Coulomb sum rule for 4 He and 16 O from coupled-cluster theory
CCoulomb sum rule for He and O from coupled-cluster theory
J. E. Sobczyk, B. Acharya, S. Bacca,
1, 2 and G. Hagen
3, 4 Institut f¨ur Kernphysik and PRISMA + Cluster of Excellence,Johannes Gutenberg-Universit¨at, 55128 Mainz, Germany Helmholtz-Institut Mainz, Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA
We demonstrate the capability of coupled-cluster theory to compute the Coulomb sum rule forthe He and O nuclei using interactions from chiral effective field theory. We perform severalchecks, including a few-body benchmark for He. We provide an analysis of the center-of-masscontaminations, which we are able to safely remove. We then compare with other theoretical resultsand experimental data available in the literature, obtaining a fair agreement. This is a first andnecessary step towards initiating a program for computing neutrino-nucleus interactions from firstprinciples and supporting the experimental long-baseline neutrino program with a state-of-the-arttheory that can reach medium-mass nuclei.
I. INTRODUCTION
Current neutrino oscillation experiments such as Mini-BooNE [1] and T2K [2] as well as future experiments suchas DUNE [3] and T2HK [4] are entering a precision phase,with an uncertainty goal of the order of a percent. Ac-curate extraction of neutrino oscillation parameters fromthese experiments requires a reliable theoretical treat-ment of the scattering of neutrinos with nuclei that con-stitute the detector material. Presently, the analysis ofdata is systematically affected by crude nuclear physicsmodels. Based on existing studies, nuclear structure un-certainties are estimated to be of the order of ten per-cent [5]. A comprehensive understanding of interactionsof neutrinos with nuclei built on a cutting edge theorywith the capability of reliably estimating uncertainties isurgently needed. Since the energy spectrum of the neu-trinos typically ranges from the MeV to the GeV scale,different mechanisms are at play: quasi–elastic scatter-ing, ∆ and π production, and deep inelastic scattering.The quasi–elastic process, which dominates the scatter-ing at sub-GeV energies, makes a significant contributionto the total cross section and is the dominant process inthe T2HK experiment, in which the neutrino beam peaksat energies of the order of 600 MeV. The quasi–elasticscattering regime is amenable to ab initio treatment ofnuclei in terms of constituent neutrons and protons, withnuclear interaction and electroweak currents grounded inchiral effective field theory ( χ EFT). At present, this ap-proach offers the best opportunity for a direct connec-tion to quantum chromodynamics and a solid uncertaintyquantification.
Ab initio calculations of neutrino interactions onmedium-mass nuclei with modern nuclear interactionshave so far been performed using the Green’s functionMonte Carlo method [6–8] and the self consistent Green’sfunction method [9]. By combining the Lorentz integraltransform (LIT) [10, 11] with coupled–cluster (CC) the-ory [12] and merging them into a method called LIT-CC,we plan to develop a new ab initio approach to com- pute general electroweak responses in medium–mass nu-clei. This paper marks a significant milestone in this di-rection. We build upon the progress made in investigat-ing the electron-scattering reactions of light nuclei withinthe LIT method (see, e.g. , Refs. [13–15]) and the stud-ies of the giant dipole resonances of light and medium-mass nuclei using the LIT-CC approach [16, 17]. Here,we calculate the Coulomb sum rule (CSR) for He and O from coupled-cluster theory. While calculations ofthe CSR for O from coupled-cluster theory already ex-ist in the literature [18], we present a different approachwith a clear goal of extending the LIT-CC formalism toneutrino scattering in the future. We identify, under-stand and tackle a problem in the LIT-CC method whichis relevant for lepton-nucleus scattering: translationallynon-invariant operators induce excitations of the nuclearcenter-of-mass (CoM) wave function which leads to spu-riosities in the calculations of electroweak responses andsum rules. We show that this CoM contamination in theCSR can be removed to a very good approximation byprojecting the final states in the matrix elements out ofthe subspace spanned by states with CoM excitations.The paper is organized as follows. In Sec. II we presentsome general formulas for the Coulomb sum rule. Sec. IIIcontains a brief introduction to coupled-cluster theory.We present several benchmarks in Sec. IV, show an anal-ysis of the spurious center-of-mass states in Sec. V andprovide a comparison with other available calculationsand experimental data in Sec. VI. Finally, in Sec. VII,we present a brief summary and outlook.
II. THE COULOMB SUM RULE
The electromagnetic sum rules have provided stringenttests for theories and experiments on electro- and photo-nuclear scattering since the early years of quantum me-chanics. The CSR, in particular, has played a majorrole in building our understanding of nuclear physics be-yond a simple model of the nucleus as a collection of non-relativistic point-like protons and neutrons bound by a a r X i v : . [ nu c l - t h ] S e p mean field [19]. It continues to guide the developmentof theories for nuclear electroweak processes (see, e.g. ,Ref. [20]) because it allows us to study the response ofthe nucleus to an external probe induced by the chargeoperator with the nuclear excitation spectrum integratedout. Formally, the CSR can be defined as an inelastic sumrule [19] as m in0 ( q ) = ∞ (cid:88) f I |(cid:104) f I | ρ I ( q ) | I (cid:105)| − |(cid:104) I | ρ I ( q ) | I (cid:105)| , = m ( q ) − Z | F el ( q ) | (1)where the first term on the right hand side is the totalsum rule of order 0, itself defined as m ( q ) = ∞ (cid:88) f I |(cid:104) f I | ρ I ( q ) | I (cid:105)| = (cid:104) I | ρ † I ( q ) ρ I ( q ) | I (cid:105) (2)and the term | F el ( q ) | = |(cid:104) I | ρ I ( q ) | I (cid:105)| /Z is the squareof the elastic charge form factor of the nucleus. The state | I (cid:105) is the intrinsic ground state of the nucleus and | f I (cid:105) runs over a complete set of states.If we consider point like nucleons and choose the di-rection of the transfer momentum q along the z -axis, thecharge operator can be written in the intrinsic frame as ρ I ( q ) = Z (cid:88) j =1 e iqz (cid:48) j , (3)where z (cid:48) j = z j − Z CoM are the particle coordinates ( z -components) relative to the CoM. In the lab system, how-ever, the charge operator is ρ ( q ) = Z (cid:88) j =1 e iqz j , (4)where z j are the lab coordinates, as used in coupled-cluster theory. A. CSR as a ground state expectation value
The charge operator written in the lab frame factor-izes into the product of an intrinsic operator and a CoMoperator since ρ ( q ) = Z (cid:88) j =1 e iqz (cid:48) j e iqZ CoM = ρ I ( q ) e iqZ CoM . (5)As a consequence, we have translational invariance for ρ I † ( q ) ρ I ( q ) = ρ † ( q ) ρ ( q ) , (6)at the operatorial level. Furthermore, assuming an ex-act factorization of the ground state wave functions intointrinsic and CoM part as | (cid:105) = | I (cid:105)| CoM (cid:105) , (7) we can formally show that a calculation of m ( q ) in thelab frame directly yields the intrinsic-frame result: (cid:104) | ρ † ( q ) ρ ( q ) | (cid:105) = (cid:104) I | ρ I † ( q ) ρ I ( q ) | I (cid:105)(cid:104) CoM | CoM (cid:105) = (cid:104) I | ρ I † ( q ) ρ I ( q ) | I (cid:105) = m ( q ) . (8)Calculating m ( q ) in this manner as the ground stateexpectation value of the two-body operator ρ † ( q ) ρ ( q ) re-veals that it is related to the Fourier transform of theproton-proton distribution function f pp ( q ) = 1 Z ( Z − (cid:104) I | Z (cid:88) j (cid:54) = k e − iq ( z k − z j ) | I (cid:105) (9)by [19] m ( q ) = Z + Z ( Z − f pp ( q ) . (10)To obtain the Coulomb sum rule from m ( q ) as inEq. (1), we need to subtract the elastic part by calculat-ing the form factor. In coupled-cluster theory, the formfactor is first obtained in the lab frame as | F ( q ) | = 1 Z |(cid:104) | ρ ( q ) | (cid:105)| . (11)Assuming again the separable ansatz of the ground stateas shown in Eq. (7) and using the factorization of thecharge operator from Eq. (5), this becomes | F ( q ) | = 1 Z |(cid:104) I | ρ I ( q ) | I (cid:105)| |(cid:104) CoM | e iqZ CoM | CoM (cid:105)| = | F el ( q ) | |(cid:104) CoM | e iqZ CoM | CoM (cid:105)| . (12)As demonstrated in Refs. [21–23], the wave-functionfactorization in Eq. (7) is valid to a high precisionin coupled-cluster theory. Furthermore, | CoM (cid:105) is theground state of a harmonic oscillator with a fixed fre-quency ˜ ω that can be determined by requiring thatthe coupled-cluster ground state expectation value ofthe CoM Hamiltonian, P / (2 M ) + 1 / M ˜ ω R − / (cid:125) ˜ ω , vanishes. One can therefore easily calculate |(cid:104) CoM | e iqZ CoM | CoM (cid:105)| . | F el ( q ) | thus obtained fromEqs. (11) and (12) is then subtracted from m ( q ) givenby Eq. (8) yielding the Coulomb sum rule m in0 ( q ). Cal-culating the CSR in this manner is denoted as “methodA” in the rest of the paper. We expect this approach tobe free of contaminations from the CoM wave function toa good level of approximation. For He, we will numeri-cally verify this by comparing calculations performed inthe lab frame using coupled-cluster theory with calcula-tions obtained from hyperspherical harmonics working inthe intrinsic frame.
B. CSR as sum over multipoles
The approach of Sec. II A is not applicable in general tothe calculation of sum rules and response functions of thelarge set of electroweak charge and current operators wewill need to include in neutrino scattering calculations.The standard approach while working with angular mo-mentum eigenstates is to perform a multipole decompo-sition of the electroweak operators, see, e.g. , Ref. [24].We therefore expand the charge operator into multipolesas ρ ( q ) = ∞ (cid:88) J =0 [ ρ ( q )] J . (13)In practice, the infinite sum over J needs to be trun-cated at a finite multipolarity. The number of multipolesneeded to achieve convergence depends on the momen-tum q and the size of the nucleus. We obtain m ( q ) as acoherent sum, m ( q ) = ∞ (cid:88) J =0 m J ( q ) , (14)where m J ( q ) = (cid:104) | [ ρ † ( q )] J [ ρ ( q )] J | (cid:105) = (cid:88) f J (cid:104) | [ ρ † ( q )] J | f J (cid:105)(cid:104) f J | [ ρ ( q )] J | (cid:105) (15)is the total strength of each multipole operator. Herewe have inserted the completeness relation in terms ofstates | f J (cid:105) labeled by total angular momentum quantumnumber J , only retained non-vanishing terms, and haverestricted ourselves to the case where the nuclear groundstate | (cid:105) has zero total angular momentum. The elas-tic part can be subtracted by simply restricting the sumover f J to | f J (cid:105) (cid:54) = | (cid:105) in Eq. (15). In this way we re-move the contribution coming from the lab form factor | F ( q ) | , which is not the same as the removal of | F el ( q ) | in method A.At this point it is important to note that, since the op-erator [ ρ ( q )] J is not translationally invariant, the states | f J (cid:105) can contain CoM excitations. Translationally non-invariant operators can generate CoM excitations by act-ing on | CoM (cid:105) even if the ground state | (cid:105) factorizes ex-actly as shown in Eq. (7). We expect such spurious ex-citations to make significant contributions to the sumrule for the lowest multipoles. In Sec. V, we will discusshow we remove these CoM excitations and demonstrateit with a practical numerical implementation. Let us notethat spurious CoM contamination could also be presentwhen using translationally invariant operators such as theelectric dipole [17] if the factorization shown in Eq. (7) isinexact. In the LIT-CC method, they can be removed inthe Lanczos algorithm, as done in Ref. [17]. In this workwe will use a similar technique (described below), andcalculating the CSR in this way will be called “methodB” in the rest of the paper. III. COUPLED-CLUSTER THEORY
In coupled-cluster theory [25–33] one uses the similar-ity transformed Hamiltonian, H N = e − T H N e T , T = T + T + . . . , (16)where H N is normal-ordered with respect to a single-reference state | Φ (cid:105) (usually a Slater determinant ob-tained from a Hartree-Fock calculation), and T is an ex-pansion in particle-hole excitations with respect to thisreference. The operator T is typically truncated at somelow rank particle-hole excitation level.Because the similarity transformed Hamiltonian isnon-Hermitian, one has to compute both the left andright eigenstates in order to evaluate expectation val-ues and transition strengths. The right ground state is | (cid:105) = | Φ (cid:105) while left ground state is obtained as (cid:104) | = (cid:104) Φ | (1 + Λ) , Λ = Λ + Λ + . . . , (17)where Λ is a sum of particle-hole de-excitation operators.In the LIT-CC method, we use the coupled-clusterequation-of-motion (EOM) method [23, 34] together witha non-symmetric Lanczos algorithm to solve for a gener-alized non-Hermitian eigenvalue problem with a sourceterm (see Ref. [35] for details). In practice one dealswith the excited states of the Hamiltonian defined as H N R µ | Φ (cid:105) = E µ R µ | Φ (cid:105) , (cid:104) Φ | L µ H N = E µ (cid:104) Φ | L µ , (18)where R µ and L µ are linear expansions in particle-holeexcitations as well. To compute an electromagnetic tran-sition strength from the ground to an excited state incoupled-cluster theory, one performs the following calcu-lation (see Ref. [35]) |(cid:104) f µ | ˆΘ | (cid:105)| = (cid:104) | ˆΘ † | f µ (cid:105)(cid:104) f µ | ˆΘ | (cid:105) = (19)= (cid:104) Φ | (1 + Λ)Θ † N R µ | Φ (cid:105)(cid:104) Φ | L µ Θ N | Φ (cid:105) , where Θ N ≡ e − T Θ N e T is the similarity transformednormal-ordered operator. In this work the operator istaken to be an electromagnetic Coulomb multipole ofrank J , namely Θ = [ ρ ( q )] J (see also Ref. [24]), and asummation on the multipoles is performed afterwards.The Coulomb multipoles are one-body operators, so thatthe Baker-Campbell-Hausdorff expansion for Θ N termi-nates at doubly nested commutatorsΘ N = Θ N + [Θ N , T ] + 12 [[Θ N , T ] , T ] . (20)Finally, the total multipole strength as calculated inmethod B is [35] m J ( q ) = (cid:104) ν JL | ν JR (cid:105) = (cid:104) Φ | (1 + Λ)Θ † N · Θ N | Φ (cid:105) , (21)where we used the closure relation. Let us label the spu-rious excitations of angular momentum J by s J . Thespurious states as well as the ground state are obtainedfrom the coupled-cluster EOM technique using an itera-tive Arnoldi algorithm. Having obtained the convergedleft (cid:104) s JL | and right | s JR (cid:105) spurious states we project themout of Eq. (21) by multiplying the left (cid:104) ν JL | and right | ν JR (cid:105) states with (cid:0) − | s JR (cid:105)(cid:104) s JL | (cid:1) to obtain¯ m J ( q ) = (cid:104) ν JL | ν JR (cid:105) − (cid:104) ν JL | s JR (cid:105)(cid:104) s JL | ν JR (cid:105) , (22)(see Sec. V below for further details).In both methods A and B, the computation of theCSR is performed in the so-called coupled-cluster withsingles-and-doubles (CCSD) approximation, where allthe particle-hole expansions mentioned above are trun-cated at the two-particle-two-hole level. While we havedeveloped the necessary technology to deal with leading-order triple corrections, in Ref. [35] we found that theyare negligible for non-energy weighted sum rules (as theCSR is), furthermore these excitations increase the com-putational cost by about an order of magnitude. Hence,we neglect them at this point and focus on studying themomentum dependence of our results. IV. BENCHMARKS
We employ Hamiltonians from χ EFT in our calcula-tions. Specifically, we employ two interactions. First, weuse the nucleon-nucleon (
N N ) chiral potential by En-tem and Machleidt [36] (labeled by N LO-EM) withoutsupplementing it with a three-nucleon (3N) interaction.This will serve to perform tests with hyperspherical har-monics expansion [37] in He and to perform an anal-ysis of the center of mass spuriosities. Second, we usethe NNLO sat interaction [38], that includes both
N N and 3 N interactions at next-to-next-to leading order in χ EFT. In this case, the parameters entering the
N N and3 N interactions were adjusted to N N phase shifts andto energies and charge radii of selected nuclei up to O.We use this interaction as it has been shown to work wellfor the nuclei studied in this paper.
Few-body benchmark for He—
We start presentingour results from coupled-cluster theory obtained frommethod A using the N LO-EM
N N interaction. InFig. 1, we show separately the elastic squared form fac-tor | F el ( q ) | and the proton-proton distribution function f pp ( q ), that are related to CSR by Eqs. (1) and (10).The calculations is performed in the CCSD scheme for amodel space of 15 major oscillator shells and an underly-ing harmonic oscillator frequency of (cid:125) ω = 20 MeV. In or-der to remove the CoM contamination from | F el ( q ) | (seeEq. (12)), an underlying CoM frequency (cid:125) ˜ ω = 20 MeVwas employed, which is known from Ref. [21]. We com-pare it to calculations performed with the hypersphericalharmonics method (HH) [39] using the same potential.Such few-body computations have been performed witha model space of K max = 16 in the HH method, wherethe accuracy is expected to be of about a percent. As onecan see, the agreement is very good over a wide range of q [MeV] | F e l ( q ) | ( a ) He HHCCSD q [MeV] f pp ( q ) ( b ) He HHCCSD
FIG. 1. Coupled-cluster calculations in CCSD comparedagainst HH using the N LO-EM interaction. The two sep-arate components making up the CSR, namely the squaredelastic form factor | F el ( q ) | (panel ( a )) and f pp ( q ) (panel ( b ))are shown as functions of the momentum transfer q . momentum transfer q , indicating that for the He andchiral potentials triples and quadrupole excitations miss-ing in the CCSD approximation can safely be neglected.Only for the function f pp ( q ), one can see a deviation ofthe two curves at the largest shown momenta. However,since this quantity is much smaller than 1 at high q val-ues, the observed alignment of the HH and CCSD curvesleads to a nice agreement at the CSR level. This allowsus to infer that the factorization of the wave function intoa CoM and an intrinsic part must hold to a large extent,since a calculation performed in the lab frame agrees withthe one in intrinsic frame. Comparison of two calculational methods—
At thispoint, we proceed with tests of the coupled-cluster com-putations, namely we want to verify if method A andB agree with each other. As mentioned above, methodB may include spurious states that need to be removed.Because identifying all spurious states by running thefull protocol described in Refs. [22, 40] is computation-ally demanding, we adopt a more pragmatic strategy inthis work. For He, we omit from the Lanczos iterations q [MeV] m i n0 ( q ) / Z ( a ) He Method AMethod B with spurious statesMethod B w/o spurious states q [MeV] m i n0 ( q ) / Z ( b ) O Method AMethod B with spurious statesMethod B w/o spurious states
FIG. 2. Comparison of method A and B: for He (panel ( a ))and O (panel ( b )) using the N LO-EM interaction. Formethod B we show the case with and without (w/o) spuriousCoM states. all states with excitation energy below 15 MeV, as theyhave to be spurious given that He has no excited statebelow the proton emission threshold. The omitted stateswere a 1 − state approximately at energy 5 × − MeVabove the ground state and a 2 + state at energy 10 MeVabove the ground state. For O, we remove a 1 − stateat 0 .
15 MeV, which also must be spurious as this nucleusdoes not have any excited states at such low energy.In Fig. 2 we show a comparison between method Aand method B for He and O using the N LO-EM in-teraction. For method B we show the curve obtained incase we do not remove the spurious states in the Lanc-zos procedure as well as the one where we remove suchstates. Calculations are performed for a model space of15 oscillator shells and a harmonic oscillator frequencyof (cid:125) ω = 20 MeV for all the three curves. In case of Owe show a band obtained by the difference between themodel spaces with 15 and 13 major shells. This gives usan idea of the uncertainty in a model space size that willbe achievable in other calculations where we will add 3 N interactions.First of all, we see that the removal of the spuriosi- ties has a much larger effect in He than in O as ex-pected, since the heavier the nucleus, the smaller theCoM contamination must be. Second, we see that the re-sults of method B w/o spurious states agree quite nicelywith those of method A for both nuclei. The removal of q [MeV] m i n0 ( q ) / Z ( a ) J = 0+1+2+3+ 8 He q [MeV] m i n0 ( q ) / Z ( b ) J = 0+1+2+3+4+ 10 O FIG. 3. Cumulative sums of the multipole expansion for theCoulomb sum rule of He (panel ( a )) and O (panel ( b ))using the N LO-EM interaction in method B w/o spuriousstates. spuriosities brings the agreement at q = 100 MeV from50%(38%) to 0 . . He( O), while at q = 200MeV, it brings the agreement from 35%(20%) to 3%(8%).We consider this comparison satisfactory and from nowon we will use method B only w/o the spurious states toperform further analysis. Convergence of multipole expansion—
A natural ques-tion in method B is how many multipoles are needed inthe expansion of Eq. (3). We show this for the N LO-EM
N N interaction in Fig. 3. These calculations areperformed for the same model space as above. Clearly,the convergence in terms of multipoles strongly dependson the momentum transfer, as expected. Furthermore,we see that the convergence is faster for the smaller Henucleus than for the larger O nucleus. For q below 200MeV only four multipoles are needed, while for q = 500MeV nine and eleven multipoles are needed to reach sat-isfactory convergence for He and O, respectively. Theinformation on the number of needed multipoles will berelevant to estimate the amount of computing time re-quested to compute response functions with the LIT-CCmethod. Otherwise, the CSR can be most efficiently com-puted with method A instead.
V. ANALYSIS OF SPURIOUS STATES
In method B we find that by far the largest spuriouscontribution that we remove, comes from the 1 − state.In order to understand the nature of the spurious stateswe removed, let us assume for the moment that the fac-torization of | (cid:105) into | I (cid:105)| CoM (cid:105) is exact, with | CoM (cid:105) being the ground state of a quantum harmonic oscilla-tor. The non-spurious excitations are those that havethe form | f J I (cid:105)| CoM (cid:105) , where | f J I (cid:105) are excitations of theintrinsic wave function, labeled by total angular momen-tum quantum number J . The simplest possible form ofthe spurious states is then | I (cid:105)| f J CoM (cid:105) , where | f J CoM (cid:105) areexcitations of the CoM wave function. The completenessrelation used in Eq. (15) then separates into the projec-tors onto the coupled-cluster ground state, the subspacespanned by non-spurious excitations and the subspacespanned by spurious excitations, as (cid:88) f J | f J (cid:105)(cid:104) f J | (cid:39) | I (cid:105)| CoM (cid:105)(cid:104) CoM |(cid:104) I | + (cid:88) f J I | f J I (cid:105)| CoM (cid:105)(cid:104) CoM |(cid:104) f J I | + (cid:88) f J CoM | I (cid:105)| f J CoM (cid:105)(cid:104) f J CoM |(cid:104) I | . (23)Inserting this relation in Eq. (15), we obtain m J ( q ) = Z | F ( q ) | δ J + [ m in0 ( q )] J + [ m sp0 ( q )] J , (24)where the first term on the right hand side is the squaredelastic form factor in the lab frame and only contributesto the monopole term, the second term is the multipolestrength of the CSR and the third term is the contribu-tion of the spurious states to m J ( q ).The Coulomb sum rule, m in0 ( q ) = ∞ (cid:88) J =0 [ m in0 ( q )] J , (25)which is calculated in practice by adding a finite numberof multipoles until convergence is achieved, is then free ofcontributions from CoM excitations. The spurious statecontribution [ m sp0 ( q )] J is given by the squared matrixelement[ m sp0 ( q )] J = |(cid:104) I |(cid:104) f J CoM | [ ρ ] J | (cid:105)| = |(cid:104) I |(cid:104) f J CoM | (cid:2) [ ρ (cid:48) ] J m ⊗ [ e iqZ CoM ] J m (cid:3) J | I (cid:105)| CoM (cid:105)| = | (cid:88) m C J J (cid:104) I | [ ρ (cid:48) ] | I (cid:105)(cid:104) f J CoM | [ e iqZ CoM ] J | CoM (cid:105)| = Z | F el ( q ) | K J CoM ( q ) , (26) where C JmJ m J m are Clebsch-Gordan coefficients and K J CoM ( q ) = |(cid:104) f J CoM | [ e iqZ CoM ] J | CoM (cid:105)| (27)depend only on the CoM.It was shown in Ref. [21] that the CoM part of thecoupled-cluster ground state, | CoM (cid:105) , is well approxi-mated by the ground state of a three-dimensional har-monic oscillator Hamiltonian with energy eigenvalues E = (cid:125) ˜ ω ( N + ), where the number of oscillator quanta N is related to the radial quantum number N r and the angu-lar momentum quantum number J by N = 2 N r + J with J ≤ N . The oscillator length parameter is b = (cid:113) (cid:125) M ˜ ω ,where M is the mass of the nucleus.Here, we make the ansatz that the CoM part of thespurious state, | f J CoM (cid:105) , is given by the excited oscillatorstate | Ψ CoM0 J (cid:105) . The CoM functions K J CoM ( q ) can then bewritten as K J CoM ( q ) = |(cid:104) Ψ CoM0 J | [ e iqZ CoM ] J | Ψ CoM000 (cid:105)| , (28)and calculated using co-ordinate representations of theoscillator states, (cid:104) R CoM | Ψ CoM N r JM (cid:105) = (cid:115) N r ! b Γ ( N r + J + 3 / (cid:18) R CoM b (cid:19) J × exp (cid:18) − R b (cid:19) L J + N r (cid:18) R b (cid:19) Y JM ( ˆ R CoM ) , (29)where L (cid:96)n ( x ) are associated Laguerre polynomials whichsatisfy L (cid:96) ( x ) = 1 for all values of (cid:96) and x . This yields K J CoM ( q ) = √ π (2 J + 1)2Γ( J + 3 / (cid:18) bq (cid:19) J exp (cid:18) − b q (cid:19) . (30)A closed-form expression for K J CoM ( q ) can also be ob-tained for a more general ansatz, | f J CoM (cid:105) = | Ψ CoM
NrJ (cid:105) ,which allows the spurious states to contain radial excita-tions with N r >
0. This derivation, along with a detailednumerical analysis of such excitations, will be communi-cated through a separate publication [41]. However, wedo find that radial excitations are negligible in the fol-lowing analysis, hence we keep our ansatz as simple aspossible.Let ¯ m J ( q ) be the multipole strength calculated by re-moving a spurious state of angular momentum J (andthe ground state for J = 0) in the Lanczos procedure,defined in Eq. (22). The difference m J ( q ) − ¯ m J ( q ) canthen be obtained in the LIT-CC method which allows usto numerically check whether m J ( q ) − ¯ m J ( q ) = Z | F el ( q ) | K J CoM ( q ) , (31)holds for the expression for K J CoM ( q ) given by Eq. (30).We thus validate all our assumptions about the nature ofthe CoM spuriosity.It is important to realize that summing Eq. (31) over J we obtain m ( q ) − ¯ m ( q ) = Z | F el ( q ) | ∞ (cid:88) J =0 K J CoM ( q ) ≈ Z | F el ( q ) | . (32)On the left hand side we have the sum of all the spuriousstates (with the ground state in the lab frame included)obtained numerically. It should correspond to the elasticform factor squared, so one can see here a direct connec-tion with Eq. (1). q [MeV] ( a ) He [ m ( q ) m ( q )]/ ZZ | F el ( q )| K ( q ); = 20MeV q [MeV] ( b ) O [ m ( q ) m ( q )]/ ZZ | F el ( q )| K ( q ); = 16MeV FIG. 4. Spurious states 1 − for He (panel ( a )) and O (panel( b )) compared with an ansatz for the CoM function fromEq. (30). The band for He is obtained with (cid:125) ˜ ω = 20 ± O the band corresponds to (cid:125) ˜ ω = 16 ± Let us note that for J = 0, we have that K ( q ) = |(cid:104) CoM | e iqZ CoM | CoM (cid:105)| , and so in Eq. (31) on the righthand side we obtain Z | F ( q ) | which stays in agreementwith Eq. (12). This way we show that Eq. (31) holds for J = 0.In Fig. 4, we plot the contributions of the 1 − spuriousstates to the sum rules for He and O, given numerically This would be exactly equal to Z | F el ( q ) | if radial excitationswere included [41]. in the LIT-CC theory by the difference m ( q ) − ¯ m ( q ),along with their K ( q ) functions given by Eq. (30)and obtain an excellent agreement. For the employed in-teraction N LO-EM, it was shown in Ref. [21] that thefrequency (cid:125) ˜ ω of | C oM (cid:105) is close to 19 MeV for He and16 MeV for O in the Λ − CCSD(T) approximation, re-spectively. We therefore varied the frequencies (cid:125) ˜ ω in arange of ± | f J CoM (cid:105) . We would liketo emphasize here that the frequency (cid:125) ˜ ω = 20 MeV for He coincides with the value employed to obtain F el ( q ),see Fig. 1. Similarly, the best agreement is obtained with (cid:125) ˜ ω = 16 MeV for O. VI. COMPARISON WITH OTHERCALCULATIONS AND EXPERIMENT
Finally, we compare the total CSR calculated fromcoupled-cluster theory with other available calculationsand experimental data. At this stage we include a smalleffect coming from the nucleon form-factor by rewritingEq. (4) ρ ( q ) = A (cid:88) j =1 G pE ( Q ) 1 + τ e iqz j + G nE ( Q ) 1 − τ e iqz j , (33)where (1 ± τ ) / Q ≈ q − ω QE with the energy transfer ω QE = q / m corresponding to the quasi-elastic peak,where m is the mass of the nucleon. We use Kelly’s pa-rameterization of G n,pE form factors [45], and also accountfor the Darwin-Foldy relativistic correction by includinga factor of 1 / [1 + Q / (4 m )], while neglecting the smallerspin-orbit contribution.In Fig. 5, we present the CSR of He and O for theN LO-EM and N LO sat interactions, respectively. TheN LO sat interaction accurately reproduces the bindingenergy and charge radius of O, and we therefore focuson this interaction when comparing with experiment andwith other theoretical approaches that use chiral
N N and3 N forces. We noticed a slower convergence with respectto the number of oscillator shells in the N LO sat calcula-tions than in the N LO-EM calculations at large values ofmomentum transfer. The N LO sat results shown in Fig. 5is for (cid:125) ω = 20 MeV and 15 major oscillator shells. By ex-trapolating the observed convergence pattern at smallermodel spaces, we expect the calculations to be convergedat q <
400 MeV. At larger values of q , the basis trunca-tion error in CSR for the N LO sat interaction is expectedto be up to a few percentage.For He we compare our CCSD calculations againstthe HH results of Ref. [15] obtained with the AV18+UIXpotential and the Green’s function Monte Carlo (GFMC) q [MeV] m i n0 ( q ) / Z ( a ) He HH; AV18 + UIXGFMC; AV18 + IL7CCSD; N LO EMCCSD; N LOsatExpt with theory tail q [MeV] m i n0 ( q ) / Z ( b ) O Mihaila et al. ; AV18 + UIXAFDMC; N LO; 1.2 fmAFDMC; N LO; 1.0 fmCCSD; N LO EMCCSD; N LO sat FIG. 5. Panel ( a ): CSR for He in comparison to other cal-culations of Ref. [15] (AV18+UIX potential) and Refs. [20](AV18+IL7 potential) together with experimental data fromRefs. [42, 43]. Panel ( b ): CSR for O in comparison to othercalculations by Mihaila et al. [28] (AV18+UIX potential) andby Lonardoni et al. [44] (with local chiral interactions, seetext for details). results of Refs. [20] obtained with the AV18+IL7 po-tential. Although Ref. [20] also included the two-bodycharge operator, it contributes noticeably to the CSR of He only beyond the range of momenta considered inthis study [46]. We see that overall the curves providea consistent trend and we note that the dependence onthe implemented Hamiltonian can be interpreted as anoverall theoretical uncertainty.Regarding He, measurements for the longitudinal re-sponse functions at intermediate momentum transferhave been performed in the past and are collected inRef. [42], while low momentum data at q = 200 and250 MeV/c are taken from Buki et. al. [43]. Since finitemaximal values of the energy transfer ω max are measuredin experimental data, the experimental CSR is obtainedas [47] m i n ( q ) = 1 Z (cid:90) ω max ω + th dω R L ( ω, q ) G P E ( Q ) + m i n , tail , (34) where m i n , tail is taken from the theoretical calculationsof He response functions of Ref. [13]. We notice that theexperimental trend is well reproduce by all calculationsof the CSR.Our results for the CSR of O are compared withother theoretical calculations in panel ( b ) of Fig. 5. Theauxiliary field diffusion Monte Carlo (AFDMC) calcu-lations [44] used local chiral interactions up to next-to-next-to-leading order (N LO) which were regularizedin co-ordinate space with two different cutoff values, R = 1 . q ∼
500 MeV region andbring it closer to ours. Further analyses that accountfor uncertainties due to the interactions and the many-body methods employed are necessary to assess whetherthe CSR obtained using different theoretical approachesare consistent with each other within those uncertain-ties. Precise experimental data, especially in case of Ofor which no data exists, can constrain Hamiltonians andmany-body methods used in theoretical calculations, andwould therefore be very useful.
VII. CONCLUSIONS
In this paper, we compute the CSR from coupled-cluster theory for He and O using χ EFT interactions.For the first time, we investigate higher momentum trans-fers regime with these potentials. Through a benchmarkwith few-body methods, we show that the coupled-clusterwave functions retain the relevant correlations across abroad range of momentum transfer, even in the CCSDscheme. This is a very promising result because accu-rate treatment of nuclear correlations in the nuclear elec-troweak currents in this range of momentum transfers isimportant for improving our understanding of neutrinoscattering.Furthermore, we devise a practical method to removethe spurious states when working in the lab frame andwe show that the method works within a precision of afew percent. This is encouraging when moving to heav-ier nuclei of interest to neutrino experiments like Arwhere center of mass effects are expected to be smaller.We provide an analysis of the dominant spuriosities weremove, and find that these are excited states of a har-monic oscillator with zero radial excitation and angularexcitation equal to J = 1 . Finally, we compare our results obtained with theN LO-EM and NNLO sat potentials with other theoreti-cal results available in the literature. In the case of He,we compare also to experimental data, which we obtainedfrom integrating the longitudinal response function pub-lished in Refs. [42, 43], and found a nice agreement. Inthe case of O no data exist for the CSR, and we findsome sensitivity to the employed Hamiltonian.This computation of the CSR in coupled-cluster theoryconstitutes, indeed, a first important step towards apply-ing this method to neutrino physics in a broader researchprogram.
ACKNOWLEDGMENTS
We would like to thank Nir Barnea, Thomas Papen-brock, and Johannes Simonis for useful discussions. Thiswork was supported by the Deutsche Forschungsgemein-schaft (DFG) through the Collaborative Research Center[The Low-Energy Frontier of the Standard Model (SFB1044)], and through the Cluster of Excellence “Preci-sion Physics, Fundamental Interactions, and Structure of Matter” (PRISMA + EXC 2118/1) funded by theDFG within the German Excellence Strategy (ProjectID 39083149), by the Office of Nuclear Physics, U.S.Department of Energy, under grants desc0018223 (NU-CLEI SciDAC-4 collaboration) and by the Field WorkProposal ERKBP72 at Oak Ridge National Laboratory(ORNL). Computer time was provided by the Innovativeand Novel Computational Impact on Theory and Exper-iment (INCITE) program. This research used resourcesof the Oak Ridge Leadership Computing Facility locatedat ORNL, which is supported by the Office of Scienceof the Department of Energy under Contract No. DE-AC05-00OR22725. 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