Final-state interactions in semi-inclusive deep inelastic scattering off the Deuteron
Wim Cosyn, Misak Sargsian
备注：
27 pages, 8 figures. Corrected typos, section II.E has been expanded a bit. Figures have been updated to conform to the publication guidelines. Results and conclusions haven't changed. Accepted for publication in PRC
Semi-inclusive deep inelastic scattering off the Deuteron with production of a slow nucleon in recoil kinematics is studied in the virtual nucleon approximation, in which the final state interaction (FSI) is calculated within general eikonal approximation. The cross section is derived in a factorized approach, with a factor describing the virtual photon interaction with the off-shell nucleon and a distorted spectral function accounting for the final-state interactions. One of the main goals of the study is to understand how much the general features of the diffractive high energy soft rescattering accounts for the observed features of FSI in deep inelastic scattering(DIS).
Comparison with the Jefferson Lab data shows good agreement in the covered range of kinematics. Most importantly, our calculation correctly reproduces the rise of the FSI in the forward direction of the slow nucleon production angle.
By fitting our calculation to the data we extracted the $W$ and $Q^2$ dependences of the total cross section and slope factor of the interaction of DIS products, $X$, off the spectator nucleon. This analysis shows the $XN$ scattering cross section rising with $W$ and decreasing with an increase of $Q^2$. Finally, our analysis points at a largely suppressed off-shell part of the rescattering amplitude.
The possibility of using a tensor polarized deuteron target in electroproduction reactions creates new opportunities for studying different phenomena related to the short-range hadronic and nuclear physics. The use of tensor polarized deuteron allows to isolate smaller than average inter-nucleon distances for the bound two-nucleon system. In this report we consider several of high $Q^2$ reactions which are particularly sensitive to the short-range two-nucleon configurations in the deuteron. The one is the relativistic dynamics of electron-bound-nucleon scattering which can be studied in both i…
The production of slow nucleons in semi-inclusive deep inelastic electron scattering off the deuteron is investigated in the region $x \gsim 0.3$ for kinematical conditions accessible at $HERA$. Within the spectator mechanism the semi-inclusive cross section exhibits a scaling property, which can be used as a model-independent test of the dominance of the spectator mechanism itself, providing in this way an interesting tool to investigate the neutron structure function. The possibility of extracting model-independent information on the neutron to proton structure function ratio from semi-inclu…
We calculate the effects of the target mass on deep-inelastic electron scattering on targets of spin one such as the deuteron, focusing on the novel structure functions that enter. It is important to understand these effects given the recent HERMES measurements of the $b_1(x,Q^2)$ structure function of the deuteron. We also derive the spin-one analogues of the Wandzura-Wilczek relations and discuss possible calculations in lattice QCD.
The outline of the presentation is as follows: I. Nucleon Distribution in the Deuteron from $e-D$ and $p-D$ Processes. II. Quark Distribution in Deuteron from its Fragmentation to Pions and Deep Inelastic $e-D$ scattering. III.Difference of These Two Distributions and Its Possible Understanding.
Due to the lack of a free neutron target the structure function of the neutron cannot be measured directly and is therefore extracted from deuteron and proton DIS data. Because the deuteron is a bound nuclear system, in order to extract the neutron structure function, one needs to apply model dependent theoretical corrections which dominate the uncertainty at the large xB region. We present here a correlation between the magnitude of the EMC effect and the amount of two nucleon Short Range Correlation (2N-SRC) pairs in nuclei. Using this correlation we propose a phenomenological procedure to e…
We investigate the results of the charged kaon multiplicities off the deuteron target from HERMES and COMPASS experiments. The discrepancy of the two data sets cannot be explained by different $Q^2$ values. Furthermore we examine the empirical parametrization of the fragmentation functions, DSS2017, and find that the agreement between the theoretical predictions and the HERMES data is less satisfactory as claimed.
We compute contributions to the parity-violating (PV) inelastic electron-deuteron scattering asymmetry arising from hadronic PV. While hadronic PV effects can be relatively important in PV threshold electro- disintegration, we find that they are highly suppressed at quasielastic kinematics. The interpretation of the PV quasielastic asymmetry is, thus, largely unaffected by hadronic PV.
We study nuclear effects on the deuteron in the deep inelastic regime using the newest available data analyzing their $Q^2$ dependence. We conclude that precise EMC ratios for large $Q^2$ (> 30, GeV$^2$) cannot be obtained without considering these nuclear effects. For this purpose we use a scheme which parametrizes these effects in a simple manner and compare our results with other recent proposals.
The detection of slow nucleons in semi-inclusive electron scattering processes off the deuteron, 2H(e, e'N)X, is a very promising tool for the extraction of model-independent information on the neutron structure function F2n(x, Q**2). The key point is the occurrence of a peculiar scaling property of the semi-inclusive cross section, which holds in the deep-inelastic regime as well as in the nucleon-resonance production regions. Moreover, the possibility to get unique information on the neutron asymmetry A1n(x, Q**2) from polarised 2H_pol(e_pol, e'N)X processes is illustrated.
The production of slow nucleons in semi-inclusive deep inelastic electron scattering off the deuteron is investigated in the region $x \gsim 0.3$. It is shown that within the spectator mechanism the semi-inclusive cross section exhibits a scaling property even at moderate values of $Q^2$ ($\sim$ few $(GeV/c)^2$) accessible at present facilities, like $CEBAF$. Such a scaling property can be used as a model-independent test of the dominance of the spectator mechanism itself and provides an interesting tool to investigate the neutron structure function. The possibility of extracting model-indepen…
A covariant field theoretical approach to deep inelastic scattering on the deuteron is presented. The deuteron structure function is calculated in terms of the Bethe-Salpeter amplitude. Numerical calculations for the nucleon contribution are made with a realistic model of the $NN$-interaction, including $π$-, $ρ$-, $ω$-, $η$-, $δ$- and $σ$-mesons, and results are compared with previous non-relativistic calculations.
Deep-inelastic scattering (DIS) from a tensor polarized deuteron is sensitive to possible non-nucleonic components of the deuteron wave function. To accurately estimate the size of the nucleonic contribution, final-state interactions (FSIs) need to be accounted for in calculations. We outline a model that, based on the diffractive nature of the effective hadron-nucleon interaction, uses the generalized eikonal approximation to model the FSIs in the resonance region, taking into account the proton-neutron component of the deuteron. The calculation uses a factorized model with a basis of three r…
We study the effect of parton angular momentum on the twist-four correction to the left-right asymmetry in the electron-deuteron parity-violating deep inelastic scattering (PVDIS). We show that this higher-twist correction is transparent to the dynamics of parton angular momentum needed to account for the Sivers and Boer-Mulders functions and spin-independent parton distribution functions. A sufficiently precise measurement of the PVDIS asymmetry may, thus, provide additional information about the parton dynamics responsible for nucleon spin.
We present calculations of deuteron elastic and inelastic breakup cross sections and angular distributions at deuteron energies below 100 MeV obtained using the post-form DWBA approximation. The elastic breakup cross section was extensively studied in the past. Very few calculations of inelastic breakup have been performed, however. We also analyze the angular momentum - energy distributions of the cross section for formation of the compound nucleus after inelastic breakup.
The spin-dependent structure function of the deuteron, g_1D, is calculated within a covariant framework. The off-shell structure of the bound nucleon gives corrections to the convolution model at a level of half a percent for x below 0.7, increasing to more than five percent at larger x. Overall, the dominant source of error comes from the lack of knowledge associated with the deuteron D-state, which may introduce an uncertainty in the neutron spin structure function, g_1n, extracted from deuterium data of up to ten percent for x around 0.2.
We explore the role of final-state interactions (FSI) in inclusive deep-inelastic scattering from the deuteron. Relating the inclusive cross section to the deuteron forward virtual Compton scattering amplitude, a general formula for the FSI contribution is derived in the generalized eikonal approximation, utilizing the diffractive nature of the effective hadron-nucleon interaction. The calculation uses a factorized model with a basis of three resonances with mass $W<2$ GeV and a continuum contribution for larger $W$ as the relevant set of effective hadron states entering the final-state intera…
Background: DIS on the polarized deuteron with detection of a proton in the nuclear breakup region (spectator tagging) represents a unique method for extracting the neutron spin structure functions and studying nuclear modifications. The tagged proton momentum controls the nuclear configuration during the DIS process and enables a differential analysis of nuclear effects. Such measurements could be performed with the future electron-ion collider (EIC) and forward proton detectors if deuteron beam polarization could be achieved. Purpose: Develop theoretical framework for polarized deuteron DIS …
The role of relativistic off-mass-shell kinematics in the deep inelastic scattering on deuteron is analyzed. It is shown that the relativistic impulse approximation reproduces effects from the binding, Fermi motion, and two-nucleon contribution. The nonrelativistic limit of the deuteron structure function is in agreement with the nonrelativistic calculations.
Inclusive $K^+$ and exclusive $K^+Y$ photoproduction on the deuteron are investigated theoretically. Modern hyperon-nucleon forces and a recently updated kaon photoproduction operator for the $γ+N\to K^++Y$ process are used. Sizable effects of the hyperon-nucleon final state interaction are found near the $K^+ΛN$ and $K^+ΣN$ thresholds in the inclusive reaction. Angular distributions for the exclusive process show clear $YN$ final state interaction effects in certain kinematic regions. Precise data especially for the inclusive process around the $K^+ΣN$ threshold would help to clarify the stre…
We give an overview of a model to describe deep-inelastic scattering (DIS) off the deuteron with a spectator proton, based on the virtual nucleon approximation (VNA). The model accounts for the final-state interactions (FSI) of the DIS debris with the spectator proton. Values of the rescattering cross section are obtained by fits to high-momentum spectator data. By using the so-called "pole extrapolation method", free neutron structure functions can be obtained by extrapolating low-momentum spectator proton data to the on-shell neutron pole. We apply this method to the BONuS data set and find …
We analyze nuclear effects in the deep inelastic scattering on deuteron in the framework of the covariant approach. It is shown that this approach gives us the way to investigate the role of relativistic effects in the deep inelastic scattering (DIS) on deuteron, such as the relativistic kinematics and the off-mass-shell behavior of the Compton amplitude of nucleon in the consistent manner. We have obtained that taking into account of the nucleon amplitude off-mass-shell behavior gives us an analog of the interaction corrections in nonrelativistic models.
We present in this paper a formalism for deuteron-induced inclusive reactions. We disentangle direct elastic breakup contributions from other processes (which we generically call non-elastic breakup) implying a capture of the neutron both above and below the neutron emission threshold. The reaction is described as a two step process, namely the breakup of the deuteron followed by the propagation of the neutron-target system driven by an optical potential. The final state interaction between the neutron and the target can eventually form an excited compound nucleus. Within this context, the dir…
The problem of the calculation of inclusive breakup cross sections in nuclear reactions is reexamined. For that purpose, the post-form theory proposed by Ichimura, Austern and Vincent [Phys. Rev. C32, 431 (1985) is revisited, and an alternative derivation of the non-elastic breakup part of the inclusive breakup is presented, making use of the coupled-channels optical theorem. Using the DWBA version of this model, several applications to deuteron and $^6$Li reactions are presented and compared with available data. The validity of the zero-range approximation of the DWBA formula is also investig…
The effects of the final state interaction in slow proton production in semi inclusive deep inelastic scattering processes off nuclei, A(e,e'p)X, are investigated in details within the spectator and target fragmentation mechanisms; in the former mechanism, the hard interaction on a nucleon of a correlated pair leads, by recoil, to the emission of the partner nucleon, whereas in the latter mechanism proton is produced when the diquark, which is formed right after the visrtual photon-quark interaction, captures a quark from the vacuum. Unlike previous papers on the subject, particular attention …
The semi-inclusive deep inelastic scattering of electrons off the deuteron ($^2H \equiv D$) and $^3He$ with detection of slow protons and deuterons, respectively, i.e. the processes $D(e,e'p)X$ and $^3He(e,e'D)X$, are calculated within the spectator mechanism, taking into account the final state interaction of the hadronizing quark with the detected protons and deuterons, respectively. It is shown that by a proper choice of the kinematics the origin of the EMC effect and the details of the interaction between the hadronizing quark and the nuclear medium can be investigated at a level which can…
We construct a language for identifying kinematical regions of transversely differential semi-inclusive deep inelastic scattering cross sections with particular underlying partonic pictures, especially in regions of moderate to low $Q$ where sensitivity to kinematical effects outside the usual very high energy limit becomes non-trivial. The partonic pictures map to power law expansions whose leading contributions ultimately lead to well-known QCD factorization theorems. We propose methods for estimating the consistency of any particular region of overall hadronic kinematics with the kinematics…
Cross sections for inclusive neutrino scattering off deuteron induced by neutral and charge-changing weak currents are calculated from threshold up to 150 MeV energies in a chiral effective field theory including high orders in the power counting. Contributions beyond leading order (LO) in the weak current are found to be small, and increase the cross sections obtained with the LO transition operators by a couple of percent over the whole energy range (0--150) MeV. The cutoff dependence is negligible, and the predicted cross sections are within $\sim 2$\% of, albeit consistently larger than, c…
By considering the $x$-dependence of $π^+$, $π^-$, $K^+$, $K^-$, $Λ$, $\barΛ$, $p$, $\bar{p}$ hadron productions in charged lepton semi-inclusive deep inelastic scattering off nuclear target (using Fe as an example) and deuteron D target, % at $Q^2=5$ GeV$^2$, we find that $(\barΛ^A/Λ^A)/(\barΛ^D/Λ^D)$ and $({\bar{p}}^A/{p}^A)/({\bar{p}}^A/p^A)$ are ideal to figure out the nuclear sea content, which is predicted to be different by different models accounting for the nuclear EMC effect.
The effects of the final state interaction (FSI) in semi inclusive deep inelastic electron scattering processes $A(e,e'p)X$ off nuclei are investigated in details. Proton production is described within the spectator and the target fragmentation mechanisms whose relevance to the experimental study of the deep inelastic structure functions of bound nucleons and the non perturbative hadronization process is analyzed. Particular attention is paid to the deuteron target within kinematical conditions corresponding to the available and forthcoming experimental data at Jlab. We argue that there are ki…
The contributions to the reaction cross section from the elastic and inelastic breakup processes, calculated within the post-form distorted wave Born-approximation theory, are used as constraints to determine the contributions to the imaginary part of the deuteron optical potentials (IPDOP) due to the breakup channels. The Coulomb part of this potential due to the elastic breakup process is seen to account for the long range absorption in the optical potential. The nuclear parts of the IPDOP due to the elastic and inelastic breakup modes peak in different regions of the nuclear surface, with t…
The semi-inclusive deep inelastic scattering of electrons off a nucleus A with detection of a slow nucleus (A-1) in the ground or low excitation states, i.e. the process A(e,e'(A-1))X, can provide useful information on the origin of the EMC effect and the mechanisms of hadronization. The theoretical description of the process is reviewed and the results of several calculations on few-body systems and complex nuclei are presented.
We discuss a hybrid model of the proton and neutron inelastic structure functions which incorporates both the partonic and the nucleon resonance structures and applicable in a wide region of the invariant mass $W$ of produced hadronic states. Focusing at the typical kinematics of JLab experiments we compute the deuteron structure functions and demonstrate good performance of the model against data. We perform systematic study of the ratios $F_2^n/F_2^d$ and $(F_2^p+F_2^n)/F_2^d$ for both, the deep-inelastic and the resonance, region in the context of recent measurements of BoNuS experiment at …
Inclusive and semi-inclusive deep inelastic leptoproduction offers possibilities to study details of the quark and gluon structure of the hadrons involved. In many of these experiments polarization is an essential ingredient. We also emphasize the dependence on transverse momenta of the quarks, which leads to azimuthal asymmetries in the produced hadrons.
In front-form dynamics the current operator can be constructed from auxiliary operators, defined in a Breit frame where initial and final three-momenta of the system are directed along the $z$ axis. Poincaré covariance constraints reduce for auxiliary operators to the ones imposed only by kinematical rotations around the $z$ axis. Elastic and transition form factors can be extracted without any ambiguity and in the elastic case the continuity equation is automatically satisfied, once Poincaré, ${\cal P}$ and ${\cal T}$ covariance, together with hermiticity, are imposed. Applications to deep in…
Background: Neutrino-nucleus quasi-elastic scattering is crucial to interpret the neutrino oscillation results in long baseline neutrino experiments. There are rather large uncertainties in the cross section, due to insufficient knowledge on the role of two-body weak currents. Purpose: Determine the role of two-body weak currents in neutrino-deuteron quasi-elastic scattering up to GeV energies. Methods: Calculate cross sections for inclusive neutrino scattering off deuteron induced by neutral and charge-changing weak currents, from threshold up to GeV energies, using the Argonne $v_{18}$ poten…
Measurement of angular distributions and energy spectra of $α$ and deuterons through breakup, transfer and incomplete fusion processes to dis-entangle their relative contributions and to investigate relative importance of breakup-fusion compared to transfer. Inclusive $α$ production cross-sections have been measured for $^6$Li + $^{51}$V system near Coulomb barrier energies. Theoretical calculations for estimation of various reaction channels contributing to $α$ production have been performed with finite range coupled reaction method using \textsc{FRESCO} code. The cross-sections from non-capt…
We summarize the discussion on the possibilities of doing inclusive and semi-inclusive deep inelastic scattering experiments at CEBAF with beam energy of the order of 10 GeV.
Using the covariant spectator theory and the transversity formalism, the unpolarized, coincidence cross section for deuteron electrodisintegration, $d(e,e'p)n$, is studied. The relativistic kinematics are reviewed, and simple theoretical formulae for the relativistic impulse approximation (RIA) are derived and discussed. Numerical predictions for the scattering in the high $Q^2$ region obtained from the RIA and five other approximations are presented and compared. We conclude that measurements of the unpolarized coincidence cross section and the asymmetry $A_ϕ$, to an accuracy that will distin…
We calculate $g_{1T}$ and $h_{1L}^\perp$, two of the eight leading twist transverse momentum dependent parton distributions (TMDs), in the light-cone quark-diquark model. The new TMDs can be measured through semi-inclusive deep inelastic scattering (SIDIS). We present predictions of the single and double spin asymmetries related to $g_{1T}$ and $h_{1L}^\perp$ in SIDIS at HERMES, COMPASS, and JLab kinematics respectively.
We investigate the sensitivity of the non-exclusive nucleon induced deuteron breakup reaction to the three-nucleon interaction and distributions of three-nucleon force effects in inclusive spectra. To this end we solve the three-nucleon Faddeev equation at a number of incoming nucleon laboratory energies using the CD Bonn nucleon-nucleon interaction alone or combined with the 2π-exchange Tucson-Melbourne three-nucleon force. Based on these solutions energy spectra of an outgoing nucleon, at a specified detection angle as well as spectra integrated over that angle, are calculated. By integratin…
We consider electron-deuteron deep-inelastic scattering (DIS) with detection of a proton in the nuclear fragmentation region ("spectator tagging") as a method for extracting the free neutron structure functions and studying their nuclear modifications. Such measurements could be performed at a future Electron-Ion Collider (EIC) with suitable forward detectors. The measured proton recoil momentum (~< 100 MeV in the deuteron rest frame) specifies the deuteron configuration during the high-energy process and permits a controlled theoretical treatment of nuclear effects. Nuclear and nucleonic stru…
We present the details of a new factorized approach to semi-inclusive deep-inelastic scattering which treats QED and QCD radiation on equal footing, and provides a systematically improvable approximation to the extraction of transverse momentum dependent parton distributions. We demonstrate how the QED contributions can be well approximated by collinear factorization, and illustrate the application of the factorized approach to QED radiation in inclusive scattering. For semi-inclusive processes, we show how radiation effects prevent a well-defined "photon-nucleon" frame, forcing one to use a t…
The helicity dependence of the inclusive $π^-$ photoproduction reaction from the deuteron in the $Δ$(1232)-resonance region is investigated with inclusion of final-state nucleon-nucleon rescattering ($NN$-FSI). For the elementary $π$-production operator an effective Lagrangian model which includes the standard pseudovector Born terms and a contribution from the $Δ$-resonance is used. The half-off-shell $NN$-scattering matrix is obtained from a separable representation of a realistic $NN$-interaction. The differential polarized cross-section difference for parallel and antiparallel helicity sta…
Inspired by a recent study of Iancu, Mueller and Triantafyllopoulos [1] and earlier papers by Golec-Biernat and Wusthoff [2,3], we propose semi-inclusive diffractive deep inelastic scattering (SIDDIS) to investigate the gluon tomography in the nucleon and nuclei at small-$x$. The relevant diffractive quark and gluon parton distribution functions (DPDF) can be computed in terms of the color dipole S-matrices in the fundamental and adjoint representations, respectively. Novel correlations from the gluon tomography in the dipole S-matrix can be experimentally studied through the DPDFs in these pr…
The differential cross-sections for elastic and inelastic neutrino-deuteron scattering are calculated analytically using nucleon-nucleon effective field theory. For elastic scattering, the deuteron axial form factor and the deuteron strange magnetic moment form factor are computed to next-to-leading order, including two-body currents. For inelastic scattering, two neutral current processes nu+d->nu+n+p, nubar+d->nubar+n+p and one charged current process nubar+d->(e^+)+n+n are computed to next-to-leading order. These depend on an isovector axial two-body matrix element whose value is yet to be …
The effects of the final state interaction (FSI) in semi-exclusive deep-inelastic scattering of electrons off the deuteron are analyzed paying particular attention to two extreme kinematical regions: i) the one where FSI effects are minimized, so that the quark distribution of bound nucleons could be investigated, and ii) the one where the re-interaction of the produced hadrons with the spectator nucleon is maximized, which would allow one to study the mechanism of hadronization of highly virtual quark. It is demonstrated that when the recoiling spectator nucleon is detected in the backward he…
Motivated by recently observed tension between $O\left(α_s^2\right)$ calculations of very large transverse momentum dependence in both semi-inclusive deep inelastic scattering and Drell-Yan scattering, we repeat the details of the calculation through $O\left(α_s^2\right)$ transversely differential cross section. The results confirm earlier calculations, and provide further support to the observation that tension exists with current parton distribution and fragmentation functions.
We extend the relativistic plane--wave impulse approximation formalism to incorporate a specific class of relativistic interference effects for use in describing inclusive electrodisintegration of $^2$H. The role of these ``exchange'' terms for the various response functions accessible in parity--conserving and --violating inclusive processes is investigated and shown, especially for the latter, to have important consequences for experiment. An extension to a simple quasi--deuteron model is also considered.
Deuteron disintegration by charged-current neutrino (CC$ν$) scattering offers the possibility to determine the energy of the incident neutrino by measuring in coincidence two of the three resulting particles: a charged lepton (usually a muon) and two protons, where we show that this channel can be isolated from all other, for instance, from those with a pion in the final state. We discuss the kinematics of the process for several detection scenarios, both in terms of kinematic variables that are natural from a theoretical point of view and others that are better matched to experimental situati…
A suggestion is made for distinguishing 2N and 6q short range correlations within the deuteron. The suggestion depends upon observing high momentum backward nucleons emerging from inelastic electromagnetic scattering from a deuteron target. A simple model is worked out to see the size of effects that may be expected.
We exploit a relationship between the Structure Functions of nucleons, the physical deuteron and of a deuteron, composed of point-nucleons to compute angular distributions of inclusive cross sections of 4.05 GeV electrons. We report general agreement with data and interpret the remaining discrepancies. We discuss the potential of the data for information on neutron structure functions $F_k^n(x,Q^2)$ and the static form factor $G_M^n(Q^2)$.
The rescattering contribution to the inclusive and exclusive deuteron electro-disintegration at the values of the Bjorken scaling variable $x=1$, as well as in the so called {\it cumulative} region ($x>1$) is calculated within a relativistic approach based on the Feynman diagram formalism taking into account colour transparency effects by the inclusion of the {\it finite formation time} (FFT) of the ejected nucleon via the introduction of the dependence of the scattering amplitude of the ejectile upon its virtuality. In the cumulative region the FFT effects which result from the real part of t…
We calculate the multiple scattering effect on single hadron production in semi-inclusive lepton-nucleus deeply inelastic scattering. We show that the quantum interference of multiple scattering amplitudes leads to suppression in hadron productions. At the leading power in medium length, the suppression can be approximately expressed in terms of a shift in $z$ of the fragmentation function $D(z)$, and could be therefore interpreted as the collisional energy loss. We compare our calculation with existing experimental data. We also discuss the effect of quark mass on the suppression. Our approac…
We present a detailed study of nuclear corrections in the deuteron (D) by performing an analysis of data from deep inelastic scattering off proton and D, dilepton pair production in $pp$ and pD interactions, and W^+- and Z boson production in pp and p pbar collisions. In particular, we discuss the determination of the off-shell function describing the modification of the parton distribution functions in bound nucleons in the context of global QCD fits. Our results are consistent with the ones obtained independently from the study of deep inelastic scattering data off heavy nuclei with $A\geq4$…
