Binding in light nuclei: Statistical NN uncertainties vs Computational accuracy
aa r X i v : . [ nu c l - t h ] A p r Binding in light nuclei: Statistical NN uncertaintiesvs Computational accuracy R. Navarro P´erez , A. Nogga, J. E. Amaro and E. Ruiz Arriola Nuclear and Chemical Science Division, Lawrence Livermore National LaboratoryLivermore, California 94551, USA Forschungszentrum J¨ulich, Institut f¨ur Kernphysik (Theorie), Institute for AdvancedSimulation, J¨ulich Center for Hadron Physics and JARA - High Performance Computing,D-52425 Julich, Germany. Departamento de F´ısica At´omica, Molecular y Nuclear and Instituto Carlos I de F´ısicaTe´orica y Computacional, Universidad de Granada E-18071 Granada, SpainE-mail: [email protected],[email protected], [email protected],[email protected]
Abstract.
We analyse the impact of the statistical uncertainties of the the nucleon-nucleoninteraction, based on the Granada-2013 np-pp database, on the binding energies of the triton andthe alpha particle using a bootstrap method, by solving the Faddeev equations for H and theYakubovsky equations for He respectively. We check that in practice about 30 samples proveenough for a reliable error estimate. An extrapolation of the well fulfilled Tjon-line correlationpredicts the experimental binding of the alpha particle within uncertainties.
Nuclear structure ab initio calculations are notoriously difficult and computationallydemanding and have thus so far been limited to light nuclei, although recently, these calculationshave been extended to more complex systems [1, 2, 3]. Besides giving important input forapplications such as astrophysically relevant nuclear reactions, these calculations are importanttests of current nuclear interactions. To this aim, not only the result itself is important but alsothe uncertainty (see e.g. the special issue [4].) From a theoretical point of view and the inferredpredictive power uncertainties can be grouped into three main categories • The input information: the basic nucleon-nucleon (NN) interaction should describe arelevant piece of the NN scattering data and the simplest two-body bound state: thedeuteron. We will call this the statistical uncertainty for reasons to be justified below. • The solution method: the way the multinucleon problem is solved once the NN interactionis represented. This requires some sufficiently high precision which makes computationscostly. We will call these the numerical uncertainty. • The representation problem: the way the input NN data are represented theoretically.Normally potentials are used, but the form of the potential in the short range region, below2 − . Presented by RNP at Workshop for young scientists with research interests focused on physics at FAIR 14-19February 2016 Garmisch-Partenkirchen (Germany) This includes in particular any theoretically based expansion of the interaction rooted or inspired by QCD suchas chiral perturbation theory or large N c expansions were some renormalization scheme dependence is unavoidable. S np0 M = 10 M = 25 0.5Bootstrap S pp0 P P P S ǫ D D D P ǫ F F D E LAB (MeV) 50 150 250 3500.17
Figure 1. (Color online) Phaseshift statistical error bands (in degrees) for the δ -shellpotential [5]. The error bands on the first two columns where obtained using a MonteCarlo familyof δ -shell potentials where potential parameters are random numbers following the multivariatenormal distribution determined by the original fit covariance matrix. The columns use a samplesize of M = 10 (left column) and M = 25 (middle column). The right column is the error barobtained from the Bootstrap to experimental data presented in [6] with M = 1000. All phaseshifts are np unless otherwise indicated.Assuming that these sources of error are independent of each other, we expect the totaluncertainty to be given, as usual, by∆ E = ∆ E + ∆ E + ∆ E (1)Clearly, the total error is dominated by the largest one. So, it makes sense either to reduce thelargest source of uncertainty or to tune all uncertainties to a similar level. This sets the limit ofpredictive power in ab initio calculations. While numerical accuracy has been a goal in itself infew-body calculations, the physical accuracy is given by all possibles sources of uncertainties.In this talk, we discuss the relation between the statistical uncertainties stemming from thefinite experimental accuracy of NN scattering data [7, 8, 9, 10] and the currently availablenumerical accuracy with which the few body problem can be solved. A pioneering work wascarried out in [11] where the so-called statistical regularization was used to evaluate the impactof errors on the binding energies of the A = 3 , χ / d . o . f . ∼ σ -self consistent database comprises 6713 np and pp scatteringdata below E LAB = 350MeV and has a χ / d . o . f = 1 .
04 [12, 5]. The procedure to propagatencertainties is based in spirit on the bootstrap analysis proposed in [6] where the 6713 npand pp scattering data are randomized and multiple ( M = 1020) χ -fits yield a multivariatedistribution of fitting parameters. This provides a sample enabling a random evaluation of anyobservable. We monitor the size M of the needed sample by looking for statistical stability ofthe output. The result for the errors in the corresponding phase shifts is compared in Fig. 1for different Monte Carlo generated sample sizes following a gaussian multivariate distributiondictated by the parameter’s covariance matrix. As we see M = 25 already gives a result ratherclose to the full bootstrap method.We have built a simple and smooth gaussian potential which can be used in most few- andmany-body calculational schemes and which provides an acceptable χ / d . o . f . = 1 .
06 [13], soit can be considered to be statistically equivalent to the original delta-shells potential [8, 5].As we will put forward here, and in agreement with previous findings using either thehyperspherical harmonics (HSH) method for A = 3 [14] and no-core full configuration shellmodel calculations [15], these estimates already suggests that the numerical accuracy is close tooptimal given the statistical uncertainty. We will use here the Faddeev equations for the A = 3case and the Yakubovsky equations for the A = 4 situation. As a first step we will consideronly NN forces explicitly and leave out 3N and 4N forces for future developments. The multipleevaluations for the triton are shown in Fig. 2. As in [14] we bin the distribution according tothe numerical accuracy, ∆ E num t ∼ .In a Monte Carlo approach many variations of the parameters produce irrelevant changes. Aprincipal component analysis looks for eigenvalues and eigenvectors of the computed observableand provides valuable information on the most relevant changes of the input parameters buthas seldomly been investigated in nuclear physics (see however Ref. [16] and references therein).In Fig. 3, we show the results of such an analysis applied to the coefficients of the gaussianpotential of Ref. [8] implemented in a Monte Carlo fashion. We found that the number ofprincipal components to obtain most of the uncertainty in E t is around 10. This indicates thatregarding ∆ E stat t a fit to the NN scattering data base could be done with less parameters, if thefit were to be designed in terms of relevant parameters only.The tiny error band suggests that the discrepancy between our number E th t = − . ± . E exp t = − . ± . M eV has to be sought in missing three-nucleon forces (3NFs). It is well known that 3NFs give an important contribution to nuclearbindings [17, 18]. This raises the question of how much of this statistical uncertainty willbe absorbed into variations in the parameters of the 3NFs. In order to implement some 3Ninformation, we invoke the empirical linear correlation displayed by the Tjon line [17] .In the Monte Carlo method, any choice of parameters p determines a value of the tritonbinding energy. Given the variations of the triton binding energy, we expect, when determiningthe α particle binding energy, a Tjon-like linear correlation of the form E α ( p ) = aE t ( p ) + b .The values found are a = 4 . b = 11 .
4. Thus we expect a Tjon-like correlation wouldgive ∆ E stat α = 4 . × ∆ E stat t = 50(5)keV which is mainly determined by the channels involvingrelative S-waves. In Fig. 4, we show our results for the H and He binding energy. The yellowband shows the fit including the uncertainty. The error bars show the numerical uncertainty.Whereas the variation of the binding energies is rather large, the linear correlation indicates thatmost of this variation will be eventually absorbed into a properly adjusted 3NFs. We take theband width as an indication for the remaining error induced by the uncertainty of NN data. Asone can see, the band width and the numerical errors are comparable. Therefore, we deduced One of the advantages of the present method over the HSH expansion employed in [14] is that the determinationof avoided crossings are difficult to identify rigorously; an effect which has to be assisted by visual inspection and,if mistaken, has a repulsive effect. This explains the longer tails of the triton binding energies distributions. Wethank Eduardo Garrido for noting this. See [19, 20] for a Similarity Renormalization Group analysis yielding the simple formula E α = 4 E t − E d . E t ( M e V ) N C o un t s E t (MeV) -7.67-7.668-7.666-7.664-7.662-7.66 0 200 400 600 800 1000 h E t i ( M e V ) Sample size00.0050.010.0150.02 0 200 400 600 800 1000 p h E t i − h E t i ( M e V ) Sample size
Figure 2. (Color online) Distribution (top left) and histogram (bottom left) representing thetriton binding energy (in MeV) for a sample of 1000 Monte Carlo generated gaussian potentialparameters. Finite sample estimates for the population mean (top right) and populationstandard deviation (bottom right) of the triton binding energy as a function of the sample size M of the gaussian potential parameters are also shown. In all panels the red band representsthe value obtained with the most likely parameters E t ( p ) = 7 . ± . ∆ E ( K e V ) NPCPotential parameters M = 1000 M = 250 M = 30 Figure 3. (Color online) Number ofprincipal components contributions ofgaussian parameters for samples of size M = 30 , , E α in harmony with the Tjon slope ∆ E num α ∼ . E num t ∼ - - - - - - - - - - - H H L H
MeV L E H H e LH M e V L - - - - - - - - - - - - - - - - - H H L H
MeV L E H H e LH M e V L Figure 4. (Color online) Tjon type analysis of the He binding energy vs the H binding energy.We show the fit to the sample of N = 30 Monte Carlo generated binding energies both in a smallscale (left panel) and extrapolated in a larger scale (right panel) compared with the experimentalpoint (blue dot). We take ∆ E num t = 1 keV and ∆ E num α = 20 keV. Acknowledgements
This work is supported by Spanish DGI with Feder funds (grant FIS2014-59386-P) and Juntade Andaluc´ıa (grant FQM225), the U.S. Department of Energy by Lawrence Livermore NationalLaboratory under Contract No. DE-AC52-07NA27344, the U.S. Department of Energy, Officeof Science, Office of Nuclear Physics under Award No. de-sc0008511 (NUCLEI SciDACCollaboration). The numerical calculations have partly been performed on JUQUEEN, JUROPAand JURECA of the JSC, J¨ulich, Germany.
References [1] Quaglioni S, Hupin G, Calci A, Navratil P and Roth R 2015 (
Preprint )[2] Meissner U G 2014
Nucl. Phys. News Preprint )[3] Dytrych T, Maris P, Launey K D, Draayer J P, Vary J P, Langr D, Saule E, Caprio M A, Catalyurek U andSosonkina M 2016 (
Preprint )[4] Ireland D and Nazarewicz W 2015 (Editors) Journal of Physics. G, Nuclear and Particle Physics [5] Navarro P´erez R, Amaro J E and Ruiz Arriola E 2013 Phys. Rev.
C88
Phys. Lett.
B738
Preprint )[8] Navarro P´erez R, Amaro J E and Ruiz Arriola E 2013
Phys. Lett.
B724
PoS
QNP2012
Preprint )[11] Adam R, Fiedeldey H, Sofianos S and Leeb H 1993
Nuclear Physics A accessed: 2016-04-04[13] Navarro P´erez R, Amaro J E and Ruiz Arriola E 2014
Phys. Rev.
C89
Phys. Rev.
C90
Phys. Rev.
C92
Nucl. Phys.
A933
Rev. Mod. Phys. Rept. Prog. Phys. Few Body Syst. Preprint1601.02360