Inclusive electron-nucleus cross section within the Self Consistent
Green’s Function approach
We compute inclusive electron-nucleus cross sections using ab initio spectral functions of He and O obtained within the Self Consistent Green’s Function approach. The formalism adopted is based on the factorization of the spectral function and the nuclear transition matrix elements. This allows to provide an accurate description of nuclear dynamics and to account for relativistic effects in the interaction vertex. Our calculations use a saturating chiral Hamiltonian in order reproduce the correct nuclear sizes. When final state interactions for the struck particle are accounted for, we find nice agreement between the data and the theory for the inclusive electron-O cross section. The results lay the foundations for future applications of the Self Consistent Green’s Function method, in both closed and open shell nuclei, to neutrino data analysis.
This work also presents results for the point-proton, charge and single-nucleon momentum distribution of the same two nuclei. The center of mass can affect these quantities for light nuclei and cannot be separated cleanly in most ab initio post-Hartree-Fock methods. In order to address this, we developed a Metropolis Monte Carlo calculation in which the center of mass coordinate can be subtracted exactly from the trial wave function and the expectation values. We gauged this effect for He by removing the center of mass effect from the Optimal Reference State wave function that is generated during the Self Consistent Green’s Function calculations. Our findings clearly indicate that the residual center of mass contribution strongly modifies calculated matter distributions with respect to those obtained in the intrinsic frame. Hence, its subtraction is crucial for a correct description of light nuclei.
The current- and next-generation of neutrino oscillation experiments require nuclear physics calculations of the structure and electroweak properties of atomic nuclei supplemented by quantified theoretical uncertainties Gran:2006jn; AguilarArevalo:2007ab; Lyubushkin:2008pe. The Deep Underground Neutrino Experiment (DUNE) will exploit Liquid Argon Time-Projection Chambers (TPCs) to test CP violation in the lepton sector and to shed light on the neutrino mass hierarchy. Hence, nuclear theories able to tackle genuine open shell nuclei, such as Argon, will be critical to the reconstruction of the initial neutrino energy.
The Self Consistent Green’s Function (SCGF) approach is an ab initio method in which the optical potential and spectral functions are calculated covering the full spectra of both nucleon attachment and removal (i.e., both close and far form the Fermi surface) Dickhoff:2004xx; Barbieri2003DRPA; Barbieri2006plbO16; Barbieri:2007zy; Barbieri:2009nx; Cipollone:2013zma. The self-consistency feature means that the input information about the ground state and excitations of the systems no longer depends on a user-defined reference state but instead it is taken directly from the computed correlated propagator. The SCGF method has recently been reformulated within Gorkov’s theory that allows to address open shell nuclei Soma:2011aj; Soma:2012zd; Soma:2013ona. Within this approach, the description of pairing correlations characterizing open shell systems is achieved by breaking particle number symmetry. The method was extended to include three-body interactions in Ref. Carbone:2013eqa; Carbone2014corr3nf. Modern two- and three-nucleon chiral forces can be fully exploited within this formalism. Because of these features, SCGF theory is a prime tool for providing the nuclear structure input necessary to calculate electroweak properties of nuclei. However, its performance in predicting lepton-nucleus reactions with chiral nuclear forces is still to be assessed.
In this work we use the saturating next-to next-to leading order (NNLO) interaction denoted as NNLO Ekstrom:2015rta and calculate the SCGF spectral functions of He and O. We obtain their point density, charge density and single-momentum distribution. All calculations are performed expanding on an harmonic oscillator basis and the dependence of the results on the oscillator parameters is investigated. For light nuclei, such as He, spurious contributions of the center of mass in the calculated wave functions can be sizable in the model spaces exploited by SCGF and other post-Hartree-Fock methods. While it is possible to show that the center of mass effectively decouples from the relative motion for large enough spaces Hagen2009PRLcom, subtracting its effect from the calculated wave function and spectral functions is a nontrivial long standing issue. Here, we address this problem by performing a Monte Carlo integration in which the center of mass component is exactly subtracted from the wave function.
In the high momentum transfer region of neutrino-nucleus scattering, the formalism based on spectral function and factorization of the nuclear transition matrix elements allows to combine a fully relativistic description of the single-nucleon interaction vertex with an accurate treatment of nuclear dynamics Benhar:2005dj; Benhar:2006wy. In order to apply any theoretical model in the neutrino data analysis, it is fundamental to validate it against the large body of electron scattering data. This work has to be considered as a first step in this direction. In fact, we present an extensive comparison with the experimental data of the electromagnetic double differential cross sections of He and O scattering, which was obtained by exploiting the corresponding SCGF spectral functions. The predictions for O are important for the data analysis of Super-Kamiokande, in which a water Cherenkov detector to study neutrinos produced from different sources is used.
In Section II we briefly review the SCGF formalism and the links of propagators with the one-body density distribution and other quantities of experimental interest. Section III is devoted to the derivation of the electron-nucleus cross section within the Impulse Approximation (IA) in which the factorization of the nuclear transition matrix elements is assumed. Final state interactions (FSI) involving the struck particle are treated as corrections. They are included using the convolution approach of Refs. Ankowski:2014yfa; Benhar:1991af. In Section IV we present results for the point density, charge density and single-momentum distribution of He and O. In addition, the inclusive electromagnetic cross sections of these two nuclei, obtained using the associated SCGF spectral functions, are compared with the experimental data and the role played by FSI is discussed. Conclusions are drawn in Section V.
Ii The Self Consistent Green’s Function Approach
The one-body Green’s Function is written as a sum of two different contributions describing the propagation of a particle and hole state Dickhoff2008book:
where and are the creation and annihilation operator in the quantum state , respectively. The so-called Lehmann representation results from inserting completeness relations in Eq. (1). This is
where () are the eigenstates and () the eigenvalues of the -body system. Introducing the transition amplitudes
and the corresponding quasiparticle energies
leads to the more compact expression
where is the unperturbed single-particle propagator and is the irreducible self-energy that encodes nuclear medium effects in the particle propagator Dickhoff2008book. The latter is given by the sum of two different terms
the first one describes the average mean field while the second one contains dynamic correlations. In practical calculations the self-energy is expanded as a function of the propagator itself, implying that an iterative procedure is required to solve the Dyson equation self-consistently. The self-energy can be calculated systematically within the Algebraic Diagrammatic Construction (ADC) method. The third order truncation of this scheme [ADC(3)] yields a propagator that includes all possible Feynman contributions up to third order but it further resums infinite series of relevant diagrams in a non-perturbative fashion Schirmer1983; Barbieri2017LNP. Two- and three-nucleon force contributions are included. A first organization of the contributions to the self-energy comes by considering the particle irreducible (PI) and skeleton diagrams. The number of Feynman diagrams entering the calculation of the Green’s Function rapidly increase when three- or many-body forces are accounted for. In order to circumvent this problem and reduce the number of Feynman diagrams to be considered, a useful strategy is to include only interaction-irreducible diagrams Carbone:2013eqa. For our calculations, we use the following medium dependent or effective one- and two-body interactions:
where and, label the matrix elements of the one-, two-, and three-body interactions, respectively. The one-body density matrix appearing in Eq. (8) reads
The use of this averaging procedure allows to retain only interaction irreducible diagrams in the effective interactions and ,
while residual contributions that include can be safely neglected Hagen:2007ew; Roth:2011vt; Barbieri2014QMBT; Cipollone:2014hfa.
The expressions of the static and dynamic self-energy up to third order, including all possible two- and three-nucleon terms that enter the expansion of the self-energy,
as well as interaction-irreducible (i.e. not averaged) three-nucleon diagrams have been recently derived in Ref. Raimondi:2017kzi.
Figure 1 displays the three simplest diagrams that enter the present calculation of the self-energy. These are taken as “seeds” for an all order resummation that eventually generates . The first contribution is at second-order while the last two are of third-order in the expansion of Eq. (7). Note that, for all the considered diagrams, the set of intermediate state configurations corresponds to two-particles–one-hole (2p1h) and two-holes–one-particle (2h1p) and that we use the two-nucleon effective interaction of Eq. (8) . Within the ADC(3) approach an infinite order summation of diagrams of Fig. 1 that includes particle-particle and hole-hole ladders as well as particle-hole rings is performed. The dynamical part of the self-energy of Eq.(7) can be rewritten in the Lehmann representation as
where are the unperturbed 2p1h and 2h1p energies, coupling matrices and C interaction matrices for the forward and backward intermediate states.
The calculations presented in this work have been performed expanding one-, two- and three-body operators on a spherical harmonic oscillator basis whose dimension and the oscillation frequency are denoted by and , respectively.
The point-proton density distribution can be readily obtained from Eq.(9) and reads
where denotes the harmonic oscillator singe-particle wave-functions and the sum includes only proton single-particle states. Analogous expressions can be written for point-neutron and matter density distributions.
The computational cost required to account for the fragmentation of the single-particle propagator into Eq. (10) rapidly increases with the size of the nucleus and of the model space. For this reason an Optimized Reference State (OpRS) approach is used to approximate the single particle propagators entering in the diagrams of Fig. 1 Barbieri:2009nx. The OpRS is taken to be an independent particle model propagator as
where represents the set of occupied states, and are the single particle energies and wave functions, respectively. The OpRS propagator is chosen to best approximate the correlated one while keeping a reduced number of poles. This is achieved by introducing the following momenta of the spectral distribution with respect to energy poles:
where and the transition amplitudes are given in Eqs. (3).
The quantities in Eq. (13) are important since they constrain the density distributions, one-body observables and the Koltun energy sum rule of the propagator Rios2017SumRules.
Hence, we obtain and by requiring that the OpRS lowest momenta of the spectral distribution reproduce those of the full calculation, i.e.
The elastic scattering of a nucleus hit by a probe and recoiling with a momentum is described by the elastic form factor . Neglecting the small spin-orbit contribution, the latter is given by
where in the laboratory frame , is the energy transfer corresponding to the elastic scattering, is the target nucleus mass and
The elastic form factor can be rewritten in terms of the Fourier transforms of the point-proton and nucleon densities as
Note that the factor in the denominator of Eq. (16) accounts for relativistic corrections to the charge operator. Assuming that , the charge distribution can be written as
The probability of finding a nucleon in the nucleus with momentum is proportional to its momentum distribution. The latter can be written in terms of the one-body density matrix of Eq. (9) as
where is the Fourier transform of the harmonic oscillator wave function
The momentum distribution is normalized as , with being the number of either protons , or neutrons .
The subtraction of the center of mass contribution from the wave function is a long standing issue affecting a number of nuclear many-body approaches relying on a single-nucleon basis expansions. Whilst for medium and heavy nuclei this correction can be safely neglected, the center of mass contribution strongly affects the results of light nuclei, such as He. In order to address this problem, we developed a Metropolis Monte Carlo (MMC) code, analogous to the one used in Variational Monte Carlo studies Carlson:2014vla, that allows us to single out the center of mass contribution to the wave function in the calculation of the charge density and the momentum distribution. The wave function we used for the MMC is the Slater determinant obtained from the OpRS calculation, . At variance with the fully correlated propagator of Eq. (5), the use of the unperturbed , (i.e. a Slater determinant) allows for a unique definition of the wave function. The spatial integrals of Eqs. (17), (19), and (20) have been performed using Metropolis Monte Carlo techniques Metropolis:1953am. A sequence of points in the 3A-dimensional space denoted by are generated by sampling from the probability distribution
At each step of the calculation the center of mass contribution to the wave function is subtracted computing the wave function and the expectation value in the intrinsic coordinates given by
Hence, the identification of the intrinsic contribution is easily achieved within Quantum Monte Carlo (QMC) algorithms Schmidt:1999lik; Pieper:1990ji; Brida:2011yp; Pastore:2012rp, since we always have access to the set of 3A-coordinates of the constituent nucleons.
Iii The Impulse Approximation and convolution approach
In the one-photon-exchange approximation, the double differential electron-nucleus cross section takes the form
where and are the laboratory four-momenta of the incoming and outgoing electrons, respectively; is the fine structure constant, , the differential solid angle in the direction of , and the four momentum transfer. The leptonic tensor is given by
The hadronic tensor is written in terms of matrix elements of the nuclear current operator between the target ground state and the hadronic final states as
where and denote the initial and final hadronic states with four-momenta and , while is the electromagnetic nuclear current operator.
At relatively large momentum transfer, MeV, the Impulse Approximation (IA) can be safely applied. Within this approximation the interaction between the struck nucleon and the spectator (A-1) particles is neglected Benhar:2006wy; Benhar:2015wva. The nuclear current operator reduces to a sum of one-body terms, and the hadronic final state factorizes as
In the above equation denotes the final-state nucleon with momentum , while describes the -body spectator system. Its energy and recoiling momentum are fixed by energy and momentum conservation, yielding
Using the factorization ansatz and inserting a single-nucleon completeness relation, the matrix element of the current operator can be written as
Substituting the last equation in Eq. (III), the contribution to the hadron tensor is given by
Momentum conservation in the single-nucleon vertex implies . The one-body current operator can be written as
and . The latter are defined in terms of the electric and magnetic form factors via
where . Finally, using the identity
we can rewrite the hadron tensor as
where the factors and have to be included to account for the implicit covariant
normalization of the four-spinors of the initial and final nucleons in the matrix elements of the relativistic current.
The hole spectral function
gives the probability distribution of removing a nucleon with momentum from the target nucleus,
leaving the residual system with an excitation energy .
Note that in Eq. (34) we neglected Coulomb interactions and the other (small) isospin-breaking terms and made the assumption, largely justified in the case of closed shell nuclei,
that the proton and neutron spectral functions are identical.
Rewriting the nuclear matrix element as
we recover the more familiar expression of the spectral function written as the imaginary part of the Green’s function describing the propagation of a hole state
In the kinematical region in which the interactions between the struck particle and the spectator system can not be neglected, the IA results have to be modified to include the effect of FSI. Following Refs. Ankowski:2014yfa; Benhar:1991af, the multiple scatterings that the struck particle undergoes during its propagation through the nuclear medium are taken into account through a convolution scheme. The IA responses are folded with a function , normalized as
The double differential cross section is then given by
In the last equation we modified the energy spectrum of the struck nucleon
by considering the real part of the optical potential derived from the Dirac phenomenological fit of Ref. Cooper:1993nx. This allows to describe the propagation of the knocked-out particle in the mean-field generated by the spectator system.
Our calculations have been performed using the NNLO chiral interaction Ekstrom:2015rta, which was specifically designed to accurately describe both binding energies and nuclear radii of mid-mass nuclei Ruiz2015Nature; Lapoux:2016exf. In Fig. 2 we analyze the convergence of the SCGF-ADC(3) point-proton densities of He with respect to the oscillator frequency () and the size of the model space (). The different lines almost superimpose, indicating that for 20 MeV and 11 the calculation converges and no longer depends on the oscillator parameters. The density calculated from the OpRS is also displayed. The nice agreement with the SCGF-ADC(3) curves follows from the requirement that the single particle energies and overlap functions in the OpRS propagator are chosen to approximate at best the true (correlated) one-body density.
The charge densities in He can be obtained from the point-proton densities through Eqs. (17) and (18). In Fig. 3 we compare the experimental charge density determined through the “Sum-of-Gaussians” parametrization given in Ref. DeJager:1987qc with those obtained from the QMC results of Ref. Lonardoni:2017egu and from the OpRS calculated in the present work. For the latter, we display both the result already shown in Fig. 2 and the distribution obtained after subtracting the center of mass effect with the MMC algorithm outlined in Sec. II. When the center of mass contamination is subtracted, we obtain the short-dashed (black) line. The comparison with the total OpRS results, corresponding to the dot-dashed (blue) line, clearly shows that for He the center of mass contribution is sizeable and can not be neglected. The use of the intrinsic wave function yields a strong enhancement of the charge density, which turns out to be very close to the QMC result. Note that the discrepancy between the experiment and the intrinsic OpRS and QMC calculations is motivated by the absence of the two-body meson exchange current contributions. These are known to have little effect on larger nuclei such as O but their inclusion is fundamental in order to correctly reproduce the He elastic form factor, from which the charge densities are extracted Lonardoni:2017egu; Carlson:2014vla; Marcucci:2015rca; Lonardoni:2018nob.
For medium-mass nuclei, the center of mass corrections are known to be less significant. Therefore, in Fig. 4 we compare the experimental charge density in O with the full SCGF-ADC(3) and the QMC calculations. There is an overall nice agreement between the theoretical curves. The SCGF-ADC(3) results perfectly reproduce the experimental points, confirming the goodness of the potential which was fitted to reproduce the experimental radius of O.
The different ADC(3) curves in Fig. 5 correspond to the single-nucleon momentum distribution of He obtained by varying the oscillator frequency and the size of the model space. As it has been already observed in Fig. 2, for these values of and the calculation is independent of the oscillator parameters. The OpRS result, corresponding to the dashed (blue) line, correctly follows that of the dressed ADC(3) propagator, although the agreement is not as close as in Fig. 2. The distribution obtained from the Hartree-Fock state are much different and they are also displayed to further emphasize the importance of nucleon-nucleon correlations. It is clearly visible that their correct inclusion leads to a strong reduction of the momentum distribution in the region of low momentum and to the appearance of a tail for values of well above the Fermi momentum.
In Fig. 6 we benchmark the intrinsic and uncorrected OpRS single-nucleon momentum distribution of He with the QMC calculation of Ref. Lonardoni:2017egu. Note that, also in this case the subtraction of the center of mass component has a sizable effect, which is crucial for recovering the agreement with the intrinsic QMC results.
The O single-nucleon momentum distributions obtained within the SCGF-ADC(3) and QMC approach are compared in Fig. 7. In the upper panel the dashed (red) line, corresponding to the QMC calculation, is found to be below the SCGF-ADC(3) results for low momenta. This is likely to be ascribed to the different choice made for the potentials. In fact, the NNLO is much softer than the AV18+UIX potential adopted in the QMC study. The use of an hard potential implies the presence of stronger high momentum components in the nuclear wave function. This becomes evident in the lower panel where the logarithmic scale has been used to show the differences in the tails. While the QMC momentum distribution exhibits a long tail extending to GeV, the use of an harmonic oscillator model space and the softer potential adopted in our calculations strongly reduce the SCGF-ADC(3) momentum distribution in the high momentum region.
Fig. 8 shows the electron-He inclusive double-differential cross sections at different values of and . The curves are obtained from the full SCGF-ADC(3) spectral function, from its OpRS approximation and from the intrinsic OpRS. The SCGF-ADC(3) cross-section represented by the dashed (red) line is quenched with respect to the solid (green) line that refers to the uncorrected OpRS. This has to be attributed to the different behavior of the curves displayed in Fig. 5. Whilst the OpRS wave functions are built to reproduce lowest energy momenta of the ADC(3) propagator—which optimises the quasiparticle energies and strength near the Fermi surface—this leaves small discrepancies in the single-nucleon momentum distribution. The comparison between the solid (green) and dashed (black) curve clearly shows that the subtraction of the center of mass component from the wave function leads to a reduction of the width and an enhancement of the quasielastic peak. Since this strongly affects the cross section in all the kinematical setups that we considered, we applied FSI corrections only to the intrinsic OpRS calculation. In order to do it, we follow the approach outlined in Sec. III, with the difference that the optical potential has been disregarded in the energy conserving -function since to the best of our knowledge neither the H-p nor the He-n optical potentials are present in the literature. The results are shown in Fig. 9. The convolution of the OpRS cross section with the folding function of Eq. (38) leads to a redistribution of the strength, which quenches the peak and enhances the tails. For MeV, , and MeV, the OpRS intrinsic calculation overestimates the data. Moreover, in all the kinematical configurations under consideration the position of the quasielastic peak is not correctly reproduced. This is likely to be ascribed to the approximate procedure we adopted to account for FSI effects, i.e. we neglected the real part of the optical potential. Its inclusion would shift the cross section towards lower values of possibly improving the agreement with the experimental data.
In Fig. 10 we compare the experimental data of the inclusive double-differential electron-O cross sections as computed from the fully correlated SCGF-ADC(3) spectral function. In the dashed (green) curve FSI effects have been implemented in full, yielding a very nice agreement with the data. In particular, the inclusion of the real part of the optical potential in the final state nucleon energy shifts the cross sections towards lower values of and the quasielastic-peak position is correctly reproduced.
We used the ab initio SCGF approach to compute the single-particle propagators of closed shell He and O nuclei. The calculations were based on the the NNLO chiral interaction since this is cable to describe simultaneously binding energies and nuclear radii of medium-mass nuclei.
We have gauged the residual center of mass contribution to the He wave function by
developing a MMC algorithm which exploits the OpRS single-particle propagator.
The resulting intrinsic charge density in He has been computed and compared
with both the QMC calculation of Ref. Lonardoni:2017egu and the experimental data.
The subtraction of the center of mass contribution turns out to be crucial in order to obtain correct predictions for this light nucleus.
The same pattern has also been observed in the single-nucleon momentum distribution; whilst the total OpRS sizably underestimates the QMC calculation,
a very nice agreement is found between the QMC and the results from the intrinsic OpRS.
In the analysis of the charge density in O, the full SCGF-ADC(3) calculation has been compared with the experimental curve. Since the radius of this nucleus has been used to fit the NNLO potential, the nearly perfect agreement with the empirical charge density is not surprising. However, the very good comparison with experimental cross sections corroborates the choice of the interaction for future studies of lepton-nucleus scattering. The origin of the discrepancies between the single-nucleon momentum distributions obtained from SCGF-ADC(3) and QMC approaches has to be attributed to the softness of the NNLO interaction. Although the two approaches provide very similar results in the region of low- and moderate-momentum, the use of an hard potential, such as AV18+UIX, implies a stronger nuclear interaction between large momenta. This manifests itself into the appearance of very high-momentum tails in the momentum distribution which are not as pronounced for the NNLO chiral force.
We employed the IA approach to perform the calculation of inclusive electromagnetic cross sections which exploits SCGF spectral functions. The electron-He double differential cross section corresponding to the intrinsic OpRS wave function sizably differs from the OpRS in which the contamination of the center of mass is still present. This indicates that the spurious effect of the center of mass can not be neglected in light nuclei. For this reason, we restricted the discussion of FSI effects to the sole OpRS intrinsic calculations. We observed that the convolution with the folding function of Ref. Ankowski:2014yfa; Benhar:1991af yields a redistribution of the strength of the cross section. However, disregarding the real part of the optical potential in the energy conserving delta function prevents a good agreement with the data for the different kinematical setup analyzed.
Fully satisfactory results have been obtained for the electron-O double differential cross section, where the IA calculation has been supplemented by FSI. Our findings indicate that the SCGF approach provides accurate predictions for medium-mass nuclei and their interaction with an electron probe. The extension to the electroweak sector will be the subject of a future work. Moreover, exploiting the Gorkov formalism we will be able to provide valuable results for open shell nuclei Soma:2013xha which will be crucial in the data analysis of future neutrino experiments, such as DUNE.
We are deeply indebted to O. Benhar, D. Lonardoni, and A. Lovato for many illuminating discussions and to P. Navrátil for providing the matrix elements of NNLO interaction. This research has been supported by the Centro Nazionale delle Ricerche (CNR) and the Royal Society under the CNR-Royal Society International Fellowship scheme NF161046 and by the United Kingdom Science and Technology Facilities Council (STFC) under Grants No. ST/L005743/1 and ST/L005816/1. Calculations were performed at the DiRAC Complexity system at the University of Leicester (BIS National E-infrastructure capital Grant No. ST/K000373/1 and STFC Grant No. ST/K0003259/1).