Yoshitaka Hatta
We discuss the orbital angular momentum of partons inside a longitudinally polarized proton in the recently proposed framework of spin decomposition. The quark orbital angular momentum defined by Ji can be decomposed into the `canonical' and the `potential' angular momentum parts, both of which are represented as the matrix element of a manifestly gauge invariant operator.
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In the last decade, it has been realized that the orbital angular momentum of partons inside the nucleon plays a major role. It contributes significantly to nucleon properties and is at the origin of many asymmetries observed in spin physics. It is therefore of paramount importance to determine this quantity if we want to understand the nucleon internal structure and experimental observables. This triggered numerous discussions and controversies about the proper definition of orbital angular momentum and its extraction from experimental data. We summarize the present situation and discuss rece…
A theoretical prediction is given for the spin and orbital angular momentum distribution functions of the nucleon within the framework of an effective quark model of QCD, i.e. the chiral quark soliton model. An outstanding feature of the model is that it predicts fairly small quark spin fraction of the nucleon $ΔΣ\simeq 0.35$, which in turn dictates that the remaining 65% of the nucleon spin is carried by the orbital angular momentum of quarks and antiquarks at the model energy scale of $Q^2 \simeq 0.3 {GeV}^2$. This large orbital angular momentum necessarily affects the scenario of scale depe…
Parallel to the construction of gauge invariant spin and orbital angular momentum for QED in paper (I) of this series, we present here an analogous but non-trivial solution for QCD. Explicitly gauge invariant spin and orbital angular momentum operators of quarks and gluons are obtained. This was previously thought to be an impossible task, and opens a more promising avenue towards the understanding of the nucleon spin structure.
The nucleon spin problem raises experimental and theoretical questions regarding the contribution of the orbital angular momentum of the quarks to the total spin of the nucleon. In this article we examine the commutation relationships of various operators that contribute to the total angular momentum of the nucleon. We find that the sum of the orbital plus gluon field angular momentum should satisfy the angular momentum commutators, at least up to the one-loop level. This requirement on the sum of these operators imposes a nontrivial restriction on the form of the color electric and magnetic f…
This is a Reply to the Comment [arXiv:0807.0699] on our paper [PRL 100:232002 (2008)].
We have calculated the Orbital Angular Momentum of quarks and gluons in the nucleon. The calculations are carried out in the next to leading order utilizing the so-called valon model. It is found that the average quark orbital angular momentum is positive, but small, and the average gluon orbital angular momentum is negative and large. We also report on some regularities about the total angular momentum of the quarks and the gluon, as well as on the orbital angular momentum of the separate partons. We have also provided partonic angular momentum, $L^{q,g}$ as a function of $Q^2$.
Using the expansion of the baryon wave function in a series of products of single quark bispinors (Dirac orbitals), the nonsinglet axial and tensor charge of the nucleon are calculated. The leading term yields ${G_A}/{G_V} = 1.27$ and in good agreement with experiment. Calculation is essentially parameter-free and depends on the string tension $σ$ and $α_s$, fixed at standard values. The importance of lower Dirac bispinor component, yielding 18% to the wave function normalization is stressed.
We address and solve the long-standing gauge-invariance problem of the nucleon spin structure. Explicitly gauge-invariant spin and orbital angular momentum operators of quarks and gluons are obtained. This was previously thought to be an impossible task, and opens a more promising avenue towards the understanding of the nucleon spin. Our research also justifies the traditional use of the canonical, gauge-dependent angular momentum operators of photons and electrons in the multipole-radiation analysis and labeling of atomic states; and sheds much light on the related energy-momentum problem in …
We introduce gauge-invariant quark and gluon angular momentum distributions after making a generalization of the angular momentum density operators. From the quark angular momentum distribution, we define the gauge-invariant and leading-twist quark {\it orbital} angular momentum distribution $L_q(x)$. The latter can be extracted from data on the polarized and unpolarized quark distributions and the off-forward distribution $E(x)$ in the forward limit. We comment upon the evolution equations obeyed by this as well as other orbital distributions considered in the literature.
Color confinement and chiral symmetry specify some important territory for the study of hadronic physics. Any hadron can be defined as a color-singlet composite system of qurks and gluons, the fundamental fields of qcd, while the landscape of the hadronic spectrum is dominated by the fact that two quark flavors, u and d, are characterized by masses small compared to the fundamental scale of this theory. Measurements sensitive to the orbital angular momenta of the color constituents of the nucleon display the interplay of chiral dynamics and confinement in a unique manner. This pageant can be e…
Scalar matrix elements involving strange quarks are studied in several models. Apart from a critical reexamination of results obtained in the Nambu and Jona-Lasinio model we study a scenario, motivated by instanton physics, where spontaneous chiral symmetry breaking is induced by the flavor-mixing 't Hooft interaction only. We also investigate possible contributions of virtual kaon loops to the strangeness content of the nucleon.
The leading order evolution of parton orbital angular momenta is treated in the probabilistic manner. As a result, the splitting probability matrix, while coinciding with the recent explicit calculations, is expressed in the terms of the standard spin-dependent and spin-averaged kernels. The implied relations between spin and orbital angular momenta evolution kernels may be interpreted as resulting from the ambiguity in the separation of the angular momenta to the spin and orbital contributions and are likely to be valid for higher orders and twists. Consequently, the orbital and total angular…
The evolution of orbital angular momentum distributions within the radiative parton model is studied. We use different scenarios for the helicity weighted parton distributions and consider a broad range of input distributions for orbital angular momentum. In all cases we are lead to the conclusion that the absolute value of the average angular momentum per parton peaks at relatively large $x\approx 0.1$ for perturbatively accessible scales. Furthermore, in all scenarios considered here the average orbital angular momentum per parton is several times larger for gluons than for quarks which favo…
Using the expansion of the baryon wave function in a series of products of single quark bispinors (Dirac orbitals), the nonsinglet axial and tensor charges of a nucleon are calculated. The leading term yields $g_A = 1.27$ in good agreement with experiment. Calculation is essentially parameter-free and depends only on the strong coupling constant value $α_s$. The importance of lower Dirac bispinor component, yielding 18% to the wave function normalization is stressed. As a check, the baryon decuplet masses in the formalism of this model are also computed using standard values of the string tens…
Analysis of the measured value of the integrated \bar{d}-\bar{u} asymmetry (Ifas = 0.147+-0.027) in the nucleon show it to arise from nucleon fluctuations into baryon plus pion. Requiring angular momentum conservation in these fluctuations shows the associated orbital angular momentum is equal to the value of the flavor asymmetry.
This is a comment on Chen et al Letter PRL,100, 232002 (2008).
Definitions of orbital angular momentum based on Wigner distributions are used as a framework to discuss the connection between the Ji definition of the quark orbital angular momentum and that of Jaffe and Manohar. We find that the difference between these two definitions can be interpreted as the change in the quark orbital angular momentum due to final state interactions as it leaves the target in a DIS experiment.
I introduce a gauge invariant decomposition of the nucleon spin into quark helicity, quark orbital, and gluon contributions. The total quark (and hence the quark orbital) contribution is shown to be measurable through virtual Compton scattering in a special kinematic region where single quark scattering dominates. This deeply-virtual Compton scattering (DVCS) has much potential to unravel the quark and gluon structure of the nucleon.
An unified scheme for describing both spin and orbital motion in symmetry-breaking chiral quark model is suggested. The analytic results of the spin and orbital angular momenta carried by different quark flavors in the nucleon are given. The quark spin reduction due to spin-flip in the chiral splitting processes is compensated by the increase of the orbital angular momentum carried by the quarks and antiquarks. The sum of both spin and orbital angular momenta in the nucleon is 1/2, if the gluons and other degrees of freedom are neglected. The same conclusion holds for other octet and decuplet …
We study the orbital angular momentum structure of the quarks inside the proton. By employing the light-cone diquark model and the overlap representation formalism, we calculate the chiral-even generalized parton distribution functions (GPDs) $H_q(x,ξ,Δ^2)$, $\widetilde{H}_q(x,ξ,Δ^2)$ and $E_q(x,ξ,Δ^2)$ at zero skewedness for $q=u$ and $d$ quarks. In our model $E_u$ and $E_d$ have opposite sign with similar size. Those GPDs are applied to calculate the orbital angular momentum (OAM) distributions, showing that $L_u(x)$ is positive, while $L_d(x)$ is consistent with zero compared with $L_u(x)$.…
We demonstrate that there exist two kinds of gauge-invariant decompositions of the nucleon spin, which are physically inequivalent. The quark and gluon orbital angular momenta (OAMs) appearing in one decomposition are basically the canonical OAMs, while the quark and gluon OAMs appearing in another decomposition are called the dynamical OAMs. It is shown that the dynamical OAMs of quarks and gluons in the nucleon can be related to definite high-energy deep-inelastic-scattering observables. On the other hand, we conjecture that the canonical OAMs of quarks and gluon in the nucleon are model-dep…
We study the chromodynamical gauge symmetry in relation to the internal spin structure of the nucleon. We show that 1) even in the helicity eigenstates the gauge-dependent spin and orbital angular momentum operators do not have gauge-independent matrix element; 2) the evolution equations for the gluon spin take very different forms in the Feynman and axial gauges, but yield the same leading behavior in the asymptotic limit; 3) the complete evolution of the gauge-dependent orbital angular momenta appears intractable in the light-cone gauge. We define a new gluon orbital angular momentum distrib…
The Sivers mechanism for the single-spin asymmetry in unpolarized lepton scattering from a transversely polarized nucleon is driven by the orbital angular momentum carried by its quark and gluon constituents, combined with QCD final-state interactions. Both quark and gluon mechanisms can generate such a single-spin asymmetry, though only the quark mechanism can explain the small single-spin asymmetry measured by the COMPASS collaboration on the deuteron, suggesting the gluon mechanism is small relative to the quark mechanism. We detail empirical studies through which the gluon and quark orbita…
Here we study the quark orbital angular momentum of the ground octet baryons employing an extended chiral constituent quark model, within which the baryon wave functions are taken to be superposition of the traditional $qqq$ and the $qqqq\bar{q}$ higher Fock components. Coupling between the two configurations is estimated using the $^3P_{0}$ quark-antiquark creation mechanism, and the corresponding coupling strength is determined by fitting the sea flavor asymmetry of the nucleon. The obtained numerical results show that the quark angular momentum of the nucleon, $Σ$, $Λ$ and $Ξ$ hyperons are …
The Goldberger-Treiman relation $M=2π/\sqrt{3}f^{\rm cl}_π$ where $M$ is the constituent quark mass in the chiral limit (cl) and $f^{\rm cl}_π$ the pion decay constant in the chiral limit predicts constituent quark masses of $m_u=328.8\pm 1.1$ MeV and $m_d=332.3\pm 1.1$ MeV for the up and down quark, respectively, when $f^{\rm cl}_π=89.8\pm 0.3$ MeV is adopted. Treating the constituent quarks as bare Dirac particles the following zero order values $μ^{(0)}}_p=2.850\pm 0.009$ and $μ^{(0)}}_n= -1.889\pm 0.006$ are obtained for the proton and neutron magnetic moments, leading to deviations from t…
I use the formalism of skewed parton distributions (SPD) to describe elastic form factors of the nucleon. The results are consistent with approximate dipole behavior of $G_{Mp}(Q^2)$ and recent JLab data for the ratio of the proton form factors $G_{Ep}/G_{Mp}$ at $Q^2\leq 3.5$ GeV$^2$. Using the Angular Momentum Sum Rule, I obtain a numerical estimate for the valence quark contributions to the nucleon spin at the level of 50%, with their orbital angular momentum {\it included}. When combined with polarized DIS measurements, this result indicated that the orbital angular momentum of quarks cont…
D-shell mixing in light baryons and its effect on the orbital motion
本文研究核子 D 壳轨道态混合，构建最大混合情形下的显式波函数，并重估磁矩与形状因子，解释了 Roper 共振特征但发现 D 态混合仍低于唯象预期。
We discuss the close relation between shadowing in deep-inelastic lepton-nucleus scattering and diffractive photo- and leptoproduction of hadrons from free nucleons. We show that the magnitude of nuclear shadowing at small Bjorken-x, as measured by the E665 and NMC collaborations, is directly related to HERA data on the amount of diffraction in the scattering from free nucleons.
The fundamental program in high energy spin physics focuses on the spin structure of the nucleon. The gluon and orbital angular momentum components of the nucleon spin are virtually unknown. The J_z=1/2 sum rule involves the integrated parton densities and can be used to extract information on the orbital angular momentum and its evolution. To avoid any bias on a model of Delta G, we assume that the gluon asymmetry, A=Delta G/G can be used to extract Delta G over a reasonable kinematic region. Combining the results for Delta G with the evolution equations, we can determine a theoretical expres…
We study the Wigner functions of the nucleon which provide multidimensional images of the quark distributions in phase space. These functions can be obtained through a Fourier transform in the transverse space of the generalized transverse-momentum dependent parton distributions. They depend on both the transverse position and the three-momentum of the quark relative to the nucleon, and therefore combine in a single picture all the information contained in the generalized parton distributions and the transverse-momentum dependent parton distributions. We focus the discussion on the distributio…
Quark spin and orbital angular momentum in the nucleon are calculated in symmetry-breaking chiral quark model. The results are compared with data and other models.
We estimate in the QCD sum rule approach the amount of the nucleon spin carried by the gluon angular momentum: the sum of the gluon spin and orbital angular momenta. The result indicates that gluons contribute at least one half of the nucleon spin at scale of 1 GeV^2.
It is shown that intrinsic orbital motion of the valence quarks has large influences on the spin-dependent as well as the spin-averaged nucleon structure functions. Its connection with the observed ``very small contribution of quark spin to nucleon spin'' and the observed violation of Gottfried sum rule is discussed.
The spin structure of nucleon at twist 2 and 3 levels is analyzed. The contribution of quark and gluon spins to nucleon spin are Lorentz invariant, while it is not sure for orbital angular momentum. The conserved fractional moments of transversity distribution are considered. The scenario of decoupled total angular momentum, determined predominantly by the unpolarized scattering, is discussed.
In a series of papers, we have established the existence of two gauge-invariant decompositions of the nucleon spin, which are physically nonequivalent. The orbital angular momenta of quarks and gluons appearing in these two decompositions are gauge-invariant dynamical orbital angular momenta and "generalized" canonical orbital angular momenta with gauge-invariance, respectively. The key quantity, which characterizes the difference between these two types of orbital angular momenta is what-we-call the {\it potential angular momentum}. We argue that the physical meaning of the potential angular …
We investigate the quark orbital angular momentum of the nucleon in the absence of gauge-field degrees of freedom, by using the concept of the Wigner distribution and the light-cone wave functions of the Fock state expansion of the nucleon. The quark orbital angular momentum is obtained from the phase-space average of the orbital angular momentum operator weighted with the Wigner distribution of unpolarized quarks in a longitudinally polarized nucleon. We also derive the light-cone wave function representation of the orbital angular momentum. In particular, we perform an expansion in the nucle…
We argue that due to Parity constraints, the helicity combination of the purely momentum space counterparts of the Wigner distributions -- the generalized transverse momentum distributions -- that describes the configuration of an unpolarized quark in a longitudinally polarized nucleon, can enter the deeply virtual Compton scattering amplitude only through matrix elements involving a final state interaction. The relevant matrix elements in turn involve light cone operators projections in the transverse direction, or they appear in the deeply virtual Compton scattering amplitude at twist three.…
Generalized transverse-momentum dependent parton distributions (GTMDs) encode the most general parton structure of hadrons. Here we focus on two twist-2 GTMDs which are denoted by $F_{1,4}$ and $G_{1,1}$ in parts of the literature. As already shown previously, both GTMDs have a close relation to orbital angular momentum of partons inside a hadron. However, recently even the mere existence of $F_{1,4}$ and $G_{1,1}$ has been doubted. We explain why this claim does not hold. We support our model-independent considerations by calculating the two GTMDs in the scalar diquark model and in the quark-…
We study the nucleon's partonic angular momentum (AM) content at peripheral transverse distances $b = \mathcal{O}(M_π^{-1})$, where the structure is governed by chiral dynamics. We compute the nucleon form factors of the energy-momentum tensor in chiral effective field theory (ChEFT) and construct the transverse densities of AM at fixed light-front time. In the periphery the spin density is suppressed, and the AM is predominantly orbital. In the first-quantized representation of ChEFT in light-front form, the field-theoretical AM density coincides with the quantum-mechanical orbital AM density…
How do the quark and gluon share the nucleon momentum? How does the nucleon spin distribute among its constituents? What means the quark and gluon momentum, spin and orbital angular momentum? These problems are analyzed and a solution is proposed based on {\it gauge invariance principle, canonical quantization rule and Poincar$\acute{e}$ covariance}.
It is shown that, based only upon two empirically known facts besides two reasonable theoretical postulates, we are inevitably led to a model-independent conclusion that the quark orbital angular momentum carries nearly half of the total nucleon spin at the low energy scale of nonperturbative QCD. Also shown are explicit model predictions for the forward limit of the unpolarized spin-flip GPDs, which are believed to give valuable information on the distributions of quark angular momentum inside the nucleon.
While the total orbital angular momentum (OAM) of a definite quark flavor in a longitudinally-polarized nucleon can be obtained through a sum rule involving twist-two generalized parton distribution (GPDs), its distribution as a function of parton momentum in light-front coordinates is more complicated to define and measure because it involves intrinsically twist-three effects. In this paper, we consider two different parton OAM distributions. The first is manifestly gauge invariant, and its moments are local operators and calculable in lattice QCD. We show that it can potentially be measured …
We calculate the hypercharge and flavor singlet axial couplings related to the spin of the nucleon in a constituent quark model. In addition to the standard one-body axial currents, the model includes two-body axial exchange currents. The latter are necessary to satisfy the Partial Conservation of Axial Current (PCAC) condition. For both axial couplings we find significant corrections to the standard quark model prediction. Exchange currents reduce the valence quark contribution to the nucleon spin and afford an interpretation of the missing nucleon spin as orbital angular momentum carried by …
The decomposition of the nucleon spin into that of quarks and gluons is related to twist-two GPDs according to Ji sum rule. Further decomposition into the helicity and the orbital angular momentum inevitably requires twist-three GPDs. In this paper we derive exact relations between twist-three GPDs and the canonical orbital angular momentum density of quarks and gluons, and check their consistency with the spin sum rule. Our work demonstrates that the complete decomposition of the nucleon spin fits well with the framework of perturbative QCD.
It is shown that, based only on two empirically known facts besides two reasonable theoretical postulates, we are inevitably led to a conclusion that the quark orbital angular momentum carries nearly half of the total nucleon spin. We also perform a model analysis to find that the quark spin fraction $ΔΣ$ is extremely sensitive to the pion mass, which may resolve the discrepancy between the observation and the prediction of the recent lattice QCD simulation carried out in the heavy pion region.
In analogy to the Altarelli-Parisi equation for the quark and gluon helicity contributions to the nucleon spin, we derive an evolution equation for the quark and gluon orbital angular momenta. The solution of the combined equations yields the asymptotic fractions of the nucleon spin carried by quarks and gluons: $3n_f/(16+3n_f)$ and $16/(16+3n_f)$, respectively, where $n_f$ is the number of active quark flavors. These are identical to the well-known asymptotic partitions of the nucleon momentum between quark and gluon contributions. We show that the axial-anomaly contribution to the quark heli…
The spin and orbital angular momentum carried by different quark flavors in the nucleon are calculated in the SU(3) chiral quark model with symmetry-breaking. The similar calculation is also performed for other octet and decuplet baryons. Furthermore, the flavor and spin contents for charm and anti-charm quarks are predicted in the SU(4) symmetry breaking chiral quark model.
Nucleon form factors play an especially important role in studying the dynamics of nucleons and explicit structure of the wave functions at arbitrary nucleon velocity. The purpose of the paper is to explain theoretically all four nucleon form factors measured experimentally in the cross section measurements (by the Rosenbluth method), yielding almost equal normalized form factors $G^p_E,G^p_M,G^n_M$, as well as in the polarization transfer experiments, where a strongly decreasing proton electric form factor has been discovered. It is shown, using relativistic hyperspherical formalism, that the…
Is gauge-invariant complete decomposition of the nucleon spin possible? Although it is a difficult theoretical question which has not reached a complete consensus yet, a general agreement now is that there are at least two physically inequivalent gauge-invariant decompositions (I) and (II) of the nucleon. %The one is a nontrivial gauge-invariant %generalization of the Jaffe-Manohar decomposition. %The other is an extension of the Ji decomposition, which allows %a gauge-invariant decomposition of the total gluon angular %momentum into the intrinsic spin and orbital parts. In these two decomposi…
We discuss in detail the spatial distribution of angular momentum inside the nucleon. We show that the discrepancies between different definitions originate from terms that integrate to zero. Even though these terms can safely be dropped at the integrated level, they have to be taken into account at the density level. Using the scalar diquark model, we illustrate our results and, for the first time, check explicitly that the equivalence between kinetic and canonical orbital angular momentum persists at the density level, as expected in a system without gauge degrees of freedom.
Parton distributions in impact parameter space, which are obtained by Fourier transforming GPDs, are exhibit a significant deviation from axial symmetry when the target and/or quark is transversely polarized. From this deformation, we present an intuitive derivation of the Ji relation. In a scalar diquark model and in QED, we compare the Ji and Jaffe-Manohar decompositions of the nucleon spin. Using the MIT bag model, we estimate spectator effects through the presence of the gluon vector potential in the definitions of the quark orbital angular momentum.
At the low energy, {\sl hadronic}, scale we calculate Orbital Angular Momentum (OAM) twist-two parton distributions for the relativistic MIT bag model and for non-relativistic quark models. We reach the scale of the data by leading order evolution in perturbative QCD. We confirm that the contribution of quarks and gluons OAM to the nucleon spin grows with $Q^2$, and it can be relevant at the experimental scale, even if it is negligible at the hadronic scale, irrespective of the model used. The sign and shape of the quark OAM distribution at high $Q^2$ may depend strongly on the relative size o…
We discuss the quark phase-space or Wigner distributions of the nucleon which combine in a single picture all the information contained in the generalized parton distributions and the transverse-momentum dependent parton distributions. In particular, we present results for the distribution of unpolarized quarks in a longitudinally polarized nucleon obtained in a light-front constituent quark model. We show how the quark orbital angular momentum can be extracted from the Wigner distributions and compare it with alternative definitions.
Chiral anomaly and nucleon spin decomposition in the chiral quark soliton model
本研究在手征夸克孤子模型中引入手征反常诱导的味单态赝标量场，计算核子自旋分解，发现该场增强夸克螺旋度贡献，增大质量则抑制增强并增加轨道贡献，但整体影响适度，不改变传统模型结论。
This note collects formulas relative to the energy dependence of the cross sections for open charm and $J/ψ$ production for $NN$-collisions at threshold. It is a basis for the best input to MonteCarlo calculations, for associative charm particle production in nucleon-nucleon, nucleon-nucleus, ion-ion and proton-antiproton collisions.
The present understanding of the spin structure of the nucleon is briefly reviewed. The main focus is on parton helicity distributions, orbital angular momentum of partons as defined through generalized parton distributions, as well as single spin asymmetries and time-reversal odd correlation functions.
This talk consists of four parts. In part one, I give an elementary discussion on constructing a Lorentz-invariant spin sum rule for the nucleon. In part two, I discuss a gauge-dependent spin sum rule, explore its relation with the polarized gluon distribution, and introduce the complete evolution equation for the spin structure. In part three, I consider a gauge-invariant spin sum rule and the related evolution equation. The solution of the equation motivates the possibility that half of the nucleon spin may be carried by gluons at low energy scales. In the final part, I discuss deeply-virtua…
We prove that the {\em gauge dependent} gluon spin, gluon and quark orbital angular momenta operators have {\em gauge invariant} expectation values on hadron states with {\em definite} momentum and polarization, therefore the conventional decomposition of nucleon spin into contributions from the spin and orbital angular momentum of quark and gluon is {\em gauge independent}. Similar conclusions apply to the {\em gauge dependent} quark momentum and kinetic energy operators, and accordingly nucleon momentum and mass structures.
We present the numerical solution of the leading order QCD evolution equation for the orbital angular momentum distributions of quarks and gluons and discuss its implications for the nucleon spin sum rule. We observe that at small-$x$, the gluon helicity and orbital angular momentum distributions are roughly of the same magnitude but with opposite signs, indicating a significant cancellation between them. A similar cancellation occurs also in the quark sector. We explain analytically the reason for this cancellation.
The proton is a composite object with spin one-half, understood to contain highly relativistic spin one-half quarks exchanging spin-one gluons, each possibly with significant orbital angular momenta. While their fundamental interactions are well described by Quantum ChromoDynamics (QCD), our standard theory of the strong interaction, nonperturbative calculations of the internal structure of the proton based directly on QCD are beginning to provide reliable results. Most of our present knowledge of the structure of the proton is based on experimental measurements interpreted within the rich fra…
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