Effective field theory in the harmonic oscillator basis
aa r X i v : . [ nu c l - t h ] A p r Effective field theory in the harmonic oscillator basis
S. Binder,
1, 2
A. Ekstr¨om,
1, 2
G. Hagen,
2, 1
T. Papenbrock,
1, 2 and K. A. Wendt
1, 2 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
We develop interactions from chiral effective field theory (EFT) that are tailored to the harmonicoscillator basis. As a consequence, ultraviolet convergence with respect to the model space isimplemented by construction and infrared convergence can be achieved by enlarging the modelspace for the kinetic energy. In oscillator EFT, matrix elements of EFTs formulated for continuousmomenta are evaluated at the discrete momenta that stem from the diagonalization of the kineticenergy in the finite oscillator space. By fitting to realistic phase shifts and deuteron data weconstruct an effective interaction from chiral EFT at next-to-leading order. Many-body coupled-cluster calculations of nuclei up to
Sn converge fast for the ground-state energies and radii infeasible model spaces.
PACS numbers: 21.30.-x, 21.30.Fe, 21.10.Dr, 21.60.-n
I. INTRODUCTION
The harmonic oscillator basis is advantageous innuclear-structure theory because it retains all symmetriesof the atomic nucleus and provides an approximate mean-field related to the nuclear shell model. However, inter-actions from chiral effective field theory (EFT) [1, 2] aretypically formulated in momentum space, while the oscil-lator basis treats momenta and coordinates on an equalfooting, thereby mixing long- and short-ranged physics.This incommensurability between the two bases is notonly of academic concern but also makes oscillator-based ab initio calculations numerically expensive. Indeed, theoscillator basis must be large enough to accommodatethe nucleus in position space as well as to contain thehigh-momentum contributions of the employed interac-tion. Furthermore, one needs to perform computationsat different values of the oscillator spacing ~ ω to gaugemodel-space independence of the computed results [3–6].Several methods have been proposed to alleviate theseproblems. Renormalization group transformations, forinstance, are routinely used to “soften” interactions [6–8], and many insights have been gained through thesetransformations [9]. However, such transformations ofthe Hamiltonian and observables [10, 11] add one layerof complexity to computations of nuclei.One can contrast the effort of computations in the os-cillator basis to, for instance, computations in nuclearlattice EFT [12]. Here, the effective interaction is tai-lored to the lattice spacing and, thus, to the ultraviolet(UV) cutoff, and well-known extrapolation formulas [13]can be used to estimate corrections due to finite latticesizes. The lattice spacing is fixed once, reducing the com-putational expenses. This motivates us to seek a sim-ilarly efficient approach for the oscillator basis, i.e., toformulate an EFT for nuclear interactions directly in theoscillator basis.In recent years, realistic ab initio nuclear computationspushed the frontier from light p -shell nuclei [14, 15] to themedium-mass regime [16–23]. At present, the precision of computational methods considerably exceeds the ac-curacy of available interactions [24], and this is the mainlimitation in pushing the frontier of ab initio computa-tions to heavy nuclei. To address this situation, sev-eral efforts, ranging from new optimization protocols forchiral interactions [25–27] to the inclusion of higher or-ders [28, 29] to the development of interactions with novelregulators [30] are under way. To facilitate the compu-tation of heavy nuclei we propose to tailor interactionsfrom chiral EFT to the oscillator basis.There exist several proposals to formulate EFTs in theoscillator basis. Haxton and coworkers proposed the os-cillator based effective theory (HOBET) [31–34]. Theyfocused on decoupling low- and high-energy modes in theoscillator basis via the Bloch-Horowitz formalism, andon the resummation of the kinetic energy to improvethe asymptotics of bound-state wave functions in con-figuration space. The HOBET interaction is based on acontact-gradient expansion, and the matrix elements arecomputed in the oscillator basis. The resulting interac-tion exhibits a weak energy dependence. The Arizonagroup [35–37] posed and studied questions related to theUV and infrared (IR) cutoffs imposed by the oscillatorbasis, and developed a pion-less EFT in the oscillator ba-sis. This EFT was also applied to harmonically trappedatoms [38]. In this approach, the interaction matrix el-ements are also based on a contact-gradient expansionand computed in the oscillator basis. Running couplingconstants depend on the UV cutoff of the employed os-cillator basis. In a sequence of other papers, T¨olle et al. studied harmonically trapped few-boson systems in aneffective field theory based on contact interactions withrunning coupling constants [39, 40].Our oscillator EFT differs from these approaches. Forthe interaction we choose an oscillator space with afixed oscillator frequency ω and a fixed maximum energy( N max + / ) ~ ω . The matrix elements of the interactionare taken from an EFT formulated in momentum spaceand evaluated at the discrete momentum eigenvalues ofthe kinetic energy in this fixed oscillator space. This re-formulation, or projection, of a momentum-space EFTonto a finite oscillator model space requires us to re-fitthe low-energy coefficients (LECs) of interactions at agiven order of the EFT. We determine these by an op-timization to scattering phase shifts (computed in thefinite oscillator basis via the J -matrix approach [41, 42])and from deuteron properties. The power counting ofthe oscillator EFT is based on that of the underlyingmomentum-space EFT. The finite oscillator space intro-duces IR and UV cutoffs [35, 43–49], and these are thusfixed for the interaction.In practical many-body calculations we will keep theoscillator frequency and the interaction fixed at ~ ω and N max , but employ the kinetic energy in larger modelspaces. This increase of the model space increases (de-creases) its UV (IR) cutoff but does not change any in-teraction matrix elements and thus leaves the IR and UVcutoff of the interaction unchanged. As UV convergenceof the many-body calculations depends on the matrix el-ements of an interaction [47], oscillator EFT guaranteesthis UV convergence by construction because no new po-tential matrix elements enter beyond N max . We stressthat our notion of UV convergence relates to the con-vergence of the many-body calculations and should notbe confused with the expectation that observables areindependent of the regulator or cutoff. Infrared conver-gence builds up the exponential tail of bound-state wavefunctions in position space, as the effective IR cutoff ofa finite nucleus is set by its radius. Thus, the increaseof the model space for the kinetic energy achieves IRconvergence. Regarding IR convergence, oscillator EFTis similar to the HOBET of Ref. [33]. In practice, IR-converged values for bound-state energies and radii canbe obtained applying “L¨uscher-like” formulas for the os-cillator basis [44].We view oscillator EFT similar to lattice EFT [12].The latter constructs an interaction on a lattice in posi-tion space while the former builds an interaction on a dis-crete (but non-equidistant) mesh in momentum space. Inboth EFTs the UV cutoff of the interaction is fixed onceand for all, and LECs are adjusted to scattering dataand bound states. The increase in lattice sites achievesIR convergence in lattice EFT, while the increase of thenumber of oscillator shells achieves IR convergence in os-cillator EFT.As we will see, the resulting EFT interaction in theoscillator basis exhibits a fast convergence, similar to thephenomenological JISP interaction [50]. From a practicalpoint of view, our approach to oscillator EFT allows usto employ all of the existing infrastructure developed fornuclear calculations.The discrete basis employed in this paper is actu-ally the basis set of a discrete variable representation(DVR) [51–57] in momentum space. While coordinate-space DVRs are particularly useful and popular in com-bination with local potentials, the results of this papersuggest that DVRs are also useful in momentum-space-based EFTs, because they facilitate the evaluation of ma-trix elements. This paper is organized as follows. In Section II weanalyze the momentum-space structure of a finite oscil-lator basis. In Section III we validate our approach byreproducing an interaction from chiral EFT at next-to-leading order (NLO). In Section IV we construct a NLOinteraction from realistic phase shifts and employ this in-teraction in many-body calculations, demonstrating thatconverged binding energies and radii can be obtainedfor nuclei in the mass-100 region in model spaces with N max = 10 to 14 without any further renormalization.We finally present our summary in Section V. II. THEORETICAL CONSIDERATIONS
In this Section we present the theoretical foundationof an EFT in the oscillator basis. We derive analyticalexpressions for the momentum eigenstates and eigenval-ues in finite oscillator spaces and present useful formulasfor interaction matrix elements.
A. Momentum states in finite oscillator spaces
The radial wave functions h r, l | n, l i = ψ nl ( r ) of oscil-lator basis states | n, l i can be represented in terms ofgeneralized Laguerre polynomials L l + / n as ψ n,l ( r ) =( − n s n !Γ( n + l + / ) b (cid:16) rb (cid:17) l e − ( rb ) L l + / n (cid:16) r b (cid:17) . (1)Here, b ≡ p ~ / ( mω ) is the oscillator length expressed interms of the nucleon mass m and oscillator frequency ω .In what follows, we use units in which ~ = 1.In free space, the spherical eigenstates of the momen-tum operator ˆ p are denoted as | k, l i with k being contin-uous. The corresponding wave functions h r, l | k, l i = r π j l ( kr ) (2)are spherical Bessel functions j l up to a normalizationfactor. These continuum states are normalized as h k ′ , l | k, l i = ∞ Z d rr h k ′ , l | r, l ih r, l | k, l i = δ ( k − k ′ ) kk ′ . (3)Introducing the momentum-space representation of theradial oscillator wave functions via the Fourier-Besseltransform˜ ψ n,l ( k ) ≡ ∞ Z d rr h k, l | r, l i ψ n,l ( r ) (4)= s n ! b Γ( n + l + / ) ( kb ) l e − k b L l + / n (cid:0) k b (cid:1) , enables us to expand the continuous momentumstates (2) in terms of the oscillator wave functions h r, l | k, l i = ∞ X n =0 ˜ ψ n,l ( k ) ψ n,l ( r ) . (5)As we want to develop an EFT, it is most impor-tant to understand the squared momentum operatorˆ p . An immediate consequence of Eq. (2) is of coursethat the spherical Bessel functions are also eigenfunc-tions of ˆ p with corresponding eigenvalues k , i.e.,ˆ p j l ( kr ) = k j l ( kr ). In the oscillator basis, the matrixrepresentation of ˆ p is tri-diagonal, with elements h n ′ , l | ˆ p | n, l i = b − (cid:20) (2 n + l + / ) δ n ′ n − p n ( n + l + / ) δ n ′ +1 n − p ( n + 1)( n + l + / ) δ n ′ − n (cid:21) . (6) We want to solve the eigenvalue problem of the op-erator ˆ p in a finite oscillator basis truncated at an en-ergy ( N max + / ) ~ ω . For partial waves with angular mo-mentum l the basis consists of wave functions (1) with n = 0 , . . . N , i.e., the sum in Eq. (5) is truncated at N ≡ (cid:20) N max − l (cid:21) . (7)Here [ x ] denotes the integer part of x . While N clearly depends on l and N max , we will suppress thisdependence in what follows. Motivated by Eq. (5) weact with the matrix of ˆ p on the component vector( ˜ ψ ,l ( k ) , . . . , ˜ ψ N,l ( k )) T . The well-known three-term re-currence relation for Laguerre polynomials (see, e.g.,Eq. 8.971(4) of Ref. [58]) implies0 = (2 n + l + / − b k ) ˜ ψ n,l ( k ) − p n ( n + l + / ) ˜ ψ n − ,l ( k ) − p ( n + 1)( n + l + / ) ˜ ψ n +1 ,l ( k ) , (8)and for our eigenvalue problem we arrive atˆ p ˜ ψ ,l ( k )...˜ ψ N − ,l ( k )˜ ψ N,l ( k ) = k ˜ ψ ,l ( k )...˜ ψ N − ,l ( k )˜ ψ N,l ( k ) + b − p ( N + 1)( N + l + / ) ˜ ψ N +1 ,l ( k ) . (9)For k = k µ,l such that ˜ ψ N +1 ,l ( k ) = 0, the second termon the right-hand side of Eq. (9) vanishes, and we obtainan eigenstate of the momentum operator (6) in the finiteoscillator space.Thus, momenta k µ,l (with µ = 0 , . . . , N ) such that k µ,l b is a root of the the Laguerre polynomial L l + / N +1 solve the eigenvalue problem of the ˆ p operator in thefinite oscillator space. We recall that L l + / N +1 has N + 1roots. Thus, in a finite model space consisting of oscil-lator functions with n = 0 , . . . , N the eigenvalues of thesquared momentum operator are the N + 1 roots k µ,l of the Laguerre polynomial L l + / N +1 . We note that k µ,l de-pends on the angular momentum l as well as N . To avoida proliferation of indices, we suppress the latter depen-dence in what follows. By construction, the basis builton discrete momentum eigenstates is a DVR [54–56].Previous studies showed that the finite oscillator basisis equivalent to a spherical cavity at low energies [44], andthe radius of this cavity is related to the wavelength ofthe discrete momentum eigenstate with lowest momen-tum. References [45, 46] give analytical results for thelowest momentum eigenvalue in the limit of N ≫
1. Theexact determination of the eigenvalues of the momentumoperator in the present work allows us to give exact val-ues for the radius of the cavity corresponding to the finite oscillator basis. This radius is relevant because it entersIR extrapolations [43, 45, 46]. Let k ,l b be the smallestroot of the Laguerre polynomial L l + / N +1 , and let z ,l de-note the smallest root of the spherical Bessel function j l .Then z ,l L ( N, l ) = k ,l (10)defines the effective radius L we seek.The radial momentum eigenfunction corresponding tothe eigenvalue k µ,l in the partial wave with angular mo-mentum l has an expansion of the form φ µ,l ( r ) ≡ c µ,l N X n =0 ˜ ψ n,l ( k µ,l ) ψ n,l ( r ) . (11)This wave function is the projection of the sphericalBessel function (5) onto the finite oscillator space. Itis also an eigenfunction of the momentum operator pro-jected onto the finite oscillator space because the specificvalues of k µ,l decouple this wave function from the ex-cluded space. In Eq. (11) c µ,l is a normalization constantthat we need to determine. In order to do so we considerthe overlap h φ µ,l | φ ν,l i = ∞ Z d rr φ µ,l ( r ) φ ν,l ( r ) (12)= c µ,l c ν,l N X n =0 ˜ ψ n,l ( k µ,l ) ˜ ψ n,l ( k ν,l )= c µ,l c ν,l p ( N + 1)( N + l + / ) × ˜ ψ N,l ( k µ,l ) ˜ ψ N +1 ,l ( k ν,l ) − ˜ ψ N +1 ,l ( k µ,l ) ˜ ψ N,l ( k ν,l )( k µ,l − k ν,l ) b . Here, we used the Christoffel-Darboux formula for or-thogonal polynomials, see, e.g., Eq. 8.974(1) of Ref. [58].As k µ,l and k ν,l are roots of ˜ ψ N +1 ,l , we confirm orthogo-nality. For k µ,l = k ν,l we use the rule by l’Hospital andfind (with help of Eq. 8.974(2) of Ref. [58]) c − µ,l = p ( N + 1)( N + l + / ) k µ,l b ˜ ψ N,l ( k µ,l ) . (13)It is also useful to compute the overlap h k µ,l , l | φ ν,l i = r π ∞ Z d rr j l ( k µ,l r ) φ ν,l ( r )= δ νµ c − µ,l . (14)This overlap vanishes for k µ,l = k ν,l , thus, the eigenstatesof the ˆ p operator in finite oscillator spaces are orthogonalto the continuous momentum eigenstates when the latterare evaluated at the discrete momenta. This exact resultis very useful for the computation of matrix elements ofa potential operator ˆ V .For arbitrary continuous momenta k we obtain fromEq. (12)˜ φ ν,l ( k ) ≡ h k, l | φ ν,l i = k ν,l /bk ν,l − k ˜ ψ N +1 ,l ( k ) . (15)The wave function (15) is the Fourier-Bessel transform ofthe discrete radial momentum wave function φ ν,l ( r ). Wenote that Eq. (11) relates the discrete momentum eigen-functions to the oscillator eigenstates via an orthogonaltransformation, implying N X µ =0 c µ,l ˜ ψ n,l ( k µ,l ) ˜ ψ n ′ ,l ( k µ,l ) = δ nn ′ , (16)and N X n =0 ˜ ψ n,l ( k µ,l ) ˜ ψ n,l ( k ν,l ) = δ νµ c − µ,l . (17)We remind the reader that the discrete set of momenta k µ,l is fixed once a particular N is chosen. Equation (16)can be used to relate oscillator basis functions to thediscrete momentum eigenfunctions (11). Thus, ψ n,l ( r ) = N X µ =0 c µ,l ˜ ψ n,l ( k µ,l ) φ µ,l ( r ) . (18) B. Matrix elements of interactions from EFT
Nucleon-nucleon (
N N ) interactions from EFT are typ-ically available for continuous momenta in a partial-wave basis in form of the matrix elements h k ′ , l ′ | ˆ V | k, l i .Numerical integration techniques are used to transformthese matrix elements into the oscillator basis. However,there is a very simple approximative relationship betweenthe matrix elements with continuous momenta and thematrix elements h φ ν,l ′ | ˆ V | φ µ,l i in the discrete momentumbasis. This relationship is motivated by EFT argumentsand we use it in our applications of oscillator EFT. Weconsider the matrix element h k ′ , l ′ | ˆ V | φ µ,l i = ∞ Z d kk h k ′ , l ′ | ˆ V | k, l ih k, l | φ µ,l i = 12 b ∞ Z dxx l + / e − x × (cid:18) h k ′ , l ′ | ˆ V | x / b − , l ih x / b − , l | x l e − x | φ µ,l i (cid:19) . (19)Here, we introduced the dimensionless integration vari-able x ≡ b k and factored out a weight function x l + / e − x from the integrand (given in brackets) in prepa-ration for the next step. We evaluate the integral us-ing ( N +1)-point generalized Gauss-Laguerre quadraturebased on the selected weight function. Thus, the matrixelement (19) becomes h k ′ , l ′ | ˆ V | φ µ,l i =12 b N X ν =0 w ν,l h k ′ , l ′ | ˆ V | x / ν,l b − , l i h x / ν,l b − , l | φ µ,l i x lν,l e − x ν,l (20)+∆ N +1 . Here, x ν,l are the roots of the Laguerre polynomial L l + / N +1 ,the weights are w ν,l ≡ Γ( N + l + / ) x ν,l ( N + 1)! h ( N + 2) L l + / N +2 ( x ν,l ) i = Γ( N + l + / ) x ν,l ( N + 1)!( N + l + / ) h L l + / N ( x ν,l ) i , (21)and the error term is∆ N +1 ≡ ( N + 1)!Γ( N + l + / )(2 N + 1)! f (2 N +2) ( ξ ) , (22)see, e.g., Ref. [59]. For the weights, we also usedEq. 8.971(6) of Ref. [58]. In the error term, f (2 N +2) ( ξ )denotes the (2 N +2)-th derivative of the integrand (givenin round brackets) of Eq. (19), evaluated at ξ which issomewhere in the integration domain.We want to estimate the order of the error term ∆ N +1 when using EFT interactions. For this purpose, we writethe potential as a sum of separable potentials V ( k ′ , k ) = X a v a g a ( k ′ ) g a ( k ) , (23)and write g a ( k ) = ( k/ Λ) l e − k b ˜ g a ( k/ Λ) . (24)Here, Λ is a high-momentum cutoff. The function˜ g a ( k/ Λ) is an even function of its arguments (see, e.g.,Ref. [2]), and can be expanded in a Taylor series˜ g a ( k ) = ∞ X n =0 ˜ g ( n ) a (0) n ! (cid:16) x Λ b (cid:17) n . (25)Here, we again used x ≡ b k . With this expansion inmind, and noting that the wave function ˜ φ ν,l can be ex-panded in terms of oscillator wave functions, the inte-grand f (given in round brackets) of Eq. (19) is a productof ˜ g a and a sum of Laguerre polynomials (from the wavefunction ˜ φ ν,l ). The ( N +1)-point Gauss-Laguerre inte-gration is exact for monomials up to x N +1 . As the wavefunction ˜ φ ν,l contains monomials up to x N , the Gauss-Laguerre integration becomes inexact for terms startingat n = N + 2 in the Taylor series (25). Thus, f (2 N +2) inthe error term (22) scales as 1 / (Λ b ) N + l +4 . In the oscil-lator EFT, typical momenta k scale as 1 /b , and the errorterm scales as ∆ N +1 = O (cid:16) ( k/ Λ) N + l +4 (cid:17) . (26)Therefore, h k ′ , l ′ | ˆ V | φ µ,l i = h k ′ , l ′ | ˆ V | k µ,l , l i c µ,l + O (cid:16) ( k/ Λ) N + l +4 (cid:17) . (27)Repeating the calculation for the bra side yields the finalresult h φ ν,l ′ | ˆ V | φ µ,l i = c ν,l ′ c µ,l h k ν,l ′ , l ′ | ˆ V | k µ,l , l i + O (cid:16) ( k/ Λ) N + l +4 (cid:17) . (28)In oscillator EFT, we will omit the correction term andset h φ ν,l ′ | ˆ V | φ µ,l i = c ν,l ′ c µ,l h k ν,l ′ , l ′ | ˆ V | k µ,l , l i . (29)We note that this assignment seems to be very naturalfor an EFT built on a finite number of discrete momen-tum states. For sufficiently large N , the difference to thematrix element obtained from an exact integration canbe view as a correction that is beyond the order of thepower counting of the EFT we build upon. In Eq. (29)the matrix elements between the discrete and continuousmomentum states simply differ by normalization factorsbecause of the different normalization (14) of discreteand continuous momentum eigenstates. Thus, partial-wave decomposed matrix elements of any momentum-space operator can readily be used to compute the corre-sponding oscillator matrix elements. In the Appendix we present an alternative motivation for the usage ofEq. (29).In the remainder of this Subsection, we give useful for-mulas that relate the matrix elements and wave functionsof the discrete momentum basis and the oscillator ba-sis. We note that the oscillator basis states | ψ n,l i are re-lated to the discrete momentum states | φ n,l i via Eq. (18).Thus, we can also give an useful formula that transformsmomentum-space matrix elements to the oscillator basisaccording to h ψ n ′ ,l ′ | ˆ V | ψ n,l i = (30) N X ν,µ =0 c ν,l ′ ˜ ψ n ′ ,l ′ ( k ν,l ′ ) h k ν,l ′ , l ′ | ˆ V | k µ,l , l i c µ,l ˜ ψ n,l ( k µ,l )+ O (cid:0) k N +2 (cid:1) . (31)This formula also reflects the well known fact that theoscillator basis mixes low- and high-momentum physics.The relation between matrix elements in the oscillatorbasis and the discrete momentum basis is given by h φ ν,l ′ | ˆ V | φ µ,l i = c ν,l ′ c µ,l N X n,n ′ =0 ˜ ψ n ′ ,l ′ ( k ν,l ′ ) h ψ n ′ ,l ′ | ˆ V | ψ n,l i ˜ ψ n,l ( k µ,l ) . (32)Finally, we discuss the inversion of Eq. (30), e.g., forsituations where scattering processes in the continuumhave to be considered for interactions based on oscillatorspaces. We obtain h k ν,l ′ , l ′ | ˆ V | k µ,l , l i = N X n,n ′ =0 ˜ ψ n ′ ,l ′ ( k ν,l ′ ) h ψ n ′ ,l ′ | ˆ V | ψ n,l i ˜ ψ n,l ( k µ,l ) (33)+ O (cid:0) k N + l +4 (cid:1) , because of Eq. (17).For arbitrary momenta, one needs to use the over-laps (15) in the evaluation of the matrix elements, andfinds the generalization of Eq. (33) as h k ′ , l ′ | ˆΠ ˆ V ˆΠ | k, l i = ˜ ψ N +1 ,l ′ ( k ′ ) ˜ ψ N +1 ,l ( k ) × N X ν,µ =0 k ν,l ′ /b ( k ′ ) − k ν,l ′ h φ ν,l ′ | ˆ V | φ µ,l i k µ,l /bk − k µ,l . (34)Here, we introduced the projection operator onto the fi-nite oscillator spaceˆΠ ≡ N X ν =0 N max − ν X l =0 | φ ν,l ih φ ν,l | . (35)We note that the projection operator acts as a UV (andIR) regulator. It is nonlocal, and can be written in manyways. Examples are h k ′ , l | ˆΠ | k, l i = N X n =0 ˜ ψ n,l ( k ′ ) ˜ ψ n,l ( k ) (36)= p ( N + 1)( N + l + / ) b × ˜ ψ N,l ( k ) ˜ ψ N +1 ,l ( k ′ ) − ˜ ψ N +1 ,l ( k ) ˜ ψ N,l ( k ′ ) k − ( k ′ ) (37)= ˜ ψ N +1 ,l ( k ′ ) ˜ ψ N +1 ,l ( k ) b × N X ν =0 k ν,l [( k ′ ) − k ν,l ][ k − k ν,l ] . (38)Here, the first identity comes directly from the defini-tion of the projector in terms of the oscillator eigenfunc-tions. The second identity follows from the calculationdisplayed in Eq. (12), while the third identity followsfrom Eq. (15).This presents us with an alternative motivation (butnot derivation) of Eq. (29). We evaluate the projectedmatrix elements (34) at discrete momenta and find withEq. (15) that h k ν,l ′ , l ′ | ˆΠ ˆ V ˆΠ | k µ,l , l i = c − ν,l ′ c − µ,l h φ ν,l ′ | ˆ V | φ µ,l i . (39)We note that h k ν,l ′ , l ′ | ˆΠ ˆ V ˆΠ | k µ,l , l i = h k ν,l ′ , l ′ | ˆ V | k µ,l , l i (40)approximately holds for the discrete momenta in the fi-nite oscillator space (cf. Eq. (28)). III. CHIRAL INTERACTIONS IN FINITEOSCILLATOR BASES
In this Section we present a proof-of-principle construc-tion of a chiral
N N interaction in the framework of theoscillator EFT. First, we study the effects that the trun-cation to a finite oscillator basis has on phase shifts ofexisting
N N interactions. Second, we demonstrate that amomentum-space chiral interaction at NLO can be equiv-alently constructed in oscillator EFT.We consider the chiral interactions N LO EM [60] andNLO sim [27]. Both interactions employ regulators of theform f ( q ) = exp " − (cid:18) q Λ χ (cid:19) n . (41)Here, q is a relative momentum, n is an integer, and Λ χ isthe high-momentum cutoff, specifically Λ χ = 500 MeV.We use n = 3 in what follows. This cutoff needs to bedistinguished from the (hard) UV cutoff [47]Λ UV ≈ p N max + / ) /b (42)of the oscillator-EFT interaction. Let us comment on using the projector (35) in combi-nation with the regulator (41). In momentum space, theprojector (35) is approximately the identity operator formomenta k, k ′ between the IR and UV cutoffs Λ IR andΛ UV of the oscillator basis. For momenta k, k ′ > Λ UV the projector (35) falls off as a Gaussian. For the reg-ulator (41) we choose Λ χ < Λ UV , which introduces asuper-Gaussian falloff for momenta Λ χ . q . Λ UV .As an example, let us consider Λ χ = 500 MeV andΛ UV = 700 MeV. Then, f (Λ UV ) ≈ × − at the pointwhere the super-Gaussian falloff goes over into a Gaus-sian falloff. Assuming a ratio q/ Λ χ = / that is typicalfor the power counting in chiral EFT, f (Λ UV ) ≈ ( / ) ,and the asymptotic Gaussian falloff is not expected to in-troduce significant contributions to contact interactionsat NLO. Eventually, one might want to consider remov-ing the regulator from an oscillator-based EFT. At thismoment however, it is also useful in taming the oscilla-tions discussed below and shown in Fig. 1. A. Effects of finite oscillator spaces on phase shifts
It is instructive to study the effects that a projectiononto a finite oscillator basis has on phase shifts. Forthis purpose, we employ the well known
N N interac-tion N LO EM [60]. First, we transform its matrix ele-ments to a finite oscillator space using numerically exactquadrature, and subsequently compute the phase shiftsusing the J -matrix approach of Ref. [42]. The two pa-rameter combinations N max = 10, ~ ω = 40 MeV, and N max = 20, ~ ω = 23 MeV respectively yield a UV cutoffΛ UV ≈
700 MeV, see Eq. (42). This considerably ex-ceeds the high-momentum cutoff Λ χ of the interaction.Figure 1 shows the resulting phase shifts in selected par-tial waves. The projection onto finite oscillator basesintroduces oscillations in the phase shifts. On the onehand, many ab initio calculations of atomic nuclei yieldpractically converged results for bound-state observablesin model spaces with N max .
14 and oscillator frequen-cies around 24 MeV. On the other hand, oscillator spacesconsisting of only 10 to 20 shells are clearly too smallto capture all the information contained in the originalinteraction. We note (i) that the N LO EM interactionis formulated for arbitrary continuous momenta whereasoscillator EFT limits the evaluation to a few discrete mo-menta, and (ii) that the present Λ UV cuts off the high-momentum tails of the N LO EM interaction.For N max → ∞ we expect to arrive at the originalphase shifts, and this is supported in Fig. 1 by the re-duced oscillations in the N max = 20 phase shifts com-pared to their N max = 10 counterparts. The periodof oscillations is approximately given by the IR cut-off. In the bottom panel of Fig. 1, the oscillator spac-ing ~ ω is increased to 80 MeV (for N max = 10) and46 MeV (for N max = 20), respectively, yielding a UV cut-off Λ UV ≈ S Λ UV = 700 MeV D Λ UV = 700 MeV − P Λ UV = 700 MeV p rel [MeV] − P Λ UV = 1000 MeV N LO EM N max = 20 N max = 10 p h a s e s h i f t [ d e g r ee s ] FIG. 1. (Color online) Computed np phase shifts of the NN potential N LO EM [60] projected onto finite oscillatorspaces with N max = 10 and N max = 20 compared to the phaseshifts from the full (i.e., not projected) interaction in selectedpartial waves as a function of the relative momentum. Inthe top three panels the oscillator frequencies were chosen as ~ ω = 40 MeV and 23 MeV, respectively, in order to obtainan oscillator cutoff Λ UV ≈
700 MeV that well exceeds thechiral cutoff Λ χ = 500 MeV. The filled circles mark the phaseshifts at the momenta corresponding to the eigenenergies ofthe scattering channel in the truncated oscillator space. In thebottom panel ~ ω is increased to 80 MeV (for N max = 10) and46 MeV (for N max = 20), respectively. This yields a UV cut-off Λ UV ≈ values of Λ UV .We note that the eigenvalues of the Hamiltonian ma-trix in ( N max + 1) oscillator shells play a special rolefor the phase shifts in uncoupled partial wave chan-nels. At these energies, the eigenfunctions are stand-ing waves with a Dirichlet boundary condition at the(energy-dependent) radius of the spherical cavity that isequivalent to the finite oscillator basis, and one can alter-natively use this information in the computation of the phase shifts [45, 61]. The filled circles in Fig. 1 indicatethe values of the phase shifts at these energies, which areclose to those of the original interaction.Subsection II B discusses two ways of projectingmomentum-space interactions onto a finite oscillatorspace. The first approach involves the determination ofthe matrix element (19) via an exact numerical integra-tion over continuous momenta. The second approach,Eq. (29), uses ( N + 1)-point Gauss-Laguerre quadratureto compute the integral in Eq. (19). This only requiresus to evaluate the interaction at those momenta that arephysically realized in the finite oscillator basis, which ismore in the spirit of an EFT. Because we want to followthe oscillator EFT approach in later sections, we studythe effect that the error term (22) associated with the( N +1)-point Gauss-Laguerre quadrature has on the pro-jected phase shifts. Figure 2 shows a comparison of pro-jected N LO EM phase shifts obtained from the two pro-jection approaches to the original N LO EM phase shifts.Overall, both versions yield very similar phase shifts. Forthe phase shifts associated with the ( N + 1)-point Gauss-Laguerre quadrature, the oscillations seem to be some-what reduced for small energies. Also, we find a notablyimproved agreement between the ( N + 1)-point Gauss-Laguerre phase shifts and the original ones at the discreteeigenenergies of the scattering channel Hamiltonians, asindicated by the full circles. From now on we exclusivelyuse the ( N +1)-point Gauss-Laguerre integration to com-pute matrix elements in oscillator EFT. B. Reproduction of the phase shifts of a NLOinteraction
In what follows we employ the oscillator EFT at NLO.All matrix elements in oscillator EFT are based onEq. (29), i.e., the matrix elements from continuum mo-mentum space are evaluated at the discrete momentaof the finite oscillator basis. At this order the chi-ral interactions depend on 11 LECs and exhibit suffi-cient complexity to qualitatively describe nuclear prop-erties. Specifically, in this Section we aim at repro-ducing the phase shifts and selected deuteron proper-ties of the chiral interaction NLO sim [27] by optimizingthese 11 LECs. Throughout this work, the χ fits eval-uate the phase shifts at 20 equidistant energies in thelaboratory energy range up to 350 MeV, with weights σ ∝ ( q/ Λ χ ) . We note that more sophisticated weights(including also the pion mass or the oscillatory patterns)would be needed for a quantification of uncertainties, see,e.g. Refs. [27, 62, 63]. In this work we only investigatethe feasibility of an oscillator EFT with regard to thecomputation of heavy nuclei.The main goal of oscillator EFT is to enable the com-putation of heavy nuclei. Therefore, we set N max = 10 forthe interaction. This allows us to perform IR extrapola-tions based on calculations in spaces with N max = 10, 12,and 14 for the kinetic energy. To determine ~ ω we study S N LO EM Gauss-Laguerreexact D
20 50 100 150 200 250 300 350 400 p rel [MeV] − P p h a s e s h i f t [ d e g r ee s ] FIG. 2. (Color online) Phase shifts of N LO EM obtainedfrom a projection onto a N max = 10, ~ ω = 40 MeV oscillatorspace using (i) exact integration in Eq. (19) (blue, dashed line)and (ii) Gauss-Laguerre quadrature Eq. (29) (green crosses),compared to the unprojected N LO EM phase shifts (black,solid line). The filled circles mark the phase shifts at themomenta corresponding to the eigenenergies of the scatteringchannel in the truncated oscillator space. phase shifts in selected partial waves in Fig. 3. The firstvalue, ~ ω = 20 MeV, again corresponds to a UV cutoffof Λ UV ≈
500 MeV and yields the familiar oscillations inthe phase shifts. The second value, ~ ω = 34 MeV, corre-sponds to a UV cutoff Λ UV ≈
650 MeV that significantlyexceeds the chiral cutoff Λ χ ≈
500 MeV. In this case, theoscillations are drastically reduced. Consequently, we use ~ ω = 34 MeV in the following.In Fig. 4 we compare further phase shifts from the os-cillator EFT to the NLO sim phase shifts. We note thatthe channel D is a prediction because there is no corre-sponding LEC at this chiral order. For completeness, andto assess the large- N max behavior, we also show the re-sults of an optimization in oscillator EFT with N max = 80and ~ ω = 6 MeV (Λ UV ≈
650 MeV) as green crosses. Ingeneral, oscillator EFT reproduces the phase shifts wellover the whole energy range up to the pion-productionthreshold.In the deuteron channel we fit not only to phaseshifts but also to its binding energy E d , point-proton ra-dius r d , and quadrupole moment Q d . For N max = 80,we reproduce the deuteron properties well, as can beseen in Table I. For the N max = 10 effective interac- − S − − P E lab [MeV] D NLO sim Λ UV = 500 MeV Λ UV = 650 MeV N max = 10 p h a s e s h i f t [ d e g r ee s ] FIG. 3. (Color online) Phase shifts of NLO interactions ob-tained from oscillator EFT through fits to NLO sim phaseshifts compared to the NLO sim data [27]. The momentum-space interaction matrix elements used in the constructionof the oscillator EFT interactions employ a fixed chiral cut-off Λ χ = 500 MeV. We fix N max = 10 and consider the twooscillator frequencies ~ ω = 20 MeV and 34 MeV, yieldingUV cutoffs Λ UV = 500 MeV and 650 MeV, respectively. ForΛ UV = 650 MeV, the oscillations in the phase shifts are no-tably reduced. tion, it becomes more difficult to simultaneously re-produce all data. Therefore, we relax the requirementto reproduce the quadrupole moment in favor of theother deuteron properties and the phase shifts. We con-verge the deuteron calculations by employing large modelspaces of 100 oscillator shells where only the kinetic en-ergy acts beyond the N max used to define the effectiveinteraction.In Table I we also compare the LECs of the effective in-teractions for N max = 10 and 80 to the LECs of NLO sim .For N max = 80, our approach quite accurately reproducesthe NLO sim LECs, with C S being the only exception.Because this LEC is associated with the deuteron chan-nel, its value is affected by the weighting of the deuteronproperties in the fit, and we expect that a better repro-duction can be achieved by assigning different weights tothe deuteron properties. More importantly, with the ex-ception of C S , the values of the LECs for the N max = 10interaction are very similar to the LECs of the NLO sim interaction. Also, all of the values are of natural size.Let us also address the sensitivity of our results to the -4004080 S -60-45-30-15 P -30-15015 P S -20-15-10-5 D -12-8-40 E P F E lab [MeV]-3-2-10 E E lab [MeV] 0.81.62.43.2 D NLO sim N max = 10 N max = 80 p h a s e s h i f t [ d e g r ee s ] p h a s e s h i f t [ d e g r ee s ] FIG. 4. (Color online) Phase shifts from oscillator EFTwith N max = 10 (full blue circles and blue line) compared toNLO sim phase shifts (black line) for partial wave channels asindicated. Also shown (green crosses) are results from oscil-lator EFT in a large model space with N max = 80. N max = 10 N max = 80 NLO sim E d [MeV] − . − . − . r d [fm] 2 .
108 1 .
962 1 . Q d [fm ] 0 .
195 0 .
259 0 . P d .
028 0 .
029 0 . C ( np ) S − . − . − . C ( pp ) S − . − . − . C ( nn ) S − . − . − . C S . . . C S − . − . − . C S − . − . − . C E .
224 0 .
232 0 . C P . . . C P .
047 1 .
023 1 . C P − . − . − . C P − . − . − . N max = 10, 80 obtained from a fitto NLO sim phase shifts and deuteron properties, comparedto NLO sim data [27]. The deuteron properties, computed inlarge N max = 100 model spaces, are the ground-state energy E d , point-proton radius r d , quadrupole moment Q d , and D -state probability P d . The oscillator frequencies ~ ω are 34MeV and 6 MeV, respectively, corresponding to a UV cutoffof Λ UV ≈
650 MeV. Units of the LECs as in Ref. [2].
UV cutoff of the employed oscillator space. For this pur-pose we keep ~ ω = 34 MeV fixed and optimize two moreinteractions defined in model spaces with N max = 12 and N max = 14, respectively. We recall that the UV cut-off of the model space increases with increasing N max ,and that N max = 10 , ,
14 corresponds to UV cutoffΛ UV ≈ sim . We see that results converge slowly towardthose of the NLO sim interaction as N max is increased. Wealso note that the few-body observables from the different N max interactions exhibit differences that are consistentwith uncertainty expectations at NLO [27, 29, 64].Clearly, the NLO interaction from oscillator EFT dif-fers from NLO sim through the complicated projectionthat introduces IR and UV cutoffs and is highly non-local, see Eq. (35). In the EFT sense the difference be-tween these interactions should be beyond the order atwhich we are currently operating. While we cannot provethis equivalence, the numerical results of this Section en-courage us to pursue the construction of a chiral NLOinteraction within oscillator EFT by optimization to thephase shifts from a high-precision N N potential in thefollowing Section.0 N max = 10 N max = 12 N max = 14 NLO sim E d -2.261 -2.227 -2.225 -2.224 r d Q d P d E H -8.944 -8.502 -8.094 -8.270 r H E He -8.169 -7.735 -7.33 -7.528 r He E He -28.736 -27.96 -27.302 -27.44 r He H, He, and He from different effective interactions, compared to NLO sim .The oscillator length ~ ω is fixed at 34 MeV, and we consider N max = 10, 12, 14, corresponding to values of the UV cutoffΛ UV ≈ IV. NLO INTERACTIONS IN OSCILLATOREFT AND MANY-BODY RESULTS
In what follows, we set N max = 10 for the interaction.On the one hand, lower oscillator frequencies correspondto larger oscillator lengths and lead to a rapid IR con-vergence. On the other hand, lower oscillator frequenciesalso correspond to lower UV cutoffs. In what followswe choose ~ ω = 24 MeV which results in a UV cutoffΛ UV ≈
550 MeV and an IR length L ≈ . χ = 450 MeV, which ensures that Λ UV significantly exceeds Λ χ .Practical calculations are performed in the laboratorysystem, and use the N max = 10 interaction togetherwith the intrinsic kinetic energy in oscillator spaces with N lab = 2 n + l ≥ N max . Here n and l are the ra-dial and angular momentum quantum numbers of theharmonic oscillator in the laboratory frame. Results for N lab = 10 , ,
14 are feasible and will allow us to performIR extrapolations for bound-state energies and radii.In this Section, we construct a chiral interaction atNLO from realistic phase shifts, and subsequently uti-lize it in coupled-cluster calculations of He, O, Ca, Zr, and
Sn. Our main objectives are (i) to presenta proof-of-principle optimization of a realistic interactionwithin the framework of the oscillator EFT, and (ii) todemonstrate that such an interaction converges fast evenin heavy nuclei. We compute the matrix elements in oscil-lator EFT, i.e., based on Eq. (29) and omit the high-ordercorrection terms.In the optimization, the low-energy coefficients are ob-tained from a χ fit to realistic scattering data (repre-sented here by phase shifts from the CD-Bonn poten-tial [65]) and deuteron properties. The fitting proce-dure is identical to the one described in Section III. Fig-ure 5 presents the resulting phase shifts for a selectionof scattering channels. We reproduce the phase shiftsover a large energy range for several partial waves. The S − − − − P CD-BonnNLO, N = 10NLO, N = 80 − −
808 3 P S − − − − D − − E P . . . . F
20 100 200 300 E lab [MeV] − . − . − . . E E lab [MeV] . . . . D p h a s e s h i f t [ d e g r ee s ] p h a s e s h i f t [ d e g r ee s ] FIG. 5. (Color online) Phase shift from oscillator EFT atNLO with N max = 10 (full blue circles and blue line) com-pared to those of the CD Bonn interaction (full black line) forpartial wave channels as indicated. The green crosses showresults from a large model space with N max = 80. N max = 10 N max = 80 experiment E d [MeV] − .
227 2 . − . r d [fm] 1 .
984 1 .
961 1 . Q d [fm ] 0 .
229 0 .
259 0 . P d .
026 0 . C ( np ) S − . − . C ( pp ) S − . − . C ( nn ) S − . − . C S . . C S − . − . C S − . − . C E .
250 0 . C P . . C P .
348 0 . C P − . − . C P − . − . N max = 10 and 80 shells ob-tained from a fit to CD-Bonn phase shifts and deuteron prop-erties. The oscillator frequencies are ~ ω = 24 MeV and 4 MeVfor N max = 10 and 80, respectively. D channel is an obvious exception, as it deviates moreclearly from the CD-Bonn phase shifts at higher energies,but we note that at NLO there is no LEC to adjust inthis channel. For E , the deviations are also consider-able, however, it is of NLO quality, see Fig. 4. For thedeuteron, we obtain a good reproduction of the bindingenergy and radius, and a reasonably well result for thequadrupole moment, as shown in Table III. The LECs,also shown in Table III, are natural in size and similar toNLO sim ; the largest deviation is for the LEC C P whichis only about / of its NLO sim counterpart.We utilize the NLO interaction in coupled-cluster cal-culations at the singles and doubles level (CCSD) [21, 66,67] of the nuclei He, O, Ca, Zr, and
Sn. Thecoupled-cluster calculations use Hartree-Fock bases thatare unitarily equivalent to the oscillator bases but leadto improved results. In these calculations, we keep theoscillator spacing fixed at ~ ω = 24 MeV in the spirit ofoscillator EFT. We employ model spaces from N lab = 10up to N lab = 16. In the oscillator basis, the potential isalways restricted to N max = 10, while the kinetic energyis used in the entire model space. The results are shownin Table IV. The N lab = 16 point is used to gauge con-vergence of the results. For the light nuclei, energies andradii are practically IR converged already for N lab = 10because r ≪ L .The convergence with respect to N lab is very fast com-pared to other interactions from chiral EFT with a cutoffof around Λ χ ≈
450 MeV [6]. For IR extrapolations ofground-state energies E we employ [44] E ( L ) = E ∞ + ae − k ∞ L , (43)where L ≈ p N lab + / ) b is the IR length [45].In the fit, we employ a theoretical uncertainty N lab 4 He O Ca Zr Sn E CCSD [MeV]10 -31.57 -142.89 -402.0 -918.4 -1230.012 -31.57 -142.92 -402.4 -923.1 -1249.314 -31.57 -142.93 -402.5 -924.6 -1255.616 -31.57 -142.93 -402.5 -925.1 -1258.3 ∞ -31.57 -142.93 -402.5 -925.4 -1260.1exp -28.30 -127.62 -342.1 -783.9 -1102.9 r [fm ]10 1.78 4.14 6.58 9.70 11.6012 1.78 4.15 6.60 9.77 11.7914 1.78 4.15 6.60 9.80 11.8516 1.78 4.15 6.61 9.82 11.88 ∞ E CCSD and squared point-proton radii r for nuclei ranging from He to
Sn fromCCSD calculations in many-body model spaces built fromsingle-particle oscillator bases with N lab = 10 , . . . ,
16. In therelative frame the oscillator EFT interaction is defined in the N max = 10 model space, and the Hamiltonian matrix ele-ments outside this space are comprised of the kinetic energyonly. Experimental data taken from [68, 69]. The experi-mental point-proton radii are extracted from charge radii bycorrecting for the finite sizes of the proton and neutron, andthe Darwin-Foldy term. σ ≡ exp ( − k ∞ L ) / ( k ∞ L ) to account for omitted correc-tions beyond the leading-order result (43). The differencebetween the IR extrapolated energies and the energy at N lab = 16 is much smaller than both, the uncertaintyof about 7% in correlation energy due to the CCSD ap-proximation, and the uncertainty expected from higherorders in the chiral expansion.Figure 6 (top) shows the result of an IR extrapolationfor the ground-state energy of Sn. Here, we plottedenergies in N lab = 10 to 16 as a function of the IR length L and also show the exponential IR extrapolation [44]The inset of Fig. 6 confirms the exponential convergenceof the energy.Figure 6 (bottom) shows the results of an IR extrapola-tion for the ground-state expectation value of the squaredradius of Sn. We plotted squared point-proton radiiin model spaces with N lab = 10 to 16 as a function ofthe IR length L and performed a χ fit to the IR extrap-olation [44] r ( L ) = r ∞ − b ( k ∞ L ) e − k ∞ L . (44)In the χ fit of the radius, we refit the con-stant k ∞ and employed theoretical uncertainties σ ≡ k ∞ L exp ( − k ∞ L ) to account for corrections to theleading-order result (44). Numerical results for the radiiare listed in Table IV. The results from IR extrapola-tions involving only the data points from N lab = 10 to14 yield E ∞ = − . r ∞ = 11 .
88 fm for Sn, which are close to the previous results includingthe N lab = 16 points.2 − − − − E [ M e V ] Sn fitdata E ∞
10 1111 12 L [fm] E − E ∞ [ M e V ] . . . . . . . . L [fm] . . . . . r [f m ] Sn fitdata r ∞ FIG. 6. (Color online) IR convergence of the ground-stateenergy (top) and squared radius (bottom) of
Sn computedfrom CCSD calculations using the oscillator EFT interaction.Data is shown as blue points, the extrapolations as red dashedlines, and the asymptotic energy and radius as solid blacklines. The inset on the top shows that the ground-state energyis approached exponentially in the IR length L . The oscillator EFT interaction at NLO overbinds nu-clei by about 1 MeV per nucleon, and radii are too small.We note that the difference between theoretical and ex-perimental values for the ground-state energies seem tobe consistent with expectations from an NLO interac-tion. We also note that three-nucleon forces entering atnext-to-next-to-leading order will be part of the satura-tion mechanism [26, 70].The rapid convergence of ground-state energies andradii in oscillator EFT suggests that chiral cutoffs ofΛ χ ≈
450 MeV are reasonable in nuclear-structure com-putations of heavy nuclei. Similar to renormalizationgroup transformations the oscillator EFT reduces themismatch between the tail of the momentum-space reg-ulator and the oscillator space.
V. SUMMARY
We developed an EFT directly in the oscillator ba-sis. In this approach UV convergence is implementedby construction, and IR convergence can be achieved byenlarging the model space for the kinetic energy only.We discussed practical aspects of the oscillator EFT and gave analytical expressions for the efficient calculation ofmatrix elements in oscillator EFT from their continuous-momentum counterparts. Within the J -matrix approachwe computed phase shifts while working exclusively inthe oscillator basis. Our results suggest that oscillationsin the phase shifts appear when the UV cutoff imposedby the oscillator basis cuts into the high-momentum tailsof the chiral interaction. To validate the oscillator EFTapproach we reproduced a chiral interaction at NLO. Fi-nally, we developed a chiral NLO interaction in oscillatorEFT by optimizing the LECs to CD-Bonn phase shiftsand experimental deuteron data. Coupled-cluster calcu-lations for nuclei from He up to
Sn exhibit a rapidconvergence for ground-state energies and radii. Theseresults suggest that the oscillator EFT is a promising can-didate to facilitate ab initio calculations of heavy atomicnuclei based on interactions from chiral EFT using cur-rently available many-body methods.
ACKNOWLEDGMENTS
We are grateful to R. J. Furnstahl and S. K¨onig forhelpful discussions and comments on the manuscript.We also thank J. Rotureau for helpful discussions. Thismaterial is based upon work supported in part by theU.S. Department of Energy, Office of Science, Officeof Nuclear Physics, under Award Numbers DE-FG02-96ER40963 (University of Tennessee), DE-SC0008499(SciDAC-3 NUCLEI Collaboration), the Field WorkProposal ERKBP57 at Oak Ridge National Labora-tory (ORNL), and under contract number DEAC05-00OR22725 (ORNL). S.B. gratefully acknowledges thefinancial support from the Alexander-von-HumboldtFoundation (Feodor-Lynen fellowship).
Appendix: Matrix elements in oscillator EFT
In this Appendix we give an alternative motivationfor the computation (29) of matrix elements in oscillatorEFT. These results are well known for DVRs, see, e.g.,Ref. [55]. The projection onto a finite oscillator space isbased on the usual scalar product h ˜Ψ f | ˜Ψ g i = ∞ Z d kk ˜Ψ f ( k ) ˜Ψ g ( k ) (A.1)for radial wave functions in momentum space. The pro-jection operator onto the finite oscillator space at fixedangular momentum l isˆΠ l ≡ N X ν =0 | φ ν,l ih φ ν,l | . (A.2)Let us alternatively consider a different projection op-erator, which is based on a different scalar product. We3write momentum-space wave functions with angular mo-mentum l as˜Ψ f ( k ) = √ b ( kb ) l e − ( kb ) f ( k b ) . (A.3)Here, f ( x ) is a polynomial in x = k b . It is clear thatany square-integrable wave function can be built fromsuch polynomials (with Laguerre polynomials L l + / n be-ing an example). We define the (semi-definite) scalarproduct ( ·|· ) of two functions ˜Ψ f and ˜Ψ g as( ˜Ψ f | ˜Ψ g ) ≡ N X µ =0 w µ,l f ( x µ,l ) g ( x µ,l ) . (A.4)Here, the weights are from Eq. (21) and the abscissas x µ,l ≡ k µ,l b are the zeros of the associated Laguerrepolynomial L l + / N +1 as demanded by ( N + 1)-point Gauss-Laguerre quadrature. Equation (A.4) defines only a semi-definite inner product, because ( ˜Ψ f | ˜Ψ f ) = 0 for any poly-nomial f with f ( x µ,l ) = 0 for µ = 0 , . . . , N .Several comments are in order. First, this semi-definitescalar product is identical to the standard scalar prod-uct (A.1) for wave functions ˜Ψ f ( k ) that are limited tothe finite oscillator space. To see this, we note that h ˜Ψ f | ˜Ψ g i = ∞ Z d kk ˜Ψ f ( k ) ˜Ψ g ( k )= ∞ Z d xx l + / e − x f ( x ) g ( x )= N X µ =0 w µ,l f ( x µ,l ) g ( x µ,l ) . (A.5)Here, we introduced the dimensionless integration vari-able x ≡ k b in the second line and employ ( N +1)-pointGauss-Laguerre quadrature in the third line. We notethat ( N + 1)-point Gauss-Laguerre quadrature is exactfor polynomials of degree up to and including 2 N + 1,i.e., it is exact for polynomials f, g spanned by L l + / n ( x )with n = 0 , . . . , N .This implies that the basis functions ˜ ψ n,l ( k ) with n = 0 , . . . , N in the finite oscillator space remain a ba-sis under the semi-definite scalar product, i.e.( ˜ ψ n,l | ˜ ψ n ′ ,l ) = δ n ′ n . (A.6)Second, we note that both scalar products also agree for N → ∞ because Gauss-Laguerre integration becomes ex-act in this limit.Rewriting the weights w ν,l in Eq. (21) as w ν,l = 2 b ( k ν,l b ) ( k ν,l b ) l e − ( k ν,l b ) ( N + 1)( N + l + / ) h ˜ ψ N,l ( k ν,l ) i = 2 b ( k ν,l b ) l e − ( k ν,l b ) c ν,l , (A.7) and using Eq. (13) yields another useful expression forthe scalar product (A.4)( ˜Ψ f | ˜Ψ g ) = N X µ =0 c µ,l ˜Ψ f ( k µ,l ) ˜Ψ g ( k µ,l ) . (A.8)A main difference between the scalar products ofEq. (A.1) and Eq. (A.8) arises when one compares thewave function h k, l | φ ν,l i = ˜ φ ν,l ( k ) (A.9)of Eq. (15) with the corresponding scalar product( k, l | ˜ φ ν,l ) = N X µ =0 c µ,l δ ( k − k µ,l ) k ˜ φ ν,l ( k ν,l )= c ν,l δ ( k − k ν,l ) k ν,l . (A.10)Here, we used Eq. (A.8) and Eq. (15) implying˜ φ µ,l ( k λ,l ) = c − λ,l δ λµ . Clearly, the Fourier-Bessel trans-form (15) of the discrete momentum eigenstate φ ν,l hasa complicated momentum dependence, while ( k, l | ˜ φ ν,l ) issimply a rescaled δ function. This simple view is consis-tent with naive expectation of a momentum eigenstate.We are now in the position to compute matrix elementsbased on the inner product (A.4). We note thatˆ V | φ µ,l ) = ˆ V ∞ Z d kk | k, l i ( k, l | φ µ,l )= ˆ V | k µ,l i c µ,l . (A.11)Here, we used Eq. (A.10). Repeating the procedure onthe bra side yields( φ ν,l ′ | ˆ V | φ µ,l ) = h k ν,l ′ , l ′ | ˆ V | k µ,l , l i c ν,l ′ c µ,l , (A.12)and this is Eq. (29). We repeat that this derivation of theinteraction matrix element is based on the scalar prod-uct (A.8) and not on the usual scalar product (A.1) forsquare integrable functions. We argue that the formerscalar product is more natural considering the discretemomentum mesh that is employed in oscillator EFT.For another view on the semi-definite scalar productwe consider the projection operatorˆ P l ≡ N X n =0 | ˜ ψ n,l )( ˜ ψ n,l | . (A.13)Indeed, ˆ P l = ˆ P l . This makes it interesting to considerthe projected wave function (cid:16) ˆ P l ˜Ψ f (cid:17) ( k µ,l ) = N X n =0 ( ˜ ψ n,l | ˜Ψ f ) ˜ ψ n,l ( k µ,l )= N X ν =0 c ν,l ˜Ψ f ( k ν,l ) N X n =0 ˜ ψ n,l ( k ν,l ) ˜ ψ n,l ( k µ,l )= ˜Ψ f ( k µ,l ) . (A.14)4Here, we evaluated the sum over n using Eq. (17) whengoing from the second to the third line. Equation (A.14)shows that the projected wave function agrees with thefull wave function at the discrete momenta k µ,l . In otherwords, the projection based on the semi-definite scalarproduct yields wave functions in finite oscillator spacesthat agree with the unprojected wave functions at thephysical momenta.Let us give another interpretation of the projection ˆ P l . One finds for the scalar product (A.8)( ˜Ψ f | ˜Ψ g ) = N X µ =0 c µ,l ˜Ψ f ( k µ,l ) ˜Ψ g ( k µ,l )= h ˜Ψ f | N X µ =0 | k µ,l , l i c µ,l h k µ,l , l | ! | ˜Ψ g i . (A.15)Thus, the scalar product ( ˜Ψ f | ˜Ψ g ) can be viewed as amatrix element of the operator in the brackets. [1] E. Epelbaum, H.-W. Hammer, and Ulf-G.Meißner, “Modern theory of nuclear forces,”Rev. Mod. Phys. , 1773–1825 (2009).[2] R. Machleidt and D.R. Entem, “Chiral ef-fective field theory and nuclear forces,”Physics Reports , 1 – 75 (2011).[3] P. Maris, J. P. Vary, and A. M. Shirokov, “ Ab initio no-core full configuration calculations of light nuclei,”Phys. Rev. C , 014308 (2009).[4] G. Hagen, T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen, “ Ab initio coupled-cluster approach to nuclearstructure with modern nucleon-nucleon interactions,”Phys. Rev. C , 034330 (2010).[5] E. D. Jurgenson, P. Maris, R. J. Furnstahl, P. Navr´atil,W. E. Ormand, and J. P. Vary, “Structureof p -shell nuclei using three-nucleon interactionsevolved with the similarity renormalization group,”Phys. Rev. C , 054312 (2013).[6] Robert Roth, Angelo Calci, Joachim Langhammer,and Sven Binder, “Evolved chiral nn + 3 n hamil-tonians for ab initio nuclear structure calculations,”Phys. Rev. C , 024325 (2014).[7] S. K. Bogner, T. T. S. Kuo, and A. Schwenk,“Model-independent low momentum nucleoninteraction from phase shift equivalence,”Physics Reports , 1 – 27 (2003).[8] S. K. Bogner, R. J. Furnstahl, and R. J. Perry, “Simi-larity renormalization group for nucleon-nucleon interac-tions,” Phys. Rev. C , 061001 (2007).[9] E. D. Jurgenson, P. Navr´atil, and R. J. Furn-stahl, “Evolution of nuclear many-body forceswith the similarity renormalization group,”Phys. Rev. Lett. , 082501 (2009).[10] A. F. Lisetskiy, M. K. G. Kruse, B. R. Barrett,P. Navratil, I. Stetcu, and J. P. Vary, “Effec-tive operators from exact many-body renormalization,”Phys. Rev. C , 024315 (2009).[11] Micah D. Schuster, Sofia Quaglioni, Calvin W. John-son, Eric D. Jurgenson, and Petr Navr´atil, “Oper-ator evolution for ab initio theory of light nuclei,”Phys. Rev. C , 011301 (2014).[12] Dean Lee, “Lattice simulationsfor few- and many-body systems,”Progress in Particle and Nuclear Physics , 117 – 154 (2009).[13] M. L¨uscher, “Volume Dependence of the Energy Spec-trum in Massive Quantum Field Theories. 1. Stable Par-ticle States,” Commun. Math. Phys. , 177 (1986).[14] Petr Navr´atil, Sofia Quaglioni, Ionel Stetcu, and Bruce R Barrett, “Recent developments in no-core shell-modelcalculations,” Journal of Physics G , 083101 (2009).[15] Bruce R. Barrett, Petr Navr´atil, andJames P. Vary, “Ab initio no core shell model,”Progress in Particle and Nuclear Physics , 131 – 181 (2013).[16] G. Hagen, M. Hjorth-Jensen, G. R. Jansen,R. Machleidt, and T. Papenbrock, “Evolution ofshell structure in neutron-rich calcium isotopes,”Phys. Rev. Lett. , 032502 (2012).[17] Jason D. Holt, Takaharu Otsuka, AchimSchwenk, and Toshio Suzuki, “Three-bodyforces and shell structure in calcium isotopes,”Journal of Physics G , 085111 (2012).[18] F. Wienholtz, D. Beck, K. Blaum, Ch. Borgmann,M. Breitenfeldt, R. B. Cakirli, S. George, F. Her-furth, J. D. Holt, M. Kowalska, S. Kreim, D. Lunney,V. Manea, J. Menendez, D. Neidherr, M. Rosenbusch,L. Schweikhard, A. Schwenk, J. Simonis, J. Stanja, R. N.Wolf, and K. Zuber, “Masses of exotic calcium isotopespin down nuclear forces,” Nature , 346–349 (2013).[19] V. Som`a, A. Cipollone, C. Barbieri, P. Navr´atil,and T. Duguet, “Chiral two- and three-nucleonforces along medium-mass isotope chains,”Phys. Rev. C , 061301 (2014).[20] T. A. L¨ahde, E. Epelbaum, H. Krebs, D. Lee,U.-G. Meißner, and G. Rupak, “Lattice ef-fective field theory for medium-mass nuclei,”Phys. Lett. B , 110 – 115 (2014).[21] G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J.Dean, “Coupled-cluster computations of atomic nuclei,”Reports on Progress in Physics , 096302 (2014).[22] H. Hergert, S. K. Bogner, T. D. Morris, S. Binder,A. Calci, J. Langhammer, and R. Roth, “ Ab ini-tio multireference in-medium similarity renormalizationgroup calculations of even calcium and nickel isotopes,”Phys. Rev. C , 041302 (2014).[23] G. Hagen, A. Ekstr¨om, C. Forss´en, G. R. Jansen,W. Nazarewicz, T. Papenbrock, K. A. Wendt, S. Bacca,N. Barnea, B. Carlsson, C. Drischler, K. Hebeler,M. Hjorth-Jensen, M. Miorelli, G. Orlandini, A. Schwenk,and J. Simonis, “Neutron and weak-charge distributionsof the Ca nucleus,” Nature Physics , 186 (2016).[24] Sven Binder, Joachim Langhammer, Angelo Calci,and Robert Roth, “Ab initio path to heavy nuclei,”Physics Letters B , 119 – 123 (2014).[25] A. Ekstr¨om, G. Baardsen, C. Forss´en, G. Hagen,M. Hjorth-Jensen, G. R. Jansen, R. Machleidt,W. Nazarewicz, T. Papenbrock, J. Sarich, and S. M. Wild, “Optimized chiral nucleon-nucleoninteraction at next-to-next-to-leading order,”Phys. Rev. Lett. , 192502 (2013).[26] A. Ekstr¨om, G. R. Jansen, K. A. Wendt, G. Hagen,T. Papenbrock, B. D. Carlsson, C. Forss´en, M. Hjorth-Jensen, P. Navr´atil, and W. Nazarewicz, “Accurate nu-clear radii and binding energies from a chiral interaction,”Phys. Rev. C , 051301 (2015).[27] B. D. Carlsson, A. Ekstr¨om, C. Forss´en, D. FahlinStr¨omberg, G. R. Jansen, O. Lilja, M. Lindby, B. A.Mattsson, and K. A. Wendt, “Uncertainty analysisand order-by-order optimization of chiral nuclear inter-actions,” Phys. Rev. X , 011019 (2016).[28] D. R. Entem, N. Kaiser, R. Machleidt, andY. Nosyk, “Peripheral nucleon-nucleon scatter-ing at fifth order of chiral perturbation theory,”Phys. Rev. C , 014002 (2015).[29] E. Epelbaum, H. Krebs, and U.-G. Meißner, “Precisionnucleon-nucleon potential at fifth order in the chiral ex-pansion,” Phys. Rev. Lett. , 122301 (2015).[30] A. Gezerlis, I. Tews, E. Epelbaum, M. Freunek, S. Gan-dolfi, K. Hebeler, A. Nogga, and A. Schwenk, “Local chi-ral effective field theory interactions and quantum montecarlo applications,” Phys. Rev. C , 054323 (2014).[31] W. C. Haxton and C.-L. Song, “Morph-ing the shell model into an effective theory,”Phys. Rev. Lett. , 5484–5487 (2000).[32] W. C. Haxton and T. Luu, “Perturba-tive effective theory in an oscillator basis?”Phys. Rev. Lett. , 182503 (2002).[33] W. C. Haxton, “Harmonic-oscillator-based effective the-ory,” in Opportunities with Exotic Beams (World Scien-tific, 2007) Chap. 13, pp. 117–131.[34] W. C. Haxton, “Form of the effective interac-tion in harmonic-oscillator-based effective theory,”Phys. Rev. C , 034005 (2008).[35] I. Stetcu, B.R. Barrett, and U. van Kolck, “No-core shell model in an effective-field-theory framework,”Physics Letters B , 358 – 362 (2007).[36] I. Stetcu, J. Rotureau, B. R. Barrett, andU. van Kolck, “Effective interactions for lightnuclei: an effective (field theory) approach,”Journal of Physics G , 064033 (2010).[37] J. Rotureau, I. Stetcu, B. R. Barrett, and U. van Kolck,“Two and three nucleons in a trap, and the continuumlimit,” Phys. Rev. C , 034003 (2012).[38] J. Rotureau, I. Stetcu, B. R. Barrett, M. C. Birse,and U. van Kolck, “Three and four harmonicallytrapped particles in an effective-field-theory framework,”Phys. Rev. A , 032711 (2010).[39] S. T¨olle, H.-W. Hammer, and B. Ch. Metsch,“Universal few-body physics in a harmonic trap,”Comptes Rendus Physique , 59 – 70 (2011).[40] S. T¨olle, H.-W. Hammer, and B. Ch. Metsch, “Con-vergence properties of the effective theory for trappedbosons,” Journal of Physics G , 055004 (2013).[41] Eric J. Heller and Hashim A. Yamani, “New L approach to quantum scattering: Theory,”Phys. Rev. A , 1201–1208 (1974).[42] A. M. Shirokov, A. I. Mazur, S. A. Zaytsev, J. P. Vary,and T. A. Weber, “Nucleon-nucleon interaction in the j -matrix inverse scattering approach and few-nucleon sys-tems,” Phys. Rev. C , 044005 (2004).[43] S. A. Coon, M. I. Avetian, M. K. G. Kruse, U. van Kolck, P. Maris, and J. P. Vary, “Convergence properties of abinitio calculations of light nuclei in a harmonic oscillatorbasis,” Phys. Rev. C , 054002 (2012).[44] R. J. Furnstahl, G. Hagen, and T. Papenbrock, “Cor-rections to nuclear energies and radii in finite oscillatorspaces,” Phys. Rev. C , 031301 (2012).[45] S. N. More, A. Ekstr¨om, R. J. Furnstahl, G. Ha-gen, and T. Papenbrock, “Universal proper-ties of infrared oscillator basis extrapolations,”Phys. Rev. C , 044326 (2013).[46] R. J. Furnstahl, S. N. More, and T. Papenbrock, “Sys-tematic expansion for infrared oscillator basis extrapola-tions,” Phys. Rev. C , 044301 (2014).[47] S. K¨onig, S. K. Bogner, R. J. Furnstahl, S. N. More,and T. Papenbrock, “Ultraviolet extrapolations in finiteoscillator bases,” Phys. Rev. C , 064007 (2014).[48] R. J. Furnstahl, G. Hagen, T. Papenbrock, andK. A. Wendt, “Infrared extrapolations for atomic nuclei,”Journal of Physics G , 034032 (2015).[49] K. A. Wendt, C. Forss´en, T. Papenbrock, and D. S¨a¨af,“Infrared length scale and extrapolations for the no-coreshell model,” Phys. Rev. C , 061301 (2015).[50] A.M. Shirokov, J.P. Vary, A.I. Mazur, and T.A. We-ber, “Realistic nuclear hamiltonian: Ab exitu approach,”Physics Letters B , 33 – 37 (2007).[51] D. O. Harris, G. G. Engerholm, and W. D.Gwinn, “Calculation of matrix elements forone-dimensional quantum-mechanical problemsand the application to anharmonic oscillators,”J. Chem. Phys. , 1515–1517 (1965).[52] A. S. Dickinson and P. R. Certain, “Calculation of matrixelements for one-dimensional quantum-mechanical prob-lems,” J. Chem. Phys. , 4209–4211 (1968).[53] J. C. Light, I. P. Hamilton, and J. V. Lill, “Generalizeddiscrete variable approximation in quantum mechanics,”J. Chem. Phys. , 1400–1409 (1985).[54] D. Baye and P.-H. Heenen, “Generalisedmeshes for quantum mechanical problems,”J. Physics A: Math. Gen. , 2041 (1986).[55] R. G. Littlejohn, M. Cargo, T. Carrington, K. A.Mitchell, and B. Poirier, “A general frameworkfor discrete variable representation basis sets,”J. Chem. Phys. , 8691–8703 (2002).[56] J. C. Light and T. Carrington, “Discrete-variable rep-resentations and their utilization,” in Adv. Chem. Phys. (John Wiley & Sons, Inc., 2007) pp. 263–310.[57] A. Bulgac and M. McNeil Forbes, “Use of the dis-crete variable representation basis in nuclear physics,”Phys. Rev. C , 051301 (2013).[58] L. S. Gradshteyn and L. M. Ryzhik, Tables of integrals,series, and products , 6th ed. (Academic Press, San Diego,2000).[59] P. Concus, D. Cassatt, G. Jaehnig, andE. Melby, “Tables for the evaluation of R ∞ x β e − x f ( x ) dx by Gauss-Laguerre quadrature,”Math. Comp. , 245–256 (1963).[60] D. R. Entem and R. Machleidt, “Accuratecharge-dependent nucleon-nucleon potential atfourth order of chiral perturbation theory,”Phys. Rev. C , 041001 (2003).[61] Thomas Luu, Martin J. Savage, Achim Schwenk, andJames P. Vary, “Nucleon-nucleon scattering in a har-monic potential,” Phys. Rev. C , 034003 (2010).[62] D. Stump, J. Pumplin, R. Brock, D. Casey, J. Hus- ton, J. Kalk, H. L. Lai, and W. K. Tung,“Uncertainties of predictions from parton distribu-tion functions. i. the lagrange multiplier method,”Phys. Rev. D , 014012 (2001).[63] R. J. Furnstahl, N. Klco, D. R. Phillips, andS. Wesolowski, “Quantifying truncation errors in effec-tive field theory,” Phys. Rev. C , 024005 (2015).[64] J. E. Lynn, I. Tews, J. Carlson, S. Gandolfi, A. Gezerlis,K. E. Schmidt, and A. Schwenk, “Chiral three-nucleoninteractions in light nuclei, neutron- α scattering, andneutron matter,” Phys. Rev. Lett. , 062501 (2016).[65] R. Machleidt, “High-precision, charge-dependent Bonn nucleon-nucleon potential,”Phys. Rev. C , 024001 (2001).[66] H. K¨ummel, K. H. L¨uhrmann, and J. G. Zabolitzky,“Many-fermion theory in expS- (or coupled cluster) form,” Physics Reports , 1 – 63 (1978).[67] I. Shavitt and R. J. Bartlett, Many-body Methods inChemistry and Physics (Cambridge University Press,2009).[68] M. Wang, G. Audi, A.H. Wapstra, F.G. Kondev, M. Mac-Cormick, X. Xu, and B. Pfeiffer, “The AME 2012 atomicmass evaluation,” Chinese Physics C , 1603 (2012).[69] I. Angeli and K.P. Marinova, “Table of experimen-tal nuclear ground state charge radii: An update,”At. Data Nucl. Data Tables , 69 – 95 (2013).[70] K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga,and A. Schwenk, “Improved nuclear matter cal-culations from chiral low-momentum interactions,”Phys. Rev. C83