Implementation of local chiral interactions in the hyperspherical harmonics formalism
IImplementation of local chiral interactions inthe hypershperical harmonics formalism
Simone Salvatore Li Muli , , ∗ , Sonia Bacca , and Nir Barnea Institut f ¨ur Kernphysik and PRISMA + Cluster of Excellence, Johannes GutenbergUniversit ¨at, 55128 Mainz, Germany Helmholtz-Institut Mainz, Johannes Gutenberg Universit ¨at Mainz, D-55099 Mainz,Germany Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
Correspondence*:Simone Salvatore Li [email protected]
ABSTRACT
With the goal of using chiral interactions at various orders to explore properties of the few-bodynuclear systems, we write the recently developed local chiral interactions as spherical irreducibletensors and implement them in the hyperspherical harmonics expansion method. We devoteparticular attention to three-body forces at next-to-next-to leading order, which play an importantrole in reproducing experimental data. We check our implementation by benchmarking the ground-state properties of H, He and He against the available Monte Carlo calculations. We thenconfirm their order-by-order truncation error estimates and further investigate uncertainties inthe charge radii obtained by using the precise muonic atom data for single-nucleon radii. Havinglocal chiral Hamiltonians at various orders implemented in our hyperspherical harmonics suitesof codes opens up the possibility to test such interactions on other light-nuclei properties, suchas electromagnetic reactions.
Keywords: nuclear interactions, hyperspherical harmonics, light nuclei, ab-initio theory, chiral perturbation theory
In 1935 the seminal idea of Yukawa [1] laid the foundation to the theory of the nuclear forces. His one-pion exchange term is nowadays known as an important contribution to the interaction among nuclei inthe long-distance range and is implemented in many nuclear interaction models. In the mid 1990s thefirst high-precision nucleon-nucleon (NN) potentials able to reproduce at the same time the deuteronproperties, the proton-proton and the proton-neutron scattering data were released. Some notable examplesof these interactions are the Argonne v (AV18) [2], the Nijmegen (Nijm93) [3] and the charge-dependentBonn (CD-Bonn) [4]. The subsequent development of three-nucleon (3N) interactions, see for instanceRefs. [5, 6], improved the description of the A > nuclear dynamics, initiating a successful theoreticalcampaign of nuclear structure and reaction predictions, see e.g., Refs. [7, 8, 9] and references therein.Despite the great success of the phenomenological interactions, there are still open questions to address,including the difficulty of providing solid uncertainty quantifications in the modeling of the forces, the lackof connection between the NN and 3N interactions and the missing direct link to quantum chromodynamic(QCD), the fundamental theory of the strong force. a r X i v : . [ nu c l - t h ] F e b i Muli et al. Running Title
An important step forward to address these issues was made when the concept of effective field theory(EFT) was introduced and applied to low-energy QCD. As suggested by Weinberg [10, 11, 12, 13], the low-energy nuclear dynamic can be described by a Lagrangian written in terms of pions and nucleons fields andconsistent with all the commonly accepted symmetries of QCD, including the (explicitly and spontaneouslybroken) chiral symmetry which strongly constrains the pion dynamics. The proposed Lagrangian containsan infinite number of terms and a systematic expansion must be introduced to make the theory applicable.Following Weinberg’s proposal, in the early 2000s modern versions of chiral-inspired nuclear interactionswere released by many groups – for a compilation of results see for instance Refs. [14, 15, 16] andreferences therein – each interaction being different by the truncation order of the chiral expansion, bythe inclusion or exclusion of the ∆(1232) -isobar, by the fitting procedure or by the regularization schemeused. Given that these interactions are derived in field theories written in momentum space, they are highlynon-local. One of the consequence is that they are difficult to implement in some of the few- and many-bodytechniques which are developed in coordinate-space representation.In recent years a new chiral-inspired set of nuclear interactions at the next-to-next-to-leading order(N2LO) has become available [17, 18, 19]. These interactions have a series of interesting properties whichmake them a promising framework for future nuclear computations. These interactions are completelywritten in coordinate space and contain only one non-local operator. Furthermore the NN and 3N termsare regularized consistently, namely the same regulator form and cut-off is used. Here, these interactionsare written for the first time as product of irreducible tensors under space rotations, a required step forthe implementation into the hyperspherical harmonics formalism. Using the method of hypersphericalharmonics, we perform benchmark tests in light-nuclear systems, where we compare to available resultsfrom the Green’s function Monte Carlo (GFMC) and the auxiliary field diffusion Monte Carlo (AFDMC)methods.This review paper is summarized as follows. In Section 2, we briefly overview the formulation ofthe hyperspherical harmonics method in coordinate-space representation. In Section 3, we present themaximally-local chiral interactions developed in Ref. [19] and rewrite the 3N force as products of irreducibletensors under space rotations. In Section 4, we show our benchmark results for H, He and He and wediscuss uncertainties. Finally, Section 5 is reserved for the conclusive remarks and the overview of futureprospects.
The hyperspherical harmonic method was firstly introduced in 1935 by Zernike and Brinkman [20],reintroduced later in the 60’s by Delves [21], Simonov [22], Zickendraht [23] and Smith [24] and it isextensively applied nowadays to the study of few-body systems. For recent reviews with applications tonuclear physics we refer the reader to the following references [25, 26]. In this work, the hypersphericalharmonic functions are constructed to form irreducible representations of the SO (3) group of spacerotations, the O ( N ) group of dynamical rotations in the space spanned by the N Jacobi vectors, and the S A permutation group of the A -particle system. The method is briefly reviewed in this section, the formalismintroduced follows closely Refs. [27, 28].We consider a system of A identical nucleons, the Jacobi coordinates { η i } are commonly introducedin order to separate the internal degrees of freedom from the center of mass. There are several ways toconstruct the set of N = A − Jacobi coordinates out of the A coordinate vectors { r i } of the nucleons. This is a provisional file, not the final typeset article i Muli et al. Running Title
One commonly used definition for the relative Jacobi vectors is η j − = (cid:115) j − j (cid:16) r j − j − j − (cid:88) i =1 r i (cid:17) ; k = 2 , ..., A. (1)From a given choice of Jacobi coordinates, the hyperspherical coordinates { ρ N , ϕ ( N ) , Ω( N ) } can beintroduced. In this notation, ρ N is the hyper-radius, Ω( N ) ≡ { Ω , ..., Ω N } where Ω j = ( θ j , φ j ) gathersthe angular coordinates of the Jacobi vectors , and ϕ ( N ) ≡ { ϕ , ..., ϕ N } is a set of hyper-angles.The hyper-radial coordinates ρ , ..., ρ N and the hyper-angular coordinates ϕ , ..., ϕ N are constructedrecursively. The transformation law for the first two Jacobi coordinates is η = ρ = ρ cos ϕ ,η = ρ sin ϕ . (2)Assuming that we already know the hyper-radial coordinates ρ , ..., ρ j − and the hyper-angularcoordinates ϕ , ..., ϕ j − the transformation law for ρ j and ϕ j reads in analogy to Eq. (2) as ρ j − = ρ j cos ϕ j ,η j = ρ j sin ϕ j . (3)The internal kinetic energy operator for the A-body system is given by the N -dimensional Laplaceoperator ∆( N ) . In terms of the hyperspherical coordinates it is written as ∆( N ) = ∆ ρ − ρ ˆ K N ( ϕ ( N ) , Ω , ..., Ω N ) (4)where the hyper-radial part is ∆ ρ = ∂ ∂ρ + 3 N − ρ ∂∂ρ , (5)while ˆ K N is the grand-angular momentum operator whose eigenfunctions are known as the hypersphericalharmonics.Denoting ˆ l j as the angular momentum operator related to η j , and ˆ L j and ˆ M j as the total orbital angularmomentum operator and z -projection of the system identified by the first j Jacobi coordinates, it is possibleto define the grand-angular momentum operator ˆ K N of the system recursively in terms of ˆ K N − and ˆ l N as[29] ˆ K N = − ∂ ∂ϕ N + 3 N − − (3 N −
2) cos(2 ϕ N )sin(2 ϕ N ) ∂∂ϕ N + 1cos ϕ N ˆ K N − + 1sin ϕ N ˆ l N (6)where ˆ K = ˆ l .The operators ˆ K N , ..., ˆ K , ˆ L N , ..., ˆ L , ˆ l N , ..., ˆ l and ˆ M N commute with each others. As aconsequence, it is possible to label hyperspherical states using the set of N − quantum numbers { K } ≡ { K N , ..., K , L N , ..., L , l N , ..., l , M N } . The hyperspherical harmonics functions Y { K N } are the Frontiers 3 i Muli et al.
Running Title eigenfunctions of the grand-angular momentum operator with eigenvalues K N ( K N + 3 N − . The explicitexpression for the resulting hyperspherical harmonics functions is given by [30] Y { K N } = (cid:34) (cid:88) m ,...,m N C L M l m ,l m C L M L M ,l m × ... × C L N M N L N − M N − ,l N m N N (cid:89) j =1 Y l j m j (Ω j ) (cid:35) ×× (cid:34) N (cid:89) j =2 N j (sin ϕ j ) l j (cos ϕ j ) K j − P (cid:2) l j + ,K j − + (3 j − (cid:3) n j (cos 2 ϕ j ) (cid:35) , (7)where C LMl i m i ,l j m j are the Clebsch-Gordan coefficients, Y l j m j (Ω j ) are the spherical harmonics associatedwith η j and N j = (cid:34) (3 K j + 3 j − n j !Γ( n j + K j − + l j + j − )Γ( n j + l j + )Γ( n j + K j − + j − ) (cid:35) (8)is a normalization constant with n j = K j − K j − − l j .In our formulation of the hyperspherical harmonics method we construct hyper-angular functions thatform irreducible tensors under the SO (3) group of spatial rotations, the O ( N ) group of kinematic rotationsand the S A group of permutations of the A nucleons. These symmetry-adapted hyperspherical harmonics, Y [ K N ] , are uniquely identified by the set of quantum numbers [ K N ] ≡ { K N , L N , M N , λ N , α N Y A , β A } .For the current purposes, it is enough to specify that λ N identifies the irreducible representation of O ( N ) , Y A is the Yamanouchi symbol which specifies the irreducible representations of the group-subgroupchain S ⊂ ... ⊂ S A presented by the appropriate Young diagrams Γ , ..., Γ A , while α N and β A areadditional quantum numbers needed to remove further degeneracies. The O ( N ) and S A symmetry-adaptedhyperspherical harmonics Y [ K N ] are constructed recursively. Assuming that Y [ K N − ] have been alreadyconstructed, the N th Jacobi coordinate is then coupled to this system, so that a state with total angularmomentum L N and grand-angular momentum K N is formed, let us call this state Y [ K N − ] ,K N L N M N . Notethat Y [ K N − ] ,K N L N M N is a irreducible tensor under O ( N − and S A − but not under O ( N ) and S A . Thestates Y [ K N ] are obtained as linear combinations of the states Y [ K N − ] ,K N L N M N , where the coefficientsof the linear combinations are labeled as (cid:104)(cid:0) K N − , L N − , λ N − , α N − ; l N (cid:1) K N L N |} K N L N λ N α N (cid:105) , (cid:104)(cid:0) λ N − Γ N β N (cid:1) λ N |} λ N Γ A β A (cid:105) and are known as hyperspherical orthogonal group parentage coefficients(hsopcs) and orthogonal group coefficients of fractional parentage (ocfps) respectively.The full expression of the symmetry-adapted hyperspherical harmonics reads Y [ K N ] = (cid:88) λ N − β N (cid:104)(cid:0) λ N − Γ N β N (cid:1) λ N |} λ N Γ A β A (cid:105) ×× (cid:88) K N − ,L N − ,α N − ,l N (cid:104)(cid:0) K N − , L N − , λ N − , α N − ; l N (cid:1) K N L N |} K N L N λ N α N (cid:105) ×× Y [ K N − ] ,K N L N M N . (9) This is a provisional file, not the final typeset article i Muli et al. Running Title
Nucleons posses also spin and isospin degrees of freedom. Because the nuclear Hamiltonian is rotationallyinvariant, nuclear states have the total angular momentum J as good quantum number. Furthermore, isospinis an approximate symmetry for the nuclear interaction with the consequence that the total isospin T of a nuclear state is a conserved quantum number. For these reasons we couple the symmetry-adaptedhyperspherical harmonics to the S A symmetry-adapted spin-isospin wavefunction χ of the A -nucleonsystem H ( K N ) = (cid:88) Y N Λ Γ A ,Y N (cid:112) | Γ A | (cid:88) M N S z C JJ z L N M N ,SS z Y [ K N ] χ [ S A ] . (10)Here ( K N ) ≡ { K N L N S N J N J N z λ N α STN Y A β A } , [ S A ] ≡ { SS z T T z Y A α STA } , Λ Γ A ,Y N is a phase factor,and | Γ A | is the dimension of the irreducible representation Γ A .Analogously to what has been done with the hyperspherical harmonics, the spin-isospin wavefunctionsare constructed recursively. Assuming that the symmetry-adapted wavefunction χ [ S j − ] have been obtained,the construction of the χ [ S j ] is done by first coupling χ [ S j − ] to the spin-isospin wavefunction of the j thnucleon, let us call this state χ [ S j − ] ,S j T j , and then taking linear combinations of χ [ S j − ] ,S j T j using thecoefficients of fractional parentage labeled as (cid:104) S j − S j T j − T j Γ j − α STj − |} S j T j Γ j α STj (cid:105) . Namely the fullexpression for χ [ S j ] reads χ [ S j ] = (cid:88) S j − T j − α STj − (cid:104) S j − S j T j − T j Γ j − α STj − |} S j T j Γ j α STj (cid:105) χ [ S j − ] ,S j T j . (11)We are finally able to expand the nuclear wavefunction in terms of hyperspherical harmonics. In practice,the expansion is performed up to a maximal value of the grand-angular quantum number K max as Ψ = (cid:88) ( K N ) R ( K N ) ( ρ N ) H ( K N ) (Ω N ) . (12)When we insert this wavefunction into the Schr¨odinger equation, an eigenvalues equation is obtained forthe hyper-radial wavefunction R ( K N ) , the eigenvalue equation is then solved by expanding the hyper-radialwavefunction in terms of an orthogonal set of functions. In this work the set is taken as the generalizedLaguerre polynomials L vn ( ρ N ) . Again, the model space is truncated to a given maximum number ofLaguerre polynomials n max R ( K N ) = n max (cid:88) n =0 C n ( K N ) L vn ( ρ N ) . (13)With the introduction of this further model space, the resulting eigenvalue equation is solved with directdiagonalization routines, or with the Lanczos method when the model space is too big for a directdiagonalization. In essence, the hyperspherical harmonics method is a powerful technique that allowsfor an exact solution of the Schr ¨odinger equation for few-body systems. In the limit where n max → ∞ and K max → ∞ the solution correspond to the exact solution to the Schr¨odinger equation. While weobserve that good convergence can be reached with n max ≤ , the convergence in terms of K max willbe carefully investigated. The uncertainty coming from the truncation of the model space, in particular of K max , can be estimated by looking at the convergence pattern of the observables of interest, for instancethe binding energy and the radius. As a consequence, the method is an excellent candidate for uncertainty Frontiers 5 i Muli et al.
Running Title quantifications in nuclear physics, with the possibility of performing tests over commonly accepted nuclearHamiltonians or making precise predictions for few-nucleon systems. Because the formulation we presenthere is developed in coordinate space, the method benefits from having local forces, such as the AV18potential. While one can formulate hyperspherical harmonics also in momentum space [31], the goal ofthis paper is to work in coordinate space and implement local-chiral interactions. To further improve theconvergence with respect to the model space, we make use of the effective interaction hypersphericalharmonics (EIHH) method. The interested reader can find more details on this approach in Ref. [32], andalso in the more recent review [33].
Nuclear physics is mainly formulated in the framework of non-relativistic quantum mechanics. The relevantdegrees of freedom are represented by the nucleons, whose interactions are remnants of the color forcesamong the quarks. In this picture, the nucleus is a compound object of A non-relativistic nucleons and thedynamic of the system is specified by the nuclear Hamiltonian operator ˆ H = ˆ T + ˆ V + ˆ W + ... = A (cid:88) i =1 ˆ T i + A (cid:88) i>j =1 ˆ V ij + A (cid:88) i>j>k =1 ˆ W ijk + ... , (14)where ˆ T is the sum of the non-relativistic kinetic energy operators of the individual nucleons, ˆ V is a sumof NN interactions and ˆ W is a sum of 3N interactions. The dots stand for higher order forces not explicitlyincluded in this work.Our goal is to solve the Sch¨odinger equation ˆ H | Ψ (cid:105) = E | Ψ (cid:105) (15)and when working with antisymmetrized wavefunctions, the expectation values of the NN and 3N termsbecome (cid:104) Ψ | ˆ V | Ψ (cid:105) = A ( A − (cid:104) Ψ | ˆ V | Ψ (cid:105) , (16) (cid:104) Ψ | ˆ W | Ψ (cid:105) = A ( A − A − (cid:104) Ψ | ˆ W | Ψ (cid:105) , where only the first two (or three) particles are involved .In the modern theory of nuclear forces, interactions are derived from the chiral effective field theory(ChEFT). In this theory, proposed first by Weinberg [10, 11, 12, 13], the chiral Lagrangian is constructedin terms of pion and nucleon fields and is consistent with the commonly accepted symmetries of QCD,including the explicitly and spontaneously broken chiral symmetry. This effective Lagrangian has infinitelymany terms, therefore one needs to introduce an ordering scheme to render the theory predictive.In ChEFT, the terms in the chiral Lagrangian are analyzed counting powers of a small external momentumover the large scale : ( Q/ Λ χ ) ν , where Q stands for an external momentum or a pion mass and Λ χ isthe chiral symmetry breaking scale, whose value is approximately given by the mass of the ρ -meson Λ χ ∼ m ρ = 770 MeV. Determining systematically the power of ν has become known as power counting. This property will be used later when we will write explicitly the form of the nuclear forces between (among) two (three) particles.
This is a provisional file, not the final typeset article i Muli et al. Running Title
The lowest possible value of ν is conventionally referred to as the leading order (LO), the second lowest isthe next-to-leading order (NLO), the third lowest is the next-to-next-to leading order (N2LO) and so on.While there are many proposed power counting schemes [34, 35, 36, 37, 38, 39], in this work we adopt theWeinberg power counting, which makes use of naive dimensional analysis [11, 12].Given that ChEFT is naturally formulated in momentum space, the derived nuclear interactions arestrongly non-local, which is a disadvantage for methods that are formulated in coordinate space. However,it has been recently found that it is possible to construct maximally local chiral interactions by regularizing incoordinate space and exploiting Fierz ambiguities to remove non-localities in the short-distance interactions[17, 18, 19].The local chiral NN forces are composed by contact ( ct ) terms and pion-exchange ( π ) terms so that theinteraction between particle 1 and 2 can be written as V = V ct + V π . (17)When working with totally anti-symmetric systems, it is possible to exploit Fierz ambiguities for removingthe non-local operators contributing to the contact NN interactions. This means that the interactions can bechosen to have the following operator structure [17] at LO V ct, LO12 = (cid:0) C S + C T σ · σ (cid:1) δ ( r ) , (18)where r is the relative distance between nucleon and nucleon , σ / are the vector-spin Pauli matricesoperating in the space of the first/second nucleon and δ is the delta function.At NLO, the following new terms enter V ct, NLO12 = − (cid:0) C + C τ · τ (cid:1) ∆ δ ( r ) − (cid:0) C + C τ · τ (cid:1) σ · σ ∆ δ ( r )+ C ∂ r δ ( r ) r L · S + (cid:0) C + C τ · τ (cid:1) × (cid:34)(cid:0) σ · r (cid:1)(cid:0) σ · r (cid:1)(cid:104) ∂ r δ ( r ) r − ∂ r δ ( r ) (cid:105) − σ · σ ∂ r δ ( r ) r (cid:35) , (19)where τ / are the vector-isospin Pauli matrices, ∆ is the Laplace operator, L and S are the total orbitalangular momentum and spin operator in the two-body system represented by the two interacting nucleons ,and the δ -function will have to be regularized. The { C i } are a set of low energy constants (LECs). Theterm proportional to the LEC C is the only non-local operator appearing in this maximally local chiralinteraction. Even though they are operators in spin space, we do not use the hat in our notation, as they are vectors, whose components are operators. We drop the hat from vectors whose components are operators.
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Following Refs. [17, 40], all the pion-exchange interactions up to N2LO can be written in a completelocal form as V π = V C ( r ) + W C ( r ) τ · τ + (cid:2) V S ( r ) + W S ( r ) τ · τ (cid:3) σ · σ + (cid:2) V T ( r ) + W T ( r ) τ · τ (cid:3) S , (20)where, S is the well known tensor operator, defined as S = 3( σ · ˆ r )( σ · ˆ r ) − ( σ · σ ) , (21)where ˆ r is the unitary vector related to the relative distance r . The local functions V C ( r ) , W C ( r ) , V S ( r ) , W S ( r ) , V T ( r ) and W T ( r ) have dependencies on the axial-vector coupling constant of thenucleon g A , on the pion decay constant F π and on the pion mass m π . These functions are evaluated at eachorder in ChEFT (LO, NLO and N2LO) and details can be found in Ref. [40]. In Ref. [19] pion loops areregularized using the spectral-function regularization (SFR) with an ultraviolet cut-off ˜Λ = 1 GeV and wefollow this prescription.The local chiral NN interactions up to N2LO are already written or can be written with minimalmodifications as irreducible tensors under space rotations. Thus, they can be easily implemented in thehyperspherical harmonics formalism in coordinate space. In fact, they have pretty much the same structureas the Argonne potential AV8’ [41]. The same does not apply to 3N interactions.
Figure 1.
Feynman diagrams of the chiral 3N force at N2LO, from the left to the right: π -term, π -termand ct -term.Three-body interactions arise at NLO in Weinberg power counting. However, at this order theircontribution is canceled out. The first non-zero contributions start at N2LO. The 3N force at this orderis composed of a two-pion ( π ) exchange, a one-pion ( π ) exchange and a 3N contact ( ct ) interaction,see Fig. 1. On the one hand, the π -term comes with the LECs c , c and c that already appear at thesubleading two-pion-exchange interaction at the NN level at the same chiral order which highlights theconsistency of the NN and 3N interactions in ChEFT. On the other hand, the one-pion exchange and the3N contact diagrams introduce two new LECs, c D and c E , which must be fitted on A ≥ observables.With respect to Ref. [19], here the 3N interaction is written for a given triplet of nucleons, since at the endwe use the fact that the wavefunction is anti-symmetric to compute the expectations values as in Eq. (16).The 3N interaction reads W = (cid:88) cyc W , = (cid:88) cyc (cid:104) W π,c , + W π,c , + W π,c , + W π,c D , + W ct,c E , (cid:105) , (22) This is a provisional file, not the final typeset article i Muli et al. Running Title where the sum runs over the cyclic permutations of the particle triplet and the notation has the intention tohighlight the symmetry of the interaction over the exchange of particles 2 and 3. Each term is denoted witha label that includes the associated LEC.The π exchange terms are given by W π,c , = AU Y U Y ( τ · τ )( σ · ˆ r )( σ · ˆ r ) ,W π,c , = B { τ · τ , τ · τ }{ χ , χ } ,W π,c , = − C [ τ · τ , τ · τ ][ χ , χ ] , (23)where the coupling constants are A = c g A m π ( (cid:126) c ) π F π , B = c g A m π ( (cid:126) c ) π F π , and C = c g A m π ( (cid:126) c ) π F π . The π -terms include the following functions Y = Y ( r ) = e − mπr r , (24) U = U ( r ) = 1 + m π r , with analogous expressions for Y and U . The operator χ (and analogously χ ) is defined as χ = X − πm π δ σ · σ = T S + ˜ Y σ · σ (25)with X = T S + Y σ · σ , (26) ˜ Y = Y − πm π δ ,δ = δ r ( r ) = 1 πn Γ( n ) r e − ( r /r ) n . In the last expression, r is the cut-off and following Refs. [19, 18] n is taken to be equal to 4.For the π -interaction terms there are two options W π,c D , = D ( τ · τ )[ X ( r ) δ + X ( r ) δ − πm π δ δ σ · σ ] (27) and W π,c D , = D ( τ · τ ) χ ( δ + δ ) , with D = c D g A m π ( (cid:126) c ) π Λ χ F π . While the difference between the two is due to regulator artifacts, in this workonly the second choice is implemented, namely W π,c D , . Frontiers 9 i Muli et al.
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For the contact term there are different options on the operator structure, which come from differentchoices in the Fierz rearrangement. In this work the following two are considered W ct,c E τ , = E ( τ · τ ) δ ( r ) δ ( r ) (28) and W ct,c E , = Eδ ( r ) δ ( r ) with E = c E ( (cid:126) c ) Λ χ F π .
3N force r c E c D c c c [fm] [GeV − ] [GeV − ] [GeV − ]N2LO (D2, E τ ) 1.0 -0.63 0.0 -0.81 -3.20 3.401.2 0.085 3.5 -0.81 -3.20 3.40 Table 1.
Fit values for the couplings c D and c E for different choices of 3N cut-offs as reported in [18, 19].The constants c , , are tuned in the pion-nucleon sector, see Ref. [15].The value of all LECs entering the 3N forces at N2LO are shown in Table 1. In Refs. [18, 19] c D and c E have been fitted in order to reproduce the He binding energy and the n - α P -wave phase shift. The above expressions for the 3N force are not written in terms of irreducible spherical tensors, so thatthey can not be implemented directly into the hyperspherical formalism. In this section we address thispoint and write the interaction in terms of irreducible spherical tensors, both in coordinate-spin space andin isospin space.For convenience, we denote the general spin space Σ λij , Σ λ, Λ ij,k and configuration space X λij , X ( λ,λ (cid:48) )Λ ij,ij irreducible tensor operators as Σ λij = [ σ i × σ j ] λ , Σ λ, Λ ij,k = (cid:104) [ σ k × [ σ i × σ j ] λ (cid:105) Λ ,X λij = [ˆ r i × ˆ r j ] λ ,X ( λ,λ (cid:48) )Λ ij,ij = (cid:104) [ˆ r i × ˆ r j ] λ × [ˆ r i × ˆ r j ] λ (cid:48) (cid:105) Λ , (29)where i, j, k are generic particle indexes and ˆ r i is the rank 1 normalized spherical tensor associated to therelative distance between particle 1 and particle i . With the notation [ˆ r i × ˆ r j ] λ we intend the two rank-onecoordinate space tensors coupled into a rank- λ tensor, and analogously for [ σ i × σ j ] λ and [ τ i × τ j ] λ inspin and isospin space, respectively. Furthermore, we define X λ ( r ij , r ij ) = [ˆ r ij × ˆ r ij ] λ , (30)where ˆ r ij is the rank 1 normalized spherical tensor associated to the relative distance between particles i and j . This is a provisional file, not the final typeset article i Muli et al. Running Title
At this point, after rearranging the couplings with a few Racah algebra steps and by using the previouslyintroduced notation, one can rewrite the 3 N interactions of Eqs. (23),(27),(28) in terms of irreducibletensors in isospin space and in the coupled spin-configuration space.The π -exchange term depending on c becomes W π,c , = AU Y U Y ( τ · τ )( σ · ˆ r )( σ · ˆ r )= −√ A [ τ × τ ] F UU (cid:0) Σ · X − Σ · X + Σ · X (cid:1) , (31)the π -exchange term that depends on c becomes W π,c , = B { τ · τ , τ · τ }{ χ , χ } = − √ B [ τ × τ ] (cid:16) Σ · (+ F T T X − √ F Y Y + F T Y + F Y T ))+ Σ · ( − F T T X )+ Σ · (+ F T T X + F T Y X + F Y T X ) (cid:17) , (32)while the term that depends on c can be expressed as W π,c , = − C [ τ · τ , τ · τ ][ χ , χ ]= 4 √ C [ τ × [ τ × τ ] ] (cid:104) Σ , , · (cid:16) − F T T X (1 , , − √ F Y Y + F T Y + F Y T ) (cid:17) + Σ , , · (cid:16) + F T T X (1 , , (cid:17) + Σ , , · (cid:16) + F T T X (1 , , (cid:17) + Σ , , · (cid:16) − F T T X (1 , , −
12 ( F T Y X + F Y T X ) (cid:17) + Σ , , · (cid:16) − F T T X (1 , , − √
32 ( F T Y X − F Y T X ) (cid:17) + Σ , , · (cid:16) + F T T X (1 , , (cid:17)(cid:105) . (33)To write the above expression in a compact form, we have introduced the following definitions F UU = U Y U Y ,F T T = 18 T T ,F Y Y = 2( ˜ Y − T )( ˜ Y − T ) ,F T Y = 6 T ( ˜ Y − T ) ,F Y T = 6( ˜ Y − T ) T . (34) Frontiers 11 i Muli et al.
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The π -exchange contribution takes the following form W π,c D , = D ( τ · τ ) χ ( δ + δ )= −√ D [ τ × τ ] (cid:104) Σ · (cid:16) − √ δ + δ ) ˜ Y (cid:17) + Σ · (cid:16) δ + δ ) T X (ˆ r , ˆ r ) (cid:17)(cid:105) , (35)while the contact terms become W ct,c E τ , = E ( τ · τ ) δ δ = −√ E [ τ × τ ] δ δ (36) and W E , = Eδ δ . We have implemented these expressions in our hypershperical harmonics codes. Since the interactionis now written in terms of irreducible tensors, the spin and isospin matrix elements can be computedanalytically. For the calculation of the spatial matrix elements one can reduce the six-dimensional integrationin the two Jacobi coordinates to a two-dimensional numerical quadrature, as explained in details in Ref. [42].Below we present the benchmark results we obtained with these local-chiral forces on few-body systemssuch as H, He and He.
In this section we show the benchmark tests of the maximally-local-chiral interactions using the EIHHmethod. We compute ground-state energies and charge radii in three- and four-nucleon systems andcompare to two Monte Carlo methods, namely the GFMC and AFDMC methods.In the computations of nuclear charge radii, we use (cid:104) r c (cid:105) = (cid:104) r pt (cid:105) + r p + A − ZZ r n + 3 (cid:126) m p c , (37)were (cid:113) (cid:104) r pt (cid:105) is the calculated point-proton radius, r p = 0 . fm [43] is the root-mean-square (rms)charge radius of the proton, r n = − . fm [43] is the squared charge radius of the neutron, and Z is the number of protons in the nucleus. The last term is the Darwin-Foldy correction to the proton-chargeradius [44] which depends on the proton mass m p . We neglect the spin-orbit relativistic contribution, sinceit is negligible in s -shell nuclei [45], as well as meson exchange currents.Keeping in mind that the goal of this work is to benchmark our expressions for the 3N forces at N2LO bycomparing to the Monte Carlo results, we have used the same numerical value for r p and r n as in Ref. [19],which follows the CODATA-2014 recommendations [43]. Hence, in a first stage we will not be using themore modern results for r p/n from Refs. [46, 47].A few words addressing the estimation of the numerical uncertainties are in line. As already said, theEIHH method allows for an exact solution of the Schr ¨odinger equation, the computed wavefunctionconverges to the true eigenfunction of the Hamiltonian operator in the limit of infinite model space. The This is a provisional file, not the final typeset article i Muli et al. Running Title model space is mostly given by the maximal number, n max , of Laguerre polynomials and the choice ofthe maximal value of the grand-angular momentum quantum number, K max , in the construction of thehyperspherical harmonics functions. It has been practically found that beyond a value n max = 50 , theexpectation values are negligibly modified. The convergence in terms of K max is more delicate, so that inorder to estimate the uncertainty coming from the truncation of the model space, we analyze the convergingpattern at increasing values of K max .To quantify our numerical uncertainty we proceed as follows. Denoting with O ( K max ) the expectationvalue of an observable ˆ O computed by setting a given maximal value of the grand-angular momentumquantum number, K max , in the wavefunction, our uncertainty in this observable is estimated by δ ( O ) = | O ( K max ) − O ( K max − | + | O ( K max − − O ( K max − | + δ res , (38)where δ res is the residual uncertainty (not due to the K max behavior) obtained by varying: the number ofradial grid points (from 70 to 90), the maximal values of the angular momentum in the construction of thetwo-body effective interaction (from 60 to 120) and the maximal number of three-body angular momentum(from 5/2 to 7/2) in the partial wave expansion of the 3N force.First, we address and discuss the benchmark of the interactions at LO and NLO, so to have a clean test onthe NN interactions. Then we move to the N2LO, where the three-body forces are included. We study the maximally-local chiral interactions for two different regulator cut-offs, indicated by r ,namely exploring the two possibilities of r = 1 . fm and r = 1 . fm. The latter gives rise to a softerinteraction compared to the first one. For the benchmarks at LO and NLO, the He nucleus is used asa testing ground. We compute point-proton charge radii, (cid:113) (cid:104) r pt (cid:105) , and ground-state energies, E , for thetwo different cut-off choices at increasing values of the grand-angular momentum quantum number andcompare to the GFMC calculations. LO NLOCut-off E (cid:113) (cid:104) r pt (cid:105) Cut-off E (cid:113) (cid:104) r pt (cid:105) [fm] [MeV] [fm] [fm] [MeV] [fm]EIHH 1.0 -42.830(6) 1.0370(3) 1.0 -21.55(4) 1.575(1)1.2 -46.6054(7) 1.01765(4) 1.2 -22.974(6) 1.5278(6)GFMC 1.0 -42.83(1) 1.02(1) 1.0 -21.56(1) 1.57(1)1.2 -46.62(1) 1.00(1) 1.2 -22.94(6) 1.53(1)Nature -28.29566 1.46(1) -28.29566 1.46(1) Table 2.
Ground-state energies and point-proton radii for the He nuclear system at LO and NLO computedwith the EIHH method. For comparison we report the GFMC results and the experimental values takenfrom Ref. [48, 49].The final results are shown in Table 2, where the uncertainty is computed as explained above usingEq. (38) with K max = 22 . An extended table with all the various K max can be found in the SupplementaryMaterial. We observe that as we enlarge the model space a nice converging pattern is obtained and ourfinal EIHH results agree with the GFMC calculations within uncertainties. By looking at the converging Frontiers 13 i Muli et al.
Running Title pattern of the studied observables as the model space is increased (see Supplementary Material), we clearlyobserve that the interaction with r = 1 . fm is much softer than the other, since the relative observablesconverge with a smaller model space. Finally, it is to note that, as shown in Table 2, the LO and NLOresults do not reproduce the measured values, but the discrepancy decreases in going from LO to NLO. We now turn to the benchmark at the next order. At N2LO we have the first appearance of 3N forces,so this will serve as a check of our irreducible tensor representation. The 3N interaction involves twonew LECs, c D and c E , coming from the π -term and from the ct -term of the 3N forces, respectively,that can not be fitted in the NN sector. In Ref. [18] these couplings have been fitted to reproduce the Hebinding energy and the n - α scattering P -wave phase shift, for which the values reported in Table 1 wereobtained. We use the same values in this work, as our goal is to perform a benchmark. In particular, herewe implement only the (D2, E τ ) 3N interactions, which we chose since the E τ term has a more generalisospin structure. Different choices of the 3N contact term have been shown to lead to different saturationproperties in neutron matter [18].As a testing ground for our N2LO Hamiltonian expressed in terms of spherical tensors outlined in theprevious section, we study the three-body He and H and the four-body He nuclear systems. We computeground-state energies, E , and charge radii, (cid:112) (cid:104) r c (cid:105) , for the two different cut-off choices r = 1 and 1.2fm and carefully study the convergence at increasing K max values. A complete table of our data is shownin the Supplementary Material. The K max convergence is also explicitly shown in a graphical manner inFig. 2, Fig. 3, and Fig. 4, where a comparison to the GFMC method is made. E G S [ M e V ] He EIHH r =1.0EIHH r =1.2GFMC r =1.0GFMC r =1.2 max r c [ f m ] Figure 2.
The ground-state energy and the charge radius of the nuclear He system as a function of thegrand-angular momentum quantum number K max . The green and blue error-bands are the GFMC resultswith the relative statistical uncertainty.As it can be seen from Fig. 2 and 3, the EIHH method is in excellent agreement with the GFMCcomputations for the three-body nuclei, for both the ground-state energies and the charge radii. The typicalnon-monotonic convergence patter of the EIHH method is observed, and a very good convergence isreached already at K max = 12 . This shows that these forces are softer than the AV18 potential, but harderthan the low-q interactions [50]. This is a provisional file, not the final typeset article i Muli et al. Running Title E G S [ M e V ] H EIHH r =1.0EIHH r =1.2GFMC r =1.0GFMC r =1.2 max r c [ f m ] Figure 3.
The ground-state energy and the charge radius of the nuclear H system as a function of thegrand-angular momentum quantum number K max . The green and blue error-bands are the GFMC resultswith the relative statistical uncertainty. E G S [ M e V ] He EIHH r =1.0EIHH r =1.2GFMC r =1.0GFMC r =1.2 max r c [ f m ] Figure 4.
The ground state energy and the charge radius of the nuclear He system as a function of thegrand-angular momentum quantum number K max . The green and blue error-bands are the GFMC resultswith the relative statistical uncertainty.For the He nucleus shown in Figure 4, we obtain a very nice agreement with the GFMC method for thecut-off value r = 1 . fm, while for the cut-off r = 1 . fm, we perfectly reproduce the charge radius, butwe observe a small deviation for the ground-state energy with respect to the GFMC.Our final EIHH results with uncertainties quantified as explained above using Eq. (38) with K max = 22 are shown in Table 3 in comparisons with the GFMC, AFDMC and the experimental data. We note thatthe small difference found for He ground-state energy is just at the level of . MeV in the centralvalues and is non-significant when the full uncertainty of the EIHH method is considered. Similar kindof sub-percentage differences between EIHH and GFMC were also observed in other benchmarks [54]
Frontiers 15 i Muli et al.
Running Title He H HeCut-off E (cid:112) (cid:104) r c (cid:105) E (cid:112) (cid:104) r c (cid:105) E (cid:112) (cid:104) r c (cid:105) [fm] [MeV] [fm] [MeV] [fm] [MeV] [fm]EIHH 1.0 -7.630(6) 1.976(7) -8.338(5) 1.759(6) -28.34(5) 1.656(6)1.2 -7.619(4) 1.974(5) -8.332(3) 1.758(5) -28.31(2) 1.651(4)GFMC 1.0 -7.65(2) 1.97(2) -8.34(1) 1.72(3) -28.30(1) 1.65(2)1.2 -7.63(4) 1.97(1) -8.35(4) 1.72(4) -28.30(1) 1.64(1)AFDMC 1.0 -7.55(8) 1.96(2) -8.33(7) 1.72(2) -27.64(13) 1.68(2)1.2 -7.64(4) 1.95(5) -8.27(5) 1.73(2) -28.37(8) 1.65(1)Nature -7.718043(2) 1.973(14) -8.481798(2) 1.759(36) -28.29566 1.681(4) Table 3.
Ground-state energies and charge radii for the nuclear He, H and He systems at N2LO in thechiral expansion computed with the EIHH, GFMC and AFDMC method. For the EIHH results, we reportthe estimation of the uncertainty coming from the truncation of the model space, the errors of the GFMCand AFDMC are statistical. Experimental values are from Ref. [51, 52, 53, 49].and can be found at this level of precision. It is to note that the cut-off r = 1 . fm leads to a harder force,where in fact, quite a large discrepancy is seen also between the GFMC and the AFDMC computations.Therefore, we do not think that this difference is significant and we consider all these results to constitute asuccessful benchmark of our implementation of 3N forces.As can be seen in Table 3, at N2LO a much improved agreement with experiment is obtained. Infact, if one compares the experimental binding energies to the LO and NLO calculations in Table 2 oneobserves that these low orders overbind (LO) or underbind (NLO) the few-body nuclei, while at N2LOnice agreement is observed. This is expected for He, given that 3N forces are fit to reproduce the Hebinding energy, however a better agreement is also found for He and H due to the strong correlationbetween the three- and four-body binding energy. Interestingly, a nice converging pattern is also found forthe nuclear charge radii.From a careful look at Table 3, one can appreciate that our EIHH calculations are more precise than theGFMC and AFDMC results in the three-nucleon sector and that our numerical uncertainty is comparable tothe experimental uncertainties for the radii. While this may be an advantage of our method, it is importantto note that the error bars quoted in this table do not include the uncertainties coming from the ChEFTexpansion, so they do not constitute the full uncertainty of the theory.We conclude this section with a further investigation on the charge radii of light-nuclear systems. InRef. [55] the proton-charge radius r p = 0 . fm and the neutron-charge radius r n = − . fm recommended by CODATA-2014 were used in the evaluation of nuclear charge radii using Eq. (37).Such single-nucleon data come from experiments that study the electron-nucleon system. Recently, thesequantities were measured more precisely by investigating muonic atoms, and one could ask what is theeffect of this increased precision in the nuclear charge radius when applying Eq. (37). To address thispoint in Table 4 we compare our results for the charge radii of He, H and He at N2LO using theCODATA-2014 single-nucleon input with the results obtained using the rms proton-charge radius comingfrom the muonic-hydrogen r p = 0 . [46] and the new value of the rms charge radius of theneutron r n = − . fm [47]. We denote the first choice with e − r c and the second with µ − r c . Thegeneral effect of using this choices of the proton and neutron charge radii amounts to a systematic reduction This is a provisional file, not the final typeset article i Muli et al. Running Title of roughly 1% of the charge radii of these light nuclei. This has to be contrasted with the full uncertainty ofthe theory that includes not only the EIHH numerical error, but also considers the uncertainty coming fromthe order-by-order chiral expansion. The latter is estimated using the algorithm proposed first in Ref. [56]and is included in Table 4.For a graphical representation of our findings, in Fig 5 we show the He nuclear charge radius atincreasing chiral orders computed for different choices for the proton and neutron charge radii. We observethat the chiral order uncertainty is of the order of 2 % , hence larger than the effect of the more precisesingle-nucleon input. Overall, we confirm the chiral oder-by-order convergence patter, already discussed inRefs. [18, 19], but there shown only for the binding energy and the point-proton radius, which does notinclude the single nucleon input.Interestingly, when comparing the He theoretical charge radius with the newest muonic atommeasurement from Ref. [57], we see that the µ − r c results are still consistent with experiment, leavinghowever space for meson exchange currents to help improving the theoretical precision, which is by farlower than the experimental one. He H HeCut-off µ -r c e -r c µ -r c e -r c µ -r c e -r c [fm] [fm] [fm] [fm] [fm] [fm] [fm]EIHH 1.0 1.96(4) 1.98(4) 1.75(3) 1.76(3) 1.64(4) 1.66(4)1.2 1.96(3) 1.97(3) 1.75(3) 1.76(3) 1.64(3) 1.65(3)Exp, electron 1.973(14) 1.759(36) 1.681(4)Exp, muon − − Table 4.
Nuclear rms charge radii for He, H and He systems at N2LO computed using either thesingle-nucleon CODATA-2014 values (columns e -r c ) or the more precise muonic atoms data (columns µ -r c ). The theoretical results are compared to data from the electron-nucleus system [51, 53] and, whenavailable, to data obtained from the muon-nucleus system [57]. In this work, the maximally local chiral interactions are implemented for the first time in the hypersphericalharmonic formalism. The benchmark tests performed in light nuclei show general agreement betweenhyperspherical harmonic results and the previously available Monte Carlo calculations. As expected, atN2LO with the inclusion of the 3N forces the experimental results are much better reproduced with respectto the LO and NLO calculations. With this study we thus confirm the nice order-by-order convergence inthe ground-state energies and in the radii that was already observed in the Monte Carlo studies.While our numerical precision of the EIHH calculations lies in the sub-percent range, we find that theuncertainties due to the chiral order expansion is higher. In case of the charge radius, we observed thatusing the most updated values of the proton and nucleon radii instead of the CODATA-2014 values leads toa variation of 1 % , which is smaller than the uncertainties found in the chiral order-by-order truncationat N2LO. Addressing first the latter by going to N3LO should be the priority if the goal is to reducetheoretical uncertainties. Frontiers 17 i Muli et al.
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LO NLO N2LO0.81.01.21.41.61.8 r c ( f m ) He exp-expe-EIHH r =1.0e-EIHH r =1.2-EIHH r =1.0-EIHH r =1.2 Figure 5.
The He charge radius computed at increasing orders of the chiral expansion. The uncertaintybars include the numerical uncertainty of the EIHH method as well as the uncertainties coming from thetruncation of the chiral expansion. The horizontal lines are the experimental values from electron scattering(solid line) [51] and and from muonic atoms (dashed line) [57].Having these new interactions implemented in our formalisms opens up the possibility of investigatingother few-body observables in the future. Our most immediate goals include the investigation of muonicatoms [33] and of the He monopole transition form factor [58] in an order-by-order chiral expansions. Wereserve these applications to future studies.
CONFLICT OF INTEREST STATEMENT
The authors declare that the research was conducted in the absence of any commercial or financialrelationships that could be construed as a potential conflict of interest.
AUTHOR CONTRIBUTIONS
All authors contributed in equal parts to this paper. N.B. derived the first expressions for the sphericaltensors, which were then checked by S.S.L. and S.B. The two-body force was implemented by S.B. whileS.S.L. implemented the expressions of the 3 N spherical tensors in the hypershperical harmonics codeand run the calculations. Results were discussed in the group at every step. All authors contributed to thewriting of the text. FUNDING
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the CollaborativeResearch Center [The Low-Energy Frontier of the Standard Model (SFB 1044)], and through the Clusterof Excellence “Precision Physics, Fundamental Interactions, and Structure of Matter” (PRISMA + EXC2118/1) funded by the DFG within the German Excellence Strategy (Project ID 39083149). Calculationswere performed on the mogon2 cluster in Mainz.
ACKNOWLEDGMENTS
S.S.L. and S.B. would like to acknowledge Joel Lynn, Ingo Tews and Diego Lonardoni for usefuldiscussions.
This is a provisional file, not the final typeset article i Muli et al. Running Title
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