MMeson-Nucleon Scattering Amplitudes from Lattice QCD
John Bulava a) CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark a) Electronic mail: [email protected]
Abstract.
Lattice QCD calculations of resonant meson-meson scattering amplitudes have improved significantly due to algorith-mic and computational advances. However, progress in meson-nucleon scattering has been slower due to difficulties in computingthe necessary correlation functions, the exponential signal-to-noise problem, and the finite-volume treatment of scattering withfermions. Nonetheless, first benchmark calculations have now been performed. The status of lattice QCD calculations of meson-nucleon scattering amplitudes is reviewed together with comments on future prospects.
Meson-nucleon scattering amplitudes are required in a variety of active research areas of nuclear and particlephysics. Near threshold they provide valuable determinations of the low-energy constants (LEC’s) of chiral pertur-bation theory involving baryons, which also govern the long-range (pion-exchange) component of the inter-nucleonforces. The first principles calculation of meson-nucleon amplitudes from lattice QCD both compliments experimentand probes the quark mass dependence, enabling studies of the convergence of the asymptotic perturbative series.Low-energy pion-nucleon scattering is also related to the pion-nucleon sigma term σ π N [1, 2], which is an input to thephenomenology of dark matter direct-detection experiments. Some tension currently exists between lattice determina-tions of σ π N and phenomenological extractions of σ π N from π N scattering data [3], so that precise lattice calculationsof the π N scattering lengths are needed. At somewhat larger center-of-mass energy, electroweak form-factors ofthe N → ∆ ( ) → N π transition are needed for neutrino-nucleus scattering experiments [4], and the nature of thelow-lying N ( ) and Λ ( ) resonances remain unsettled.Although lattice QCD is well established as a reliable non-perturbative approach to calculating the properties ofsingle hadrons, lattice QCD studies of hadron scattering amplitudes have proven more difficult. Increased develop-ment of the formalism for relating finite-volume lattice observables to infinite-volume amplitudes, the rapid growthof computer power, and novel efficient numerical algorithms have combined to advance the state-of-the-art. Thisreview highlights the substantial recent progress achieved in lattice QCD calculations of hadron scattering ampli-tudes, in particular on the nascent subfield of meson-nucleon systems. A related review of lattice QCD calculationsof meson-baryon scattering amplitudes is given in Ref. [5].A detailed introduction to lattice QCD is beyond the scope of this work [6, 7], but several main points are worthyof mention. Lattice QCD employs the path integral formulation of QFT regulated on a finite space-time lattice of ( L / a ) × ( T / a ) points, where a denotes the lattice spacing and L ( T ) the spatial (temporal) extent. After analyticallyintegrating out the quark fields, Markov chain Monte Carlo methods sample the remaining path integral over thegluon field (denoted U ) to produce stochastic estimates for Euclidean n -point correlation functions. Employing thetime-momentum representation for a correlation function between hadron interpolating fields O i and O j results in thespectral decomposition C i j ( p , τ ) = (cid:104) O i ( p , τ ) O † j ( ) (cid:105) U = ∑ n (cid:104) | ˆ O i ( p ) | n (cid:105) (cid:104) n | ˆ O † j ( ) | (cid:105) e − E n τ + O ( e − MT ) (1)where (cid:104) . . . (cid:105) U denotes the Monte Carlo average over suitably distributed gluon fields and the sum runs over finite-volume Hamiltonian eigenstates ˆ H | n (cid:105) = E n | n (cid:105) . The form of the exponentially suppressed finite- T effects depends onthe temporal boundary conditions and the interpolating operators. Evidently the large-time limit of these Euclidean n -point functions is dominated by the lowest contributing eigenstates and analyses of C i j ( p , τ ) treating the firstfew terms in the sum enable a determination of the low-lying finite-volume energies and matrix elements [8, 9, 10].Such determinations are hampered by the exponential signal-to-noise degradation typically present as τ → ∞ so thatin practice finite-volume levels are isolated successfully only if the employed set of operators { O i } has significantoverlap onto them.As is evident from Eq. 1, lattice QCD simulations are performed in Euclidean time τ = it . The usual determinationof scattering amplitudes in real time relies on an asymptotic formalism, such as the well-known LSZ [11] and Haag-Ruelle [12, 13] approaches, where the asymptotic time limits t → ± ∞ of correlation functions isolate the desired in andout states. Although an analogous asymptotic formalism has been proposed for Euclidean time in Ref. [14], Maianiand Testa [15] proved that the naive large-separation limit of Euclidean correlators does not (in general) yield on-shellscattering amplitudes. Fortunately, a work-around was developed by Lüscher [16] in which the finite-volume is used a r X i v : . [ h e p - l a t ] S e p s a tool to probe hadron interactions. In this approach the signal consists of the deviation of finite-volume two-hadronlevels from their non-interacting values. Although originally formulated for total momentum P = E Λ L ( P ) and infinite-volumetwo-to-two scattering amplitudes is given bydet [ K − ( E cm ) − B ( P ) ( L , E cm )] + O ( e − ML ) = , E cm = (cid:113) ( E Λ L ( P )) + P (2)where the exponentially suppressed corrections are ignored in practical applications and Λ denotes an irreduciblerepresentation (irrep) of the relevant finite-volume symmetry group. This ‘quantization condition’ yields informationabout the infinite-volume K -matrix in the form of a determinant over all possible total angular momenta ( J ), total spincombinations ( S ), and two-hadron scattering channels ( a ). Encoding the reduced symmetry of the cubic volume inlattice QCD simulations, the (known) B -matrix mixes the infinite number of possible J while the K -matrix is diagonalin J but dense in S and a . The practical application of Eq. 2 requires a block diagonalization in the basis of finite-volume irreps, and a truncation to the limited set of contributing partial waves below some maximum orbital angularmomentum (cid:96) max . Only in the single-channel, single-partial wave approximation is there a one-to-one correspondencebetween E Λ L ( P ) and K − ( E cm ) . In other cases a global fit is performed to all energies resulting in a model-dependentparametrization of K − ( E cm ) .We turn now to the lattice QCD calculation of finite-volume two-hadron energies { E Λ L ( P ) } from the two-point cor-relation functions in Eq. 1. Correlation functions between two-hadron interpolating operators in which each hadronhas definite spatial momentum are most effective in isolating two-hadron states. However, the measurement of suchcorrelation functions on an ensemble of gauge field configurations has long been a computational challenge. Sincethe Grassmann-valued quark fields are integrated out analytically in the lattice QCD path integral, Wick’s theorem isemployed to express hadron correlation functions in terms of quark propagators. On a finite discrete Euclidean lattice,the quark propagators between space-time points x and y is given as the inverse of the large, sparse,ill-conditionedDirac operator M − ( x , y ) . This matrix is so large that its inverse computed only by solving linear systems M φ = η forsome ‘sink’ φ ( x ) given a ‘source’ η ( y ) . Projection onto definite spatial momentum however requires knowledge ofthe entire matrix inverse, so-called all-to-all quark propagators. The calculation of all required elements of this matrixinverse on each gauge configuration by solving one linear system for each space-time point is prohibitively expensive,so alternative algorithms are required. One approach that has been particularly successful is the Laplacian-Heavisidemethod (LapH) [57, 58] in which the all-to-all quark propagator is projected onto the subspace spanned by the N ev lowest modes of the gauge covariant Laplace operator, reducing its dimension significantly and enabling the compu-tation of the smeared all-to-all propagator. This method has the added benefits of affecting a form of quark smearing,whereby the overlap onto the high-lying states in Eq. 1 is suppressed, and reducing the problem of correlation functionconstruction to the contraction of individual hadron tensors, for which significant optimizations can be applied [59].Although the LapH approach still requires a large number of inversions on each gauge configuration, advances inalgorithms [60, 61] for solving linear systems involving the Dirac matrix M have improved the computational costsignificantly.Complimentary developments in the finite-volume formalism and in the computation of two-hadron energies inlattice QCD have driven recent progress in calculations of two-hadron scattering amplitudes. Although much of thisprogress has been in amplitudes with two pseudoscalar mesons, first meson-nucleon calculations have been performed.It is these calculations that are highlighted here, with work on threshold N π amplitudes summarized separately fromfirst calculations of the low-lying ∆ ( ) resonance. Finally, a new approach for determining scattering amplitudesfor spectral functions without employing the finite volume is reviewed, before concluding with a summary and futureprospects. .01.21.41.61.82.0 E [ G e V ] Exp. N N , N π - - FIGURE 1.
Left : the low-energy K + p scattering amplitude from Ref. [64] on a single ensemble with N f = + m π = Right : low-lying finitevolume spectrum in the I = / G u irrep from Ref. [65] on a single N f = m π = N ∗ ( ) and N ∗ ( ) resonances and the second column is the lattice QCD finite volume spectrumdetermined using only single-hadron operators. The third column is the spectrum determined accurately using both single-hadronand N π interpolating operators, illustrating the need for a complete basis in practical calculations. THRESHOLD SCATTERING AMPLITUDES
The natural first application of the methods described above is to near-threshold meson-nucleon amplitudes, wherethe effective range expansion of (cid:96) = p cm cot δ ( p cm ) = − a + r p + P r p + O ( p ) (3)defines the scattering length a , the effective range r , and the shape parameter P . These are of course parametrization-independent constants that characterize the low-energy interaction and thus of phenomenological significance. Onthe finite-volume formalism side, truncating Eq. 2 to s -wave contributions only yields a one-to-one correspondencebetween lattice QCD energies and scattering amplitudes. Although not required in practical applications, the thresholdexpansion of Eq. 2 is instructive and yields ∆ E ≡ E G g mN ( L ) − M m − M N = − π a µ mN L (cid:104) + c a L + c (cid:16) a L (cid:17)(cid:105) + O (cid:18) L (cid:19) (4)where E G g mN ( L ) is the finite-volume energy of the lowest lying meson-nucleon energy in the G g irrep, µ mN the reducedmass of the meson-nucleon system, and c and c are known constants [62]. Eq. 4 illustrates that finite-volume energyshifts indeed constitute the signal for scattering amplitudes, and that sensitivity to higher-order terms in Eq. 3 occursonly at O (cid:0) L (cid:1) .Several low-energy s -wave meson-nucleon scattering amplitudes have been calculated using this approach inRefs. [63, 64]. The quantum numbers of the overall system are chosen to ensure the absence of ‘same-time’ diagramswhich contain quark propagators that start and end at the same time. Since only a single ground state is determined,projecting onto definite momentum everywhere is not strictly required, although it does enhances the overlap ontothe desired level. Refs. [63, 64] exploit these simplifications and do not employ the LapH approach for the requiredquark propagators. Ref. [64] employs ensembles of N f = + m π = , K + p scattering are shown in Fig. 1.Ref. [65] computes the lowest three I = / N π finite-volume energy levels in the G u irrep for which the leadingpartial wave approximation yields the near-threshold s -wave amplitude. Since this system has non-maximal isospin, - - p m N -m cm E d c o t p m c m q g = 0, H P = 1, G P = 3, F P = 3, F P = 4, G P s ( MeV )04590135180 / , [] m = 1414(36) MeV g N = 26(7) O Dh , H g C D v , GC D v , G J=3/2, P-wave Analysis
FIGURE 2.
Left : the elastic I = / p -wave N π scattering amplitude from Ref. [66] on a single ensemble of N f = + m π = ∆ ( ) resonance is located near the N π threshold. Individual bootstrap samples are shown for each point to illustrate correlations and the lines indicate a Breit-Wignerfit. Ref. [66] also justified the single-partial wave approximation by expanding the determinant condition in Eq. 2 to include allcontributing d -waves, finding their contribution negligible. Right : Preliminary I = / p -wave N π scattering phase shift fromRef. [68] on a single ensemble of N f = + m π = same-time diagrams are required and the LapH method is used to efficiently determine the (smeared) all-to-all quarkpropagators. Definite three-momentum projection is performed for all hadrons and appropriate interpolators for eachof the three lowest-lying states are included. Results for the three finite-volume energy levels for a single ensembleof N f = m π = N π operator. Due tothe exponentially degrading signal-to-noise ratio, there is a finite time range over which the signal can be tracked andthe lowest-lying level is determined correctly only if the N π is included. Although only the first level is relevant forthe near-threshold scattering amplitude, the next two are in qualitative agreement with the experimentally determined N ∗ ( ) and N ∗ ( ) resonances. RESONANT AMPLITUDES
As is evident in Fig. 1, both single-hadron and two-hadron interpolating operators are required to determine the low-lying spectrum in the presence of resonances. Below three or more hadron thresholds, this spectrum can be interpretedaccording to Eq. 2 to obtain information about meson-nucleon scattering amplitudes above threshold. The benchmarkexample of such an analysis is the I = / p -wave elastic N π scattering amplitude, which contains the narrow ∆ ( ) resonance. This analysis must include single-hadron ∆ ( ) interpolating operators as well as N π operators withappropriate overall quantum numbers.The first published results on the ∆ ( ) resonance appeared in Ref. [66], although a report on preliminaryearlier work is found in Ref. [67]. Ref. [66] employs a single ensemble of N f = + m π = m u , d + m s = const . to the physical value. Because thedegenerate light quark masses are larger than their physical values, the ∆ ( ) is approximately stable and locatednear E thresh = m N + m π . This hampers a determination of the energy dependence of the amplitude, as is evidentin Fig. 2. Nonetheless, a Breit-Wigner fit to p cot δ yields m ∆ = ( ) MeV and g BW ∆ N π = . ( . ) which areconsistent with phenomenological determinations from experimental data. Although Ref. [66] employs the singlepartial wave approximation, the influence of d -waves is checked explicitly by enlarging the determinant in Eq. 2 usingthe formulae and computer programs published in Ref. [22]. IGURE 3.
From Ref. [70], preliminary results for the elastic I = / p -wave N π scattering amplitude on a single ensembleof N f = + m π = Left : the finite volume spectra in irreps containing both s - and p -wavecontributions. Right : a plot of p -wave dominated irreps for which the single-partial wave approximation is applied together with aBreit-Wigner fit to all levels where the s -wave is modeled by a constant. Preliminary results on the ∆ ( ) resonance are also found in Ref. [68] on a single ensemble of N f = + m π = g ∆ N π compatible with Ref. [66]and the preliminary determination in Ref. [69].In both Refs. [66] and [68] energies are calculated in a number of kinematic frames, but with finite-volume irrepsjudiciously chosen so that (cid:96) = s - and p -waves contribute, enabling a larger number of constraints on the K -matrix due to the increased number offinite-volume energies. However, a more complicated analysis is required where s - and p -waves are fit simultaneouslyto Eq. 2. First preliminary results from an analysis of this type [70] are shown in Fig. on a single ensemble of N f = + m π = s - and p -wave dominated irreps provides a numberof additional finite-volume energy levels which better constrain the p -wave Breit-Wigner fit parameters. The s -waveis modeled using the leading term in the effective range expansion from Eq. 3. SCATTERING FROM SPECTRAL FUNCTIONS
Although it continues to be successful where applicable, the finite-volume formalism employed above has severalshortcomings. Chiefly among them is the restriction of Eq. 2 to energies below three (or more) hadron thresholds,preventing the study of many interesting systems including excited nucleon resonances. If the three-hadron formalismis fully developed and applied, finite-volume energy levels up to four (or more) hadron thresholds may be interpreted,but no general approach exists for levels above arbitrary inelastic thresholds. Further limitations of the finite-volumeformalism include the inability to directly calculate inclusive rates such as the purely hadronic process p + p → X ,where X denotes a sum over all hadronic final states.An alternative approach to determining real-time scattering amplitudes from Euclidean correlation functions ismotivated by expressing the two-point correlation function in Eq. 1 as C i j ( p , τ ) = (cid:90) ∞ dE ρ i j ( p , E ) e − E τ , ρ i j ( p , E ) = ∑ n δ ( E − E n ) (cid:104) | ˆ O i ( p ) | n (cid:105) (cid:104) n | ˆ O † j ( ) | (cid:105) (5)in terms of the spectral function ρ i j ( p , E ) . These spectral functions are independent of the metric signature andtherefore contain real-time information. However, C i j ( p , τ ) is determined from the lattice Monte Carlo calculationat a finite number of τ , each with a statistical error. The reconstruction of ρ i j ( p , E ) from such data is therefore anill-posed problem. p E/m (E) e r = 0.61 p /m e L = 2.88 e L = 3.84 e - p E/m (E) e r = 0.61 p /m e L = 3.84, e data(770) r (1450) r FIGURE 4.
Spectral reconstruction of the smeared isovector vector correlator from Ref. [81] for N f = + m π = a = . Left : ρ ε ( E ) at two volumes illustrating the rapid onset of the infinite volume limit at fixed ε . Right : Individual state contributions of a fit to the data to Eq. 8 with N pole = ρ ( ) andthe ρ ( ) . A more modest goal is the determination of the smeared spectral function ρ ε i j ( p , E ) = (cid:90) ∞ d ω δ ε ( E − ω ) , lim ε → + δ ε ( x ) = δ ( x ) (6)where the smearing kernel δ ε ( E − ω ) approaches a Dirac- δ function as the smearing width is reduced. While thesmearing is a necessary limitation introduced by the nature of the problem, it is advantageous in bridging the gapbetween finite and infinite volume. Although unsmeared spectral functions are very different in finite and infinitevolume, at finite (fixed) smearing width ε the infinite-volume limit is well defined. Furthermore, recent improve-ments [71] to the Backus-Gilbert algorithm [72, 73] for spectral reconstruction enable the efficient determination of ρ ε with an input functional form for the smearing kernel δ ε ( E − ω ) . These methods provide the exact smeared spec-tral function, but smeared with a (known) smearing kernel ˆ δ ε ( E , ω ) that is only approximately equal to the desiredone δ ε ( E − ω ) . An ideal choice for the smearing kernel [74, 75] is the real part of the standard i ε -prescription forcausal propagation πδ ε ( E − ω ) = Re iE − ω + i ε = ε ( E − ω ) + ε . (7)which has desirable analyticity properties in ε , unlike (for instance) a gaussian smearing kernel. Physical on-shellscattering amplitudes are then obtained from the ordered double limit lim ε → + lim L → ∞ . This approach was proposedfirst for inclusive processes mediated by external currents in Ref. [76] and extended to arbitrary scattering amplitudesin Ref. [74].As a illustrative application, consider the total rate for the inclusive process ˆ J → hadrons, which ˆ J is an externalcurrent. An example of such a process is the ratio R had ( s ) = σ ( e + e − → hadrons ) / πα em ( s ) s with the electromagneticexternal current, which is relevant for the phenomenology of ( g − ) µ [77]. Fictitious inclusive processes can also beconsidered, in which the currents do not correspond to real-world external probes, but can be an arbitrary hadronicinterpolator ˆ O had . In order for such processes to act as novel probes of QCD, these hadronic interpolators must berenormalized to possess a well-defined continuum limit. The renormalization of arbitrary hadron interpolators can beaccomplished either using the Wilson flow [78, 79] or the approach of Ref. [80], but indicative preliminary results areshown with a smeared unrenormalized I = σ ( O had → hadrons )( E ) ∝ lim ε → + ρ ε ( E ) is determined at finite ε and multiple volumes using the freely-available lattice correlator data fromRef. [81].Two different spatial volumes illustrate the rapid approach to the L → ∞ limit at finite ε , so that an infinite-volume ansatz for ρ ε ( E ) is appropriate. In this channel peaks associated with the ρ ( ) resonance as well as excited I G ( J P ) = + ( − ) resonances are expected. However, the smearing width ε = . m π = ansatz that treats them as infinitesimally narrow is appropriate. To fit ρ ε ( E ) , we employ the /m e e re M = 1.5m max e L = 4, min e d = 0, /m ε ε re M d = 2 FIGURE 5.
Preliminary results for the real part of the elastic I = O ( ) model with mL = .
4. For zero total momentum, results for M ε ( E ∗ ) from Eq. are shown for zero units of relative momentum( left ) and two units of relative momentum ( right ). For each case, the order double limit is performed by extrapolating linearly to ε → [ ε min , ε max ] . The horizontal solid line corresponds to the exact results from Refs. [9, 83], which is consistentwith the extrapolated values. Deviations from linearity are evident for ε < ε min illustrating the importance of the ordered doublelimit. model ρ ε ( E ) = N pole ∑ n = A n δ ( E − E n ) (8)to describe the data in Fig. 4. The N pole = χ / d . o . f . = .
8, but using the four-parameter N pole = ansatz reduces this to χ / d . o . f . = . E = ( ) MeV and E = ( ) MeV which are consistent with theexperimentally determined masses of the ρ ( ) and ρ ( ) .While this inclusive example models the smeared total rate at finite ε , exclusive amplitudes are can be recoveredin the ε → + limit using the LSZ reduction approach of Ref. [74]. As a first demonstration of using spectralfunctions to obtain exclusive amplitudes [82], consider the 1 + O ( ) model employed in Ref. [9]. LikeQCD, this model is asymptotically free and has a mass gap, but the elastic two-to-two scattering amplitude is knownanalytically [83] so it is an ideal test case. Furthermore the global O ( ) symmetry is reminiscent of isospin symmetryin QCD, so that two-to-two scattering proceeds in one of three ‘isospin’ channels with I = , , ρ ε ( E ) . The two-to-two scattering amplitude M ( E ∗ ) is then computed from the ordered double limit M ( E ∗ ) = lim ε → + lim L → ∞ M ε ( E ∗ ) , M ε ( E ∗ ) = Z − ε ρ ε ( E ∗ ) (9)where E ∗ is the total (lab frame) energy and Z the interpolator overlap factor. Note the factor of ε which performsthe ‘amputation’ of the on-shell pole. Since only the real part of the amplitude is considered here, the smearing kernelfrom Eq. 7 is employed.The double limit in Eq. is performed numerically in Fig. 5 at a fixed mL = . ε = ε min < ε < ε max , with ε min L = ε max = . m toremain in the linear regime. As is evident in Fig. 5, this extrapolation results in scattering amplitudes consistent withthe known analytic result. Deviations from linearity are also apparent in Fig. 5 for ε < ε min , which illustrate the needfor the appropriate ordered double limit. ONCLUSIONS
This review surveys the current state of lattice QCD calculations of meson-nucleon scattering amplitudes, which isan emerging subfield. The finite-volume formalism of Eq. 2 has been successful in first calculations of two-to-twoamplitudes below three or more hadron thresholds. These include near-threshold studies which determine parametersof the effective range expansion as well as first studies of the ∆ ( ) resonance across the entire elastic region.All meson-nucleon calculations to date have also treated only elastic scattering, although Eq. 2 applies also to cou-pled two-hadron scattering channels which are relevant for the Λ ( ) resonance. Furthermore, extension of thefinite-volume formalism to treat meson-meson-baryon channels and the corresponding lattice QCD computation ofthe spectra have not yet been completed, although first calculations of three-pion scattering amplitudes are encourag-ing [59, 84, 85]. Nonetheless, the range of impact of the two-to-two scattering amplitudes can be increased by usinglattice QCD scattering data as an input to effective field theories and EFT-inspired models. Work in this direction withnucleons has already been performed [86, 87].In general progress in meson-baryon scattering on the lattice has been slower than meson-meson calculations dueto several difficulties, such as the increased signal-to-noise problem, complications in correlator construction due tothe extra quark, and complications in the finite-volume formalism from the inclusion of non-identical particles withnon-zero overall spin. Despite these difficulties, it is reasonable to assume that (in analogy with the meson-mesonsector) precise results for elastic amplitudes and coupled meson-baryon scattering channels will be produced in thenear future. Making further comparisons with meson scattering amplitudes, it is likely that calculations of meson-baryon amplitudes mediated by external electroweak currents ˆ J ew are also possible, such as N + J ew → N + π and N + π + J ew → N + π , which have a number of phenomenological applications. There has been substantial progresson analogous amplitudes in the meson-meson sector, such as γ ∗ → ππ in Refs. [36, 81] and π + γ ∗ → ππ in Refs. [88,89].The final topic discussed in this review is a novel approach to computing scattering amplitudes in lattice QCDsimulations based on spectral functions. This does not employ the finite volume whatsoever and as such requireslarge volumes to saturate the L → ∞ limit. Because of this, larger volumes are required than in typical lattice QCDsimulations, although some preliminary indicative results for inclusive decays on existing lattice QCD data are en-couraging. The LSZ reduction approach of Ref. [74] has not yet been employed in lattice QCD but preliminary resultsin a 1+1 dimensional toy model reproduce the known elastic two-to-two scattering amplitude. Overall, if feasible theinfinite-volume formalism based on spectral functions has the potential to circumvent many of the limitations presentwhen using the finite-volume, such as partial wave and coupled-channel mixing as well as the difficulty in goingabove arbitrary inelastic thresholds. These issues are of course relevant when studying excited nucleon resonances.However, the degree to which Refs. [74] and [76] can be applied to lattice QCD simulation data is an open question. REFERENCES
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