The falsification of Chiral Nuclear Forces
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The falsification of Chiral Nuclear Forces
E. Ruiz Arriola , a b , J. E. Amaro , and R. Navarro Perez Departamento de Física Atómica, Molecular y Nuclear andInstituto Carlos I de Física Teórica y Computacional, Universidad de GranadaE-18071 Granada, Spain Nuclear and Chemical Science Division, Lawrence Livermore National LaboratoryLivermore, California 94551, USA
Abstract.
Predictive power in theoretical nuclear physics has been a major concern inthe study of nuclear structure and reactions. The Effective Field Theory (EFT) based onchiral expansions provides a model independent hierarchy for many body forces at longdistances but their predictive power may be undermined by the regularization scheme de-pendence induced by the counterterms and encoding the short distances dynamics whichseem to dominate the uncertainties. We analyze several examples including zero energyNN scattering or perturbative counterterm-free peripheral scattering where one wouldexpect these methods to work best and unveil relevant systematic discrepancies when afair comparison to the Granada-2013 NN-database and partial wave analysis (PWA) isundertaken.
Nuclear Physics has always been characterized by the fact that experiment is much more precisethan theory. For nuclear masses one has ∆ M ( Z , N ) exp < (cid:28) ∆ M ( Z , N ) th but it is unclear whatthe theoretical uncertainty is. Traditionally, the theoretical and reductionist predictive power flow isexpected to be from light to heavy nuclei form a Hamiltonian with multinucleon forces H ( A ) = T + V N + V N + V N + . . . → H ( A ) Ψ n = E n ( A ) Ψ n . (1)In the absence of ab initio determinations, phenomenological V nN interactions are adjusted to NNscattering and light nuclei binding energies. The chiral approach, originally suggested by Weinbergin 1990 [1] (see e.g. [2–4] for reviews) to nuclear forces provides a power counting in terms of thepion weak decay constant f π , with the appealing feature of systematically providing a hierarchy V χ N (cid:29) V χ N (cid:29) V χ N (cid:29) . . . (2) a e-mail: [email protected] b Speaker at XIIth Conference on Quark Confinement and the Hadron Spectrum.Work supported by Spanish Ministerio de Economia y Competitividad and European FEDER funds (grant FIS2014-59386-P),the Agencia de Innovacion y Desarrollo de Andalucia (grant No. FQM225), the U.S. Department of Energy by LawrenceLivermore National Laboratory under Contract No. DE-AC52-07NA27344, U.S. Department of Energy, Office of Science,Office of Nuclear Physics under Award No. de-sc0008511 (NUCLEI SciDAC Collaboration) a r X i v : . [ nu c l - t h ] N ov PJ Web of Conferences
Because the pion mass is so small, chiral approaches are unambiguous at long distances via1 π ,2 π ,3 π , . . . exchanges for relative distances above a short distance cut-off r c , V n π ( r c ) ∼ e − nr c m π .For instance, NN chiral potentials, constructed in perturbation theory, are universal and contain chiralconstants c , c , c , . . . which can be related to π N scattering [2–4]. At long distances we have V χ NN ( r ) = V π NN ( r ) + V π NN ( r ) + V π NN ( r ) + . . . r (cid:29) r c , (3)whereas they become singular at short distances V χ NN ( r ) = a f π r + a f π r + a f π r + . . . r (cid:28) r c , (4)and some regularization must be introduced in any practical calculation. Thus, they trade the “old”model dependence for the “new” regulator dependence. What is the best theoretical accuracy we canget within “reasonable” cut-offs ? What is a reasonable cut-off ? Can the short distance piece beorganized as a power counting compatible with the chiral expansion of the long distance piece ?The huge effort which has been carried out over the last 25 years ellapsed since the seminal workof Weinberg, harvesting over 1000 citations, proves the computational feasibility of the chiral nuclearagenda requiring large scale calculations and many CPU computing hours. Here, we depart from themain streamline and wonder if chiral nuclear forces can be falsified or validated and, if yes, if they areuseful for nuclear structure applications from the point of view of the predictive power.Of course, all this has to do with proper assessment and evaluation of uncertainties of any sort andin particular in the NN interaction. Our original and simple estimates [5, 6] of ∆ B th / A ∼ . ∼ ∆ B th ( O ) / ∼ ∆ B sem / A ∼ . V π NN , corresponding to chiral 2 π exchange ( χ TPE).
From our point of view, making first a fair statistical treatment is a prerequisite to credibly aim at anyprecision goal in low energy nuclear physics where information is extracted by fits. We remind thefact that least squares χ -fitting any (good or bad) model to some set of data is always possible andcorresponds to just minimizing a distance between the predictions of the theory and the experimentalmeasurements. How can we disentangle between true and false models?.The well-known statistical approach, to which we stick, provides one probabilistic answer anddepends on the number of data, N Dat , the number of parameters determined from this data, N Par , andthe nature of experimental uncertainties. The natural question is: What is the probability that giventhe data the theory is correct ? This corresponds to the Bayesian approach which requires some apriori expectations on the goodness of the theory regardless of the data and is dealt with often byaugmenting the experimental χ with an additive theoretical contribution χ . However, it can beproven that when N Dat (cid:29) N Par one can ignore these a priori expectations since χ ∼ N Dat (cid:29) χ ∼ N Par and proceed with the frequentist approach where just the opposite question is posed: what isthe probability of data given the model ?. In our analysis below, where we have N Dat ∼ One could stay Bayesian if some relative weighting of χ and χ is implemented (see [10, 11] and references therein). ONF12 N Par ∼
40, we expect no fundamental differences. We thus simply ask: what is the probability q thatthe the model is false ?. The p-value is p = − q and if p is smaller than a predetermined confidencelevel we will not trust the model and we will declare it to be false. Note that 1) we can never be surethat the model is true and 2) any experiment can be right if errors are sufficiently large and the theorycannot be falsified. This said, p = .
68 when χ / ν = ± (cid:112) / ν with ν = N Dat − N Par .In general we expect discrepancies between theory and data and, ideally, if our theory is an ap-proximation to the true theory we expect the optimal accuracy of the truncation to be comparable withthe given experimental accuracy and both to be compatible within their corresponding uncertainties(see [12] for a Bayesian viewpoint). If this is or is not the case we validate or falsify the approximatedtheory against experiment and declare theory and experiment to be compatible or incompatible respec-tively. Optimal accuracy, while desirable, is not really needed to validate the theory. In the end largesterrors dominate regardless of their origin; the approximated theory may be valid but inaccurate.How should the discrepancies or residuals be interpreted ? Statistics has the obvious advantagethat if we have no good reasons to suspect the theory we can test if residuals behave as, often gaussian,fluctuations and determine a confidence interval for fitting parameters within these fluctuations.
The NN scattering amplitude has 5 independent complex components for any given energy, whichmust and can be determined from a complete set of measurements involving differential cross sectionsand polarization observables. From this point of view it is worth reminding that phase shifts obtainedin PWA are not data by themselves unless a complete set of 10 fixed energy and angle dependent mea-surements have been carried out, a rare case among the bunch of existing 8000 np+pp scattering databelow 350 MeV LAB energy and which corresponds to a maximal CM momentum of p maxCM = − .In order to intertwine all available, often incomplete and partially self-contradictory, information someenergy dependence interpolation is needed. We assume a potential approach inspired by a Wilsonianpoint of view where we take a grid of equidistant radial “thick” points in coordinate space separatedby the finite resolution given by the shortest de Broglie wavelength, ∆ r = ¯ h / p maxCM ∼ . r c = π exchange gives the entire strong contribution. Thecounting of parameters [13] yields about 40 “thick” points, which can be represented by delta-shells(DS) [14] as originally proposed by Avilés [15]. The whole procedure needs long distance electro-magnetic and relativistic contributions such as Coulomb, vacuum polarization and magnetic momentsinteractions. This approach allows to select the largest self-consistent existing NN database with atotal of 6713 NN scattering data driven by the coarse grained potential [16, 17] with the rewardingconsequence that statistical uncertainties can confidently be propagated. Precise determinationsof chiral coefficients, c , c , c [18, 19], the isospin breaking pion-nucleon [20, 21], and the pion-nucleon-delta [7] coupling constants have been made. The questions on the cut-of r c raised above were answered by separating the potential as follows [19] V ( r ) = V short ( r ) θ ( r c − r ) + V χ long ( r ) θ ( r − r c ) , V short = µ ∑ n λ n δ ( r − r n ) , (5)with r n = n ∆ r . Several fits varying r c and E maxLAB were performed. The results were checked to bestatistically consistent and are summarized in Table 1. It is striking that D -waves, nominaly N3LO and PJ Web of Conferences
Table 1.
Fits of chiral TPE potentials depending on the cutoff radius and the maximum fitting energy [19].
Max T LAB r c c c c Highest χ / ν MeV fm GeV − GeV − GeV − counterterm350 1.8 − . ( ) − . ( ) . ( ) F − . ( ) . ( ) . ( ) F − . ( ) − . ( ) . ( ) D − . − .
89 4 . P − . ( ) . ( ) . ( ) P r c < . χ -potentials [22, 23] take r c = . − . δ short , .i.e. the phase shifts stemming solely from V short compatible with zero within uncertainties, i.e. | V short | < ∆ V ?. This corresponds to check what partial waves fullfill | δ short | ≤ ∆ δ stat when r c = . χ TPE makes peripheral phases (large angular momentum) to besuitable for a perturbative comparison without counterterms [24–26]. However, one should take intoaccount that 1) peripheral phases can only be obtained from a complete phase shift analyses and 2)their uncertainties are tiny [16]. The analysis of [26] just makes an eyeball comparison which looksreasonable but the agreement was not quantified. We find that peripheral waves predicted by 5th-order chiral perturbation theory are not consistent with the Granada-2013 self-consistent NN database | δ Ch , N4LO − δ PWA | > ∆ δ PWA , stat . (6)Sometimes we get even 3 σ discrepancies. More details on this peripheral analysis will be presentedelsewhere. Of course, one may thing that 125 MeV is too large an energy. We find that when we godown to 40 MeV, the χ TPE potential becomes invisible being compatible with zero [13, 27].The chiral potential (including ∆ -degrees of freedom) of Ref. [23] explicitly violates Weinberg’scounting since it has N2LO long distance and N3LO short distance pieces, and residuals are notgaussian. More recently, the local short distance components of this potential have been fitted up to125 MeV LAB energy [28] improving the goodness of the fit, similarly to [19] (see also table 1). The low energy threshold parameters allow to probe the structure of chiral potentials against the NNinteraction. The current approach to chiral interactions is to incorporate the χ TPE tail and includeshort range counterterms fitted to pp and np phase-shifts or scattering data [29, 30]. However, theseapproaches are subject to strong systematic uncertainties since a fit to phase-shifts may be subjected tooff-shell ambiguities and so far low energy chiral potentials fitted to data have not achieved gaussianresiduals [30] or even have huge [22] or moderate [23] χ / ν values. To avoid these shortcomings weuse χ TPE [13, 32] with a simpler short range structure inferred from low energy threshold parame-ters [7] with their uncertainties inherited from the 2013-Granada fit [16]. This corresponds to zeroenergy renormalization condition of the counterterms. This was done using the SAID database (http://gwdac.phys.gwu.edu/), a 25 σ incompatible fit with p (cid:28) In momentum space counterterms corresponds to coefficients of polynomials, see e.g. [31], which can be fixed by lowenergy threshold parameters by implicit renormalization.
ONF12
Table 2.
Delta-Shell parameters located at r = . r = . χ TPE potential (see main text). S P P P S ε D P λ -0.572(7) − − − -0.368(9) -0.706(7) -4.15(1) − λ -0.201(3) -0.033(3) 0.103(7) 0.221(2) -0.246(4) -0.386(7) 0.35(1) -0.125(1)One could naively expect to be able to set any number of short range counterterms to reproducethe same number of low energy threshold parameters. Actually, in order to have as the 9 countertermsdictated by Weinberg to N2LO as in [29] we need to fix α and r for both S and S waves, the mix-ing α ε and α for the P , P , P , P [7]. In practice this turned out to be unfeasible in particular forthe J = a and r . If instead one includes two countertermsin each partial wave in the J = a and r matrices. With this structure we have a total of 12 short range parameters set to reproduce12 low energy threshold parameters from [7], and not the 9 expected from N2LO [29]. Statisticaluncertainties can be propagated by making fits to each of the 1020 sets of threshold parameters thatwere calculated from the bootstrap generated DS potentials [33]; this directly takes into account anystatistical correlation between low energy parameters. Table 2 lists the resulting 12- λ i parameters. InFigure 1 we show the phase-shifts corresponding to the DS- χ TPE potential with the parameters ofTable 2 and compare them to the DS-OPE potential [16, 34]. We observe a good agreement betweenboth representations up to a laboratory energy of 20 MeV.
Figure 1. χ TPE zero energy renormalized np phase-shifts fixing the low energy threshold parameters (see maintext) [7] compared with the phases obtained from the fit to the 2013-Granada database [16].
Chiral nuclear forces have been massively implemented in Nuclear Physics in the last 25 years withthe legitimate hope of providing a unified description of nuclear phenomena more rooted in QCDand less model dependent than most of the phenomenological approaches. This huge effort provesthat they are not only calculable but also that they can be used in light nuclei studies, but their indis-pensability remains to be established. Their systematic uncertainties may be large and they might notbe necessarily more predictive than the usual phenomenological and non-chiral approaches. Withinthe EFT approach there is a residual model dependence regarding the finite cut-off regularization
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