$^{15}$C: from Halo-EFT structure to the study of transfer, breakup and radiative-capture reactions
Laura Moschini Jiecheng Yang Pierre Capel
nucl-th

  
   
    [1] 
    [1] 
     [1]  [1]  [1] #1et al. [1] (<ref>) [1] Eq. (<ref>) [1] Ref. <cit.> [2] Refs. <cit.> [1] Sec. <ref> [1] Chapter <ref> [1] Appendix <ref> [1] Table <ref> [1] Fig. <ref> [1] ^#1Schrödinger  [2] ⟶_#1→#2 [1]#1 [1]#1[2]∂ #1∂ #2[2]d#1d#2
    
    laura.moschini@ulb.ac.bePhysique Nucléaire et Physique Quantique (C.P. 229)
    
    Université libre de Bruxelles (ULB), 50 avenue F.D. Roosevelt, B-1050 Brussels, Belgiumjiecyang@ulb.ac.bePhysique Nucléaire et Physique Quantique (C.P. 229)
    
    Université libre de Bruxelles (ULB), 50 avenue F.D. Roosevelt, B-1050 Brussels, BelgiumAfdeling Kern-en Stralingsfysica, Celestijnenlaan 200d-bus 2418, 3001 Leuven, Belgiumpcapel@uni-mainz.deInstitut für Kernphysik, 
    Johannes Gutenberg-Universität Mainz,
    Johann-Joachim-Becher Weg 45, 
    D-55099 Mainz, GermanyPhysique Nucléaire et Physique Quantique (C.P. 229)
    
    Université libre de Bruxelles (ULB), 50 avenue F.D. Roosevelt, B-1050 Brussels, Belgium
      Background 
    Aside from being a one-neutron halo nucleus, ^15C is interesting because it is involved in reactions of relevance for several nucleosynthesis scenarios.
    
      Purpose
    The aim of this work is to analyze various reactions involving ^15C, using a single structure model based on halo effective field theory (Halo EFT) following the excellent results obtained in [P. Capel, D. R. Phillips, and H.-W. Hammer, Phys. Rev. C  98, 034610 (2018)].
    
      Method 
    To develop a Halo-EFT model of ^15C at next to leading order (NLO), we first extract the asymptotic normalization coefficient (ANC) of its ground state by analyzing ^14 C(d,p)^15 C transfer data at low energy using the method developed in [J. Yang and P. Capel, Phys. Rev. C  98, 054602 (2018)].
    Using the Halo-EFT description of ^15C constrained with this ANC, we study the ^15C Coulomb breakup at high (605 MeV/nucleon) and intermediate (68 MeV/nucleon) energies using eikonal-based models with a consistent treatment of nuclear and Coulomb interactions at all orders, and which take into account proper relativistic corrections. 
    Finally, we study the ^14 C(n,γ)^15 C radiative capture.
    
      Results 
    Our theoretical cross sections are in good agreement with experimental data for all reactions, thereby assessing the robustness of the Halo-EFT model of this nucleus. 
    Since a simple NLO description is enough to reproduce all data, the only nuclear-structure observables that matter are the ^15C binding energy and its ANC, showing that all the reactions considered are purely peripheral.
    In particular, it confirms the value we have obtained for the ANC of the ^15C ground state: C^2_1/2^+=1.59±0.06 fm^-1. 
    Our model of ^15C provides also a new estimate of the radiative-capture cross section at astrophysical energy: σ_n,γ(23.3  keV)=4.66±0.14 μb.
    
    
      Conclusions
    Including a Halo-EFT description of ^15C within precise models of reactions is confirmed to be an excellent way to relate the reaction cross sections and the structure of the nucleus.
    Its systematic expansion enables us to establish how the reaction process is affected by that structure and deduce which nuclear-structure observables are actually probed in the collision.
    From this, we can infer valuable information on both the structure of ^15C and its synthesis through the ^14 C(n,γ)^15 C radiative capture at astrophysical energies.
    
    
    ^15C: from halo effective field theory structure
     to the study of transfer, breakup and radiative-capture reactions
        Pierre Capel
        December 30, 2023
    ===================================================================================================================
    
    
    
    § INTRODUCTION
    
    The nucleus ^15C is interesting for various reasons.
    On a nuclear-structure viewpoint, ^15C is one of the best known one-neutron halo nuclei <cit.>.
    Due to its small one-neutron separation energy [S_n(^15 C)=1.218 MeV], 
    the ground state of ^15C is mostly described as a two-body structure, 
    in which the valence neutron is loosely bound in a 1s_1/2 orbital to a ^14C in its 0^+ ground state.
    Thanks to its loose binding and the fact that it sits in an l=0 orbital, the valence neutron exhibits a high probability of presence at a large distance from the other nucleons.
    It therefore forms like a diffuse halo surrounding a compact core <cit.>.
    The existence of halos in some nuclei challenges our view of the nucleus, which is usually seen as a compact object with a nucleon density at saturation.
    Halo nuclei, including ^15C, are thus the focus of many experimental and theoretical studies <cit.>.
    
    The study of ^15C has also applications in nuclear astrophysics.
    Its synthesis through one-neutron radiative capture by ^14C has been suggested to be part of neutron-induced CNO cycles, which take place in the helium-burning zone of asymptotic-giant-branch (AGB) stars <cit.>.
    This ^14C(n,γ)^15C reaction is also the doorstep to the production of heavy elements in inhomogeneous big-bang nucleosynthesis <cit.> and it has been shown to be part of possible reaction routes in the nuclear chart during the r process in Type II supernovæ <cit.>.
    It is therefore necessary to have a reliable estimate of the cross section for this radiative capture at astrophysical energy, and hence to better understand the structure of ^15C.
    
    Because ^15C exhibits a short lifetime, its structure cannot be probed with usual spectroscopic techniques.
    This nucleus is therefore mostly studied through reactions.
    Transfer, such as (d,p), measured in both direct and inverse kinematics, has been used to infer the single-particle structure of ^15C <cit.>.
    In breakup, the lose binding of the valence neutron to the core is broken up during the collision of the nucleus on a target, hence revealing its internal core-n structure.
    Various experimental campaigns have been set up to measure the inclusive breakup—also known as knockout—of ^15C on light targets at intermediate beam energies <cit.>.
    In these measurements, only ^14C is detected after the reaction, and information pertaining to the single-particle structure of ^15C is inferred from the analysis of the parallel-momentum distribution of the core.
    In Refs. <cit.>, the Coulomb (exclusive) breakup of ^15C has been measured.
    In that case, both the ^14C core and the halo neutron are detected in coincidence after the dissociation of the ^15C projectile on a Pb target.
    Being dominated by the Coulomb interaction, this reaction process is rather clean as it exhibits little dependence on the choice of the optical potentials used to describe the nuclear interaction between the projectile constituents (core and n) with the target.
    
    In addition to its interest in the study of the halo structure of ^15C, Coulomb breakup has also been suggested as an indirect method to deduce the cross section for the ^14 C(n,γ)^15C radiative capture at low energies <cit.>.
    The idea behind the Coulomb-breakup method is that this dissociation, which is often described as resulting from the exchange of virtual photons between the projectile and the heavy target <cit.>, can be seen as the time-reversed reaction of the radiative capture, where a (real) photon is emitted following the capture of a neutron by the core.
    Later analyses have shown that the breakup process is not that simple and that higher-order effects spoil this nice picture <cit.>.
    However, it has been suggested that the Coulomb-breakup measurements could be used to infer the asymptotic normalization coefficient (ANC) of the ^15C ground-state wave function <cit.>.
    However, due to the aforementioned higher-order effects, a precise model of the reaction is needed in the analysis of the reaction <cit.>.
    Because the radiative capture ^14 C(n,γ)^15C is a purely peripheral process <cit.>, a reliable estimate of this ANC can then be used to compute its cross section.
    Following 15CANC_2006, it has also been suggested to rely on the strong sensitivity of transfer reaction to the single-particle structure of the nucleus to measure the ANC of the ^15C ground-state wave function for that purpose <cit.>.
    
    Since the radiative capture ^14 C(n,γ)^15C has been measured directly by Reifarth  <cit.>, the ^15C case provides the opportunity to test the validity of the different indirect methods listed above.
    
    In the present work, we reanalyze the transfer <cit.>, Coulomb-breakup <cit.> and radiative-capture <cit.> measurements using one single description of the one-neutron halo nucleus ^15C.
    For this, we follow the recent idea developed in CPH18 
    and include, within precise models of reactions, 
    a description of the nucleus based on halo effective field theory (Halo EFT) <cit.> 
    (see Ref. HJP17 for a recent review).
    Halo EFT exploits the natural separation of scales that is observed in halo nuclei—viz. the difference between the small size of the core R_ core and the large extension of the halo R_ halo—to build an effective Hamiltonian constructed as an expansion in powers of the small parameter R_ core/R_ halo.
    This allows us to introduce, order by order, the different nuclear-structure parameters in the description of the nucleus within the reaction models, and thereby to deduce how each of them affects the reaction processes.
    This puts a strong constraint on what can be learned about the structure of ^15C from transfer and breakup experiments and how this nuclear-structure information relates to the direct radiative-capture capture measurement of Reifarth2008.
    
    This article is structured as follows. 
    In sec2 we introduce the Halo EFT description of ^15C and explain how it is fitted at next to leading order (NLO).
    Using this description, we reanalyze transfer measurements at E_d=14<cit.> and 17.06 MeV  <cit.> in Sec. <ref>.
    In sec3 we use the same ^15C structure to study its breakup at high (605 MeV/nucleon <cit.>) and intermediate (68 MeV/nucleon <cit.>) energy. 
    In sec4, we study the ^14 C(n,γ)^15 C radiative capture <cit.>.
    Finally, in sec5, we summarize our results and provide the outlook for future work.
    
    
    
    § HALO-EFT DESCRIPTION OF ^15C 
    
    
     §.§ Single-particle structure of ^15C 
    
    Being a one-neutron halo nucleus, ^15C can be modeled as a neutron loosely bound to a ^14C core.
    With the assumption that the ^14C core is in its ground state (0^+),  the ^+ ground state (g.s.) of  ^15C can be described by a ^14C(0^+)⊗ 1s_1/2 configuration  and its ^+ excited state (e.s.) by a ^14C(0^+)⊗ 0d_5/2.
    These states have an energy relative to the one-neutron threshold of E_ g.s. = -1.218 MeV and E_ e.s. = -0.478 MeV, respectively.
    
    To model this system, the core A of mass m_A and charge Z_A e is assumed to be of spin and parity 0^+ and we neglect its internal structure.
    The halo nucleus B=A+n is thus of mass m_B=m_A+m_n, with m_n the neutron mass, and charge Z_B e=Z_A e.
    Such a two-body structure is described by the internal Hamiltonian
    
    H_0=-ħ^2Δ/2μ_An + V_An(r),
    e0
    where r is the A-n relative coordinate, μ_An = m_A m_n /m_B is their reduced mass, and V_An is the effective potential simulating their interaction.
    In partial wave ljm, the eigenstates of H_0 read
    
    H_0 φ_ljm(E_lj,r) = E_lj φ_ljm(E_lj,r),
    e0a
    where j is the total angular momentum resulting from the coupling of the orbital angular momentum l with the spin of the halo neutron and m is its projection.
    The eigenstates of H_0 of negative energy E_n'lj are discrete and correspond to the bound states of the 
    two-body model of the projectile B. These include physical A-n bound states of the system 
    as well as Pauli forbidden states, which simulate the presence of neutrons within the core A.
    We enumerate them by adding the number of nodes in the radial wave function n' to the other quantum numbers.
    They are normed to unity and their reduced radial wave function behaves asymptotically as
    
    u_n'lj(r)r∞b_n'lj  ik_n'ljr h_l^(1)(ik_n'lj r),
    e0b
    where ħ k_n'lj=√(2μ_An|E_n'lj|), whith |E_n'lj| the A-n binding energy, and h_l^(1) is a spherical Bessel function of the third kind <cit.>.
    The single-particle asymptotic normalization constant (SPANC) b_n'lj defines the
    strength of the exponential tail of the A-n bound-state wave function <cit.>. 
    This SPANC will vary with the geometry of the potential used to simulate the A-n interaction <cit.>.
    The asymptotic behavior (<ref>) is universal, therefore it exists also in the actual structure of the nucleus, viz. in the overlap wave function obtained within a microscopic calculation of the nucleus <cit.>. 
    Being affected by the inherent couplings between the different configurations in the actual structure of the nucleus, in particular those involving the core in one of its excited states, the true asymptotic normalization constant (ANC) of the overlap wave function of the physical state of spin and parity J^π corresponding to the configuration in which the core is in its 0^+ ground state, C_J^π, differs from the SPANC b_n'lj obtained in the effective single-particle description considered here <cit.>.
    
    The positive-energy states describe the A-n continuum, i.e. the broken-up projectile.
    Their reduced radial parts are normalized according to
    
    u_kljr∞ kr[cosδ_lj j_l(kr) + sinδ_lj n_l(kr)]
    e0c
    where δ_lj is the phaseshift at energy E_lj and ħ k = √(2μ_AnE_lj); 
    j_l and n_l are spherical Bessel functions of the first and second kinds, respectively <cit.>.
    
    As mentioned above, the A-n interaction is described by an effective potential V_An.
    In this study, following the idea developed in CPH18, this potential is built within a Halo-EFT 
    description of the nucleus <cit.>.
    At the leading order (LO), this interaction consists of a simple contact term within the sole s wave.
    As usual, this interaction is regularized with a Gaussian
    
    V_An^ LO(r) = V_0^s1/2 e^-r ^2/2 r_0 ^2.
    e-LO
    The range of the Gaussian r_0 corresponds to the scale of the short-range physics neglected in this Halo-EFT description.
    Changing its value will enable us to generate different single-particle wave functions to describe the 
    ^14C-n system and hence test the sensitivity of our reaction calculations to the internal part of the wave function of the projectile.
    At LO, the only free parameter V_0^s1/2 is adjusted to reproduce E_ g.s.=-1.218 MeV within a 1s_1/2 orbit.
    
    At next-to-leading order (NLO), the interaction is extended up to the p waves and contains, 
    in addition to the contact term its second-order derivative.
    For simplicity, we follow Ref. <cit.>
    and use the equivalent following parametrisation of the interaction
    
    V_An^ NLO(r) = V_0^lj e^-r ^2/2 r_0 ^2+V_2^lj r^2e^-r ^2/2 r_0 ^2.
    e-1_1
    To constrain the potential parameters V_0^s1/2 and V_2^s1/2 in the s wave, we need two structure observables: in addition to the binding energy of the state, we also use its ANC. 
    Various groups have estimated this ANC from reaction data <cit.>. 
    In this work, we use the method presented in yang-capel to deduce this ANC from low-energy transfer data selected at forward angle (see transfer). 
    
    Unlike ^11Be, ^15C does not exhibit any low-lying bound or resonant ^- or ^-  states to which we could fit the effective interaction e-1_1 in the p waves.
    Therefore, true to the spirit of Halo-EFT, we set this interaction to 0 in the p_3/2 and p_1/2 partial waves.
    Interestingly, this treatment is in agreement with preliminary results obtained in an ab initio calculation of ^15C performed within the no-core shell model with continuum (NCSMC), which predicts negligible phaseshifts at low ^14C-n energies in both p waves <cit.>.
    
    
    
    
    
    At NLO, the interaction V_An is nil in higher partial waves.
    Since the ^+ excited bound state of ^15C plays a role in the radiative capture (see sec4), we follow the idea of CPH18 and go beyond NLO to include a 0d_5/2 state at E_ e.s.=-0.478 MeV.
    The potential in that partial wave is chosen similar to that of e-1_1.
    We fit the depths V_0^d5/2 and V_2^d5/2 to reproduce the experimental binding energy of the ^+ state and the ANC deduced from transfer data.
    
    
    
    
     §.§ Extraction of the ANC of the ^15C bound states from the analysis of low-energy transfer reactions 
    
    To obtain a reliable estimate of the ANC of both bound states of ^15C, we follow the idea developed in Ref. <cit.> and reanalyze ^14C(d,p)^15C transfer data.
    In that reference, it was found that (d,p) transfer reactions are purely peripheral when they are performed at low beam energy (viz. E_d≲15 MeV) and when the data are selected at forward angles.
    Within these experimental conditions, the transfer cross section scales perfectly with the square of the final state ANC C_J^π^2.
    That value can then be reliably extracted from a comparison between reaction calculations performed using a single-particle description of the nucleus similar to the one presented in 15Cstructure and experimental data <cit.>.
    
    We therefore need ^14C(d,p)^15C transfer data measured at low energies, and which contain enough data points at forward angles for this extraction of the ANC of ^15C to be statistically meaningful.
    Two experiments satisfying the low-energy condition have been performed: one at the University of Notre Dame at E_d=14 MeV <cit.>, and another at the Nuclear Physics Institute of the Czech Academy of Sciences at E_d=17.06 MeV <cit.>.
    Unfortunately, the former contains only one point at θ<15^∘, which we deem not enough for this extraction.
    Fortunately, although performed at a slightly higher energy, the latter experiment contains six points at θ<12^∘, which seems enough to constrain the ANC within proper peripheral conditions (see below).
    
    Following the method presented in yang-capel, we couple a leading order (LO) Halo-EFT description of ^15C with a finite-range adiabatic distorted wave approximation (FR-ADWA) model <cit.>.
    This model provides a reliable description of transfer reactions at these energies <cit.>.
    As in yang-capel, we consider the CH89 global potential <cit.> to generate the optical potentials in the incoming (d-^14C) and outgoing (p-^15C) channels.
    The Reid soft core potential <cit.> is used to compute the deuteron bound state. 
    The deuteron adiabatic potentials are obtained with the frontend code of TWOFNR <cit.>
    and the transfer calculations are performed using FRESCO <cit.>.
    We illustrate here the results for the ground state, the method to extract the ANC of the excited state is analogous, though less efficient because it corresponds to a d^14C-n bound state (see Ref. <cit.> for the details).
    
    We first build eight Gaussian potentials at the LO of Halo-EFT [see e-LO] considering different ranges r_0 between 0.6 fm and 2.0 fm. 
    For each width the depth V_0^s1/2 is adjusted to reproduce the neutron binding energy in the ^15C final state (see Table <ref>).
    These potentials provide different single-particle radial wave functions u_1s1/2 
    with very different SPANCs b^(r_0)_1s1/2, but also a significant change in the surface part of the nucleus, 
    i.e. in the range 2  fm≲ r≲ 4 fm, see f-0_17MeV.
    This is the corner stone of the method developed in yang-capel, because it is known that transfer reactions can be sensitive to that region <cit.>.
    Using single-particle wave functions that strongly differ, not only in their SPANC, but also in their shape within that surface region will enable us to accurately determine the conditions under which the reaction is purely peripheral, and thus under which a reliable estimate of the actual ANC of the nucleus can be inferred.
    
    
    With this input, we compute within the FR-ADWA <cit.> the corresponding theoretical differential cross section dσ_ th/dΩ for the transfer to the ^15C g.s. at E_d=17.06 MeV <cit.>, expressed as a function of the relative direction Ω = (θ, ϕ) between the proton and the ^15C in the outgoing channel.
    These results are displayed in f-1_17MeV(a) for the eight g.s. wave functions shown in f-0_17MeV.
    At forward angles, the cross sections exhibit a huge sensitivity to the choice of the ^14C-n wave function.
    They seem to scale with the square of the SPANC, as one would expect if the process were purely peripheral <cit.>. 
    To confirm this,  we have plotted the transfer cross section scaled by b_1s1/2^2 in f-1_17MeV(b). 
    In this way, the spread in the results is significantly reduced at forward angles. 
    
    
    
    To precisely determine within which angular range the data should be limited to select strictly peripheral conditions, we remove the major angular dependence by considering the ratio
     R_r_0/1.4  fm(θ) = ( b^(1.4  fm)_n'lj/b^(r_0)_n'lj)^2 dσ^(r_0)_ th/dΩ/dσ^(1.4  fm)_ th/dΩ-1,
    e-1_2
    where the transfer cross section computed using the ^14C-n Gaussian potential of range r_0, scaled by the square of the corresponding SPANC b_1s1/2^(r_0), is divided by the result obtained with r_0 = 1.4 fm, which is at the center of the range in r_0.
    The results are displayed in f-1_17MeV(c).
    We see that all ratios R_r_0/1.4  fm fall very close to one another at small angles, confirming the peripherality of the reaction when data measured at low beam energy are selected in the forward direction.
    To define an angular range in which the reaction can be considered
    as peripheral, we consider a maximum of 5% difference [horizontal black dotted lines in f-1_17MeV(c)]. 
    In this case, this happens only at very forward angles, viz. when θ < 12^∘.
    There are six data points within this angular region in this experiment <cit.>.
    Note that there is no data available within this angular range in the case of the experiment performed at the lower energy E_d=14 MeV <cit.>.
    
    Having determined the angular region within which the process is purely peripheral, we extract the value of the ANC C_1/2^+(r_0) for each of the single-particle wave function shown in f-0_17MeV.
    This is done by scaling, through a  χ^2 minimization, the corresponding theoretical cross section to the data selected at θ < 12^∘<cit.>. 
    The ANCs C_1/2^+(r_0) obtained in this way are shown in f-3 as a function of the potential width r_0.
    The error bars correspond to the uncertainty in the χ^2 minimization.
    Despite the huge changes in the radial wave functions observed in f-0_17MeV, the ANCs extracted are nearly independent of r_0; they fall within 4% from each other.
    This is similar to what was obtained for ^11Be (see Fig. 8 of yang-capel), hence confirming the validity of the method.
    
    To deduce an estimate of the actual ANC C_1/2^+, we average the C_1/2^+(r_0) results and get C_1/2^+=1.26±0.02 fm^-1/2 (C^2_1/2^+=1.59±0.06 fm^-1) displayed as the horizontal red dashed line and gray band in f-3.
    Following the same process, we obtain for the e.s. an estimate of the ANC of C_5/2^+=0.056±0.001 fm^-1/2. 
    
    
    
    We compare our estimate with values extracted from the analysis of other experiments in tab3.
    Though on the lower end of the range, the ANC we obtain agrees with most of the others.
    Our value is within the uncertainty band of the ANC extracted from knockout measurements in 15CANC_2002, 
    which is not surprising because that reaction is mostly peripheral <cit.>.
    Compared to the value extracted from the width of the ^+ ground state of the proton-unbound mirror nucleus ^15F, our C_1/2^+ seems too low.
    However, as explained in Muk10, that resonant state being quite broad, its width used in this analysis might be marred with significant uncertainty.
    In 15CANC_2007, Pang  have used the aforementioned ^14C(d,p)^15C transfer data measured at E_d=14 MeV <cit.>, which have not enough points at forward angles to be purely peripheral.
    Its large value is most likely due to that issue.
    Note also that the normalization of the E_d=14 MeV data has been questioned in 17MeVdp.
    Interestingly, we are in excellent agreement with the value obtained by Summers and Nunes in their analysis <cit.> of the Coulomb breakup cross section of ^15C measured at RIKEN <cit.>.
    Since this reaction is very peripheral <cit.>, this is not surprising (see breakup68AMeV).
    Our ANC is also perfectly compatible with the value extracted from the same data at E_d=17.06 MeV in 17MeVdp.
    The C_1/2^+ we have obtained is on the lower end of the uncertainty range of the value extracted from the ^13 C(^14 C,^15 C)^12C and d(^14 C,p)^15C transfer experiments in 15CANC_2014. 
    However, these experiments have been performed at energies corresponding to E_d≈24 MeV, where the reaction is not fully peripheral <cit.>, which may explain the slight disagreement with our ANC.
    
    
    
    The value we have obtained from the method developed in yang-capel is therefore in good agreement with most of the values cited in the literature, and the differences we observe with previous analyses can be explained from uncertainties in these analyses.
    Incidentally, as was observed in our previous analysis of the ^10 Be(d,p)^11 Be transfer <cit.>, this ANC for the ground state of ^15C is in excellent agreement with the C_1/2^+^2=1.644 fm^-1 obtained by Navrátil  in the aforementioned ab initio calculation of this one-neutron halo nucleus <cit.>.
    The present work will therefore provide a stringent test of the value predicted in that NCSMC calculation.
    
    
    
     §.§ Halo-EFT description of ^15C at NLO
    
    Having inferred a reliable value of the ANC for the ^15C g.s., we can now proceed as suggested in CPH18 and adjust a NLO Halo-EFT potential e-1_1 to describe this nucleus within our reaction models.
    In the s_1/2 partial wave, the two depths of the Gaussian potential are fitted to reproduce the experimental binding energy of the halo neutron to the core and our ANC.
    As in Refs. <cit.>, we perform this fit for three different ranges r_0 to test the sensitivity of our reaction calculations to the short-range physics of the ^14C-n overlap wave function. The depths obtained by these fits are listed in Table <ref>.
    
    
    
    As mentioned earlier, the interaction in the p wave is set to zero, in agreement with preliminary results of the ab initio calculations <cit.>.
    In tab4, we also provide the depths for ^14C-n potentials in the d_5/2 partial wave, which are fitted to reproduce the binding energy and ANC of the ^+ excited bound state of ^15C.
    This goes beyond the NLO of Halo EFT, but it will enable us to check the influence of the presence of that state in the ^15C spectrum in reaction calculations <cit.>.
    
    Figure <ref> displays the 1s_1/2 single-particle radial wave functions generated by the three potentials of tab4.
    By construction, they exhibit the identical behavior in the asymptotic region, viz. for r≳ 4 fm.
    However, as expected, the three wave functions exhibit significant differences at short distances, which will enable us to test the sensitivity to the short-range physics of ^15C of the various reactions we consider in the following.
    
    
    
    § TRANSFER REACTION ^14C(D,P)^15C 
    
    We start our analysis of the reactions involving ^15C using the NLO description developed in 
    potentials by looking at how it behaves in transfer reactions.
    We consider the low-energy reactions measured at E_d=17.06 MeV <cit.> and E_d=14 MeV <cit.>.
    We use the same FR-ADWA model <cit.> and potentials employed to extract the ANC in the previous section.
    
    Figure <ref> displays the cross sections for the ^14 C(d,p)^15C transfer reaction obtained at (a) E_d=17.06 MeV and (b) E_d=14 MeV.
    The results of the FR-ADWA calculations for each of the three ranges of the Gaussian NLO potential e-1_1 are shown in the same colors and line types as the corresponding radial wave functions in f-WF.
    The green band shows the uncertainty in the cross sections, obtained with the Gaussian potential of range r_0=1.5 fm, 
    related to the uncertainty in the ANC we have extracted in transfer.
    For comparison, we also show the results obtained with the LO description of ^15C using r_0=1.4 fm (purple dashed line).
    
    
    
    At E_d=17.06 MeV, without much surprise, the agreement of our NLO calculations with the data is perfect at forward angle since this is the region within which the fit has been performed in transfer.
    The transfer cross section obtained with the LO description of ^15C misses the data 
    by a factor that corresponds to the value of the ANC, which is not fitted at this order. 
    This confirms the importance of fitting both the energy and the ANC of the bound state to correctly reproduce the data.
    All three NLO ^14C-n potentials provide the same cross section in the angular range of peripherality of the reaction, viz. θ<12^∘.
    The agreement between the different wave functions actually extends beyond that range.
    At larger angles, however, the transfer cross sections obtained with the three different single-particle 1s_1/2 wave functions differ from one another, confirming that, at large angles, the reaction is sensitive to the short-range physics in ^15C.
    The uncertainty band encompasses the error bars of the forward-angle data, but cannot explain the discrepancy between our calculations and the experimental points at large angles.
    This shows the limit of the present approach: Halo-EFT provides a proper low-energy—viz. large distances—description of the projectile, but, by construction, does not account for the details of the internal part of the ^15C wave function.
    Hopefully, including a more precise wave function of the projectile could improve the description of the data at large angles.
    This could be done, e.g., using the overlap wave function provided by the ab initio calculation of Navrátil  <cit.>.
    Alternatively, one could use a more elaborated two-body model of ^15C, e.g., including core-excitation <cit.>.
    
    
    
    
    § COULOMB BREAKUP OF ^15C 
    
    
    We now turn to the Coulomb breakup of ^15C.
    As mentioned in the Introduction, this reaction has been measured on a lead target twice at two different energies.
    First at GSI at 605 MeV/nucleon by Datta Pramanik  <cit.> and second at RIKEN at 68 MeV/nucleon by Nakamura and his collaborators <cit.>.
    These two experiments are similar to those performed previously on the one-neutron halo nucleus ^11Be <cit.>, which were recently successfully analyzed using a Halo-EFT description of  ^11Be <cit.>.
    We therefore follow these references and apply the same models of the reaction using the NLO description of ^15C 
    detailed in potentials.
    
    
    
    
     §.§ Breakup of ^15C on lead at 605 MeV/nucleon 
    
    
    To analyze the breakup cross section of ^15C measured on Pb at GSI at 605 MeV/nucleon <cit.>, we follow what we did in MC19 and use an eikonal-based model of the reaction <cit.>, which properly accounts for special relativity.
    
    In that model, the projectile B is described by the two-body system introduced in sec2: a core A, to which a neutron n is loosely bound, and which interact through the NLO Halo-EFT potential adjusted in potentials.
    The target T is seen as a structureless body of mass m_T and charge Z_Te, which interacts with the projectile constituents A and n through the potentials V_AT and V_nT, respectively.
    We solve the problem within the Jacobi set of coordinates composed of the internal coordinate of the projectile r [see e0] and the relative coordinate of the projectile center of mass to the target R.
    The latter is explicitly decomposed into its longitudinal Z and transverse b components relative to the incoming beam axis.
    
    At this high beam energy, the use of the eikonal approximation is fully justified as well as the usual adiabatic—or sudden—treatment of the projectile dynamics during the reaction, i.e., we neglect the change in the projectile internal energy in comparison with its kinetic energy.
    To properly account for special relativity, we follow Satchler <cit.> and derive the eikonal wave function, which describes the projectile-target relative motion, from the Klein-Gordon equation expressed within the B-T center-of-momentum (CM) frame <cit.>.
    Within this description of the reaction, the three-body wave function exhibits the following asymptotic behavior
    Ψ^(m_0)(R,r)Z+∞ e^iK_0Z e^iχ(b,r)φ_n'_0l_0j_0m_0(r),
    e1
    where ħ K_0 is the initial B-T momentum, χ is the eikonal phase that accounts for the interaction between the target and the projectile constituents, and φ_n'_0l_0j_0m_0 is the wave function of the projectile ground state, in which it is assumed to be initially.
    Formally, the eikonal phase χ reads <cit.>χ(b,r)=-1/ħ v∫_-∞^∞[ V_AT(R,r)+V_nT(R,r)] dZ,
    echi
    where v is the B-T relative velocity.
    This phase can be interpreted semi-classically by seeing the projectile B following a straight-line trajectory at fixed impact parameter b along which its wave function accumulates a complex phase due to its interaction with the target.
    It is composed of three terms: χ=χ_BT^C+χ^C+χ^N.
    The first χ_BT^C(b)=2ηln(K_0b), with η=Z_BZ_Te^2/4πϵ_0ħ v, the Sommerfeld parameter of the reaction, simply describes the Coulomb scattering of the projectile by the target <cit.>.
    It does not depend on r, and hence does not contribute to the breakup of B.
    The second
    χ^C(b,r)=η∫_-∞^∞(1/|R-m_n/m_Br|-1/R) dZ
    echiC
    is the Coulomb term that contributes to the excitation of the projectile.
    This phase diverges because the infinite range of the Coulomb interaction is not compatible with the sudden approximation, which assumes that the collision takes place in a short time.
    To solve this issue, we use the Coulomb correction to the eikonal model (CCE) detailed in Refs. <cit.>.
    In that correction, the diverging eikonal Coulomb phase echiC is replaced at the first order by the first order of the perturbation theory <cit.>
    e^iχ^C→ e^iχ^C-iχ^C+iχ^ FO.
    eCCE
    For the first-order estimate of the Coulomb phase, we consider the relativistic expression limited to the E1 term <cit.>χ^ FO(b,r)=-ηm_n/m_B2ω/γ v[K_1(ω b/γ v)b·r/b+i1/γ K_0(ω b/γ v)Z],
    eFO
    where γ=1/√(1-v^2/c^2)[Note the difference with MC19, where we had considered for the calculation of γ the velocity of the projectile in the CM rest frame. 
    Note also the correct formulation of our equation (<ref>) with the 1/γ factor (check Eq. (2.15) of Ref. <cit.>). 
    These corrections have little effect on our results.].
    
    
    The third term of the eikonal phase χ^N corresponds to the nuclear interaction.
    At low and intermediate energies, it is usually described by optical potentials fitted to reproduce elastic-scattering cross sections.
    At high energy, and especially for exotic nuclei, it is difficult to find appropriate potentials.
    Therefore, we rely on the optical limit approximation (OLA) of the Glauber theory <cit.>, which has been successfully used in previous studies <cit.>.
    In that approximation, the nuclear eikonal phase is obtained by averaging a profile function Γ_NN, which simulates the nucleon-nucleon interaction, over the density of the colliding nuclei
    χ^OLA_xT(b_x)=  i ∬ρ_T(r')  ρ_x(r”) Γ_NN(b-s'+s”)dr” dr',
    e8
    where x stands for either A or n, the two constituents of the projectile, and where s' and s” are the transverse components of the internal coordinate of the target (r') and x (r”), respectively.
    In our three-body model of the reaction, the nuclear eikonal phase thus reads
    χ^N(b,r) = χ^OLA_AT(b_A) + χ^OLA_nT(b_n).
    e9
    We consider the usual form of the profile function
    Γ_NN(b)=1-i α_NN/4πβ_NNσ^ tote^-b^2/2β_NNe10
    where  σ^ tot is the total cross section for the NN collision,  α_NN corresponds to the ratio of the real to the imaginary part of the NN-scattering amplitude, and β_NN is the slope of NN elastic differential cross section.
    These parameters are isospin dependent, which means that, in practice, the OLA phase e8 splits into four terms.
    For the parameters of Eq. (<ref>) we use the values provided in Ref. <cit.> for an energy of 650 MeV.
    The densities used in e8 for the ^14C core and the ^208Pb target are approximated by the two-parameter Fermi distributions of Ref. <cit.>, in which the authors study a systematization of nuclear densities based on charge distributions extracted from electron-scattering experiments as well as on theoretical densities derived from Dirac-Hartree-Bogoliubov calculations.
    For ρ_n, we consider a Dirac delta function.
    
    The breakup cross sections obtained with this model of reaction are displayed in f2b as a function of the relative energy E between the ^14C core and the neutron after dissociation.
    To enable the comparison with the experimental data of Dat03, all theoretical cross sections have been folded with the experimental energy resolution, which we have considered identical to the one provided by Palit  in the analysis of the Coulomb breakup of ^11Be measured at GSI <cit.>.
    The calculations performed with all three ^14C-n potentials listed in tab4 are shown.
    The sensitivity of our calculations to the uncertainty in the ^15C g.s. ANC extracted in transfer is shown by the green band.
    The result of the calculation obtained without relativistic corrections is displayed as the purple dashed line.
    This clearly demonstrates the significance of these corrections at this beam energy.
    
    
    
    Let us first note that our theoretical predictions are in excellent agreement with the data at all energies.
    As expected, we do not note any appreciable difference between the calculations performed with the different Halo-EFT wave functions 
    (see f-WF).
    This result confirms that this reaction is purely peripheral, in the sense that it is sensitive only to the tail of the projectile wave function and not to its interior.
    The excellent agreement with the data observed in this reaction observable suggests that the ANC we have extracted from the transfer data, combined with the choice of a nil interaction in the p^14C-n partial waves, is valid structurewise <cit.>.
    Accordingly, the predictions of the ab initio calculations of Navrátil  seem correct <cit.>.
    
    
    In a subsequent test, we have analyzed how the inclusion of the ^15C e.s.—described here as a 0d_5/2 bound state (see 15Cstructure)—affects our breakup calculations.
    The presence of that state in the ^15C spectrum has no significant effect upon this reaction process; calculations performed with the Halo-EFT descriptions of ^15C beyond NLO, which include this state, are nearly identical to those shown in f2b.
    This is reminiscent of what has been observed in CPH18 in the analysis of the RIKEN Coulomb breakup experiment of ^11Be <cit.>, in which the presence of the ^+ resonance, also described within the d_5/2 partial wave, is barely noticeable in the cross section.
    This result is not surprising in a reaction that is strongly dominated by an E1 transition from the s bound state towards the p continuum.
    The existence of a d state in the low-energy spectrum of the projectile is more clearly seen in nuclear-dominated reactions, where quadrupole transitions are more significant <cit.>.
    Therefore, for this Coulomb-dominated reaction, a Halo-EFT expansion limited to NLO is sufficient: 
    the d bound state would actually appear only at the next order (i.e. next to next to leading order, N^2LO), and it has nearly no influence in our breakup calculations.
    This hence suggests that staying at NLO with a potential fitted to the ANC and binding energy of the g.s. in the s wave and a nil potential in the p wave, is enough to describe the experimental energy distributions for the breakup of ^15C.
    
    
    
    
     §.§ Breakup of ^15C on lead at 68 MeV/nucleon 
    
    The Coulomb breakup of ^15C has also been measured on Pb at RIKEN at 68 MeV/nucleon by Nakamura  <cit.>.
    To reanalyze these data using the Halo-EFT description of ^15C developed in potentials, we consider the dynamical eikonal approximation (DEA) <cit.>.
    This model of reaction is also based on the eikonal approximation, however, it does not include the usual adiabatic approximation, which means that it properly includes the dynamics of the projectile during the collision, which has been shown to matter at this intermediate beam energy <cit.>.
    Besides having proved to be very efficient in the description of various observables measured in the breakup of one-neutron <cit.> and one-proton <cit.> halo nuclei, the model has been shown to be in excellent agreement with other breakup models on this very reaction <cit.>.
    
    Following CPH18, we include the ^14C-n Halo-EFT potentials within the DEA and compute the breakup cross section at the RIKEN energy.
    To describe the nuclear interaction between the projectile constituents and the target, we follow CEN12 and consider optical potentials found in the literature.
    The ^14C-Pb potential is obtained from the scaling of an ^16O-Pb potential fitted to reproduce the elastic-scattering cross section of these nuclei at 94 MeV/nucleon <cit.>.
    We simply scale the radius of the potential by 0.987 = (14^1/3 + 208^1/3)/(16^1/3 + 208^1/3) to account for the mass difference between ^16O and ^14C and ignore the difference in beam energy.
    We use the Bechetti and Greenlees global nucleon-target optical potential to simulate the n-Pb interaction <cit.>.
    Note that the details of these interactions are provided in the supplemental material of CEN12.
    
    The results of these calculations are shown in f4 as a function of the ^14C-n continuum energy E.
    We consider the two angular cuts under which the experimental data have been measured, i.e., θ<6^∘, which includes the entire significant angular range, and θ<2.1^∘, the forward-angle selection.
    To allow for a direct comparison with the data of Nak09-15C, the results of our calculations have been folded with the experimental energy resolution.
    The green band shows the effect of the uncertainty on the ANC. 
    
    As in our analysis of the GSI experiment <cit.>, we obtain an excellent agreement with the data on the whole energy spectrum.
    All three NLO ^14C-n potentials lead to identical cross sections showing that, at this energy also, the reaction is purely peripheral and that the ANC we have extracted from the low-energy transfer data and the nil phaseshift in the ^14C-np waves are consistent with this other set of data.
    Our analysis hence independently confirms the value of the ANC extracted by Summers and Nunes from this same Coulomb breakup cross section <cit.>.
    The slightly larger ANC they have obtained (see line 4 of tab3) is probably due to their use of a non-zero interaction in the p wave, which tends to reduce these contributions to the breakup <cit.>.
    Since there is no experimental observable upon which to constrain the phaseshift in these partial waves, we have to rely on theoretical hypotheses.
    We have made a choice consistent with what we have done in the ^11Be case <cit.> and with preliminary ab initio predictions <cit.>.
    As shown in CN17, for the Coulomb breakup of loosely bound s wave nuclei, it is the combination of ANC in the g.s. and phaseshift in the p continuum that matters, especially at low energy E in the ^14C-n continuum and forward scattering angle.
    The excellent agreement with the data displayed in Figs. <ref>, <ref>, and <ref> justifies our choice.
    However, the uncertainty in the data is not sufficiently small to disprove the choice made in SN08.
    Using their choice of ^14C-n potentials would most likely provide as good an agreement with experiment as ours.
    Incidentally, this also confirms the ab initio prediction of Navrátil  for the ANC of the ^15C g.s.
    
    In addition to these NLO calculations, we have also performed another set of calculations going beyond NLO by including the e.s. in the ^15C spectrum as a 0d_5/2 bound state.
    The results, not shown here for clarity, are identical to those displayed in f4, confirming that in Coulomb-dominated reactions the details in the description of the d waves are irrelevant, and that an NLO Halo-EFT description of the projectile is sufficient.
    
    
    
    § RADIATIVE CAPTURE ^14 C(N,Γ)^15 C
    
    
    As mentioned in the Introduction, the radiative capture of a neutron by ^14C to form a ^15C nucleus [^14 C(n,γ)^15 C] plays a significant role in various astrophysical sites, from the possible inhomogeneous big-bang nucleosynthesis <cit.> to neutron-induced CNO cycles in AGB stars <cit.> and possible role in Type II supernovæ <cit.>.
    It is therefore useful for models of these astrophysical phenomena to have a reliable estimate of this reaction rate.
    Unfortunately it is difficult to measure directly: both reactants are radioactive and, although ^14C targets can be provided, obtaining purely monochromatic neutron beams is not simple.
    This is why indirect techniques, such as the Coulomb-breakup method <cit.>, have been proposed.
    Nevertheless, recently, Reifarth  have taken up the gauntlet and performed a direct measure of this radiative capture <cit.>.
    
    In sec3, we have shown that the Halo-EFT description of ^15C at NLO was sufficient to describe the breakup cross sections measured at GSI <cit.> and RIKEN <cit.>.
    As expected from the analyses published in Refs. <cit.>, this model of ^15C should also provide a good estimate for the radiative-capture cross section at low energy.
    In this section, we compare our prediction with the data of Reifarth  <cit.>.
    
    The radiative-capture ^14 C(n,γ)^15 C is dominated by the E1 transition from the p waves in the ^14C-n continuum towards the 1s_1/2 ground state of ^15C.
    A small contribution comes also from the capture from the p continuum waves to the 0d_5/2 excited state of the nucleus.
    Since these two contributions cannot be disentangled in the experiment of Reifarth  we use the Halo-EFT description of ^15C beyond NLO to include this excited state in our model of the reaction.
    To perform the calculations, we proceed as in CN17.
    
    
    
    
    The radiative-capture cross section obtained in this way is displayed in f5 as a function of the relative energy E between the neutron and the ^14C nucleus in the entrance channel.
    The three ^14C-n Gaussian potentials provide identical cross sections, confirming that this reaction is purely peripheral <cit.>.
    The effect of the ANC uncertainty is shown by the green band.
    The contribution due to the capture towards the 0d_5/2 e.s. is, as observed elsewhere <cit.>, of the order of 5%.
    The details of the description of this state, and especially the accuracy of its ANC extracted from transfer data, are thus completely negligible in this analysis.
    We have checked that the contribution of the E2 term to the radiative capture is orders of magnitude lower than the E1.
    The cross section displayed in f5 is in excellent agreement with prior predictions <cit.> 
    and the ab initio prediction of Navraátil  <cit.>.
    It is however slightly lower than what has been obtained in the analysis of the direct experiment <cit.>.
    
    
    To properly confront these results with the data measured by Reifarth  <cit.>, we need to account for the distribution of the neutron energy in the incoming beam <cit.>.
    The values averaged over the neutron distributions shown in Fig. 3 of Reifarth2008 are provided in tab5 alongside the experimental data.
    
    
    The experimental values are the ones provided in Table V of Reifarth2008.
    The theoretical cross sections are the one obtained using the ^14C-n potentials listed in tab4 of the present article.
    These values include the small contribution of the capture to the 0d_5/2 bound state that simulates the ^+ e.s. of ^15C.
    The uncertainty provided for the theoretical value corresponds to the uncertainty on the ANC of the g.s. of ^15C.
    The sensitivity to the range r_0 of the Gaussian potential e-1_1 is smaller than the precision provided here.
    
    
    Our theoretical predictions are usually in good agreement with the experimental values of Reifarth  <cit.>.
    The only significant difference is observed at the lowest energy point, where our prediction lies two sigma lower than the measured cross section.
    This seems to be an issue for most of the indirect estimates of this cross section <cit.>.
    Therefore, either there is some new physics not considered in the single-particle descriptions used in these references and in the present study, or there is some systematic uncertainty, which has not been well accounted for in the analysis of the experiment.
    The cross section we derive from our Halo-EFT description of ^15C at the single astrophysical energy E=23.3 keV is σ_n,γ(23.3  keV)=4.66±0.14 μb, which is slightly lower than what other groups obtain <cit.>.
    
    Within our study, this is the only one oddity in the analysis of various reaction observables, which are all peripheral, and in particular with Coulomb-breakup cross sections, which are sensitive to the same nuclear-structure observables as the radiative capture, viz. the ANC of the g.s. of ^15C and the phaseshift in the ^14C-np waves <cit.>.
    We therefore believe that they are well constrained within our model of ^15C.
    The E1 strength this model predicts, and upon which both the Coulomb-breakup and the radiative-capture cross sections depend, should thus be quite reliable.
    Figure <ref> provides this dB( E1)/dE as a function of the relative energy E between the ^14C
    and the neutron in the continuum. The value we obtain from our NLO ^14C-n potentials are compared with the E1 strength 
    inferred from the Coulomb-breakup measurement by Nakamura  <cit.>. 
    We observe that the latter is systematically lower than the dB( E1)/dE deduced from our Halo-EFT model of ^15C, 
    even though we are in perfect agreement with their Coulomb-breakup cross sections (see Fig. <ref>). 
    This difference is due to higher-order effects, which are neglected in the analysis of the RIKEN data. 
    As already shown in Refs. <cit.>, these effects are significant and cannot be ignored in the reaction model. 
    This is the reason why the RIKEN prediction of the cross section for the radiative capture  ^14C(n,γ)^15C 
    underestimates the direct measurement or Reifarth  (see Fig. 3 of Ref. <cit.>). 
    A comparison with that observable within the ab initio model of  Navrátil  would be interesting to confirm our prediction.
    
    
    
    § SUMMARY AND OUTLOOK 
    
    The exotic nucleus ^15C raises interests in various fields.
    It exhibits a one-neutron halo <cit.>, and its synthesis through the radiative capture of a neutron by ^14C takes place in various astrophysical sites <cit.>.
    It is therefore interesting to better understand its structure and to provide astrophysicists with reliable cross sections for the radiative capture ^14 C(n,γ)^15 C at low energies.
    
    In this work, we have reanalyzed various reactions involving ^15C using one single description of that nucleus.
    Following the work initiated in CPH18, we have considered a Halo-EFT description of that one-neutron halo nucleus.
    Once coupled to a precise model of reactions, this very systematic expansion enables us to accurately determine the observables that affect the reaction process and hence, which can be probed through experimental measurements <cit.>.
    
    Using a LO Halo-EFT Hamiltonian e-LO, we have reanalyzed the ^14 C(d,p)^15 C transfer data at low energy <cit.> within the framework of the FR-ADWA <cit.>.
    Following the results of yang-capel, focusing on the forward-angle region, enables us to select purely peripheral data, from which a reliable estimate of the ANC of the g.s. of ^15C has been inferred.
    The value obtained C_1/2^+=1.26±0.02 fm^-1/2 (C^2_1/2^+=1.59±0.06 fm^-1) is in good agreement with previous work <cit.> and with preliminary ab initio predictions <cit.>.
    
    The ANC hence obtained coupled to the binding energy of the valence neutron to the ^14C provides us with two nuclear-structure observables, upon which we have constrained a Halo-EFT Hamiltonian at NLO.
    This Hamiltonian has then be used within precise models of reactions to reanalyze transfer data <cit.>, Coulomb-breakup cross sections measured at high <cit.> and intermediate <cit.> energies, and cross sections for the radiative capture ^14 C(n,γ)^15 C<cit.>.
    In all cases, we observe a very good agreement with experiment without the need for any additional adjustment.
    
    By showing that all these experiments can be described at the NLO of the Halo-EFT expansion, these analyses indicate that the core-neutron binding energy and the ground-state ANC are the sole nuclear-structure observables that need to be constrained to reproduce these data.
    These reactions are therefore purely peripheral, 
    in the sense that they probe only the tail of the projectile wave function and not its interior.
    Especially, no need is found for a renormalisation of the projectile wave function, confirming that no spectroscopic factor can be extracted from such measurements <cit.>.
    Going beyond NLO, we have found that the presence of the bound excited state of ^15C in its description has no effect in Coulomb-breakup calculations.
    
    From this NLO description of ^15C we have been able to infer a reliable estimate of the E1 strength from the ^+ ground state of ^15C to its ^14C-n continuum.
    This dB( E1)/dE leads to excellent agreement with the measurements of both the ^15C Coulomb breakup <cit.> and the radiative capture ^14 C(n,γ)^15 C<cit.>.
    Accordingly, we suggest as a cross section for the latter process at astrophysical energy the value σ_n,γ(23.3  keV)=4.66±0.14 μb.
    
    The excellent results obtained within this framework confirms the interest of coupling a Halo-EFT description of the nucleus to existing precise models of reactions <cit.>.
    They also drive us to extend this idea to other reactions, like knockout <cit.>.
    Hopefully, the model developed herein and in CPH18 will enable us to reproduce existing data on ^15C and ^11Be <cit.>.
    We also plan to apply this model to other halo nuclei, like ^19C and ^31Ne.
    
    
    This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 654002,
    the Deutsche Forschungsgemeinschaft within the Collaborative Research Centers 1044 and 1245, 
    and the PRISMA (Precision Physics, Fundamental Interactions and Structure of Matter) Cluster of Excellence.
    J. Y. is supported by the China Scholarship Council (CSC).
    P. C. acknowledges the support of the State of Rhineland-Palatinate.
    
    
    
    
    
    
    
    
    
    73
    fxundefined [1]
     ifx#1
    fnum [1]
      #1firstoftwo
     secondoftwo
     
    fx [1]
      #1firstoftwo
     secondoftwo
     
    
    
    
    
    
    noop [0]secondoftwo
    
    ref[1]@startlink#1@href
    href[1]#1@endlink
    anitize@url [0] `
    12 `$12
      `&12 `#12 `1̂2 `_12 `%12
    startlink[1]
    endlink[0]
    
    rl [1]href #1
    
    
    
    
    
    
    
    
    
    
    
    
    @bib@innerbibempty
    
     [Tanihata(1996)]Tan96
      authorauthorI. Tanihata, titletitleNeutron halo nuclei,http://stacks.iop.org/0954-3899/22/i=2/a=004journaljournalJ. Phys. G volume22,pages157 (year1996)NoStop
     [Riisager(2013)]Rii13
      authorauthorK. Riisager, titletitleHalos and related
      structures, http://stacks.iop.org/1402-4896/2013/i=T152/a=014001journaljournalPhys. Scr. volume2013, pages014001 (year2013)NoStop
     [Hansen and Jonson(1987)]HJ87
      authorauthorP. G. Hansen and authorB. Jonson, titletitleThe neutron halo of
      extremely neutron-rich nuclei, https://doi.org/10.1209/0295-5075/4/4/005journaljournalEurophys. Lett. volume4, pages409 (year1987)NoStop
     [Wiescher et al.(1999)Wiescher, Görres, and Schatz]WGS99
      authorauthorM. Wiescher, authorJ. Görres, and authorH. Schatz, titletitleBreak-out reactions from
      the cno cycles, http://stacks.iop.org/0954-3899/25/i=6/a=201journaljournalJ. Phys. G volume25, pagesR133 (year1999)NoStop
     [Kajino et al.(1990)Kajino,
      Mathews, and Fuller]KMF90
      authorauthorT. Kajino, authorG. J. Mathews, and authorG. M. Fuller, titletitlePrimordial nucleosynthesis
      of intermediate-mass elements in baryon-number-inhomogeneous big bang models:
      observational tests, @noop journaljournalAstrophys. J. volume364, pages7 (year1990)NoStop
     [Terasawa et al.(2001)Terasawa, Sumiyoshi, Kajino, Mathews, and Tanihata]Ter01
      authorauthorM. Terasawa, authorK. Sumiyoshi,
      authorT. Kajino, authorG. J. Mathews, and authorI. Tanihata, titletitleNew nuclear reaction flow during r-process
      nucleosynthesis in supernovae: Critical role of light, neutron-rich nuclei,https://doi.org/10.1086/323526journaljournalAstrophys. J. volume562, pages470 (year2001)NoStop
     [Goss et al.(1975)Goss,
      Jolivette, Browne, Darden,
      Weller, and Blue]14MeVdp
      authorauthorJ. D. Goss, authorP. L. Jolivette,
      authorC. P. Browne, authorS. E. Darden, authorH. R. Weller, and authorR. A. Blue, https://doi.org/10.1103/PhysRevC.12.1730journaljournalPhys. Rev. C volume12, pages1730 (year1975)NoStop
     [Cecil et al.(1975)Cecil,
      Shepard, Anderson, Peterson,  and Kaczkowski]Cec75
      authorauthorF. Cecil, authorJ. Shepard,
      authorR. Anderson, authorR. Peterson, and authorP. Kaczkowski, titletitleCharged particle reaction studies on 14c, https://doi.org/10.1016/0375-9474(75)90676-4journaljournalNucl. Phys. volumeA255,pages243  (year1975)NoStop
     [Murillo et al.(1994)Murillo, Sen, and Darden]MSD94
      authorauthorG. Murillo, authorS. Sen, andauthorS. Darden, titletitleA study of the reactions14c(→d, d′)14c and
      14c(→d, p)15c at 16.0 mev, https://doi.org/10.1016/0375-9474(94)90797-8journaljournalNucl. Phys. volumeA579,pages125  (year1994)NoStop
     [Mukhamedzhanov et al.(2011)Mukhamedzhanov, Burjan, Gulino,
      Hons, Kroha, McCleskey,
      Mrázek, Nguyen, Nunes,
      S̆. Piskor̆, Romano, Sergi, Spitaleri, and Tribble]17MeVdp
      authorauthorA. M. Mukhamedzhanov, authorV. Burjan, authorM. Gulino,
      authorZ. Hons, authorV. Kroha, authorM. McCleskey, authorJ. Mrázek, authorN. Nguyen, authorF. M.Nunes, authorS̆. Piskor̆,
      authorS. Romano, authorM. L. Sergi, authorC. Spitaleri, and authorR. E. Tribble, https://doi.org/10.1103/PhysRevC.84.024616journaljournalPhys. Rev. C volume84, pages024616 (year2011)NoStop
     [Tostevin et al.(2002)Tostevin, Bazin, Brown, Glasmacher, Hansen, Maddalena,
      Navin, and Sherrill]Tos02
      authorauthorJ. A. Tostevin, authorD. Bazin,
      authorB. A. Brown, authorT. Glasmacher, authorP. G. Hansen, authorV. Maddalena, authorA. Navin, and authorB. M. Sherrill, titletitleSingle-neutron removal reactions from ^15C
      and ^11Be: Deviations from the eikonal approximation, https://doi.org/10.1103/PhysRevC.66.024607journaljournalPhys. Rev. C volume66, pages024607 (year2002)NoStop
     [Sauvan et al.(2004)Sauvan,
      Carstoiu, Orr, Winfield,
      Freer, Angélique, Catford,
      Clarke, Curtis, Grévy,
      Le Brun, Lewitowicz, Liégard, Marqués, Mac Cormick,
      Roussel-Chomaz, Saint Laurent, andShawcross]Sau04
      authorauthorE. Sauvan, authorF. Carstoiu,
      authorN. A. Orr, authorJ. S. Winfield, authorM. Freer, authorJ. C. Angélique, authorW. N. Catford, authorN. M.Clarke, authorN. Curtis, authorS. Grévy,
      authorC. Le Brun, authorM. Lewitowicz, authorE. Liégard, authorF. M. Marqués, authorM. Mac Cormick, authorP. Roussel-Chomaz, authorM.-G. Saint Laurent, and authorM. Shawcross, titletitleOne-neutron removal reactions on light neutron-rich
      nuclei, https://doi.org/10.1103/PhysRevC.69.044603journaljournalPhys. Rev. C volume69, pages044603 (year2004)NoStop
     [Fang et al.(2004)Fang,
      Yamaguchi, Zheng, Ozawa,
      Chiba, Kanungo, Kato,
      Morimoto, Ohnishi, Suda,
      Yamaguchi, Yoshida, Yoshida,  and Tanihata]Fang04
      authorauthorD. Q. Fang, authorT. Yamaguchi,
      authorT. Zheng, authorA. Ozawa, authorM. Chiba, authorR. Kanungo, authorT. Kato, authorK. Morimoto, authorT. Ohnishi,
      authorT. Suda, authorY. Yamaguchi, authorA. Yoshida, authorK. Yoshida, and authorI. Tanihata, titletitleOne-neutron halo structure in ^15C, https://doi.org/10.1103/PhysRevC.69.034613journaljournalPhys. Rev. C volume69, pages034613 (year2004)NoStop
     [Pramanik et al.(2003)Pramanik, Aumann, Boretzky, Carlson, Cortina, Elze, Emling, Geissel, Grünschloß,
      Hellström, Ilievski, Kratz, Kulessa, Leifels, Leistenschneider, Lubkiewicz, Münzenberg, Reiter, Simon,
      Sümmerer, Wajda, and Walus]Dat03
      authorauthorU. D. Pramanik, authorT. Aumann,
      authorK. Boretzky, authorB. Carlson, authorD. Cortina, authorT. Elze, authorH. Emling, authorH. Geissel,
      authorA. Grünschloß,
      authorM. Hellström, authorS. Ilievski, authorJ. Kratz, authorR. Kulessa, authorY. Leifels, authorA. Leistenschneider, authorE. Lubkiewicz, authorG. Münzenberg, authorP. Reiter, authorH. Simon, authorK. Sümmerer, authorE. Wajda, and authorW. Walus, titletitleCoulomb
      breakup of the neutron-rich isotopes 15c and 17c, https://doi.org/10.1016/S0370-2693(02)03016-2journaljournalPhys. Lett. volumeB551,pages63  (year2003)NoStop
     [Nakamura et al.(2009)Nakamura, Fukuda, Aoi, Imai, Ishihara, Iwasaki, Kobayashi, Kubo, Mengoni, Motobayashi, Notani, Otsu, Sakurai, Shimoura, Teranishi, Watanabe, and Yoneda]Nak09-15C
      authorauthorT. Nakamura, authorN. Fukuda,
      authorN. Aoi, authorN. Imai, authorM. Ishihara, authorH. Iwasaki, authorT. Kobayashi, authorT. Kubo, authorA. Mengoni, authorT. Motobayashi, authorM. Notani,
      authorH. Otsu, authorH. Sakurai, authorS. Shimoura, authorT. Teranishi, authorY. X.Watanabe, and authorK. Yoneda, titletitleNeutron
      capture cross section of ^14C of astrophysical interest studied
      by coulomb breakup of ^15C, https://doi.org/10.1103/PhysRevC.79.035805journaljournalPhys. Rev. C volume79, pages035805 (year2009)NoStop
     [Baur et al.(1986)Baur,
      Bertulani, and Rebel]BBR86
      authorauthorG. Baur, authorC. Bertulani,  and authorH. Rebel, titletitleCoulomb dissociation as a source of information
      on radiative capture processes of astrophysical interest, https://doi.org/10.1016/0375-9474(86)90290-3journaljournalNucl. Phys. volumeA458,pages188  (year1986)NoStop
     [Baur et al.(2003)Baur,
      Hencken, and Trautmann]BHT03
      authorauthorG. Baur, authorK. Hencken, andauthorD. Trautmann, titletitleElectromagnetic dissociation as a tool for nuclear
      structure and astrophysics, https://doi.org/10.1016/S0146-6410(03)90006-8journaljournalProg. Part. Nucl. Phys. volume51, pages487  (year2003)NoStop
     [Winther and Alder(1979)]winther-alder
      authorauthorA. Winther and authorK. Alder, titletitleRelativistic coulomb
      excitation, https://doi.org/10.1016/0375-9474(79)90528-1journaljournalNucl. Phys. volumeA319, pages518  (year1979)NoStop
     [Esbensen et al.(2005)Esbensen, Bertsch, and Snover]EBS05
      authorauthorH. Esbensen, authorG. F. Bertsch, and authorK. A. Snover, titletitleAccurate theory for
      determining S_17 by Coulomb dissociation, https://doi.org/10.1103/PhysRevLett.94.042502journaljournalPhys. Rev. Lett. volume94,pages042502 (year2005)NoStop
     [Capel and Baye(2005)]CB05
      authorauthorP. Capel and authorD. Baye,titletitleCoupling-in-the-continuum effects in
      Coulomb dissociation of halo nuclei, https://doi.org/10.1103/PhysRevC.71.044609journaljournalPhys. Rev. C volume71, pages044609 (year2005)NoStop
     [Summers and Nunes(2008a)]SN08
      authorauthorN. C. Summers and authorF. M. Nunes, titletitleExtracting (n,γ)
      direct capture cross sections from coulomb dissociation: Application to
      ^14C(n,γ)^15C, https://doi.org/10.1103/PhysRevC.78.011601journaljournalPhys. Rev. C volume78, eid011601 (year2008a)NoStop
     [Summers and Nunes(2008b)]SN08err
      authorauthorN. C. Summers and authorF. M. Nunes, titletitleErratum: Extracting (n,
      ) direct capture cross sections from coulomb
      dissociation: Application to ^14C(n,
      )^15C [phys. rev. c 78, 011601 (2008)],https://doi.org/10.1103/PhysRevC.78.069908journaljournalPhys. Rev. C volume78,pages069908 (year2008b)NoStop
     [Esbensen(2009)]Esb09
      authorauthorH. Esbensen, titletitleCoulomb dissociation of
      ^15C and radiative neutron capture on ^14C,https://doi.org/10.1103/PhysRevC.80.024608journaljournalPhys. Rev. C volume80,pages024608 (year2009)NoStop
     [Capel and Nollet(2017)]CN17
      authorauthorP. Capel and authorY. Nollet,titletitleReconciling coulomb breakup and
      neutron radiative capture, https://doi.org/10.1103/PhysRevC.96.015801journaljournalPhys. Rev. C volume96, pages015801 (year2017)NoStop
     [Timofeyuk et al.(2006)Timofeyuk, Baye, Descouvemont,
      Kamouni, and Thompson]15CANC_2006
      authorauthorN. K. Timofeyuk, authorD. Baye,
      authorP. Descouvemont, authorR. Kamouni, and authorI. J. Thompson, https://doi.org//10.1103/PhysRevLett.96.162501journaljournalPhys. Rev. Lett. volume96,pages162501 (year2006)NoStop
     [Reifarth et al.(2008)Reifarth, Heil, Forssén, Besserer, Couture, Dababneh, Dörr, Görres, Haight, Käppeler, Mengoni, O'Brien, Patronis, Plag, Rundberg, Wiescher, and Wilhelmy]Reifarth2008
      authorauthorR. Reifarth, authorM. Heil,
      authorC. Forssén, authorU. Besserer, authorA. Couture, authorS. Dababneh, authorL. Dörr, authorJ. Görres, authorR. C.Haight, authorF. Käppeler, authorA. Mengoni,
      authorS. O'Brien, authorN. Patronis, authorR. Plag, authorR. S. Rundberg, authorM. Wiescher, and authorJ. B. Wilhelmy, https://doi.org//10.1103/PhysRevC.77.015804journaljournalPhys. Rev. C volume77, pages015804 (year2008)NoStop
     [Capel et al.(2018)Capel,
      Phillips, and Hammer]CPH18
      authorauthorP. Capel, authorD. R. Phillips, and authorH.-W. Hammer, https://doi.org/10.1103/PhysRevC.98.034610journaljournalPhys. Rev. C volume98, pages034610 (year2018)NoStop
     [Bertulani et al.(2002)Bertulani, Hammer, and van
      Kolck]BHK02
      authorauthorC. Bertulani, authorH.-W. Hammer, and authorU. van
      Kolck, https://doi.org/10.1016/S0375-9474(02)01270-8journaljournalNucl. Phys. volumeA712, pages37 (year2002)NoStop
     [Hammer et al.(2017)Hammer,
      Ji, and Phillips]HJP17
      authorauthorH.-W. Hammer, authorC. Ji, andauthorD. R. Phillips,titletitleEffective field theory description of
      halo nuclei, http://stacks.iop.org/0954-3899/44/i=10/a=103002journaljournalJ. Phys. G volume44, pages103002 (year2017)NoStop
     [Abramowitz and Stegun(1970)]AS70
      authorauthorM. Abramowitz and authorI. A. Stegun, @noop titleHandbook of Mathematical
      Functions (publisherDover, addressNew-York, year1970)NoStop
     [Blokhintsev et al.(1977)Blokhintsev, Borbei, and Dolinskii]ANC77
      authorauthorL. D. Blokhintsev, authorI. Borbei, and authorE. Dolinskii, @noop journaljournalFizika Ehlementarnykh Chastits i Atomnogo Yadra volume8, pages1189 (year1977)NoStop
     [McCleskey et al.(2014)McCleskey, Mukhamedzhanov, Trache,
      Tribble, Banu, Eremenko,
      Goldberg, Lui, McCleskey,
      Roeder, Spiridon, Carstoiu,
      Burjan, Hons, , and Thompson]15CANC_2014
      authorauthorM. McCleskey, authorA. M. Mukhamedzhanov, authorL. Trache, authorR. E. Tribble,
      authorA. Banu, authorV. Eremenko, authorV. Z. Goldberg, authorY.-W. Lui, authorE. McCleskey, authorB. Roeder, authorA. Spiridon, authorF. Carstoiu, authorV. Burjan, authorZ. Hons, , and authorI. J.Thompson, https://doi.org/10.1103/PhysRevC.89.044605journaljournalPhys. Rev. C volume89, pages044605 (year2014)NoStop
     [Belyaeva et al.(2014)Belyaeva, Perez-Torres, Ogloblin,
      Demyanova, Ershov, and Goncharov]BPODEG14
      authorauthorT. L. Belyaeva, authorR. Perez-Torres, authorA. A. Ogloblin, authorA. S. Demyanova, authorS. N. Ershov, and authorS. A. Goncharov, https://doi.org/10.1103/PhysRevC.90.064610journaljournalPhys. Rev. C volume90, pages064610 (year2014)NoStop
     [Timofeyuk(2014)]Tim14
      authorauthorN. K. Timofeyuk, https://doi.org/10.1088/0954-3899/41/9/094008journaljournalJ. Phys. G volume41, pages094008 (year2014)NoStop
     [Capel and Nunes(2006)]CN06
      authorauthorP. Capel and authorF. M. Nunes, titletitleInfluence of the projectile
      description on breakup calculations, https://doi.org/10.1103/PhysRevC.73.014615journaljournalPhys. Rev. C volume73, pages014615 (year2006)NoStop
     [Capel et al.(2010)Capel,
      Danielewicz, and Nunes]CDN10
      authorauthorP. Capel, authorP. Danielewicz,  and authorF. M. Nunes,titletitleDeducing spectroscopic factors from
      wave-function asymptotics, https://doi.org/10.1103/PhysRevC.82.054612journaljournalPhys. Rev. C volume82, pages054612 (year2010)NoStop
     [Trache et al.(2002)Trache,
      Azhari, Carstoiu, Gagliardi,
      Mukhamedzhanov, Tang, Tribble, and Zhou]15CANC_2002
      authorauthorL. Trache, authorA. Azhari,
      authorF. Carstoiu, authorC. A. Gagliardi, authorA. M. Mukhamedzhanov, authorX. D. Tang, authorR. E. Tribble, and authorS. Zhou, http://cyclotron.tamu.edu/progress-reports/2001-2002/PDF
      journaljournalTex. A & M Cyclotron Prog.
      Rep. volumeI, pages16 (year2002)NoStop
     [Pang et al.(2007)Pang,
      Nunes, and Mukhamedzhanov]15CANC_2007
      authorauthorD. Y. Pang, authorF. M. Nunes,  and authorA. M. Mukhamedzhanov, https://doi.org//10.1103/PhysRevC.75.024601journaljournalPhys. Rev. C volume75, pages024601 (year2007)NoStop
     [Yang and Capel(2018)]yang-capel
      authorauthorJ. Yang and authorP. Capel,https://doi.org/10.1103/PhysRevC.98.054602journaljournalPhys. Rev. C volume98,pages054602 (year2018)NoStop
     [Navrátil(2018)]Nav18p
      authorauthorP. Navrátil, @noop howpublished(private
      communication) (year2018)NoStop
     [Johnson and Tandy(1974)]JT74
      authorauthorR. Johnson and authorP. Tandy, http://www.sciencedirect.com/science/article/pii/037594747490178XjournaljournalNucl. Phys. A volume235, pages56 (year1974)NoStop
     [Nunes and Deltuva(2011)]NunesDeltuva2011
      authorauthorF. M. Nunes and authorA. Deltuva, https://doi.org//10.1103/PhysRevC.84.034607journaljournalPhys. Rev. C volume84, pages034607 (year2011)NoStop
     [Upadhyay et al.(2012)Upadhyay, Deltuva, and Nunes]Upadhyay2012
      authorauthorN. J. Upadhyay, authorA. Deltuva,  and authorF. M. Nunes, https://doi.org//10.1103/PhysRevC.85.054621journaljournalPhys. Rev. C volume85, pages054621 (year2012)NoStop
     [Varner et al.(1991)Varner,
      Thompson, McAbee, Ludwig,  and Clegg]VTMLC91
      authorauthorP. R. Varner, authorW. Thompson,
      authorT. McAbee, authorE. Ludwig, and authorT. Clegg, http://www.sciencedirect.com/science/article/pii/037015739190039OjournaljournalPhys. Rep. volume201, pages57 (year1991)NoStop
     [Reid(1968)]Reid68
      authorauthorR. V. Reid, http://www.sciencedirect.com/science/article/pii/0003491668901267journaljournalAnn. Phys. (N. Y.) volume50, pages411 (year1968)NoStop
     [Igarashi and Toyama(2008)]IT08
      authorauthorM. Igarashi and authorM. Toyama, @noop journaljournalComputer program TWOFNR, University of Surrey version  (year2008)NoStop
     [Thompson(1988)]FRESCO
      authorauthorI. J. Thompson, http://www.sciencedirect.com/science/article/pii/0167797788900056journaljournalComput. Phys. Rep. volume7, pages167 (year1988)NoStop
     [Hebborn and Capel(2019)]HC19
      authorauthorC. Hebborn and authorP. Capel, https://arxiv.org/abs/1906.07660howpublishedarXiv:1906.07660 [nucl-th] (year2019),notesubmitted to Phys. Rev. C.Stop
     [Mukhamedzhanov et al.(2010)Mukhamedzhanov, Irgaziev, Goldberg,
      Orlov, and Qazi]Muk10
      authorauthorA. M. Mukhamedzhanov, authorB. F. Irgaziev, authorV. Z. Goldberg, authorY. V. Orlov, and authorI. Qazi,titletitleBound, virtual, and resonance
      s-matrix poles from the schrödinger equation, https://doi.org/10.1103/PhysRevC.81.054314journaljournalPhys. Rev. C volume81, pages054314 (year2010)NoStop
     [Capel and Nunes(2007)]CN07
      authorauthorP. Capel and authorF. M. Nunes, titletitlePeripherality of breakup
      reactions, https://doi.org/10.1103/PhysRevC.75.054609journaljournalPhys. Rev. C volume75, pages054609 (year2007)NoStop
     [Moschini and Capel(2019)]MC19
      authorauthorL. Moschini and authorP. Capel, titletitleReliable extraction of the
      db(e1)/de for 11be from its breakup at 520 mev/nucleon, https://doi.org/10.1016/j.physletb.2019.01.041journaljournalPhys. Lett. volumeB790,pages367  (year2019)NoStop
     [Gómez-Ramos et al.(2015)Gómez-Ramos, Moro, Gómez-Camacho, andThompson]GMG15
      authorauthorM. Gómez-Ramos, authorA. M. Moro, authorJ. Gómez-Camacho, and authorI. J.Thompson, titletitleTransfer
      induced by core excitation within an extended distorted-wave born
      approximation method, https://doi.org/10.1103/PhysRevC.92.014613journaljournalPhys. Rev. C volume92, pages014613 (year2015)NoStop
     [Palit et al.(2003)Palit,
      Adrich, Aumann, Boretzky,
      Carlson, Cortina, Datta Pramanik, Elze, Emling, Geissel, Hellström, Jones, Kratz, Kulessa, Leifels, Leistenschneider, Münzenberg, Nociforo,
      Reiter, Simon, Sümmerer,  and Walus]gsi_exp
      authorauthorR. Palit, authorP. Adrich,
      authorT. Aumann, authorK. Boretzky, authorB. V. Carlson, authorD. Cortina, authorU. Datta Pramanik, authorT. W. Elze, authorH. Emling, authorH. Geissel, authorM. Hellström, authorK. L.Jones, authorJ. V. Kratz, authorR. Kulessa,
      authorY. Leifels, authorA. Leistenschneider, authorG. Münzenberg, authorC. Nociforo, authorP. Reiter, authorH. Simon, authorK. Sümmerer, and authorW. Walus (collaborationLAND/FRS
      Collaboration), titletitleExclusive
      measurement of breakup reactions with the one-neutron halo nucleus
      ^11Be, https://doi.org/10.1103/PhysRevC.68.034318journaljournalPhys. Rev. C volume68, pages034318 (year2003)NoStop
     [Fukuda et al.(2004)Fukuda,
      Nakamura, Aoi, Imai,
      Ishihara, Kobayashi, Iwasaki,
      Kubo, Mengoni, Notani,
      Otsu, Sakurai, Shimoura,
      Teranishi, Watanabe, and Yoneda]Fuk04
      authorauthorN. Fukuda, authorT. Nakamura,
      authorN. Aoi, authorN. Imai, authorM. Ishihara, authorT. Kobayashi, authorH. Iwasaki, authorT. Kubo, authorA. Mengoni, authorM. Notani,
      authorH. Otsu, authorH. Sakurai, authorS. Shimoura, authorT. Teranishi, authorY. X.Watanabe, and authorK. Yoneda, titletitleCoulomb
      and nuclear breakup of a halo nucleus ^11Be, https://doi.org/10.1103/PhysRevC.70.054606journaljournalPhys. Rev. C volume70, pages054606 (year2004)NoStop
     [Glauber(1959)]Glauber
      authorauthorR. Glauber, @noop titleLectures in Theoretical
      Physics, edited by editorW. Brittin and editorL. Dunham, Vol. volume1 (publisherInterscience, addressNew York, year1959) p.pages315NoStop
     [Baye and Capel(2012)]BC12
      authorauthorD. Baye and authorP. Capel,titleBreakup reaction models for two- and three-cluster
      projectiles, in @noop booktitleClusters in
      Nuclei, Vol. 2, seriesLecture Notes in Physics, Vol.volume848, editoredited by editorC. Beck (publisherSpringer, addressHeidelberg, year2012)notepp. 121–163NoStop
     [Satchler(1992)]satchler
      authorauthorG. Satchler, titletitleLocal potential model
      for pion-nucleus scattering and π^+π^- excitation ratios, https://doi.org/10.1016/0375-9474(92)90173-HjournaljournalNucl. Phys. volumeA540,pages533  (year1992)NoStop
     [Pang(2014)]pang
      authorauthorD.-Y. Pang, titletitleEffects of relativistic
      kinematics in heavy ion elastic scattering, http://stacks.iop.org/1674-1137/38/i=2/a=024104journaljournalChin. Phys. C volume38,pages024104 (year2014)NoStop
     [Bertulani and Danielewicz(2004)]bertulani-danielewicz
      authorauthorC. Bertulani and authorP. Danielewicz, @noop titleIntroduction to
      Nuclear Reactions (publisherInstitute of Physics
      Publishing, addressBristol, year2004)NoStop
     [Margueron et al.(2003)Margueron, Bonaccorso, and Brink]bonaccorso-brink
      authorauthorJ. Margueron, authorA. Bonaccorso, and authorD. Brink, titletitleA non-perturbative approach
      to halo breakup, https://doi.org/10.1016/S0375-9474(03)01092-3journaljournalNucl. Phys. volumeA720, pages337  (year2003)NoStop
     [Capel et al.(2008)Capel,
      Baye, and Suzuki]CBS08
      authorauthorP. Capel, authorD. Baye, andauthorY. Suzuki, titletitleCoulomb-corrected eikonal description of the
      breakup of halo nuclei, https://doi.org/10.1103/PhysRevC.78.054602journaljournalPhys. Rev. C volume78, eid054602 (year2008)NoStop
     [Note1()]Note1
      noteNote the difference with Ref. <cit.>, where we had
      considered for the calculation of γ the velocity of the projectile in
      the CM rest frame. Note also the correct formulation of our equation (<ref>) with the 1/γ factor (check Eq. (2.15) of Ref. <cit.>). These corrections have little effect on our
      results.Stop
     [Horiuchi et al.(2010)Horiuchi, Suzuki, Capel, andBaye]31Ne
      authorauthorW. Horiuchi, authorY. Suzuki,
      authorP. Capel, and authorD. Baye, titletitleProbing the weakly-bound neutron orbit of
      ^31Ne with total reaction and one-neutron removal cross
      sections, https://doi.org/10.1103/PhysRevC.81.024606journaljournalPhys. Rev. C volume81, pages024606 (year2010)NoStop
     [Abu-Ibrahim et al.(2008)Abu-Ibrahim, Horiuchi, Kohama, andSuzuki]OLA_prof-func_param
      authorauthorB. Abu-Ibrahim, authorW. Horiuchi, authorA. Kohama,  and authorY. Suzuki,titletitleReaction cross sections of carbon
      isotopes incident on a proton, https://doi.org/10.1103/PhysRevC.77.034607journaljournalPhys. Rev. C volume77, pages034607 (year2008)NoStop
     [Chamon et al.(2002)Chamon,
      Carlson, Gasques, Pereira,
      De Conti, Alvarez, Hussein,
      Cândido Ribeiro, Rossi, and Silva]OLA_densities_param
      authorauthorL. C. Chamon, authorB. V. Carlson,
      authorL. R. Gasques, authorD. Pereira, authorC. De Conti, authorM. A. G. Alvarez, authorM. S. Hussein, authorM. A. Cândido Ribeiro, authorE. S. Rossi, and authorC. P. Silva, titletitleToward a global description of the nucleus-nucleus interaction,https://doi.org/10.1103/PhysRevC.66.014610journaljournalPhys. Rev. C volume66,pages014610 (year2002)NoStop
     [Baye et al.(2005)Baye,
      Capel, and Goldstein]BCG05
      authorauthorD. Baye, authorP. Capel, andauthorG. Goldstein, titletitleCollisions of halo nuclei within a dynamical
      eikonal approximation, https://doi.org/10.1103/PhysRevLett.95.082502journaljournalPhys. Rev. Lett. volume95,pages082502 (year2005)NoStop
     [Goldstein et al.(2006)Goldstein, Baye, and Capel]GBC06
      authorauthorG. Goldstein, authorD. Baye,  and authorP. Capel, titletitleDynamical eikonal approximation in breakup
      reactions of ^11Be, https://doi.org/10.1103/PhysRevC.73.024602journaljournalPhys. Rev. C volume73, pages024602 (year2006)NoStop
     [Goldstein et al.(2007)Goldstein, Capel, and Baye]GCB07
      authorauthorG. Goldstein, authorP. Capel,  and authorD. Baye, titletitleAnalysis of Coulomb breakup experiments of
      ^8 B with a dynamical eikonal approximation, https://doi.org/10.1103/PhysRevC.76.024608journaljournalPhys. Rev. C volume76, pages024608 (year2007)NoStop
     [Capel et al.(2012)Capel,
      Esbensen, and Nunes]CEN12
      authorauthorP. Capel, authorH. Esbensen,  and authorF. M. Nunes,titletitleComparing nonperturbative models of
      the breakup of neutron-halo nuclei, https://doi.org/10.1103/PhysRevC.85.044604journaljournalPhys. Rev. C volume85, pages044604 (year2012)NoStop
     [Roussel-Chomaz et al.(1988)Roussel-Chomaz, Alamanos, Auger,
      Barrette, Berthier, Fernandez, Papineau, Doubre, andMittig]Rou88
      authorauthorP. Roussel-Chomaz, authorN. Alamanos, authorF. Auger,
      authorJ. Barrette, authorB. Berthier, authorB. Fernandez, authorL. Papineau, authorH. Doubre, and authorW. Mittig, titletitle16o elastic scattering at elab = 94 mev/nucleon, https://doi.org/10.1016/0375-9474(88)90324-7journaljournalNucl. Phys. volumeA477,pages345  (year1988)NoStop
     [Becchetti and Greenlees(1969)]BG69
      authorauthorF. D. Becchetti and authorG. W. Greenlees, titletitleNucleon-nucleus
      optical-model parameters, https://doi.org/10.1103/PhysRev.182.1190journaljournalPhys. Rev. volume182, pages1190 (year1969)NoStop
     [Rupak et al.(2012)Rupak,
      Fernando, and Vaghani]RFV12
      authorauthorG. Rupak, authorL. Fernando,  and authorA. Vaghani,titletitleRadiative neutron capture on carbon-14
      in effective field theory, https://doi.org/10.1103/PhysRevC.86.044608journaljournalPhys. Rev. C volume86, pages044608 (year2012)NoStop
     [Capel and Nollet(2018)]CN18
      authorauthorP. Capel and authorY. Nollet,titletitleErratum: Reconciling coulomb breakup
      and neutron radiative capture [phys. rev. c 96, 015801 (2017)], https://doi.org/10.1103/PhysRevC.98.019906journaljournalPhys. Rev. C volume98, pages019906 (year2018)NoStop
