Γ − Γ + (cid:18) m π + q (cid:19) ( γ + γ + G + G ) − ( m π + q ) (cid:0) K + K (cid:1)(cid:21) i ( (cid:126) σ + (cid:126) σ ) · ( (cid:126) q × (cid:126) p ) (cid:27) , (3.32)and V ( (cid:48)(cid:48) ) med = g A π M f π (cid:26)(cid:20) Γ + Γ + γ + p γ + γ + p γ + G ∗ + G ∗ − q ( γ + γ ) − (cid:18) m π + p + q (cid:19) ( γ + γ + G + G )+ (cid:18) m π + q (cid:19) (cid:18) ( m π + q + p ) K + K − K ∗ − K ∗ (cid:19)(cid:21) × i ( (cid:126) σ + (cid:126) σ ) · ( (cid:126) q × (cid:126) p )+ (cid:20) ( m π + q + p ) K − K ∗ − γ − G (cid:21) ( q (cid:126) σ · (cid:126) σ − (cid:126) σ · (cid:126) q (cid:126) σ · (cid:126) q )+ (cid:20) ( m π + q + p ) (cid:18) K + K + K (cid:19) − γ − γ − γ − G − G − G − ( K ∗ + K ∗ + K ∗ ) (cid:21) (cid:126) σ · ( (cid:126) q × (cid:126) p ) (cid:126) σ · ( (cid:126) q × (cid:126) p ) (cid:27) .(3.33)The present calculation has revealed a typing error G ∗ + G ∗ → G ∗ + G ∗ in the fifthline of eq. (45) in ref. [3] that is corrected in eq. (3.33) above.Let it be noted that once again, the quadratic spin-orbit operator (cid:126) σ · ( (cid:126) q × (cid:126) p ) (cid:126) σ · ( (cid:126) q × (cid:126) p ) appears directly in the course of the calculation in both of the contributions above.The newly introduced loop-functions K ν , ν =
0, . . . , 3 are defined by Fermi sphere28 . Resulting Contributions to the In-Medium Neutron-Neutron Interaction integrals over the symmetric sum of the product of two different pion propagators,one of which is squared. The K ν are discussed in more detail in the appendix, andthey also exhibit an additional dependence on the momentum-transfer q . Lastly, a newnotation is employed, whereby an additional factor of l in the Fermi sphere integral isdenoted by an asterisk ∗ in the subscript of the loop-function. Here, l stands for the3-momentum modulus of neutrons in the Fermi sea.This concludes the chapter on effective in-medium nn-potentials, resulting from closingone neutron line in the short-range terms and relativistic corrections to the 3n-force atN LO in Chiral Effective Field Theory.In the next chapter, a summarizing conclusion is given, followed by an outlook tofurther research, in order to build upon the results of this and previous works innuclear many-body calculations. 29 . Conclusion and Outlook
At its starting point, this work deals with three-nucleon forces of Chiral Effective FieldTheory, which are crucial for achieving an accurate description of nuclear phenomena.Specifically, a previously developed method to efficiently include 3N-forces in nuclearmany-body computations through a density-dependent potential V med is employed. V med is calculated from the 3N-forces by closing one nucleon line and integrating overthe filled Fermi sphere. Making use of this approach, the contributions to an in-mediumneutron-neutron interaction representing the corresponding subleading chiral 3n-forces,namely short-range terms and relativistic corrections, have been calculated.The contributions to V med are given as explicit expressions, some of which dependingon loop-functions that are detailed in the appendix, either in closed analytical form, orthrough a one-parameter radial integral.Thus, it is shown that while many contributions of the in-medium nn-interaction are -apart from a constant factor - identical to the terms in isospin-symmetric matter, somediffer drastically, and previously vanishing terms from self-closings now yield non-zerocontributions.Therefore, it is evident that in some cases, the contributions to V med in pure neutronmatter have to be computed explicitly, while others may be easily adapted from previ-ously calculated terms in isospin-symmetric matter.Consequently, the in-medium nn-potential V med detailed in this work is henceforthavailable for implementation in e.g. many-body calculations of the equation of state ofpure neutron matter.Nevertheless, there are still some unknowns before moving forward. It is unclearat this stage which terms give rise to significant alterations to ordinary two-bodynn-interactions, and which are possibly negligible. Therefore, a detailed partial-waveanalysis of V med has yet to be carried out, in order to determine the size of its contribu-tions.Lastly, since this work only deals with the short-range terms and relativistic correctionsat N LO, the computations of V med from the intermediate- and long-range contributionsat the same order, as well as higher order interaction terms of the chiral 3n-forces, stillhave to be performed in the future. 30 ppendix . Loop-Functions On the following pages, the loop-functions Γ ν ( p , k n ) , γ ν ( p , k n ) , ν =
1, . . . , 5 and G ν ( p , q , k n ) , K ν ( p , q , k n ) , ν =
1, 2, 3 used previously to write down the in-medium po-tentials V med , are specified. The dependencies of Γ ν , γ ν on p , k n , and the dependenciesof G ν , K ν on p , q , k n are suppressed for the sake of notational simplicity. Furthermore,the relevant integrals are shown as a function of (cid:126) p , but yield the same results whencomputed as a function of (cid:126) p (cid:48) , since only on-shell scattering with | (cid:126) p | = p = | (cid:126) p (cid:48) | isconsidered.The loop-functions are obtained by employing the methods detailed in section (2.2)of chapter 2, most notably by making use of the symmetry under the exchange ofthree-momentum indices. Γ ν Functions
The Γ ν functions - with ν =
0, 1, . . . , 5 - are given by Fermi sphere integrals over asingle pion propagator with the additional tensorial factors of 1 ( ν = l i ( ν = l i l j ( ν =
2, 3) and l i l j l k ( ν =
4, 5), where i , j , k =
1, 2, 3. The decompositions of therelevant integrals read: (cid:90) | (cid:126) l | < k n d l π { l i , l i l j , l i l j l k } m π + ( (cid:126) l + (cid:126) p ) = (cid:110) Γ , p i Γ , δ ij Γ + p i p j Γ , ( p i δ jk + p j δ ik + p k δ ij ) Γ + p i p j p k Γ (cid:111) , (A.1)leading to the following ( p , k n )-dependent functions Γ = k n − m π (cid:20) arctan k n + pm π + arctan k n − pm π (cid:21) + m π + k n − p p × ln m π + ( k n + p ) m π + ( k n − p ) , (A.2)32 . Loop-Functions Γ = k n p ( m π + k n + p ) − Γ − p [ m π + ( k n + p ) ][ m π + ( k n − p ) ] × ln m π + ( k n + p ) m π + ( k n − p ) , (A.3) Γ = k n + ( k n − m π − p ) Γ + ( m π + k n − p ) Γ , (A.4) Γ = k n p − m π + k n + p p Γ − m π + k n + p p Γ , (A.5) Γ = m π Γ + k n (cid:20) p − m π − k n + p ( k n − k n m π − m π ) − ( k n + m π ) p (cid:21) + p [ m π + ( k n + p ) ][ m π + ( k n − p ) ] × (cid:104) ( k n + m π ) + p ( k n + m π ) − p (cid:105) ln m π + ( k n + p ) m π + ( k n − p ) , (A.6) Γ = − Γ + k n (cid:20) + p ( k n + m π ) + k n + m π p + p ( k n + k n m π + m π ) (cid:21) − p (cid:2) m π + ( k n + p ) (cid:3) × (cid:2) m π + ( k n − p ) (cid:3) (cid:104) ( k n + m π ) + p ( k n + m π ) + p (cid:105) × ln m π + ( k n + p ) m π + ( k n − p ) . (A.7) γ ν Functions
The γ ν functions - with ν =
0, 1, . . . , 5 - are given by Fermi sphere integrals over asquared pion propagator with the additional tensorial factors of 1 ( ν = l i ( ν = l i l j ( ν =
2, 3) and l i l j l k ( ν =
4, 5), where i , j , k =
1, 2, 3. The decompositions of the33 . Loop-Functions relevant integrals read: (cid:90) | (cid:126) l | < k n d l π { l i , l i l j , l i l j l k } (cid:2) m π + ( (cid:126) l + (cid:126) p ) (cid:3) = (cid:110) γ , p i γ , δ ij γ + p i p j γ , ( p i δ jk + p j δ ik + p k δ ij ) γ + p i p j p k γ (cid:111) . (A.8)Obviously, the relation γ ν = − ∂ Γ ν / ∂ m π is fulfilled by definition. With that knowledge,the analytical form of these ( p , k n )-dependent loop-functions is obtained as γ = m π (cid:20) arctan k n + pm π + arctan k n − pm π (cid:21) − p ln m π + ( k n + p ) m π + ( k n − p ) , (A.9) γ = − γ − k n p + p + k n + m π p ln m π + ( k n + p ) m π + ( k n − p ) , (A.10) γ = k n p ( p − k n − m π ) − m π γ + p (cid:2) ( p + k n + m π ) − p ( p + m π ) (cid:3) ln m π + ( k n + p ) m π + ( k n − p ) , (A.11) γ = γ + k n p ( p + k n + m π ) − p (cid:2) ( k n + m π ) + p ( k n + m π ) + p (cid:3) ln m π + ( k n + p ) m π + ( k n − p ) , (A.12) γ = m π γ + k n p (cid:20) ( k n + m π ) + p ( k n + m π ) − p (cid:21) + p − k n − m π p (cid:2) p + p ( k n + m π ) + ( k n + m π ) (cid:3) × ln m π + ( k n + p ) m π + ( k n − p ) , (A.13) γ = − γ − k n p (cid:20) p ( k n + m π ) + + k n + m π p (cid:21) + p (cid:2) p ( k n + m π ) + p + p ( k n + m π )( k n + m π )+ ( k n + m π ) (cid:3) ln m π + ( k n + p ) m π + ( k n − p ) . (A.14)34 . Loop-Functions G ν Functions
The G ν functions - with ν =
0, 1, 2, 3 - are given by Fermi sphere integrals over twodifferent pion propagators with the additional tensorial factors of 1 ( ν = l i ( ν = l i l j ( ν =
2, 3), where i , j =
1, 2, 3. The decompositions of the relevant integralsread: (cid:90) | (cid:126) l | < k n d l π { l i , l i l j } (cid:2) m π + ( (cid:126) l + (cid:126) p ) (cid:3)(cid:2) m π + ( (cid:126) l + (cid:126) p (cid:48) ) (cid:3) = (cid:110) G , ( p i + p (cid:48) i ) G , δ ij G + ( p i + p (cid:48) i )( p j + p (cid:48) j ) G + ( p i − p (cid:48) i )( p j − p (cid:48) j ) G (cid:111) , (A.15)where the symmetry under (cid:126) p ↔ (cid:126) p (cid:48) has been exploited. Although it is required for theconstruction of G and G , the function G itself does not appear in the contributionsto the in-medium nn-potential V med and does not need to be specified further.However, contrary to the previous loop functions, the integrals cannot be solvedanalytically. Instead, they are obtained by first computing the radial one-parameterintegrals G
0, 0 ∗ , ∗∗ = q k n (cid:90) dl { l , l , l } (cid:112) B + q l ln q l + (cid:112) B + q l √ B (A.16)numerically, where B = (cid:2) m π + ( (cid:126) l + (cid:126) p ) (cid:3)(cid:2) m π + ( (cid:126) l + (cid:126) p (cid:48) ) (cid:3) . The asterisks in the sub-script indicate additional factors of l in the corresponding Fermi sphere integral.Subsequently, it is necessary to solve a system of linear equations with the result G = p − q (cid:2) Γ − ( m π + p ) G − G ∗ (cid:3) , (A.17) G ∗ = p − q (cid:2) Γ + p Γ − ( m π + p ) G ∗ − G ∗∗ (cid:3) , (A.18) G = ( m π + p ) G + G ∗ + G ∗ , (A.19) G = p − q (cid:20) Γ − ( m π + p ) G − G ∗ − G ∗ (cid:21) . (A.20)35 . Loop-Functions K ν Functions
The K ν functions - with ν =
0, 1, 2, 3 - are given by Fermi sphere integrals over thesymmetric sum of a squared pion propagator multiplied by a different pion propagator,with the additional tensorial factors of 1 ( ν = l i ( ν =
1) and l i l j ( ν =
2, 3), where i , j =
1, 2, 3. The decompositions of the relevant integrals read: (cid:90) | (cid:126) l | < k n d l π (cid:32) (cid:2) m π + ( (cid:126) l + (cid:126) p ) (cid:3) (cid:2) m π + ( (cid:126) l + (cid:126) p (cid:48) ) (cid:3) + (cid:2) m π + ( (cid:126) l + (cid:126) p ) (cid:3)(cid:2) m π + ( (cid:126) l + (cid:126) p (cid:48) ) (cid:3) (cid:33) × { l i , l i l j } = (cid:110) K , ( p i + p (cid:48) i ) K , δ ij K + ( p i + p (cid:48) i )( p j + p (cid:48) j ) K + ( p i − p (cid:48) i )( p j − p (cid:48) j ) K (cid:111) , (A.21)where the symmetry under (cid:126) p ↔ (cid:126) p (cid:48) is used once again. As before, K is relevant for theconstruction of K and K , but does not itself appear in the nn-potentials and is thus ofno further interest here.Moreover, it can be easily verified that the K ν satisfy K ν = − ∂ G ν / ∂ m π . Just as for the G ν functions, the K ν are not analytically calculable and have to be obtained by firstcomputing the radial one-parameter integrals K
0, 0 ∗ , ∗∗ , ∗∗∗ = k n (cid:90) dl m π + l + p B + q l (cid:34) lB + q (cid:112) B + q l ln q l + (cid:112) B + q l √ B (cid:35) × { l , l , l , l } (A.22)numerically, and then solving a system of linear equations with the result36 . Loop-FunctionsK = p − q (cid:2) γ + G − ( m π + p ) K − K ∗ (cid:3) , (A.23) K ∗ = p − q (cid:2) γ + p γ + G ∗ − ( m π + p ) K ∗ − K ∗∗ (cid:3) , (A.24) K = ( m π + p ) K − G + K ∗ + K ∗ , (A.25) K = p − q (cid:104) γ − ( m π + p ) K + G − K ∗ − K ∗ (cid:105) , (A.26) K ∗∗ = p − q (cid:2) G ∗∗ − ( m π + p ) K ∗∗ − K ∗∗∗ + γ ∗∗ (cid:3) , (A.27) K ∗ = ( m π + p ) K ∗ − G ∗ + K ∗∗ + K ∗∗ , (A.28) K ∗ = p − q (cid:20) ( γ + p γ ) − ( m π + p ) K ∗ + G ∗ − K ∗∗ − K ∗∗ (cid:21) . (A.29)For the sake of simplicity, a new loop function γ ∗∗ - defined as the Fermi sphere integralover a squared pion propagator multiplied by l - is introduced in eq. (A.27).Its analytical form reads: γ ∗∗ = (cid:90) | (cid:126) l | < k n d l π l (cid:2) m π + ( (cid:126) l + (cid:126) p ) (cid:3) = ( p + m π − p m π ) γ + k n (cid:18) p + k n − m π (cid:19) + m π − p p × ln m π + ( k n + p ) m π + ( k n − p ) . (A.30)37 eferences [1] J. W. Holt, N. Kaiser, and W. Weise. “Density-dependent effective nucleon-nucleoninteraction from chiral three-nucleon forces.” In: Phys. Rev. C81
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